Foundations-of-colloid-science.pdf

Foundations of

Colloid Science SECOND EDITION

Robert J. Hunter School of Chemisty University of Sydney

OXFORD UNIVERSITY PRESS

OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford 0x2 ~ D P Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogoti Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris S5o Paulo Singapore Taipei Tokyo Toronto Warsaw Oxford is a registered trade mark of Oxford University Press in the U K and in certain other countries Published in the United States by Oxford University Press Inc., New York 0 Robert J. Hunter, 2001 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2001 Reprinted 2002, 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data applied for) ISBN 0 19 850502 7 Typeset by EXPO Holdings, Malaysia Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk

PREFACE TO THE SECOND EDITION It is now over ten years since the second volume of the first edition went to press and much has happened in colloid science in the interim. Such a short time frame has not, however, significantly affected the foundations of the subject. Why then go to the trouble of preparing a second edition? Whence comes the motivation and opportunity? The motivation came from my feeling that the two volume work, as presently constructed, is less effective as a teaching tool than it might be. Books of this kind are seldom read from cover to cover but when an author takes sole responsibility for the content of a book the natural tendency is to try to tell a coherent story and to do so in as linear a fashion as possible. Because of the collaborative nature of the exercise, that was not always possible with the first edition. In 1998, through the good offices of my friend and colleague Professor Sture Nordholm, I had the opportunity of trialing the entire scope of the book in a graduate course in the University of Goteborg. A group of about 40 Scandinavian students from various universities and industries submitted themselves to a 90 lecture presentation over a period of about two months. In this I was helped by Professors John Gregory (UC London), Roland Kjellander (Goteborg), Stjepan Marcelja (ANU), Ron Ottewill (Bristol), Dimo Platikanov (Sofia), The0 van de Ven (McGill and PPRIC), and Lee White (Carnegie-Mellon). Although the first edition was the reference text, the lecturers were, of course, free to develop their material in whatever way they chose. I have taken the opportunity to incorporate some of their many ideas into this revised version of the text. The editors of Oxford University Press were enthusiastic about the possibility of a second edition, especially if it could be confined to a single volume. I therefore undertook to revise the manuscript completely and to pare it down to around 900 pages, whilst at the same time updating the sections where change had been most significant. Inevitably some material had to go and rather than try to make small excisions everywhere I opted to simply remove a whole section from the scope: the material on thin films and emulsions. That is by no means to minimize its importance but rather to acknowledge that it is in many ways a separate study. Certainly those subjects draw on common material, from thermodynamics and double layer overlap to name just two, but there is also a good deal of material which is unique to the behaviour of emulsions. I was also influenced by the appearance of new texts which provide a much more detailed treatment of these subjects than I could hope to do in the few pages that could be made available to them. I refer to the books by D. Exerowa and P.M. Kruglyakov Foam and foamJilms (Elsevier 1998) and by B.P. Binks (ed.) Modern aspects of emulsion science (Royal SOC.Chem. (London) 1998). Those familiar with the first edition will notice the extensive changes in the order of treatment of the subject. I have used one rule throughout: treat each subject as soon as possible in the context of what has already been covered, with a minimum of forward

vii

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I

P R E F A C E TO THE SECOND EDITION

referencing. That means that thermodynamics (Chapter 2) and transport properties (Chapter 4) appear much earlier whilst van der Waals forces (Chapter 11) and stability theory (chapter 12) are treated much later. I have also added material on surface characterization (Chapter 6) and have substantially augmented the material on particle sizing, since there have been many developments in those areas in the last ten years. My collaborators on the first edition have been content to leave this rewrite to me. They no doubt felt they had already done more than enough towards the success of the enterprise in their earlier contributions. I have tried to weed out all those minor typographical errors which can hinder understanding and will be grateful for any help in eliminating any that remain. I can only hope that in my further attempts to make the text more accessible I have not unwittingly introduced any debilitating misconceptions. We can all be sure that the coming century will, however, bring many new insights to the subject as it continues to draw on a wider range of techniques in the exploration of the fascinating world that lies between the atomic and the macroscopic. R.J.H. [email protected] Sydney,Janua y 2000

Solutions to the problems are available by email from the author at the above address.

Collaborators on the material from the first edition Derek Y.C. Chan (Mathematics, University of Melbourne) Len R. Fisher (Physics, University of Bristol-formerly

CSIRO, Sydney)

Franz Grieser (Chemistry, University of Melbourne) John B. Hayter (Oakridge National Laboratory, Tennessee) Donald. H. Napper (Chemistry, University of Sydney) Richard W. O’Brien (Colloidal Dynamics Pty Ltd.-formerly South Wales)

University of New

Norman Parker (CSIRO, Sydney) Richard M. Pashley (Chemistry, Australian National University, Canberra) Lee R. White (Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Penn. (formerly University of Melbourne) Solutions to the problems are available by email from the author at the above address.

CONTENTS 1

NATURE OF COLLOIDAL DISPERSIONS 1.1 Introduction 1.2 Technological and biological significance of colloidal dispersions 1.3 Classificationof colloids 1.4 Some typical colloidal dispersions 1.5 Brownian motion and diffusion 1.6 Electrical charge and colloid stability 1.7 Effect of polymers on colloid stability

2 THERMODYNAMICS

OF SURFACES

2.1 Introduction 2.2 Surface energy and i t s consequences 2.3 Thermodynamics of surfaces 2.4 The Gibbs adsorption equation 2.5 Thermodynamic behaviour of small particles 2.6 Equilibrium shape of a crystal 2.7 Behaviour of liquids in capillaries 2.8 Homogeneous nucleation 2.9 Limits of applicability of the Kelvin and Young-Laplace

equations 2.10 Contact angle and wetting behaviour 2.11 Measurement of surface tension and contact angle

3

1 4 5 6 24 33 40

RESPONSE TO EXTERNAL FIELDS AND STRESSES 3.1 Response to gravitational and centrifugal fields 3.2 Response of a dielectric material to an electric field 3.3 Response to electromagnetic (light) waves 3.4 Response to a mechanical stress

45 45 56 63 72 81 84 93 97 100 112

116 124 133 144

4 TRANSPORT PROPERTIES OF SUSPENSIONS 4.1 Introduction 4.2 The mass conservation equation 4.3 Stress in a moving fluid 4.4 Stress and velocity field in a fluid in thermodynamic equilibrium 4.5 Relationship between the stress tensor and the velocity field 4.6 The Navier-Stokes equations 4.7 Methods for measuring the viscosity 4.8 Sedimentation of a suspension 4.9 Brownian motion revisited 4.10 The flow properties of suspensions

IX

157 158 160 162 164 167 170 178 181 188

X I CONTENTS

5 PARTICLE SIZE AND SHAPE 5.1 General considerations 5.2 Direct microscopic observation 5.3 Particle size distribution 5.4 Theoretical distribution functions 5.5 Sedimentation methods of determining particle size 5.6 Electrical pulse counters 5.7 Light scattering methods 5.8 Hydrodynamic methods 5.9 Acoustic methods 5.10 Summary of sizing methods

201 204 213 221 226 232 236 246 250 255

6 ADSORPTION ONTO SOLID

SURFACES 6.1 Vacuum characterization methods 6.2 Some non-vacuum techniques 6.3 Adsorption and desorption a t the solid-gas interface 6.4 Adsorption a t the solid-liquid interface 6.5 Adsorption of neutral polymers

262 269 277 287 293

7 ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER 7.1 The electrostatic potential of a phase 7.2 The mercury-solution interface 7.3 Potential distribution a t a flat surface -the

Gouy-Chapman model 7.4 Comparison with experiment 7.5 Adsorption of (uncharged) molecules a t the mercury-solution interface 7.6 Limitations of the Poisson-Boltzmann equation 7.7 The silver iodide-solution intetface 7.8 Other Nernstian surfaces 7.9 Mechanisms of surface charge generation 7.10 The double layer on oxide surfaces 7.11 The double layer around a sphere 7.12 The double layer around a cylinder

305 309 317 328 341 342 344 355 356 361 365 369

8 ELECTROKINETICS AND THE Z E T A

POTENTIAL 8.1 Introduction 8.2 Equilibrium double layer theory of electrokinetics 8.3 Reciprocity relations 8.4 The surface of shear 8.5 Measuring electrokinetic properties 8.6 Limitations of the elementary theory 8.7 The standard double layer model 8.8 Double layer dynamics 8.9 Electrokineticeffects in thin double layer systems 8.10 Numerical solutions of the linearized electrokinetic equations 8.11 Electrokinetics in alternating fields 8.12 Validity of the electrokinetic equations

374 375 384 384 387 393 395 400 408 41 5 416 426

CONTENTS

9 ASSOCIATION

COLLOIDS 9.1 The critical micellization concentration (c.m.c) 9.2 Factors affecting the c.m.c. 9.3 Equilibrium constant treatment of micelle formation 9.4 Thermodynamics of micelle formation 9.5 Spectroscopic techniques for investigating micelle structure 9.6 Micellar dynamics 9.7 Molecular packing and i t s effect on aggregate formation 9.8 Statistical thermodynamics of chain packing in micelles

435 438 443 450 460 466 472 476

10 ADSORPTION AT CHARGED INTERFACES 10.1 Introduction 10.2 Adsorption of potential determining ions 10.3 Detection of Stern layer adsorption 10.4 The oxide-solution interface 10.5 Adsorption of multivalent ions 10.6 Surfactant adsorption

482 485 490 501 509 518

1 1 THE THEORY OF V A N

DER WAALS FORCES 11.1 Introduction 11.2 London theory 11.3 Pairwise summation of forces (Hamaker theory) 11.4 Retardation effects in Hamaker theory 11.5 The Deryaguin approximation 11.6 Modern dispersion force theory 11.7 Numerical computation of interaction energy 11.8 Influence of electrolyte concentration 11.9 Theoretical estimation of surface properties

533 536 539 547 549 552 563 571 574

12 DOUBLE

LAYER INTERACTION AND PARTICLE COAGULATION 12.1 Surface conditions during interaction 582 12.2 Free energy of formation of a double layer 584 586 12.3 Overlap of two flat double layers 594 12.4 Interaction between dissimilar flat plates 598 12.5 Interaction between two spherical particles 601 12.6 Total potential energy of interaction 12.7 Experimental studies of the equilibrium interaction between diffuse double layers 604 12.8 Kinetics of coagulation 616 12.9 Effect of polymers on colloid stability 628

13 INTRODUCTION

TO STATISTICAL MECHANICS OF FLUIDS 13.1 Introduction 13.2 Molecular interactions 13.3 The structure of liquids 13.4 The potential of mean force

638 639 641 646

Ixi

xii I

CONTENTS

13.5 Time-dependent correlation functions 13.6 Applications of the pair distribution function 13.7 Measurement of correlation functions 13.8 Calculation of distribution functions

652 654 657 663

14 SCATTERING

STUDIES OF COLLOID STRUCTURE 14.1 Introduction 14.2 Relating potential to structure 14.3 Use of scattering to measure structure 14.4 Structure of concentrated isotropic dispersions of spherical particles 14.5 Neutron reflectivity

669 676 684 698 705

15 RHEOLOGY O F COLLOIDAL

DISPERSIONS 15.1 Introduction 15.2 Behaviour of time-independent inelastic fluids 15.3 Behaviour of time-dependent inelastic fluids 15.4 Visco-elastic fluids 15.5 Measurement of rheological properties of inelastic fluids in Couette flow 15.6 Capillary viscometer 15.7 Cone and plate or cone and cone viscometer 15.8 Time-dependent inelastic behaviour 15.9 Microrheology 15.10 Microscopic basis of rheological models

APPENDICES Appendix A1 Calculation of the allowed surface interaction modes in modern Dispersion Force Theory Appendix A2 Evaluation of the sum of the roots of the dispersion relation Appendix A3 Vector calculus and Poisson’s equation Appendix A4 Fourier transforms Appendix A5 Elementary thermodynamic relationships in the absence of surface contributions Appendix A6 Electrical units INDEX

714 71 5 721 724 728 734 739 740 741 749

767 768 770 777 780 784 787

Nature of Colloidal Dispersions 1.IIntroduction 1.2 Technological and biological significance of colloidal dispersions

1.3 Classification of colloids 1.4 Some typical colloidal dispersions 1.4.1 Preparation of colloidal dispersions 1.4.2 Monodisperse sols 1.4.3 Association colloids 1.4.4 Emulsions 1.4.5 Clay minerals 1.5 Brownian motion and diffusion 1.5.1 The one-dimensional random walk 1.5.2 The phenomenology of diffusion 1.5.3 Time-dependent diffusion processes 1.5.4 The Einstein-Smoluchowski equation 1.6 Electrical charge and colloid stability 1.6.1 The electrical charge a t a surface 1.6.2 Observation of coagulation behaviour 1.6.3 Coagulation by potential control 1.6.4 Coagulation by electrolyte addition 1.6.5 The critical coagulation concentration 1.6.6 Forces between colloidal particles 1.7 Effect of polymers on colloid stability 1.7.1 Steric stabilization 1.7.2 Polymer flocculation

1.IIntroduction In the normal solutions encountered in chemical situations, the solute and solvent molecules are of comparable size and we normally assume that the solute molecules are, on average, dispersed uniformly throughout the (continuous) solvent. There is an important class of materials, however, in which the kinetic units that are dispersed through the solvent are very much larger than the molecules of the

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1: NATURE OF COLLOIDAL DISPERSIONS

solvent. Such systems are called colloidal dispersions and they may arise in a variety of ways. If a substance, A, is insoluble in substance B it will usually be possible to break A down into very small particles that can be distributed more or less uniformly through the substance B. Substance A is then called the disperse phase and substance B, the dispersion medium. In general, A and B may be either solids, liquids, or gases so the dispersion could be regarded as a state of matter, accessible to any substance A, given the appropriate temperature and pressure and a means of producing and maintaining small discrete lumps of A distributed throughout B. If A is a solid, the particles may be produced by crushing and grinding a macroscopic piece of pure A, by growing small crystals of A by some chemical reaction, or by controlled crystallization of A from some solvent. Just how one goes about distributing A through the medium B (which might be a solid, a liquid, or a gas) and maintaining the discrete nature of the A particles (i.e. preventing aggregation) forms a considerable part of the theory and practice of colloid science. The lower limit of size for dispersions of this kind is around 1 nm. Smaller particles would ultimately become indistinguishable from true solutions. The upper limit is normally set at a radius of 1 p m but there is no clear distinction between the behaviour of particles of 1 p m and the somewhat larger particles often encountered in emulsions, in mineral separation processes, and in ceramic engineering. There are, of course, some molecules that are individually larger than 1 nm in size. These ‘macromolecules’ can often be uniformly dispersed through a fluid medium and they then form a colloidal solution or dispersion. Proteins, polysaccharides (such as starch), and many synthetic polymers fall into this category. It was just such a substance (a naturally occurring gum) that suggested the name ‘colloid’ (from the Greek word for glue) to the pioneer investigator in the field, Thomas Graham, in the 1860s. Th e field of polymer science has now developed into an entirely separate discipline with a vast specialized literature of its own. We will, therefore, treat only those few parts of it that are of most significance in colloid science. A third class of colloidal dispersions arises when a number of molecules of normal size associate together to form an aggregate. Soap molecules, for example, if they are at a sufficiently high concentration in a suitable solvent, can associate together to form micelles. These structures are of colloidal dimensions and the resulting system is referred to as an association colloid. (The term colloidal electrolyte is also used for ionic soap and detergent systems.) The distinguishing feature of all colloidal systems is that the area of contact between the disperse particles and the dispersion medium is relatively large. The energy associated with creating and maintaining that interface is significant, so the study of interfacial (or surface) chemistry is an integral part of the study of colloids. We will, however, limit ourselves to those aspects of surface chemistry of direct relevance to colloids. It is, incidentally, not necessary that all of the dimensions of the disperse particle be very small for the system to be of interest to colloid scientists. Some important colloidal systems have particles that can readily be seen with the aid of an ordinary microscope, such as textile fibres or the cellulose fibres used in making paper. It is sufficient if one of the characteristic dimensions of the particle (in this case the

INTRODUCTION

13

diameter) is small. The gas bubbles that make up a foam are usually larger than colloid size but in that case the thin lamellae of liquid between the bubbles have a thickness of the order of colloidal dimensions. They, therefore, also fall within the purview of colloid science. Most of the examples quoted above refer to dispersions of solids in liquids and this is the area with which we will be principally concerned. Many of the conclusions are, however, applicable to liquids in liquids (emulsions) and gases in liquids (foams). Dispersions in which the continuous (dispersion) phase is solid are becoming increasingly important in the field of material science where many new composites are being developed, but this area remains outside the scope of the present volume (see, for example, Evans and Langdon 1976 and the monographs by Schwartz 1996, 1997). Likewise, dispersions of solids and liquids in gases will not be treated in any detail. Such aerosols as they are called, are certainly colloidal systems but the theoretical concepts required to describe their behaviour, although similar in some respects, differ quite significantly from those with a liquid dispersion medium. There are, in addition, a number of recent works concerned with the solid-gas interface (e.g. Hercules 1992, Somorjai 1994) and the most recent edition of Adamson’s long-standing text (Adamson and Gast 1997) gives a strong coverage of that area. Table 1.1 lists the various types of colloidal system with common examples of each. Note that all possible combinations can be realized except that of gas in gas.

Table 1.7 The various types of colloidal dispersion with some common examples. The nomenclature is adapted fiom Ostwald (1 907) Disperse phase

Dispersion medium

Notation

Technical name

Examples

Solid

Gas

S/G

Aerosol

Smoke

Liquid

Gas

L/G

Aerosol

Hairspray ,mist, fog

Solid

Liquid

S/L

Sol or dispersion

Printing ink, paint

Liquid

Liquid

L/L

Emulsion

Milk, mayonnaise

Gas

Liquid

G/L

Foam

Fire-extinguisher foam

Solid

Solid

s/s

Solid dispersion

Ruby glass (Au in glass), some alloys

Liquid

Solid

L/S

Solid emulsion

Bituminous road paving, ice cream

Gas

Solid

G/S

Solid foam

Insulating foam

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1: NATURE OF COLLOIDAL DISPERSIONS

Exercises 1.1 . I Give further examples of each of the colloidal systems listed in Table 1.1 and

state their approximate composition. 1.1.2 Starting with a cube of solid, 1 cm along each edge, what is the total surface area when the solid is subdivided into cubes lop4 cm on each edge? Repeat the calculation for lop5 cm and lop6 cm cubes. Calculate the surface energy per particle in each case, assuming the surface energy is 70 mJ m-' and compare this with thermal energy ( k T )at room temperature (25 "C). What is the total surface energy for each system? 1 . I .3 Show that the surface area per unit mass, A,, of particles of density p is given by: A, = k'/pr where Y is some characteristic dimension, and k' = 3 for spheres, 2 for thin cylindrical discs and long rods, and 4 for long square prisms. (Am is called the specific surface area and typical values for a colloidal material fall in the range 1-lo3 m2 g-I.1 1.1.4 Chemical bonding energies are commonly of the order 100 kJ. Show that surface energies of particles will approach this value for sizes of about 1 nm. (Assume the surface energy is 0.1 J m-2 and take reasonable values for the density and molar mass.) 1.1.5 A mineral oxide of density 2.8 g cmP3is broken up into colloid sized particles in a ball mill. Calculate the total surface area of the crushed material (m2 g-') when the average particle radius is 5 x loP4 cm. What is the total area when the particle radius is 500 A?

1.2 Technological and biological significance of colloidal dispersions Almost all of the ancient and modern craft industries draw much of their technical expertise from colloid science. In paper-making, apart from the cellulose fibres used as the meshwork, there are clay particles used to improve opacity and polymer latex particles to bind the components together and they are all colloidal, as are the calcite and clay particles used as coatings to produce a shiny texture. The inks used in ball point pens, in xerography, and in high-speed printing presses each owe their special properties to their colloidal character, as do also the many varieties of paints and cosmetics. Ceramic products from expensive china to building bricks are make from clay/water sols and modern colloid techniques are currently being used to develop a new generation of very tough (fracture resistant) ceramic materials for use in rocket nose cones, car engines, and in medical prostheses (Evans and Langdon 1976). Colloid science is important in extracting oil from geological deposits (Littman 1997), in converting oil to petroleum, and in making rubber tyres as well as in mineral extraction. The aerosols for dispensing domestic products such as shaving cream and deodorants have their agricultural counterparts in the sprays used for dispensing weedicides and insecticides. On the debit side, the same techniques are used for making defoliants, and the gels and dispersions used in flame throwers, napalm, and riot control gases.

CLASSIFICATION OF COLLOIDS

15

Apart from the widely recognized colloidal nature of protein and polysaccharide solutions there are many other biological systems that have been studied using the methods of colloid science. The flow properties of blood are best understood in terms of its being a colloidal dispersion of (deformable) flat plates (the red corpuscles) in a liquid (Goldsmith and Mason 1975). The flow properties of faecal material must sometimes be modified by colloid chemical techniques to avoid unpleasant physiological consequences. The greater part of the food processing, preserving, and packaging industry rests heavily on colloid chemistry (see, for example, Ritsch and Reineccius 1995), and agricultural scientists require a knowledge of the colloidal properties of soils in order to induce optimum plant growth. Medical practitioners have used a colloidal dispersion of gold (potable gold) since the Middle Ages for treating a variety of ailments. Modern colloidal microcapsule techniques allow controlled release of a drug (Lee and Good 1987, De and Maitra 1997) and, in some cases, accurate targeting onto a particular organ. More routine applications of colloid chemical principles crop up in the preparation of emulsions and suspensions that must remain homogeneous for long periods on the shelf (or at least be readily redispersed on shaking). Aqueous emulsions of perfluoro-hydrocarbons have also been developed as (temporary) blood substitutes (Riess and Le Blanc 1978). Apart from its contributions to engineering, agriculture, biology, and medicine, colloid science also has an important role to play in reducing the harmful effects of technological development. Many pollution problems are due to the presence of unwanted colloidal materials, and their removal (from air or waterways) calls for the application of colloid chemical techniques. The specific adsorptive properties of colloids can also be used to remove, to concentrate, and possibly to recover, industrial products (especially metal ions) from air and water.

1.3 Classification of colloids Freundlich, in his classical text on the subject (1926), suggested that colloidal dispersions could be divided into two classes, called lyophilic (solvent loving) and lyophobic (solvent hating) respectively, depending on the ease with which the system could be redispersed if it was allowed to dry out. As with most such dichotomies, further study has revealed a complete range of intermediate types, but it is still useful to distinguish between the extremes. Kruyt (1952) uses the same classification and refers to them also as reversible and irreversible systems, respectively. This terminology expresses more clearly the real nature of the distinction because the ultimate test of whether a system is lyophilic is to determine whether the dispersion process occurs spontaneously when the solvent is added to the colloid. In the grey area between the two extremes lie systems that can exhibit both forms of behaviour. For example, the clay mineral montmorillonite (Section 1.4.5) will disperse spontaneously in water if its negative charge is neutralized by strongly hydrated cations (e.g. Li+) but not if the cation is poorly hydrated (e.g. Cs+) or highly charged (Ca2+). One contributing factor to the difference in behaviour between reversible (lyophilic) and irreversible (lyophobic) systems is the extent to which the dispersion medium (solvent) is able to interact with the atoms of the suspended particle. If the solvent can

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I : N A T U R E OF COLLOIDAL DISPERSIONS

come into contact with all or most of those atoms then solvation energy will be important and the colloid should be lyophilic (reversible) in some suitable solvent. If the solvent is prevented, by the structure of the suspended particles (i.e. the disperse phase), from coming into contact with any but a small fraction of the atoms of those particles then the colloid will almost certainly be lyophobic (i.e. irreversible) in its behaviour, even if the surface atoms interact strongly with the solvent. When the dispersion medium is water, the terms hydrophilic and hydrophobic are used; the great majority of the present work is devoted to the examination of hydrophobic sols. The lyophilic colloid solution is thermodynamically stable since there is a reduction in the Gibbs free energy when the ‘solute’ is dispersed. The strong interaction between ‘solute’ and solvent usually supplies sufficient energy to break up the disperse phase ( A H < 0) and there is often an increase in entropy as well; any reduction in solvent entropy due to the interaction with ‘solute’ is usually more than compensated by the entropy increase of the ‘solute’. For the lyophobic colloid, the Gibbs free energy increases when the disperse phase is distributed through the dispersion medium so that it is a minimum when the disperse phase remains in the form of a single lump. A lyophobic colloid can, therefore, only be dispersed if its surface is treated in some way that causes a strong repulsion to exist between the particles. In this way the particles can be prevented from aggregating (or coagulating) for long periods, although it must be emphasized that they are still thermodynamically unstable and the barrier to coagulation is merely a kinetic one+. Given enough time they will ultimately form an aggregate. There is a well developed theory to describe the interaction between particles of a lyophobic colloid but the behaviour of lyophilic colloids is more difficult to describe. The reason for this is that all of the forces involved in lyophobic systems are also important for lyophilic systems but in addition, for the lyophilics, there are very strong specific solvent effects that are difficult to predict. We will, therefore, spend quite some time developing the theory of lyophobic colloids, choosing as typical examples the silver halide, clay mineral, metal oxide, and polymer latex sols, since these have been well characterized and much studied. The principles that emerge from those studies will then be applied to a number of more practical systems.

1.4 Some typical colloidal dispersions 1.4.1 Preparation of colloidal dispersions The general methods of preparing colloidal dispersions have been known for a long time and are adequately discussed in the older literature (Svedberg 1928; Weiser 1933; Alexander and Johnson 1949). We will describe these only briefly and then proceed to the more recent developments in which careful control of the growth process has led to the production

+Strictly speaking one should distinguish the aggregate of particles from the bulk solid. The dispersion is metastable with respect to the bulk solid.

SOME TYPICAL COLLOIDAL DISPERSIONS

17

of dispersions in which the particles all have almost the same size and shape. Such systems are ideal for testing aspects of the theory of dispersions, they have some interesting properties in their own right, and they may even offer some special advantages in certain technological processes. We will also examine some naturally occurring colloidal systems that offer special advantages for testing various theoretical analyses. It will come as no surprise to learn that the theoretical description of the equilibrium, kinetic, and transport properties of colloidal systems is almost always confined to certain simple geometric shapes; usually the sphere or infinite flat plate but sometimes the cylindrical rod or disc and more rarely the spheroid. T o adequately test the validity of these descriptions one must have available colloidal dispersions in which the particle shapes conform as accurately as possible to these simple geometric types and that has only recently become possible. Only by improving our understanding of such model systems can we hope to improve our descriptions of the behaviour of real colloidal systems. Svedberg (1928) divides the preparation of colloidal dispersions into two categories: dispersion and condensation, of which the latter is probably more important for fine material. In the dispersion methods, a sample of bulk material is broken down to colloidal dimensions by some kind of mechanical process. The most direct method is by grinding in a colloid mill. This device subjects a coarse suspension of particles to a very high shear field by forcing it into a narrow gap between two surfaces that are rotating rapidly with respect to one another. The particles are then torn apart by the shearing process and a colloidal dispersion results, provided that the solution contains a suitable dispersing agent to prevent the small particles from aggregating together (see Section 1.4.3 below). A similar effect can be achieved, especially with liquid-in-liquid dispersions (emulsions), by subjecting a mixture of the two phases to a high frequency sound wave (-20 kHz). This process, known as ultrasonication, also requires the presence of a dispersing agent if a stable sol is to result. Sols can also be formed by passing an electric arc between two wires placed under the surface of a liquid (Bredig’s procedure). This is also an example of a dispersion method in that some of the sol almost certainly results from pieces of metal being torn from the surface of the wire; there is, however, some condensation from the vapour also involved. Condensation methods are much more numerous and more diverse. They may involve dissolution and reprecipitation, condensation from the vapour or chemical reaction. In the first category is the formation of a solid paraffin sol in water. This can be done by dissolving paraffin wax in ethanol and pouring a little into a large volume of boiling water. The ethanol rapidly boils off leaving an opalescent dispersion of the paraffin. The second type is exemplified by the spontaneous formation of a mist or fog from a supersaturated vapour; provided the degree of supersaturation is sufficiently high (Svedberg suggests a vapour pressure more than eight times the equilibrium value) the formation of many droplets of very small size (< 1 pm) is assured. The chemical methods may involve reduction, oxidation, or double decomposition. Metal sols can be produced by reduction (gold by reducing chlorauric acid, HAuC4, with hydrogen peroxide or red phosphorus; silver from silver nitrate and ferrous citrate). The particles are usually very small (<< 100 nm) and with some recipes are so well stabilized that they behave as though they were lyophilic (Frens and Overbeek 1969).

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I : N A T U R E OF COLLOIDAL DISPERSIONS

Oxidation can be used to produce sulphur sols either from hydrogen sulphide:

or from thiosulphate solutions:

S20:-

+ H20 + S + SO:- + 2H+ + 2e-.

The latter reaction occurs at a well defined rate, determined by the pH and S20:concentration and is the basis of one of the well known ‘clock reactions’ for demonstrating kinetic concepts in elementary chemistry courses. It has been studied in detail by Johnston and McAmish (1973), who suggest that the rate-determining step is probably:

HS20,

+ S20:-

+ S2 + HSO,

+ S@-.

We will return to this reaction shortly to consider the formation of monodisperse sulphur sols. Many inorganic compounds that are insoluble in water can be induced to form colloidal dispersions if they are formed by mixing fairly concentrated reagents, especially in the presence of a dispersing agent. Thus barium sulphate sols can be prepared by mixing Ba(SCN)2 and (NH4)2SO4 in the presence of a little potassium citrate to act as dispersant. Arsenious sulphide sols are formed by bubbling H2S through a solution of As203 while silver halide sols are readily prepared by mixing silver nitrate and alkali halide solutions. Indeed, it is often necessary in gravimetric analysis procedures to treat a precipitate (e.g. BaS04) in a special way to prevent the formation of a colloidal sol or to aggregate a sol, once formed, in order to filter it effectively. There are other types of chemical reaction that are useful in certain cases but of limited applicability. Photodecomposition is, for example, important in the formation of silver particles from AgBr in photographic processes. Hydrolysis is also an important technique for preparing sols of the transition metal oxides and hydroxides; it has been very effectively exploited by Matijevic for the preparation of highly monodisperse systems (i.e. systems of uniform particle size), the subject to which we will now address ourselves.

1.4.2 Monodisperse sols Overbeek (1981) points out that monodisperse (or homodisperse or isodisperse) systems have played an important part in the development of our understanding of colloidal sols and, more importantly, have allowed colloid science to make essential contributions to our understanding of the behaviour of matter. In Section 1.5 we will see how Perrin used them to establish the experimental basis of the kinetic theory of matter (and indeed the very existence of molecules) and the value of the Avogadro constant. Likewise, the proof, by Svedberg, that proteins were well-defined molecules with precise molar masses was essential to the development of modern biochemistry. The key to formation of monodisperse sols is illustrated in Fig. 1.4.1. The chemical reaction by which the sol material is being formed must proceed at a suitable rate (and this can be controlled by temperature and concentration conditions). Precipitation does not occur as soon as the concentration of the product exceeds its saturation (i.e.

SOME TYPICAL COLLOIDAL DISPERSIONS

19

maximum equilibrium) concentration in the solution. Rather, it is necessary for a certain level of supersaturation to be reached before there are formed the nuclei on which crystal growth can subsequently occur. Conditions must be arranged so that nucleation occurs in a single short burst so that all subsequent deposition occurs on those initial nuclei. The nuclei grow rapidly at first and it is this that causes the concentration to fall below the nucleation level. The formation of new material must be maintained at a rate that keeps the solution concentration between the horizontal broken lines so that no new nuclei can form. Zaiser and La Mer (1948) used this technique to produce highly monodisperse sulphur sols from dilute (- 0.003 M) thiosulphate solution in acid (see Exercise 1.4.2 below). Reiss and La Mer (1950) subsequently showed that so long as the rate-determining step is diffusion of material to the growing surface (and not the incorporation step), the rate of change of surface area with time is the same for all the particles (though it may vary with time). Overbeek (1981) shows that for such a system the particle size distribution must become more narrow with increasing time. He also shows that, even if it is the incorporation step which is rate determining, the particle size distribution will narrow with time if either (i) the rate of incorporation is the same for all particles or (ii) the rate of incorporation is proportional to the surface area of the particles. Only in the less likely event of the incorporation rate being proportional to particle volume does the size distribution fail to become sharper with time, though even then it remains constant. Instead of relying on the natural formation of nuclei, an alternative approach is to bring the concentration to the region between saturation and nucleation and then add some very small seed crystals of the desired product or some other material of the same crystal habit. The small differences in size between these initial seeds are rapidly smoothed out as the crystals grow. Zsigmondy used this method very effectively to produce monodisperse gold sols from very fine (3 nm) gold seed particles obtained by reduction of a gold salt with red phosphorus (the Faraday sol). Matijevic and his co-workers have prepared monodisperse samples of a wide variety of transition metal oxides and hydroxides using a controlled hydrolysis technique, usually under fairly acid conditions, in the presence of certain complexing ions, such as sulphate and phosphate. The precursor in these cases is probably a basic complex Nucleation period

----

Nucleation concentration

Time

Fig. 1.4.1 Illustrating the production of a monodispersesol by confining the formation of nuclei to a very short period, so that the particle number remains constant and all grow together to the same size. (After Overbeek 1981.)

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1: NATURE OF COLLOIDAL DISPERSIONS

species like M,(OH)b(S04)~-b-2c))+(where x is the oxidation number of M), from which the nuclei form. Because these metals have a marked propensity for forming polynuclear complexes of varying molar mass, the exact mechanism of nucleus formation and growth will probably never be fully known since the solution composition depends markedly on the pH and concentration of the various species present. Nevertheless some attempts are being made to reach an understanding of these systems (Matijevic and Bell 1973). Some examples of the sols grown by Matijevic and his co-workers are shown in Fig. 1.4.2. It should be noted that the very regular character of these particles may be misleading in some cases: these micrographs would not reveal whether the particles are porous. Adsorption studies suggest that some of them are. (Fuerstenau 1982, personal communication.) Another important colloidal material, from the commercial point of view, is silicon dioxide (silica), which is sold as Ludox, Aerosil, etc. for use as a catalytic support, as a rubber reinforcing agent, a filler for paint, and for more specialized applications such as antireflecting coatings and encapsulating compounds for electronic components. Silica sols can also be used for making synthetic opal since they form the basis of natural opal. Monodisperse sols can be prepared from silicic acid (see, for example, Iler 1979, p. 312) or by hydrolysis of ethyl orthosilicate (Stober et al. 1968) and their highly spherical appearance suggests that the solid is an amorphous inorganic polymer in this case. Organic polymers can also be prepared as monodisperse spheres in water by the method of emulsion polymerization; these dispersions are milky in appearance and are called latices (by analogy with the natural rubber latex). One of the earliest and most detailed studies was of a carboxylate latex by Ottewill and Shaw (1967). Polystyrene and poly(methy1 methacrylate) (PMMA) latices have been widely used as models in colloid chemical studies because they are easy to prepare as monodisperse spheres (see Liu and Krieger 1978). A range of monodisperse latices with varying densities of negatively and positively charged groups on their surfaces has also been prepared by Homola and James (1977). These are often referred to as amphoteric latices but should more properly be called zwitterionic by analogy with the behaviour of proteins in solutions of varying pH. A note of caution should be sounded here. The various monodisperse latex preparations that have been used as models to test colloid chemical theories are usually assumed to consist of smooth spheres with occasional electrically charged (negative) groups firmly embedded in the surface. There is a growing body of experimental evidence to show that many of these systems are far from ideal in their behaviour (Napper and Hunter 1975, McDonogh and Hunter 1983). The zwitterionic latices, with many charged groups in the surface, are even more likely to show such effects. (See also Healy et al. 1978.)

1.4.3 Association colloids The term soap is applied to the sodium or potassium salts of long-chain fatty acids that are but one example of a general class of substances called amphiphalest. These are substances whose molecules consist of two well-defined regions: one which is oilsoluble (lipophilic, oleophilic or hydrophobic) and one which is water-soluble +Theyare sometimes called amphipathzc molecules, which refers to their ambivalence about what they hate rather than what they like - a bit like the optimist (pessimist) with the half full (empty) wine glass.

SOME TYPICAL COLLOIDAL DISPERSIONS

I 11

Fig. 1.4.2 Monodisperse inorganic colloids: (a) zinc sulphide (sphalerite);(b) cadmium carbonate; (c) a-Ferric oxide (haematite); (d) basic ferric sulphate (alunite).(Photographscourtesy of Professor E. Matijevic, Clarkson University, N.Y.)

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1: NATURE OF COLLOIDAL DISPERSIONS

Fiq. 1.4.2 continued

SOME TYPICAL COLLOIDAL DISPERSIONS

I 13

Air

Aqueous solution

Micelle

Fig. 1.4.3 (a) Conventional representations of a surfactant molecule as a rod or a flexible tail (the hydrocarbon chain) and a head group. The chain is not infinitely flexible but is limited by the C-C-C bond angle. Fairly free rotation can occur except at double bonds. The cross-sectional area of the paraffin chain is about 0.2 nm2 when fully extended and this is comparable to the head group size for -OH and -NH2, but smaller than -SO;. (b) Schematic arrangement of amphiphile (soap or detergent) molecules at low concentration in water. (Note that counterions are not shown.) (c) The situation above the critical micellization concentration (c.m.c.). Note that adsorption will also occur on the walls of the vessel. The arrangement there is not shown because it is more problematical and is, in any case, dependent on how hydrophilic the surface of the vessel is.

(hydrophilic) (Fig. 1.4.3(a)). The hydrophobic part is non-polar and usually consists of aliphatic or aromatic hydrocarbon residues. The hydrophobic character is not much affected by introducing halogens and similar groups (Laughlin 1981). The hydrophilic part consists of polar groups which can interact strongly with water (especially hydroxyl, carboxyl, and ionic groups). The fatty acid soaps are typical examples: CH3(CH2),COO-Na+ sodium stearate (n = 16)

CH3(CH2),CH =CH(CH2),COO-NaS sodium oleate (n = 7 and is)

sodium palmitate (n = 14) as are also the common anionic detergents: CH3( C H Z ) O.SOzO-Na+ ~ sodium dodecyl sulphate or SDS' alkylbenzene sulphonate (sodium salt) sodium tetradecyl sulphate

R is a (preferably linear) alkyl chain

'Sometimes referred to as sodium lauryl sulphate. Lauryl alcohol, from which that name is derived, is the C12 alcohol but lauric acid is the C12 carboxylic acid so it has a C11 chain. Somewhat confusing.

(NC12)

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1: N A T U R E OF COLLOIDAL DISPERSIONS

The most significant characteristic of this type of amphiphile is the tendency to adsorb very strongly at the interface between air and water (Fig. 1.4.3(b)); in doing so, the hydrophobic part of the molecule can escape from the aqueous environment whilst the hydrophilic head group can remain immersed in the water. Such substances are said to be strongly surface-active because they lower the surface (or interfacial) tension, y. They therefore make the formation of new surface easier and are widely used as foaming and dispersing agents. Commercial surface active agents (or surfactants) are used for a variety of purposes: as cleaning agents (detergents), colloid chemical stabilizers, and wetting agents. We will return to these matters in Chapter 2. For the moment we are concerned with another of their properties. At very low concentrations (< lop4 M, say) many surface active agents are soluble in water to form simple solutions; if they are ionic, like the fatty acid soaps or the alkyl sulphate detergents they will be dissociated as weak or strong electrolytes, respectively. Some of the molecules will also be preferentially adsorbed at the surfaces of the solution (i.e. the air-solution interface if there is one, and at the walls of the container) (Fig. 1.4.3(b)). As the concentration rises this adsorption becomes stronger until saturation is reached when the molecules are packed close together with strong lateral interactions occurring between the hydrophobic chains, which tend to stick up out of the water (Fig. 1.4.3(c)). [Note that some soap molecules will almost certainly be adsorbed on the walls of the beaker. The arrangement there is not shown because it is more problematic and depends to a considerable extent on how hydrophobic the surface is.] Another aggregation process, which often occurs at about the same surfactant concentration, is the formation of micelles (Fig 1.4.3(c)). These are structures in which the hydrophobic portions of the surfactant molecule associate together to form regions from which the solvent, water, is excluded. T h e hydrophilic head groups remain on the outer surface to maximize their interaction with the water and the oppositely charged ions (called counterions). A significant fraction of the counterions remains strongly bound to the head groups so that the lateral repulsive force between those groups is greatly reduced. T h e precise structure of the micelle depends upon the temperature and concentration but also on the details of the molecular structure: size of head group, length and number of hydrocarbon chains, presence of branches, double bonds or aromatic rings, etc. We will deal with those matters in Chapter 9. For the moment we restrict attention to the simplest amphiphiles: those with a polar (hydrophilic) head group at one end and one or two straight hydrocarbon chains attached. This includes the simple soaps and detergents and some natural lipids. These substances form micelles of colloidal size and, as noted in Section 1.1, are called association colloids or, more rarely, colloidal electrolytes. The concentration at which micelles first form in the solution is called the critical micellization concentration (c.m.c.). It is marked by quite sharp changes in slope when various transport and equilibrium properties (like electrical conductivity and surface tension) are plotted against concentration. The initial suggestions on micellar structure by Hartley and by McBain have been refined by the work of Stigter (1967) and many others. Suffice it to say at this stage that the long chain fatty acid soaps and simple detergents like sodium dodecylsulphate initially form micelles that are spherical in shape and have a fairly well defined

SOME TYPICAL COLLOIDAL DISPERSIONS

I 15

Fig. 1.4.4 A sodium dodecyl sulphate micelle. The more detailed picture, which emerges from a statistical mechanical calculation of the likely structure. (To be discussed in more detail in Chapter 9.) (Drawn by Dr J.N. Israelachvili from calculations by Dr D.W.R. Gruen, Australian National University.)

aggregation number (- 50 molecules for sodium stearate) (Fig. 1.4.4). They are, therefore, monodisperse. As the surfactant concentration is increased above the c.m.c., the initially spherical micelles become more distorted in shape, forming cylindrical rods or flattened discs (Fig. 1.4.5). Ultimately, at high ratios of soap to water they form liquid crystals and other so-called ‘mesomorphic phases’, a discussion of which would take us beyond the scope of this book. An interesting recent development is the use of such surfactant phases as templates for the formation of microporous solids of well-defined structure. These are in great demand in molecular separation processes (Ciesla and Schuth 1999). Under other circumstances amphiphilic substances can form two dimensional membranes, or bilayers to separate two aqueous regions, very similar to a biological membrane (Fig. 1.4.5(B)). If the bilayer is continuous and encloses an aqueous region the result is a more or less spherical vesicle (Fig. 1.4.6(a)) or a microtubule in which a double layer of surfactant encloses a cylindrical water region. This type of structure, including the liquid crystal, provides a number of possible models for investigating colloid chemical behaviour under controlled conditions. In non-aqueous media, small amounts of water are able to act as nuclei for the formation of inverse micelles in which the surfactant head groups point inwards to stabilise the water droplets.

Fig. 1.4.5 (A) A cylindrical micelle and (B) a bilayer. (Reproduced from Evans and Wennerstrom 1999 with permission.)

1.4.4 Emulsions Emulsions, like solid dispersions in liquids, can be formed by either condensation or dispersion methods. The use of mechanical dispersion methods is more common in this case because the energies involved are generally smaller. Apart from the ultrasonication technique mentioned above (Section 1.4. I), high-speed stirring or shaking of a two-phase liquid mixture can often induce emulsification, especially if a dispersing agent is present. The resulting droplets are, of course, spherical, provided that the interfacial tension (i.e. surface energy) is positive and sufficiently large. Many two-phase systems are able to undergo spontaneous emulstJication especially if a third component is present. This term applies strictly only to those systems in which no mechanical energy at all is required, though it is sometimes applied to systems that are simply easy to emulsify. In some cases, the spontaneous emulsification occurs because of the presence of a surface active agent, which lowers the interfacial tension essentially to zero. Negligible energy is then required for the formation of the

SOME TYPICAL COLLOIDAL DISPERSIONS

I 17

Water

Water

Water

oil

Fig. 1.4.6 (a) Section through the centre of a bilayer vesicle. The wall thickness is about twice the chain length and there is water on both sides of the surfactant. The conformation of the chains is close to that of the liquid hydrocarbon. (b) A surfactant micelle swollen by the presence of some solubilized oil. The molecules of oil are all in intimate contact with surfactant hydrocarbon chains. Incorporation of more oil would lead to an oil in water microemulsion (c). In that case there would be a separate pure oil region (containing no surfactant) in the interior. The microemulsion systems are usually generated by using two surfactants -often an ionic detergent (filled circles) together with a neutral dipolar compound of similar chain length (open circles) (called a co-surfuctunt).

emulsion. A special case of this sort is the formation of ‘microemulsions’, in which the droplet size is very small (-10 nm). Such systems have attracted a good deal of attention recently because of their technological significance. Although the surface tension is essentially zero, the microemulsion droplets are spherical and almost monodisperse (Overbeek 1980). The oily interior of a micelle can be used to take up more oil (a process called sobbilkation, which is important in detergency). There is then no sharp dividing line between an oil-swollen micelle (Fig. 1.4.6(b)) and a

18 I

1: N A T U R E OF COLLOIDAL DISPERSIONS

microemulsion of oil in water (Fig. 1.4.6(c)); nor, for that matter between a microemulsion and a normal emulsion. In some cases of spontaneous emulsification, the interfacial tension (or energy) remains positive but the energy necessary for emulsification is supplied by the redistribution of a solute between the two phases. Davies and Rideal (1963) give a detailed account of such a phenomenon (with photographs), when a solution of toluene in alcohol is mixed with water. The mechanism of emulsification is best described as ‘diffusion and stranding’. Toluene and alcohol diffuse simultaneously into the aqueous phase and as the soluble alcohol diffuses ahead, the insoluble toluene is left stranded in the aqueous phase as small droplets. Provided the interfacial tension or energy, y, is sufficiently high, the emulsion droplets must be spherical and this should make emulsion systems ideal candidates as model systems for testing colloid theories. Unfortunately, they have some drawbacks: the droplet size distribution is often rather wide, the kinetic behaviour of the droplets is sometimes affected by the fact that the interior is fluid and therefore potentially mobile and the possibility of coalescence leading to a change in droplet size is a further complication. The microemulsions referred to earlier, being spherical and almost monodisperse offer special advantages as model systems.

1.4.5Clay minerals The inorganic fraction of soils and most natural sediments consists almost entirely of silica and the various silicates. The term clay is used in soil science and agriculture to mean any material of particle size less than 2 p m but the term clay mineral refers to specific groups of silicate minerals. Some clay minerals have long been used in the ceramic industry because their plate-like crystal habit and ability to bond to one another when heated to high temperatures makes them suitable for making bricks, earthenware, and pottery (including china). Clays are also used extensively as fillers in making paper, paint, and rubber tyres, among other things. Indeed, so extensive is the use of these materials in industry that clay minerals rank second only to oil in terms of tonnages used. We are concerned here with the special property of silicon, when bonded in a certain way with oxygen, to form extensive flat plates or sheets?. When combined with similar flat sheets of an aluminium oxide they can produce layered crystals, which in favourable cases can be cleaved to yield surfaces that are believed to be atomically smooth and flat over relatively large distances (of the order of a few square millimetres). Such smooth solid surfaces have made it possible in recent years to make measurements of the forces between solid particles with an accuracy and reliability not previously possible (see Chapter 12). Space does not permit a detailed discussion of the structure of the layer silicates. The reader is referred to the texts of Grim (1953) or van Olphen(1977). We will consider only the basic structures (talc, pyrophyllite, and kaolinite) and the ways in

tCrystallographers refer to these as layers rather than sheets. We will use ‘sheets’ to avoid confusion with the electrical double layer formed when the particles are dispersed in water (section 1.6).

SOME TYPICAL COLLOIDAL DISPERSIONS

I 19

Fig. 1.4.7 Arrangement of silica tetrahedra in hexagonal rings to form a layer. Only the oxygens are visible; a silicon sits at the centre of each tetrahedral arrangement of oxygens. The apical oxygens (top layer) are shared with the adjoining alumina layer. Note the hole in the centre of the hexagonal ring in the lower layer of oxygens. The counterions which balance the crystal charge can, in some cases (e.g. K+) sit in those holes.

which the first two are modified to yield vermiculite, mica, and montmorillonite, since these are the systems that have been most extensively used as colloid chemical models. The basic silicon-oxygen unit is a tetrahedron with four oxygens surrounding the central silicon. The bonds are approximately 50 per cent ionic and 50 per cent covalent in character and in the clay minerals of interest to us the tetrahedra are linked to form hexagonal rings (Fig. 1.4.7). This pattern can be repeated indefinitely in two dimensions to form the sheet. Aluminium in combination with oxygen (and hydroxyls) forms an octahedron with the aluminium at the centre and again these octahedra can be linked to form a more closely packed two-dimensional sheet. In the kaolinite crystal a sheet of alumina octahedra sits on top of a sheet of silica tetrahedra with the apical oxygen atoms from the silica being shared with the aluminium atoms of the upper layer (Fig. 1.4.8(a)). This ‘ideal’ structure would be completely uncharged and the perfect kaolinite crystal would be built up by laying these double sheets one on top of another. The bonding of one double sheet to the next occurs partly through van der Waals forces and partly through hydrogen bonds from the OHs of the octahedral sheet to the oxygens of the next silica sheet (Fig.1.4.8(b)). These bonds are so numerous that only rather drastic treatments are able to prise open the structure. The crystals are normally hexagonally shaped discs with an axial ratio of

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1: NATURE OF COLLOIDAL DISPERSIONS

Fig. 1.4.8 (a) A diagrammatic sketch of the 'ideal' kaolin layer [Al(OH)2]2.O.(SiO2)2. One hydroxyl ion is situated within the hexagonal ring of apical, tetrahedral oxygens and there are three others in the uppermost plane of the octahedral sheet. The two sheets combined make up the kaolin layer. (b) Simplified schematic diagram of the kaolinite structure. Note that the upper and lower cleavage surfaces in the perfect crystal are quite different (but see text). A typical crystal would have about 100 or so layers. (c) A typical kaolinite crystal of aspect ratio ( d u )about 0.1. Note the negative charges on the basal planes (perpendicular to the c-axis of the crystal) and positive charges around the edges. The latter are eliminated at pHs above about 7.

the order of 1O:l (Fig. 1.4.8(c)) and they seldom grow to sizes of more than a few micrometres. The real crystals are usually considered to carry a negative charge on the basal surfaces (i.e. the larger flat surfaces). It cannot be attributed to dissociation of the few Al-OH groups (since they would produce either positive or negative charges depending on the pH). In any case, recent evidence (Ma and Eggleton 1999a) suggests that in most kaolinite deposits the outer planes are modified so that only the silica surfaces are exposed. This is also the probable result of the traditional preparation procedure (Posner and Quirk 1964) which is designed to remove any contaminating aluminium from the basal planes. It is generally assumed that the crystal charge is due to substitution of aluminium for silicon in the tetrahedral layer with a consequent imbalance of negative charge. The edges of the crystal, where imperfections necessarily occur because of bond breakage, carry a positive charge at low pH and this decreases to zero as the pH is raised to about 7 (Schofield and Samson 1954).The positive charge is generally considered to be due to dissociation from the aluminium octahedra: )Al-OH +)Al+ +OH-, at low pH and is partly compensated by negative charges from the silica tetrahedra. Recently Ma and Eggelton (19996) have argued that in the kaolinites which they have studied the crystal charge is confined solely to the edges and there is no isomorphous

SOME TYPICAL COLLOIDAL DISPERSIONS

121

substitution. Whether this is always so is an open question; it has always been difficult to verify the substitution because it does not influence the X-ray data and reverse compensations (substituting silicon for aluminium) can also occur in the body of the crystal. Kaolinite is called a 1:l non-swelling dioctahedral clay because (i) it has one silica layer to one alumina layer and (ii) the double sheets do not separate from one another under any normal conditions. The term dioctahedral refers to the fact that only two out of every three possible sites in the octahedral layer are occupied by aluminium ions. The other clay minerals of interest to us are all of the 2:l type (i.e. two sheets of silica to one of alumina or two of silica to one of magnesium oxide). The two parent materials arepyrophyllite (with alumina in the central layer) and talc (with magnesia in the central layer) (Fig. 1.4.9a). Pyrophyllite is again a dioctahedral whilst talc is a trioctahedral mineral, since all three possible sites must be occupied by M$+ to obtain charge balance. One important difference between the 1:l and 2:l minerals is that in the 2: 1 minerals there is no possibility of hydrogen bonding between successive triple sheets. The basal oxygen planes can interact with each other only by way of van der Waals forces. They are, therefore, very easily cleaved along this plane. The parent minerals pyrophyllite and talc are not of much interest to us but if they are modified by the substitution of some of the tetrahedral silicon by aluminium, they develop very interesting new properties. Replacing one quarter of the silicons by aluminium in pyrophyllite generates muscovite or white mica (Fig. 1.4.9(b)). This confers a very large negative charge on each sheet and that charge must be balanced; in muscovite it is balanced by the presence of potassium ions, which can fit snugly in the hexagonal hole of the silica sheet shown by the unbroken lines in Fig. 1.4.9(b). This greatly strengthens the bonding between each triple sheet so that mica does not tend to expand in water. A similar replacement of silicon by aluminium in the talc structure generates another form of mica called phlogopite. A further substitution of some of the octahedral magnesium in phlogopite by other divalent metal (usually Fe2+) ions produces another mica called biotite. Mica crystals can be very large (many centimetres across) and can be readily cleaved in air or vacuum. Good specimens when carefully cleaved can yield the macroscopic atomically smooth surfaces referred to earlier. Montmorillonite can also be related to the pyrophyllite structure by the substitution of approximately one in six of the aluminium ions in the octahedral layers by magnesium or other divalent ions. Again this generates a negative charge throughout each triple sheet and this must be compensated by the presence of cations in the interlayer region (Fig. 1.4.10). The material called Wyoming bentonite, which finds many uses (as a filler, a catalytic support, and in drilling muds), consists of a mixture of montmorillonite with a related material called beidellite in which aluminium ion is isomorphously substituted for silicon in the tetrahedral layers. When dry montmorillonite is placed in a moist atmosphere, it is able to take up water vapour by adsorbing it between the triple-sheets (that is, in the interlayer region) and the shape of the water vapour adsorption isotherm suggests that about four layers of water can be taken up into this region. The same behaviour occurs when the clay is immersed in concentrated salt solution (-1 M) (Norrish 1954). The spontaneous hydration is presumably associated with the presence of the cations, since the parent material (pyrophyllite), which has no such ions, is quite hydrophobic.

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1: NATURE OF COLLOIDAL DISPERSIONS

Fig. 1.4.9 (a) A sketch of the ideal 2:l layer silicate. The trioctahedral mineral talc (MgO)z.Mg (OH)z.(Si02)4] has all three octahedral sites occupied by Mg2+. In pyrophyllite [(AlO(OH))z. (Sioz)~]only two of the three octahedral sites are occupied by the counterions required to balance the crystal charge after isomorphous replacement. (b) Schematic diagram of the white mica structure. The potassium ions are shown (large hatched circle in (a)) and they balance negative charges in the silica layers caused by the substitution of about a quarter of the silicon ions by aluminium. The ‘ideal’ formula is: K+[(AlO(OH))z .(AlSi308)]-.

In salt solutions, another important phenomenon occurs and one which has been studied in clay mineral systems for over a century. This is the process of cation exchange, which can occur between ions in solution and the ions in the interlayer region (i.e. those on the basal cleavage planes). The process is made easier if the electrolyte concentration is lowered because the triple-sheets (or platelets) can then separate from one another to allow easier access to the adsorbed cations. The separation of the platelets occurs more readily if the interlayer cations are monovalent and strongly hydrated (Na+ or Li+ for example) because the inter-platelet repulsion is stronger in that case, as we will show in Chapter 12. The number of cations adsorbed on the clay (the cation exchange capacity) is an important characteristic of the material and is a direct measure of the degree of isomorphous substitution that has occurred in the

Next Page SOME TYPICAL COLLOIDAL DISPERSIONS I23

Fig. 1.4.10 Schematic diagram of the montmorillonite structure. It is very similar to that of mica (Fig. 1.4.9(b)) but the negative charges are now in the central (octahedral) layer and are fewer in number (about one in six aluminiums is replaced by Mg2+ to give: Na033[(A11.67Mgo.33) (0(OH))2 (Sioz)~]).The lower charge density, larger distance between the positive and negative charges, and the poor fit of the counterion (when hydrated) make it possible for sodium montmorillonite to expand readily when placed in water.

pyrophyllite structure. A typical value for montmorillonite would be about one mole of univalent charge per kilogram of clay (see Exercise 1.4.7). All clay minerals that have crystal structures with unbalanced charge exhibit ion exchange behaviour (e.g. kaolinite, vermiculite), which is of particular importance in determining the retention and availability of plant nutrients in the soil. Indeed, the strong affinity of the potassium ion for certain clay minerals is, to a large extent, responsible for the relative dominance of the sodium ion in sea water. The final clay mineral we wish to discuss is vermiculite. Its structure is derived from that of talc by the substitution of about one in six of the tetrahedral silicons by aluminium. The balancing (exchangeable) cation is often magnesium but it may be replaced by other divalent or monovalent ions. Vermiculite, like mica, can form large sheets that show a pronounced cleavage parallel to the plane of the sheets. Apart from the exchange of simple cations, there has been a very large amount of work done on the adsorption of more complex ions and molecules onto montmorillonite and vermiculite or their intercalation into the interlayer regions (see, for example, Weiss 1963). Before leaving the subject of clay minerals it should be emphasized that the chemical compositions given above apply to the ideal (or idealized) crystals. The real materials, as they occur in nature, seldom conform to these idealizations and that must be taken into account in interpreting their behaviour. It should also be mentioned that there are other geometries available in naturally occurring clay minerals (e.g. the thin cylindrical rods of attapulgite; van Olphen 1977, p. 7), which have so far not been much exploited for the testing of physical models of colloidal systems (though, see Buscall 1982).

r

Exercises 1.4.1 Prepare a paraffin wax sol in water by the method described above. Use about 5-10 ml ethanol saturated with paraffin wax and pour it into about 600 ml of boiling distilled water in a 1 L beaker. (Careful: the alcohol boils very vigorously.) Use this sol to examine the general statements made in the text. Note, for example, the Tyndall effect when a light beam is passed through the sol. Put a drop on a microscope slide and examine it in dark field illumination.

The rmody namics of Surfaces 2.1 Introduction 2.2 Surface energy and its consequences 2.2.1 Surface tension and surface free energy 2.2.2 Molecular origins o f surface tension 2.2.3 Pressure differences across curved surfaces equation

- the Young-Laplace

2.3 Thermodynamics of surfaces 2.3.1 Mechanical work done by a system with a surface 2.3.2 Surface excess quantities 2.3.3 Fundamental equations o f surface thermodynamics 2.4 The Gibbs adsorption equation 2.4.1 The relative adsorption 2.4.2 The general form o f the y-In c relation 2.4.3 Particular forms o f the Gibbs equation 2.4.4 Two liquid phases in contact 2.5 Thermodynamic behaviour of small particles 2.5.1 The Kelvin equation 2.5.2 Applications of the Kelvin equation (a) Drops of liquid in a vapour (b) Bubbles in a liquid 2.5.3 Effect of temperature on vapour pressure -the Thomson equation (a) Liquid drop suspended in i t s vapour (b) Bubble immersed in a liquid 2.5.4 Application of the Kelvin and Thomson equations t o solid particles 2.6 Equilibrium shape of a crystal 2.7 Behaviour of liquids in capillaries 2.7.1 Capillary pressure 2.7.2 Capillary condensation 2.7.3 Capillary rise in a powder 2.8 Homogeneous nucleation 2.9 Limits of applicability of the Kelvin and Young-Laplace equations 2.10 Contact angle and wetting behaviour 2.10.1 Adhesion, cohesion, and wetting

44

SURFACE ENERGY AND ITS CONSEQUENCES

145

2.10.2 Meniscus shape and wetting 2.10.3 Sessile and pendant drops and bubbles 2.10.4 Heterogeneous nucleation 2.10.5 Contact angle on heterogeneous and rough surfaces (a) Chemical heterogeneity (b) Surface roughness 2.1 1 Measurement of surface tension and contact angle

2.11 .I Contact angle hysteresis

2.1 Introduction Lyophobic colloids are thermodynamically meta-stable (Section 1.6). Nevertheless, the time-scale of many of the molecular exchanges occurring within a lyophobic colloidal suspension is very short compared with the lifetime of the suspension. Processes involving such molecular exchanges may, therefore, be treated by equilibrium thermodynamics. In particular, adsorption equilibrium at the particle surface is rapidly established, and so thermodynamics may be used to describe the effects of surface active materials, including ions, on the properties of the suspension. Such thermodynamic descriptions are an essential underpinning for the more detailed molecular descriptions with which much of this book is concerned. The analysis here follows the lines laid down in the introductory text by Aveyard and Haydon (1973) and especially the more extensive analysis provided by Defay et al. (1966), as translated by Everett. We begin with a discussion of the important concepts of surface tension and surface free energy and their origins at the molecular level. The consequent pressure difference across a curved interface is then calculated (the Young-Laplace equation). This gives rise to some important effects, which will be taken up after the basic equations of surface thermodynamics have been introduced. The thermodynamic treatment given here is intermediate between the rather oversimplified procedures, which merely demonstrate the reasonableness of the key results, and the rigorous procedures that are needed to cover all contingencies. In most cases we have chosen to simplify the notation by sacrificing a little generality rather than by sacrificing rigour on the grounds that the former is easier to recover than the latter. The consequences of pressure differences across curved interfaces are particularly relevant to colloidal particles (Sections 2.5 and 2.6) and are also important in capillary phenomena (Section 2.7). A number of specific problems are then discussed, finishing with some references to methods for measuring surface tension and contact angle (Section 2.11).

2.2 Surface energy and its consequences 2.2.1 Surface tension and surface free energy The existence of surface tension can be expected from the difference in energies between molecules at the surface and molecules in the bulk phase of a material.

46 I

2: THERMODYNAMICS OF S U R F A C E S

Consider first a homogeneous liquid or solid consisting of molecules of type A, in equilibrium with its vapour. Suppose that VAA(Y) is the potential energy of interaction of two molecules of type A separated by distance r, when the potential energy of an isolated molecule of A is taken as zero. Assuming that nearest neighbour interactions are dominant in a condensed phase, and potential energies of interaction are pairwise additive, the energy per molecule in the bulk phase becomes: EA,bulk

iZAA,bulkvAA(%b)

(2.2.1)

where ZU,bulk is the number of molecules in the shell of nearest neighbours in the bulk phase and q, is the average distance of these neighbours from the central molecule. The energy EA,Sper molecule at the surface is, similarly:

where we expect r, M r b and ZU,S % iZAA,bulk. Remembering that VAA is negative, it is clear that there is an increase in potential energy on taking a molecule from the bulk to the surface i.e. work must be done in creating a new surface. If the interface is between two condensed phases (say two liquids) consisting of molecules of types A and B respectively, then a molecule of A will lose about half of its interactions with A, but gain about an equal number of interactions with B, in moving from the bulk liquid to the interface. So

where the meaning of the subscripts is self-evident. T o a first approximation, provided the molecular species have similar sizes: ZAA,S

+ ZAB,S = ZAA,bulk = ZBB,bulk

(2.2.4)

so that

but

A similar argument applies to molecules of B. To create new surface, molecules of A and B must be brought to the interface. If the overall energy change

is positive, the interface will tend to shrink to its minimum possible area. However, if SE < 0, the interface will tend to grow and the phases will tend to dissolve in each other. [Strictly, we should deal with the free energy that would include an entropy contribution favouring dissolution even if E were positive, provided it was not too large.]

SURFACE ENERGY AND ITS CONSEQUENCES

147

The number of molecules in the surface is generally a small proportion of the number in the bulk; for example, a spherical droplet of water, of volume 1 cm, has only a fraction (2 x lo-’) of its molecules in the surface. Thus the energy of the surface molecules will make an important contribution to the total energy only for (a) processes where there is no change in the bulk energy, or (b) systems that are so subdivided that the surface energies are, in any case, comparable to bulk energies. Case (b) does not apply to most lyophobic colloidal systems though it does apply to lyophilic systems. For lyophobic systems, case (a) applies; i.e. the surface energy is important because the bulk energy is substantially unaffected by most colloid processes. From the above argument, it is clear that the work, 6w, required to create new surface is proportional to the number of molecules brought from the bulk to the surface, and hence to the area, 6 A , of the new surface:

6wcc6A

or

6w = y 6 A

(2.2.6)

where y, the proportionality constant, is deJined as the surface energy (Linford 1978) or the specific surface free energy (de Bruyn 1966). Note that it has dimensions of energy per unit area or force per unit length and for a pure liquid it is numerically equal to the surface tension. T o see the relation between these two concepts, consider the following thought experiment. If an arbitrary surface is extended as in Fig. 2.2.l(a), the increase in area 6 A is given by:

6 A = 16x.

(2.2.7)

The work done, 6w, in increasing the area is

6w = y 6 A = y16x.

(2.2.8)

I

Interface Total foyce

f Fig. 2.2.1 (a) Increasing the area of a surface of arbitrary shape. (b) The definition of surface tension. 6A is an element of area in the surface bounded by a perimeter of length I (see text).

48 I

2: THERMODYNAMICS OF S U R F A C E S

Table 2.7 Surface and interfacial tensions of some liquids (in mN m-‘) at 293 K (firom Aveyard and Haydon 1973, p . 70, with permission)

Liquid-vapour

Water-liquid

4y/d T (liquid-vapour) (mN m-l K-l)

Water

72.75

-

Octane

21.69

51.68

0.095

Dodecane

25.44

52.90

0.088

Hexadecane

27.46

53.77

0.085

Benzene

28.88

35.00

0.13

Carbon tetrachloride

26.77

45.0

Mercury

476

375

0.16

-

dy/dT for the hydrocarbon-water interface is 0.09 mN rn-’ K-’

If that work is done by a force F, which is applied to the perimeter, then:

6w = Fax

(2.2.9)

and so, comparing eqns (2.2.8) and (2.2.9): y = F/l.

(2.2.10)

That is, y acts like a force per unit length of the perimeter opposing any attempt to increase the area, i.e. like a restoring force or tension (Section 3.4 below). More formally (Fig 2.2.l(b), if we consider an area element in the surface, 6A bounded by a perimeter of length 1 then y is conceived as a force per unit length exerted by the region outside 6A to keep it from shrinking. y is called the surface (or interfacial)+tension, and is generally expressed as mN m-l (millinewton per metre, equivalent to dyne cm-’ in cgs units). For a liquid-liquid or liquid-vapour interface, the equilibrium value of y is independent of the orientation of the line element 61, so that the surface is in a state of uniform tension in every direction if the surface is quiescent. Typical values for the surface tensions of some pure liquids and interfacial tensions for liquid-liquid systems are given in Table 2.1. An important element in the above argument about ‘extending a surface’ or ‘creating fresh surface’ is that we mean new surface with the same properties as the original surface. In other words, the surface is created by adding new molecules from the bulk, maintaining the same properties of the different molecules in a mixed system, and with the molecules in their equilibrium configuration. If the increase in surface area were to be achieved by increasing the average distances between the molecules (in the surface +It is common usage to use the term ‘surface tension’ if one phase is a gas, but otherwise to use ‘interfacial tension’.

SURFACE ENERGY AND ITS CONSEQUENCES

149

and bulk), then the extension of the surface would be accompanied by a change in the bulk energy of the system. This would be an elastic deformation of the body; it would, in general, require more energy input and the work done would almost certainly depend non-linearly on the area increase, 6A. For a pure liquid in contact with its own vapour or with another pure, immiscible liquid, the surface tension is numerically equal to the excess surface free energy, as noted above. This is not the case if adsorbed species are present at the interface, or if the surface is solid. The surfaces of solids also possess a tension and confer an excess free energy on the solid. Nevertheless, it is both experimentally and conceptually difficult to treat the thermodynamics of solid surfaces in a similar way to that of liquid surfaces. The basic problem is that a freshly cleaved, or indeed very aged, solid surface is not in equilibrium, since the atoms are not, in general, free to move to positions of lower free energy. In a liquid, on the other hand, dynamic equilibrium is established very quickly. Thus, for example, a drop of spilt mercury will round up in a fraction of a second, whereas the pyramids have retained their shapes (except for a minimal dynamic interaction with tourists) over thousands of years.

2.2.2 Molecular origins of surface tension In the previous section we deduced the existence of surface tension from the difference in energies between a molecule in the bulk of the liquid and a similar molecule at the surface of that bulk phase. That argument is a convincing proof that the properties of a surface can be represented by a uniform tension in the surface. Equivalent presentations are common in the literature and in texts concerning surface phenomena. By contrast, there have been very few attempts to explain how an ‘unbalanced’ intermolecular force normal to the surface can be responsible for a stress parallel to that surface. In this section we will present a qualitative outline of Orowan’s (1970) more mathematical argument for the existence of a surface tension as a consequence of the attractive (and repulsive) forces between molecules. An understanding of this material is not required for what follows in the rest of the chapter, and readers content with the energy approach may wish to proceed directly to Section 2.2.3. Orowan’s argument depends upon the balance of forces on infinitesimal cubic elements of fluid near the surface. The forces arise from the pressure within the fluid so we first need to understand the molecular basis of pressure. It will be shown how this pressure is anisotropic near the surface and that this must lead to a tension in the surface. The pressure in a fluid in equilibrium is the time-averaged normal force per unit area exerted by all the molecules on one side of an imaginary test surface on all the molecules on the other side of the test surface. This pressure can be separated into two parts: 1. The kinetic contribution due to the transport of momentum by molecules moving across the test surface. Its value is given by:

pk = m k T

(2.2.11)

where PN is the number density of the molecules and k and T have their usual significance. This is the familiar pressure term in the kinetic theory of a perfect gas. It is the same for a liquid and is positive.

50 I

2: THERMODYNAMICS OF S U R F A C E S

2. The cohesive contribution due to the time average of the net attractive and repulsive forces between molecules on opposite sides of the imaginary test surface in the body of the fluid. This second contribution, called the static pressure, p', is normally negative (i.e. the attractive force dominates) and it is particularly important for dense gases or liquids. The total pressure in a liquid is thus less than the kinetic pressure, and must be equal to the applied pressure (i.e. the vapour pressure for a one-component system) despite the fact that the kinetic pressure is so much higher in the liquid (because p~ is higher). T o simplify the argument we will neglect the repulsive forces when considering the static pressure; the repulsive forces have a much shorter range than the attractive forces and are important only at extremely high external pressures. The intermolecular forces can be considered to have a sphere of influence beyond which they are negligible (an idea introduced by Laplace in 1806). For molecules further from the surface than the diameter of this sphere, the pressure must be isotropic because of the symmetry of their surroundings. Near the surface between the two bulk phases, the tangential and normal contributions to the static pressure are not the same because of the asymmetric distribution of molecules within the sphere of influence. When the sphere of influence cuts the surface (Fig. 2.2.2) there are fewer and fewer pairs of molecules attracting across the test plane as the centre of the sphere approaches the surface and hence the magnitude of the static pressure contribution decreases for both the pressure across a plane parallel to the surface (Fig. 2.2.2(a)) and for the pressure across a plane normal to the surface (Fig. 2.2.2(b)). It is apparent that when the test plane is in the surface (Fig. 2.2.3), the decrease in magnitude of the normal static pressure is greater than the decrease in magnitude of the tangential static pressure because of the smaller number of interactions remaining in the former case. Now consider the consequences of the fact that the forces on opposite sides of an infinitesimally small cube must be equal and opposite for mechanical equilibrium. The net pressure normal to the surface must be constant right through the surface. The contribution from the kinetic and static pressures normal to the surface are shown schematically in Fig. 2.2.4(a). The normal pressure is a constant equal to PO, the pressure in the bulk phases. (We are neglecting here the hydrostatic pressure at different depths in the liquid.)

Fig. 2.2.2 Sphere of influence around an imaginary test plane near a liquid surface. (a) With test plane parallel to the surface; (b) with test plane normal to surface. In both cases the attraction between molecules on opposite sides of the test plane must be less than in the bulk of the liquid because of the deficiency of molecules in the region of vapour in the sphere of influence.

SURFACE ENERGY AND ITS CONSEQUENCESI 5 1

Fig. 2.2.3 Sphere of influence around imaginary planes in the liquid surface. (a) With test plane parallel to the surface; (b) with test plane normal to the surface. Note the attraction between molecules in quadrants (I 11) and (I11 IV) in (a) will be very small because there are very few molecules in quadrants I11 and IV. Attraction between molecules in quadrants (I IV) and (I1 111) in (b) will be much greater as quadrants I and I1 are densely populated.

+

+

+

+

4 Pressure k

P" Liquid Vapour

- - - - - - - - Pn'Po

-

(a>

Surface region

A Pressure

P: Liquid Vapour \

I

*Z PS

(b)

, , Surface region

Fig. 2.2.4 (a) Variation of the kinetic (a:), the static (a:) and the net (p,) pressures normal to the vapour-liquid interface. (b) Variation of kinetic (a:), static (as), and net (at), pressures tangential to the vapour-liquid interface as a function of position.

52 I

2: THERMODYNAMICS OF S U R F A C E S

A similar graph for the pressure tangential to the surface cannot give a constant net pressure because the kinetic contribution is identical to that for the normal pressure k

k

pt = p , = PNkT while the static contribution is different (Ps # p i ) . The resultant pressure, p,, is less than $0 on passing through the surface. The resulting net stress is the surface tension and is equal to the integral of the deficit of the tangential pressure through the surface layer: (2.2.12) -co

It is not, of course, an unbalanced force, but the balancing force must be applied externally, for example by elastic deformation of the walls of the container. There may appear to be no balancing force available for isolated liquid droplets, but we will find that there is an increased hydrostatic pressure within the droplet, resulting from surface tension, and this provides the necessary restoring force.

2.2.3 Pressure differences across curved surfaces -the Young-Laplace equation The tension in a surface must be balanced by some equal and opposite force. For an isolated particle, or droplet, the balancing force must come from stresses generated within the particle or droplet by the surface tension itself. The stresses so generated? depend, for liquids, on the surface tension and on the curvature of the surface. (The corresponding case for solids will be discussed later (Section 2.6).) This simple experimental fact was discovered by Hauksbee in 1709, but nearly a century passed before Young (1805) deduced the correct theoretical relationship between capillary pressure and the surface curvature. According to Rayleigh (191l), Young's result was 'rendered obscure by his scrupulous avoidance of mathematical symbols' and it was left to Laplace (1806), working independently, to publish the first algebraic equation (as an appendix to a ten-volume work on astronomy!) linking capillary pressure, p c , and meniscus curvature (Klein 1974):

where y is again the surface tension, and rl and r2 are the radii of curvature of any two normal sections of the surface perpendicular to one another. This equation is now known as the Young-Laplace equation (sometimes, unfairly, the Laplace equation (Pujado et al. 1972)) and is easily derived as follows (Defay et al. 1966). Consider a small part ACBD of the surface of a static liquid drop (Fig. 2.2.5(a)). The drop need not be spherical, and the circumference of the selected part of the surface is simply defined as a line in the surface at a constant distance, d, from a chosen point X on the surface. Through X draw any pair of orthogonal lines AB and CD in the surface. ?Known generically for liquids as the 'capillary pressure'

SURFACE ENERGY AND ITS CONSEQUENCES

'

\

\

\

153

$1

'\

\

Fig. 2.2.5 (a) Equilibrium of a liquid cap. (b) A curved surface with no pressure drop across it because 1/& = - (1/Rz). (After Adamson and Gast 1997, p. 9.) (See Exercise 2.10.6.)

For a sufficiently small value of d, AB, and CD can be considered as parts of circles with radii ~1 and r2 respectively. It is known from the differential geometry of surfaces (Weatherburn 1930) that the directions of AB and CD can always be chosen so that ~1 is a maximum (R1) and r2 a minimum (Rz).Furthermore, 1

-

r1

+ 1 = 1 + 1 =J -

-

-

r2

R1

R2

(a quantity independent of the orientation of the axes) (2.2.14)

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2: T H E R M O D Y N A M I C S OF SURFACES

(3is often referred to as the mean curvature of the surface.J-’is the harmonic mean of R1 and R2.) Now consider the forces exerted by the cap of liquid surface ACBD. The surface has a tension y , so that, for example, an element 61 of the boundary at A parallel to XY and given by exerts a resolved downward force Fvert Fvert = ySl sin 4 + y(d/Rl)Sl as d + 0

(2.2.15)

The total downward force exerted by four similar elements at A, B, C, and D is

ySl

2d 2d -+(Rl

R2) -2dySl -

-+- $,) .

(di

Since this expression is independent of the choice of AB and CD, it can be integrated around the circumference (one quarter of a revolution, since there are four segments) to give the total downward force due to surface tension of

But the drop is not in motion, so there must be an upward force to balance this downward force. The upward force is provided by a pressure difference p” -p’ between the inside and outside of the drop. Equating the two forces: (2.2.16)

or

Ap(= p”

-

p’) = y

(dl + dJ -

-

(2.2.17)

which is the Young-Laplace equation. In qualitative terms, the surface tension tends to compress the droplet, increasing its internal pressure. The opposite situation arises with a concave liquid surface, as occurs with a bubble in a liquid or the meniscus of a wetting liquid in a capillary. Here the pressure in the liquid is lower than that outside it. The Young-Laplace equation caters for this situation since R1 and R2 are now negative, so that Ap is also negative. The radius is taken as positive if the corresponding centre of curvature lies in the phase in which p” is measured, and negative in the converse case. For a liquid in equilibrium Ap must be constant across all parts of its free surface, otherwise liquid flow would occur down the resulting internal pressure gradient unless an external field (e.g. gravity) were balancing the pressure change. Since y is also a constant, it follows that, in the absence o f externalJields all liquid surfaces are surfaces of constant mean curvature 1/R1+ 1/R2. For an open film (e.g. a soap film on a wire frame) Ap is necessarily zero. This could mean rl = r2 = 00 (a flat film), but could also mean r1 = 9 2 at all points on the surface, even though both rl and r2 change from point to point. This situation is illustrated in Fig. 2.2.5(b) which shows the shape expected for a soap film pulled

SURFACE ENERGY AND ITS CONSEQUENCES

155

between two open cylindrical pipes. Similar considerations apply to a liquid drop whose shape is unaffected by gravity, a situation that can be achieved experimentally by floating the drop in an immiscible liquid of equal density. The Young-Laplace equation (eqn (2.2.17)) has been introduced at this stage to complete the argument of Section 2.2.2 and to provide us with the important relation (see Exercise 2.2.2):

(d, d,)

dA= - + - d V

(2.2.18)

which will be used in several contexts in our discussion of surface thermodynamics. Equation (2.2.17) is actually a complicated differential equation, since R1 and R2 are second-order differential functions of the Cartesian coordinates and its solution under appropriate boundary conditions describes the shape of surfaces like that in Fig. 2.2.5(b) or the shape of liquid menisci (Section 2.10.2).

r

Exercises 2.2.1 Calculate the number of molecules in 1 cm3 of water from the molar volume (18 an3).If it is in the shape of a 1 cm cube show that the ratio of surface energy to bulk energy is approximately lop7. How big a collection of molecules would be needed so that the surface energy was the same as the bulk energy? 2.2.2 Consider the increase dA in area of a small surface element and the concomitant volume, dvtraversed by the surface (Fig. 2.2.6). Show that dA = (1/R1 1/R2)dV where R1 and R2 are as defined in Section 2.2.3 above.

+

Volume dV

/

Fig. 2.2.6 Relation between increase in volume and surface area for a surface of arbitrary curvature. Both radii increase by d R at constant angle 681, 60,.

2.2.3 Use the result derived in Exercise 2.2.2 to establish the Young-Laplace equation using an energy minimization argument. 2.2.4. Calculate the excess pressure inside drops ofwater of radius lop5cm and lop6cm, respectively. (Take y = 70 mN m-'.)

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2: THERMODYNAMICS OF S U R F A C E S

2.3 Thermodynamics of surfaces The presence of a surface introduces an additional factor to be considered in the thermodynamics of such a system, since changes in the surface area imply that there is work being done either on the system or by the system on its surroundings. The equations of bulk phase thermodynamics thus need modification if surface changes are contributing significantly to the total energy changes in the system. A summary of useful bulk phase thermodynamic relations is given in Appendix A5.

2.3.1 Mechanical work done by a system with a surface If a one component, one-phase system expands by a volume d V against an external pressure p, the mechanical work done on the system is given by

(2.3.1)

dw = -pdV

(i.e. the work done by the system is a negative contribution to the internal energy.) If, however, the system contains two phases, we know from the Young-Laplace equation (2.2.17) that, unless the interface between the phases has zero curvature, the pressures p' and p" in the two phases will be different. Hence the total work done by the system will be dw = -p'd V' - p"d V"

+ ydA.

(2.3.2)

Consider, for example the work done during evaporation of a liquid droplet having 1/&) (Fig. 2.3.1). If the some arbitrary shape of constant mean curvature (1/R1 V" and total volume of the system is V and the external pressure is p' then V = V' d V = d V' d V". The work done on the system by its surrounding is:

+

+

+

P' .dV

Vapour

P' Liquid@'' V'y

V'

Fig. 2.3.1 Evaporation of a droplet whose surface has an arbitrary (but constant mean) curvature.

THERMODYNAMICS OF SURFACES

157

which may be written (2.3.3) Introducing eqn (2.2.17): dw = -9’d V’ - p”d V”

+ y(1/R1+ 1/Rz)d V”

(2.3.4)

using eqn (2.2.18). One can thus identify a mechanical work term for each of the phases and a ‘surface work’ term for the interface, without having to introduce a hypothetical surface piston or the stretching effect depicted in Fig. 2.2.1. It is useful at this stage to consider the droplet in Fig. 2.3.1 as a sphere of radius r. Equation (2.3.4) then becomes: dw = -p’d V’ - p”d V”

+ (2y/r)d V”.

(2.3.5)

There is, however, a gradient in density at the surface of the droplet, and for very small droplets the distance over which the gradient extends may actually be a significant fraction of the total radius of the droplet. It could reasonably be argued that the droplet ‘surface’ is anywhere within the region of changing density, and each arbitrary choice would give a different value of r (and of d V’ and d V”). Fortunately, however, we know that V” = 4nr3/3 so that dV“ = 4nr2dr and

A = 4nr2 so that dA = 8nrdr = (2/r)dV”.

(2.3.6)

It seems, then, that eqns (2.3.2) and (2.3.5) are equivalent, no matter where we choose to place the surface. Equation (2.3.5), however, incorporates the Young-Laplace equation, which defines the value of r to be used, sincep“-p’ would be different for any other value of r. The imaginary, infinitesimally thin surface defined by this value of r is called the surface of tension. What we have done is to replace the real droplet, with its rather fuzzy radius, with an imaginary droplet having a sharp boundary and whose mechanical properties are equivalent to those of the real droplet.

2.3.2 Surface excess quantities The presence of the surface affects virtually all of the thermodynamic parameters of a system. It is very convenient to think of a system containing a surface as being made up of three parts: two bulk phases, of volumes V’ and V”, and the surface separating them. Any extensive thermodynamic property, like the energy, U, of the system, can then be apportioned between these parts as follows. If the energies per unit volume in the two phases are u’ and u”, then the total energy of the system due to contributions

58 I

2: THERMODYNAMICS OF S U R F A C E S

from the bulk phases must be u' Vf u" must then be given by:

+ u" V". The energy to be ascribed to the surface, (2.3.7)

Other surface quantities such as the surface entropy, S" surface Helmholtz free energy, P and surface Gibbs free energy, G" may be similarly defined. It must be clearly understood that these surface quantities can only be deJined in terms of a particular model system and so their values will always depend on the model chosen. The model described above (and indicated by superscript 0)is called the Gibbs convention. Note that by dividing the total volume V into the precise volumes V'and V" we have, in effect, constructed an imaginary system having the same thermodynamic properties as the real system but with the two phases separated by an infinitesimally thin dividing surface and having constant densities up to that surface. This ideal system replaces the real system, with its finite dividing region where the density is rapidly, but not discontinuously, changing. As with the droplet radius (Section 2.3.1), the only reason for replacing the real system with a model is because it is easier to think of such extensive quantities as energy and volume in discrete lumps rather than as continuously varying quantities. It also means that intensive quantities like pressure and density are given definite values in each phase even when the interface is curved. The procedure was invented by Gibbs, and the imaginary dividing surface is called the Gibbs dividing surface (Gibbs 1874-8). Other procedures are possible (e.g. Guggenheim 1976) for planar surfaces, but generally become impossibly cumbersome if the surface is curved. Melrose (1968) presents a very full and clear discussion of the Gibbs treatment of curved surfaces and its integration with hydrostatic treatments of the stress in the surface. The presence of the interface also affects the molecular composition. In a twophase multicomponent system we can write L and c" as the concentrations of component i in each phase and if the volumes of the phases are again Vf and V"(where V' V" = V) then the numbers of moles of component i in each phase are:

+

nif = ciI Vf

and

ny =

4V".

(2.3.8)

Once again, the extra amount of i that can be accommodated in the system because of the presence of the interface is evidently:

(2.3.9) where ni is the total number of moles in the whole system. Note that all of the quantities on the right-hand side of eqn (2.3.9) are unambiguously defined. A surface concentration can be defined as: n y / A = ri.

(2.3.10)

The notation r; for the surface (excess) concentration of component i is used almost universally and we will use it henceforth. Note that the value of ri may change dramatically not only in magnitude but even in sign, as the chosen dividing surface is

THERMODYNAMICS OF SURFACES

159

Phase 11

-----Interfacial region

---- - - Phase I 0 c: c:

Ci

Fig. 2.3.2 The concentrations of a component i across an interface in a mixture (a) can be plotted and will generally show rapid changes through the interfacial region (b). The Gibbs model (hatched regions in (b)) ascribes constant compositions to both phases up to an arbitrarily defined interface. The excess material (dotted region in (b) is ascribed to the infinitely thin Gibbs surface GG', (c). A different arbitrary choice of Gibbs surface (d) can reduce the surface concentration to zero (note equality of dotted and hatched areas in (d)) or even make it negative. In practice, it is useful to choose GG' so that ny for the solvent is zero. The concentrations of all other components must then be referred to this same surface. Note that for curved surfaces this choice of dividing surface is very unlikely to coincide with the surface of tension (Melrose 1968), and so the model system would not be mechanically equivalent to the real one.

varied even by a fraction of a nanometre (Fig. 2.3.2). For a curved surface the area, A, also depends on the choice of dividing surface and this has a further effect on ri.

2.3.3 Fundamental equations of surface thermodynamics A general infinitesimal reversible process in a two-phase system will result in infinitesimal changes to the various thermodynamic parameters. Thus U+ U+dU S+S+dS ni

+ ni + dni

V' V"

+ V' + dV' + V" + dVff

A+A+dA.

(2.3.11)

For the bulk phases (cf. eqn (A5.1)):

(2.3.12) (2.3.13)

60 I

2: THERMODYNAMICS OF S U R F A C E S

while for the Gibbs surface:

Adding the above three equations, and remembering that u” = U - U‘ - Uf,S“ = S Sf - S“, we obtain a major governing equation in surface thermodynamics: d U = TdS -p’dV’ -p”dV”

+ ydA + Ci(p :d n : + pydny + pPdnP).

(2.3.15)

The first term on the right is the heat absorbed by the system, the next three terms are the mechanical work done on the system, while the last term is the chemical work done on the system. Equation (2.3.15) leads to some interesting results. First, consider a situation in which the infinitesimal processes occur in a constant total volume, V, at fixed temperature, T. Then the change, dF, in the Helmholtz free energy of the system is given by: d F = d(U - TS) = d U - TdS - -pfd

vf

-

p“dV“ + ydA

+

i(p:dn:

+ pydny + pPdnP).

(2.3.16)

But the system does no mechanical work at constant volume, so the sum of the first three terms on the right must be zero. Also, since dV’ = -dV” we have: -pNdVff -p’dV’

+ ydA = 0 = (p’ -p”)dV” + ydA.

(2.3.17)

Now using eqn (2.2.18): (p”

-

p’)d V” = y( 1/R1

+ 1/Rz)d V”

(2.3.18)

from which the Young-Laplace equation follows immediately. Note that it is here derived on purely thermodynamic grounds. Returning to eqn (2.3.16) we can introduce the condition dn:

+ dny + dn;

=0

(2.3.19)

to rewrite it in the form:

(2.3.20) Since F is a minimum in a closed isothermal system at fixed volume (i.e. d F = 0) we have (2.3.21) pi = py = &’(= pi, say) for all i. We thus reach the very reasonable conclusion that, at equilibrium, the chemical potential of any component is the same in each bulk phase and at the surface.

THERMODYNAMICS OF SURFACES

I61

We can now derive a series of formulae, of varying usefulness, relating to the other thermodynamic parameters. From eqn (2.3.14):

(2.3.22) This one is not very useful since there is no practical way to hold 3' and n: constant whilst varying u" . The corresponding result from eqn (2.3.15):

Y=

(E)

(2.3.23)

S,V', V",[nJ

is rather more useful but a better definition flows from the Helmholtz free energy. Recall that for a system consisting of a single phase (eqn (A5.21)):

u = TS

-

PV

+ C pin;.

(2.3.24)

The integration procedure? that leads to this equation (see Appendix AS) can be applied to each of the phases (') and (") and to the surface to yield:

U" = TS"

+ YA+

c

pin:.

(2.3.25)

F = U - TS = Ff +F" +I;"

(2.3.26)

Then from the definition of F

we have (Exercise 2.3.1):

I;" = U"

-

TS" = y A + c p i n p

and, hence (Exercise 2.3.1): dF" = dU" - TdS" - Y d T = -S"dT

+ ydA +

(2.3.27)

c

pidn:

(2.3.28)

We can, therefore define y thus:

(2.3.29) Alternatively, since (Exercise 2.3.1): d F = -SdT

-

p'd V'

-

+

pffdVff ydA

+

c

pidni

(2.3.30)

we have:

(2.3.3 1)

+The integration procedure corresponds to increasing the size of the system whilst keeping all the intensive properties of the system constant (in particular keeping the composition of the system constant). It is only in this case that the integrated form of the equation makes sense.

62 I

2: THERMODYNAMICS OF S U R F A C E S

which is probably the most useful definition since the subscripted variables can readily be held constant experimentally. Note also that (from eqn 2.3.27):

where ri is defined in eqn (2.3.10) and f " is the Helmholtz free energy per unit area of the surface. This equation shows that f " is not equal to the surface tension, y, except for a specific choice of the Gibbs dividing surface, namely that where C piri = 0; this is a very easy and natural choice for a one-component system, but highly unusual for multicomponent systems. This is a convenient point to clear up a problem that often arises in the literature. It is frequently stated that surface tension and 'specific surface free energy' (or some such term) are equal only for a one-component system. Such statements can be properly interpreted only when the terms are carefully defined (Morrow 1970). The surface tension is defined by eqn (2.3.31) and if y is independent of area (which will be so if there is an infinite reservoir of any adsorbing species) then this equation can be integrated to give: W ) T , V / , V y n ; ]= YAA (2.3.33) or

Y = (AF/AA)T,,,,V.,[ni].

(2.3.34)

Thus, the change in Helmholtz free energy of the system (model independent) per unit change in surface area is numerically equal to the surface tension, y, and this is true for a system with any number of components provided y is independent of area. On the other hand, for the model system

[2.3.32] and the specific Helmholtz free energy ascribed to the surface (and model dependent) is only equal to y for a particular choice of dividing surface, such that C piri = 0. This choice of dividing surface is a natural one only for a one component system.

Exercises 2.3.1 Establish eqns (2.3.27), (2.3.28), and (2.3.30). 2.3.2 The surface excess Gibbs free energy may be defined either as

G=U"-TS"+~V"

or

G = U" - Ts" +pV" - yA.

Show that the latter definition leads to

G" =

C/.& i

which is analogous to the bulk phase equation (eqn (A5.25)). [Note that V" = 0 in the Gibbs model.]

G I B B S ADSORPTION E Q U A T I O N

I63

2.4 Gibbs adsorption equation The single most valuable equation in surface thermodynamics is the Gibbs adsorption isotherm. It is derived for surfaces in the same way that the Gibbs-Duhem equation (eqn (A5.23)) is derived for bulk phases. Briefly, integration? of eqn (2.3.14) yields:

(2.4.1) Differentiating eqn (2.4.1) and comparing it with eqn (2.3.14) we have the Gibbs adsorption equation: YdT

+Ady + C npdpi = 0.

(2.4.2)

We can then write, at constant temperature, the Gibbs adsorption isotherm:

(2.4.3) This equation is one of the most widely used expressions in surface and colloid science and we will explore its meaning at some length. It can be applied to systems where the surface tension can be measured (e.g. those containing liquid-liquid or liquid-vapour interfaces) in order to calculate the surface concentration of the adsorbed species causing the surface tension change. Equally, if the surface concentration can be measured directly but y cannot (as occurs in many solid-gas and solid-liquid systems), the Gibbs adsorption equation can be used to calculate the lowering of y (i.e. the spreading pressure, ll)from the measured adsorption. Unfortunately, the absolute value of ri is extremely dependent on the choice of dividing surface (see Fig. 2.3.2). We have already noted though (Fig. 2.3.2) that the Gibbs dividing surface is normally chosen so that n: and, hence, I': for the solvent are equal to zero so that all other components are measured with reference to that surface, giving the relative surface concentrations. We will see shortly how to define the relative surface concentration operationally so that its value is independent of the position of the dividin plane. The resulting (experimentally useful) quantity, ri,1, (sometimes written ri ) is the surface excess of i relative to the solvent (1) and its value is numerically equal to I'y with the convention I'y = 0. The superscript (T in this case refers only to the use of a Gibbs dividing surface. We will also use the convention that r? = 0 for the solvent, so our ri,1will always be equal to rjl).

cB

2.4.1 The relative adsorption The Gibbs-Duhem equation (eqn (A5.23)), when applied to the two bulk phases at constant temperature, gives (remember pi is the same in both phases): N

V'dp' =

N

nidpi 1

and

Vf'dpf' =

nbdpi 1

+Recallthat this integration process is done whilst keeping the intensive variables (and in particular the composition) constant.

(2.4.4)

64 I

2: THERMODYNAMICS OF S U R F A C E S

and so, since the concentration cj is nJ V in each phase: N

N

dp' = x c : d p i

and

(2.4.5)

dp" = x ( d p i .

1

1

If the interface is planar or if its curvature does not change so that (p" - p') is constant, then dp" = dp' Equations (2.4.5) then give?, for the variation in p1:

(2.4.6) We can now use this expression to eliminate the unknown adsorption equation to obtain (Exercise 2.4.3):

p1

from the Gibbs

(2.4.7) The quantity inside the square brackets is defined for each component i as the relative adsorption of that component and is written:

ri,l= ri - rl-ACj

Ac1

(2.4.8)

4 4.

where Ac, = Although it is not essential, it is customary in the case of a liquid-vapour interface to take the solvent as component 1 and to compare the adsorption behaviour of all other components to it, since this choice leads to the definition of quantities that are directly accessible from experiments. The definition of ri,1brings with it several advantages:

1. The value of r,,lis independent of the arbitrary choice of dividing surface. This is obvious from our derivation of eqn (2.4.7) since it depends only on general thermodynamic considerations. In a two-component system the quantity

is a direct measure of -dy/dp2 and since this is a physically defined quantity it cannot depend on the (arbitrary) choice of dividing surface. T o make this point clear we will analyse in detail the expression for r2,1. Using the Gibbs convention: ny = n, - VI cjI - V' I c.N by definition = n; - 4)

~4+ ~"(4

+Ifthe curvature changes these equations become more complicated.

(2.4.9)

G I B B S ADSORPTION E Q U A T I O N I65

where V’ and Y” are the volumes of phases (’) and (”) respectively and Y = Y’ V” is the total volume. ny and ni are, as before, the number of moles ascribed to the surface and total number of moles in the system respectively. For component 1, eqn (2.4.9) gives:

+

ny = nl - Vci

+ V’’
(2.4.10)

and eliminating V’between the two eqns (2.4.9) and (2.4.10) (Exercise 2.4.3):

np - ny(Ac;/Acl) = (ni - Vd)

-

(nl

-

Yc‘,)(Aci/Ac1).

(2.4.11)

None of the quantities on the right-hand side of eqn (2.4.11) depends on the choice of dividing surface, and if eqn (2.4.11) is divided by the surface area, A, we recover the right-hand side of eqn (2.4.8), which defines rj,1.For a curved surface it may be shown that A is the area of the surface of tension (Buff 1956). 2. r;,l has an intuitively appealing physical meaning. Since ri,1 is independent of the choice of dividing surface, we are at liberty to pick any dividing surface that suits us. In particular, we can choose the surface where rl = 0 (without needing to be able to specify its position experimentally to a fraction of a nanometre). For that surface, ri,l = r,.In other words, the relative adsorption of component i with respect to component 1 isjust the surface excess concentration of component i at the surface where the adsorption of component 1 is zero. As a simple example, consider adsorption at the gas-liquid interface (e.g. in a foam or aerosol). If phase (”) is the gas phase, then cy= c;= 0, and so

where x; are mole fractions. The same equation holds for adsorption from a solution onto a solid surface. In particular, if the solute is very dilute then

ri,l= ri = n p / A = (ni

-

(V’)/A

(2.4.13)

i.e. one can identify the relative adsorption or the surface excess simply by determining the number of moles of solute that appear to have been removed from the bulk solution and dividing by the surface area of the solid. Such direct measurements of adsorption can be done using radio-isotopes or by measurement of concentration changes in the bulk solution when the solid adsorbent is added. 3. r;,l can be introduced into the Gibbs adsorption equation, and hence related to experimentally measurable quantities. We restrict ourselves to the simplest case, that of a two- component system, for which, from eqn (2.4.3):

This equation describes how the surface tension of a solution of, say, component 2 in solvent 1 is changed as the activity of substance 2 (and hence p2) is altered, at constant temperature.

66 I

2: THERMODYNAMICS OF S U R F A C E S

If we choose the dividing surface so that rl = 0 then

so that

:

p2

rz = r2,1and so: (2.4.15)

-dY = F2,1dP2. We can also write

r2 =

= p!

+ RTIna2

dp2 = RTd In a2

[ 1.5.121

(2.4.16)

where a2 is the activity of component 2. It follows that:

(2.4.17) If the activity coefficient of component 2 is y2, so that a2 =~ 2 x 2we may write eqn (2.4.17) as (Exercise 2.4.2):

(2.4.18) where x2 is the mole fraction of component 2. For ideal solutions:

(2.4.19) and if the solution is fairly dilute so that x2 a 62 (the molar concentration) then

(2.4.20) which is more obviously an adsorption isotherm. It gives the relationship between the amount adsorbed and the solution concentration, 62, in terms of the effect which c2 has on the surface tension. Substances which lower the surface tension will have positive values of r; they are said to be surface active. Some typical values of y as a function of solution concentration for various aliphatic alcohols are given in Fig. 2.4.1. Note how rapidly y decreases with increase in solution concentration, indicating that r2,l is large in these cases. Note also the increase in adsorption as the chain length increases. These alcohols would be described as moderately surface active. We will find that, by contrast, electrolytes tend to raise the surface tension of water, indicating that they are negatively adsorbed at the air-water interface (i.e. they tend to be repelled towards the bulk of the liquid). The Gibbs adsorption equation has been tested experimentally by McBain and Humphreys (1932), who used a flying microtome to scoop thin (0.1 mm), hopefully uniform, surface layers from a series of aqueous surfactant solutions. The relative

G I B B S ADSORPTION E Q U A T I O N I67

Fig. 2.4.1 Surface tension of aqueous solutions of n-aliphatic alcohols. A-E: G-Clo primary alkanol. (After Defay et al. 1966, with permission).

adsorptions r 2 , 1 of the surfactant (relative to the water) were calculated from the mole fractions of surfactant and water in the scooped-up and bulk phases, together with the volumes of scooped-up liquid and the surface areas. Comparison of these experimental values of r 2 , l with those calculated from eqn (2.4.20)showed good agreement (Defay et ul. 1966). The remaining discrepancies are no doubt attributable to the experimental difficulties rather than the thermodynamics.

2.4.2 The general form of the y - In c relation Figure 2.4.2 gives examples of the most commonly observed behaviour of the surface tension of a solution as a function of solute concentration. Curve I is the behaviour that was displayed in an alternative form in Fig. 2.4.1. It is characteristic of solutions of substances that are to some extent lyophobic towards the bulk solvent, so that the change in y with increasing concentration is (from eqn (2.4.20)), negative. Such substances will tend to accumulate at the surface in preference to remaining in the bulk. If the bulk solvent is water then most polar organic molecules, if they are reasonably soluble, will behave like curve I. By contrast curve I1 is the behaviour expected of a lyophilic solute. The most obvious examples in aqueous solution are the ionic salts which are negatively adsorbed at (i.e. repelled away from) the air-water interface and so raise the surface tension. The same behaviour is also observed with hydrophilic solutes like sucrose. Curve I11 shows the behaviour of a highly surface active substance - a true surfuctunt- such as a detergent (i.e. an amphiphilic substance (Section 1.4)). Even at very low concentrations such solutes have a profound effect on y and they have a great many uses in colloid science as a consequence. These we will discuss in more detail

68 I

2: THERMODYNAMICS OF S U R F A C E S

Y I1 Pure solvent

'"W

1

1

I11

I C .m.c.

Concentration of solute

Fig. 2.4.2 The three types of curves commonly observed when the surface tension of a solution is plotted against concentration of solute. (After Davies and Rideall963.) The minimum in curve I11 is due to impurities and disappears after rigorous purification.

later. At quite modest concentrations (often -lop3 - lO-'M) the decrease in surface tension ceases with the formation of micelles. The critical micellization concentration (c.m.c, Section 1.4.3) is indicated in the figure. The minimum shown in the figure is caused by the presence of impurities? which are usually present in commercial preparations. It disappears on rigorous purification.

2.4.3 Particular forms of the Gibbs equation We have already shown in eqns (2.4.12) and (2.4.13) the appropriate forms of the Gibbs equation for adsorption of a simple solute at the liquid-gas or liquid-solid surface where there is no penetration of the solute or the solvent into the second phase. These equations become even further simplified if nearly all of the solute is adsorbed, as is the case with insoluble monolayers, for then

For surface active electrolytes (e.g. an anionic detergent like R.S030-Na+ an alkylsulphate) adsorbing at the air-water interface we have: NaA % Na+

+ A-

where A- = R.S030-. Assuming the salt is fully ionized, then

+Sodium dodecyl sulphate, for example, is commonly contaminated with the dodecyl alcohol which is formed by hydrolysis.

G I B B S ADSORPTION E Q U A T I O N

I69

Using the Gibbs convention ( r H z O = 0) and assuming that the interface as a whole is electrically neutral we have r N a + , l = r A - , 1 and so:

or

dy = - 2 r ~ - , l R T d l n a *

(2.4.24)

where we have introduced the mean ionic activity defined by a: = ayay-

(2.4.25)

+

with v = v+ v-, the number of ions produced when the molecule dissociates into v+ cations and v- anions. For a dilute solution in which the activity coefficienty (compare eqn 2.4.18) is close to unity we have: (2.4.26) Note that the reduction in y for a given (molar) adsorption is doubled in this case: it is a colligative property like osmotic pressure and freezing point depression and depends on the number of solute particles in the interface. If a large excess of a simple sodium salt (say NaCl) is added in addition to the surfactant then the system can be held at constant ionic strength. The chloride ion can be assumed (Fig. 2.4.2, curve 11) to have negligible surface activity (ra-M 0) and the chemical potential of both the chloride and sodium ions can be held essentially constant as the concentration of NaA is altered. In that case the counterpart to eqn (2.4.23) is:

so that

(2.4.27) Note that the factor 2 has disappeared in this case. This difference in behaviour of a surfactant in the absence and in the presence of a swamping concentration of electrolyte is quite significant.It can be investigated with the use of radiotracers such as tritium (3H)that undergo p-decay. The p-radiation is rapidly absorbed in water so that only radiation originating in the surface layers can escape. Using appropriately labelled surfactants one can directly measure the amount of surfactant in the surface layers (see Hiemenz 1977, p. 282).

2.4.4 Two liquid phases in contact The phase rule applied to a system containing only plane interfaces reads (Defay et al. 1966, p. 77):

70 I

2: THERMODYNAMICS OF S U R F A C E S

f=Z+C-R-P-$+S

(2.4.28)

where f is the number of degree of freedom, C the number of components, R the number of chemical reactions between components, P the number of phases, $ the number of surface phases, and S the number of surfaces. Assuming $ = S (the usual case) and R = 0 we obtain the more usually stated form of Gibbs’ phase rule:

f =2+c-P.

(2.4.29)

The number of variables which must be specified in order to specify the state of the phase (in the absence of external fields) is the usual 2 (temperature and pressure) plus one each for the chemical potentials of each of the constituents (i.e. 2 C). There is, however a Gibbs-Duhem equation (see Appendix A5.3) relating all the chemical potentials of the constituents of each phase so this gives P restraints; the result is equation (2.4.29). We have, for two phases in contact:

+

f = C Even this very simple form of the phase rule reveals a curious paradox. In a dispersion of solid particles in a liquid medium we often refer to the surface area of the interface between the two so we are inherently regarding the system as consisting of two phases. Yet a similar dispersion of a protein in water would naturally be regarded as a single phase system. How can it be possible to regard one dispersion as a two-phase and the second as a one-phase system, and how does this affect the application of the Gibbs phase rule? Kruyt (1952, p. 11) discusses this question and shows that one can choose to treat a colloidal system as either one- or two-phase, whichever is most convenient. The usual lyophobic colloid is best treated as a two-phase system since the disperse phase has a negligible effect on the properties of the dispersion medium. If one chooses to regard it as a single phase system one would have to acknowledge that the chemical potential of the sol is not effectively a variable so that both C and P in equation (2.4.29) are reduced by one and the value off remains unchanged. The difference in behaviour of the two types of system can be illustrated by considering the vapour pressure of the dispersion medium as a function of temperature. For the lyophobic system, the vapour pressure is essentially unaffected by the concentration of the colloid whereas the lyophilic colloid will lower the solvent vapour pressure at any temperature, to an extent depending on its concentration. A more complete resolution of this problem can be given using the theory of the thermodynamics of small systems (Hill 1967). Hall and Pethica (1967) show how that theory is applied to micellar systems and demonstrate the superiority of the one-phase description in that case. The simple case of a two-component solution in contact with its vapour is treated extensively in most textbooks. The two independent variables in that case are usually chosen to be the temperature and the concentration of solute. For an n-component mixture, the temperature and the concentration of (n- 1) components are independent

G I B B S ADSORPTION E Q U A T I O N I71

and one can eliminate the solvent activity (PI) using the Gibbs convention. The pressure in this case is a dependent variable because it is determined by the vapour pressures of the n components at the temperature in question. For two liquids in contact the more appropriate choice of independent variables is T,p , and (n- 2) of the component concentrations because in this case the pressure and temperature can be arbitrarily altered. What the phase rule tells us is that we can then set the surface concentration of two components to zero. The most appropriate choice is the two liquid solvents that form the bulk of the contacting phases. One then has two Gibbs dividing surfaces and the surface concentration of the other (n - 2) components is measured relative to both components 1 and 2. It can be shown that in that case:

In effect, one calculates the surface concentration of any component (from 3 to n) by adding the surface excess concentrations from the two phases (Exercises 2.4.4-8). It is not so easy to relate the result to the concept of the Gibbs dividing surfaces (because there are now two of them) but the surface excess can be given a simple physical interpretation using the alternative concept of a surface phase (Exercise 2.4.8). Defay et al. 1966, p. 89 show that the difference between ri,12and ri,l is usually negligible.

r

Exercises 2.4.1 Imagine a sharply defined surface layer of adsorbed material at the liquid-vapour interface. By placing the Gibbs dividing surface at the boundary between the adsorbed layer and the liquid, use eqn (2.4.8) to show that the relative adsorption ri.1is zero when

i.e. when the components i and 1 are present in the same molar ratio in the surface layer as they are in the liquid. Show also that ri,l is positive if the surface layer is relatively richer in i, negative if the surface layer is relatively poorer in i. 2.4.2 Establish eqn (2.4.18). 2.4.3 Establish eqn (2.4.7) and (2.4.11). 2.4.4 If two phases, (’) and (”) are in contact at constant T and p, the Gibbs-Duhem equation requires that 1nidpi = 0 in each phase. Consider the situation where solute 3 is distributed between the bulk solvents 1 and 2. Show that the surface excess of solute 3 relative to 1 and 2 is given by:

(Hint: refer back to eqn (2.4.3) and use the Gibbs-Duhem equations to eliminate dp1 and dpz. Note the similarity to eqn (2.4.30).)

72 I

2: THERMODYNAMICS OF S U R F A C E S

2.4.5 Show that if the two solvents 1 and 2 in Exercise 2.4.4 are mutually insoluble then

2.4.6 Consider the system indicated below where the data for the surface region is obtained from a sample which contains some bulk phase:

Bulk oil Surface region Bulk water

Oil 0.96 0.84 0.36

Water 0.09 1.02

0.24 0.51

1.08

0.18

Solute

mole p mole mole

Calculate the surface excess of solute in this case, relative to both oil and water, using the relation in Exercise 2.4.4. (Assume the area of the interface from which the sample is taken is 250 cm’.) 2.4.7 What is the surface version of the Gibbs-Duhem relationship (cf. eqn. A5.23)? 2.4.8 Repeat the calculation in Exercise 2.4.6 using the Guggenheim (1976) concept of a surface phase which has a finite thickness and is defined by boundaries placed inside both the bulk phases. T o solve the problem, ensure first that you place the boundaries in each phase so that the surface excess of both water and oil is zero. Hence calculate the surface excess of the solute and verify that the result is the same as in Exercise 2.4.6.

2.5 Thermodynamic behaviour of small particles 2.5.1 The Kelvin equation The existence of a pressure difference across a curved interface (governed by the Young-Laplace equation (2.2.17)) has a number of important colloid chemical consequences. For very small particles (droplets or bubbles) the pressure difference may be so great that the chemical potential of the material is affected. Taking y = 70 m N m-l and a spherical drop radius of 50 nm, eqn (2.2.17) gives for the pressure difference (2 x 70 x 1OP3/5 x lo-’) Pa = 28 x lo5 Pa 28 atm. Such a pressure, applied to a liquid will raise its chemical potential (and hence its vapour pressure) by a measurable amount. The same excess pressure inside a gas bubble can be interpreted as a reduced pressure in the adjoining liquid and will cause a lowering of its vapour pressure. If the curvature is caused by the fact that the liquid-vapour interface is being formed as a meniscus in a small capillary, the same lowering of vapour pressure occurs, with important consequences for adsorption in porous solids (see capillary condensation in Section 2.7 below).

THERMODYNAMIC BEHAVIOUR OF SMALL PARTICLES

173

Fig. 2.5.1 Capillary rise of a wetting liquid in a cylindrical capillary tube contained in a closed isothermal chamber. The space above the liquid column contains only the vapour.

An idea of the magnitude of the vapour pressure changes involved can be gained from Kelvin's original, elegant physical dcrivation in which he considered a capillary rise experiment in a closed, isothermal vessel (Fig. 2.5.1; Thomson 1870, 1871). For thermodynamic equilibrium to hold, the liquid and the vapour must be in equilibrium both at the flat liquid interface and the curved upper meniscus. Since the hydrostatic pressure (in the vapour phase) is lower at the upper meniscus by p,gh than it is at the plane liquid surface, it follows that the equilibrium vapour pressure at the meniscus is lower than that at the plane liquid surface by this amount (p, is the density of the vapour.) The Young-Laplace equation (eqn (2.2.17)) gives, at the meniscus:

so that

p,gh = y(;)

(A)

(2.5.2)

(2.5.3)

where p' is the equilibrium vapour pressure at the meniscus and po is the equilibrium vapour pressure above the flat liquid surface. Equation (2.5.3) is the original (approximate) form of the Kelvin equation. A more rigorous analysis of the effect of surface curvature on p is given by Defay et al. (1966, Chapter 15), from which a few results are summarized below. For a single pure substance the fundamental equations are, for a sphere of radius r:

Ap = p" - p' = 2 y / ~

(2.5.4) (2.5.5)

74 I

2: T H E R M O D Y N A M I C S OF SURFACES

where, by convention, (") and (') refer to the phase on the concave and convex sides of the interface, respectively. Note that eqn (2.5.4) can be generalized to any curved 1/Rz) where R1 and Rz are the principal radii of surface by replacing 2/r by (1/R1 curvature and may have the same or different signs. The superscript convention used here allows the equations to be applied to both menisci and droplets. Note that r is positive for droplets which, therefore, have an increased vapour pressure; it is negative for a concave meniscus (Fig. 2.5.1). For an infinitesimal process applied to a system initially at equilibrium, we have

+

dp" - dp' = d(2y/r)

(2.5.6)

dp" = d p f = dp .

(2.5.7)

and

The Gibbs-Duhem equation (eqn (A5.23)) in each phase becomes:

(2.5.8) when applied to a pure substance (where S = S/n; and 7= V/ni, are the molar entropy and volume respectively). Equations (2.5.6-8) are the fundamental relations from which a large number of important results can be deduced. For example, at constant temperature we have (from eqns (2.5.7) and (2.5.8)):

and so, from eqn (2.5.6)

(2.5.10) Equation (2.5.10) may be regarded as the most general form of the Kelvin equation as it applies to spherical interfaces.

2.5.2 Applications of the Kelvin equation (a) Drops of liquid in a vapour In this case ( ' I ) refers to the liquid and 7" << F'(w RT/p'). Then from eqn (2.5.10):

(2.5.1 1) Integrating from r = 00 (a flat surface where the equilibrium vapour pressure is p o ) to some finite radius where the equilibrium vapour pressure is p' we have (Exercise 2.5.2):

THERMODYNAMIC BEHAVIOUR OF SMALL PARTICLES

-

+

(PI

lnp’/pO% 2 y V f f / r R T

or

1

-p o )

(droplets)

if 7‘‘ << 7‘.

175

(2.5.12) (2.5.12a)

Equation (2.5.12) is the exact form of Kelvin’s equation. It shows that the equilibrium vapour pressure of the drop is higher than that of the flat liquid surface. It follows, therefore, that if a number of droplets of uniform radius are initially in equilibrium with a surrounding vapour (of infinite volume) that equilibrium must be unstable because if condensation occurs on a drop its radius will increase, its equilibrium vapour pressure will decrease, and it will continue to grow. Conversely, if a little evaporation occurs from a droplet, its radius decreases, its equilibrium vapour pressure increases, and it continues to evaporate. If the droplets are initially of different size, then the large ones will grow at the expense of the smaller ones. This argument applies if there is a sufficient reservoir of vapour, so that droplets do not change the partial vapour pressure as they evaporate or grow. The important point to note is that, for a given vapour pressure there is a critical size, r,, equal to the size in equilibrium with the vapour pressure, above which the droplets will grow but below which they will evaporate. When a vapour is cooled so that it becomes supersaturated, condensation to the liquid cannot occur until there are formed some nuclei of liquid of a size that can continue to grow. We will return to the study of this nucleation the0y in Section 2.8. An expression analogous to eqn (2.5.12) can be written for the solubility of a solid crystal in a surrounding liquid:

(2.5.13) where C(r), C , are the solubilities of the particles and the bulk solid respectively and y is the interfacial surface energy per unit area. The relation between the ‘effective’ radius, r, and the dimensions of the solid crystal will be examined shortly (Section 2.6). Again, eqn (2.5.13) suggests that large crystals will grow at the expense of smaller ones - a phenomenon known as Ostwald ripening (see Kahlweit 1975). It should be noted, however, that this can only occur at an appreciable rate if the solubility of the substance in the surrounding liquid is high enough. In many cases this is not so and a suspension of particles of many different sizes can coexist in quasi-equilibrium for very long times indeed. (b) Bubbles in a liquid In this case, the phase (”) is a gas and so we have (Exercise 2.5.2):

po

(bubble or concave meniscus)

RT or

p”

ln-w Po

2yVf rR T

-~

(ifV‘

<< V”).

(2.5.14)

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2: THERMODYNAMICS OF S U R F A C E S

Table 2.2 Influence of radius of curvature on the equilibrium vapour pressure above a spherical water surface, calculated from eqns (2.5.12) and (2.5.14), assuming that y is independent of r r (nm)

p '/Po (droplet)

p "/Po (bubble)

1000

1.001

0.999

100

1.011

0.989

10

1.115

0.897

1

2.968

0.337

Hence, the vapour pressure inside a bubble is smaller than the value for a flat surface. This explains the phenomenon of superheating of liquids above the normal boiling point. Remember that there has to be an excess pressure inside a bubble and yet the vapour pressure is actually lower than that over a flat surface at the same temperature. Boiling can only occur if either (i) the liquid can vaporize into a pre-existing bubble of reasonable size or (ii) the temperature is raised sufficiently high so that the equilibrium vapour pressure even in a small bubble is large enough to allow it to continue to grow. Case (i) is more usual, where the pre-existing bubble is, say, an air bubble previously released from the liquid. Such bubbles form on the walls of the container as the liquid is heated and dissolved gases become less soluble. Alternatively, they can be introduced deliberately by adding porous solids ('boiling chips'); gas contained in the pores expands as the temperature rises and produces seed bubbles into which vaporization can occur. Case (ii) is called homogeneous nucleation (Section 2.8) and it can only occur at temperatures well above the normal boiling point. It turns out that a temperature of almost 200 "C is required to induce boiling of water at atmospheric pressure if rigorous efforts are made to exclude any extraneous nucleation bubbles and one relies entirely on the formation of bubbles of the pure vapour. It should be noted that the changes in vapour pressure due to surface curvature and predicted by the Kelvin equation are negligible except for very highly curved surfaces (Table 2.2). It should also be noted that the integrations in eqns (2.5.12-14) are carried out with respect to the variable y / r . They remain valid, therefore, even if y changes with radius and pressure, provided that the value substituted for is the appropriate one for the value of r, and p' (or p") concerned.

2.5.3 Effect of temperature on vapour pressure - The Thomson equation We now want to calculate the effect of temperature change on the vapour pressure inside a bubble. We must, therefore, derive the analogue of the Clausius-Clapeyron equation for phase equilibrium across a curved interface. The resulting equation is attributed to J.J. Thomsont (the discoverer of the electron). In deriving Thornson's +Not to be confused with William Thomson who became Lord Kelvin and made the other important contribution in this area.

THERMODYNAMIC BEHAVIOUR OF SMALL PARTICLES

177

equation we will assume that the latent heat of vaporization AH, for the liquid is independent of curvature. Although there is some slight dependence at very high curvature it may generally be neglected. Defay et al. (1966) give a full discussion of this question (pp. 23CL9) and point out that even at r = 1 nm the decrease in AHvapfor water is only about 6 per cent. With this approximation we can easily determine the effect of change in radius of curvature (r) of a drop or bubble on the temperature required to establish equilibrium across the interface, if the external pressure is kept constant. (a) Liquid drop suspended in its vapour From eqn (2.5.8), with (”) referring to the liquid, we have dp’ = 0 (since the external pressure is constant) and so:

(3’- 3 ” ) d T + /”do” and

=0

(2.5.

dp” = d(2y/r)

Setting 3’ - 3” = AHvap/Tand integrating from obtain:

(2.5. 5)

Y

= 00 to some finite value r we

(2.5.17)

which is the Thomson equation (Thomson 1888). We have assumed here that 7”and AH,, are unaffected by the temperature change. This is a reasonable assumption in the present case because the temperature range involved in practice is fairly small (see Table 2.3). In this case, T < TO(since AH,, is always positive). Thus as Y decreases the equilibrium temperature becomes lower and lower. In order to induce a vapour to condense onto a small droplet of the liquid phase it is necessary to cool it down to a temperature below the normal condensation temperature T, corresponding to the external pressure p‘. This is the phenomenon calling supercooling. Supercooled (i.e. supersaturated) vapours are used in the Wilson cloud chamber for the detection of radioactive particles. The particles ionize the air in the chamber and this aids the formation of nuclei of sufficient size to permit further deposition (i.e. drop growth) at the temperature of the chamber. It should be noted that the integration process leading to eqn (2.5.17) corresponds to a gradual (reversible) bending of the interface, with a corresponding reduction in temperature to maintain equilibrium between vapour and liquid at each stage. As the bending progresses the pressure inside the droplet increases and in order to maintain the chemical potential constant we make a corresponding reduction to the temperature. The actual condensation process that occurs when nucleation is induced is, of course, a highly irreversible process. The vapour deposits rapidly because as the droplet grows the equilibrium temperature rises so the driving force towards condensation tends to increase. This is offset, however, by the release of the latent heat of condensation, which tends to raise the temperature of the whole system.

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2: THERMODYNAMICS OF S U R F A C E S

(b) Bubble immersed in a liquid In this case (”) refers to the gas phase and dp’ = 0 again. Assuming ideal behaviour for = R T ) and using the Young-Laplace equation (eqn (2.2.17)), we obtain the gas (p”7” from eqn (2.5.8):

(2.5.18) (In the integration process, AHvapis assumed constant with respect to temperature but this assumption can easily be dispensed with (Exercise 2.5.3).) This form of Thornson’s equation gives the temperature at which a bubble of vapour of radius Y can exist in equilibrium inside a liquid. For small values of Y it follows that T >TOand, indeed, the temperature can be much higher than the normal boiling point as we noted above. Recall that there has to be an excess pressure inside the bubble so eqn (2.5.18) calculates how high the temperature must be raised to achieve that higher vapour pressure. Table 2.3 gives values for the equilibrium temperature for drops and bubbles of various sizes. Notice that the effect on the equilibrium of bubbles is much larger than that of drops. That is not surprising when one recalls that the effect is due to the excess pressure inside the sphere: the effect of pressure on the chemical potential of a gas is much greater than its effect on a liquid. Note also that the more elaborate equation derived in Exercise 2.5.3 has been used to estimate T for bubbles because significant changes occur in AHvapover the temperature range involved.

2.5.4 Application of the Kelvin and Thomson equations t o solid particles We have already noted (Section 2.5.2) that small solid particles may be treated as spheres with an equivalent radius. The resulting excess pressure experienced by the solid will increase its chemical potential and Defay et al. (1966) list the following consequences:

“1. The vapour pressure of small crystals is greater than that of large crystals. In the presence of vapour, large crystals will grow at the expense of small crystals. 2. Small crystals will melt at a temperature lower than the normal melting point. The m.p. of a small crystal will be given by

(2.5.19) where TOis the normal melting point at the same external pressure, y s ~is the is the molar volume of the solid and AHf,, is the molar interfacial energy . . .7, heat of fusion . . . . 3. The melting point of a substance solidified in the pores of an inert material will depend upon the size of the pores . . . . 4.Small crystals may have a heat of fusion and a heat of sublimation smaller than the value for bulk solid.”

Table 2.3 Influence of curvature on the equilibrium temperature of droplets and bubbles. (Adaptedfrom D e f y et al. (1966) p . 242).

Droplet of water in water vapour at 1 atm

T/ To

rI0

y(mNm-')

T/ To

rI0

y(mN m-')

t ("C)

A&,(kJ

1

373

55.46

1

373

55.46

100

40.40

10000

0.9999946

372.998

55.46

1.0088

376

54.75

103.3

40.23

1000

0.999946

372.980

55.46

1.0574

394.3

50.85

121.4

39.26

100

0.99948

372.807

55.46

1.2037

449

39.10

176

36.29

10

0.99485

371.08

55.8

1.461

545

18.44

272

31.00

5

0.98906

368.92

56.34

f-

(nm)

00

-

Bubble of water vapour in water at 1 atm

V " has been taken as 1.043 x 18 cm3 mol-'

mol-')

.

U

CD

Next Page 80 I

2: THERMODYNAMICS OF S U R F A C E S

Point 1 has already been alluded to above. Point 2 is very important in the field of ceramics where finely powdered materials are heated to high temperatures and sintering occurs (i.e. fusion and partial inter-diffusion occurs at localized centres where the radius of curvature is very small). Point 3 has important consequences in tertiary oil recovery and the treatment of oil sands and shales to recover high molar mass hydrocarbon fractions (i.e. heavy crude) since the melting point of such material is affected by the capillary pressure to which it is subjected in the pores. (See Section 2.7.1 for further discussion of capillary pressure.) Equation (2.5.13) for the effect of particle size on solubility can be made a little more precise by using activities rather than concentrations: (2.5.20) where y and yo are activity coefficients. This formulation has the advantage that it can be extended to apply to electrolyte solutions, by the introduction of the mean ionic activity:

a t = a7.a"-

[2.4.251

where v is the number of ions produced when the salt dissociates. Equation (2.5.20) then becomes (Defay et al. 1966, p. 272): (2.5.21)

Fig. 2.5.2 Effect of particle size on the Ni-Ni(CO), equilibrium. (From Defay et al. 1966, with permission.)

Response to External Fields and Stresses 3.1 Response to gravitational and centrifugal fields 3.1.1 Settling under gravity 3.1.2 Sedimentation equilibrium under gravity 3.1.3 Settling in a centrifugal field 3.1.4 Sedimentation equilibrium in a centrifugal field

3.2 Response of a dielectric material to an electric field 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

Static electric fields Response o f a bound electron t o an alternating field The dielectric response function The phase lag between D and E The shape o f E ( W )

3.3 Response to electromagnetic (light) waves 3.3.1 Scattering by small particles (Rayleigh scattering) 3.3.2 Rayleigh-Cans-Debye (RGD) scattering 3.3.3 Mie scattering 3.4 Response to a mechanical stress 3.4.1 The rheology of colloidal materials 3.4.2 Ideal solids and liquids 3.4.3 The general response to a shearing stress (a) Stress relaxation after a sudden strain (b) Stress relaxation after cessation o f steady shear fl o w (c) Creep after a sudden stress 3.4.4 Response t o an oscillating shear field 3.4.5 Viscous shear behaviour

In this chapter we explore the response of a suspension of colloidal particles to external fields and forces of various kinds. That response will tell us a good deal about the internal nature of the suspension and its particles. Understanding the relation between an external influence and the resulting response will also enable us to better predict the likely response of a colloidal material to one or more influences applied singly,

115

116 I

3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

simultaneously, or serially to the suspension. The ideas are not elementary and the observed behaviour can be quite complex but we are interested here in giving an overview so that the vocabulary connected with the basic features of colloidal systems can be introduced. The most pervasive effect is that of the Earth’s gravitational field and the extension from gravitational to centrifugal fields is a natural one for colloidal systems, especially when the particle size is less than about 1 pm. The response of a dielectric material to an alternating electric field introduces some important concepts which can be applied immediately to the subject of light scattering by colloidal particles but that will be applied in a more definitive way in Chapter 1 1 to develop the theory of the attractive forces between colloidal particles (long range van der Waals forces). It is recognized that many students of colloid science have only passing acquaintance with the mathematical tools used in this section so we have included a number of practice problems to bring out the main features of the use of complex functions to describe elastic and inelastic responses to an applied stress. There follows an introduction to light scattering, which is a technique widely used for investigating particle size (Chapter 5), especially in its more recent manifestation (called photon correlation spectroscopy). The elementary functions of a complex variable are again used to describe the response of a system to a mechanical stress. An introduction to that theory is also presented at this stage to bring home the similarities between this and the dielectric response and to provide a sound basis for the following chapter on the viscous behaviour of colloidal systems. Some of that material will not be needed again until Chapter 15 when we take up the study of the rheology of suspensions.

3.1 Response to gravitational and centrifugal fields 3.1 .I Settling under gravity When a particle of mass m begins to settle through a fluid under the influence of gravity (Fig. 3.1.1) it is initially acted upon by three forces: the gravitational force, mg, the upthrust due to the displaced fluid, m‘g, and the frictional force,&, due to the (viscous) drag of the surrounding fluid. From Newton’s Law the net force is given by:

mg - m ‘g -& = m duldt

(3.1.1)

where g is the acceleration due to gravity and u is the velocity at time t. The frictional force increases with the particle velocity and for colloidal particles settling in a dense medium (like water) it very quickly balances the net downward force (see Exercise 3.1.1). The acceleration is then zero and the particles travel with their terminal velocity ut. For a particle of reasonably regular shape, the drag force is given by eqn (1.5.15) and for a rigid spherical particle of radius Y we can again set the constant B = 6nyr as in eqn (1.5.19). Thus if ps and p1 are the densities of the solid particles and the liquid respectively we have, for the Stokes settling radius, r: (ps - p1).4nr3g/3 = b n y r ~ ,

(3.1.2)

RESPONSE TO GRAVITATIONAL AND C EN T R IF U GA L F I E L D S

I 117

Fig. 3.1.1 Forces on a particle settling under gravity.

so that

(3.1.3) In principle, it would be possible to study sedimentation by measuring ut directly with an ultramicroscope, but most suspensions consist of particles with a range of sizes and settling velocities will therefore vary widely. It is more usual to follow the changes in particle concentration at a certain depth, h, below the surface of the suspension. After time t, all particles for which ut(r) 2 h / t will have settled beyond this depth and by following the concentration as a function of time, a particle size distribution can be built up (Section 5.5). For non-spherical particles the value of B can sometimes be estimated theoretically, but only for simple shapes. Thus, for an oblate spheroid (Fig. 5.1.1(a)), the correction to eqn (1.5.19) is simple for small departures from sphericity:

B = 6nva(l

-

~ / 5 ) ;( E + 0)

(3.1.4)

where E , the eccentricity, is (a - b)/b. Allen (1975, p. 169) gives a few similar relations for regular shapes, but these are of limited value for colloid settling because they require a knowledge of the initial orientation of the particles with respect to the gravitational field. Unless one or other of the principal axes of symmetry (assuming they exist) is aligned with the field, the particle’s motion is very irregular and certainly not confined to a vertical direction. Fortunately, as the particle size decreases, these considerations become rather less important because the particles become subject to increasingly vigorous Brownian motion, which has two important effects. For

118 I

3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

anisometric particles the orientation becomes increasingly randomized and tends to simplify the problem but the increasing translational Brownian motion (i.e. diffusion) interferes with the gravitational settling process. We have already shown that when Brownian diffusion is large enough to be measured, the friction coefficient can be calculated directly from the diffusion coefficient using Einstein’s equation:

B = kT/V

[1.5.18]

When that occurs, however, the settling rate is significantly affected by the thermal diffusive motion of the particles. The r.m.s. displacement for a sphere is (from eqns (1.5.18), (1.5.19), and (1.5.25)): (F)l/’ = (kT/3nyr)’/’&

(3.1.5)

which for a 1 p m particle in water would be about 0.7 p m in 1 s. The gravitational settling by a particle of density 2 x lo3 kg mP3 in the same time would be only about 2 pm . For colloidal particles, then, gravitational settling is of limited use except for very dense particles. Apart from the interference of Brownian motion, there are considerable practical difficulties in ensuring the necessary degree of temperature stability over long settling periods; very small convection currents can easily vitiate the results. Settling is, however, widely used in the construction industry and agriculture for characterizing suspensions of fairly dense solids towards the upper end of the colloid size range. Even if the particles are not spherical and they are too large to have a measurable Brownian diffusion coefficient one can still obtain an estimate of the distribution of the equivalent settling radius, r, which is undoubtedly a useful comparison measurement between different samples of the same material. For any particular particle, r is the radius of the sphere of the same density which settles at the same rate. Jennings and Parslow (1988) give a detailed account of how r is related to the actual dimensions of particles of different shapes, for which explicit settling formulae have been derived. There are some obvious limitations to the above analysis. T o begin with, eqn (3.1.3) is derived for a single particle settling all alone in an infinite expanse of fluid. It can only be correct when the streamlines around the separate particles do not interfere with one another (i.e. in the absence of hydrodynamic interaction). It should be applied only to the sedimenting of dilute suspensions (< 1 per cent); for higher concentrations a rather more elaborate treatment (Chapter 4)is necessary. Equation (3.1.3) also requires that the particles and the fluid be characterized by their densities, pp and p1. For large particles (> 1 pm) this presents little problem but for small colloidal particles the possibility exists that rather special effects occur at the interface between the particles and the liquid. The particle may have, associated with it, a few molecular layers of liquid that move with it as a single kinetic unit. This solvated particle will then have a density intermediate between ps and p1; even the density of this absorbed liquid layer may differ slightly from that of p1 in the bulk. The extreme case of solvation occurs, of course, with lyophilic colloids, but such substances do not usually settle appreciably under gravity. A more common occurrence with particulate dispersions is that the

RESPONSE TO GRAVITATIONAL AND CENTRIFUGAL FIELDS

I 119

particles are permeable to the fluid or they are aggregated into flocs, which contain trapped fluid. The sedimentation characteristics are then those of the composite. Equation (3.1.3) is also derived for solid particles, with the condition that there is no slip between the particle surface and the fluid (i.e. the fluid velocity is the same as that of the adjoining point on the particle at each point on the surface). For emulsion droplets (which usually rise in a gravitational field) and air bubbles, the drag force can be shown to be (Frumkin and Levich 1946): (3.1.6) where ~1 and 72 are the viscosities of the drop (or bubble) and the fluid respectively. Equation (3.1.6) assumes that the interface is perfectly fluid so that it cannot support 1< ~ 2 the ) drag is then reduced by one an applied stress. For air bubbles in water ( ~ < third. In actual practice, this seldom occurs because the drop or bubble surface almost always has an adsorbed layer of a surfactant (whether by accident or design) and this tends to make the interface rigid so the correction is not applicable (Levich 1962) and eqn (3.1.3) still holds.

3.1.2 Sedimentation equilibrium under gravity When particle size and density are such that the effect of the gravitational field is comparable to the effects of thermal diffusion it becomes possible for an equilibrium particle distribution to be established as a function of height in the suspension. It was this possibility which led to another of the classical methods for determining the Avogadro number. In the absence of any external forces, the composition of a phase at equilibrium is uniform throughout (eqn (2.3.21)) and is characterized by a unique value of the chemical potential of each of the components, pi. If external fields are important they must be incorporated into the definition of p. For example, to consider the composition of the earth’s atmosphere as a function of height above the ground we could introduce the gravito-chemical potential, p;,defined by:

and at equilibrium. Here. Then since:

djZ; = dp; +Mid$ = 0

(3.1.7)

4 is the gravitational potential (= gh) and Mi is the molar mass.

assuming ideal behaviour, we could write eqn (3.1.7) so as to determine the concentration ci as a function of height: d lnci =

~

-M;gdh RT .

(3.1 .S)

120 I

3: R E S P O N S E TO E X T E R N A L F I E L D S AND S T R E S S E S

In colloidal dispersions, gravitationalforces are much more significant than they are for molecular solutions or gases. In fact it is possible to observe the operation of gravity directly over very small distances and this was the second method exploited by Perrin (1909) (c.f. Section 1.5.4) to obtain an estimate of the Avogadro constant. Perrin and his collaborators prepared dispersions of highly monodisperse spheres of a natural colloid called gamboge. (This in itself was a very considerable feat in those days: in his account of Perrin’s work, Overbeek (1982) points out that Perrin and his students would start with 1 kg of gamboge and after several months of patient fractional centrifugation might finish up with a few hundred milligrams of monodisperse particles.) When this suspension was mounted on a vertical microscope slide (Svedberg 1928) the number of colloidal particles could be directly counted at different heights and from the resulting data a value of the Boltzmann constant, k, could be calculated (Exercise 3.1.4). Then since k = R / N A , where R is the universal gas constant, Avogadro’s constant NA could be obtained. Perrin obtained a value of 6-7 x mol-’ in his experiments and subsequent measurements of Westgren gave 6.05 x (See Svedberg very close to the presently accepted value of 6.022 x 1928, p. 101 for some discussion of this work.) This procedure relies on the notion that the microscopically visible particles are being continually bombarded by the surrounding molecules and so come to have the same average translational energy (3k T/2). Their distribution in the gravitational field, therefore, reflects the energy of the surrounding molecules. These two sets of experiments by Perrin provided some of the very first direct evidence for the real existence of atoms, which were regarded by many scientists at the time (ca. 1900) as no more than convenient figments of the theoreticians’ imagination.

3.1.3 Settling in a centrifugal field The time required, even for a large colloidal particle to settle through a reasonable distance under the influence of gravity alone (Exercise 3.1.2) makes that procedure rather limited. In most cases it is necessary to increase the sedimentation rate by subjecting the particles to centrifugation. Apart from the saving in time, there is then less danger of convection currents upsetting the results and the distance moved by sedimentation can be much greater than the Brownian motion.

Fig. 3.1.2 Forces on a particle in a centrifuge tube. The outward force is an apparent (virtual) force invoked to explain the motion of the particle with respect to a coordinate frame attached to the rotor and moving with it.

RESPONSE TO GRAVITATIONAL AND CENTRIFUGAL FIELDS

I 121

Consider a particle immersed in a liquid at a distance x from the axis of the centrifuge head (or rotor), Fig. 3.1.2. If it were to stay at that distance as the head revolved, it would have to be acted on by a (centripetal) force directed towards the centre of the rotor and forcing the particle to travel in a circle. The magnitude of that force is equal to (m- m’)w2x, where w is the angular velocity (radians s-l) of the rotor and (m - m’) is the apparent mass (corrected for buoyancy) of the particle. In the absence of that force, the particle moves away from the axis of rotation as if it were acted on by a force of that magnitude acting outwards. Again it is retarded by a frictional force that is proportional to its velocity and again it takes only a very short time before these two forces are balanced (Exercise 3.1 .l): (m

-

m‘)m2x = Bu(x) = B dxldt.

(3.1.9)

Note, however, that in this case the velocity is not constant but increases as the particle moves towards the outer end of the tube. The quantity: S = u(x)/w2x = (m - m’)/B

(3.1.10)

is called the sedimentation coefJicient and is an important characteristic of the material. For polymeric materials (including proteins) a more appropriate form is obtained by considering the molar mass, M, of the solute, so that:

1

m - m’ = -(M - 7

2 ~ 1= ) M(l - ~ZPI)/NA

NA

where v 2 is the (partial) molar volume of the polymer and VZ is the volume per unit mass (= pi’); NAis the Avogadro constant. The sedimentation coefficient can then be written (using eqn (1.5.18)):

MD RT

S = M(l - ZZ~ZP~)/NAB = -(l

- Vzpl)

(3.1.11)

and, hence, a knowledge of S and the diffusion coefficient, D, can be used to estimate the molar mass (or molecular weight) of the polymer. S is obtained by writing eqn (3.1.9) in the form: dx/x = m[(l - pl/pS)w2/B]dt = S w2dt and on integration:

In(x2/xl) = s w2(t2 - tl).

(3.1.12)

Plots of In x as a function of time at known rotation speeds can therefore be used to determine S. For colloidal particles where the notion of molar mass is inappropriate, the apparent mass of the particle can be obtained using eqn (3.1.10) and the friction factor estimated from diffusion experiments (eqn (1.5.18)). Of course, if the particles are known to be spherical one may use eqn (1.5.19) to estimate B directly and hence determine the radius:

(3.1.13)

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3: R E S P O N S E TO E X T E R N A L F I E L D S AND S T R E S S E S

For non-spherical particles of regular shape, the corresponding equivalent sphere can be calculated using the methods set out by Jennings and Parslow (1988). Values of S range from about 1 ps for large colloidal particles down to values of the s for proteins. The time unit s is called 1 Svedberg, in recognition of order the man who developed the ultracentrifuge and the above procedure for establishing protein molar masses. The simplest application of eqn (3.1.12) is the two-layer sedimentation technique in which the colloidal suspension is initially placed as a thin layer on top of the clear suspension medium. As centrifugation proceeds the components of that layer travel down the centrifuge tube at rates determined by eqn (3.1.12) and in favourable cases their individual progress can be followed, usually by an optical procedure. This method is particularly suited to the separation and characterization of mixtures of distinct materials with well-defined mass and density characteristics, such as mixtures of proteins. It does, however, have some problems (in particular the phenomenon of ‘streaming’) and more detailed treatments should be consulted (Allen 1975, pp. 26673) for methods of minimizing their effects.

3.1.4 Sedimentation equilibrium in a centrifugal field The Brownian motion of colloidal particles can still affect the sedimentation process, even in a centrifugation experiment. Indeed, for very small particles, like protein molecules, the Brownian diffusion can oppose the sedimentation so effectively that the sedimentation appears to cease and an equilibrium (or steady-state) situation is reached in which the concentration profile of particles down the sedimentation tube remains constant with time. The situation is entirely analogous to the distribution of gas molecules in the Earth’s atmosphere or the gravitational sedimentation equilibrium (Section 3.1.2). The corresponding expression for the concentration of the colloid, as a function of distance from the axis of the centrifuge rotor, is:

(3.1.14) where the potential energy of molecules in the Earth’s gravitational field (mgh) is replaced by the particle potential energy in the centrifugal field. (See Exercise 3.1.7 for an informal proof of this equation.) The distance x in eqn (3.1.14) is measured from some arbitrarily chosen reference line where the concentration is Co. Note also that the exponent is positive in this case since the centrifugal field acts to concentrate the particles at larger values of x. Equation (3.1.14) may be written:

(3.1.15) which is the form used to calculate the molar mass (Mi) of a protein or polymer from centrifugation measurements. Note that it requires no assumptions about the friction factor B. Equation (3.1.15) does, however, assume ideal behaviour for the sedimenting

RESPONSE TO GRAVITATIONAL AND C EN T R IF U GA L F I E L D S

I 123

material (Exercise 3.1.6). A more elaborate analysis, taking account of departures from ideality leads to:

(3.1.16)

d(x2) - 2RT

where yi is the activity coefficient. This same activity coefficient correction can also be applied to equation (3.1.1 1) in more concentrated polymer or protein systems. The notion of activity coefficient is not really appropriate to particulate dispersions so we will not discuss this procedure further. It is more relevant to the behaviour of lyophilic than lyophobic colloids.

Exercises 3.1.I Consider a spherical particle settling under gravity according to eqn (3.1.1). Show that during the period before it reaches terminal velocity its equation of motion is:

where p = (ps- pl)/ps and G = 9~/2p,?. Hence show that u = (pg/G )[1- exp ( - Gt )] during this period. How long does it take for a particle of radius 1 p m and density 3 x lo3 kg mP3 to reach 99 per cent of its terminal velocity in water (density = lo3 kg mP3, viscosity = lop3NmP2 s)?Repeat the calculation for a radius of 0.1 p m and 0.01 pm. (Allen 1975, p. 158). 3.1.2 A suspension of silica particles (ps= 2.8 g cmP3)in water is allowed to settle in a cylinder at 20 O C. Calculate the time required for a particle of 2 p m radius to settle a distance of 20 cm, assuming it is spherical. (Take v = lop2g cm-’ s-l = 1 centipoise.) Convert the data to SI units, and repeat the calculation. 3.1.3 Show that for colloidal particles dispersed in a liquid, the equilibrium number of particles, N , at a height, k above a reference level, Izo is given by:

N = No exp[-(m - m’)g(h - ho)/kir] where NOis the number of particles at height ko and m’ is the mass of fluid displaced by a particle of mass m (k is the Boltzmann constant). 3.1.4 Svedberg (1928, p. 101) gives the following table of Westgren’s date for the number of particles at different heights in a gold sol at sedimentation equilibrium under gravity. Height (pm) Number

0 889

100 692

200 572

300 426

400 357

500 253

Height Number

600 217

700 185

800 152

900 125

1000

1100

108

78

.

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

Assume the particles have radius 21 nm and density 19.3 g cmP3 and the temperature is 20 "C. Estimate k from the equation derived in Exercise 3.1.3 and then calculate NA assuming R = 8.31 J K-' mol-'. Repeat the calculation with a radius of 22 nm and note how sensitive the answer is to this variable. 3.1.5 The time taken for a particle to reach its terminal velocity under gravity is about 5 / G (Exercise 3.1.1). Show that the sedimentation coefficient, S,is also an approximate measure of this time. 3.1.6 In a sedimentation equilibrium experiment, Svedberg found the following concentration ((7-depth ( x ) profile for carboxyhaemoglobin: 4cm)

C(%)

4.61 1.220

4.56 1.061

4.51 0.930

4.46 0.832

The rotor speed was 8710 r.p.m. and the temperature was 20.3 "C. Take the density of the water as 0.9988 g cm-3 and that of the dry protein as 1.338 g cmP3 and estimate the molar mass of the protein, assuming that the molecules sediment as individual particles. 3.1.7 An informal demonstration of the reasonableness of eqn (3.1.14)can be given by balancing the sedimentation force, (m - m')m2x against the driving force for the diffusion process (eqn (1.5.11)).Derive eqn (3.1.14) by this method; then derive eqn (3.1.15). 3.1.8 Calculate the centrifugal acceleration at a distance of 7 cm from the axis of an ultracentrifuge rotor travelling at 20 000,40 000, and 60 000 r.p.m. and compare this with the gravitational acceleration, g.

3.2 Response of a dielectric material to an electric field If a sinusoidally varying electric field is applied to a material, the electrons in the atoms of that material will be induced to oscillate in response to the field and the analysis of their response provides valuable information about the structure of the material. The actual forces experienced by a particular electron are not simply related to the magnitude of the applied field at any time because the electron responds to the local field and that is influenced by the surrounding material. The previous history of the electric field may have led to the surrounding material being polarized in a certain way and, if the frequency of the field is high, those 'memories' may still be able to exert an influence on the electron under consideration. T o discuss the response of a dielectric material to a time-varying field we will need to be able to take account of such effects. Before we develop the tools for that purpose we must briefly review the behaviour in a static electric field.

3.2.1 Static electric fields Consider a dielectric material confined between two flat plates that have charges per unit area +a0 and -0, respectively (Atkins 1978, p. 747; 1982, p. 768). In the absence of a dielectric material the magnitude of the electric field E, between the plates is ~ O / E O ,

RESPONSE OF A DIELECTRIC MATERIAL TO AN ELECTRIC F I E L D

I 125

tad

-a&

Fig. 3.2.1 Polarization of the molecules of a dielectric between two plates of area A on which is a charge per unit area +(Tand d. (After Atkins 1978.)

where €0 is the dielectric permittivity of a vacuum (€0 = 8.85 x lo-'' F m-l or C V-' m-'). With the dielectric material in place, the field drops to Q / E = a0/erc0,where E, is a characteristic of the material (the relative dielectricpermittivity, E / E O , sometimes called the dielectric constant). Note that E , > 1; the field is always reduced because the dielectric material aligns itself as shown in Fig. 3.2.1 and this partly cancels the applied field. The capacity of this parallel plate condenser is equal to the ratio of charge to potential difference. If the space between the plates is empty, then:

and when the dielectric is present it increases to:

C = Eoc,d/d

(3.2.2)

so the ratio of the capacitances (C/Co) can be used to measure E,. Thepolarization, P, is in this case equal to the charge density on the surface of the dielectric adjacent to each plate (Fig. 3.2.1). The total charge of one such face is P d and on the other - P d . We can regard this as a large dipole, separated by a distance d so the dipole moment is P d d and the dipole moment per unit volume is P d d / d d or P. In more general terms we can write for the polarization vector at any point in a dielectric

where p is the local dipole moment of the molecules at r and p~ is their local number density. P is again the dipole moment per unit volume. The induced charge f P d near the plates reduces the effective charge to a value (00 - P ) d so that the new field has a magnitude (from eqn (3.2.1)):

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

Alternatively we can evaluate E from the effect of the permittivity: E = ao/crco. Eliminating 00 between these two relations gives:

P = EOE(E,- 1)

(3.2.4)

P = EE- roE.

(3.2.5)

The vector EE is called the dielectric displacement D, and in general

D = EOE+ P .

(3.2.6)

We will find in Chapter 7 that the fundamental equations of electrostatics give an expression for the distribution of the free charges in a dielectric medium in terms of D rather than E (as would be the situation in vacuo).

3.2.2. Response of a bound electron to an alternating field In order to get an idea of the way a charge distribution responds to a time-varying electric field we consider the simplest possible situation. Suppose (Fig. 3.2.2) a negative charge -4, of mass m, is attached by a spring, of force constant k, to a positive charge (the spring represents the attractive interaction between the two and is introduced so that the motion of the electron will be simply harmonic). Newton's law gives, for the equation of motion (Richmond 1975):

(3.2.7) where E(t) is the magnitude of the electric field which is assumed to be oscillating with a frequency w (rad s-'). This could be represented by

(3.2.8)

E(t) = Eo cos wt but, for convenience, we write it as

(3.2.9)

E(t) = EO exp (-;at)

+

-a

-4

+ I L

I I

I q

-

1

I

I

Fig. 3.2.2 Model of a charge undergoing simple harmonic motion.

RESPONSE OF A DIELECTRIC MATERIAL TO AN ELECTRIC F I E L D

I 127

where it is understood that only the real part of eqn (3.2.9) has any direct physical significance. (E(t) = Re[Eoe-”O“] = Eo cos(wt) since efie = cos 8 f isin 8. There are two reasons for making this substitution: (a) exponentials are rather easier to handle in integration processes than are trigonometric functions; and, more importantly, (b) the use of the complex number procedure allows us to separate out the elastic (storage) effects during the interaction from any dissipative (frictional or viscous (lossy)) processes like absorption. The form of eqn (3.2.7) suggests solutions of the form:

x = xo exp (-iwt)

(3.2.10)

and direct differentiation and substitution in eqn (3.2.7) gives (for x = xo, t = 0): -mw 2 xg = -koxo

+ qE0.

(3.2.11)

The polarization of the system is, in this simple case, equal to the dipole moment: P(t) = 9x6) so that:

P(t) = Poexp (-;cot)

(3.2.12)

Po = qxo.

(3.2.13)

where:

From eqns (3.2.11) and (3.2.13) (Exercise 3.2.1) we have: 2 2 ( ~ 0 w )Po

=

~02

Eo

(3.2.14)

where wo 2 = ko/m

and aowi =

2/m.

Provided w is not too close to 9 we can write

(3.2.1 5 ) The constant of proportionality between PO and EO is, in this case, the (frequency dependent) polarizability, a, which measures the ease with which electrons can be displaced by an applied field. This is so because the polarization is here equal to the induced dipole moment p and a is deJined by the relation:

p=aE

(3.2.16)

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

It is obvious from eqn (3.2.15) that something rather dramatic occurs when w = wo since

Po would then increase without limit. This is the natural resonancefrequency of the system, and at that frequency the dipole would be able to absorb energy from the field during every oscillation and so gradually increase its amplitude until it was infinitely large. There would be, at this frequency, an extremely sharp line in the absorption spectrum. In practice, of course, apart from the quantum effect, the spring would not remain harmonic in its action and in a real material other constraints would limit the motion of the charge so that the absorption peak would be reduced in height and broadened. The result is a Lorentzian function. (See Exercises 3.2.6 and 3.2.7) In order to complete the solution of eqn (3.2.7) to find P(t) as a function of E(t) we would have to introduce the notion of a complex polarizability (a=a’ ia”)and solve for a’ and a”(Richmond 1975). Rather than do that we will now examine the more general relation between the vectors P and E, which applies when a collection of interacting dipoles is involved. The vector P still measures the dipole moment per unit volume but its relation to E requires further discussion. wo = ( & / m

+

3.2.3 The dielectric response function The more general relation between P and E when the field varies with time has to take account of the past influence of the field on the present state of polarization and that involves the idea of a memory function. In place of eqn (3.2.3) we now write: (3.2.17) t> = PNP(I, t ) W

-

9

and the dielectric displacement becomes (compare eqn (3.2.6)):

D(r, t ) = EOE(T, t )

+ P(r, t).

(3.2.18)

Because the induced polarization takes a finite time to decay, the total polarization at time t is dependent on the electric field at all previous times t - t (0 C t C 00). The most general (linear) relation for P (I,t) is then (Landau and Lifshitz 1960):

s

M ..

P(r, t ) = €0

m(t)E(r, t

- t)dt

(3.2.19)

0

where m ( t ) is the memory function, which tells us the contribution to P(r, t ) at time t because of an electric field E(r, t-t), applied at a time t before t (Parsegian 1975). We will encounter a similar memory function in Section 3.4 in connection with the mechanical (rheological) properties of a colloidal material. [Such linear relations between an imposed field and the resulting response can only be expected to hold for low to moderate fields. Ultimately, at sufficiently high field strengths, the material will respond non-linearly. This will introduce further complications that are beyond the scope of the present treatment.] The function m ( t ) describes the decay of an induced polarization with time and, of physical necessity, must tend to zero for sufficiently large t: lim(as t + 00) m ( t ) = 0

(3.2.20)

RESPONSE OF A DIELECTRIC MATERIAL TO AN ELECTRIC F I E L D

I 129

Parsegian (1975) gives a useful discussion on m (t)in which he draws attention to the fact that it must also be finite for all t (since otherwise one would have an infinite polarization produced by a finite field. Recall the problem with the undamped oscillating charge at the resonance frequency wo in Section 3.2.2.). Note also that t must never be negative since that would imply that the field at some future time was affecting the present polarization (i.e. the cause coming after the effect). (Such violations of the causality principle may be quantum mechanically possible (Wheeler and Feynman 1945) but will not concern us here.) The function m ( t ) contains all the information about local electronic, vibrational, rotational and translational relaxationst of the component molecules of the dielectric material and for most purposes can be taken to be independent of the position Y within the material. From eqns (3.2.18) and (3.2.19):

and, once again, we represent E as a complex function (compare eqn (3.2.9)):

(3.2.22)

~ ( rt ) ,= Eo(r)eiwf. We can then write, for the (complex) displacement vector: m(t)Eo(r)exp(-iw(t

-

t)dt

[I 1

or D(r, t) = ~ & ( rt), 1

+

1

m(t)e"'dt

(3.2.23)

since t is constant inside the integral. Alternatively, returning to the original definition of D (= &) in eqn (3.2.5) we can write:

where: 00

E,(w)= 1

+

I

m ( t ) exp(iwt)dt

0

and ~ ( w= ) EOE,(W)as before.

+Relaxationis the process whereby a system which has been perturbed in some way returns to its equilibrium condition

(3.2.25)

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3: RESPONSE TO E X T E R N A LFIELDS AND S T R E S S E S

Note that in this formulation E,(w) is a dimensionless generalization of the relative permittivity or dielectric constant, E,, appearing in eqn (3.2.4) and:

s

00

E,

= E,(O) = 1

+ 0 m(t)dt.

(3.2.26)

The dielectric response function E ( W ) is simply the extension of the familiar static relationship between D and E to take account of time-varying electric fields.

3.2.4 The phase lag between D and E Since the polarization at time t is the residual effect of the electric field at all previous times (and not just E(t)),both P(t)and D(t)tend to be out of phase with E(t)and this is taken into account by the complex character of E(w). We can represent E ( W ) by real (E’(w))and imaginary (E”(w))parts so that: E(W)

= €’(W)

+ ZE”(W)

(3.2.27)

and, then, from eqn (3.2.25):

(3.2.28) and 00

E”(w)= €0

J

m ( t ) sin ~t d t

(3.2.29)

0

(Note that ~ ( 0 = ) ~ ’ ( 0 and ) ~ ” ( 0= ) 0.)

(3.2.30)

T o explore the phase lag, E ( W ) can be written in the alternative form (Exercise 3.2.3):

where

(3.2.32) and

(3.2.33) Substituting eqn (3.2.31) in (3.2.24) and taking the real part of the resulting equation we see that if the physical field E ( I , t) has the form of eqn (3.2.22) then the displacement vector has the form (Exercise 3.2.4):

D(I, t ) = IE(w)IEo(I)cos [ ~- t6(0)].

(3.2.34)

RESPONSE OF A DIELECTRIC MATERIAL TO AN ELECTRIC F I E L D

I 131

The effect of the complex dielectric response is to cause a phase difference, 6, between the field E and the displacement vector. The angle, 6, is a direct measure of the energy dissipation which occurs in the dielectric as the field passes through it. If 6 = 0, the field passes through the dielectric without loss and the medium is said to be transparent to the field at that frequency. The maximum loss angle, 6 = n / 2 , corresponds to total absorption of the electric field energy by the medium. We will see in Section 3.4 that it corresponds, for a mechanical system, to the viscous dissipation of energy when a shear wave passes through a pure liquid. From eqn (3.2.32) it is clear that it is the imaginary part, E”, of the dielectric response that measures the extent of this loss, or absorption, of energy. If E” = 0 then E ( W ) = ~ ’ ( wis ) purely real and there is no dissipation of energy from the field at all. At very high (optical) frequencies, the behaviour remains the same but, for historical reasons, it is described in rather different terms. In place of E’ and E” we have a refractive index nl(w) and an absorption (or extinction) co-efficient, K(W) which are related to er(w)by (Exercise 3.2.9): (a1

+ ZK)2 = E, = EL + ZE:

(3.2.3 5 )

so that 2

- K 2 = EL

and 2nlK = E:.

(3.2.36)

The optical spectrum normally records K as a function of w (or the wavelength, A, of the light) and we normally measure nl at frequencies somewhat removed from an absorption line, where the material can behave elastically,but eqns (3.2.35) and (3.2.36) apply over the whole frequency range. Note again that E” is directly proportional to the absorption coefficient.

3.2.5The shape of

E(O)

So far we have said nothing about the explicit mathematical form of E ( W ) except what is contained in eqn (3.2.25). It is possible to show from eqn (3.2.25), using Laplace transforms (Parsegian 1975), that any physical material can be adequately represented by a sum (or integral) of terms of the form (3.2.37)

where the constants&, hj, gj, and WQ must be chosen to fit the experimental data. In practice, one can limit the types of response to just two: (a) a Debye relaxation term to characterize the response of a rotating dipole, and (b) a Sellmeier damped-oscillator form to characterize infra-red, visible, and ultraviolet absorptions. For the Debye relaxation there is no restoring force, since the dipole simply rotates (restrained by friction) to follow the field. (This assumes that all of the dipoles can behave independently which is obviously an approximation.) Since such a rotator has no resonance frequency ( 9 j = 0) eqn (3.2.37) gives for such a system: hi

di

(3.2.38)

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

For the infra-red and ultraviolet absorptions a suitable form is (see Exercise 3.2.8):

.6

(3.2.39)

+ gj(-Zw) + (-Zw)2

and we could expect the following expression to represent a (non-conducting) polar dielectric material over the entire frequency range:

Exercises 3.2.1 Establish eqn (3.2.14). 3.2.2 Draw an Argand diagram to represent the functions (3.2.31)- (3.2.33). 3.2.3 Establish eqns (3.2.31) - (3.2.33). 3.2.4 Establish eqn (3.2.34). 3.2.5 Consider a capacitor (for which C ( w ) = ~ ( w ) d / dsubjected ) to an alternating voltage V = VOcos wt. The resulting current is the real part of I = V / Z where Z is the capacitive impedance or reactance and Z = -l/ZwC. (This is the a.c. analogue of Ohm’s Law.) Show that I = (od/d)VO[O[E”COS wt - &sin ot] where d is the capacitor plate area. By considering the integral 0 j2nI(wt).V(wt)d(wt) show that the power dissipated in the capacitor is entirely determined by 6”: Power dissipated per cycle = (nd/d)V;E”(O) whereas E’ relates to the storage of electrical energy during the cycle. 3.2.6 Consider the extension of the problem treated in Section 3.2.2 to include a friction (damping) term to the electron’s motion. The equation of motion will then be:

m -d2x =-k dt2

OX

-

B -dx dt

+ qE(t).

Take E = Eo exp(-iwt) and x = A exp(-iw t ) where A, the amplitude of the motion, can be complex in this formulation. Verify by direct substitution in the equation of motion that:

Show that A can be written in the form x’ functions and then:

+ i x” where x’ and +02B2].

X” = qE0 wB/[m2(wi- w2)2

X”

are real

R E S P O N S E TO ELECTROMAGNETIC (LIGHT) W A V E S

I 133

3.2.7 Show that in the neighbourhood of the resonance frequency (w % 0 0 ) (Exercise 3.2.6):

where Q = B/2m. The plot of X” as a function of w is called a Lorentzian distribution. It is commonly used to represent the shape of the lines in an absorption spectrum (see Fig. 5.7.4). Again it is the imaginary part of the displacement function which has the form of the absorption (dissipation) curve. Since the dipole moment is qx this measures the magnitude of the dipole moment out of phase with the field. 3.2.8 Show that if (eqn (3.2.39)): E(O)

=

€4 w& +gj(-iw)

+ (-iw) ’ = d(w) + id’(@)

then the imaginary part 6’’ has exactly the same form as d’obtained from Exercise (3.2.6).Thus eqn (3.2.39)can be expected to represent the behaviour of a bound electron moving against viscous friction in response to an applied field. 3.2.9 An electromagnetic wave of velocity c has wave vector k where k = d c . The wave vector in any medium is related to the medium properties by k’ = c(w),u (w)w’ where p(w) is the magnetic permeability of the medium. We also know that the velocity in vucuo is given by co = (rope)-'/'. The refractive index, n and absorption coefficient, K , are dejined by the relation (n+ ~ K ) O / C O= k. Use these relations to justify eqn (3.2.36)for non-magnetic materials (for which p

= Po).

3.3 Response to electromagnetic (light) waves When electromagnetic radiation strikes a particle it may be absorbed, transmitted, scattered, refracted, or diffracted. Absorption gives rise to a range of spectral data which must be interpreted using the usual methods of chemical quantum theory. In this respect colloidal systems behave like other chemical systems and we will have something to say about such spectra in later chapters. For the moment we are more interested in phenomena which are more characteristic of colloidal systems. We noted earlier (Exercise 1.4.1) that colloidal particles when immersed in a fluid are able to scatter a beam of light (the Tyndall effect). The scattering pattern (i.e. the intensity of the scattered light as a function of 8, the angle between the incident beam and the scattered beam) depends very strongly on the particle size and on the wavelength of the light. The spectral colours that are sometimes generated have fascinated investigators for centuries. It is only very recently, however, that the full potentialities of the study of light scattering have been realized, with the introduction of lasers to give coherent, monochromatic, intense and narrow

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incident beams, together with sensitive and stable photon-detection apparatus and rapid data analysis by computer. The theory of light scattering has been extensively described by van de Hulst (1957) and by Kerker (1969). The following analysis owes much to recent reviews by Sorensen (1997) and by Kralchevsky et al. (1997). A proper treatment of the interaction between an electromagnetic wave of wavelength h and a particle consisting of many thousands of atoms would require the formalism of quantum field theory and would, in fact, be insoluble by present methods. Fortunately, however, it turns out that the problem can be tackled from a classical point of view, and was indeed solved in fairly general fashion before the quantum theory gained much credibility (Mie 1908; Debye 1909). Indeed, an approximate analysis, valid for very small particles was given by Lord Rayleigh in 1871.

3.3.1 Scattering by small particles Particles are considered to be small if the characteristic dimension is small compared with the wave length, A,of the light. For spherical particles of radius a, we can define a dimensionless size parameter B = 2na/h and confine ourselves initially to the case B << 1. For present purposes we need only concern ourselves with the electrical component of the electromagnetic wave. The general representation of the electric field associated with a travelling light wave of frequency w is:

E = Eo exp i(k.r - wt)

(3.3.1)

where k is the wave vector which has a magnitude k = w/c = 2n/h where w is the frequency. The relation between the incident wave and a wave scattered in any direction is indicated in Fig 3.3.l(a). The incident wave is characterized by a wave vector ki and the scattered wave by k,.In most practical applications of light scattering, the observation of the scattered wave is confined to the horizontal plane (4 = 90°) and we are concerned with elastic scattering so that the magnitude of the wave vector is unaffected (ki = k, = 2n/h). We must also consider the polarization of the incident and the scattered light (i.e. the plane in which the electric field vector is considered to be oscillating). T h e state of polarization can be described in terms of two independent polarization states: horizontal (H) and vertical (V). For an initial wave travelling along the x axis as in Fig. 3.3.l(b), we will take the vertical plane to be defined by they and z axes and the horizontal plane by the x and y axes. Normal (incoherent) light from the sun or from an incandescent lamp is said to be unpolarized which means that the plane of polarization is randomly oriented. Such a source can be regarded as providing equal intensities of vertical and horizontal polarization. In principle it would be possible to use light which was initially polarized in the H or V direction and to study the scattered light in either H or V polarization. We would then have to consider the intensities:

R E S P O N S E TO ELECTROMAGNETIC (LIGHT) W A V E S

I

I 135

/r

Y

Fig. 3.3.1 (a) The incident wave is characterized by a wave vector ki and the scattered wave by wave vector k,. The plane P is perpendicular to the scattered wave and the amplitude measured in that plane will be proportional to sin 4 and to cos 8.

Fig. 3.3.1 (b) The usual configuration for light scattering in which the incident and scattered beam lie in the xy plane which is horizontal. 8 is the scattering angle and in most modern systems the incident beam is provided by a laser source which is vertically polarized (V).

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In most early studies the light source was an incandescent lamp with unpolarized ( I H= Iv) light but in more recent apparatus the light is supplied by a laser, and is almost invariably vertically polarized. Th e scattered light is also usually viewed in vertical polarization so we will concentrate, unless it is stated otherwise, on IVV which, for very small particles, is independent of the scattering angle 0 in the horizontal plane. The theory of light scattering has been developed along two distinct lines which look somewhat different but which are in fact identical. When light passes through a non-absorbing, perfectly homogeneous medium, the electric field vector causes the molecules of the medium to be polarized. The fluctuating dipoles which result will radiate energy in the form of light but this radiation is uniform in all directions and cancels itself in the body of the medium; the light therefore appears to move through the medium unchanged. In a real medium, however, there are always small fluctuations occurring in the local molecular density and these local density differences result in incomplete cancellation and, hence, some very weak scattering. It is this weak scattering which gives rise to the colour of the oceans and the sky. Theoretical descriptions can be given in terms of (spherical) solid particles of known refractive index immersed in a medium of a different refractive index, or they can be couched in terms of the density fluctuations occurring in a solution with a refractive index which differs from that of the solvent; we will generally use the former treatment. Rayleigh’s initial (1871) analysis concerned the scattering from a collection of very small, non-conducting, and non-absorbing particles acting independently of one another. In that case we have

p = 2xa/h << 1 and n p << 1

(3.3.2)

where n = %,/no is the refractive index of the particles relative to that of the surrounding medium. The first condition ensures that the whole of the particle is subjected to the same electric field strength at each instant in time so there is a negligible difference in the phases of the scattered wave from different regions of the particle. The particles must also be separated from one another so that the scattering from one does not interfere with that from other particles. Rayleigh argued that the amplitude of the scattered wave, relative to that of the incident light would fall off inversely with distance, r from the scatterer. The electric field of the incident light would produce a dipole of moment p which would be proportional to the polarizability, a of the scattering material. The field of the scattered wave produced by this oscillating dipole would, at any distance, also be proportional to p and, hence, to a. Furthermore he knew that a was proportional to the volume, V,of the scattering particle (Exercise 3.3.2). The relative amplitude of scattered and initial electric fields is, of course, a dimensionless number and, apart from depending on the ratio V/r could also depend on such parameters as the wavelength, the permittivity of the medium, the density of the scatterer etc. But, apart from the wavelength, all of these other parameters would involve dimensions of mass and/or time which are apparently not involved. Only the wavelength can be involved and that as the inverse square so that the relative amplitude of scattered and incident waves is proportional to

R E S P O N S E TO ELECTROMAGNETIC (LIGHT) W A V E S

I 137

the dimensionless ratio V/rA2 or a/rA2. The intensity of the light beam is proportional to the square of the amplitude and so the relative intensity should be given by:

I/Io

c(

a2/r2h4.

In terms of the magnitude of the wave vector, k, and using SI units, Rayleigh’s equation becomes:

1 6n4

2

(3.3.3)

where the quantity in square brackets has the dimensions of volume and is the polarizability of the scatterer, as that term was understood by Rayleigh. [The 4n€o term is a consequence of the use of rationalized SI units. (Appendix A6)] Rayleigh’s eqn (3.3.3) is particularly applicable to the scattering from molecules and it is invoked to explain why the sky is blue: scattering from small fluctuations in the density of molecules in the air is strongest for light of short wavelength so it is the blue end of the visible spectrum that we see. At sunset it is the transmitted (red) end of the spectrum that is most obvious, and the scattering is augmented by the presence of dust particles and water droplets in the lower parts of the atmosphere. A more familiar form of Rayleigh’s equation is obtained by expressing the polarizabilities in terms of the refractive indices of the scatterers. It can be shown, using the continuum theory of dielectrics (Section 3.2) that the polarizability of a dielectric is given by the Clausius-Mossotti equation (see e.g. Atkins 1978, p. 756; 1982, p. 7 7 3 ) (Exercise 3.3.2):

(3.3.4) where Nm is the number density of molecules of polarizability a! and radius, R . The same relation can be assumed to hold for a small spherical particle. Then using eqn (3.2.36) for the dielectric permittivity in the absence of absorption ( E n2):

(3.3.5) So the scattering from N (independent) particles, each of volume v would be:

(3.3.6) where N is the number of particles in the scattering volume, V, and N = N p V, where N p is the number of particles per unit volume. This is the more usual formulation of Rayleigh’s equation. In order to obtain an expression for the scattered intensity which is independent of the geometry of the instrument, the usual procedure is to calculate the Rayleigh ratio (R

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(3.3.7) where the function P(@)depends on the polarization of the incident and scattered light. By considering the projections of the electric field vector onto the scattering plane (Fig. 3.3.1) it can be seen that if 4 = 90":

Pw = 1;

PHH

= cos2 0;

P V H

= PHv = 0.

(3.3.8)

If the incident light is unpolarized (U) it can be regarded as a 50:50 mixture of vertically and horizontally polarized light and:

Pw =

1 2 i; Pm = ?cos 8

and Pw = $(l

+ cos2@).

(3.3.8a)

Few colloidal particles conform to the requirements of the Rayleigh model (j3 << 1) but eqn (3.3.6) is important because it emphasizes the strong dependence of scattering on particle size, and wavelength. Since the scattering is proportional to v2Np/h4at any angle it is apparent that for any particle size (v constant) the scattering increases directly with particle concentration and for a given mass concentration (vNp constant) of the particles, the scattering will increase with particle size. This second property is extensively used to follow the process of particle aggregation (called coagulation or flocculation (Section 1.6)). T o use eqn (3.3.4) for the excess polarizability of a suspension we require knowledge of the ratio n (= np/no) of the refractive index of the particles to that of the surrounding medium and must also assume that the particles are spherical. A slightly different formulation (Kralchevsky et al. 1997) follows if the scatterers are treated as forming a solution of refractive index n, compared with the refractive index of the medium, no (Exercise 3.3.3):

where n, is the refractive index of the suspension of particles of mass concentration C = N p M / N ~We . can assume that the refractive index increment (n, - no)/Np can be approximated by the derivative dn,/dNp. Substituting this expression in eqn (3.3.3) gives, for the Rayleigh ratio (Exercise 3.3.3):

R(@) = CKMP(@) where K =

(3.3.10)

This relation allows one to determine, in principle, the molar mass, M, from the scattering behaviour. Note that this analysis applies only to very small particles (such as protein molecules) for which the molar mass is a more appropriate size parameter than the radius. As the particle size increases and a more elaborate theory is required, it is only the detailed @-dependencethat changes; the dependence of I on h and the number and size of the scatterers remains as indicated in eqn (3.3.7) and (3.3.10).

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I 139

3.3.2 Rayleigh-Gans-Debye (RGD) scattering When the particles are too large to satisfy the condition /3 << 1, Rayleigh (1910) suggested a strategy which was developed by Debye (1915) and Gans (1921). It depends on the assumption that the incident light beam, which generates the dipoles is not affected (either in magnitude or in phase) by the presence of the particles. That will be so for small particles but also for larger ones if the following conditions are met: In - 11 << 1

and 2/3ln - 11 << 1.

(3.3.1 1)

Note that the second condition allows the particles to be larger than for Rayleigh scattering, provided the refractive index difference between particle and medium is sufficiently small. As the particle size increases the scattering pattern ceases to be symmetrical and the amount of forward scattering (0" c 0 c 90") increases at the expense of back scattering (90°C 0 C 180").The waves which are scattered in the strictly forward direction (0 = 0") remain in phase and the intensity in this direction is a maximum (P (0 = 0) = 1) and so from eqn (3.3.10) we can set R(0) = CKM. Then we can define P(0) as (Kralchevsky et al. (1997):

P(0) = R(O)/R(O).

(3.3.12)

Expressions for P(0) can be calculated from purely geometrical considerations and this has been done for a number of particle shapes. (See, for example, Kralchevsky et al. (1997) Table 4.) The intensity of the scattered wave in any direction, 0, depends on the phase relationships of the waves coming from different parts of the scattering particle which will depend on the path differences involved. The results are therefore quoted in terms of the scattering vector, Q, defined as shown in Fig. 3.3.2. As noted above, the scattering is elastic so ki = ks = 2n/h and so we have:

Q = 2k sin 0/2 = [4x/h] sin 0/2 = [4nno/ho] sin 0/2

(3.3.13)

where ho is the wavelength in vacuo and h is the wavelength in the medium. The theoretical form of P(0) for homogeneous spheres is fairly simple (Pusey 1982) and is derived in exercise (14.3.8b):

(3.3.14) and analogous expressions can be written for rods and Gaussian coils. Expansion of eqn (3.3.14) in powers of Qa gives (Exercise 3.3.4):

P(0) = 1 - (Q&5

+ .....

(3.3.1 5)

and it can be shown that for particles of arbitrary shape, satisfying the RGD criteria, Pusey (1982):

P(6) = 1 - (@G)'/3

+ 0(QUG)4.....

(3.3.16)

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

Fig. 3.3.2 The geometry of a scattering event. Path difference =a

+ b = (ki - ks).ri,= -Q.rij

where UG is the radius of gyration of the particle (Exercise 3.3.5). T o estimate UG we use eqn (3.3.10) to write (3.3.16) in the form:

+ O(Q4) ]

KC = [MP(0)]-' = R(0) M

-

(3.3.17)

so a plot of KC/R (0) against sin2(0/2) for small values of 0 will give values of both the molecular weight and the radius of gyration of the scattering entities. This is called a 'Guinier plot'. Equation (3.3.16) can be used for particles in the range 20 < ac/(nm) < 100. The technique has become much more useful with the advent of lasers because they can be sharply collimated so measurements of scattering intensity can be made at very small angles to the propagation direction. The technique is known as low angle laser light scattering (LALLS). The values of P(0) used in eqns (3.3.12-17) are for vertically polarized initial and scattered light but the relationships in eqn (3.3.8) still hold and can be used to evaluate the other contributions once Pv(0) is known. Thus we have (Kralchevsky 1997):

The asymmetry in the scattering pattern for particles in the RGD region makes it possible to obtain information about particle size and, in some cases, shape. Values of P (0) as a function of the scattering vector are shown in Fig. 3.3.3 (a) for particles of various shapes. Even the ratio of forward to back scattering, as measured by the dissymetry ratio: RD = 14511135

(3.3.19)

can give information on shape if the size is known and vice versa (Figure 3.3.3(b)).

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I 141

3.3.3 Mie scattering As the particle size increases, the scattering pattern becomes still more complicated as the spherical waves from each scattering centre in the particle increasingly interfere with one another so that the intensity shows pronounced maxima and minima at particular angles, 13, determined by the size parameter /3 = 2 n d h and the particle refractive index (or polarizability). If white light is used for illumination of monodisperse sols, the result is the appearance of strong beams of light of particular colours (chiefly green and red) at particular angles. These are called higher-order Tyndall spectra (HOTS) and their analysis can lead to data on particle size.

Fig. 3.3.3 (a) Scattering function P(0) for macromoleculesof various shapes. (After Marshall 1978, p. 477.) The abscissa represents Qd/2no where Q is the scattering vector and d is the diameter for the sphere or disc, and the length for the rod. For a random coil, the abscissa represents Q d / J 6 n o where d is the r.m.s. end-to-end distance.

Spheres

54-

3-

21

0.1

0.2 L I/?

0.3

0.4

Figure 3.3.3 (b) The dissymmetry ratio RD as a function of (relative) characteristic length, for spheres, random coils, and rods. (From Kerker 1969, p. 432, with permission.)

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

-

In this region (d A) the complete theory developed by Mie (1908) must be used to properly describe the scattering pattern. Mie solved the general problem for a sphere of any size when the refractive indices of both the particle and the medium can be complex (i.e. for the case where the light wave may be absorbed as well as scattered (Section 3.2)). We will not describe that result in detail. Suffice it to say that as the wave vector increases above the Guinier regime, the intensity falls off generally as (Qu)-~ but with a pronounced pattern of maxima and minima as shown in Fig. 3.3.4. The precise positions of the maxima and the depths of the minima depend on the details of refractive index and size. A good general description of this region is provided by Sorensen (1997). When the system is moderately polydisperse, the maxima and minima from different size spheres tend to smooth one another out but one still observes the general fourth power fall off in the intensity with increase in the relative size (Qu). The region where In I versus In (Qu) has slope -4 is called the Porod region. T he success of the Mie theory may be judged from Fig. 3.3.5. That data was obtained using an ingenious device designed by Gucker e t ul. (1973) and able to determine the complete scattering pattern from a single spherical particle over

I 0-5 0.1

I

I

1

10

Fig. 3.3.4 Mie scattering intensity for vertically polarized incident and detected light for spheres of size parameter B = 0.4, 1,2,4, and 8 and refractive index n = 1.33 as a function of the dimensionless factor Qa. For polydisperse systems the rapid variations in the region Qa > 2 tend to cancel one another and the intensity falls smoothly with slope 4 in this log-log plot. (This is the Porod region.) (From Sorensen 1997 with permission.)

R E S P O N S E TO ELECTROMAGNETIC (LIGHT) W A V E S

I 143

-Experimental

--- Theoretical

0

360

Degree

Fig. 3.3.5 Comparison of experimental data and theoretical (Mie) curves for scattering from a single polystyrene sphere. 0.771 is the least squares deviationbetween theory and experiment. (From Marshall et al. 1976, with permission.)

360" in about 20 ms. Several scans of the same particle can be done consecutively while it remains essentially stationary in the laser beam. Only symmetric patterns (Fig. 3.3.5) are accepted and stored for analysis. (See also Davis and Ray 1980.) We will examine the application of Mie theory to particle size determination in Chapter 5.

Exercises 3.3.1 Describe briefly what occurs when visible light is absorbed by a particle. In what way is the energy usually stored? Why does re-radiation usually occur at a longer wavelength? When is this not the case? If re-radiation does occur at the incident wavelength is the effect on the incident beam noticeable? Why? 3.3.2 Use the definitions of a and P from Section 3.2 to find a link between a and E (w). In a region where there is no absorption, show that

a

€0

(n; - 1)/N

for a collection of gas molecules of refractive index nl with N molecules per unit volume. Show that this can be reconciled with eqn (3.3.4) for a condensed medium (where N = (47ta3/3)-') if cr is related to the refractive index. 3.3.3 Establish eqns (3.3.9.) and (3.3.10). 3.3.4 Use the standard expansions: sin x = x - x3/3!

+ x5/5!..... and cos x = 1

to establish eqn (3.3.15).

-

+

x2/2! x4/4! - .....

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

3.3.5 Show that the radius of gyration of a spherical particle of uniform density is related to its radius by: ac = &3/5) and (3.3.16) for a sphere.

a.

Note that this reconciles eqns (3.3.15)

(Hint: UG = (moment of inertia/total

3.4 Response to a mechanical stress 3.4.1 The rheology of colloidal materials Rheology is the study of the deformation that occurs when a material is subjected to a stress. The stress (force per unit area) can be applied in various ways: as a compression, as a tension, or as a shearing process (Fig. 3.4.1). In compression and tension, dilute dispersions behave very much like simple liquids, especially if the particles are rigid and/or incompressible. Only in highly concentrated dispersions does one encounter unusual behaviour under tension, while under compression most condensed materials (solid or liquid) behave rather similarly. On the other hand, even quite dilute colloidal dispersions can exhibit very unusual behaviour when subjected to a shearing stress. In particular, the simple distinction between solid (elastic) and liquid (viscous) materials becomes blurred and a whole range of intermediate behaviour patterns is exhibited. Indeed, it is often these very unusual deformation properties that are sought after in the application of a colloidal dispersion. Consider, for example, the way the ‘apparent viscosity’ of a paint changes during its application: it is high when the paint is held on the brush but flows freely when sheared against the surface to be painted; it must quickly increase in viscosity so that it does not run down (drip or sag) under gravity but must flow sufficiently to eliminate the brush marks. The dependence of the viscosity on time and the shearing stress to which it is subjected determine the success or otherwise of the paint. Even more stringently controlled flow characteristics are required for high speed processes such as newspaper printing, paper making, electronic component dipping and encapsulation, and the preparation of photographic film and magnetic tape etc. In all of these situations (and many more) it is the rheological character of a colloid dispersion that is important. Consider the simple shearing regime illustrated in Fig. 3.4.1. The lower plate is held stationary and the upper plate is pulled by a force, .F acting in the x direction over an area d.The force per unit area or traction? (shearing stress) applied to the +The quantity F / d is usually refered to as the shearing stress and we will use that expression henceforth. Strictly speaking, however, the traction is a vector quantity whilst the stress at a point is a tensor that is the aggregate of all tractions acting on the surface elements of different orientation that contain that point (Reiner 1960, p. 5). We are assuming then that we can consider only one component of the stress on the upper surface. (The stress tensor will be introduced in the next chapter.)

RESPONSE TO A M E C H A N I C A L STRESS

0

I 145

XI

Fig. 3.4.1 Application of a shearing stress S (= g/d) to a material, produces a strain y = tan a.

material between the plates will cause a deformation (or strain), y. When the force is removed, we find that either: (a) the material returns to its original shape (elastic recovery); (b) the material remains in the new position @ow has occurred); or (c) some partial recovery occurs. These three behaviour patterns are characteristic of solids, liquids, and plastic materials, respectively, but in practice, most materials can exhibit any or all of them, depending on the time scale involved in the application of the stress and the measurement of its effects. The time scale is measured by the Deborah number (Harris 1977, p. 21):

DN =

relaxation time of material time of observation .

(3.4.1)

As DN + 0, materials tend to behave more like fluids and as DN + 00 they behave like solids. Thus geologists can speak of the ‘flow’ of rocks over geological time, while a person falling from a great height into water encounters its solid-like characteristics when deformed over short time intervals.

3.4.2 Ideal solids and liquids The ideal behaviour for an elastic solid experiencing a tensile (stretching) stress ST,is described by Hooke’s Law:

where Y is called the Young’s modulus of the material. The strain, y, is in that case the relative change in length. The corresponding behaviour under a shearing stress is described thus:

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

F l d a y or F / d = S = G y

(3.4.2)

where G is the shear modulus of the material and y is defined in Fig. 3.4.1. Many solids conform to eqn (3.4.2) for small stresses and provided the stress remains below some upper limit S L , they will recover their original shape completely when the stress is removed. If S > SLthe material suffers permanent deformation, i.e. someflow or creep occurs and the solid has begun to exhibit some of the characteristics of a plastic or a liquid. Ideal liquid-like behaviour is described as Newtonian behaviour and in that case the applied shearing stress is directly proportional to the time rate of strain or rate of shear ( y = dy/dt):

S a y or S = q y

(3.4.3)

where the proportionality constant, q, is the (first coefficient of shear) viscosity. This equation was proposed by Newton to describe the flow behaviour of simple fluids (gases and liquids like water) undergoing steady shear. Consider the system shown in Fig. 3.4.2 where a simple liquid is confined between two plates. The lower plate is stationary and the upper plate is being pulled at a velocity v by the application of a force per unit area, S. It may be assumed that the liquid in contact with the lower plate remains stationary whilst that in contact with the upper plate must move with the velocity v. Between the two a gradual variation in velocity occurs. The rate of shear, 9, for this simple shear regime is equal to the velocity gradient v/h and is normally measured in s-'. In the more general case i. = dv/dxz.

3.4.3 The general response to a shearing stress The study of colloidal dispersions (including the more concentrated systems described as slurries or pastes) reveals that ideal behaviour (either solid-like or liquid-like) is the exception rather than the rule. Even quite dilute dispersions can show departures from Newtonian liquid behaviour (eqn 3.4.3),especially if the particles are anisometric (like,

Fig. 3.4.2 Deformation (flow) of a liquid under an applied shearing stress, S. If the velocity of the upper plate is z, then i. = d(tan a)/dt = ( l / h ) dxl/dt) = v/h. (This coordinate system is used throughout the next chapter.)

RESPONSE TO A M E C H A N I C A L STRESS

I 147

for example, the clay minerals (Section 1.4.5)).Measurements at higher concentration or on shorter time scales (higher Deborah number) may begin to reveal evidence of solid-like (elastic) behaviour in what otherwise appears to be a (viscous) liquid. Such materials are said to be vasco-elastic and they may be intrinsically solids or liquids, depending on which of the two characteristics is dominant. For materials of this sort, the entire deformation history may, to some extent, influence its present structure. Once again we encounter the need for a memoy function (as in Section 3.2.3) to properly describe the behaviour. An introduction to the theory is given by Ferry (1980) from which the remainder of this section is taken. Again we restrict ourselves to the linear theory (compare eqn (3.2.19))in which the effects of sequential changes in strain are assumed to be additive: S(t)=

i

(3.4.4)

G(t - t') p(t') dt'.

--oo

Note that in this formulation of the memory process t - t' = t,so that t is the total elapsed time and the integration is carried out over all past times up to the current time t. G(t) is called the (shear) relaxation modulus of the material. If the function G(t) approaches zero for very large t, then the substance is liquidlike and an alternative formulation of eqn (3.4.4) can be given in terms of the strain, rather than the rate of strain (Exercise 3.4.1):

s I

S(t)= -

(3.4.5)

m(t - t')y(t, t')dt'

-ca

in which m(t), the memory function is -dG(t)/dt. Equation (3.4.5) is arrived at by integrating eqn (3.4.4) by parts, using the fact that y(t, t') = f p ( P ) dt" and taking the state of the material at time t"= t as the reference state (Bird et al. 1977). If G(t) remains finite for large t then the substance is solid-like and eqn (3.4.5) contains less information than eqn (3.4.4) and must be augmented by another term. Equations like (3.4.4)and (3.4.5)are called constitutive equationsand, as in the case of the dielectric response function, they can in principle contain all the information about the relation between an imposed simple shear stress and the resulting strain. Another form of constitutive equation links the resulting strain to the history of the time derivative of the stress:

1 t

y(t) =

J(t - t

')i(t ')dt'

(3.4.6)

--oo

where S = dS/dt and J(t) is called the creep compliance. A knowledge of either G(t), m(t), orJ(t) can be used to predict the effect of some imposed shear stress, provided it is not too large, nor applied too rapidly. A few typical experimental situations will now be examined (Ferry 1980): (a) Stress relaxation after a sudden strain If a sudden strain, y, is imposed on a material over a short period of time, (Fig. 3.4.3) then, from eqn (3.4.4):

<

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

‘I,

Y

Time

Time

t0

Time

Fig. 3.4.3 Time profiles for a simple stress relaxation experiment following a sudden strain. The behaviour of S for - 6 < t < to is not usually accessible but S(t) can be followed for t > to.

S(t) =

1

G(t - t’)(y/$) dt’

(3.4.7)

t0-c

since y is zero outside of this range. The mean value theorem tells us that the value of the integral can be written (Fig. 3.4.4 and Exercise 3.4.2): Y S(t)=-.&-G(t-to+(l-e)$)

for O l e 5 1.

&-

For to = 0 we then have, writing e‘ = 1 - e:

S(t) = yG(t

+ e’ $)

M

yG(t)

(3.4.8)

Fig. 3.4.4The mean value theorem establishes that there is a horizontal line which can be drawn so as to equalize the shaded areas. Then .[,?Gdt = area ABCD = tG(tl+et) for some e < 1.

RESPONSE TO A M E C H A N I C A L STRESS

I 149

for times that are long compared to (. The ratio of a stress to the corresponding strain is called a modulus (Section 3.4.2) and for a perfectly elastic body, the equilibrium shear modulus G = S / y from eqn (3.4.2). G(t) is then the time-dependent analogue of G, measured in an experiment with this sort of time pattern. Provided the strain y is not too large, the value of G(t) should be independent of y and it is only then that this simple procedure is of value. The concept can be applied to liquid-like and solid-like materials. (b) Stress relaxation after cessation of steady shear flow As noted earlier, liquid-like materials can be grossly deformed and still retain some structural characteristics. They can be deformed at a constant strain rate 9, under a steady shearing stress S where S = Yqo. (qo is used for the viscosity to signify that it is measured at sufficiently low shear rates for the behaviour to be linear). If the flow is suddenly stopped, the shearing stress decays with time and it can be shown (Exercise 3.4.3) that, for to = 0 (Fig. 3.4.5): 00

(3.4.9)

(c) Creep after a sudden stress The opposite experiment to that in (a) above is to apply a sudden stress to a material and then to hold it constant and to follow the resulting strain as the material accommodates itself (Fig. 3.4.6). One can again use the mean value theorem on eqn (3.4.6) to show that (for to = 0) (Exercise 3.4.4):

y(t) = SXt

I

Stress

+ e '6)

SXt)

(3.4.10)

,

t0

Time

Fig. 3.4.5 A simple shear stress relaxation experiment, following cessation of flow at time t = to.

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3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

m m

Y

2

; ; ;I I

Time

Time

t0

Fig. 3.4.6 A creep compliance experiment in which a sudden stress is imposed and then maintained constant while the resulting strain is measured.

so that J(t) is the reciprocal of a modulus. It is a monotonically non-decreasing function of time and for a perfectly elastic solid _7 = 1/G. However, J(t) # 1/G(t) because the time course of the two experiments is different. There are many other types of transient experiment that can be performed and they all yield some information about the mechanical response of the material. Choice of the most appropriate measurement depends on what information is sought. Another type of experiment that provides useful information on these viscoelastic materials is one in which an oscillating shear regime is imposed on the system. Considering the discussion in Section 3.2 it should come as no surprise to

ok I

I I

I

I

I

I

I

I

I

I

I

I

I

I

I

Time

Fig. 3.4.7 Stress-strain relationship for an ideal solid and an ideal liquid subjected to a sinusoidal stress. Note that for the liquid it is the strain rate (p) that is in phase with the stress and this is shown by the full curve (which also corresponds to the strain for an elastic solid). The stress in the liquid leads the strain by n/2.

RESPONSE TO A M E C H A N I C A L STRESS

I 151

find that once again the use of a complex variable enables us to keep track simultaneously of the elastic (storage) and viscous (dissipative) characteristics of the material (Fig. 3.4.7).

3.4.4 Response to an oscillating shear field Suppose a material, described by eqn (3.4.4), is caused to undergo a periodic strain of frequency w (rad s-'):

(3.4.11)

y = yosin wt

where yo is the maximum amplitude of the strain. Then y=wy 0

(3.4.12)

COSWt

and substituting in eqn (3.4.4) with t - t' = t we have

s

00

S(t) =

(3.4.13)

G(t)wyocos[w(t - t)]d t

0

[1

= yo w

1

G(t)sin w t d t sin wt

+

1

00

+ yo

w

G(t)cos w t d t

COSWt

0

(since cos (A - B) = cos A cos B sin A sin B and t is a constant in the integration process). The integrals will converge if G + 0 as t + 00, and this will be so if the material is liquid-like since, by definition, a liquid cannot permanently support a shearing stress. The terms in square brackets are functions of w but not o f t and we can write

S(t) = yo(G 'sin wt

+ G "cos wt)

(3.4.14)

where G'(w) represents a modulus that measures the ratio of the in-phase stress to the strain. This is the shear storage modulus. The quantity G"(w) likewise measures the ratio of the stress to the strain which is 90" out of phase; it is the shear loss modulus. The nomenclature here reflects the result obtained in Exercise 3.2.5: the lossy (dissipative) part of the process is represented by G". For a purely elastic solid the stress and strain remain in phase and so G" = 0 and G' = G. For a purely viscous liquid it is the rate of strain y that remains in phase with the applied stress, according to eqn (3.4.3), and the material has no elastic (storage) character so G' = 0. Then

S / y = S / [ w y0cos wt] = G " / w = r].

(3.4.15)

If one wants to characterize a viscoelastic material in terms of viscosity, it is necessary to use a complex viscosity function: r]

= r]'

+ iv"

(3.4.16)

152 I

3: RESPONSE TO EXTERNAL FIELDS AND STRESSES

and then in eqn (3.4.15) q = q’ for purely viscous behaviour. For the viscoelastic material TI‘= G“/w

and

(3.4.17)

q” = G‘/w.

An alternate method of describing a viscoelastic fluid (Fig. 3.4.8) is in terms of the phase lag, 6, between stress and strain. Writing

s = Sosin (wt + S) = SO

coss sin wt

+ SO

sins cos wt

(3.4.18)

and comparing with eqn (3.4.14) we see that

SO

SO G” = -sin6 Yo

G’ = -cos6;

YO

(3.4.19)

and

(3.4.20)

G“/G‘ = tan 6.

An oscillating measurement at frequency w corresponds to a transient measurement over a time t = l/w and the result obtained gives two pieces of information: the ratio of the amplitudes of stress to strain (S ‘ / y o ) and the phase lag, 6, or alternatively the values of G’ and G” or of fand q’ (from eqn (3.4.17)). The alternative representation in terms of a complex strain y* = yOexp(zwt)and the corresponding complex stress S* = P e x p [z(wt+6)] yields a complex modulus G*= S*/y* where G* =GI iG”and I G* I = (Gf2 G’”)~. Polymer solutions show very pronounced viscoelastic behaviour and much experimental work has been done on the phenomenon (see Bird et al. 1977; Ferry 1980). The elastic component is much smaller in most colloidal dispersions but coagulated colloidal sols do show some viscoelasticity (van de Ven and Hunter 1979) and we will return to that behaviour in Chapter 15.

+

+

m I

/ I

I

!L I

I

I I

I I I

I I I

Fig. 3.4.8 Stress-strain relationship for a visco-elastic fluid. S = 0 corresponds to elastic solid and 8 = n/2 corremonds to viscous liauid.

RESPONSE TO A M E C H A N I C A L STRESS

I 153

3.4.5 Viscous shear behaviour For many colloidal dispersions, the elastic effects play a rather secondary role in the behaviour, especially if the system is being sheared very strongly. Thus, in the pumping of a slurry or the high speed extrusion of magnetic iron oxide paste to make recorder tape, we are more concerned with the viscous (dissipative) aspects of the flow behaviour, even though the storage or elastic properties have to be recognized. The fundamental assumptions of linearity, small strains, and small strain rates on which the memory equations (eqns (3.4.4)-(3.4.5)) are based no longer apply. In one sense this makes the analysis easier because the majority of the time effects may then have disappeared. In many such situations it is sufficient, at least as a first step, to investigate the relationship between shear stress and shear rate, to obtain a more general form of eqn (3.4.3)in which the viscosity, q, is no longer a constant. In the simplest case, when the experimental measurement is conducted on a time scale which is long compared to the relaxation time of the system, the viscosity becomes independent of time, though in general it will depend on the shear rate. We can then define an apparent viscosity, qappas

or a dafferential viscosity: rdiff ( Y )

= d S/ d j.

(3.4.22)

Some of the behaviour patterns commonly exhibited by colloidal dispersions are shown in Fig. 3.4.9. If rappand rdiff both decrease regularly with shear rate (curve 2) the behaviour is called pseudoplastic and if they both increase (curve 3) it is dilatant. If the material behaves like a solid until a certain value of stress is reached and then deforms like a Newtonian liquid obeying:

S-SB=VPLY

(3.4.23)

where T]PL is constant, this is called Bingham behaviour and it is the ‘ideal’ standard for L called the plastic viscosity and SBis called the Bingham yield plastic behaviour. ~ P is value. In this case rdiff is constant but qappdecreases continuously from its zero shear value (TO + 00) to some limiting (infinite shear) value qoo. This type of behaviour is observed in, among other things, concentrated dispersions (slurries) of coal in water at fairly high volume fractions and low shear rates. Newtonian behaviour is observed in dilute stable dispersions of spherical particles. Pseudoplastic behaviour occurs even in dilute dispersions of anisometric particles because the increasing shear rate tends to orient the particles along the fluid streamlines and this decreases the viscosity. Dilatancy is common in concentrated dispersions; in this case it is the flow of (lubricating) fluid between the particles that dominates the behaviour and the shearing process tends to drive particles together which constricts the flow channels. Non-ideal plastic behaviour (Fig. 3.4.9, curve 5) is characteristicof fairly dilute (-10%) coagulated colloidal dispersions in which every collision between two particles results in the formation of a (temporary or permanent) link. The extrapolated value of stress obtained from the linear high shear rate behaviour is again called a Bingham yield value and

154 I

3: R E S P O N S E TO EXTERNALFIELDS AND S T R E S S E S

Fig. 3.4.9 Common forms of flow behaviour for colloidal dispersions: (1) Newtonian; (2) shear thinning (pseudoplastic); (3) shear thickening (dilatant); (4) (ideal) Bingham plastic; (5) non-ideal plastic. So,the primary yield value, is a well-defined quantity for concentrated dispersions and pastes (Nguyen and Boger 1983) but for dilute systems its value depends on the previous shear history of the sample and erroneous values can easily be obtained as instrument artefacts.

the differential viscosity at high shear rate can be called a plastic viscosity. We will discuss these forms of flow behaviour, and their interpretation in more detail in Chapter 15. If the time scale on which the measurements are conducted is sufficiently short (that is, comparable with or shorter than the characteristic relaxation time for the structure of the flow units) then one may also observe changes in the apparent or differential viscosity as a function o f time even at constant shear rate. The behaviour shown in Fig. 3.4.9 can then be regarded as the steady-state behaviour, arrived at after the system has had sufficient time to relax (i.e. to establish a (dynamic) structure in response to the shearing stress). Some systems show a gradual decrease and some a gradual increase in apparent or differential viscosity with time at a fixed rate of shear. These behaviour patterns are referred to as thixotropy and rheopexy respectively. Thixotropy occurs commonly in dispersions of very anisometric particles (e.g. montmorillonite (bentonite), Section 1.43, especially at moderate and high shear rates. The increasing shear rate requires an appreciable time to break down the particle linkages to establish the steady-state structure for that shear rate. Rheopexy usually occurs under conditions of gentle stirring where the slight degree of agitation apparently allows particles to establish extensive structures, which resist the shearing action.

Exercises 3.4.1 Establish eqn (3.4.5) from eqn (3.4.4). 3.4.2 Establish eqn (3.4.8). 3.4.3 Show that in the stress relaxation experiment of Fig. 3.4.5 S(t) = 1;

I t-to

3.4.4 Establish eqn (3.4.10).

G ( t ) dt.

REFERENCES

References Allen, T. (1975). Particle size measurement. In the Powder technology series (ed. J.C. Williams). Chapman and Hall, London. Atkins, P.W. (1978). Physical chemistry. (2nd edn 1982), (3rd edn 1986). Oxford University Press, Oxford. Bird, R.B., Armstrong, R.C., and Hassager 0.(1977). Dynamics ofpolymerjuids, Vol. 1, p. 277. Wiley, New York. Davis, E.J. and Ray, A.K. (1980). 3. Colloid Interface Sci. 75, 566. Debye, P. (1909). Ann. Phys. 30 (4), 57. Debye, P. (1915). Ann. Phys., 46, 809. Ferry, J.D. (1980). Viscoelasticproperties of polymers (3rd edn). Wiley, New York Frumkin, A.N. and Levich, V.I. (1946). Acta physiochim. U R S S 21, 193. Gans, R. (1921). Ann. Phys. 65, 97. Gucker, F.T., T h n a , J., Lin, H.M., Huang, C.M., Ems, S.C., and Marshall, T. R. (1973). Aerosol Sci. 4, 389. Harris, J. (1977). Rheology and non-Newtonian j o w . Longmans, London. Jennings, B.R. and Parslow, K. (1988). Proc. Roy. Soc. Lond. A419, 13749. Kerker, M. (1969). The scattering of light and other electromagnetic radiation. Academic Press, New York. Kralchevsky, P.A., Danov, K.D., and Denkov, N.D. (1997). Chapter 11 in Handbook of surface and colloid chemistry (ed. K.S. Birdi). CRC Press, New York. Landau, L.D. and Lifshitz, E.M. (1960). Electrodynamics of continuous media. Pergamon, New York. Levich, V.I. (1962). Physicochemical hydrodynamics.Prentice Hall, Englewood Cliffs, N.J. Marshall, A.G. (1978). Biophysical chemistry -principles, techniques and applications. Wiley. New York. [An excellent treatment giving valuable insights into physical models.] Marshall, T.R., Parmenter, C.S., and Seaver, M. (1976). 3.Colloid Interface Sci. 55, 624. Mie, G. (1908). Ann. Phys (Leipzig) 25, 377. Nguyen, Q D . and Boger, D.V. (1983). Acta Rheologica 27, 321; 29 (1985), 335. Overbeek, J.T.G. (1982). Adv. Colloid Interface Sci. 15, 251-77. Parsegian, V.A. (1975). Long range van der Waals interactions. In Physical chemistry: enriching topicsfrom colloid and surface science (ed. H. van Olphen and K. J. Mysels) Chapter 4.Theorex, La Jolla, California Perrin, J. (1909). Ann. Chim. Phys. 18 (8), 5 Pusey, P.N. (1982). Light scattering. In Colloidal dispersions (ed. J.W. Goodwin) Chapter 6. Royal Society of Chemistry, London. Rayleigh, Lord (1871). Phil. Mag. 41, 107, 274, 447. Rayleigh, Lord (1910). Proc. Roy. Soc. (London) A84, 25. Reiner, M. (1960). Deformation, strain andjow. H.K. Lewis, London Richmond, P. (1975). The theory and calculation of van de Waals forces. In Colloid science (ed. D.H. Everett) Volume 2, Chapter 4. Specialist Periodical Report, Chemical Society, London. [There are a number of typographical errors in the treatment of the problem given there.] Sorensen, C.M. (1997). Scattering and absorption of light by particles and aggregates. In Handbook of surface and colloid chemistry (ed. K.S. Birdi). CRC Publishing, New York.

I155

156 I

3: R E S P O N S E TO E X T E R N A LFIELDS AND S T R E S S E S

Svedberg, T. (1928). Colloid chemistry. Chemical Catalog, New York. van de Hulst, H.C. (1957). Light scattering by smallparticles. Wiley, New York van de Ven, T.G. and Hunter, R.J.(1979).3. Colloid Interface Sci. 68, 135. Wheeler R. and Feynman, R.(1945). Rev. mod. Phys. 17, 156; 21,424 (1949).

Transport Properties of Suspensions 4.1 Introduction 4.2 The mass conservation equation 4.3 Stress in a moving fluid 4.4 Stress and velocity field in a fluid in thermodynamic equilibrium 4.5 Relationship between the stress tensor and the velocity field 4.5.1 The rate of strain tensor e 4.5.2 Physical significance of e 4.5.3 Relationship between stress and strain rate in suspensions 4.6 The Navier-Stokes equations 4.7 Methods for measuring the viscosity 4.7.1 The Couette (cylinder-in-cylinder)viscometer 4.7.2 The Ostwald viscometer 4.7.3 The cone and plate viscometer 4.8 Sedimentation of a suspension 4.8.1 The Stokes equations and the sedimentation coefficient 4.8.2 Sedimentation in a concentrated suspension 4.9 Brownian motion revisited 4.9.1 Gradient diffusion in a concentrated suspension 4.9.2 Self-diffusion in a concentrated suspension 4.9.3 The Langevin equation 4.9.4 The Brownian motion of non-spherical particles 4.10 The flow properties of suspensions 4.10.1 The macroscopic flow field 4.10.2 The macroscopic stress tensor 4.10.3 The effective viscosity of a dilute suspension of spheres 4.10.4 Dilute suspensions of spheroidal particles 4.10.5 Concentrated suspensions

4.1 Introduction In this chapter we will describe the calculation of three of the transport properties introduced in Chapters 1 and 3: the sedimentation coefficient, the Brownian diffusivity,

157

158 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

and the effective viscosity of a colloidal dispersion. For simplicity we will concentrate initially on the case of a suspension of rigid, force-free particles; this is an idealization that can be approximated in practice by a suspension in which the double-layer thickness is much smaller than the particle radius, and the repulsive forces are large enough to prevent coagulation. For most separations the colloidal forces are then unimportant. To calculate colloidal transport properties it is necessary to determine the way in which the solvent flows around the suspended particles. In preparation for this analysis we will devote the first portion of the chapter to a discussion of the relevant aspects of fluid mechanics. Although earlier colloidal texts were able to get by with a very elementary treatment of this subject, research in the field of transport properties has now reached a level where it is no longer possible, even in a qualitative sense, to understand the problem, let alone the solution, without some basis in fluid mechanics. It is hoped that the first few sections of this chapter will provide such a basis. In this description it will be assumed that the reader is familiar with the notation and elementary methods of vector calculus; a few revision notes are provided on this subject in Appendix 3.

4.2 The mass conservation equation In our study of fluid motion the molecular nature of the fluid will be neglected and it will be treated as a continuum. For this approximation to be valid, attention must be restricted to regions of the fluid which contain many molecules. On the assumption that colloidal particles are much larger than the solvent molecules, we will use the terms ‘fluid particle’ and ‘point in a fluid’ to refer to regions that are much smaller than the colloidal particles, but much larger than the intermolecular spacing. In general, the velocity v at a point in the fluid will depend on the position of the point and on the time; so v(x,t) = v (XI,x2, xg, t) where XI,x2, xg are the Cartesian coordinates of the point, and x is the ‘position vector’ from the origin to the point. These quantities are illustrated in Fig. 4.2.1. The components of v in the direction of the coordinate axes will be denoted by 01, 02,and v3 respectively. The aim in this and the following sections is to set up the differential equations that must be solved in order to determine, for example, the flow around a colloidal particle. As the form of these equations does not depend on the presence of the colloidal particles, we will for the moment neglect the suspended particles and concentrate on the case of a pure solvent, since this simplifies the derivation. The density of water changes by only 0.01 per cent if the pressure is increased from 1 to 2 atm; we can, therefore, treat water (and most other solvents for that matter) as incompressible. This incompressibility property, together with the principle of mass conservation, can be used to set up the first of the differential equations for v. Theflux of fluid,j (i.e. the mass of liquid that flows into a volume over any period of time) must be balanced by the amount flowing out. The situation is analogous to that shown in Appendix A3.3 for the flux of the electric field and the same argument leads to the conclusion that the flux of material out of any volume element (that is the difference between what flows out and what flows in) is:

THE MASS CONSERVATIONEQUATION

I159

Fig. 4.2.1 The small rectangular volume referred to in the derivation of the continuity eqn (4.2.3).

P has coordinates (XI,

XZ, ~ 3 ) .

The total flux per unit volume out of the fluid element at point x is therefore

(4.2.1) In this case the obvious measure of the flux into or out of a fluid element is the velocity of the fluid and so eqn (4.2.1) can equally be written:

avi C,,=O

or

A A ~ = O

(4.2.2)

t=l

which is called the continuity equation. The second form is read as div v (short for the divergence of v). This relation is quite general; the flow lines do not need to be parallel to the coordinate axes and the sample volume can have any shape. The same argument can be used to establish that, if some component, i, of the system (say a solute species) is moving with the fluid and diffusing in response to a concentration gradient, say, it will have a flux in each direction through the volume element and the net accumulation of the material in the volume element as a function of time must be given by:

ani/& = -divj,

= -V .jj

(4.2.3)

where ni is the number of molecules of species i per unit volume. The part of the flux which is due to hydrodynamic flow is n p (compare Fig. 1.5.2)and the net contribution of this component will be zero but there will be an additional component due to the diffusion process which may well be non-zero. Equations (4.2.1-3) are all consequences of the notion of mass conservation.

160 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

( Exercise

I

4.2.1 Establish eqns 4.2.2 and 4.2.3 using the sort of argument developed in Appendix 3.3.

4.3 Stress in a moving fluid The remaining equations for the fluid velocity are obtained by applying Newton’s Second Law of motion to a small block of fluid. In preparation for this step we first discuss the nature of the forces that act in a fluid, and the way these forces are related to the local velocity field. In general the forces can be labelled as either short-range or long-range, where the short-range forces are those that act over molecular distances and the long-range ones include colloidal dispersion forces and gravity. In this and the following two sections we concentrate on the short-range forces and their relationship to the local velocity field. Consider the forces acting across an area element AA that is translating and rotating with the local fluid at x (Fig. 4.3.1). T h e unit vectors along each of the three coordinate axes are E l , E2, and E3 respectively and the orientation of the element of surface can be defined by specifying the vector normal to the surface, fi. For a small block, the force, AF, which acts on this area creates a stress S = ( A F / AA) which is represented as S(x,fi). As the parcel becomes smaller and smaller, the long-range forces decrease in proportion to the volume of the parcel whereas these surface stresses decrease as the total surface area. Likewise the inertia forces due to the mass of the parcel decrease as the volume so they too become less important as the parcel shrinks in volume. In the limit of a very small parcel of fluid, Newton’s Second Law requires that all these surface stresses be balanced to zero. We can then write S(X,6)AA

+ CjS(x,-2j)AAj

=0

(4.3.1)

Fig. 4.3.1 The small block of fluid used in the derivation of the formula (4.3.5) for the stress on a plane BDC of arbitrary orientation.

I

STRESS I N A M O V I N G FLUID

I161

where the summation is over the three orthogonal faces. This equation can be simplified using the geometrical expression (Exercise 4.3.1): =AA2j

AAj

together with the relation

S(x, 2j ) = -S(x,

’

(4.3.2)

-2j),

(4.3.3)

which follows from Newton’s third Law (action = reaction). Combining eqns (4.3.2 and 3) with (4.3.1) gives: S(x, ii) =

cis(x,

2j)

. ii .

(4.3.4)

This formula enables us to calculate the stress on an arbitrary plane through x,given the stress on the three planes with unit normals il, iz, and i3. It is customary to write the components of the stresses S (x, 4) on these three orthogonal planes as aij where: S(x, 2j ) = Xi 0ij (x) 2i

(4.3.5)

the zth component of the stress on the plane with unit normal 4. Equation (4.3.4) can be written more compactly as aij is

S(x, ii) = (x) . ii

(4.3.6)

where is a 3 x 3 (Cartesian) tensor whose elements can be represented in the form of a matrix (see Fig. 4.3.2):

[2 : 21. 011

012

013

(4.3.7)

Since students often encounter difficulties with tensors, we should emphasize that this is simply a convenient piece of notation. The important thing to bear in mind about the stress tensor is its physical significance; that is that the components aij are the components of the stresses on three orthogonal planes through the point. The net (turning) moment due to the stresses on any small block of fluid is approximately zero+ so, (from Fig. 4.3.2), it is clear that: 0.. g - 0.. Jr.

(4.3.8)

for all i andj. For this reason the stress tensor is said to be ‘symmetric’. Finally, for future reference we note that the net force due to the stresses over a macroscopic surface A is obtained by adding the contributions from the surface elements. On taking the limit of very small area elements and using the formula (4.3.6) we find that the net stress force is

f* *ii dA. t We are still assuming that the parcel of fluid is so small that inertial effects are negligible.

(4.3.9)

162 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Fig. 4.3.2 An illustration of the stresses acting on a small rectangular block with faces parallel to the coordinate planes. The shear stress, S, depicted in Fig. 3.4.2 would correspond to 012. (Note that the origin of the stresses labelled 0i3 lies in the plane of the front face of the block.)

Exercises 4.3.1 Derive the formula (4.3.2)for the area of the orthogonal faces of the block shown in Fig. 4.3.1 using t%foll%g information. By the definition of the crossprod%, AA fi =(; BC x BD) where AA is the area of t h e 9 i n g face and BC and BD are vectors along the e dge s3hat face. Show that BC = 1 3 4 - 1 2 4 and + BD = l l i l - 1 2 4 . Calculate BC x BD and use the result to verify eqn (4.3.2).

4.4 Stress and velocity field in a fluid in thermodynamic

equilibrium By thermodynamic arguments it can be shown (Landau and Lifshitz 1969, section 12) that the stress in a fluid in equilibrium is simply that due to the hydrostatic pressure: S(X, fi) = -p(x)fi

(4.4.1)

where p is the pressure, the minus sign being included to indicate the compressive nature of the stress. Recalling the definition (4.3.6) of the components of the stress tensor, we see that = -pz

in a fluid in equilibrium, where Z is the ‘unit tensor’, which has components

[k K 81

(4.4.2)

STRESS AND VELOCITY FIELD IN A FLUID I N THERMODYNAMIC EQUILIBRIUM

I 163

If a fluid in equilibrium is in motion, it must move as a rigid body, for this is the motion that maximizes the entropy (Landau and Lifshitz 1969, section 10). It can be shown (Meriam 1966, section 30) that the velocity field for an arbitrary rigid body motion must have the form

v(x) = v + x x

(4.4.3)

where V and are independent of x, and are the translational and angular velocities respectively. Those readers who are unfamiliar with this formula may find it helpful to consider the example of a glass of liquid placed on the centre of a steadily rotating turntable, as shown in Fig. 4.4.1. In a frame of reference that moves with the turntable the fluid appears to be at rest. In the laboratory frame the fluid particles will move steadily around circles centred on the axis of rotation with a speed of 2nr/T, where T is the time for each rotation and Y is the distance of the particle from the axis of rotation. The components of the fluid velocity in the laboratory frame of reference are therefore given by

-2nr . -2nx2 sin8 = T T ’ 2nr 2nx1 v2 = -cos8 = T T ’ and v3 = 0. V]

=-

~

~

These component expressions can be written in the vector form (Appendix A3):

v = ( 2 n & / T ) x x.

(4.4.4)

Comparing this result with the general formula (4.4.3) for rigid body motion, we see that the translational velocity is zero, while the angular velocity is given by:

Fig. 4.4.1 An illustration of a fluid that is in equilibrium and in motion: a glass of liquid on the centre of a rotating turntable.

164 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Exercise 4.4.1. Establish eqn (4.4.4).

4.5 Relationship between the stress tensor and the velocity field In a fluid that is undergoing deformation, frictional stresses are set up which tend to retard the deforming motion. Our aim in this section is to find the relationship between these frictional stresses and the deforming motion. Although the equilibrium formula (eqn (4.4.2)) for the stress tensor is not valid here, it is convenient to define the pressure in a deforming fluid by

P = -(m

+ a22 + (733)/3

(4.5.1)

and to write the stress tensor as

=-jZ+

D

.

(4.5.2)

The quantity is known as the ‘deviatoric stress tensor’, since it represents the deviation of the stress tensor from the equilibrium form (eqn (4.4.2)). will depend on the history of the motion of the fluid, being zero for a fluid which has been in steady rigid body motion for a sufficiently long time. Since the stresses arise from short-range forces in the liquid, the deviatoric stress tensor at a point will only depend on the history of the motion of the fluid in the neighbourhood of that point. Furthermore, since the fluid molecules jiggle around and rearrange themselves very rapidly, the effect of past motions will soon fade. For most liquids the time for this molecular rearrangement is much smaller than the time required for the macroscopic velocity v (x, t) to change significantly. Thus, to a good approximation, D(x, t ) will only depend on the instantaneous velocity field in the neighbourhood of x,since this is the quantity that characterizes the recent deformation history. We do not expect to depend on the absolute velocity of the neighbouring fluid particles, but only on their velocity relative to the particle at x . If the distance Ax between neighbouring fluid particles is sufficiently small we may approximate this relative velocity with the aid of the formula for a total differential, viz. V(X

+ aX,t )

-V(X,

t ) M Ej(a~/axj)Axj

(4.5.3)

where the derivatives are evaluated at (x, t). Thus the local relative velocity field is determined by the three derivatives &/ax,, and therefore D(x,t ) will be a function of the nine scalar quantities avilaxj. In fact, because of the isotropic nature of simple fluids, it turns out that we can express in terms of just two of these terms:

8 rl = q(avj/axj + avj/axi)

(4.5.4)

RELATIONSHIP B E T W E E N THE STRESS TENSOR AND THE VELOCITY F I E L D

I 165

where q is the shear viscosity. For the simple shear flow shown in Fig. 3.4.2 we can write: v1

= yx2,

and then eqn (4.5.4) shows that

v2

012

= 0,

v3 = 0,

= qp, where

is the shear rate.

4.5.1 The rate of strain tensor e The expression (eqn 4.5.4)) for the stress tensor is usually written in the form

where e- -1

(-hi+z). avj

ax,

-2

(4.5.6)

The nine quantities eij can be regarded as the components of a tensor called the ‘rate of strain tensor’, denoted by e. In this notation eqn (4.5.5) becomes = 2qe.

Combining this formula with the expression (4.5.2) for the total stress tensor, we get = -pZ

or

*ij

= -p6ij

+ 2qe + 2qeij

(4.5.7) (4.5.8)

where 6 is the ‘Kronecker delta’, which is 1 when i = j and 0 otherwise. Unfortunately there is no direct way of testing to see if a particular fluid does in fact satisfy the above relations, for it is not possible to set up a flow field in which all the components of the rate of strain tensor can be varied independently. It is possible, however, to test the validity of these formulae indirectly by comparing measurements made in various flow fields with predicted values obtained from solutions of the differential equations for the velocity and pressure fields. Since these equations, which will be set out in the following section, are derived using the formula (4.5.8), the measurements provide a test of the formula. For many commonly occurring liquids it is found that eqn (4.5.8) is valid over the entire range of strain rates encountered in normal practice. Such liquids are said to be ‘Newtonian’. The viscosities of some of these liquids are listed in Table 4.1 Since viscosity decreases rather rapidly with increase in temperature it is important to note the temperature at which the measurements were made: in this case 15 “C. Table 4.2 gives the viscosities of water for a range of temperatures. Some experimental techniques for measuring viscosities will be described in Section 4.7.

4.5.2 Physical significance of e Before proceeding to the derivation of the differential equations for the velocity field referred to above, we pause briefly to discuss the physical significance of the rate of

166 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Table 4.7 The viscosities of commonly occurring liquids at 15 “C and I atm. The unit in the SI system is I N m-2 s = I Pascals = 1 Pa s = 10 poise. (I poise = 1 dyne cm-2 s = I g cm-’ s-‘)

@a

4

Water

Mercury

Ethyl alcohol

Carbon Olive tetrachloride oil

Glycerol

0.00114

0.00158

0.00134

0.00104

2.33

0.099

Table 4.2 The viscosity of water over a range of temperatures Temperature (“C)

0

5

10

15

20

25

dmPa s)

1.787

1.514

1.304

1.137

1.002

0.891

strain tensor e. With the aid of the formula (4.5.6)for the components of e we can write the expression (4.5.3) for the relative velocity in the component form

where e i is evaluated at the point x. The first two terms on the right-hand side of this formula are the components of the vector expression

v(x)+(x) x x

(4.5.10)

where the components of the ‘local angular velocity’ (x) are related to the quantities: ;[avi/axj - avj/axi]

(see Exercise 4.5.1)

On comparing eqn (4.5.9)with the general formula (eqn (4.4.3))for rigid body motion, we see that the first two terms on the right-hand side of eqn (4.5.9) correspond to a translation and a rigid body rotation of the fluid. Thus the local rate of deformation or ‘strain’ of the fluid in the neighbourhood o fx is entirely determined by the last term in eqn (4.5.9), which in turn depends only on e(x). This is the reason why e is known as the ‘rate of strain tensor’.

4.5.3 Relationship between stress and strain rate in suspensions Although we will be studying the rheology of suspensions in detail in Chapter 15, it is appropriate at this point to describe the limitations of the above arguments when applied to suspensions. In Section 3.4.3we noted that most suspensions behave, from the macroscopic point of view, like non-Newtonian liquids, even if the suspending liquid is Newtonian. To understand why this is so it is important to appreciate the fact that the macroscopic stress tensor represents an average of the stress over regions containing a large number of

THE NAVIER-STOKES E Q U A T I O N S I167

colloidal particles, and that this average therefore depends on the particle configuration. For the case of a dilute suspension of rod-like particles for example, this configuration is characterized by the particle orientation distribution. In general the macroscopic flow tends to give the particles a ‘preferred’ configuration, as for example in the case of a shear flow, where rod-like particles tend to be aligned with the streamlines (Section 4.10.4). This ‘ordering’ tendency is opposed by the Brownian motion, and the final particle configuration is determined by the balance between these two opposing forces. In the limit of weak strain rates the Brownian motion dominates, and the configuration becomes statistically isotropic. Thus in this limit the Newtonian form (eqn (4.5.4))applies. At higher strain rates this is not generally the case, and as a result the rheological behaviour is characterized not by one, but by a number of viscosity coefficients, and these in turn usually depend on the strain rates. For unsteady flows the situation can be even more complicated, because a change in the imposed strain rates leads to a change in particle configuration, a change that may take a significant time. In a suspension of rod-like particles this ‘relaxation time’ is determined by the time required for Brownian motion to reorient the particles, which for a 1 p m long particle is of the order of 1 s. If the imposed strain rates change significantly over this period, the assumption that the stress depends on the instantaneous rate of strain breaks down, and to determine the stress it is necessary to look at the recent history of strain rates. Clearly the combination of rigid particles and Newtonian liquid can lead to some formidable complications!

I

Exercises 4.5.1 By writing out the components of the expression (4.5.10)for the local velocity field explicitly, and comparing the result with the first two terms of eqn (4.5.9), show that the components of the angular velocity are of the form i ( h i / a x j -

&,,axi).

4.6 The Navier-Stokes equations In this section we describe the origins of the remaining differential equation for the velocity field in a Newtonian liquid. This equation is obtained by applying Newton’s Second Law of motion to a small rectangular block of liquid centred on the point x. Rather than repeat the detailed description of the stresses, which we considered in Section 4.3-5, we will restrict attention to a much simpler problem to illustrate the physical significance of the terms. We consider a unidirectional flow which is everywhere parallel to the x1 axis with the velocity depending only on x2 (Fig. 4.6.1).In this case the shear force on the top of the block is

I

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

xz

t P is the point (x,,

x3

Fig. 4.6.1 Coordinate system for a uniform flow field.

Evaluating the force on the bottom of the block and then subtracting, we find that the net force per unit volume is @Vl *

r12e1.

8x2

If v1 also depends on x3 there will be additional shear forces on the faces with unit normals f&,leading to an extra force @Vl*

r12e1.

8x3

Finally, in the general case, where v1 also depends on XI, and the components v2 and v3 are non-zero, there will be additional normal stresses on the f 21 faces, which lead to a force per unit volume:

This force, which cannot be obtained from the simple Newtonian expression (eqn (3.4.3)), arises not from shearing of the liquid but from stretching in the x1 direction. Additional shear stresses in the x1 direction due to the velocity gradients &/ax1 and av3/axl lead to a force per unit volume

which, by the incompressibility constraint, can be written as

THE NAVIER-STOKES E Q U A T I O N S I169

Thus on summing these results, we find that the net force in the x1 direction is, per unit volume:

with similar results for the x2 and x3 direction. The viscous forces on this parcel of fluid can then be represented in general by the term qV2v. In addition there is the force produced by the local gradient of the pressure so the total stress force per unit volume is:

-vp + qv2v.

(4.6.1)

In addition to the pressure gradient and the viscous force there will also be a contribution from the long-range forces. We let F denote the total long-range force per unit volume. If gravity is the only such force then:

F = pg

(4.6.2)

where g is the gravitational vector, directed vertically down. The total force per unit volume on a Newtonian liquid is therefore given by the sum

F

-

Vp

+ qV2v.

(4.6.3)

The required differential equation for v is obtained by equating this term to the mass times the acceleration per unit volume of the block and writing the acceleration in terms of the velocity field. During a time At the velocity of a particle originally at x changes by V(X

+ AX,t + At) - V ( X , t).

If At is sufficiently small we can approximate this difference by the total differential formula

and hence obtain for the acceleration (Exercise 4.6.1): &/at

+v . v v .

(4.6.4)

Although there will be some slight variation in the acceleration over the block, we can to first approximation take the acceleration to be uniform. By combining the above result with the expression (4.6.3) for the force per unit volume on the block, we obtain

:(

p -+v.VV

)

=F-Vp+qV2v.

(4.6.5)

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

This equation and the incompressibility constraint

v-v=o

(4.6.6)

are known as the Navier-Stokes equations. These are the equations that govern most of the commonly occurring fluid flows, from the large scale flows in the oceans down to the small scale flows of interest here. In most applications the long-range force distribution F(x) is given+ and the unknowns are the velocity and pressure fields. T o determine these quantities for any situation it is necessary to solve the Navier-Stokes equations subject to conditions on the boundary of the fluid. These conditions depend on the nature of the boundaries, which in our case are the solid particle surfaces. Experimental observations indicate that the layer of fluid next to the surface sticks to that surface; thus the particle and the fluid adjacent to the particle move with the same local velocity. This is referred to as the ‘no-slip’ boundary condition.

Exercises 4.6.1 Establish eqn (4.6.4) using the fact that Axi = viAt if At is small.

4.7 Methods for measuring the viscosity In this section we will study some simple devices for measuring viscosity, namely the Couette, the Ostwald, and the cone/plate viscometers. We will not attempt to present a detailed description of the experimental procedures, but will concentrate instead on the theory behind these devices. The discussion is made simpler by the fact that most viscometers are designed to operate with very simple (unidirectional) flow regimes. We can therefore analyse the flow behaviour by simple force balance arguments, which are more transparent than the formalism involving the general stresses on a fluid element.

4.7.1 The Couette (cylinder-in-cylinder) viscometer The Couette viscometer consists of two coaxial cylinders, as shown in Fig. 4.7.1. The space between the cylinders is filled with liquid. One of the cylinders (preferably the outer one) is rotated at a constant angular velocity (strad s-l) and the other is held in place by a torsion wire. After a very short time, a steady flow pattern is developed in which the fluid elements move in circular paths around the central axis; each layer of fluid imparts a torque to the layer closer into the centre. The viscous drag of the fluid against the surface of the (inner) cylinder then causes it to twist against the torsion wire

t In flows around charged colloidal particles, F depends on the distributionof ions in the liquid; in that case the Navier-Stokes equations must be supplemented by equations for the ion distribution and electrical potiential.

I

METHODS FOR MEA SU R IN G THE VISCOSITY

I171

Torsion

wire

L

Bob

(a)

Drive shaft

Fig. 4.7.1 (a) A vertical cross sectional sketch of a Couette viscometer. The liquid occupies the shaded area between the two cylinders. (b) Isometric and cross-sectional views of a Couette viscometer. Ri and R, are the radii of the inner and outer cylinders respectively.

and the cylinder comes to rest when the torque imparted to it by the moving fluid is equal to the restoring torque in the wire. When the inner cylinder stops turning the flow velocity of the fluid varies from the inner wall (v(r) = 0) to the outer wall (v(r) = QR,) where R, is the radius of the outer cylinder (Fig. 4.7.l(b)). The shear rate in the gap is not simply dv/dr (compare Fig. 3.4.2) for we must make allowance for the fact that some part of the fluid motion is a rigid body rotation which does not shear the fluid. We then have Shear rate = y = dv/dr - w = dv/dr - v/r = rdw/dr.

(4.7.1)

The resulting shear stress S over the area of each cylinder of fluid generates a torque, T given by

T = S x area of cylinder x radial distance from axis = 2nr2LS

(4.7.2)

where L is the length of the cylinder immersed in the fluid. Under steady flow conditions, the torque on every cylinder is the same and equal to that on the inner cylinder, T;: (4.7.3) This expression can be integrated with respect to r (Exercise (4.7.1) to give: (4.7.4)

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

where Ri is the radius of the inner cylinder. On the outer cylinder, v must be equal to Ro!2 and substituting this in eqn (4.7.4) gives:

(4.7.5) where C is an instrument constant. The torque on the inner cylinder causes it to turn through an angle 0 and it can usually be assumed that the angle is linearly related to the torque even for rotations approaching 0 = n.So T; = k0 where k is another instrument constant. The plot of 0 against !2 for a Newtonian liquid should be linear with slope Cy/k and given the data for a liquid of known viscosity (usually water) one can then determine the viscosity of other fluids. The shear rate at different points in the gap is given by the bracketed term in eqn (4.7.3) and, hence, is equal to T;/2ngL? so for a viscometer in which R, = 2R; the shear rate at the inner wall would be four times that at the outer wall. Such a device would not be very suitable for studying the behaviour of a liquid for which the viscosity varied with shear rate (Fig. 3.4.9). In that case the above analysis would be invalid since g would vary across the gap. Although it is possible to modify the analysis to take into account the shear rate dependence of the viscosity (Krieger and Maron 1952, 1954), the problem is usually overcome by making the gap width Ro-Ri much smaller than the cylinder radius. In that case the shear (or strain) rate is approximately uniform across the gap (Exercise 4.7.3) and so the Newtonian analysis can be applied. The above results can, as we mentioned earlier, be derived directly from the NavierStokes equations. Normally, a problem such as this would be solved by first writing the Navier-Stokes equations in cylindrical coordinates in order to take advantage of the symmetry of the problem. The appropriate forms of these equations in cylindrical and other curvilinear coordinate systems are given in standard fluid texts (see for example Appendix 2 of Batchelor 1967). The assumption that the flow is circumferential then leads to three differential equations. The first equation is equivalent to eqn (4.7.3), while the remaining two have the form (Landau and Lifshitz 1959, section 18)

(4.7.6) The last equation simply represents the hydrostatic variation of pressure with depth x, assuming the axis of the apparatus to be vertical, while the first expression represents the balance between centrifugal forces and radial pressure gradients in the liquid. For the case of a rotating inner cylinder, this balance breaks down at high speed and a steady radial flow pattern develops. The speed at which this flow begins can be predicted from stability analysis of the Navier-Stokes equation. For the fixed inner cylinder device the circumferential flow pattern breaks down at much higher speeds; in this case the breakdown leads to a turbulent flow. The speed at which the turbulence begins cannot as yet be predicted theoretically. Since the measured torques can only be interpreted theoretically for the circumferential flow regime it is important to test any device to ensure that torque and speed are linearly related over the experimental range of speeds; this will ensure that there is no radial flow. Clearly, Couette devices with a rotating outer cylinder are preferable, since the circumferential flow assumption breaks down at high speeds for these devices. It should be noted, however, that this problem

METHODS FOR MEA SU R IN G THE VISCOSITY

I173

is not so serious for liquids of high viscosity and most commercial instruments actually use the stationary outer cylinder configuration, presumably because it is easier to manage mechanically. One final point on the Couette viscometer. The flow regime in the bottom of the cylinder is different from that in the annulus and an end correction may, therefore, be necessary. It can be assessed (Alexander and Johnson 1949) by filling the cylinder to two different depths, L1 and L2 and determining the difference in deflection (61 - Q2) for a given rotational speed (Exercise 4.7.5). Hunter and Nicol (1968) in their experiments shaped the bottom of the two cylinders in the form of cones calculated to give approximately the same shear rate as that in the annulus. This seems to eliminate the end effect quite satisfactorily. (See Van Wazer et al. 1963, pp. 68-72 for details.)

4.7.2 The Ostwald viscometer Another device which is commonly used by colloid chemists is the Ostwald viscometer, shown in Fig. 4.7.2(a). This viscometer consists essentially of two reservoirs linked by a fine capillary tube. Fluid is drawn up to the top mark of the left-hand tube in the diagram and allowed to drain out while the tube is held vertical. The time required for the level to drop to the lower mark is related to the viscosity; the more viscous the fluid the longer it takes to drain. In order to find the precise form of this relationship we must determine the flow field in the capillary. We take the x1 axis of our coordinate system to coincide with the centre line of the capillary tube, with x1 increasing down the tube. It is assumed that the fluid flows in the x1 direction only. The continuity equation therefore reduces to avl/axl = 0, implying that the flow profile is independent of distance down the tube. Writing out each component of the remaining Navier-Stokes equation (4.6.5) separately, we get av1

aP

at

8x1

p-=pg--+q

ap/ax2 = o

a2vl (ax;

a2vl ax:)

(4.7.7)

ap/ax3 = 0.

(4.7.8)

-+-

and

For most viscometers of this type, the inertia term in eqn (4.7.7) may be neglected, and we may treat the flow as time independent (Exercise 4.7.12). From eqn (4.7.8) it follows that the pressure is uniform over the cross-section of the tube. Thus the eqn (4.7.7) contains a term, ap/axl, which is independent of x2 and xg, while the remaining terms are independent of XI. It follows that

(4.7.9) where G is a constant. Since the pressure gradient is uniform down the tube, we can write: G = ($2

-

PlVL - Pg

(4.7.10)

where pl and p2 are the pressures at the top and bottom of the capillary and L is the tube length.

174 I

4:TRANSPORT PROPERTIES O F S U S P E N S I O N S

K

i

Fig. 4.7.2 (a) The Ostwald viscometer. The volume of liquid used is such that the height of the right hand reservoir moves symmetrically about the centre of that reservoir as the height in the left hand arm falls from the upper to the lower mark. The constriction makes timing more accurate.

Ii

I

-+

Ar' I I

Fig. 4.7.2 (b) Forces on a cylinder of fluid moving through a cylindrical capillary.

METHODS FOR MEA SU R IN G THE VISCOSITY

I175

Since the tube has a circular cross-section, v1 can only depend on distance r from the centre line. Thus eqn (4.7.7) reduces to the ordinary differential equation (see Exercise 4.7.6) (4.7.11) As in the previous section, this differential equation for the velocity can also be obtained by combining the Newtonian expression for the stress tensor with a force balance on a suitably chosen volume of fluid. In this case the appropriate volume is a thin cylinder co-axial with the capillary tube (Fig. 4.7.2(b) and Exercise (4.7.13)). The solution of eqn (4.7.11) is readily found to be (Exercise 4.7.7) v1

= --(uG

2

--Y

4rl

2

).

(4.7.12)

This parabolic flow profile is referred to as ‘Poiseuille flow’. It applies not just to capillaries but to flow down any circular tube, provided the flow rate is not greater than the value at which turbulence sets in; for capillaries that is not usually a problem. Integration of the velocity over the tube cross-section leads to the Hagen-Poiseuille expression for volume flow rate, Q, viz (Exercise 4.7.7):

[

a

nGa4 npga4 hl l+-. 8rl 811

Q = /2nrvldr = -~ 0

~

;

hz]

(4.7.13)

The Ostwald viscometer is very useful for obtaining accurate measurements of the viscosity of Newtonian fluids. Its major limitation in applications to colloidal systems is that the shear rate varies so widely across the capillary. It is necessarily zero at the axis and at the wall it can be as high as 2000 s-l. It would therefore seem to be impossible to use such a device for measuring the viscosity of a non-Newtonian liquid, but correction procedures are available and these will be discussed in Chapter 15. (See, for example, Maron e t ul. 1954.)

4.7.3 The cone and plate viscometer Another interesting device for measuring the flow behaviour of non-Newtonian fluids is the cone and plate viscometer (Fig. 4.7.3). It is obvious from the figure that this instrument cannot be used for liquids of very low viscosity because the only method of restraining the fluid is through the surface tension generating a Laplace pressure (Section 2.7.1) across the curved meniscus around the edges. The analysis of the flow in this device is simplified by the fact that the angle between the cone and the plate is normally of the order of a degree or so, and thus the opposing surfaces are nearly parallel. By using order of magnitude estimates of the various terms in the NavierStokes equations for this situation it can be shown that the flow is locally the same as that between two infinite parallel plates, separated by the local gap thickness h. This is a result which is not just limited to cone and plate flow, but applies to any flow between

176 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Fig. 4.7.3 The cone and plate viscometer. The angle is typically very small (<3" and often 4"). The cone is usually rotated but this is not always so. Rotating the plate allows the cone to be more easily suspended on a torsion wire. In most modern instruments the upper element is attached to a piezo-electric crystal which permits the measurement of the normal stress generated by nonNewtonian fluids. (See Chapter 15 for more details.)

closely spaced nearly parallel solid surfaces and forms the basis of most studies on lubricant flow (Batchelor 1967, section 4.8). For this reason this parallel-plate approximation is called the lubrication approximation. In this case one of the parallel plates is fixed while the other moves tangentially. Thus the flow is locally a simple shear flow with shear rate V / h where V is the local cone or plate velocity, depending on which of the surfaces is fixed. By using the fact that

V=rQ

and

hmra!

(4.7.14)

! is the angular velocity, we see where r is the distance from the axis of rotation and 2 that the local shear rate is Q/a.Thus the shear rate is uniform in this device, and it is therefore well suited to the study of non-Newtonian liquids. The torque or moment M due to the uniform shear stresses on the cone or plate is readily found to be

(4.7.15) where R is the cone/plate radius. The configuration of this rheometer makes it particularly suitable for measuring the normal stresses to which we referred earlier (Section 4.6), and which are an important characteristic of many polymer systems.

Exercises 4.7.1 Derive eqns (4.7.4) and (4.7.5). [Hint: Consider r(d(v/r)/dr).] 4.7.2 By solving eqn (4.7.5) show that the velocity field due to a rotating cylinder of radius R1 in an infinite liquid is given by v = R : / r where Q is the angular velocity of the cylinder. Calculate the torque per unit length required to rotate the cylinder.

METHODS FOR MEA SU R IN G THE VISCOSITY

I177

4.7.3 Derive an expression for the strain rate in the gap of a Couette viscometer in terms of the inner and outer radii, R; and R, and the angular velocity, Q of the outer cylinder. Show that for very small gap widths, d, this can be reduced to strain rate = Q(R;/d). 4.7.4 If the radius of the cylinders in a Couette viscometer is large compared with the gap width, d, it is possible to approximate the flow in the gap by assuming it occurs between two parallel flat plates. The bottom plate, at R1, is stationary and the top plate, at R2, moves with velocity QR2. (a) Show that in that case the continuity equation gives (&l/axl) = 0, v2 = vg = 0 and that the Navier-Stokes equation reduces to

(b) Hence show that (d2vl/dx;) = 0 and so vl(x2) = (QR2 x2)/d. (c) Calculate the torque on the inner cylinder and verify that the result is in agreement with the general formula (eqn (4.7.5)) in this limit. 4.7.5 Show that if the cylinders in a Couette viscometer are filled to two different depths, L1 and L2 then even in the presence of 'end effects' we can still write:

where $1 and $2 are the corresponding angles of twist of the torsion wire, and K' is an instrument constant. 4.7.6 Assuming that the velocity v1 in eqn (4.7.9) depends only on the distance Y from the centreline of the capillary tube, show that this equation reduces to the form (4.7.11). [Hint: first show that

ax;

(

-

d2v %)'+** dY2 ax2 drax;

and

ar/ax2 = x2/r]

4.7.7 Establish the results (4.7.12) and (4.7.13) for Poiseuille flow assuming that the pressures, pi are purely hydrostatic. 4.7.8 A certain Ostwald viscometer has a capillary tube of length 5 cm and diameter 0.5 mm. Calculate the time required for the level of the water in the top reservoir to drop by 1 cm, assuming both reservoirs have a circular crosssection of diameter 1 cm. The temperature of the water is 20 "C. You may assume that the term (hl - h2)/L << 1 in eqn (4.7.13) for the flow rate.

4.7.9 Calculate the shear rate at the wall of the capillary tube for the viscometer described in the previous question.

4.7.10In applying the hydrostatic formula (eqn (4.7.13)) for the pressure in the Ostwald viscometer, we have, in effect, neglected the vV2v term in the equation of motion in the portion of the apparatus beyond the capillary tube. We aim to estimate the resulting error in the formula (4.7.13) for the flow rate. The order of magnitude of the neglected qV2v term in this region is V / ( U ' )where ~ Vand

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

a’ are the typical velocity and radius of the tube. These viscous forces must be balanced by an extra pressure gradient. With the aid of this estimate, show that the relative error in the final formula (4.7.13) for Q is of the order of (u/u’)~(L’/L) where L’ is the length of the viscometer tube beyond the capillary. 4.7.1 1 Hiemenz (1977) gives the following expression for the torque on the cone of a cone/plate viscometer

M=

t a n a +$[(I

4 -nR3qS’2cos a 3 sina)/(l - sina)]}cosa‘

+

Show that this reduces to eqn (4.7.15) for small a.(You will need the approximation In (1 x ) x for small x . ) 4.7.12 Use the kind of argument displayed in Exercise (4.7.10) to show that for an Ostwald viscometer with a flow time of order 100 s, the inertial term (i.e. the L. H.S.) in the Navier-Stokes equation may safely be ignored. 4.7.1 3 Establish eqn (4.7.11) using a force balance argument on a cylinder coaxial with the capillary tube (Fig. 4.7.2(b)).

+

4.8 Sedimentation of a suspension 4.8.1 The Stokes equations and the sedimentation coefficient T o determine the transport properties of a suspension, it is necessary to solve the NavierStokes equations for the flow field around the individual particles. This problem is greatly simplified by the fact that the flow field in this case has a very small length-scale, and as a result it is usually possible to neglect the inertia terms in the equation of motion. In the neighbourhood of the particle the fluid velocity is expected to vary with distance on a length-scale of the order of the particle radius a. The velocity gradients should therefore be of order V / a , where V is the particle velocity. Hence the magnitude of the inertia term pv .Vv in the equations of motion will be of the order of p( V 2/a). By similar arguments the magnitude of the viscous force term is estimated to be q V/a2,and thus the ratio

va

Ipv. Vvl/lqV2vl= p-.

(4.8.1)

rl

This non-dimensional quantity is known as the Reynolds number. For most macroscopic flows, such as the flow around a tennis ball, the Reynolds number is very large, but for colloidal flows it is usually very small, thanks to the small particle radius and velocity. For example, for a particle of 0.5 p m radius moving with a typical velocity of m s-l in water at 20 “C the Reynolds number is 5 x lo-’. Thus for colloidal scale flows we may neglect the pv .Vv term in the equation of motion (eqn (4.6.5)), which reduces to

av

p- = F - Vp at

+ qV2v.

(4.8.2)

SEDIMENTATION OF A SUSPENSION

I 179

Unlike the original equation, this equation is linear, and hence the sum of two solutions is also a solution (see Exercise 4.8.1). We can therefore analyse a combination of effects such as Brownian motion and sedimentation, by first analysing each component in isolation and then superposing to obtain the combined effect. In this section we will begin by studying sedimentation in the absence of Brownian motion. In a frame of reference which moves with the steadily descending particle, the fluid velocity field is independent of time, and thus the equations of motion reduce to:

rV2v = Vp - F

and

V.v=O

(4.8.3)

These are known as the Stokes equations. They are the equations which must be solved for the calculation of the sedimentation coefficient and the viscosity of a suspension of particles. For a sedimenting particle, the boundary conditions are that v = 0 on the particle surface r = a, and v + -V far from the particle. The sedimentation velocity V is determined from the constraint that the net force on the particle is zero. The solution to the problem, which is described in standard texts (see for example section 4.9 of Batchelor 1970 or section 20 of Landau and Lifshitz 1959) is given by

-1

3a la3) +----Vcos8r+ 1 2 r 2r3 (

n

------

:ar

: ar33 )

Vsin88

(4.8.4)

where 8 is the angle between the direction of particle motion and the radius vector to the point in question. Note that the disturbance velocity v +V due to the presence of the particle decays at large distances like r-l (see Exercise 4.8.2). As we shall see, this relatively slow drop-off leads to significant particle interactions which limit the above analysis to very dilute suspensions. The force on the particle due to the fluid is obtained by integrating the fluid stresses over the particle surface which produces two terms: the bouyancy force in the direction of particle motion and the Stokes expression (6nqaV)for the viscous drag on the particle. The final expression for the particle sedimentation velocity is then (compare eqn 3.1.3):

(4.8.5) where it is assumed that the body force on the particle is (p,/p)F, pp being the particle density. The Stokes equations have also been solved for ellipsoidal particles; in this case the sedimentation velocity depends on the particle orientation. Perrin (1934) has calculated the average sedimentation velocity for a dilute suspension of ellipsoids on the assumption that the Brownian motion has given the particles a uniform orientation distribution.

4.8.2 Sedimentation in a concentrated suspension In a concentrated suspension the sedimentation velocity of a particle is affected by its hydrodynamic interaction with neighbouring particles. Since the disturbance

180 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

velocity due to an isolated particle drops off on a length-scale of the order of the particle radius, the hydrodynamic interaction between a pair of particles will only be significant if their separation is of the order of a or less. Thus at low particle volume fractions 4, the fraction of interacting particles should be proportional to 4, and of these the vast majority will be interacting with only one neighbour, since the probability of finding two (or more) particles within one or two radii of a given particle is proportional to 8. The average sedimentation velocity (V) for such a suspension will therefore be given by an expression of the form

(V) = Vo(1 + a4 + 842 + ......)

(4.8.6)

where Vo is the sedimentation velocity of an isolated particle, and the coefficient a can be obtained from the solution of Stokes equations for a pair of interacting particles, averaged over all possible separations. The difficulties involved in the solution of this Stokes flow problem are so great that at present it is only possible to calculate a for spherical particles. For the important case of force-free spheres for which Brownian motion has made all separations equally likely it has been found that a = -6.55 (Batchelor 1972). In Fig. 4.8.1 the measured sedimentation velocities are shown for latex suspensions over a range of volume fractions. The broken line represents the approximate formula

(V) = (Vo)(l - 6.554)

(4.8.7)

obtained by truncating the series eqn (4.8.6) at the 4 term, and using Batchelor’s value for a. From the figure it can be seen that the approximation is accurate to about 5 per cent if 4 < 0.05. At higher volume fractions the O(4’) term in eqn (4.8.6) becomes significant, reflecting interactions between groups of three or more particles. The fact that these interactions are significant at such a low volume fraction can be attributed to the slow l / r drop-off in the velocity field of a sedimentary particle. The unbroken line in Fig. 4.8.1 represents the formula (Ekdawi and Hunter 1985)

[

( V ) =vo 1 - -

T’*

(4.8.8)

with the parameters k1 and p set equal to 5.4 and 0.585 respectively, in order to fit the data. The origins of this type of representation are discussed briefly at the end of Section 4.10. T he above formulae for average sedimentation velocity are only valid over regions in which the suspension is macroscopically homogeneous, because in an inhomogeneous suspension the Brownian motion, which we have so far neglected, can lead to a flux of particles in addition to that due to the sedimentation force. T h e problem of calculating this diffusive flux in a dilute suspension has been discussed in Section 1.5. In the following section we will look at the case of concentrated suspensions.

BROWNIAN MOTION REVISITED

0.0 0.0

I

I

0.1

0.2

0.3

0.4

4

I181

0.5

Fig. 4.8.1 The average sedimentation velocity ( V ) is a function of particle volume fraction. The points were obtained from measurements on latex suspensions and the curve represents eqn (4.8.8) with k = 5.4 and p = 0.585. (From Buscall et al. 1982, with permission).

r

Exercises 4.8.1 Let v ~ , pand ~ ,v2,p2 be two solutions to the continuity equation and eqn (4.8.2). Verify that the sum v1 v2, p l p 2 also satisfies the equations. (This is a consequence of the linearity of the equations.) Show that this result is not true if eqn (4.8.2) is replaced by the full Navier-Stokes equation. 4.8.2 Satisfy yourself that the disturbance velocity v V = v V (cos 6' i. - sin 6'6) where ? and 6 are unit orthogonal vectors with i. directed radially from the particle centre. Hence show that, for large distances:

+

+

+

+

v + V = (3~/2)[V+(V/2)sin6'6].(l/r)

4.9 Brownian motion revisited 4.9.1 Gradient diffusion in a concentrated suspension Gradient diffusion refers to the motion of particles as a consequence of a concentration gradient. The diffusive flux in an inhomogeneous concentrated suspension can be calculated by an extension of the argument used by Einstein in his original study of dilute suspensions (see Einstein 1956). The argument centres on the case of a suspension in equilibrium under an applied field. The field is assumed to act only on the particles, giving rise to a forcef (x), where x denotes the position of the particle

182 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

centre. As a result of this field the equilibrium particle density n(x) will be nonuniform. There will therefore be a flux due to Brownian motion, which will be balanced by the sedimentation flux. As mentioned in the previous section, these two fluxes can be calculated separately and superposed. The sedimentation flux is obtained by multiplying the concentration by the velocity (recall Fig. 1.5.3):

where n is the particle number density and K(4)/6nqa is the average sedimentation velocity in a suspension in which the particles are acted on by a unit force. Comparing with the formula (4.8.7) for the sedimentation velocity for rigid spheres in nonconcentrated suspensions, we find

(4.9.2) For more concentrated suspensions (4 > 0.05), the empirical expression (4.8.8) can be used for calculating K. The Brownian flux will presumably depend on the local variations in particle number density, variations that are characterized by the quantity Vn. By similar arguments to those that lead to the Newtonian formula (eqn (4.5.7)) between stress and rate of strain it can be shown that the flux density due to Brownian motion has the form -2)Vn for a locally isotropic suspension. This is identical to Fick's first law (eqn (1.5.17)), but now we must allow for the fact that 2) depends on the local particle volume fraction. In equilibrium, the net particle flux is zero, so

(4.9.3) We also have the constraint that the net body force -n(x) f per unit volume of suspension must be balanced by the surface forces on the volume.+ Since the suspension is in equilibrium, the surface forces must take the form of an osmotic pressure ll, acting normal to the surface. The force balance equation therefore takes the form -n(x)f = Vll or, since the local osmotic pressure depends only on particle density in this case:

-n(xy =

~

dll Vn. dn

(4.9.4)

Using this equation to eliminate the nfterm in eqn (4.9.3) we find that the diffusivity is given by

(4.9.5)

t We are speaking here of a volume containing many particles

BROWNIAN MOTION REVISITED

I183

Thus the diffusivity can be calculated from sedimentation and osmotic pressure data for the suspension. On replacing K(4)in the above expression by the formula (4.9.2) and using the approximate result dll/dn = kT(1+

84)

(4.9.6)

(see Batchelor 1976) we find that

D/DO = 1

+ 1.454

(4.9.7)

where Do = kT/6nrla is the Einstein formula for the diffusivity in a dilute suspension, and terms of order have been neglected. It can be seen that the effects of particle interaction on sedimentation velocity and osmotic pressure nearly cancel out, leaving a relatively weak diffusion-concentration dependence. The experimental verification of eqn (4.9.7) is described in Russel’s (1981) review article. In a non-equilibrium suspension, differences between the sedimentation and diffusion fluxes lead to variations in the concentration n, variations that can be calculated using the equation an = V.[- K(4) (-Vn at 6nqa dn

-

-

nF)]

(4.9.8)

where F is the net sedimentation force on the particle. The derivation of this equation follows similar lines to that of eqn (1.5.20) for one-dimensional diffusion in a dilute suspension. For suspensions of non-spherical particles, and for suspensions of spheres at high concentrations, the quantities K(4) and dll/dn must be determined experimentally. Once these quantities are known, eqn (4.9.8) can be used for the prediction of sedimentation behaviour in any situation. This approach has been successfully applied to a number of colloids by Philip and Smiles (1982).

4.9.2 Self-diffusion in a concentrated suspension The formula

(x2) z1 = (2Dt)

[ 1.5.251

for the root-mean-squared displacement of an isolated spherical particle can also be applied to concentrated isotropic suspensions, with the proviso that the quantity D is not in general equal to the gradient diffusivity studied in the previous section. The quantity D in the above expression is usually referred to as the ‘tracer’ or ‘self diffusion coefficient. It measures the movement of an individual particle surrounded by a (uniform) collection of like particles. T o calculate this quantity it is necessary to solve the Stokes equations for the velocity of the tracer particle moving under the steady force in the presence of force-free neighbouring particles, and then average over all particle configurations. As in the case of gradient diffusion, the exact theoretical analysis is limited to the low concentration range where pair interactions dominate.

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

The formula for the tracer diffusion coefficient in a suspension of spheres is found to be (Batchelor 1976) 2, = (1 - 1.83@)kT/6nva.

(4.9.9)

Not surprisingly, the Brownian motion of a particle is hindered by interactions with its neighbours (Russel 1981, p. 437).

4.9.3 The Langevin equation In Chapter 1 the Brownian motion of an isolated particle was treated as a random walk, with the direction of each step being independent of previous steps. This assumption of independence places a lower limit on the step time t, for t must be large enough to allow the particle to ‘forget’ its previous movements. In order to estimate this lower limit for t, and to obtain other details of the dynamics of the Brownian motion, it is customary to use the Langevin equation mdV/dt

+ 6nqaV = F(t)

(4.9.10)

for the velocity V of an isolated spherical particle. This is simply Newton’s second law for a particle of mass m acted upon by a fluctuating Brownian force F(t),with the liquid drag represented by the Stokes formula -6nqaV. Since this formula was derived in Section (4.8.1) for a particle in steady motion, its application to the present problem can only be justified if the inertia term in the equation of motion of the liquid (eqn (4.8.2)) is negligible. After we have obtained our solution to the Langevin equation, we must therefore check for consistency by estimating the size of this neglected inertia term. Since the particle is much more massive than the surrounding molecules, its response time will be much greater than the time scale of the fluctuating force F(t), which will presumably be of the order of the relaxation time for water molecules (-10-13 s). Thus the velocity of a particle at any time is determined not just by the instantaneous force on the particle, but by the history of that force over several particle relaxation times. In order to estimate this relaxation time, we take the average of the Langevin equation over all those particles that have a given velocity V at t = 0. By the above argument, we expect that the particle velocity and the force will be statistically independent. Hence the average of F(t) over this group of particles will be zero, and the average of eqn (4.9.10) becomes md(V)/dt

+ 6nva(V) = 0.

(4.9.1 1)

The solution to this equation is (Exercise 4.9.1):

where t,, = m/6nqa. From eqn (4.9.12) it can be seen that M 0 if t is large compared with t,,. Since the average velocity of a random sample of particles is zero, t,, provides a measure of the time required for a particle to ‘forget’ its initial velocity. For a 0.1 p m radius particle of density 2 g cm-3 in water at 25 “C, t,, = 5 x lop9 s. For the

BROWNIAN MOTION REVISITED

I185

random walk analogue of Chapter 1 to be valid, the step times t must be much greater than this relaxation time. We are now in a position to test the validity of the Langevin equation by estimating the size of the neglected ‘inertia’ term p(av/at) in the equation of motion (eqn (4.8.2)) of the liquid. Since the particle velocity decays from its initial value VOover a time t,,, this inertia term is of order pVo/t,,. The viscous term qV2v will be of order qVo/a2, and so the ratio

(4.9.13) From eqn (4.9.12) we see that this ratio is of order p/p,, where p,, is the particle density. So for cases of practical interest, in which the particle density is not large compared to the fluid density, the Langevin equation is invalid! The correct analysis must be based on the equation of motion (eqn (4.8.2)), which takes into account the inertial forces in the liquid, together with a fluctuating ‘body force’ F,representing the effects of Brownian motion. As in the Langevin equation, this fluctuating term can be removed by averaging. The solution of these equations for the average velocity (V) yields a similar estimate for to,but the decay in (V) at large t is found to have a t-3/2form, instead of the exponential form found earlier (Russel 198 1, p. 428). Thus while the Langevin equation may not be strictly valid, it provides qualitatively correct results, and serves as a simple model for illustrating the techniques used in the analysis of the complete equations describing Brownian motion dynamics.

4.9.4 The Brownian motion of non-spherical particles In this section we will study the effect of Brownian motion on the orientation and position of spheroidal particles in a dilute suspension. The rotational diffusion of these particles can be analysed along similar lines to the translational diffusion discussed in Chapter 1. In particular the motion can be treated as a random walk, with the result that (compare eqn (1 5.25)): (02(t))= 4Drt

(4.9.14)

where 6 is the change in the particle orientation during time t, and Dr is termed the rotational diffusion coefficient. This result is valid provided the change in orientation is small (see Exercise 4.9.2). The effect of Brownian motion on the particle orientation distribution in a dilute suspension is described by the differential equation (McQuarrie 1976, p. 398). aR

-

at

[

= Dr

(.

1 a sin0-a sin 0 29

~-

~ +-)

1 a sin20

2~]

(4.9.15)

where the angles 0 and 4 used to specify the orientation of the particle axis are illustrated in Fig. 4.9.1. By definition, the quantity R(0, 4, t) sin 0 A0 A 4 is equal to the fraction of particles whose orientation lies in the range (0 f (A0/2), 4 f A4/2)).

186 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Fig. 4.9.1 The angles 0 and 4 used to specify the orientationof a particle, relative to a fixed set of Cartesian coordinates.

By multiplying eqn (4.9.15) by cos 8 and integrating over all orientations it can be shown that

(cos8) = exp ( - ~ D J )

(4.9.16)

for a suspension in which all the particles are initially aligned in the 0 = 0 direction. Equation (4.9.16) refers to what is called Debye relaxation, since it was used by Peter Debye (1929) to describe the relaxation behaviour of molecular dipoles. From eqn (4.9.16) it can be seen that the time required for the particle distribution to become isotropic (i.e. Ccos 8> M 0) is of order l/D,.. The quantity Dr can be calculated using a variation of the Einstein argument described in Section 4.9.1, that is by considering a suspension in equilibrium under the action of an applied field which in this case tends to align the particles. The applied field leads to an extra term in eqn (4.9.15) corresponding to the rotation of the particles in the absence of Brownian motion. The solution of eqn (4.9.15) for this equilibrium problem yields:

Dr = k T / B ,

(4.9.17)

where B, is the friction factor for particle rotation, equal in magnitude to the torque required to rotate the particle with unit angular velocity, in the absence of Brownian motion. From the solution of the Stokes equations around a rotating spheroid it is found that (Perrin 1934):

for a cigar-shaped (prolate) spheroid, where q = b/u is the ratio of the length of the minor axis to the major axis (the axis of rotation of the particle). The formula for discshaped spheroids is given in Exercise 4.9.5.

BROWNIAN MOTION REVISITED

I187

The problem of translational diffusion can be treated in a similar manner: for a suspension with a uniform orientation distribution, the translational diffusion coefficient is given by

where BIIand BI are the friction coefficients for motion parallel and perpendicular to the particle axis respectively. The formulae for these quantities are given in Perrin’s (1934) paper. With the aid of these formulae and the expression for the rotational diffusivity it is possible to estimate the particle size and shape from combined translational and rotational diffusivity measurements on dilute suspensions. The results relate of course to the size of the particle in solution, and this may differ from that of the dry particle as a result of solvent absorption or adsorption. This method has mainly been applied to protein and virus particles, but some work has also been done on montmorillonite suspensions (Shah 1963). The extension of theoretical and experimental studies of translational and rotational Brownian motion and diffusion to intermediate and higher concentrations has to date been limited to spherical particles (Degiorgio and Piazza (1996) and Degiorgio et al. (1994)).

r

Exercises 4.9.1 Establish eqn (4.9.12). 4.9.2 By using a suitable approximation for small values of 8 in eqn (4.9.16),derive eqn (4.9.14) for the mean squared orientation change of a particle during a small period t. 4.9.3 Calculate the value of Vr, for values of u / b from 1 to 20 and a = lop7 m using eqn (4.9.18). Show that for large a/b, this formula takes the approximate form

Compare values obtained with this formula with the exact value obtained earlier. 4.9.4 Alexander and Johnson (1949; p. 400) quote a value of Vr = 7 s-l obtained by Edsall for the rotary diffusion constant of rabbit myosin at T = 276 K. Taking the viscosity of water as 1.6 x lop3 Pa s at this temperature and a high value (say u / b M 100) for the axial ratio, estimate the length of the myosin molecule. (Independent measurements of the particle volume would permit a further refinement of this result.) 4.9.5 For an oblate (disc-shaped) particle the rotational diffusivity Vr,is given by RT v - 16nyab2(1 [3&(2 - 8) -

-

&)(q2 - 1)-f arctan(& - 1)f

Show that for q >>1 (i.e. a flat disc): V,.% 3kT/32yb3

+3

1

188 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

4.10 The flow properties of suspensions One of the most important transport properties of a colloidal dispersion is its viscosity. The problem of estimating the influence of the particles on the macroscopically observed viscosity has attracted much attention from some eminent scientists. Undoubtedly the most famous result is that derived by Einstein for the effect of volume fraction of particles on viscosity (eqn 4.10.9) below). It is a deceptively simple expression that represents quite well the limiting behaviour of smooth spheres at low concentrations in a Newtonian fluid. We will not follow Einstein’s method here, but a rather more general procedure that proves to be more productive in the long run.

4.10.1 The macroscopic flow field In our analysis of fluid motion we neglected the molecular nature of the fluid, with the proviso that attention be restricted to volumes of fluid containing large numbers of molecules. By the same token, a suspension can be treated as a one-component continuum, provided we only consider volumes containing many particles. ‘Local properties’ such as the velocity and density for this continuum can be defined as averages over volumes containing large numbers of particles. For example, the average velocity is defined as 1

P

(4.10.1) V

where the sample volume Vis centred on the point x.In a flowing suspension this average velocity will vary on a macroscopic length scale L determined by the apparatus; in flow down a pipe for example, L will be of the order of the pipe radius. Thus in order to give formula (4.10.1) an unambiguous meaning we must specify that the sample volume Vbe much smaller than L3,while still being large enough to contain many particles. Differential equations for the macroscopic velocity field v(x) can be derived in a similar manner to the Navier-Stokes equations for the local velocity field; that is by applying the Principle of Conservation of Mass and Newton’s Second Law to a rectangular block of fluid. However, whereas in the earlier sections we took this block to be much larger than the molecules and much smaller than the particles, the block in this case is taken to have the dimensions of a sample volume, as defined above. The most difficult step in the derivation of these differential equations for the macroscopic velocity field is the determination of the relationship between the shortrange forces that act on the surface of the block and the macroscopic velocity field; that is, the analogue of the Newtonian expression (eqn (4.5.8)) for the microscopic tensor in a liquid. The remainder of this section is concerned with the derivation of this relationship.

4.10.2 The macroscopic stress tensor The volume average value of the macroscopic stress tensor is defined as 1

() = - /d V.

V. v

(4.10.2)

THE FLOW PROPERTIES OF S U S P E N S I O N SI189

As in the case of the microscopic stress tensor, we can write (cf. eqn 4.5.2):

+ (D)

() = -PI

where P is an average pressure. CD> is the part of <> that depends on the motion. By using the fact that = 2qe where e is the rate of strain tensor in the liquid, (c.f. eqn (4.5.8)) we can write eqn (4.10.2) in the form ( ) = - P Z + 3V/ e d V + d / D d V VI

(4.10.3)

VP

where V, is the volume occupied by the particles in V, and 6 is the liquid volume. The rate of strain tensor e is zero inside the rigid particles (see Exercise 4.10.1), and so the integral over 6 in eqn (4.10.3) can be formally extended over the particle volume as well, resulting in () = -PI

+ 2q(e) + n(S)

(4.10.4)

where

(e) = JV/ e d V

(4.10.5)

V

is the average rate of strain tensor. As before, n denotes the particle number density, and <S> denotes an average, over the particles in V, of the quantity

/

Dd V

(4.10.6)

V’

where Vi is the volume of the ith particle in V. <S> is referred to as the average particle dipole strength. The term ‘dipole strength’ arises from the application of this volume averaging technique to the electrical transport properties of suspensions. There each uncharged particle behaves, when viewed from a distance, like an electric dipole with a dipole strength given by an integral form analogous to eqn (4.10.6).The average dipole strength in (4.10.4)represents the particle contribution to the stress. T o calculate this term we must solve the Stokes equations around the particles. In the following section we will outline the solution to this problem for a dilute suspension of spheres.

4.10.3 The effective viscosity of a dilute suspension of spheres For a dilute suspension we can treat each particle as being alone in an infinite liquid. T o determine the dipole strength in this case, in the absence of Brownian motion, we must solve the Stokes equations for the velocity and pressure fields around the isolated particle, subject to the usual boundary conditions. (The velocity far from the particle is the same as it would be in the absence of the particle and the velocity of the fluid at the

190 I

4: T R A N S P O R T P R O P E R T I E S OF S U S P E N S I O N S

surface of the particle is the same as the particle velocity - the no-slip boundary condition.) In this case it turns out that a spherical particle will remain at rest in the ambient field and if it is rigid, the resulting value of the dipole strength is given by (Landau and Lifshitz 1959):

S = (20/3)m3q(e).

(4.10.7)

In the absence of particle interaction, the dipole strength is the same for each sphere in the sample volume, and the formula (4.10.4) becomes () = -PI

+ 2q[l + 5@](e).

(4.10.8)

This has the same form as the expression (eqn (4.5.8)) for the stress in a Newtonian liquid. Thus from the macroscopic point of view, a dilute suspension of spheres behaves as a Newtonian liquid with viscosity q* where q* = q(l

+ 2.5$).

(4.10.9)

This result was obtained by Einstein? from a calculation of the energy dissipation in the suspension. This dissipation method suffers from the disadvantage that it yields only a single number, namely the rate of energy dissipation. While this may be adequate for isotropic suspensions, which can be characterized by a single viscosity, it fails when applied to a non-isotropic suspension such as a suspension of rod-like particles aligned by a flow. Such suspensions require a number of ‘viscosities’ to describe their flow properties. For this reason, the recent papers on this subject have adopted the volume average approach described here, an approach first devised by Landau and Lifshitz (1959).

4.10.4 Dilute suspensions of spheroidal particles The calculation of the flow behaviour of dilute suspension of spheroids is both more complicated and more interesting than the problem for spheres, for as we shall see, these suspensions exhibit non-Newtonian effects that are common to a large class of suspensions of this kind. We can, therefore, gain some insight into the factors that are responsible for non-Newtonian behaviour in more complicated suspensions. Although the mathematical details of the calculation of <> for a dilute suspension of spheroids are beyond the scope of this volume, we have established a sufficient foundation in the chapter to enable us to give a description of the methods used, and some of the results that have so far been obtained. We will examine some of the effects in a little more detail in Chapter 15 but for a more complete description of the many results which have already been obtained in this area one should consult a more appropriate source such as the monograph by van de Ven (1989) (p. 149 et seq.). The formula (4.10.4)relating the macroscopic stress to the average dipole strength is

t Eistein initially calculated the coefficient of q5 as unity but one of Perrin’s students measured the effect and found it fell between 2 and 3. Perrin alerted Einstein who revised his analysis and obtained 2.5.

T H E FLOW P R O P E R T I E S OF S U S P E N S I O N S

1191

valid for those suspensions, and since the suspensions are dilute, we can again calculate the dipole strength of each particle as if it were alone in an infinite liquid. This calculation involves the solution of the Stokes equations subject to the same boundary conditions as for the spherical particle. As before, the dipole strength due to the rigid body component of the ambient field is zero and the particle simply translates and rotates with the field. Unlike the spherical particle however, the dipole strength of the spheroid in the ambient straining field is found to depend on its orientation relative to the field, and furthermore this straining field induces an angular velocity in the particle that depends on particle orientation. This rotation is superposed on the rigid body rotation described earlier. Thus spheroidal particles do not simply rotate with a uniform angular velocity like spheres, but move in a more complicated fashion that depends on the type of flow. For example, in a shear flow it is found (Okagawa et al. 1973) that cigar- shaped particles rotate in a periodic fashion, with the minimum rate of rotation occurring when the particle is most nearly aligned with the streamlines. In other flows such as the extensional flow of a stream of liquid from a hole in the bottom of a container, there is a preferred orientation (also along the streamlines in this case) at which the particles have zero angular velocity. Such flows tend to align the particles in the preferred direction. Since the average dipole strength depends on the orientation distribution, R (0, $), the same suspension may behave quite differently for these different flows. T o calculate the distribution function R for any flow it is necessary to solve eqn (4.9.15) with an extra term due to the rotation of the particles by the ambient flow. Since the rotation rate is non-uniform, the ambient flow tends to make the distribution function non-uniform. For example, in the shear flow case described above, the particles move more slowly when they are aligned with the streamlines, and so R will be larger for the aligned orientation. This aligning tendency is opposed by the Brownian motion term in eqn (4.9.15), and the final form of the distribution function in a steady flow is determined by the relative magnitude of these two opposing terms. In a shear flow this quantity is characterized by a ‘Piclet number’ p/Dr. When the ratio is small, the Brownian motion dominates and the orientation distribution is approximately uniform. At large Piclet numbers the distribution is quite non-uniform, but rather surprisingly the Brownian motion still plays an important part in the shear flow case, for in the absence of Brownian motion the particles simply rotate in a periodic fashion leading to an orientation distribution R that varies periodically. In such a situation the effect of a small amount of Brownian motion over a long period of time results in a steady-state orientation distribution. The orientation distribution has been calculated for a number of limiting cases such as nearly spherical particles at low or high PCclet numbers, for a range of flows (Leal and Hinch 1973). Once the orientation distribution is known, the average dipole strength can be calculated directly, but even at this stage there is an unexpected complication, for in addition to the dipole strength obtained from the solution of the Stokes equations in the ambient flow there is a direct ‘contribution’ from the Brownian motion. The dipole strength of a particle undergoing Brownian motion will, like its angular velocity, depend on the history of the fluctuating torque on the particle over the previous few relaxation periods. In a suspension in which the orientation distribution is non-uniform, the particles that instantaneously have a given orientation represent a biased sample, since more will have come (via Brownian rotations) from

192 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

neighbouring orientations where R is large than from those where R is small. Thus on average the particles will have experienced more torques in one direction that in another, and as a result their average dipole strength due to these torques is non-zero. Although this extra dipole strength is the consequence of fluctuating torques it can in fact be calculated from the solution of Stokes’ equations around a spheroid rotating under the action of a steady ‘thermal torque’ analogous to the thermal force introduced in Section 1.5.2 (see Hinch and Leal 1972). Since the orientation distribution depends on the type of flow, it is difficult to make general statements about the flow properties of these suspensions; consequently we will limit our attention to the results obtained for the important case of a steady shear flow. Even in this restricted case it is not possible to describe the flow properties in terms of a single viscosity, for when a suspension is subjected to the idealized shear flow of Fig. 3.4.2, the top plate would experience both a shear force and a normal force, both depending on the shear rate. We will concentrate here on the calculation of the ‘shear viscosity’, which represents the ratio of shear stress to shear strain rate. It is found that the shear viscosity of the suspension, r]*, decreases with increasing shear rate. Although this viscosity dependence cannot be calculated for arbitrary particle shapes, the limiting viscosities at small and large Piclet numbers have been ) and Leal 1972). These results are calculated for arbitrary aspect ratios ~ ( = a / b (Hinch illustrated in Fig. 4.10.1, when [ r ] ] is the ‘intrinsic viscosity’ defined by (4.10.10) The broken lines represent approximate formulae valid for disc-shaped (la<< l), and rod-like ( ~ > > l particles. ) With the aid of these results it should be possible to obtain information about particle shape from measurements of the high and low shear rate viscosities.

I 300 -

I

I

I

I

I

I

1

/

100 -

I

I

I

32 1 _._

I

0.01 0.03

I

0.1

I

I

0.3 1 w=llq

I

I

I

3

10

30

1

Fig. 4.10.1. The high and low shear rate intrinsic viscosities of a suspension of spheroidal particles. From Figs 5 and 8 of Hinch and Leal’s (1972) paper.

THE FLOW PROPERTIES OF S U S P E N S I O N SI193

4.10.5 Concentrated suspensions As with the other transport properties, theoretical studies in this area have been limited to the case of suspensions of spheres, and attention has centred on the case of semidilute suspensions, in which pair interactions are dominant. For such suspensions, the average dipole strength can be approximated by the formula

where SOis the dipole strength of an isolated sphere, and S1 is related to the dipole strength of one of a pair of particles in the ambient flow, averaged over all pair configurations. T o calculate this average it is necessary to determine the pair probability function p(r), where p(r) is the probability of finding a particle with its centre in the small volume A V around the point r, given that there is a sphere at the origin. p(r) is the analogue of the orientation distribution R (@,$)for the dilute spheroid problem and, like the orientation distribution, p(r) depends on the type of flow. In Fig. 4.10.2 we show the relative trajectories of pairs of spheres in a shear flow (Batchelor and Green 1972~).The lines represent the paths traced out by the centre of one member of the pair relative to the other for the case of pairs lying in the plane of the flow. Since the path lines are symmetrical about the xz and XI axes, only one quadrant has been shown in the figure. If the ambient shear rate av?/axz is negative, the second particle will move from right to left along the path lines in the diagram. The quantity RZrepresents the xz coordinate of the particle far upstream, where the path lines are parallel to the XI axis. From the diagram it can be seen that the flow tends to bring sphere pairs together. For the case Rz/a C 1, the pairs are brought into such proximity that even a slight attractive force may be enough to bring about ‘shear-induced’ coagulation.

32 R;la2=9

I

2

I

2

A 1

0

1

2

xlla

3

4

5

Fig. 4.10.2 The trajectories of one member of a pair of spheres in a shear flow, relative to the other sphere, for pairs lying in the plane of the flow. The circle of radius 2 is the trajectory for touching particles. (From Batchelor and Green 1972a, with permission.)

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

Those trajectories that lie below the R2 = 0 line represent pairs in closed orbits. In the absence of Brownian motion the pair distribution function in that region will vary periodically with time, like the orientation distribution for the spheroid problem. For such particles Ri/a2 < 0 so there are no (real) values of R far upstream of the particle. The difference between open and closed streamlines is shown more clearly in Fig. 4.10.3 which again has the origin at the centre of particle 1. It shows the limiting trajectory and the minimum separation distance which can be very small (< 1 nm). This distance of closest approach can be used to estimate the roughness of the particle surface (Arp and Mason 1977). In extensional flows there is also a tendency for pairs to be brought together, but there are no closed orbits. These flows lead to an increase in the probability p(r) at small separations, a tendency that is of course opposed by the Brownian motion. For a given flow the form of p(r)is determined by the relative magnitude of these two opposing terms, characterized by the Piclet number j.a2/D, where is a

Fig. 4.10.3 Equatorial trajectories of two spheres in simple shear flow (schematic). The solid lines are the relative trajectories of a sphere of radius a2 with respect to the reference sphere. The minimum approach distance, &in, may be less than 1 nm. (After van de Ven 1989, p. 362 with permission.)

THE FLOW PROPERTIES OF S U S P E N S I O N SI195

0.00

0.05

0.10

0.15

0.20

Volume fraction Fig. 4.10.4 The effect of particle crowding on viscosity. The broken line is drawn according to the Einstein eqn (4.10.9) and indicates that these silica spheres show the theoretical intrinsic viscosity of 2.5. The points are experimental and the full curve is drawn using the polynomial expression (4.10.12) with b = 5 and c = 53. (From Jones et al. 1991 with permission.)

typical macroscopic strain rate and D is the translational diffusivity of an isolated sphere. At low Piclet numbers the Brownian motion dominates and p(r) is approximately uniform. In this case it is found that the suspension behaves as a Newtonian liquid with viscosity q* = q(l

+ 2.54 + 6.24’)

(4.10.11)

(Batchelor 1977). The theoretical coefficient of the 42 term is certainly close to the experimental value since Jones et al. (1991) fitted their experimental results to a polynomial of the form: q* = q(l

+ 2.54 + b42+ ~

4 ~ )

(4.10.12)

and achieved best fit with b = 5 and c = 53. The fit (Fig. 4.10.4) what is more, is excellent up to about 4 = 18%. At high Piclet numbers the problem of determining p(r) in a shear flow is complicated by the closed trajectories referred to earlier, for in the region of closed trajectories a small amount of Brownian motion can have a significant effect on p(r),in the same way that the Brownian motion has a significant effect in the spheroid case at high shear rates. In the extensional flow, where there are no closed trajectories, the effect of the Brownian motion is unimportant. The calculation of the bulk stress in this flow is described by Batchelor and Green (19726). For more concentrated suspensions there are a number of semi-empirical formulae that can be used for calculating the shear viscosity. The data for the limiting shear

196 I

4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

viscosities at high and low shear rates (that is, high and low Piclet numbers) can be adequately represented by the Dougherty-Krieger formula (4.10.13) where [ q ] is the intrinsic viscosity (eqn (4.10.10)) and p is an adjustable parameter. This formula can be derived by considering the change in the viscosity Sq* caused by an increase in the volume fraction 64. If the suspension before the addition of particles is treated as a Newtonian liquid with viscosity q*, then by the Einstein formula Sq* = 2.5q*.S@.

In order to take into account the fact that the suspension is rigid when the particles are closely packed, S4.is replaced in the above formula by S$/( 1 - K4) where 1 - K+ is the volume available for the added particles. Integration of the resulting formula yields (Exercise 4.10.2): (4.10.14)

0.0

0.1

0.2

0.3 0.4 Volume fraction

0.5

0.6

0.7

Fig. 4.1 0.5 The low shear rate viscosities of concentrated suspensions of the same silica spheres as shown in Fig. 4.10.4. The curve represents the Krieger-Dougherty eqn (4.10.13) with [qlp = 2 and p = 0.631. Note that the relative viscosity is here shown on a log scale to accommodate values up to 100 000. The intrinsic viscosity used here is 2/0.63 = 3.17 which is a little higher than the measured value (Fig. 4.10.4) as is indicated by the fact that the curve runs a little above the data points at the lowest volume fractions. The fit is, however, remarkably good. (Data from Jones et al. 1991.)

THE FLOW PROPERTIES OF S U S P E N S I O N SI197

where p = 1/K is expected to be approximately equal to the volume fraction for close packing. In practice, p is treated as an adjustable parameter, and the 2.5 is replaced by the measured intrinsic viscosity [q], which can differ from the Einstein value due to effects such as non-sphericity, particle charge, and the presence of small numbers of permanent doublets. Equation (4.10.14) is an example of an effective medium relationship. There are a number of these very useful formulae which can be used to describe the phenomenology of, for example, flow and sedimentation, where the more formal theoretical relations are very limited in their range. From Fig. 4.10.5 it can be seen that the formula (4.10.13) provides an excellent fit of the measured low shear rate viscosities of a suspension of silica spheres up to concentrations near to the close-pack limit. In this case the value of [qbwas 2 and p = 0.631. The same relation was used by Krieger to describe successfully his low and high shear rate viscosity measurements for latex suspensions. In both cases [q] = 2.67, while p = 0.57 for the low shear and 0.68 for the high shear viscosity. For such suspensions, the viscosity at intermediate shear rates can be calculated using an equation of the form (Krieger 1972)t:

-.=(I+%) rl* - rT r12

-

IS1

-I

(4.10.15)

rll

where r l l * and rl2 * are the high and low shear limiting viscosities respectively. S is the shear stress, and Si is a characteristic shear stress related to the particle diffusivity (Exercise 4.10.4):

where a is an adjustable parameter, independent of 4 and strain rate. Considering the fact that there are only two adjustable parameters in these formulae, viz. p and a,the fit of the data is most impressive. Figure 4.10.6 shows the measured shear viscosities for latex suspensions at 4 = 0.35 and 0.45. The lines represent the Krieger formula (4.10.15)with a = 0.431, and the rli * values calculated from eqn (4.10.13) with the [ q ] and p values given earlier. There is an obvious similarity between eqn (4.10.13) and eqn (4.8.8) for the sedimentation velocity of a concentrated suspension. The latter equation gives a very good description of the sedimentation velocity at concentrations far in excess of those for which the more exact theory is applicable (Fig. 4.8.1). It can be rationalized by treating the sedimentation of each particle as though it were obeying eqn (4.8.5) but assuming that the surrounding medium has the average density and the average shear viscosity of the suspension (not the liquid). This is what is meant by the term effective medium theory mentioned above. We will have more to say about such descriptions in Chapter 15.

+ See Chapter 15 for more details.

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4: TRANSPORT PROPERTIES O F S U S P E N S I O N S

- Predicted 10 ,P

b

9

8 7

0.01 0.03

0.1

0.3

0.01 0.03 0.1

0.3

1.0

3.0

10

30

1.0

3.0

10

30

3.5

S, Fig. 4.10.6 Measured shear viscosities of concentrated latex suspension as a function of nondimensional shear stress. Si = Sa3/kT.The volume fraction is 0.35 for (a) and 0.45 for (b). The lines represent eqn (4.10.15)with a = 0.431. (From Krieger 1972, with permission.)

Exercises 4.1 0.1 In deriving the formula (4.10.4) for the macroscopic stress tensor in a suspension of rigid particles, we used the fact that e = 0 inside the particles. Establish this result by calculating the strain tensor for the rigid body velocity field V + x x. 4.1 0.2 Establish eqn (4.10.14) using the suggested integration procedure. 4.10.3 Show that Si in eqn (4.10.15) is proportional to Q / a 2 where V is the particle diffusion coefficient. 4.10.4 Calculate S, for the suspensions shown in Fig. 4.10.5 assuming the particle radius is 100 nm and temperature is 25 "C.At what shear rate is the suspension viscosity half way between ql * and q 2 * ? What do ~ l and * QZ * represent?

References Alexander, A.E. and Johnson, P. (1949). Colloid science. Oxford University Press, Oxford.

Arp, P.A. and Mason, S.G. (1977). J. Colloid Interface Sci. 61, 44. Batchelor, G.K. (1967). An introduction to fluid dynamics. Cambridge University Press, Cambridge.

REFERENCES

Batchelor, G.K. (1970). J Fluid Mech. 41(3), 545. Batchelor, G.K. (1972). J. Fluid Mech. 52, 245. Batchelor, G.K. (1976). J. Fluid Mech. 74, 1. Batchelor, G.K. (1977). J Fluid Mech. 83, 97. Batchelor, G.K. and Green, I.T. (1972~). 3. Fluid Mech. 56, 375. Batchelor, G.K. and Green, I.T. (19726). 3. Fluid Mech. 56,401. Buscall, R., Goodwin, J.W., Ottewill, R.H., and Tadros, Th. F. (1982). J. Colloid Interface Sci. 85, 78. Debye, P. (1929). Polar molecules. Reinhold, New York. Degiorgio, V., Piazza, R., and Jones, R.B. (1994). Physics Rev. E 52,2707-17. Degiorgio, V. and Piazza, R. (1996). Current Opinion in Colloid and Interface Sci. 1 [l], 11-16. Einstein, A. (1956). Investigations on the theory of the Brownian movement. Dover, New York. Ekdawi, N. and Hunter, R.J. (1985). Colloids and surfaces 15, 147-59. Goodwin, J.W. (1975). The rheology of dispersions. In Colloid science (ed. D.H. Everett), Vol. 2, pp. 246-93. Chemical Society, London. Hiemenz, P.C. (1977). Principles of colloid and surface chemistry, p. 59. Marcel Dekker, New York Hinch, E.J. and Leal, L.G. (1972). J Fluid Mech. 52, 683-712. Hunter, R.J. and Nicol, S.K. (1968). J. Colloid Interface Sci. 29, 250. Jones, D.A.R., Leary, B., and Boger, D.V. (1991). J. Colloid Interface Sci. 147, 479-95. Krieger, I.M. (1972). Adv. Colloid Interface Sci. 3, 111. Krieger, I.M. and Maron, S. (1952). 3. appl. Phys. 23, 147-8. Krieger, I.M. and Maron, S. (1954). 3. appl. Phys. 25, 72-5. Landau, L.D. and Lifshitz, E.M. (1959). Fluid mechanics. Pergamon Press, Oxford. Landau, L.D. and Lifshitz, E.M. (1969). Statisticalphysics. Pergamon Press, Oxford. Leal, L.G. and Hinch, E.J. (1973). Rheol. Acta 12, 127. Maron, S.H., Krieger, I.M., and Sisko, A.W. (1954).J appl. Phys. 25,9714. Meriam, J.L. (1966). Dynamics. Wiley, New York. McQuarrie, D.A. (1976). Statistical mechanics. Harper and Row, New York. Okagawa, A., Cox, R.G., and Mason, S.G. (1973).J Colloid Interface Sci. 45,303. Perrin, F. (1934). 3. Phys. Radium 5(7) 497. Philip, J.R. and Smiles, D.E. (1982). Adv. Colloid Interface Sci. 17, 83. Russel, W.B. (1981). Ann. Rev. Fluid Mech. 13,425-55. Shah, M.J. (1963).3. Phys. Chem. 67,2215-19. Van de Ven, T.G.M. (1989). Colloidal hydrodynamics, pp. 582. Academic Press, London. Van Wazer, J.R., Lyons, J.W., Kim, K.Y., and Colwell, R.E. (1963). Viscosity and flow measurement. Interscience, New York.

11%

Particle Size and Shape 5.1 General considerations 5.2 Direct microscopic observation 5.2.1 Optical (light) microscopy 5.2.2 The ultramicroscope 5.2.3 The transmission electron microscope 5.2.4 The scanning electron microscope 5.2.5 The 'size' of irregular particles 5.3 Particle size distribution 5.3.1 The mean and standard deviation 5.3.2 Moments of a distribution 5.3.3 The continuous distribution function 5.3.4 Logarithmic distributions 5.3.5 The geometric mean 5.3.6 The measure of polydispersity 5.4 Theoretical distribution functions 5.4.1 The normal distribution 5.4.2 The log-normal distribution 5.4.3 Other distributions

5.5 Sedimentation methods of determining particle size 5.5.1 Sedimentation under gravity (a) Time dependent settling (b) Sedimentation equilibrium in a gravitational field 5.5.2 Centrifugal sedimentation (a) Time-dependent behaviour (b) Determining size in the ultracentrifuge

5.6 Electrical pulse counters 5.6.1 Theory of the Coulter counter 5.7 Light scattering methods 5.7.1 Intensity methods 5.7.2 Instrumentation and data treatment 5.7.3 Dynamic (quasi-elastic) light scattering (also called photon correlation spectroscopy)

200

GENERAL CONSIDERATIONS

1201

5.8 Hydrodynamic methods 5.8.1 Capillary hydrodynamic fractionation (CHDF) 5.8.2 Field flow fractionation (FFF) 5.9 Acoustic methods 5.9.1 Velocimetry and attenuation 5.9.2 Electroacoustics

5.10 Summary of sizing methods

This chapter concerns itself first with the direct observation of the sizes and shapes of particles commonly found in colloidal systems. Since the range of sizes in a suspension is often very considerable we next consider how best to represent the distribution of particle sizes by some convenient algebraic formula. Some of the more common methods of particle size analysis are then described. The theoretical basis of each method is given in sufficient detail to appreciate its scope and limitations but to obtain practical experimental details of any particular method the reader should consult the more specialized manuals (Kissa 1999, Allen 1990, Barth 1984) or reviews (Miller and Lines 1988). This is an area of active instrument development and the range of instruments and their characteristics are constantly changing. It is hoped that the principles developed below will help the reader to safely navigate between the claims and counterclaims for the various technologies.

5.1 General considerations The most significant characteristics of many colloidal dispersions (especially aerosols and the dispersions of solids in liquids) are the size and shape of the particles, since most other properties of the system are influenced to some extent by these factors. The idealized systems of monodisperse or highly regular particles discussed in Section 1.4 are of great importance in the testing of fundamental physical models of colloid behaviour, but it must be recognized that the majority of colloidal dispersions of scientific and technological interest consist of particles that differ markedly in size (sometimes over several orders of magnitude in characteristic dimension) and may be of very irregular shape. T o treat such systems at all we may have to adopt some rather drastic assumptions. In theoretical analyses they are usually approximated as spheres, or more rarely as spheroids (i.e. the solid body obtained by rotating an ellipse around one of its axes (Fig. 5.1.1)) or cylinders. Disc-shaped particles may be regarded as cylinders of very small height, or as oblate spheroids, or sometimes as ‘infinite’ flat plates. Octahedral, rhomboidal, or cubic crystals, especially if they are small, will often behave like spheres, and long parallelepipeds can usually be approximated as cylindrical rods. Particles produced by dispersion methods have shapes that depend partly on the natural cleavage planes of the crystal but that also reflect the presence and disposition of imperfections, cracks, and other flaws, which offer points or lines of weakness at which the imposed stress tends to concentrate. Fracture can then produce very sharp edges and asperities on the particle surface.

202 I

5 : PARTICLE SIZE A N D SHAPE

I A X~S

ofrotation

t h

Fig. 5.1 .I (a) An oblate spheroid, obtained by rotating an ellipse around its short axis. The crosssection in the plane where b is measured is circular. (b) A prolate spheroid, obtained by rotating an ellipse around its long axis (like a football). Sections parallel to b are circular. (c) A disc may be approximated as an oblate spheroid, a cylinder (d), or an ‘infinite’ flat plate (e).

As the dispersion process is continued down to colloidal dimensions (say, in a colloid mill (Section 1.4.1)) this effect is, to some extent, reduced because very sharp asperities have a very small radius of curvature and hence tend to dissolve preferentially, as would be expected from the analogue of the Kelvin equation (eqn (2.5.13)). Very small colloidal particles (< 100 nm) therefore, often appear to be rather less jagged in outline than the larger (microscopic) particles obtained by simple grinding and crushing operations. Particles produced by condensation have shapes that depend upon the rate of growth of different crystal faces. If thermodynamic equilibrium is maintained during crystal growth we noted in Section 2.6 that the shape is determined by the condition that the sum CAiyi is a minimum at constant volume of the crystal. Ai and yi are the area and surface energy of the zth face, respectively. (The surface energy of the different faces of a crystal (001, 110 etc.) is slightly different because of the differences in packing density of the atoms.) In many cases, however, the growth rate of a face is influenced by kinetic factors (e.g. rate of diffusion to the face, or rate of incorporation in it) rather than thermodynamic (equilibrium) ones. It is also observed that certain substances can be preferentially adsorbed onto particular crystal faces, changing the surface energy and profoundly altering the shape (or habit) of the crystal (Fig. 5.1.2).

GENERAL CONSIDERATIONS I203

Fig. 5.1.2 (a) Electron micrograph of calcium carbonate (calcite) crystals (normal habit). (b) Crystals of the same calcium carbonate obtained from a solution containing an organophosphorus crystal habit modifier. (Photographs obtained from the work of Leonard Dubin 1980, of Nalco Chemical Company, Illinois, which markets these modifying compounds.)

A very stimulating and original treatment of the concepts of particle shape (including an introduction to the use of the mathematical theory offuzzy setst) is given by Beddow (1980, Chapter 6) and the interested reader is referred to that work. Rather

+These are sets for which the concept of membership is fuzzy (i.e. one may not be certain whether an object does or does not belong to the set). Nevertheless, they can be handled logically with the help of fuzzy set theory.

204 I

5 : PARTICLE SIZE AND SHAPE

more conventional treatments of the subject are given in the standard works by Orr and Dallavalle (1959) and Herdan (1960). A brief description of the more sophisticated techniques for describing shape is given by Sutton (1976); the use of Fourier series is described there and by Beddow et al. (1977) while Allen (1981) and Kissa (1999) describe the commonly used shape coefficients and shape factors. For more regular particles (discs, rods, and ellipsoids), Jennings and Parslow (1988) have given a general account of the relations between the dimensions measured by different analytical techniques, and the equivalent spherical diameter of the particle (Section 3.1.1). An excellent introduction to the notion of fractals and their significance in chemistry (including some reference to particle size and surface area measurement) is given by Harrison (1995). Space here permits only a very brief outline of the methods of treating irregular particles. Finally, it must be emphasized that different methods of particle size analysis will almost certainly yield different results but that may not mean that one method is superior to the other. It may reflect the fact that different characteristics of the particles are being measured. Combining the results from different measurements can yield more significant information about the particles. A most striking example is provided by Jennings (1993) who combined data from rotary diffusion and sedimentation to estimate the axial ratio of disk-shaped particles of kaolin.

5.2 Direct microscopic observation The most significant development in this area this century has, of course, been the transmission electron microscope (TEM), and more recently the scanning electron microscope (SEM). Since colloidal particles generally fall below the limit of resolution of the optical microscope, the early colloid chemists were forced to rely on indirect evidence to obtain an idea of particle shape and, in the absence of any evidence to the contrary, tended to assume that all particles were roughly spherical. There were, of course, some exceptions: the crystal structure of the common clay minerals suggested that they should be plate-like (lath-shaped) and vanadium pentoxide sols showed striking optical and viscous properties that indicated that they were composed of long rods. Thanks to the electron microscope, however, we can now determine the shape of many colloidal particles with very little residual uncertainty. Solid crystalline particles present very little problem because they are substantially unaffected by the normal methods of specimen preparation. Some of the softer polymeric materials, like poly(methy1 methacrylate) do, however, have a tendency to melt in an intense electron beam. Even more difficult to deal with are systems in which the sub-microscopic structures are very sensitive to the presence of the solvent, and likely to be destroyed by the drying process which is essential in electron microscopy (since the specimen must ultimately be placed in a vacuum for viewing with the electron beam). Even here, however, the modern techniques developed for the examination of biological structures (such as freezefracture) are increasingly being used to determine the size and shape of colloidal structures. Before dealing with these developments we will briefly examine the older techniques of optical and ultramicroscopy.

D I R E C T MICROSCOPIC OBSERVATION

I205

5.2.1 Optical (light) microscopy The observation of colloidal particles with an optical microscope is limited by the resolving power of the microscope. This refers to the ability to discriminate between two closely spaced points in the field of view of the microscope. The trains of light waves emanating from two neighbouring points can interfere with one another to produce a diffraction pattern of alternate light and dark bands. The effect is particularly marked if the distance between the points is comparable with the wavelength, A, of the light being used. If the two points are the opposite edges of a particle, then the particle will appear as an object of indeterminate shape surrounded by a halo of alternate light and dark rings, the intensity of which decreases rapidly with distance from the particle centre. It may be shown by the methods of geometrical optics that the resolving power, dp, of a microscope is given by (Feynmann et al. (1963)):

dp

A/[no sin 01

(5.2.1)

where no is the refractive index of the medium and 20 is the angle subtended by the microscope objective at the focal plane (Fig. 5.2.l(a)). Resolving power can evidently be improved (i.e. dp decreased) by reducing the wavelength, or increasing no and 8. In practice only visible light is used (A-500 nm), but no can be increased by filling the region between the lens and the sample with a transparent oil (no % 1.5) instead of air. (A drop of immersion oil is placed on top of the microscope cover slip and the ‘oil-immersion’ lens lowered into it.) Wide angle lenses are also helpful (i.e. increased 0) but the angle is limited by other optical problems like spherical and chromatic aberration. In effect the lower limit for dp is about 0.2 pm, so optical microscopy is limited to the upper end of the colloidal size range. There are now available numerous commercial devices for automatically scanning and analysing images obtained from light (and other forms of) microscopy. See Allen (1997, pp. 13740) and also Kissa (1999) Chapter 10.

I Microscope lens - objective -

I I

+d+

Fig. 5.2.1 (a) Resolving power of a microscope (see text).

206 I

5 : PARTICLE SIZE AND SHAPE

d a C

b

e

Fig. 5.2.1 (b) The slit ultramicroscope. An intense beam of light (from an arc, Xenon lamp, or laser) a, emerges through a slit, b, and is focused by a lens, c, into a chamber, d, containingthe colloidal sol. The light scattered from the particles can be viewed though the microscope, e, against a dark background.

5.2.2 The ultramicroscope The Tyndall effect (Exercise 1.4.1) can be used to make very small colloidal particles visible, not as well defined shapes, but as pin points of light against a dark background, in the ultramicroscope. Th e principle of the method is illustrated in Fig. 5.2.l(b). Although the particles are not directly visible, it is possible to obtain some idea of their relative size from the amount of light scattered by each pin-point. The amount of light scattered depends not only on the particle volume but also on its relative refractive index, the wavelength of the light, and the angle of observation (Section 3.3). The minimum size of metal sols that can be seen in the ultramicroscope is about 5-10 nm but for sols of lower refractive index it is rather higher (- 50 nm). This is, nevertheless, a considerable improvement on the simple light microscope, especially with the advent of laser light sources. It is also possible to infer something of the particle’s shape in the ultramicroscope. Particles that are highly anisometric are continually changing their orientation with respect to the incident light and the observer, as a result of their rotational Brownian motion (Section 4.9.4). This rapid fluctuation in orientation produces a twinkling appearance in the light spots as the scattered intensity varies. By contrast, spherical particles show a steady light, although their translational Brownian motion is still readily visible, especially if they are small. It is difficult to determine the size of a colloid particle directly in the ultramicroscope (but see Davidson and Haller 1976 and Cummins et al. 1983). An indirect measurement can, however, be made by determining the number concentration of particles (see Exercise 5.2.1), either by direct counting or, more conveniently, using automatic counting devices (see Sections 5.2.5 and 5.6 below).

D I R E C T MICROSCOPIC OBSERVATION

I207

5.2.3 The transmission electron microscope This device depends for its operation on the wave nature of the electron and the fact that electric and magnetic fields of suitable geometry are able to function like lenses to refract, deflect, and focus an electron beam. The ultimate limit of resolution of an electron microscope is determined by the electron wavelength but, in practice, the limit is set by the performance of the magnetic lenses and the maintenance of stable magnetic fields (see Exercise 5.2.2). Currently, the best instruments can resolve down to below 0.2 nm. For a detailed description of the apparatus and techniques one should consult a standard text. The following description of the basic apparatus is taken from Silverman et al. (1971, p. 111 et seg.). The electron beam is produced by thermionic emission from a heated tungsten cathode, C, and is accelerated towards an aperture in the anode, A (Fig. 5.2.2(a)). It is then focused by a lens L1, and passes through the sample, which is mounted on a transparent grid, G. Electrons are absorbed or scattered by various parts of the

Fig. 5.2.2 (a) Schematic diagram of the electron microscope. (For description see text.)

208 I

5 : PARTICLE SIZE AND SHAPE

Fig. 5.2.2 (b). Shadowing (or shadow-casting) of a spherical or cylindrical particle produces a characteristic intensity pattern in the transmitted beam, since the shadowing material (usually a metal) is a stronger absorber - scatterer of electrons. I is the intensity (or number) of transmitted electrons.

specimen in proportion to the local electron density, and the remainder are transmitted. An electromagnetic objective lens, L2, collects the transmitted electrons and magnifies the image of the specimen 10 to 200 times, into the object plane of a magnetic projector lens system (L3), which induces a further magnification of 50 to 400 times as it projects the electrons onto a fluorescent screen, S. There the image may be viewed directly or photographed with a fine-grained film to be enlarged a further 5 to 10 times. The overall magnification factor ranges from about 100 to 1 000 000 times. Since the human eye can discriminate between points separated by about 0.2 mm, the lower limit of observation is then from 2 p m down to about 0.2 nm, so the instrument can resolve individual atoms in favourable cases. The image formed by this procedure is a two-dimensional representation of the actual structure, and in some cases this is all that is required. In many cases, however, it is helpful to have an idea of the surface topography and this is most readily achieved by shadow-casting. A beam of metal atoms is fired, En vucuo, at an angle to the sample and the deposit modifies its electron transmission characteristics. Figure 5.2.2(b) shows schematically how the intensity might be expected to vary in the case shown. Although the human eye will quickly interpret the resulting image in terms of a particle shape it is important to realize that the pattern of lightness and darkness on the screen or photograph is the result of a complex sequence of events and an exact shape analysis may call for more detailed consideration of the influence of shadowing angle 0 and direction on the apparent shape. Metals such as gold, platinum, palladium, nickel, and chromium may be used for shadow casting.

D I R E C T MICROSCOPIC OBSERVATION

I209

The main problems encountered in electron microscopy concern the preparation of the sample in a way that permits it to be transferred to the evacuated (lop4 mm Hg) chamber of the microscope and to be bombarded with electrons of high energy without undergoing changes in structure. Problems of electrostatic charging, melting, evaporation, and decomposition in the beam can be minimized by careful sample preparation (Anderson et al. 1992). The method of preparation of specimens and the supporting films on which they are placed is described by Allen (1997 pp. 142-5) who also describes the use of the replication techniques first introduced to study biological samples. By using a suitable material to form the replica, a very labile surface can be converted into one of the same shape which will withstand the rigours of electron bombardment (see, for example, McDonald et al. 1977). The technique is particularly useful for studying surface films, and is used for preparation of the final image in freeze-fracture methods.

5.2.4 The scanning electron microscope Even with the use of replication and shadowing techniques, the transmission electron microscope is limited in its ability to show particle shape. It cannot, for instance, readily give information about re-entrant surfaces, though stereoscopic methods can give a three-dimensional effect. The scanning electron microscope (SEM) is able to provide quite remarkable images, which are interpreted by the eye as truly threedimensional (Fig. 5.2.3). In this instrument (see Johari 1968) an electron beam is focused to about 5-10 nm and deflected in a regular manner across the surface of the sample, which is held at an angle to the beam. The low velocity secondary electrons that are emitted as a result are drawn towards a collector grid and fall onto a sensitive detector. The output from this detector is used to modulate the intensity of an electron beam in a cathode ray tube (CRT). The beam itself is made to scan the surface of the CRT in synchronism with the scanning of the sample by the primary electron beam. The result is a reconstructed image on the CRT much like a T V picture. The big advantage of the SEM is that the secondary electrons are emitted at low voltage and so can be easily deflected to follow curved paths to the collector. The electrons emerging from parts of the surface that are out of the line-of-sight are also collected (though at lower intensity) and it is this that contributes most to the striking realism of the three-dimensional image. The depth of field is also very large (some 300-500 times that which is available in a light microscope at the same magnification) so that the SEM is often used to examine the fine detail of quite large structures. The limit of resolution is, however, somewhat larger than for the transmission electron microscope. The scanning electron beam can also be used to provide detailed information on the surface composition of the sample. The instrument is coupled to a solid-state X-ray detector, capable of determining the intensity and wavelength of the characteristic Xrays emitted by the surface atoms when bombarded with electrons. This is called electronprobe microanalysis and it is particularly useful for the study of composite materials. It is not very sensitive to the elements of low atomic number (2 < 12) but these can be detected using Auger? electron spectroscopy (Section 6.1). +PronouncedOh-zhay.

210 I

5 : PARTICLE SIZE A N D SHAPE

Fig. 5.2.3 Scanning electron micrograph of small (-0.1 pm) polystyrene (PS) particles adsorbed on larger PS particles (-1 pm). (Photograph courtesy of Professor Brian Vincent, University of Bristol.)

A more recent development is the scanning transmission electron microscope (STEM) in which a fine beam of electrons is rastered over the surface of the specimen and the transmitted electron beam is displayed on a cathode ray oscilloscope screen, as in SEM. The result is much like TEM except that it is much more sensitive. Since the beam is being scanned it has a much less damaging effect on the specimen (Allen 1997, p. 148). The scanning tunnelling microscope (STM) is a further development of the technology which is discussed briefly in the next chapter. The use of multiple exposure techniques with SEM, with coloured overlays, can produce images of remarkable detail and great beauty (Ward 2000).

5.2.5 The 'size' of irregular particles In the course of this chapter we will examine quite a number of ways of estimating the size of colloidal particles. The most appropriate method in any situation depends on why one requires the size. For smooth spheres there is only one size characteristic but for most other shapes there is anything up to an infinite number of choices of 'size'. It

DIRECT MICROSCOPIC OBSERVATION

I 21 1

is important, therefore, to choose a method of size measurement which is likely to reflect the aspect of the particles which is of most interest. In one case, it might be the surface area but in another case it might be more important to know the ‘settling radius’. We will begin by looking at the direct visual measurements and then introduce the elementary statistical procedures needed to handle them before going on to consider other methods of size measurement. Whether derived from a light microscope or an electron microscope, the photographic image of the particles will represent a sample of their cross-sectional areas. For highly irregular particles there are several possible measurements (outlined by Allen 1981, p. 104). Of these the most commonly used are (Fig. 5.2.4):

1. Martin’s diameter (&): the length of the line that bisects the image of the particle. The lines may be drawn in any direction, but the direction must be maintained constant for all the image measurements.

2. Feret’s diameter (df):the distance between two tangents on opposite sides of the particle, parallel to some fixed direction.

3. The projected area diameter (d,): the diameter of a circle having the same area as the particle, viewed normally to a plane surface on which the particle is at rest in a stable position. (This is usually assumed to be the case for electron micrographs, since the drying process would be expected to favour a stable particle orientation.) There are differences of opinion over the usefulness of the Martin and Feret diameters, although various relationships have been derived or experimentally established between them and d,, for different materials. Thus, the ratio dJdf has characteristic values (a little greater than one) for different ground solids and can be used as an empirical measure of shape. For very irregular particles, these diameters will depend strongly on the orientation of the particles with respect to the choice of measurement direction.

Fig. 5.2.4 (a) Martin’s diameter shown (d,) Feret’s diameter shown (df). (b) The projected area diameter, d,

212 I

5 : PARTICLE SIZE A N D SHAPE

The projected area diameter, d,, (Fig. 5.2.4) can be fairly readily measured, either using a graticule or a semi-automatic procedure. The semi-automatic procedures, though still tedious, are somewhat faster. The Zeiss-Endter particle sizer (Allen 1975, p.141), for example, is best suited to analysing photomicrographs. It projects onto the photograph a circle of light, the diameter of which is determined by an iris diaphragm controlled by the operator. When the circle is of the correct size the operator depresses a switch, which actuates one of a number of counters, each of which is associated with a pre-set size range. The machine also marks the particle with a pin-hole to avoid double counting. The entire image is projected onto a frosted screen for easier viewing. The main advantage claimed for this instrument and the other similar ones described by Allen (1981) is that they permit the operator to exercise some judgement, both in particle selection (in heterogeneous systems) and in deciding what to do about overlapping particles. Modern automatic imaging procedures measure d, and df and other length characteristics in many directions; the average and the scatter then give much more information about the particles. This is clearly an area where the h t h e r development of microprocessor discrimination and control will prove increasingly effective. A related procedure is that of chord measurement in which a laser beam traverses a sample of particles (usually in air) and the length of the chord cut by the beam is calculated from the back-scattered light. The mean chord length, C, is related uniquely to the area-toperimeter ratio for any particle outline or m a y of outlines (Scarlett 1997):

C = RA/P.

(5.2.2)

The three dimensional equivalent of this equation relates the average chord length (measured in random directions) to the volume to surface ratio:

c = 4 v/s.

(5.2.3)

This relation applies to any array of particles with any shape and is a convenient way of measuring parameters such as V / S . This ratio can also be estimated by combining a sizing method, such as light scattering which depends on volume, with one which depends on surface area (like gas adsorption).

Exercises 5.2.1 Show that the average radius, of spherical particles is given by: ?;

= (3cV/4npN3

where p is the particle density, c is the concentration of the sol (by mass) and it is found to contain N particles in a volume V. 5.2.2. The wavelength associated with an electron is given by the de Broglie relation, h = h / p = h/(2meE)4, where h is Planck’s constant, p is the electron momentum, meits mass, and E its kinetic energy. Estimate the wavelength of an electron that has been accelerated through a voltage of 10 kV so that it has acquired an energy, E, of 10 keV. [A typical acceleration voltage would be of this order.]

PARTICLE SIZE DISTRIBUTION

I213

5.3 Particle size distribution Whenever we are confronted with the problem of describing the particle size of a system that is heterodisperse or polydisperse (i.e. contains many different sizes of particle) we resort to breaking the range of sizes up into convenient steps or classes, and recording the number of particles in each class. Consider, for example, the data in Table 5.1, which might represent the diameters of a sample of particles produced by a precipitation reaction. If the observed particle sizes ranged from 65 nm to 0.6 pm one might choose to break that range into 11 steps of 50 nm, as shown, and to record the number of particles in each class. The resulting data can then be plotted as a histogram or as a smoothed curve (Fig. 5.3.l(a)), or as a curve showing the cumulative percentage equal to or smaller than a given size (called the ‘per cent undersize’). Rather than go to

i 4

rI

-7-

I

!/

II

Fig. 5.3.1 (a) The frequencyhistogram can be replaced by a smooth curve. Note that the modal size is the most common one. (b) The continuous distribution function, F (di) and its relation to the frequency histogram.

214 I

5 : PARTICLE SIZE AND SHAPE

Table 5.1 Class range Midpoint of (nm) class range di (nm)

Number of particles ni

Fraction in Total this class number with A d C di

Cumulative percentage d C di

51-100

75

29

0.012

29

1.2

101-150

125

109

0.044

138

5.5

151-200

175

211

0.084

349

14.0

201-250

225

372

0.149

721

28.8

251-300

275

558

0.223

1279

51.2

301-3 50

325

440

0.176

1719

68.8

351400

375

307

0.123

2026

81.0

401450

425

223

0.089

2249

90.0

451-500

475

139

0.056

2388

95.5

501-550

525

81

0.032

2469

98.8

551400

575

31

0.012

2500

100.0

C n, = N = 2500. C J = 1.00

the trouble of plotting data each time, it is often sufficient to specify the main features of the distribution using a few numbers.

5.3.1 The mean and standard deviation The most important characteristics of a distribution are the mean, which measures the central tendency, and the standard deviation, which measures the spread of the data. The mean diameter is defined as: (5.3.1)

whereA is the fraction in class i (Table 5.1). (To distinguish it from other mean diameters we should strictly refer to this as the number length mean diameter.) The standard deviation, (T,is defined as: (5.3.2)

Note that just as in the random walk (eqn 1.5.9), we do not use (di - 2) as a measure of the spread because it can be positive or negative, and for a symmetrical distribution C ( d i - 2) = 0 even though the distribution might be quite broad.

PARTICLE SIZE DISTRIBUTION

I215

The quantity inside the main brackets is called the variance (= a’) of the population. We take the square root of the sum of the squares so that (T can be more readily compared with the mean. It is easy to show (Exercise 5.3.2) that (T

= [d2 - (Z)2]1

(5.3.3)

which is often easier to compute than eqn (5.3.2). (d2 is the average value of the squares of the sample diameters.)

5.3.2 Moments of a distribution It is useful at this stage to introduce the concept of a moment of a distribution about a point. T he j t h moment of the distribution of dj about the point do is defined by the relation:

(5.3.4) i

From eqn (5.3.1) it is apparent that the first moment about the origin (do = 0) is the mean whilst from eqn (5.3.2) the second moment about the mean is the variance (0’). The third moment about the mean is a measure of the skewness of the distribution. Since it is an odd function (f(x) = -f(-x)) it will be zero for a perfectly symmetric distribution and its magnitude measures the departure from symmetry. The fourth moment about the mean measures the kurtosis. It very heavily weights the points far away from the mean and so is a measure of the length of the tail of the distribution. These ‘moments’ may be interpreted in quite a different way in polydisperse (heterodisperse) systems. When, for example, a measurement is made of the total surface area, As, of a polydisperse system of spherical particles we can write:

As = C n j n d2j .

(5.3.5)

(This total area can be determined by measuring the capacity of the solid, S, to adsorb a gas.) If the total number of particles, N , is also known then the number area mean diameter, ZNA is defined as the diameter of the sphere for which

so that

(5.3.6) Obviously, a system of N uniform spheres of diameter ~ N has A the same surface area as the original sample, S. Note that the number area mean diameter+ is the square root of the second moment of the distribution of dj about the origin. In a similar way one can define a series of different average sizes many of which are directly accessible to measurement (Table 5.2). +This is often simply called the surface (or area) average diameter.

216 I

5 : PARTICLE SIZE AND SHAPE

Table 5.2 Some possible average dimensions for colloidal particles

Name

Symbol and Definition

Quantity averaged

Weighting factor

(i)

Number length mean diameter

C nidi d(ordm)=C n,

Diameter

Number in each class

(ii)

Number area mean diameter

(iii) Number volume mean diameter (iv) Mass area mean diameter

Number in each class -

(d33 or

=

(c;~!)’ ~

-

Particle volume

Number in each class

Particle area

Mass in each class

This leads naturally to the concept of weighting a distribution (i.e. treating some particles as more important than others). Table 5.2 lists only a few of the possible ways of weighting a distribution. The first is weighted by number and length and the second is weighted by number and area, since

Instead of weighting by volume and number as in distribution (iii) we can equally well weight the distribution by mass and area as in (iv) or by mass and volume:

where mi is the mass of material in class i that is characterized by a volume Y,. The constants &’ and K’’ in these relations are geometric factors that may be calculated for simple geometries. (k’= n and k” = n/6 for spheres if we retain the definition of dab suggested in Table 5.2; they are often ignored.)

5.3.3 The continuous distribution function Figure 5.3.1 shows that the histogram can be replaced by a smooth curve but it is important to note that this is not a plot of ni against di. For the continuous curve, F(d,), the number of particles dni in the range di to di ddi is given by:

+

dnj = F(dj) dd,

(5.3.7)

where the function F(di) is called the (number) distribution function for dj. In Fig. 5.3.l(a) it will be noticed that all of the classes have the same width, so that the height of the rectangles is proportional to nj. In the more general case, we may choose

PARTICLE SIZE DISTRIBUTION

I217

to vary the width of the classes and in that case it is preferable to draw rectangles whose areas reflect the numbers of particles in the class. Then we can readily relate the distribution function, F , to ni using eqn (5.3.7). It is apparent from Fig. 5.3.l(b) that

and the area under the F (di) curve will give the total number of particles. For the data in Table 5.1 the value of F (di) is related to that o f 5 by a constant factor since f

n, F(d$d, --=-

“N

N

.

One could generate the distribution function (in nm-’), then by multiplying column 4 by (2500/50) = 50 and it would have the same shape as the broken line in Fig. 5.3.l(a).

5.3.4 Logarithmic distributions We noted earlier (Section 5.1) that size distributions often extend over several orders of magnitude. Figure 5.3.2 shows a possible distribution that could be obtained by a direct counting operation. It extends over only two orders of magnitude but shows a very asymmetric distribution typical of material produced by grinding. In principle it is possible to describe such a curve with any desired degree of accuracy, by using a sufficient number of parameters (e.g. the mean and the second, third etc. moments). In practice, it is much better to transform the distribution into a more symmetrical shape so that two parameters still give a reasonable description. A common procedure for curves like Fig. 5.3.2 is to convert the length to a logarithmic scale so that the spread of values is more easily accommodated.Note also that the width of the rectangles in Fig. 5.3.2 is varying with d and the height measures the number, ni, in each class. T o convert ni to an area basis we would set F(d,) = n,/6d, as before. In this case, however, we wish to calculate the distribution function for In di so we set

dn, = F(dJ d l n d , =

F(dj)ddj di

~

(5.3.8)

so that

6dj n, M F(dJ -. di

(5.3.9)

In Table 5.3 the calculation is taken a stage further by dividing through by N to obtain

J; and converting this to a percentage (column 6) and this function is plotted in Fig. 5.3.3. It is called (Allen 1981) the relative percentagefrequency distribution function and is useful for comparing different samples over the same size range, since the calculation takes account of differences in the sample size, N.

5.3.5 The geometric mean The mean of the values of In di is defined in the usual way (cf. eqn (5.3.1)):

Ind = C J l n d j i

(5.3.10)

218 I

5 : PARTICLE SIZE AND SHAPE

t

I

l

l

I

I

I

1500

L

II

I

2000

2500

1

t

3000

3500

Fig. 5.3.2 A possible size distribution produced by grinding.

and this quantity is related, not to the arithmetic mean of the diameters, but to the dg,which is the Nth root of the product of all the diameters:

geometric mean

1IN

(5.3.11)

Table 5.3 Size range (nm)

Interval 6di (nm)

Average size di

Frequency Fraction

L

ni

100FIN =

~

lOOL d, 6di

ln(di) in nm

50-80

30

65

3

0.004

0.833

4.17

80-100

20

90

15

0.019

8.65

4.50

100-140

40

120

38

0.049

14.62

4.79

140-200

60

170

81

0.104

29.4

5.14

200-300

100

250

163

0.209

52.2

5.52

300420

120

360

143

0.183

55.0

5.89

420-600

180

510

108

0.138

39.2

6.23

600-800

200

700

83

0.106

37.2

6.55

800-1100

300

950

63

0.081

25.6

6.86

1100-1500

400

1300

42

0.054

17.5

7.17

1500-2000

500

1750

23

0.030

10.32

7.47

2000-2700

700

2350

13

0.017

5.60

7.76

2700-3500

800

3100

5

0.006

2.48

8.04

N =C

ni

=780 C L = 1.00

PARTICLE SIZE DISTRIBUTION

I219

Fig. 5.3.3 The relative percentage frequency distributionfunction plotted against In (di/nm). (Data from Fig. 5.3.2.)

(The symbol l7i here requires the arithmetic product to be taken of the i terms.) Although in these definitions it is usual to use number averaging there is no reason, in principle, why they cannot be extended to mass, area, or volume averages (Section 5.3.2) so that

1 n(dF) =-xtzjlndi ( j N i =CJIlndi=m.

'-h

lnd--1n

)

(5.3.12)

i

5.3.6 The measure of polydispersity We noted earlier (Section 5.3.2) that the spread of a distribution can be described with any desired degree of accuracy by calculating its various moments (eqn (5.3.4)). In many experimental situations however, we do not have access to the entire distribution but may only have estimates of various possible mean values. Because these are related to the moments they can be used to estimate the spread of the distribution or degree of polydispersity, as indicated by eqn (5.3.3) (and Exercise 5.3.2). Thus one can calculate the standard deviation from a knowledge of the area mean and length mean diameters. Alternatively, one can use the ratio of these two quantities as a measure of polydispersity, P (Exercise 5.3.3):

(5.3.13)

220 I

5 : PARTICLE SIZE A N D SHAPE

The spread of diameters in a polymer latex system is often characterized by the coefficient of variation, defined by: Coefficient of variation = (a/d)x 100%.

(5.3.14)

Typically, a system would be regarded as monodisperse if the coefficient of variation were less than 5 per cent (or, at most 10 per cent).

Exercises 5.3.1 Calculate the mean, standard deviation, and the variance of the distribution shown in Table 5.1. What is the difference between the mean and the mode in this case? 5.3.2 Establish eqn (5.3.3) from (5.3.2). 5.3.3 Calculate the number area mean diameter, N A of the particles described in Table 5.1 and compare it with the number length mean diameter. Show that &A = (a2 and check this with the result obtained in Exercise 5.3.1 (note the relevance to eqn (5.3.13)). 5.3.4 The accompanying Fig. 5.3.4 shows a simple distribution function which can be approximated by a parabola:

+ z2)e

Show that

F=

3Ndj ~

2N2

(1 -

2)

where N is the number of particles. [Hint: First show that F = bu( 1 - u/2) where u = dJd.1

Fig. 5.3.4 Parabolic distribution (Exercise 5.3.4).

Verify that 2 = j di dni/jdni. What is the maximum value of F?

THEORETICAL DISTRIBUTION FUNCTIONS

I 221

5.3.5 Calculate the number (arithmetic) mean of dj (2 or 2m) and its standard deviation and do the same for In di for the distribution in Table 5.3. What is the (number length) geometric mean diameter of these particles? 5.3.6 Calculate the number volume mean diameter of the particles in Table 5.1 and compare it with the number area mean diameter calculated in Exercise 5.3.3. Why is it larger? Is this always true? is the ratio of the fourth to the Show that the mass average diameter, third moment of the distribution.

zd,

5.4 Theoretical distribution functions 5.4.1 The normal distribution Various functions have been proposed to describe the distribution function, F, obtained in particle size analysis. Some, like the Nukiyama-Tanasawa equation (Cadle 1965, p. 36) are completely empirical, while others (e.g. the Rosin-Rammler equation; Herdan 1960, p. 86) have some theoretical basis. By far the most commonly used, and most firmly based general relationship is the standard (normal or Gaussian) distribution, which we have already encountered:

[ 1.5.91 This is shown again in Fig. 5.4.l(a). Note that, in principle, the deviations from the mean extend to infinity in both directions but in practice the bulk of the distribution (68.2%) lies within f a of the mean and less than 0.3% lies outside the range X f 3 a . The normal distribution function is closely related to the error function and lies at the heart of the statistical treatment of errors. The justification for applying it to particle size distributions is that the positive or negative dtflerences from the mean value that occur can be assumed to be caused by the operation of a large number of uncontrolled (and uncontrollable) influences. T o the extent that those influences operate in a random fashion, eqn (1.5.9) ought to be applicable (Herdan 1960, p. 73). In fact, we know that the distribution is often very different from eqn (1.5.9) but the distribution of In d may then be close to normal (Fig. 5.3.3). We will, therefore, concentrate attention on the normal and the log-normal distributions, and discuss some others only briefly. The significance of the variance (0’)is clear from eqn (1.5.9). It is the scaling parameter that determines how rapidly the exponential function drops to zero. The pre-exponential factor is introduced so that:

(5.4.1) -co

222 I

5 : PARTICLE SIZE A N D SHAPE

t 3.08 2.33 1.28 0.84 0.53 0.25

n

-0.25 -0.53 -0.84 -1.28 -2.33 -3.08

Fig. 5.4.1 The normal (Gaussian) distribution and the cumulative distribution curves.

(This is the normalizing condition, and when used in this form fG gives the fraction of the population in each interval. Then dni = NfG dx.) The distribution can also be represented by the cumulative distribution curve, which of material that is less than a given size (Fig. 5.4.l(b)). represents the fraction, 9, Setting t = (x - ??))/(Twe have:

(5.4.2)

This function is closely related to the error function (erf t), which measures the area under the error curve between the mean and some particular value oft:

(5.4.3)

THEORETICAL DISTRIBUTION FUNCTIONS

k+

I 223

In terms of the error function: 9 (t 5 ti) = erf ti. (Note that the integral in eqn (5.4.3) is negative for t i < 0 (i.e. x < X) so that 9(t 5 ti) is then less that Values of the error function are given in standard texts (e.g. Herdan 1960, p. 77 or the CRC Handbook). The fraction of material oversize is, of course, 9 ( t p ti) = 1 - 9 (t 5 ti) = - erf ti. The relation between the normal curve and the cumulative curve is illustrated in Fig. 5.4.1. It is possible to purchase special graph paper ('probability paper') on which the ordinate scale varies in such a way as to convert the sigmoidal cumulative distribution function (Fig. 5.4.l(b)) into a straight line (5.4.1(c)). Such paper is useful for testing whether a particular set of data does or does not approximate to a normal distribution. If it does, then the standard deviation can be read from the graph by identifying the size that is bigger than 15.9 per cent (i.e. (50- 68.2/2) per cent) of the sample and subtracting this from the mean. What, then, is the relation between the standard deviation as it appears in eqn (1.5.9) and the formula given earlier for a discrete distribution (eqn (5.3.2))? We can replace the right hand side of eqn (5.3.2) by its counterpart for a continuous variable:

i.)

i

[

]'=[k

f (d Sdn - z)2dn

f(d -z)2dn]f

(5.4.4)

and substitute dn = Nf dd =

~

N

ex.[ -i(T)2]dd d-d

0 d W to show that the expression (5.4.4) is indeed equal to 0.This is exactly analogous to the demonstration (using eqn (1.5.24)) that the r.m.s. displacement of a diffusing particle is equal to 0 (Exercise 1.5.5). Any of the expressions given for discrete distributions can be converted in the same way to treat the continuous distribution function (see Exercise 5.4.2).

5.4.2 The log-normal distribution We have noted already (Fig. 5.3.3) that the particle size produced by grinding often follows an approximately log-normal distribution. This is the expected outcome if ratios of equal amount greater than or less than the mean are of equal likelihood rather than differencesfrom the mean (Herdan 1960, p. 81). Making the transformation x = In d we then say that d is log-normally distributed if x has the distribution function: (5.4.5) In that case the distribution of d is: (5.4.6)

224 I

5: PARTICLE SIZE AND SHAPE

where d, is the geometric mean of the values of d, and a, is the geometric standard deviation of the distribution of ratios around the geometric mean. Note that, by analogy with eqn (5.3.12),the logarithm of a, is equal to the standard deviation of In d. The number of particles between dl and d2 in a log-normal distribution is (Herdan 1960, p. 81):

1

d2lP

n

.

- L ' -P & 4

exp[-$(

lnd

-

lnd,

)']d(lnd)

(5.4.7)

di If

where p = In a,.

5.4.3 Other distributions Although in many cases a particle size distribution can be transformed in such a way as to make it approximate a normal distribution, there are, of course, situations in which this is impossible. The most obvious case is when the sample has more than one modal size (Fig. 5.4.2). Bimodal or even polymodal distributions can occur in the preparation of a colloidal sol by a condensation method (Section 1.4.1) if there are two different nucleation periods separated by a significant time interval. Although it is, in principle, possible to represent such a shape using a polynomial: m

f =c a n d n n=O it is much better to attempt to resolve the underlying distributions, as suggested in Fig. 5.4.2. This is called &convolution and it is a problem often encountered in the interpretation of spectral data; computer programs can be written to perform the task if the underlying distributions have a known form ( e g Gaussian or Lorentzian). The Gaudin-Schuman distribution:

(5.4.8)

Fig. 5.4.2 A bimodal distribution resulting from the overlap of two 'normal' distributions having modes dM1 and d M 2 .

THEORETICAL DISTRIBUTION FUNCTIONS

I 225

is a particularly simple one, derived by comparing the various sizes with that of the largest particle. This produces a linear log-log plot with a slope, m, which is small for widely spread populations, and larger for narrow spreads (Scarlett 1997). The RosinRammler distribution:

(5.4.9) also compares all sizes with a characteristic size (dc) and is commonly used for describing the results of a comminution (grinding) process. The function can be plotted linearly on a lnln versus In plot and again a wider distribution has a smaller value of n (Scarlett 1997). The Schulz distribution:

( z + 17 exp) [?] a(z + 1)

~ ( a= )

z+l

(5.4.10)

2

is related to the Gaussian but becomes progressively narrower as the parameter z increases and becomes a delta function at a = ii as z+ 00. It is mentioned in Chapter 14. One final word about the normal distribution. When one first encounters the use of error concepts in the physical sciences one is tempted to regard the physical measurement (say the length of a piece of wire) as the important quantity and the associated error or standard deviation as at best merely an indication of reliability and at worst an unavoidable nuisance. It should be clear from the discussion of particle size distribution that the standard deviation is just as important as the mean in specifying the population, since in that case there is no single 'true size' that the measurement is attempting to estimate. There are in fact many situations in which the deviation is more important than the mean. The translational diffusion of a colloidal material (Section 1.5) is an obvious example. At a more fundamental level, however, the normal curve stands as one of the basic reference curves to which other natural distributions can be compared. If they correspond reasonably closely then the whole panoply of statistical methods can be applied with confidence. A few measurements then serve to define the whole population. It has been rightly said that the normal curve is to the statistician what the straight line is to the physicist.

r

Exercises 5.4.1 Show that the normal distribution curve has inflexions on either side of the mean and the distance between them is 20. 5.4.2 (i) Show that, for the distribution described in Exercise 5.3.4, the standard deviation is given by a = d / f i . (ii) What fraction of the material lies between d f a in this case? Compare this with the normal distribution.

226 I

5 : PARTICLE SIZE AND SHAPE

5.5 Sedimentation methods of determining particle size There are many indirect methods of estimating the particle size of colloidal dispersions, of which the oldest and most widely used depend on the principle of determining the sedimentation rate.

5.5.1 Sedimentation under gravity (a) Time dependent settling The underlying theory was treated in Section 3.1.1 and many of the problems of the method, as they apply to colloidal systems have already been alluded to. Only rather dense materials (such as mineral particles) show sufficient sedimentation under gravity to be easily measurable, and then only for particle sizes of order 1 pm. The method is applied in several distinct forms:

(i) Measurement of the concentration of particles at a certain height as a function of time. In this method the concentration may be determined by (a)sampling with a pipette; or (b) measuring the concentration by light absorption, X-ray absorption (or even neutron or p r a y absorption or backscattering of Brays); (ii) detecting the change in suspension density at a certain height using an hydrometer or a Cartesian diver (Allen 1981 p. 290; 1997 p. 309); (iii)determining the hydrostatic pressure at that height using a Pitot tube or a transducer; (iv) measuring the accumulated deposit as a function of time, and (v) the two layer method in which the suspended material is confined initially to a thin layer above the suspension medium. One can then apply one of the other methods to follow the sedimentation. Method (i) depends on the fact that, as sedimentation proceeds, the particles which fall out of the measuring zone are replaced by other particles coming from above, until the time when the largest particles have fallen down through that level. Thus the particle concentration remains constant for some little time and then falls gradually to zero. For successful measurement the temperature must be maintained constant and the particle concentration must be very accurately assessed. Using a pipette to withdraw a sample creates some disturbance which limits accuracy. Using light absorption requires that the particle concentration be very low so that there is no multiple scattering and the obstruction caused by each particle is additive. Such low concentrations limit the accuracy of the procedure. The use of X-ray absorption allows higher particle concentrations to be used, at the expense of some particle interference during settling. The same procedures can, of course, be applied to centrifugal settling and that is more productive for most colloidal systems. In method (ii) the density is directly related to particle concentration. The main problem is that the large size of the normal hydrometer and its peculiar shape makes it difficult to locate the height to which the measurement refers, and extensive corrections are necessary (Orr and Dallavalle (1959). The diver method has the advantage that a number of different divers with different densities can be used to make measurements at different stages in the sedimentation process.

SEDIMENTATION METHODS OF DETERMINING PARTICLE S I Z E

0.7 0.6

I227

-~~

0.5 --

0.4

0.3

~~

--

0

1

2

3

4

5

In t

Fig. 5.5.1 (a) The mass accumulated on the balance pan as a function of time in a gravity sedimentation analysis. (b) Alternative method of analysing the data from Fig. 5.5.l(a).

In method (iii) the pressure is generated by the entire column of suspension above the measurement level so it gradually falls as sedimentation proceeds. See Allen (1981 p. 314) for details. In method (iv) the material is allowed to accumulate on a balance pan suspended in the dispersion. The method was introduced by Oden (1926) and improved on by Bostock (1952). Figure 5.5.1 shows a plot of the accumulated mass, M , of sediment on the pan as a function of time. This is made up of two contributions: (a) the mass, m due to particles (for which d d l ) large enough to have sedimented through the entire depth; and (b) a fraction of the smaller particles that have fallen a shorter distance:

1 4

M(t) = m

+

&I,

1

?M(d) dd = h

4

1 k

&ax

M(d) d d +

&in

;M(d) dd

(5.5.1)

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5 : PARTICLE SIZE AND SHAPE

where v is the velocity of particles of diameter d and h is the total depth of the suspension. Differentiating eqn (5.5.1) with respect to t gives:

dM

-=

dt

/

iM(d)dd

(5.5.2)

We can then write eqn (5.5.1) in the form:

(5.5.3)

A plot of m as a percentage of the total mass of material available for settling will have the same shape as the cumulative oversize distribution function. T o see that the second term (dM/dln t) measures the contribution of the smaller particles, consider Fig. 5.5.l(a). The slope dM/dt at any time, t, measures the rate of incidence of particles smaller than the cut-off size corresponding to that time, since all larger particles have already deposited. But this process has been going on at the same rate for the entire time, so it contributed a mass t dM/dt to the pan. Getting rn by the method implied in Fig. 5.5.l(a) has some problems, however, because drawing tangents to experimental data is never a very precise procedure. T h e use of the final logarithmic form (Fig. 5.5.l(b)) is said to involve less uncertainty (Lloyd 1997) since the tangents can be drawn more precisely at low values o f t . Only data of very high accuracy can produce a reliable size distribution using this method, but it is usually possible to determine the modal value with fair accuracy. T h e time axis must, of course, be transformed into the value of d using eqn (3.1.3) (see Exercise 5.5.1):

(5.5.4) One basic limitation of this technique is that the region immediately below the pan becomes depleted of particles and so has a lower density than the solution immediately above the pan. This results in a convective motion of the suspension medium, which interferes with the sedimentation process to some extent. A possible correction procedure is described by Allen (1975 p. 225). Method (v) in which the suspension is initially placed in a thin layer on top of the suspension medium (called the line start method) makes the analysis of results simpler. It is commonly used in centrifugation and was discussed briefly in Section 3.1.3. For more details see Chapter 9 of Kissa (1999) and for its application in the disc centrifuge see Allen (1997 p. 320). (b) Sedimentation equilibrium in a gravitational field As noted in Section 3.1.2 this was the method used by Perrin to determine the Avogadro number using a suspension in which the particle size was known. It does not seem to have been developed for the reverse procedure but the corresponding

Adsorption onto Solid Surfaces 6.1 Vacuum characterization methods for solid surfaces 6.1.1 The field emission and field ion microscopes 6.1.2 LEED and RHEED 6.1-3 Auger electron spectroscopy (AES) 6.1.4 X-Ray photoelectron spectroscopy (XPS) 6.1.5 Secondary ion mass spectrometry (SIMS) 6.1.6 Scanning tunnelling microscopy (STM)

6.2 Some non-vacuum techniques 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5

The surface force apparatus (SFA and MASIF) The atomic (or scanning) force microscope (AFM or SFM) Total internal reflectance microscopy (TIRM) Attenuated total reflectance spectroscopy (ATR) Total internal reflectance fluoroscopy (TIRF)

6.3 Adsorption and desorption at the solid-gas interface 6.3.1 6.3.2 6.3.3 6.3.4

The mechanics of adsorption The Langmuir adsorption isotherm The BET adsorption isotherm Temperature programmed desorption (TPD) and thermal desorption mass spectrometry (TDMS)

6.4 Adsorption at the solid-liquid interface 6.4.1 Adsorption from dilute solution 6.4.2 The Langmuir isotherm 6.4.3 The Freundlich isotherm 6.4.4 Thermodynamics of adsorption

6.5 Adsorption of neutral polymers 6.5.1 Polymer chains 6.5.2 Polymer chains in solution 6.5.3 Limitations of the simple free volume theory 6.5.4 General aspects of polymer adsorption

The surface structure of solids has been an important subject of scientific study for much of the twentieth century, initially because of its significance in understanding the

259

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6 : ADSORPTION ONTO SOLID SURFACES

behaviour of solid catalysts. The early work on the subject was, therefore, heavily influenced by the adsorption behaviour of various gases on solids, a subject which was given a dramatic impetus by the introduction of gas warfare in World War I. There were, however, many limitations in these studies, due firstly to the very small numbers of molecules involved in surface reactions and hence the high sensitivity needed for detection. The other problem was contamination. It was not until the development of ultra-high vacuum technology (as part of the space research program in the 1960s) that it became possible to deal with truly clean surfaces on a more or less routine basis. Using the well established techniques of the electron microscope (Section 5.2.3) and a variety of special spectroscopic methods, the study of the details of surface atomic structure expanded to become a highly specialized field from the late 1960s onwards. The scientific, technological, and commercial driving forces for these developments came from many sources: new catalyst requirements, harder wearing surfaces and better lubrication (tribology), fracture resistant ceramics, chemical sensors, magnetic tape and other memory devices, dry paper copying, and a host of others, including the ever increasing demands of the computer micro-chip industry. The detailed study of these solid surfaces can involve the application of a barrage of techniques to a carefully prepared surface under the most stringent purity conditions. Many colloidal solids would, however, change structure irreversibly under high vacuum and our primary interest in hydro-colloids also makes much of the high vacuum work of only marginal relevance, because of the overwhelming influence of water on the structure of the solid-water interface. Water does not merely orient itself at the interface, it often interacts chemically with the surface atoms and may even penetrate to some extent into the surface layers. We will, therefore, concentrate on techniques which can throw some light on those processes and leave the many other fields for the specialist monographs (for example Somorjai 1994, Morrison 1990). Fortunately, there still remain a large number of spectroscopic and microscopic techniques, as well as theoretical modelling and computational methods which can throw light on the solid-liquid interfaces of most interest to us. The following introduction is based on the specialist works referred to above, the recent review by Soriaga et al. (1995) and Chapter 8 of the recent revision of Adamson's well known text (Adamson and Gast 1997). One of the more interesting developments of recent years was the construction of complex instruments which enabled a surface to be examined in an aqueous and also a vacuum environment. Figure 6.1.1 shows such a device. The details need not concern us but note that the left hand side of the instrument has a number of manipulators which can be used to subject the sample to electrochemical tests in an aqueous environment. The sample can then be transferred through the gate to the right hand side of the instrument which is the high vacuum chamber (at a pressure of order 5 x lo-'' mbar) with a central platform for the sample manipulator and an array of emitters and receivers on the extreme right. These can be used to fire electrons, ions, or laser or X-ray beams at the sample and to collect the resulting emission from the sample for analysis. The sample can then be returned to the aqueous environment for a further check without being exposed to the normal laboratory atmosphere. Soriaga et al. (1995) give an example in which 3-pyridylhydroquinone adsorbed on the (1 11) face of a platinum crystal is examined with cyclic voltammetry. The sample is then examined under high vacuum and returned to the electrochemical cell where it shows almost the identical cyclic voltammagram.

TO ROUGHING PUMP

MANIPULATOR

r - 4 U

Fig. 6.1.1 Schematic drawing of an experimental arrangement which permits both aqueous electrochemical and ultra high vacuum study of the same surface without exposure to the normal laboratory atmosphere. (Reproduced from Hanson and Yeager 1988.) 0American Chemical Society.

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6 : ADSORPTION ONTO SOLID SURFACES

The number of techniques which can be employed in the ultrahigh vacuum chamber gives rise to a bewildering variety of acronyms which are summarized in a ten page table by Somorjai (1990). They run from AES (Auger electron spectroscopy) through LEED (low angle electron diffraction) to XPS (X-ray photoelectron spectroscopy) of which only a few will be examined below. More recent developments in this area have made the complexities of Fig. 6.1.1 unnecessary, except for highly specialized purposes. A single device now makes it possible to obtain direct visual information about the disposition of surface atoms and adsorbed molecules in aqueous solutions before and after some electrochemical oxidation or reduction procedure. These electrochemical scanning probe microscope (ECSPM) techniques will be discussed in Section 6.2.

6.1 Vacuum characterization methods for solid surfaces The table referred to above (Somorjai (1990) Table 1.1 pp. 19-28) provides references to the more important techniques, as does a similar table (VIII-1) in Adamson and Gast (1997). Most methods involve aiming a narrow beam of photons or particles (electrons or ions) at the surface and analysing the resulting particles or photons to determine their nature, and their angular distribution and/or energy. The main problem is the low density of surface atoms (of order 10’’ per cm’) compared with the usual bulk concentrations of loz3 atoms per cm3. That means that the detection systems must be very sensitive. It is also important that the method examines only the surface atoms (or some controlled depth into the surface). Photons do not penetrate far into metal surfaces but they may penetrate through several layers of adsorbed molecules. Electrons and ions may have much greater penetration depths but the penetration may be reduced by using a very low angle of incidence. The ultra-high vacuum is required to ensure that small quantities of a contaminant do not adsorb on the exposed surface and vitiate the results. (See Exercise 6.1.1.) We consider first a procedure involving an electron beam. When the primary beam strikes a solid surface it gives rise to some back scattered (primary) electrons and also an emitted (secondary) electron beam, with a characteristic energy distribution (Fig. 6.1.2). The very large peak on the left consists of the true secondary electrons, produced by multiple inelastic collisions between the impinging electrons and the bound electrons in the solid. These are the electrons which are harnessed in the scanning electron microscope (Section 5.2.4). The large peak on the extreme right, E,, is due to elastic scattering (i.e. no energy loss) of the primary electrons and corresponds to only a few percent of the original number. Just to the left of this peak are a number of small peaks which correspond to electrons which have lost some energy through interaction with the vibrational energy levels of the surface atoms. The spectrum in this region is used to identify the bonding arrangements in surface atoms. Much further back from E,, corresponding to much greater energy loss, is a region where the Auger electrons (Section 5.2.4) are found (see Section 6.2.1).

6.1 .I The field emission and field ion microscopes One of the earliest instruments to enable a direct visualization of surface molecular structure was the field emission microscope (Muller 1943). It did not require the

VACUUM CHARACTERIZATION METHODS FOR SOLID SURFACES

1(

.-I

x 1200

400

300

200

100

I 263

I 0

ENERGY LOSS (mev)

200 ENERGY (eV)

Fig. 6.1.2 Intensity versus energy of scattered electrons from a (rhenium) metal surface covered with a chemisorbed monolayer. (a) Auger electron spectrum and (b) a high resolution electron energy loss spectrum. (c) The LEED pattern. (Reproduced from Somorjai and Bent 1985 with permission.)

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6 : ADSORPTION ONTO SOLID SURFACES

V

Fig. 6.1.3 The field emission microscope. A is an anode, connected to the screen, S. Electrons emitted from the tip, T are attracted to the screen. The (glass) envelope, E, encloses a high vacuum, V. (From Gomer 1955. 0Academic Press.)

production of an incident beam. Rather, it relied on the emission of electrons from a very fine metal tip placed in a high voltage field (Fig. 6.1.3). If the tip is fine enough 0.1 pm) and is placed at the centre of a hemispherical shell which is (radius a charged (positively) to, say, V =10 000 volt, then the field strength at the tip can be shown to be approximately V / u = lo9 V/cm. At such high field strengths, electrons are torn from the metal surface and accelerated in straight lines towards the outer fluorescent screen. If a molecule, like a polynuclear hydrocarbon, is adsorbed on the tip, its outline appears on the fluorescent screen since it affects the ease of release of the electrons. The field ion microscope is a variation of this device in which a gas (e.g. helium) is introduced into the field emission microscope at low pressure. As the atoms of the gas are adsorbed on the tip and move over the surface they become ionized and are then accelerated towards the (negatively charged) fluorescent screen. Ionization occurs preferentially near protruding atoms (those on the edges of step dislocations) so the resulting spot pattern reflects that of the atoms on the metal surface. (See Atkins 1982 pp. 1010-12.)

-

VACUUM CHARACTERIZATION METHODS FOR SOLID SURFACES

I 265

,4 k9

voltage Screen

Fig. 6.1.4 Schematic arrangement of the apparatus for Low Energy Electron Diffraction (LEED). (From Sanchez 1992.)

6.1.2 LEED and RHEED The low energy electron diffraction (LEED) technique arose out of the very early experiments of Davisson and Germer in 1927, which demonstrated for the first time the wave nature of the electron (Atkins 1982, p. 395). Those early experiments were confused by the presence of adsorbed gas molecules on the surface being investigated, because the vacuum was only about lop6mbar. Modern instruments use vacuums of 1 OW'' mbar to ensure that no surface contamination occurs until the desired adsorbate is introduced (Exercise 6.1.1). A schematic view of the apparatus is shown in Fig. 6.1.4. Electrons from a hot filament are uniformly accelerated in a beam as they pass through the narrow gap between the first, second, and third anodes (which are charged to successively higher positive voltages). They then strike the (crystal) sample normal to its surface. The (elastically) scattered electrons are collected on the screen after passing through the suppressor grids. These grids (i.e. gauze electrodes) are negatively charged to voltages which prevent all but the most energetic electrons (i.e. the elastically scattered ones) from getting through to the screen. The pattern on the screen consists of a series of

266

I 6 : ADSORPTION ONTO SOLID SURFACES

spots (like an X-ray diffraction pattern) corresponding to the reciprocal lattice of the surface atoms (Fig. 6.1.2(c)). From this pattern the surface disposition of the atoms can be deduced and it is often slighty different from what would be expected from the known bulk structure. It is also possible to obtain a pattern after the adsorption of a gas and to infer its disposition on the surface. A closely related technique is W E E D (reflection high energy electron diffraction) which differs mainly in using high energy (30-100 keV) electrons in the primary beam. At these energies a direct beam would penetrate deep into the sample surface so glancing incidence must be used to prevent that occurring. Only a small fraction of the initial beam is back scattered onto the collecting screen so higher sensitivity collectors or longer times are required. The main advantage is that the voltages on the suppressor grids are not so critical because the inelastic and elastic scattered electrons differ greatly in energy. W E E D is useful for studying the morphology of thin films because the use of glancing incidence means that the frontal view of the sample is unimpeded and can be used to accommodate an electron source or the analyser or a film deposition source.

6.1.3 Auger electron spectroscopy (AES) In this technique, probe electrons of energy 2-3 keV are fired at the surface. When a primary electron displaces an electron from one of the inner shells of a surface atom, an electron from an outer shell drops down to take its place. When this happens, all of the excess energy is taken up by another outer shell electron which is then emitted from the atom with an energy which is characteristic of the atom and independent of the energy of the incident beam. This is the Auger electron and its energy can be used to identify the surface atom (Fig. 6.1.2). If, for example, the initial expelled electron was from the K-shell and it is replaced by an LI shell electron, then the net energy available is (EK - ELI) and if this is used to expel an electron from the LII shell then its energy will be (Ek - ELI) - ELII.The LI to K transition is the most common (Adamson and Gast 1997) so the Auger spectrum of an electron of typej normally records the energy: (EK - ELI) - Ej. It is common practice to combine both LEED and AES in the same apparatus. LEED uses normal incidence and AES uses grazing incidence to maximize the effect of surface atoms and adsorbed molecules. The main problem in AES is to discriminate accurately between electrons of slightly different energies. One procedure (called the retarding field analyser) involves arranging the voltages on the supressor grids (Fig. 6.1.4) to make that discrimination. A more recent method called the cylindrical metal analyser (Fig. 6.1.5) uses two coaxial metal cylinders to achieve the same result. The voltages on the two cylinders are adjusted so that only those electrons with exactly the right kinetic energy are able to pass through both slits and land on the detector.

6.1.4 X-Ray photoelectron spectroscopy (XPS) This procedure was formerly called ESCA - electronic spectroscopy for chemical analysis. XPS is closely allied to AES (Section 6.1.3), but uses a monochromatic beam of X-rays to dislodge electrons from the inner (K- and L-shells) of the surface atoms and then analyses the energy of the emitted electrons (including Auger electrons)

VACUUM CHARACTERIZATION METHODS FOR SOLID SURFACES

I 267

Sample

Fig. 6.1.5 Schematic arrangement of the cylindrical mirror analyser for Auger electron spectroscopy. The voltage on the two coaxial cylinders is arranged so that the curved path of the chosen electrons allows them to pass through both slits and land on the detector. (After Soriaga et al. 1995.)

directly. It is much more precise than Auger and can be used to detect changes in valence states of adsorbed species. Adamson and Gast (1997) give an example of the obvious change in the XP spectrum when a clean aluminium surface is allowed to oxidize; a distinct increase occurs in the ratio of the height of the A13+ peak to the Al0 peak. The kinetic energy of the emitted photo-electron is determined in the spectrometer and it is given by: K.E. = hu - BE - ecjsp where u is the frequency of the initial X-ray, BE is the binding energy of the electron, and cjsp is the work function of the spectrometer (in volts) which must be known. The binding energy is influenced principally by the nature of the atom from which it came, but is affected also by the valence state of the atom and, to some extent, by the disposition and polarity of the bonds from adjacent atoms. These ‘chemical shifts’ can also be used to identify more subtle effects in adsorbate molecules. The kinetic energies of the emitted photo-electrons are normally very low so only electrons from the upper 1-5 nm are able to escape. The method is therefore very sensitive to surface

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6 : ADSORPTION ONTO SOLID SURFACES

structure. Most elements (except for hydrogen and helium) have practical detection limits of about 1-10 Yo of a monolayer.

6.1.5 Secondary ion mass spectrometry (SIMS) This procedure is used for identifying atomic surface constituents. A primary beam of ions is aimed at glancing incidence onto the surface and the secondary particles (ions with different composition and charge) are collected and focussed in a mass spectrometer, and identified in the usual way by their charge to mass ratio. O+, Ar+, and Ga+ with T he primary beam species are ions such as Cs+, 02+, energies in the range 1 to 30 keV, so the process is usually much more energetic than electron bombardment. This results in sputtering of the surface (i.e. a gouging of the atoms out of the surface and incorporation of some of the primary ions into the solid). Th e implantation of primary ions can occur to depths of up to 10 nm. T he secondary beam consists of monatomic and polyatomic particles of sample material and resputtered primary ions, along with electrons and photons with kinetic energies in the range from zero to several hundred electron volts. The primary beam can be focussed to less than 1 micron in diameter and can be scanned across the surface to yield a microanalysis of the surface structure. Alternatively, it can be trained on a certain area of the sample and will gradually gouge out the surface and generate a depth profile of the material in the surface layers; this process is called dynamic SIMS. At the very lowest (least energetic) level in scanning or microanalysis mode (static S I M S ) it is possible to do an analysis which consumes much less than a single monolayer. As little as of a monolayer can be detected by this method which is the most sensitive surface analysis technique available (Morrison 1990, p. 106). Its main drawback is that it destroys the surface it is analysing.

6.1.6 Scanning tunnelling microscopy (STM) This method relies on the measurement of the electron current which passes between the sample surface and a microtip which passes over the surface in a raster pattern (as in a cathode ray tube). The current flows by electron tunnelling through the gap between the tip and the surface and that current falls off exponentially with the width of the gap (of order 1 nm). The tip position is controlled by a feedback mechanism to keep the current (and hence the distance to the surface) constant. As the surface is rastered below the tip in the x-y direction, the position of the tip (in the x-direction) provides a record of the surface topography. T he method requires an initial flat conducting surface on which the surface layers are built up. Adamson and Gast (1997) provide some examples of its use in the study of crystal defects on silicon and the electrochemical roughening of a Au-Ag alloy, together with a striking picture of a cluster of CO molecules on a surface. The clarity of the image which can be obtained with a suitable adsorbate is illustrated in Fig. 6.1.6. Some striking images are also provided in a recent review by Rabe (1999).

SOME NON-VACUUM TECHNIQUES I269

Fig. 6.1.6 STM image of a pyridine substituted porphyrin derivative on iodine modified Au (111) in 0.1 M HClO4. Obtained at 0.82 V w.r.t. the reversible H electrode. (14nm image obtained by K. Itaya, Tohuku University, taken from the Digital Instruments brochure on ScanningTunnelling Microscopy.)

Exercises 6.1 .I The number of collisions which gas molecules make with unit area of the container walls is given by Atkins (1982, p. 874): Z, = p/(2~rmk7)i where p is the gas pressure and m is the mass of the gas molecule. Calculate the number of collisions per cm2 on any surface in contact with oxygen gas at 25 "C and a pressure of lop6mbar. (Take 1bar = 1 atm and check the unit system before you begin.) The surface can be assumed to contain some 1015 atoms per cm2. To what pressure must the oxygen gas be reduced in order for the collision rate per surface atom to be of the order of once per hour?

6.2 Some non-vacuum techniques T o study the surfaces of hydrocolloidst with least disruption means studying the solid surface in contact with water. Studies of relatively concentrated systems involving ?Any dispersion of a solid, liquid or gas in water is a hydrocolloid, though the term is most commonly applied to dispersions of solid in water, or in predominantly aqueous dispersion medium.

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6 : ADSORPTION ONTO SOLID SURFACES

large surface areas were relatively straightforward, because the problems of contamination were then easily manageable. Such systems were, however, only capable of providing information on average surface properties and there were usually formidable theoretical problems in dealing with the multiple interactions which occur in concentrated systems. As the demand for more precise information grew, it became necessary to concentrate on very small areas of well-defined surface (much as was done in the high vacuum studies). For many years it proved impossible to achieve the necessary levels of cleanliness to do such studies with any certainty. Even Deryaguin and his colleagues, who had shown the superiority of their methods of measuring the forces between macroscopic objects in the 1950s were to fall foul of impurities in the study of minute amounts of ‘polywater’ in the 1960s. By the 1970s, however, the problems had been ironed out to the point where it became possible to undertake the detailed study of small areas of the solid-water interface on a more or less routine basis, with a gradual development of commercial instruments for the purpose. Here we consider a few techniques which are related to the vacuum techniques but which can function effectively in aqueous media.

6.2.1 The surface force apparatus (SFA) The first such instrument was the surface force apparatus designed by Israelachvili (see Israelachvili and Adams 1978) as a development of the corresponding instrument for measurement in vacuo, designed by Tabor and Winterton (1969). This instrument was specifically designed to study the interactions between approaching solid surfaces as a means of testing the fundamental (DLVO) theory of colloid stability (Section 1.6.6). The SFA made it possible to bring together, under water, two macroscopic flat surfaces, and to measure accurately the forces of attraction and repulsion generated between them (from lo-’ to lop4 N) as a function of distance. The atomically flat surfaces were obtained by exfoliating natural muscovite mica (Section 1.4.5). The use of a crystalline layered silicate material allowed the measurement of forces down to very small separations (initially about 1 nm but later down to 0.1 nm), whereas even the well-polished surfaces of quartz available in the 1970s were found to be rough on the microscale, having a random distribution of hills of the order of at least 5 nm, which affected the measured, short-range interaction. A schematic diagram of the force measuring apparatus is shown in Fig. 6.2.1. Thin, parallel-cleaved mica sheets (1-3 p m thick) are silvered on the back face with a 50 nm layer and are then glued down onto curved transparent silica discs which are polished to give a single radius and positioned to orientate the surfaces in the geometry of crossed cylinders, the thin sheets following the curvature of the discs. White light passing through the thin sheets is multiply reflected between the silvered back surfaces such that only certain wavelengths (FECO fringes, i.e. fringes of equal chromatic order) are transmitted and these can be measured in a spectrometer. Analysis of the shift in wavelength from when the mica surfaces are in molecular contact to when they are moved apart gives an accurate measure of the separation. Using the apparatus illustrated in Fig. 6.2.1 the lower mica surface can be moved towards the upper one in a well-controlled motion via the lower rod, which compresses the fairly weak helical spring. This spring in turn acts on the left-hand side of the much more rigid doublecantilever spring and hence the lower disc. Since the latter spring is about 1000 times stiffer than the helical spring a movement of, say, 10 p m in the lower rod moves the

SOME NON-VACUUM TECHNIQUES I271

light to spectrometer

micrometer rod

microscope objective

piezoelectric tube

thermistor

-moin

conductivit

SIJpport

.... __ . .

-. _- cantilever spring

spring

-

0

cm

white

5

u

', water inlet water outlet via pH cell

Fig. 6.2.1 Schematic drawing of apparatus to measure long-range forces between two crossed cylindrical sheets of mica (of thickness 1 p m and radius of curvature -1 cm) immersed in liquid. (From Israelachvih and Adams 1978, with permission.)

-

lower disc only about 10 nm. A final positional adjustment can be made by applying a varying voltage to the piezoelectric tube on which the upper disc in mounted. The separation can be accurately monitored from the corresponding shift in fringe wavelength. The forces were calculated from the observed deformation of the spring and the known spring constant. Measurement of the interaction forces under a wide range of solution conditions can give a very good test of double-layer theory. Since forces can be measured down to small separations, both the attractive van der Waals force and any solvent structural forces can also be measured. We will discuss the results of such measurements in detail in Chapter 12. The application of the SFA to the study of the adsorption of surfactants has recently been reviewed by Claesson and Kjellin (1999). A variation of this instrument, called the MASIF (for Measurement and Analysis of Surface Interaction Forces) is described by Parker (1994). It uses a bimorph strip (i.e. two slices of piezo-electric material sandwiched together) to sense the pressure between the approaching surfaces The distance between the surfaces is controlled by a translation stage fitted to a piezo-electric displacement transducer which allows fine adjustment to about 0.1 nm. The apparatus is very simple in principle, has a small volume (about 10 mL) and can be used with non-transparent specimens, whereas the original SFA requires transparency for the development of the interference fringes. It is not without its difficulties, however, the most important of which are discussed by Claesson et al. (1995).

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6.2.2 The atomic (or scanning) force microscope (AFM or SFM) The AFM (also called the Scanning Force Microscope) is closely related to the S T M but can be used to study surfaces under similar conditions to those used with the SFA. Figure 6.2.2 shows a schematic arrangement of the instrument (DiNardo 1994, Claesson et al. 1995). It functions by moving a very fine tip in a raster pattern across a surface whilst holding the tip at a constant distance from the surface, as in the S T M (Section 6.1.6). In this case the tip-to-surface distance is controlled by maintaining a constant force between the tip and the surface (rather than a constant tunnelling current as in the STM). It can function in any fluid medium provided the viscosity is not too large. The specimen sits on a calibrated piezo-electric crystal and the probe tip is mounted on a sensitive cantilever. The laser beam is reflected from the top of the probe to fall evenly on a pair of diodes. If the probe is displaced up or down as it moves over the surface, the out of balance current from the diodes is used to lift or lower the specimen by changing the voltage applied to the piezo crystal until the cantilevered probe returns to its normal deflection. The voltage required to do that is a measure of the height of the surface specimen at that point in the raster pattern. The original device used a pyramidal silicon nitride tip which was able to trace the topography of an atomic surface on, say, a pure silicon crystal and determine the position of steps and individual molecular or atomic clusters. The ‘distance’ between the tip and the substrate was in this case essentially zero and the probe was operating in the ‘contact’ mode where the force between tip and surface is the Born repulsion. The same configuration is used for the study of adsorbed layers of surfactant molecules (Scales 1999) which we will discuss in Chapter 10.

Split Photodiode Laser

-

Voltage deflection of cantilever To computer

for force-distance display Voltage gives relative position of surfaces

Cantilever SiN tip

Colloidal Particles Calibrated Piezo Crystal

Fig. 6.2.2 Schematic diagram of the Atomic (Scanning) Force Microscope (AFM or SFM) when used in non-contact (colloid probe) mode. In contact mode the colloidal particles would be absent and the Si3N4 tip would make direct contact with the specimen on the upper surface of the piezo crystal. (After Claesson et al. 1995.)

SOME NON-VACUUM TECHNIQUES I273

For fundamental colloid science studies where one is seeking the relation between force and distance over a range of separations (the 'non-contact' mode), the interaction between the very fine Si3N4 tip and the underlying surface would change with separation. T o overcome this effect Ducker et al. (1991, 1992) showed that it was possible to attach a small spherical colloidal particle (of order 5 p m in diameter) to the tip and to study the interaction between that sphere and another surface attached to the bottom piezo-electric plate. That procedure has greatly expanded the range of surfaces for which fundamental forcedistance information can now be obtained. For a description of how the measurements are transformed into data on force versus distance, see Claesson et al. (1995). The more recent devices of this type are similar in principle to the SFA and the AFM and are referred to as electrochemical scanning probe microscope (ECSPM) devices. They can be used directly in aqueous solution or in the high vacuum mode discussed in Section 6.1. They also incorporate advanced techniques for studying the electrochemical reaction properties of the adsorbed molecules.

6.2.3 Total internal reflectance microscopy (TIRM) This procedure was developed by Prieve and his colleagues and his review (Prieve 1999) should be consulted for details. A light beam is totally internally reflected from an interface between glass (or similar material), of refractive index nl, and a solution of refractive index n2 (Fig. 6.2.3). Total internal reflection occurs when nl > n2 and the angle of incidence, 81, is greater than some critical value (usually about 65" for aqueous systems against glass). [Note that in geometrical optics, the angle of incidence is measured with respect to the normal so glancing incidence means 81 M 9OO.l Although the reflection process in the glass is described as total, the combination of the incident and reflected wave generates what is called an evanescent wave in the adjoining medium. This is a standing wave which dies off in intensity exponentially with distance from the interface (Fig. 6.2.3). In T I M , microscopic ( w 10 p m diameter) glass or plastic spheres are allowed to settle towards a glass plate under gravity, where their positions are determined by the balance of the gravitational force with the repulsive force between the (charged) particle and the plate (Section 1.6). One sphere is selected for study and its height (h) above the surface is monitored by measuring how it scatters the evanescent wave. The intensity of the evanescent wave

Incident light

Aqueous solution (n2)

E"

Fig. 6.2.3 The evanescent wave formed by total internal reflection from a glass-water interface. (After Lassen and Malmsten 1996.) dp is the characteristic penetration depth of the wave.

274 I

6 : ADSORPTION ONTO SOLID SURFACES

decreases with distance from the interface and the scattering from the sphere follows a similar exponential relation:

I@) = I0 exp(-gh)

(6.2.1)

-

with g-’ 100 nm. The scattered intensity can be measured to about 1% so the precision of the position measurement is about 1 nm. The particle undergoes Brownian motion so it moves erratically about its most favoured position but is restricted in lateral movement by a light trap.+ At low electrolyte concentrations (< 1 mM) the repulsive double layer force dominates and so the separation distance is about 7-10 times the Debye length. The method is extraordinarily sensitive (being able to weigh a single particle) and gives results in close accord with the expectations of electrical double layer theory (Chapter 12) when the particle is well separated from the surface. There remain, however, some discrepancies at higher electrolyte concentrations, when the particle sits closer to the surface. Estimates of the attractive (van der Waals) force do not agree with those obtained with the surface force apparatus (Section 6.2.1). Prieve (1999) suggested that this discrepancy was due to ‘surface roughness’ since neither the sphere nor the glass plate could be guaranteed to be smooth to the same degree as the mica sheets in the SFA apparatus. That proposal has since been confirmed by Bevan and Prieve (1999). The TIRM has also been used to show how strongly the diffusion coefficient is altered by proximity of the sphere to the surface (N 25-fold reduction) and also to investigate the interaction between protein covered surfaces (Liebert and Prieve 1995) and vesicles. The most powerful aspect of the technique stems from the fact that the particle is essentially unconfined (at least in the vertical direction). That could well prove to be vital for the study of the interaction between surfaces covered with protein or polymer. Whereas in the SFA and ATM apparatus the surfaces are forced together, the TIRM allows the particle to sense the presence of the polymer chain as it approaches; its own motion will then reflect the relaxation of the polymer chain.

6.2.4 Attenuated total reflectance spectroscopy (ATR) There are few spectroscopic techniques which can be applied directly to the study of the colloid/aqueous solution interface and indeed few which have the sensitivity to give information about any aqueous interface, mainly because of the small volume of the interfacial region. One of the most promising is Attenuated Total Reflectance, because it can be concentrated on the interfacial region alone, and can be arranged to sample that region several times in a single measurement. Using a thin glass block, the ATR beam can be bounced several times off the interface during a single pass, and at each encounter, the reflectiodpenetration depends on the properties of the interfacial region. The final beam therefore has a signature which contains information on the adsorbed material in the interface. The visible/W method had been used particularly for the study of surfactant, polymer, and protein adsorption at the glass (or quartz)aqueous solution interface. t A powerful laser beam is used to apply a constraining force on the particle.

SOME NON-VACUUM TECHNIQUES I275

Y’I 122;

1095 I

1300

1100

U

3900

3260

2z20

1680

1440

700

900

WAVENUMBER (cm-1) Fig. 6.2.4 ATR-CIR (cylindrical internal reflectance) spectra showing the influence of phosphate adsorption on goethite in aqueous suspension. (a) pH 4, lo-’ M NaC1, no phosphate. (b) pH 4, lo-’ M NaCl, 48 pmol of phophate per g of goethite; (c) pH 4, lo-’ M NaCI, 100 pmol of phosphate per g of goethite; (d) spectrum of aqueous solution of NazHF’O4 at pH 4 and I = lo-’ M in NaC1. (From Tejedor-Tejedor and Anderson 1986 with permission.)

For their infra-red (ATR-FTIR) studies of the goethite-solution interface, Tejedor-Tejedor and Anderson (1986) fired the infra-red beam into a cylindrical crystal (80 mm long x 6 mm diameter) of zinc selenide immersed in a fairly concentrated suspension (100 g of goethite per litre of aqueous solution). The crystal is transparent to IR and the beam is internally reflected over twenty times as it passes through the crystal, with five reflections occurring in contact with the solution. The IR spectrum is analysed in Fourier transform mode, with corrections for the empty cell and using the difference between the signal before and after the addition of the adsorbate. Figure 6.2.4 shows the spectra obtained before and after addition of phosphate ion at pH 4 in a background sodium chloride solution. In the case of visible/UV radiation the aim is to get the bulk of the light absorption process occurring in the interface, so most studies are of the interface between the glass (or quartz) and the adjoining aqueous solution. With FTIR, the ATR mode can be used in the same way to study the surface near where the reflection is occurring (Baty

276 I

6: ADSORPTION ONTO SOLID SURFACES

et al. 1996) but that is not its only advantage. Since water has such a very strong absorption in the IR region, the IR spectra of aqueous systems are normally studied in extremely short path-length cells. The ATR method is a way of circumventing that problem, especially for suspensions, since such cells cannot be easily filled or cleaned and would be easily plugged with the solid (Tejedor-Tejedor and Anderson 1986). The paper by Baty et al. (1996), referred to above, is a good example of the way in which application of XPS (Section 6.1.4), ATR-FTIR, and AFM (Section 6.2.2) can be applied to the same system to reveal details of the adsorption behaviour of a protein at an interface.

6.2.5 Total internal reflectance fluoroscopy (TIRF) This technique is closely related to ATR but instead of studying the absorption due to the evanescent wave, the excitation due to that wave produces a fluorescent emission which is collected in a monochromator and passed on to a photomultiplier tube for analysis (Fig. 6.2.5). The exciting laser beam is chopped at a characteristic frequency so the fluorescent emission is modulated at that frequency and this aids in the detection process. After suitable calibration the TIRF signal can be used to measure the adsorption of a protein at the glass-water interface. (Lassen and Malmsten 1996).

COMPETTllVE PROTEIN ADSORPTION

f

4\

JI

c \

1

\

Fig. 6.2.5 Schematic illustration of the TIRF apparatus. The excitation side consists of the laser with a neutral density filter (OD), a shutter, S, a chopper& and two mirrors, MI and Mz. On the emission side are two lenses, the monochromator, the photomultipliertube (PMT), a fast amplifier, a photon counter (PhC), and personal computer (PC) for data analysis. (After Lassen and Malmsten 1996.)

ADSORPTION AND DESORPTIONAT THE SOLID-GAS INTERFACE

I277

6.3 Adsorption and desorption at the solid-gas interface We noted earlier some of the technological imperatives which drove the early studies of gas adsorption on solid surfaces and how those studies have led in recent years to a wide variety of investigative techniques. One of the most successful researchers in those early years was Irving Langmuir whose work has proven to be of particular value in colloid and surface science. Langmuir’s studies of the adsorption of gases on metals were initially undertaken to improve the qualities of the incandescent electric light bulb. The early bulbs produced by Thomas Edison consisted of a carbon filament inside an evacuated glass bulb but they were superseded by bulbs with a metal (tungsten) filament filled with a gas (initially nitrogen but later mainly argon with a little nitrogen) to reduce the metal evaporation at the high temperatures involved. Some discussion of gas adsorption on metals can be found in the usual physical chemistry textbooks (Atkins 1982, Chapter 29) but we will examine the salient features here to serve as a basis for a more extended study of adsorption processes in subsequent chapters. Before it became possible to study the solid surface with the sophisticated microscopic techniques which have been discussed above, quite a lot of information had been gleaned from macroscopic studies of the adsorption and desorption behaviour of various gases. The speed of adsorption and the amount adsorbed as a function of temperature and pressure allow us to distinguish two distinct types of adsorption process: physical and chemical (referred to as physisorption and chemisorption respectively). In physisorption, the gas molecule normally remains intact and interacts with the surface through van der Waals forces (dipole interactions if it is a polar molecule and/ or temporary dipole-induced dipole (i.e. dispersion) interactions if it is not). In chemisorption, a chemical bond is formed between the gas and the metal, usually after the gas has broken down into fragments under the influence of the surface forces. The energies involved in chemisorption are comparable with chemical bond energies (i.e. several hundred kilojoules per mole). For physisorption the energy is more likely less than 50 kJ/mole. Physisorption normally increases at low temperatures when the kinetic energy of the gas molecules is too low for them to escape the surface bonding. Chemisorption decreases at low temperatures because there is then insufficient activation energy to break the chemical bonds in the adsorbing molecule prior to its bonding with the surface.

6.3.1 The mechanics of adsorption The number of collisions occurring between gas molecules and a solid surface can be calculated from the kinetic theory (Exercise 6.1.1). At normal temperatures and pressures it is not difficult to show that each surface atom is struck by a gas molecule about 10’ times per second. In some of those encounters the gas atom simply bounces back without adsorbing but a certain fraction will remain on the surface. Once adsorbed, the atom or molecule, if it is only physically adsorbed, will be able to move more or less freely around the surface and will tend to come to rest at a kink site or step dislocation (Atkins 1982 p. 1003.) where it can interact more strongly with the surface

278 I

6 : ADSORPTION ONTO SOLID SURFACES

atoms. There is thus a tendency for these more highly active sites on the surface to be filled up first. The extent of adsorption is measured by the coverage, $, defined by:

‘

Number of surface sites occupied = Total number of surface sites *

(6.3.1)

The rate at which the first adsorbed layer is built up decreases as the layer nears completion but the time required for equilbrium to be reached is expected to be less than one second, unless the gas is able to penetrate into the interior of the solid. Experimental data on gas adsorption is collected by placing a solid surface (often in the form of a powder) in an evacuated chamber and heating it for some time, whilst continuing to pump out any evolved gases. When the surface is considered to be ‘clean’, the sample is taken to the (usually lower) temperature of the experiment. Known amounts of the adsorbing gas are then allowed into the sample chamber and the residual gas pressure is measured after equilibrium has been established. This allows one to estimate the amount adsorbed as a function of the equilibrium pressure at the temperature of the experiment. The resulting plot is called an adsorption isotherm. Theoretical descriptions of the adsorption process are usually given in terms of adsorption isotherms but, as Adamson (1990 Chapter 16) points out in his very detailed discussion of this area, mere agreement with an isotherm does not tell the whole story. T o obtain a proper understanding of adsorption one must also examine the energy involved in the adsorption and desorption process. Here we confine ourselves to a few of the basic isotherms and only briefly describe the technique used to estimate the desorption energy. For more details see Adamson (1990) or Adamson and Gast (1997).

6.3.2 The Langmuir adsorption isotherm The amount of material which is adsorbed on a surface, at a particular temperature, depends on the amount of that substance in the gas phase. For surface area determination we are normally concerned mainly with the laying down of the first layer of adsorbate. That is a process which involves interaction between different species. If more than one layer is to be adsorbed, the process involved for the second and subsequent layers is much like condensation of the gas to a liquid. It is normally a physical adsorption process and usually occurs only at pressures close to the normal equilibrium vapour pressure of the liquid. The Langmuir adsorption isotherm gives the relation between the coverage of the first layer, $, and the gas pressure at a particular temperature. Langmuir’s treatment assumes that all the adsorption sites are equivalent and the ability of the adsorbate to bind there is independent of whether adjacent sites are occupied or not. The adsorbed molecules are assumed to be in dynamic equilibrium with the molecules in the surrounding gas: A(g)

+ M(surface) + AM

(6.3.2)

and the rate coefficients for the adsorption and desorption process are k., and k d respectively. The rate of adsorption is proportional to the pressure of A, and the number of sites available on the surface, N(1- 6) where N is the total number of sites.

ADSORPTION AND DESORPTIONAT THE SOLID-GAS INTERFACE

1.0r

I279

...

. _ . . . . . .-

0

2

4

6

8 10 12 Pressure (am)

14

.

16

Fig. 6.3.1 The Langmuir adsorption isotherm (eqn 6.3.8) for various values of the constant K.

Therefore: Rate of adsorption = kalpaN(l - 6).

(6.3.3)

The rate of desorption is proportional to the amount of gas adsorbed: Rate of desorption = kdN6.

(6.3.4)

At equilibrium the rates are equal and by equating eqns (6.3.3 and 4 ) and solving for 6 we arrive at the Langmuar Isotherm:

where K = k,/kd. The form of the isotherm for various values of K is shown in Fig. 6.3.1. Higher values of K mean a higher afinity of the gas for the solid. Note that the coverage increases with pressure and eventually reaches a maximum, corresponding to a monolayer, if K or the pressure is large enough. The quantity K is the equilibrium constant corresponding to the reaction (6.3.2) and it depends on the temperature. For physical adsorption, lower temperatures favour the adsorption process and so K is expected to increase as the temperature falls. Since K is an equilibrium constant we can use the usual equations of chemical thermodynamics to relate the temperature dependence of K to the enthalpy of adsorption. The appropriate equation is usually referred to as the van't Hoff isochore:

280 I

6 : ADSORPTION ONTO SOLID SURFACES

The subscript .$ means that the equilibrium constant at each temperature is measured at constant coverage. Under those conditions, K = const. x (l/pa) and so:

(6.3.7) (Note the change in sign.) This value of A g d , is called the isosteric heat of adsorption referring to the fact that it applies to a certain value of the coverage.+ The Langmuir model implies that A g , , should be constant but it is more likely to be a function of coverage: at low .$ because of the preferential filling of highly active sites and at high 6 because of lateral interactions between the adsorbed molecules as they approach close pack on the surface. Allen (1997, Vol. 2 p. 46) gives examples of related isotherms based on alternative assumptions about the dependence of A H on coverage. A logarithmic relation between A H and ,$,for example, gives rise to the Freundlich isotherm (6 = kpl’q) which we will discuss in Section 6.4.3 and Chapter 10. The Langmuir isotherm can be tested by rearranging it into a linear form, e.g.:

This would be most suitable if we had a direct method of determining the coverage. More usually we have measurements of the volume, v of gas which is taken up by a given mass of solid, m, at different pressures. Then, if V, is the maximum uptake, assumed to correspond to total coverage, we would have .$ = v / V, and so, v / ( V, - v ) = Kpa which can be rearranged to give (Exercise 6.3.1):

Thus, if K is independent of coverage, a plot of p , / v against pa should be linear with a slope from which the maximum adsorption capacity (V,) could be estimated. The intercept would then allow K to be estimated at that temperature. If we know how much area the molecules occupy on the solid surface, the value of V, can be used to estimate the surface area of the solid (Exercise 6.3.2). That method is particularly useful for adsorption from solution (Chapter 10) where the same equations hold, except that the gas pressure is replaced by the solute concentration. Figure 6.3.2 shows a plot of this sort for the adsorption of CO onto charcoal. Note that the line has a slight curvature which would be expected if K were not strictly constant. Using eqn (6.3.7) to obtain the isosteric heat of adsorption is not too difficult for physically adsorbed species because equilibrium is usually fairly quickly attained. For chemisorbed species that is unlikely. Observations must be made at constant coverage and it is necessary to establish the equilibrium adsorption condition by showing that the adsorption isotherm at each temperature is the same as the desorption isotherm at the same temperature.

6.3.3. The BET adsorption isotherm The Langmuir isotherm is suitable only for situations where the adsorption is limited to a monolayer (and not always even then). When multilayer adsorption is possible, t Isosteric comes from the Greek meaning ‘same space’.

ADSORPTION AND DESORPTIONAT THE SOLID-GAS INTERFACE

200

0

400

600

I281

800

plmmHg Fig. 6.3.2 Test of the Langmuir isotherm for CO adsorbed on charcoal. (From Atkins 1982 with permission.)

leading ultimately to condensation of the vapour as a liquid onto the solid, a better description is provided by the isotherm developed by Brunauer, Emmett, and Teller (1938) and universally known by their initials. The lirst layer in that case may sometimes (though rarely) involve chemisorption but all subsequent layers will be physisorbed. A derivation of this equation is given by Atkins (1982 p. 1026) and we will not repeat it here. It is an extension of Langmuir’s argument with successive layers being formed independently. An arriving molecule may land on a bare spot or one on which there is already one or more molecular layers stacked up. The final result, for the volume of gas adsorbed, Y compared to the amount, Vmonrequired to form a monolayer is:

(6.3.10) where p* is the equilibrium vapour pressure of the adsorbate and c is a constant related to the adsorption and desorption rate constants. The coverage as a function of relative pressure is shown in Fig. 6.3.3. As the value of c increases one can see the clear development of a knee in the curve corresponding to the formation of a monolayer. As the vapour pressure approaches the value of the equilibrium vapour pressure for the condensed liquid, the adsorption increases exponentially and ultimately a macroscopic film of liquid is formed on the surface. Equation (6.3.10) can be written (Exercise 6.3.4): 2

(1

- z)Y

-- 1 - CVmon

+-(c-

1)x

CVmm

P P*

where x = -.

(6.3.11)

Knowing the vapour pressure of the adsorbate allows one to plot the function x/( 1 - x) V as a function of x to establish the validity of the isotherm, and to obtain the

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6 : ADSORPTION ONTO SOLID SURFACES

values of c and V,,, from the intercept and slope. Most systems give a linear plot up to at least p/p* = x = 0.3 (Fig. 6.3.4). A suitable gas for determining the area of a solid will have a high value of c and in that case the intercept is zero and the slope is l/Vmon.This allows one to obtain an estimate of the area from a single point measurement (accurate to better than 5% if c > 100) in the relative pressure region 0.14.3 (Exercise 6.3.3). The usual gas used is nitrogen at a temperature near to the boiling point (-196 "C)but krypton is a common alternative, particularly useful for samples of small surface area. Although for many solids, the shape of the adsorption isotherm is very much like that shown in Fig. 6.3.3 for some value of c, the agreement is by no means exact, as is evident from the fact that the data begin to depart from the linear plot in Fig. 6.3.4 for values of x (= p/p*) greater than about 0.3. There is a variety of reasons for this sort of 'non-ideal' behaviour. So far we have treated the solid as a smooth uniform surface to which the gas all has ready access. That is not always the case, especially if the solid is finely divided and/or porous. In that case the shape of the isotherm may become very different from that shown in Fig. 6.3.3, especially for high values of z where the gas is approaching the point at which it can condense into a liquid. It may also show some hysteresis effects, that is, the curve may be different depending on whether the pressure is being increased or decreased (Fig. 2.7.4). One reason for this has already been discussed in Section 2.7.2. As the gas approaches its saturation vapour pressure it becomes possible for condensation to occur in fine capillaries in the solid even before its saturation value is reached, if the resulting radius of curvature of the liquid surface is small enough. Allen (1997 Vol. 2) discusses other possible reasons for observed departures from the BET isotherm. The most obvious is the assumption that the heat of adsorption is independent of coverage and is the same for all layers except the first. There are two errors here: the heat of adsorption of molecules in the first layer depends on coverage because the first adsorbate molecules occupy the more energetic sites. For the second

Relative pressure ( p/p*) Fig. 6.3.3 The BET adsorption isotherm for various values of c (= 0.1,0.5, 1,2,5, 10, 100, 1000.)

ADSORPTION AND DESORPTIONAT THE SOLID-GAS INTERFACE

///

0.06 h

0.05 -

0.0

I283

0.1

0.2 0.3 Relative pressure (z=p/p*)

I

I

0.4

0.5

Fig. 6.3.4 The BET adsorption isotherm for two kaolinite samples, plotted according to eqn (6.3.11).

and subsequent layers this is not expected to be significant since they are occurring on a layer of the adsorbate molecules; the energy of adsorption of these layers is generally assumed to be closely related to the liquefaction energy. The effect of lateral interaction will, however, be important for both the first and the subsequent layers. When the layer is approaching saturation the number of near neighbour interactions increases rapidly with (. Equation 6.3.10 is, then, unlikely to be very satisfactory for very low (C0.05) and very high (-1) values of x. There is some confusion concerning the actual model used in deriving the BET equation. Some texts suggest that the derivation implies that atoms or molecules can be stacked on top of one another in columns of arbitrary height and still yield the normal adsorption energy. There is, however, no reason to assume such unlikely stacking arrangements. It is sufficient if one permits the formation of islands of the first layer to be used as substrates for the formation of the second layer and for islands of the second layer, before that layer is completed, to function as substrates for the third layer and so on. This is the picture suggested by Atkins (1982 Fig. 29.21) and supported by the calculations of Lowell (1975) who showed that, when the total amount of adsorption corresponded to a monolayer, the fraction of the surface covered by molecules i layers deep was:

(6.3.12) For c = 100 the values of 2

(Ci;.)m

0 0.0909

are as follows: 1 0.8264

2 0.0751

3 0.00683

4 0.00062

5 0.00006

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6 : ADSORPTION ONTO SOLID SURFACES

So for this value of c, corresponding to a modest degree of affinity, the surface is about 9% bare at monolayer coverage, less than a tenth of the first layer is covered with a second layer of molecules, and likewise for the third and subsequent layers. When a molecule hits the surface and sticks, whether on the bare surface or on a previously adsorbed film, it may diffuse around the surface and, in that case, will eventually meet an island of adsorbed molecules if one exists. Whether it stays associated with that group or remains independent will depend on the temperature, and the balance between the energy associated with the lateral interactions and the entropy gain of remaining free to move over the entire surface. This picture seems straightforward and one would expect to be able to distinguish relatively easily between a process in which the adsorbate is attached firmly to fixed sites and one which is mobile. In multilayer adsorption it would seem to be intuitively obvious that the molecule would be fairly mobile, but for the first layer Adamson (1990 Chapter XVI Section 3B) examines the question at some length using a statistical mechanical argument. He concludes that the distinction is not at all easy to make, even from measurements of the entropy of the adsorption process. Whether the first layer of adsorbate is fixed or mobile, the most important prediction of the BET equation is the monolayer volume, V,,. For the high values of c for which the method is most satisfactory, Vmo, corresponds to a low value of the relative pressure, x (C0.4). The values given by the equation for values of x greater than about 0.4 should be taken as only a rough guide to the anticipated adsorption behaviour. A critical review of the theory is given by Dollimore et al. (1976) and Allen (1997 Vol. 2) gives a discussion of the BET equation and its modifications, including a description of the n-layer BET model which can be used to treat adsorption in restricted spaces, such as occur in microporous solids (Exercise 6.3.6). Adamson (1990) also gives a derivation of the BET equation and describes briefly some of the important modifications. He also introduces Polanyi’s theory of the adsorption potential and describes the isotherms which result from that approach, together with the general thermodynamic theory of adsorption. Most commercial adsorbents are porous and this is an important aspect of their nature. They are characterized by their specific surface area (area per unit mass of solid) and classified as macroporous (pores > -50 nm), mesoporous (pores between 2 and 50 nm) or microporous (pores < 2 nm). These limits are somewhat flexible since the detailed behaviour depends on pore shape and the size of the adsorbing molecule. Macroporous solids behave much like simple surfaces. Mesoporous solids show the phenomenon of capillary condensation referred to above. Microporous systems show capillary condensation at very low pressures so the isotherm may look like a high affinity Langmuir, but the plateau often slopes upwards as the external surfaces become covered with more than one layer at higher gas pressures. It is important to recognize the microporous behaviour because the plateau in that case corresponds not to a monolayer but to the micropore volume. The effective micropore volume should be quoted for the particular adsorbate, say nitrogen at 77 K, since it will be influenced by the accessibility of the pores and the molar volume of the adsorbate, which may differ from its bulk value. For a description of the procedures for determining pore volume and pore size, see Kissa (1999 Chapter 3) or Allen (1997 Vol. 2) who also includes a listing of commercial devices for the measurement.

ADSORPTION AND DESORPTIONAT THE SOLID-GAS INTERFACE

I285

6.3.4 Temperature programmed desorption (TPD) and thermal desorption mass spectrometry (TDMS) These are valuable techniques for studying both adsorption and desorption kinetics. They are thus used to infer information about the adsorbent surface and, in particular, the nature of adsorption sites. A clean metal filament or a surface is exposed to a known (very low) pressure of a gas that flows steadily over it (Adamson 1990 p. 690). At the usual pressures (- lo-' mmHg) the establishment of a monolayer may take some minutes. The surface is then heated, either by a laser beam or by passing a current through the filament. If the heating is slow, the gas pressure may be monitored directly

4

0

0.5 L

200

400

Temperature ("C) Fig. 6.3.5 TPD spectra for hydrogen chemisorbed on flat (1 ll), stepped (557), and kinked (12,9,8) single crystal surface of platinum. Curves correspond to exposure to different volumes of the gas [in litres (L)]. (From Somorjai 1994, p. 350 with permission.)

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6 : ADSORPTION ONTO SOLID SURFACES

to give a programmed desorption curve (Fig. 6.3.5). The temperature/desorption pattern can distinguish gas adsorbed on flat surfaces, at step dislocations or at kink sites. Adamson (1990 p. 692) shows how temperature programmed desorption can provide information on the desorption energy, E, using the Redhead equation (Exercise 6.3.7): (6.3.13) where T, is the temperature of the maximum desorption rate, /3 is the rate of temperature rise, and A is the frequency factor for desorption (usually taken as 1013 s-'). If the gas is able to react with the surface (chemisorption) it is more usual to use a rapid heating process (flash desorption) and then to analyse the products with a mass spectrometer. (This is TDMS.)

Exercises 6.3.1 Establish eqns (6.3.5) and (6.3.9). 6.3.2When the Langmuir equation is used to measure the surface area of a solid, S, (m2/g) it is assumed that the area occupied by a molecule of the adsorbing gas (nitrogen) at its normal boiling point (-196 "C) is 0.162 nm2. Show that S, = 4.35 V, where V, is the volume corresponding to a monolayer of gas. [ V, is usually corrected to standard temperature and pressure (i.e. 1 atm and 0 "C) (Allen (1997) Vol. 2 p. 44).] 6.3.3Atkins (1982) gives the following data for the pressures of CO required to cause adsorption of 10.0 cm3 of gas to be adsorbed onto a sample (3.022 g) of charcoal. (All volumes corrected to 1 atm. pressure and 273 K.) Show from eqn (6.3.7) that the enthalpy of adsorption can be obtained from a plot of In p as a function of 1/T. What is the value of AH,, at this coverage? This is the isosteric heat of adsorption. T/K p/mm Hg

200 210 220 230 240 250 30.0 37.1 45.2 54.0 63.5 73.9

Assuming that a CO molecule occupies an area of 0.2 nm2 and that the coverage in this case is 0.4, estimate the area of the solid (m2/g). 6.3.4(a) Derive eqn (6.3.11) from (6.3.10). (b) The following table gives the volume (Vin cm3)of nitrogen gas (corrected to 1 atm pressure and 273 K) adsorbed on a carbon black sample (mass 1 g) as a function of pressure (p (mm of Hg)). Plot the volume as a function of pressure. The lowest point of the linear part of the curve is usually identified with monolayer formation. Replot the data in terms of eqn (6.3.11) assuming that p* is 760 mm (i.e. the measuring temperature is the temperature of boiling liquid nitrogen). Do the estimates of monolayer coverage agree? Take the area of a nitrogen molecule as 0.16 nm2 and estimate the surface area of the sample of carbon black (m2/g).

ADSORPTION AT THE SOLID-LIQUID INTERFACE

17 15

V

P

22 20

25 50

28 80

30 100

37 210

47 320

54

440

I287

75 525

6.3.5Estimate the surface areas of the kaolinite samples in Fig. 6.3.4 assuming that the mass of solid is 1 g in each case and the volume is measured in cm3 corrected to 1 atm pressure and 273 K. 6.3.6 The n-layer BET equation is derived on the assumption that adsorption is limited to a maximum of n layers. It can be written (Allen (1997) Vol. 2 p. 56): V

v,

-

+

+

cz 1 - (n 1)zn nz”+l (1 - 2) 1 (c - 1)x - czn+l

+

Show that this reduces to the Langmuir equation for n = 1. How is c related to the value of K in the Langmuir equation? 6.3.7The rate of desorption, D may be written: D = -df/dt =Af exp (-E/RT) where E is the desorption energy and A is the frequency factor. Assuming the temperature is rising linearly with time: T = TO Pt, establish eqn (6.3.13) by determining the maximum desorption rate. Use the data from Fig. 6.3.5 for the desorption of hydrogen from the Pt (111) face to estimate the energy of desorption from that face assuming that the heating rate was 8 K/s (Adamson 1990 p. 692.)

+

6.4 Adsorption at the solid-liquid interface When adsorption occurs from (say, aqueous) solution onto a solid, liquid, or gaseous surface, it is important to recognize that the adsorbate is competing against the solvent (water) molecules for a place at the interface. We discussed in Section 2.4 the thermodynamic relation between the tendency of a substance to adsorb into the interfacial region and the effect of that substance on the interfacial tension (or surface free energy), y. Substances which lower the surface tension are said to be ‘surface active’; they accumulate at the interface because in so doing they lower the total energy of the system. The result was the Gibbs adsorption isotherm (eqn (2.4.3)), an equation of very considerable importance for the study of the surface properties of liquidvapour and liquid-liquid systems. The main application of that equation stems from the fact that the value of y is relatively easily measured in such systems and the results (as functions of solute concentration) provide a model-free estimate of the amount of adsorption, especially for simple, two-component systems. Thus, surface tension measurements at the air-water interface provide a wealth of data on the behaviour of soaps, detergents, and other surface active materials (lecithins, proteins, etc.). Measurements of y for the mercury-aqueous solution interface (Section 7.2) provide the basis for the study of electrocapillarity which underpins most of our ideas of the structure of the electrical double layer. The corresponding measurements are also of great value in the study of emulsion systems.

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6 : ADSORPTION ONTO SOLID SURFACES

Unfortunately, for the solid-liquid systems with which we are principally concerned, eqn (2.4.3) is of rather limited value, due to the problems associated with the measurement of the surface tension (or surface free energy) of solids (Section 2.2.1). Indeed, in this case the equation is used in the opposite manner, to obtain estimates of changes in the surface energy from independent measurements of the amount of adsorption. Because of the large surface areas which commonly occur in colloidal systems, the amount of adsorption can often be determined directly from the depletion of the adjoining liquid phase, especially if the adsorbate is present in fairly dilute solution (Section 7.7). Furthermore, it often turns out that we are most interested in the adsorption properties of relatively minor components of the liquid phase, especially if they are surface active (i.e. strongly adsorbed). Such substances are widely used to modify and control the equilibrium and kinetic (transport) properties of colloidal dispersions. In this section, therefore, we will confine attention almost entirely to the adsorption of components which are relatively dilute (on a mole fraction basis). The adsorption behaviour of colloidal dispersions in mixed solvent systems is discussed by Everett (1981) and by Lyklema (1995 Chapter 2) and that ground will not be traversed here. There is, of course, much literature on the general theory of adsorption from mixed solvents onto solid surfaces. Fortunately, that has also been very ably reviewed by Everett et al. in a number of papers in the series Colloid Science (Everett 1973: Brown and Everett 1975; Everett and Podoll 1979; Davis and Everett 1983). Lyklema (1995) also provides a comprehensive coverage of this area.

6.4.1 Adsorption from dilute solution As noted above, adsorption of a solute from dilute solution onto a solid surface must be seen as an exchange process between the solute in solution and one or more molecules of the solvent at the surface. That may occur because of a positive affinity of the solute for the surface or as a result of the rejection of the solute by the solvent. In the case of surface active agents, it is often a combination of both such effects. The fact that the adsorption occurs from a dilute solution makes its detection and measurement relatively easy, especially as the area for adsorption can usually be made large enough to produce a significant change in the solute concentration. In setting up possible theoretical isotherms for the adsorption process we need to be aware of the influence of the surface solvent on the mobility of surface adsorbed solute molecules, the interactions between the solute and solvent and between solute molecules, both in the bulk and at the surface and, of course, all the problems associated with surface roughness, porosity, and heterogeneity, as mentioned in respect of the gassolid interface. It should be noted, however, that there are some compensations: water is such a very active molecule both in its interactions with itself and with other materials that it tends to smoothe out some of the heterogeneities on solid surfaces. The dilute solution restriction also makes the positioning of the Gibbs dividing surface (Section 2.4) a great deal simpler in this case than it is for a general liquid-liquid interface. Lyklema (1995) distinguishes the six most common isotherm types as shown in Fig. 6.4.1. The linear form is usually only observed over a very short concentration range, since it will inevitably give way to some curvature as the surface fills with adsorbate. The Langmuir form is commonly observed, and does not always mean that

ADSORPTION AT THE SOLID-LIQUID INTERFACE

r

r

/ Linear

f

c F-Freundlich C

C

C

r

L-Langmuir

r

I289

i

7 H-High affinity c

S- Sigmoidal c

C

Fig. 6.4.1 Phenomenologicalclassification of isotherms from dilute solution, G is the concentration of solute in the liquid (After Lyklema 1995 p. 2.65)

the solute is behaving according to the postulates of Langmuir’s theory (Section 6.3.2). Freundlich’s isotherm is also commonly observed and, as noted in Section (6.3), can be explained on the basis of a logarithmic variation in the energy of surface sites. The high affinity isotherm is normally regarded as a variation of the Languir in which the initial region shows such strong adsorption that none of the adsorbate remains in solution until the surface nears saturation. The sigmoid suggests that some nucleation process must first occur either in solution or, more likely, on the surface, before adsorption can proceed. The step isotherm is sometimes interpreted in terms of bilayer formation but may also be due to the reorientation of the surface layer to accommodate more material (e.g. the change from horizontal to vertical orientation of a long chain molecule). Note that none of the isotherms shows a maximum. Lyklema (1995 p. 2.66) points out that such behaviour is the result of complications such as the appearance of new phases, scavenging of the adsorbate by micelles, competition from spurious contaminants (including homologues of the adsorbate), or simply analytical artefacts. A true maximum in the surface excess, r, with increase in solution concentration is impossible since it would indicate a decrease in chemical potential with increasing concentration and that is thermodynamically impossible. We study isotherms in order to understand the mechanisms and the forces involved in the adsorption process and to gain an understanding of the structure of the adsorbed species. Control of the extent of adsorption and the orientation of the adsorbate can be useful for controlling the transport and equilibrium properties of a suspension. In addition to the study of the isotherm at various temperatures, it is also valuable to have supporting calorimetric and spectroscopic information.

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6 : ADSORPTION ONTO SOLID SURFACES

6.4.2 The Langmuir isotherm The adsorption of a neutral molecule (A) onto a solid surface can be represented as an exchange process between that molecule and an adsorbed solvent molecule (S): A(liq)

+ S(surface) 2A(surface) + S(1iq)

(6.4.1)

and if the molecules are of similar size, the thermodynamic equilibrium constant, K, is given by (Everett 1965):

(6.4.2) where fA and f s are (rational) activity coefficients of the adsorbing solute (A) and the solvent (S) in the liquid phase (1) and at the surface (a)respectively; x is the mole fraction of A in the liquid phase and $ is the coverage (i.e. the fraction of the surface sites occupied by A ) . The standard free energy of adsorption per molecule is given by:

AG;,

= -kTlnK.

(6.4.3)

In the most elementary treatment, we set the activity coefficients to unity for both the liquid phase and adsorbed species so that (Exercise 6.4.1), for dilute solutions (x << 1):

(6.4.4) where n i is the number of molecules adsorbed per unit area and N, is the total number of sites available per unit area. Equation (6.4.4) is a form of the Langmuir adsorption isotherm suitable for solutions. Note its similarity to eqn (6.3.5) for gases. Despite its simple form, eqn (6.4.4) has been successfully used in a variety of different situations. Kitchener (1965) has given a good introductory account of the conditions under which we can expect to draw meaningful conclusions from the shape of the adsorption isotherm. The simple Langmuir form is expected to hold for clean, smooth, nonporous surfaces, showing reversible, physical adsorption of a pure solute, if the adsorption occurs uniformly over the surface and the adsorbed molecules interact laterally merely as hard spheres (Fig. 6.4.1). The amount of adsorption (in moles of adsorbate per unit mass of adsorbent) is

(6.4.5) where A is the specific surface area (i.e. the area per unit mass) and N A is the Avogadro number. If mi is the maximum value of mA (corresponding to monolayer coverage) then mi = N,A/NA and

mA KX KC; K'c; .$-=--mi - 1 Kx 55.5 Kcj 1 K'cj

+

+

+

(6.4.6)

where c; is the molar concentration and eqn (6.4.6) is written for aqueous solution. It follows from eqn (6.4.6) that a plot of c;/mA against cj should be linear for a system

ADSORPTION AT THE SOLID-LIQUID INTERFACE

I291

following a Langmuir isotherm (Exercise 6.4.2 and 3) and this is the way the data is usually tested. The plot enables one to evaluate both the maximum adsorption, mi, and the equilibrium constant, K’. At low concentrations (when K’ci << l), eqn (6.4.6) would suggest a linear relation between mA and ci (or x) and this describes the first adsorption isotherm, which corresponds to Henry’s law for dilute solutions. Th e concentration at which the surface is half covered ( x 1 / 2 ) is inversely related to the equilibrium constant (Exercise 6.4.1) and K (or x1/2) may, therefore, be taken as the measure of the affinity of the surface for the adsorbate.

6.4.3 The Freundlich isotherm If the surface is patchwise homogeneous, having, say, two different types of surface with significantly different affinities for the adsorbate, the behaviour may be as shown in Fig. 6.4.2 (where the overall isotherm (111) is the sum of two Langmuir expressions with different K values). Th e extreme form of this behaviour is shown in Fig. 6.4.2(b) where the system obeys the Freundlich isotherm:

1.6

I

I

C

0 .*

1

0.8

0

cn

.-e

d 0

0.2 0.4 0.6 0.8 1 X

0.6 0.4

0.2 0 4 0

0.4 0.6 0.8 Solution Concentration

0.2

1

Fig. 6.4.2 Surface heterogeneity and the Freundlich isotherm. Curve (a) shows the result of combining two Langmuir isotherms with very different affinities. In the extreme case of sites with many different affinities, the result is the Freundlich isotherm (b).

292 I

6 : ADSORPTION ONTO SOLID SURFACES

where q is an empirical constant (usually lying between 2 and 10). [Davies and Rideal (1963) point out that this isotherm was actually proposed by Kuster and initially rejected by Freundlich, presumably because it does not have the correct limiting behaviour at low and high concentration. Freundlich did, however, demonstrate its applicability to a number of systems.] As noted above, eqn (6.4.7) can be shown (see Adamson 1990, p. 425) to result from a modification to the Langmuir derivation in which the affinity of the surface sites (as measured by the heat of adsorption, Q) varies continuously, in accordance with an expression of the form:

where a is a constant related to the constant KF.

6.4.4 Thermodynamics of adsorption As in the case of gas adsorption, the equilibrium constant in eqn (6.4.6) can be related to the Gibbs free energy, entropy, and enthalpy of the adsorption process:

K' = exp(-AGo/RT) = exp(ASo/R).exp(-AHo/RT)

(6.4.9)

where A€#' = -Q As before, (compare eqn (6.3.7)) we can obtain a measure of A€#' by measuring the concentration C at which the coverage has a certain value, as a function of temperature:

(aln c / ~ T ) = ( -AHO,,/RT~

(6.4.10)

and A q d , is the isosteric heat of adsorption.

t

6 I-

2 1

0

1

2

3

4

5

6 7 x2/10-3

8

1

2

3

4

5

A ad,H(ref.)/Jg1

Fig. 6.4.3 Adsorption of long chain alkanes from n-heptane on various grades of graphite. Left: Enthalpy isotherms for n-docosane (n-C22H%);right: correlation with the enthalpy of adsorption for a reference sample. (From Lyklema 1995, with permission. Redrawn from Kern and Findenegg 1980.)

ADSORPTION OF NEUTRAL POLYMERS

I293

Lyklema (1995, p. 2.71-3) describes the thermodynamic data for a series of nalkanes on carbon (various graphites including Vulcans and Graphon)+, studied by Kern and Findenegg (1980). Although the isotherms and the plots of AH,d, against solution concentration are sigmoidal (Fig. 6.4.3), the values of AH,,, for the different adsorbates show the same proportionality to one another at all solution concentrations, suggesting that all of the surfaces are quite homogeneous. This proportionality also enabled the surface areas of the different solids to be compared and the results were in excellent agreement with those obtained using the BET method (Section 6.3). Unfortunately this is not usually the case.

r

Exercises 6.4.1Establish eqn (6.4.4)and (6.4.6).Show that the concentration when the surface is half covered is given by x1/2 = K-'.

6.4.2Derive an appropriate form of eqn (6.4.4) to test it by a linear plot. How do you evaluate the constants appearing in the equation?

6.4.3Check the following data for conformity to the Langmuir isotherm and estimate the area of the solid assuming that the cross-sectional area of the adsorbate molecules is 0.20 nm2. ci(mrno1e Lp') ni (mmole gp')

(Note that

ci

0.20 0.035

0.81 0.081

1.20 0.105

1.70 0.102

2.00 0.103

is the equilibrium concentration after adsorption is complete.)

6.4.4Show that the Langmuir equation (6.4.6) can be put in the form ln[t;/(l

-

t;)] - In ci = In K'.

6.5 Adsorption of neutral polymers The polymers which are used for modifying the behaviour of colloidal systems are almost invariably linear (i.e. with little or no cross-linking) and we will confine ourselves to them. They may be regarded as long chains of atoms with side groups in a repetitive pattern, usually attached to every second or third atom in the backbone: R

I

-(B-B-)n

T h e backbone atoms may all be carbon or silicon but carbon can alternate in a regular pattern with oxygen or nitrogen and silicon can alternate with oxygen. T h e presence of the side chains and their composition has some effect on the behaviour

t A form of graphite produced by high temperature ( t

2700 "C) treatment.

294 I

6 : ADSORPTION ONTO SOLID SURFACES

because they affect the flexibility of the chain but for the usual polymers with which we are concerned the restrictions are not very important and the polymer can be treated as highly flexible. Synthetic polymers are sometimes made from a single monomer and sometimes with a mixture of two or three monomers. The latter may be randomly mixed in the final chain or, more commonly, consist of blocks of one monomer unit interspersed with blocks of the other monomer. The block copolymers have some useful properties which are exploited to provide colloid stabilization (Section 12.9). The adsorption behaviour of polymers at interfaces is not unlike that of the nonionic surfactants treated in Section (6.4.5), though the greater size of the polymer molecules makes for even more varied behaviour. Mention was made in Section 1.7.2 of the notion that polymers are adsorbed at the solidsolution interface in the form of trains, loops, and tails (Fig. 1.7.1) (Jenkel and Rumbach 1951). That arrangement is a compromise forced by the usual balance between energy and entropy considerations. In this case the relative interaction energy between the polymer segments and (i) the solvent and (ii) the sites on the surface must be balanced with the very large entropy effects which result from adsorption: the reduction in the entropy of the polymer and the increase in the entropy of the solvent as many molecules of solvent are released for each molecule of polymer adsorbed. The treatment given here is drawn from the much more extensive discussion given by Lyklema (1995, Sections 5.3-5.9). We will discuss briefly the theory of polymer solutions and then extend the ideas in a more qualitative manner to the adsorbed state.

6.5.1 Polymer chains The simplest model for a polymer in solution is closely related to the random walk model we used for the diffusion process (Section 1.5.1). The positions of the centres of each successive atom in the chain are calculated by allowing each atom to be randomly placed but at a fixed distance (the bond length) from its neighbour on the chain (Fig. 6.5.1). Such an arrangement makes no allowance for the finite volume occupied by the atoms or for any restrictions on bond rotation but it does capture the major feature.

r

Fig. 6.5.1 The polymer chain (in this case in two dimensions) can be modelled as a random walk with a common step length, 1, which results in a fluctuatingend-to-end length r after N steps. (The atoms in this case are assumed to have zero volume.)

ADSORPTION OF NEUTRAL POLYMERS

I295

The end-to-end distance r is a characteristic feature of the chain and the mean square value of r is given by Exercise (6.5.1):

(2)= N l2

(6.5.1)

where N is the number of bonds in the chain and 1is the length of the bond. The mean end-to-end distance (r2)ican be taken as a measure of the mean coil diameter. A rather better measure is the radius of gyration, aG (Exercise 3.3.5) which is given by a& = N 12/6 for a long, perfectly flexible chain and measures the r.m.s. distance of the segments from the centre of mass of the molecule. Real polymers are not quite so flexible and so the chain is not quite so tightly coiled. The effect is accounted for by a persistence parameter, p so that:

(2)= 6 p N 1 2

or

aG 2 = p N l 2.

(6.5.2)

p is then 1 / 6 for a highly flexible chain and typical values for real polymers fall in the range 0 . 5 4 . It is possible to treat any long chain, no matter what its p value is as though it were flexible by grouping a sufficient number of segments together (Kuhn 1934) into what are called Kuhn segments. The orientation of these segments with respect to one another will be random, provided the group is large enough. The number, n of bonds in such a statistical chain element (s.c.e) will be larger the more rigid the chain is. The real chain of N bonds is then modelled as an ideal chain of Nk (= N / k ) beads, each of length lk (= bl) where k and b are greater than unity. Then (?) = Nklt = ( b 2 / k )N12 = 6pN12 = 6ak since b and k must satisfy the condition b 2 / k = 6p so that p remains the parameter describing the stiffness. T o specify the system completely we require another condition on b and k and that can be obtained by requiring that the contour length of the Kuhn chain (= NkZk) is the same as the contour length (=N I) of the real chain. The result is k = b = 6p. The typical Kuhn segment contains 5 to 20 backbone atoms, depending on the flexibility of the chain (Lyklema 1995 p. 5.5). An important consequence,of this analysis is that the ‘size’ of the polymer molecule d or to MT where M is the molar mass of the polymer. is proportional to i

6.5.2 Polymer chains in solution The above discussion takes no account of the finite size of the atoms in the polymer chain. Including that volume has the effect of expanding the size of the polymer. The interaction between solvent and polymer also has an effect: strong attraction will increase the amount of bound solvent and so increase the effective volume of the polymer. Poor solvent/polymer interaction will lead to a tighter polymer chain and lower polymer volume. Lyklema (1995 p. 5.6) gives a brief discussion of how these effects can be incorporated into the simple statistical models discussed in Section 6.5.1. Here we will use the alternative approach of describing the interaction initially in terms of the volume contributions of polymer and solvent. The statistical calculation of those volumes can be left for the time being. The thermodynamics of polymer solutions was developed independently by Flory and by Huggins in the early 1940s (Flory 1941; Huggins 1941). The theory estimates

296

I 6 : ADSORPTION ONTO SOLID SURFACES

the free energy of mixing of pure amorphous polymer molecules with pure solvent by separately calculating the entropy of mixing (which is a combinatorial term) and the energy of mixing (which measures the energy of interaction between polymer and solvent when they are in contact). These two terms are then combined in the usual manner: A F = ~ auM- T A P

(6.5.3)

where the superscript M denotes mixing. The combinatorial entropy was originally calculated by Flory using a lattice approach but it can be more simply derived from thefree volume of the molecules. The free volume of a substance represents that fraction of the total (external) volume which is not occupied by the geometric volumes of the constituent molecules. In what follows, the solvent and the polymer will be denoted by subscripts 1 and 2 respectively. We will also assume, for simplicity, that both the polymer and the solvent have the same free volume fraction,fv, and that this remains the same on mixing. (i.e. there is no overall change in volume on mixing.) VVis a dimensionless relative volume; the actual volume of empty space associated with a polymer molecule is of course much larger than that associated with a molecule of solvent.] The free volume accessible to nl molecules of solvent and n2 molecules of polymer V The volume V , ( z = 1, 2) is the before mixing is nl V l f , and n2 V ~ respectively. is the partial molar measured external volume per molecule (= / ~ / N Awhere volume and NA is the Avogadro number). After mixing, the total accessible free volume becomes (nl Vl n2 V+V. By analogy with the treatment for gases, the thermodynamic probability, W which appears in the Bolzmann equation S = k In W is proportional to the free volume accessible to the centres of mass of the species in the system. The entropy change of the solvent on mixing with the polymer is therefore:

+

where v1 is the volume fraction of solvent in the polymer solution. Similarly, the entropy change of the amorphous polymer molecules on mixing with the solvent is n2k In v2. Hence the combinatorial entropy of mixing of the polymer and solvent is: ASM = -R{nl In v1

+ n2 In v2)

(6.5.4)

which is positive, since v1, v2 C 1. The enthalpy of mixing is evaluated in the Flory-Huggins treatment using a quasichemical reaction approach. The solvent-solvent, polymer-polymer, and solventpolymer contacts are represented: 1-1 + 2 - 2

+2(1-2).

(6.5.5)

For these contacts, an interaction parameter x1 is defined such that x1 kT is the difference in energy of a solvent molecule (hence the subscript) when immersed in pure polymer compared with that in pure solvent. For nl solvent molecules, each

ADSORPTION OF NEUTRAL POLYMERS

I297

immersed in pure polymer, the energy change would be nlXlkT. In a polymer solution, the probability of a solvent molecule being in contact with a polymer segment is simply vz. It follows that the change in the contact dissimilarity energy, is given by: AUM = nlvzXlkT.

(6.5.6)

The total free energy of mixing according to the Flory-Huggins theory is then (from eqns 6.5.3, 4, and 6):

AFM = kT{nl In v1

+ nz In v2 + n l v z x l } .

(6.5.7)

Since x1 is often positive for non-aqueous solvents (i.e. mixing is endothermic), the contact dissimilarity term often opposes mixing. In contrast, both the combinatorial entropy terms are negative and promote mixing. When the polymer is of high molecular weight, nz is comparatively small and the dominant term promoting mixing is nlkT In V I . The primary reason why polymer and solvent molecules mix to form polymer solutions is now apparent: the entropy of the solvent molecules is increased as a result of the additional space available when the domains of the polymer molecules become accessible to the solvent. The detailed structure of the polymer is irrelevant according to the precepts of the Flory-Huggins theory; rods are just as effective as coils in providing space for the solvent molecules (Flory 1970). This point is stressed because it is sometimes erroneously asserted that polymer molecules dissolve primarily because of an increase in their configurational entropy. Note that although x1k T was originally introduced as a change in internal energy, the arguments presented would be essentially unchanged if XlkT were a free energy change. In this way X I ,as determined experimentally, may incorporate both energy and non-combinatorial entropy contributions. Finally it may be noted that relation (6.5.7) for the free energy of mixing is very similar to the Bragg-Williams equation for the mixing of small molecules:

(6.5.8) The only difference is in the use of volume fraction statistics for the polymer solution instead of the mole fraction statistics used for molecules of comparable size.

6.5.3 Limitations of the simple free volume theory Since its publication, the Flory-Huggins theory has been widely, and in many respects successfully, used to account for the behaviour of polymer solutions. Indeed the theory has been referred to as a ‘paradigm of polymer science’ (Derham et al. 1974). It does, however, have a number of significant shortcomings which can be pointed out from three experimental observations. First, experimental studies of the temperature-dependence of XI allow it to be resolved into its energy and entropy components. Although XI was originally introduced into the theory as an energy term, experiments show that for many nonaqueous polymer-solvent systems, the positive values for XI are determined primarily

298 I

6 : ADSORPTION ONTO SOLID SURFACES

by entropic considerations. The corresponding energy terms are relatively small and of variable sign. The experimental results imply that there is a non-combinatorial entropy change that opposes the mixing of polymer and solvent and is not accounted for by the Flory-Huggins theory. Second, x1 is found experimentally to depend on the polymer concentration. Usually XI,which has values lying in the range 0.1- 0.5, becomes more positive as the polymer concentration increases, (e.g. poly(iso-butylene) in benzene) but some exceptions are known (e.g. polystyrene in toluene). Third, the Flory-Huggins theory predicts that, as mixing of polymer and solvent is an entropically driven process, it should be favoured as the temperature is increased. Yet experimentally it is found that most, if not all, polymer solutions can be induced to undergo phase separation as the temperature is raised to near the critical point of the solvent. In this regard, polymer solutions differ significantly from solutions of small molecules. The phase separation which occurs near the critical point gives an important clue to the origin of the problem. This is the failure of the theory to account properly for the difference in the free volume of the solvent compared to the polymer. This difference is dramatically magnified near the critical point of the solvent when the solvent molecules become gas-like, with a large free volume. The polymer molecules, in contrast, undergo a relatively small change in free volume since they are constrained by their connectivity. In effect, the polymer segments cause the gas-like solvent molecules to undergo ‘condensation’ (Patterson 1969). The resulting decrease in entropy of the solvent on contact with the polymer can outweigh the combinatorial entropy of mixing. It is important to note that this difference in free volume can persist down to room temperature. Aqueous polymer solutions often show phase separations on heating but at temperatures well below the critical temperature of water. The explanation for this phenomenon must be sought elsewhere. It appears to be associated with the directionality of the hydrogen bonds formed between water and the polymer but that is far from certain. For further discussion of this point and Flory’s (1970) more elaborate analysis of the free volume problem, the reader is referred to Napper (1983). Lyklema (1995) also takes the discussion much further, extending the dilute solution theory into the region of strong polymer overlap.

6.5.4 General aspects of polymer adsorption Because of the extended nature of the polymer molecule, the description of its adsorption is rather different from that for simple molecules. A knowledge of the number of polymer molecules adsorbed as a function of concentration is not sufficient to predict the resulting behaviour. The adsorbed layer of polymer, as we noted in Section 1.7.1 consists of trains, loops, and tails and to describe the interfacial region in detail we need information on the number of polymer segments as a function of distance from the adsorbent surface. This segment density distribution would typically look like that shown in Fig. 6.5.2 and it can be measured by techniques like neutron scattering and neutron reflection (Section 14.5). The segment density falls off monotonically with distance to its bulk solution value. There is also present a certain

ADSORPTION OF NEUTRAL POLYMERS

i

I299

z

Fig. 6.5.2 Schematic representation of a polymer concentrationprofile as a function of distance z from the interface.The upper curve gives the overall segment concentrationc(z). the lower curve the concentration cf (z) due to non-adsorbed chains. Both curves approach the bulk solution concentration cb at large z. The hatched area is the polymer surface excess r"; the sum of the shaded and hatched areas represents the total adsorbed amount ra.The difference between F a and rex is denoted rd(shaded area). [Redrawn from Lyklema 1995, Fig. 5.6.1

concentration of segments which arise from polymer molecules which are not adsorbed. At low polymer concentration, the adsorption profile tends to be steeper, with most of the segments close to the adsorbing surface. As the concentration rises, the loops and tails become more extended so the profile decreases more gently with distance. The higher the polymer molecular weight the more closely it tends to sit to the surface but this is also influenced by the relative size of polymer-surface, solvent-surface, and polymer-solvent interactions. An alternate measure of the profile is the bound fraction or train fraction which measures the amount of polymer within one segment length, 1, of the surface, divided by the total adsorbed amount. It can sometimes be estimated by spectroscopic methods, if the polymer has some appropriate markers, but results are often difficult to interpret. The thickness of the polymer layer can be comparable to the radius of gyration, aG, of the polymer and this can be detected in hydrodynamic studies (see, for example, Firth et al. 1974). The result obtained is again a little difficult to interpret because the apparent thickness of the adsorbed layer depends on the degree to which the solvent can drain through the polymer layer. The amount of adsorption can, in principle, be either positive or negative (depletion) but we will leave the latter possibility until Chapter 12. [It is important at rather high particle concentrations when the spatial constraints prevent the polymer from entering the regions between two particles.] The most important parameters of

300 I

6 : ADSORPTION ONTO SOLID SURFACES

positive polymer adsorption are illustrated in Fig. 6.5.2. The polymer excess concentration rexis represented by the upper shaded area in the figure, corresponding to the amount which is in excess of the bulk concentration. For the dilute solutions with which we are concerned, it can be assumed that the bulk concentration is near enough to zero and then rexcan be equated to the total adsorption, r. The bound fraction discussed above will be given by p = 1 c(l)/I'. Despite the complexity of the adsorption behaviour of polymers, they commonly exhibit a Langmuir isotherm of the high affinity type, with a plateau at about 1 mg mP2.The solution concentration at which this occurs depends on the surface area to be covered but equilibrium solution concentrations of order 0.1- 1 mg/L are typical. The free energy of adsorption depends on the competition between the polymer segments and the solvent for sites on the surface. With solid substrates, the van der Waals forces may be augmented by hydrogen bonding or more specific bonding interactions. It may also be affected by hydrophobic bonding, in which case the polymer adsorption may have more to do with rejection of polymer by the solvent than positive interaction with the substrate. The other notable feature of polymer adsorption is that it may require quite some time to come to equilibrium, or to reach some sort of steady state. This is in contrast to the behaviour of small molecules on non-porous surfaces for which adsorption is near enough to instantaneous (equilibrium in times of order milliseconds). The polymer molecule will diffuse more slowly to the region of the interface and once there it will begin a process of segmental adsorption and desorption. Segment attachment points might even diffuse slowly over the surface, if the attachment is purely physical. Thus it may take some time before it has achieved a true free energy minimum. A further problem is caused by the fact that the polymer is usually polydisperse so that some partitioning of the molecules occurs in the adsorption process. The lower molecular weight members can diffuse to the surface faster but the final equilibrium adsorption usually favours the higher molecular weight species. The establishment of that equilibrium can be very slow indeed. The energetic considerations discussed above can be made a little more quantitative by introducing a dimensionless adsorption energy parameter xs (Silberberg 1968) defined by:

where A UI is the adsorption energy for a solvent molecule and subscript 2 refers to a polymer segment. xs will be positive if there is net segment-surface interaction, since both A U terms are negative. For negative values of xs there will be no adsorption because, apart from the unfavourable energetics, the entropy effect is also unfavourable. When the polymer molecule gets close to the surface, half of the configurational space becomes unavailable to it so its configurational entropy decreases. For long polymer chains, accumulation occurs at an interface only if the energy effect ( f ) exceeds a critical value, of the order of a few tenths of KT. Once this value is exceeded, however, adsorption is of the high affinity type described above, because of the possibility of making many contacts.

REFERENCES

I301

T h e theoretical treatment of the polymer conformation near an adsorbing surface, taking account of the polymer's own volume constraints and the special attraction of the surface, is a formidable problem and even a qualitative description of the current attempts to tackle it would take us too far afield. Lyklema (1995 section 5.4) provides an overview of the important contributions of de Gennes and of Scheutjens and Fleer in this area.

Exercise 6.5.1 Use the equations of Section 1.5 to derive eqn (6.5.1). 6.5.2 Calculate the relative contributions of each of the terms in eqn (6.5.7) for the mixing of poly(methy1 methacrylate) (5.0 g) of molecular weight 500 000 with toluene (200 g) at 20 'C, assuming that the interaction parameter is 0.45 under these conditions. Take the densities of the polymer and solvent as 1.190 and 0.866 g ~ m - respectively, ~, and assume ideal mixing. Specify which terms favour and which disfavour mixing.

References Adamson, A.W. (1990). Physical chemistry of surfaces. Interscience, New York. Adamson, A.W. and Gast, A. (1997). Physical chemistry of surfaces (8th edn). Interscience, New York. Allen, T. (1990). Particle size measurement. 4th edn. [Powder Technology series (ed. J.C. Williams)]. 5th edn. (1997). In two vols. Chapman and Hall, London. Atkins, P.W. (1982). Physical chemistry 2nd edn. Oxford University Press, Oxford. Baty, A.M., Suci, P.A., Tyler, B.J., and Geesey, G.G. (1996).J. Colloid Interface Sci. 177, 307-15. Bevan, M. and Prieve, D.C. (1999). Langmuir 15,7925-36. Brown, C.E. and Everett, D.H. (1975). Adsorption at the solid/liquid interface. In Colloid science (ed. D.H. Everett) Vol. 2, Chapter 2. Chemical Society, London. Brunauer, S., Emmett, P.H., and Teller, E. (1938).J. Amer.Chem. SOL.60,309. Claesson, P.M., Ederth, T., Bergeron, V., and Rutland, M.W. (1995). Surface force measurements. I n Trends in Physical Chemistry 5, 161-94. Claesson, P.M. and Kjellin, U.R.M. (1999). Studies of interactions between interfaces across surfactant solutions employing various surface force techniques. In Modern characterization methods of surfactant systems (ed. B.P. Binks). Chapter 8. Marcel Dekker, New York. Davis, J. and Everett D.H. (1983). Adsorption from solution. In Colloid science (ed. D.H. Everett) Vol. 4, Chapter 3. Chemical Society, London. Davies, J.T.and Rideal, E.K. (1963). Interfacial phenomena, 2nd edn. Academic Press, New York and London. Derham, K.W., Goldsbrough,J., and Gordon, M. (1974). Pure appl. Chem. 38,97. DiNardo, J. (1994). Nanoscale characterization of surfaces and interfaces. VCH, Weinheim. Dollimore, D., Spooner, P., and Turner, A. (1976). Surface Technol. 4[2], 121-60.

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Ducker, W.A, Senden, T.J., and Pashley, R.M. (1991). Nature, 353,239; Langmuir 8, 1831 (1992). Ertl, G. and Kuppers, J. (1985). Low energy electrons and surface chemistry. VCH Publishers, New York. Everett, D.H. (1965). Trans. Faraday. SOL.61, 2478. Everett, D.H. (1973). Adsorption at the solid/liquid interface: non-aqueous systems. In Colloid science (ed. D.H. Everett) Vol. 1, Chapter 2. Chemical Society, London. Everett, D.H. (1981). In Colloid dispersions (ed. J.W.Goodwin) Chapter 4, pp. 71-97. Royal Society of Chemistry, London. Everett, D.H. and Podoll, R.T. (1979). Adsorption at the solid/liquid interface. In Colloid science (ed. D.H.Everett) Vol. 3, Chapter 2. Chemical Society, London. Firth, B.A., Neville, P.C., and Hunter, R.J. (1974). J. Colloid Interface Sci., 49, 214-20. Flory, P.J. (1941). J. Chem. Phys. 9, 660. Flory, P.J. (1970). Disc. Faraday SOL.49, 7. Gomer, R. (1955). Adv. Catalysis 7, 93. Hanson, M.E. and Yeager, E. (1988). Electrochemical surface science. Molecular phenomena at electrode surfaces (ed. Manuel P. Soriaga). ACS symposium series, No. 378 American Chemical Society, Washington, DC. Huggins, M.L. (1941).J Chem. Phys. 9,440. Israelachvili, J.N. and Adams, G.E. (1978). 3’.Chem. SOL.Faraday I 74, 975 Jenkel, R. and Rumbach, B. (1951). 2. Electrochem. 55, 612. Kern, H.E. and Findenegg, G.H. (1980).J.Colloid Interface Sci. 75, 346. Kissa, E. (1999). Dispersions. Characterization, testing and measurement. Surfactant Science Series 84, (ed. A.T. Hubbard) Marcel Dekker, New York. Kitchener, J.A. (1965). J. Photograph. Sci. 13 152-9. Kuhn, W. (1934). Kolloid 2. 68, 2. Lassen B. and Malmsten, M. (1996). J.Colloid Interface Sci. 179, 470-7. Liebert, R.B. and Prieve, D.C. (1995). Biophys. J. 69,66. Lowell, S. (1975). Powder Technol. 2, 291-3. Lyklema, J. (1995). Fundamentals of interface and colloid science. Vol 11: Solid-liquid interfaces. Academic Press, New York and London. Muller, E.W. (1943). Z.Phys3. 120, 266 & 270; 131 (1951), 136. Morrison, S.R. (1990). The chemicalphysicsof surfaces, 2nd edn. Plenum Press, New York & London. Napper, D.H. (1983). Polymeric stabilization of colloidal dispersions, pp. 428. Academic Press, London and New York. Parker, J.L. (1994). Prog. Surface Sci. 47, 205. Patterson, D. (1969). Macromolecules 2, 672. Prieve, D.C. (1999). Adv. Colloid interface Sci. (Feb) 82, 93-125. Rabe, J.P. (1999). Scanning tunnelling microscopy at solid-liquid interfaces. In Modern characterization methods of surfactant systems (ed. B.P. Binks) Chapter 2. Marcel Dekker, New York. Sanchez, I.C. (1992). Physics of polymer surfaces and interfaces, ButterworthHeinemann. Scales, P.J. (1999). Atomic force microscopy. In Modern characterization methods of surfactant systems (ed. B.P. Binks) Chapter 3. Marcel Dekker, New York. Silberberg, A. (1968). J Chem. Phys. 48,2835. Somorjai, G.A. (1990). Introduction to surface chemistry and catalysis 2nd edn. (1994). John Wiley, New York.

REFERENCES

Somorjai, G.A. and Bent, B.E. (1985). Prog. Colloid Polymer Scz. 70, 38. Soriaga M.P., Harrington, D.A., Stickney, J.L., and Wieckowski, A. (1995). In Modern Aspects of Electrochem. [28] (ed. B.E. Conway) Plenum Press, New York. Tabor, D. and Winterton, R.H.S. (1969). Proc. Roy. Soc. (London) A312,435-50. Tejedor-Tejedor, M.I. and Anderson, M.A. (1986). Langmuir 2, 203-10.

I303

Electrified Interfaces: The Electrical Double Layer 7.1 The electrostatic potential of a phase 7.1 .I The outer (Volta) and inner (Galvani) potential of a phase 7.1.2 The potential difference between two phases 7.2 The mercury-solution interface 7.3 Potential distribution at a flat surface -the Gouy-Chapman model 7.3.1 The Debye-Huckel approximation 7.3.2 Solution of the complete Poisson-Boltzmann equation 7.3.3 The diffuse layer charge 7.3.4 The inner (compact) double layer (a) Charge-free inner region (b) Adsorbed charge in the inner region

7.4 Comparison with experiment 7.4.1 Presence or absence of specific adsorption 7.4.2 No specific adsorption 7.4.3 Interpretation of specific adsorption 7.4.4 The discreteness of charge (or adsorbed ion self-atmosphere)effect 7.5 Adsorption of (uncharged) molecules at the mercury-solution interface

7.6 Limitations of the Poisson-Boltzmann equation 7.7 The silver iodide-solution interface

7.7.1 Potential-determiningions 7.7.2 The completely reversible electrode 7.7.3 Determining the point of zero charge 7.7.4 Determination of surface area (a) Positive adsorption (b) Negative adsorption 7.7.5 The capacitance of the Agl interface 7.7.6 The suspension effect 7.8 Other Nernstian surfaces 7.9 Mechanisms of surface charge generation 7.9.1 Dissociation of a single site 7.9.2 Two-site dissociation models

304

THE ELECTROSTATICPOTENTIAL OF A PH A SE

I305

7.10 The double layer on oxide surfaces 7.10.1 Clay mineral systems 7.11 The double layer around a sphere 7.11.1 Charge in t h e diffuse layer 7.1 1.2 Behaviour at high potentials 7.12 The double layer around a cylinder

7.1 The electrostatic potential of a phase T o describe the behaviour of ionic components in the neighbourhood of a charged interface it is first necessary to clarify some basic concepts in the theory of electrostatics. This discussion is based on some fundamental distinctions introduced by Lange and described by Overbeek (1952) and developed in further detail in a review by Parsons (1954). A summary was given by Hunter (1981) from which the following treatment is derived. At the surface of any phase, even a pure metal in vacuo, there is a separation of positive and negative charge components so as to create a region of varying electrical potential which extends over distances of the order of one or more molecular diameters. The electrostatic potential difference generated over those layers is commonly of order 1 volt. When two phases come into contact there is a similar tendency for ions, electrons, and dipolar constituents to arrange themselves in the neighbourhood of the interface in order to minimize their free energy. T h e resulting electric field may also cause polarization effects in neighbouring molecules. In any case, all of these effects result in a difference of electrical potential between the interiors of the two phases. This difference is called the ‘inner’ or Galvani potential difference and it is generally given the symbol A$. Despite its theoretical significance and universal occurrence it is a melancholy fact that A$ cannot be measured unequivocally except when the two phases are identical, when most of the interesting interfacial effects disappear. The potential difference A$ would be expected to measure the total work done in bringing a test charge from the interior of one phase into the interior of the other, passing through the interface on the way. The problem is that the work done would depend on the nature (particularly the size) of the charge used. The theoretical charge is assumed to be sufficiently small (infinitesimal if necessary) so that it does not influence the arrangement of ions and dipoles during its passage. In practice, however, the smallest charge we can actually use is an electron and it is certainly large enough to influence the orientation of charges and dipoles in the interface. There is no problem about measuring the difference in $ between different regions of the same homogeneous phase, since it is assumed that the test charge affects all such regions in a similar fashion. The distinctions introduced by Lange, and elaborated by Guggenheim (1929) and Parsons (1954) allow us to see which potential differences can be measured and which cannot. It must be said, however, that some ingenious approaches have been made to arrive at very plausible estimates of these ‘inaccessible’ quantities.

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7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

7.1.1 The outer (Volta) and t h e inner (Galvani) potential of a phase The electrostatic potential near an isolated (macroscopic) charged object in vacuo is a well-defined quantity which measures the work done in bringing a unit charge from infinity up to the neighbourhood of the surface. It is the potential difference between two points in the vacuum and so can be easily measured. The only restriction is that one must not approach the surface too closely since then the test charge may begin to interact directly with the phase or may affect the arrangement of charges near the surface. If, for example, the macroscopic object is a metallic sphere of radius a then the charge, Q o n it will be confined to its surface and the potential will fall off with distance from the surface in accordance with Coulomb’s Law: Y=

Q 4n€o(a

+

(7.1.1) Y)

where €0 is the permittivity of the vacuum. The plot of Y as a function of r for a = 1 cm shows (Exercise 7.1.1) that, in the near neighbourhood of the surface ( lop5 cm < r < lop2 cm) the potential in essentially constant and equal to Yo = Q/4n€oa. The magnitude of the plateau potential is independent of the test charge (so long as it is small) and depends only on the sphere and its charge. It is called the outer (or Volta) potential of the phase. If the test charge is taken too close to the surface, however, (r < lop6 cm) then the measured potential depends on the details of the interaction of the charge with the surface. In order to understand the nature of the inner (or Galvani) potential, 4 of a phase we will use the procedure suggested by Parsons (1954). We note first that the work done in transferring a charged particle i from infinity into the interior of a phase, a is equal to (See Appendix A5.2 the electrochemical potential of the particle i in phase a,i.e. and eqn (A5.20) for a definition of the electrochemicalpotential.) Ideally we would like to break that up into a ‘chemical’ and an ‘electrical’ part but, as we noted above, that is not possible because some of the ‘chemical’ effects are electrical in nature. There is, however, a useful distinction to be made between the interactions which the charged particle makes with the phase as a whole and the other interactions due to the charge and dipole arrangements at the surface of the phase. We suppose that the phase is a sphere of material and that its charge and surface dipole layers are confined to a thin shell. We can then replace the original sphere by two objects: (i) a thin empty shell of the same size as the original sphere, containing the

Fig. 7.1 .I (a) The original sphere with its surface layer; (b) A sphere of homogeneous composition; (c) Spherical shell with charge and dipole layers.

THE ELECTROSTATICPOTENTIAL OF A PH A SE

I307

same arrangement of charge and dipole material and (ii) a sphere of uniform composition with no charge or dipole layers (Fig. 7.1.1). The total work done in transferring a particle i of charge zie from infinity to the point B, in the interior of the The first sphere, can be broken down into two contributions ( W = W' W" = part ( W ') is the work involved in bringing the charge to the point B' in the interior of the homogeneous sphere (Fig. 7.1.1(b)). This measures the way in which the charge interacts with the bulk of phase a and we will equate that with the chemicalpotential of the particle i in phase a, py. It should be made clear, however, that this will depend on the size and charge of the particle i. If i is a small sphere of radius r, this term will contain a term of the form

+

x).

due to the polarization of the medium by the charged sphere (the Born effect) (Smith 1973). The other term, W" measures an electrical potential in the interior of a and this is = py zie4". It the inner (or Galvani) potential, 4". It is in this sense that we write: is convenient to break 4" down further into a part due to the overall net charge on the phase (Y")and a part called the chi- orjump-potential, xffdue to the arrangement of dipoles and any charge separation which may occur at the surface:

+

4"

= Y"

+ x".

(7.1.2)

Although Y" can in principle be measured, x" can only be estimated on the basis of a model.

7.1.2 The potential difference between two phases When two phases are in contact, the difference in their outer potentials, "ABY, can be measured simply by bringing probes up to the near neighbourhood of the two surfaces (Fig. 7.1.2) and measuring the voltage difference with a device which draws a negligible current. In the case of two metals in contact the result is a measure of the relative affinity of the two metals for electrons (called the contact potential). The surfaces need to be clean but the contact between the metals need not be perfect as long as electrons can travel freely between them. The contact potential difference between two metals measures the difference in their work functions; electrons flow from one to the other to establish a sufficient voltage difference so that the electron potential is the same in both metals (Grahame 1947). The difference in the inner potential, "As$, (the Galvani potential difference) between two phases, on the other hand, can be measured only when the two phases are of identical composition. It is then obtained from the difference in electrochemical potential of the electron (2) in the two phases [compare Appendix AS, eqn (AS.19)]:

-B pi

-

= ( pBi

+ zie4B) - (p~.g+ zje4") = zie(qV - 4")

(7.1.3)

because that is experimentally measurable using a potentiometer (a device for measuring the difference in voltage between two wires made of the same metal). When

308 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

III

V Fig. 7.1.2 Measurement of the contact or Volta potential difference between two metals.

the two phases are not the same we cannot assume that the ‘chemical’ part of is the same in both phases. As Guggenheim (1929) points out, the problem stems from the very notion of an electrostatic potential at a point. It measures the work down in bringing an injinitesimal charge (not an electron) from infinity up to the point in question. Equation (7.1.2) implies that the difference in the inner potential “As4 can be written:

and it is then tempting to think of the first term as representing the effect of charge which has moved entirely from one phase to the other. The second term would then be due solely to the disposition of dipoles at the interface. Unfortunately that is not a distinction which can be made with any certainty. In the case of a macroscopic metallic electrode in contact with an aqueous solution, it is still useful to break the term A 4 into two parts: a contribution due to the separation of the free charges (ions and electrons) (A$) and a contribution (Axdipole)due to the orientation of dipolar molecules at the interface. (Note, however, that in general A$ # A’€’ and A x # Axdipolein eqn (7.1.4).) The potential due to the ion distribution in the aqueous phase can then be assumed to change from zero, in the bulk, to some value $0 on the electrode surface with $0 M A$ and:

‘4

= $0

+ A Xdipole.

(7.1.5)

is often referred to as the surface potential by colloid chemists. The surface potential, as that term is used in surface chemistry, is measured by bringing a probe up near a water surface before and after spreading some surfactant material at the interface. Its relationship to the potentials discussed here is examined by Davies and Rideal(l963, pp. 64-79). Essentially it is a measure of the change in Volta potential caused by a controlled degree of contamination (the surface film). The significance of the Volta potential, AY, is much smaller in the case of a disperse phase since it is no longer possible to apply the argument of eqn (7.1.2). The particles $0

THE MERCURY-SOLUTION INTERFACE

I309

are so small that it makes no sense to speak of a macroscopic charge separation that is measurable from outside the solid phase. In that case it is again preferable to treat the entire double layer region as an arrangement of charges and dipoles that is uncharged overall. It is, however, still profitable to separate out the contribution due to the disposition of free charges from that due to dipole orientation (eqn (7.1.5)).

f

I

Exercise 7.1 . I . Sketch the curve of Y/Yo against loglo Y from eqn (7.1.1) for lo-’ < r(cm) < 100 and a = 1 cm.

7.2 The mercury-solution interface We noted in Section 1.6 that many important properties of colloidal systems are influenced by the electric charges on the particle surface. When immersed in an electrolyte solution a charged colloidal particle will be surrounded by ions of opposite sign so that, from a distance, it appears to be electrically neutral. The surrounding ions are, however, able to move under the influence of thermal diffusion so that the region of charge imbalance, due to the presence of the particle, can be quite significant, relative to the size of the particle itself. Indeed, for very small particles (-50 nm) the disturbance it creates can stretch out to several particle diameters. The arrangement of electric charge on the particle, together with the balancing charge in the solution, is called an electrical double layer, and it has been studied on various surfaces for about 200 years. There are many recent reviews of the structure of the double layer, since it is the basis of the entire field of electrochemistry: the same double layer forms at the surface of an electrode and determines its behaviour in an electrochemical cell, an electrolysis apparatus or an electroanalytical chemical device. We undertake a review here to provide a coherent description on which to base the remainder of our work in colloid chemistry. More detailed treatments are given in the texts by Bockris and Reddy (1970), Delahay (1966), Sparnaay (1972), Hunter (1981), Bockris et al. (1980) and volume I1 of the treatise by Lyklema (1995). The most reliable information on the components of charge at an electrified interface comes from the study of the mercury-aqueous electrolyte solution interface. A small drop of mercury issuing from a glass capillary under the surface of an electrolyte solution is probably as close as we can get to an ideal system for study (Fig. 7.2.1). The mercury is contained in a reservoir, M, and as it drops from the lower end of capillary %2 a new (and clean) surface is continually created. T h e surface is also molecularly smooth, which makes interpretation of results easier. If the solution contains no ions that can readily undergo oxidation or reduction then there is no mechanism for transport of electric charge across the mercury interface. The mercury is then said to be perfectly polarized.? It is then possible to adjust the electrical t The word polarization is used by electrochemists to mean any process that leads to a limitation in current flow.

I

I

310 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

B

Fig. 7.2.1 Schematic arrangement for determining the electrical capacitance across the surface of a mercury drop D (see text). G is an electrode made of, say Pt gauze and R is a H2/Pt reference electrode.

potential difference across the mercury interface by altering the setting on the potentiometer, P. The potential drop between the interior of the reference electrode, R, and the solution is determined by the activity of H+ ions in solution (and the pressure of the Hz gas). Any change dE in the setting of the potentiometer, P is, therefore, immediately transmitted to the surface of the drop. Since no current can flow through the drop interface, the only effect is a gradual build up of charge on the drop, with a counterbalancing charge in the surrounding solution. This double layer of charge (compare Fig. 1.6.1) behaves like an electrical capacitor and the magnitude of the capacitance can be measured with a suitable instrument at terminals AB in Fig. 7.2.1. The system shown in Fig. 7.2.1 could be used to study the behaviour of, say, HC104 or HNO3 solution, since it turns out that the H+ ion is not very easily reduced on the mercurysolution interface (i.e. negligible current flows through the interface so long as the mercury is not made too negative with respect to the reference electrode). Typical

THE MERCURY-SOLUTION INTERFACE

I311

experiments of this simple kind take two forms. Either the bulk activity of HC104 is kept constant and the setting, E, on the potentiometer is varied, or E is kept fixed and the activity (or concentration) of HC104 in the solution is changed. In both cases we want to know how the surface tension of the mercury-solution interface is affected. We will find that such studies provide a wealth of information about adsorption in charged systems and they can, in some important cases, be extended to the solid-solution interfaces of interest in colloid chemistry. The thermodynamics of such a charged interface can be analysed using the form of the Gibbs adsorption equation appropriate to charged species (Exercise 7.2.1):

The electrochemical potential pi, is defined in eqn (A5.19) where it is pointed out (Appendix 5) that in systems involving electric charges, the equilibrium condition between two phases at constant temperature requires equality of the electrochemical potential for any species (including electrons) that has access to both phases. In particular it should be noted that the potentiometer P measures the difference in the electrochemical potential of the electrons in the wires Cu( 1) and Cu(2):

E = -F1[Pe(Cu(l) = --P-'[Pe(M)

- PeCu(2)]

(7.2.2)

- Pe(Pt)]

(7.2.3)

where F is the Faraday of charge. Equation (7.2.3) involves the electron equilibrium with the metal electrodes. We now seek to apply eqn (7.2.1) to the simple system of HC104 in aqueous solution in contact with the mercury surface. The procedure is that of Parsons (1975~). The components in the mercury may be taken to be Hg2+, e-, and Hg. They are not independent, of course, and equilibrium requires that

Likewise in the water: P+

+ P-

= Psalt-

2H+

+ 2e-

2 Hz(g)

-

(7.2.5)

Also at the refereence electrode

and so

From eqn (7.2.1) we have

+

dPHgz+ redpe(M) -dy = r@+

+ r+dP+ + r-dp- + ~

H

~~ POH ~(7.2.7) o

312 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

and now we need to introduce the equilibrium conditions in both phases to eliminate quantities which are not accessible to independent measurement. We note that: dpHg2+= dpHg - 2dpe(M) = -2dpe(M) 1 djZ - -dpHz +-2 and

-

djZe(Pt) = -djZe(Pt)

dp- = dpsdt - dP+.

(7.2.8) (7.2.9) (7.2.10)

(Equations (7.2.8) and (7.2.9) assume that in the experiments referred to above the activity of the mercury and the Hz gas are kept constant. The subscript salt in this case refers to HC104 .) Substituting eqns (7.2.8-10) in eqn (7.2.7) and rearranging terms (Exercise 7.2.2) gives:

The coefficient of djZe(M) measures the excess of electrons in the interface above that required to neutralize the Hg2+ in the interface. It corresponds, therefore, to -ao/F where (TO is the charge per unit area on the mercury surface. Likewise (r+- r - ) F is the balancing charge in the aqueous solution (where F = 96 485 C mol-'). Substituting these quantities in eqn (7.2.11) and using eqn (7.2.3) we have: (7.2.12) + F-dPsaIt + ~ H dPH,o ~ O = oodE+ + r?dpsait where the subscript + on E refers to the fact that E is measured in a system with a -dy = nodE+

reference electrode reversible to the cation. If the reference electrode is made reversible to the anion, a similar argument gives:

(The superscript (T on r refers to the use of the Gibbs convention with r H z O set equal to zero to get relative surface excesses of cations and anions (Section 2.4.1). We will drop it in future to avoid confusion with (TO.) Equation (7.2.12) leads directly to the Lippmann equation: (7.2.14) which was put forward over a century ago and forms the basis of the study of electrocapillarity. The interfacial tension can be measured in a variety of ways (Section 2.11) of which the most common for the dropping mercury electrode (DME) is the drop weight method. The high surface tension of mercury makes the drop almost exactly spherical in shape (Perram et al. 1973a) and by weighing a

THE MERCURY-SOLUTION INTERFACE

I313

known number of drops the (relative) value of y can be estimated. T o obtain exact data, however, a numerical solution of the curvature equations is required and that . the is beyond the scope of the present discussion (see Perram et al. 1 9 7 3 ~ )In capillary electrometer (Fig. 7.2.2), the mercury-electrolyte interface is formed in a tube of varying radius. As y varies with E the hydrostatic pressure required to return the interface to the same position in the capillary is measured. Since that is determined only by the diameter at that point (and y ) one can calculate y (or a relative y ) from the Young-Laplace equation (Section 2.2.3). For details of more recent apparatus see, for example, Mohilner and Kakiuchi (1981), which references earlier work on computer-controlled devices. Note that when there is no charge on the mercury surface (ay/aE), = 0 and at this value of E the interfacial tension is a maximum. Figure 7.2.3 shows some typical plots of y against the applied potential difference. When E is made negative (cathodic polarization) the mercury surface is negative with respect to the solution. The predominant counterions on the solution side are then the cations and Fig. 7.2.3 shows that Na+ and K+ behave essentially identically with respect to y. On the other hand, under anodic polarization, the counterions are anions and it is clear from the figure that even halide ions behave very differently from one another. It is perhaps not surprising that anions show some of their chemical character against a metal surface whereas cations tend to behave more like simple positive charges with no special affinity for the surface. We will discuss this behaviour in more detail below (Section 7.3). The most striking feature of the curves in Fig. 7.2.3 is the maximum in y, called the electrocapillay maximum (e.c.m.). The maximum value identifies the point at which a0 = 0 (from eqn (7.2.14))and so is also referred to as the point of zero charge (P.z.c.). Putting a charge on the mercury surface, whether positive or negative, has the same effect as does a surfactant at other liquid surfaces (Section 2.4.2); it makes it easier to expand the surface because that reduces the lateral repulsion between the charges. The charge at any other value of E can be obtained by differentiation of the y versus E curve. At the dropping mercury electrode, however, it is more usual to measure the electrical capacitance of the drop surface by surrounding it with a

Mercury

Microscope Aqueous solution

Fig. 7.2.2 Principle of the Lippmann capillary electrometer.

314 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

I

-

420

-

400

-

I

I

I

I

I

I

I

I

1

I

l

l

1

I

1

f

I

I

I

-

380 360 -

-

-

$ 340 -

-

320 300

-

-

-

-

280 -

260 -

-

1

1

0.6

1

1

1

1

1

1

1

1

1

1

1

1

1

~

1

I

0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 Relative potentiometer setting ( E (calomel) +0.48)V

0.4

1~

1 I

.

-1.4

Fig. 7.2.3 Electrocapillary curves (after Grahame 1947). The potentiometer reading E, has been adjusted so that E = 0 at the electrocapillary maximum for sodium fluoride (see text). (Copyright American Chemical Society.)

counter-electrode (G in Fig. 7.2.1), usually made of platinum gauze. The total impedance between the mercury and the electrode G is made up of capacitive effects at each electrode plus a resistive component, R, through the solution. Since the electrode capacitances are in series the large area of G makes its contribution negligible and provided the electrolyte concentration is not too low (so that R is not too high), an accurate determination of the capacitance, C, of the dropping mercury electrode can be made. C is measured by imposing a small a.c. signal across the electrode system and the measured value, therefore, corresponds to a dafferential capacitance:

C=

(2)

(7.2.15)

iu.

which, from eqn (7.2.14) is equal to -d2 y/dE2. The importance of the capacitance lies in the fact that, being a differential quantity, it contains a great deal more detailed information than does the surface tension y. T o see this more clearly we note that the curves in Fig. 7.2.3 are very nearly parabolic. If they are represented by an equation of the form

Y = Yem - b ’ ( ~- EemJ2

(7.2.16)

THE MERCURY-SOLUTION INTERFACE

I315

(where the subscript ecm refers to the electrocapillary maximum) we would have (from eqn 7.2.14): -(dy/dE) = 2b'(E

-

EKm)= DO

and C = dao/dE = 2b' (i.e. a constant). The actual experimental values of C are far from constant as is shown in Fig. 7.2.4. This data is for sodium fluoride, which is now recognized to be about the simplest possible electrolyte behaviour to interpret. We will attempt some interpretation in Section 7.3. The data in Fig. 7.2.4 can be used to determine the charge on the electrode at any value of E since (from eqn (7.2.15)):

00 =

f

CdE.

(7.2.17)

E=E,,

A second integration can be used to evaluate y from the Lippmann equation: E

/

y-yem=-/crodE=-/

C(dE)'

(7.2.18)

n

and this value can be compared with the direct measurements as a check on the reliability of the data.

I

0.8 0.4

I

I

I

I

I

0 -0.4 -0.8 -1.2 -1.6 -2.0 E - E c c m (V)

Fig. 7.2.4 Differential capacitance at the DME in contact with NaF solutions at 25" C (After Grahame 1947.) (Copyright American Chemical Society.)

316

I 7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

If measurements are done at a variety of salt concentrations (as shown in Fig. 7.2.4) one can determine the values of r+and r- from eqn (7.2.12) or (7.2.13): (7.2.19) Only one of these is needed since the electroneutrality condition requires that F ( r - - r+)= 00. In this way a huge body of data has been built up on the relation between applied e.m.f. and interfacial adsorption for a large number of systems. The (perfectly polarized) mercury-solution interface has been examined in some detail here because it is important to establish the sound thermodynamic basis of the experimental work in that area: it is the underpinning for the models of the electrical double layer to be discussed below (Section 7.3). In colloid systems the presence of the solid-solution interface introduces all the problems concerning the quantity y s ~ (Section 2.5.4). They can be circumvented almost entirely, however, by appeal to basic as a thermodynamic relationships like eqns (7.2.12-19) with the interpretation of y s ~ surface free energy (Section 2.2.1). The surface charge density, 00, is a relatively easy quantity to measure in a solid-liquid system. This does not mean that our descriptions of the solid-liquid interface rest solely on the certainties of classical thermodynamics. An understanding of the microscopic structure at the interface can only come from the introduction of extra-thermodynamic assumptions. The thermodynamics assists in manipulation of experimental data without introducing new and possibly erroneous assumptions. In that respect, modern electrochemistry may be regarded, in the words of one of its most respected modern exponents as 'the most remarkable of the applications of classical thermodynamics' (Parsons 1980). In the derivation of the important expressions (7.2.12) and (7.2.13) it was tacitly assumed that each of the phases in contact could be characterized by their inner (Galvani) potential, 4 so that the electrochemical potential of any species in that phase was fixed by eqn (A5.19). As we noted in Section 7.1, the difference A 4 in Galvani potentials between two phases can only be directly measured if the phases have the same chemical composition. The system shown in Fig. 7.2.1 can be represented by: CU(2) I H2(g>,Pt, I HC104(ad I Hg I CU(1)

45

44

43

42

41

When a measurement is made of the difference in voltage between the two copper wires (using either some electronic measuring device or a potentiometer) the quantity which is measured is the difference in the Galvani potential in the two wires:

E = 41 - 45

(7.2.20)

and this is possible because the two copper wires have the same chemical composition. The quantity which is being measured is the difference in the electrochemicalpotential of the electron in the two wires:

E = F-'{jZe(Cu(2))

-

jZe(Cu(l))} = FP1{pe(Cu(2)) (7.2.21)

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I317

where pe is the chemical part of the electrochemical potential of the electron in the copper. Since the chemical composition is the same in the two copper wires this term cancels and eqn (7.2.20) is reconciled with eqn (7.2.2). Now writing:

E = (41 - 42) + (42 - 43) + (43 - 44) + (44

- 45)

(7.2.22)

we see that the first and last terms are in the nature of metal-metal contact potentials that are characteristic of the metals and are, therefore, constant. Likewise (43 -44) is determined by the activity of H+ in the solution and if it is kept constant while the potentiometer is adjusted we have:

where A 4 is the potential difference between the interior of the mercury (or its surface) and the interior of the aqueous solution. It is this potential A 4 which we will wish to investigate at the microscopic level in the next section.

Exercises. 7.2.1 The Helmholtz free energy function for an electrified surface phase can be written (compare eqn (2.3.28)):

where 4 C d q = 4 ziF Cdni". Use this, together with the usual f integration procedure (Appendix AS) to obtain eqn (7.2.1). [Note: 4 is the electrostatic potential characterizing the surface and the 4 C d y term represents the work done in placing charged species in the interface.] (Don't confuse F' with the Faraday, F.) 7.2.2 Establish eqn (7.2.11).

7.3 Potential distribution at a flat surface - the GouyChapman model The earliest theoretical studies of the behaviour of an electrified interface were made by Helmholtz well over a century ago. His equations were interpreted by Perrin as implying a simple charge distribution in the solution, consisting of a plane layer of charge opposite to that on the solid. The equations for the electrical potential as a function of distance into the solution can readily be solved for this simple model of the double layer and they were able to explain some features of the behaviour of doublelayer systems. The success of the kinetic theory of molecular behaviour made it clear, however, that the Helmholtz model was unrealistic - especially in the treatment of the electric charge in the solution. Since the metal is an electronic conductor it is reasonable to

318 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

assume that the charge on it is confined to the surface and that that surface can be regarded as a surface of constant potential. In the solution, on the other hand, ions of opposite sign predominate over ions of the same sign but the latter are not completely excluded from the surface region (Fig. 1.6.1). That surface region is also of significant thickness. We will anticipate the final result and state (as we did in Section 1.6) that this region of varying charge density stretches over distances of order 100 nm in dilute electrolyte solution and rather less at higher concentrations. The curvature of the mercury drop is of the order of 100 p m so it can be reasonably approximated as flat so far as the electrical double layer is concerned. We must now determine the electrical charge and potential distribution in this diffuse charge region by solving the relevant electrical and statistical thermodynamic equations. The problem was first tackled by Gouy (in 1910) and, independently, by Chapman (1913) and the result is referred to as the Gouy-Chapman model. Solutions of that model are available in the standard texts (e.g. Adamson 1967; Overbeek 1952) and reviews (Grahame 1947). They depend on the solution of what is called the PoissonBoltzmann equation, one of the most important equations of statisticalphysics. Although some criticism can be levelled at this equation on strictly statistical mechanical grounds it has been shown to be remarkably accurate in its representation of the diffuse double layer so we will save the criticism for later (Section 7.6). Rather than repeat the treatment so readily available elsewhere we will discuss a rather better model than the simple one proposed by Gouy and Chapman. We will recognize, from the outset, that the ions in the solution (whether bare or hydrated) have a finite size and so are not able to get closer than a certain distance from the metal surface. There is, therefore, a charge-free region near the surface, which must be treated differently from the rest of the double layer. The thickness of the charge-free region varies from about one bare ion radius (say 0.1 nm) up to or a little beyond one hydrated ion radius (- 0.5 nm). The transition from $2 to $3 across the interface occurs over a finite distance and it may be assumed that $2 remains constant almost right up to the interface (Fig. 7.3.1).

I

O

d

X

Fig. 7.3.1 A possible potential distribution across the metalkelectrolyteinterface. The region between x = 0 and x = d is assumed to be free of any charges and to have a permittivity~i which differs from the

bulk value E , and may be a function of position. Outside that layer, E takes its bulk values in each phase. (The potential distribution in the inner region (0 < x < d) will be considered later.)

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I319

This is a reasonable assumption for a metal. The potential at the plane x = 0 is q5M (= 42) and for x > 0 it is determined by the Poisson equation (see Appendix A3 for a discussion of the meaning of this equation): div D = div EE = p

(7.3.1)

where D is the dielectric displacement vector (eqn (3.2.6)) and p is the local volume density of charge (i.e. the number of charges per unit volume). Now

E = -grad

4

(7.3.2)

and so from eqns (7.3.1) and (7.3.2): div For the region x > d, where E (=

(E

grad 4) = -p.

(7.3.3)

is constant:

E,)

div grad 4 = V2 4 = - p / ~ , = - ~ / E o E ,

(7.3.4)

where V2 is the Laplace operator (Appendix A3) and E, (= E , / E ~ ) is the (dimensionless) relative permittivity (Section 3.2). In the region x > d, the ions are influenced by the local electrostatic potential. If the metal surface is charged there will be an accumulation of oppositely charged ions given by the Boltzmann equation: nj = nj 0 exp(-zoi/kT)

(7.3.5)

where wi represents the work done in bringing an ion i up from the bulk solution (where 4 = 4 3 ) to a point in the double layer where the potential is, say, 4 and np is the bulk concentration of ions of type i. As a first approximation, we assume that wi = zie(4 - 43) = z;e+

where

+

= 4 - 43.

(7.3.6) (7.3.7)

In other words, the only work done in bringing the ion near the surface is the electrical work done on or by the ion as it moves in response to the field. This ignores the energies involved in moving aside other ions or creating a hole in the solvent, or any effect which the ion might have on the local structure of the solvent or the distribution of other ions. The ion is simply treated as a point charge. The volume density of charge, p, is given by p = xinixie

(7.3.8)

where the summation is over all the species of ion present and the valency, xi, may take positive or negative values. From eqns (7.3.4-8), assuming 43 is constant:

v + = --1E

njO x;e exp ( -zie+ F).

2

EOEr

i

(7.3.9)

320 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE L A Y E R

This equation is the Poisson-Boltzmann equation referred to earlier; it is one of the most important equations we will encounter since it is the basis of our understanding of electrolyte solutions, electrode processes, colloid interaction, membrane transport, nerve conduction, transistor behaviour, and even plasma physics.

7.3.1 The Debye-Huckel approximation If the electrical energy is small compared to thermal energy ( I x,e$ I < k 7 ) it is possible to expand the exponential in eqn (7.3.9) neglecting all but the first two terms, to give: xien!

xte2ny$/kT

-

i

1

.

(7.3.10)

The first summation term must be zero to preserve electroneutrality in the bulk solution, so: (7.3.11) where

K

= [e2Cn!zt/ckTlf.

(7.3.12)

This simplification of assuming $ to be very small is called the Debye-Huckel approximation because it was used by those two workers in their theory of strong electrolytes. The solution of eqn (7.3.1 1) is of the form $ = const. exp(-Kx) (Exercise 7.3.3.). The quantity K (which has the dimensions of (length)-' is called the DebyeHiickelparameter and it plays a prominent part in the theory of the double layer. The extent of the double layer is measured by the size of 1 / ~ the : region of variable potential shown in Fig. 7.3.1 (from x = d out to the bulk solution) is of the order of 3 / ~ to 4 / ~ Note . that, apart from some fundamental constants, K depends only on the temperature and the bulk electrolyte concentration. At 25 "C in water the value of K is given by (Exercise 7.3.1): (7.3.13) = 3.2881/I(nm-l)

(7.3.14)

where F is Faraday's constant and I is the ionic strength (= C ci xi2 where ci is the M 1:l aqueous electrolyte solution 1 / = ~ 9.6 ionic concentration in mol L-'). In nm and for the systems of most interest in colloid science 1 / ranges ~ from a fraction of a nanometre to about 100 nm.

7.3.2 Solution of the complete Poisson-Boltzmann equation Unfortunately, in most situations of interest in colloid science and electrochemistry it is not possible to assume that Ize$l < k T . The range of values of E shown in Figs 7.2.3 and 7.2.4 suggest that ($2 -4s) will be of order one volt so that e$ 1.6 x

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I321

J which is about 40 KT at room temperature. Under these conditions the complete Poisson-Boltzmann eqn (7.3.9) must be solved. Fortunately, for the case of a flat surface that is relatively straightforward. T o simplify the algebra we set zi = z+ = - z- = z so that the analysis is limited to a symmetric z:z electrolyte. It turns out that this is not a very serious restriction because in most situations of interest in colloid science, the behaviour of the system is governed almost entirely by the ions of sign opposite to that of the surface (see, for example, Section 1.6.5). Equation (7.3.9) can then be written (Exercise 7.3.4): d2$ dx2

2n0ze ze$ sinh E kT

-- -

(7.3.15)

using the identity sinh p = (exp p - exp(-p))/2. This can be integrated by multiplying both sides by 2(d$/dx): d$d2$ 2--=dx dx2

4n0ze . ze$d$ sinh--. E kT dx

(7.3.16)

The left-hand side is the derivative (with respect to x) of (d$/dx)2. Integrating with respect to x then gives:

1"

( 3 ) 2 d x = /-sinh-d$. 4n0ze dx dx E

ze$ kT

(7.3.17)

Integrating from some point out in the bulk solution where $ = 0 and d$/dx = 0 (see Fig. 7.3.1) up to some point in the double layer (x>d), we have (Exercise 7.3.4):

(g)2=

4nokT [ c o s h g - 11

or (Exercise 7.3.5) -

d$= dx

']

8nokT

[

-~

E

. ze$ 2 ~ k T ze$ = -sinh sinh2kT ze 2kT

(7.3.18)

(7.3.19)

from eqn (7.3.12). Note that the negative sign is chosen so that d$/dx is always negative for $ > 0 and positive for I) < 0. This ensures that I $1 always decreases going towards the bulk solution and becomes zero far from the surface. Equation (7.3.19) can be integrated from a point in the double layer up to the plane x = d to give (Exercise 7.3.6). tanh (ze$/4kT) = tanh(ze$d/4kT) exp[-lc(x - d)].

(7.3.20)

For very low potentials the substitution tanh p = p can be made and eqn (7.3.20) reduces to

322 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

which is the solution to the linear equation (7.3.11) (see Exercise 7.3.3). A comparison between eqns (7.3.20) and (7.3.21) is shown in Fig. 7.3.2. Note that for X$d = Ze$d/ kT < 2 (i.e. z$rd < 51.4 mV at room temperature) the approximate expression is reasonably accurate. Note also that far out in the double layer, where the potential is low, we can substitute tanh p M p and then: $=

4k T Z exp[-lc(x ze

~

- d)]

(7.3.22)

where

(7.3.23) Since Z approaches unity for high values of z$d one can expect the potential far from the wall to resemble that for a wall of potential $d = 4kT/ze irrespective of the actual potential provided it is sufficiently high (compare eqn (7.3.22) with (7.3.21)). In colloid situations, any measurement of a highly charged system at ordinary temperatures will suggest that $rd M (lOO/z) mV if the measuring method only samples the outer region of the double layer. Likewise, if one wants to predict the behaviour of a highly charged system in a situation in which only the outer part of the diffuse layer is important, the approximation $rd M (lOO/z) mV should be a good one.

Fig. 7.3.2 Electrical potential in the diffuse double layer according to the Gouy-Chapman mpdel. Full curves are from eqn (7.3.20) and broken lines from eqn (7.3.21) for z$,j = 2 and 4. ($ is a dimensionless quantity called the reduced potential.) (After Overbeek 1952.)

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I323

7.3.3 The diffuse layer charge The total charge, per unit area of surface, in the diffuse layer is given by:

1 00

=

p dx

(7.3.24)

d

and substituting for p from eqn (7.3.4) (since d24/dx2 = d2$/dx2):

(7.3.25)

and from eqn (7.3.19):

~ K ~ .T Eze$d 4n0ze . ze$d sinh= -~ sinh 2kT * xe 2kT K

ad = -~

(7.3.27)

Note that the sign of a d is opposite to that of $d (since z > 0). For a symmetric electrolyte in water at 25 "C, eqn (7.3.27) gives (Exercise 7.3.8): ad

= -11.74~sinh(19.46)x$d)

(7.3.28)

for a in pc cmP2 when $d is in volts and c is in mol L-'. For very small potentials, where the linear eqn (7.3.11) can be used, a similar analysis leads to (Exercise 7.3.10): ad

= -6K$d

(7.3.29) (7.3.30)

The quantity Kd is called the integral capacitance of the (diffuse) double layer and eqn (7.3.30) shows that, for low potentials, the diffuse layer behaves like a parallel plate capacitor with a spacing of 1 / between ~ the plates, a charge of + a d and - a d on them, and a potential difference of $d. For low potentials then, the Helmholtz model is quite satisfactory for many purposes. [It may be noted in passing that in this case also the integral capacitance is equal to the differential capacitance (-dad /d$d).] One further quantity of importance is the differential capacitance of the diffuse double layer, Cd, defined by (from eqn (7.3.27)):

(7.3.31)

324 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

I

-

0.1M 0.01M 0.001M 0.0001M -

60

2ol , 10

0 -1.0

-0.5

0 Vdv)

+0.5

+1.0

Fig. 7.3.3 Differential capacity of the diffuse double layer (from eqn (7.3.31)).

in water at 25 "c,for c in mole L-' and $d in volts. Values of c d as a function Of $d at different electrolyte concentrations are given in Fig. 7.3.3. Note the similarity in the shape of C about the point $d = 0 and the experimental curve for c d at E E,, and low electrolyte concentration in Fig. 7.2.4.

7.3.4 The inner (compact) double layer We must now consider the potential distribution in the region 0 < x < d. In the mercury-aqueous solution system we will find that d is of the order of 0.5 nm so this region can accommodate only a few layers of solvent molecules. Nevertheless we assume, at least for the moment, that the concept of a dielectric permittivity remains valid and that Poisson's equation (7.3.3) is satisfied in this region. There are several models of varying complexity that can now be investigated, and their predictions compared with experiment. (a) Charge-free inner region From eqn (7.3.3) we have: div

(-~i

grad$) = 0

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I325

or d -(c;grad$)=O dx

so

d$ ~(x)-==Ql dx

(7.3.32)

where QJ is a constant. T o evaluate QJ we note that, at x = d using eqn (7.3.26):

(7.3.33) (The symbol d- means that d $/dx is evaluated on the left-hand side of the line x = d whilst d+ is evaluated on the right-hand side.) At points where the permittivity changes value there is a discontinuity in the derivative of $. The quantity on the left of eqn (7.3.33) is fixed throughout the region 0 < x < d and so: 0

11.0

1

d+=

11.d

d

(7.3.34)

where E; is an average permittivity over the inner layer region defined by

(7.3.35)

The capacity of this inner layer region is then given by:

(7.3.36) where 00 is the charge on the metal (which balances the charge in the diffuse layer). Note that when ai = 0, both the integral and differential capacitance of the inner region can be treated as parallel plate capacitors in series with the corresponding diffuse layer capacitances to form the total capacitance. (b) Adsorbed charge in the inner region The simplest model to account for the possibility of additional charge adsorbed in the inner region is that proposed initially by Stern (1924) and refined by Grahame (1947). All of the ions are assumed to be confined to a layer at a distance x = b from the metal surface and again they are treated as point charges, in a first approximation (Fig. 7.3.4). Since the regions 0 C x < b and b < x C d are again free of any charge, the same procedure can be used to arrive at expressions for the potential drop across each region: $0 -

h = aob/G

(7.3.37)

326 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

(7.3.38) with average values of permittivity defined in each region. Charge balance also requires that

It is common practice to assume that €1 and €2 are constant so that eqns (7.3.37) and (7.3.38) predict a linear change in potential in each region but the above derivation shows that this is not necessary. If E;(x) is allowed to vary smoothly from x = 0 to x = d, then $ will also vary smoothly, rather than discontinuously. Such models have been examined by Buff and Goel (1969), Levine (1971), and Robinson and Levine (1973). A complete solution of the potential and charge distribution in the double layer would give values for $0, $i, $d, ao,ai, and a d . s o far we have four equations (7.3.27), (7.3.37), (7.3.38), and (7.3.39) - and six unknowns. In the case where there is no charge in the plane x = 6, there are four unknowns (@o, $d, 00 and a d ) and three equations (eqns (7.3.27), (7.3.34 or 36), and (7.3.39) with ai = 0). One further equation comes from the imposed external e.m.f. E. Using eqn (7.1.5) we have

I I

/

IHP

OHP

I I

Lous solution

I I I I I I

b d

Fig. 7.3.4 The Gouy-Chapman-Grahame model of the electrical double layer. The plane where the diffuse layer begins (at x = d) is called the outer Helmholtz plane (OHP) and additional adsorbed charge is assumed to be confined to another plane called the inner Helmholtz plane (IHP). At the mercuryaqueous solution interface the IHP is assumed to be the locus of the centres of adsorbed (dehydrated) anions whilst the OHF' is the plane of closest approach of (hydrated) cations.

POTENTIAL DISTRIBUTIONAT A FLAT SURFACE - T H E

GOUY-CHAPMAN M O D E L

I327

and for the moment we will ignore the last term. Then, using eqn (7.2.23): dE = d(42

-

43) = d$o.

(7.3.41)

The assumption that (Axdipole)is constant would appear at first sight to be a difficult one to justify. We will use eqn (7.3.41) to develop eqn (7.4.9), relating the measured capacitance of the mercury drop to the capacitances Ci and c d of the model. Fortunately, the error introduced by this procedure is small because the contribution of the dipoles to the capacity of the interface is so large that, in a series arrangement, as we assume, the effect on the total capacitance is negligible. (We will return to this point in Section 7.4.) The four equations would appear to be sufficient to completely describe the system if q = 0. But what if a; # O? That means that some ions are sitting very close to the mercury surface. It is usually assumed (Grahame 1947) that the plane x = d is the plane of closest approach of hydrated ions, so any ion in the plane x = b must be dehydrated (at least on the side next to the mercury surface). It can be in that position only if the free energy of adsorption more than compensates for the work done in dehydrating it. T o describe such adsorbed ions requires an isotherm of the form: ai

= xienf = xief( N s ,ai,$i, Oi)

(7.3.42)

where ny is the number of ions adsorbed per unit area, which is expected to be a function of (a) the number of adsorption sites on the surface ( N J ,(b) the activity of ion i in the solution (ai),(c) the local electrostatic potential, and (d) 8i, an extra free energy term that takes into account all other special effects that the ion experiences when it is in the plane x = b. This will involve its special interaction with the metal, which will include purely physical interactions (like the image force)+ but may also involve 'chemical' effects (i.e. effects that are only described in terms of molecular orbital overlap and bond formation). t The image force of a charge or dipole near a conductor is an attractive force generated by the interaction between the charge and its (oppositely charged) image in the conductor.

Exercises 7.3.1.Establish eqns (7.3.13) and (7.3.14) using the fact that F = 96 485 C mol-l, R = 8.31 J K-' mol-',

€0

= 8.85 x

F m-' (i.e. CV-' m-l) and cr = 78.5.

7.3.2 Calculate the value of 1 / in ~ each of the following solutions: (i) lop2 M KC1; (ii) lop4M KCl; (iii) lop6M KCl; (iv) lop3M NaCl + M Na2S04; (v) lop3M K2SO4; (vi) 5 x lop3 M MgS04. 7.3.3 Show that the solution to eqn (7.3.11) for a flat surface can be written:

328 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

7.3.4 Establish eqns (7.3.15) and (7.3.18). 7.3.5 Establish eqn (7.3.19) using the identity coshp = 2 sinh2(p/2) + 1. 7.3.6 Establish eqn (7.3.20) using the fact that sinh p = 2 sinh p/2.cosh p/2; sech p = 1/ coshp and d tanhpldp = sech'p.

7.3.7 Show that for an asymmetric electrolyte:

and so

where (sgn @d)is the algebraic sign of the potential (i.e. sgn @ = @/

I @I ).

7.3.8 Establish the value of the conversion factors in eqns (7.3.28) and (7.3.31). 7.3.9 The diffuse layer charge a d can be broken up into a contribution :a from the cations and ad from the anions. Use the fact that, for a symmetric electrolyte,

to show that :a ={2zen0/~}[exp(- .z@d/2k7) - 11 and a ; = {2zen0/~) [l - exp(ze@d/2k7)] 7.3.10 Establish eqn (7.3.29) using the result obtained in Exercise 7.3.3.

7.4 Comparison with experiment 7.4.1 Presence or absence of specific adsorption The similarity in the electrocapillary curves on the cathodic side (Fig. 7.2.3) for sodium and potassium salts suggests that these two ions behave in the same way at the interface. They are said to be indtfferent ions and they are not specifically adsorbed at the interface. This can be explained by assuming that they remain hydrated as they approach the mercury surface and never get closer than the plane x = d. Their adsorption is described purely by their response to the electric field in the diffuse part of the double layer. By contrast the monovalent anions all behave as individuals. T h e standard (Grahame 1947) model assumes that this is because such ions are able to penetrate into the inner region, presumably because their hydration energy is lower than that of cations, but also, possibly, because anions can interact more effectively with the

COMPARISON WITH EXPER IMEN T

I329

metal. They are then said to be speczjically adsorbed. Not all anions are able to do this, however; the evidence given below suggests that the fluoride ion, at least at modest concentrations, is not able to penetrate into the inner region. The question is: can we identify when specific adsorption is occurring? It would be preferable if we could do this in a completely model-independent way. Although that is not quite possible it is certainly possible to offer an experimental criterion that can distinguish those situations in which there is no need to invoke specific adsorption. If this criterion is obeyed it means that the adsorption behaviour can be quite adequately accounted for by appealing to diffuse double layer theory alone. The test for the absence of specific adsorption relies on an examination of the dependence of the electrocapillary maximum (e.c.m.) or point of zero charge (P.z.c.) on the concentration (or activity) of the background electrolyte solution.The influence of electrolyte on the e.c.m. was studied by Esin and Markov (1939) and the coefficient, B, defined by

(7.4.1)

B= F ( E ) 00

is called the Esin and Markov coefficient. When it is evaluated at the e.c.m. (i.e. where 00 = 0), it is found to have a value of zero in some cases, and for such electrolytes we can assert that there is no need to invoke specific adsorption to account for their behaviour in the neighbourhood of the e.c.m. The behaviour can be quite adequately described in terms of the ion concentrations in the dzfluse part of the double layer. Of course, we are interested in the behaviour for all values of 00 and the complete analysis of the problem is not restricted to the immediate neighbourhood of the e.c.m. Delahay (1966) sets out the development, as it was applied by Parsons (1957), and we will follow that approach here. T he e.m.f., E, appearing in the expression (7.4.1) for /3 is normally measured in a cell with a liquidjunction. (That means that the test solution is separated from the reference electrode solution by a membrane or porous plug which permits ion transport but prevents physical mixing). Th e Galvani potential of the reference electrolyte is then different from that of the test solution and the difference is called the liquid junction potential. Considerable effort goes into minimizing (or at least maintaining constant) this potential difference. The exact thermodynamic analysis of the Esin and Markov effect is best done using a slightly modified coefficient, PO,involving a reference electrode without a liquidjunction (like the one treated in Section 7.2). It is not difficult to show from eqn (7.2.12) that (Exercise 7.4.1):

PO = F($),=

.

-F(>

(7.4.2)

CL

The coefficient thus measures how the surface charge is balanced by an excess of anions or a deficit of cations. In the absence of specific adsorption the balance is

determined solely by the Poisson-Boltzmann equation and it can be shown (Parsons 1957) that, in that case (Exercise 7.4.3):

Here ad is the amount of negative charge per unit area in the diffuse layer and a+ is the mean ionic activity of the electrolyte. B is a constant (=2zen0/~from Exercise 7.3.9). This equation can be applied to data over the entire range of p and E+ values for a particular salt to determine whether its behaviour can be explained without invoking specific adsorption. (See Delahay 1966, p. 55). In particular, it can be seen from eqn (7.4.3) that as the surface is polarized more positively or more negatively, the limiting values of Po are (Exercise 7.4.2):

Parsons (1957) shows that for sodium fluoride on mercury, eqn (7.4.3) is valid for all of the data obtained by Grahame. In colloid systems it is rare to apply such a strong test. More usually we examine the behaviour near the point of zero charge (or the e.c.m.). From eqn (7.4.3), the Po coefficient is then - and it is independent of electrolyte concentration. In simple physical terms this means that the diffuse layer charge is made up of a certain quantity of anions and an equal deficit of cations (or vice versa). This is a direct consequence of the linear form of the Poisson-Boltzmann equation for low potentials. Now suppose that instead of using a reference electrode which is reversible to the cation, as we have done so far, we use a reference electrode of fixed electrolyte activity (like a calomel electrode) with a liquid junction leading to the aqueous solution. The e.m.f. imposed is then say E,, where (Exercise 7.4.7):

i

E, = E+

+ [RTIzF] In a+ + a constant

(7.4.5)

and the constant includes the liquid junction potential difference between the reference electrode and the aqueous solution. Equation (7.4.2) now becomes

That is Thus, at the e.c.m. where Po =

($I,=

& [PO.I

4,the quantity F(aEr/ap),=o

= P' = 0.

COMPARISON WITH EXPER IMEN T

I331

That is to say, the point of zero charge is not affected by the electrolyte concentration i f it is measured with respect t o a reference electrode with a liquid junction. This is the usual test that is used in colloid chemistry. If the point of zero charge can be determined at a number of electrolyte concentrations and it turns out to be independent of the electrolyte concentration, this serves to establish that the electrolyte being used is an indifferent one (i.e. is not specifically adsorbed). How is this definition related to our ideas about the nature of specific adsorption? If an ion is specifically adsorbed it has a special relation with the surface and we expect some adsorption to occur even when the charge on the metal is zero (at the e.c.m. or P.z.c.). Suppose it is the anion that is specifically adsorbed (as is often the case on mercury). Then the presence of anions near the surface tends to drive electrons back into the bulk mercury and in order to restore the condition 00 = 0 it is necessary to give the mercury a more negative polarization. In other words, the position of the e.c.m. shifts to more negative values. As the activity of that ion in the aqueous solution is increased, the tendency to adsorb is increased and the e.c.m. moves further to the cathodic side. If there is no specific adsorption, this effect is absent and so: (7.4.8)

The difference in the surface tension behaviour is illustrated schematically in Fig. 7.4.1. Note that if there is no specific adsorption, ye.c.m. is unaffected by the salt concentration whereas when 0; # 0 the maximum is lowered and shlfted as p increases and so specific adsorption is readily identified.

I I

I I I

I

q=o

I I

re

0 E-Eecm

-ve

+ve

0 E-E,,(at

-ve

10%)

Fig. 7.4.1 Illustrating the effect of specific adsorption of an anion on the electrocapillary curves at different salt concentrations. (a) No specific adsorption. (b) With specific adsorption.

332 I

7: ELECTRIFIED INTERFACES: THE ELECTRICAL DOUBLE LAYER

7.4.2 No specific adsorption T he behaviour of sodium fluoride solutions can be analysed on the assumption that cri = 0 (i.e. neither the sodium nor the fluoride ion is specifically adsorbed at the mercury-solution interface). In that case 00 = -(Td so that (from eqns 7.3.31, 37, and 42):

(7.4.9)

where CT is the total differential capacity of the double layer. The system behaves as a pair of capacitors in series. (This is obvious when 0; = 0 but is not strictly true for o;# 0 (see Exercise 7.4.4).) It is apparent from eqn (7.4.9)that the smaller of the two capacitances on the right is the one that determines the observed value CT. Incorporating into the model an inner double layer region of limited capacitance has the effect of limiting CT so that the very large values calculated for Cd (Fig. 7.3.3) are not observed. In fact, Cd only influences and then only at low electrolyte the observed CT in the neighbourhood of E, concentrations (Fig. 7.4.2). A more detailed examination of the data in Fig. 7.2.4 shows that C; is not constant, as is suggested in Fig. 7.4.2, but depends upon the charge on the metal surface. Grahame (1947) made the important suggestion that Ci could be estimated from the total capacitance at high electrolyte concentration (-1 M) when c d is so large that CT M C;. He then calculated the expected capacitance at lower concentrations, from eqn (7.4.9),assuming only that C; was the same when the charge on the metal was the same. I

I

I I

I

I

I

I

I

I d‘

I

I

I

I I

I

Fig. 7.4.2 Schematic diagram of the effect of adding a capacitances in series: at low concentrations and near the e.c.m the differentialcapacitance determines the behaviour but in other situations Ci is more important.

COMPARISON WITH EXPERIMENT

1333

Figures 7.4.3 and 7.4.4 show that these calculations go a long way towards accounting for the capacitance data for the mercurysodium fluoride solution interface. T he value of C; can be calculated as a function of a0 at various temperatures (Fig. 7.4.5). T o interpret these in terms of eqn (7.3.37) we need some idea of the distance d which we have earlier suggested is of the order of the radius of a hydrated cation. If d = 0.5 nm then a Ci value of 32 pF cmP2 corresponds to ~i =EOE, = 32 x lop6 x lo4 F m-2 x 5 x lo-'' m = 1.6 x lo-'' F m-l so that E , = 18 compared with the normal value for bulk water of about 80. Is it reasonableto expect such a low value for the relative permittivity in this region? The high value of E, in bulk water is due to the ability of the water molecules to orient themselves in an applied field (Section 3.2) and to reorient themselves to follow the field if it is changing. The measurement of Ci is done with an alternating applied field (Section 7.2) but the water molecules near the mercury surface are not able to follow that field as easily as those in bulk water because they are already oriented to a considerable extent by the very high electric field near the surface. The potential drop across the inner layer is, from eqn (7.3.36), equal to ao/Ci; for C; = 32 pF and a0 = 16 pC mP2 this has a value of 0.5 volt. If that potential drop occurs across a distance of 0.5 nm the field strength is lo9 Vm-'. The energy of a dipole in an electric field is given byp.E wherep is the dipole moment which, for water, is 6 x lop3' C m. Its potential energy in the field is, therefore, of order 6 x lop2' J which is about 1.5 kT. Although this effect alone might not be expected to lower E, from 80 to 18 there are other effects (including the local field of the ions in the OHP and the image force in the mercury) which further restrict the orientational motion of the water molecules in response to the field. In the extreme case of a completely oriented layer, the anticipated value of E, is about 6 for water, so a mean value of 18 is not unreasonable. At least in this respect the model is self-consistent. It should also be noted that since q = 0, and 00 = -ad the potential in the double layer can be calculated from eqn (7.3.27). It turns out that, even at extreme polarizations, I $rd I is never more than about 0.2 V, and then only at very low electrolyte concentrations (Exercise 7.4.6). As the electrolyte concentration increases

-

O.01MNaF

-Observed

..... Calculated

- -

14 t

i

8

1

1

1

l

1

1

1

I

1

1

1

1

1

1

1

1

1

1

I

GI

-0.8 -1.2 -1.6 Volt Potential relative to the calomel electrode (with 1M KCl) 0

-0.4

Fig. 7.4.3 Comparison of calculated and experimental differential capacitances at 25 "C in water using Grahame's method. (After Payne 1972, with permission.)

334 I

7: E L E C T R I F I E D I N T E R F A C E S : T H E E L E C T R I C A L D O U B L E L A Y E R

25 23

T

8

21 19

’

v

17 15 13

I

12

8

I

I

0 -4 -8 -12 Surface charge, o0@Ccrn-’)

-16

I

I

I

I

4

I

i 1

Fig. 7.4.4 Differential capacitance as a function of surface charge calculated by Grahame’s method at two different temperatures. (After Payne 1972, with permission.)

I $d I falls to less than 50 mV in 1 M solution. This is why the diffuse layer capacitance is well described by the simple Poisson-Boltzmann equation, at least at modest electrolyte concentrations. Returning now to the neglect of the Axdipoleterm in eqn (7.3.41) we can see how Grahame’s procedure (Figs 7.4.3-5) largely circumvents the problem. Using the more exact expression for dE in eqn (7.4.9) we have: 1 dE -- - d[$O CT

do0

+ Axdipole]

do0 - d(’h - $d) do0

+-d$d do0

(7.4.10)

and the first term can still be identified with l/Ci. Grahame’s procedure amounts to assuming that the charge on the metal is much more important than, say the electrolyte concentration, in determining the detailed structure (including dipole orientation) of the inner layer. It does not assume that Axdipoleis constant under all conditions but only that at any particular value of no, Axdipole is unaffected by the electrolyte concentration. Figure 7.4.5 shows clearly that the inner layer capacitance varies significantly with the charge on the metal and if the variation is attributed largely to dipole orientation (so that the thickness parameters, b and d, are constant) then Axdipolewill also vary significantly with charge. Even at the e.c.m. in the absence of specific adsorption when $0 = 0 (from eqns (7.3.27) and (7.3.36)), the value of ( 4 2 +)ecm = Axdipolemay be quite large (of the

COMPARISON WITH EXPERIMENT I

I

I

I

I

I

I

I

I

1335

I

Fig. 7.4.5 Capacity of the inner region of the double layer on mercury in the presence of NaF

order of tens or hundreds of millivolts). Further discussion of this point can be found in Bockris and Reddy (1970) and Sparnaay (1972). It is of particular importance in the study of the adsorption of uncharged (organic) molecules because they usually act as dipoles, which compete with water molecules for sites at the surface and hence profoundly affect the x (‘chi’) potential. A complete molecular model of the interfacial region would require a description, in molecular terms, of the permittivity E; and distance d as functions of the polarization (00) and temperature. Quite a lot of work has been done in this area. The early models are reviewed by MacDonald and Barlow (1964), by Levine et al. (1967), and by Bockris and Reddy (1970), while Sparnaay (1972, pp. 92-104) gives a very good description of the models current up to that time. More recently, Parsons (19756) and Oldham and Parsons (1977) have developed a four-state model for the water molecules, based on an earlier suggestion of Damaskin and Frumkin (1974). Salem (1976) and Damaskin (1977) also offer simple models of the same system. All are attempting, with varying degrees of success, to describe the dependence of E; on 00 and temperature. A more general description would need to take account of the metal surface (Trasatti 1971; Gardiner 1975). The subtlety of the behaviour even when q = 0 makes it obvious that a complete description of the inner region when q # 0 would be a very difficult task indeed. A good model description should also allow one to transfer the calculation to other solvents (on which there is also a great deal of data) with a realistic adjustment of the distance parameter d and the use of independently determinable properties like dipole moment.

336 I 7: ELECTRIFIED

INTERFACES: THE ELECTRICAL DOUBLE LAYER

In colloid chemical systems it is rare to find solid surfaces that are atomically smooth. There is, therefore, seldom much point in attempting a detailed description of ci and di n the inner region. It often suffices to postulate a value for the capacitance C; and to use that as a fundamental parameter of the system. The main point we will take from this analysis is that a good deal of information about the nature of a charged interface can be obtained from the study of the adsorption of non-specifically adsorbed ions. These are called indafferent electrolyte ions and a systematic study of any colloidal system always begins with a study of its behaviour towards such electrolytes (commonly the alkali nitrates on silver iodide or alkali halides on other systems). Indifferent ions are assumed to interact with the surface only in response to the ‘longrange’ forces that operate beyond the Outer Helmholtz Plane (OHP) (Fig. 7.3.4). Any ion that can penetrate into the inner region becomes subject to other short range and much more highly specific interactions. Because they are more strongly hydrated, cations tend to be indifferent on the mercury surface whereas anions are often specifically adsorbed.

7.4.3 Interpretation of specific adsorption Figure 7.2.3 suggests that the e.c.m., when referred to the same reference electrode, is different for the different halides of sodium. Measurements on the chloride, bromide, and iodide all give curves like those shown in Fig. 7.4.l(b), indicating that for all of these systems, in contrast to the fluoride, the anion is specifically adsorbed. The amount of specific adsorption of the anion can be determined if it can be assumed that the cation is not specifically adsorbed. Measurements of r+( = < a y / a p ) ~ - ) can then be set equal to od+/zF where oi is the contribution of cations to the diffuse layer charge. This allows calculation of ‘$d (from Exercise 7.3.9) and hence od.Then since 00 = -(q od)we can obtain o i (which will be negative in this case). Values of oi for a variety of anions (including NO;, CNS-, and the halides) on the mercury surface are

+

15 h

Tg u

3

10

G I

5

0 0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 Volt E (relativeto normal calomel electrode,i.e. Calomel electrodewith 1M KCl)

Fig. 7.4.6 Amount of specifically adsorbed anions in various electrolytes (0.1 M) in contact with mercury at 25 C. Curves computed by Parsons from Grahame’s data. Vertical lines indicate P.Z.C. (From Mott and Watts-Tobin 1961, with permission.) O

Next Page COMPARISON WITH EXPERIMENT

1337

shown in Fig. 7.4.6. Note that, except for Br- and I-, the amounts of anions adsorbed at the e.c.m. are quite small ( 5 2.5 pC cm-’). A great deal of work has been done on the development of adsorption isotherms to describe curves of this sort using equations of the form of (7.3.42). Since our primary concern is with colloidal systems we will not examine this material in detail because it is adequately treated elsewhere. (See, for example, Delahay 1966.) There are, however, some general ideas that come out of this work and that have a bearing on the treatment of colloidal systems. Most, if not all, treatments assume that the adsorbed ion is, in effect, in a separate phase and that its electrochemical potential is equal to that in the bulk. They differ only in the degree of sophistication that is brought to bear in calculating for the specifically adsorbed ions. The original Stern isotherm can be derived as an extension of the Langmuir isotherm (eqn (6.4.6)) by incorporating a suitable expression for the adsorption equilibrium constant, K (per ion) = exp (-AG :d,/kT). We can then write: a;= x;eN,( =

x;eN,x; exp(- A G:,/k T) 1 xi exp(-AG:&/kT)

+

(7.4.11)

where .$ is the coverage, xi is the mole fraction of i in the bulk solution, and N, is the number of available adsorption sites for the ion, per unit area. The term in the denominator accounts for the effect of ions already present in the layer and is important when the sites are almost fully occupied. This is seldom the case for adsorbed charges (because of lateral repulsion) so Grahame (1947) neglects that term and uses a simpler expression: oi = 2xiern0 exp(-AGi,/kT)

(7.4.12)

where r is the radius of the adsorbed ion. This amounts to estimating N, on the surface as equal to 2rN, where N, is the number of water molecules per unit volume. Both of these isotherms when expressed in terms of the coverage, can be put in the form (Exercise 6.4.4):

In ( + h(() = In a; - AG,O,,/kT

(7.4.13)

where h(6) = -ln(l - () for the Stern (Langmuir) isotherm and a; (the activity of ion i in the bulk) is usually set equal to xi. Improvements in the description can then involve either the function A((), which is an entropic correction for ion size (put equal to zero in eqn (7.4.12)) or in the calculation of Actd,. The simplest analysis would begin with:

(7.4.14) where $i is the electrostatic potential in the Inner Helmholtz Plane (IHP) (called the macropotential) and 8; incorporates all interactions other than the ‘macroscopic’ electrical one. When this is done it is found that 0; depends upon the state of charge or polarization of the surface (but see Section 7.4.4 below). When testing a particular isotherm (i.e. the

Electrokinetics and the Zeta Potential 8.1 Introduction 8.2 Equilibrium double layer theory of electrokinetics

8.2.1 Electro-osmosis 8.2.2Streaming potential 8.2.3 Electrophoresis: the Smoluchowski and Huckel formulae 8.2.4The Henry formula 8.3 Reciprocity relations 8.4 The surface of shear 8.5 Measuring electrokinetic properties

8.5.1 Streaming potential and streaming current measurement 8.5.2 Electrophoresis 8.6 Limitations of the elementary theory 8.7 The standard double layer model

8.7.1 The electrokinetic equations 8.7.2 Boundary conditions 8.8 Double layer dynamics

8.8.1 Development of a double layer on a conductor 8.8.2 Double layer due to ion sources a t a dielectric surface 8.8.3 Application to a colloidal problem 8.8.4Check of the linearization approximation 8.9 Electrokinetic effects in thin double layer systems 8.9.1 Limitations of Smoluchowski's formula 8.9.2 Dukhin's analysis 8.9.3 Solution for an isolated spherical particle 8.9.4 Extension to other electrokinetic calculations 8.10 Numerical solutions of the linearized electrokinetic equations 8.11 Electrokinetics in alternating fields

8.11 .I Low frequency behaviour 8.11.2 High frequency conductance (or dielectric dispersion) 8.11.3 Electroacoustics 8.12 Validity of the electrokinetic equations

373

374 I

8: E L E C T R O K I N E T I C S A N D T H E Z E T A P O T E N T I A L

8.1 Introduction In the previous chapter we saw how some of the main features of the electrical double layer could be examined experimentally. Our models of the interface are attempts to provide a detailed picture of the charge and potential distribution in the neighbourhood of the electrically charged surface. Since it is not possible to probe the electrostatic potential directly, we must subject the proposed models to as rigorous a testing program as is possible. One of the most fruitful methods of obtaining further information on the structure of the electrical double layer is the use of electrokinetic procedures. Electrokinetics refers to all those processes in which the boundary layer between one charged phase and another is forced to undergo some sort of shearing process. The charge attached to one phase (say the solid) will then move in one direction and that associated with the adjoining phase will move (more or less tangentially) in the opposite direction. Th e relative motion can be analysed and from this it is possible to infer something of the way the double layer reacts to the shearing regime. In favourable cases it is then possible to calculate how the moving charge is distributed between the two phases. If our models of double layer structure are correct we should expect the electrokinetic data to be reconcilable with the equilibrium data referred to above. That expectation is met in some important cases but in others there remain significant discrepancies which are the object of current research. The shearing process referred to above occurs even when a particle is undergoing its normal Brownian motion or when a colloidal suspension is made to flow. A complete treatment of those processes for charged particles would therefore require a proper consideration of the electrokinetic effects. We will find, however, that these effects are more clearly defined when the particles are studied in a particular way. When, for example, a suspension of positively charged particles is subjected to an electric field, the particles will move towards the negatively charged cathode whilst the surrounding double layer ions will be drawn towards the anode. That process is called electrophoresis. It was one of the first of the electrokinetic effects to be studied (Reuss 1809) and remains one of the most important manifestations. In other cases, the solid may remain stationary but the charges in the adjoining fluid may move when the electric field is applied. This is what happens when a capillary, or a porous medium, containing an electrolyte solution, is subjected to an electric field; the process is called electro-osmosis. If, instead of applying an electric field, we force the electrolyte solution through a porous medium, (or a capillary) under a hydrostatic pressure then there is generated an electrical potential between the ends of the capillary (or porous medium). This is called the streaming potential. Finally, if a suspension of charged particles is allowed to settle, the resulting particle motion causes the development of a potential difference between the upper and lower parts of the suspension. The process is called the Dorn effect and it gives rise to the sedimentation potential. There are various other effects which come under the general purview of electrokinetics but these are its most important manifestations.

EQUILIBRIUM DOUBLE LAYER THEORY OF ELECTROKINETICS

1375

8.2 Equilibrium double layer theory of electrokinetics In formal mathemical terms the problem can be stated quite simply. When a suspension of charged particles, with their surrounding double layers, is in motion, whether simply as a result of thermal agitation or the imposition of some external force, the local motion in the neighbourhood of the particle surface will be governed by the Stokes equations (4.8.3) with the body force, F,in this case given by -peV$ where $ is the local electrostatic potential (Appendix A3.2): qv2v - vp = peV$

(8.2.1)

v.v=o

(8.2.2)

and where pe is the volume density of charge. [We use the subscript e to avoid confusion with the normal density of the fluid.] The treatment assumes that inertia forces are negligible. In general the ion density pe is unknown, and eqn (8.2.1) must be supplemented by a set of ion-conservation equations, one for each species of ion. In certain circumstances, however, the disturbing influence (e.g. an applied electric field) does not affect the ion density and thus pe can be approximated in eqn (8.2.1) by its equilibrium value, obtained from the solution of the Poisson-Boltzmann equation (Section 7.3). In this initial analysis we will concentrate on problems of this type.

8.2.1 Electro-osmosis This term refers to the motion of liquid induced by an applied electric field. Such a motion occurs when an electric field is applied across a porous plug, but we begin with the much simpler case of the flow induced in a capillary tube by an electric field E parallel to the tube axis. The flow results from the presence of a double layer at the tube wall. For a glass capillary containing a simple aqueous electrolyte solution, the charge on the tube wall arises from the dissociation of surface silanol groups (-SOH) or the preferential adsorption of OH- ions and is almost invariably negative. It will be balanced by an equal and opposite charge in the electrolyte. Application of the electric field causes the ions in the double layer to move towards one electrode or the other. Since the ions are predominantly of one sign their motion gives rise to a body force on the liquid in the double layer, and it is this body force which sets the liquid in motion. In most cases, the tube radius is much larger than the double-layer thickness, and we can analyse the flow in the double layer on the assumption that the surface is locally flat. Since the applied field is parallel to the tube surface, the resulting ion migration will not affect the charge density pe. (As ions move towards the electrode they are replaced by ions of the same sign from further along the tube.) Thus the body force on the liquid may be written as

F = peE = --EoE,V2 $eE

(8.2.3)

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I 8 : ELECTROKINETICS AND THE ZETA POTENTIAL

I

+ve

Fig. 8.2.1 The local Cartesian coordinate system used in the analysis of electro-osmotic flow near a solid boundary with the velocity, P, and potential, $, superimposed.

where we have used Poisson’s equation (eqn (7.3.4)) relating pe to the equilibrium potential @e. Note that it is only the external field, E which drives the flow. In analysing the flow we use the Cartesian coordinates shown in Fig. 8.2.1, where the x1 axis is parallel to the applied field, and the x2 axis is normal to the local tube surface. In this coordinate system the forcebalance equation (8.2.1) becomes d2vl ap q---=-peE dx; axl

(8.2.4)

and -

(8.2.5)

In setting up these equations we have assumed that the fluid flows parallel to the tube. From the continuity equation (8.2.2) it then follows that 711 is independent of XI.It is further assumed that none of the quantities will depend on xg, and for this reason we have only shown the x1 and x2 components of the force balance equation. In the absence of an applied pressure gradient and assuming that gravity is unimportant, p will be independent of x1 and eqn (8.2.4) reduces to:

(8.2.6)

EQUILIBRIUM DOUBLE LAYER THEORY OF ELECTROKINETICS

1377

where we have replaced the electrical force by the formula (8.2.3).The first integration of this equation yields:

A second integration with respect to x2 and using the no-slip boundary condition, gives (Exercise 8.2.1):

where A1 is a constant of integration, and { is the equilibrium potential at the ‘plane of shear’, where the liquid velocity is zero. This represents the effective location of the solid-liquid interface and the fluid velocity increases from that point as we move away from the surface (Fig. 8.2.1.) Since the equilibrium potential may be a rapidly varying function of position near the particle surface, the value that is assigned to 5 will be very sensitive to the position of this plane of shear. The plane will presumably be displaced out from the tube surface by a distance of the order of the thickness of the adsorbed ion layer on the surface, so that 5 M $d, the diffuse double layer potential, but the exact position of the plane is still the subject of some debate (see for example, Hunter 1981). We will take up this matter again in Sections 8.4 and 10.2.4. The quantityA1 in eqn (8.2.7) is related to the velocity gradient beyond the doublelayer (where the gradient in lcr, is negligible). In the absence of an applied pressure gradient A1 must be zero, for the velocity gradients and the associated shear stresses in this region only arise in order to balance the pressure forces on the liquid. Thus from eqn (8.2.7) we see that the fluid velocity rises from zero at the plane of shear to a limiting value v, beyond the double layer, where

From the macroscopic point of view the fluid appears to slip past the surface with this velocity, hence the subscripts (Fig. 8.2.1 and Fig. 8.5.2). Since only a small fraction of the total fluid volume lies in the double layer, the total flow rate is approximately given by v,A, where A is the cross-sectional area of the tube. Thus with the aid of the formula (8.2.8) it is possible to calculate the zeta potential (5) from measurements of the electro-osmotic flow rate. Such measurements provide useful information about the charging process at the glass-solution interface, and they will be discussed further in Chapter 10. Equation (8.2.8) was first obtained by Smoluchowski in 1903. Since that time the analysis has been extended to the case of capillaries in which the radius is not large compared with K - ~ (Hunter 1981, section 3.5) and, more importantly it has also been extended to the case of porous plugs with thin double-layers. In the latter case the local analysis of the velocity field given here is still valid, but the local applied field E is distorted by the presence of the particles. Overbeek (1952) has shown that the velocity is given by eqn (8.2.8) everywhere in the pores beyond the double layer, provided E represents the local electric field.

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

The justification for this surprising observation is quite straightforward. Substituting the form:

v = -kC/rllE

(8.2.9)

in the Stokes equations, and using the fact that, V . E = 0 we see that the continuity equation (8.2.2) is automatically satisfied while the force balance equation (8.2.1) is satisfied beyond the double layer with zero pressure gradient. Thus the formula (8.2.9) is the required solution to Stokes’ equations with zero applied pressure gradient. The macroscopic electro-osmotic velocity in a porous plug is therefore given by

(8.2.10)

where the integral extends over the liquid in the sample volume Vand the contribution from the double layer is assumed to be negligible. The integral in this expression can be evaluated by measuring the macroscopic electric current density, given by

(8.2.1 1)

V v, where K, is the electrolyte conductivity. Combining these equations we get

(8.2.12) an equation which can be used for the calculation of zeta potentials in porous plugs, provided that the contribution to the conductivity from the double layer is negligible.

8.2.2 Streaming potential When the liquid in a capillary tube or porous plug is set in motion by an applied pressure gradient, the double-layer charge moves with the surrounding liquid, giving rise to an electric current. The resulting transfer of charge downstream leads to an electric field in the opposite direction that tends to reduce the current until, after a very short time the current due to the pressure gradient is balanced by the current due to the back electric field. The potential drop associated with this electric field is called the ‘streaming potential’. The calculation of the streaming potential for a cylindrical capillary tube is carried out as follows. The velocity due to the applied pressure difference is given by the Poiseuille formula (eqn (4.7.12)), which in this case takes the form

(8.2.13)

EQUILIBRIUM DOUBLE LAYER THEORY OF ELECTROKINETICS

1379

where Ap is the pressure difference across the tube, L is the tube length, and a is the radius. The electric current due to convection of the charge with the flow is:

I1 =

]

~J~Y~,(Y)v~(Y)~Y.

(8.2.14)

0

This integral is dominated by the contribution from the double layer where pe is nonzero. In this region the formula (8.2.13) for the velocity can be approximated by (Exercise 8.2.2): VI %

[Ap . a/2qL](a

-

Y)

(8.2.15)

and the formula (8.2.14) therefore reduces to

where y = a - Y. Using Poisson's equation (eqn (7.3.4)) to replace pe by -d2$,/dy2 and integrating by parts we find (Exercise 8.2.3):

(8.2.16) This current is balanced by the current due to the induced field E,, a current that is approximately given by

(8.2.17)

I2 = K, na2E,

in the case when the double-layer contribution can be neglected. This assumption is valid if

exp(xe{/2kT) Ka

<<

1

(8.2.18)

for a symmetrical electrolyte. Assuming that this constraint is satisfied, we find that the condition of zero net current yields:

and hence the potential difference across the tube is

(8.2.19)

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

Overbeek (1952, p. 204) has shown that this equation can be extended to porous plugs, provided the constraint (8.2.18) is satisfied, where a is the particle radius. Equation (8.2.19) is most likely to fail when the electrolyte concentration is low (< 5 x lop3 M) since K is then small and the zeta potential is likely to be large. The problem lies mainly in the assumption that the back conduction occurs only through the bulk electrolyte. It can be improved significantly by including a term for the conductivity through the double layer or surface regions. This is accounted for by the surface conduction, Ks which commonly has values of order lop9 !X1. K, is measured per unit length of the perimeter of the tube so the total conduction is given by na2Ke 2na K,. Applying this correction to eqn (8.2.19) gives:

+

(8.2.20) Rutgers (1940) showed that the true zeta potential could be estimated using capillaries of different radii to eliminate Ks from eqn (8.2.20) assuming it to be the same at the same electrolyte concentration. An alternative, and much simpler, procedure is to measure the conductivity in the actual cell (or in the porous medium) rather than relying on the use of the bulk conductivity of the electrolyte (Briggs 1928). One measures the actual resistance and compares this with the of the electrolyte in the capillary or porous medium (Rexp) value expected from measurements at high electrolyte concentration, when the effects of surface conduction should be negligible. Equation (8.2.19) then becomes:

(8.2.21) Very accurate estimates of 5 in vitreous silica capillaries have been obtained by Wood and Robinson (1946) using this relation.

8.2.3 Electrophoresis:the Smoluchowski and Huckel formulae The term electrophoresis refers to the motion of suspended particles in an applied electric field. Experimentally it is found that the particle velocity is proportional to the applied field strength. For spherical particles this relationship takes the form

v = PE E

(8.2.22)

where PE is called the electrophoretic mobility of the particle. In this section we will discuss the link between electrophoretic mobility and (-potential. In all cases it will be assumed that the particle can be treated as being alone in an infinite liquid. The earliest solution to the problem was given by Smoluchowski (1921) for the K a >> 1 (thin double-layer) case. Smoluchowski reasoned that the problem of determining the local flow in the double layer in this case is the same as the electro-osmosis problem described in Section 8.2.1, provided we take a frame of reference which moves with the particle. Although this is a reasonable assumption for particles with a dielectric constant which is much less than that of water, it is hard to justify in the general case when the local electric field can have a component directed into the surface. As it turns out, this objection is unimportant, for the electrophoretic mobility has been shown to be independent of particle dielectric constant (O’Brien and White 1978), and thus results obtained using Smoluchowski’s argument for the zero dielectric constant case can be extended to the

EQUILIBRIUM DOUBLE L A Y E R THEORY OF ELECTROKINETICS I381

general case. Smoluchowski concluded that the relationship between particle velocity and electric field should have the form = WVlE

and hence that the electrophoretic mobility is given by pE

(8.2.23)

= €
The justification for this step, which was given by Overbeek in (1952) (p. 207) follows very similar lines to the analysis of the electro-osmotic flow described in Section 8.2.1. The formula (8.2.23) can be applied to a particle of arbitrary shape provided the particle dimensions are much greater than the double-layer thickness and that the constraint (eqn (8.2.18)) is satisfied (O’Brien 1983). Since the double layer is thin this constraint can only be violated at high 5 potentials (Exercise 8.2.3). Huckel ( 1924) solved the electrophoresis problem for the opposite extreme condition of very thick double layers (KU << 1). In that case, the field lines are almost unaffected by the particle (Fig. 8.2.2) and the electrical force on the particle (Q . E) is balanced by the viscous drag of the fluid (eqn 1.5.19) and so (Exercise 8.2.4):

+

p~ = v / E = Q/bnqa = [ 2 ~ < / 3(1 ~ ] ~ a M) 2 ~ < / 3 ~

(8.2.24)

The discrepancy between these two relations (23 and 24) was resolved by Henry (1931) by taking proper account of the way the particle influences the electric field lines in its vicinity.

8.2.4 The Henry formula T o study the effect of varying double-layer thickness on mobility, Henry (1931) calculated p~ for a spherical particle with arbitrary double-layer thickness on the assumption that the charge density is unaffected by the applied field. This assumption is valid provided the <-potential is sufficiently low. T o determine the electrophoretic mobility of a particle with low <-potential it is therefore necessary to solve the Stokes equations ((8.2.1 and 2)), subject to the constraints that the velocity tends to zero far from the particle, and that the net force on the particle is zero. The techniques for solving problems of this sort will be described below. The resulting formula for the electrophoretic mobility is

Fig. 8.2.2 Effect of a non-conducting particle on the applied field. (a) K a << 1; (b) K a >> 1. The ~ the particle surface. broken line is at a distance of 1 / from

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

wherefi ( K U ) is a monotonically varying function which increases from 1.0 at KU = 0 to 1.50 at KU = 00. Obviously, at the lower limit we regain the Huckel equation (8.2.24) and at the upper limit, the Smoluchowskiequation (8.2.23). At high K a the force on the particle is almost solely due to the electrophoretic retardation: the ions in the double layer drag the fluid with them and the particle moves in the opposite direction. At very low KU the double layer is still subjected to that force but practically none of it is transferred to the very small particle. Instead the particle is restrained by the viscous drag of the fluid. A graph offi(Ku) is given in Fig. 8.2.3. Henry presented his results in the form of two power series: one for low and one for high KU and admitted that there was a hiatus between their regions of applicability. Ohshima (1994) has produced a single equation which gives a very good representation of the functionfi over the whole range of K a . His equation is:

(8.2.26) He has also given a corresponding relation for a cylindrical particle (Ohshima 1996). Figure 8.2.4 gives an idea of the range of validity of Henry's formula. His results are compared with some curves of mobility versus (-potential obtained from a computer

10-2

lo-'

1

10'

102

103

Ka

Fig. 8.2.3 The value of Henry's (1931) functionh ( K U ) for the effect of the particle size and double layer thickness on the electrophoretic mobility. Recalculated function using the formula of Ohshima (1994) with permission.

EQUILIBRIUM DOUBLE L A Y E R THEORY OF ELECTROKINETICS I383

solution of the exact equations for the flow and the ion densities around a sphere (O'Brien and White 1978). From these curves it can be seen that the Henry formula, which represents the tangent at the origin, is valid for a range of {-potentials that increases with KU, from a value of about 50 mV at KU = 1 to the value indicated by the constraint (8.2.18) at large KU values.

Exercises 8.2.1Establish eqn (8.2.7). Calculate the electro-osmotic velocity for a capillary tube using eqn (8.2.8), for a (-potential of 45 mV, and an applied field of 1000 V m-l. The water temperature is 20 "C. [Check the unit system.]

8.2.2Establish the approximate relation (8.2.15) for vl and then establish eqn (8.2.16). 8.2.3For KU = 50, determine the value of < at which [exp(e{/2kg/~u = 1; the Smoluchowski formula is invalid for such

< potentials (see Section 8.2.4).

8.2.4Establish eqn (8.2.24).

0

e[lkT=r

-[=e[lkT 5

10

Fig. 8.2.4 Computed electrophoretic mobilities for a spherical particle in a KCl electrolyte. The ordinate is a non-dimensional mobility, given by [ 3 q e / 2 ~ k T ] p(a) ~ . Small KU. (b) Large Ka. The broken lines are discussed in Section 8.9. [(a) reproduced from O'Brien and White, 1978 with permission and (b) from O'Brien and Hunter 1981.1

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

8.3 Reciprocity relations Comparison of eqns (8.2.12) and (8.2.19) reveals that the phenomena of electroosmosis and streaming potential are connected by the relation

(8.3.1) where A p is the applied pressure difference, VTis the total volume flow per unit time, and i is the current carried. This is known as Saxen's relation and it has been recognized for a long time (Saxen 1892). It is one of a general class of relations that can be derived from Onsager's reciprocity relations, using the thermodynamics of irreversible processes. Similar relations can be established between, for example, electrophoretic mobility and the sedimentation potential. Mazur and Overbeek (195 1) showed that, for porous plugs as well as single capillaries, eqn (8.3.1) can be supplemented by:

(8.3.2)

These equations apply even in systems involving surface conductance or doublelayer overlap when the simple relations like eqns (8.2.12) and (8.2.19) will certainly fail. They demonstrate that agreement between the various experimental procedures (e.g. electro-osmosis and streaming potential) for a given surface system can be expected to occur even if the equations for the zeta potential are quite erroneous. The theory of irreversible processes provides some important unifying equations for electrokinetics but it must be admitted that its application beyond the sort of relation set out above (eqns (8.3.1 and 2)) has not so far proved very profitable.

8.4 The surface of shear In the above analysis we have assumed that both the viscosity and the permittivity of the medium are unaffected by the electric field in the double layer and take their bulk values right up to some imagined surface, close to the real particle surface. The position of that surface o f shear has not been exactly specified yet but in Chapter 10 we will see that, at least for systems involving smooth surfaces and simple ions, the <-potential must be close to, if not coincident with, the diffuse double-layer potential (i.e. { w $rd). There remains, however, a question regarding the basic assumption that the viscosity of the solvent medium (say water) retains its bulk value right up to that plane and then suddenly becomes infinite. Fortunately, it is not too difficult to relax this restriction, at least for the situations where the Smoluchowski equation holds. If we assume that both E and q may be functions of the distance from the particle

THE SURFACE OF SH EA R

I385

surface then the charge density should be represented by the more general expression (compare eqn (7.3.3)): div (E grad $re) = -pe.

(8.4.1)

The corresponding form of eqn (8.2.8) then becomes (Overbeek 1952, p. 199):

(8.4.2) and after integration:

(8.4.3)

The electric field in the double layer is expected to reduce the value of E and increase the value of r] so the overall effect is to limit the mobility. When that mobility is then substituted in the usual Smoluchowskiequation (8.2.23), the magnitude of the (-potential is also limited, no matter how large the difise double layer potential, I $d I may be. Significant effects are expected to appear at field strengths of order lo7 V/m and such values are expected to o m at distances within 1 nm of the surface (Fig. 8.4.1). Davies and %deal (1963) made some estimates of the ratio E / V and concluded that the shear plane would be of order 0.3 nm from the plane of the head groups in their model for a surfactant stabilized oil-in-water emulsion system. A better comparison would, however, be with the diffuse layer potential and this was done analytically by Lyklema and Overbeek (1961). They considered the case where only the viscosity was assumed to vary according to an expression suggested by Andrade and Dodd (1946, 1951):

(8.4.4) wherefis called the viscoelectric coeficient. Values off are not available for water (since it breaks down under high electric fields) sofwas estimated from model considerations and given a value of mP2 V2 which is about 3 to 5 times larger than the measured values for typical organic solvents. The field strengths, E, in the double layer were calculated from eqn (7.3.19) and substituting this in the integral in eqn (8.4.3) gives (Exercise 8.4.1):

(8.4.5)

where A,, = 8000cRTf/~ where c is the molar electrolyte concentration. The integration can be done analytically but Lyklema and Overbeek presented their results as a comparison of $d with the apparent (-potential (as calculated using the Smoluchowski formula (8.2.23). The result is shown in Fig. 8.4.2. The limiting value

386 I

8 : ELECTROKINETICS AND THE ZETA POTENTIAL

x2

x2

-

x.2

-

Fig. 8.4.1 Viscosity (top) and tangential velocity profile (bottom) for a real double layer (left) and a layer containing a discrete slip plane at a distance 6 from the surface. The surface is moving upward in the x1 direction with respect to the stationary liquid. (Redrawn from Lyklema 1995, p. 4.40.)

for the <-potential decreases as the electrolyte concentration increases, since the electric field in the double layer then increases. Lyklema (1995) shows that such limiting behaviour is commonly observed, at least on solid surfaces (Fig. 8.4.3). Note that the comparison is made in terms of the electrokinetic charge and the surface charge (determined by titration) since one cannot unequivocally estimate the surface potential. Although there are other possible explanations for the limiting of <-potential this is the most plausible (Lyklema 1995 p. 4.43). The data on some oil-water systems suggest (Hunter 1966) that the estimates off used by Lyklema and Overbeek may be too high for those systems and when reduced to values more consistent with the zeta data would require both E and q to be varied. These two effects are more than additive and the end result is qualitativelymuch the same as Fig. 8.4.3 but the plateaux occur at higher concentrationson the oil-water interface. The problem is discussed at some length in Hunter 1981 (section 5.3). Israelachvili (1991) presents a variety of evidence for the proposition that the ordering and increased viscosity of water is stronger near a solid surface than near a deformable one and that appears to be borne out by the electrokinetic data. We will therefore take the Lyklema and Overbeek model, with their estimate of the viscoelectric coefficient, as a good basis for interpreting the <-potential for the hydrophilic oxidesolution interface. For some oil-in-water emulsion systems (Hunter

MEASURING ELECTROKINETIC PROPERTIES

I 387

180 160 140 120 100 c = lCFz(A< 1)

_--

80

c = 3.5 x lCFz (A = 1)

60 40

c = lCF1 (A > 1)

20 1

0

1

40

1

1

80

1

1

120

I

l

l

160

I

200

I

I

240

I

mV

“kd

Fig. 8.4.2 Apparent electrokinetic potential as a function of the OHP potential, @d when the slip process is determined by the viscoelectric effect. G is the molar concentration. (From Lyklema 1995, Fig. 4.12 with permission.) [A=A,]

and O’Brien 1997) and some latex systems (Midmore et al. 1996) values of I 5 I in excess of 150 mV at 0.001 M suggest that the viscosity effect is much smaller in those cases. In any case, the region over which the viscosity changes abruptly to high values is usually small compared to the diameter of a water molecule (Exercise 8.4.1) so that the concept of the plane of shear remains a reasonable one.

Exercise 8.4.1 The viscosity of water in the double layer may be affected by the high electric fields there. The magnitude of the effect may be estimated from the relation (8.4.4): q = qo[l +f ( d @ / d ~ ) ~Use ] . the value forfestimated by Lyklema and Overbeek (1O-l’ VP2m2).Devise a spreadsheet program to estimate the values of q for various @d values from 200 mV down to O mV in solutions of concentration lop2, and 10-1 M (1:l electrolyte) near a flat charged electrode. Calculate the corresponding distances from the plane @d and hence plot q/qo as a function of distance for each concentration out to about 5 nm. Estimate the distance between the point where the viscosity is twice its bulk value to that where it is ten times its bulk value, as a function of electrolyte concentration. Comment on the result.

8.5 Measuring electrokinetic properties The various experimental methods for determining such quantities as the electrophoretic and electro-osmotic mobility and the streaming potential and current

388 I

8 : ELECTROKINETICS AND THE ZETA POTENTIAL

I

-3

*ek = *O

N

I

TiO,, 1 6 , KNO,

$ ,.

U

3

3

-2 TiO,, 1 6 , KNO,

-1 10-2

F

FeOOH, l e 3

6

4

2

-2

2-47

FeOOH, l e 3

\

4

4

-8

-10 -12 a o / p Ccm-*

TiO,, 10-3 or 1e2 KNO,

- FeOOH, 1 t 2 Fig. 8.4.3 Observed electrokinetic charge as a function of the surface charge for a number of systems. (From Lyklema 1995 with permission.)

were reviewed by Hunter (1981 Chapter 4) and Lyklema (1995 section 4.5) gives a brief update. Rather than repeat that material we will concentrate attention on the important new developments in the measurement of electrophoretic mobility, streaming potential, and current. The dielectric dispersion and ultrasonic or electroacoustic effects will be discussed after we have dealt with the dynamics of the double layer. Phenomena like sedimentation potential and electro-osmosis have important theoretical implications but they have not so far found much use as methods for the measurement of zeta potential except in a few research situations. The theory of streaming potential and current was developed first for capillaries of regular shape and many measurements have been done on capillary tubes made from suitable materials like glass, silica, and quartz. Measurements between flat plates of materials like mica (Section 1.4.5) are also possible (Lyons e t al. 1981), but the most flexible arrangement is the packed bed which allows almost any particulate material to be studied. The analysis of the results in terms of zeta potential is, however, made very difficult if the pores between the particles are not sufficiently large to permit full development of the double layers. For more or less spherical particles the pore size ( r )

MEASURING ELECTROKINETIC PROPERTIES

I 389

is of order 15% of the particle size (a). Since the double layer extends for about 3 / ~ from the surface (Section 7.3),in order to avoid overlap one needs KT >3 or Ka > -20. This thin double layer region is of special interest and we will examine it in some detail in Section 8.9.

8.5.1 Streaming potential and streaming current measurement The measurement of streaming potential is relatively straightforward. The double layer ions are carried downstream by the flow and their accumulation there generates a field which causes a back conduction. When the forward and back currents are equal, the potential difference across the capillary is the streaming potential which was discussed in Section 8.2.2. The potential must be measured with an electrometer which draws a minimum of current so that the current flow, and hence the field in the cell is not disturbed. Instruments with input impedances of lO"S2 or more (similar to those used for glass electrode p H measurement) are used and the corresponding pressure can be measured with a suitable transducer or in laboratory work with a simple manometer. The main requirement is to use a good electrode system which will respond reversibly to the current flows. Either platinized platinum or silver-silver chloride electrodes are normally used. Any asymmetry in the electrode system can be eliminated either by backing off the potential at zero flow or, more reliably, reversing the flow and averaging the streaming potentials obtained in both directions (Hunter and Alexander 1962). In streaming current measurements, the electrodes need to be able to collect all of the current caused by the flow so the electrodes again are the main limiting factor. Two electrodes, at either end of the capillary, are connected by a low resistance (i.e. much lower than the solution resistance) in order to short-circuit the back current. The current through this resistor is measured (Fig. 8.5.1) as a function of the applied pressure difference (A$). Provided the electrodes are non-polarizable (Section 7.2) they will collect all of the charge which is pushed through the capillary by the flow field. The current through the resistor will be given by eqn (8.2.14)and so the potential can be obtained without any assumptions about the back- conduction path through the capillary. Details of the procedure used in the measurement of these effects are given by Hunter (1981 Chapter 4). The more recent development of oscillating pressure systems (in the streaming current device or SCD) is dealt with below as an aspect of the behaviour at alternating frequencies.

<-

8.5.2 Electrophoresis The standard methods for determining p~ are (i) micro-electrophoresis and (ii) moving boundary electrophoresis. In the micro-electrophoresis procedure, the particles, in very dilute suspension, are placed in a closed capillary tube of circular or rectangular cross-section and viewed under ultra-microscope conditions (Fig. 5.2.l(b)). An electric field is applied at the ends of the tube and the velocity of the particles determined by following their progress against a grid in the ocular lens system of the microscope. The traditional process was tedious, time consuming and not very accurate since only a limited number of particles (of order 20) would normally be timed. The main problem, however, was that the charge on the walls of the capillary

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

103-104 Q

YzZlAmplifier

.u.

Recorder

Fig. 8.5.1 Measurement of streaming current. The electrodes are normally platinized platinum gauze to avoid polarization and impedance to liquid flow. The resistor should have a resistance small compared with that of the cell and the microvolt amplifier must have an impedance approaching 1 Ma.

caused the fluid in the cell to flow by electro-osmosis as soon as the field was applied (Fig. 8.5.2(a)). In a closed cell, that flow produces a build up of pressure which causes the fluid to flow back down the centre of the tube (Fig. 8.5.2(b)). This flow and counter flow causes the particles to move with different velocities at different depths in the tube and it is necessary to concentrate attention on particles at a particular point (called the stationary level), where the fluid velocity is zero, in order to determine the true electrophoretic mobility. Since the apparent velocity of the particles varies significantly across the cell it is necessary to locate the stationary level with some precision. The initial commercial devices concentrated on (i) improving the illumination and the precision of location of the stationary layer and (ii) devising methods to sample the motion of a large number of particles. In the Malvern instrument (Fig. 8.5.3) two coherent beams of red light, derived by splitting the output from a low powered H e N e laser, are made to cross at the stationary level in the capillary cell. The resulting interference process causes bands of high and low intensity illumination and the particles are drawn across this pattern by the field. The scattered light shows a similar fluctuation and the frequency of the fluctuations is related to the speed of the particles. The scattered light is collected by a photomultiplier and analysed by a digital correlator which is able to extract the frequency component due to the particle mobility, and hence to construct a distribution function of particle mobilities. T o determine the sign of the particle charge, one of the mirrors is set into oscillatory motion. When the particles are moving in the opposite direction to the mirror they appear to move faster and this can be correlated with the field direction. In an alternative procedure (Sephy, France) the motion of the particles in the field is subjected to

MEASURING ELECTROKINETIC PROPERTIES

-31~

+iY-

I 391

-31~

-cI

PI

Counter

I

Fig. 8.5.2 Velocity profile in a capillary during (a) electro-osmosis and (b) electro-osmotic counter pressure measurement or closed tube electro-osmosis. The thickness of the layer of varying velocity at the wall has been exaggerated. It is not visible with an ordinary microscopeso the liquid right up to the wall appears to move with velocity v, = v, (the slip velocity). 1 / is~ the double layer thickness.

standard optical image analysis to determine the distribution of particle mobilities and, hence, the distribution of zeta potentials. We should note in passing that the phenomenon of electrophoresis is widely used in biochemical analysis for the separation and (partial) identificationof proteins. Differences in the electrophoretic mobility of proteins, either through a gel or over a wet paper surface are due in part to the <-potential.They are, however, also influenced by, for example, the differential adsorbability of the different proteins onto the paper surface and how this is affected by pH. These methods are therefore not suitable for determining absolute mobilities of protein molecules and we will not discuss them further. The moving boundary method of determining electrophoretic mobility can, in principle, be used to determine the absolute mobility of proteins and was once the only way of doing so. It involved creating a boundary between the suspension and the clear suspension medium, usually in a U-tube. An electric field was then applied and the velocity of the boundary was measured as a function of the applied field strength. It has now been almost entirely superseded by the laser Doppler light scattering procedure. This is an extension of the dynamic light scattering (DLS, QELS, or PCS) procedure discussed in Section 5.7 for determining particle size. If an electric field is applied to the particles in a DLS experiment, the particles will all move in a particular direction in response to the field. This drift velocity is superposed on that of the normal Brownian diffusion so that the spectrum of scattered light is not only broadened by diffusion (Fig. 5.7.4), but shifted in frequency by a certain amount. The autocorrelation function in this case is given by:

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

I

Moving mirror

dtionary level

Fig. 8.5.3 Schematic arrangement of the Malvern Zetasizer IIc system.

where q = p ~E .is the particle velocity in response to the field and Q is the magnitude of the scattering vector [= (4nno/ho) sinf3/2]. [The g function was introduced in Section 5.7 and will be treated more formally in Chapters 13 and 14.1 The intensity spectrum is the same shape as in Fig. 5.7.4 but shifted along the frequency axis by an amount (Ware 1974):

Q . Y E = l Q q cos (8/2) = Q ~ E cos E (8/2)

(8.5.2)

where f3 is the scattering angle (Section 3.3). The correlation function, instead of simply decreasing exponentially to zero with the parameter t (Fig. 5.7.6), oscillates (like the cosine function) with a decreasing amplitude until it reaches zero. The cosine function shows that the shift becomes smaller at larger scattering angles. The resolution of the method, R is defined as the ratio of the Doppler shift (Q . V E ) to the Doppler broadening ( D @ ) and so:

for small f3 values (tan f3 M 0). T o obtain sufficient resolution (R M 20 say) for small particles (high D values) requires rather high field strengths (>lo0 V/cm). Such field strengths can cause polarization and heating problems even at moderate salt concentrations and it is usually necessary to use a pulsed or alternating rather than a steady field. Again we will find it useful to examine the dynamics of the double layer (Section 8.8) before exploring this matter further. Suffice it to say at this stage that small colloidal particles are able to move in phase with an applied field, and with their d.c. mobility values, at frequencies well into the kilohertz range.

LIMITATIONS OF THE ELEMENTARY THEORY

1393

8.6 Limitations of the elementary theory In the elementary theory of electrokinetics (Section 8.2) we discussed only the very simplest situation where the externally applied field does not affect the ion density in the region near the surface. The double layer is then assumed to retain its equilibrium charge distribution even when the field is applied. In the remainder of this chapter we aim to remove that restriction, but before doing so we will outline some of the other extensions of the theory developed in Section 8.2. In electro-osmotic flow of a fluid past a charged surface, the mobile ions in the diffuse double layer respond to the externally applied electric field and this generates a body force on the fluid causing it to move. The fluid velocity rises from its value of zero in the plane of shear, to some limiting value, v,, just outside the double layer (Fig. 8.2.1). If the solid surface is in the form of a rectangular or cylindrical tube of macroscopic dimensions, the double layers are confined to sub-microscopic regions near the walls and the fluid moves through the tube as a plug (Fig. 8.5.2(a)). The mathematical problem is simplified in that case because the potential varies only very near to the wall and then only in one dimension. As noted above, the local ion density in the double layer remains unchanged by the flow field. If the capillary is closed at its ends, or the outflowing liquid is allowed to develop a pressure head, there will be a back flow of liquid which counteracts the electro-osmotic flow (Fig. 8.5.2(b)). Even in this case the charge distribution at the walls remains unaffected and the flow behaviour of the bulk fluid can be treated by the Poiseuille equation (Section 8.2). In some cases an external pressure is applied to counteract the osmosis entirely and its magnitude is used to determine the zeta potential. Similar considerations apply to the phenomena of streaming potential and its analogue the streaming current. The main problems which have arisen in the treatment of electro-osmosis and streaming potential and current concern the application of the theory to situations where the flow is either: (i) through a capillary in which the radius is not large compared with the double layer thickness; or (ii) through a porous plug or membrane system. In (i) it can no longer be assumed that the charge distribution in the double layer is unaffected by the applied field. For very fine capillaries the electrostatic potential, even in the absence of the applied field, may be difficult to calculate because the double layers on opposite walls overlap or cannot develop fully. One then needs the theory of Chapter 12 to estimate the equilibrium situation, to say nothing of the effect of the applied field. In case (ii) it has been customary to replace the porous medium by a bundle of parallel capillaries or a seriedparallel combination with some account taken of tortuosity of the flow lines. These developments have been treated in some detail by Hunter (1981) and will not be examined here. The more recent treatments (e.g. O’Brien and Perrins 1984) suggest that, for packed beds of unconsolidated granular material, it may be preferable to use a cell model to account for the transport properties. We will refer to that procedure further in Section 8.11. Whereas one can go quite some distance in discussing electro-osmosis, and streaming potential and current using the simple Smoluchowski theory of Section 8.2, the treatment of the electrophoresis problem almost immediately brings into focus the

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8: E L E C T R O K I N E T I C S A N D T H E Z E T A P O T E N T I A L

complexities of the more general electrokinetic processes. Only in the case of very large particles with very thin double layers (KU > 250) can one confidently apply the Smoluchowski equation (8.2.23)and then only for I { I 5 100 mV. For lower values of KU and modest I {I potentials (-75 mV) the error in that procedure can be 30% or more. The first effect to take account of is the size of the particle, since the field lines must bend around it. That effect was discussed in Section 8.2.3. Provided that the 5potential is not too high it is still possible to assume that the charge density in the double layer is unaffected by the applied field and by the resulting fluid flows around the particle. The electrophoretic mobility for a given zeta potential then becomes a complicated function of the particle size (eqn (8.2.25 and 26)). This behaviour is described by the limiting slope of the curves in Fig. 8.2.4 near the origin. T o lift the restriction on the zeta potential and permit the examination of the general electrokinetic problem, even for the simplest geometry (an isolated sphere), requires us to examine the physical processes involved in some depth, because the imposition of the field will then alter the charge distribution in the double layer. The principles thus established will provide the groundwork for a better study of the traditional areas of electro-osmosis and streaming potential, as well as for electrophoresis and the sedimentation potential. They will also provide insight into the effect of electric charge on the viscosity of suspensions (the primary and secondary electroviscous effects) (Hunter 1981, Chapter 5). In addition, they provide an understanding of the electrical conduction and dielectric impedance behaviour of colloidal suspensions and the complex interactions which occur when an ultrasonic pressure wave passes through a colloidal suspension (O’Brien 1988). The calculation of these effects involves the solution of a set of partial differential equations, known as the ‘electrokinetic equations’. The unknowns in these equations are the ion number densities, the electrical potential, and the pressure and velocity fields in the electrolyte surrounding the individual particles. These equations involve a number of local transport properties, such as the ionic mobilities and fluid viscosity in the double layer, properties which cannot be directly measured at present. Hence the theoreticians have had to adopt a number of ad hoc assumptions in setting up the electrokinetic equations. In the following section, we will outline these assumptions, and describe a procedure for testing their validity. The test involves the calculation, and measurement of a number of electrokinetic effects on a well characterized suspension. The methods for carrying out such calculations are outlined in Sections 8.9-8.10. In Section 8.7 we set out the arguments used in the derivation of the electrokinetic equations. The calculation of any electrokinetic effect involves the solution of these equations subject to certain boundary conditions (Section 8.7.2), the form of which depends to some extent on the transport property of interest. In order to understand the physical processes responsible for the various electrokinetic effects, we need to have some grasp of the time scales on which processes occur in double layer systems. There has been a considerable volume of work in this area (relating particularly to the electrical impedance of colloidal dispersions as a function of frequency) but the insights gained from those studies have not previously been applied in a coherent fashion to the electrokinetic effects. Here (Section 8.8) we present two solutions of the (time-dependent) electrokinetic equations for a flat double

THE STANDARD DOUBLE LAYER MODEL

1395

layer and then show how those results shed light on the electrokinetic problems. In Section 8.9 we describe the theoretical techniques that have recently been developed for the solution of the electrokinetic equations in thin double layer systems, and we describe applications of these techniques to transport property calculations. If the double layer is not thin in comparison with the particle radius, it is usually necessary to resort to computer solutions of the electrokinetic equations in order to calculate transport properties. In Section 8.10 we outline the computer solutions that have so far been obtained. In Section 8.11 we look at some low and high frequency electrokinetic processes. Finally, in Section 8.12, results are described of the few tests which have so far been carried out to determine the validity of the electrokinetic equations. Some of the material in the remainder of this chapter calls for more than the normal level of mathematical analysis. My attempts to make it accessible will not always succeed so it should be emphasized that it is possible to gain a quite effective conception of the application and utility of the zeta potential without having a full understanding of the mathematical theory behind its calculation. For many applications, all one needs is the basic concept of the slipping plane and the electrostatic potential characterizing that plane. That is so if one is concerned only with the use of zeta in exploring equilibrium adsorption behaviour at charged interfaces. Even in many kinetic processes, like the collisions occurring during coagulation, the ‘equilibrium’ zeta potential seems to govern the behaviour. The moral is: ‘don’t give up too easily, but don’t despair if it all looks like more than you wanted to know. Zeta can still prove useful’.

8.7 The standard double layer model In his pioneering study of the electrophoresis of a spherical particle, Overbeek (1943) made a number of simplifying assumptions which have formed the basis for nearly every subsequent electrokinetic study. In addition to the usual assumptions made in the derivation of the Poisson-Boltzmann equation (see for example Section 7.6), he assumed that the fluid viscosity and the ionic mobilities in the diffuse double layer were the same as in the bulk electrolyte. Recent measurements of the electrical repulsion between charged mica surfaces (see Chapter 12) have provided support for the Poisson-Boltzmann assumptions, and the assumption concerning fluid viscosity in the diffuse part of the double layer has also been tested for mica surfaces approaching in water (Chan and Horn 1985; Israelachvili 1986). In the standard electrokinetic model of the double layer it is further assumed that the hydrodynamic particle surface, or ‘shear plane’ as it is called, is homogeneous, smooth, and impervious. Any adsorbed ions or water molecules lying within (i.e. behind) the shear plane are taken to be fixed to the particle. The Russian school of Dukhin and his coworkers have relaxed that restriction and considered the consequences of Stern layer conduction, even though there is usually assumed to be no liquid flow in that layer. Lyklema (1995, Chapter 4) gives a brief history of the notion of ‘surface conduction’ and argues that Stern layer conduction should be regarded as essentially universal. Evidence is now accumulating that the Stern layer

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

conduction process is certainly fairly common, and the mobility of ions in that layer is often close to that of the diffuse layer ions (Lyklema and Minor 1998). Nevertheless, the fact remains that in the classical analyses of the electrophoresis problem by Overbeek (1943), Booth (1950), Wiersema et al. (1966), and O’Brien and White (1978) the assumption was made that surface conduction was confined to the ions above the shear plane. The initial theoretical analysis of surface conduction itself, which was elaborated by Bikerman (1935, 1940), was also confined to conduction in the diffuse layer. We will,therefore, continue to regard the conduction process in the Stern layer as anomalous surface conduction and look at its effects after we have examined the primary problem. Lyklema refers to this classical treatment as referring only to rigid particles but that would seem to imply that Stern layer conductance is in some way connected with Stern layer fluidity or distortability which is certainly not the intention. Although it is not possible to directly test all the features of this classical model, there is an obvious indirect test: by comparing an electrokinetic measurement on a well-characterized suspension with the corresponding theoretical value obtained with the standard model it is possible to determine the equilibrium particle charge or {potential on the shear plane. If the standard model is correct, such comparisons should yield the same potential for every type of electrokinetic measurement carried out on that suspension. It will be clear from the results of Section 8.3 that a true test of consistency must involve correlations other than those which can be predicted from the Onsager reciprocity relations, since such correlations (e.g. between streaming potential and electro-osmosis (eqn (8.3.2)) remain true even when the simple model breaks down. One valuable insight provided by the argument over surface conduction is the realization that when anomalous surface conduction is occurring it leads to an overestimate of 5 from conduction and dielectric dispersion type meansurements and an underestimate from electrophoresis measurements. If, as seems to be the case, that discrepancy can be removed by the introduction of Stern layer conduction, then that would seem to be a vindication of the approach. In any case, in order to apply a consistency test it is necessary to have theories for a number of different electrokinetic effects. In the following two sections we will set out the mathematical problem that must be solved in the development of such theories.

8.7.1 The electrokinetic equations In the absence of any chemical reactions in the electrolyte, each ionic species satisfies an ion conservation equation of the form

an,lat = -v . f j

(8.7.1)

where nj is the number of ions of type j per unit volume, and& is the flux density (i.e. the number of ions per unit area of type j passing through the surface of a volume element). The term on the left represents the rate of change of the ion number in a unit volume while the term on the right is the nett rate at which ions enter that volume (see Fig. 1.5.3 and Appendix A3). The local flux density& is related to the local fluid velocity and gradients in ion density and electrical potential. It is in the formulation of this relationship that

THE STANDARD DOUBLE LAYER MODEL

1397

assumptions must be made about the local transport properties of the double layer. In the standard double layer model, the flux density is assumed to be given by

fI.= - Dj[Vnj (diffusion)

+

(.~jenj/kT)V+l (conduction)

+

njv (convection)

(8.7.2)

+

where D, and zje are the diffusivity and charge of the ion, is the electrical potential, and v the fluid velocity. The terms on the right of this expression represent the contributions from Brownian motion, the local electric field, and convection with the surrounding fluid respectively. The combination of eqns (8.7.1) and (8.7.2) yields a set of differential equations, one for each species of ion, in which the unknowns are nj, and v. The quantities nj and are also related by Poisson’s equation (Section 7.3):

+,

+

..N

v2+= - C zieni/r

(8.7.3)

i= 1

where the permittivity of the electrolyte E is the same everywhere. The sum in this expression represents the local free charge density, and the summation is carried out over the N ionic species in the electrolyte. In most applications the electrolyte can be treated as incompressible, in which case the mass conservation equation takes the form:

[8.2.2]

v . v = 0.

The set of equations is completed by Newton’s Second Law, applied to unit volume of fluid, viz (compare with eqn (4.6.5)): N

pfavlat = qv2v - VP -

C njzjev+.

(8.7.4)

j= 1

Here p is the pressure, and pf and q the fluid density and viscosity respectively. In accordance with the standard double layer model (Section 8.6) q is assumed to be uniform across the double layer. The terms on the right hand side of the equation represent the nett frictional force, the pressure force, and the electrical force per unit volume respectively. The inertia term on the left hand side has been linearized on the grounds that the Reynolds number of the flow (eqn (4.8.1)) is small for colloidal systems. At equilibrium the fluid velocity and ionic fluxes are identically zero. The combination of eqns (8.7.2) and (8.7.3) under these conditions yields the familiar Poisson-Boltzmann equation (7.3.9) while eqn (8.7.4) gives the osmotic pressure distribution in the electrolyte (Exercise 8.7.2). The solution of the electrokinetic equations in a non-equilibrium situation is a formidable problem, thanks to the presence of the non-lineart terms nj v and njV+. T o + A non-linear term involves the product of the unknown quantities.

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overcome this problem we consider systems which are only slightly disturbed from the equilibrium state; these terms can then be approximated by linear expressions. This approximation, which greatly simplifies the calculation is common to nearly every theoretical study in this area. We use the symbol to denote an equilibrium quantity and a prefix 6 to denote the small disturbance. Hence the ion density is written as nj = nj 6nj where 6njln; is assumed to be much less than unity. On substituting this expression and a similar form for in eqns (8.7.1), and (8.7.2) and combining, we get (Exercise 8.7.1)

+

+

In formulating this equation we have neglected the products of small disturbances, such as SnjVS+ and Gnjv, and we have used eqn (4.6.6) together with the fact that the are solutions to eqns (8.7.1) and (8.7.2). equilibrium quantities nj and With a similar approximation, eqn (8.7.4) becomes (Exercise 8.7.1):

+’

(8.7.6)

Since this set of equations is linear, a sum of solutions is also a solution. This property greatly simplifies the calculation of electrokinetic effects and we will concentrate on such linearized equations in this chapter.

8.7.2 Boundary conditions In the standard model it is assumed that the particle and any adsorbed ions and bound water molecules move as a single rigid body. The boundary between this body and the electrolyte is assumed to be smooth and impervious. It is further assumed that the fluid adjacent to the particle surface moves with that surface. In a frame of reference moving with the particle this ‘no-slip’ boundary condition reduces to

(8.7.7)

v=o

on the particle surface. Since the surface is impervious to ions the component of the ion flux normal to the surface is also zero, that is

(8.7.8)

fj.i=o

where & is the unit normal directed outwards from the surface. The final boundary condition at the particle surface arises from electrostatics, viz (compare with eqn (7.3.33)):

€(Z),€.(Z)

= 6a

P

(8.7.9)

THE STANDARD DOUBLE LAYER MODEL

1399

where a/& is the direction normal to the surface and 6a is the change in surface charge density due to the external disturbance. The subscripts p and f denote the values on the particle side and the fluid side of the interface respectively. [Note that we will also be using the symbol n for number of ions per unit volume but it will always be subscripted or superscripted.] The remaining boundary conditions vary from one problem to the next. In the electrophoresis problem, for example, we can say that with increasing distance from the particle: 6nj + 0 f o r j = 1,2, ...N

and

VSy9 + -E

(8.7.10)

where E is the applied electric field. In addition, the fluid motion caused by the particle movement approaches zero at large distances. Thus in a frame of reference moving with the particle, the liquid velocity far from the particle?, ~(oo),is opposite to that of the particle:

where p is the electrophoretic mobility; this extra unknown is determined by the constraint that the nett force on the particle is zero. In the case of a dilute suspension of sedimenting particles, the outer boundary conditions are 6nj + 0 and v

(00)

+ -U

where U is the sedimentation velocity, while Vy9 tends to the uniform value required to balance the current flow caused by the sedimentation process (Saville 1982). It is this field which gives rise to the ‘sedimentation potential’, the voltage difference between the top and bottom of a sedimenting charged suspension. For concentrated suspensions and porous media it is not possible to treat each particle as being alone in an infinite liquid. In that case the above outer boundary conditions are replaced by the requirements that the volume averages of VGnj,V6$, and v(00) or Vp take the prescribed macroscopic values. For example, in the case of a porous medium subjected to a steady macroscopic field E, and a zero applied pressure gradient, the appropriate boundary conditions are (VSnj)= 0,

( V p )= 0,

and

(VSy9)= -E,

where ( ) denotes a volume average over a representative portion of the plug away from the boundaries (where some electrolyte inhomogeneities may have built up). +Weassume, to simplify the exposition, that the particle moves in the direction of the applied field without rotating.

400 I

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Exercises

+

+

8.7.1 Establish eqns (8.7.5) and (8.7.6). [Note that V(# @) = V# V@ and v . (#v) = v -V# #V. v.] 8.7.2 Show that when a flat plate is immersed in an electrolyte, the equilibrium ion distribution (from eqn (8.7.2)) is a Boltzmann distribution.

+

8.8 Double layer dynamics In this section we will study two solutions of the electrokinetic equations for timedependent flat double layers. It might, at first sight, seem unnecessary to consider time-dependent solutions of the equations when most electrokinetic processes are studied under steady state conditions. It is, however, only by examining the timedependent behaviour that we gain a clear insight into the characteristic time scales involved in electrokinetic problems. Knowing this we can confidently specify the conditions under which the a/& terms in the electrokinetic equations can be ignored. The solutions will also provide us with estimates of the typical magnitudes of the disturbances in ion density and electrical potential, estimates which can be used to check the validity of the linearization procedure in the previous section. More importantly, these solutions provide some insight into the various physical processes that give rise to electrokineticeffects. They will also provide the basis for the application of the theory to situations involving rapidly alternating fields. This is becoming an increasingly important area of activity, with the expansion of dielectric dispersion studies (O’Brien 19863) and the advent of ultrasonic analysis (O’Brien 1988).

8.8.1 Development of a double layer on a conductor The first problem is concerned with the development of a double layer at the interface between an infinite metal plate and an electrolyte. Initially, the plate is uncharged and the system is at rest. At time t = 0, a switch between the plate and a d.c. potential source is closed and a charge (0per unit area) flows onto the plate. The aim is to determine how the double layer develops with time. From the symmetry of the problem and the linearity of the electrokinetic equations it follows that the only allowable fluid motion must be normal to the plate. Such a motion would, however, violate the incompressibility constraint (4.6.6). The fluid therefore remains at rest. Since the quantities V@’ and @n! are zero in this case, the ion conservation eqn (8.7.5) reduces to (Exercise 8.8.1): (8.8.1)

Eliminating &r!, with the aid of Poisson’s equation (8.7.3), we get

(8.8.2)

DOUBLE LAYER D Y N A M I C S

I401

In order to remove unnecessary complications we will make two assumptions at this point: firstly, that the time required for the charge to develop on the plate is much smaller than the time required for the double layer to develop; in effect the charge density jumps instantaneously to the uniform value (T and thereafter remains constant. Secondly, it is assumed that the electrolyte is symmetric, containing two ionic species, both with diffusivity D The coupling between the eqns (8.8.2) for each species can be removed in this case by the introduction of the variables 1 h = -(an1 2

+ 6nz)

1 and g = -(an1 2 - 6nz).

(8.8.3)

Note that the charge density is 2gze where z is the valency of ion type 1. The function g thus monitors the local changes in charge density whilst h monitors the local changes in salt concentration. In terms of h and g the eqns (8.8.2) for both ion species become (Exercise 8.8.2):

ahlat = Dv2h

(8.8.4)

aglat = v(v2g - K’g)

(8.8.5)

and

where K’

= 2 n0z2e2/EkT

a0 being the equilibrium density of either ion species. [In this case a0 = noo, the ion concentration in the bulk electrolyte, so K has its usual significance (Section 7.3. l).] By symmetry, h and g must be the same for all points at a given distance from the plate, so h and g depend only on t and x and they must both decay to zero far from the plate. The remaining boundary conditions are

(8.8.6) The solution for h is simply h = 0. That is to say, the charging of the plate produces symmetric and opposite changes in the concentrations of the two ions at each point in the solution near the plate. This symmetry was noted earlier (Section 7.4.1). It is a consequence of the linearization of the Poisson-Boltzmann equation for small perturbations. This amounts to the introduction of the Debye-Huckel approximation. The solution to the problem for g is found, using Laplace transform techniques (see, Stephenson 1977 Chap 7, 8), to be [this is a correction to the earlier text provided by Dr M. Minor]: K(T

g(x, t) = -{exp(-Kx)erfc[X - T] - exp(lcx)erfc[X 4ze

+ q)

(8.8.7)

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8 : ELECTROKINETICS AND THE ZETA POTENTIAL

where the error function, erfc, is defined by [compare with eqn (5.4.3)]:

and X = x/[24(Dt)];T = J ( d D t ) . The error function complement falls monotonically from unity at y = 0 to 0 as y approaches infinity. From eqn (8.8.7) it can be shown that, as t + 00, (Exercise 8.8.3): g(x, t ) M ( ~ o / 2 z e exp ) (-Kx).

The time required for the double layer to reach this equilibrium form can be seen from Fig. 8.8.1, where g(x, t) is plotted as a function of KX for three different values of K2Dt. The equilibrium form is indicated by the broken line. Clearly the double layer is approximately in equilibrium if t is larger than a few multiples of ~ / K ~This D . quantity therefore provides a measure of the double layer relaxation time and for lop3 M KC1 the time is approximately lo-’ seconds. In most electrokinetic studies the time scale of the macroscopic disturbance is much greater than 1//c2D,and hence the double layers are at each instant in local equilibrium with the surrounding electrolyte. This observation forms the basis of much of the recent work on thin double layer electrokinetics (Section 8.9). We will find (Section 8.8.3) that when a double layer system (such as a colloidal suspension) is subjected to a rapidly varying electric field

1

0.9

+k2Dt=O.l

0.8

--t k2Dt=0.5

0.7

.-

d

\

0.6 0.5

a J fi 0.4 cu 0.3 0.2 0.1 0

0

0.5

1

1.5

2

kx Fig. 8.8.1 The form of the charge density as a function of distance from a flat conducting surface at a number of instants after the appearance of a uniform charge on the plate. [Figure kindly provided by Dr M. Minor (2003 priv. comm.).] The abscissa is KX and the ordinate is 2zeg/~awhilst the parameter is T2 = 2 D t .

Association Colloids 9.1 The critical micellization concentration (c.m.c) 9.2 Factors affecting the c.m.c. 9.2.1 Effect of head group and chain length 9.2.2 Effect of counterion 9.2.3 Effect of temperature and pressure 9.2.4 Effect of added salt 9.2.5 Effect of organic molecules

9.3 Equilibrium constant treatment of micelle formation 9.3.1 The closed association model 9.3.2 Multiple equilibrium models (a) Dimerization (b) K,, values of similar magnitude (c) Strong dependence of K,, on tz 9.4 Thermodynamics of micelle formation 9.4.1 Estimation of 9.4.2 Estimation of z ( h g ) 9.4.3 Choice of standard state and concentration units 9.4.4 Enthalpy and entropy of micelle formation

ae

9.5 Spectroscopic techniques for investigating micelle structure 9.5.1 Methods for determining the c.m.c. (a) Solubilization of additives (b) Spectral change of additives 9.5.2 Fluorescence methods for determining aggregation numbers (a) Steady-state emission quenching method (b) Time-resolvedemission quenching method 9.5.3 Interfacial electrostatic potential of micelles using solubilized pH indicators 9.5.4 Polarity of the micelle-water interface

9.6 Micellar dynamics 9.6.1 Kinetics of micelle formation 9.6.2 Residence times of probe molecules in micelles

434

THE CRITICAL MlCELLlZATlON CONCENTRATION (C.M.C.)

I 435

9.6.3 Determination of microfluidity using fluorescence probes 9.6.4 Other spectroscopic techniques 9.7 Molecular packing and its effect on aggregate formation

9.8 Statistical thermodynamics of chain packing in micelles

9.1 The critical micellization concentration (c.m.c.) We noted in Section 1.4.3 that certain molecules (called amphiphiles) are able to form aggregates called micelles in aqueous solution, provided their concentration is sufficiently high. The concentration at which this micelle formation occurs is usually fairly sharply defined and it can be identified by observing the behaviour of any one of a number of equilibrium or transport properties of the solution (Fig. 9.1.1), each of which undergoes a rather abrupt change in concentration dependence at much the same point (called the critical micellization concentration or c.m.c.) We do not propose to review the many methods which have been used to detect the onset of micellization although some reference will be made to the recent spectroscopic procedures in Section 9.5.1. Close examination reveals that, in some cases, different methods of measurement would yield c.m.c. values varying by almost 50 per cent (see, for example, Kresheck 1975, Fig. 1) and the same method in different hands can produce a similar spread. Some of the variation may be due to the presence of impurities, small amounts of which are known to have a significant effect on the c.m.c. Some variation can be traced to uncertainties in the extrapolation procedures used to define the c.m.c. (The value

c.m.c.

Concentration of surfactant

Fig. 9.1.I Schematic representation of the concentration dependence of some physical properties for solutions of micelleforming amphiphiles. (After Lindman and Wennerstrom 1980, with uermission.)

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9: ASSOCIATIONC O L L O I D S

derived from the same conductance data can vary significantly, depending on whether a plot is made of the molar conductance against C’/’ or the specific conductivity (conductance) against log C.) Despite these limitations the concept of critical micellization concentration remains an important one. It can be defined in terms of one or other of the properties suggested by Fig. 9.1.1 but a more general definition is (Phillips 1955): (9.1.1) where q5 is any one of the properties (Exercise 9.1.1) and CT is the total concentration of the amphiphile or surfactant. Micelle formation is only one of a number of characteristic aggregation phenomena which amphiphilic molecules undergo and one might well ask what causes this behaviour pattern, intermediate as it is between true solution and a separation of the components into two distinct phases. After all, it does not occur with long chain alcohols, amides, or amines which also have a polar head group and a lipophilic or hydrophobic chain attached. It seems to be necessary to have either a charged head group (carboxylate, sulphate, or quaternary ammonium), a zwitterionic group, or a rather bulky oxygen-containing hydrophilic group (polyoxyethylene, phosphine oxide, amine oxide, or a sugar residue) able to undergo significant hydrogen bonding as well as dipolar interaction with water. (We concentrate mainly on the formation of micelles in aqueous solution because the characteristics of micelles formed in non-aqueous solutions are less well established. For strongly hydrogen bonded solvents (e.g. formamide and hydrazine), however, the behaviour is similar in many respects to water (Kresheck 1975). Evidently, the balance of forces which leads to micelle formation is a subtle one. If the hydrophilic effect is sufficiently strong the molecule can enjoy complete solution; if it is too weak the substance is merely insoluble (e.g. octadecanol). The structure of micelles has been a subject of controversy for many years, since the pioneering works of Hartley (1936) and McBain (1950) arguing the cases for spherical and lamellar structures respectively. The spherical (or near-spherical) form was generally accepted as the dominant species in dilute aqueous solutions until recent times. Tanford (1980) in his extensive review of the mechanism of micelle formation prefers the description ‘disc-like’ since the weight of evidence from transport and equilibrium properties suggests that the structures are better described as oblate spheroids (Fig. 5.1.1). This view is not, however, universally accepted and the most elaborate theoretical calculations (to be discussed in Section 9.8) are based on a spherical shape, at least for the dodecyl sulphate micelle near its c.m.c. Throughout the following discussion it would be wise to bear in mind the fact that the forces controlling micellar structure are delicately balanced so that distortions of the shape (either due to the shearing processes or fluctuations) may occur fairly easily (but see Section 9.7). Furthermore, any attempt to represent the structure can at best be a statement of the average shape over some discrete time interval. The exchange that occurs between monomer molecules and those in the micelle occupies a time-scale of the order of 1-10 ps (Aniansson 1978) and the entry or exit of a monomer presumably occurs by a series

THE CRITICAL MlCELLlZATlON CONCENTRATION (C.M.C.)

I 437

of diffusive steps, one methylene group at a time. The surface must, therefore, be somewhat rough on the nanosecond time scale (Section 9.6). More detailed thermodynamic analysis (Section 9.4) reveals some, at first sight, quite surprising results. For one thing, the aggregation process in water at room temperature is accompanied by a significant increase in entropy, which is the main contribution to the negative AGO value for micellization. The A@ value is usually small, at least for systems with small degrees of aggregation (50-loo), and often slightly positive. The traditional view of the mechanism of micelle formation has been based on the study of the (very slight) solubility of hydrocarbons in water, and what has come to be known as the hydrojhobic effect (Tanford 1980).It seems that, at room temperature, the presence of a hydrocarbon molecule in water causes a significant decrease in the (partial molar) entropy of the water, suggesting that it induces an increase in the degree of structuring of the water molecules. The isolated hydrocarbon molecule forms a cavity in the water structure and the walls of that cavity are lined with water molecules with a bonding pattern that differs, on average, from the bulk pattern and that, furthermore, varies in a complex and subtle way with change in temperature. The predominant effect of a hydrocarbon molecule is to increase the degree of structure in the immediately surrounding water and this is one of the main features of the hydrophobic effect. The other major effect is to disrupt the extensive hydrogen bonding pattern in the water. Evidently the entropy increase associated with this latter process is outweighed by the energy increase involved and its contribution to AGO is again positive (Frank and Evans 1945). When hydrocarbon residues aggregate in aqueous solution to produce a micelle, the reverse process occurs: the hydrogen bonding structure in the water is, to a large extent, restored. For this process, both enthalpy and entropy changes are negative. The 'melting' of the cavities that surrounded the hydrocarbon chains gives rise to an entropy increase in the water that more than compensates for the decreased randomness of the hydrocarbon chains as they enter the micelles. This view of the micellization process, in which the principal driving force is the partial molar entropy increase of the water, has been strongly challenged by some studies of aqueous systems at high temperatures (up to 166 "C) and micellization in pure hydrazine solutions (Ramadan et al. 1983) and we will return to this discussion when we have established the basis for a thermodynamic analysis (Section 9.4). Since the head group remains surrounded by water its contribution to the energetics of micellization is much less but its role is essential. It is the nature of the head group and the interactions that occur between head groups that, in principle, determine the size and shape of the aggregate structure (Israelachvili et al. 1976). We will however, defer more detailed discussion of these matters until later (Section 9.7). We will first look at the phenomenology of micelle formation - what effect factors like temperature, pressure, and salt content have on the c.m.c. We will then examine some of the simple mass-action models of micelle formation and discuss in more detail the thermodynamics of the process. We then consider some of the methods (especially spectroscopic procedures) that have been used to study other features of micellar structure and dynamics. T h e chapter concludes with a brief description of the current ideas about the geometric factors that determine aggregate shapes and the statistical mechanical description of a typical micelle. More detailed discussion of these matters will be deferred to

438 I

9: ASSOCIATIONC O L L O I D S

Chapter 14, when the theory of liquid structure and scattering behaviour can be brought to bear on the problem. Binks (1999) provides a survey of the current microscopic, optical and non-optical techniques being applied to the characterization of surfactant systems. These include, among others, S T M (Section 6.1.6) and AFM (Section 6.2.2), surface light scattering (Section 3.2 and 5.7) and reflection and non-linear optical techniques as well as X-ray and neutron reflectivity (Section 14.5), electroacoustics and ultrasonics (Section 5.9).

Exercises 9.1 . I Show, using appropriate sketches, that eqn (9.1.1) is a general statement of the fact that the plot of the property ~ ( C Tagainst ) CT undergoes a sharp change in slope at CT = c.m.c.

9.2 Factors affecting the c.m.c. A very large and valuable collection of data on the c.m.c. of various surfactants has been compiled by Mukerjee and Mysels (1971) from which a number of important generalizations emerge. A more recent compilation of the physico-chemical properties of various anionic, cationic, and non-ionic surfactants has now been provided by van 0 s et al. (1993) and Clint (1992) has reviewed surfactant aggregation.

9.2.1 Effect of head group and chain length For surfactants with a single straight hydrocarbon chain, the c.m.c. is related to the number of carbon atoms in the chain (m)by:

(9.2.1)

loglo(c.m.c) = bo - blmc

where bo and bl are constants. Figure 9.2.1 illustrates this point for a number of typical ionic and non-ionic surfactants. Some of the reasons for the differences in bo and bl values revealed by this figure and Table 9.1 will be discussed below (Section 9.4.2). It is hardly surprising that the nature of the head group should effect the value of bo, but it is also apparent that it profoundly effects bl as well. It should also be noted that the non-ionics usually have much lower c.m.c.s than the ionics despite their generally larger bo values. Most ionics of given chain length show very similar values for c.m.c. Lindman and Wennerstrom (1980) quote, for the straight chain Clz (dodecyl) surfactants the following values (with the counterion in brackets):

8 mM 15 mM 20 mM

for for for

-0.S0, (Na+) -NHl (Cl-) -N+ ((333)s (Cl-)

10 mM 12 mM

for for

-SO, (Naf) 40; (Kf)

Modifications to the hydrocarbon chain (such as introducing branching, or double bonds, or polar functional groups along the chain) usually lead to increases in the

FACTORS AFFECTING THE C.M.C.

1439

.z -7 h VJ

5

F

.Ei * -6 0

4

i2

-5 -4

?

c!

-3 v

M

2 -2 6

8

10

12

14

16

18

Fig. 9.2.1 Plots ofloglo c.m.c. (in mole fraction units) versus m,, the number of carbon atoms in the alkyl chain. (Temperature in general 25 "C.) (a) Alkyl hexaoxyethylene glycol monoethers. (b) Alkyl trimethylammonium bromides in 0.5 M NaBr. (c) N-alkyl betaines. (d) Sodium alkyl sulphates (40 "C). (e) Sodium alkylcarboxylates. (From Lindman and Wennerstrom 1980, with permission.)

Table 9.1 Values of bo and bl in eqn (9.2.1) for various surfactants. (Selectedfrom Kresheck 1975.) Surfactant

Temperature ("C)

bo

bi

Na carboxylates

20

2.41

0.341

K carboxylates

25

1.92

0.290

Alkane sulphonates

40

1.59

0.294

Akyl sulphates

45

1.42

0.295

Akylammonium chlorides

25

1.25

0.265

Akyltrioxyethylene-glycol monoether

25

2.32

0.554

Akyldimethylamine oxide

27

3.37

0.57

+These values are likely to be p H dependent because this amphiphile becomes cationic at low pH.

c.m.c., but the introduction of a benzene ring is equivalent to adding about 3.5 methylene groups to the chain length. Introducing fluorine in place of hydrogen has a quite marked effect, at first increasing and then decreasing the c.m.c. (ultimately to less than 10 per cent of the hydrocarbon value), as the proportion of fluorine is increased towards saturation (Mukerjee and Mysels 1975).We will return to this point in Section 9.6.4.

9.2.2 Effect of counterion It should come as no surprise to learn that counterion valency has a strong effect on c. m.c. and for various ions of the same valency, the lyotropic series has a role to play in

440 I

9: ASSOCIATIONC O L L O I D S

the explanation of smaller variations. [The lyotropic series is an ordering of ions of the same valency according to their size. We will have a little more to say about it in connection with the coagulation behaviour of salts (Chapter 12).] The values for bo and bl quoted in Table 9.1 for the sodium and potassium salts of carboxylic acids are, however, rather more difficult to rationalize. Small differences in bo can be attributed to differences in degree of hydration and cation binding to the head group, but it seems surprising that the bl value should vary so much in this case.

9.2.3 Effect of temperature and pressure One of the most surprising things about micellization is the very weak temperature and pressure dependence of the c.m.c., considering that it is an association process (Lindman and Wennerstrom 1980). This is a reflection, of course, of the very subtle changes in bonding, heat capacity, and volume that accompany the micellization process. It seems likely that if a wide enough temperature range were accessible, all amphiphile systems would show a temperature at which the c.m.c. was a minimum (Kresheck 1975). Raising the temperature has a quite different effect on ionic and non-ionic surfactants. For ionics, there is a temperature (called the KrafJttpoint)below which the solubility is quite low and the solution appears to contain no micelles. Above the Krafft temperature, micelle formation evidently becomes possible and there is a rapid increase in solubility of the surfactant. It is significant that surfactants are usually much less effective (as, for example, detergents) below the Krafft point. Non-ionic surfactants tend to behave in the opposite manner. As the temperature is raised, a point may be reached at which large aggregates of the non-ionic separate out into a distinct phase and the temperature at which this occurs is referred to as the cloud point. It is usually less sharp than the Krafft point (Leja 1982). We will postpone further discussion of temperature and pressure effects until the thermodynamics of micellization have been examined in more detail.

9.2.4 Effect of added salt Adding an indifferent electrolyte (Section 1.6) to an amphiphile/water system has a pronounced effect on the c.m.c., especially for ionics. For non-ionics the effect is smaller but still significant and the difference between the two is dramatically demonstrated by the difference in the functional dependence of c.m.c. on salt concentration, C log(c.m.c) = b2

+ b3C (non-ionic)

(9.2.2)

log(c.m.c) = b4

+ b5 log C (ionic).

(9.2.3)

and

The constants, bi, depend upon the nature of the electrolyte (Fig. 9.2.2). For the ionics, the principal effect of the salt is to partially screen the electrostatic repulsion between the head groups and so lower the c.m.c. Values of b5 of from -0.6 to -1.2 are given (Kresheck 1975) for the influence of sodium salts on sodium carboxylates. The more subtle influences of salts of the same valence type are again usually discussed in terms of the lyotropic series.

FACTORS AFFECTING THE C.M.C.

c.m.c+CNaCl (mol

1441

L-’1

Fig. 9.2.2 The effect of added salt on the c.m.c. of SDS and dodecylaminehydrochloride(DHC). (From Stigter 1975a, with permission.)

For the non-ionics the concentrations of salts required to produce significant effects are much higher and the discussion of such behaviour introduces the notion of ‘salting in’ and ‘salting out’ of non-electrolytes by the electrolyte (Ray and Nemethy 1971). This amounts to a description of the competition between the surfactant (chiefly the head group) and the electrolyte for the opportunity to associate with the water. T o put it more precisely, the activity coefficient of the monomer surfactant changes as the electrolyte concentration and type alters. If the monomer is salted out by electrolyte then micellization is thermodynamically favoured and the c.m.c. is reduced. The reverse situation applies if the monomer is salted in.

9.2.5 Effect of organic molecules Quite small amounts of organic material can have a significant influence on the c.m.c., and the properties of micellar solutions. Recall Fig. 2.4.2, where the presence of small amounts of a certain impurity caused a minimum in the plot of surface tension against surfactant concentration in the neighbourhood of the c.m.c. The classical example of this behaviour occurs in aqueous solutions of sodium dodecyl sulphate (SDS), where the presence of dodecanol (a hydrolysis product) causes a minimum in the surface tension due to the competing effects of adsorption of dodecanol at the aqueous solution-air interface and solubilization in the SDS micelles. Such anomalous behaviour is to be expected between supposedly identical industrial surfactants, as a consequence of the presence of impurities or manufacturing by-products. It is an important aspect of the behaviour of micelles that they are able to act as sites for the dissolution of lipophilic (i.e. fat-soluble) molecules. The use of surfactants as detergents, stabilizers and dispersing agents depends on this property, known as solubilization.It is characterized by a dramatic increase in the solubility of the lipophilic material at concentrations of the surfactant above the c.m.c. It is, in fact, used as a means of detecting the onset of micellization. One of the problems with this method, however, is that the lipophilic material itself may influence the value of the c.m.c. by contributing to or opposing the aggregation forces.

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9: ASSOCIATIONC O L L O I D S

It is common practice to divide organic compounds into two main groups, depending on their mode of action in influencing the c.m.c. Group A is composed of molecules (like alcohols with moderate to long hydrocarbon chains) that appear to be adsorbed in the outer regions of the micelle, forming a palisade (i.e. fence-like) structure with the surfactant molecules. This lowers the free energy of micellization to more negative values and so reduces the c.m.c.; such molecules can also influence the micelle shape. Straight chain molecules have the most marked effect, the latter reaching a maximum when the length of the hydrophobic chain of the additive is about the same as that of the surfactant. A decreased electrostatic repulsion between ionized head groups, and reduction in steric hindrance for non-ionic surfactants have been proposed as likely explanations for these observed effects. Group A compounds are generally effective at quite low bulk concentrations. They behave in an analogous fashion in other areas such as in the addition of small quantities of non-ionic molecules to flotation pulps (Fuerstenau 1976), in the penetration of insoluble charged monolayers by compounds such as hexadecanol (Gaines 1966) and in enhancing foam stability (Kitchener 1964). Group B materials alter the c.m.c. at substantially higher bulk concentrations and probably exert their influence through modification of the bulk water structure.+ The effect is usually discussed in terms of whether the additive is a (water) structure maker or a structure breaker. Typical ‘structure makers’ are xylose and fructose (Schwuger 1971) and ‘structure breakers’ are urea and formamide (Schick 1967). Structure breakers increase the c.m.c. of surfactants in aqueous solution, exerting their strongest influence on non-ionic surfactants of the polyethyleneoxide type. Presumably the presence of a structure breaker reduces the amount of water structure that the hydrophobic residues of the surfactant can induce. T h e entropy increase on micelle formation is thus reduced and so the c.m.c. is raised. T h e concept is not, however, a very straightforward one to apply. Even where the solute is able to interact very strongly with water its effect may be overall structure breaking, firstly because it has to pull water from its existing structure and secondly, because the resulting entity may substantially disrupt the remaining water structure. For a description of these complexities, see the treatment by Franks (1983, Chapters 9 and 10). It cannot be emphasized too strongly that the interactions between organic molecules and water are subtle and complex. The enthalpic and entropic contributions are often finely balanced so that the free energy of solution suggests a simple pattern that cannot be sustained on deeper analysis. Furthermore, the variations of these thermodynamic parameters with temperature and with composition are often so complex and difficult to explain that many researchers in the field would reject the dichotomy into ‘makers’ and ‘breakers’ of structure as being too simplistic. It remains, however, a useful notion in the extreme case to distinguish the increased interactions that occur between water molecules because a hydrocarbon moiety is present and those that occur between solute and water (usually hydrogen bonding) that disturb the normal (very strong) hydrogen bonding of the water itself.

The short-chain alcohols like ethanol and methyl propanol-2 can act as both group A and B materials.

EQUILIBRIUM CONSTANT TREATMENT OF MICELLE F OR M A T ION

1443

Exercises 9.2.1 Estimate the c.m.c.s for the sodium salts of the CS, C10, C12, and C14 straightchain alkyl sulphates from the data in Table 9.1. The experimental values at 25 "C are: 130.3 33.0 8.08 and 2.05 mM and at 40 "C are: 136 33.5 8.7 and 2.21 mM, respectively. Comment on the relation between the results.

9.3 Equilibrium constant treatment of micelle formation As Tanford (1977) has pointed out, the formation of micelles can be treated in a formally rigorous way in terms of all of the possible equilibria: K2 K3 Z+Z$Z,+Z$Z,

...

Kn $Z,

+ z p ...

(9.3.1)

with equilibrium constants K, for n = 2 - 00. The various thermodynamic parameters (AGO, AH(', AS()) for the aggregation process could then be expressed in terms of the K,. Unfortunately, it is not possible to measure the individual equilibrium constants, and recourse must be had to one of a number of models to simplify the situation. We will describe here two of the simplest possible models, each of which finds some applications. They are the closed association model (Section 9.3.1) and the multiple equilibria model (Section 9.3.2) in its three manifestations: (a) dimers dominate; (b) all K, of equal size; and (c) one K, much larger than the rest. The last of these is an improvement on the closed association model.

9.3.1 The closed association model Observations of the size of more or less spherical micelles in the neighbourhood of the c.m.c. (such as those of sodium dodecyl sulphate (SDS)) suggest that the size range is very limited. The simplest assumption to make in treating eqn (9.3.1) is, therefore, that only one of the K, values is important. (For SDS it would be about K60 at 25 "C.)In that case the micelle formation is represented as:

n monomers p micelle or nZ p M for which the equilibrium constant, K , is:

K=

[micelles] -C, monomer^]^ C; .

(9.3.2)

The inherent assumption here is that the activities may be replaced by concentrations. For the monomer this amounts to assuming that the only departure from ideal behaviour is the aggregation process. It could, in principle, be removed by estimating other activity corrections from solution theory. Assuming ideal behaviour for the

444 I

9: ASSOCIATION COLLOIDS

micelles is more problematical because of the large size difference between monomers and micelles. The micelles will also interact strongly. For ionics the interaction will become very significant as soon as the mean separation is less than about (8-10)/~, where K is the Debye-Huckel parameter (Section 7.3), and that occurs at surfactant concentrations not far above the c.m.c. (Exercise 9.3.1). It must also be noted that when ionic micelles are formed there is a strong tendency for the counterions to be associated closely with the head groups, because of the high electrostatic potential in that region. This is a further source of non-ideality, which is discussed in Section 9.5 (Evans and Ninham 1983). From eqn (9.3.2) we have (Exercise 9.3.4):

A@

= -RT

In K = -RT In C,+nRT In C,

(9.3.3)

and

-AGO RT = -AGO = -lnC, n n ~

- RTlnC,.

(9.3.3a)

At the c.m.c., we set C, = Co and the total surfactant concentration, CT, above this point is:

+

(9.3.4)

CT = CO nC,. From eqns (9.3.2) and (9.3.4): n

K=

Lm

(CT - nC,)"

(9.3.5) *

Mukerjee (1975) shows how eqn ( 9 . 3 4 , with a K value of unity and n = 100 gives rise to a sharp transition from a system in which all of the surfactant is present as monomer

Fig. 9.3.1 Variation of dC,,,/dCT with total surfactant concentration for different values of the aggregation number, n. COis the critical micellization concentration and C, the concentration of micelles.

EQUILIBRIUM CONSTANT TREATMENT OF MICELLE F OR M A T ION

1445

to one in which the monomer concentration remains essentially constant above the c.m.c. and all additional surfactant goes into micelle formation (Exercise 9.3.2). As an alternative approach (Exercise 9.3.3), eqn (9.3.5) can be differentiated to obtain:

(9.3.6) The plot of dC,,,/dCT against concentration for K = 1 and various values of n is shown in Fig. 9.3.1. When it is realised that for most of the commonly used surfactants the value of n is at least 50 it becomes clear that the concept of a critical micellization concentration (being the concentration in the neighbourhood of which, micelle formation begins) is a reasonable one. As n+ 00 the transition becomes sharper and ultimately approaches the behaviour expected of a first-order phase transition. In this extreme case it is possible to treat the micelle formation as a phase separation. For smaller values of n there are problems involved in the phase separation model (Hall and Pethica 1967) and these can only be properly resolved by resorting to the formalism of small system thermodynamics (Hill 1963).

9.3.2 Multiple equilibrium models The closed association model (Section 9.3.1) is not physically appealing. If, for example, n = 50, it is difficult to see why the addition of one extra monomer to the aggregate of 49 should drastically reduce the free energy of the aggregate. And why should it be difficult, or impossible, to add an additional monomer? It may be argued that a certain number of monomers is required to build a complete structure and certainly one can see that some minimal number is required to produce a structure in which the head groups can effectively shield the hydrocarbon residues from the water. If the aggregate were crystalline there might be geometric reasons for some rather closely specified aggregation number, but there is abundant evidence (see, for example, Phillips 1955; Fisher and Oakenfull 1977) that the interior of most micelles (at least those formed from long-chain surfactants) is liquid-like. T o obtain a physically reasonable model, it is therefore necessary to write out the full equilibrium between monomer and micelles of all sizes and then to seek a physically reasonable basis for defining a relationship between the equilibrium constants. Such a relationship should predict the observed facts of micelle size (or size distribution) and the thermodynamic parameters, while still having only a small number of adjustable parameters. The treatment here follows that of Mukerjee (1975) with some modifications. Consider again the equilibria in eqn (9.3.1). For any n-mer, the stepwise association constant is:

(9.3.7) The overall association constant, then, for the formation of x, from xi (i.e. n q $ x,) is:

*K --"I'

' - [XI]"

where *K, = n K , , 2

(9.3.8)

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9: ASSOCIATIONC O L L O I D S

(i.e. "K, is the product of all stepwise association constants K2, K3,. . ... up to K,.). Again in defining "K, we replace activities with concentrations with all of the uncertainties previously referred to. The concentration of all solute species, S, (in mol dmP3) is:

(9.3.9) and, expressed in terms of the total monomer concentration, M I , this becomes:

(9.3.10) Then the number average degree of association of all species (including the monomer) is (see Section 5.3.2):

(9.3.11) The mass average is given by:

(9.3.12) We normally concern ourselves only with associated species and exclude the monomer. In these terms, the corresponding values of number and mass average are:

(9.3.13)

(9.3.14) It is evident from eqn (9.3.8) that, as [XI] increases, the concentration of each associated species increases. The value of n influences each associated species and the percentage increase of each increases with n. From eqns (9.3.11)-(9.3.14) both number average and mass average increases. On the other hand, with increasing dilution A41 + 0, N,+ 1, + 1 and M I M [XI]. The individual K, values define the concentrations at which particular n-mers at any become most important and, hence, they control the values of Enand concentration. The products of association and their size distribution depend on concentration and on K,(n), i.e. how K, depends upon n, which is a reflection of the molecular architecture of the associated species. Three main types of association behaviour may be identified:

xw

ww

(a) simple dimerization (K2 dominant); (b) formation of micelles with a wide range of aggregation numbers (all values of K, approximately equal); and (c) formation of micelles with a narrow size range (strong dependence of K, on n).

EQUILIBRIUM CONSTANT TREATMENT OF MICELLE F OR M A T ION

1447

The first two cases are rather trivial, while the third covers most of the interesting micelle-forming compounds and will, therefore, be discussed in more detail. (a) Dimerization The formation of dimers will of course take place in all self-associating systems. Whether or not the process is limited to dimer formation or continues predominantly on to multimers will be decided by the value of KZin comparison with other K, values. Dimerization appears to be restricted to dilute aqueous solutions of some flexible chain surfactants, such as carboxylic acids (Oakenfull and Fenwick 1974) and to solutions of some bile salts, notably sodium cholate (Small 1968; Oakenfull and Fisher 1977). Both of these are rather special cases, in which the structure of the interacting molecules favours formation of a ‘closed’ dimer. The carboxylic acids for example, show cooperative hydrogen bonding to form a cyclic structure:

Bile salts may show a similar interaction, with three cooperative hydrogen bonds, although the evidence for this has been disputed (Zana 1978; Oakenfull and Fisher 1978). (b) K, values of similar magnitude For the equilibrium between monomers and micelles (eqn. (9.3.l)), let

. . .K,=K.

K2=K3=

(9.3.15 )

The total concentration, S, can now be directly related to the equilibrium monomer concentration [XI]. Defining X = K [XI] then

+ + [ ~ g+] . ...[x,]. . .

S = [XI]

[~2]

+ + + . . . . . . .Kn-’[~1IM-’) = [X1](1 + x + x2+ x3+ . . . . x n - l ) = [ ~ l ] ( l K[x~] K2[x1I2

IX1l

--

1-x

(9.3.16)

for large n,

if X < 1, which it must be in real systems that conform to the association scheme outlined above. MI, the total monomer concentration, now becomes (Exercise 9.3.5):

(9.3. 7) and thus

(g) 112

= 1 - K[xl]

(9.3.

448 I

9: ASSOCIATIONC O L L O I D S

so that if [XI] is measured experimentally (Mukerjee and Ghosh 1970), K may be evaluated. N,, the number average degree of association of all species is:

N,= Ml/S = 1/(1

-

X)

(9.3.19)

and the mass average, Xw, is (Exercise 9.3.5):

Nw = Z/Ml

= (1

+ X)/( 1

-

X).

(9.3.20)

It is not difficult to establish the following expressions (Exercise 9.3.6):

and

z - [XI] X(4 - 3 x + X2) -

~

[Xll

(1 - x ) 3

(9.3.21)

from which the modified number and mass average degrees of association (eqns (9.3.13) and (9.3.14)) follow (Exercise 9.3.6):

(9.3.22) The index of polydispersity is given by (recall Section 5.3.6). -

=*NW =1+ *Nn

X (2 - x ) 2

(9.3.23)

which approaches 1 for small X and 2 as X approaches 1. The breadth of the size distribution thus increases with the degree of association. It is not always broad as is suggested by Mukerjee (1975). This model often gives a good description of the behaviour of molecules that are rigid and flat, with faces of approximately equal hydrophobicity. Such molecules associate by a simple stacking arrangement as had been demonstrated with the cationic dye, methylene blue (Mukerjee and Ghosh 1970). As the size of the stack increases, charge repulsion builds up and K, gradually decreases at higher concentrations. This association model is also a fairly good representation of the behaviour of many nucleosides and of the stability of double stranded DNA (Ts’o 1968). In both cases, stacking interactions occur between the organic bases. (c) Strong dependence of K, on n. This is the situation that was crudely examined in the closed association model. The delicate force balance involved in micelle formation leads, not to an increase or decrease in K, with n, but rather to a value of n for which K, is a maximum. The values of K,-1 and K,+1 will be of comparable magnitude but one will expect to find the resulting size distribution to be fairly narrow.

EQUILIBRIUM CONSTANT TREATMENT OF MICELLE F OR M A T ION

1449

As noted earlier (Section 9.1), the existence of a preferred micelle size is easily explained on the basis of the competing requirements for bringing the hydrocarbon chains into intimate contact, away from the aqueous environment, while maintaining the head groups as far apart as possible. There is very little energy change in separating the hydrocarbon chains from the water (Exercise 9.1.2) and the negative free energy associated with that process stems largely from the concomitant entropy increase in the water. In the case of an ionic surfactant, as each additional monomer is added to the micelle, the contribution to the free energy change becomes less negative because the developing micellar charge causes an increasing (positive) free energy change, reflecting the repulsion between the head groups. Even when n is large, a relatively broad maximum in the values of K, may produce a narrow size distribution. The free energy change involved in the formation of the nmer from the monomer is, from eqns (9.3.3) and (9.3.8):

~q= -RT

In *K,

(9.3.24)

and only a slight minimum in the plot of AGt/n against n is required to produce narrow size distributions of micelles, as Stigter and Overbeek (1957) have shown. Mukerjee (1975) gives a striking illustration of this point. Using the following empirical expression for In *K,: In *K, = 2(n

0

-

25

1) ln(n - 1)

50

-

0.02(n - 1)’

n

75

+ 2.7896(n

-

1)

(9.3.25)

100

Fig. 9.3.2 Variation in the concentration of monomers existing in the form of micelles (n[x,]), as a function of the number of monomers per micelle, n, for an assumed free energy profile, AG:/n. (From Mukerjee 1975, with permission.)

450 I

9: ASSOCIATIONC O L L O I D S

which exhibits a very broad maximum (Fig. 9.3.2) Mukerjee calculated the values of n[x,] from eqn (9.3.8) (i.e. the concentration of n-mers expressed in terms of monomer concentration), for a monomer concentration of 4.11 x mol L-' (i.e. ln[xl] = -10). The result is a narrow size distribution (Fig. 9.3.2) peaking at n = 97 and with a half-width of less than 10, even though A e / n changes by less than 2 per cent over the whole range from 70-120 for n. This figure illustrates clearly why the calculation of the anticipated size distribution in a particular case requires accurately measured "K, values near the maximum or good estimates of A G . We will examine the extent to which A G can be estimated in the next section. When the monomer chains are very long, very large polydisperse aggregates apparently form, even in dilute solution (Debye and Anacker 1951). These micelles are thought to be flexible cylinders and may be described by a self-association model similar to that discussed above (Mukerjee 1974). The molecular requirements of micelles of various geometries will be discussed in Section 9.7. Our immediate aim is to estimate the value of AG: as accurately as possible and, from this, to determine the micelle size distribution and other properties.

Exercises 9.3.1 Estimate the mean centre-to-centre distance between the micelles of sodium dodecyl sulphate (SDS) when the surfactant concentration is 5 x lo-' M. Assume that the c.m.c. is 8 mM. Show that this corresponds to about 4 / when ~ K is calculated on the basis of the residual monomer concentration. (Note that the minimum micelle radius is about 2.5 nm, corresponding to a stretched C12 alkyl chain.) 9.3.2Take K = 1 and n = 100 in eqn (9.3.5) and discuss the variation of C, (=CT - nC,) and nC, with C,/Co in the range 0 5 CT/CO5 3. (Assume that in this formulation the concentration of micelles is expressed in moles of micelles per litre.) 9.3.3Establish eqn (9.3.6). 9.3.4Integrate eqn (A5.11) to establish eqn (9.3.3) for the case where only P,Vwork is involved. (Use the fact that AG = 0 at equilibrium if P and Tare constant.) 9.3.5Establish eqn (9.3.17) using the series expansion for (1-a-'. Establish eqns (9.3.19) and (9.3.20) by showing that Z = [xl](l 9.3.6Establish the expressions (9.3.21) and use them to derive (9.3.22).

+ a/(]a3.

9.4 Thermodynamics of micelle formation Careful analyses of the thermodynamics of micelle formation have been given by a number of authors, including Hall and Pethica (1967) and Tanford (1980). We will follow the latter treatment, with some modifications. The aim is to relate the chemical potential of an amphiphile or surfactant in free solution with that of the same molecule in a micelle of arbitrary size. In an isothermal system at equilibrium this quantity must be constant throughout the system.

T H E R M O D Y N A M I C S OF MICELLE FORMATION

I451

Tanford distinguishes what he calls the 'cratic' contribution to the chemical potential from the intrinsic contribution, which is due to local (chemical and physical) interactions. The cratic part is that due to the entropy of mixing and so for any particular size of micelle, is equal to:

RT In (mole fraction of micelles of size n). This would give the contribution per mole of micelles of size n, assuming ideal behaviour. (The use of mole fractions is connected with the most appropriate choice of standard state and will be discussed in more detail in Section 9.4.3.) Again (cf. Fig. 9.3.2) it is more convenient to express this contribution in terms of the concentration of the monomeric surfactant: RT In (X,/n) where X , is the mole fraction of monomer in micelles of size n. The cratic contribution per mole of monomeric amphiphile is l / n of this and SO:

(9.4.1) [Do not confuse X , with [x,] as used in Section 9.3; the latter is the concentration expressed in terms of moles of n-mers.] Note that in this formulation, each of the micellar sizes is treated as a separate component, with its own standard state chemical potential. Equating pmic,,with the value for the free surfactant gives: pkic., - py = RT In a1 - (RT/n) In {X,/n} 0

or

In X , =

-n(pmic,n - P!) RT

+ n In a1 + In n

(9.4.2) (9.4.3)

where a1 is the activity of the monomer. Equation (9.4.3) gives explicitly the distribution function for micelles of different size, in terms of the quantity (-npLc,, - np!) which is the value of A G for the reaction in which an n-mer is formed from monomers. An optimal size n" can be defined as that value of n for which X , is a maximum at the particular surfactant activity:

(8 In &/an),,

=0

(n = n*)

(9.4.4)

and Tanford (1980) points out that if the size distribution is reasonably narrow, the value of n" is experimentally indistinguishable from the number average or mass average micellar size. All can then be set approximately equal to a mean size, n. At this level of approximation, all micelles are treated as the same and lumped together with a standard chemical potential, pLic:

(9.4.5) The activity of the free surfactant a1 is, of course, given by ylX1 where y1 is the activity coefficient and it is tempting to assume that y1 M 1, since X I is usually fairly

452 I

9: ASSOCIATIONC O L L O I D S

small. (Even above the c.m.c. the concentration of the free surfactant remains close to the c.m.c. value.) The problem with this procedure is that even at the concentrations normally encountered, ionic surfactants exhibit activity coefficient effects due to interactions other than micelle formation. Some account could be taken of such effects using, say, the extended DebyeHiickel theory but the experimental procedure of extrapolating to infinite dilution breaks down in this case because the presence of the micelles may itself influence the interactions between the free surfactant molecules. This effect may not be as serious as Tanford (1980) suggests, however, since the free surfactant ions will tend to be excluded from the double layer regions surrounding the micelles. For non-ionic surfactants the activity coefficient correction can probably be dispensed with altogether with negligible error (Desnoyers et al. 1983). The relation between the standard chemical potential change and the c.m.c. (Xo)can be obtained from eqn (9.4.5) by introducing the ratio cr = Xmic/Xoand recognizing that, at the c.m.c., XI = XO- Xmic.We then have (Exercise 9.4.2): (piic- &/RT

= [(E - l)/?i]lnXo

+ lnyl + ln(1 - o)+ (1/E) In @/a).

(9.4.6)

T o estimate values of (pLic- p!) thus requires a knowledge of the c.m.c. (XO)and the mean aggregation number, E. The value of cr can be taken to be anywhere from 0.01 to 0.1 with negligible effect on the result but, of course, one has to either assumeyl = 1 or make some correction for it. This latter correction should be relatively unimportant if - p;) as the chain length changes. one wishes only to evaluate the change in Indeed, for many purposes, the following approximation, valid for large ki and small cr (Exercise 9.4.3) is sufficiently accurate:

@Lie

0

-

pIL;, - p1 = AGO= R T In

x,.

(9.4.7)

Note that this is identical to eqn (9.3.3a) if n is very large and the surfactant concentration is expressed in mole fraction terms. It cannot be applied in this form to ionic surfactants, for which a further correction is essential. (See Section 9.5 below.) Further discussion of the thermodynamic determinants of micellar size and shape is given by Missel et al. (1980). See also Ekwall et al. (1971). ~

9.4.1 Estimation of AGO It seems reasonable to assume that the free energy change associated with the transfer of one mole of monomer from free solution into a micelle: (9.4.8) could be broken into various contributions: ~~

AGO = AGo(CH3)+(m

~

-

~

l)AGo(CH2)+AGo(hg) = AGIl,

+ AGo(hg)

(9.4.9)

where hg represents a (hydrophilic) head group. This kind of breakdown is suggested by the solubility and vapour pressure behaviour of homologous series of organic molecules, in which a constant increment or

T H E R M O D Y N A M I C S OF MICELLE F O R M A T I O N

1453

decrement is noted for each additional - -CH2 group. Studies of the solubilities of alkanes in water suggest values for AGO(CH3) of-8.8 kJ mol-' or -3.5 RT, whilst the free energy of transfer of a mole of methylene groups from water to a hydrocarbon environment (obtained from studies of adsorption at the oil-water interface) is about -1.4 RT. Tanford (1980) quotes the work of Swarbrick and Daruwala (1969, 1970) on the N-alkyl betaines (R-N(CH3); CH2COO-) for which ?z data are available, so that AGO can be estimated from eqns (9.4.6) or (9.4.7). The change in AGO for each additional CH2 in the group R i s 2 0 6 kJ/mol or -1.23RT, in reasonable agreement with the value quoted above for AGO(CH2). ~

9.4.2Estimation of AGO(hg) ~

The contribution of the hydrophilic head group AGo(hg) to the free energy of micellization is invariably p o s i t i v e d s o opposes the process. Very little progress has been made on the calculation of AGo(hg) for non-ionic molecules. It is assumed to arise from steric interactions as the large head groups crowd the surface but beyond that there is little that can be said from a theoretical point of view. The contributions of different non-ionic head groups can, however, be evaluated from experimental data. By contrast, more progress has been made on the calculation of AGo(hg) for ionic surfactants, assuming that it is dominated by the electrostatic effects predictable from the Gouy-Chapman theory of the double layer. We consider the energy change involved in establishing an n-mer (of charge q = ne). The electrical contribution per mole of monomer is calculated by imagining small increments of charge dq to be transferred to the initially uncharged spherical n-mer:

where q' and $0' refer to the charge and potential on the micelle surface during the charging process. The problem is that, for spherical particles, there is no explicit relation between the surface charge, q, and the surface potential. Only at low potentials (strictly $0 < 25 mV) can the DebyeHuckel relation (eqn (7.11.6)) be invoked:

' = 4nE$o

'a(1

+

(9.4.11)

Ka)

where a is the micelle radius and then (Exercise 9.4.4) a ( h g ) = $oe/2 = ne2/8nca(l

+~

a )

(9.4.12)

where e is the proton charge. Unfortunately, this approximation is rarely, if ever, valid in micellar systems. Stigter (19754 has shown, however, that a more exact calculation, using the computer-calculated relation between $0 and q for spherical particles, can satisfactorily account for the variation of c.m.c. with salt concentration in SDS solutions at 25 "C (Table 9.2) over the range from 0 to 0.2 M sodium chloride. In this region, the micellar aggregation number changes over less than a factor of two (from 65-1 10) and the concentration of free surfactant remains approximately equal to the c.m.c.

454 I

9: ASSOCIATION COLLOIDS

Using eqns (9.4.7) and (9.4.9) we can then write ~

S In Xo = 6 In Co = S[AGO(hg)/RT]

(9.4.13)

where 6 measures the change compared to the value in the absence of salt. Equation (9.4.13) assumes that CO (mol L-l) c( XO at low concentrations. The agreement between columns 5 and 6 is excellent, suggesting that double-layer theory gives a good account of the effect of added salt, at least up to 0.2 M. The remaining discrepancy is of the order of the activity coefficient correction for unassociated ions and this has not been included. The DebyeHuckel procedure would obviously give a very poor representation, being some 5&65 per cent higher at all concentrations. Note that in this approach, no attempt was made to relate the absolute magnitude of AGO(e1ec) to the experimental estimates of A@(hg). That would require an assumption about the degree of ion binding and a choice of the appropriate value of 11.0 (or 11.d) to characterize the surface. More elaborate calculations of this nature are given by Stigter (19756). In Table 9.2 it is assumed that ion b i n d i n d n o t affected over the concentration range involved. A more recent calculation of AGO(elec)along similar lines is given by Evans and Ninham (1983). At somewhat higher salt concentrations (above about 0.45 M) the aggregation number of SDS increases dramatically to over 1000 and a more elaborate model is required (see, for example, Mazer et al. 1977 and Gunnarson et al. 1980). The ionic strength effect (Table 9.2) has another important consequence that was noted in connection with Table 9.1. The constant 61, which measures the effect of each additional CHZ- group on the c.m.c., is much smaller for most ionics than it is for nonionics. This may be partly due to the much better shielding of the hydrocarbon chains from the water in the case of non-ionics. In the ionics, the repulsion forces prevent the head groups from packing close together so that there is always a significant hydrocarbon-water interface (Fig. 1.4.4). Tanford (1980), however, claims that the ~

~

Table 9.2 Calculation of the electrical contribution to the free energy of sodium dodecyl sulphate (SDS) micelles at 25 "C. (From Stigter 1975a, with permission.)

0

8.12

7.96

4.85

0

0

0.01

5.29

7.28

4.39

0.43

0.46

0.03

3.13

6.04

3.81

0.95

1.04

0.05

2.27

5.70

3.62

1.27

1.23

0.1

1.46

4.80

3.21

1.72

1.64

0.2

0.92

4.15

2.85

2.18

2.00

Column 3 uses the DebyeHuckel approximation (eqn (9.4.13)) while column 4 uses the 'exact' $0' - q' relation from Gouy-Chapman (G-C) theory. A@ values in units of RT.

T H E R M O D Y N A M I C S OF MICELLE F O R M A T I O N

1455

difference in bl values can be almost entirely accounted for by the ionic strength effect. As the number of carbon atoms decreases, the c.m.c. tends to increase but for ionics this is partly offset by the increasing ionic strength due to the surfactant itself. The resulting tendency to lower c.m.c. (Table 9.2) thus opposes the effect of shortening the chain. This explanation is supported by the behaviour of the n-alkytrimethylammonium bromides in 0.5 M salt solution. (Exercise 9.4.6).

9.4.3 Choice of standard state and concentration units For a detailed discussion of this question the reader is referred to specialized treatments such as those of Shinoda (1963,1978), Phillips (1955), Anacker (1970), and Kishimoto and Sumida (1974). The choice of mole fraction as the concentration unit implies that the standard state for the surfactant is the pure material but physically it is more reasonable to interpret it in terms of a totally hydrated state. Consider the following schematic diagram: Monomeric surfactant hydrate (H~btion) (unit mole fraction) l

Solid surfactant

5

*

*

Micellization

under standard

Micellar surfactant

hydrate (unit mole fraction)

state conditions

(Formation of micelle hydrate)

Surfactant solution (at c.m.c.)

Micellar

3 surfactant 4 - - - - - - - - -

(Equilibrium

between monomers,

solution (at c.m.c.)

counterions, and micelles.)

AG5 may be identified with the standard free energy change in terms of unit mole fraction, so that AG5 becomes A@ and may be calculated through eqn (9.4.2) (with AGO = pii, - py). The condition under which micelles are formed spontaneously is given by:

-

AG1

+ AG2 5 0.

(9.4.14)

For compounds such as alcohols, amides, and like substances, the free energy of solution, AG1, is positive to the extent that AGl G2 > 0 and no micelles are formed. AG1 is reduced by the presence of an ionic head group, or a strongly polar head group of the ethylene oxide variety and micelles will begin to form when AG1 AG2 = 0. As the temperature is lowered, the positive entropy of micelle formation means that at some point (called the Krafft temperature) the overall free energy change for micelle formation is no longer negative and the solubility of the surfactant decreases dramatically. (See Section 9.2.3.)

+

+

456 I

9: ASSOCIATIONC O L L O I D S

9.4.4 Enthalpy and entropy of micelle formation From eqn (9.4.5) we can write, for the free energy change on micelle formation: = p:,

M

RT In a1 - -lnX,i, n

- py = R T

RTln(c.m.c) -

RT lnX,i, n

(9.4.15)

~

neglecting the (n-' In n) term. The temperature and pressure derivatives of eqn (9.4.15) give the standard enthalpy change A H o and volume change A V o ,per mole of monomer (Kresheck 1975; Exercise 9.4.5):

~ = - R T z (aln(c.m.c) aT ) p + TR( T T) ~a l n ~ , ~ , and *=R7.(

(9.4.16)

P

)

a In (c.m.c) ap ) T - T (R T a Inap Xmic .

(9.4.17)

T

-

A 9 can be obtained from: -

AsO = (S -B ) / T .

-40 I -20

(9.4.18)

I

I

I

0

20

40

TAZ(H rno1-l)

Fig. 9.4.1 Compensation plot of data for a variety of ionic and non-ionic surfactants in various liquids. Open circles are for water, with or without additives. Filled circles are for benzene and filled triangles for formamide. (Modified from Kresheck 1975, Fig. 9 with permission.)

T H E R M O D Y N A M I C S OF MICELLE F O R M A T I O N

1457

In the neighbourhood of the c.m.c. the value of Xmicis small, and since n is not always known, it is common practice to n&ct the second term on the right of each of these expressions and to estimate A@ from the approximate eqn (9.4.7). Although Kresheck (1975) defends this procedure, chiefly on the grounds that the AHo data obtained agree with the calorimetric estimates, Muller (1977) has called it into question, on account of the large changes in aggregation number that can occur with temperature. Despite some reservations then, we will examine the resulting A H o and ASo data because they have been used to develop a deeper insight into the micelle formation process. Th e most striking feature is the relation between A@ and T A P , which is shown in Fig. 9.4.1. Taking T = 300 K gives a slope of unity, so that this is called the compensation temperature. It is very significant that this compensation (making AGO = 0) should occur so close to room temperature, emphasizing how delicate the balance between energy and entropy must be in the micellization process. As noted earlier it has, until very recently, been assumed that the major factor driving the surfactant molecules into aggregation in water is a positive entropy change, presumably associated with breakdown of the structured water which surrounds the hydrocarbon chain in the unassociated species. Such an interpretation carries over readily to the formamide system in which some structuring by dissolved hydrocarbon also occurs. T h e more extensive studies of Evans et al. (1984) on the alkyltrimethyl ammonium bromides in water (from 25 "C to 166 "C) and in hydrazine (Ramadan et al. 1983) suggest that this interpretation is erroneous or, at any rate, misleading. At high temperatures (90 "C) water loses most of its peculiar structural properties and the formation of structured water in the walls of the hydrocarbon cavities is no longer possible. Neither is it expected to occur in hydrazine and yet both of these systems exhibit micellization phenomena. T he difference is that ASo for the process is now negative, as is also A€$'. Evans et al. argue that, in such circumstances, it is not sensible to attribute the micellization at room temperature to the positive entropy change. That change is made up of two parts: a large positive part due to removal of water from around the hydrocarbon and a (smaller) negative part due to transfer of the hydrocarbon (and counterions) into the micelle. Since the first water structure part is not present in hydrazine or hot water it cannot be the general driving force for micellization. Rather it is the second part (for which AH(' is also negative, due to some extent to the re-establishment of the hydrogen bonds in the solvent) that must be the usual driving force. Evans and Ninham (1986) have also applied this analysis to changes in protein conformation, to vesicle formation and fusion and to other biochemically important self-assembly processes. The limited amount of data on other solvents also presents a complex picture. Why should T A P be positive (although smaller than the water value) for the formation of (presumably inverse) micelles of dodecyl-ammonium alkanoates in benzene? The problem of interpretation is highlighted by the comparison of the behaviour of dodecylammonium octanoate in benzene and cyclohexane (Kresheck 1975, taken from Kitahara 1967):

~

~

~

~

~

458 I

9: ASSOCIATIONC O L L O I D S

-

AHo

TZF

kJ mol-’

kJ mol-’

In benzene

-5.4

+10.0

In cyclohexane

-32

-12.1

Although these data were derived at different temperatures (299 K and 313 K, respectively) it is obvious that no nave interpretation will ‘explain’ such a difference. This could be a situation in which the changes in aggregation number over the temperature range are impossible to ignore, as Muller (1977) has argued. Certainly it is important to recognize that these molecules, when dissolved in benzene at low concentrations (before they aggregate), are present as ion pairs, so the starting point for the aggregation processes is very different from that in aqueous solution. Indeed, Kertes (1977) has called into question the whole concept of micelle formation as applied to solutions in aprotic nonaqueous media (such as benzene). It should also be noted that the presence of trace amounts of water has a profound effect on the micelliiation process. One thing is clear from the minimal amount of data considered here: the very simple behaviour of the free energy function often masks a very complex shift in A S and AH values as temperature changes. Sometimes the study of these detailed shifts provides deeper insights into the process. In the present case it may well have led to an overemphasis of the role of ‘structured water’ in the phenomenon referred to as ‘the hydrophobic bond’. For a further discussion of the problems involved in the interpretation of AG in terms of A H and A S in this case see Evans and Wennerstrom (1999) section 5.6. Tables 9.3 and 9.4 provide some illustrative data on micellization for the common ionic surfactants. The degree of ionization is the ratio of the apparent charge on the micelle (p) to the aggregation number.

Exercises

+

9.4.1 Note that as % + 00 eqn (9.4.5) reduces to pLi, = py RT In a l . How would this be interpreted in terms of a phase separation model? 9.4.2 Establish eqn (9.4.6). 9.4.3 Show that a crude estimate of AGO for the micellization process can be obtained from: ~

~

AGO

M

RT In Xo ~

+

where X O= c.m.c. Show that, in water: In CO A @ / R T In 55.5 where COis the c.m.c. expressed in mol L-’, provided the c.m.c. occurs at low concentration. 9.4.4 Establish eqn (9.4.12). 9.4.5 Show that T[a(AG/T)/arJp= -AH/T. Hence show that, when applied to the components of a reaction: d(AGo/q / d T = -A@/ T ’. Use this to establish eqn (9.4.16). Also establish eqn (9.4.17).

T H E R M O D Y N A M I C S OF MICELLE F O R M A T I O N

I

1459

9.4.6 Discuss the relation between the bl values obtained for alkyltrimethylammonium bromides in the presence of salt (Fig. 9.2.1) and those for the corresponding chlorides in the absence of salt (Table 9.1).

Table 9.3 Critical micelle concentrations, aggregation numbers, effective degree of ionization of micelles (p/n), and free energies of micelleformation for various ionic surfactants. (Philldps 1955; Ford et al. 1966). (Bracketed values (fir SDS) are more recent values from Kratohvill980.) ~

Material

Solvent

A. Sodium dodecyl sulphate

Water 0.02 M NaCl 0.03 M NaCl 0.10 M NaCl 0.20 M NaCl 0.40 M NaCl

c.m.c(mM) n

8.1 3.82 3.09 1.39 0.83 0.52

p/n

~

AQ RT

SO(58) 94 100 112(91) 118(105) 126(-129)

0.18 0.14 0.13 0.12 0.14 0.13

-15.8 -16.0 -16.2 -15.9 -15.8 -15.7

Dodecylamine hydrochloride

Water 0.0157 M NaCl 0.0237 M NaCl 0.0460 M NaCl

13.1 10.4 9.25 7.23

56 93 101 142

0.14 0.13 0.12 0.09

-15.2 -15.1 -15.0 -15.2

Decyltrimethyl ammonium bromide

Water 0.013 M NaCl

68.0 63.4

36 38

0.25 0.26

-11.3 -11.3

Dodecyl Water trimethyl 0.013 M NaCl ammonium bromide

15.3 10.7

50 56

0.21 0.17

-14.3 -14.6

Water 0.02 M KCl 0.05 M KCl 0.08 M KCl

14.7 11.3 8.46 6.88

207 377 487 497

0.22 0.25 0.22 0.23

-13.8 -13.6 -13.7 -13.6

Water 0.02 M KBr 0.04 M KBr 0.06 M L B r 0.06 M KBr 0.06 M RbBr 0.08 M KBr 0.10 M KBr

11.6 7.32 4.88 3.96 3.96 3.35 3.36 2.74

58 80 95 87 95 98 95 139

0.20 0.19 0.15 0.17 0.18 0.18 0.21 0.19

-14.9 -14.7 -15.1

Water 0.0025 M KI 0.0050 M KI 0.0100 M KI

5.60 4.53 3.87 2.94

87 90 94 124

0.13 0.11 0.09 -0

-16.7 -17.2 -17.2 -18.0

_____

B. Dodecyl pyridinium chloride Dodecyl pyridinium bromide

Dodecyl pyridinium iodide

+These data from Ford et al. (1966) seem to be on the low side.

-

-14.9 -

-14.6 -14.8

460 I

9: ASSOCIATIONC O L L O I D S

Table 9.4 ~~

~

AGO, ASo, and AHo contributions to micellization (Kishimoto and Sumida 1974; Tori and Nakagawa 1963; Corkill et al. 1964.) ~

Compound

Solvent

Temperature("C)

~

A@

~

AHo TASO (kJ per mol)

9.5 Spectroscopic techniques for investigating micelle structure Spectroscopic techniques based either on optical absorption or emission of light from some 'probe' molecule are now well established for investigating a wide range of physical properties of micellar solutions. A brief outline of several of these techniques is given in the following discussion. For a general description of the principles involved see Turro et al. (1977) and Somasundaran et al. (1999).

9.5.1 Methods for determining the c.m.c. (a) Solubilization of additives The fact that micelles can solubilize relatively large amounts of sparingly water-soluble compounds has been used to measure the onset of micelle formation (Mukerjee and Mysels 1971). The method is to measure the concentration of a chosen sparingly water-soluble substance, possessing a convenient UV-visible absorbing chromophore, in the presence of increasing amounts of surfactant. Below the c.m.c., the concentration of the solubilizate in solution is the same as in aqueous solution in the absence of surfactant. Above the c.m.c., the total amount of the additive in solution increases sharply as the total micelle concentration increases. (b) Spectral change of additives Some dyes such as Pinacyanol and Rhodamine 6G show changes in their absorption spectrum when solubilized by micelles (Mukerjee and Mysels 1971). Other aromatic organic molecules such as naphthalene, anthracene, and, in particular, pyrene (Almgren et al. 1979a) show similar changes but also show changes in their fluorescence spectra when associated with micelles. These spectral changes have been used to monitor the c.m.c. in the same way as the solubilization procedure (see Fig. 9.5.l(a) and (b)). An important feature of the fluorescence method is that, since it is far more sensitive than optical absorption, lower concentrations of probe molecules can be used. This avoids the problems of the additive influencing the c.m.c. of the surfactant, which can occur at high probe concentrations.

SPECTROSCOPIC T E C H N I Q U E SF O R INVESTIGATING MICELLE STRUCTURE

0

(b)

5

10 C, (mol L-')

15

I 461

20x 10-~

Fig. 9.5.1 (a) Pyrene monomer fluorescence in aqueous sodium dodecyl sulphate (SDS) solutions, at SDS concentrationsbelow and above the c.m.c. (After Kalyanasundaramand Thomas 1977, with permission. Copyright American Chemical Society 1977.) (b) Variation of the ratio of intensity of peaks I11 and I from Fig. 9.5.1 (a) as a function of the SDS concentration.

9.5.2 Fluorescence methods for determining aggregation number (a) Steady-state emission quenching method In 1978 Turro and Yekta presented an extremely simple method for determining the number average aggregation number, n, of micelles. The method is based on the quenching of a luminescent probe by a known amount of quencher molecules. The method relies on several assumptions:

462 I

9: ASSOCIATIONC O L L O I D S

(i) both the probe and quencher are completely associated with the micelles; (ii) both the probe and quencher remain attached to the micelle for times much longer than the unquenched lifetime of the luminescent probe; (iii) the quenching in or on a micelle containing both a probe and a quencher molecule is much faster than the emission lifetime of the probe so that emission is observed only from micelles without quenchers; (iv) the distribution of the probe and quencher among the micelles is known; in practice, a Poisson distribution is assumed (Exercise 9.6.1). If the above assumptions are valid, the relative intensity of the fluorescence of a probe is related to the quencher concentration and micelle concentration by the relationship.

I = I0 exp [-Cq/Cm]

(9.5.1)

where I0 is the emission intensity in the absence of quencher (C, = 0) and I the emission in the presence of quencher. The micelle concentration, C,, is given by:

C, = [CT - c.m.c]/n

(9.5.2)

where CT is the total surfactant concentration. Rearranging eqns (9.5.1) and (9.5.2) yields (Exercise 9.6.2): ln(Io/I) =

c, .?z CT - c.m.c'

(9.5.3)

Thus a plot of In (Io/I) as a function of the quencher concentration allows Ei to be determined. For the probe [Ru(bipy)] and quencher [9-methyl anthracene] used by Turro and Yekta for SDS micelles, the range of validity of eqn (9.5.3) has been investigated by Almgren and Lofroth (1980). Using a time-resolved procedure (discussed in the next section) they showed that the method is quite accurate up to aggregation numbers of about 120. Beyond this size the micelles become sufficiently large that condition (iii) above is no longer valid. However, the method would still be applicable if a longer lived probe, such as a phosphorescent molecule, were to replace the ruthenium tris-(bipyridyl) ion. (b) Time-resolved emission quenching method If a probe molecule in a micellar solution is excited by a short radiation pulse (from a laser, say) then, in the presence of quencher molecules, its emission intensity will decay with time according to an expression of the form (see, for example, Almgren and Lofroth 1980):

(9.5.4) where k, (s-l) is the quenching rate constant in the micelle, and to the emission lifetime of the solubilized excited probe in the absence of a quencher.

SPECTROSCOPIC TECHNIQUESFOR INVESTIGATING

00

MICELLE STRUCTURE

I463

It is instructive to consider the form of eqn (9.5.4) at long and short times. For t + we have

(9.5.5) and for t

+ 0 when exp(-k,t)

1 - R,t:

(9.5.6) Thus, the logarithmic emission decay curves for different Cq/Cm (changing C, rather than Cm),will be a family of curves that have the same slope at long times equal to - l / q . However, at short times the slopes depend on C, according to eqn (9.5.6). An example of such a set of emission-time quenching curves is presented in Fig. 9.5.2. Computer fitting routines are generally used to determine k, and Cq/Cm (and, subsequently, n ) from the emission decay curves. A simple method of determining n is to extrapolate the long-time slopes to t = 0 and, if the family of curves has been standardized, the intercept gives Cq/C, from eqn (9.5.5) and % follows from eqn (9.5.2) at the known C,. The advantage of the time-resolved method over the steadyisithat it can be used over a much wider range of state method for determining ? micelle sizes.

I 0

I

I

200

I

400

600

800

Time (ns)

Fig. 9.5.2 Semilog plots of the fluorescence quenching of excited pyrene in 0.05 M SDS as a function of Cu2+ concentration. (From Grieser and Tausch-Treml 1980, with permission. Copyright American Chemical Society 1980.)

464 I

9: ASSOCIATIONC O L L O I D S

9.5.3 Interfacial electrostatic potentials of micelles using solubilized pH indicators T o determine the electrical potential at the interface of a charged micelle, such as SDS or cetyltrimethylammonium bromide (CTAB), use can be made of pH indicators that are bound to the micelle surface. This method was originally explored by Hartley and Roe (1940). They observed a shift in the pK of the indicator in micellar solutions compared to pure aqueous solutions. This shift they attributed to a change in the 'local interfacial' proton concentration HT at the surface of the micelles compared to that in bulk solution, Hb+.The relation between the proton concentration at the interface and that in the bulk solution is given by the Boltzmann equation (Section 7.3):

W+I, = W + l b exp (-e$rlkT)

(9.5.7)

where $r is the interfacial potential and the other constants have their usual meaning. However, the shift in the pK of the indicator in micellar solutions may be caused not only by the electrostatic potential but also by a different local environment at the micellar surface, e.g. by a lower dielectric constant as compared to bulk water. Mukerjee and Banerjee (1964) pointed out that to measure the electrostatic contribution to the pK shift, the intrinsic interfacial pK, pK', must be known. The 'apparent' pK (pKa) is related to pK' by: pK" - pK' = -e$r/2.3 kT

(9.5.8)

Fernandez and Fromherz (1977) in an excellent study on the surface potential of micelles, titrated alkyl coumarins (the fluorescence intensity of these molecules is pHsensitive) in charged and neutral micelles. Figure 9.5.3 shows the characteristic

1.0 0

.-* .+

s u

0.5

O J

B 0

3

4

5

6

7

8 9 Bulk pH

1 0 1 1

1 2 1 3

Fig. 9.5.3 Degree of dissociation of acid pH indicator versus bulk pH. The figure compares the titration of hydroxycoumarin chromophore, hydrophobically bound to positively (CTAB) and negatively (SDS) charged micelles in 24 mM surfactant concentrationin water. (From Fernandez and Fromherz 1977, with permission. Copyright American Chemical Society 1977.)

SPECTROSCOPIC T E C H N I Q U E SF O R INVESTIGATING MICELLE STRUCTURE

I 465

titration curves obtained. T o calculate an electrostatic potential for the charged system, the apparent pK in the neutral micelles was taken as the intrinsic interfacial pK'. Their assumption that the intrinsic pK is similar in the charged and uncharged interface was supported by other experimental data. The extent of the pH shift due to the zerocharge surface environment can be seen in Fig. 9.5.3, where the titration curve of the coumarins in water is shown alongside the titration curve in neutral Triton X-100 micelles. The surface potentials calculated using eqn (9.5.8) for CTAB and SDS were +148 mV and -134 mV respectively. The potentials are sensitive to the total ionic strength of the solutions and the background electrolyte, but these values are very reasonable estimates of the surface potential, $0, expected for spherical micelles of the size suggested in Table 9.3 (Exercise 9.6.3). It appears, therefore, that the probe molecule is in this case sampling the potential in the plane of the head groups and is not merely being affected by the diffuse layer potential, which would be much lower in this case.

9.5.4 Polarity of the micelle-water interface As already indicated in the previous section, molecules located at the micelle-water interface appear to sense an environment which is neither completely water-like nor completely hydrocarbon-like. An exact description of the environment is impossible, but it is possible to relate the characteristics to an apparent dielectric permittivity. This approach has been taken in several investigations of the micelle-water interface. Molecules that show wavelength changes in their spectral absorption or emission bands are commonly used. The method is to measure the position of the spectral feature as a function of the known dielectric permittivity of a solvent. Usually these are made up of dioxanewater, or alcohol-water mixtures. The position of the spectral

Table 9.5 Effective permittivity at the surface of various micelles.

C12 trimethylammonium chloride (DTAC)

40

36

30

ClzTA bromide

35

33

29

Cetyl (C16)TA chloride (CTAC)

31

31

28

30

30

27

Sodium decyl Sulphate (SDeS)

51d

55

Sodium dodecyl sulphate (SDS)

51d

55

Triton X-100

30

30

27

Brij 35

28

29

27

C12 (ethylene oxide)g

28

29

27

c 1 6

TA bromide (CTAB)

Effective permittivity based on aethanol/water mixtures; bdioxane/water mixtures; 'n- alcohols; dmethanol/water mixtures.

466

I 9: ASSOCIATIONC O L L O I D S

band of the molecule in the micelle is compared to its position in the calibration solvent, and an effective permittivity is obtained. Various molecules have been used as dielectric probes including pyrene carboxaldehyde, pyridinium N-phenol betaine, dodecylpyridinium iodide, and benzophenone. The most extensive study of the effective interfacial permittivity was made by Zachariasse et al. (1981). Some of the permittivities they determined for micelles using pyridinium-N-phenol betaine as a probe are listed in Table 9.5. Although there are some differences evident in Table 9.5, depending on the reference solvent, the results show a remarkable degree of consistency. It should be noted, however, that the value obtained is crucially dependent on the structure of the probe molecule, which will presumably tend to sample the environment that is most energetically favourable, and that will depend on the disposition of its own hydrophilic and lipophilic parts. For further discussion see Mukerjee et al. (1977).

9.6 Micellar dynamics As mentioned earlier in this chapter micelles are dynamic units, constantly forming and dissociating on a time-scale in the microsecond to millisecond range. The kinetics of micelle formation and breakdown have mainly been studied by fast relaxation methods (temperature-jump, pressure-jump, ultrasonic absorption, shock-tube methods, etc; see Aniansson et al. 1976 and Muller 1977). In these methods a system is subjected to a sudden perturbation and it is then monitored as it returns to equilibrium or departs from it. Such measurements reveal the presence of two? relaxation processes in the perturbed system -a fast step (nanosecond to microsecond time-scale), related to the exchange of monomers between the bulk aqueous phase and micelles of different sizes, and a slower relaxation (microsecond to millisecond) related to the formation or break-up of micelles. The types of equilbria that exist in micellar solutions can be divided into the following (Muller 1977): A. Ionization

Mn Xfi

+ MS Xfi-l+

X.

(9.6.1)

Here Mn is a micelle with n monomers, and p counterions, X, of either positive or negative charge.

B. Monomer exchange

(9.6.2)

where S is the surfactant monomer.

C. Formation/dissolution process

nS + M,.

+In some cases three relaxation processes have been observed for ionic surfactant solution -a very fast process (ca.50 ns) has been attributed to counter-ion relaxation in the system (see Section 8.8).

(9.6.3)

MICELLAR D Y N A M I C S

D. Partial breakdown and reformation ( a is

Mn f t Ma

+ Mn-a

I467

(9.6.4)

an integer not greatly different from n / 2 ) .

E. Size redistribution (number of micelles unchanged) (9.6.5)

(b is small such that Mm-b is still considered a micelle). F. Size redistribution (number of micelles changed)

It is interesting to note that although a non-equilibrated system can relax by all of the above processes, only two well-defined relaxation times are consistently observed. This is partly a consequence of the widely varying rates involved in the above reaction steps. As already mentioned, process A is very rapid, and generally outside the time regime studied. Process B is the only other process that can occur rapidly and in a single step, and hence it is assigned to the fast relaxation time. (An independent pulse radiolysis study (Almgren et al. 1979b) has confirmed that the fast reaction is due to monomer exchange, at least in the case of SDS.) Process C is actually a shorthand representation of a mechanism with ( n - 1) steps, 1.e.

This sequence has been identified with the slower relaxation event. It has been argued (Muller 1977) that although reactions (9.6.4) and (9.6.6) may also be slow, because they must also proceed through a sequence such as (9.6.7), the relative concentration of species involved is small and therefore not easily recognized.

9.6.1 Kinetics of micelle formation Although there are some differences of opinion on the most appropriate theoretical treatment of micelle formation (see, for example, Muller 1977; Kahlweit 1981), the most widely accepted approach is that of Aniansson and Wall (1974). Their analysis is based on the consideration that there is a distribution of aggregates and micelle sizes in any micellar system. A schematic form of such a distribution is given in Fig. 9.6.1 (after Kahlweit (1981)). In the relaxation treatment of Aniansson and Wall the fast relaxation time constant tl is given as 1

k-

--tl 02

+ k-(CT

-

C,)/%C,

(9.6.8)

where k- is the reverse rate constant for reaction (9.6.2), and C, is the monomer concentration at equilibrium following the perturbation. This result is in accord with

468 I

9: ASSOCIATIONC O L L O I D S

A

n-

Fig. 9.6.1 Micelle size distribution.M,, is the number of aggregates of size a. The aggregates on the left side of the minimum (L) are called submicellar, those on the right-hand side (proper) micelles with mean size of ?i and the width of their size distribution is given as 0.

the observed linear rise of l/tl with CT, the total surfactant concentration, noted in most experiments. A brief discussion of the derivation of this kind of relaxation is given by Vold and Vold (1983,pp. 612-14). The derivation of the slow relaxation process involves a number of assumptions and approximations. The reason for this can be readily appreciated when it is remembered that n steps are required, the rate constants of the individual steps are not all equal, and the concentrations of the intermediate sized aggregates need to be specified. Based on a model of mass flow, the simplifications made by Aniansson and Wall allowed a single relaxation time to be derived and it is given by,

1

(9.6.9)

R1 in the above equation is a function related to the restrictions on flow of monomers from the aggregates based on the mass flow model. R1 itself is dependent on COin a complex way, part of which is due to the fact that the theory does not take into account the redistribution of free counterions, i.e. the theory is a better description for nonionic surfactants. Advances on the Aniansson and Wall model by Chan, Kahlweit and co-workers (1977) resulted in a theory specifically designed for ionic surfactants. Although there are weaknesses in the model, it predicts that, for ionic surfactants, a plot of l / q against Co should exhibit a maximum, while for non-ionics the relaxation rate should be a monotonically increasing function of Co, in good agreement with observed results. Some examples of the relaxation times observed for alkyl sulphates are given in Table 9.6,showing the time scales involved in the fast and slow processes.

MICELLAR D Y N A M I C S

I469

Table 9.6 Relaxation times tl (ps) and

Surfactant

t2

(ms) for some sodium alkyl sulphates.

Temperature ("C)

Concentration (mM)

tl

rz

(w)

(ms)

1

760

350

2.1 2.1 2.1 3

320 245 155 125

41 19 7 34

-

1.8 50

10 50

-

The relaxation times are, of course, related to the rate constants for association ( K + ) and dissociation ( K - ) as represented by reaction (9.6.2) earlier. The rate constants for a series of sodium alkyl sulphates are given in Table 9.7. The obvious pattern shown in Table 9.7 is that, as the alkyl chain becomes larger, the exit rate of monomer becomes appreciably slower, as one would expect with increasing hydrophobic character of the monomer. The association rate constant also decreases although only slightly, reflecting basically a diffusion controlled rate step with some electrostatic repulsion involved between the charged micelle and the anionic monomer (Aniansson e t al. 1976). In summary it may be said that although some uncertainties remain in understanding the results of the relaxation processes in micellar systems, a reasonably good picture of the equilibria has been established.

Table 9.7 Kinetic parameters of association and dissociation of alkyl sulphatesfrom their micelles (from Aniansson et al. 1976).

NaC6so4

17

420

1320

3.2

0.0405

NaC7 SO4

22

220

730

3.3

0.1

NaCsSO4

27

130

100

0.77

0.207

NaC9SO4

33

60

140

2.3

0.55

NaCll SO4

52

16

40

2.6

3.25

NaC12 SO4

64

8.2

10

1.2

7.80

NaC14SO4

80

2.05

Note: The equilibrium constant K = k+ii/k-

0.96 %

0.47

39

ii/(c.m.c); ii is the average aggregation number.

470 I

9: ASSOCIATIONC O L L O I D S

Table 9.8 Residence times in micelles (ps).

Probe molecule

SDS

CTAB

Solubility in water ( p mol L-')

59

303

0.22

243

588

0.6

10

62

41

Naphthalene

4

13

220

Benzene

0.23

Anthracene Pyrene Biphenyl

1.3

2.3

104

9.6.2Residence times of probe molecules in micelles Like the surfactant monomer, the micelle-solubilized additive is not rigidly fixed in the micelles. Not only can it move about within the micelle, but it is in constant dynamic equilibrium with the bulk aqueous phase. The residence time of small molecular substances, such as those commonly used as probes, is an important consideration when interpreting dynamic results such as fluorescence quenching behaviour. Since most hydrophobic probes have residence times that are longer than their fluorescence lifetimes, fluorescence quenching techniques are unsuitable for determining probe residence lifetimes in micelles. However, it was shown by Almgren et al. (19796) that if phosphorescence is used, residence times can be determined. In Table 9.8 several probe molecules are listed with their residence times? and solubility in water. As can be seen from Table 9.8, apart from anthracene, the residence times of the various probes decrease with increase in their solubility in water. Also, the residence times are dependent on the micelle type, which in part may be due to weak complexes forming between the quaternary ammonium head group of CTAB and the solubilized additive. The existence of finite residence times means that over long periods of time the additive is uniformly distributed among the micelles in solution. However, at time intervals less than their residence time the distribution of the additives will be statistical, having a form given by the Poisson distribution (Exercise 9.6.1). Virtually all kinetic studies to date have arrived at this conclusion.

9.6.3Determination of microfluidity using fluorescence probes The most common method for studying fluidity of micelles has been fluorescence depolarization, which measures the resistance of a probe molecule to rotational or reorientational motion in its local environment. The fluorescence emission intensity of a solubilized probe is measured at crossed (11)and parallel (Ill), positions of the polarizers. The degree of polarization is given by:

(9.6.10) +Thisis the time taken for a fraction (1 - e-'), i.e. 63 per cent, of the probe molecules to escape.

MICELLAR D Y N A M I C S

I471

The Perrin (1932) equation relates the degree of polarization d to other parameters of the system by:

(9.6.11) where do is the degree of polarization in an extremely viscous solvent, TF is the average lifetime of the fluorescent molecule, VOis the effective volume of the molecule, and q is the viscosity of the environment. The more viscous the environment, the larger the value of d. An alternative method (Zachariasse 1978) for measuring microfluidity is based on the formation of intramolecular excimers of molecules with two chromophores (such as dipyrenyl alkanes). Th e relative yield of excimer to monomer (obtained from their fluorescence emission) is viscosity dependent, so by comparing this ratio in micellar solutions with a reference solvent of known viscosity the microfluidity can be found. The results obtained using different methods and different probes (Somasundaran et al. (1999) are, however, highly variable (ranging from 4-50 centipoise for SDS), which is probably a reflection of the fact that the effective viscosity depends on position in the micelle -varying from high fluidity in the interior to more viscous at the interface. Such variations with position in the micelle can be monitored by n.m.r. spectroscopy (see next section).

9.6.4 Other spectroscopic techniques The most important technique remaining is that of n.m.r. spectroscopy, which can be used in a variety of modes and with a number of probe nuclei (hydrogen, the halides, the alkali metals) to study the phenomena of hydration, ion binding, and the mobility of segments of the hydrocarbon chain. The interpretation of results is, however, rather too complicated to go into here and the reader is referred to more specialized reviews (e.g. Lindman et al. 1977; Wennerstrom and Lindman 1979). Unfortunately, one of the earliest applications of n.m.r. (using fluorine-substituted hydrocarbon chains) led to the conclusion that there must be significant penetration of water into the core (Muller and Birkhahn 1967; Muller and Platko 1971), a result that is now generally regarded as erroneous (Mukerjee and Mysels 1975). That work did, however, ultimately point up the anomalous nature of the fluoro-hydrocarbon mixing behaviour as indicated in Section 9.2.1. Evidently a terminal -CF3 group on a hydrocarbon chain spends much more time near the surface of the micelle than does the corresponding - C H 3 terminal group. Studies of the alkali metal ions by n.m.r. generally reveal that, even when bound to the head groups, they retain their primary hydration sheath. This is consistent with the usual picture of the Stern layer. The n.m.r. data also confirm the notion, mentioned above, that the rotational freedom of the segments of the hydrocarbon chain increases as one moves away from the head group. The ‘microviscosity’ likewise tends to be higher in the neighbourhood of the head groups. The most convincing evidence for a liquid-like structure of the hydrocarbon core also comes from the n.m.r. studies. Wennerstrom et al. (1979) used TI relaxation times of the I3Cnuclei along the chain to

472 I

9: ASSOCIATIONC O L L O I D S

obtain an estimate of the (rotational) correlation time. Values of around 10 ps were obtained, in close agreement with measurements on liquid hydrocarbons. Electron spin resonance (e sr.) spectroscopy has also been used in micellar studies by Stilbs and Lindman (1974). They investigated the mobility of the vanadyl ion (V02+) as a counterion on the micellar surface, and found its mobility to be very high. The rotational correlation time was -70 ps, consistent with a fairly loose association with the head groups. Substitution of a nitrosyl group (-NO) along the hydrocarbon chain could also provide an unpaired electron, from which the e.s.r. signal would give information on the local environment in the core. Such procedures, however, can lead to equivocal results because the presence of the probe group can modify the mobility of the chain to which it is attached. For a recent review, see Somasundaran et al. (1999). r

Exercises 9.6.1. The probability of finding a micelle with i solute species is given by the Poisson relationship, -.

pi= S"XP(-S) i!

-

where S =

[solute] [micelles].

Given that [solute] = lop3M and [micelles] = 5 x lop4,lop3 M, and 2 x lop3 M, calculate the probability of micelles containing 0, 1, 2, 3, and 4 solute molecules. Plot a graph of probability vs. number of solute molecules for each micelle concentration; note the relative proportion of molecules per micelle. Is it misleading to think that if [solute] = [micelle] each micelle contains one solute molecule? 9.6.2Establish eqn (9.5.3) assuming that both probe and quencher are spread amongst the micelles in a Poisson distribution. 9.6.3Calculate the charge, Q, on a micelle of radius 2.5 nm in 24 mM solution if the surface potential is -1 38 mV, using the semi-empirical Gouy-Chapman expression (eqn (7.11.11)). Compare your result with the approximate value from eqn (9.4.12). (Note that eqn (7.11.11) is quite accurate for such large potentials and small a values. The error, compared with the exact computer solution for @ = 150 mV and KU M 1 is only about 1 per cent (Loeb et al. 1961).) 9.6.4Addition of indifferent electrolyte usually lowers the estimated surface potential of a micelle. Fernandez and Fromherz (1977) found that, for 1:l electrolyte, d@/ d loglo C M -60 mV. Show that this is the expected result for a system where the diffuse layer charge remains constant. (Use the equations of Section 7.3. for a flat double layer.)

9.7 Molecular packing and its effect on aggregate formation So far, we have dealt in detail with both thermodynamic and kinetic considerations, but have assumed for the most part that the micelles are roughly spherical in shape. We

MOLECULAR PACKING A N D ITS EFFECT O N AGGREGATE F OR M A T ION

1473

have made little allowance for the existence of rod-like micelles, vesicles, or bilayers but Tanford (1972), Mitchell and Ninham (1981), and Israelachvili et al. (1976, 1977, 1980) have revived an idea originally proposed by Hartley (1941), viz. that molecular packing plays a crucial role in determining allowed structures, at least in the case of dilute surfactant solutions. Aggregation is, as outlined previously, described by either the mass action or phase models. Typically, chemical potentials of the monomer in the aggregate and in solution at equilibrium are equated and the normal relationships are derived. In this section we wish to examine the contributions to the standard chemical potential or molar free energy per surfactant molecule in the aggregate, &: p,0 = p,B

+ pS,+ p: + molecular packing term.

(9.7.1)

These terms may be further described, referring to Fig. 9.7.l(a) for clarification. The bulk term, p:, is a constant and is a measure of the free energy change involved in removing hydrophobic tails from an aqueous environment into the micelle interior. The latter is assumed to be liquid-like which, as we have already seen, is a sound assumption (Vikingstad and Hoiland 1978).The surface term, pk,includes a quantity yA to account for the fact that the hydrophobic tails have some residual contact with the aqueous phase. A is the area per surfactant head group and y is the hydrocarbonwater interfacial tension. Head group interactions, which may be due to steric, hydration, electrostatic, and other forces contribute a repulsion energy. Their quantitative description is still elusive; however, for electrostatic repulsion, an energy contribution varying inversely with A would be expected. pk then takes the form: (9.7.2) reaching a maximum at 2yAo where A0 is a limiting or optimal area per head group. The curvature term, pk,accounts for reductions in the effective surface tension and alterations in the electrostatic energy when a spherical surface is formed, rather than a planar one. Generally speaking these corrections may be ignored, except for very special cases (Israelachvili et al. 1976). Lastly the packing term needs to be considered. One needs to account for the fact that the interior of the aggregate is fluid-like and incompressible. The aggregate radius or radii, hydrophobic chain volume, and surface area per head group are of paramount importance here, as discussed below. In more concentrated systems, where other interactions take place, eqn (9.7.2) will contain additional contributions. For the present, however, Israelachvili and Ninham (1977) have shown that when &, is decomposed into the first three terms, the correct dependence of c.m.c. on hydrocarbon chain length is predicted as are the effects of temperature and ionic strength. However the existence of rod-like micelles or bilayers suggests certain geometric constraints and it is with these that we shall now concern ourselves, i.e. the nature of the molecular packing term. For simplicity, consider firstly a spherical micelle (Fig. 9.7.l(a)). The radius, R", surface area per head group A and hydrophobic chain volume V are linked by

V / A = R"/3.

(9.7.3)

474 I

9:ASSOCIATION COLLOIDS

The radius of a spherical micelle cannot be greater than a specific critical length I, slightly less than the fully extended length of the hydrocarbon chain, if it is assumed that the head group never moves into the core. Clearly, then, when V/A& > 1/3

/Area

A

Volume

Fig. 9.7.1 (a) Schematic representation of a model spherical micelle of core radius R”. (After Mitchell and Ninham 1981, with permission.) (b) Geometric packing of a a hydrocarbon region of volume Vand surface area A, at a surface with two radii of curvature, RI and Rz. Equation (9.7.7) gives the relation between V ,A, R1, and Rz and the length, 1, of the hydrocarbon. For both R1, Rz > 1 a void region is formed behind the hydrocarbon region. (After Israelachvili et al. 1976, with permission.)

MOLECULAR PACKING A N D ITS EFFECT O N AGGREGATE F OR M A T ION

1475

spherical micelles will not form unless A > Ao. The critical condition for the formation of spheres is

1

VIA01 --. “-3

(9.7.4)

Similarly, for cylindrical micelles, it may be shown that

1

VIA01 - “-2

(9.7.5)

V/AOlC= 1.

(9.7.6)

and for planar bilayers

Any aggregated structure must satisfy two basic criteria: (a) no point within the aggregate can be farther from the surface of tension than I,; (b) the total hydrocarbon must approximatelysatisfy V / Z = &/A0 core volume, V ,and the total surface area, d, = n, the aggregation number. Here V is the volume occupied by a single hydrocarbon molecule of the appropriate chain length in the liquid state. This criterion is only an approximate one since the average surface area per surfactant head group is assumed to be equal to Ao. Between a sphere and a cylinder, one might reasonably expect to find a variety of transition shapes. Thus if we consider a surfactant (Fig. 9.7.l(b))with a head group of surface area A and liquid hydrocarbon core volume V in a micelle or bilayer vesicle where the local radii of curvature are R1 and R2, the following equation results:

(9.7.7) where 1 is the length of the hydrocarbon region of the amphiphile. Equation (9.7.7)is exact for spheres ( R I = R2), cylinders (R2 = co),and planar surfaces (R1 = R2 = 00) and is accurate to within one per cent for other cases (Israelachvili et al. 1976). It may then be predicted that bilayers or vesicles exist when

1 < V/AOl, < 1 2

-

(9.7.8)

and inverted structures occur when

V/AOl, > 1.

(9.7.9)

This deceptively simple packing model allows many physical properties of micelles and vesicles such as size, shape, polydispersity, etc. to be predicted. The interested reader should consult the original references for more detailed discussions.

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9.8 Statistical thermodynamics of chain packing in micelles In order to model, theoretically, the hydrocarbon chains in micelles, it is first necessary to understand several points. Fully saturated alkyl chains are very flexible. Each C-C bond in a fully extended alkyl chain exists in a trans state but each bond can also exist in two other conformations - the gauche+ and gauche- states, which can each be formed at an energy of -0.8 KTat room temperature. Therefore, in a fully saturated alkyl chain of length 12 (a dodecylsulphate chain for instance) there are a very large number of fairly low energy states of the chain. It is possible to imagine that fully saturated alkyl chains can pack in a frozen array of all-trans chains, and indeed this is the packing found in solid bulk n-alkane. However, without considerable (energetically unfavourable) hydrocarbon-water contact, it is impossible to see how such an array could exist in a micelle. Elementary geometrical considerations (Section 9.7) and the experimental evidence discussed in Section 9.6 suggest that the interior of a micelle is liquid, and that the alkyl chains are conformationally disordered. In a non-polar liquid (such as the micellar interior), the packing density is determined by the combined action of very short-range intermolecular repulsive forces (arising from overlap of electron clouds) and longer range van der Waals attractions. Since the chains are chemically identical to n-alkane chains, we should expect a packing density almost identical to that found for bulk liquid n-alkane, as noted in Section 9.7. There is little doubt that, for an ionic surfactant, the head groups are almost completely excluded from the hydrophobic core of a micelle. There was, however, some disagreement about the extent of water penetration in the core, as noted in Section 9.6.4. Experiments in which probe molecules chemically bonded to different parts of the alkyl chains were observed to behave as if they sat in a partly hydrophilic environment have, in the past, been interpreted as implying extensive water penetration in the core (see, for example, Menger et al. 1978). We have seen earlier, however, that free energies of micellization are comparable with free energies of transfer of alkyl chains from water to bulk n-alkane which implies that the core of a micelle must be almost devoid of water. Neutron scattering experiments (Bendedouch et al. 1983; Cabane et al. 1983) and n.m.r. relaxation experiments (Halle and Carstrom 1981) have given unequivocal evidence that water penetration in the core is minimal and that the hydrocarbon-water interface is smooth, with an average roughness of the order of the diameter of a water molecule. A theoretical model has been developed in the light of all this evidence by Gruen (1981). The model assumes the existence of a hydrophobic core that neither head groups nor water can enter. The amphiphile chains are allowed to exist in all their possible conformations (trans, gauche+, and gauche- states for each C-C bond). On average, the chains are constrained to pack into the hydrophobic core of the micelle at liquid alkane density throughout. T o a limited extent they may also exist outside the hydrophobic core, but they are then subject to an increased free energy. Each chain conformation is assigned its appropriate Boltzmann factor and thermodynamic averages over all conformations are evaluated. Several illuminating conclusions emerge from the study. They may be illustrated by considering a spherical micelle formed from a surfactant with an alkyl chain of length 12 (e.g. a dodecylsulphate). If the hydrophobic core of the micelle has a radius equal to

STATISTICAL T H E R M O D Y N A M I C S OF C H A I N PACKING I N MICELLES

1477

the length of a fully extended chain (1.67 nm), it will contain approximately 56 chains. The model predicts that the free energy cost of packing the chains into this structure is less than 0.5 kT per chain. By comparison, the free energy gained when a C12 chain is transferred from water to bulk n-alkane is 20kT (recall Section 9.4 above). In the micelle one or two chains must be completely straight (in their all-trans state) in order to fill the volume at the centre of the micelle, but no more than one or two. On average, the model predicts a loss of only 0.2 gauche bonds per chain on transfer from a bulk n-alkane environment to the micelle. The fact that half the volume of the hydrophobic core occurs within 0.34 nm of its surface has important consequences. The mean position of each segment in the chain is nearer to the surface of the aggregate than to the micelle centre. The terminal CH3 group, although on average closer to the centre than any other group, sits a mean distance of 1.04 nm from the centre (and only 0.63 nm from the core surface). Because of the liquid-like micelle interior, and the flexibility of the chains, all segments sample the surface of the micelle. Even the terminal CH3 group is in contact with the surface (and hence in contact with a partly hydrophilic environment) approximately 20 per cent of the time. (This observation explains why probes attached to any part of the chain behave as though they were in a partly hydrophilic environment.) The model suggests that the hydrocarbon-water interface is fairly sharp. 0.2 nm beyond the core surface, the average hydrocarbon volume fraction is 0.02 and falling fast. It may seem extraordinary that a structure as dynamic as a micelle (with monomers being associated with the micelle for only lop6- lo-’ s) could have such a smooth surface. It occurs because the hydrophobic effect is very strong and so the vast majority of time that any monomer is associated with the micelle it sits with almost all of its chain inside the hydrophobic core. It is this picture of the SDS micelle that was used in Fig. 1.4.4 for the spherical micelle structure. Each of the five spherical shells contains approximately the correct number of chain segments to ensure an even packing density throughout. Note particularly the large fraction that is in the outermost shell. This model is capable of reconciling a considerable body of experimental evidence but there remain some disagreements over details. Hayter and Penfold (1981), for example, argue that the neutron scattering results suggest a slightly rougher hydrocarbon-water interface, though the difference is not great.

-

References Almgren, M., Grieser, F., and Thomas, J.K. (1979a).J.Am. Chem. Soc. 101,279. Almgren, M., Grieser, F., and Thomas, J.K. (19796).J. Chem. Soc. Faraday Trans. I 75, 1674. Almgren, M. and Lofroth, J.E. (1980).J Colloid Interface Sci. 81,486. Anacker, E.W. (1970).Micelle formation of cationic surfactants in aqueous media. In Cationic surfactants (ed. E. Jungermann). Marcel Dekker, New York. Aniansson, E.G. (1978).J Phys. Chem. 82,2805. Aniansson, E.G. and Wall, S.N. (1974).J.Phys. Chern.78, 1024. Aniansson, E.G., Wall, S.N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R, Lang, J., and Tondre, C. (1976).J Phys. Chem. 80,905.

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Bendedouch, D., Chen, S.-H., and Koehler, W.C. (1983).3. Phys. Chem. 87,153. Bids, B.P. (ed.) (1999). Modern characterization methods ofsurfactant system, pp. 601. No. 83 in Surfactant science series. Marcel Dekker, New York. Cabane, B., Duplessix, R, and Zemb, T. (1983). In Surfactants in solution (ed. K. L. Mittal and B. Lindman). Plenum Press, New York. Chan, S.-K. and Kahlweit, M. (1977). Ber. Buns. Phys. Chem. 81, 1294. Chan, S.-K., Herrman, U., Ostner, W., and Kahlweit, M. (1977). Ber. Buns. Phys. Chem. 81,60 and 396. Chan, S.-K., Herrman, U., Ostner, W., and Kahlweit, M. (1978). Ber. Buns. Phys. Chem. 82, 380. Clint, J.H. (1992). Surfactant aggregation, pp. xi 283. Chapman and Hall New York. Corkill, J.M., Goodmann, J.F., and Harrold, S.P. (1964). Trans. Faraday SOL. 60, 202. Debye, P. and Anacker, E.W. (1951). 3. Phys. Colloid Chem. 55, 644. Desnoyers, J.E., Caron, G., DeLisi, R., Roberts, D., Roux, A., and Perron, G. (1983). 3. Phys. Chem. 87, 1397406. Ekwall, P., Mandell, L., and Solyom, P. (1971). J. Colloid Interface Sci. 36, 519-28. Evans, D.F. and Ninham, B.W. (1983). J. Phys. Chem. 87, 5025-32. Evans, D.F. and Ninham, B.W. (1986). J. Phys. Chem. 90,226-34. Evans, D.F., Allen, M., Ninham, B.W., and Fouda, A. (1984).3. Solution Chem. 13, 87-101. Evans, D.F. and Wennerstrom, H. (1999). The Colloidal Domain (2nd edn), pp. 632. Wiley-VCH, New York. Fernandez, M.C. and Fromherz, P. (1977). 3. Phys. Chem. 81, 1755. Fisher, L.R. and Oakenfull, D.G. (1977). Chem. Soc. Rev. 6,25. Ford, W.P.J., Ottewill, R.H., and Parriera, H.C. (1966) 3. Colloid Interface Sci. 21, 522. Frank, H.S. and Evans, M.W. (1945). 3. Chem. Phys. 13, 507. Franks, F. (1983). Water. Royal Society of Chemistry, London. Fuerstenau, M.C. (1976). Flotation, Vols 1 and 2. Am. Inst. MMPE, New York. Gaines, G.L. (1966). Insoluble monolayers at liquid-gas interfaces. Interscience, New York. Grieser, F. and Tausch-Treml, R. (1980). 3. Am. Chem. SOL.102, 7258. Gruen, D.W.R. (1981).J. Colloid Interface Sci. 84. 281. Gunnarson, G., Jonsson, B., and Wennerstrom, H. (1980). J. Phys. Chem. 84, 3 114-21. Hall, D.G. and Pethica, B.A. (1967). Thermodynamics of micelle formation. In Nonionic surfactants (ed. M. Schick), Chapter 16. Marcel Dekker, New York. Halle, B. and Carlstrom, G. (1981). 3. Phys. Chem. 85, 2142. Hartley, G.S. (1936). Aqueous solutions ofparaf3n chain salts. Herman et Cie, Paris. Hartley, G.S. (1941). Trans. Faraday Soc. 37, 130. Hartley, G.S. and Roe, J.W. (1940). Trans. Faraday SOL.36, 101. Hayter, J.B. and Penfold, J. (1981).3. Chem. SOL.Faraday Trans. 177, 185143. Hill, T.L. (1963). Thermodynamics of small systems, Vol. 1. Benjamin, New York. Israelachvili, J.N. and Ninham, B.W. (1977). 3. Colloid Interface Sci. 58, 14-25. Israelachvili, J.N., Mitchell, D.J., and Ninham, B.W. (1976). 3. Chem SOL. Faraday Trans. 2 72, 1525. Israelachvili, J.N., Mitchell, D.J., and Ninham, B.W. (1977). Biochem. Biophys. Acta 470, 185.

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Israelachvili, J.N., Marcelja, S., and Horn, R.G. (1980).Q Rev. Biophys. 13,121. Kahlweit, M. (1981). Pure Appl. Chem. 53,2069. Kalyanasundaram, K. and Thomas, J.K. (1977).J. Am. Chem. Soc. 99,203944. Kertes, A.S. (1977). Aggregation of surfactants in hydrocarbons. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) Vol. 1, pp. 445-54. Plenum Press, New York. Kishimoto, J. and Sumida, K. (1974). Chem. Pharm. Bull. (Japan) 22, 1108. Kitahara, A. (1967). In Nonionic surfactants (ed. M.J. Schick), p. 289. Marcel Dekker, New York. Kitchener, J.A. (1964). In Recent progress in surface science (ed. J.F. Danielli, K. Pankhurst, and A.C. Riddiford) Vol. 1. Academic Press, New York. Kratohvil, J. (1980). 3. Colloid Interface Sci. 75, 271-5. Kresheck, G.C. (1975). Surfactants. In Water - a comprehensive treatise (ed. F. Franks) Chapter 2, pp. 95-167. Plenum Press, New York. Leja, J. (1982). Surface chemistry offroth flotation, pp. 284-6. Plenum Press, New York. Lindman, B., Lindblom, G. Wennerstrom, H., and Gustavsson, H. (1977). Ionic interactions in amphiphilic systems studied by n.m.r. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) pp. 195-227. Plenum Press, New York. Lindman, B. and Wennerstrom, H. (1980). Topics in current chemistry 87, pp. 1-83. Springer, Berlin. Loeb, A.L., Wiersema, P.H., and Overbeek, J. Th. G. (1961). The electrical double layer around a spherical colloid particle, p. 37. M I T Press, Cambridge, Mass. Mazer, N.A., Carey, M.C., and Benedek, G.B. (1977). The size, shape and thermodynamics of sodium dodecyl sulphate (SDS) micelles using quasielastic light-scattering spectroscopy. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) Vol. 1, pp. 359-81. Plenum Press, New York. McBain, J.W. (1950). Colloid science. D.C. Heath, Boston. Menger, F.M., Jerkumica, J.M., and Johnson, J.C. (1978).J. Am. Chem. Soc. 100,4676. Missel, P.J., Mazer, N.A., Benedek, G.B., Young, C.Y., and Carey, M.C. (1980). J. Phys. Chem. 84, 1044-57. Mitchell, D.J. and Ninham, B.W. (1981). 3’. Chem. Soc. Faraday Trans. 2 77, 601. Mukerjee, P. (1974). J Pharm. Sci. 63,972. Mukerjee, P. (1975). Differing patterns of self-association and micelle formation. In Physical chemistry, enriching topics from colloid and surface science (ed. H. van Olphen and K.J. Mysels) Chapter 9; IUPAC Commission 1.6. Theorex, La Jolla, California. Mukerjee, P. and Banerjee, K. (1964). 3’.Phys. Chem. 68, 3567. Mukerjee, P. and Ghosh, A.-K. (1970).3. Am. Chem. Soc. 92, 6419 Mukerjee, P. and Mysels, K.J. (1971). Critical micelle concentrations of aqueous surfactant systems. NSRDS-NBS 36, National Bureau of Standards. US Government Printing Ofice, Washington, D. C. Mukerjee, P. and Mysels, K.J. (1975). Anomalies of partially fluorinated surfactant micelles. A.C.S. symposia (ed. K.L. Mittal) Series 9, p. 239. American Chemical Society, Washington, D.C. Mukerjee, P., Cardinal, J.R., and Desai, N.R. (1977). The nature of the local micro-environments in aqueous micellar systems. In Micellization, solubilization and micro- emulsions (ed. K.L. Mittal) pp. 24141. Plenum Press, New York.

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Muller, N. (1977). Errors in micellization enthalpies from temperature dependence of c.m.c.s. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) Vol. 1, pp. 229-39. Plenum Press, New York. Muller, N. and Birkhahn, R.H. (1967). J. Phys. Chem. 71,957. Muller, N. and Platko, F.E. (1971). J. Phys. Chem. 75, 547 Ninham, B.W. (1981). Pure App. Chem. 53,2135. Oakenfull, D.G. and Fenwick D.E. (1974). J. Phys. Chem. 78, 1759. Oakenfull, D.G. and Fisher, L.R. (1977).J. Phys. Chem. 81, 1838. Oakenfull, D.G. and Fisher, L.R. (1978). J. Phys. Chem. 82,2443-5. Os, N.M. van, Haak, J.R., and Rupert, L.A.M. (1993). Physico- chemicalproperties of selected anionic, cationic and non-ionic surfactants, pp. viii 608. Elsevier, Amsterdam and New York. Perrin, F. (1932). Ann. Phys. Paris. 17, 283. Phillips, J.N. (1955). Trans. Faraday Soc. 51, 561. Ramadan, M.S., Evans, D.F., and Lumry, R. (1983).J Phys. Chem. 87,453843. 93,6787. Ray, A. and Nemethy, G. (1971). J. Am. Chem. SOC. Schick, M. J. (ed.) (1967). Nonionic surfactants. Marcel Dekker, New York. Schwuger, M. (1971). Ber. Buns. Phys. Chem. 75, 167. Shinoda, K. (1963). Colloidalsurfactants: some physico-chemical properties. Academic Press, New York. Shinoda, K. (1978). Principles of solution and solubility. Marcel Dekker, New York. Small, D.M. (1968). Adv. Chem. Ser. 84, 31. Somasundaran, P., Huang, L., and Fan, A. (1999). Fluorescence and ESR spectroscopy. In Modern methods of surfactant characterization. Chapter 7. pp. 213-54. (ed. B.P. Binks) Surfactant Science Series, 83. Marcel Dekker, New York. Stigter, D. and Overbeek, J.Th.G. (1957). Proc. 2nd Internat. Congress on Surface Activity. 1, p. 311. Stigter, D. (1975~).Electrostatic interactions in aqueous environments. In Physical chemistry: enriching topics from colloid and surface science (ed. H. van Olphen and K.J. Mysels) Chapter 12; IUPAC Commission 1.6. Theorex, La Jolla, California. Stigter, D. (19756). J. Phys. Chem. 79, 1015-22. Stilbs, P. and Lindman, B. (1974). J. Colloid Interface Sci. 46, 177. Swarbrick, J. and Daruwala, J. (1969). J. Phys. Chem. 73, 2627. Swarbrick, J. and Daruwala, J. (1970). J. Phys. Chem. 74, 1293. Tanford, C. (1972). J. Phys. Chem. 76,3020. Tanford, C. (1977). Thermodynamics of micellization of simple amphiphiles in aqueous media. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) pp. 119-32. Plenum Press, New York. Tanford, C. (1980). The hydrophobic effect. Formation of micelles and biological membranes (2nd edn). Wiley, New York. Tori, K. and Nakagawa, T. (1963). Kolloid-Z. 2. Polym. 188,47; 189, 50. Ts’o, P.O.P. (1968). In Molecular associations in biology (ed. B. Pullman). Academic Press, New York. Turro, N.J., Geiger, M.W., Hautala, R.R., and Schore, N.E. (1977). Fluorescent probes for micellar systems. In Micellization, solubilization and microemulsions (ed. K.L. Mittal) pp. 75-86. Plenum Press, New York. 100, 5951. Turro, N.J. and Yekta, A. (1978).J. Am. Chem. SOC. Vikingstad, E. and Hoiland, H. (1978). J. Colloid Interface Sci. 64, 510.

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Vold, R.D. and Vold, M.J. (1983). Colloid and interface chemistry. Addison-Wesley, Reading Mass. Wennerstrom, H. and Lindman, B. (1979). Phys. Rep. 1, 1. Wennerstrom, H., Lindman, B., Soderman, O., Drakenberg, T., and Rosenholm, J.B. (1979). 3’.Am. Chem. Soc. 101, 6860. Zachariasse, K.A. (1978). Chem. Phys. Lett. 57,429. Zachariasse, K.A., Nguyen van Phuc, and Kozankiewicz, B. (1981).J. Phys. Chem. 85, 267&83. Zana, R. (1978). 3’.Phys. Chem. 82,2440-3.

I481

Adsorption a t Charged Interfaces 10.1 Introduction 10.1 .I The Gouy-Chapman-Stern-Grahame

(GCSG) model revisited

10.2 Adsorption of potential determining ions 10.2.1 Surface charge density and surface potential 10.2.2 Surface complexation models (a) Single-site model (b) Two-site model 10.2.3 Position of the plane of shear 10.3 Detection of Stern layer adsorption

10.3.1 Modificationsto the Stern layer model 10.4 The oxide-solution interface 10.5 Adsorption of multivalent ions 10.5.1 Adsorption of hydrolysable metal ions onto oxide surfaces 10.6 Surfactant adsorption 10.6.1 Ionic surfactant adsorption on a hydrophobic surface (Agl) 10.6.2 Ionic surfactant adsorption on hydrophilic (oxide) surfaces 10.6.3 Adsorption of ionic surfactants on other surfaces 10.6.4Adsorption of non-ionic surfactants

10.1 Introduction We discussed the adsorption of uncharged molecules at the liquid-liquid interface briefly in Section 2.4 and at the solid-liquid interface more extensively in Chapter 6. In this chapter we will concentrate attention on the adsorption of charged species at charged interfaces, in an attempt to reconcile the data obtained from equilibrium (usually titration) studies with those obtained from electrokinetics. The classical work in this field is on the mercury-solution interface, on which there is much literature. Delahay (1965) (Chapter 5) lists (after Parsons 1958) the various isotherms which have been used to describe the adsorption of a solute onto a charged surface. At the mercurysolution interface the analysis can be taken quite a long way and one can compare the various possible isotherms (Henry, Volmer, Langmuir, Frumkin, Temkin, etc.) to determine which one best fits the data. From the

482

INTRODUCTION

I483

corresponding equation of state for the ‘pressure’ of the adsorbed species (i.e. the spreading pressure (Section 2.10.1)) one can then attempt to draw some inferences about the nature of the interaction between the adsorbed molecules (both with one another and with the surface). Surveys of the isotherms commonly used in colloid and surface chemistry have been given by, for example, Davies and Rideal 1963 (Chapter 4), Adamson 1969 (Chapter 8), Vold and Vold 1983 (Chapter 4) and Dobias and Rybinski (1999). Some of these are purely empirical relationshipsand some have a more or less solid theoreticalbasis. Here we are concerned primarily with adsorption of ionic species onto surfaces which are usually themselves charged by ionic adsorption or dissociation processes. Coulombic interactions will always be important and are often difficult to separate from other ‘chemical’ bonding interactions. The isotherm most commonly used is the Langmuir equation (6.4.6) and its extension to charged interfaces, the Stern equation which was introduced in Section 7.4.3. Adsorption from fairly dilute solutions onto the solid-liquid or liquid-liquid (emulsion) interface is usually limited to, at most, a monolayer, and the Stern model can usually give a good account of the behaviour. The Stern model is, however, most readily applied to a metallic conducting interface where the surface charge can be regarded as smeared out uniformly and the surface itself is at a constant potential, $0. The classical silver iodide surface (Section 7.7) could be reasonably approximated by a constant potential surface and one might expect the counterions to be restricted in their approach to a very smooth AgI surface in somewhat the same way as is observed at the mercury-solution interface. That surface is also hydrophobic like mercury. But for most colloid surfaces the surface potential develops as a result of the presence of individual ionic groups which can be quite well separated one from another. Furthermore, the intrinsic surface roughness means that the locus of the centres of counterions may be very close to that of the surface ionic charges. The first layers of the interfacial region may then be likened to a very concentrated electrolyte solution where ionic activities may be far removed from their bulk solution values and ion pairing may be more the rule that the exception. The current models of the oxidesolution interface attempt to take these effects into account by dealing specifically with the chemical reaction equilibria which are involved. These are the bases of the sitedissociation-site-binding models.

10.1.I The Gouy-Chapman-Stern-Grahame

(GCSG) model revisited

We noted in Section 7.3.4 that the complete solution of the GCSG model required the solution of six simultaneous equations to obtain the values of $0, $i, $d, Go, Oi, and Od. Those equations are (7.3.27, 7.3.37, 7.3.38, 7.3.39, and 7.4.11) and we require a final one linking the surface potential to the concentration (strictly activity) of the potential determining ions. For the silver-silver iodide surface we use the Nernst equation (7.7.5).Equations (7.3.37 and 38) can be replaced by estimates of the capacitance of the inner and outer compact layers, Kl and K2 respectively, or by fixing K1 and the ratio of the distances d and b (Fig. 7.3.4). Lyklema (1995, Figs 3.22 to 3.25) shows how reasonable estimates of those parameters, along with some appropriate values for the Stern layer parameters (Nsand the equilibrium constant K ) can produce a wide range of isotherms (showing the relation between surface charge and surface potential for various electrolyte concentrations).

484 I

10: ADSORPTION AT CHARGED INTERFACES

I!

+4

f 2

I

I

9

10

PH

Fig. 10.1.I Sum of diffuse layer charges (oa)and Stern layer charge (q for i = Na+, Cl-) with the measured surface charge (no) for a sample of titania. (From Foissy et al. 1982, with permission.)

T o test the model we require data on the surface charge (by titration, say) and the components of charge in the diffuse layer and the Stern layer. The diffuse layer charge can be determined using electrokinetic data, provided we can assume that 5 $d and we will use that approach in the discussion below. That procedure was used by Foissey et al. (1982) who determined not only the titration charge and the diffuse layer (5) potential but also followed the adsorption of Na+ and C1- on a titanium dioxide surface using radio-isotopes. They were able to build up a complete picture of the double layer charge (Fig. 10.1.1) which showed that only about 10% was in the diffuse layer. Na+ was adsorbed into the Stern layer at pH values above the p.z.c (about pH 6.3) and C1- was adsorbed at lower pHs. The required adsorption potentials (eqn 7.4.14) were not very large (0 = 3.31 kT for Na+ and -2.85 kT for Cl-) and were independent of pH (and hence of the state of charge of the surface). The Stern theory was able to quantitatively account for the entire charge distribution with an outer layer capacitance of 22 pF crnp2.The important point about this result is that there did not appear to be any significant amount of ‘superequivalent’ adsorption (i.e. Stern layer adsorption of an ion of the same sign as the surface charge). This behaviour is consistent with the low adsorption potentials and contrary to that normally expected of ‘specifically adsorbed’ ions. It is referred to by Lyklema (1995 p. 3.64) as speczjic adsorption of the second kind and provides a different, and much simpler, picture to that suggested by the site-dissociation-site-binding models to be considered below (Section 10.2 and 10.4). It may well be much more common in colloidal systems than has previously been recognized.

ADSORPTION OF POTENTIAL DETERMINING I O N S

I485

10.2 Adsorption of potential determining ions In the classical studies of adsorption at the mercury-solution interface, described in Section 7.4, the electrical state of the surface can be described unequivocally in terms of the charge per unit area on the mercury, since the point of zero charge (P.z.c.) can be identified with the electrocapillary maximum (e.c.m.). Although there remains some uncertainty in the absolute values of the potential difference across the interface (Section 7.1), the changes in potential difference can be specified with respect to the potential applied at the e.c.m. Important insights were gained in that system by the study of the adsorption behaviour of indifferent electrolyte ions (i.e. ions whose response to the electrical state of the interface could be described purely in terms of the electrical interactions incorporated in the Poisson-Boltzmann equation). Indeed, it is doubtful whether any progress could have been made in the study of specifically adsorbed ions without the bench-mark provided by the indifferent ions. It is important to note that at the mercury-solution interface, the indifferent ions are assumed to be confined to the diffuse part of the double layer although, strictly speaking, we have no independent assessment of the diffuse layer potential in that case. In colloidal systems, where the {-potential provides such an assessment, this seems to be no longer the case. The surface charges involved are much higher and, as we saw in Fig. 10.1.1, good descriptions of the behaviour can be had by allowing the counterions to penetrate into the inner Helmholtz plane (Fig. 7.3.4) simply in response to the high electrostatic potentials there. We saw in Section 7.7 how the concept of potential determining ions (p.d.i.) was introduced to allow control of the surface state of the classical silver iodide colloid. That concept was extended in Section 7.9 to the H+ and OH- ions which control the charge status of many other surfaces, including the oxides (Section 7.10). In the classic studies on these systems outlined in Chapter 7, a great deal of significance was attached to the distinction between (i) potential determining, (ii) indifferent, and (iii) specifically adsorbed ions. It was formerly assumed that a type (ii) electrolyte could be identified by its effect on the plot of surface charge versus p.d.i concentration. If the curves for different electrolyte concentrations (Fig. 7.7.1) showed a common intersection point (c.i.p.) it was assumed that the electrolyte was indifferent and the c.i.p. corresponded to a point of zero charge (P.z.c.). We now know that this is not necessarily so. It seems that if the cation and anion of the electrolyte both adsorb to comparable extents (0, M 0-) then a plot of the apparent surface charge as a function of p.d.i. concentration still looks like Fig. 7.7.1. Such a c.i.p. might not then correspond to a P.Z.C. Fortunately, for colloidal systems, we can also assess the iso-electric point (where 5 = 0) and if this is independent of electrolyte concentration and also coincides with the c.i.p. then it can confidently be regarded as a P.Z.C. The notion of ‘indifferent electrolyte’ in colloid chemistry would then be one requiring a near-zero value of the specific adsorption potential as defined in eqn (7.4.14).

10.2.1 Surface charge density and surface potential The surface charge density, DO, is the net charge per unit area generated by the potential determining anion and cation. In the case of silver iodide:

486 I

10: ADSORPTION AT CHARGED INTERFACES

DO

= F(rAg+

-

fi-1 = p ( r A g N O 3

-

rKd

(10.2.1)

and for oxides:

The formulation in terms of the excess of neutral species is perhaps more appropriate because the double layer region is neutral overall. In practice, however, we will distinguish the actual potential determining ions, Ag+ and I- from their partners because measurements are normally done in the presence of very small concentrations of the p.d.i, with a swamping concentration of an (indifferent) electrolyte (KNO3 in this case). The concentrations of the partner ions (K+ and NO, in eqn (10.2.1)) are then negligible compared to their concentrations in the remaining indifferent electrolyte. In the simplest models of the silver iodideaqueous solution interface one would assume that the excess of Ag+ or I- ions (depending on whether the surface was positive or negative) would be uniformly distributed over the surface and would be indistinguishable from the lattice ions. At the point of zero charge there would be large but equal numbers of Ag+ and I- ions on the surface and as the surface potential was built up by adding more of one or other species, the additional ions would remain a negligible fraction of the total number of such ions on the surface (Exercise 7.7.1). In such circumstances, the chemical activity of those surface ions will be independent of the state of charge of the surface.+ It should be recalled that that was the crucial assumption involved in establishing (Section 7.7.1) the Nernst equation (from eqn (7.7.5)): [7.7.5] for the surface potential of silver iodide at (or $0 (volt) % 0.0592 loglo([Ag+]/[Ag+],,) 25 "C. There is reason to believe that the charge is not confined to the surface but extends some distance into the bulk solid, in the form of a diffuse defect structure (see Sparnaay 1972). This diffuse structure in the solid has been invoked to account for certain features of colloid behaviour (see, for example, Napper and Hunter 1972 and Hunter 1981 pp. 54-5 for brief reviews of that work). The more recent consensus, however, is that its effect is principally on the position of the point of zero charge, which is usually taken to be an experimentally measured quantity (Levine et al. 1970). We will, therefore, assume that the silver iodide surface can be treated as a nonconducting hydrophobic plane with a potential given by eqn (7.7.5) and a uniform distribution of charges confined to its surface. For many colloidal systems, especially those for which H+ and OH- are the potential determining ions there are very few charged groups present at the point of zero charge. Most of the ionisable groups are in the form -A-H and it can no longer be t If the lattice ions are considered to be able to establish an equilibrium with the surface ions the point can be made much more strongly.

ADSORPTION OF POTENTIAL DETERMINING I O N S

I487

assumed that the charges which bring about the surface potential enjoy a constant chemical environment, independent of the state of charge of the surface. The analogue of eqn (7.7.5): 11.0 = (2.303

kT/e)(pHpzc

-

PHI

(10.2.3)

is then no longer valid and the more elaborate analysis of Section 7.9 and 7.10 becomes necessary. (Semiconductor oxides such as RuOz seem to obey eqn (10.2.3) quite well and Ti02 departs from it only to the extent of about lo%, but for SiOz it is a very poor representation indeed.)

10.2.2 Surface complexation models (a) Single-site model In Section 7.9 we treated the single site dissociation model (appropriate for some polymer latices) and showed (Exercise 7.9.1) that, for such a system, the fraction a- of (negatively) charged sites was given by eqn (7.9.4). An approximate expression for 11.0 was then derived (eqn (7.9.7)) which clearly showed a significant departure from eqn (10.2.3). It is important to note that even in this simple case it is no longer possible to write an explicit expression for 11.0 (like eqn (10.2.3)) and that further assumptions, concerning the double layer structure and diffuse layer charge, are required before 11.0 can be estimated. It turns out that this simple site-dissociation model:

-SOH f t - SO-

+ H+

with

K, = {SO- HH+> {SOH)

(10.2.4)

is not adequate to describe the behaviour of polymer latex systems. T o get an accurate representation of the potentiometric (and conductimetric) titration curves for, say, a carboxylate latex, it seems to be necessary to introduce the possibility of site-binding i.e. the occurrence of reactions like:

-SO-

+ Na+ f t SO-Na’.

(10.2.5)

Such a reaction makes possible the development of a significant surface charge without pushing the electrostatic potential in the surface regions up to unrealistic values. The counterions ‘screen’ the developing surface charge. The problem is that it now becomes necessary to evaluate, in addition to K,, the equilibrium constant for eqn (10.2.5): KNa

=

{-SO -Na +} {-SO -}{Na +}

(10.2.6)

Unfortunately, as it stands such a quantity will not be constant as the surface is charged up, unless the quantities in braces are interpreted as electrochemical activities. The problem is to find a sufficiently general procedure to describe the electrical and chemical effects at the charged interface. The most common procedure has been to try to separate out the chemical and electrical effects which contribute to this exchange process by modelling it in terms of our current pictures of the structure of the electrical double layer (Fig. 7.3.4). The ‘chemical’ aspect of the binding process is subsumed into a so-called intrinsic surface binding constant aames and Parks 1982):

488 I

10: ADSORPTION AT CHARGED INTERFACES

[-SO-Na+] rG:= [-SO -]"a +Is .

(10.2.7)

where the quantities in square brackets are now chemical activities and the subscript 's' refers to the surface. In practice, these activities are usually replaced by concentrations, which implies a simple random mixing model for the surface species [SOH] and [SO-Na+]. For computational purposes it also turns out to be easier to replace eqn (10.2.5) by an exchange reaction: -SOH

+ Na:p

-SO-Na+

+ H:

(1 0.2.8)

for which

(10.2.9) The surface activities or concentrations of H+ and Na+ ions are estimated from the Boltzmann equation: [Ion], = [IonIb exp(-ze$/kT)

( 10.2.10)

where z is the valency of the ion and $ is the mean electrical potential which it experiences in the surface. In the models discussed below we will find that $ is sometimes taken as the same for all adsorbed ions, ($0 = $i = $d) and sometimes equal to the electrokinetic potential, (5) and sometimes given different values for the potential determining ions ($0) and the 'specifically adsorbed' ions (h).The most elaborate (triple-layer) model, used by James et al. (1978) (Fig. 7.3.4) to describe the titration data of Stone-Masui and Watillon (1975) on polystyrene uses the latter procedure so that

(10.2.11) where $ = e$/kT is the reduced (i.e. dimensionless) surface potential and b refers to bulk concentration. (b) Two-site model In Sections 7.9 and 7.10 we also treated the two-site models, including the zwitterionic and the amphoteric models. In order to solve these models it is again necessary to have an alternative expression for ao.If there is no inner (Stern) layer adsorption we have 00 = - a d and eqns (7.9.5) and (7.9.6). T o close the equations we have from eqn (7.3.36) the relation a0 = Ki ($0 - $d) where K; is the (integral) capacitance of the inner (compact) part of the double layer. The detailed procedure for fitting the electrokinetic data is given by Rendall and Smith (1978) and outlined by Hunter (1981 pp. 2 7 6 8 ) . It need not concern us here. Suffice it to say that the fit to 5 data for a nylon surface is very good (Fig. 10.2.1),and is obtained with reasonable values of the integral capacitance, Ki (25 pF cmP2) and the

ADSORPTION OF POTENTIAL DETERMINING I O N S

-

2

4

6

o 8

6

10

I489

a

PH

Fig. 10.2.1 Comparison of experimental {-potential data with theoretical curves calculated from a two site model for nylon at 25 "C.Electrolyte concentration:0 M; 0: M; A:5 x lop3 M; 0: lop2 M. Model parameters: Ns+ e = 0.62 pC cmp2; Ns- e = 1.42 pC atp2; i.e. p. = 5.4; pKp = 5.5; KA = 8.6; K, = 13.2; Ki = 25 pF cm-'. (After Rendall and Smith 1978 with permission.)

acid and base dissociation constants of the zwitterionic model. Unfortunately, it was not tested directly with titration data to determine how well it fits the surface charge. It is perhaps for this reason that it was not necessary for Rendall and Smith to invoke the notion of site binding (but see Section 10.3 below). When an effort is made to reconcile the titration and electrokinetic data on oxide surfaces with the site dissociation model, it certainly seems to be necessary to invoke site binding (see Section 10.4.)

10.2.3 Position of the plane of shear In the development and testing of models of the solid-solution interface, the electrokinetic (5-) potential has played a rather equivocal r81e, due principally to some uncertainty in determining the position of the plane of shear at which 5 is measured. In recent years, a considerable body of circumstantial evidence has accumulated that 5 is equal to, or very close to, the potential characterizing the diffuse part of the double layer ($rd). The evidence is summarized by Hunter (1981 pp. 210-16) but one aspect of the analysis is worth describing here. That is the technique introduced by Smith (1973). Smith examined the behaviour of the (-potential in the neighbourhood of the isoelectric point (where 5 = 0). If the shear plane is a distance A from the outer

490 I

10: ADSORPTION AT CHARGED INTERFACES

Helmholtz plane (Section 7.3) where the potential is $d then, for such low potentials (compare with eqn (7.3.21)):

where K is the DebyeHuckel parameter (Section 7.3). Considering, for example, the silver iodide surface, the slope of the
(10.2.13) The quantity (d$o/dpAg), = N , is the Nernst factor (obtainable from eqn (10.2.3)) and is equal to 59.2 mV at 25 "C, whilst the quantity (d$d/d$o)K, in the absence of specific (inner layer) adsorption, can be written in terms of the capacitances of the inner and diffuse layers (Kiand c d respectively) (Exercise 10.2.1). Substituting in eqn (10.2.13) then gives:

I:

S-'= N-' 1 + - exp(KA).

[

(10.2.14)

Smith (1973) gives a plot of the experimental values of S-' against c d (= E K ) for AgI in KN03(aq) at 25 "C. The plot is linear with the expected intercept at C d = 0 giving N = 59.2 mV and from the slope, the value of K; is 30 pF cmP2. Since this latter figure can be confirmed by direct titration measurements, the result implies that exp ( K A )is very close to unity. Smith concludes that A must be less than 0.1 nm in this system. He has applied the same approach to a number of other systems (see for example Smith 1976) with similar results. This conclusion is of very great importance; if it is accepted, it makes the {-potential a much more valuable and important quantity for testing double layer models, as we shall see in the next section.

Exercises 10.2.1 Use the definitions of Ki and

c d , for an electrical double layer with no inner layer adsorption, from Chapter 7 and show that (d$d/d$o) = Ki/(Ki Cd). Show also that c d = EK at low potentials, where E is the permittivity.

+

10.3 Detection of Stern layer adsorption From the discussion in Section 10.2.2 it is clear that the mere presence of a common intersection point in the potentiometric titration curves at different salt concentrations cannot be used to infer that Stern (inner) layer adsorption is absent. A sound

DETECTION OF STERN LAYER ADSORPTION

I491

thermodynamic approach to the problem of detecting inner layer adsorption has, however, been provided in a series of papers by Hall (1978, 1980) and his collaborators (Hall and Rendall 1980; Hall et al. 1980) based on an approach developed first by Smith (1973). Although the early analysis was based on the Grahame model of the double layer (Section 7.4) Hall’s later (1980) paper shows it to be independent of the detailed structure of the interface. It does, however, depend heavily on the use of the electrokinetic (<-) potential (Chapter 8), especially in the region near where = 0 (the isoelectric point or i.e.p.). In fact, it makes no direct use of the potentiometric titration data at all, although in favourable cases (namely when there is no inner layer adsorption) the surface charge and potential can be calculated. We will deal here with the simplest case where there is no inner layer adsorption and establish a test of congruence for the experimental data. If the data fails that test then we may assume that inner layer adsorption is occurring. In the case of the oxide surface, the failure is so catastrophic that inner layer adsorption seems much the most likely possibility. It must be emphasized however, that such adsorption can occur even when Oi M 0 in eqn (7.4.14) so it can be non-spec&. T o keep the argument as straightforward as possible we will work in terms of the Grahame (1950) model, rather than in the more abstract thermodynamic terms of Hall’s (1980) paper. The assumptions are: (i) there is a region (the compact or inner double layer region of Section 7.3.4) which is free of any counterions or co-ions (i.e. ai = 0 in Fig. 7.3.4); (ii) the Poisson-Boltzmann equation applies in the solution outside of this layer; (iii) a procedure is available to calculate the {-potential; and (iv) the plane of shear coincides with the outer boundary of the region in (i) (i.e. = $d). Assumption (ii) was discussed in Section 7.6 and (iii) in Chapter 8. Assumption (iv) was discussed in Section 10.2.3. With these assumptions, a simple double layer, generated by a surface excess, ri, of potential determining ions, of valence z;, in the presence of a z:z supporting electrolyte, will be governed by the following equations (compare with Section 7.3)

<

<

00

-ad

and

= z; efi

(10.3.1)

= [2€kT ~ / z e ]sinh (ze$d/2 kT)

(10.3.2)

=0

(10.3.3)

b pi - zi e$d

(10.3.4)

a0 + a d

Now we define pi

nb is the bulk concentration of component i and p! is the standard state value of the chemical potential. Hall (1978) shows, by a general thermodynamic argument, that for all points with the same ri, the value of pi - p! is constant. It follows then (Exercise 10.3.1) that at constant 00:

zi e$rd = constant - 2.303 kT p X

(10.3.6)

492 I

10: ADSORPTION AT CHARGED INTERFACES

where X is a potential determining ion and the constant is a function of 00. The (integral) capacity of the compact layer is given by eqn (7.3.36) and for the case where the surface potential is given by the Nernst equation (Exercise 10.3.1):

+d=[

2.303 kT zie

2.303 kT zie PX

(10.3.7)

where pX' is the value of p X at the P.Z.C. For the zwitterionic surface of Section 7.9.2 (iii), eqn (10.3.7) is modified by the inclusion of an additional term (Exercise 10.3.2):

@d=[

2.303 k T 00 kT pX' ----ln(Y/Y') Ki 2zie zie

PX

(10.3.8)

where Y = [( 1- a-)/a-][a+/( 1 - a+)]and X' and Y' are the values of X and Y at the P.Z.C. In the absence of inner layer adsorption, constant values of 00 correspond to constant Od and, for systems which are expected to obey the Nernst equation (10.2.3), eqn (10.3.7) can be tested by plotting values of $fd (= {) for a fixed a d against ( p x pX') = ApX. The resulting plot should be linear with the Nernst slope (59.2/2; mV) and Hall and Rendall (1980) show that, for the limited amount of data available, that is so for the calcite (CaC03) and Ca3(P04)2 surface. For the AgI surface it is true for low to moderate electrolyte concentrations but some evidence emerges for inner layer adsorption at higher electrolyte concentrations (> 0.1 M KN03). For zwitterionic surfaces, this test is not so easy to apply since the function Y / Y' must first be evaluated. That has been done for the nylon sol (Rendall and Smith 1978), and Hall and Rendall (1980) show that, using those values in eqn (10.3.8) gives a plot of ( against pH which has the Nernst slope. The conclusion is that the nylon system does not exhibit inner layer adsorption of the supporting electrolyte (NaCl). The same conclusion is reached from the congruence test, which is an alternative method of using eqns (10.3.7) or (10.3.8). In the congruence test a plot is made of Od against the function pX* = p X

+ z; e+d/2.303 kT = p X + z; e(l2.303 kT.

(10.3.9)

If eqns (10.3.7) or (10.3.8) are valid then all of the data at different electrolyte concentrations should fall on a common curve and Hall and Rendall (1980) show that this is so for the nylon sol and is also true for AgI provided the salt concentration is less than 0.1 M as before. (Note that to apply this test to nylon it is not necessary to evaluate the departure from Nernst behaviour.) Using the Nernst equation, eqn (10.3.7) can be rearranged to read (Exercise 10.3.3): Od

= (2.303 kT/zi e)Ki ApX*

ApX* = ApX

where

+ z;e(/2.303 kT

( 10.3.10)

DETECTION OF STERN LAYER ADSORPTION

1493

I

-2 -1 l t I

1 ApAg+(zle<12.303 kT)

Fig. 10.3.1 Congruence test for electrokinetic data on silver iodide. Results from titration (open symbols) and electrokinetics (filled symbols). The unbroken line corresponds to an integral capacitance Ki of 26.4 pF cm-’. The broken line is for Ki = 18 pF cm-’. The linearity indicates conformityto the Nernst equation (10.2.9). The ionic strength ranges from lo-’ ( 0 )to lop3M(0). (Modified from Hall and Rendall 1980.)

Figure 10.3.1 shows a plot of od against ApAg%for the silver iodide surface, from which the integral capacitance, Ki can be estimated. The full line, for positively charged surfaces The broken line gives a rather better description of corresponds to Kj = 26.4 pF mp2. the data for negative surfaces and corresponds to K ; = 18 pF cmp2. This was the value used by Hunter (1981) to represent the {-potential data on silver iodide (Fig. 10.3.2)over the range -100 mV < { < 100 mV. (The model used for that analysis corresponds exactly with that given above.) The variation in capacitance of the inner layer for positive and negative surfaces is a familiar feature of the mercurysolution interface (Fig. 7.4.5) and is attributed to the closer approach distance of (unhydrated) anions compared to (hydrated) cations. [Strictly one should not compare these integral capacitance values with the differential capacitance but in regions where K; is constant they are identical (Exercise 7.4.5) and even when Ci is changing fairly quickly near the p.z.c (Fig. 10.3.3) they differ by only a few pF cmP2 (Exercise 10.3.4)]. The data shown in Fig. 10.3.1 suggest that over the range 2 > (oo/pC cmp2) > -2, different but constant values of Ki can be used on either side of the point of zero charge. These values are compared in Fig. 10.3.3 with the estimates of the dzfferentiul capacity of the inner layer, obtained from potentiometric titration data by Lyklema and Overbeek (1961). It appears from this figure that a good description of the charge behaviour at higher values of I a0 I could not be given with these same K; values. Unfortunately there is no reliable electrokinetic data on this system for values of a0 outside the range indicated on Fig. 10.3.1 and 2.

494 I

10: ADSORPTION AT CHARGED INTERFACES

Fig. 10.3.2 {-potential of AgI in 0.001 M KNO3; KU = 1 and T = 293 K. 0 :calculated using the Henry equation (8.2.25), from data of Osseo-hare et al. (1978). 0: obtained from the complete numerical treatment of O'Brien and White (see Section 8.10). The full line is the theoretically predicted value of $d, in the absence of specific adsorption, for & = 18 pF m-'. (From Hunter 1981.)

-KF

I

l

l

1

+2

0

1

-2

1

1

1

-4

c cm-')

uo(p

Fig. 10.3.3 Comparison of inner layer differential capacitance, Ci, of AgI as determined by titratable charge (Lyklema and Overbeek 1961) with estimates of Ki obtained (A) by Hall and Rendall (1980) and (B) by Hunter (1981) using electrokinetic data. Values of Ci for the mercury-solution interface are included for comparison.

DETECTION OF STERN LAYER ADSORPTION

1495

Fig. 10.3.4 The congruence test applied to the silica-solution interface. The catastrophic failure is here attributed to counterion adsorption into the Stern layer. (From Hall and Rendall 1980.)

When the congruence test is applied to electrokinetic data on silica it fails in a very obvious fashion (Fig. 10.3.4). No minor shift in the position of the plane of shear can account for such behaviour. What Fig. 10.3.4 does not show, however, is the very large increase which occurs in 00 as K increases (see Figs 10.1.1 and 10.4.1). T o reconcile the resulting large 00 values with the quite modest measured values of { it is reasonable, therefore, to assume that one (or perhaps both) ions of the electrolyte are, in this case, adsorbed into the inner layer, even though the potentiometric titration curves exhibit a common intersection point (c.i.p.). We would have to postulate that cations and anions are specifically adsorbed in approximately equal amounts (or not at all) at the P.Z.C.in order to understand why the c.i.p. occurs. Indeed it transpires that the parameters introduced to describe the adsorption of these simple ions (Na+, K+, C1-, NO,) on oxide surfaces do correspond to almost exactly equal adsorption of each ion type at the P.Z.C.(Section 10.4).

10.3.1 Modifications t o the Stern layer model In Section 7.4.3 we discussed the Stern layer model as it has been developed to describe the behaviour of the mercury-solution interface. In its most elementary form (the ‘Zeroth Order Model’ as Healy and White (1978) call it) there is no charge in the compact region (q= 0) and we can use eqn (7.3.36) in the form 00 = K; (+0- +d) to describe the potential drop. This is the form in which it was used in Section 10.3 to examine systems with no inner layer adsorption. When this same ‘correction’ is applied to the simple site dissociation model (eqn 7.9.4) it has a profound effect on the relation between surface charge and potential. Figure 10.3.5(a) shows that if one introduces values for the integral capacitance, K;, similar to those observed on AgI and nylon (-20 pF cmP2) the expected charge for a given pH is dramatically reduced. The

496 I

10: ADSORPTION AT CHARGED INTERFACES

a

3

4

5

6

7

8

9

10

Fig. 10.3.5 (a) The effect of introducing a charge free inner layer into the single site dissociation model. (From Healy and White 1978, with permission.) Approach of a cation to the plane of the surface charge need not be restricted to one atomic radius as in (b) but may occur as in (c). This would correspond to very high values (-100-250 pF cm-’) of the inner layer capacitance Ki.

experimental data of Ottewill and Yates (1975) on a carboxylate latex show no such reduction and can be best represented by a very high Ki value. Considering eqn (7.3.36) in the form Ki = Ei/d and recognizing that the dielectric permittivity Ei is, if anything, significantly less than the bulk value, it follows that the thickness, d, of the compact layer must be very small (< 0.3 nm and probably < 0.05 nm) in these systems. This means that the counter ions can approach very close to the plane of the surface charge groups. They are evidently not subject to the same restriction as applies on the mercury, nylon, and AgI surface. It should not be

DETECTION OF STERN LAYER ADSORPTION

1497

0

II

+

CH2-0-R’

(CH,),N-CH CHa-R”

II

CH2-0

0 -P

II I

I

+ --O -CH2 -CH2 4 H R -NH3 -NH3

0Phosphatidyl serine (PS) R = COOPhosphatidyl ethanolamine (PE) R=H

I

R ” 4 4 H R’-&CH, Phosphatidyl cholines (PC) (Lecithins)

Fig. 10.3.6 The common phospholipids. They are tri-esters of glycerol. The third acid residue is an organic amine-phosphate. R ’ and R ” are long-chain fatty acid residues (CH2,+zCO- or &Hz,CO-) with n = 13-20. The commonest acids are hexadecanoic (palmitic) (a = 15) and cis-9octadecenoic (oleic) (n = 17).

surprising that in these carboxylate latices the surface charge groups do not generate an impenetrable plane with the counterions constrained to remain at least one ion radius distant from the head-group plane (Fig. 10.3.5(b)). Rather one would expect that the arrangement shown in Fig. 10.3.5(c) would be more energetically favourable. This conclusion had already been reached by Davies and Rideal (1963) for spread monolayers at the air-water interface. It appears to be a quite general phenomenon for surfaces in which the charge is confined to particular chemical groups (although nylon seems to be an exception). A model of this sort was used by Hunter (1966) to account for the electrokinetic behaviour of surfactant stabilized emulsions. In that case $0 and 5 had been estimated independently (Haydon and Taylor 1960) and at ‘low’ charge densities the two were identical, suggesting that the diffuse double layer could be assumed to start from the plane of the headgroups. (‘Low’ in this context (-10 p C cmP2) is actually higher than the maximum values attained by carboxylate latices.) More recently, McLaughlin and his collaborators, in an interesting series of papers on phospholipid vesicles (see for example Eisenberg et al. (1979)) have also used a model in which the Poisson-Boltzmann equation is assumed to hold right up to the plane containing the head group. In their systems the surface charge density on the vesicle can be varied by varying the proportion of phosphatidyl choline (PC) to phosphatidyl serine (PS), because PC is dipolar whilst PS is negatively charged (Fig. 10.3.6). T o analyse their results, McLaughlin et al. introduce a modified form of Stern layer which is essentially of zero thickness, by allowing counterions to adsorb onto the headgroups in accordance with a Langmuir isotherm (compare with Sections 7.4.3, 10.1.1, and Exercise 10.3.5):

498 I

10: ADSORPTION AT CHARGED INTERFACES

On

(10.3.11)

where 00 is the maximum charge density (due to the total PS- concentration in the surface), O, is the net charge after adsorption of Na+, and [Na+Is is the surface is the intrinsic association constant.) They then concentration of the counterion. assume that the plane of shear (Section 8.2.1) lies at a distance A ( m 0.2 nm) from the charge groups and calculate the potential there ($) from (compare eqn (7.3.20):

(ek

tanh z$/4 = (tanh z q0/4)exp (-KA).

( 10.3.12)

The value of q0is calculated from the usual Gouy-Chapman expression (eqn (10.3.2)) assuming that the surface charge density which determines $0 is equal to a,. The agreement between (at A = 0.2 nm) and the measured <-potential is excellent (Fig. 10.3.7) and the model can be checked using two independent methods: fluorescence probes (Section 9.5.2) adsorbed at phospholipid bilayers and electrical conductance measurements. Both give values for $0 in close accord with those estimated from the model. The use of eqn (10.3.12) to estimate implicitly assumes the validity of the PoissonBoltzmann equation even in that first 0.2 nm between the head groups and the shear plane. Although there is sufficient room for the necessary counterions to be accommodated, it must be noted that the effective local average concentration in that

+

<

0.114 c

g -100 -120

-

0.015

- 140 0.0031

-160 / 1 1 , 1 1 1 1

0.01

I

I

1111,1I

,

I

1 1 ( 1 1 1 1

1 Surface concentration of PS (nrn-') 0.1

I

10

Fig. 10.3.7 {-potential of vesicles formed from phosphatidylcholine (PC) and phosphatidylserine (PS) mixtures. The total negative charge, no, is determined by the PS content since PC is zwitterionic. Some of the charge is balanced by Na+ ions in the surface layer and the zeta potential is calculated some little distance from the head groups using the Poisson-Boltzmann equation with @o estimated from the nett surface charge. (From Eisenberg et al. 1979.)

DETECTION OF STERN LAYER ADSORPTION

1499

layer can be very high -up to -9 M in the most unfavourable case (Exercise 10.3.6). The simple Poisson-Boltzmann equation must be treated with considerable suspicion at such high electrolyte concentrations. The value of 4:; = 0.6 M-' required by Eisenberg et al. (1979)to fit their data corresponds to a chemical free energy of adsorption of the sodium ion to the headgroups of = 0; = -kTln(55.5 x K & ) = -3.5

AG$,

kT

(10.3.13)

if it is interpreted in terms of the Stern theory (Section 7.4.3). A very similar procedure was used by Kamo et al. (1978)to treat the electrokinetic behaviour of neutral liposomes (covered with PC (Fig. 10.3.6))to which an anionic fluorescent probe had been adsorbed. The probe was l-anilino-naphthalene-8sulphonate (ANS-) which was shown by X-ray analysis to sit in the plane of the headgroups. In this case it is the probe which is responsible for the surface charge, and Kamo et al. used the Langmuir isotherm (equation (6.4.6))to describe its adsorption to the surface. Coupling this with the usual expression for the free energy of adsorption

(1 0.3.14) leads immediately to the expression derived earlier for the Stern layer charge (eqn (7.4.11)) but in this case it is actually the surface charge (Exercise 10.3.7):

where -S is a surface site and the 55.5 in the denominator converts from mole fraction to molar concentration units. (Note that K A Nis~not an intrinsic dissociation constant.) In this case the amount of ANS- bound to the surface of the liposome (i.e. [-%ANSI can be estimated from the fluorescence intensity, I,so that (Exercise

10.3.8):

1

-I - 4[-S

1

1 -

ANS-]

(1 0.3.16)

where 4 is a proportionality constant. Kamo et al. introduced the approximation @o % and obtained a good linear relation between 1/I and (ci)-' exp(- e
<

K' = exp(-@i/kT).

(1 0.3.17)

For all of the salt systems studied (NaC1, KC1, CaC12, MgC12) the value of Q; was

-(16.3 f 0.2) kT o n phosphatidyl choline and -15 kT on phosphatidyl ethanolamine. We may conclude then that in systems where the surface charge is produced by groups which can be expected to be fairly mobile in the interface, the 'Stern Layer' is

500 I

10: ADSORPTION AT CHARGED INTERFACES

z -l

Fig. 10.3.8 Plot of I-' against the <-potential/concentration function suggested by eqn (10.3.16). (From Kamo et al. 1978, with permission.)

very thin, the integral capacitance is very large and, to all intents and purposes, the Poisson-Boltzmann equation appears to hold essentially right up to the plane of the headgroups.

Exercises 10.3.1 10.3.2 10.3.3 10.3.4

Establish eqns (10.3.6) and (10.3.7). Establish the modified form of eqn (10.3.7) assuming @O = 0 at the P.Z.C. Establish eqn (10.3.10). Fig. 10.3.3 shows a plot of the differential capacitance of the inner layer on AgI:

+

If it is represented by the empirical expression Ci = 3 1 5Do d(@O- @d)' one can estimate the potential drop (@o- @d) when a0 = 1 pC cmP2 (by integration). Use this estimate to calculate the integral capacity and compare it with the average value of Ci over the range (31 < (Ci/pF cmP2) < 36). 10.3.5 For the reaction involved on the phospholipid vesicle surface: ci

=

+ Na+ +PS- Na+. the equilibrium constant q: = [PS-Na]/[PS-][Na+],. PS-

-

Use this to derive eqn (10.3.11). (PS- stands for phosphatidyl serine, Fig. 10.3.6) 10.3.6 Eisenberg et al. (1979) give for in eqn (10.3.11) the value 0.6 M-l. Take some typical t values from Fig. 10.3.7 and estimate: (i) @o; (ii) a,;(iii) ao;and (iv) the average concentration of Na+ in the first 0.2 nm of the solution adjacent

THE OXIDE-SOLUTION INTERFACE

I 501

to the head groups. (The ionic strength may be assumed to be equal to the Na+ molarity shown on the figure.) [Note that the log scale in Fig 10.3.7 must be read as the reciprocal in nm2 per molecule.] 10.3.7 For the adsorption of ANS- on liposomes, Kamo et al. (1978) consider the equilibrium

where -S is a surface site. The equilibrium constant is KANS= [-S-ANS-]/[S] [ANS-1. Show that the number of adsorbed ANS- molecules per unit area is

where x is the mole fraction of ANS- in the bulk solution. Hence derive eqn (10.3.15) assuming that AGS, = -kT In KANS= - (e@o - Oi). 10.3.8 Establish eqns (10.3.16) and (10.3.17).

10.4 The oxide-solution interface We are now in a position to examine in more detail the current models of the oxidesolution interface. We noted in Section 7.10 that they are based on the amphoteric version of the two-site dissociation model. The low values of (-potential in these systems, coupled with extremely high values of titratable charge led to the suggestion that: (i) the surface potential was significantly different from the Nernst value (Hunter and Wright 1971) and depended on the total electrolyte concentration; and (ii) a significant proportion of the titratable (surface) charge must be balanced off with counterions inside the shear plane (Fig. 10.1.1). The GCSG model of Section 10.1.1 can describe the phenomenon very well, but that model does not take explicit account of the charge generation process at the interface. Of the various attempts to include dissociation we will examine, in detail, only the most highly developed. The site-dissociation-site-binding model which grew out of the early analyses of Yates et al. (1974) and Bowden et al. (1977), was developed by Davis et al. (1978) and was reviewed in detail by James and Parks (1982). The dissociation of the amphoteric group is represented as follows:

-SOH p

-

SO-

+ H+;

.

KEt =

[-SO-][H+],

[-SOHI

(10.4.2)

with the constants formulated as acid dissociation constants. Counterions are then assumed to adsorb onto the resulting charged groups:

502 I

10: ADSORPTION AT CHARGED INTERFACES

-SO-

+ Na'

+ - SO-Na+

and

-SOH$

+ C1-

p -SOH:Cl-

(10.4.3)

For purposes of computation, these equilibria are written as exchange reactions:

and

-SOHlCl-

f t -SOH

+ H+ + C1-; (10.4.5)

so that (Exercise 10.4.1):

The protons involved in these reactions are assumed to lie in the surface plane so that [H+], is given by eqn (10.2.10) with $ = $0, whilst the counterions reside in the Stern plane where the potential is $i. The intrinsic constants can then be written (Exercise 10.4.1): (10.4.7)

(10.4.8) where c is the bulk concentration of the Na+ and C1- ions. The total number of surface sites per unit area, N,, the surface density of titrable charge, a0 and the amount of adsorbed charge, a; are then given by:

+

Ns = NA([SOW [SOH:]

+ [SO-] + [SOHlCl] + [SO-Na']),

00 = eN~([soH,+] - [SO-]

and

+ [SOH$Cl-]

-

[SO-Na+]),

(10.4.9) (10.4.10)

a; = eN~([so-Na+]- [SOH:Cl-]).

Combining these with the expressions for the integral capacities of the two parts of the inner (compact) layer:

together with the relation (7.3.27) between $d and a d and (7.3.39) between the charge densities provides the complete set of equations for the model. The six unknowns: $0,

THE OXIDE-SOLUTION INTERFACE

I 503

y!ri, y!rd, 00, q,and Od can be solved for in terms of the independent variables, c and pH, provided suitable values are chosen for the parameters: * pNa, t

* P$, K F and K Z , N,, Kl and K2

with the proviso that Kal and Ka2 are linked to the

P.Z.C.

(Exercise 10.4.2):

(10.4.12) There are several possible procedures for solving the model. The first step is the which Davis et al. (1978) and estimation of the intrinsic complexing constants, James and Parks (1982) estimate using a double extrapolation technique [extrapolating to zero surface charge and zero electrolyte concentration]. Unfortunately, the procedure is somewhat ambiguous in this case because of the possible co-existence of sites of opposite sign at a given pH. Davis et al. assume that at any pH (not too close to the P.z.c.) only one of the possible sites is present and the procedure is then essentially the same as for a single site model. This assumption is, however, strictly correct only when the acid dissociation constants of the surface sites are very different, that is for rather large values of A p e ( > 4). The oxide which they treat in detail (TiO2) has the smallest ApPft value (-2 according to Fig. 7.10.2). In fact, their procedure produces a rather larger value of A p P t t (-4) which may well be an artefact. If one uses an values without assuming that only one optimization procedure to evaluate the *Pt sign of charged site is present at any pH, one obtains smaller values of Ap@ft for TiO2, closer adherence to the Nernst equation, and larger numbers of bound counterions (Koopal, personal communication). The complete amphoteric site-dissociation-site-binding model has been solved using a programme called MINEQL and applied to the oxide-solution interface (Davis et al. 1978) and the polymer-latex-solution interface (James et al. 1978).The results of a typical matching process for the Ti02 are shown in Fig. 10.4.1. Note that the required values of p * p n tfor the cation (7.2) and the anion (4.2)correspond to values of the chemical adsorption potential, f3ion, of (Exercise 10.4.3):

*e,!,,

&,, = -kT ln(55.5 Kint)

(10.4.13)

so that OK+ = -8.4 KT and @NO; = -7.5 kT. It is this relatively small difference in adsorption potential between the cations and anions which leads to a common intersection point. As Johnson (1984) has shown in his computer simulation studies, quite large differences are required between 0, and 8- to produce a significant shift in the P.Z.C.(These simple ions would, therefore, be regarded as specifically adsorbed at the oxide-solution interface.) The amount of specific adsorption occurring near the P.Z.C.is very large indeed. It is given by (Exercise 10.4.4):

lGl N,e

c exp(-&/kT) 55.5 c exp(-&/kT)

+

(10.4.14)

504 I

10: ADSORPTION AT CHARGED INTERFACES

-20

c

Fig. 10.4.1 Surface charge density (Yates 1975) and <-potential (Wiese 1973) of a Ti02 dispersion as a function of pH at various concentrationsof KNO3 and at 25 "C. Solid lines are calculated from the site binding model with constants indicated on the figure. (From James and Parks 1982.)

so for c = 0.1 mol L-' and 13, M -8 kT,o*/eN, is about 0.84; i.e. 84% of all sites have counterions of each sign associated with them!! At such levels this simple expression is obviously inaccurate since it does not take account of the area occupied by both ions. A more accurate (but still very approximate) estimate would include a factor of 2 in the second term of the denominator (Exercise 10.4.4) and this would imply a figure of 46% occupancy for each ion (which is at least possible). The point to be noted is simply that on the Stern model of the double layer the amount of counterion adsorption is extremely high. For c = 0.01 M, the fraction is 0.35 (0.26) and for c = 0.001 M it is 0.051 (0.049). The figures in brackets are obtained from the modified formula and these lower values are likely to be much more reliable estimates. and the resulting I& I values are It should be noted that these estimates of much larger than those obtained by Barrow et al. (1980) using the model introduced by Bowden et al. (1977). The discrepancy is, however, probably more apparent than real. The Bowden model postulates that a neutral site should be defined by the group OHM-OHz so that only one charged site (either or -) can develop for every two surface oxygens. Their N , value is, therefore, perhaps half of that used by James and Parks (1982). Such a change would be expected (from eqn (10.4.14), to require a larger I f3+I to achieve a given value of I o+ I . In the Bowden model, however, a separate estimate is

+

THE OXIDE-SOLUTION INTERFACE

I 505

made of the maximum number, NT, of adsorption sites for each anion and cation (based on the plateau of the adsorption when the electrostatic potential is favourable). Some of the ‘chemical affinity’ is, therefore, subsumed into this revised estimate of NT for each ion. (Recall Exercise 10.3.7 and note that the product N s K ~ occurs ~ s in the numerator.) Also, in the James and Parks model, ions such as Na+ and C1- can only adsorb onto previously charged sites (- SO- and - SOH: respectively) which requires a bigger I 8+ I to achieve the same level of saturation. This postulate can be regarded as a crude means of separating out, to some extent, chemical effects from those related to the state of charge of the surface; it should not, therefore, be lightly discarded. The nature of the interaction between simple ions such as K+, Na+, C1-, and NO, and the oxide surface is discussed in some detail by Sposito (19816). He regards them as forming ‘outer sphere’ complexes with the appropriate surface site, rather than the ‘inner sphere’ complexes one would normally expect for a specifically adsorbed ion. This means that the adsorbed ion retains its hydration sheath. In terms of the double layer model developed on mercury, such an ion would not be regarded as specifically (i.e. chemically) adsorbed, but on the oxide surface this is no longer the case. Even with its hydration layer, the ion is involved in a ‘chemical complexation’ reaction, although the large values of inner layer capacitance (Kl M 200 pF cmP2) suggest that these complexed ions lie near to the plane of the surface charge. The question whether ions form inner-sphere or outer-sphere complexes is not simple. In their studies on the adsorption of multivalent ions onto phospholipid vesicles, McLaughlin et al. (1981) used n.m.r. measurements to distinguish the two possibilities for the Co2+ ion. They found that only about 10 per cent of the cobalt was bound in inner complexes whilst the remainder was outer sphere. It should be noted, however, that the distinction between the two is perhaps not so important on these surfaces. As noted above (Section 10.3.1), the adsorbed counterions are able to sit in or close to the plane of the head-groups whether or not they retain their hydration sheaths. The difference between inner and outer sphere complexes should not, therefore, be regarded as synonymous with adsorption in the inner and outer Helmholtz plane (Fig. 7.3.4). The difference, presumably, lies principally in the magnitude of the chemical adsorption potential (being more negative for inner sphere complexes). It should be made clear that the site-binding model estimates of adsorption of simple ions like Na+ and C1- on oxide surfaces are demonstrably excessive. The radiotracer studies of Foissy et al. (1982) discussed in Section 10.1.1 may not give exact values but there can be no doubt that, if there were such large amounts of salt adsorption at the P.z.c., those studies would have shown them. It seems much more likely then that the salt adsorption is a fallacy and that the picture revealed by Fig. 10.1.1 is nearer the truth. How then do we reconcile the results? The large values of 13i which appear in the site-binding models occur because it is necessary to pull large amounts of counterion charge into the inner layer to balance the high surface charge with a small <-potential. Because the adsorption is confined to existing charged sites, it is necessary to postulate a high 8 value (eqn 7.4.14) for the cation in order to get enough adsorption. This, in turn, produces a large adsorption density of the cation at the p.z.c and some adsorption of this same ion when the surface changes sign. In turn that requires more adsorption of the anion to produce the same effect on the diffuse layer charge. Every increase in the ‘specific’ adsorption potential of one ion produces a counterbalancing increase for

506 I

10: ADSORPTION AT CHARGED INTERFACES

Fig. 10.4.2 Plots of NS-' against C,j for: (A) AgI, (B) TiOz, and (C) SiOz using KNO3 as indifferent electrolyte. Curve D shows the effect of putting A = 0.5 nm with the same initial slope as (C). (From Smith 1976, with permission.)

the other in order to satisfy the requirement that there should be no diffuse charge at the P.Z.C.These high values for both ions are thus artefacts of the fitting procedure. The picture can be examined further using Smith's (1976) analysis of the electrokinetic data on oxide surfaces. Using a somewhat more elaborate version of the technique described in Section 10.2.3, Smith shows that, if inner layer adsorption is occurring, then the expression for (d$ro/d$d)K becomes (Exercise 10.4.5):

(10.4.15) It is, of course, no longer possible to identify (d$ro/dpX) in eqn (10.2.13) with the Nernst slope, N, but instead one must use eqn (7.10.6).Smith then arrives at (compare with 10.2.14):

where K, = 2N,e240/kT. Plots of NS-' against c d (= E K ) are shown in Fig. 10.4.2 for AgI, TiOz, and SiOz. For the AgI system, inner layer adsorption is absent near the P.Z.C.(a;= 0) and $0 is large so that eqn (10.2.14) is recovered and the system shows the Nernst slope. For Ti02 and SiOz the slope is different but the fact that the plots are still linear and pass through NS-' = 1 suggests that A is small (< 0.5 nm). Smith analyses the case KZ >> K; (corresponding to a shear plane close to the IHP but a significant separation between this and the plane of the head groups). This leads to:

THE OXIDE-SOLUTION INTERFACE

I 507

(10.4.17) The linearity and the intercept at NS-' = 1 then would require that (da;/d$d) be small compared with c d for d l c d (which seems difficult to understand when c d approaches zero). These results can be more readily reconciled with the ideas developed earlier in this section if we assume that the capacitance of the inner part of the compact layer is high, as is normally required in the models (Kl M 200 pF cmP2).In that case K;/K2 % 1 and eqn (10.4.17) is replaced by:

1 + (ki-+- k> ki:)

NS- = 1

cd

exp @A).

(10.4.18)

If in the neighbourhood of the P.Z.C. we set 0; equal to its maximum value (a;% ao) (Fig. 10.1.1) and use the experimental values of dao/d$d % dao/d( we find that the term K;'(dai/d$d) is very small compared with unity for both Ti02 and Si02 (Exercise 10.4.6). The linear relationship, the intercept, and the altered slope are then all easily understandable. Note that the high slope for Si02 can be accounted for in terms of a decrease in the apparent integral capacitance of the inner layer, KA: K i l = KLl

+ IY-1.

(10.4.19)

This corresponds to a small value for K,, and hence a small q50 (from the definition of K, in eqn (10.4.16)) and a large ApK!tt value as noted earlier (Fig. 7.10.2). There are not many systems for which both titration (DO)and (-potential data are available outside the Ti02 system shown in Fig. 10.4.1. T h e charge data is well described in that figure but the agreement with the (-potential is rather poor. The impression is often given (see for example, Morel et al. 1981) that any one of the usual complexation models can give a satisfactory description of titration data, with a suitable choice of parameters. That is certainly not the case, however, if one demands a simultaneous description of 00 and (. Unfortunately, that more stringent test has rarely been applied. Smit et al. (1978) have, for example, made some direct radiotracer estimations of the amount of sodium adsorbed into the compact layer of (non-porous) vitreous silica and, like Foissey et al. (1982) have been able to account satisfactorily for both the electrokinetic (with ( = $d) and titration data on that system. The new electro kinetic methods of measurement, involving high frequency conductance and mobility, are also providing some fresh evidence on this question as we noted in section 8.11.

I

Exercises 10.4.1 Verify eqns (10.4.6), (10.4.7), and (10.4.8). 10.4.2 Establish eqn (10.4.12).

Next Page 508 I

10: ADSORPTION AT CHARGED INTERFACES

10.4.3 Show that when the sitedissociation-sitebinding model is interpreted in terms of the Stern equation (compare with eqn (10.3.15)) then eqn (10.3.13) holds so that, for example,

& = -kTln(55S*Kg/C)

and

&03 = kTln(55.5

K:/*K&)

(Hint: assume UK = N A e [SO-K+] and @i M @o).What are the QN, and f3cl values corresponding to the Barrow et al. (1980) estimates of Kit = 0.72 L mol-' and = 0.1 1 L mol-' on goethite? Note that Davis et al. used pK'At = 4.9; pK'G' = 10.7 and p*K'g = 6.6 to describe their data on goethite. To what value of 8- does this correspond? 10.4.4 Verify eqn (10.4.14). The more general expression for a;,taking some account of surface coverage by both ions is:

K't:

with a similar expression for a-, where g* = exp (-A G!*,da,/kT) = exp[(x*e@i 8*)/kT] and x is the mole fraction of the ion in solution. Show that this implies the correction factor 2 (at the P.z.c.) referred to in Section 10.4 under eqn (10.4.14). 10.4.5 Use the definitions of K1 and Kz:

+

with the charge neutrality condition, to show that for the double layer model shown in Fig. 7.3.4

Hence or otherwise show that

+

if c d = dad/d@d and KF1 = KT1 KF'. (Assume that independent of @d when I 5 I is small.) 10.4.6 Use the data given in Fig. 10.4.1 to show that

K2

and Ki are

for the Ti02 system. Take N, = 5 x lo'* m-', and get &J from Section 7.10 and the ApK value. [(dao/d<)s+o is even lower for the SiOz system.]

The Theory of Van Der Waals Forces 11.1 Introduction 11.1.1 Interactions between molecules 11.2 London theory 11.3 Pairwise summation of forces (Hamaker theory) 11.3.1 Interaction of a molecule with a macrobody 11.3.2 Interaction of two macrobodies 11.3.3 Effect of the suspension medium 11.4 Retardation effects in Hamaker theory

11.5 The Deryaguin approximation 11.6 Modern dispersion force theory 11.6.1 Interaction between two flat semi-infinite bodies across a vacuum 11.6.2 ~ ( wrevisited ) 11.6.3 The dispersion relation method 11.6.4 Modern theory for planar half-spaces 11.7 Numerical computation of interaction energy 11.7.1 Construction of .$
11.8 Influence of electrolyte concentration 11.9 Theoretical estimation of surface properties 11.9.1 Surface and interfacial tension and energy 11.9.2 Contact angle of liquids on low-energy solids

11.IIntroduction A treatment of the theoretical calculation of the van der Waals or Hamaker function (A) will necessarily take us into mathematical areas that many will find unfamiliar. We will make every effort to explain the analysis as clearly as possible but will concentrate on those aspects that are most germane to the practical problem of calculating the value of A for a particular material. It must be admitted that not every colloid scientist will choose to undertake such a calculation but we will show how, at least for simple

533

534 I

1 1 : THE THEORY OF V A N DER WAALS FORCES

systems, it can be easily programmed on a standard spreadsheet. If van der Waals functions are to be used with confidence it is important that workers in the field are aware of the data on which they are based. If it does nothing else this should stimulate the collection of more accurate data of the kind needed to make more reliable assessments of A for different situations. A number of reviews of the subject have appeared in recent years and reference will be made to them in what follows. We take the opportunity to develop a fairly detailed account of the theory here, partly because of its intrinsic scientific interest, but particularly because it ties this aspect of colloid science in with the mainstream of physical chemistry. Some of the more technical aspects of the theory have been relegated to appendices but the bulk of it is treated in easily digestible steps with short exercises that are intended to keep the reader involved. They require only simple algebraic manipulations or, at most, some elementary calculus, and attention is drawn to them at the appropriate stage in the text. We begin with a discussion of the microscopic theory of London and then take up the more modern Lifshitz macroscopic treatment. The microscopic theory is based on the assumption that interactions between pairs of molecules can be added together to obtain the total interaction, i.e. that the interaction between a molecule in one colloidal particle and a molecule in another particle is unaffected by the presence of all of the other molecules. That is a gross assumption that the macroscopic approach seeks to circumvent. The microscopic theory begins with the calculation of the potential energy between two molecules (Section 11.1) then between a molecule and a large (colloidal) body (Section 11.3.1) and then between two colloidal bodies (Section 11.3.2). A number of simple geometries can be studied (flat plates, spheres, and cylinders), but more complicated shapes rapidly generate very considerable algebraic complexity. We next consider the effect of placing the particles in a dispersion medium like water (Section 11.3.3). If the particles are separated by distances larger than about 50 nm it can be shown that the interaction energy is reduced because the electric field can only be propagated from one molecule to another at the speed of light. This is the retardation efect and it is discussed in Section 11.4. In Section 11.5 we introduce the Deryaguin approximation, which is a useful device for calculating interactions between large colloidal particles (where the radius of the particle is large compared with the range of the interaction force). The macroscopic (Lifshitz) approach is introduced (Section 11.6) by treating a very simple problem that illustrates the basic ideas embodied in the theory and, in particular, the concept of a dispersion relation. Some further remarks are then made because it is the property that is of about the dielectric response function, fundamental significance in the theory. No attempt is made to derive the basic equations of the theory, rather, we try to exhibit their reasonableness in the light of the ideas developed thus far. Section 11.7 outlines the procedure required to actually calculate the Hamaker function, A, while Section 11.8 introduces, very briefly, some considerations relating to the effect of an intervening electrolyte. Finally, in Section 11.9 we use the theory to calculate some surface thermodynamic quantities like surface tension and contact angle.

~(zc),

INTRODUCTION

1535

Many of the exercises in this chapter are designed to show the reader how much of the mathematical manipulation should be accessible with only the tools of elementary calculus and a little algebra. It is not the intention to expose the assumptions of the macroscopic theory to rigorous analysis but rather to indicate the nature of the experimental data that is currently used to evaluate the interaction energy. Hopefully, a wider knowledge of the limitations of that data will lead to the collection of much more useful information on the important colloid chemical materials. The range of values of the Hamaker function (at short distances where retardation is unimportant) for substances immersed in water can be summarized as follows:

A+O<

10

metds

-3 1

0.3 x lop2' J

oxides and halides

hydrocarbons

These figures reflect the differences in the polarizabilities of the various materials, on which A largely depends. For approximate calculations, the above information may be sufficiently accurate. For more exacting analyses, the more detailed considerations of this chapter must be invoked.

11.I .I Interactions between molecules The existence of long-range attractive forces between atoms can be inferred from the observation of first order phase changes (e.g. condensation of gases to liquids) and, less spectacularly perhaps, from the deviations from the perfect gas laws of Boyle and Charles. In 1873, van der Waals proposed the famous equation of state of a gas:

(p+$)(V-nb)=nRT

(1 1.1.1)

in which the constant b accounted for the finite volume occupied by the molecules of the gas and the constant a was directly related to the strength of the intermolecular attractive forces. The success of the van der Waals equation in summarizing the properties of gaseous systems and their phase behaviour stimulated theoretical attempts to discover the origin of the forces responsible. It is interesting to note that, even in 1873 with only a vague notion of the origins of intermolecular forces, van der Waals was already separating the short-range repulsive forces that give rise to the excluded volume term b from the long range attractive forces that account for the constant a. It has been pointed out (Sparnaay 1983) that as long ago as 1686, Isaac Newton in his Principia discussed the attraction between two bodies separated by a distance, R, in terms of a force, proportional to R-", where n > 4.Though he could not discuss the possible origins of such a force he could show that n must exceed four or the interaction between a small particle and a large plate would become infinitely large. In the early years of the present century, several workers (van der Waals 1909; Reinganum 1912; Thomsonl914;Keesom 1921) sought an explanation of the 'van der Waals' forces by postulating the existence of a permanent electric dipole in each atom or molecule of the gas. By suitably averaging the permanent dipoledipole interaction energy over all orientations of the dipoles at a fixed separation distance R, an attractive potential energy proportional to 1/R6 was obtained.

536 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

This simple theoretical explanation suffered from the unfortunate fact that the fundamental postulate could be tested experimentally and the necessary permanent dipole moment was found to be absent in most of the simplest molecules. Debye (1920) proposed a less easily tested model in which atoms and molecules were assumed to possess a permanent quadrupole moment. An attractive interaction energy (al/R9) was obtained by orientation averaging the permanent quadrupole-induced dipole interaction energy. It was, however, only with the advent of quantum mechanics that a satisfactory explanation of the origin of van der Waals forces was forthcoming (London 1930).

11.2 London theory The explanation given for the van der Waals, or dispersion, force in elementary physical chemistry texts is that it results from the interaction between a temporary dipole on one molecule and the induced dipole on a neighbouring molecule. A proper calculation of that interaction requires the use of second-order perturbation theory but the general form of the result can be obtained by the following very simplistic argument (Israelachvili 1974). In the Bohr model of the hydrogen atom, the electron is regarded as travelling in well-defined orbits about the nucleus. The orbit of smallest radius a0 is the ground state and Bohr calculated that a0

= e2 / 8 x ~ o h ~

(1 1.2.1)

where e is the proton charge, €0 is the permittivity of free space, h is Planck’s constant, and u is a characteristic frequency (in Hz) associated with the electron’s motion around the nucleus ( u = 3.3 x l O I 5 s-’ for the Bohr hydrogen atom). (Note that the value of a0 given by eqn (1 1.2.1) corresponds to the maximum value in the electron density distribution, 111.0 I 2, in the electronic ground state of hydrogen, as calculated by quantum mechanics.) The energy -hu is the energy of the electron in the ground state (relative to the separated particles) and so is equal to the ionization potential of the H atom. Although the H atom has no permanent dipole moment it can be regarded as having an instantaneous dipole moment, p, or order p l m aoe. The field of this instantaneous dipole, at a distance R from the atom will be of the order:

(11.2.2) If a neutral atom is nearby it will therefore be polarized by this field and acquire an induced dipole moment of strength p2: p2 = aE

m aaoe/(4neoR3)

(1 1.2.3)

where a is the atomic polarizability of the second atom. This measures the ease with which the electron distribution can be displaced (Section 3.2) and is proportional to the

LONDON THEORY

1537

volume of the atom (a M 4n€oa;). The potential energy of interaction between the dipoles $1 and p2 is then, using eqn (11.2.3):

Knt(R) = -$lp2/4n€oR3

M

- ~ a ; e ~ / ( 4 n € o ) ~= R ~-2a2hv/[(4nc,)2R6].

(1 1.2.4)

The more exact expression arrived at using perturbation theory is:

and

3 , m

= (Em

-

Eo)~

(11.2.6)

is the frequency (rad s-l) of electromagnetic radiation which would cause the transition from the ground state to the excited state llr, in the isolated molecule. The corresponding oscillator strength for this transition is given by:

(11.2.7) where Ip, I is the magnitude of the transition dipole, h = h/2n and me is the electron mass. fOm measures the probability of the transition occurring and is, therefore, the measure of the intensity of the absorption band. The similarities between eqns (11.2.5) and (11.2.4) are obvious. Apart from the 1/R6 dependence we see that the magnitude of the interaction energy is determined by the ease with which the electrons in the two atoms are able to undergo displacements, since this is, in effect, what the terms in the summation calculate. The connection between Knt and the absorption spectrum of the material is already apparent and we will return to that point again. The oscillator strengths, fOm obey a useful sum rule (Dalgarno and Lynn 1957) derived from quantum mechanics:

Elirn =N

(11.2.8)

m

where N is the number of electrons in the molecule. The London theory of the van der Waals force does not require the existence of permanent molecular multipole moments and demonstrates the universality of this long range attractive force. If it can be assumed that only one frequency dominates in the interaction it is possible to write the constant CAB in eqn (11.2.5) in the form:

(11.2.9)

538 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

where a; is the (experimentally measurable) zero frequency polarizability of molecule A and N A is the number of electrons it contains. Since the dominant frequency is usually that of an outer orbital electronic transition, a better approximation results by taking NA to be the number of electrons in the outer orbital only. Since NA enters to the half power in eqn (1 1.2.9) the choice of this value is not too critical and eqn ( 1 1.2.9) (with this modification for NA) produces satisfactory estimates of CAB(Pitzer 1959; Wilson 1965). The magnitude and range of the van der Waals force between molecules follows easily from eqn (1 1.2.9). Let us define the length Ro by: ( 1 1.2.10)

and a frequency, 00 by:

(1 1.2.11)

With these definitions the van der Waals interaction energy (eqn (1 1.2.5)) becomes:

3V,,(R) = - - h*(Ro/R)6 4

( 1 1.2.12)

which is usually quoted as London’s equation. Typical values ofa0/(4n~o)range from 1 to 20 x lop3’m3 (McClellan 1963), which 10l6 rad s-’. Thus yields Ro 0.2 nm and 00

-

-

l&(R)

-

-

M

[0.2/R(nm)l6 Joules.

-

( 1 1.2.13)

For R 0.4 nm at T 300 K, we see that Knt 2 kT,which emphasizes the fact that van der Waals forces are thermodynamically important forces under normal experimental condition. It should be remembered, however, that other long-range interactions between molecules can be important, especially if the molecules have a permanent dipole moment. The importance of the London interaction for colloid science lies in the fact that, when interactions between large collections of molecules are treated, it is the London dispersion force that dominates over the dipole-dipole (Keesom) and dipole-induced dipole (Debye) forces (if the molecules are uncharged). This is because the summation assumes that the interactions are pairwise additive and this is a rather better assumption for the London forces than for the others (Overbeek 1952, p. 265). The modern Lifshitz theory avoids this problem and calculates the change in the (Helmholtz) free energy due to all of the interactions at the same time.

PAIRWISE SUMMATION OF FORCES (HAMAKER THEORY)

1539

r

Exercise 11.2.1. Establish eqn (11.2.4). [Note that the polarizability is sometimes defined in terms of the equationp = a@ so that it has dimensions of volume. (Atkins 1978, p. 751; 1982, p. 772).] 11.2.2 Establish eqn (11.2.12) and note the similarity to eqn (11.2.4). Only the

constant is missing from the simple formula.

11.3 Pairwise summation of forces (Hamaker theory) Following London’s explanation of the origins of van der Waals forces, several workers (Kallmann and Willstatter 1932; Bradley 1932; Hamaker 1937) were quick to realize that such universal long-range intermolecular forces could give rise to the long-range attractive forces between macroscopic objects that must be invoked to explain the phenomenon of colloid coagulation (Section 1.6). The simplest procedure for calculating the van der Waals force between macrobodies is to use the method of pairwise summation of intermolecular forces. T o elucidate the technique let us consider a set of Nmolecules at positions Rj (i = 1,2, . . ..,N). The separation distance of molecules i and j is R i given by

Rc = IRj - Rjl.

(1 1.3.1)

The interaction energy of the N-body system, in the pairwise summation method, is taken to be (1 1.3.2)

?it

(R)is the interaction energy of molecules i andj separated by distance Rq in where the absence of any other molecules. For van der Waals forces derived from second-order quantum perturbation theory, eqn (11.3.2) is only an approximation. The internal states of molecules i and j will be modified by the presence of all the other molecules of the system and they do not, therefore, interact with each other in the way that they would if the other molecules were not present. Clearly the pairwise summation method will be least in error when the molecules are far from one another, so that the individual pair interactions are relatively unaffected by the presence of other molecules. 11.3.1 Interaction of a molecule with a macrobody In the pairwise summation approximation let us calculate the van der Waals interaction energy of a molecule of type 1 at position Y with a body of arbitrary shape comprised of molecules of type 2 at number density p2. This would correspond to the energy of physical adsorption of molecule 1 on adsorbent 2. Consider a volume element dV’ at position r/ of the body containing p2dV’ type-2 molecules (see Fig. 11.3.1(a)). The

540 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

(a)

Fig. 1 1.3.1 (a) The geometry of a type 1 molecule interacting with a body comprised of type 2 molecules at density p2. (b) Coordinate system for a Hamaker summation in planar half-space. z is the radius of a ring of the material in the solid.

interaction energy of molecule 1 with the molecules of the volume element in the pairwise approximation is: (1 1.3.3) The total van der Waals interaction energy of the molecule 1 with the body 2 is obtained by integrating eqn (11.3.3) over the volume of the body, viz. @12(r)

= -C12P2

Ir -

(1 1.3.4)

Body 2

For the case of usual interest (a molecule at a distance D from a planar half-space)+ we may write, using Pythagoras' theorem (see Fig. 11.3.1(b) and Exercise 11.3.1): 0

0

0

0

ti.e. a body of infinite extent bounded by a plane surface.

P A I R W I S E SUMMATION OF FORCES (HAMAKER THEORY)

I541

Note how the summation process leads to a much longer range interaction (DP3) compared with that for moleculemolecule interaction (rP6).

11.3.2 Interaction of two macrobodies If the molecule 1 is part of another body comprised of molecules of type 1 at number density p1, then the interaction energy of the molecules of type 1 in a volume element d V at position r is just

where $12 (r) is given by eqn (11.3.4). The total van der Waals interaction energy of body 1 with body 2 is then VA = p1

s

A12

Body 1

dV' Body 1

(11.3.7)

Body 2

where we introduce the Hamaker constant A12 defined by (11.3 .S)

A12 = 2PlP2C12.

In the limit as the bodies are separated by a distance large compared to the largest dimension of each body, the distance Ir - r' I can be replaced by R, the distance between the centres of mass of the two bodies, and eqn (11.3.7) reduces to (11.3.9) where V1 and V2 are the volumes of the bodies. Let us now consider some specific cases of interest in colloid science where analytic solutions of eqn (11.3.7) may be obtained. For the case of two plane parallel half-spaces separated by a distance L, we may write eqn (11.3.7) as 00

A12 vA(L) = -~ 67c

(11.3.1 0) Surface of body 1

0

with the aid of eqn (11.3.5). In this case the volume element has been chosen as dxdS where x is a coordinate measuring normal distance into body 1from its surface and dS1 is an element in that surface. Performing the x integration and evaluating the interaction per unit area of the surface of body 1 we have EA(L) = -A12/12nL2.

(1 1.3.11)

Note that the range of the interaction energy is even longer in this case, compared with that in Section 11.3.1.

542 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

The Hamaker summation for more complicated geometries is, as one might expect, a mathematically involved operation. One other geometry leads to a relatively simple formula, viz. the spheresphere interaction. When body 1 is a sphere of radius a1 and body 2 is a sphere of radius a2, with centre-centre distance R, eqn (1 1.3.7) reduces to (Hamaker 1937):

(1 1.3.12) In the limit where R >>a1 +a2, this result yields:

(1 1.3.13) In terms of the distance of closest approach of the spheres H (= R (1 1.3.12) becomes (Exercise 11.3.2):

VA(H) =

--[-+

A12 1 12h1 1 + h 2

hl

+2hlln(hl(

1 + h i +h3

1+h2

1

+ hl + h3

-

u1-a2), eqn

)}]

(11.3.14)

In the limit as H << 5,we have, from eqn (11.3.14) (Exercise 11.3.3):

12H VA(H) = -&[I

+t(l -

The case of a sphere of radius a at distance H from a planar half-space is obtained from eqn (11.3.15) by setting a1 = a and taking the limit as a2 + 00. We obtain (Exercise 11.3.3):

(1 1.3.16) Several other geometries yield analytic expressions for VA. Among these are multilayers, and parallel and crossed cylinders. (See Mahanty and Ninham 1976, pp. 13-21; also Fig. 11.3.2.)

11.3.3 Effect of the suspension medium The van der Waals interaction energy VA,as calculated by the Hamaker summation method of the previous section, is of limited use in colloidal problems where the macrobodies (particles) are surrounded not by a vacuum, as assumed in eqn (11.3.2),

PAIRWISE SUMMATION OF FORCES (HAMAKER THEORY)

1543

Two spheres al,a2>>H

P--l

Sphere and half-space

a<
I

Parallel cylinders

VA K -

-length H5

length)

0.c) ---

Crossed cylinders VA

V A =-

(al a2) A/6H

(per unit = -At12 nL2 area)

Fig. 11.3.2 Distance dependence of the non-retarded potential energy between pairs of solid surfaces of various geometries. The retarded interaction has a distance dependence 1/H times that and the proportionality constant then changes. In some cases the integration process yields quite messy results and these have not been included explicitly. Mahanty and Ninham (1976) remark that numerical integration over the particle volumes is usually easier and more satisfactory than seeking analytical solutions for all but the simplest geometries.

but by a fluid medium whose molecules interact with the other molecules of the system. The presence of these extra interactions can drastically diminish the value of VA,and even reverse its sign in some circumstances. T o see why, consider bodies 1 and 2 immersed in suspension medium 3. If they are brought from infinite separation to some distance D apart, the free energy change (the interaction energy) is not as large as in the case where medium 3 is a vacuum. When isolated, body 1 is interacting with its environment - a universe of medium 3. When

544 I

1 1 : THE THEORY OF V A N DER WAALS FORCES

Fig. 11.3.3 Thermodynamic path for calculating the interaction energy V132(D)of bodies 1 and 2 in a fluid medium 3.

brought close to body 2 it is interacting with a very similar environment, the only difference being that a number of molecules of medium 3 have been replaced by the molecules of body 2. The change of energy is less than that of the vacuum situation where the environment of body 1 originally contains no molecules and is then modified by the addition of the molecules of body 2. T o calculate the van der Waals interaction energy in the presence of a suspension medium, we consider the following thermodynamic path (see Fig. 11.3.3). Consider an initial state in which bodies 1 and 2 are infinitely separated in medium 3. We may regard the molecules of 3 in the volume that will finally be occupied by body 1, to be a ‘body’ 3 as shown. Let us remove body 1 and body 3 from the medium to vacuum. The change in free energy in this first step is just

where D is the distance from ‘body 3’ to body 2 and Fi is the interaction energy of the isolated body i with a universe of medium 3. The energy change in removing body 3 is not - F3 but -[F3 -V33 ( 0 ) V32 (D] since its environment is not all medium 3. The energy V33 (D)- V32 (D)represents the change in the interaction energy of body 3 with its environment, when the molecules of 3 that would have occupied the position of body 2 are replaced by body 2. Note in this argument we are using the notation that V&(0)represents the (vacuum) interaction energy of a body the size and shape of

+

PAIRWISE SUMMATION OF FORCES (HAMAKER THEORY)

1545

body 1 comprised of k-type molecules with a body the size and shape of body 2 comprised of j-type molecules at separation distance D. The second step is to return body 3 to the medium into the hole occupied originally by body 1 and to return body 1 to the hole originally occupied by body 3. The energy change in this second process is

As before, the energy Vlz - V13 represents the change in the interaction energy of body 1 with its environment when the molecules of 3 that would have occupied the position of body 2 are replaced by body 2. The interaction energy Vl32 (D)of bodies 1 and 2 at separation D in bathing medium 3 is given by

Vl32(D)= A P + AF' = Vlz(D)

+ V33(D)- Vl3(D)- V&(D).

(11.3.19)

By the nature of the pairwise summation method we may write (see eqn (11.3.7))

Vkj(D)= -AkjV(D)

(1 1.3.20)

where V(D)is a positive function only of the geometry of the system and independent of the composition of the bodies 1 and 2 and Ak, is the vacuum Hamaker constant given by eqn (11.3.8). Thus eqn (11.3.19) becomes:

where the effective Hamaker constant A132 is given by

A132 = A12

+A33 - A13 - A32.

(1 1.3.22)

The presence of a bathing medium does not change the distance dependence of the van der Waals force but alters its magnitude through A132. With an obvious notation, we may write Alv2 = A12 when bodies 1 and 2 are separated by a vacuum and from eqn (11.3.22), it is apparent that

+

A32 which is usually so. As Fig. 11.3.4 clearly shows, this provided A33 < A13 reduction in Hamaker constant in a bathing medium can be one of even two orders of magnitude. Obviously for All = A33, the function A131 is reduced to zero. Several other generalizations can be made. In particular we note that Ajkj > 0 so that like particles always attract one another in any medium. That is not necessarily the case for unlike particles since A132 can be negative. A reasonable approximation for the interaction function between different materials is: (1 1.3.23)

546 I

I I : T H E THEORY OF VAN D E R WAALS FORCES

h

0 N 0 r

2s.

A11(x 1OA2O) joule

Fig. 11.3.4 The effect of an intervening medium (3) on the interaction between two semi-infinite bodies. The value O f A33 is taken as 3.70 x lopzoJ corresponding to liquid water. (See Exercise 11.3.4.)

and this can be used to express eqn (11.3.22) in the more convenient form (Exercise 11.3.8):

(11.3.24) It should be noted, though, that the usefulness of such approximations for accurate research purposes is fast disappearing with the ready availability of high speed computers and the rise of modern dispersion force theory.

Exercises 11.3.1 Establish eqn (11.3.5) 11.3.2 Establish eqn (11.3.14) from (11.3.12). Also show that for two equal spheres of

radius a:

where s = R/a. 11.3.3 Establish eqn (11.3.15) and (11.3.16). 11.3.4 Verify Fig. 11.3.4 using A33 = 3.70 x

J for water.

RETARDATION EFFECTS I N HAMAKER THEORY

1547

1 1.3.5 Plot the potential energy of a molecule as a function of distance D from an infinite flat plate, using eqn (1 1.3.5) in the range 0 < D < 25 nm. Take the molar

volume as 20 cm3 and estimate C12 from eqn (1 1.2.4) using a reasonable value for the frequency u. 1 1.3.6 Plot the potential energy per unit area as a function of separation L between J in a fluid two semi-infinite flat plates of a material 1 for which A = 6.0 x 2 for which A = 4.5 x lop2’ J. (Make 1 < L < 25 nm.) 11.3.7 Plot the function VA(EJ)/A12 for two spheres of radius 100 nm and 150 nm in the range 0 < H < 25 nm. 11.3.8 Establish eqn (11.3.24).

11.4 Retardation effects in Hamaker theory All the preceding calculations for the interaction energy of macrobodies are based on eqn (11.2.5) - the inverse sixth power law dependence of the long-range intermolecular attraction. This, in turn, is based on the quantum perturbation theory treatment of the interaction energy, which can be represented by a set of Coulomb interactions between the electrons and nuclei of the molecules. This Coulombic form for F,,, is an electrostatic approximation which is valid only if the molecules are sufficiently close to one another. T o understand this let us consider how the attractive interaction energy arises. The molecular charge distribution A is constantly varying due to its internal electronic (and nuclear) motions and this variation propagates a complex electromagnetic field into the surrounding space, the frequencies of which are those of the fundamental intramolecular motions. The field travels through space at the speed of light, c, until it reaches molecule B which is then polarized by the field. The oscillating dipole induced in B (together with the higher multipoles) re-radiates an electromagnetic field which is propagated back to A and interacts with it. If the time between A’s radiating of the electromagnetic field and absorbing its reflection from B is negligible compared with the characteristic time for the internal motion, then A will be substantially in the same configuration on absorption as it was on emission, and the reflected field will, on average, be parallel to the emitting dipole moment. This will produce maximal lowering of molecule A’s energy. If, on the other hand, the propagation time is comparable with the characteristic time for internal motion, the instantaneous dipole of A will have substantially altered its orientation by the time the reflected field is received at A. The dipole moment component parallel to the reflected field will, on average, be smaller than in the non-retarded case and the energy decrease of molecule A will therefore be less. The propagation time is -R/c and the molecular characteristic time is - ( 2 x / 9 ) . Thus, provided

the propagation can be regarded as instantaneous (non-retarded) and our present electrostatic analysis is valid. When R becomes a significant fraction of the

548 I

I I : T H E THEORY OF VAN D E R WAALS FORCES

characteristic wavelength ho for the internal molecular motion, the finite propagation time (retardation) must be allowed for and the interaction energy will be smaller than that predicted by the electrostatic theory. By modifying Vn, to account correctly for the finite propagation time of the electromagnetic interaction between the molecular charge distributions, Casimir and Polder (1948) calculated the correct behaviour of Vint(R)for all R. In particular for R << ho the RP6 form (eqn (11.2.5)) is valid and for R >> ha:

(1 1.4.2)

The major consequence of the retardation effect is to limit the range of macroscopic (i.e. ‘long-range’) van der Waals forces. Overbeek (1952, p. 266) represents the Casimir and Polder correction function in the following way (compare eqn (1 1.2.12)):

where f ( p ) = 1.01 - 0 . 1 4 ~ for

2.45 and f ( p ) = - - -

P

2.04

P2

1

(11.4.3)

for p > 3 .

Correction factor

Fig. 1 1.4.1 Retardationcorrection factor to the London attraction energy between two flat plates of infinite thickness at a separation L. Note that althoughfb) = 0.59 for p = 3, the total attraction energy is down by a factor of about four at that point. (From Overbeek 1952, p. 269, with permission.)

THE DERYAGUIN APPROXIMATION

1549

Here p (= 2nR/ho) expresses the separation in terms of the characteristic wavelength (eqn (11.4.1)). Overbeek used eqn (1 1.4.3)to calculate the interaction energy between two semiinfinite flat plates (Fig. 11.4.1)and Hunter (1963)(and also Vincent 1973)extended the calculation to plates of infinite area but finite thickness. Detailed calculations of the retarded interaction between two equal spheres and a sphereplate combination have been provided by Clayfield et al. (1971)and that work has been summarized by Gregory (1 98l), who introduces some new approximation formulae and compares his results with the ‘exact’ Hamaker calculations. All of these approaches, however, suffer from a number of fundamental limitations. Apart from the breakdown of the additivity assumption it is obvious that the use of a single ho value can at best only represent one of the characteristic modes of vibration of the electric charge in the medium. As we shall see when we deal with the Lifshitz (macroscopic) approach, every possible vibrational mode needs to be examined separately since each has its own characteristic retardation behaviour. The approximation formulae developed by Gregory can then best be used to assess which modes need to be most carefully considered in the light of the distances and geometry of the problem. Since the retardation effect is automatically incorporated into the Lifshitz approach we will postpone further discussion on this point until the modern theory has been developed in Section 11.6.

Exercise 1 1.4.1 Plot the correction factorfb) (eqn (11.4.3))as a function ofp for 0 < p < 10.

Note the difference between this and Fp of Fig. 11.4.1.

11.5 The Deryaguin approximation When dealing with interactions between macrobodies, it is often the case that the range of the interaction is such that the two bodies do not interact significantly until the distance of closest approach is small compared to the radii of curvature of the bodies. Under these circumstances, a very useful approximate expression for the interaction energy of the bodies can be derived from the corresponding interaction energy per unit area of plane parallel half-spaces. The approximation is due, originally, to Deryaguin (1934). A suitable coordinate system is shown in Fig. 11.5.1. The bodies are assumed smooth and quadratically curved in the neighbourhood of 0 and 0’. Consider an element of area d S at the point P(x, y, z ) on the surface of body 1. Provided the radii of curvature of bodies 1 and 2 are large compared to the separation distance H, then the area element can be approximately regarded as a surface element of a plane half-space parallel to and separated a distance D (see Fig. 11.5.1) from another planar half-space. If E(D)is the energy per unit area of half-space 1 interacting with the half-space 2, then, in this approximation, d V = E(D)dSl

(1 1.5.1)

1

550 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

Fig. 1 1.5.1 Geometry of two interacting bodies illustrating appropriate principal coordinate axes for each body. The insert shows the orientation of the coordinate axes at 0 and 0’, looking along the line of centres between the two bodies.

is the interaction energy. The distance D increases as the area element is taken further from the origin 0 and the associated energy d V decreases appropriately. In the Deryaguin approximation, the total interaction energy of bodies 1 and 2 is obtained by integrating d V over the surface of body 1, viz. V(H) =

/

dSIE(D).

(1 1.5.2)

Surface of body 1

The energy E(D) must decay sufficiently rapidly with distance D so that contributions to V(H) are insignificant from area elements very far from the centre 0 where curvature effects are large and eqn (11.5.1) is a poor approximation. Deryaguin’s original analysis was applied to two spheres of large radius of curvature, a, and in that case: 00

V(H) = 2 n / E(D)ydy.

(1 1.5.3)

0

However, from Fig. 11.5.1. (D - H)/2= a - (a2 - y2). Hence 2 y dy = a (1 -?/a2) dD M a dD for y << a. In that case

(1 1.5.4) H

White (1983) has shown that the general equation for V(H)where the approaching surfaces may have arbitrary orientation and curvature (provided the latter is large) is:

THE DERYAGUIN APPROXIMATION

I551

M

(1 1.5.5)

where

h.lh.2=

(d,-+rd,)(d2-+- d;)

+sin

4

(d, d& ---

--a) ; ;

(11*5*6)

The R s and R's are the principal radii of curvature of the two surfaces respectively. The angle (see Fig. 11.5.1) is the angle between the principal axes on the two surfaces. Obviously, 4 is immaterial when either of the bodies is a sphere and for two equal spheres h.1h.2 = 4/a2, which reduces eqn (11.5.5) to (11.5.4). From the definition of the force F(H) exerted by one body on another we have that (1 1.5.7)

Equation (1 1.5.5) is a valid approximation for any type of interaction energy and need not be restricted to the van der Waals interaction of this chapter. T o apply eqn (1 1.5.5) with confidence to any particular interaction it is sufficient to require that:

Lo/Ro << 1 and H/Ro << 1

(1 1.5.8)

where LOis the length scale on which the interaction decays to zero and Ro is the smallest radius of curvature of the system. One geometry of particular experimental importance is that of crossed cylinders of radii a1 and 4.In this case R1 = a,, R2 = 00, R', = a2, Ri = 00 and 4 = n/2. For this geometry

ala2= i/ala2

(1 1.5.9)

and the Deryaguin approximation is just

1 00

V ( H )= 2x&a2)

dL E(L).

(1 1.5.10)

H

Thus the force between crossed cylinders at separation distance H is

Experimentally, the measurement of F(H)between crossed cylinders is used to obtain directly the interaction energy per unit area of planar half-spaces (Israelachvili and Tabor 1972). (See Chapter 12.) The Deryaguin procedure is useful for dealing with interactions between large spheres at close separations or a large sphere near a flat plate. It becomes increasingly inaccurate as the sphere size decreases and relative separation increases so that the curvature of the sphere becomes more significant in determining the strength of the

552 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

interaction. A much more useful procedure called surface element integration (Bhattacharjee and Elimelech 1997) is then more appropriate. When applied to the calculation of the van der Waals interaction between a small sphere and a flat plate it is capable of yielding results in quantitative agreement with the exact numerical calculation. One interesting aspect of the calculation is that there is a discernible (though obviously small) contribution from interactions involving the elements on the side of the sphere away from the flat plate.

11.5.1. Use eqn (11S . 5 ) to show that the van der Waals interaction energy of two large spheres of radii a1 and a2 in the limit H << ai is given by

Note that this corresponds to taking only the first term in the expression ( 11.3.15).

11.6 Modern dispersion force theory The Hamaker treatment of van der Waals forces between macrobodies suffers from two restrictions: (a) the assumption of pairwise additivity of molecular interactions of the London type; and (b) the neglect of the retardation effect. Both of these defects are remedied in the modern theory of van der Waals (dispersion) forces in macrosystems. The original theory is due to Lifshitz (1956) and was generalized by Dzyaloshinski et al. (1961). These treatments involve advanced statistical mechanical and quantum field theoretical arguments and cannot be repeated here. The interested reader is referred to Landau and Lifshitz (1969). Fortunately, the results of the theory can be arrived at by a much simpler and more readily understandable approach, which was first suggested by Casimir and Polder (1948) and subsequently developed by van Kampen et al. (1968). It is, in effect, an extension of Planck’s original treatment of a black body radiator. An excellent introductory discussion of this approach is given by Parsegian (1975) and reviews at various levels have been given by Gregory (1969), Visser (1972), Israelachvili (1973a, 1974), Richmond (1975), Mahanty and Ninham (1976), and Hough and White (1980). In what follows we will try to indicate the main physical features of the simplified approach and skirt around the mathematical tricks that are used to solve the problem, giving only a hint as to the formalism involved. It is important to realize, from the outset, however, that this so-called ‘heuristic’ approach can ultimately be justified by appeal to the much more rigorous methods of quantum field theory, at least for simple geometries. The advantages of the approach are that it can be applied to many situations in which the corresponding quantum problem would be quite intractable.

MODERN DISPERSION FORCE THEORY

1553

The fundamental idea that lies behind the method is that the interaction we are trying to calculated is propagated as an electromagnetic wave from one body to another over distances that are large compared to atomic dimensions. Furthermore, the frequencies with which the electrons in these bodies are able to move most readily (wg < lo1*s-l) correspond to wavelengths which are also large compared with atomic dimensions. It should be possible, therefore, to analyse the propagation of these waves by appeal to the classical equations of wave motion (i.e. Maxwell’s equations). The medium could be treated as a continuum and its bulk properties introduced through , in Section 3.2. The method the permittivity (or dielectric response), ~ ( w )introduced may break down at very close separations, where the graininess of the matter becomes evident, but that turns out to be of little importance. Our analysis of the London dispersion interaction (Sections 11.1-1 1.3) makes clear the importance of the fluctuating dipoles in two interacting atoms. The macroscopic approach draws heavily on the idea of correlated fluctuations, that is, the idea that the charge movements that occur on one atom are influenced by the movements occurring on neighbouring atoms and even those that are further away. The macroscopic body is considered to be made up of many local oscillating dipoles that are continuously radiating energy (as any vibrating dipole must do according to classical electromagnetic (e.m.) theory). These dipoles are also continuously absorbing energy from the e.m. field generated by all of their neighbours. Obviously, they are best able to absorb energy at frequencies corresponding to one of their natural resonances (recall Exercises 3.2.5-7) and these are also the frequencies at which they are most easily polarized and those at which they radiate energy. Two macroscopic bodies thus ‘see’ or ‘feel’ one another across an intervening gap as a result of the electromagnetic waves which emanate from one to the other. Most of the energy of these correlated fluctuations remains inside the body and is part of its cohesive energy, i.e. what holds it together. The part that is of most concern to us is carried by the electromagnetic waves that escape from the surface of the two neighbouring bodies. This is the energy contained in the surface vibrational modes, and our problem is to determine how that electromagnetic field is propagated across the gap between the two bodies. We will show how that is done for the very simplest possible situation: two semi-infinite flat plates (i.e. two half-spaces) separated by a vacuum. That calculation will introduce most of the important concepts, which can then be generalized to more important situations. It should be obvious that, in considering how one material responds to the electromagnetic field generated by a neighbouring material, we will draw heavily on the dielectric response function (Section 3.2). That quantity contains all the important information about how the substance responds to an alternating electric field. It should also be clear that such information is implicitly contained in the electromagnetic absorption spectrum of the material because that measures exactly the same thing: what frequencies of e.m. energy can be taken up by the material and to what extent? We will find that the function C ( W ) can, in principle, be constructed from a knowledge of the absorption spectrum, but that it is a difficult task requiring an extraordinary level of information over the whole frequency range (from microwave to far ultraviolet). Fortunately, the solution of the van der Waals problem requires much less information -merely the values of the function C ( W ) for purely imaginay values of the frequency (i.e. ~(zt) where t is a real number). At first sight this may seem a rather

554 I

1 1 : THE THEORY OF V A N DER WAALS FORCES

strange and ‘unphysical’ quantity but it should be noted that if we substitute w = i( in eqn (3.2.25) we have: (1 1.6.1)

so that e(i() may be regarded as the measure of the response of the system to an exponentially decaying (rather than an oscillating) field. 1/t is then the time-constant.

11.6.1 Interaction between two flat semi-infinite bodies across a vacuum This problem is treated in an elementary fashion by Hunter (1975) and somewhat more formally by Richmond (1975) and Mahanty and Ninham (1976). We will adopt the elementary approach here, again because we want only to indicate how the final formalism arises. (See Fig. 11.6.1.) We wish to find the frequencies of electromagnetic radiation that emanate from surface (1) and are propagated across the vacuum to ( 2 ) and which satisfy the Maxwell equations in the gap. For the moment we assume that the velocity of light c = 00 so that we are ignoring retardation effects. The result will therefore be valid only for small values of L. The waves that can ‘fit in’ to this space are the analogues of the standing waves that ‘fit in’ to the black body radiator. If we can establish which frequencies (wj) produce these standing waves at a given separation, then the interaction energy can be calculated from the change in the zero point energy of the electromagnetic field:

We assume that this is the only contribution to the energy change. Calculating the internal energy change corresponds to London’s and Hamaker’s calculation. Equation

Fig. 1 1.6.1 Two semi-infinite materials separated by a vacuum. The approximate shape of the lowest permitted frequency wave function is shown. (Note the change in slope at the interface due to the change in permittivity.)

MODERN DISPERSION FORCE THEORY

1555

(1 1.6.2) is true only at the absolute zero of temperature. The temperature must be introduced by calculating the free energy change A F rather than AU and we will introduce that correction later. As the gap width diminishes, there is a decrease in the number and frequency of the modes which can satisfy Maxwell’s equations in the gap and hence the (zero point) internal energy of the system decreases. It is shown in Appendix A1 that the frequencies which have the necessary characteristics are defined by what is called a dispersion relation? which takes the form:

+

where A,, = (€1 - E O ) / ( E I €0) and R is the magnitude of the wave vector (eqn (3.3.1)). The values of w(= wj) which satisfy eqn (11.6.3) are the frequencies which can be propagated across the gap. For very large L values, the exponential term may be neglected and wj (00) must then satisfy:

If we take for q ( w ) the simplest form suggested by eqn (3.2.40), corresponding to a single undamped absorption mode at frequency w0

(1 1.6.5) then eqn (11.6.4) is satisfied by (Exercise 11.6.1): Wj(c0)

= fi/2

+ w$.

(1 1.6.6)

Thus the absorption mode gives rise to a permitted frequency that is just a little higher than the absorption frequency. Furthermore, if we add more terms in the series to represent more absorption frequencies in eqn (11.6.5), each one will give rise to an additional wj value like that in eqn (1 1.6.6),since each of the terms is important only in the neighbourhood of the absorption frequency. In this way we could construct all the wj s required to calculate li wj(00), in eqn (1 1.6.2), provided the absorptions were well separated. At finite distances the solution would be a little more difficult but two things can still be said: to each absorption frequency there still corresponds an wj that will satisfy eqn (1 1.6.3) but its numerical value is lower than wj(00) and this will ensure that VA is negative. Also, it is not too difficult to see that for finite L, eqn (11.6.3) has solutions only when €1 (w) is negative. From eqn (1 1.6.5) we see that this occurs at values of w just above a resonance frequency (Fig. 11.6.2).

$x

+The word ‘dispersion’ refers to the process whereby light of the different wavelengths is dispersed on going through a triangular prism. Equation (1 1.6.3) gives the condition for propagation of a wave without hindrance (as distinct from absorption).

556 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

/ /

Fig. 11.6.2 The dielectric response function E ( W ) for a single undamped oscillator of resonance frequency W O . We will discuss the function +$) later, but note how simple it is in form compared with the behaviour of E' in the neighbourhood of an absorption.

It might be possible, in principle, to solve some simple problems by this tortuous procedure: finding the wj s that satisfy the dispersion equation for infinite separation and again for finite L and substituting in eqn (1 1.6.2). Fortunately, however, that is not necessary. By virtue of a very powerful theorem in the theory of functions of a complex variable (namely Cauchy's theorem) it is possible to evaluate the sum function in eqn (1 1.6.2) without knowing the separate values of wj. The mathematical technique introduces no new physical principles so we will not pursue it here. It is outlined in a simplified fashion in Appendix A2 in order to show why it is that the solution involves only the values of E for purely imaginary values of the frequency (i.e. €(it)). We will shortly examine (Section 11.6.3) the more general problem of two arbitrary materials separated by a medium of different characteristics, using more rigorous procedures. Before doing so, however, we note a few more subtleties about the dielectric response function ~ ( w ) .

11.6.2 E(W) revisited The simple form for ~ ( wsuggested ) by eqn (11.6.5) is fundamentally deficient. It is obviously a real function of the real variable w. How then can it properly represent an absorption when we know that the dissipative (absorptive) part of c is contained in &'(Exercise 3.2.5), which is zero in this case? The difficulty arises from the use of an undamped oscillator (gj = 0). The form of ~ ( w= ) E' i E" is shown in Fig. 11.6.2, where E" appears as a delta function (i.e. an absorption with zero line width). We will ) with respect to find that the calculations involve integrals of the ~ " ( w function frequency and it is a property of the delta function that such an integral is non-zero. (In a crude way we may say that the height of the absorption peak is infinitely large and this , the dielectric offsets, to some extent, its zero line width.) The real part, ~ ' ( w )of response measures the transmission properties of the medium through its relation to the refractive index (eqns 3.2.35 and 3.2.36).

+

MODERN DISPERSION FORCE THEORY

1557

Fig. 11.6.3 (a)A schematic plot of ~ ” ( was) a function of frequency. (b) Schematic plots of s ’ ( w ) (dotted curve) and €(it)(smooth curve) as functions of frequency. Note the coincidence of the two functions in the ‘flat’ regions between absorption frequencies.

It should be noted that our earlier expression for ~ ( w(eqn ) 3.2.40) is in the form of a real function of the complex variable zw. It can readily be converted into the form E = E’(w)+zd’(w)(i.e. to a complex function of the real variable w ) and that is the form we are presently discussing (see Exercise 11.6.2). That is the form that makes physical sense, since the separate components E’ and E” are readily measurable. The expressions (3.2.28) and (3.2.29)for E’ and E” give rise to an important relation between them. This is one of the Kramers-Kronig relations and it readst (Carrier e t al. 1966)

€:(Lo) =

1

+-

(11.6.7) 0

Eqn (11.6.7) means that if E”(w) (i.e. the absorption spectrum) is known for all frequencies then ~ ’ ( w can, ) in principle, be calculated from eqn (1 1.6.7) (see Fig. 11.6.3 (b)).

+This integral diverges near x = w so that region has to be deleted from the integration process.

558 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

We have already assumed, in writing expressions like eqn (3.2.39), that the function can take complex values of its argument but the results of Exercise 3.2.8 show that this is merely a mathematical manipulation of the function ~ ( w=) c' Ed' of a real variable. For later use we must now formally extend the definition of c:

+

m(t)exp(iwt)d t

[3.2.25]

+

to allow w to take general complex values (w = 26). There are formal mathematical tests that can be applied to establish the conditions under which this can be done. In particular, we will be concerned with values of c for purely imaginary values of w (i.e. ~(9)) as defined for 6 > -60 (where m ( t ) M exp(-tot) for large t) when the integral certainly converges. It is possible, however, to derive a Kramers-Kronig relation ) eqn (1 1.6.1) (Landau and Lifshitz 1969) from the definitions (eqn 3.2.29) for ~ " ( wand for E(Et)), namely:

(1 1.6.8) by eliminating the unknown m(t ). This equation connects the function €(it)to the imaginary part of the real (i.e. experimentally measurable) dielectric response. Note the close similarity of eqn (11.6.7) defining E' (w) and eqn (11.6.8) defining E(E6) as functionalst of E;(W). We will make use of this similarity to construct €(it)from experimental data (see Section 11.7). For the present, it is sufficient to note that €(it) defined by eqn (1 1.6.8) is an even function of t. The Kramers-Kronig relation has served to extend the definition of e(i6) onto the negative 6 axis.

11.6.3 The dispersion relation method The concept of the individual molecular charge distribution acting as an antenna, emitting and receiving e.m. radiation generated by the intra- (and inter-) molecular motions has been introduced already (Section 11.6). That these molecular antennae are densely packed in most materials does not change this fundamental physical picture although the e.m. fields are now propagated through a dielectric medium rather than free space. Provided we are concerned with radiation of wavelength much larger than intermolecular spacings (w < 10l8 rad s-'), the effect of the medium can be described by the dielectric response function ~ ( wof) the medium. That is, an e.m. field (with electric and magnetic field components E and H) of frequency w propagating in the

+A functional is a mathematical device that converts a function (in this case ~ ~ " ( x ) ) into a single number. In this case a new number is generated for each new value of 6 and those are the (real) values of the function e(i6). $A brief outline of the formalism of vector calculus with definitions of the operators V. and V2 is given in Appendix A3.

MODERN DISPERSION FORCE THEORY

1559

uniform dielectric material must satisfy the macroscopic equations1 of Maxwell (Landau and Lifshitz 1960) viz:

v .(E(W)E)

=0

(1 1.6.9)

v .(P(4H) =0

(1 1.6.10)

+

(1 1.6.1 1)

V2E w 2 p ( w ) ~ ( w )= E0

and an identical equation to (1 1.6.11) with E replaced by H. The role of p(w) in the induction of magnetic dipole moment density in the medium by the magnetic field H is exactly analogous to the role of E ( W ) in electric dipole but, ) induction. We could discuss the properties of p(o)in an identical manner to ~ ( o since the magnitude of the induced magnetic dipole moment is small in most materials it is usually an excellent approximation to write p(w) po, the vacuum magnetic permeability. The error is of the order of one part in lo6.The only exceptions to this approximation are the ferromagnetic materials, where p(w) can be -103p0 or more. Recall (Exercise 3.2.9) that the velocity of light in vucuo is given by c = (Eopo)-i

(1 1.6.12)

a result we will use later. Thus modern dispersion force theory starts with the concept of a randomly fluctuating electromagnetic field pervading the material system. This field is driven by the motions of the individual molecular charge distributions, propagates with frequencies characteristic of these motions and is constrained to obey the macroscopic Maxwell’s equations given above. In the presence of dielectric boundaries (interfaces), where ~ ( win) particular suffers a sudden change in value over a length scale (the width of the interface) small compared to the wavelength of the e.m. radiation, the usual e.m. boundary conditions must be satisfied (compare Section 11.6.1). That is, we impose continuity of (a) the normal components of the dielectric displacement D(= EE)and of p H , and (b) the parallel components of E and H across the interface. These boundary conditions serve as constraints on the allowed frequencies of propagation. (A communications engineer would regard the system as a dielectric waveguide and, using conventional waveguide theory, would subsume the constraints imposed by the dielectric boundaries of the system into a single dispersion relation.) That is, for a system of given geometry and dielectric properties we can find a function D(w) such that the allowed frequencies of e.m. propagation in the system are given by

D(w) = 0

(1 1.6.13)

just as we did for the simple problem in Appendix A l . In that case (Exercise A1.2) the magnetic field vector was ignored and only the continuity of potential and dielectric displacement were required. Clearly the roots of eqn (11.6.13), wj (j= 1, 2,. . .), are functions of the geometry and dielectric properties of the system. T o make the connection with quantum mechanics, the fluctuating e.m. field is regarded as a collection of photons. If o,is an allowed frequency (D(wj)= 0) then the

560 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

+

energy associated with this mode is simply (n 1/2)h o j , where n is the number of photons in the mode. As photons are emitted and absorbed by the molecules of the system, the number of photons in any given mode fluctuates along with the energy contained in that mode. The important quantity for present purposes is the Helmholtz free energy of the mode, 4 - the energy available to do work on the system’s environment. From statistical thermodynamics (Landau and Lifshitz 1969) we have (11.6.14) = kgTln[2 sinh @@j/2kgT)]

(1 1.6.15)

where Zj is the partition function for the modej (Exercise 11.6.3) and the summation is over all of the permitted quantum states of the modej. (We use k g here for the Boltzmann constant to avoid any confusion with the magnitude of the wave vector.) The total free energy of the e.m. field is obtained by summing the free energies for each mode

F = kg T

cjln[2sinh

@Wj/2kg T ) ] .

(11.6.16)

However, to evaluate the wj values for eqn (1 1.6.16) explicitly is not feasible in general. It is possible to circumvent this difficulty by using the same mathematical device that was alluded to earlier (see Ninham et al. 1970). The free energy of the e.m. field can be written as:

(11.6.17) where

tn= [2ltkgTfln.

(11.6.18)

The summation is from n = 0 to 00 and the prime on the summation sign indicates that the n = 0 term must be divided by two. Equation (11.6.17) is derived from (11.6.16) using Cauchy’s theorem (Appendix A2) and replaces the difficulty of explicitly obtaining the zeros of D(w) with the difficulty of evaluating the function ID(;() at an infinite set of discrete values (Langbein 1974). If the system comprises two dielectric bodies immersed in a third dielectric fluid phase, the dispersion relation for the system will change as the separation of the bodies is altered. The allowed modes of the system will change correspondingly, with a consequent change in the e.m. free energy of the system. The van der Waals interaction energy V(D)of the two bodies is just the work done to bring them from infinite separation to the separation distance D. Assuming that the work done manifests itself only in the change in e.m. free energy during the process, we can write:

where :

MODERN DISPERSION FORCE THEORY

I561

The dispersion relation method is a physically appealing approach to modern dispersion theory and provides an adequate rationale for the form of the final In particular we see how a sum over imaginary frequencies is expression for VA(0). involved, why the theory depends on a knowledge of the €(it)function for each of the dielectric materials involved, and how the temperature of the system enters, both explicitly in eqn (11.6.19) and implicitly through the definition of Cn.

11.6.4 Modern theory for planar half-spaces One geometry for which the dispersion relation can be explicitly calculated is that of two plane parallel half-spaces of material 1 and 2 immersed in, and separated by, a thickness L of fluid 3. The result is algebraicallyrather complicated but its relationship to the expressions derived earlier will become clearer as we proceed. The interaction energy per unit area of the half-space surface is (Ninham et al. 1970): 00

E132(L) = kBT C ' J [ k dk/2n] ln{D,D,}. n=O

(1 1.6.20)

0

The dispersion relation in this case is D E = 1 - A13A23 exp(-x)

(1 1.6.21)

with an analogous expression for V Mand

(1 1.6.22)

x; = 4(kq2

x = x3

+ xo[-]€ 3 P3 and

0' = 1,2, 3)

xo = 2Lt&3

~ 3 1 ~ .

(1 1.6.23) (11.6.24)

All E and p functions are evaluated at o = iCn. Recall that for all except the ferromagnetic materials we may make the replacement pj = PO with negligible error and then DM = 1. Let us examine the distance dependence of E132(L). Changing to variable x (where x2 = 4k2L2 x;), eqn (1 1.6.20) becomes:

+

x'/ 00 n

xln{DEDM}dx

E132(L) = [kgT/8nL2]

n=O

(1 1.6.25)

0

The function [&PI; yppearing in eqn (11.6.24) for xo is of order l / c (see equation 11.6.12).Thus L[EP]? is a measure of the time of propagation of an e.m. field over the distance L. When this time is small compared to the period of the radiation ((J'

562 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

there will be negligible phase lag. In this limit (often called the c+

00

limit) both L and

xo tend to zero and x1 = x2 = x. The non-retarded interaction energy (c = 00) is

(1 1.6.26) where the constant A132 is given by:

(The transformation from the integral to the summation in eqn (1 1.6.27) is considered in Exercise 11.6.4. The ~ivalues are again evaluated at zt,. In practice one usually needs only a few terms in the series for s to obtain a very accurate assessment of the integral.) Thus the non-retarded dispersion energy has exactly the same form as obtained by the Hamaker summation (eqn (1 1.3.11)). Lifshitz theory in the non-retarded limit still produces a Hamaker constant, but one that has a more complicated density than that obtained in the simpler Hamaker theory. Note dependence (via the €(it)) also that, in modern theory, the bathing medium is automatically included. Lifshitz theory and Hamaker theory agree to leading order in the density and will give comparable results when the €(it)values for the material of the system are close to unity. It is precisely in this dilute regime that pairwise summation would be expected to be a valid approximation. Indeed, the importance of many-body effects can be gauged directly by noting the significant deviations of c(ic) from €0 that occur in liquid and solid dielectrics. In many systems of colloidal interest, Lifshitz theory and Hamaker theory can predict Hamaker constants differing by an order of magnitude or more. It is this quantitative discrepancy that has reduced Hamaker theory to a minor role in modern colloid science. We note that the n = 0 term in the expression (11.6.25) for El32 is always nonretarded, since xo is zero for n = 0 regardless of the value of c. It follows that even for finite c we can separate out an unretarded term like eqn (11.6.26) where the van der Waals constant is the zero frequency contribution:

Various experiments have been performed to test the validity of modern dispersion force theory (Sabisky and Anderson 1974; Israelachvili and Tabor 1972) with considerable success. On the theoretical side, other geometries, e.g. spheres or cylinders, have been examined (see Mahanty and Ninham 1976 for details). Generally speaking, the explicit expressions so obtained for V*(H)are cumbersome and often difficult to compute. The planar half-space problem dealt with here is, however, a

NUM E RICAL COMPUTATION OF INTERACTION E N E R G Y

I 563

relatively straightforward calculation that all modern colloid scientists should be able to appreciate (Section 11.7.2). Useful approximate results in other geometries at small separation distances can be obtained by applying the Deryaguin approximation (eqn 11.5.3) to the planar result. A quasi-empirical procedure that yields a strikingly accurate approximation to the interaction energy of two bodies, VA(H), over the entire separation distance range, is the replacement of the Hamaker constant in the Hamaker summation expression (eqn (1 1.3.21)) by the quantity (12nH2 E132(H)) calculated from eqn ( 1 1.6.20) (Pailthorpe and Russel 1982). An excellent compilation of Hamaker constants for a variety of materials and combinations, especially those of interest in the application of atomic force microscopy, has been given by Bergstrom (1997).

Exercises. 11.6.1 Verify that eqn (11.6.6) is a solution of eqn (11.6.4) if ~ ( wis) given by eqn (1 1.6.5). 1 1.6.2 Consider the dielectric response function:

+ +

and convert it into E,(o) = E:(w) i~r(o) for real o.(E’ and E” must both be real functions.) 11.6.3 Establish eqn (11.6.15) from eqn (11.6.14). 00

1 1.6.4 Use the series expansion for ln(1

-

p) = c ( - p ’ / s ) for 0 < p < 1 1

to show that

7

xln[l

c 00

-

aexp(-x)]dx = -

0

s=

as/s3

1

(Compare eqn 11.6.30).

11.7 Numerical computation of interaction energy Whether performing the relatively straightforward numerical task of calculating a Hamaker constant via eqn (1 1.6.27) or the more difficult one of evaluating El32 ( L )via eqn (1 1.6.20), it is obvious that the functions ~(26)must be known at the points 6 = (a given by eqn (11.6.18). Before discussing the calculation of E132(L), we must first may be constructed from the available experimental data. demonstrate how ~(i6) 11.7.1 Construction of E(<)

The quantity €(it) was defined for a uniform dielectric in Section 11.6.2. The operational definition is the Kramers-Kronig relation, eqn (1 1.6.8), relating €(it)to the

564 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

experimentally measurable e"(w). Equation (1 1.6.8)serves to connect the function E(<) to the physical world. Its calculation requires a knowledge of the absorption spectrum of the material over the entire real frequency range, 0 5 w 5 00. Where such information exists, eqn (11.63) will give exactly the function required by Lifshitz theory. Modern electron loss spectroscopy is capable of yielding E:(w) over a very large frequency range in the ultraviolet regime and this has been used recently to construct e(i<) (Chan and Richmond 1977). With only a few exceptions complete curves of e:(w), however experimentally determined, are unavailable, and approximate methods must be used. It is in this spirit that the method of Ninham and Parsegian (1970) was developed. ) ~ " ( w )the , quantity €(it)is a very It should be understood that, unlike ~ ' ( wand unspectacular function of its argument. T o see this we note that e,(iO) = e:(O) = e,(O)

e(ico) = €0.

and

(1 1.7.1)

Furthermore, we note from eqn(ll.6.8) that, since e:(w) is everywhere positive, e(i<)is a monotonic decreasing function of .$.Thus e(i<) decreases steadily from the static ) = 0 to €0 at $ = 00 (see Figs 11.6.2 and 11.6.3). It is this dielectric constant ~ ( 0 at well-behaved nature of €(it) that enables its construction from the minimum of experimental information as we shall show below. It should be pointed out that the behaviour of e(z<)in the ultraviolet frequency range (< > 10l6 rad s-') is of paramount importance in the calculation 0fA132 or E132(L). The ~ reason for this is that the frequencies tnoccur at equally spaced intervals of 2 n k T/h (-3x 1014 rad s-' at T - 300 K). In the microwave (6 10" rad s-l) and infra-red (6 1014 rad s-l) regions, there are very few sampling points. For example, there are approximately 30 terms in the frequency sum (eqn (1 1.6.20)) in the region < 10l6 rad s-l compared with 300 terms in the region 10l6 <<< 10'' rad s-l, where e(i.$)is still reasonably large (Fig. 11.6.3). This picture is somewhat modified in cases when the intervening medium 3 has dielectric properties and relaxation frequencies comparable to one or both of the half spaces 1 and 2. Then the ultraviolet terms in the frequency sum do not contribute as much to the sum because either or both of A13 and A23 are small. In such cases, the ultraviolet representation is not quite as critical as it would be if medium 3 were a vacuum. In all cases, however, the ultraviolet region makes a major contribution to the total frequency sum. The relative unimportance of the infra-red contribution to €(it)allows one to ignore, in all but an average sort of way, the complicated fine structure of relaxations in this region. Moreover, the UV absorption spectra of materials are usually simple if somewhat broad. If these UV and IR spectra are known, then one can easily extract from the data the relaxation frequencies Wk corresponding to the important absorption peaks in the spectra. The function <(w) can then be represented in a simple way by a discrete set of peaked functions of various heights (and widths) centred on each of the frequencies

<

-

-

<

N

(11.7.2) where Fk(w - wk) is a function peaked about wk that tends to zero for I w - wk I >> 0. The area under each of these peaks is a measure of the strength of the absorption. We

NUM E RICAL COMPUTATION OF INTERACTION E N E R G Y

I 565

would need to specify the functional form of Fk(W - W k ) in order to use the fundamental equation (11.63) to construct ~(26).Since the function e(z6) is a simple function, it is possible to ignore the detailed shape of the peak and to replace the function Fk by the zero width, infinite height delta function without introducing too much error. Thus, we make the approximation, that

/

00

where

fk =

Fk(W

- Wk)

dm

(11.7.4)

0

is the oscillator strength of the absorption at Wk (The oscillator strengthsfk of the dielectric are intimately related to the molecular oscillator strengthsfo, defined in eqn (11.2.7) but modified to take account of intermolecular interactions.) We may now write, from eqns (11.7.2) and (11.7.3):

Substituting eqn (1 1.7.5) into the fundamental relation (11.63) and making use of the delta function property:

7

6(W - W k ) G ( W ) d W

= G(Wk)

(11.7.6)

<wL!

we obtain (Exercise 11.7.1): (11.7.7) in terms of Equation (1 1.7.7) is the Ninhm-Parsegian representation of ~(4) experimentally accessible quantities, viz. relaxation frequencies Wk and oscillator strengthsfk. Note (from eqn (11.7.1) that ck must satisfy the relationship: N k=l

We note here the close relation between E ( z 6 ) and E:(w)as evidenced by eqns (11.6.7 and 11.6.8) connecting these functions to E:(w).If the representation (11.7.5) for E:(w) is used in eqn (11.6.7) we obtain (Exercise (11.7.1): (11.7.9)

566 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

These two functions E(i$) and ~ ' ( ware ) plotted schematically in Fig 11.6.3(b). We see ) ~(ie)are that when the relaxation frequencies are widely spaced, both ~ ' ( w and characterized by large regions where those functions are essentially constant and coincident. This would occur in cases where there was only one significant UV relaxation and where IR relaxations were either unimportant (and so could all be lumped together as a single relaxation at some average frequency containing all the oscillator strength) or dominated by a single IR relaxation. Between the significant relaxations E;(O) is essentially zero. There E:(w)is simply the square of the refractive index n(w) (eqn (3.2.36)), which is usually tabulated in data handbooks together with the dielectric constant. If we denote by €,(before w ) and €,(after w ) the values of the relative dielectric response (n2)in the flat regions before and after the relaxation w, then obviously: E,

(after

WN)

( 11.7.10)

= r(oo)/€o= 1

(since OIN is defined as the largest relaxation frequency) and E,

(1 1.7.1 1)

(before wk+l) = c, (after wk).

If we evaluate eqn (11.7.9) in the flat region before the last frequency W N but after the frequency W N - ~ ,we obtain, E,

(before LON) = €, (after W N - ~ ) M 1

+ CN

( 11.7.12)

since, for w in this region and all frequencies wk widely spaced,

>>

( w / w ~ ) ~1 (k = 1,2,3,

....., N

- 1.)

and

<< 1.

(w/wN)~

Therefore

CN = E , (before LON) - E, (after W N )

(1 1.7.13)

using definition (11.7.9). Similarly, between W N - ~ and w N - 2 we have E,

(after wN-2) = E , (before W N - ~ )= 1

+ CN-1 + C N .

Using eqn. (11.7.12), we obtain = E, (before w N - 1 ) - E, (after

CN-1

WN-~).

( 11.7.14)

Proceeding in this manner it is a simple matter to show (for widely spaced relaxation frequencies) that

Ck

E,

(before wk) - E , (after wk)

(1 1.7.15)

for all k. Thus the Ninham-Parsegian representation becomes:

(

c(i$) = €0 1

- €,(after wk) + C €,(before1wk) + ($/%I2 k=l

( 11.7.16)

NUM E RICAL COMPUTATION OF INTERACTION E N E R G Y

I 567

The detailed application of this approach to the calculation of E(z$) for fused quartz is given by Hough and White (1980) and will not be repeated here, but a simpler problem is discussed below. It should be noted that when the wk values are close together, no flat regions between relaxations occur. Thus it is only possible to find E, (before) and €,(after) listed in the literature when the significant relaxation frequencies are well spaced. If one has a system where the IR relaxations are likely to be important (due to similarities in dielectric response in the UV region) and several IR frequencies are significant, one must use data on the relative strengths of the absorptions. The total oscillator strength CIRto be assigned to the IR absorption can still be computed because it is given by:

c, = r,(O) - n12

(1 1.7.17)

where nl is the refractive index in the visible region. An inspection of the IR spectrum suffices to estimate the relative strengths of the different bands among which this is to be split. (See Hough and White 1980 for details.) Unfortunately, experimental data in the literature for the UV region are more scanty. Where evidence exists as to a complicated UV absorption spectrum the technique of using more than one frequency with appropriate sharing of the available oscillator strength (a2 - l), as outlined above, is clearly applicable in the absence of direct knowledge of E ; ( W ) . At this juncture it is worth making a few remarks about the microwave contribution to €(it).In some polar substances there is a significant permanent dipole relaxation at microwave frequencies (-10'' rad s-'). For example, in water the dielectric response rad s-'). drops from about 80 at zero frequency to about 4 at IR frequencies This relaxation has sometimes been included in the ~ ( i $construction ) (eqn (1 1.7.7)) by a Debye term (Ninham and Parsegian 1970) (compare eqn (3.2.38)):

Such a term is numerically dangerous, since it becomes the dominant term in E,(i() in the far UV, where it has no right to exist. The microwave relaxation would typically ' ~ s-'. As the occur at 5x10'' rad s-' and would certainly be complete by 5 ~ 1 0 rad function E ( i ( ) is sampled first at $ = 0 rad s-' and next at $ = 3x 1014 rad s-', we need only a construction that gives ~(20)and ~ ( i correctly (See Fig. 11.7.1). That ~ ( i $ ) is not equal to E ( i is immaterial provided that these values are correct. One would need to be interested in a dipolar substance at T 2 K before the behaviour in the microwave region was even faintly important. (Only then would the n = 1 sample point fall in the microwave region.) The reader is therefore advised to omit such a term from the E ( Z ( ) representation. The value of the UV relaxation frequency is a critical parameter in the calculation of A132 and E132(L). In the absence of UV spectral data, a frequency w = 11/h corresponding to the first ionization potential 11 of the material is often taken. This can be a serious error and should be avoided. In the absence of knowledge of <(w) over the frequency regime w > 1015 rad s-l, it is still possible to obtain a reasonably accurate construction of €(it)in the UV region provided the absorption is simple; that is, only one relaxation is important.

-

568 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

c(rad sC1)

Fig. 1 1.7.1 A schematic representation of the function ~(26)for a substance with a single relaxation in the microwave region (q,,ia0), a single IR absorption peak and a broad absorption in the UV. Note the positions of the sampling points t,, for T = 300 K.

For ( in the visible region (provided ( >> w ~we) can write (from eqn (11.7.7)): ( 1 1.7.18)

Similarly, for ~ ' ( win) the visible region we can write, from eqn(11.7.9):

( 1 1.7.19) Rearranging eqn.(l1.7.19)we obtain: n2 - 1 = ( n 2 - 1)w2/(wv)2

+c,.

(1 1.7.20)

Therefore a plot of [n2(w)- 11 against [n2(w)- 1]w2should yield a straight line of slope (1 /-)2and intercept CUV.Experimentally it is relatively straightforward to measure the refractive index n as a function of wavelength h (= 2nc/w) in the visible region and such information is tabulated in the literature for most common substances. The plot indicated in eqn (1 1.7.20)(called a Cauchy plot) is shown for some common materials in Fig. 11.7.2).Table 11.1 also lists the wv, CUV,data so obtained along with IR and zero frequency data. Table 1 1.1 enables an accurate construction of the e(i() function in most cases, though for water the more elaborate procedure of Gingell and Parsegian (1972)is preferable because of the significant IR contributions. In the literature, use has sometimes been made of an ultraviolet interpolation to connect the near UV representation of € ( i t ) (eqn 11.7.7) to the far UV (plasma) representation. For a full discussion of this procedure see Hough and White (1980).

NUMERICAL COMPUTATION OF INTERACTION ENERGY

100

0

d(nZ-l) x

-

1 569

2 0

Fig. 11.7.2 Cauchy plots (from Hough and White 1980),with permission. (a) Sapphire; (b) Calcite (0-ordinary ray; e- extraordinary ray); (c) Crystalline quartz; (d) Fused quartz and fused silica; (e) Calcium fluoride; (0 Water.

We advise strongly against the use of such procedures, which are unnecessary and often misleading.

11.7.2 Representation of

~(26)for metals

For substances that possess conduction electrons, the dielectric response is dominated by their presence, because they are so mobile and so easily polarizable. The behaviour of the metals is well approximated by that of a dilute electron plasma. Thus for these is the plasma response function: systems, a suitable representation of ~(zt)

where the plasma frequency is given by (Jackson 1975): 2

2

mp = Pele

/me60

where pel is the number density of conduction electrons and me is the electron mass. Typical wp values are rad s-l, so that €(it)for metals is extremely large until the ultraviolet region is reached. Consequently, interaction energies are much larger than for dielectric systems. The screening effect of an intervening dielectric is not as important for the interaction of metal half-spaces as for dielectric half-spaces.

570 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

Table 11.1 Dielectric data f o r c(i6) construction f o r a variety of common substances.

Material

~(0)

ni

CIR ( ~ ~ (0 ni) )

WIR

cuv

w

( x 1014rads-l)

(ni-1)

(x 1016rad spl)

Alkanes n=5

1.844

1.819

0.025

5.540

0.819

1.877

6

1.890

1.864

0.026

5.540

0.864

1.873

1.899

0.025t

5.540

0.898

1.870

7 8

1.948

1.925

0.023

5.540

0.925

1.863

9

1.972

1.947

0.025

5.540

0.947

1.864

10

1.991

1.965

0.026

5.540

0.965

1.873

11

2.005

1.979

0.026

5.540

0.979

1.853

12

2.014

1.991

0.023

5.540

0.991

1.877

13

2.002

0.025t

5.540

1.002

1.852

14

2.011

0.025t

5.540

1.011

1.846

15

2.019

0.025t

5.540

1.019

1.845

16

2.026

0.025t

5.540

1.026

1.848

80.10

1.755

Gingell and Parsegian (1972)

0.755

1.899

ord. ray

4.27

2.350

1.92

2.093

1.350

2.040

Xord. ray

4.34

2.377

1.96

2.093

1.377

2.024

average

4.29

2.359

1.93

2.093

1.359

2.032

Fused quartz

3.80

2.098

1.70

1.880

1.098

2.024

Fused silica

3.81

2.098

1.71

1.880

1.098

2.033

ord. ray

8.0

2.683

5.3

1.683

1.660

Xord. ray

8.5

2.182

6.3

1.182

2.134

average

8.2

2.516

5.7

2.691

1.516

1.897

caF2

7.36

2.036

5.32

0.6279

1.036

2.368

Water Xtalline quartz

Calcite

Sapphire

11.6

3.071

8.5

1.880

2.071

2.017

PMMA

3.4

2.189

1.2

5.540

1.189

1.915

PVC

3.2

2.333

0.9

5.540

1.333

1.815

PS

2.6

2.447

0.2

5.540

1.424

1.432

Poly-(isoprene) 2.41

2.255

0.16

5.540

1.255

1.565

2.10

1.846

0.25

2.270

0.846

1.793

PTFE

PMMA: Poly (methyl methacrylate); PVC Poly(viny1 chloride); PS: Poly(styrene); PTFE: Poly (tetrafluorethylene). +Assumedvalue.

I N F L U E N C EOF ELECTROLYTE CONCENTRATION

11.7.3 Numerical evaluation of

I 571

E132(L)

In the non-retarded (small L) region, it is sufficient to calculate the Hamaker constant A132 in order to evaluate E132(L). This is a simple numerical procedure involving the calculation of ~l(zt,), ~~(zt,) and ~3(2$,)for each of the values of tn(n = 0, 1,2,. . .) and the subsequent evaluation of the sums in eqn (11.6.27). The s sum is very rapidly convergent and only a few terms (less than ten and sometimes only three of four are needed to obtain the necessary accuracy). The n sum usually requires a few thousand terms for a precise vale 0fA132, but is easily set up on a spreadsheet. In Table 11.2 we list some typical Al3zvalues calculated from €(it)functions constructed from the data in Table 11.1. Example: Calculation of the Hamaker constant for the dodecane I air I dodecane system from the data in Table 11.1 using any spreadsheet programme. We are here calculatingA121 where 1 is the dodecane and 2 is air (i.e. vacuum). The value of the A13A23 function simplifies to Ai2 and since E ~ / Q = 1 for all frequencies we can write A12 = [(EI/Eo-~)/(E~/Eo+~)]. The function q / ~ isg given by: El/EO

=1

0.023

+ +

0.991

l2

[

5.54 tnx 1014I 2 + l + [ 1.877 x 10l6

where t,, = [4n2 k~T/h]n= 2.4513 x lOI4 n rad/s. Step 1. Set up the first column with n = 0 to 2000. Set up column 2 with values of = 2.4513 n. Step 2. Set up column 3 with E I / E O = 1 0.023/[1+(~012/5.54)~] 0.991/[1+(~012/ 187.7)2]. Step 3. Set up column 4 as f = [(co13-l)/(c013+1)]~. Step 4. Set up column 5 as f f 2/8+ f 3/27 f 4/64 (more than enough terms). Step 5. Add all these terms in column 5, except the first one, then add it in after multiplying by 0.5. This is the final frequency sum. T o get A121 you now only have to J. Table 11.2 gives multiply by 1.5 kBT and you should get 5.036 x 5.04 x lop2' J.

+

+

+

+

(Exercises 11.7.1 Establish eqns (11.7.7), (11.7.8), and (11.7.9).

11.8 Influence of electrolyte concentration One of the most important applications of the theory of van der Waals forces in colloid science is to the interaction of two materials separated by an aqueous electrolyte solution. We must now ask what effect the presence of the ions has on the propagation of the electromagnetic interactions which give rise to van der Waals forces. Although

II

572 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

Table 11.2 Hamaker constants (x l@20)/Jforsome common materials.

Material (M)

M I air IM

M I water I M

M I water I air

M I air I water

water I M I air

3.63

0.108

Alkanes n =5

3.75

0.336

6

4.07

0.360

-0.368 x lo-’

3.78

0.285

7

4.32

0.386

-0.118

3.89

0.423

8

4.50

0.410

-0.200

3.97

0.527

9

4.66

0.435

-0.275

4.05

0.624

10

4.82

0.462

-0.344

4.11

0.719

11

4.88

0.471

-0.368

4.14

0.751

12

5.04

0.502

-0.436

4.20

0.848

13

5.05

0.504

-0.442

4.21

0.855

14

5.10

0.514

-0.464

4.23

0.886

15

5.16

0.526

-0.490

4.25

0.923

16

5.23

0.540

-0.518

4.28

0.964

Fused quartz

6.50

0.833

-1.01

4.81

Xstal. quartz

8.83

1.70

-1.83

5.59

Water

3.70

0

0

3.70

Fused silica

6.55

0.849

-1.03

4.83

2.23

-2.26

6.00

1.04

-1.23

5.06

5.32

-3.78

7.40

Calcite Calcium fluoride Sapphire

10.1 7.20 15.6

0.153

PMMA

7.11

1.05

-1.25

5.03

PVC

7.78

1.30

-1.50

5.25

PS

6.58

0.950

-1.06

4.81

Poly (isoprene)

5.99

0.743

-0.836

4.59

PTFE

3.80

0.333

0.128

3.67

Mica (brown) Mica (green)

-

1.98

-

-

-

2.14

-

-

this question cannot be answered completely we can make a few observations; a more complete description of the situation was given by Mahanty and Ninham (1976) (Chapter 7) who made it clear that there are profound problems involved in the general treatment of two charged interfaces separated by an electrolyte solution.

I N F L U E N C EOF ELECTROLYTE CONCENTRATION

I 573

Richmond (1975) provides a simple analysis of the effect of an aqueous electrolyte solution between two uncharged surfaces. The important point to note is that ionst are so large that they are able to respond only to the lowest frequencies of the e.m. field. In fact, the main effects will be on the zero frequency (to) term. This is quite a significant term for systems containing water -especially two organic materials separated by an aqueous film (or vice versa) - because the W contributions are similar for the two different materials and tend to cancel one another. The zero frequency term can then dominate and some early calculations suggested that it might be responsible for a quite large unretarded attraction, of importance in many biological situations. It now appears that the presence of the electrolyte very considerably damps that attraction and A' (from eqn (11.6.28)) is not unduly large. If the approaching surfaces are uncharged, it can be shown that the charge density fluctuation caused by the electric field is: p(r, t ) = -c(w)z4(r, t ) (for w

-

0) and p(r, t) = 0 (for w 2 109s-'),

where K is the DebyeHuckel parameter that depends on the electrolyte concentration (Section 7.3.1). The dispersion relation is sampled at w = 0 and multiples of 2 n k ~ T / h which is of order 3 x 1014 s-l for T 300 K so only the w = 0 term is affected by the presence of the electrolyte. Then:

-

=-

4

7

dx x In (1

-

813823 exp(-s))

(1 1.8.1)

0

where s2 = x2 + ~ ( K L )and ~

6. j3

cj(0)x - ~(0)s

- €j(o)X

+ €3(O)S

Once ~ K becomes L greater than unity, the n = 0 term will be screened and will approach zero. Thus the presence of the electrolyte ensures that the n = 0 term is no longer the dominant term in the estimation of E132(L) for large L. Comparison of eqn (11.8.1) with the first term of eqn (11.6.27) shows that the effect is to make the argument of the logarithm function approach unity more quickly so that Ao is reduced in magnitude. An effect of order 50% is quite possible. The more realistic problem of two approaching charged surfaces has been solved only under conditions that Mahanty and Ninham (1976) regard as too restrictive to furnish unequivocal results. The theoretical point at issue is the question of whether it is permissible to treat the attraction and repulsion potential energies as separable. The success of the DLVO theory (which assumes such a separation is possible), and the lack of any viable alternative, gives us no choice. It is not the first time (nor will it be the last) where the limited insights of an approximate theory are used until a more rigorous development becomes available. TMahanty and Ninham point out that this statement may not hold for the proton in water.

574 I

1 1 : THE THEORY OF V A N DER WAALS FORCES

11.9 Theoretical estimation of surface properties 11.9.1 Surface and interfacial tension and energy The discussion in Section 2.10 shows that many surface properties can be drawn into a unified conceptual framework if we can estimate, with reasonable accuracy, the , and n v since this would allow a calculation of the interfacial energies, y s ~ ysv, contact angle 8, the work of adhesion and cohesion, and the spreading coefficient. The most effective pioneering work in this area was done by Fowkes who extended an earlier suggestion of Girifalco and Good (1957). In his introductory review of the subject, Fowkes (1965) shows the value of extracting from the surface or interfacial energy the component due to dispersion (van der Waals) interaction, yd. He assumes that at the hydrocarbon-vapour interface it is the only contribution:

(1 1.9.1) Fowkes used the rather crude Hamaker theory to estimate surface free energies but with the analysis given above we should be able to improve on that. In order to calculate y& from the dispersion equations we have only to imagine the reverse process to that shown in Fig. 2.10.l(a). That is, we bring together two semi-infinite blocks of hydrocarbon (of unit area) from infinity until they are (almost) touching H H stands (separation L,) and calculate the interaction energy per unit area, E ~ where for hydrocarbon and V for vacuum (or vapour), from eqn (11.6.26). Then

(11.9.2) Unfortunately, to get a finite value out we must postulate a minimum (non-zero) value for L, corresponding to the ‘distance’ between the bodies when they are in contact. We noted in Section (11.6.4) that the divergent behaviour as L approaches zero is a consequence of the failure of the heuristic method of van Kampen to take account of the ‘graininess’ of matter and that it is absent from the exact Lifshitz theory. Intuitively, we expect L, in the more simplified theory to be of the order of a molecular calculated from spectral data for various hydrocarbons are radius. The values of Agiven in Table 11.2 (in the column headed M I air I M) and using those values in eqn (1 1.9.2) with the experimental ~ / H Cvalues we find the values of L, listed in Table 11.3. They are close to being constant and are certainly of the correct order of magnitude. The slight decrease with increasing chain length reflects the increase in density of hydrocarbons with molar mass. Hough and White (1980) show that the critical length, L,, calculated in this way is inversely proportional to the square root of the liquid density. This relationship can be exploited to obtain the values calculated in the last column of the table with only one adjustable parameter, the value of L, for some reference hydrocarbon, calculated from the measured surface tension. In Table 11.3 we have used the value for decane and the calculated values are then given by: YHV

=

AHVH x P = 6.78 x 1014Ap 24n x (0.1638 x x 0.7300 x lo3

(11.9.3)

THEORETICAL ESTIMATION OF SURFACE PROPERTIES

I 575

Table 17.3 Estimation of surface tension. (All but the last column come from Hough and White 1980, with permission.)

Alkane

0) 1

(Table 11.2)

(J mp2)

~ ~ L, (nm) (kg mp3)

C5H12

3.75

16.05

0.6262

0.1758

15.90

C6H14

4.07

18.40

0.6603

0.1712

18.20

CsH1s

4.50

21.62

0.7025

0.1660

21.40

C10H22

4.82

23.83

0.7300

0.163(8)

(23.83)

C12H26

5.03

25.35

0.7487

0.1622

25.50

C14H30

5.05

26.56

0.7628

0.1594

26.09

C16H34

5.23

27.47

0.7733

0.1587

27.39

IO~OAWH

0

~

i03mc (eqn (1 1.9.2)) (theory?)

+Calculatedusing L, = (L,)n=10x (p/p,,=lo)f.

for A in joules and p in kg m-3. Hough and White (1980) apply a less objectionable procedure for avoiding the L + 0 problem but it is a good deal more difficult to apply than eqn (1 1.9.3). In order to estimate the dispersion contribution to y at other interfaces, Fowkes argues as follows. When two immiscible liquids (say mercury and a hydrocarbon) are in contact, the dominant interaction between them is the dispersion (van der Waals) interaction and in the case of a hydrocarbon-hydrocarbon interface it may be regarded as the only important interaction. The surface tension of the pure hydrocarbon is reduced by the interaction that the hydrocarbon molecules have with the adjacent mercury atoms. Likewise, the surface tension of the mercury is reduced from its value in air ( y ~ to) some lower value due to interactions between the mercury atoms and the hydrocarbon molecules. Fowkes estimates the reduction in each case from the geometric mean of the dispersion components (compare eqn (1 1.3.23)): For the hydrocarbon and for mercury

0 d ~ H V M= yHv - (yHv 0

y&)i

d

(11.9.4)

~MVH = yM - (yHvy&)i.

Then the interfacial tension between mercury and hydrocarbon is: 0

0

m c = YM + YHV

d - 2(yHVy&)'.

(1 1.9.5)

Since y& = y h this equation contains only one unknown y$ and by comparing the interfacial tensions of a series of hydrocarbons against mercury it is possible to estimate the dispersion component of the surface tension of mercury. Fowkes finds a value of y$ = 200 f 7 mJ mP2. The remainder of y~ (284 mJ mP2) would be attributed to metal bonding in this case.

576 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

The same argument can be applied to the water-hydrocarbon surface and a value for y i z 0 of 21.8 f 0.7 J m-’ is obtained (Exercise 11.9.2). The remarkable thing about

these results is that if we now apply them to the mercury-water interface, assuming again that the only important interaction is the dispersion force, we find: YMW

0

= yM

d d f + YW0 - 2(YMYW)

= (484

+ 72.8

-

2(200 x 21.8)i = 424.8 mJ m-’

which is very close to the experimental value of 426-427 mJ m-’. There is, therefore, surprisingly little contribution from the permanent dipole moment of the water molecule. This empirical finding suggests that reliable estimates of interfacial tension can be obtained directly from the dispersion energy, even for water films where hydrogen bonding might be expected to play a role. Unfortunately, a more detailed examination (Hough and White 1980) shows that one can be very easily led into false conclusions regarding the behaviour of water films if only the dispersion contribution is examined.

11.9.2 Contact angle of liquids on low energy solids According to eqn (2.10.7):

Normally the presence of the n, term makes theoretical estimation of the contact angle from eqn (1 1.9.6) quite impossible. In the case of very-low-energy-solids like polyethylene, however, it is found that ne = 0. That is to say, the vapour of high surface energy liquids will not adsorb onto a low-energy solid since this will not lower the surface energy. Neither will the vapour of a hydrocarbon liquid adsorb to any appreciable extent. Low-energy solids are substances for which AG,d, is small for any adsorbate. They are usually substances with little or no dipole moment and very low polarizability so that they can interact with adsorbates only by van der Waals forces, and then only slightly. The classic example is the non-stick frying pan coated with poly-( tetrafluorethylene). If we again assume that the interaction between the liquid and the low-energy solid is predominantly due to dispersion forces, we can then substitute for y s ~from the analogue of eqn ( 1 1.9.5) and obtain:

(11.9.7) Fowkes (1965) shows that a plot of the cosine of the measured contact angle against ( y ; ) f / y for ~ a number of liquids on low-energy surfaces (like polyethylene and paraffin wax): Fig. 11.9.1 does obey eqn (11.9.7), from the slope of which an estimate of the dispersion contribution to the surface energy of the solid (y,”) can be obtained. Note that again water appears to behave like other liquids in this plot.

THEORETICAL ESTIMATION OF SURFACE PROPERTIES

I 577

YP

40

+1

20

30

10 20 30 40

50 60 70

m $ 0

u

80

d

90

2

100

g

0

2 5

u

110 120 130 140 150 -1

0.1

0.2

0.3

180

(YLd)flYL

Fig. 11.9.1 Contact angle of a number of liquids on low energy surfaces. V: polyethylene; 0: paraffin wax; 0: c36H74; . fluorodecanoic : acid monolayer on platinum. All contact angles below the arrow are with water. (Copyright American Chemical Society. Reproduced with permission from Fowkes 1965.) -

The value of y,d for the four solids in Fig. 11.9.1 should be calculable from the van der Waals constants using (compare eqn (1 1.9.2)):

(11.9.8) if a suitable value of L, is chosen. Taking L, = 0.2 nm and the extrapolated value of Asvs for C36H74 from Table 11.2 (ASVS= 6.0 x lop2' J) we obtain y,d = 20 mJ mP2,in quantitative agreement with the value obtained for paraffin wax from Fig. 11.9.1. Although this agreement is to some extent fortuitous it is very encouraging. The other values show the tendency one would expect (increasing y,d with increasing molar mass) and they suggest that Asvs for polyethylene should be about 37/20 x 6 x loP2' = 1.1 x J: not an unreasonable figure. The behaviour of alkanes on poly-(tetrafluorethylene) (PTFE) should be amenable to more exact analysis, and this has been done by Israelachvili (1973b), whose work is reviewed by Hough and White (1980). Using eqn (2.10.8) we have: YSL

= Y,o

+ Yh+ ESVH(0)

(11.9.9)

578 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

50

rn

-

v G)

M

:

*

cd 0 a4

8 0

5

10 No. of carbon atoms

15

Fig. 11.9.2 A comparison of the theoretical (a) and experimental (b) curve for contact angles of alkanes on PTFE. Wetting would correspond to 6' = 0 and is expected to occur for n 5 5. (From Hough and White (1980) with permission.)

where EsvH(O)is the van der Waals energy of interaction per unit area of the solid with the liquid hydrocarbon across a gap of essentially zero width. This term is assumed to represent the work of adhesion between the solid and the liquid. Again, we have to make a cut-off approximation and set

(1 1.9.10) where, for simplicity, we assume that the critical separation L, is the same as for the liquid (eqn (1 1.9.2)). Using eqns (11.9.2), (11.9.6), and (11.9.10) we have:

cose = -1

- 2[AsvH/A-].

(1 1.9.11)

Hough and White (1980) recalculated cos 8 using their more accurate assessments of A ~ V and H A ~ forHa number of liquid hydrocarbons on PTFE and obtained the results shown in Fig. 11.9.2. A cautionary note is in order at this point. The use of L, as an adjustable parameter is not always successful. Ninham (1980) points out that it fails for liquid argon. The fact that it works reasonably well for liquid hydrocarbons must therefore be to some extent fortuitous. Using the estimates of van der Waals constants given by Bergstrom (1997) should allow techniques like those of Fowkes, even with their limitations, to provide a more systematic approach to the problem of wetting and contact angle. Progress involves the combined efforts of experimentalists and theoreticians but the general procedures are now clear. There remain enough observed anomalies to keep this area an active and fruitful one and to ensure the value of further research on both experimental and theoretical fronts.

REFERENCES

1579

Exercises 1 1.9.1 Use the data in Table 11.2 to estimate theoretical values of cos 6 for hydrocarbons with 6, 8, 10, 12, 14, and 16 carbons and compare your results with those shown in Fig. 11.9.2. 1 1.9.2 Given the following data for the interfacial tension ( n z ) between water and various hydrocarbons, estimate y:. n=

6

7

8

10

14

Yl(Cn&n+2) (d m-’) y d m N m-’)

18.4 51.1

20.4 50.2

21.8 50.8

23.9 51. 2

25.6 52.2

References Atkins, P.W. (1978). Physical chemistry (1st edn) (6th edn 1998). Oxford University Press, Oxford. Bergstrom, L. (1997). Adv. Colloid Interface Sci. 70, 12549. Bhattacharjee, S. and Elimelech, M. (1997).J. Colloid Interface Scz. 193,273-85. Bradley, R.S. (1932). Phil. Mag. 13,853. Carrier, G.F., Crook, M., and Pearson, C.E. (1966). Functions of a complex variable. McGraw-Hill, New York. Casimir, H.B.G. and Polder, D. (1948). Phys. Rev. 73, 360. Chan, D.C. and Richmond, P. (1977). Proc. Royal. Soc. Lond. A353, 163. Christenson, H.K. (1983). Ph.D. thesis, Australian National University. Clayfield, E.J., Lumb, E.C, and Mackey, P.H. (1971).J Colloid Interface Sci. 37, 382. Dalgarno, A. and Lynn, N. (1957). Proc. Roy. Soc.Lond. A70,802. Debye, P. (1920). Phys. 2. 21, 178. Deryaguin,B.V. (1934). Kolloid-2. 69, 155. Dzyaloshinski, I.E.. Lifshitz, E.M., and Pitaevski, L.P. (1961). AdnPhys. 10, 165. Fowkes, F.M. (1965). Attractive forces at interfaces. In Chemistry and physics of interfaces (ed. D.E. Gushee), pp. 1-12. American Chemical Society, Washington. Gingell, D. and Pargsegian, V.A. (1972).J Theor. Biol. 36,41. Girifalco, L.A. and Good, R.J. (1957). J Phys. Chem. 61, 904. Gregory, J. (1969). Adv. Colloid Interface Sci., 2, 3 9 H 1 7 . Gregory, J. (1981). 3’.Colloid Interface Sci. 83, 138. Hamaker, H.C. (1937). Physics, 4, 1058. Hough, D.B. and White, L.R. (1980). Adv. Colloid Interface Sci. 14, 3-41. Hunter, R.J. (1963). Austral. 3’.Chem. 16, 774. Hunter, R.J. (1975). Electrochemical aspects of colloid chemistry. In Modern aspects of electrochemistry (ed. B.E. Conway and J.O’M Bockris) Vol. 11, Chapter 2, pp. 33-84. Plenum Press, New York. Israelachvili, J.N. (1973~).Q Rev. Biophys. 6, 341. Israelachvili, J.N. (19736).J. Chem. Soc. Faraday Trans. 2 69, 1729 Israelachvili, J.N. (1974). Contemp. Phys. 15, 159.

580 I

I I : T H E THEORY OF VAN D E R WAALS F O R C E S

Israelachvili, J.N. and Tabor, D. (1972). Proc. Roy. Soc. Lond. A331, 19. Jackson, J.D. (1975). Classical electrodynamics (2nd edn). Wiley, New York. Kallmann, H. and Willstatter, M. (1932). Naturwissenschaften 20, 952. Keesom, W.H. (1921). Phys. 2. 22, 129 and 643. Landau, L.D. and Lifshitz, E.M. (1960). Electrodynamics of continuous media. Pergamon, London. Landau, L.D. and Lifshitz, E.M. (1969). Statistical physics. Pergamon, Oxford. Langbein, D. (1974). Theory of van der Waals attraction. Tracts in modern Physics. Springer, Berlin. Lifshitz, E.M. (1956). Sov. Phys.JETP., 2, 73. London, F.(1930). 2. Phys. 63,245. Mahanty, J. and Ninham, B.W. (1976). Dispersion forces. Academic Press, London. McClellan, A.L. (1963). Tables of experimental dipole moments. Freeman, San Francisco. Ninham, B.W. (1980). J. Phys. Chem. 84, 1423. Ninham, B.W. and Parsegian, V.A. (1970). Biophys. J. 10,646. Ninham, B.W., Parsegian, V.A., and Weiss, G. (1970).J. Statist. Phys. 2, 323. Overbeek, J. Th. G. (1952). In Colloid Science (ed. H.R. Kruyt) Vol.1, p. 265. Elsevier, Amsterdam. Pailthorpe, B.A. and Russel, W. (1982). J. Colloid Interface Sci. 89, 563-6. Parsegian, V.A. (1975). Long range van der Waals forces. In Physical chemistry: enriching topicsfrom colloid and surface science (ed. H. van Olphen and K.J. Mysels) Chapter 4. IUPAC Commission 16. Theorex, La Jolla, California. Pitzer, K.S. (1959). Adv. Chem. Phys. 2, 59. Reinganum, M. (1912). Ann. Phys. 38,649. Richmond, P. (1975). In Colloid science (ed. D.H. Everett) Vol. 2, Chapter 4. Chemical Society, London. Sabisky E.S. and Anderson, C.H. (1974). Low temperature physics - LT13 Proceedings of the 13th International Conference on Low Temperature Physics (1972), Vol. 1, pp. 206-10. Sparnaay, M.J. (1983). J. Colloid Interface Sci., 91, 307. Thomson. J.J. (1914). Phil. Mag. 27, 757. van der Waals, J.H. (1873). Ph.D. Thesis, University of Leiden. van der Waals, J.D. Jnr. (1909). Amsterdam Acad. Proc. pp. 132, 315. van Kampen N.G., Nijboer, B.R.A., and Schram, K. (1968). Phys. Lett. 26A, 307. Vincent, B. (1973). J. Colloid Interface Sci. 42, 270. Visser, J. (1972). Adv. Colloid Interface Sci. 3, 33143. White, L.R. (1983). J. Colloid Interface Sci. 95, 286-8. Wilson, J.N. (1965). J. chem. Phys. 43,2564.

Double Layer Interaction and Particle Coagulation 12.1 Surface conditions during interaction 12.2 Free energy of formation of a double layer 12.3 Overlap of two flat double layers 12.3.1 Approximate relations for overlap o f flat double layers 12.4 Interaction between dissimilar flat plates 12.4.1 Approximation formulae for the interaction between dissimilar plates 12.5 Interaction between two spherical particles 12.5.1 For large values of Ka 12.5.2 For small values of Ka 12.6 Total potential energy of interaction 12.6.1 The Schultz-Hardy rule 12.7 Experimental studies of the equilibrium interaction between diffuse

double layers 12.7.1 Adsorbed liquid films 12.7.2 Soap films 12.7.3 Swelling of clays 12.7.4 Direct force measurements between mica surfaces 12.8 Kinetics of coagulation 12.8.1 Rate o f rapid coagulation 12.8.2 Aggregate formation 12.8.3 Coagulation under shear 12.8.4 Rate o f slow coagulation 12.8.5 The fractal character of aggregates 12.9 Effect of polymers on colloid stability 12.9.1 Steric stabilization 12.9.2 Polymer bridging 12.9.3 Charge effects 12.9.4 Depletion interactions

581

582 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

In this chapter we consider the various forces that come into play when two colloidal particles or two double-layer systems approach one another. The behaviour is determined to some extent by how rapidly the approach occurs, since the double layers may take a significant time to adjust to the new situation. Equilibrium behaviour may then be somewhat different from transient behaviour. The repulsive interaction that occurs between double layers of like sign can be analysed by two alternative procedures by considering: (a) the free energy change involved when interaction occurs; or (b) the osmotic pressure generated by the accumulation of ions between the particles. The case of large flat plates can be treated fairly rigorously but some useful approximation formulae are also derived for the case of low potentials or for small degrees of double layer overlap. We then look at situations where the potentials on the two approaching surfaces are different. This is relevant to such processes as flotation, mixed colloidal suspensions, and the drainage of wetting films on solids. The theory for spherical particles is more complicated algebraically and will not be discussed in great detail. The experimental evidence in support of the DLVO Theory (Section 1.6) comes from a variety of sources but the most definitive results have been obtained since about 1976, using atomically smooth sheets of mica, interacting in an aqueous electrolyte solution. That work clearly establishes the general validity of the theory presented below, at the same time suggesting that when particles approach one another very closely there are quite dramatic new effects that come into play. The chapter concludes with a brief description of the kinetics of coagulation, the fractal dimensions of colloidal aggregates, and the effect of polymers on colloid stability.

12.1 Surface conditions during interaction When two colloidal particles (or two charged interfaces, in general) approach one another so that their electrical double layers begin to overlap the result is usually a repulsive force, which tends to oppose further approach (Section 1.6). For flat plates, this effect can be understood in terms of the osmotic pressure created by the difference in ion concentration in the region between the two approaching surfaces compared with the bulk (or reservoir) concentration (Fig. 12.1.1). T o simplify the exposition we will initially assume that the surfaces are planar and that the Poisson-Boltzmann equation holds over the entire region between them. In most interaction situations it is only the diffuse layers that interact so that the boundary electrostatic potential (Section 7.3) is $rd rather than $0, but we will leave that aside for the moment. When two surfaces approach one another there are several possible situations that might arise. The approach may be slow, so that equilibrium can be established between the ions on the surface and in the bulk. For silver iodide particles under those conditions one would expect the surface potential to remain constant during

SURFACE CONDITIONS DURING INTERACTION

y/,c

I 583

.........\..*

I" Fig. 12.1.1 The overlap of two diffuse double layers. The potential distribution in the neighbourhood of a single double layer is shown The full line is the anticipated potential distributionfor the pair of particles. llr,is the potential at the minimum (which in this case is also in the median plane). The high potentials between the plates (llr,> 0) lead to high counterion concentrationsand this in turn leads to a large osmotic pressure, tending to push the particles apart.

the approach. On the other hand, if the particle charge is caused by built-in crystal defects, as in some clay minerals, it might be more sensible to assume that the surface charge is constant during approach. In the case of oxide surfaces, the interaction may itself influence the degree of dissociation of surface groups (Section 7.10) so that neither $0 nor 00 is constant. T h e condition known as charge regulation (Ninham and Parsegian 1971) may then be more appropriate. There are other possibilities if specific adsorption in a Stern layer is involved. In general, however, these three possibilities can adequately cover most experimental situations. The constant potential case was extensively studied by the early workers in the field and is the basis of the DLVO theory of colloid stability (Section 1.6) described in detail in the monograph by Verwey and Overbeek (1948). T h e other possibilities are best understood as fairly straightforward generalizations of the constant potential case. We will examine first the case where the approaching surfaces are identical and subsequently deal with the case of unlike systems (heterocoagulation). The repulsion can be calculated either from the osmotic pressure, as noted above, or from the increase in Helmholtz free energy, AF, that occurs as two double layers overlap. T he osmotic pressure method cannot be applied to the approach of spherical particles because another factor (the Maxwell stress, Section 12.3) enters in that case. T he free energy method is rather more general in its application. We will begin, therefore, with an examination of the free energy of a single (i.e. isolated) double layer.

584 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

12.2 Free energy of formation of a double layer When a silver iodide crystal is immersed in an aqueous solution, the double layer forms spontaneously on its surface (Section 7.7). The free energy of formation (AG or AF) is, therefore, negative. It can be calculated by determining the effect of the double layer on the capacity of the system to do work and that in turn can be calculated by determining the work that would have to be done to establish the double-layer structure from an initially uncharged system. The double layer arises because of a preferential adsorption of potentialdetermining ions of one sign over those of the other. The adsorption occurs against an increasingly unfavourable electrostatic potential because the chemical potential of the ion (Ag+ or I-) is lower on the surface than it is in the bulk solution. If we begin with an initially uncharged system and imagine the ions being adsorbed one at a time, then for each ion adsorbed there is a reduction, A p , in the chemical potential. This ‘chemical’ part of the free energy change can be evaluated by noting that, when the last ion goes on, the chemical free energy decrease, A p , is exactly compensated by the electrical free energy increase: e$o per ion or oo$o per unit area of surface, i.e.

Apchern = -00df0= AFchern (per unit area of surface)

(12.2.1)

(This inherently assumes that A F is independent of the state of charge of the surface. It is strictly true only at low charge density and corresponds to the condition introduced in Section 7.7.1 to derive the Nernst equation. Chan and Mitchell (1983) show how to remove this restriction to generate a more general expression for A F but we will not need that in the present treatment.) During the charging process, as the double layer builds up, the electrical work done can be calculated by imagining the transfer of infinitesimal amounts of charge d o (per where Yo increases from zero to its final value $0 unit area) through a potential when the last piece of charge is installed. The electrical work done is then:

vo

AFelec=

1

$;do.

(1 2.2.2)

0

(This is analogous to the electrical work done in charging a capacitor.) There is another ‘work’ term involved in the arrangement of the charges in the solution, after each step in the charging process, so that the balancing charge in the solution can be established. It is not difficult to show, however, (Exercise 12.2.1) that this does not involve any further change in free energy if the equilibrium ion distribution obeys the Boltzmann equation. The total free energy change involved in establishing the double layer is then

(1 2.2.3)

F R E E ENERGY OF FORMATION OF A DOUBLEL A Y E R

I585

(Note that the chemical term is always larger in magnitude than the electrical term so that AF is negative since 00 and $0 always have the same sign.) Equation (12.2.3) may also be derived directly from the Lippmann equation (7.2.14), expressed in the form (ay/avo), = do. Then:

/

11.0

AF= AF = yo y o- yy== / d/ yd = y - j= o h-d0I &ohdI&. . 0

The free energy of a single (isolated) diffuse double layer is readily calculated from eqn (12.2.3) because 00 can be expressed as a function of $0 using the appropriate form of eqn (7.3.27). Then (Exercise 12.2.2):

(12.2.4) (Note that no distinction is made between AF and AG in these systems: the pV contribution to the work done in the charging process is negligible.) Some simple expressions for AF can be established when the potential on the surface is low (Exercise 12.2.3). We can now readily see why the interaction is repulsive. As the particles approach under constant potential conditions, the potential profile (Fig. 12.2.1) becomes increasingly shallow. The absolute value of d$/dx at the surface, therefore, decreases. This can be interpreted as a decrease in the surface charge density since (compare eqn (7.3.26)): DO =

-E(d$/dX),,o.

(12.2.5)

As the particles approach, the surface potential can only remain constant if the potential determining ions are gradually driven off the surface until the double layer finally disappears on contact. Since the double layer was initially formed spontaneously, this forcible discharge of the plates results in an increase in free energy. The repulsive potential energy VR,due to approach of the two plates is thus given by:

VR= [AF(D)- AF(co)]

(12.2.6)

where AF(o0) is the free energy (per unit area) associated with the isolated double layer and AF(D) is the corresponding free energy when the plates are at a distance D apart. The corresponding expression for the repulsive potential energy, evaluated from the osmotic pressure, 3, between the plates is: D

VR= - S p d D

(12.2.7)

586 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

Fig. 12.2.1 As two plates approach at constant potential the slope d$/dx at x = 0 decreases in absolute value, indicating a progressive discharge of the surface potential-determiningions.

where ji is the dtflerence in the osmotic pressure midway between the plates and in the bulk solution. Both the free energy and the osmotic pressure method can be used to calculate VR for flat plates and each has advantages in different situations. Given VR,the total potential energy of interaction VT can then be obtained by adding VA(Chapter 11) and VT can then be used to describe the stability behaviour (Section 1.6).

r

Exercises 12.2.1 Show that the free energy change involved in arranging the diffuse counter

charge is zero if the ions obey the Boltzmann equation. 12.2.2 Establish eqn (12.2.4) using the appropriate form of eqn (7.3.27). 12.2.3 Show that at low potentials, where the Debye-Huckel approximation holds, the free energy of a flat double layer is A F = - ~ c K & and of a sphere is -2ma (1 + .a,&.

12.3 Overlap of two flat double layers It was noted in Section 1.6 that the stability behaviour of colloidal sols was determined almost entirely by the concentration of the counterion. It is possible, therefore, to restrict attention to symmetric electrolyte systems because only the valency of the counterion is of any great importance. A salt like MgCl2 will then be expected to behave more like a 1:l electrolyte with respect to a positive surface and more like a 2:2 electrolyte with respect to a negative surface. The potential profiles drawn in Figs 12.1.1 and 12.2.1 must still satisfy the PoissonBoltzmann equation (7.3.15) which, for a symmetric electrolyte, can be written. 0 xe$ 2n xesinh-

KT

- C-

d2$ = 0. dx2

(12.3.1)

O V E R L A P OF TWO FLAT DOUBLE L A Y E R S

I587

This equation can be integrated (for a fixed value of D) as before to obtain (compare eqn 7.3.18): =p ,

(a constant).

(12.3.2)

The first term on the left is related to the osmotic pressure between the plates. That pressure varies from point to point because of the variation in $ (and, hence, of the local ion concentration). It can be most readily evaluated at the midplane between the particles, because at that plane (d$/dx) = 0 and so the osmotic pressure is equal to 9,:

p , = 2nOkTcosh

kT

The higher osmotic pressure at other points between the plates is partly offset by the other term on the left of eqn (12.3.2) (= $ IE I ’), which is called the Maxwell stress. The actual force per unit area exerted on the plates is given by the difference in osmotic pressure between the solution in the midplane between the plates and that outside (in the reservoir of electrolyte):

3 = KT(n+ + n-

-

2%0) = 2nokT(coshy,

-

1)

(12.3.4)

where ym = ze$,/kT is a dimensionless potential. T o calculate VRas a function of D from eqns (12.2.7) and (12.3.4) we would need to , as an explicit function of D. Unfortunately that is only possible if certain know $ assumptions are made (see Section 12.3.1). For spheres, there is no plane where the Maxwell stress is zero; the osmotic pressure method is then inapplicable. T he alternative method for flat plates, using eqn (12.2.6), also involves a similar problem. In that case we have an expression for AF(o0)in eqn (12.2.4) but AF(D) depends on the surface charge 00 at each value of D and, again, it is not possible to obtain an analytical expression for that. It is, however, possible to solve eqn (12.3.2) and to express the potential profile in terms of elliptic integrals of the first kind. It is then possible to calculate AF(D) and, hence, VR in terms of elliptic integrals of the second kind. Th e resulting expressions are rather lengthy and will not be repeated here. Th e complete analysis is given by Verwey and Overbeek (1948), and the final result is quoted by Overbeek (1952). An outline of the procedure for finding the potential profile is given by Hunter (1981) (Appendix 5), while a detailed discussion of the procedures is given in the compilation of tables by Devereux and de Bruyn (1963). For our purposes, we need only record the final result, in the form of Table 12.1 (from Verwey and Overbeek 1948) which gives ,, and KD. From this values off(ym,$o) = (z’/K)VRfor various values of $0, $ table it is possible to calculated VRas a function of D for any given $0 value and electrolyte concentration (Exercise 12.3.1). The ready availability of computing facilities makes it worthwhile to explore the use of direct numerical procedures to evaluate VRaccurately as a function of D. The

Table 12.1 f(pm,@Q) = ( . z ~ / K ) V Rin units of

3 m - ’ f i r K in m-’. Corresponding values of K D are also given f i r dtfferent values ofyo (= ze@o/kT). The mid-plane potential $rm = kTy,/ze can also be read offfir any yo and KD. (Note that D is the total separation between the plates.) (Adaptedfrom Overbeek 1952, p . 254, with permission.) The temperature is taken as 25 “C and the relative dielectric permittivity 6, = 78.55. (KD values obtained by doubling Overbeek’svalues and so may contain some round-off error.)

10 f

KD

268.3 228.2 0.00868 0.0000

9

f 161.5 KD 0.0000

8

f

192.6 0.01672

160.0 0.0268

135.2 0.0146

115.2 0.0276

95.6 0.0442

127.1 0.0408

75.4 0.0874

44.1 0.1626

25.4 14.1 0.286 0.488

7.36 0.824

3.42 1.380

1.26 2.296

0.26 3.924

0.06 5.442

0.015 6.880

0.0023 8.732

76.3 0.0674

44.3 0.1442

25.4 0.268

14.1 0.472

7.36 0.806

3.42 1.358

1.26 2.278

0.26 3.906

0.06 5.424

0.015 6.862

0.0023 8.714

44. 8 0.111

25.4 0.238

14.1 0.442

7.36 0.776

3.42 1.330

1.26 2.248

0.26 3.878

0.06 5.396

0.015 6.834

0.0023 8.686

25.8 14.17 0.183 0.392

7.36 0.728

3.42 1.282

1.26 2.202

0.26 3.830

0.06 5.348

0.015 6.786

0.0023 8.636

14.38 7.39 0.3018 0.646

3.42 1.202

1.26 2.122

0.26 3.752

0.06 5.270

0.015 6.708

0.0023 8.560

KD

96.52 0.0000

80.56 0.0242

68.56 0.0454

56.60 0.0728

7

f KD

57.13 0.0000

47.46 0.0398

40.18 0.075

32.89 0.120

6

f KD

33.27 0.0000

27.47 0.0654

23.04 0.1236

18.66 0.198

5

f KD

18.83 0.0000

15.32 0.1082

12.69 0.2036

10.07 0.3264

7.52 3.43 0.4976 1.066

1.26 1.990

0.26 3.622

0.06 5.140

0.015 6.580

0.0023 8.430

4

f

10.13 0.0000

8.07 0.1782

6.51 0.336

4.97 0.5384

3.50 0.821

1.26 1.768

0.26 3.404

0.06 4.924

0.015 6.362

0.0023 8.214

4.962 0.0000

3.793 0.2942

2.913 0.5548

2.061 0.891

1.291 1.362

0.26 3.036

0.06 4.560

0.015 5.996

0.0023 7.848

1.993 0.0000

1.413 0.487

0.966 0.9286

0.584 1.502

0.265 2.356

0.06 3.916

0.015 5.360

0.0023 7.216

0.4682 0.0000

0.271 0.8712

0.135 1.710

0.0348 3.064

0.063 2.558

0.015 4.070

0.0023 5.942

KD 3

f

KD 2

f

KD 1

f

KD

O V E R L A P OF TWO FLAT DOUBLE L A Y E R S

I589

following method was described by Chan et al. (1980). Writing eqn (7.3.15) in terms of the dimensionless variables y = x e @ / k T and X = KX we have (Exercise 12.3.2): d2y/dX2 = sinhy.

(12.3.5)

This can be integrated once to give (Exercise 12.3.2): dy/dX = sgnO/,)Q = sgnO/,)[2(cosh y - coshy,)$

(12.3.6)

where the coordinate system is placed with the origin for X in the midplane. The function sgn(y) = y / Iy I (= f l ) gives the algebraic sign to be attached to dy/dX so that there is a minimum in I 11.1 at the midplane. Now note that (Exercise 12.3.3): d Q - sinhy - sgnO/,) dY Q Q

[( Q2 I + ~oshy,)~-l]'

(12.3.7)

and hence (Exercise 12.3.3): (12.3.8) This equation can be used to calculate the relation between X and ym provided the value ofQon the surface, is known. The variable Qis defined by eqn (12.3.6) and if the surface potential, 11.0, is known then

a

Qs = [ ~ ( c o s ~~ coshy,)]f o

(12.3.9)

where yo = xe@o/kT. Given any arbitrary ym(with I ymI C I yo I ) one can integrate (numerically) eqn (12.3.8) from the midplane where X = 0 and Q = 0 to obtain corresponding values of Qand X. When Qreaches the value the corresponding X value corresponds to half the separation between the plates (i.e. X (Q= Qs)= K D / ~ ) for the given initial value of y,. By choosing different ym values from near zero (corresponding to large separations) up to near yo (small separations) one can obtain y, as a function of separation D for the given surface condition (Qs). Since the force per unit area between the plates is then given by eqn (12.3.4), the potential energy of repulsion can be calculated by numerical integration using eqn (12.2.7). If the interaction occurs under conditions of constant surface charge, a0 then Qs is given by:

a,

(from eqn (12.2.5)). Again, the integration of eqn (12.3.8) is pursued until the value of Qs is reached and the corresponding value of X is recorded for subsequent evaluation of V, from eqn (12.2.7).

590 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

The same procedure can be used for the condition of charge regulation. In that case the surface groups undergo some form of dissociation and the degree of dissociation is influenced by the interaction. Using the single-site model of Section 7.9.1 it is necessary to ensure that the surface condition satisfies eqn (7.9.4). For a positive surface that releases a negative ion (A-) into solution the corresponding relation is:

Hence, when two such identical surfaces interact this condition must be fulfilled at all separations. In order to apply the numerical algorithm described in the previous section the finishing condition Qmust always comply with eqn (12.3.11). Calculation of the repulsive interaction with this surface condition gives the regulating case where both surface potential and charge will vary with separation in order to satisfy the surface equilibria. The precise regulation of the interacting surface will depend on the concentration of adsorbing ions, the dissociation constant, and the site density. An example is shown in Fig. 12.3.1. It is found that the regulation case must always lie between the limits of constant charge and potential (Chan and Mitchell 1983). For some systems, the difference in interaction energy between the range of surface conditions is slight. However, for spherical colloids near the point of coagulation the effect of regulation can be of the ultmost importance. Under these conditions it is necessary to obtain values for the surface dissociation parameters, e.g. site density and dissociation or binding constants.

1.0 0.8 -

VR

mJ m-’

Fig. 12.3.1 Comparison between the constant surface potential (lowest curve), constant surface charge (uppermost curve) and charge regulation (CR) calculation of VR.The calculation is for a M in 1:l electrolyte. (From Healy et al. 1980, surface of ApK = 6 (see Section 7.10) at pH 7 and with permission.)

O V E R L A P OF TWO FLAT DOUBLE L A Y E R S

I591

This same numerical procedure can also be used for calculating the interaction 1 and a 2 between two dissimilar plates. Two separate integrations are done up to a and the resulting values of D1 and D2 are added together to obtain the total separation between the plates.

12.3.1 Approximate relations for overlap of flat double layers If the potential in the midplane is sufficiently small, the cosh term in eqn (12.3.4)can be expanded to obtain (Exercise 12.3.4): (12.3.12)

For small degrees of double-layer overlap (i.e. D > 1 / ~ the ) midplane potential can be approximated by adding the potentials due to each plate. If x = 0 on one plate, then:

$, = 2$(x = D/2). Under these conditions, eqn (7.3.22)is a very good approximation for $ and so: $m

(12.3.13)

= [8kT/xe]Zexp(-~D/2)

where Z = tanh ze$0/4kT. The repulsion potential energy is then given by (Exercise 12.3.4):

/ D

V$ = -

jdD = -

00

-

/ D

64nokTZ2exp(-KD)dD (12.3.14)

00

64n0k TZ2 exp( -KD). K

This is one of the most widely used approximate expressions for VR.It is also valid for conditions of constant charge, since little discharge occurs if the degree of double layer overlap is small. If the potential in the region between the plates is small everywhere (i.e. I yo I < 1) so that the DebyeHuckel equation (7.3.11) holds, then it can be shown (Exercise 12.3.5) that the midplane potential is given by: $rn

(12.3.15)

= $o/ C O S ~ ( K D / ~ ) .

Then using the appropriate form for the osmotic pressure when I $,,, I is small (eqn (12.3.12))it can be shown that (Exercise 12.3.6):

v$=,-

2nokT

tanh(~D/2)]=

4nokT ~

K

exp(-(KD) yo 1 exp(-KD)

+

(12.3.16) *

592 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

Comparing this with eqn (12.3.14)it would seem that a very good approximation, valid for all potentials provided the interaction is not too strong, would be (Verwey and Overbeek 1948, p. 97) (Exercise 12.3.7):

32n'kT VZ = y Z2[1 - tanh(~D/2)].

(12.3.17)

Figure 12.3.2 shows a comparison between the 'exact' value of VR calculated from Table 12.1 (or by numerical integration) with that obtained using eqn (12.3.14). The log-linear relation between VRand D is evident over a considerable range of KDvalues, though the approximate equation tends to overestimate the value of VR. The corresponding expression for VR when the plates interact under conditions of constant charge Vg rather than constant potential is given by Parsegian and Gingell(l972):

vg =

(12.3.18)

sinh KD

Note that under low potential conditions where 00 = EK$O and small degrees of double layer overlap (exp(-KD) << 1) this expression reduces to:

(12.3.19)

V; = ~ C K & exp(-KD)

which is the same as the result obtained under constant (low) potential conditions (Exercise 12.3.9).This equality of V i and Vg for low potentials holds only at large D. At small D values the repulsion energy is different in the two cases (though the force is the same).

t

2o llRV'O 5

2 1

0.5

0.2 0.1 0.05 I

I

I

0

1

2

3

4

5 KD-

6

Fig. 12.3.2 Repulsive potential energy on a logarithmic scale against distance separating the plates. Full curves:Table 12.1. Dotted curves;eqn (12.3.14).The energy scale is in mJ mp2for K = lo9 mpl and in pJ m-' for K = lo6 m-'. (After Venvey and Overbeek 1948, p. 85, with permission.)

OVERLAP OF TWO FLAT DOUBLE LAYERS

1593

A more elaborate expression, valid for higher surface potentials, has been derived by Gregory (1973a). He calculated the volume density of charge between the plates by adding together the effects of (i) the compression process limiting the available volume and (ii) the presence of the other plate with its counterions. The potential in the midplane can then be written:

sinhy, = yo cosech K D / ~

(12.3.20)

where again, yo = ze+o/kT and $0 is the potential when the plates are infinitely far apart. The repulsion potential energy under constant charge conditions is then:

(12.3.2 1)

+

where B = [l y; cosech2 (~D/2)]f.This gives values in close agreement with the exact calculation.

7

Exercises 12.3.1

12.3.2 12.3.3 12.3.4 12.3.5

12.3.6 12.3.7 12.3.8

Calculate the potential energy of repulsion V, as a function of separation (0 5 D 5 20 nm) for two flat plates of surface potential 51.4 mV in a 2:2 electrolyte at a concentration of lop4 M in water at 298 K. Establish eqn (12.3.5) and integrate it to obtain eqn (12.3.6) using the boundary condition dy/dX = 0 at X = 0 and y = ym. Establish eqns (12.3.7) and (12.3.8) Expand the cosh term in eqn (12.3.4) as a power series to establish eqn (12.3.12). Hence establish eqn (12.3.14). Integrate the Debye-Huckel equation d2@/dx2= K ~ @with the appropriate boundary conditions to obtain d@/dx = K(@ Use the substitution @ = &,cosh w to perform the second integration and hence show that @O = @m cosh K D / ~ . Use eqn (12.3.12) along with the result ofExercise (12.3.5) and eqn (12.2.7) to establish eqn (12.3.16). Calculate the value of VRfrom eqn (12.3.17) for the same conditions as in Exercise 12.3.1 and plot the results on the same graph. Show that for small @O (in the DebyeHuckel approximation) the potential profile between two flat plates may be written (for large D):

em)lf2.

coSh[~(D/2- x)] where x = 0 on one plate. COS~(KD/~) @(') = @O 12.3.9

Show that if the Debye-Huckel approximation holds throughout the region between two approaching flat plates then VR= 2r~&exp(-~D)if the surface potential is constant during the interaction.

594 I

12: DOUBLE LAYER INTERACTION A N D PARTICLE COAGULATION

12.3.10 Use eqns (8.7.3) and (8.7.4) to derive eqn (12.3.2) for the osmotic pressure between two identical charged flat plates in the form: p-p, = (~/2)(d@/dx)~ where p , is the pressure in the midplane and p is the osmotic pressure elsewhere.

12.4 Interaction between dissimilar flat plates It was noted in connection with eqn (12.3.10) that the numerical procedure for evaluating VRcan also be used in the case where the plates have different potentials. The potential profile then is as shown in Fig. 12.4.l(a) and the two parts ($01 - )$ , and ($m - $02) can be treated separately.

I\

I

.

/-

Fig. 12.4.1 (a) The potential profile between two dissimilar plates A and B of surface potential @01 and $002. Note that the minimum no longer occurs in the midplane. (b) Forceedistance relation for the interaction between two plates of unequal (but constant) potential; both potentials have the same sign.

INTERACTION BETWEEN DISSIMILAR FLAT PLATES

I 595

It is also possible to calculate the 'exact' value of VR from the information in Table 12.1 by using the method of isodynamic curves as suggested by Deryaguin (1954). The method is described in some detail by Hunter (1975). It depends on the fact that the quantity = (Pm - 2nOk7') = p - $(d$/dx)2

(12.4.1)

is constant everywhere between the plates (eqn (12.3.2)). Here p = 2nokT (cosh ze$/kT - 1) a n d j is the value o f p for $ = $m (as defined by eqn (12.3.4)). The potential profile between the plates A and B is simply a part of the profile between two plates of equalpotential, if the second plate is placed at the appropriate position (C in Fig. 12.4.la). A plate B of surface potential $02 placed as shown in Fig. 12.4.la is repelled by the same force as would be exerted on a plate, C, of potential $01 placed further away, at a distance D from the first plate. The total potential energy of repulsion between the plates A and B is then given by:

VR(AB) = iVR(AC)

+ ~VR(EB)

(12.4.2)

and the quantities on the right depend only on interactions between identical plates. If the second plate were placed at the position E in Fig. 12.4.l(a) it would still experience the same force if its potential were $02 but this would correspond to a plate with a charge of opposite sign to the first plate (A). [Note that (d$/dx) at the plate E has the opposite slope to that at plate B.] Thus it is possible for plates of opposite sign of charge to repel one another, if they are interacting under constant potential conditions. The force between the two plates A and B at separation h is determined by adjusting the position of the plate C (called the control plate) until the potential profile between A and B passes through $02 at a distance h from plate A. Then, from eqn (12.4.2):

The force between the two plates, for any particular values of $01 and $02 has the same magnitude at two different values of h (corresponding to position B and E in Fig. 12.4.1(a)). The plot of force as a function of h is shown schematically in Fig. 12.4.l(b). Notice that it rises to a maximum which must correspond to the situation when plate 2 is a distance D/2 from the first plate and $02 = lcr, (where lcr, is the potential midway between two plates of potential $01 and separation 0). Note also that when plate B is in that position (Fig. 12.4.l(a)) the function d$/dx = 0 at the surface of B and so B must be uncharged (from eqn (7.3.26)). The maximum pressure is then given by eqn (12.3.4). This at first sight rather surprising result (that the force is a maximum when the plate B is uncharged) can be readily understood when one realizes that constant potential plates, as they move through the position of the maximum, change their surface charge from positive to negative values (assuming $01 and $02 are both positive to begin with). Indeed, if the charge on the second plate is made sufficiently negative then the repulsion force diminishes to zero. This corresponds to a situation where the

596 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

potential on the second plate at distance ho is exactly equal to the potential at a distance ho from an isolated plate of potential $01 (see Exercise 12.4.5): In tanh[xe$02/4kT] = In tanh[xe$ol/4kT]

-K

~ O

(12.4.4)

The potential profile between the plates is then identical to that of an isolated plate and there is no force (Fig. 12.4.1(b)). The charge on the second plate compensates for all the charge which would have been associated with plate 1 in the region ho < x < 00. Finally, it should be noted that if the second plate is moved even closer to the first plate, then its potential can only be kept constant by piling even more negative charge onto it. (Note the higher slope of (d$/dx) for x < ho.) This gives rise to an attractive force between the plates even though they have the same sign of potential. The converse behaviour is not possible -if both surfaces have the same sign of charge then the interaction is always repulsive. The above analysis assumes that the surface potential is constant on each plate. For a discussion of the constant charge behaviour see Jones and Levine (1969). The charge regulation behaviour is treated in Chan et al. (1976)

12.4.1 Approximation formulae for the interaction between dissimilar plates So far we have used the osmotic pressure method to evaluate the repulsive potential energy in most cases. Returning to the free energy method it should be noted that eqn (12.2.3) remains valid for the free energy associated with a pair of dissimilar interacting double layers, provided the integration is done for both plates. The problem is that this is rarely possible because a0 is not known as an explicit function of D for constant $0. There is, however, one situation in which it is possible and that is when the potential is sufficiently small for the DebyeHuckel approximation to hold between the plates. We (compare eqn (7.3.30)) and substituting in eqn (12.2.3) can then set q0equal to QEK and integrating gives (Exercise 12.2.3): A F = -$000/2.

(12.4.5)

Hogg et al. (1966) used this procedure with two dissimilar double layers of (constant) potential to obtain:

where the surface charges 01 and a 2 are, of course, functions of the separation distance

D.Then if we assume that the DebyeHuckel equation (7.3.11) applies: d2$/dx2 = K2$,

(12.4.7)

the potential between the plates can be written (Exercise 12.4.2):

$(x) = A1 cosh KX

+ A2 sinh

KX

(12.4.8)

INTERACTION BETWEEN DISSIMILAR FLAT PLATES

I 597

where A1 and A2 are constants to be obtained using the boundary conditions given by the two potentials $01 and $02. Thus at x = 0 and x = D we obtain the results: $01

= AI

and

$02

= A1 cosh KD

+A2 sinh KD.

(12.4.9)

Hence combining (12.4.9) with eqn (12.4.8) (Exercise 12.4.2):

(12.4.10) which describes the potential distribution as a function of distance x from the $01 potential surface. Using eqn (12.4.10) we can now obtain by differentiation the potential gradient at each surface (Exercise 12.4.2):

(12.4.11) and

9 1 dx

= -K($OI

cosech KD- $02 coth KD).

x=D

Hence the double-layer charge on each surface is given by+:

and

01

= --EK($o~ cosech KD- $01 coth KD)

02

= +EK($o~ coth KD- $01 cosech KD).

(12.4.12)

Using eqn (12.4.6) we obtain: AF(D) = $~[2$ol$o2

cosech KD -

+ &2) coth KD]

(12.4.13)

and if D is very large (no interaction):

(12.4.14) Hence using the interaction energy definition (eqn (12.2.6)):

Thus we obtain an explicit expression for the potential energy of repulsion between two planar surfaces of constant unequal (low) potential. When $01 = $02 and the interaction is weak, this equation reduces to the relation derived in Exercise 12.3.9. (But see Exercise 12.4.6.) Equation (12.4.15) allows computation of the effect of unequal potentials on the interaction, which corresponds to the case of heterocoagulation. Under conditions of constant dissimilar potential we noted in Section 12.4 above that it is possible to obtain + Note that on plate 1 the charge is -E(d+/dx)

whereas on plate 2 it is +E(d+/dx).

598 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

attractive double-layer forces below some particular separation. The separation (in terms of K - l ) at which the interaction energy VRbecomes zero can easily be obtained for particular values of $01 and $02 using eqn (12.4.15) (Exercise 12.4.4). For example, for $01 = 25 mV and $02 = 50 mV, VR = 0 at 0.2 K - ’ . (The separation at which an energy maximum occurs can similarly be found from eqn (12.4.15).) The separation at which the force between the plates changes from being positive (repulsive) to negative (attractive) is a rather more significant quantity and that is examined in Exercise 12.4.5. As noted above (Section 12.4) there is no force between the plates when the potential of the second plate satisfies $01 = $02 exp(-KD) so that the profile between the plates is the same as for an isolated double layer at low potential. For an extension of this treatment to higher potentials, see Ohshima et al. (1982, 1983). In summary, if the charge on the plates is fixed during an interaction then the interaction free energy is always positive (repulsive) if the signs are like and negative (attractive) if they are unlike. If the potential is constant during an interaction and is different on each plate then an attraction can occur at small separations, even when the potentials have the same sign.

Exercises 12.4.1 If the approaching plates are of opposite sign of potential the control plate must

12.4.2 12.4.3 12.4.4 12.4.5

have potential -@o and the potential in the midplane is then zero. Discuss the resulting forces in terms of the Maxwell stress in the region between the plates (refer to Hunter 1975). Verify that eqn (12.4.8) is a general solution for eqn (12.4.7). (Compare the result of Exercise 12.3.5.) Derive eqns (12.4.9-1 1). Establish eqns (12.4.13-15). Verify that @ = 0 for @01 = 25 mV and @02 = 50 mV for KD 0.2. The force between two flat plates is given by F = -d&/dD. Use eqn (12.4.15) to show that:

+ G2)

F = - ~ E K ~ [ ( G ~ cosech KD- 2@01@02 coth KD]/ sinh KD

) plate 1 cannot ‘see’ and hence prove that F = 0 when @02 = @01 exp ( ~ K D(i.e. plate 2 if the potential there is the same as it would be for an isolated plate). 12.4.6 Show that for any value of KD, eqn (12.4.15) reduces to V$ = a&[1

-

tanh(~D/2)]when

@01

= @02 = @o.

12.5 Interaction between two spherical particles 12.5.1 For large values of

Ka

When two identical spherical particles of radius a approach under conditions of constant potential, the repulsive potential energy can be calculated using the

INTERACTION BETWEEN TWO SPHERICAL PARTICLES

I 599

Deryaguin procedure (Section 11.5) if K a is sufficiently large. We then write (Verwey and Overbeek 1948, p. 138; compare eqn (11.5.4)): M

(12.5.1) Introducing eqn (12.3.16) as an approximation for (Exercise 12.5.1):

@ (valid for low potentials) then

00

V$ = n a m & /[l - tanh(~D/2)]dD= 2nca&j ln[l

+ exp(-~H)]

(12.5.2)

H

where H is the distance of closest approach (Fig. 11.5.1). If the centre-to-centre distance is Y and we set s = r / a and t = tca, eqn (12.5.2) can be expressed in the alternative form:

A more accurate calculation of VR can be made from eqn (12.5.1) by a numerical integration procedure in which the ‘exact’ value for VRis used at each separation. In actual practice, Verwey and Overbeek (1948) used the approximate formula (eqn (12.3.14)) for VR (flat plate) and integrated this directly to get an approximate expression for VR(spheres) like eqn (12.5.3). They could then introduce a correction function and integrate the correction numerically. In this way only a small part of the final result is involved in the rather tedious numerical integration process. With modern computer facilities the whole process can be done quite quickly and accurately. Wiese and Healy (1970) have shown that, for low potentials, the repulsion energy under conditions of constant charge is given by (Exercise 12.5.1):

V i = V$ - 2xca& ln[l

- exp(-2~H)] = -2nca&ln[l-

exp(-~H)]. (12.5.4)

The Deryaguin method may be used provided a is reasonably large (say K a > 10) and the results of ‘exact’ and approximate calculations of VRare shown in Fig. 12.5.l(a).

12.5.2 For small values of

Ka

When the double layer around the particles is very extensive (Ka C 5) the Deryaguin procedure begins to break down and an alternative approach is necessary. Verwey and Overbeek (1948) developed an approximate procedure for low surface potentials but the resulting expressions are not simple. They did, however, show that an approximate value for VRcan be calculated from:

600 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

Z2VR

(pJ $l)

0.3

0.2

L

\\\

//LKa=2.0

0.1

0.0 0

1

2

KH

Fig. 12.5.1 (a) The repulsive potential energy, VRbetween two large spherical particles when the exact expression for VR(flat plates) is used in eqn (12.5.1). Broken lines refer to the approximate expression (eqn (12.5.4)). (Adapted from Venvey and Overbeek 1948, p. 141, with permission.) (b) The potential energy of repulsion for approach at constant surface potential when a is small. (Adapted from Overbeek 1952. p. 260, with permission.)

TOTAL POTENTIAL ENERGY OF INTERACTION

I601

if an error of up to about 40 per cent can be tolerated. The results of the more complete calculation for V i are shown in Fig. 12.5.l(b). This expression is valid only for low surface potentials but a number of other approximate expressions have been developed by Honig and Mu1 (1971) for small and large separations, and moderate potentials or charges, on the basis of both the constant potential and constant charge assumption. An alternate procedure is the surface element integration method of Bhattacharjee and Elimelech (1997) referred to briefly in Section 11.5. That can give reasonable estimates of interaction energies for small values of KU with no restrictions on double layer potential.

[ Exercise

I

12.5.1. Establish eqns (12.5.2) and (12.5.3).Also establish the identity of the alternate expressions in eqn (12.5.4).

I

12.6 Total potential energy of interaction We have already discussed, in a qualitative way, the total potential energy of interaction between surfaces in Fig. (1.6.2). The total potential energy of interaction between a pair of approaching particles is:

where VAis obtained using the appropriate expression for the non-retarded van der Waals interaction (Chapter 11) between two infinite flat plates (compare eqns (1 1.3.11 and 11.6.26)):

The repulsive term due to the double layer interaction between identical surfaces, VR, is given, approximately, for the case of weak interaction (KD> l), by eqn (12.3.14). The combination of these two functions is the essence of the DLVO theory. (The particular (approximate) expressions used here give rise to the curves shown in Fig. 1.6.2. It should be noted that the van der Waals attraction always dominates at both large and small separations; in the former case however, it may be too weak to be of significance. At small separations VR must approach a finite magnitude, whereas I VA I increases very markedly and hence is expected to pull the surfaces into a deep attractive well, called the primay minimum.This well is not infinitely deep, as expected from the equation for VA,because of a very steep, short-range repulsion between the atoms on each surface (see Fig. 12.6.1).The secondary minimum that occurs at larger ~ ) for a number of important effects in colloidal distances (- 7 ~ is ~responsible suspensions and these will be alluded to later. Experimental investigation of the coagulation properties of a wide range of colloidal solutions suggest that not all systems can be explained using the DLVO theory. Many

I

602 I

1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

Fig. 12.6.1 Total potential energy of interaction VT,where Vs is the potential energy of repulsion due to the solvent layers. Vs is assumed to be negligible until D < -10 nm.

experiments now indicate that there is an extra, so-called structural term that must be included in eqn (12.6.1). This term arises because of the influence of a surface on adjacent solvent layers. Depending on the type of surface this can give rise to either repulsive or attractive forces. We may therefore generally define the total interaction as:

where Vs is the solvent structural term. The early evidence for the existence of a Vs term came from the observation that some colloids (e.g. silica) could not be coagulated even at very high electrolyte concentrations, where the double layer should be completely compressed. Also, the phenomenon of repeptization (i.e. redispersal of coagulated particles by dilution of the electrolyte) cannot be explained by the simple DLVO theory (eqn (12.6.1)), since an increase in VRby diluting the coagulating electrolyte cannot significantly affect the depth of the primary minimum. Evidence has been obtained for both repulsive and attractive solvent mediated forces which are significant at separations of up to about 5 nm (see Section 12.7). This force may well be responsible for repeptization and for the stability of some surfaces at high electrolyte concentrations. Its magnitude can be estimated, at least for the mica surface (Section 12.7.4) on the assumption that the DLVO theory, which holds well for moderate and large D (>lo nm), can also describe

TOTAL POTENTIAL ENERGY OF INTERACTION

I603

the VAand VRterms for small values of D. The discrepancy, for 0 < (D/nm) < 5 is then attributed to Vs. In aqueous media, positive values of Vs indicate the presence of 'hydration forces', whereas negative, attractive values are believed to be due to the 'hydrophobic interaction'. Both types of interaction apparently decay exponentially with decay lengths typically of the order of one nanometre (Fig. 12.7.10).

12.6.1 The Schultz-Hardy rule In Section 1.6.5we discussed the concept of a critical coagulation concentration (c.c.~): that is the concentration of indifferent electrolyte that induces rapid coagulation. The data in Table 1.2 indicate that the C.C.C.is a very strong function of the counterion valency, an observation known as the Schultz-Hardy rule. It was one of the early triumphs of the DLVO theory that it was able to account for that strong valence dependence on the basis of equations like (12.6.2) and (12.3.14). It is not difficult to show (Exercise 12.6.1)that the potential energy barrier that prevents rapid coagulation is reduced to zero when KD= 2 (curve b of Fig. 1.6.2). Substituting this value into the expression for VT (eqn (12.6.1)):

(12.6.4) allows an estimate of the C.C.C.(Exercise 12.6.1);

(12.6.5) where NAis the Avogadro number and E, is the relative dielectric permittivity. The quantities on the right are in SI units. (For c.g.s. units the constant is 1.07 x lo5 and 4n€o = 1 statfarad cm-'.) At 25 "C in water (taking E, = 80) if the potential is high (2 = 1): c.c.c.(mol L-I) = 87 x 10-40/[z6A2]

(12.6.6)

where A is in joules. The observed variation of C.C.C.with valency does depend approximately on the inverse sixth power of z. The data for As2S3 in Table 1.2 for example, give the ratio (for z = 1, 2, and 3):

50 : 0.7 : 0.09 w 1 : 0.014 : 0.0018 compared to the 'theoretical' 1 : 0.016: 0.0014. Although this agreement is impressive there is some doubt as to its significance. For one thing, the values of van der Waals or Hamaker constant estimated from eqn (12.6.6) are rather high (A w [87 x lop4'/ 0.05]'/2 = 4 x J). More seriously, it must be noted that eqn (12.6.6)is derived on the assumption that the surface potential is high ( Z M 1). This latter condition is very difficult to reconcile with the general experimental observation that coagulation usually occurs between low

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1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

potential surfaces. If the more realistic assumption of low potentials is used, for example, the result obtained is (Exercise 12.6.2): c.c.c.(mol L-') a [ + ~ / x 2 ]

(1 2.6.7)

where now we have a less dramatic dependence on the valency and a strong dependence on the surface potential. The experimentaldata could easily be explained by this result if +o a l/x. That is, if the surface potential was reduced by, for example, adsorption of oppositely charged ions, where the adsorption energy increases with valency of the ion. At present there is some interest in applying ion adsorption models to quantitative coagulation studies in a similar manner to the model used in the study of surface regulation (Section 12.3). This raises the question of which potential is to be used in the equations for the repulsive potential energy. Since the interaction occurs between the diffuse layers it would seem to be more reasonable to assess the magnitude of VRfrom the value of the diffuse layer potential, +d (Section 7.3). That potential can be estimated from the electrokinetic (or zeta-) potential ({), which was discussed in Chapter 8. A large body of experimental evidence shows that when rapid coagulation occurs in a colloidal sol, the {-potential is commonly around 25-50 mV, which is indeed too small to assume Z M 1 if we define Z in terms of +d (eqn (12.3.13)). Such a result is, however, quite consistent with eqn (12.6.7) above, since { would measure the potential, +d, after any counterion adsorption in the Stern plane (Section 7.4)

Exercises 12.6.1 The condition for the potential energy barrier to just disappear is that ( VA+VR)= 0 and d( VA+VR)/dD = 0 simultaneously. Show that this occurs when KD= 2. Hence establish the relations (12.6.4) and (12.6.5). 12.6.2 Use the approximate expression (12.5.5) together with an appropriate attractive energy for spheres of radius a (Chapter 11) to establish the relation (12.6.7) for the C.C.C.under conditions of low surface potential. 12.6.3 Calculate the repulsive potential energy (under constant potential conditions) between two spherical particles of radius 0.5 pm, of surface potential 35 mV, when the electrolyte concentration is (a) lop4 M NaCl and (b) lo-' M NaC1. (Use H values from 0 to 20 nm.) 12.6.4 Calculate the attraction potential energy between the particles in Exercise 12.6.3 (assuming that A = 5 x lop2' J) as a function of Hfor 0 < H < 20 nm. Combine this with the curves for V, found in Exercise 12.6.3 to produce curves for V, and comment on the result. (Ignore Vs.)

12.7 Experimental studies of the equilibrium interaction between diffuse double layers The theory developed in Sections 12.1-12.6 has been applied to a large variety of problems over the past 60 years. Since the late 1970s it has become possible to set up

EXPERIMENTAL STUDIES OF THE EQUILIBRIUMINTERACTION

I605

systems that are sufficiently well-defined and controllable to provide quantitative tests of the interaction force or energy as a function of distance between the approaching surfaces. In this section we will examine some of the experiments conducted on double layers that are approaching sufficiently slowly to enable equilibrium to be maintained. We will also restrict attention here for the most part to the interaction of macroscopic surfaces. Studies of the equilibrium interaction between microscopic (colloidal) particles can be conducted by a variety of methods [osmotic pressure measurement (Barclay and Ottewilll970; Barclay et al. 1972), light scattering and neutron scattering (Ottewill 1982), centrifugation (El-Aaser and Robertson 1971, 1973)] and general agreement is obtained between the experimental results and the DLVO theory. A proper discussion of those experiments requires, however, a knowledge of the average particleparticle distances in concentrated suspensions. The treatment uses the notion of distribution functions which are discussed in Chapter 13 and their measurement using scattering theory (Chapter 14). The aim in such experiments is to compare the theoretical shape of the energy barrier with the experimental observations. One can also study the kinetics of the coagulation process, which are particularly sensitive to the height of the barrier, but little influenced by other features of its shape. That material is discussed briefly in Section 12.8. In Chapter 15 the DLVO theory is applied to the interpretation of rheology (flow) experiments on coagulating colloidal suspensions. Kinetics experiments are not always conducted under equilibrium double layer conditions. Whether double layer equilibrium can be assumed during the interaction depends on the velocity of approach (e.g. the shear rate in a rheology experiment) and the rate of adsorption/desorption of the potential determining ions. The general validity of the DLVO theory was established in the period from 1940 to 1980 by the study of (a) adsorbed liquid films, (b) soap films, (c) swelling of clay minerals, and (d) interaction between immersed solid bodies. Clay minerals are, of course, microscopic particles but the distance between them can be estimated from macroscopic observations (assuming that they are aligned parallel to one another) and confirmed by low angle X-ray diffraction without recourse to radial distribution functions.

12.7.1 Adsorbed liquid films In 1938 Langmuir applied double-layer theory to explain the so-called ‘Jones-Ray effect’ which refers to the fact that small additions of electrolyte cause a reduction in the apparent surface tension of water when it is measured by the capillary rise method. Langmuir ascribed this to the presence of a water film of varying thickness on the wall of the capillary and calculated its thickness using the osmotic pressure method which leads to eqn (12.3.4).More recent studies (see, for example, Adamson 1967, p. 78) cast doubt on this explanation of the Jones-Ray effect, but the existence of thin aqueous films on glass, stabilized by double-layer forces, is not in question. These thin film experiments were extended by Deryaguin and Kussakov (1939) to the study of the film of liquid between an air bubble and a flat glass surface immersed in an electrolyte (Fig. 12.7.1). The excess pressure in the film is given by the Laplace pressure in the bubble 2y/r (Section 2.2.3) and the corresponding equilibrium film thickness can be measured by optical interference techniques. Analysis of the measurement for the case of water on glass showed rough agreement with the

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1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

Fig. 12.7.1 Study of thin liquid film by the captive bubble method.

Langmuir equation (12.3.4). Addition of electrolyte (NaC1) produced, as in the JonesRay effect, a marked reduction in film thickness, but this was not as great as expected for reasonable values of the surface potential (i.e. $0
12.7.2 Soap films Soap films are very easily formed by bubbling gas through surfactant solution and are stabilized by the repulsive forces between layers of surfactant molecules adsorbed at the air-solution interface. These repulsive forces are often sufficiently strong to prevent drainage of the water layer which is favoured by the combined action of attractive van der Waals forces and gravitational forces. For films with a water-layer thickness greater than about 10 nm the dominant force is, for the case of ionic surfactants, due to double-layer repulsion; it is usually balanced by hydrostatic pressure (see Fig. 12.7.2(a)). A soap film can also be simply formed by drawing a wire frame vertically upwards through the surface of a surfactant solution. At each height H above the solution the double-layer pressure must be balanced by the hydrostatic pressure ( H p g) tending to drain the film (Fig. 12.7.2(b)). Hence, in a region of film 10 cm above the solution there must be a repulsive double-layer pressure of about lo3 N m-’. If the water-layer thickness D at this height can be measured we are then in a position to study the double-layer interaction. The film thickness can be measured by the observation of reflected light which produces colours by interference between the front and rear surfaces. To obtain equilibrium measurements it is necessary to carefully control the environment in which the film is drawn with regard to temperature, humidity, and vibration. Deryaguin and Titievskaya (1957) were the first to suggest that soap films could be used to investigate the forces that stabilize hydrophobic colloids, and they obtained reasonable film thicknesses for soap films in the pressure range 30-200 N m-’. They showed that soap films of greater than about 20 nm thickness were stabilized by the overlap of diffuse double layers, with surface potential of about 30 mV. Although the potential could not be obtained independently, the results gave clear

EXPERIMENTAL STUDIES OF THE EQUILIBRIUMINTERACTION

Soap solution

I607

I

Fig. 12.7.2 (a) Soap film stabilized by cationic surfactant. (b) Soap film produced by drawing a wire frame out of a solution.

evidence for the validity of the double layer model. Scheludko, Lyklema, and Mysels continued this work to investigate more comprehensively the effect of salt concentration on the equilibrium film thickness. That work is reviewed by Lyklema (1967). Th e agreement between calculated and experimental film thickness (Fig. 12.7.3) is very reasonable. Donners et al. (1977) have used light scattering from soap films in order to investigate surface forces. By measuring the power spectrum of light scattered from ripples or fluctuations at the soap-film interface, information can be obtained both from single interfaces (i.e. surface tension and viscosity data) and from interacting charged soap layers. Their results indicate that for CTAB (cationic cetyl trimethylammonium bromide) films the great majority of the head group charge (98 per cent) is balanced by counterions in a compact layer, and the appropriate potential determining the magnitude of VR is the diffuse layer potential, +,J.

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1 2 : D O U B L E L A Y E R I N T E R A C T I O NAND PARTICLE COAGULATION

I 0-3 1 0-2 lo-' Counterion concentration (mol L-')

10"

Fig. 12.7.3 Calculated and experimental thickness of soap films as a function of ionic strength (OAmerican Chemical Society). (From Lyklema and Mysels 1965, with permission.)

12.7.3Swelling of clays The structure of clay minerals was described in Section 1.4.5 where we noted their significance as model systems for the study of double-layer interaction. The actual swelling behaviour of clay is, of course, very important in agriculture, and in civil engineering (dam, road, and building construction) and in the making of ceramics and other clay products. In montmorillonite and vermiculite the alumino-silicate sheets are separated by water layers whose thickness varies with the concentration and type of electrolyte. In

o 0 h

I

z. E z.

0.03 M LiCl

lo.': lo.'

v

tiL Y

Increasing 1 Decreasing J pressure

-

O

DLVO theory -

\

5ti

P

0

4n

loJ: loJ;

4 4

6

8 10 Plate separation (nm)

12

1

Fig. 12.7.4 Swelling of lithium vermiculite under pressure in 0.03 M LiCl. The theoretical line is calculated from the total crystal charge, which is almost certainly much higher than the diffuse layer charge in this system. (From Norrish and Rausell-Colom 1963, with permission.)

Introduction to StatisticaI Mec ha nics of Fluids 13.1 Introduction 13.2 Molecular interactions 13.3 The structure of liquids 13.4 The potential of mean force 13.5 Time-dependent correlation functions 13.6 Applications of the pair distribution function 13.7 Measurement of correlation functions 13.7.1 Static structure factor 13.7.2 Dynamic structure factor 13.8 Calculation of distribution functions

13.1 Introduction Many of the properties of colloidal systems can be quite accurately described by essentially continuum theories. In such theories, the properties of the solvent or dispersion medium are conveniently characterized by some bulk macroscopic parameters such as the dielectric permittivity or shear viscosity coefficient. Where the molecular nature of the solvent has to be included, it is introduced through the use of ‘cut off parameters or similar conceptual devices (like the quantity L, in Table 11.4). While it may be aesthetically and philosophically more satisfying to describe a physical system in terms of the molecular properties of the constituent molecules at the outset, such an approach will be impossible to implement in practice and the details may even obscure the more interesting features of the system. Nevertheless it is clear that experimental techniques have become sufficiently refined directly to detect, for instance, molecular granularity in surface force measurements (Fig. 12.7.8) and the theoretical framework required to understand such results already exists in the field of liquid state physics. Also in studies of concentrated dispersions (e.g. polymer latices, silica particles, micelles, and microemulsions) by conventional and dynamic light scattering as well as

638

MOLECULAR I N T E R A C T I O N S I639

neutron scattering it is possible to regard the dispersion as a ‘liquid’ in which the colloidal particles play the role of the ‘molecules’. (See Chapter 14.) As a consequence, some acquaintance with the theory of the liquid state is necessary to keep abreast of more recent developments in colloid science. The aim of this chapter is to introduce the basic ideas and concepts in a statistical mechanical description of liquids in an heuristic and non-rigorous fashion. It is intended as an informal introduction to the subject matter, which should allow the reader to gain some ideas about its capabilities and limitations. Detailed treatments of the subject may be found in a number of text books (McQuarrie 1977; Egglestaff 1967; Hansen and McDonald 1976).

13.2 Molecular interactions Statistical mechanics is concerned with deducing the macroscopic properties of a system in terms of the molecular properties of the constituent molecules, but due to the large number of particles in the system, only a description in the statistical sense is possible. Indeed it is quite unnecessary to have an exact knowledge of the position and velocity of every particle at all times. For most liquid systems of interest to the colloid and surface chemist, quantum effects do not have to be considered. We shall further assume that there is no coupling between the intermolecular and the intramolecular degrees of freedom. As a result, the translational motion of the molecules can be considered separately from the vibrational motions. In order to be able to treat the translational motion classically, the de Broglie wavelength of the molecules (Exercise 13.2.1), h = h/(mkT$, must be small compared to the mean interparticle spacing, p-’/3 where p is the number of molecules per unit volume. In other words, the molecules should have a ‘particle-’ rather than ‘wave-’ like character. When intermolecular interactions are negligible (for instance at high temperatures and low densities) only the kinetic energy associated with the motion of the centre-ofmass of each molecule contributes to the intermolecular mode of motion. This kinetic energy will give rise to equilibrium bulk behaviour of the system as though it is a monatomic ideal gas at the same temperature and density. The inclusion of intermolecular interactions will result in ‘excess’ properties of the system over and above that of an ideal gas.+ The fundamental property that determines the existence of various states of matter of a given substance is the intermolecular interaction. The potential energy of a system of Nmolecules in a container of volume Vdepends on the location of all the molecules and can be written as U(rl,rz,.. ., rN) where rj (i = 1,2,. . ., N) denotes the position of the ith molecule. Note that for the case in which the N ‘molecules’ are colloidal particles in a dispersion, the quantity U(r1,rz,. . ., rN) should be the free energy of that configuration of colloidal particles. T o proceed it is often assumed that the

t Note that in many cases this ‘excess’ quantity can be negative so that the relevant property of the real gas is smaller than that of the ideal gas.

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

total potential energy of the system can be simply written as a sum of interactions between every pair of molecules in the system, that is ~ ( r 1l 2,,

.....,r-N) = C

u(ri, rj).

(13.2.1)

pairs

It is important, however, to recognize the existence of three-, four-, and higher-body interactions. For instance, even for a simple substance such as argon, three-body interactions are needed to explain the temperature dependence of the third virial coefficient. Unfortunately, for more complicated molecules little is known about the nature of the many-body interactions so one simply chooses the parameters in the pair potential that would go some way towards compensating for these effects. In fact, in the case of concentrated colloidal dispersions stabilized by electrical double layer repulsion, the effective pair potential between the colloidal particles will, in general, contain manybody effects, in that it will vary with the concentration of colloidal particles in the system. In the following discussion, it is not necessary for eqn (13.2.1) to be valid, although it is useful and indeed sufficiently accurate in most cases to think in terms of pair interactions. For molecules of a simple monatomic liquid the interaction energy between a pair of molecules has the general form shown in Fig. 13.2.1. The interaction only depends on the separation between the centres of the molecules. At large separations, Y > 6 the interaction is attractive, u(r) C 0. For monatomic molecules this is due to the London-van der Waals attraction, which falls off with separation like rP6 (Chapter 11). At small separations, overlap of the electron orbitals of the two molecules gives rise to a repulsion which increases very rapidly as the separation decreases. A common analytical representation of these two effects is the Lennard-Jones formula (see Fig. 13.2.1): u(r) = k[(;)I2-(;)"]

(13.2.2)

Fig. 13.2.1 The interaction energy between two monoatomic molecules as a function of the separation between the molecules.

THE STRUCTURE OF L I Q U I D S

I641

The two parameters 6 and E provide a measure of the size and strength of the interaction between the molecules (see Exercise 13.2.2). [Note that E in this chapter is an energy and not the permittivity. The permittivity will always be written with a subscript.] The pair potential illustrated in Fig. 13.2.1 shares certain features with the DLVO potential between two spherical colloidal particles (see Chapter 12) where the attraction is again due to van der Waals interactions. However, the effective size of the colloidal particles is determined by the distance at which the repulsive interaction exceeds a few LT. For colloidal systems at low salt concentrations this effective size can be much larger than the actual physical size of the particles (for instance when K a << 1). The presence of the primary minimum (Fig. 12.6.1) can be ignored provided the height of the primary maximum is much greater than KTbecause in that case particleparticle contact becomes highly improbable. The interaction between two polyatomic molecules will depend on the separation as well as the orientation and conformation of each molecule. Apart from the obvious steric effects, interactions involving dipoledipole, dipole-induced dipole, etc., are responsible for the orientational dependence in the pair potential (Hirchfelder et al. 1954). As we shall see, the effective size or excluded volume between molecules plays an important role in determining the structure of liquids as well as concentrated colloidal dispersions.

Exercises 13.2.1 The de Broglie wavelength for translational motion of a molecule was defined as h = I z / ( r n I ~ 7 ) ~What / ~ . is the corresponding translational velocity of the molecules? What is the kinetic energy (in terms of K T ) ? 13.2.2 Verify that the depth of the energy minimum in Fig. 13.2.1 is E where E is defined in eqn (13.2.2). Where is the minimum, relative to S ? Plot the function u(r) for 0 5 r 5 1 nm using 6 = 0.3 nm, E = 2 KT.

13.3 The structure of liquids Of the three states of matter, the properties of the liquid state are probably the least easy to calculate. In the gaseous state where the number density is low and intermolecular interactions are infrequent, one has the ideal gas to serve as a good starting point to describe the system. Effects of intermolecular interactions can be accounted for by the usual virial expansion (Exercise 13.3.1). In the crystalline state, one can take advantage of the regular arrangement of molecular centres and use the harmonic perfect lattice to model its properties. The liquid state has all the difficulties of a high density system without the benefits of long range order in the form of a well defined crystal structure. None the less, one can still speak of the short range structure of a liquid in a statistical sense.

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

Consider a liquid of N monatomic molecules in a container of volume V. The probability of finding a particular molecule in a volume element A V is just (A V/ V). The number of molecules one expects to find in A V is therefore

(13.3.1)

N(AV/V)= pAV,

where the factor N arises because we can select from any one of N molecules. The above expression is valid provided we are dealing with a uniform system. For nonuniform systems such as a liquid near the interfacial region, the average number of molecules at position r is given by p(')(r) d V where d V is the differential volume element located at r. The quantity p(')(r) is called the one-particle number density. In a crystalline solid p(')(r) is specified by the lattice structure but in a bulk liquid p(')(r) = p = N / V (a constant). T o characterize the structure in a bulk liquid, one would have to speak of joint probabilities of finding two or more molecules at different points in the liquid. In the absence of intermolecular forces, each molecule moves independently of the remaining molecules in the system. The joint probability of finding one particular molecule in the volume element AV1 located at 11 and another particular molecule in the volume element A V2 at 1-2 is just (A Vl/ V)(A V2/ V). Thus the number of molecules in these volume elements is N(N-1) x ( A Vl/ V)(A V2/ V). The factor N(N- 1) follows from the fact that there are N choices for the first molecule and (N-1) choices for the second. In the thermodynamic limit (N + 00, V + 00 but (N/ V ) = p, a constant) this number becomes p2 A V1 A V2. When molecular interactions are taken into consideration, the presence of a molecule at one point will influence the probability of finding another molecule at a nearby location. We quantify this effect as follows: joint probability of observing molecules in dV1 at rl and in dV2 at r 2

(1 3.3.2)

= P2g(rl r2)d Vl d Vz 7

The function g(r1, r2) is the pair correlation function or pair distribution function (this is a standard notation). As the name suggests, it accounts for the influence of intermolecular forces on the likelihood of simultaneously observing two molecules at positions rl and 1-2.From eqn (13.3.3) we can see from the discussion in the previous paragraph that ( p dV1) is the probability of observing a molecule in the volume element dV1 located at rl. Therefore (pg(r1, r2) dVz) is the conditional probability of observing a molecule in the volume element dVz located at rz, given that a molecule is already in dV1 at 11. In a bulk liquid the pair distribution function can only be a function of the distance between the points rl and r2, hence

Therefore pg(r) may be regarded as the local density of molecules given a molecule is located at the origin of the axis system, or in other words 4n?pg(r) dr is the number of molecules located within a spherical shell of inner and outer radii Y and r dr centred

+

THE STRUCTURE OF L I Q U I D S

I643

about a given molecule. As a consequence g(r) is also known as the radial distribution function. Since pg(r) is the local number density, it must approach the bulk value, p, as r becomes large since the effects of the molecule at the origin must become negligible far from this molecule. Thus we must have the result g(r) + 1

as

r + 00.

(1 3.3.5)

Furthermore as a result of the repulsion between two molecules at small separations, we must have g(r) + 0

as

r +0

(1 3.3.6)

which simply says that excluded volume effects prevent two molecules from being at the same location. From the very general physical constraints on the behaviour of g(r) given in eqns (13.3.5) and (13.3.6) we can make a qualitative argument about what g(r) should look like for a liquid. We know that the number density of molecules of a material in the liquid state is high, similar to that in the solid state, since the change in density upon melting is 10 - 20 per cent. Therefore the average distance between molecules is close to the molecular size. (We take 6 to be a rough measure of molecular size.) Due to the strong repulsion that exists between molecules when their separation is less than 6, the probability of finding a pair of molecules separated by much less than 6, (i.e. the value ofg(r)) must be very close to zero. But in order to maintain the high molecular number density of a liquid this geometric constraint implies that there must be a ‘shell’ of molecules at a distance slightly greater than 6 from any given molecule. The existence of such a shell prevents there being any other molecules with centres between one and two diameters from any given molecule. By a similar argument, this shell ofJiYSt nearest neighbour molecules also forces the existence of a shell of second nearest neighbour molecules. However, as one goes further from the central molecule, correlations between the centres of other molecules and the central molecule diminish rapidly. At positions corresponding to the location of the shells of neighbouring molecules, the local density is higher than the average bulk density; hence at these points g(r) > 1. Between these shells, the local density is lower than the bulk density and hence g(r) C 1. The magnitudes of these local density fluctuations decrease with increasing distance from the central molecule i.e. as r increases; g(r) must therefore be an oscillatory function of r where the amplitudes of the oscillations decrease with increasing r and g(r) + 1 as Y + 00. It is important to remember that in the above discussion on the form of g(r), we are dealing with the structure of the liquid at a given instant in time. The ‘shells’ of molecules about a given central molecule are not meant to imply the existence of permanent structural entities. Indeed molecules belonging to different ‘shells’ are in constant dynamical exchange. In a monatomic liquid the lifetime over which any given pair of molecules can be regarded as nearest neighbours is of the order of a few picoseconds. (See Section 13.5.) Some examples of g(r) are given in Figs 13.3.1 and 13.3.2. The result for argon (Fig. 13.3.1) is typical of that for simple monatomic liquids. For a liquid of polyatomic

-

644 I

1 3 : INTRODUCTION TO STATISTICAL M E C H A N I C S OF FLUIDS

molecules, the pair correlation function will depend on the separation as well as the orientation of each of the molecules. In the case of water (Fig. 13.3.2(a)) this orientation dependence is equivalent to saying that there are three different atomic correlation functions between two molecules: the oxygen - oxygen, goo; the oxygen hydrogen, and the hydrogen - hydrogen, gHH. However, g o o is very nearly equal to the pair correlation function between the centres-of-mass of two water molecules. Note that there are no density correlations around a given water molecule beyond about 0.8 nm. The pair correlation for a hard sphere (Fig. 13.3.2(b)) fluid serves to illustrate the observation that the characteristic form of g(r) is due to the repulsion between the molecules. The sharp discontinuity of g(r) at Y = 6 is a consequence of the abrupt nature of the hard sphere interaction, namely

u(r) = 0 for r > 6

and

u(r) = 00

for r < 6

(13.3.7)

From a knowledge of g(r) one may try to infer the coordination number, N, (the number of nearest neighbours) of a given molecule. However, there are a number of ways of defining N, in terms ofg(r), and these estimates can differ by up to 30 per cent (Pings 1968). For instance, in liquids of noble gases below the critical temperature, N, varies from about 3.7 f0.5 at the critical point to about 10 f2 at the triple point. The value of N , varies with the density but is fairly insensitive to temperature variations. For liquid water, the coordination number is about four and is almost independent of temperature in the range 0-200 "C (Narton and Levy 1972). Another familiar example of pair correlation functions is that between ions in a bulk electrolyte. Even if the solvent is regarded as a continuum, we must have at least a two component system, namely anions and cations. The ionic pair correlation function between species of type i a n d j is written as gc(r). In the linear Debye-Huckel theory g i (r) is approximated by (for r >6, the ionic diameter) (Exercise 13.3.2): gq = exp[-zje&(r)/kT]

( )

M

1 - [z+$i(r)/kT]

exp[-K(r - a)] 4n€o €,kT (1 K6)T .

= 1 - z.z. L

e2

~

(13.3.8)

+

For r < 6 we have gc(r) = 0. Notice that in this example the pair correlation function does not have the characteristic oscillatory form of that in a dense liquid. The reason is

Fig. 13.3.1 The pair correlation function for liquid argon at 143 Kand 0.91 g crnp3,which is about half way between the critical and the triple point near the vapour pressure curve (Pings 1968).

THE STRUCTURE OF L I Q U I D S

I645

that the electrolyte, which may be viewed as an ionic liquid, is at a very low density and interacts via long range Coulombic forces. Although we have only considered pair correlation functions so far, it is straightforward to generalize to the concept of triplet, four-body, etc., correlation r 2 , q ) is the joint functions. For instance, the triplet correlation function g(3)(rl, probability of observing three molecules at positions rl, r2, and r3. In the next section, we shall have occasion to use this quantity.

\ 2-

h

v L.

$ 1

-

0 Y...............J

2

3

r/6 Fig. 13.3.2 (a) The oxygen-oxygen correlation function for water at 20" C. (After Narten and Levy 1971.) (b) The pair correlation function for a hard sphere fluid at a volume fraction of 45%.

646 I

1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

r

Exercises 13.3.1 Departures from ideal behaviour in gases can be represented by a virial expansion, as for example:

where B, Care the second and third virial coefficients, which are functions only of the temperature. Show that if the van der Waals equation (see Exercise (12.9.1) is written as a virial expansion then B (b - a/RT) i.e. the second virial coefficient incorporates both the attractive and repulsive components of the departure from ideal behaviour. 13.3.2 (a) Use eqn (7.11.7) for y9i to establish eqn (13.3.8). (b) The number density of ions of typej at r, given there is an ion of type i at the origin is nj (r) = nj g i (r)where nj is the average number density of ions of typej in the system. The excess density of ions of typej is then Anj (r) = nj ( r )- nj = nj [gii ( r ) - 11. Therefore, the number of excessj ions in a spherical shell of radius r and thickness dr centred on ion i is: Anj(r)41tr2dr = F(r)dr. Show that F(r) is a maximum at r = 1 / ~ .

13.4 The potential of mean force In the previous section we have seen that the pair distribution function g(r) can be interpreted as the probability of finding another molecule at a distance r from a given molecule. In a dilute gas, this probabilistic interpretation means that g(r) must have the form of a Boltzmann factor

where u(r) is the interaction potential between two molecules. For a dense liquid, it can be shown that g(r) may be represented as a power series in the number density of molecules and eqn (13.4.1) is just the first term in such an expansion. Since u(r) becomes larger and positive as r + 0 (repulsive interaction) eqn (13.4.1) automatically satisfies the constraint (13.3.6). In general we may write

where W(r)is the reversible work needed to move two molecules through a liquid from some initial large separation (say infinity) to a separation r. In a dilute gas this is just the work done against the interaction potential between two molecules since the effect of a third molecule on this process is negligible at low densities.

THE POTENTIAL OF M E A N FORCE

I647

In a dense liquid the situation is quite different. A relative displacement of two molecules will result in uncontrolled changes in the surrounding molecules. Some molecules have to move aside to create space while others move in to fill in any empty spaces left behind. If the displacement of one molecule relative to another is carried out sufficiently slowly, and equilibrium is maintained at all times, any rearrangement of all the other molecules will be reversible. Thus if FAV(?') is the average force between two molecules separated by a distance r (averaged over all configurations of the rest of the molecules in the system) then the reversible work, W(r) needed to move the two molecules from a large separation to some separation Y is just

s

v'=v

W(r)= -

FAv(r')dr'

(13.4.3)

#=oO

where the integral is to be taken along any path between some point far away (r' = 00) and the point r' = r. Since the process is reversible (i.e. the moving process is not dissipative) the integral is independent of the actual course of the path taken. An equivalent statement of eqn (13.4.3)is that the x-component of the average force between two particles is

with similar expressions for they- and z-components. Thus W(r) has the role of a potential energy from which the components of the average force may be obtained by taking the appropriate spatial derivative. W(r)is therefore known as the potential of averageforce or potential of mean force. It should be evident from the above discussion that W(r) is in fact the change in thefree energy in bringing two molecules from infinity to a separation r. For example, the total interaction free energy between two colloidal particles (van der Waals and electrical double layer interaction) is a potential of mean force. In a dense liquid, the potential of mean force W(r) depends on the temperature and the density since it is associated with the average force between particles, averaged over all other particles in the system. To see how this comes about in more detail, consider the x-component of the average force on a molecule at rl due to the presence of another molecule at r2. From eqn (13.4.4) this force is

There are, of course, similar equations for the y- and z-components of the average force. The first term on the right hand side of eqn (13.4.5) is the 'direct' force on the molecule due to the pair interaction with the molecule at r2. The second term accounts for the effects of all other molecules in the system. This term says the following: let the conditional probability of finding a molecule at r3, given two molecules are already at rl and r2 be P(r3 I 11, r2). Under the same conditions, the number of molecules in a volume element d r 3 , located at r3, is therefore pP(r3 lrl,r2)dr3 and each of these

648 I

1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

molecules exerts a force on the molecule at rl, the x-component of which is (-du( 1 rl I)/dxl). The total effect of all other molecules in the system can be obtained by integrating (or summing) over all volume elements. The conditional probability, P(r3 1 rl,r2) is related to the triplet and pair distribution function by: 13

g(3)(rl,r2, r3) =

r2)P(r3Irl, r2)

(1 3.4.6)

which states that d3)(rl, r 2 , r3), the joint probability of observing molecules at rl, 1-2, and r3 is equal to the probability of finding molecules at rl and rz (namely g(r1, r2)) multiplied by the conditional probability P(r3 1 rl, r ~of) observing a molecule at r3 given that molecules already exist at rl and rz. By combining eqns (13.4.2), (13.4.5), and (13.4.6) we find (Exercise 13.4.1):

where Y = Irl - r 2 I. This exact result was first obtained by Born, Green, and Yvon (BGY) and is known as a member of the BGY hierarchy equation, which provides a relation between the pair and triplet correlation functions. The equation may be solved (though not easily) for the pair distribution function g(r) provided we have some additional information about the triplet distribution function g(3)(rl,r - 2 , ~ ) . Unfortunately little is known about g(3)(rl,rz, r3) except that, like g(r), it can also be written as g(3)(rl,r2, r3) = e x p [ - ~ ( ~ ) ( r2, r ~ r, 3 ) i k ~ ]

(1 3.4.8)

with m3)(r1, 1-2, 13) being the triplet potential of mean force for assembling three molecules at rl, 1 2 , and 1 3 from infinity. One possibility, known as the superposition approximation, is to approximate M3)by a sum of potentials of mean force between each pair of molecules in the triplet, that is, W(3)(r1~r2~r3)W(lri

-r31)+

W(lrz

-r31)+

w(lr3 -111).

(13.4.9)

Note that this remains an approximation even for systems in which the molecules interact solely via two-body potentials. Using eqns (13.4.2) and (13.4.8), eqn (13.4.9) is equivalent to the assumption of independent probabilities g(3)(rl,r2, r3)= g(b-1 - ~

~ I M I -~ r3Z 1)g(1.3 - 1 1 I).

(13.4.10)

By combining eqns (13.4.7) and (13.4.10) we obtain a non-linear integro-differential equation for g(r) and hence W(r)(so called because the unknown functions occur in a differential as well as under the integral sign). The solution of such an equation is not a straight exercise in numerical analysis. Another feature of the potential of mean force at liquid densities is that it has a more complex structure than the interaction potential between the molecules. This can be illustrated by considering a hard sphere fluid. From Fig. 13.4.1 we can see that, although two hard spheres do not interact when their centres are separated by more than one diameter, the potential of mean force exhibits regions of attraction and repulsion. It is fairly easy to see how this comes about in general. Imagine the situation when two molecules are far apart. Each molecule suffers collisions with all other molecules in the system equally

THE POTENTIAL OF M E A N FORCE

I649

Fig. 13.4.1 The pair potential, the potential of mean force, and the average force for a hard sphere fluid at a volume fraction of 45%.

on all sides, and consequently the average force between the two molecules is zero. Consider now the configuration in which the two molecular centres are 1 to 1; diameters apart. In this situation there is insufficient room for a third molecule to fit in between the two molecules (see Fig. 13.4.2). As a result collisions with all other molecules in the system which are nearly in line with the intermolecular axis will have a net effect of pushing the two molecules together. In other words, the molecules experiencean effective attractive force (< 0) between them. When the two molecular centres are 1; to 2 diameters apart, collisions with other neighbouring molecules will now tend to force the two molecules apart in order to fit in a third molecule. As a consequence the two molecules experience an effective repulsive

650 I

1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

Fig. 13.4.2 A schematic illustration of the distribution of hard spheres in the neighbourhood of two spheres held at a fixed separation.

force (> 0) between them. Thus even when there is no direct interaction between two hard sphere molecules the presence of other molecules can generate an eflective attractive and repulsive interaction between the two molecules. While the above discussion only exploits the excluded volume and associated geometric constraints, the presence of attractive interactions in real liquids will only modify the details of the effective interaction. The general structure of a liquid is determined almost entirely by the repulsive (excluded volume) forces in the system. [Note the similarity between this description and that of the depletion effects of non-adsorbing polymers (Section 12.9.4).] A striking illustration of the form of the potential of mean force across liquids may be found in direct measurements of the force between crossed mica cylinders across organic liquids (Horn and Israelachvili 1981). The apparatus used in these experiments is that shown in Fig. 6.2.1. In the example given in Fig. 13.4.3 the liquid is octamethyl cyclo-tetrasiloxane, whose molecule is nearly spherical in shape with a mean diameter of 1 nm. Direct measurement of the force between the cylinders as a function of separation can in this case reveal the graininess of the liquid because the molecules are so very large. Recall that it has more recently proved possible to detect the graininess of water molecules in this apparatus (Fig. 12.7.10) (Israelachvili and Pashley 1983). A more familiar example of a potential of mean force may be found in the theory of the electrical double layer at a charged surface. The density of ions of species i near a flat surface may be written as Pi(x) = pi

exp[-W,(x)lkTI.

(13.4.11)

(In the notation of Section 7.3 pj(x) = ni and pi = np.) In the Gouy-Chapman theory the potential of mean force of ionic species i, Wi(x) is approximated by the charge on the ion (xje) multiplied by the mean electrostatic potential +(x) (Section 7.3), i.e.

W;:(x)M zje$(x).

(13.4.12)

The more elaborate treatments referred to in Section 7.6 attempt to improve on eqn (13.4.12) by introducing corrections for the various effects: the finite ion size, the work done in moving water molecules aside to make way for the ion, the effect of the ion on its neighbours and on the structure of the surrounding (dipolar) liquid, the effect of the local field on the volume occupied by the water molecules (electrostriction), etc.

THE POTENTIAL OF M E A N FORCE

I651

T I I I

I I

I

I I I I

I I I

T I

I I

I I

I I I I

I I

I I

I

I

I

I

I

I

I

,

,

2

3

4

5

6

7

8

9

Distance (nm)

Fig. 13.4.3 Measurement of force as a function of separation between two mica plates in octamethyl-cyclotetrasiloxane showing the solvent structure. (After Horn and Israelachvili 1981 with permission.)

Fortunately, as noted in Section 7.6, some of these effects operate in opposite directions to other effects and the overall correction is fairly small provided the electrolyte concentration and the potential are not too high.

Exercises 13.4.1 Establish eqn (13.4.7).

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

13.5 Time-dependent correlation functions In Section 13.3 we dealt with spatial correlations in a liquid at some given instant in time. As the molecules in a liquid are in constant thermal agitation, we need some further concepts to describe how a system evolves in time, under equilibrium conditions. Such ideas are important because they can give us some feeling for the lifetime of the type of liquid structures discussed earlier. Recent studies on concentrated colloidal systems, using dynamic light scattering, have revealed a wealth of information on the transport properties e.g. diffusion coefficients (Section 1 S.2) of colloid dispersions. An appreciation of these advances will also require some familiarity with the concepts to be discussed in this section. Recall that the pair distribution function g(r) is the probability of finding a molecule at a distance Y from a given molecule. In this definition we have tacitly assumed that the attempts to observe the two molecules are to be carried out at the same time. Clearly one can ask the question: given a molecule is at some point (the origin say) at time 0, what is the density of molecules at a time t later at a distance Y away? The required density is called the time-dependent pair correlation function G(r,t ) so that G(r,t ) d r is the number of molecules at time t in the volume element dr located at r given that there is a molecule at r = 0, at t = 0. The function G(r, t ) is also known as the van Hove correlation function. As for g(r), the time- dependent pair correlation function depends only on Y = Ir I in an isotropic liquid. Now it is clear that given there is a molecule at the origin at t = 0, the molecule observed at Y at time t can be any molecule in the system -including the molecule that was originally at the origin. Thus G(r, t ) can be divided into two parts: a ‘self part involving the same molecule and a ‘distinct’ part involving two dzfferent molecules. In the obvious notation, we have

G(r,t ) = G&, t )

+ Gd(r, t).

(13.5.1)

The self part, G&, t)is obviously related to processes such as self diffusion (Section 4.9.2) while the distinct part, Gd(r,t ) Will give us a measure of the life time of liquid structures. In fact we must havetfor an isotropic liquid: pg(Y)

Gd(Y,t = 0).

(1 3.5.2)

In Figs 13.5.1 and 13.5.2 we show some examples of the distinct part, Gd(r, t ) of the time-dependent pair correlation function. These results are obtained by computer simulation of a system of particles or molecules that interact via some model potential (see Section 13.8). The parameters are chosen to mimic the properties of argon (Rahman 1964) and a suspension of polystyrene latices (Gaylor et al. 1980, 1981). The rate at which the structure in Gd(Y,t ) decays in time is indicative of the lifetime of the local environment around a given molecule and this changes as a result of the thermal motion of the molecules. With the passage of time or with increasing separation, we have the expected limit Gd(Y, t ) + p as t + 00 or Y + 00 which implies that all Some authors have defined G(r,t) with the bulk density normalized out so that Gd(r,t=O)=g(r).

TIME-DEPENDENT C O R R E L A T I O NF U N C T ION S

I653

Fig. 13.5.1 The distinct time dependent correlation function G&, t ) for a Lennard-Jones liquid ( c / k = 20 K 6 = 0.34 nm) which closely models liquid argon at 94.4 K and 1.374 g cmp3. (c is the well depth shown in Fig. 13.2.1.) (After Rahman 1964.)

2h

r=

v

$1'Q

0

I

I

500

1000

Fig. 13.5.2 The distinct time dependent correlation function, G&, t) for a colloidal dispersion of sphericalparticles (radius 23 nm, surface potential 150 mV, M 1:l electrolyte,volume fraction 4.4 x lop4)interacting in accordance with the linear DebyeHuckel formula (section 12.5) for electrical double layer interaction (Gaylor et al. 1980). Full line t = 0; . . . t = 5 x 10p4s;- - - - t = 2 x 10p3s.

654 I

1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

correlations eventually fade away and there is equal probability of finding a second molecule anywhere. [Note the very large difference in the time scales for these two systems.]

13.6 Applications of the pair distribution function We have already seen that the pair distribution function g(r) contains information about the structure of molecules, the co-ordination number, and the average interaction free energy. A further use for g(r) is in evaluating certain averages which allow important connections to be made between statistical mechanics and thermodynamics. We will now examine the three most important of these. Suppose we want to know the average internal energy of a liquid due to intermolecular forces. If the N molecules are located at position {rl, rz,. . ., r ~the} internal energy of this particular configuration is (assuming pair interactions, u(r):

(13.6.1)

The first term is the kinetic energy due to translational motion of the molecules. The notation in eqn (13.6.1) means that the two summations run over all values of i = 1, 2,. . ., N a n d j = 1,2,. . ., N , but all terms with i = j are omitted (since a molecule does not interact with itself). The factor 1/2 is to correct for double counting in the two summations. T o obtain the average internal energy ( U ) however, we need to average over all possible configurations of the molecules. We can accomplish this as follows: choose a molecule and set up a coordinate system at the centre of this molecule. The number of molecules in a spherical shell of radius Y and thickness dr around this molecule at the origin is 4x2pg(r) dr. The total interaction energy between the central molecule and all other molecules may be obtained by multiplying the number in the spherical shell by u(r) and integrating over I from r = 0 to r = 00. The total interaction energy between all molecules is therefore 00

3 ( U ) = - NkT 2

+ $Np

(1 3.6.2) 0

The factor N arises because there are N choices for the molecule at the origin and the factor 1/2 corrects for the double counting, as each molecule has been counted once as the central molecule and again as one of the molecules in the shell. One can also obtain a similar expression for the pressure by differentiating the free energy function (p = -(W/~V)T)but we will not go into the details here. The result (McQuarrie 1977, p. 262) is an expression in terms of g(r):

7 M

2x p = pkT - -pz 3

d4r) dr

y 3 - g ( ~ )d ~ .

0

(1 3.6.3)

APPLICATIONS OF THE PAIR DISTRIBUTION FUNCTION

I655

Equations (13.6.2) and (13.6.3) thus provide possible links between statistical mechanics and thermodynamics. For example, ifg(r) and hence C U> are known functions of p and T, all other thermodynamic quantities can be calculated in the usual way. Since the pair correlation function, g(r) is related to the local density, it is possible to show from the theory of linear response (see Section 3.2.3) that g(r) also characterizes the change in the local density as a consequence of an external perturbation (Hansen and McDonald 1976). This is explained as follows. Consider a bulk liquid which has u(ri).That is, the external field acts been subjected to an external field of the form on each molecule of the liquid and gives it a potential energy, u, which depends only on the position, r;, of that molecule. The potential u(r)may be the gravitational field, or the external radiation field in a scattering experiment (light, X-rays, or neutrons) or it may even be that due to the effect of a colloidal particle on the neighbouring dispersion medium, or on the ions in that medium.The change in local density A&) is then given by (Exercise 13.6.1):

A&)

= p(r) - p = -[pu(r)/kT]

s

- (p2/kT)

h(r - #)u(#)~I'

(13.6.4)

where we have used the standard notation h(r) = g(r) - 1. This result is strictly valid for weak external fields (u/KT C 1) since terms of order [ u / k g 2 and higher powers have been omitted from the general expression for A&). The notion ofg(r) as a response function which characterizeschanges in local density as a result of an external perturbation can be extended by introducing the idea of the susceptibility, 2 of a system. [A familiar example of a susceptibilityis the conductivity of a circuit element, K(w),which relates the response [namely, the current I(@), through the circuit] to the perturbation [in this case the applied voltage, V(w)]:

K ( 4 = I(@)/ V(w)

(= 1/Z(4) where Z is the impedance, introduced in Exercise 3.2.5. [In a biological situation one would speak of the susceptibility of an organism to a stimulus so producing a response. In linear theory the response is proportional to the stimulus and the proportionality constant is the susceptibility.] Ideally, we would like to know the value of the ratio g(r)/u(r) but the form of eqn (13.6.4), in which both g and u appear in an integral, poses a problem. Fortunately, the difficulty can be circumvented by introducing a mathematical device called a Fourier transform which is described briefly in Appendix A4. The Fourier transform of eqn (13.6.4) is given by (see Exercise 13.6.2):

(1 3.6.5) Then, if x(Q) is the susceptibility, defined by:

[ + 2[ r

- kT

This final form for

00 00

1

1

r sin(Qr)h(r)dr] .

2 (Q)is derived in Exercise 13.6.3.

(1 3.6.6)

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

The conductivity K(o)characterizes the response of the electrical circuit element as a function of the angular frequency. By analogy, (Q) in eqn (13.6.6) characterizes the response of the system to external fields of spatialfrequency Q = 2n/h(where h is the wavelength). Those values of Qfor which the susceptibility (Q) is large (and hence the response is large) correspond to resonance spatial frequencies which are in turn related to the structure of the system. For X-rays interacting with a crystal these spatial frequencies are related to the lattice spacings of the atoms. (See Atkins 1978, p. 710 et seq.) For the more general case of scattering from, say, a liquid or a colloidal dispersion, the resonance spatial frequencies of the system relate to the local ordering in the liquid, or the particles respectively. As we shall see in Section 13.7, scattering experiments provide a direct measure of the structure factor, S ( a , which is related to the Fourier transform of the pair correlation function, rather than to g(r) itself. However, for a long wavelength (>> molecular size) perturbation (e.g. a gravitationalfield) the only important values of Q are Q m 0 (recall that Q IX A-') and hence:

X(Q

= 0) = -(P/KT)[l

+ ph(Q

= O)] 00

= -(p/kT)[l

+ 4 n p I ?h(r)dr].

(1 3.6.7)

0

The function inside the square brackets is of fundamental importance since it can be shown that it is directly related to the isothermal compressibility of the fluid:

(1 3.6.8)

Hence

X(Q = 0) = --p

2K T .

(1 3.6.9)

[In view of the probabilistic interpretation of g(r) one might think that the integral 4n ?g(r) dr should be equal to N-1 rather than be related to the compressibility as indicated in eqn (13.6.9).This result for g(r) is valid befire the thermodynamic limit ( V + 00, N + 00 with N / V = p) is taken. However, to relate the integral in eqn (13.6.7) to a thermodynamic quantity, one must first obtain the function h(r) in the thermodynamic limit before performing the integral. This somewhat subtle point is discussed in detail in Hill 1956.1

I

Exercises. 13.6.1 Equation (13.6.4) has a rigorous derivation based on statistical mechanics. Here we consider a simplified heuristic derivation. The local density at r in an external field u(r)can be written as P ( 4 = P e v - W(r)/KT)

I

MEASUREMENT OF CORRELATION FUNCTIONS

I 657

where W(r)is the free energy [relative to a point where u(r) = 01 of putting a molecule at r. The quantity W(r)has two contributions: (a) the direct interaction between the molecule and the external field, i.e. u(r) and (b) the effect which the presence of a molecule at r will have on the local density of molecules around r. The resulting excess of molecules (which may be positive or negative) due to (b) also interacts with the external field and contributes to W(r).For a weak external field this second contribution has the form

where [pg(r - r‘) - p] dr‘ is the excess number of molecules at the volume element dr‘ located at r‘ given that there is a molecule at r. Start with the above argument and derive eqn (13.6.4) assuming that u/kT << 1. 13.6.2 Obtain (13.6.5) using the three dimensional convolution theorem (Appendix A4). (The symbol above a function denotes its Fourier transform.) 13.6.3 Derive eqn (13.6.6).

-

Hints :

/f(r)dr =

7 7[ d@

0

exp(-kr) = exp(-zkr cos 8) and

0

sin 8d8 g f ( r ) - dr

0

1 sin x = -[exp(zx) - exp(-zx)].

2i

13.7 Measurement of correlation functions 13.7.1 Static structure factor The structure of a crystal can be determined, for instance, by X-ray diffraction experiments. When the wavelength of the radiation is comparable to the interatomic spacings in a crystal, constructive and destructive interference between scattered waves originating from different atoms generate the characteristic sharp Bragg diffraction pattern. The same principle can be applied to determine molecular correlations in a liquid. In a liquid, however, the intermolecular spacings are distributed over a range of values and hence the diffraction pattern will be diffuse, as there will be some scattering at all angles. There are two possible processes that can give rise to a diffraction pattern: interparticle and intraparticle scattering. We use the term particle to mean either a molecule in a liquid or a colloidal particle in a dispersion. A single particle in isolation can scatter radiation with a scattering power which is proportional to the square of the volume of the particle [eqn (3.3.5)]. We have seen that a study of the scattering process by single independent particles provides a method for determining the particle size (Section 5.7.1).

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

When there is interaction between the particles, the positions of neighbouring molecules become correlated and can therefore give rise to diffraction patterns. Under certain conditions, the study of interparticle scattering can yield information about the structure of a liquid or colloidal system. In most experimental situations, we can assume that the scattering process only changes the direction and not the energy or frequency of the radiation. This assumption (of elastic scattering) is valid if the energy of the radiation is much larger than the average kinetic energy of the scattering particle i.e. hu >> kT,so that there is negligible energy lost to or gained from the scatterers (Section 3.3). Furthermore, in order to be able to analyse the scattering pattern, the concentration of scattering particles must be ‘sufficiently’ dilute so that the incident radiation can at most only suffer a single scattering event before reaching the detector; multiple scattering events are negligible. The definition of ‘sufficiently’ dilute varies with the type of radiation used. For instance, X-rays and visible light are scattered by the electron clouds in the molecules whereas neutrons are scattered by the nuclei which are lo4 times smaller in size than the electronic distribution. Neutron scattering can, therefore, under appropriate circumstances, be applied effectively to more concentrated dispersions. Recall the discussion of the scattering from two particles at ri and rj described in Fig. 3.3.2). The incident beam propagates along the vector ki and the scattered beam along k,. The assumption of elastic scattering implies that Iki I = Ik, I = 2n/h where h is the wavelength of the radiation. For scattering in a colloidal dispersion, h is the wavelength in the dispersion medium which is related to the wavelength in vacuo, ho, and the refractive index of the medium, no, by ho = noh. The phase dzfference A& between the scattered beams from particles i and j was shown to be equal to 9.r-iwhere the scattering vector Q is defined by (compare eqn (3.3.13)):

Q = k,

-

ki and lQl = 21kl sin(8/2) = ( 4 4 h )sin(O/2)

(13.7.1)

from Fig. 3.3.2 with kj M ks.The angle 8 through which the incident beam has been deflected is known as the scattering angle. At an observation point, r far from all scattering centres (a condition well satisfied in an experimental setup), the scattered field from the ith particle can be shown to be: (13.7.2)

where A is the scattering amplitude and c j j the phase. (Compare this with equation (5.7.15) where the light scattered by a single colloidal particle is calculated.) The total scattering field from all particles, assumed identical and hence having the same scattering amplitude, is N

A N

(1 3.7.3) i= 1

MEASUREMENT OF CORRELATION FUNCTIONS

I 659

Since the observation point is far from the scattering volume, we can, to an excellent approximation, replace II - ri I by I I I = r. T o obtain the final expression we have used the expression for the phase difference to measure all phases relative to particle 1. The scattered intensity is (Exercise 13.7.1)

(1 3.7.4)

I

J

In the above discussion, the positions of the particles in the system are assumed to be fixed in a given configuration. This is quite reasonable as the time taken for radiation to pass through the entire scattering volume is much smaller than the time scale of molecular motion (Exercise 13.72). However, experimental measurement of the intensity is really averaged over many particle configurations. In other words, we require the average (I,)which is given by:

r')]

+p

s

I'

dRh(R) exp(iQ.R)

(1 3.7.5)

Since the quantities to be averaged involve only pairs of particles, the pair correlation can be used to effect this average (compare Section 13.6). The first term on the right hand side corresponds to scattering from N independent particles. The second term corresponds to scattering from the surface of the scattering volume. For a macroscopic sample the magnitude of this term is negligible except in the forward direction (Q = 0) which is difficult to study with light waves and X-rays (though it is studied by low angle laser light and small angle neutron scattering (LALLS and SANS)). This term will, therefore, be omitted from the remainder of this chapter. The third term accounts for interference effects due to correlations in the positions of the scattering molecules, distance R apart. So far, we have treated the particles as point scatterers. The general result for the average scattered intensity from particles of finite size should read ( I, ) = constant x f(0) x P(Q)S(QJ

(1 3.7.6)

in which f(0) depends on the polarisation of the incident and scattered radiation and although it may depend on 0 it will be unity for the usual laser light system (see Sections 3.3.1 and 3.3.2). Theform factor P(QJ is a property of the shape and size of the

660 I

1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

scattering particle and was introduced in eqn (3.3.14). The (static) structure factor S(Q) is [compare with eqn (13.6.6)]: M

Q

r sin@ h(r)dr = 1

+ ph(Q)

(1 3.7.7)

0

and is proportional to the susceptibility i ( Q ) described in Section 13.6. Thus we see that scattering measurements provide access to the Fourier transform (see Appendix A4) of the pair correlation function g(r) = 1 h(r). In principle, given the function S(Q) which has been measured over a sufficiently large range of Qvalues, it is possible to invert eqn (13.7.7) to get g(r). Unfortunately this inversion process involves serious technical and numerical complications which will not be discussed here (see Pings 1968). If a light scattering study is carried out to measure the colloidal structure factor S(Q) of a dispersion, an extrapolation to Q = 0 will yield the value of the osmotic compressibility. This quantity has been extensively studied by Ottewill and his coworkers (see Ottewill 1982). From eqns (13.6.7) and (13.7.7) we have

+

On the other hand, in the limit of large Q, it can be shown from eqn (13.7.7) that S(Q) + 1 as Q+ 00 because h(r) + 0 (i.e. g(r) + 1) as r+ 00. Physically we can see how the large Qlimit comes about. From eqns (13.6.6) and (13.7.7) we see that S(Q) is proportional to the susceptibility X ( Q ) which characterizes the density response of the system to an external perturbation of spatial frequency Q o r wavelength h = 2n/Q. As Q+ 00 (A + 0), the wavelength will become much smaller than the interparticle spacing and the external field would have undergone many oscillations between every pair of particles. Consequently there can be no correlations in the local density in response to such a perturbation, as each particle is in effect subject to uncorrelated perturbations. At intermediate values of Q, the structure factor S(Q) is an oscillatory function. The maxima of S(Q)are analogues of Bragg peaks in the crystal diffraction pattern and the locations of these maxima are roughly at values of Qfo r which the product Qd is an integral multiple of 2n, where d is the mean spacing between particles (Brown et al. 1976). Since S(Q) is proportional to i ( Q ) , the susceptibility of the local density to an external perturbation of spatial frequency Q (wavelength h = 2n/Q),we therefore expect that external fields of wavelength d will generate large density fluctuations as the positions of the particles in the system are ‘in phase’ with the applied field. In the dilute limit ( p + 0) we have S(QJ = 1 [see eqn (13.7.7)] and this limit is used to determine the prefactors of S(Q) in eqn (13.7.6) which are independent of number density. As discussed in Section 13.3, the form of the pair distribution g(r) of molecules in a liquid or that of particles in a stable colloidal dispersion is controlled mainly by the excluded volume effect due to the repulsive interaction between the particles or molecules. The form of the structure factor S(Q) is therefore also determined by similar factors. Indeed experimentally determined S(Q) for a variety of systems can be

MEASUREMENT OF CORRELATION FUNCTIONS

I 661

Fig. 13.7.1 A comparison of the structure factor of a Lennard-Jones fluid at ~6~ = 0.844, kT/e = 0.72, which is near the triple point (full line) with the structure factor of a hard sphere fluid (points). (After Verlet 1968.)

fitted to the structure factor of a hard sphere fluid, provided a suitable hard sphere diameter is used as illustrated in Fig. 13.7.1. In reality this only poses a new question as to how to determine a priori the appropriate hard sphere size. A corollary of this observation is that attempts to fit experimental structure factors do not provide very stringent or sensitive methods of determining the interparticle interaction. An analytic expression for S(@ for a hard sphere fluid has been obtained (Baxter 1968) in the Percus-Yevick approximation (see Section 13.8) which for most practical purposes is sufficiently accurate:

1 ~

S(Q)

=1

+ 244 {a(sinx x3 ~

+ 12cp2a { 24/x3 + 4( 1 x3 ~

-

-

xcosx)

+ b[(2/x2

6 / x 2 )sin x - (1

-

-

1)xcosx

12/x2

+ 2 sinx

+ 24/x4)x cos x]}

where 6 is the hard sphere diameter, p the number density, x

-

2/x]} (1 3.7.9)

= QS; 4 = npS3/6 and (13.7.10)

13.7.2 Dynamic structure factor With the advent of photon correlation spectroscopy (PCS) it has been possible to probe the dynamic properties of a colloidal system under suitable conditions (Section 5.7.3). In dynamic light scattering the fluctuation in the intensity of the scattered beam is monitored (Pusey and Tough 1982). Such a measurement yields the time dependent analogue of the static structure factor S(@. In fact the quantity measured is known as

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13: INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

the intermediate scattering function, F(Q,t) which is related to the time dependent (van Hove) pair correlationfunction, G(r,t) by

F(Q, t ) = 1

+

s

drexp(-iQ.r)[G(r, t ) - p]

(13.7.11)

and F(.Q t= 0) = S(QJ (Exercise 13.7.3). However the experimental results are presented in a normalized form in terms of the field autocorrelationfunction g(')(Q, t ) (compare eqn (5.7.15):

The function g(')(Q, t) is a smooth decreasing function of time. For non-interacting particles we have (compare with eqn (5.7.16)):

g(')(Q, t ) = exp(-DoQ2t)

where DO= kT/6nr]a

(13.7.13)

is the diffusion coefficient of the free particle of radius a in a medium of shear viscosity r]. In general when Ink(') t)]is plotted as a function of (@t) the rate of decay is slower than that of DO due to interparticle interaction (Fig. 13.7.2) and a central problem in studying the dynamics of colloidal systems is the interpretation of the form of g(')(Q, t ) (Pusey and Tough 1982).

(a

OC

(a

Fig. 13.7.2 The field autocorrelation function g(') t ) at Q = &, the first maximum of the structure factor, for a dispersion of deionized polystyrene latex spheres (radius 25 nm, volume fraction -lop3.) (Redrawn from Pusey and Tough 1982.)

C A L C U L A T I O N O F DISTRIBUTION FUNCTIONS

I

I663

Exercises 13.7.1 Justify the final form for I, in eqn (13.7.4) from the previous equation. 13.7.2 Compare the time scale of molecular motion with the time taken for light to

pass through a scattering cell. 13.7.3 Show that F ( Q t = 0) = S(QJ using eqn (13.7.11).

13.8 Calculation of distribution functions We have already seen in Section 13.4 that, by considering the average force between two molecules, one can obtain an exact equation (eqn 13.4.7) for the equilibrium pair distribution function g(r) in terms of the triplet distribution. By making certain approximations about the latter (such as the superposition approximation, eqns (13.4.9) or (13.4.10)) we can obtain a rather difficult equation to be solved for g(r). The resulting (integro-differential) equation is based on the concept that the total average force between two molecules is made up of a direct force due to the intermolecular potential and an indirect force due to the presence of other molecules. Another way of representing correlations between molecules in a liquid also uses the idea of ‘direct’ and ‘indirect’ effects. The function h(r) = g(r)-l measures the total correlation between two molecules. This correlation vanishes at large separations since g(r) + 1 as r + 00. We define the total correlation to consist of a sum of two contributions: a ‘direct’ correlation transmitted between the two molecules, characterized by a function c(r), the direct correlation function, plus a contribution that is transmitted via a third molecule. The latter is made up of a direct correlation between molecules 1 and 3 multiplied by the total correlation function h(r) = g(r) - 1, between molecules 2 and 3. The final result has to be integrated over all possible positions of the third molecule. Adding these two contributions we obtain the equation, known as the Ornstein-Zernike equation,

In order to solve this single equation, which contains two unknown functions, we need some extra information. It turns out that from a detailed analysis of the statistical mechanics of liquids, it is possible to give a formal prescription for the direct correlation function c(r) in terms of the intermolecular potential, the temperature, density, and the total correlation function h(r). This prescription follows from a formal density expansion of the potential of mean force in powers of the density (Hansen and MacDonald 1976). Unfortunately, like the virial equation of state, the expression for c(r) cannot be written down in closed form. However, by omitting unmanageable terms, one can obtain an approximate expression for c(r). This procedure generates a class of approximation schemes with names such as the Hypernetted Chain (HNC), the Percus-Yevick (PY), the Mean Spherical Approximation (MSA), etc. One simple rigorous result about the function c(r) is

I

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1 3 : INTRODUCTION T O STATISTICAL MECHANICS OF FLUIDS

that for systems that are characterized by a pair potential u(r), it has the limiting form ~ ( r )+ -u(r)/kT,

as

r

+ 00.

(1 3.8.2)

Indeed the familiar deb ye-Huckel theory of electrolytes is equivalent to assuming that eqn (13.8.2)is valid for all r and not just in the limit r + 00. T o see the origin of the various approximations mentioned above, we begin with the formally exact result (compare eqn (13.3.8)):

The function B(r) represents contributions from so called Bridge functions due to their appearance in a graphical representation. They are the terms that are difficult to handle. In the hypernetted chain approximation one assumes B(r) = 0 and obtains g(r) = exp[-u(r)/kT

+ h(r) - c(r)].

(1 3.8.4)

The Percus-Yevick approximation is equivalent to assuming that [h(r) - c(r)] < 1 so that eqn(13.8.4) can be expanded to give

In the mean spherical approximation the exponential function in eqn (13.8.4)is linearized to give (Exercise 13.8.1): c ( r ) = -u(r)/kT.

(1 3.8.6)

It is easy to see that for hard sphere interaction (c.f. eqn (13.3.7))the Percus-Yevick equation implies that c(r) = 0 for r > 6 while the hard sphere exclusion imposes the exact condition g(r) = 1

+ h(r) = 0

for

r < 6.

(1 3.8.7)

For hard sphere fluids, the PY approximation turns out to be quite accurate in addition to the fact that it has an analytic solution. [See eqns (13.7.9 and lo)]. Unfortunately, this appears to be a coincidence due to a cancellation of errors. In general, it is not possible to obtain a good a priori estimate of the errors in the approximations described above. However, experience has shown that for systems involving Coulombic interactions, the hypernetted chain approximation (which is in fact a slightly more sophisticated version of the Poisson-Boltzmann theory) seems quite accurate. A feature of these approximate integral equation methods that should be remembered is that because the pair distribution functions are not given exactly, the different possible methods of obtaining the thermodynamics (namely via the energy equation (eqn (13.6.2))or the pressure or virial equation (13.6.3)or the compressibility equations (13.6.8)and (13.7.8)will in general not yield identical results. For instance if

C A L C U L A T I O N O F DISTRIBUTION FUNCTIONS

I665

one solves the Ornstein-Zernike equation (13.8.1) in the Percus-Yevick approximation one obtains two different expressions for the pressure:

p" " p k T - (1 -4)3 +

9" - 1 + 2 4 + 3 4 2 PkT(1 -4)2

(compressibility equation)

(virial equation)

(13.8.8)

(13.8.9)

where 4 = nps3/6 is the hard sphere volume fraction. It turns out that for the hard sphere fluid the compressibilityresult is an overestimate of the pressure while the virial result is an underestimate. An heuristic equation constructed by Carnahan and Starling (see Hansen and McDonald 1976) (13.8.10) gives excellent agreement with the exact results for hard spheres. For a general liquid, experience has shown that the energy tends to give the most accurate results. For a hard sphere system, the energy equation cannot be used (Exercise 13.8.2). A more systematic way of determining the structure and thermodynamic properties of a simple liquid is by perturbation methods. These methods are based on the observation that at liquid densities the repulsive interaction between the molecules plays a dominant role in determining the basic structure of a liquid. The idea then is to separate the intermolecular potential, U ( Y ) into a sum of attractive, UA(Y) and repulsive, U R ( Y ) parts

The structure of the liquid is assumed to be dictated by UR(Y);the attractive interaction, UA(Y), is treated as a perturbation. Furthermore, the repulsive interaction is replaced by an effective hard sphere system. For instance, in the Barker-Henderson theory (McQuarrie 1977; Hansen and McDonald 1976) which is one of the more successful descriptions, the effective hard sphere diameter is given by 00 n

(13.8.12) 0

There are many other theories that exploit essentially the same ideas though differing in the choice of the decomposition of the intermolecular potential (see eqn (13.8.11)) as well as in the detailed treatment of the attractive and repulsive part of the pair potential. A direct method of determining liquid structure is by computer simulations. Given the form of the intermolecular potential, one carries out a computer 'experiment' to see how such a 'model' liquid would behave. There are two methods of simulation. In the

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1 3 : INTRODUCTION TO STATISTICAL M E C H A N I C S OF FLUIDS

Monte Carlo method, the idea is to generate a large number (-lo6) of possible configurations of the system (positional, and where necessary orientational, coordinates of the molecules) and the configurations are weighted according to the Boltzmann factor, (exp(-U/k7)) of the whole system. (In practice, this weighting is done by a clever method called significant sampling, the details of which need not concern us here.) These configurations are then used to obtain averages of equilibrium properties. In implementing this scheme, one chooses a cubical cell containing 102-103 particles. T o simulate an infinite system, this cell is replicated as neighbouring cells of a central cell. The number of particles in the cell is limited by computer memory, time, and cost. However, for short-ranged potentials (e.g. the Lennard-Jones potential) and for state points that are not too close to the critical point where one might expect long range fluctuations, the above constraints are not serious. Indeed, Monte Carlo studies have been carried out to simulate a dispersion of colloidal particles interacting via DLVO potentials (Snook and van Megan 1976). However, for the case of an electrolyte solution with ions and solvent molecules, computer simulations still pose an enormous problem since to simulate a 0.1 M electrolyte with 200 ions, one would need several thousand solvent molecules. In order to obtain time dependent properties, a different simulation method known as molecular dynamics has to be used. In this case the idea is to treat the collection of particles in a cell as a many-body problem in classical mechanics and to solve Newton’s laws of motion to determine the position and velocity of each particle as a function of time. The effective size of the cell is increased enormously by the use of ‘periodic boundary conditions’. In effect, this means that when the trajectory of a particle takes it towards a cell boundary it is allowed to leave the system and a new particle is introduced on the opposite side (Fig. 13.8.1) with the appropriate velocity. This method is a more efficient simulation technique and equilibrium properties can also be extracted by using a time average. Examples of results of molecular dynamics simulations have been given in Section 13.5. In this regard we note that in simulating the dynamical properties of a colloidal system it is obviously not feasible to study the time evolution of the colloidal particles as well as the solvent molecules. Due to the very large difference between the mass of a colloidal particle and a solvent molecule

Fig. 13.8.1 An illustration of periodic boundary conditions.

REFERENCES

I667

the characteristic time scale associated with the evolution of each species is very different. Indeed one is only interested in the solvent in that it provides the ‘driving force’ for the random Brownian motion of the colloidal particle. Consequently one regards a colloidal dispersion simply as a collection of colloidal particles in which the motion of each particle will have a random component of the appropriate statistical property to simulate the effects of thermal agitations due to the solvent. Hydrodynamic interactions due to the motion of the colloidal particles are treated by classical continuum hydrodynamics (compare Section 4.9). This simulation method is known as Brownian dynamics and its potentialities have yet to be fully explored. For a resumi of advances in the study of concentrated dispersions see the 1983 Faraday Discussion of the Royal Society of Chemistry (No. 76).

Exercises 13.8.1 Verify eqn (13.8.6). 13.8.2 Why is it not possible to use the energy equation to obtain the thermodynamics of a hard sphere fluid? 13.8.3 Plot the functionspC/pkTandpU/pkTfor0 5 4 5 0.5 and compare them with the Carnahan and Starling eqn (13.8.10) which represents the hard sphere fluid very well.

References Atkins, P.W. (1978). Physical Chemistry. Oxford University Press. Baxter, R.J. (1968). Aust. J Phys. 21, 563. Brown, J.C., Goodwin, J.W., Ottewill, R.H., and Pusey, P.N. (1976). In Colloid and Interface Science (ed. M. Kerker) Vol. 4, p. 59. Academic Press, New York. Egglestaff, P.A. (1967). A n introduction to the liquid state. Academic Press, New York. Gaylor, K., Snook, I., van Megan, W., and Watts, R.O. (1980).3. Chem. Soc. Faraday 2, 76, 1067. Gaylor, K., Snook, I., and van Megan, W. (1981).J. Chem. Phys. 75, 1682. Hansen, J.-P. and McDonald, I.R. (1976). Theory of simple liquids. Academic Press, New York. Hill, T.L. (1956). Statistical mechanics. McGraw-Hill, New York. Hirchfelder, J.O., Curtis, C.F., and Bird, R.B. (1954). Molecular theory of gases and liquids. John Wiley, New York. Horn, R.G. and Israelachvili, J.N. (1981).3’.Chem. Phys. 75, 1400. Israelachvili, J.N. and Pashley, R.M. (1983). Nature 306,249, McQuarrie, D.A. (1977). Statistical mechanics. Harper and Row, New York. Narten, A.H. and Levy, H.A. (1971).3. Chem. Phys. 55,2263. Narten, A.H. and Levy, H.A. (1972). In Water: a comprehensive treatise (ed. F. Franks). Plenum, New York. Ottewill, R.H. (1982). Concentrated systems. In Colloidal dispersions (ed. J.W. Goodwin). Royal Society of Chemistry, London.

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Pings, C.J. (1968). Structure of simple fluids by X-ray Diffraction. In Physics of simple juids (eds H.N.V. Temperly, J.S. Rowlinson, and G.S. Rushbrooke) Chapter 10, pp. 3 8 7 4 5 . North Holland, Amsterdam. Pusey, P.N. and Tough, R.J.A. (1982).Adv. in Colloid Interface Science 16,143-59. Snook, I. and van Megan, W. (1976). In Colloid and surface science 4, (ed. M. Kerker). Academic Press, New York. Rahman, A. (1964). Phys. Rev. 136A, 405-1 1. Verlet, L. (1968). Phys. Rev. 165, 201-14.

Scattering Studies of Colloid Structure 14.1 Introduction 14.2 Relating potential to structure 14.3 Use of scattering to measure structure 14.3.1 Contrast matching of core-shell systems 14.3.2 Structure of adsorbed surfactant layers 14.4 Structure of concentrated isotropic dispersions of spherical particles 14.5 Neutron reflectivity 14.5.1 Reflectivity theory 14.5.2 Application to the solid-liquid interface.

14.1 Introduction We begin this chapter with a discussion of the structure of concentrated dispersions and how that can be elucidated using the technique of neutron scattering. We define a colloidal dispersion as concentrated if its properties are influenced by interactions between the constituent particles. In the case of a sterically stabilized suspension, for example, the primary interaction may be due to excluded volume, which merely reflects the fact that particles cannot pass through one another. This is the so-called ‘hard-sphere’ interaction of Chapter 13. We can obtain a feeling for the magnitude of the effect of this interaction by computing the osmotic compressibility, K T , of a hard sphere suspension, which from eqns (13.7.8), (13.7.9), and (13.7.10) is given in the Percus-Yevick approximation by+

+

(14.1.1) @z/(l 24912 where p is the number density of particles of diameter 6 in the suspension, and PkTKT = [(I

-

4 = xpS3/6

(14.1.2)

is the fraction of the total volume occupied by particles (i.e. the volume fraction of the suspension). The right-hand side of eqn (14.1.1) is unity at infinite dilution, and has only fallen to half of its infinite dilution value at a volume fraction of 8.8%, which is a + Don’t confuse KT with the Debye parameter K of Section 7.3.

669

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

fairly high solids content in suspension. In the case of hard spheres, therefore, our concept of ‘concentrated’ parallels the intuitive idea of a high concentration of dispersed material. For many of the most interesting colloidal systems, however, this is not the case. Dispersions of sulphonated polystyrene latex particles, which we shall consider in some detail later, provide an important class of examples. These particles interact through a screened Coulomb potential whose range is the Debye length, h.d = K1,of the suspension (Section 7.3). As a first crude estimate of the effect of adding such a potential to the excluded volume interaction already present, we may treat the particles as if they had an effective diameter of (6+2hd). The effective volume fraction of such a suspension is, from eqn (14.1.2),

(14.1.3) so that if h.d is only two particle diameters, the behaviour of the 8.8% hard-sphere suspension considered earlier will occur in this suspension at 4 = 0.07%. In this example, the suspension shows concentrated behaviour, even though it contains very little material. We shall see in Section 14.2 that the effective hard-sphere approximation used above is in most cases not even qualitatively correct as a means of representing dispersions with real finite-ranged potentials. The physics displayed in the above example is quite general, however. This may be seen by the following thought experiment, which also allows us to quantify the notion of ‘concentrated’. By definition, the average volume per particle is l/p, where p is the number density of particles in the suspension. Let this volume be represented by a cube of side d, so that d would be the average separation between particles if they were equally spaced. Then since p = l/d3, we may write eqn (13.6.8) as (Exercise 14.1.1): KT

= -3(8 In d/8p),.

(14.1.4)

The osmotic compressibility is thus a measure of the resistance of the dispersion to any attempt to squeeze its particles closer together, and anything which increases that resistance will lower the compressibility. Clearly, adding a repulsive potential has this effect, and the longer the range of the potential, the lower the density at which the compressibility will begin to diminish. Imagine first starting with a hard-sphere system, in which the range of the interaction is the diameter, 6, of the spheres. From the above discussion, we expect this system to exhibit features associated with concentrated behaviour once d becomes comparable with the range of the potential, or when 6/d M 1. Now let the spheres develop a charge, for example by ionization, with the resulting total ionic strength of the suspension corresponding to a Debye screening length Ad. In this case, the onset of concentrated behaviour will occur when

(14.1.5) Equation (14.1.5) shows that in a charged colloidal dispersion there may, in fact, be three regimes of concentration, in each of which we may expect different behaviour: Ad >>6, Ad 6; and Ad <
INTRODUCTION

I671

Instead of a repulsion, we could equally well have switched on an attractive potential between the spheres in our thought experiment, so that it would become easier to reduce the average distance between particles. In this case, the compressibility will increase as the attraction becomes stronger. In a stable colloid, the thermal energy of the particles will stop them from coagulating, despite this ‘stickiness’, by breaking up pairs before they have time to become clusters. If the potential is sufficiently attractive, however, the system will collapse to a different phase. As this point is approached, the compressibility will increase without bound, diverging at the phase transition. The presence of several length scales, each of which may be independently varied, will be of fundamental importance in the discussions of both theory and experiment throughout the remainder of this chapter. This variety of length scales is, in fact, probably the single most important feature which distinguishes the physics of a colloidal dispersion from that of the simple liquids discussed in Chapter 13. A case of particular importance is when one of the length scales dominates all the others. This length may be thought of as the ruler with which we measure distances in the dispersion, and apparently different dispersions may often have identical structures when each is measured with its own particular ruler. We have already seen an example of this in the hard sphere structure factor, eqn (13.7.9). The ruler length is 6, and all hard sphere liquid structures of a given volume fraction are the same when Q is measured in units of 1/6 (or Y in units of a), regardless of the actual value of 6. In Section 14.2 we shall show how the interactions which determine the length scales in the dispersion may be related theoretically to the dispersion structure, and in Section 14.3 we shall see how scattering techniques which select information on these length scales may be used to measure structure experimentally. Dispersions in which the particles are (approximately) spherical are well-understood, and some examples will be given in Section 14.4. T he structure of colloidal dispersions is usually liquid-like, with the important consequence that descriptions of structure must generally be in probabilistic terms, for example using the pair distribution function, g(r), which was introduced in Section 13.3. The ‘structure’ which we may observe in an experiment is in fact the mean of many specific structures, averaged over the time taken to perform the

Fig. 14.1.1 Stereo pair showing a colloidal crystal structure with face centred cubic symmetry. (Stereo viewers are usually available in the crystallography department.)

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

experiment. During the measurement, structures in the dispersion are constantly being destroyed and reformed by the thermal energy of the particles. If the repulsion between particles is sufficiently strong (relative to the thermal energy), however, the particles will crystallize onto a lattice as each attempts to maximize its distance, d, from its neighbours. Such colloidal crystals may be formed, for example, in polystyrene sulphonate latex suspensions which have been deionized to the point where h d rel="nofollow">> d, with p chosen to make d >> 6. The particle spacing is usually of the order of the wavelength of visible light so they give rise to spectacular Bragg diffraction with colourful visual effects. Since there is no preferred orientation for spheres, these crystal structures are usually cubic, d being the lattice parameter. A typical structure is shown in Fig. 14.1.1. This is an example of forming an anisotropic dispersion structure from components which are spherical and which interact through a spherically symmetric potential. Colloidal crystal structures may be melted by reducing the range of the potential between spheres, for example by adding sufficient salt to make hd C d, and Fig. 14.1.2 shows the type of liquid structure which results. (This figure was generated by the Brownian dynamics method discussed in Section 13.8.) T o the eye, the second figure shows effectively random structure, compared with the obvious organization in the first. The liquid structure is, however, far from random - so far, in fact, that it is on the point of crystallizing into the first structure! The reason for this apparent discrepancy is that, while the eye is a remarkable tool for detecting symmetry, it is almost completely ineffective in observing correlations once global symmetry has been lost. The figure gives no apparent indication of the fact that there is actually a highly preferred specific distance between each particle and its neighbours. (This should be kept very much in mind when looking at electron micrographs, for example.) A figure with particles at randomly generated coordinates would in practice look much the same as Fig. 14.1.2 (although close examination would probably show some overlapping particles in the random structure).

Fig. 14.1.2 Stereo pair showing a snapshot of a colloidal dispersion of polystyrene latex particles at 13% volume fraction in M salt; the surface potential is 50 mV. This structure is almost at the point of crystallizing. A small increase in surface potential would convert it to the structure in Fig. 14.1.1.

INTRODUCTION

I673

How, then, can we know that Fig. 14.1.2 represents a structure with nearly as high a degree of organization as Fig. 14.1.1?The answer is to use some method which tells us directly about correlations in the system. One method is to construct g(r), by measuring all possible distances r between pairs of particles in the box and plotting the number of times a given distance occurs as a function of that distance, normalized to 4nP. (To obtain reasonable accuracy, we should either have a very large number of particles in the box, or we should repeat the procedure for a large number of boxes observed at different times.) The g(r) so obtained is shown in Fig. 14.1.3, which should be compared with g(r) = 1 for a random structure. A more efficient method, applicable to real samples, is to measure directly the amplitude density of particle-particle separations, by performing a scattering experiment to measure the Fourier transform (see Appendix A4) of g(r), namely S(Q) (see eqn (13.7.7)). Anticipating the results of the next section, this quantity modulates the intensity of radiation scattered by the dispersion, and we expect to see maximum scattered intensity at Q m 2n/ro, where YO is the most probable pair separation between particles. Now there is no preferred direction in our isotropic dispersion, so S(QJ will depend only on the scattering angle, 8,and not on the scattering direction (recall that Q= (4n/h) sin Q / 2 at wavelength A). We thus expect to see a DebyeScherrer cone of maximum intensity, and this will indeed be observed in a normal scattering experiment. In contrast, the crystal structure of Fig. 14.1.1 will scatter sharply defined Bragg spots in directions determined by the orientation and symmetry of the crystal. On some distance scale, however, there must be strong local symmetry even in the isotropic sample. Consider a pair of particles held fixed and imagine bringing a third particle into their proximity; the preferred position will be on the perpendicular bisector of the line joining the first two,

Fig. 14.1.3 The pair correlation function, g(r), as a function of r (in units of diameter, 6 ) calculated from many snapshots of the dispersion shown in Fig. 14.1.2.

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

since the interactions are screened Coulomb (Exercise 14.1.2). It is easy to see that the approach trajectory of a new particle entering the space near any accidental grouping of a few particles will be strongly influenced by the instantaneous orientation of the group. Furthermore, this may be true for many groups at any given time, but the orientation of one group relative to another will be random (which is why the dispersion is isotropic as a whole). The lifetime of these locally symmetric groups is governed by the time it takes a particle to move a distance of order the size of the cluster. In a simple (monatomic) liquid, this is only a few picoseconds, but in a colloidal dispersion it may be many

LASER

Fig. 14.1.4 (a) The experimental apparatus of Clark et al. (1983). (b) Scattered intensity pattern (averaged over 30 ms) observed using a large scattering volume. (c)-(e) The patterns observed at three different times when only -25 particles were illuminated.

INTRODUCTION

I675

milliseconds (Exercise 14.1.4) or even seconds. Clark et al. (1983) recognized that these times were sufficiently long to perform a laser-light scattering measurement, and undertook a series of experiments which demonstrate graphically the above concepts. The experimental arrangement (Fig. 14.1.4(a)) was extremely simple: a laser was shone on a colloidal latex dispersion which was almost at the point of crystallization, and the scattered light was detected by a T V camera. The image was recorded on videotape, which could be played back frame-by-frame, each frame corresponding to an average over 30 milliseconds of observation. As expected from the previous discussion, a cone of scattering was observed in each frame (Fig. 14.1.4(b)). Next, the laser was focused tightly, so that only a small volume of sample, containing about 25 latex particles, was illuminated. While the real-time image on the T V screen was unchanged (still corresponding to the same cone of scattering), the single frame images were dramatically different (Figs 14.1.4~-e), showing well-defined diffraction spots being scattered from the liquid. These spots have symmetries and orientations which vary with time, and correspond to the observation of just one fluctuating cluster in the dispersion; they give a net cone of scattering when averaged over time (by viewing the tape at normal speed) or over a large number of clusters (using an unfocused laser beam). This experiment shows how a liquid structure can exhibit different types of behaviour when viewed on different space-time scales, and emphasizes the need to decide which features of the structure are appropriate to observe when designing experiments to measure the structure of concentrated systems. We shall return to this in Section 14.3, after discussing the theory necessary for an understanding of scattering experiments.

Exercises 14.1.1 Establish eqn (14.1.4). 14.1.2 Consider three latex particles interacting through a screened Coulomb pair potential U = [exp(-rij/hd)]/rc, where rc is the distance between particles i and j. Fix two of the particles, and let the third move on a locus at any fixed distance measured from the midpoint of the first two. Show that the sum of the pair potentials is a minimum when the centres of the three particles form an isosceles triangle, and is a maximum when the particles are colinear. 14.1.3 Use the equipartition theorem to estimate the average thermal velocity, u (m/s), of particles moving in a given direction in a dispersion in terms of their mass, m (kg) and the temperature, T (K). 14.1.4 Use the previous result to find the time taken by a polystyrene sphere of diameter 234 nm to travel 1 diameter in a suspension at T = 298 K, assuming it moves in a straight line. (Take the density of polystyrene to be 950 kg/m3.) Now calculate the time the particle takes to travel 5.3 pm, assuming it takes a random walk with step-length equal to: (a) the diameter; and (b) 1/4 diameter. Discuss your answers with reference to the time-scales which occur in simple liquids, and compare them with those observed in the experiments of Clark et al. (1983), to which these numerical values correspond.

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14.2 Relating potential to structure We have already seen in Section 13.4 that many-bodied systems are fundamentally different in nature from few-bodied systems. Consider, for example, the three particles of Exercise 14.1.2. These, if unconstrained, would clearly fly apart. Now we know that dispersions of particles interacting through screened Coulomb potentials exist, and are stable at densities where the average inter-particle distance is still small enough for pairwise repulsion to dominate the van der Waals attraction (Exercise 14.2.1). How can this be? A qualitative idea may be obtained by comparing the case of three isolated particles with that of three neighbouring particles chosen in the middle of a suspension; in the latter case, although the three test particles repel each other, they lie between many pairs of diametrically opposed surrounding particles, and the repulsion from each such pair is trying to compress them together. Although the surrounding particles may lie at a greater distance from the test particles (with a correspondingly diminished potential) than the test particles do from each other, the number of particles contributing to the effect grows as 2, and the net effect is to prevent boundless expansion of the test triplet, as long as the other particles surround it. What guarantees the latter, however? The answer is electroneutrality. We can clearly choose a spherical region around the test particles which contains exactly the same number of positive and negative charges, so that overall it is neutral. If the entire dispersion could be sub-divided into such neutral regions, Newton’s theorem tells us that they will have no electrostatic interaction with one another, and hence form a stable system. This can indeed be proved, although it is non-trivial to do so. (The basis for the proof relies on dividing the system into ‘holes’ of many different sizes, for which reason the proof is known as the ‘Swiss cheese’ theorem; see Lebowitz and Lieb 1969.) While the above argument tells us that there will be no correlations between particles sufficiently far apart in a suspension of charged particles (simply by choosing particles in separate neutral ‘holes’, for example), it is equally clear that close contact between neighbouring particles will be extremely unlikely. (It is not impossible, however. Since charged particles offer finite resistance to being squeezed together, they are often referred to as forming ‘soft sphere’ systems.) More formally, we expect the pair correlation function, g(r), to be smallest for small interparticle separations. In contrast, in a neutral system where excluded volume effects dominate, contact is the most likely configuration, because it is the only time the particles are aware of each other’s existence; there is nothing else to prevent the thermal energy from throwing particles together. For hard spheres, therefore, g(r) is largest at small distances. Comparison of Figs. (13.3.2(b)) and (14.1.3) shows the fundamentally different nature of the pair correlation functions in the two types of system. We now see the reason for the statement in Section 14.1 that effective hard-sphere models are not generally useful when discussing soft-sphere systems; no change in the length scale used to measure r (e.g. that used in eqn (14.1.3)) will convert the form ofg(r) shown in Fig. (13.3.2(b)) to that shown in Fig. (14.1.3). The central task of liquid-state theory, begun in Chapter 13, is to quantify this intimate relationship between g(r) and the interparticle potential. We shall continue that task here with a very specific restriction in mind, namely the need to produce theory which may be usefully connected with the scattering experiments to be discussed in the next section. For the experimentalist, this excludes from practical

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consideration any technique, such as simulation, which requires computational times long compared with typical measurement times. (We shall see, however, that simulation can be of vital use in deciding whether an apparently useful approximate theory is also sufficiently accurate to be of practical interest.) In this context, the most useful approaches have been based on the integral equation approximations introduced in Section 13.8. These, in turn, all derive from the equation of Ornstein and Zernike (1914), which we shall refer to as the OZ eqn. For a system containing one species it is eqn (13.8.1). In a colloidal dispersion, there may be a number of species present, and we must generalize the OZ equation to a set of coupled equations which describe correlations between all possible pairs of species. In the general case, orientational as well as positional correlations must be taken into account, and the notation quickly becomes sufficiently complicated to obscure the physical content of the theory for the non-specialist. In this and the next section, we shall restrict discussion to the case of spherical (non-orientable) particles, which introduces most of the essential physics. The reader is recommended to read the article by Blum and Torruella (1972) for other examples of the more general case. In an obvious extension of the notation of Section 13.3, we shall write the pair correlation function between a particle of type i and another of ty p e j as gi(r), with a similar notation for the direct correlation function, ci(r). In a dispersion containing s species of spherical components, with species k having number density P k , the OZ eqn (13.8.1) becomes the set of eqns:

Since the order in which pairs are taken is irrelevant (;jis equivalent toji), there will be s(s+1)/2 simultaneous equations to be solved for the s(s+l) correlation functions hi(+ cij(r). This is clearly not possible (since there are fewer equations than unknowns), and eqns (14.2.1)must be supplemented with another set of s(s 1)/2 equations, known as the closure conditions, before the system can be solved. [Note: the symbol d3t indicates a vector integration in three dimensions. Exercise 13.6.3 indicates how this is done.] The OZ equations essentially define the cc(r) in terms of the hg(r), and do not in themselves contain any specific information about the potentials between particles in the fluid. The need for closure relations provides the opportunity to add this information about any particular system. Note, however, that the OZ equations do require the number densities to be specified, and hence require as input the average interparticle distances. Solving for h(r) will then tell us how the particle positions fluctuate about this mean value (compare with Section 13.3). If we could calculate, say, the cq(r) by some independent method, the hi(r) would follow directly from eqn (14.2.1). Unfortunately, however, there is no simple physical interpretation of the direct correlation function, cq(r), for values of r inside the particle. Statistical mechanics at present yields an a priori estimate of this function only outside the particle. A common feature among the various types of commonly used closure schemes which were introduced in Section 13.8 is therefore to partition space into two regions, r > and r < <,where $ is an appropriate length, and to assume some knowledge of cij(r) in the first region and of h&) in the second. For the latter, the most

+

<

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

convenient assumption is to specify a distance of closest approach, say &, for any given pair of particles; by definition we then have hq(r) = gi(r) - 1 = -1,

for r <
(14.2.2)

This relation is obviously exact if there is, in fact, a distance of closest approach, and it is an excellent approximation for all dispersions in which the particles may be taken as having hard cores, regardless of other details of the interaction potential. This is especially useful when the core diameter can be identified with the physical diameter of the particle, &, in which case $q = (Si

+ Sj)/2.

(14.2.3)

For the remaining spatial region, from the choices presented in Section 13.8, we shall restrict our discussion to the generalization of eqn (13.8.6), the mean spherical approximation (MSA): cq(r) = -Uq(r)/kT,

forr >
(14.2.4)

From a purely theoretical viewpoint, there are better approximations available than the MSA, and the choice of the latter is mainly pragmatic. The MSA (and certain other approximations based on the MSA) may be solved analytically for several realistic model potentials, and these analytic results have provided the first formal framework within which to analyse scattering experiments on colloids. (The term ‘analytic’ solution is often used by liquid-state theorists to mean the reduction of the OZ integral equation to a set of simultaneous, non-linear algebraic equations which must still be solved numerically. We shall call the latter ‘semi-analytic’ solutions, and reserve the term analytic for solutions specified in closed form in terms of standard functions.) T he Percus-Yevick hard-sphere system is a special case of the MSA, in which Uq = 0 unless the particles are in contact. Th e solution to the OZ equation for this case has been given for monodisperse spheres in Section 13.7. An analytic solution for a mixture of two sizes of hard spheres has been given by Lebowitz (1964) and Ashcroft and Langreth (1967). Th e general polydisperse hard-sphere case has been considered numerically by van Beurten and Vrij (1981). A remarkable feature of the binary mixture case is the large effect which the presence of a small concentration of a second component can have on the structure of a hard-sphere fluid. This is a purely geometric effect, related to the fact that in a neutral system, a hole formed in a cluster of large spheres may be filled by a smaller sphere. In a dispersion of charged spheres, the smaller sphere is unlikely to approach a cluster of similarly charged spheres in the first place, and quite different packing results. For this and other reasons (some of which we have already considered), hardsphere fluids are somewhat unique, and we shall need to employ a rather more realistic potential to handle the general case of dispersions of charged particles (or, more succinctly, charged dispersions). For a dispersion of uniformly charged spheres, Newton’s theorem lets us write the electrostatic pair potentials exactly as the Coulomb interactions between equivalent point charges, so that the MSA becomes

R EL A T IN G POTENTIALTO STRUCTURE

I679

together with eqns (14.2.1-14.2.4). Here, xie is the charge on species i, and EOE, is the dielectric permittivity of the dispersing medium. In the simplest case of (partially) ionized monodisperse colloidal particles (z = 1) and counterions (z = 2) without added salt, there will thus be a system of three equations to solve. This is the so-called primitive model (PM), which may be solved semi-analytically (Blum and Heye 1977). It shows a very interesting limiting feature (Medina-Noyola and McQuarrie 1980): when the counterions are taken to have negligible size, the potential of mean force (compare with Section 13.4) between two isolated colloidal particles may be explicitly calculated as (14.2.6)

which is just the pair potential derived initially by Verwey and Overbeek (1948) on the basis of the Poisson-Boltzmann eqn (7.3.11) (see Exercise 14.2.3). The above result suggests a further simplification of the system of OZ equations for monodisperse charged dispersions. Although there are several lengths in the problem (ad and the &), an obvious feature of the colloidal state is that the colloidal particles are usually very much larger than any other ion in solution, so that, among the particle dimensions, 81 is by far the most significant (the colloidal particles in a monodisperse system will always be labelled i = 1, with z = 2 labelling the counterions and i 2 3 referring to any added salt). Whenever this is the case, we may concentrate our attention on the particles alone, and treat them as if they were the only macroscopic component present (Hayter and Penfold 1981a,b; Hayter 1983). This is the ‘onecomponent macrofluid’ approach (OCM), in which all correlations are neglected except those between the macroions, which are taken as interacting through the potential of eqn (14.2.6). The counterions thus provide a neutralizing background whose quantitative effect on the structure now appears indirectly, through the screening length (Ad) in the potential. The main features of the OCM approach are that it preserves those length scales which dominate the physics, and the correlations calculated are those of most interest, namely those between the colloidal particles themselves. The OCM is thus equivalent to the primitive model (PM) in the limit 82 + 0, and since it is numerically no more difficult to solve the P M for finite counterions than in this limit (see, for example, Naegele et al. 1985), we must look at the underlying mean spherical approximation (MSA) to see why the OCM is of interest. Consider the apparently straightforward problem of using the MSA to determine correlations in a dilute system. In the limit Pk + 0, eqn (14.2.1) reduces to

(14.2.7) From eqns (14.2.4) and (14.2.5), the MSA in this case thus amounts to gG(r)

1 - Ui,.(r)/kT, r >
(14.2.8)

so that whenever Uq(r) > KT, g&) < 0, which is not permitted for a probability function. In the low-density limit, therefore, the MSA yields unphysical results, and it

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

is clear that this becomes worse as the potential becomes more strongly repulsive. Numerical calculations show that this breakdown of the MSA may occur at volume fractions as high as 20% in many representative charged dispersions, so that the P M is not generally useful. We shall now see, however, that the OCM may be reformulated in a way that avoids this problem without adding any further mathematical complications. T o this end, we return to our earlier discussion of the scaling of physical lengths in the system. In the case of hard spheres, we have seen that apparently different suspensions have equivalent structures at equal volume fractions, as long as the sphere diameter is used as the unit of length when describing the structure. This means that, if we know g(r) for a fluid of hard spheres of diameter 60, we may trivially calculate the structure of a fluid of hard spheres of diameter 61 at the same volume fraction by simply rescaling g(r) to g(rSo/ 61). In the case of charged hard spheres, a second length, h d , enters the physics, and as we saw in Section 14.1 this introduces concentration regimes in which the behaviour may cross over from being dominated by excluded volume to being dominated by electrostatics. The former must clearly be the case at high density, simply because the particles are crowded together and their size plays an important role. At the other extreme of low density but long-ranged interaction, however, the size of the particles must be irrelevant (except in so far as it influences the net charge), since at long range the interactions will appear to be between point charges. Such a system will have the characteristics of a one-component plasma (see, for example, Baus and Hansen 1980), in which the density (i.e. particle concentration) rather than the particle diameter dominates the structure, so that the appropriate length-scale for correlations is the ion-sphere radius, a, defined by (4n/3)a3 = l/p

(14.2.9)

a = 6/(2&).

(14.2.10)

from which (Exercise 14.2.4)

Consider next how the potential scales with particle size. Rewriting eqn (14.2.6) in terms of x = r/6 and dividing by k T (omitting subscripts, which will not be needed for the OCM), we obtain the general dimensionless form for the potential V(x)/kT = y exp(-K ’x)/x, where and

y = z2e2 exp(~’)/[4neoe~6 kT(l

x 2 1

(14.2.11a)

+~’/2)~]

(14.2.1 1b)

I( = 6/hd (= K 6 ) .

(14.2.11~)

The potential (in k T units) when two particles are in contact (x = 1) is given by y exp (-K’). For x < 1, we may complete the potential by writing V(x)/kT = 00

for x < 1

which is simply another way of expressing the condition (14.2.2).

(14.2.11d)

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I681

We may equally well write eqn (14.2.6) in terms of y = r/2a = ~ I ’ / ~ xwhich , is the appropriate dimensionless scale in the plasma limit. Instead of eqn (14.2.1l), we then have the equivalent forms I

VCy)/kT = Y ’exp(--K”A/y, VCy)/kT = 00

where y‘ =

dy

and

y 2

$7

(14.2.12a)

Y

4

(14.2.12b)

K”

= I(/($:)

(14.2.12~)

When this form of the pair potential is plotted for several volume fractions (Fig. 14.2.1), a remarkable feature emerges: apart from the position of the hard core, eqns (14.2.12) describe a universal potential for an infinite family of colloidal systems having common values of y’ and K” at their particular volume fraction. We now ask, under what conditions will members of this family have a universal structure (when measured with a ruler of length a (eqn (14.2.10))? This question is equivalent to asking when the hard core can influence the structure, since all other aspects of the potential are common to all members of the family. Now, by ‘structure’ we mean an average representation of the configurations which the particles in the dispersion can adopt (compare with Section 13.1). Since interparticle distances y for which the Boltzmann factor exp[-V(y)/kT] is small are improbable, the presence of a hard core will have negligible influence on the structure provided YO, = $‘/3) >> 1, since particles will rarely climb high enough up the potential curve to sample it.

Fig. 14.2.1 Plot of the potential function V(y)/kTagainst y for three dispersions (volume fraction > 4 2 > 43) with common values of y’ and K ” .

41

682 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

Thus, although we send the potential infinite at that point, to provide a convenient mathematical closure to the OZ equation, in physical terms this can at most represent a small perturbation on the system. (This is just another way of saying that the particle size no longer has a determining influence on the structure.) In particular, we expect all of the curves of Fig. 14.2.1 to lead to the same dispersion structure, in the same way as common volume fractions lead to the same structure in hard sphere suspensions. This is an important result, because it tells us how to compute the structure for a whole class of suspensions if we know the structure of any one of them, simply by a rescaling operation. It is a generalization of a well-known result in plasma physics, namely that systems with the same Coulomb coupling constant, defined by

r = 2Y(r = 2a)/kT

(14.2.13)

all have the same structure. The central condition under which the above argument is valid is that the Boltzmann factor should be negligible for configurations in which the particles are (nearly) in contact. This will clearly never be the case when the interaction is attractive, since contact configurations are then highly probable, so it will never be applicable to the PM, because g12 must always describe correlation between ions of opposite charge. The OCM, on the other hand, only attempts to find correlations between colloidal particles which interact repulsively. T o emphasize this feature, we write the neglect of counterion-colloid and counterion
for

r >0

(14.2.14)

which simply states that counterion positions are random with respect to either colloidal particle or other counterion positions. The main correlations of this type present in the real system are those required to satisfy the Poisson-Boltzmann equation, and are already included in the Debye length in the potential. These additional closure conditions are defined throughout space, and so two of the three OZ equations are solved (by definition). The OCM thus reduces to the MSA defined by eqns (14.2.14) for the sole case i = j = 1. This may be solved in closed analytic form, but still suffers from the defects of the MSA discussed earlier. Now, however, we have a model system to which our scaling argument applies, since all interactions are repulsive. Remembering that the MSA gives good results at high density, we thus solve the low density case simply by constructing an equivalent high density system, using eqn (14.2.12) to specify the potential in the new system. The analytic solution to the MSA gives the structure accurately at the new density, and we simply rescale the length-scale of this structure to obtain the desired result. This is the rescaled MSA (RMSA) proposed by Hansen and Hayter (1982), and it provides a powerful means of computing correlations in charged colloidal systems of any particle concentration. One final refinement is needed to relate it to real systems, however. The closure (14.2.14) is not quite the same as taking the PM with 82 + 0, because it allows the counterions to appear anywhere in space, including inside the colloidal particles. This

R EL A T IN G POTENTIALTO STRUCTURE

I683

2.0

1.0 h

k

v

a0

0

-1.0

0

10

20 r/6

30

40

3

Fig. 14.2.2 Comparison of g(r) against r/6 calculated by two different methods for a latex dispersion at volume fraction qj = 1.2 x lop4 in lop6 M salt. The latex diameter is 6 = 90 nm and ) RMSA. The surface potential is 230 mV. Data computed by (m) Brownian dynamics; ( simulation took about 6 hours on a desk top computer (IBM-XT), compared with 2 seconds for the RMSA.

is known as a penetrating background, and is easily handled. In a suspension of volume fraction 4 in a uniform penetrating background, a fraction (1 - 4) of the background lies inside the particles, which therefore have their effective charge reduced by this amount. T o compute correlations for a real system of particles having charge 2, the calculation must be performed for particles carrying an effective charge zeff

= Z/(1 - 4)

(14.2.15)

to allow for the charge compensation by the neutralizing background. T o keep the structures in the effective and real systems equivalent, we must still work at fixed Coulomb coupling (eqn (14.2.13)), which yields the effective screening corresponding to eqn (14.2.15) (Exercise 14.2.6):

Equations (14.2.15) and (14.2.16) complete the specification of the use of the analytic RMSA calculation to compute correlation functions in the OCM approximation. The validity of this approximation has been thoroughly tested by comparison with computer simulation results, which are essentially exact, and it proves to be extremely accurate; the agreement shown in Fig. 14.2.2 is typical throughout the entire range of particle concentrations and potentials found in charged colloidal systems.

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

Exercises 14.2.1Why must van der Waals attraction always dominate screened Coulomb repulsion at sufficiently large distances? Is this still the case for a bare (unscreened) Coulomb potential? At large enough distances, both potentials may be very small compared with kT,in which case the question is of little or no significance; can you give an example of a system where it matters in practice? 14.2.2F. Zernike was awarded the Nobel Prize in Physics in 1953 for the invention of phase-contrast microscopy, but is probably better known for his contribution to liquid state physics. The Ornstein-Zernike eqn was published in 1914, but not solved for any liquid until 1962. The original paper, which is reprinted in Frisch and Lebowitz (1964), was published, inter aha, to correct some misconceptions which Einstein had about correlations in the liquid state! Look it up, and compare the original derivation with that outlined in Section 13.8. 14.2.3Show that eqn (14.2.6) is identical to the repulsion energy calculated by Verwey and Overbeek (1948) for approaching spheres in the low potential approximation. 14.2.4Derive eqn (14.2.10) from eqns (14.1.2) and (14.2.9). 14.2.5Find expressions for y (equivalent to eqn 14.2.11b) when the inter-particle potential is expressed in terms of (a) the surface potential or (b) the surface charge density, rather than the total charge. 14.2.6Use eqns (14.2.10), (14.2.11(a)), and (14.2.13) to show that eqn (14.2.16) is implied by eqn (14.2.15) when the coupling, r, is held fixed.

14.3 Use of scattering to measure structure We have seen already (Sections 14.2, 14.3, and 13.7) that the interaction between electro-magnetic radiation (light or X-rays) and the electrons in a sample leads to scattering of the radiation, whenever the local electron density fluctuates away from its mean value in the sample; that is, radiation is scattered by refractive index variations in the sample. This result is general for any type of radiation, and is the basis for various scattering techniques, such as electron diffraction, in which matter waves, rather than electromagnetic waves, may be scattered. Once we are able to calculate the refractive index for each given type of radiation, a common scattering formalism applies. The practicality of different types of scattering experiment is, however, another question. Electrons, for example, interact very strongly with other electrons, and are therefore strongly scattered by fluctuations in electron density. Since strong scattering means a strong signal, this would appear to be an advantage, but consideration of a practical experiment shows that, in fact, it is a major drawback for any sample (such as a colloidal dispersion) which must be contained. The scattering from the minor imperfections always present in the container proves to be so strong that electrons of suitable energy would never get to the sample. In the case of light or X-rays, the interactions are weaker, and the container problem can be overcome in most cases. The interaction with the sample is still sufficiently strong, however, that multiple scattering becomes a

USE O F SCATTERING TO MEASURE STRUCTURE

I685

major problem in dense samples, unless they are so thin that they may exhibit artefacts induced by interactions with the walls of the container, and hence no longer represent the bulk dispersion. Since the mid-l960s, therefore, concentrated colloidal dispersions have been increasingly studied by a form of radiation which appears to interact only very weakly with matter, namely neutrons. Neutron scattering has proved to be such an important tool in materials science in general that most neutron scattering centres in the world now make their facilities available to scientists from other laboratories on a routine basis, and neutrons are more accessible to the colloid chemist than, for example, synchrotron X-rays. The technique is, however, much less readily available than a light scattering or X-ray camera in one’s home laboratory, and its popularity is due to several unique scientific features which we shall explore in the remainder of this chapter. Neutrons are produced by well-known processes in reactors (fission) or accelerators (spallation), and the interested reader is referred to any nuclear physics text for details. As produced, they are too energetic for use in scattering experiments, and must be slowed down in a moderator until their kinetic energy corresponds to about room temperature (thermal neutrons) or less (cold neutrons), at which point their de Broglie wavelengths h = h/mnv are comparable with typical interatomic distances (Exercise 14.3.1); here, h is the Planck constant, m, is the neutron mass, and v the neutron velocity. In the study of colloids, we shall be interested in correlations over many atomic distances, or conversely in Fourier components of the structure having small Qvalues, so we shall mostly be interested in cold neutron scattering. Th e neutron has a magnetic moment, but no measurable charge or electric dipole moment, so it has negligible interaction with electrons unless they are unpaired. Th e only scattering we will consider will be due to interactions with the nuclei of the atoms in the dispersion. These interactions are in fact very strong, but of very much shorter range m) than interatomic distances. Now, it should already be clear from the previous discussion (Sections 14.2 and 14.3 and Chapter 13) that scattering is concerned with phase shifts between scattered waves, and the way in which waves of various phases arrive at different atoms in the sample. This means that we do not need to know the amplitude and phase of any wave right at the point of scattering, as long as we can calculate these quantities when the wave arrives at another atom. In the case of neutrons, this is at a large distance from the point of scattering, compared with the range of the scattering interaction. We may thus use a device which is common in scattering theory. We replace the real interaction potential which scatters the neutron by any fictitious potential which gives the correct answer for scattered amplitude and phase at any distance which will appear in the scattering calculation. Such potentials are known as pseudopotentials, and the appropriate pseudopotential for neutron scattering is (14.3.1) where fi = h/2n, Rj is the position of thejth nucleus, and the amplitude bj has the dimensions of length. (Note that 6 is here the Dirac delta function as defined in Chapter 11 (eqn (11.7.3)).)

686 I 14:SCATTERING STUDIES OF C O L L O I D STRUCTURE

Table 14.1 Neutron scattering amplitudes, b, for selected nuclei. (See Lovesey 1984 for a more extensive listing.)

Element

‘H

b(lOPl5m) -3.741

*D

C

N

0

Na

A1

Si

C1

Ti

6.674

6.648

9.36

5.805

3.63

3.449

4.149

9.579

-3.30

Table 14.1 summarizes some neutron scattering amplitudes for nuclei typically found in colloidal dispersions. Two features are noteworthy. First, there is no systematic variation with atomic number (unlike the case for X-rays), and light atoms may be as easily studied as heavy atoms. Second, certain values are negative, so that cancellation of amplitudes may occur. (Recalling that exp(in) = -1, a negative value merely indicates a phase shift of n on scattering.) The refractive index, n, of any material for a particular radiation is defined as the ratio of the momentum of the radiation in vacuo, po, to its momentum, p , in the bulk material. Since po = mnvO and the corresponding kinetic energy is m,vi/2, we may write the momentum in vucuo in terms of the kinetic energy: po = d 2 m n E o ) .At any point, r, in the material, the energy will be EO U(r),where U(r)is given by eqn (14.3.1). We may calculate the mean value, U, of this potential by averaging over the volume, V, of the sample:

+

U = (1/V)

s

U(r)d3r =

-x2d2

bj

mn

V

(14.3.2)

where the integration follows immediately from the properties of the &function. The quantity Cbj/V is the scattering amplitude density, B, in the material (Exercise 14.3.3) and corresponds to electron density in light or X-ray scattering. Clearly, the sum need only be evaluated over a representative formula unit, using the corresponding volume, V = 1/ N , where N is the number density of such units in the material, so we may write B = N C b, provided it is understood that the sum is over atoms in the formula unit whose density is N . Since the scattering amplitude of vacuum is null, we thus have

Note that an- 1 is very small [since U << Eo, and may have either sign (compare with Exercise 14.3.3)]. In terms of the de Broglie wavelength, A, an equivalent form for the refractive index is : n, = 1 - BA2/n.

(14.3.4)

Two materials with scattering amplitude densities B1 and B2 will have the same refractive index if B1 = B2. The difference B1 - B2 is called the contrast between the two materials, and refractive index matching thus corresponds to null contrast. This is

USE O F SCATTERING TO MEASURE STRUCTURE

I687

often loosely referred to as ‘contrast matching’. One very important application is to match one region of a particle, such as the core of a coated colloid, to the solvent, so that neutrons will only be scattered by the remaining part, which may be studied separately (see Section 14.3.1 and Exercise 14.3.5). Contrast matching may often be achieved simply by varying the ratio of H20 to DzO in an aqueous suspension. B varies from -0.56 x 1014 mP2 in pure H20 to 6.34 x 1014 mP2 in pure D20. The values of neutron scattering amplitude listed in Table 14.1 enable us to calculate the (coherent) neutron scattering amplitude density+,B for various molecules (Exercise 14.3.3). This is the analogue of the electron density for a light scattering event and is given by B = C b;/Vwhere Vis the molecular volume. The monomer unit for poly(styrene) for example is [(C6Hs)CH.C&] or CsHs. The value of x b i is therefore 8 x (6.648-3.741) x lo-’’ m = 23.26 x lo-’’ m. The molar mass of this unit is 104.1 daltons and taking the density as 1054 kg m-3 the value of B is

23.26 x lo-’’ x 6.023 x

x 1054/104.1 = 1.418 x lOl4mP2.

Values for water, DzO, and a number of polymers are given in Table 14.2. Thermal neutrons, by definition, have energies comparable with those of dynamic thermal excitations in the sample, and a major use of neutron scattering is to measure such excitations. An introduction to the measurement of time-dependent phenomena with neutrons is given by Lovesey (1977). In this chapter, we shall only be concerned

Fig. 14.3.1 The rotating helical monochromator. Only neutrons with a particular velocity can traverse the slot and they form the incident beam.

+ Often called the scattering length density. We consider only the coherent scattering. An incoherent scattering component is also present but it is isotropic and can be subtracted from the scattered signal (eqn 14.3.10). It is cause by the presence of different nuclear isotopes of the component atoms which have different nuclear crosssection.

K

J

B

Lower monochromator housing Upper monochromator housing Graphite monochromatingcrystals Cold beryllium filter Collimator and neutron beam guide Sample chamber 58 cm gate valve 20 m by 152 cm dim vacuum flight path 28 cm wood shielding Detector carriage 112 cm diam twodimentional position sensitive detector (64 by 64 cm active area) Data acquisition system

Fig. 14.3.2 T h e 30 m SANS camera layout at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory. Neutrons from the reactor (off the figure at top left) are reflected by two sets of graphite crystal monochromators A and B into the neutron guide/collimator system, C and then onto the sample at F. Scattered neutrons are detected by the 64 x 64 cm2 two-dimensional position-sensitive detector K, which is mounted on a chariot, J. The chariot moves on rails in the evacuated tube H allowing the detector to be positioned anywhere from 1.5 m to 20 m from the sample.

USE O F SCATTERING TO MEASURE STRUCTURE

I689

with time-averaged structures, and hence with static experiments, in which no energy analysis is performed. Neutrons of a particular wavelength [or velocity] are selected from a beam, either by Bragg reflection from a suitable crystal (such as graphite) or by passage through a rotating helix whose speed and pitch are adjusted to allow transmission only of neutrons of the desired velocity (Fig. 14.3.1). The monochromatic beam is then collimated by appropriate slits or diaphragms before impinging on the sample, and the intensity scattered in a given direction is measured by a detector of known efficiency. In the case of scattering from colloids, we are interested in scattering at small Q (corresponding to large distances), which means small angles and long wavelengths. Typically, 0.4 CA/(nm) C 1.2 and 0.05 C B/(deg) C 5.0, corresponding to 0.01 C Q/ (nm-’) C 3. This type of experiment is known as small angle neutron scattering or SANS. SANS cameras of all types share several common features, exemplified in the practical instrument shown in Fig. 14.3.2. The basic optical geometry is that of a pinhole camera, in which the source (usually a circular hole at the entrance to the collimating system) is imaged on a two-dimensional position-sensitive detector through the ‘pinhole’ at the sample position. The latter is usually at least 1cm in diameter, to allow studies of bulk samples, and the overall size of the instrument is sufficiently large (several tens of metres long) to maintain apparent pinhole focusing. The beam is transported in vacuo where possible, to minimize air scattering, but windows may be used as dictated by different sample containment requirements, since many common materials (e.g. quartz or aluminium) are effectively totally transparent to neutrons of long wavelength. The sample-to-detector distance may be varied by moving the detector on rails inside the evacuated tube after the sample. Optimal focussing requires a fixed ratio between the source-to-sample and sample-to-detector distances. This is maintained by transporting neutrons to the source via a variable-length neutron guide, which operates in a manner exactly analogous to a light-pipe. Reference to eqn (14.3.4) shows that the neutron refractive index is less than unity for any material with a net positive mean scattering amplitude density, so that vacuum is optically dense relative to such a material. In this case, total external reflection is possible at the interface and neutrons are transported inside the evacuated guide tube by successive reflections. One of the most important features of SANS instruments is that they are situated at large central facilities, and are used on a shared basis by many groups, most of whom are not specialists in scattering. This has led to the development of highly automated equipment, with standard procedures for correcting and normalizing the data, so that the outcome of a SANS measurement is the absolutely scaled intensity, I(!.?),scattered coherently by correlations in the sample being studied. We may thus, without loss, bypass technical details of the corrections (in particular, the removal of solvent scattering and the corrections for single-atom, or incoherent, effects), and proceed directly to a consideration of the information obtainable from the corrected coherent intensity, I(!.?),which corresponds to the scattering from the colloidal particles alone. We have seen that the refractive index, which corresponds to scattering at Q = 0, depends on the quantity N C b. At finite we must allow for the phase at each atom when summing the scattered amplitudes. The square of the resultant total amplitude is the scattered intensity (compare with Section 13.7) for any given configuration of the

a

690 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

Table 14.2 Coherent neutron scattering amplitude (length) densitiesfor various substances.

Molecule

Water D20 h-PS d-PS

h-PMMA d-PMMA h-PAN Cl2H26 C12D26

[B/lO"]mP2

-0.56 6.40 1.42 6.47

1.07

7.03

2.27

-0.46

6.71

(PS: Poly(styrene);PMMA poly(methylmethacry1ate);P A N poly(acrylonitrile).)

atoms. Averaging over all possible atomic configurations then yields the measured intensity (Exercise 14.3.7):

(14.3.5)

(14.3.6) Here, Rj is the position vector of thejth atom in the sample, and ( ) represents the configurational(ensemble) average. Equation (14.3.6) shows that whenever atomsj and k are well-separated, the phase factor will oscillate through many cycles and will average to zero over small variations in R, unless I Q I is small; conversely, the phase factor will remain essentially unity (the refractive limit) at small Q unless large correlation distances are involved. SANS thus has the desirable feature of being sensitive to the colloidal structure while not inundating us with detail on the scale of atomic distances (which we usually know from separate, high Q diffraction experiments). We are considering scattering by the colloidal particles alone, the solvent scattering having been measured separately and subtracted from the data. (We shall return to this point shortly.) The double sum in eqn (14.3.6) is thus over all possible pairs of atoms in the dispersed particles. Any given pair of atoms either lies in the same colloidal particle, say particle n, or there is one atom from the pair in each of two colloidal particles, say particles n and m. The summation can be broken up into a sum term involving atoms of n and another term involving the ensemble average of scattering from pairs of particles with different centre-to-centre distances. The resulting scattering thus depends intimately on the probability of a given interparticle distance, and hence on the pair distribution function, g(R,) (Sections 13.8 and 14.2). Before further discussion of this point, however, it is worth examining that part of the scattering which does not depend on interparticle correlations. T o this end, let us consider the dilute case, when particles n and m are uncorrelated, so that only the scattering from a single particle need be considered. Then [compare with eqn (14.3.6)]

USE O F SCATTERING TO MEASURE STRUCTURE

I691

where the form factor for a given particle is defined as bjexp(zQ.rj).

F(Q) =

(14.3.8)

j

The form factor gives the amplitude and phase of a wave scattered by a given particle. The scattered intensity is thus the square of the modulus of the form factor, and the total scattered intensity in this case is just the sum of the intensities scattered by the individual particles. Form factors are readily calculable for various common geometrical shapes (Exercise 14.3.8). Since SANS is insensitive to detail on the atomic scale, individual atomic scattering lengths may be replaced by local averages over, for example, functional chemical groups of several atoms. This lets us work in terms of the local scattering amplitude density B(r)at position r in the sample, and has the considerable advantage that we need only know the local chemical composition, without knowing the detailed local atomic structure. In this case, it is convenient to replace the sum over atoms in a particle by an integration of the scattering amplitude density over the volume of the particle, so that we shall generally write

s

F(Q) = [B(r)- Bm]exp(zQ.r)d3r

(14.3.9)

where we have now explicitly allowed for the fact that scattering from the same volume of the suspension medium (of scattering amplitude density B,) has been subtracted. The form factor will in general depend on the relative orientation of the particle and the momentum transfer, Q. When B(r) depends only on Y = Irl we can use the technique outlined in Exercise (13.6.3) to transform eqn (14.3.9) into the Debye result, familiar in the theory of crystallography (see Atkins 1982 p. 755) (Exercise 14.3.8): sin(@) [B(r) - Bm]r2dr. Qr

(14.3.10)

In the particular case where the particle is a sphere of uniform scattering amplitude density, Bo, the integral can be evaluated explicitly to give (Exercise 14.3.8(b)):

where Y is the volume of the sphere. In the case of scattering from an isotropic dispersion, the intensity must also be averaged over all possible particle orientations:

1

2n n

4n

I(Q)= 1

0

sI(Q)sinB dQd$

(14.3.12)

0

where (B,$) are the polar angles defining the particle orientation relative to Q. (This is irrelevant for spheres, of course, since they have no orientation.)

692 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

The form factor used in eqn (14.3.11) is related to the earlier form factor introduced in connection with light scattering (Section 3.3) simply by P = F2 and so we see the relation between this and the earlier eqn (3.3.14). Some information about the shape of anisotropic particles is lost in the angular averaging, but that can be avoided. The analogue of the Rayleigh-Gans-Debye equation for neutron scattering is then:

where the B factor takes the place of the refractive index [eqn (14.3.4)] and Npis the number of particles of volume V,. If a model for the particle shape is available, scattering measurements on a dilute dispersion will provide detailed quantitative information on the geometry. For cylinders of length 21 aligned parallel to the direction of Q for example the form factor is

F(Q) = CV sin (Ql)/(QI).

(14.3.14)

More importantly, we may obtain certain information modelfree. In the limit Q = 0, eqn (14.3.9) is simply the integral of the scattering amplitude density over the volume of the particle; that is, the total scattering amplitude:

I(0) = A(B,

-

4

Vp

(14.3.15)

where 4 is the particle volume fraction. The value of B, can be varied over a wide range using mixtures of D20 and H20 and using eqn (14.3.15) in the form [compare Cebula et al. (1978)l: h h ( 0 ) = (B,

-

B,)[$A Vp]4

we see that a plot of J I ( 0 )against B, (Fig. 14.3.3) will pass through zero at the point where B, = Bp allowing us to check experimentally the estimates in Table 14.2. The slope and intercept of these plots is given by -[$A V,]Jand Bp [4AVp]4 respectively, so the ratio: intercept/slope = -Bp

(14.3.16)

which gives a further check on Bp.Furthermore, if the instrument constant A, and the volume fraction are known then obviously one can evaluate V, and, hence, the radius of the particles. An alternative formulation allows one to calculate the molecular weight (via the known chemical composition and scattering amplitude values). For this reason, the ability to measure absolute intensities easily is an important feature of SANS. A second general result for dilute systems is Guinier’s law which we met in Section 3.3 in connection with light scattering. In the limit of small Q (see Exercises 3.3.4 and 3.3.5):

USE O F SCATTERING TO MEASURE STRUCTURE

PS

\Hydrogenated

I 0

1

I693

I 2 B, W 2 )

3

4

5

6

7 xi014

Fig. 14.3.3 The function d I ( 0 )plotted against scattering length density of the dispersion medium, B, for polystyrene and deuterated polystyrene in H20/D20 mixtures. (After Ottewill.)

where the radius of gyration, a c , is defined for a particle of any shape by aG =

S?B(r)d3r B(r)d3r

s

(14.3.18) *

Thus a characteristic dimension of the particle may be obtained from the initial slope of a plot of ln[l(QJ] vs @. (This is called a Guinier plot as we noted in Section 3.3.)

14.3.1 Contrast matching of core-shell systems Table 14.1 shows that hydrogen and deuterium have neutron scattering amplitudes of opposite sign. We noted earlier that this raises the possibility of making mixtures of water and deuterium oxide (D20) which would have a wide range of effective neutron 'refractive indices'. It is thus possible to make up a mixture with the same effective scattering properties as, say, a hydrocarbon chain. The hydrocarbon chain would then appear to be invisible to neutrons when immersed in such a mixture. This technique is called contrast matching. It can be used to explore, for example, the structure of a spherical particle made of a core of one material with a shell of a distinct material (Fig. 14.3.4).As a particular case it can also be used to determine the disposition of an adsorbed layer of a surfactant on a particle surface or the segment density distribution of polymer chains near a surface (Section 14.3.2).Livsey and Ottewill(l991) show that a similar procedure can be used in light scattering, where core-shell systems show extreme sensitivity to the refractive index of the medium. Consider the case of a spherical core particle of radius a with a concentric outer shell of thickness b suspended in a solvent. Integration of eqn (14.3.9) then gives for the form factor of the particle (Markovid and Ottewill (1986)):

694 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

(4

(b)

Fig. 14.3.4 Illustrating the technique of contrast matching. (a) B, = Bp (b) B, = B, (c) B, # B, # B,. Subscript is medium, ,is particle, and is shell.

where Vp is the volume of the core particle, Vis the total volume of the core plus shell, Bp, B,, and B, are respectively the scattering length densities of the core particle, shell, and solvent medium, and (from eqn.14.3.11): f ( x ) = 3(sin x - x cos x)/x3.

(14.3.20)

If b = 0 so there is no shell present we have:

which reduces to eqn (14.3.15) for [f(Q=O)] = 1. The same result is obtained ifwe set B, = B,. Thus if the shell and the solvent are matched we see only the central core particle. If the core is contrast matched by setting Bp = B, we obtain: (14.3.22) This will be the scattering from the shell. It is also possible to work under conditions such that B, = 0 in which case the scattering intensity is given by:

This formulation is particularly useful for studying the adsorption of surfactants onto the surface of polymer latex particles which we will now examine.

14.3.2 Structure of adsorbed surfactant layers Even with small angle neutron scattering (SANS) it is not possible to measure at zero scattering angle. It is, however, possible to use the Guinier Law (eqn (14.3.17)) to obtain accurate values of I(0) and to use these to simplify the treatment of data for

USE O F SCATTERING TO MEASURE STRUCTURE

I695

mixed systems, since as Q+ 0 the functionf(x) + 1. Combining this result with that in equation (14.3.23) gives:

I(0) = AN,[BP VP

+ B, VSl2

(14.3.24)

where V, = Y- Yp is the volume of the shell which can also be written: (14.3.25)

V, = (b - a)S,

where b is the (core + shell) radius and S, is the surface area of the particle. We can also set B, = b,/ V, where b, is the scattering length density of the adsorbed molecule and V, is its molar volume. The number of adsorbed molecules per particle, n is V,/ V, and if r is the number adsorbed per unit area then:

We can now set

B, Y, = [b,/ V,] V, = [b,/ Vm](b

- a)Sp = b

J sp

(14.3.27)

and so, eqn (14.3.24) becomes:

I(O)= AN,[B,

v, + b,rsPl2.

1

2

(14.3.28)

3

~2 I 10-4 A-2

Fig. 14.3.5 Guinier plots of (a) bare polystyrene latex; (b) latex at 4 x 10p3M total d23dodecanoate concentration; (c) latex at 1.2 x 10p2M total dz3-dodecanoate concentration. (After Harris et al. 1983.)

696 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

So for a latex particle with an adsorbed layer we can write: I

Tp+s=

(a)Bp'+ rSp =

ANP

Vp

(14.3.29)

b,

p+s

whilst the corresponding scattering function for the uncoated particle is, from eqn (14.3.24):

'-

T - (%)I=AN,

'

(14.3.30)

BpVp.

The ratio of the two scattering functions is, therefore: (14.3.31) This relation has been used by Harris et al. (1983) to measure the adsorption of d23dodecanoate ions onto a polystyrene particle at pH 8.1. A 0.2% dispersion of the latex

I

0.5

1.o

Equilibrium concentration/

1.5

mol dm-3

Fig. 14.3.6 Adsorption isotherm of d23-dodecanoic acid on polystyrenelatex at pH 8.1 and 22" C at a total electrolyte concentrationof 2.2 x lO-*M. A: Calculated by fitting a spherical cell model, with error bars. 0 :Data obtained by Guinier analysis (Fig. 14.3.5). (After Harris et al. 1983.)

USE O F SCATTERING TO MEASURE STRUCTURE

I697

was made up in 8% D20/92’/0 H20 (for which B, = 0). The scattering behaviour at low angle was as shown in Fig. 14.3.5 and by using the Guinier Law it was possible to extract I(0)values as a function of added surfactant. T h e resulting isotherm is shown in Fig. 14.3.6.

Exercises. 14.3.1Given the neutron rest mass is m, = 1.67494 x lopz7kg, calculate the velocity of a neutron whose thermal energy corresponds to a temperature (a) T = 293 K, (b) T = 20 K. Are relativistic effects important? Calculate the de Broglie wavelengths corresponding to these velocities, and compare the neutron energies with those of X-ray photons at the same wavelengths. 14.3.2Calculate the electron densities (e-/m3) in (a) CsH170H, (b) CsH18, (c) CzHsOH, (d) A1203,(e) SiO2, (f) H20, and (g) D20. Take the respective densities as 825, 704, 790, 3970, 2655, 1000, and 1100 kg/m3. 14.3.3Use Table 14.1 to calculate the neutron scattering amplitude densities (m/m3) corresponding to the electron densities found in the previous exercise. Compare the relative light scattering contrasts (scattering density differences) between the various materials with those for neutron scattering. How does the mean potential (14.3.2) compare with kT for each material at room temperature? 14.3.4Use the results of Exercises 14.3.3 to evaluate eqn (14.3.9) at Q= 0 for a uniform silica particle in H20. Show how a measurement of this intensity can be used to find the mass of the particle. Does the result depend on particle shape? [Note that this procedure can be used to evaluate the molar mass of any particle, e.g. a protein in dilute solution.] 14.3.5What mixture of H20 and D20 would make the A1203 invisible to neutrons in a mixed A1203 dispersion, so that SiOz-Si02 correlations could be studied directly? (Assume ideal mixing and ignore hydration effects.) Is there an equivalent solvent mixture for light scattering? 14.3.6The refractive index, normally thought of as a ‘wave’ property, was derived in eqn (14.3.3) using a purely particle kinetic energy. See, for example, Bacon (1975) for a derivation based entirely on wave mechanics. (This is a rather striking example of wave -particle duality.) Many examples of ‘neutron optical’ experiments, such as interferometry, are given by Klein and Werner (1983). 14.3.7Satisfy yourself that the eqns. (14.3.5) and (14.3.6) are equivalent. 14.3.8(a) Show that, when B(r) depends only on Y = Irl ,eqn (14.3.8) reduces to the Debye result

s

F ( P ) = 4n [B(r)- Bm]3sin (Qr)/(Qr) dr. [Hint: Choose spherical polar coordinates centred on the particle and evaluate the anpdar integrals using the hints suggested in Exercise (13.6.3).]

698 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

(b) Use part (a) to derive the form factor for a sphere of radius a and uniform scattering amplitude density Bo: F(Q) = 3 V(BO

-

B,)[sin(Qu)

- ~u

cos(~a)]/(~a)~

where Vis the volume of the sphere. 14.3.9 Justify the Guinier plot of In I(QJ against @ for determining the radius of gyration of a polymer molecule using the series expansions for sin and cos in

eqn (14.3.11).

14.4 Structure of concentrated isotropic dispersions of spherical particles In the case of spherical particles, there is no particle orientation to be considered, and eqn (14.3.7) can be extended to take account of the correlations between particles:

where the form factors now only depend on the magnitude of Q since the particles are isotropic. If all the particles are not the same size, there will be an obvious correlation between the distance of closest approach of any pair of particles and their form factors, which depend on the particle sizes. Calculation of the ensemble average will require knowledge of all of the partial pair distribution functions g,, (compare with eqns 14.2.2-3)). This calculation is possible numerically for hard sphere potentials (see van Beurten and Vrij 1981), but becomes intractable for more general potentials. If the dispersion is monodisperse (or may be approximated as monodisperse), however, Fn(QJ no longer depends on n, so that eqn (14.4.1) reduces to

where F(QJ is the form factor of any sphere. The expression in [ ] has already been discussed at length in Section 13.7; it is the static structure factor, S ( g , for the monodisperse, isotropic dispersion. Thus the scattered intensity in this very particular case reduces to

In the case of a non-interacting (dilute) dispersion, S(QJ = 1, and this result reduces to the monodisperse form of eqn (14.3.7), as expected. In the general case, the Q + 0 limit is of special interest. Recalling the discussion at the start of this chapter, and the relation (13.7.8) between osmotic compressibility and S(O), the sign of the mean

STRUCTURE OF CONCENTRATED ISOTROPICDISPERSIONSOF SPHERICAL PARTICLES

I 699

interparticle potential may be obtained by comparing the measured scattered intensity from a concentrated suspension with that calculated from a measurement on a dilute dispersion, scaled by the concentration ratio. The scattering from the concentrated system will be lower than the scaled value if the potential of mean force is repulsive, and higher if it is attractive. S(Dfor other values of Q in a monodisperse system can readily be evaluated in a similar way. Since S(@ + 1 in dilute systems, we can use eqn (14.3.13) to write for concentrated and dilute dispersions:

so that

S(Dis given by: (14.4.6)

Figure 14.4.l(a), which shows S(@ for a number of charged latex dispersions, illustrates these features. The data were obtained from SANS intensities via eqn (14.4.3). In Fig. 14.4.l(a), the Debye length and particle size are held constant, while the dispersed volume fraction is increased from 1% to 13%. As anticipated in the presence of a repulsive interaction, the compressibility [reflected in S(O)]diminishes with increasing concentration, while the degree of structural order [as evidenced by the height of the peak in S(@] increases. Similar effects are observed when the range of the repulsion is increased (by lowering the ionic strength) at constant concentration (Fig. 14.4.1(b)).

Fig. 14.4.1 (a) Plot of S(QJ against Qfor a polystyrene latex in lop4 mol dmp3 NaCl solution, at volume fractions of: ( 0 )1%; (0) 4%; (A)8%; and (0) 13% (Goodwin et al. 1985).

700 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

Fig. 14.4.1 (b) Plot of S(QJ against Q for a polystyrene latex of 4% volume fraction, at ionic strengths corresponding to: ( 0 )5 mM NaCI; (A) 1 mM NaU, (0) deionized suspension (Goodwin et al. 1985).

The opposite effect may be seen in Fig. 14.4.2, which plots the observed SANS intensity from a non-ionic micellar dispersion, in which the short-ranged interaction between particles becomes increasingly attractive as the temperature is raised. Reference to eqn (14.4.3) shows that the scattering at high Q will become the same as

I

I

-

-

-

L

Fig. 14.4.2 Plot of scattered intensity against Q for a dispersion of non-ionic micelles at the indicated temperatures. The solid lines are fits to eqn (14.4.3) using an attractive short-ranged potential to generate S(QJ(Hayter and Zulauf 1982).

STRUCTURE OF CONCENTRATED ISOTROPICDISPERSIONSOF SPHERICAL PARTICLES

I 701

that from a non-interacting system, since S(QJ then tends to unity; this is evident in the fact that the scattering becomes independent of the temperature as Qincreases. At low on the other hand, increasing the attraction by raising the temperature leads to much higher forward scattering, since the compressibility increases. (The uppermost curve is very close to the lower consolute temperature, at which the phase transition will be accompanied by infinite compressibility). It should be emphasized that eqn (14.4.3), which is exact, is valid only under the restrictive conditions posed, namely monodisperse, spherical particles, and is usually not even a rough approximation otherwise. The reason is that we have used the condition of spherical symmetry in its derivation, and ‘slightly broken’ symmetry is usually meaningless. There is one case, however, in which an approximation with some physical validity is possible, and that is the case of a dilute dispersion of charged particles at low ionic strength. The average interparticle distance in such a dispersion is large, relative to the particle diameter, because of the long range of the potential under conditions of low screening (see Section 14.1). Since the interaction between charged spheres at a distance is identical to the interaction between equivalent point charges, and the spheres in this case will effectively never come into contact, this type of suspension is in the plasma-like limit (discussed in Section 14.2). If the charge on each sphere were independent of diameter, the diameter would not be a relevant physical length in the problem. In this limit, we may use the so-called decoupling approximation. The basis of the decoupling approximation is to assume that the correlations in eqn (14.4.1) do not depend on the particular pair of particles involved (i.e. that gmn(r)does not depend on rn, n). In turn, this means that the form factors are uncorrelated, and so they may be averaged over sizes, independently of the centre-to-centre correlations involving R,,. Denoting a size average by a horizontal bar, eqn (14.4.1) becomes in this approximation

(14.4.7) which again reduces to the size-averaged dilute result when S(QJ = 1. Analytic expressions are available (Hayter 1985) for the size-averaged form factors for sphere/ shell particles in the case of a Schulz size distribution (eqn (5.4.10)). In physical terms, the decoupling approximation is equivalent to constructing snapshots of the dispersion by the following two-step process: (i) place points in space such that their pair correlation function is the same as would be calculated for a monodisperse suspension of spheres of the average size; (ii) randomly place the actual polydisperse spheres on these points. It is easy to see that the procedure is reasonable only if there is effectively no chance of placing two spheres such that they overlap, so that it can only be attempted under conditions of low particle concentration and long ranged repulsive interaction. Even when these conditions are satisfied, it must still be remembered that the arguments leading to it are based on the assumption that the diameter plays no role. How good an assumption this is depends on how much the interparticle potential varies with the particle diameters.

702 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

Returning to our conceptual two-step construction of the dispersion, variation of potential with diameter requires that we generate the equivalent pair correlation function for a dispersion of spheres which have not only the average size, but also the average interaction. This has an interesting consequence. We first note that, by analogy with eqn (14.4.3),the decoupling approximation may be written as

I(QJ = N m S e , ( Q J

where Se,(Q)

=1

+ [S(Q)

-

1 ] m

+ m2 (14.4.8)

so that .Ye,(&)) may be obtained by measuring the intensity scattered per particle by concentrated and dilute dispersions, respectively, and taking the ratio [compare eqn

(14.4.6)]. Consider the data of Fig. 14.4.3,which shows Se,(QJ measured at two volume fractions of a dilute latex dispersion at low ionic strength. (To emphasize our earlier comments about the universality of the general scattering formalism these are in fact light scattering data, due to Brown et al. 1975.) The ionic strength and volume fractions are known experimental parameters, and the surface potential in this system was evaluated by extensive computer simulation modelling of the data (van Megan and Snook 1977). Further, the particle size distribution was obtained from electron microscopy, so that all parameters of the RMSA are known. The solid lines are direct RMSA calculations of S(QJ,assuming a monodisperse system of the mean particle size and surface potential. The data at low Qare not well predicted. These dispersions are precisely those where the decoupling approximation may be expected to work, but before applying it we must decide how to choose the mean interaction potential which will be used to calculate S(QJ for the equivalent monodisperse system in eqn (14.4.8). From Section 14.2we expect the polydisperse and equivalent monodisperse structures to correspond when the Coulomb coupling, r, is the same in the two systems. Under reduce to three choices conditions of small screening (K M 0), eqns (14.2.11)and (14.2.13) for the size- averaged coupling, depending on whether we average at constant charge density, 00, constant surface potential, $0, or constant total charge, Z: -

r

Z2

o<

$is2 [T

-

r

-

r

s

[T c(

c(

IX

22/s] c(

Constant total charge

(14.4.94

+is] Constant surface potential

(14.4.9b)

[T c( 00231

Constant chargedensity

(14.4.96)

where is the mean particle diameter. The potential used to calculate S(QJ in eqn (14.4.8)thus depends on different moments of the size distribution, depending on which aspect of the potential is held constant during the interaction. The best match to the data of Fig. 14.4.3 is obtained using eqn (14.4.96)in the decoupling approximation (dashed curve). Since there are no free parameters, this is an experimental indication that these latex systems are characterized by constant surface charge density. There are very few polydisperse systems for which the decoupling approximation is a reasonable approximation, and it should only be applied with extreme care, keeping

STRUCTURE OF CONCENTRATED ISOTROPICDISPERSIONSOF SPHERICAL PARTICLES

I 703

5

Fig. 14.4.3 S,c (@ measured by light scattering for deionized latex dispersionsat volume fractions: (a) 1.66 x lop4; (b) 4.85 x lop4 (Brown et al. 1975). The mean particle diameter is 90 nm, with a )Calculated S(@ assuming a monodisperse suspension. standard deviation of 19%. ( (- - --) calculated in the decoupling approximation at constant surface charge density. ~

in mind the conditions of its derivation. For systems which are (at least nearly) monodisperse, however, eqn (14.4.3) gives an excellent description of the scattering from a wide variety of concentrated colloidal dispersions, and allows direct observation of the dispersion structure factors (eqn 14.4.6). Mixed monodisperse colloids of different sizes provide a particular limiting case of polydispersity. In the special case of a binary mixture of colloidal particles, the reduction of eqn (14.4.1) to an analytic form analogous to eqn (14.4.3) has been given by Ashcroft and Langreth (1967). Designating the smaller particles as type 1 (diameter Sl), and the larger as type 2 (diameter 62), the result is

(14.4.10) where x is the relative number concentration of particles of type 2, F,(QJ is the form factor for a particle of typej (j= 1,2), and the Sg (QJ are the partial structure factors for correlations between particles of type i and typej, corresponding to the partial pair may be evaluated from eqns correlation functions hy(r) of eqn (14.2.1). The Sg (14.2.1-4) in the case of hard sphere potentials, using a solution due to Lebowitz

(a

704 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

(1964) (see also Baxter 1970), so that eqn (14.4.10) is fully analytic in the case of neutral colloids. The most remarkable feature of these mixed systems is the large effect which a small amount of a second component may have on the scattering from the first, especially at low Q(Fig. 14.4.4(a)). This is particularly pronounced in the case of neutral colloids, where very strong cross-correlations may develop between the sizes and positions of the particles. This is evident in Fig. 14.4.4(b), which shows that the addition of the

Q

1.2

0.8 h

0)

v

0.4

0

L

/ /-

-0.41-< 0

I

1

I

I

I

2

I

3

I

I I 4 (x 10-2)

Q Fig. 14.4.4 (a) Plot of I(QJ against Q for a 30% volume fraction dispersion of 25 nm particles ), compared with that from the same dispersion containing an added 2% volume fraction of 50 nm particles (- - --). (b) Plot of the particle structure factors corresponding to the mixture shown in (a) the 25 nm particles are labelled type 1 and the 50 nm, type 2. Sll(QJ is shown both in the presence (A) and in the absence (B) of the second component. (

~

NEUTRON REFLECTIVITY

I705

second component of Fig. 14.4.4(a) has very little effect on the correlations (Sll)which were present between particles in the original (unperturbed) structure, and that the but that strong added particles remain almost uncorrelated with one another (SZZ), correlations (S12) develop between the two different types of particle. One would expect the small particles to tend to fit into the spaces between the larger ones. We have discussed scattering techniques which are either time independent, or which observe structural features which vary on macroscopic time scales, for example under the influence of viscous shear. The extension of SANS to timedependent measurements on the microscopic (t > 1 ns) timescale has become possible in recent years using a technique known as neutron spin-echo (NSE), which is the neutron analogue of photon correlation spectroscopy. T h e interested reader is referred to Hayter (1985) for an introduction to the application of NSE to colloidal systems.

14.5 Neutron reflectivity The interference patterns formed by the reflection of visible light from thin liquid (oil) films on water are familiar to us all. The resulting spectral patterns have intrigued mankind since ancient times and the explanation of those effects was one of the early triumphs of the wave theory of light. The measurement of neutron reflectivity is comparatively recent, having come into its own only in the 1990s, but it shows great promise of being a very powerful technique. When a neutron beam is reflected at low angle, 0, from a film of thickness d, the wave trains from one surface of the film travel further than those from the other surface. The path difference is (2d sine) and the interference between the wave trains will give rise to constructive peaks at distances defined by:

2 d sin 0 = n h

(14.5.1)

where n is an integer and h is the neutron wavelength. We have alluded to this phenomenon in discussing ATR spectroscopy (Section 6.2) and in the use of thin soap films for the investigation of the interaction between double layers in Chapter 12 but in the latter case the procedure is used only to determine the actual thickness of the soap film. (Some attempts are made to correct for the composite nature of the film (to separate the soap from the water) but it is not a well-defined procedure.) The advantage of neutron reflectivity is that the wave lengths are very much smaller (by about 1OOOx) so that one can probe much thinner films, down to molecular dimensions. Also, the relation between neutron reflectivity (strictly refractive index) and composition is much more direct (eqn 14.3.4) and better understood than is the case for light. Coupling these advantages with the possibilities for index matching (Section 14.3.1) provides one of the most powerful tools for investigating the structure and conformation of adsorbed surfactants and macromolecules at interfaces. The field has been reviewed by Thomas (1999) and that work should be consulted for more details. It is important to note that the neutron refractive index calculated from eqn (14.3.4) is very near to unity. For a 1 nm wavelength, n, is 1.000 009 for H20 and 0.999 90 for

706 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

DzO. Reflection is therefore only possible at glancing incidence and much of the beam still goes into the substrate from which some neutrons will be scattered back into the detector. The background scattering is, fortunately, almost independent of Q and can be readily corrected for. Figure 14.5.1 shows the calculated reflectivity of 0.4 nm wavelength neutrons from a 2 nm (full line) and a 5 nm surfactant layer in a system where the contrast has been adjusted so that only the surfactant is visible to the

0 2 4 6 Distance normal to surfacehm Fig. 14.5.1 (a) Calculated neutron reflectivity (continuous line) for a typical 2 nm thick surfactant layer at the air-water interface (h. = 0.4 nm). The horizontal dashed line is the background scattered from the bulk solution. The dashed reflectivity profile is for a 5 nm layer of constant scattering length density (Fig. 14.5.1 (b)). The underlying liquid has been adjusted to zero reflectivity [the NRW (null reflecting water) condition]; this makes the air-water interface invisible but some scattering still occurs from the bulk water. (From Thomas 1999 with permission.)

NEUTRON REFLECTIVITY

I707

neutrons. Note the constancy of the background scattering and the interference effects for the thicker layer. T he advantage of observing the interference fringes directly is that the thickness of the layer can be determined directly from eqn (14.5.1) independently of any information about the composition of the layer. Even for the thinner (-2 nm) films the shape of the reflectivity profile can be used to estimate the ‘thickness’ of the adsorbed film but the method is unable to distinguish between different atom density profiles (say, the constant density profile of Fig. 14.5.1 and a Gaussian) until the thickness exceeds about 3 nm. The striking ability of the method to distinguish different aspects of the adsorbed layer is shown in Fig. 14.5.2. The reflectivity profile from a surface layer of fully deuterated CTAB is there compared with profiles from CTAB in which deuterium replaces hydrogen in the head group, and in the chain. In this way one can determine the thickness of the chain, the head group and the overall molecule. The values obtained (1.65, 1.4, and 1.9 nm, respectively) are far from simply additive, suggesting that the structure differs from the simplistic picture usually displayed. Thomas shows how Lu e t al. (1995) used progressive substitution of the chain with deuterium to provide a more plausible description involving the notion of ‘roughness’ of the adsorbed film, with the chains tilted from the vertical (but see Section 14.5.2). This sort of technique can obviously also be applied to study the interaction between a surfactant and its co- surfactant or between two different surfactants using differential deuterium labelling.

Fig. 14.5.2 Neutron reflectivity profiles of three labelled cetyl trimethylammonium bromides:(0) [C16D33N(CD3)31+Brp; (+) [C16D33N(CH3)31+Brp; and (X) [C16H33N(CD3)31+ Brp. All measurements done in NRW with the surfactants at their c.m.c. (Data from Lu et al. 1994).

708 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

14.5.1 Reflectivity theory The neutron reflectivity profile from a flat surface is given approximately by (Crowley 1993):

(14.5.2) where, in this case, the factor, F, is the Fourier transform of the mean scattering length density distribution perpendicular to the interface (compare eqn 14.3.7):

1

00

F(Q) =

B(z) exp(-zQz)dz.

(14.5.3)

-ca

For a uniform monolayer of surfactant (s) between media 1 and 2 this becomes:

R = [16d/Q4]{(&

-

+

B I ) ~ (B2 - B,)2

(14.5.4)

+2(B, - Bl)(B2 - B,) cos Qd}

where d is the layer thickness. If the two media are matched by deuterium substitution to the same refractive index (B1 = B2 = B) this reduces to (Exercise 14.5.1):

R

= {[8n/Q2](Bs - B)~in(Qd/2)}~

(14.5.5)

which represents a set of regular interference fringes of intensity (B, - B)’ superimposed on a signal which decays very quickly as Q increases. This is the behaviour displayed in Figure 14.5.1. Rearranging eqn (14.5.5) into the form:

Q2R = 16dd2(B, - B)2y

o

1

2

(14.5.6)

gives the limiting value:

lim Q2R= 16dd2(B, - B)2.

Furthermore, at the air/NRW surface

Q-0

where

B = 0 lim Q 2 R = 16dd2b2p2= 16db2r2

(14.5.7)

Q-0

with b the scattering amplitude and p the number density of atoms; r is, as before, the surface coverage. This result makes it possible to derive the surface coverage independently of the thickness, by extrapolating the reflectivity function @R to Q= 0. The theory behind the analysis of the deuterated CTAB example shown in Fig. 14.5.2 follows along similar lines. Consider a molecule which is made up of two parts, A and B which can be separately labelled. The molecule is assumed to be adsorbed at the water (W) surface. The scattering length density profile along the normal to the surface can be written:

NEUTRON REFLECTIVITY

I709

and substituting this into eqn (14.5.2) gives, for the reflectivity (Thomas 1999):

where the h factors are the self and partial structure factors, defined by:

hii = IF(Q)I2 and

hi; = Re[Fi(Q).F;(Q)]

(14.5.10)

and, again, the F factors are the one-dimensional Fourier transforms of the number densities along the normal to the surface. Then the reflectivities of each of the labelled species: dA-OB, OA-dB and dA-dB at the NRW surface will be:

respectively. Measurement of these three reflectivities can therefore give all the necessary information to recover the Fourier transform of the scattering length density for each sample. One can then deduce the actual density profile with a minimum of uncertainty. Thomas (1999) discusses a number of aspects of surfactant systems, including some rather surprising effects of certain large counterions (tetramethyl ammonium and p-toluene sulphonate), at the water-air interface and the reader is referred to that work for details.

14.5.2 Application to the solid-liquid interface Many solids are sufficiently transparent to neutrons to make it possible to measure reflectivities at the solid-water interface, with the beam approaching the surface through the solid. The most commonly used systems are silicon and silica (as both quartz and in the amorphous form). Unfortunately, the mica surface, which is so useful for providing an atomically flat surface, is not so suitable because it strongly diffracts a neutron beam which is at glancing angle to the basal plane. The scattering length densities of silica, quartz and amorphous silica are 2.1, 4.2, and 3.4 x lop4 nmP2 respectively and this means that different reflectivity patterns are obtained for a given surfactant, depending on whether or not it is deuterated and whether the study is done in water or D20. Figure 14.5.3 shows, for example, the reflectivity of C&,j on the quartz surface.+The thickness measurement (derived from the interference pattern) shows that a bilayer is involved but more detailed analysis [Lee et al. (1989) and McDermott et al. (1995)l shows that it is an incomplete layer containing a considerable amount of D20. This picture of flattened micelles is in accord with measurements using the atomic force microscope in non-contact mode (Ducker and Grant 1996). C& is a surfactant consisting of a twelve unit hydrocarbon chain with 6 ethylene oxide residues attached. The latter is the hydrophilic head group.

710 I

14: SCATTERING STUDIES OF COLLOID STRUCTURE

Fig. 14.5.3 Reflectivities of C12E6 on the bare quartz surface. The compositions are: (0) hC12hE6 in D20; (O)dCl~hE6in water, and (+)hC,zhE6 in water. The continuous lines are calculated for a bilayer 4.3 nm thick. The points at a reflectivity of lo-’ correspond to negative values after background subtraction and should be taken to indicate that the reflectivity is lower than about 2 x lo-’. (From Thomas 1999, with permission.)

Fragneto et al. (1996) interpret their measurements on partially labelled C16TAB molecules as evidence that the adsorption occurs as a bilayer. The series of isotopic species represented as C16-,d&dTAB with m = 4,8, and 12 give distinct reflectivity profiles which can be fitted well with a bilayer model. Although this is the commonly assumed model for the adsorbate above the c.m.c, some recent studies by Schulz et al. (1999) using the atomic force microscope (Section 6.2.2) indicate that TTAB (the C14 analogue) adsorbs on quartz as spheres or cylindrical micelles depending on the salt concentration, just as it does in bulk. Their neutron reflectivity measurements on the same system were consistent with the micellar arrangement and not with a bilayer, though they could not distinguish unequivocally between spheres and cylinders. Interestingly, when they adsorbed didodecyldimethylammoniumbromide on quartz, it formed a bilayer, just as it does in bulk, and this was evident in both the AFM and the neutron reflectivity measurements. The silica surface can readily be made hydrophobic by treatment with alkyl trichlorosilanes. Adsorption of surfactants is then expected to occur as a monolayer with the hydrophobic chain down on the surface and the head group facing the water. For most simple surfactants that yields a layer which is rather too thin for the usual estimation of thickness using the interference pattern. Thomas (1999) shows how this drawback can be overcome, at the cost of some increase in complexity, by studying the behaviour of films made with successive layers of surfactants of different type. He also discusses the use of special reagents for creating and studying surface adsorbed layers with unique functionality, designed to act as specific adsorption substrates for other species.

NEUTRON REFLECTIVITY

Exercise 14.5.1 Establish eqn (14.5.5).

References Ashcroft, N.W. and Langreth, D.C. (1967). Phys. Rev. 156,685. Atkins, P.W. (1982). Physical chemistry 2nd (student) edn,. p. 755. Oxford University Press, Oxford. Bacon, G.E. (1975). Neutron dzffraction (3rd edn). Clarendon Press, Oxford. Baus, M. and Hansen, J.-P. (1980). Phys. Rep. 59, 1. Baxter, R.J. (1970). J. Chem. Phys. 52, 4559. Blum, L. and Torruella, A.J. (1972).j? Chem. Phys. 56,303 (and references therein). Blum, L. and W y e , J.S. (1977).J. Phys. Chem. 81, 1311. Brown, J.C., Pusey, P.N., Goodwin, J.W., and Ottewill, R.H. (1975).J Phys. A, 8, 664. Cebula D.J., Thomas, R.K Harris, N.M, Tabony, J., and White J.W. (1978). Faraday Soc. Disc. 65,7691 Clark, N.A., Ackerson, B.J., and Hurd, A.J. (1983). Phys. Rev. Lett. 50, 1459. Crowley,T.L. (1993). Physica A, 195, 354. Ducker, W. and Grant, L.M. (1996). J. Phys. Chem. 100, 3207. Fragneto, G., Thomas, R.K., Rennie, A.R., and Penfold, J. (1996). Langmuir 12, 6036. Frisch, H.L. and Lebowitz, J.L. (1964). The equilibrium theory of classicalfluids. Benjamin, New York. Goodwin, J.W., Ottewill, R.H., Owens, S.M., Richardson, R.A., and Hayter, J. B. (1985). Makromol. chem. Suppl. l O / l l , 499. Hansen, J.-P. and Hayter, J.B. (1982). Mol. Phys. 46,651. Harris, N.M., Ottewill, R.H. and White, J.W. (1983). In Adsorptionfrom Solution (R.H. Ottewill, C.H. Rochester, and A.L. Smith (eds) 1982 Symposium Proc., pp.139-154. Academic Press, London. Hayter, J.B. (1983). Faraday Discuss. Chem. Soc. 76, 7. This volume is devoted to concentrated colloidal dispersions and contains many papers of interest. Hayter, J.B. (1985). In Physics of amphiphiles: micelles, vesicles and microemulsions (ed. V. Degiorgio and M. Corti), p. 59. North Holland, Amsterdam. Hayter, J.B. and Penfold, J. (1981~).Mol. Phys. 42, 109. Hayter, J.B. and Penfold, J. (1981b).J. Chem. SOL.Faraday Trans I , 77, 1851. Hayter, J.B. and Zulauf, M. (1982). Colloid Polymer Sci. 260, 1023. Klein, A.G. and Werner, S.A. (1983). Rept. Prog. Phys. 46, 259. Lebowitz, J.L. (1964). Phys. Rev. A, 133, 895. Lebowitz, J.L. and Lieb, E.H. (1969). Phys. Rev. Lett. 22, 631. Lee, E.M., Thomas, R.K., C u m i n s , P.G., Staples, E.J., Penfold, J. and Rennie, A.R. (1989). Chem.Phys. Letters 162, 196. Livsey, I. and Ottewill, R.H. (1991) Adv. Colloid Interface Sci. 36, 173-184. Lovesey, S.W. (1977). In Dynamics of solids and liquids by neutron scattering (ed. S. W. Lovesey and T. Springer), Chapter 1. Springer, New York. Lovesey, S.W. (1984). Theory of neutron scattering from condensed matter. Clarendon Press, Oxford.

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14: SCATTERING STUDIES OF COLLOID STRUCTURE

Lu, J.R., Hromadova, M., Simister, E.A., Thomas, R.K., and Penfold, J. (1994).J. Phys. Chem. 98, 11519. Lu, J.R., Li, Z.X., Smallwood,J., Thomas, R.K. and Penfold, J. (1995)J. phys. Chem. 99, 8233. Markovi;, I. and Ottewill, R.H. (1986). Colloid and Polymer Sci. 264, 65-76. Medina-Noyola, M. and McQuarrie, D.A. (1980). J. Chem. Phys. 73, 6279. McDermott, D.C., Lu, J.R., Lee, E.M., Thomas, R.K., and Rennie, A.R. (1992). Langmuir 8, 1204. Naegele, G., Klein, R, and Medina-Noyola, M. (1985).J. Chem. Phys. 83,2560. Ornstein, O.S. and Zernike, F. (1914). Proc. K. Ned. Acad. Wet. 17,793. (Reprinted in Frisch and Lebowitz, 1964.) Schulz, J.C., Warr, G.G., Butler, P.D., and Hami1ton.W.A. (1999). Adsorbed layer structure of cationic surfactants on quartz. Submitted to Phys. Rev. Let. Thomas, R.K. (1999). Neutron reflectivity at interfaces” In Modern characterization methods of surfactant systems (ed. B.P. Binks). Marcel Dekker, New York. van Beurten, P. and Vrij, A. (1981).J. Chem. Phys. 74,2744. van Megan, W. and Snook, I.K. (1977). 3. Chem. Phys. 66, 813. Verwey, E.J.W. and Overbeek, J. Th. G. (1948). Theory of stability of lyophobic colloids, pp. 205. Elsevier, Amsterdam.

Rheology of Colloidal Dispersions 15.1 Introduction 15.2 Behaviour of time-independent inelastic fluids 15.2.1 Bingham plastic behaviour 15.2.2 Pseudoplastic behaviour 15.2.3 Dilatant behaviour 15.3 Behaviour of time-dependent inelastic fluids 15.4 Visco-elastic fluids 15.4.1 Macroscopic flow behaviour of elastic fluids 15.4.2 Describing the dynamic (oscillatory) behaviour of visco-elastic fluids 15.4.3 Steady shear flow of visco-elastic fluids 15.4.4 Comparison between dynamic and steady shear flow properties of visco-elastic fluids 15.5 Measurement of rheological properties of inelastic fluids in Couette flow 15.5.1 The stress-strain rate relationship 15.5.2 Sources of error in the Couette viscometer (a) End effects (b) Wall slip effects (c) Temperature control (d) Taylor vortex development 15.6 Capillary viscometer 15.6.1 Flow rate versus pressure drop 15.7 Cone and plate or cone and cone viscometer 15.8 Time-dependent inelastic behaviour 15.9 Microrheology 15.10 Microscopic basis of rheological models 15.10.1 Flow behaviour of a dispersion of hard spheres 15.10.2 Flow of systems with anisometric particles 15.10.3 Kinetic interpretation of non-Newtonian flow 15.10.4 Flow of coagulated colloidal sols 15.10.5 Time-dependent systems: kinetic interpretation of thixotropy 15.10.6 Elastic behaviour of concentrated sols

713

714 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

15.1 Introduction Rheology is the study of the deformation and flow of materials under the influence of an applied stress. Some of the basic elements of this study were introduced in Section 3.4. Colloidal materials, especially gels and pastes, commonly exhibit both solid-like (elastic) and liquid-like (viscous) behaviour and a complete treatment ought to take both such aspects into account simultaneously. Unfortunately, the theoretical analysis of such materials requires a rather extensive array of mathematical tools and is, even so, restricted in most cases to the limit of small strains and strain rates (the region of linear visco-elastic theory). In order to make any real progress towards the solution of practical problems it will, therefore, be necessary to adopt a very pragmatic approach. In some cases the theoretical constructs will guide us in the choice of suitable measurement procedures whilst in other cases the phenomenology may be so complex that the best one can hope for is to be able to provide some understanding of the macroscopic behaviour in terms of what can be assumed to be occurring at the microscopic (particleparticle interaction) level. We have already seen (Section 4.10.4) that a considerable departure from Newtonian liquid behaviour is expected to occur when anisometric particles are immersed in a Newtonian liquid. Such behaviour is predicted (Fig. 4.10.1) even when the particles are in such dilute suspension that their mutual interactions may be ignored. How much more complex, then, is the range of anticipated behaviour when the particles can interact with one another. For modest concentrations of spherical (or near spherical) particles whose long range van der Waals attraction has been suppressed, the effect is principally on the viscous behaviour (Fig. 3.4.9) and such systems show little elasticity. As the number of particle contacts increases, however, so that contact is essentially continuous, the colloidal sol begins to take on the properties of a gel; its elasticity then increases markedly and it begins to behave as a (possibly rather easily distorted) solid. For extremely anisometric particles (e.g. the long, thin rods of the clay mineral attapulgite, vanadium pentoxide, or laponite (synthetic aluminium silicate)) such gel formation can occur at very low particle concentrations (< lob). If van der Waals attraction is allowed to dominate over repulsive forces, the systems can show gel-like and elastic properties even if the primary particles are spherical; long chains of spheres are probably involved in the gel structure in that case. Linear visco-elastic theory, appropriate to the treatment of small strains, has been widely applied to the behaviour of polymer solutions and has yielded some useful insights. In the treatment given here we will make some reference to that area but will concentrate most attention on the behaviour of suspensions undergoing continuous strain (i.e. flow) where elastic effects, though important, play a secondary role in many respects. Even with this limitation, the difficulties of providing a fundamental theoretical base remain formidable and we shall often have to settle for a semiempirical approach. The first problem which must be addressed is the question of how to collect unambiguous rheological information. Once one introduces the notion of a variable viscosity, the discussion of viscometer devices in Section 4.7 is no longer adequate. For time-independent systems in which the viscosity, q, depends only on the shear rate,

BEHAVIOUR OF TIME-INDEPENDENT IN EL A ST IC F L U I D S

I715

the Couette viscometer can give reliable data if the gap width is small (recall Exercise 4.7.4) but for the Ostwald (capillary) viscometer, in which the shear rate necessarily varies from zero in the centre to a maximum at the wall, an alternative approach is necessary. We will discuss the usual approach to such problems in Section 15.6.

15.2 Behaviour of time-independent inelastic fluids We noted in Section 3.4.1 that any system will exhibit elastic behaviour if it is examined on a sufficiently short time scale, and likewise will show flow behaviour over sufficiently long times. Nevertheless, it is convenient to distinguish those systems which do not show any appreciable elastic behaviour and for which there is essentially no time dependence in their viscosity, at least on the time scale of the experiment. These are characterized by low Deborah number, & (eqn (3.4.1)); that is, their characteristic relaxation time is short compared to the times involved in the experiment. The Newtonian fluid falls into this category but so, too, do a lot of colloidal systems which are decidedly non-Newtonian, and it is often the departure from Newtonian behaviour which is the important and valuable characteristic of the colloidal suspension. There are also many systems in which the material behaves like an elastic solid for sufficiently small stresses but once some critical stress is exceeded the material ‘yields’ and the resulting flow behaviour may exhibit very little elasticity. For present purposes we will treat such systems as inelastic since it is the region above the yield value (SO) which is normally of most interest. In Chapter 4 we developed in some detail the equations for the viscous behaviour of a Newtonian liquid, which is characterized by a proportionality between the shearing stress and the consequent rate of strain. In the most general flow situation this is expressed in the tensor equation:

[4.5.5] where D: is the deviatoric stress tensor, r] is the viscosity, and eq is the rate of strain tensor. For the simple flow regimes with which we will be primarily concerned here (Fig. 15.2.1) the shearing stress is applied in only one direction and the observed flow occurs in the same direction. In such cases a set of coordinates can be chosen so that the velocity field takes the simple form (Boger et al. 1980): u1 =f(coordinate 2 only);

uz = u3 = 0

(15.2.1)

where subscript 1 indicates the direction of flow and 2 is at right angles to that direction and also to the boundary surface. The more common flow regimes of this sort are illustrated in Fig. 15.2.2, and they are all exploited in different forms of viscometer; they are, therefore, referred to as vzscometrzcflows. The general relation between the deviatoric stress tensor, u:, and the rate of strain tensor, eij, can then be greatly simplified. For an inelastic fluid, in these simple flow conditions, the normal stresses ( ~ i i ) are , all equal to the local hydrodynamic pressure, so that 011

= 0 2 2 = 0 3 3 = -p

(15.2.2)

716 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

and the only non-zero components of the shearing stress are q equal) so that

2

and a21 (which are also

(1 5.2.3)

where S2l is the externally applied stress. This was the situation treated in Section 4.7. Similarly, the only non-zero component of the rate of strain tensor is el2 where:

2e12

avl =-+-. ax2

avz axl

(1 5.2.4)

In these viscometric flows, it is conventional to replace 2e12 by the rate of strain, j which is also a function of only one coordinate (though it does not appear so when expressed in terms of Cartesian coordinates as in eqn (15.2.4)). We can, therefore, write:

where S (= Sl2 = S21) is the externally applied stress. Whereas in Section 4.7 we regarded q as a constant (as it is for a Newtonian fluid) we must now explore the possibilities when it depends on shear rate. Typical shear rate dependences were depicted in Fig. 3.4.9. When defined in this way, q ( j ) is often referred to as the apparent viscosity, to distinguish it from the dzfferential viscosity (dS/ d j ) . Though it is the apparent viscosity which emerges as the more important characterizing parameter for the liquid, we will sometimes find it easier to develop a model description of the macroscopic behaviour using the differential viscosity. We are now in a position to discuss the general phenomenology of time independent inelastic fluids. We begin by describing the empirical relationships which are used to represent the shear stressshear rate behaviour. It is usually necessary to have some (preferably model independent) representation of the behaviour before one can extract from the experimental data an S versus j plot (usually called a rheogram or basic shear diagram).

Fig. 15.2.1 Stress components in a simple shear flow.

BEHAVIOUR OF TIME-INDEPENDENT IN EL A ST IC F L U I D S

I717

Fig. 15.2.2 (a) Poiseuille flow in a tube. (b) Couette flow in the annulus between two cylinders. (c) The cone and plate configuration. Note that Om, is normally only a few degrees (or less) so that the system is much more like two parallel plates. (From Boger et al. (1980) with permission.)

15.2.1 Bingham plastic behaviour As noted in Section 3.4, this type of material behaves as a solid for small applied stresses (S< SB)and then flows with a constant differential viscosity ( q p ~= d S / d j ) for higher shear stresses (curve 4 of Fig. 3.4.9). SBis called the Bingham yield value and: S =S B

+

VPL j.

[3.4.23]

Concentrated slurries of coal dispersed in water exhibit this type of behaviour (Rowel1 1983 personal communication) essentially exactly, as do some food products (Halmos and Tiu 1981). It is of more general significance because many engineering designs are based on this expression since the departure of the system from this ‘ideal’ behaviour is often sufficiently small to be neglected. It is, however, more usual to observe a non-linear relation between shear stress and shear rate for shear stresses above the yield value. The simplest model used to fit such systems is the Herschel-Bulkley model (Exercise 15.2.1):

S=So+Kj”

(1 5.2.6)

718 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

where So is referred to as the primary yield value. This behaviour would be detected by a linear plot of log (S- So) against log exhibiting a slope, n, different from unity. This is, of course, purely an empirical relation which has no underlying theoretical base. It is, therefore, unwise to use it outside the region over which it has been established experimentally. Th e physical behaviour of these fluids can be understood in terms of a three dimensional structure which has sufficient strength to prevent flow if the applied stress is less than So. For S > So the structure may collapse suddenly to produce flow units which are not subsequently affected by the shear regime. This would give rise to ideal Bingham plastic flow and then So = SB. More usually the breakdown is progressive and the flow units become smaller, and/ or more compact, and/or more closely aligned with the stream lines as the shear rate increases: the result in any case is a decrease in differential viscosity (dS/dy) with increasing shear rate. An intermediate behaviour pattern has been observed in the flow of coagulated dispersions of small particle size (< 1 pm) (Hunter 1982). Such systems show a non-linear S versus p relation at low shear rates (1-500 s-l) which becomes strictly linear at higher shear rates (500 - 3000 s-'); (see Section 15.10.4). This is, however, a relatively restricted range of p and the behaviour outside that range has not been well characterized. At very low shear rates (< 1 s-l) the shear stress is very low but it may not be zero when = 0 (curve 5 of Fig. 3.4.9). T h e intercept on the stress axis from the linear part of the curve at high shear rate (SBin Fig. 3.4.9) is again called the Bingham yield value since eqn (3.4.23) above is obeyed for high values of p (> 500 s-'). Yield strength is a very important characteristic of such widely different materials as oil-well drilling muds, paint, toothpaste, paper pulp, crude oil and many food products, pharmaceutical preparations, and cosmetics.

-

15.2.2. Pseudoplastic behaviour For substances which show a negligible yield value (So + 0) but a varying differential viscosity, it is often possible to represent the behaviour by a power law relation:

S=Kj."

(15.2.7)

known as the Ostwald-de Waele model (Skelland 1967). Obviously the HerschelBulkley eqn (15.2.6) is a simple extension of this expression. The main advantage of eqn (15.2.7) lies in its simple two-parameter form and the ease with which it can be differentiated and integrated. Even if it applies only over a limited range of 9, it can be used to extract meaningful rheological data from viscometric flow situations provided that it holds for all shear rates being experienced by the material. Once the S vs. p data have been acquired they can then be fitted with a more elaborate expression. There are a number of two, three, and even four parameter expressions which have been developed on the basis of more or less elaborate microscopic models. Some of them are listed in Table 15.1, which is modified from the compilation provided by Skelland (1967).

BEHAVIOUR OF TIME-INDEPENDENT IN EL A ST IC F L U I D S

I719

Table 15.I Models relating shear stress

(S)to shear rate

y, forJuids without a yield stress.

Model

Equation

Empirical parameters

Power Law or Ostwald-de Waele"

S=Ky"

Bingham

s = S B + VPL?

Ellisb

s/Y = rl = ao/[l+ ( S / S ~ ) ~ - ~ I

de Haven (1959)

Sly = 9 = qo/[l

Prandtl-Eyring

S = A arcsinh [ y / B ]

+ Cs"]

Adapted from Skelland (1967). "Ostwald (1926), Reiner (1949); bReiner (1960); 'Eyring (1936), Prandtl(l928); dChristiansenet al. (1955); eBoger et al. (1980), Meter (1964).

T he more elaborate Ellis and Meter models recognize the fact that the shear stress-shear rate curve is often linear at very low and again at very high shear rates so that it is possible to define two limiting viscosities:

lim

S

Y+O

and

y

= qo = zero shear viscosity

. s= roo= infinite shear viscosity Y+m y

(15.2.8)

lim

and the apparent viscosity falls smoothly from ~0 to roo as p increases. Since the dimensionless parameter a in these models is often close to 2, the parameter S; can be roughly identified as the shear stress at which ~0 has fallen half way to its final value. Figure 15.2.3 shows how the introduction of these extra parameters improves the fit for an extensive range of viscosity data compiled by Boger (1977). Both of these models have a form similar to that used by Krieger (1972), to which we alluded in Section 4.10.8 (Exercise 15.2.2):

~

r-roo = - roo

ro

[

1+-

71-l

(15.2.9)

where r] is the apparent viscosity. Note that this corresponds to a = 2 in the Ellis and Meter models. Boger found a = 2.43 to give the best fit to his data but no doubt a reasonable fit could be obtained using a = 2, which reduces the Meter model to only three parameters. An alternative representation can be given in terms of the shear rate (Carreau 1968): (1 5.2.10)

720 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Meter model \ \

1u--

1U”

10-

Shear rate

Ellis ‘,model

IU’

(s-I)

Fig. 15.2.3 Viscosity as a function of shear rate for a poly(acry1ic acid) solution in water (from Boger 1977, with permission.) Note that only the Meter model fits the data over the entire range of shear rates.

where h is an empirical (time) parameter. This has some advantages since, in the normal Couette viscometer in which these systems are studied, it is the shear rate rather than the shear stress which is the independent variable. The de Haven model is obviously identical to the Ellis model with appropriate relations between the parameters (Exercise 15.2.2). The Eyring models are based on the concept of activated transport and are more likely to be relevant to the viscous behaviour of molecular fluids, including (possibly) dissolved polymers, but they have been used in the description of the visco-elastic behaviour of latex suspensions at very low electrolyte concentrations (Goodwin et al. 1982). We will return to discuss the microscopic origins of these sorts of expressions in Section 15.10.

15.2.3 Dilatant behaviour The term ‘dilatant’ refers to the fact that when a system, consisting of irregularly shaped particles of solid in a liquid, is sheared it increases in volume. It is also observed that the resistance to shear increases (usually fairly dramatically) with increase in shearing rate (curve 3 of Fig. 3.4.9). Thus there is in this case a coincidence of volume expansion and an increased viscosity with shear rate. There is, however, no necessary connection between these two phenomena and it is possible for each effect to exist in the absence of the other. The increase in viscosity with shear rate should, therefore, be referred to as rheologzcal dilatancy to be more precise. It is usually attributed to the fact that in these systems the liquid is effectively acting as a lubricant between the particles. If one attempts to shear too quickly the particles are pushed more closely together in some regions (though separated in other regions). The overall effect is to reduce the free movement of the fluid and make the whole system more resistant to shear. Dilatant behaviour is not as common as pseudoplasticity and is usually confined to certain ranges of particle size and concentration. It is most common for high particle

BEHAVIOUR OF T I M E - D E P E N D E N T I N E L A S T I CF L U I D S

I721

concentrations and is often reversible. The dependence of S on y can usually be represented by the Power Law (eqn (15.2.7)) but with n > 1. Despite its technological significance, dilatancy has not been studied in anything like the detail of pseudoplasticity (and thixotropy) and we will not examine it further here. r

Exercises 15.2.1 The following shear stress-shear rate data were obtained on a meat extract at 77 "C (Boger et al. 1980). (The first element of each number pair is shear rate (s-l) and the second is shear stress (N m-2).): 0.111, 18.04; 0.140, 18.32; 0.176, 18.65; 0.222, 19.09; 0.279, 19.62; 0.352, 20.31; 0.443, 21.16; 0.557, 22.24; 0.702, 23.60; 0.883, 25.30; 1.112, 27.45; 1.400, 30.16; 1.762, 33.56; 2.218, 37.85; 2.793; 43.25; 3.516, 50.05; 4.426, 58.60; 5.576, 69.41; 7.015, 82.94; 8.831, 100.01; 11.117, 121.50; 14.00, 148.60. Plot S versus 1; on linear paper and on log-log paper and hence determine the characteristics of this fluid in terms of eqns (15.2.6) and (15.2.7). 15.2.2 Show that the Meter model ofTable 15.1 corresponds to eqn (15.2.9) with a! = 2. What are the relations between the de Haven and Ellis parameters?Why does the Ellis model break down at very high shear rates? 15.2.3 The following shear stress-shear rate data were obtained for an aqueous solution of methyl cellulose at 18 "C (Boger et al. 1980). (The first figure is shear rate (s-') and the second is shear stress (N mP2).) 0.1400, 0.117; 0.1762, 0.141; 0.2218, 0.169; 0.2793, 0.211; 0.3516, 0.281; 0.4426, 0.352; 0.5572, 0.446; 0.7015, 0.563; 0.8831, 0.687; 1.1117, 0.847; 1.400, 1.076; 1.762, 1.305; 2.218, 1.625; 2.793, 2.010; 3.516, 2.53; 4.426,3.08; 5.572,3.79; 7.015,4.68; 8.831,5.41; 11.117,6.53; 14.000,8.11; 17.620, 9.46; 22.18, 11.49; 27.93, 13.52; 35.16, 16.22; 44.26, 18.92; 55.72, 22.10; 70.15, 26.13; 88.31, 30.00; 111.17, 34.80; 140.00, 40.0; 176.2, 45.7; 221.8, 52.5; 279.3, 57.6. See how well they fit a power law (log-log) plot over limited ranges of shear rate. State the shear ranges over which a reasonable straight line fit is obtained and estimate the parameters K and n applicable to those regions. 15.2.4 Recast the Ellis model in the form 1; = a(S bSm)and show that it gives a good fit to the data of Exercise 15.2.3 over the whole range of 1; using the parameters ~0 = 0.794 N mP2 s, S i = 21.554 N mP2 and a! = 2.027. Compare this with the fit using eqn (15.2.9) with a suitable choice of roo.(That is the Meter model with a! = 2.)

+

15.3 Behaviour of time-dependent inelastic fluids It sometimes happens that the apparent viscosity of a suspension depends on both the shear rate and the time for which the system has been sheared. This is an indication of the fact that the relaxation time for the material is of the same order as

722 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

the time over which the system is being studied. In other words, the ‘structure’ in the system, which determines the variable shear viscosity, is being altered at a rate which is observable over the time period of the measurement. As we noted earlier, it is always possible to encounter this sort of phenomenon if one chooses the right time scale but it is sometimes very difficult to avoid. If the system has a short relaxation time (say
Shear rate (f)

Fig. 15.3.1 (a) Relation between shear stress and shear rate for a thixotropicfluid. t is the time taken between successive changes in the shear rate. The material is assumed to have a relaxation time of about 10-20 seconds and it is assumed that after each successive change of shear rate, the shear stress is measured after an interval o f t seconds and the resulting points are joined by a smooth curve.

BEHAVIOUR OF T I M E - D E P E N D E N T I N E L A S T I CF L U I D S

0

I723

Time

Fig. 15.3.1 (b) The dependence of shear stress on time at a given shear rate for a thixotropic material.

Figure 15.3.2 also shows the expected behaviour of a negatively thixotropic material. This phenomenon is not common in particulate systems. It occurs more readily in polymer systems, however, where rapid shearing can induce considerable entanglement. Such systems will usually relax back to a lower viscosity when left undisturbed as the shear induced ‘structure’ breaks down due to thermal motions of the polymer. Thixotropy has been observed in crude oils, in bentonite (montmorillonite) clays (Section 1.4.5), and in some food, pharmaceuticals, and cosmetic products. It is also a common and increasingly important characteristic of paints. In engineering situations (e.g. pipe-line flow) the thixotropy can often be ignored since its effects become less important if the system is undergoing continuous shear. It

Shear rate (f)

Fig. 15.3.2 Hysteresis loops for thixotropic fluids (both positive and negative).

724 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

is, however, important in the starting up phase of a pipeline flow. (See Govier and Aziz 1972 for further discussion of this point.) Obviously, the collection of meaningful rheological parameters in thixotropic materials is a difficult task. At the very least it is important to ensure as far as possible that these parameters are measured on a sample which has been subjected to a shear and thermal history which is appropriate to the problem being studied. If, for example, one is concerned with the slow flow of a suspension from a storage tank into a pipe one would examine the suspension after storage under zero shear conditions, making sure that the sample was not sheared during transfer to the measuring apparatus. If, on the other hand, the material was to be pumped from the storage tank, the infinite shear or steady state (long shear time) data would be more appropriate.

15.4 Visco-elastic fluids 15.4.1 Macroscopic flow behaviour of elastic fluids Fluids which have some elastic character exhibit a number of quite dramatic flow effects. Indeed, in several respects their behaviour is opposite to that of a purely viscous material. When, for example, a viscoelastic material is stirred with a rod (Fig. 15.4.l(a)) the fluid moves towards the rod and begins to climb up it, in contrast to the viscous liquid which tends to form a vortex. T h e same effect (called the ‘Weissenberg effect’) is observed in Couette flow: the viscoelastic fluid tends to accumulate near the inner cylinder rather than being thrown towards the outer cylinder, as occurs for a purely viscous fluid at sufficiently high rotational speeds. When a stream of viscous fluid emerges from a vertical pipe its diameter tends to be smaller than that of the pipe, whereas a viscoelastic fluid in the same circumstances has a larger diameter than the pipe (Fig. 15.4.1(c)). (This phenomenon is known as die swell since it often occurs when polymeric solutions or melts or colloidal suspensions or pastes are extruded from a die.) When a disc is rotated in a viscoelastic fluid the entire flow pattern is opposite to that exhibited by a Newtonian fluid (Figs 15.4.l(d) and (e)). T h e same is true of the flow pattern near a cylinder oscillating in the fluid. Other effects include the development of vortices when the fluid flows from a wide tube into a narrow tube, even when the flow is slow. An elastic fluid can even exhibit the ‘tubeless syphon’ effect, in which the syphon continues to function when the upstream end is removed some little distance from the liquid (Fig. 15.4.1(0). One can observe such elastic behaviour when pouring hair shampoo and other surfactant systems, but some polymer systems of very high molar mass exhibit these effects to a quite extraordinary degree. They have, therefore, been studied in considerable detail and have generated a vast literature. Photographs and discussions of these effects (and others) are given in Lodge 1964; Fredrickson 1964; Walters 1975; Huilgoll975; Han 1976; and a particularly clear discussion is given by Bird et al. 1977. See also the brief review article by Bird and Curtiss (1984).

VISCO-ELASTIC F L U I D S

Newtonian

Viscoelastic

u Newtonian

I725

Viscoelastic

Fig. 15.4.1 Flow behaviour of visco-elastic fluids. (After Bird and Curtiss 1984.) (a) The Weissenbergeffect; (b) Couette flow; (c) Die swell; (d) and (e) influence of a rotating disc on the flow pattern; and (f) the open channel syphon.

15.4.2 Describing the dynamic (oscillatory) behaviour of visco-elastic fluids We described in Section 3.4.4 some of the mathematical techniques which are used to characterize a visco-elastic substance which is being subjected to an oscillating shear field. In particular, we noted that it was possible to define a complex viscosity function:

726 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

q* = q’

+ iq“

[3.4.16]

where q’ is called the dynamic viscosity and is a function of frequency, w , in the same way as the steady shear viscosity, q, is a function of shear rate. Recall that from eqn (3.4.17)for a linear visco-elastic material, q’ = G”/w where G” is called the loss modulus of the material. T he imaginary part of q*, (q”) measures the elastic response of the material and we will see that it is related to the normal stress differences which give rise to the Weissenberg effects and the other peculiar manifestations of visco-elastic flow. It is also related to the storage (or dynamic rigidity) modulus, G’ by eqn (3.4.17):~” G‘/w.

15.4.3 Steady shear flow of visco-elastic fluids In a simple (viscometric) shear flow, such as is shown in Fig. 15.2.1, eqn (15.2.2)is no longer valid for a visco-elastic fluid. We noted in Section 4.5 that a completely general specification of the stress tensor requires a large number (actually 54) of unknown constants but for a simple isotropic fluid this reduces to just one: the shear viscosity, q. The question is then: how much more complicated must we make the stress tensor to adequately describe a visco-elastic fluid undergoing a viscometric flow? It turns out that the observed behaviour can be explained by postulating that the flow regime gives rise to dzfferent normal stresses in each of the three directions so that the stress tensor is now given by

and the deviatoric stress tensor is: (compare eqn 15.2.3):

! I

=

s12

(1 5.4.2)

P22

0

P33

[Note that the normal stress, p l l , p 2 2 , p 3 3 , is opposite in direction to the hydrodynamic pressure. It is a tensile stress when it is positive and a compressive stress when negative.] The hydrostatic pressure, p , is still given by eqn (4.5.1) and, hence, it follows that PI1

+ + P22

p33

= 0.

(15.4.3)

Thus, only two of the normal stresses are independent. The flow of an incompressible visco-elastic fluid can therefore be characterized by a measurement of the shear stress and two of the three deviatoric normal stresses. In practice, one measures the shear stress, S l 2 and the two normal stress differences: ( p l l - p 2 2 ) and ( p 2 2 - p 3 3 ) of which the first is the more important.

VISCO-ELASTICF L U I D S

I727

T o understand how these normal stresses cause the Weissenberg (or climbing rod) effect note that with reference to Fig. 15.2.1, the axis of rotation of the rod is perpendicular to the 12 plane. If the component of the normal stress, 011, in the direction of flow is greater in magnitude than the components perpendicular to it (022, 033) there will result a tension in the direction of flow which increases towards the axis of rotation (as the shear rate increases). This tension is equivalent to a pressure acting uniformly on each concentric fluid layer, increasing towards the axis of rotation and forcing the liquid to climb the rod.

15.4.4 Comparison between dynamic and steady shear flow properties of visco-elastic fluids It can be shown on theoretical grounds (No11 1958) that in the limit of zero shear rate, the viscosity, and the first normal stress difference, * I ,as measured in a steady shear experiment, are related to the dynamic properties of the material:

~0

and

=

G”

= lim - = lim v’ -0 w w-0

P l l -P22

YL

. 2G‘ wow2

= limp=

(15.4.4)

. 24‘ lim-.

w o w

(15.4.5)

Figure 15.4.2 shows some comparative data for a sample of polystyrene in which eqn (15.4.4) appears to be satisfied. Equation (15.4.5) is presumably only satisfied for frequencies and shear rates below the experimentally accessible range. The first normal stress difference is relatively easy to measure at low shear rates and is always positive and, typically, for polymer solutions and melts it is greater than the shear stress at the same shear rate. For most colloidal suspensions, W1 is much smaller than the shear stress; only in gels and pastes does one expect the development of high degrees of elasticity. Even for polymeric liquids it seems that the second normal stress difference (pzz p 3 3 ) is small, and usually negative, compared to (pll - pzz). It is, therefore, usually possible to assume it to be zero (‘the Weissenberg hypothesis’) and this will certainly be true for colloidal suspensions. Although ( p l l - p 2 2 ) and S12 are relatively easy to measure, results for polymer solutions have been limited to the region below Y = 10 s-l in conventional viscometers, presumably because of the development of flow instabilities in the viscometer (Boger et al. 1980). This, of course, is much lower than the shear rates commonly used in polymer technology, so some estimate is needed of these parameters at higher shear rates. Two important approximate procedures used to obtain such estimates are: (1) the Cox-Mertz rule; and (2) the determination of the first normal stress difference from simple shear data. The Cox-Mertz rule states that the curve of versus 9 should be superposable on the curve of ( q ’2 q ’’2)1/2 against w. (That is r] (Y) and qrn (w) are essentially identical.) Bird and Curtiss (1984) have used a very general statistical mechanical

-

+

728 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

lo5-

k c

- 106

-

E ‘ -lo5

104-

g is 2

v

M -lo4

lo3-

lo2

I

I

I

I

I

<

lo3

Fig. 15.4.2 Comparison of dynamic (q’, G’) data with steady shear data [q, (all - p 2 2 ) ] for a sample of polystyrene (Styron 686 from Dow Chemical Co.) Mass average and number average molar masses were 2.89 x lo5 and 1.02 x lo5 daltons respectively (Section 5.3.6). (From Han et al. 1975, with permission.)

model to derive a constitutive eguationt for a polymer solution or melt and established that the Cox-Mertz rule should be true at least for small shear rates (hj.or hw < 50 where h is a characteristic relaxation time of the material). It is, however, used over a much wider range of j . (and w ) than can be strictly justified theoretically. The estimation of the first normal stress difference from simple shear measurements also depends on the postulation of a particular sort of constitutive equation. One popular procedure is that described by Abdel-Khalik et al. (1974) based on the Goddard-Miller (1966) constitutive equation. There is, however, little point in pursuing these developments until we have constitutive equations which are more suited to the description of colloidal dispersions.

15.5 Measurement of rheological properties of inelastic fluids in Couette flow We return now to the question of how to obtain meaningful rheological data from viscometric flow situations when the fluid is inelastic but non-Newtonian. In general the shearing stress and rate of shear will depend upon the position in the viscometer so

The notion of a constitutive equation was introduced in Section 3.4. It is a statement, in mathematical terms, of the way the elements of the stress tensor depend on the properties of the material. Ideally those properties are expressed in terms of the various elastic and viscous moduli (i.e. the measurable rheological properties) of the material, but as physical chemists we would ultimately hope to estimate those latter quantities from a general statistical mechanical model of the system - as Bird and Curtis (among others) have attempted.

MEASUREMENT OF RHE O LO G ICAL P R O P E R T I E S O F INELASTIC F L U I D S I N COUETTE F L O W

I729

comparisons must be made at the same place. For the coaxial cylinder (Couette) viscometer (Section 4.7) this is normally the wall of the inner cylinder. Before examining the situations in detail we recall that the shear rate in a viscometric flow is not necessarily the same as the velocity gradient. In the Couette viscometer, for example, we can see from eqn (4.7.1) that for a Newtonian fluid the shear rate is given by

(15.5.1) and not simply by duQ/dr. (Here w(r)is the angular velocity = v ~ / r .The ) second term on the left (= w ) accounts for that part of the motion which is simply a rigid body rotation and does not contribute to the shearing process.

15.5.1 The stress-strain rate relationship The following analysis draws on the excellent treatments by van Wazer et al. (1963) and Bird et al. (1960) which should be consulted for more details. It should be noted that the distribution of stress in the gap does not depend on the properties of the fluid. (The flow regime in each cylinder of fluid in the annulus will adjust itself so that the torque is constant from the inner to the outer cylinder.) The experimental data consists of measurements of applied torque as a function of rotational speed (see Section 4.7). When the system is undergoing steady shear, the torque, T, on a cylinder of fluid of radius r and length L, in the annulus between inner and outer cylinders and coaxial with them is (Fig. 4.7.l(b)):

T = 2Rr2Ls.

[4.7.2]

This torque is constant across the gap and equal to the external torque, TO,on the stationary cylinder (Section 4.7). The determination of the shear rate is, however, a little more difficult since the velocity distribution across the gap depends on the properties of the fluid. Fortunately, one can determine those properties from the experimental data using a procedure developed by Krieger and Maron (1952, 1954) based on earlier work by Mooney (1931) and Saal and Koens (1933) (also see Reiner 1960). Since eqn (15.2.4) still holds for the rate of strain of an inelastic non-Newtonian fluid, we can still use eqn ( 1 5 5 . 1 ) for the shear rate. We assume that this shear rate is some arbitrary function of the stress: -Y

dwldr =f (S)

(1 5.5.2)

and obtain a general relation between the rate of rotation and the measured torque: -rdw/dr =f(S)=f(To/2&L).

(1 5.5.3)

From eqn (4.7.2) we have (Exercise 155.1): dr/r = -dS/2S

(1 5.5.4)

730 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

and so

do=

if (S)dS/S

(1 5.5.5)

Assuming that the outer cylinder is stationary, and integrating from the bob, where S = S b (= S;) to the cup where S = S, (=So),and from w = Q to 0, we obtain (Exercise 15.5.1):

(1 5.5.6)

This equation can be handled in a number of ways to yield solutions for coaxial cylinder viscometers and these are discussed by van Wazer et al. (1963 pp. 57-61). The simplest solution applies for the case where the outer cylinder is of infinite radius (i.e. the fluid is driven by the motion of a bob immersed in a large volume of liquid, as in the Brookfield viscometer). In that case S, is zero, and differentiation of eqn (15.5.6) with respect to S b gives

f(&)

= 2(dQ/d In &).

(1 5.5.7)

Thus f (&) can be obtained directly from a plot of Q versus In S b or Q versus In TO. This solution is valid for any substance which does not have a yield value. For substances with a yield value there will be no motion of the fluid at distances r > ro where the stress has fallen below the yield value. Since this then becomes the effective outer cylinder wall, eqn (15.5.7) breaks down. When the fluid has a yield value (Fig. 3.4.9 curve (4) or (5)) the simplest 1) so that the procedure is to use a viscometer with a narrow gap width (Rb/R, shear rate remains essentially constant across the gap. There are, however, situations in which that is not possible and one must then resort to one of two possible procedures: (1) assume a particular (empirical) relationship for the fluid and analyse the flow regime in the gap to arrive at the S(p) relation from the measured To(S2)curve; or (2) use a general asymptotic estimate of the shear rate at the surface of the bob, valid for any fluid. The best known example of the first procedure is the Reiner-Riwlin (1927) equation based on the assumption that the fluid is an ideal Bingham plastic (Table 15.1). They derive (Exercise 15.5.2):

(1 5.5.8) provided that all of the material in the annulus is undergoing shear. If the gap width is fairly large, this will not be the case at lower shear rates. There will develop a region near one of the walls in which the material moves as a rigid body with the same velocity as the wall so that the effective gap width is reduced. The effect of this is to reduce the slope of the Q versus To curve which may be erroneously interpreted as due to an increase in the differential viscosity at low shear rate. van Wazer et al. (1963 Fig. 2.5) give a specific example of how an ideal Bingham plastic material can appear to be nonideal (Fig. 3.4.9 curve (5)) or even pseudoplastic (curve (2)), if examined over a narrow

MEASUREMENT OF RHE O LO G ICAL P R O P E R T I E S O F INELASTIC F L U I D S I N COUETTE F L O W

I731

range of shear rates in a wide gap viscometer. They also show how the more elaborate models of Table 15.1 can be used to derive reliable rheological data on non-Newtonian fluids. The second procedure is more general and depends upon the use of more or less exact estimates of the shear rate at the stationary cylinder. Various procedures have been developed by Krieger and Maron (1952, 1954), Krieger and Elrod (1953), and Calderbank and Moo-Young (1959) with different levels of approximation using either: (1) a multiple bob method; (2) a two-bob approximation of (1); and (3) a single bob method which depends upon eqn (15.5.6). These methods are discussed in some detail by van Wazer et al. (1963 pp. 5 7 4 1 ) and we will not examine them further. Suffice it to note that they make it possible to obtain reliable values of S versus 9 (within 1-1S0/o) even with viscometers of large gap width, although the errors do increase with increase in the ratio R,/Rb(= R,/Ri). (See Fig. 15.5.1.) It should be pointed out that there is a limit to the extent to which the gap can be reduced. T o treat the fluid between the cylinders as a homogeneous material it is necessary to have a gap which is much larger (-1OOx) than the size of the largest particles or aggregates in the suspension. T o satisfy this requirement, most commercial viscometers tend to have rather large gaps so that the ratio R,/RI, departs significantly from unity. Rayner et al. (1979) solved the problem of measuring the flow behaviour of coarse coal slurries (particle size -50 pm) by simply increasing the size of both Rb and R, so that the rheological problem was replaced by an engineering one.

-

4

8

12 16 20 Angular velocity, 8 (s-l)

24

28

!

Fig. 15.5.1 The shear stress-shear rate curve in a wide gap Couette viscometer for an ideal Bingham plastic in the case where some of the material in the gap is not undergoing shear. The stress required to establish flow across the entire gap is shown on the shear stress axis and in the inset figure. (After van Wazer et al. 1963, p. 64.)

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

15.5.2 Sources of error in the Couette viscometer (a) End effects So far we have assumed that the inner cylinder is very long so that the peculiar flow regime in the bottom of the viscometer can be ignored. We examined very briefly (Exercise 4.7.6) the procedure for dealing with this effect in the case of a Newtonian fluid. The usual experimental procedure in a wide-gap viscometer is to fill the viscometer to different heights and to plot the function To/Q against depth of immersion (L). If this plot is linear then the extrapolation back to the L axis when TQ/Qis zero gives the effective length LOwhich must be added to L to take account of the extra torque transmitted to the bob by the flow occurring in the base of the viscometer. As noted earlier (Section 4.7), the end effect can be reduced considerably by shaping the bottom of the bob in the form of a cone which runs inside a conical depression in the base of the cup (see,for example, Rayner et al. (1979). An alternative method of reducing or even eliminating the end effect is to use a bob with a hemispherical recess in the base. An air bubble can be trapped in the recess and this limits the transfer of momentum from the fluid to the bob. (b) Wall slip effects In the derivation of the above equations it is assumed that the fluid in contact with the inner and outer cylinders moves with the same velocity as the adjoining solid. This is the expected behaviour for simple fluids because the size of the molecules is much smaller than the asperities and cavities on the (atomically)rough surface of the cylinders. Even a well-machined surface will trap some fluid in these crevices and will cause it to move with the same velocity as the solid surface. For a colloidal dispersion this is not necessarily so. If the particles are larger in size than the roughness of the surfaces, it is possible for a phase separation to occur so that the surface is covered with a thin layer of suspension medium which may act as a lubricant, allowing the suspension to ‘slip’ with respect to the solid surface. The problem can be alleviated by deliberately roughening the cylinder surfaces or, in extreme cases, introducing knurls, ridges, or even spikes. (c) Temperature control The viscosity of a molecular fluid normally decreases approximately exponentially with temperature so it is important to control the temperature in any viscometry. For suspensions the viscosity may be a more complicated function of temperature but control is still usually necessary. Not only is it essential to protect the suspension from heat gain or loss due to ambient temperature fluctuations but it is also necessary to make provision for the heat generated by the flow process itself, especially at high shear rates. Since all of the mechanical energy put into the fluid ultimately appears as heat or as a rise in temperature this may be a major design consideration. Obviously, a narrow gap (I?,/& M 1) is again an advantage, and constructing the outer cylinder of metal with a surrounding thermostatted jacket is the normal procedure. Failure to control these temperature effects will generally result in a decrease in apparent viscosity with time of shearing. This could easily be confused with the thixotropic breakdown effects discussed in Section 15.3. (d) Taylor vortex development The equations of motion for the fluid in a Couette viscometer are based on the assumption that the flow is laminar and occurs in a circular path around the axis. As

MEASUREMENT OF RHEOLOGICAL PROPERTIES OF INELASTIC FLUIDS I N COUETTE FLOW

1733

noted in Section 4.7 this is no longer true if the rotational velocity becomes too high. Just as the flow in a pipe becomes turbulent if the Reynolds number (Re) (Section 4.8.1) exceeds about 2000, so too does it become turbulent in a Couette viscometer when Re exceeds a certain figure (Exercise 15.5.6). If the outer cylinder is rotating and the inner one is stationary, this instability does not develop until Re exceeds about 50 000 so it is seldom of any consequence. For many commercial instruments, however, the opposite configuration is used and instability can then occur at quite low speeds (as little as a few hundred reciprocal seconds). This instability is caused not by turbulence but by the onset of a radial flow pattern (generated by centrifugal effects) and was studied by Taylor (1923). T h e phenomenon is ascribed to the occurrence of ‘Taylor vortices’ which are circular flow patterns occurring in certain parts of the annulus. T h e critical Reynolds number at which this is expected to happen is given approximately by (van Wazer 1963 p. 85):

Considering the low shear rates at which this behaviour can occur (see Exercise 15.5.6) it is surprising that so many commercial viscometers use this configuration.

7

Exercises 15.5.1 Establish eqns (15.5.4) and (15.5.6). 15.5.2 Assuming that for an ideal Bingham fluid one can write shear rate = - Y d d d r = (S- SB)/~PL derive the Reiner-Riwlin equation (15.5.8) assuming that the fluid is moving with angular velocity Q at the inner cylinder (Y = Rb) and zero only at the outer cylinder (Y = Rc). 15.5.3 Show that if the Bingham yield value, SB,is such that only some of the fluid is moving (S, < SB< S b (where and b refer to outer (cup) and inner (bob) cylinders) then the critical radius beyond which the fluid is stationary is Rcrit= Rb(Sb/&$. Hence show that the critical stress &it (measured at the bob) which must be exceeded so that flow occurs throughout the annulus is given by Scrit

= SB(Rc/Rb)2.

15.5.4 Show that the Reiner-Riwlin equation (15.5.8) can be written in the form

where k1 and k2 are apparatus constants. (The data in the linear regime (where TOcx Q is treated by plotting this function against S b (= To/2nR;L) and determining the slope (which is q p ~ ) . ) 15.5.5 For the power law fluid S = Kl(-rdw/dr)” in a coaxial cylinder viscometer we have

To = S.2ny2L = K1(-rdo/dr)”.2nr2L.

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Integrate this expression between the limits w = 0 and Q and Y = R, and Rb (i.e. stationary outer cylinder) to obtain:

Hence show that

(A plot of In Q against In &, (or In TO)gives a slope from which n is obtained and so K1 can be evaluated from the general expression for S (Table 15.1).) 15.5.6 According to eqn (4.8.1),the Reynolds number is given by Re = p V a / q which for the Couette viscometer with a rotating bob becomes (&),it = [ v b (I?,- Rb) p/q],,.it where v b is the (linear) velocity of the bob (= u ~ )Estimate . the velocity gradient at which one would anticipate that water would exhibit instability in a Couette viscometer with Rb = 3 cm, R, = 3.1 cm. [Take q = lop3N m-’s and p = lo3 kg m-3.] (Hint: refer to Exercise 4.7.4.)

15.6 Capillary viscometer We discussed the behaviour of a Newtonian fluid in a capillary (Ostwald) viscometer in Section 4.7.2. For a non-Newtonian fluid it is convenient to pursue the analysis by way of the stresses rather than the velocity field. In a circular tube, the equations of motion for a (possibly viscoelastic) fluid, in cylindrical coordinates reduce to (Boger et al. 1980 p. 37): 8% a w - - 0 0 (a) Y component : -0 ~

ay +

(6) 8 component : 2 = 0;

ae

Y

p = f ( r , x)

(15.6.1)

ap 1 a (c) x component : ax = --(yorz) ray from which it is not too difficult to show (Exercise 15.6.1) that for this arbitrary fluid the stress components are: r dP (a) a,, = -2 dx

(1 5.6.2)

CAPILLARY VISCOMETER

1735

For the inelastic fluids with which we are chiefly concerned, the pressure is a function of z only and p, = pee = p, = 0 so that these equations reduce to (Boger et al. 1980): YdP or, = -- YAP. (15.6.3) a, = -p(O, z ) = a,, = -p(z). 2dz 2L ' ~

The shear stress at the wall is, from eqn (15.6.2a):

S, = iadp/dz = aAp/2L

(15.6.4)

where a is the capillary radius and L is its length. Hence, from eqn (15.6.2a): ar, = S,r/a

(15.6.5)

which shows that the shearing stress increases linearly from its value of zero at the tube axis to a maximum value, S, at the wall. This relationship is valid regardless of the flow regime (laminar or turbulent) and is equally valid for inelastic and viscoelastic fluids and for fluids having a yield stress. The relation between wall shear stress and pressure drop (eqn 15.6.4) permits the development of a general relation between volume flow rate and the pressure over the capillary length (Rabinovitch 1929).

15.6.1 Flow rate versus pressure drop The following analysis follows that of Boger et al. (1980). The volume flow rate, is related to the fluid velocity, v,, by (compare eqn 4.7.13): d Q = v,2nrdr

(15.6.6)

which can be integrated by parts to give: Q = n[v,g

-

/gd.]".

(15.6.7) 0

Assuming no slip at the tube wall (see Section 15.5.2b) so that v, = 0 for Y = a: a

Q = -n/r2dv,.

(15.6.8)

0

The shear rate for fully developed laminar flow in a tube is in this case (- dv,/dr) (compare eqn (15.5.1)) and this will be a function of the shear stress: -dv,/dr

=f (arz).

(15.6.9)

(The negative sign is introduced because v, decreases from its maximum (on the axis) as r increases.) Combination of eqns (15.6.5), (15.6.8), and (15.6.9) then leads to (Exercise 15.6.2): (15.6.10)

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Just as in the case of the Couette viscometer (Section 15.5) this expression can be used in two different ways: (1) it can be integrated for a particular fluid model (Table 15.1) to obtain the volumetric flow rate/pressure drop relationship for flow through a circular pipe; or (2) it can be differentiated to obtain an expression for shear rate at the wall, independent of the fluid model. If, for example, we choose a power law model for the fluid:

(15.6.1 1)

a,, = K(-dv,/dr)"

so thatf(o,,) = - dv,/dr = (o,/K)'f/"then, it is not difficult to show that (Exercise

15.6.3):

~

Q -

(15.6.12)

where dp/dz = A p / L is the pressure gradient along the length of the pipe. Corresponding expressions for other fluid models are given in Table 15.2. Notice that the quantity 8V/D (= 4Q/nu3) is equal to the shear rate at the wall (S,/q) for a Newtonian fluid and that it appears as a natural parameter for the other flow models. It is common practice to plot this function (or its logarithm) against the pressure drop in order to represent the flow behaviour (see eqn (15.6.15) below). It turns out (Exercise 15.6.3) that, for a power law fluid (especially if n is small), the pressure drop is much less sensitive to flow rate than it is for a Newtonian fluid. For this reason, viscometers which rely on the measurement of differential pressure drop across capillaries are not very satisfactory for examining the behaviour of pseudoplastic fluids. The flow profile for such fluids is very different from that shown by a Newtonian fluid (Fig. 15.6.1(a)). As n decreases below unity the velocity becomes almost constant across a large part of the tube. For dilatant systems (n > 1) the profile is

Table 15.2 Relation between Qand S, or Ap for laminarjow of various time-independent inelasticjuids in tubes. Fluid

Relation

Newtonian

8V/D = 4Q//na3 = S,/q (Hagen-Poiseuille eqn.) (D= tube diameter; V = average fluid velocity = Q / x a 2 )

Ideal Bingham plastica

(SB/SW)~] 4Q//na3 = [Sm/T]pI,][1 - 4SB/3Sw (Buckinghan-Reiner eqn.) For SB< s, this reduces to: 8V/D = [S, - ~ S B / ~ ] / ~ P L

Ellis Law (Table 15.1)

8V/D = (S,/q~){l 4(a+ 3)-'[SW/Si]"-'}

+5

+

For a Bingham plastic, the shear stress near the axis of the tube ( r << a) does not exceed the yield stress (SB) so that all of that material moves as a solid (that is, as an unsheared plug) with a resulting velocity and shear rate profile as indicated in Fig. 15.6.l(b). Wazer et al. (1963 pp. 194-7).

CAPILLARY VISCOMETER

1737

8 9 .Y

n Velocity ( V )

Relative distance from axis (rlR)

Fig. 15.6.1 Velocity profiles and approximate shear rate distribution for power law fluids. (a) Velocity profiles corresponding to the same average velocity. (b) Reduced shear rate as a function of reduced radius. (After van Wazer et al. 1963, p.198.)

sharper than the parabolic result obtained for a Newtonian fluid. The corresponding distribution of shear rate is also shown in Fig. 15.6.l(b). The expressions in Table 15.2 need to be manipulated a little to obtain values of, for example, apparent viscosity as a function of shear rate. For the power-law model that problem is examined in Exercise 15.6.5. Returning to the second procedure for dealing with eqn (15.6.10), we can multiply both sides by S: and differentiate with respect to S, to obtain (Boger et al. 1980):

wheref(S,)

= (-duz/d&a

is the shear rate at the wall (from eqn (15.6.9)).

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Then, from eqn (15.6.4), dS,/S, rate at the wall is:

= d In S, = d In Ap = (Ap)-’d(Ap) so the shear

(15.6.14)

This is called the Rabinowitch-Mooney equation (Mooney 1931). It enables the calculationof the wall shear rate from the measurable quantities a, Q, and Ap. It is valid for time independent inelastic and viscoelastic fluids, in fully developed flow with no slip at the wall. Equation (15.6.14) may be simplified somewhat in form by introducing the quantity

b = d 1n(4Q/na3)/d ln(aAp/2L)

(15.6.15)

and then (Exercise 15.6.4):

yw = (3 + b)Q/na3 = $3

+ b](SV/D).

(15.6.16)

The value of b is first evaluated from the slope of the log-log plot (eqn (15.6.15)). It may be necessary to determine several different b values for different ranges of shear rate. One can then evaluate YW and S, and hence the apparent viscosity from the ratio of the two. Note that this procedure requires no assumptions about the viscous behaviour of the fluid. For a power law fluid (eqn (15.2.7)) it is apparent from eqn (15.6.15) that b = l/n (Exercise 15.6.4). Capillary viscometers may be constructed of glass with the flow occurring solely as a consequence of a difference in hydrostatic head between inlet and outlet (Section 4.7.2). For concentrated colloidal suspensions or polymer solutions, however, it is usually necessary to employ gas pressure or a moving ram to drive the fluid through. There are various problems associated with this procedure and a number of significant sources of error which are discussed in the standard texts (see,for example, van Wazer et al. (1963 pp. 199-215) and Sherman (1970 pp. 3849)). Some of these errors can be estimated and corrected for directly whilst some can be minimized by proper design. End effects can, for example, be corrected by making measurements in capillaries of different length and using an extrapolation procedure. The kinetic energy effect, at least for near-Newtonian liquids, can be minimized by the use of capillaries with tapered ends (Caw and Wylie 1961). We will not examine these questions in any further detail. The problems caused by particle migration away from the tube walls are, however, briefly considered in Section 15.8.

15.6.1 Derive eqn (15.6.2(a))assuming that dp/dx is constant for fully developed flow. Rewrite eqn (15.6.la) in terms of the deviatoric stresses p , and $00 and hence show that (Boger et al. 1980):

i

P, - 8 + d o , 4 - (Pll -poo)dlnr 0

Hence derive eqns (15.6.2b) and (15.6.2~).

= 0.

CONE AND PLATE OR CONE AND CONE VISCOMETER

1739

15.6.2 Derive eqn (15.6.10) using eqns (15.6.5) and (15.6.9). 15.6.3 Derive eqn (15.6.12). What is the effect on the pressure gradient of increasing the velocity fourfold when the flow behaviour index (n) is 0.3? Compare this with a fourfold increase in velocity for a Newtonian fluid. 15.6.4 Derive eqn (15.6.16) and then show that, for a power law fluid with index n, b = l/n. 15.6.5 Show that eqn (15.6.12) can be written in the form (Boger et al. 1980):

The plot of In Qversus In (Ap/L) (which corresponds to In (8V/D) versus In (aAp/2L) should give a straight line of slope l/n. The intercept then gives a value of K. Show that the apparent viscosity, q, is q = K'/"S('-'/") which can, hence, be evaluated for any wall stress S, = aAp/2L. 15.6.6 The power law parameters for a low density polythene at 190 "C are n = 0.72 and K = 13400 N sn mP2 and are valid for 0.0214 5 9 5 0.427 s-'. Calculate the minimum and maximum volume and mass flow rates which this material experiences at these extreme shear rates when flowing through a long cylindrical die of diameter 4 cm. (The density of the polyethylene at 190 "C is 900 kg mP3.) (Mass flow = W = pVA where A is the cross-sectional area of the die.) (Hint: check Exercise 15.6.4). 15.6.7 Note that from eqn (15.6.16) the quantity 4Q/na3 = 8 V / D is equal to the wall shear rate only if b = 1 (Newtonian fluid). Show that the approximate apparent viscosity qa = Sw+ (8V/D) based on this approximate estimate of the wall shear rate is related to the true apparent viscosity, q, for an inelastic fluid by the expression:

This equation (due to Philippoff 1942) can be used to correct data (qa) based on uncorrected shear rates to those ( q ) based on true values (van Wazer et al. (1963) p. 193).

15.7 Cone and plate or cone and cone viscometer This instrument was discussed briefly in Section 4.7.3 in the context of Newtonian fluid behaviour. For small gap angles, a,the shear rate is almost constant over the entire gap so it turns out that the behaviour of inelastic fluids can be adequately determined using the theory developed previously. For elastic fluids, the cone and plate viscometer is widely used for determining the magnitude of the first normal stress difference (eqn (15.4.5)): = [pll - p2~]/(?12)~. This can be done directly in instruments like the Weissenberg Rheogoniometer in

740 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

which the upper plate is attached to a torsion bar and the lower cone is rotated. The cone is able to move downward in response to the normal stress effect and the instrument measures the force, F, necessary to return the cone to its initial position relative to the upper plate. This reaction force can be directly related to the first normal stress difference (Lodge 1964): (15.7.1) where R is the plate radius. If two flat plates are used, instead of a cone and plate, the force required to keep the gap space constant is given by: (15.7.2) For parallel plate geometry, the shear stress and shear rate are given by: (15.7.3) where A is the gap space between the plates (assuming it is small) (compare with eqn (4.7.4) and Exercise 4.7.4). Thus a comparison of the cone and plate and the parallel plate behaviour enable a complete characterization of the fluid, assuming that it obeys the Weissenberg assumption ($22 % $33).

15.8 Time-dependent inelastic behaviour T o study fluids with time-dependent flow characteristics (either thixotropic or rheopectic) it is preferable to use a viscometer which imposes a constant shear rate on the entire sample. Most work on these systems is therefore done with a Couette or a cone and plate viscometer. These have the additional advantage that the entire measurement is conducted on a single sample of the material so that its history can be examined over a prolonged period. The most important class of materials are the thixotropic fluids, for which the apparent viscosity decreases with time at a given shear rate, presumably because the shear regime imposes stresses on the flow units (particles or flocs) which cause them to undergo structural breakdown, leading to the lower viscosity. (Reorientations or minor rearrangements are less likely to produce measurable thixotropy since they would be expected to occur on too short a time scale.) We have already alluded to the occurrence of hysteresis loops (Section 15.3) for such materials. Some possible behaviour patterns for common materials like paints and printing inks are shown in Fig. 15.8.l(a). A rather better procedure for investigating this type of behaviour is illustrated in Fig. 15.8.l(b), for systems which respond reasonably quickly to an imposed shear regime. At the points marked 1, 2, 3, etc. the shear rate is increased in a step-wise manner and the shear stress monitored until it settles to a steady state value. These steady state stress values as a function of shear rate usually follow a pseudoplastic curve (Fig. 3.4.9 curve (2)),for the reasons described in Section 15.10.5.

MICRORHEOLOGY

I741

*

i2

c rn

1 8

c m

Shear rate

(9 )

Fig. 15.8.1 (a) Possible hysteresis behaviour for the case where (i) yield stress is fixed and (ii) yield stress is reduced by shearing. (b). Shear stress-time behaviour for a material subjected to increasing strain rates at points 1,2,3, etc. The stress required for that strain rate rises sharply at first and then falls exponentially to a steady state value as the ‘structure’ of the fluid breaks down to some extent. This method is one of a number introduced by Oersterle.

It is important to note that true thixotropic behaviour is a result of structural changes in the body of the material. It must be distinguished from other time dependent effects on the apparent rheological behaviour, such as those caused by sedimentation, phase separation, or temperature effects. ‘Syneresis’, in which a gel contracts and exudes a small amount of the dispersion medium, can produce an apparent reduction in viscosity with time if the exuded fluid forms a lubricating layer around the viscometer walls (see Section 15.5.2b).

15.9 Microrheology T o understand the macroscopic rheological behaviour of a colloidal dispersion it would be helpful to know what happens at the microscopic level when a colloidal dispersion

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

undergoes a shearing process. We discussed this question briefly in Section 4.10, for the case of spherical and spheroidal particles at low concentrations and for spheres at higher concentrations. There is a very large amount of literature on this subject, most of it couched in the language of tensor calculus so we will be able to discuss only a small part of it here. A much more detailed discussion is provided by van de Ven (1989) and by Russel et al. (1989). The shear behaviour of spheroidal particles at low concentration has been investigated both theoretically and experimentally in considerable detail, especially for larger particles which are microscopically visible and, hence, essentially unaffected by Brownian motion. The theoretical equations of Jeffery (1922) which describe the detailed trajectories of the particles and the motion of their axes of symmetry have been amply confirmed by the elegant experimental investigations carried out by Mason and his collaborators. The early work is described in an extensive review by Goldsmith and Mason (1967) from which a few illustrative examples will be described below. We will then describe a few of the many later investigations in this area by the same group. When a dilute suspension of rods or discs is subjected to a Couette type flow, the individual particles describe a rather complicated pattern of rotations induced by the shear field. The motion of the axis of revolution of the spheroid appears very irregular but it can be analysed exactly using Jeffery’s theoretical equations. Using the coordinate system shown in Fig. 15.9.1, the rate of change of the angle $1 with time is given by:

Here p is the shear rate and q is the (equivalent) particle axis ratio. Integration of this expression with respect to time (assuming p is constant) gives (Exercise 15.9.1): tan41 = 4-l tan[?t/(q

+ 1/q)].

(1 5.9.2)

Thus when viewed along the XI direction, the axis of revolution of the particle rotates with a period of rotation given by:

and

tan $1 = q-’ tan(2nt/T).

(1 5.9.4)

From eqn (15.9.1) it can be shown (Exercise 15.9.2) that d$l/dt is a maximum for ($1 = 0, n)or for ($1 = n/2,3n/2) depending on whether the spheroid is prolate (q< 1) or oblate ( q > 1). That is, of course, what one would expect. The rod ( q < 1) will be twisting more quickly when it is oriented at right angles to the streamlines and least quickly when it is parallel to the streamlines. The same is true for the disc, but for the disc the broad face is oriented across the stream lines when the axis of revolution is oriented along the streamlines ($ = n/2, 3n/2). The excellent agreement between theory and experiment for this simple situation is shown in Fig. 15.9.2. The end of the rod when viewed in the direction of X2 appears to oscillate backwards and forwards along an arc of an ellipse which may be more or less eccentric. Again Jeffery’s equations are able to quantitatively describe the behaviour but a curious feature arises. If Brownian motion and particleparticle interactions are unimportant, and the

MICRORHEOLOGY

1743

Fig. 15.9.1 Orientation of a spheroidal particle in a shear field. AB is the major semi-axis of the spheroid and the shear field is represented by a linear increase in the velocity in the X3 direction as X2 increases (i.e. simple Couette flow).

suspending liquid is Newtonian, the actual characteristics of this ellipse are quite arbitrary; different particles will have ellipses of different eccentricity, depending on their initial orientation, but will retain them so long as the flow is slow and steady. Interactions, or Brownian motion or non-Newtonian characteristicsof the suspending fluid, all tend to push the particle into orbits which exhibit the minimum rate of energy dissipation. This is but one example of a large number of similar studies undertaken by Mason and his colleagues to relate the theoretical and experimental behaviour of rods, discs, and spheres in shear flow and sometimes in the presence of other (e.g. electric) fields.

tlT

Fig. 15.9.2 Measured 41 as a function of time for a rod (q = 0.122) and a disc ( q = 4). The points are experimental, and lines are calculated from eqns (15.9.3) and (15.9.4). (From Goldsmith and Mason 1967 with permission.)

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

The experiments involve microscopic observation of the suspended particles, usually with a cine-camera. Various arrangements are used to keep the particles located in the field of view of the microscope: for Couette flow the inner and outer cylinders are rotated in opposite directions (and their speeds can be controlled independently). The stationary fluid layer can thus be positioned at any point in the annulus and by making it correspond with the centre of the particle selected for viewing, the translational motion of the particle can be reduced to zero (Mason and Bartok 1959). For Poiseuille flow the ‘travelling capillary’ tube is used. In this device the capillary is driven by a motor in the direction opposite to the flow direction at a rate which reduces to zero the apparent velocity of the particle under observation. Particles can then be viewed continuously as they travel for distances up to 50 cm. (Goldsmith and Mason 1962; Vadas et ul. 1973). Moving on from the rigid rods and discs discussed above, the behaviour of deformable rods and fluid drops provides even more complications. Goldsmith and Mason (1967) discuss the fluid circulation inside a drop in Couette flow. Where flow can occur inside the drop, the drop itself causes less disturbance to the stream lines in the surrounding fluid and so the viscosity of an emulsion is lower than the Einstein value. Taylor (1932) gives, for droplets held spherical by surface tension, the relation:

(1 5.9.5) where h is the ratio of the viscosities of suspended phase to continuous medium and 4e is the volume fraction of spheres. Thus q varies from the Einstein value for rigid spheres (h + 00) to qo(1 @J as h approaches zero. This result is valid only if the interface is clean. In the presence of surfactants (or surface active impurities) the rigidity of the interface prevents transfer of momentum to the fluid inside the drops and the system behaves as a suspension of rigid spheres (Nawab and Mason 1958). The stress imposed by the shear field on a fluid drop can cause it to be deformed and if the deformation is sufficiently pronounced the drop may split into two, usually with formation of some very small satellite drops. The conditions under which such ‘bursting’ occurs are discussed by Goldsmith and Mason (1967) who also discuss the way in which a uniform shear field can bend a flexible rod or a linear set of loosely interconnected spheres. When the shear field is non-uniform, as occurs in Poiseuille flow through a capillary, the detailed rotational behaviour of the individual particles is similar to that observed in Couette flow, if the flow is slow and the particles are small compared to the tube radius. Isolated rigid spheres, for example, rotate with a constant velocity equal to half the local shear rate just as they do in Couette flow (Exercise 15.9.3). Since the shear rate varies linearly across the tube, from its value of zero on the axis, the rate of rotation (my = y/2) likewise varies linearly with I as is shown in Fig. 15.9.3. Such spheres, provided they are small enough, show no tendency to depart from the stream lines of the fluid. When the particle radius, R, becomes significant compared to its distance from the wall ( R / ( u y )> 0.1 for spheres), the particle rotates more slowly near the wall (as also

+

MICRORHEOLOGY

1745

1.2

l.o[

rlR

Fig. 15.9.3 Measured angular velocity of neutrally buoyant small rigid spheres in a suspension undergoing Poiseuille flow, as a function of the radial distance from the tube axis. The lines drawn are calculated from the equation wo = 2Qr/na4 where the volume flow rate Q is doubled in going from curve 3 to 2 and doubled again from curve 2 tol. (From Goldsmith and Mason 1967, with permission.)

do rods and discs). Furthermore, for particle Reynolds numbers? (Section 4.8.1) even as small as lop4,the interaction between the particle and the wall results in a migration of the particle (Exercise 15.9.4) away from the wall, creating a particle free zone which is of lower viscosity and which can lead to errors in the estimation of viscosity using the capillary flow technique (Section 15.6). Even rigid spheres will migrate across the stream lines to regions of lower shear rate, especially if the suspension concentration is higher so that particle interactions are common: deformable particles or drops of any size will migrate away from the wall, even at low concentrations, and will continue to migrate, at gradually decreasing velocity, until they reach the axis. For more recent developmets in this area see Davis (1993). The next important step (Mason 1977) was the study of the trajectories of pairs of particles in a shear field. This was discussed briefly in Section 4.10.5 where we introduced the idea of open and closed orbits. According to the analysis of Batchelor and Green (1972) two spheres approaching one another from infinity (in the absence of Brownian motion, external forces or many-body interactions) are unable to penetrate into the region of closed stream lines (Fig. 4.10.2 with Ri/a2 C 0). They are, therefore, unable to approach closer than some minimum surface-to-surface distance, H, which depends upon the relative size of the spheres. For spheres of equal size the minimum value of H/a (where a is the particle radius) is 4.2 x so for the particles usually studied (a 3 pm), the minimum approach distance is well inside the double layer region. A very small

-

+ In this case the particle velocity is zero with respect to the surrounding fluid so the Reynolds number of Section 4.8.1 does not apply. The internal (particle) Reynolds number is now R2Ppo/qo where R is the radius.

746 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

4

Weak attraction

Fig. 15.9.4 Observed trajectories of equal sized latex spheres (in Poiseuille flow) in 50% aqueous glycerol. (a) 4.0 p m spheres in 1 mM KCl. (b) 2.6 p m spheres in 10 mh4 KCl. (From van de Ven 1982 with permission.)

particle (<< 1 pm) cannot approach so close but even the smallest particle can move to within a few hundred nanometres of this ‘large’ (3 pm) particle. When double layer forces, Brownian motion, or third body interactions, are present (van de Ven 1982) the trajectory is slightly modified and careful measurements (see Takamura et al. 1981a,b) can distinguish the occurrence of double layer repulsion and attraction (Fig. 15.9.4).Indeed, the measurements can be sufficiently precise to enable an estimate to be made of the van der Waals constant and the effective surface potential (Fig. 15.9.5). The former turns out to be concordant with theoretical estimates and the latter agrees with the measured (-potential on the particles. It is impossible to do justice here to the enormous body of work in this area, especially that from Mason and his colleagues. The more recent material on interacting spheres is reviewed by van de Ven (1982, 1989) who discusses in some detail the difference between ‘primary’ and ‘secondary’ doublets. Spheres which are interacting at the primary minimum can be distinguished from those interacting at the secondary potential energy minimum (see Fig. 1.6.2) by examining the rate at which they rotate around each other in the shear flow. It is even possible to distinguish between non-touching and

I

I

(4

v0=-45 mV &=lo0 nm

8-

i o Z 1 J; ~(

I

I

I

I

-5

0

5

10

I

I

I

I

(b)

\tr

A

&=lo0 nm A=6x1dZ1J

8 .-

4-

1

I

'?

\ I

*0(mV)

I

.i:

1

I

I

I

1

0

5

10

Reduced time Fig. 15.9.5 Plots of trajectories of the spheres comprising a doublet showing the fit to theoretical curves as the various parameters are varied. Note that for J. &(h) is the total interaction Doublet 8 (figs a, b, and c) a good fit is obtained with ho = 100 nm, @O = 4 5 mV, and the Hamaker constant, A = 6 x here, calculated from DLVO theory, whilst ho is the retardation parameter of Fig. 11.4.1. (From Takamura et al. 1981a, with permission.)

748 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

d2

‘PI

-d2

d2

cp1

0

I

I

-5

0

-d 2

,

Reduced time

Fig. 15.9.6 Trajectories of colliding polystyrene latex spheres in 50% aqueous glycerol forming permanent doublets. The upper curve is for a system with added cationic flocculating agent and the asterisk marks the point at which it is suggested that the spheres are pulled into permanent contact by a polymer bridge. The system shown in the lower figure has no added flocculant and the doublet here is ascribed to van der Waals forces. (From Takamura et al. 1981b, with permission.)

touching primary doublets, by more careful observation of their rotational behaviour. It seems that, as a suspension ages, the number of non-touching primary doublets decreases, presumably because the primary energy minimum becomes deeper, either through impurity adsorption or restructuring of the surface or because the surfaces rotate into a most favourable orientation. It is even possible, in favourable cases (Takamura et al. 1981a,b) to detect the effect on the rotation of a pair of spheres caused by the establishment of a polymer bridge between them (Fig. 15.9.6). At the calculated minimum separation distance (70 nm), the energy of interaction as calculated from DLVO theory (Chapter 12) would be zero but the study of these rotating doublets clearly reveals the presence of long range interactions which can best be explained as the result of the formation of a flexible polymer bridge between the particles.

Exercises 15.9.1 Integrate eqn (15.9.1) to establish eqn (15.9.2). 15.9.2 Establish that the maximum values of d&/dt occur at 41 = 0 and n if q < 1 (a cigar shaped particle) and at n/2 and 3n/2 if q > 1 (a disc).

MICROSCOPIC BASIS OF RHEOLOGICAL MODELS

1749

15.9.3 Show that a rigid sphere in Couette flow rotates with a constant velocity equal to half the local shear rate. 15.9.4 Estimate the particle Reynolds number for a system of neutrally buoyant spheres of 100 nm radius suspended in water in a capillary viscometer of length 20 cm, for which the efflux time is 100 seconds. Problems arise in the measurement of viscosity of such systems (due to creation of a particle free zone near the wall) if this Reynolds number exceeds about lop5.At what particle size does this occur?

15.10 Microscopic basis of rheological models We must now examine some of the physico-chemical descriptions which have been proposed to underpin the mathematical models of flow behaviour displayed in Table 15.1. T he early literature on the subject has been reviewed by Sherman (1970) and by Goodwin (1975) (among others) from which some of Sections 15.10.1-1 5.10.3 are taken. Most discussions of pseudoplastic or plastic behaviour assume that the decrease in viscosity with increasing shear rate is due to a gradual reduction in the amount of ‘structure’ in the system. They differ mainly in the way in which the applied stress is partitioned between the structural (breakdown) effects and the viscous effects. The original analysis of Williamson (1929) has been refined by Goodeve (1939) and Gillespie (1960) and this forms the basis of most treatments of pseudoplasticity or Bingham behaviour. Before describing that model, however, we should re-examine, briefly, the behaviour of a suspension of hard spheres, since as we have already seen (Section 4.10) even that system can display pseudoplastic behaviour.

15.10.1 Flow behaviour of a dispersion of hard spheres We have already discussed the Einstein relation (Section 4.10.3) for the viscosity of a dilute suspension of hard spheres. The extension of that relation to higher volume fractions, using the methods of Mooney (1951) and Dougherty (1959) was also discussed in Section 4.10.5. It should be noted, however, that when hydrodynamic interactions become important, the system no longer behaves as a Newtonian fluid because the shear field is able to change the equilibrium statistical distribution of particles. The time taken for the system to return to its ‘equilibrium’ distribution after a disturbance is of order (Exercise 15.10.1): t

N

~IT~I~LZ~/KT

(15.10.1)

and if this is long compared to l / p the system will show shear thinning behaviour (Goodwin 1982). This effect has been demonstrated by Krieger who has written an excellent review of the area (Krieger 1972) drawing attention to the utility of describing the flow behaviour in terms of various dimensionless groups. In addition to

750 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

the obvious ones of relative viscosity (v/vo),volume fraction (4 = 4nNa3/3), and relative density, pr, he uses a reduced time (compare with eqn (15.10.1)): t,. = kTt/Vo a3

(1 5.10.2)

a reduced shear stress (S,.= Sa3/kT),and an internal Reynolds number, Re = a 2 j p o / ~ o . We showed earlier (Fig. 4.10.6) that for slow (Re + 0), steady (t,. + 00) flow of a dispersion of neutrally buoyant (pr = 1) particles, the function r], = f ( 4 , S,) was well described by eqn (4.10.15)which is of the form derived by Krieger and Dougherty (1959) for this situation. The fact that it is necessary to use such a relationship evenfor hard spheres should indicate that the ‘structure’ which is often alluded to in the derivation of such relations, need not correspond to actual bond formation in the normal sense. The variation in viscosity in these systems is, however, quite small-f order 15-30% say, whereas there are many systems in which the viscosity changes over several orders of magnitude between its zero shear and high shear limits. It is to these that we must now address ourselves.

15.10.2 Flow of systems with anisometric particles Even in the absence of colloidal interactions, a collection of anisometric particles can exhibit quite complex flow behaviour as the shear field interacts with the particle Brownian motion. The orientation of the particles with respect to the stream-lines depends on the relative magnitudes of the two effects, as measured by the PecGt number (a2y/D,.)where D,.is the rotational Brownian diffusion coefficient. We discussed this behaviour briefly in Section 4.10, with particular reference to the behaviour at infinite particle dilution (i.e. the effect on the intrinsic viscosity of the suspension). Although there are formidable obstacles to the extension of that theory to higher concentrations, it is possible to treat more concentrated systems of anisometric particles using the semi-empirical procedures described in the next section. T he ‘bonds’ which are postulated to be formed and broken by the shearing process will often be more a convenience than a reality but the resulting descriptions may still prove useful for obtaining an insight into what is happening at the microscopic level.

15.10.3 Kinetic interpretation of non-Newtonian flow Williamson (1929) made the first quantitative attempt to describe shear thinning behaviour in terms of structural breakdown. His method of partitioning the shear stress between this breakdown process and the maintenance of viscous flow is described by Sherman (1970) in some detail and has been extended by Goodeve (1939) and Gillespie (1960). It is most appropriate for systems for which the S - Y curve becomes strictly linear above some critical shear rate (say Fc) and has been used by Ekdawi and Hunter (1983) for describing the flow behaviour of coagulated sols at low shear rate. The energy dissipated per unit volume per unit time is equal to the product S j and this is the basis of the partitioning procedure which is illustrated in Fig. 15.10.1. The stress required to support the viscous flow Svis assumed to be given by the Newtonian expression: s v

= VPLY

(15.10.3)

MICROSCOPIC BASIS OF R H E O L O G I C A LM O D E L S

I751

Fig. 15.10.1 Partitioning the shear stress between viscous flow and structural breakdown. On the curvilinear part of the S-9 relation at, say, G, the stress S, required to support the viscous flow is subtracted from the total stress to obtain the broken line OBC which therefore represents the stress, ST involved in structural breakdown, as a function of 9.

for all values of

i. and this is represented by the line OC. The total stress is then

This can be written: (1 5.10.5)

where ST(00)is the value of STas + 00 and C is a measure of the curvature of the S - function at low shear rates. For C = 0 the expression obviously simplifies to the Bingham relation (eqn (3.4.23) and Table 15.1) with &(00) = Sp,whilst for ST(00) = 0 it represents simple Newtonian behaviour. The next step was to find some rationale for the function C and this was provided by Goodeve (1939) with his impulse theory which was modified by Gillespie (1960) to incorporate the effect of Brownian motion. Essentially these developments sought to y), by examining the rate of 'link' formation and describe the structural effects, ST( breakdown during shear. A similar approach was adopted by Casson (1959), by Denny and Brodkey (1962)' and later by Cross (1965). The arguments are treated at some length by Sherman (1970) so we will describe only the Cross development as indicative of the procedure. Cross assumed that the suspension consisted of chains of particles with an average of L links per chain. The links were formed by Brownian motion with a rate constant kz and were ruptured by both Brownian motion (rate constant ko) and by shear (with rate constant k l j " ) . The rate of change in the number of links is then given by: dL/dt = k2P - (ko

+ k1j")L

(15.10.6)

752 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

where P is the number of particles. [Note that buildup of structure by the shearing process (rheopexy) is denied in this approach.] In the steady state, dL/dt = 0 and so L = k2P/(ko kly).If L = LOwhen = 0, then LO= k2P/ko and so:

v

+

( 15.10.7)

where K = kl/ko. In order to connect the number of links, L, to the viscosity, Cross appealed to Bueche’s (1952) analysis of the viscosity of a polymer solution and wrote:

r]=r],+BL where B is constant. Then putting

r]

= r]o when L = LOgives r

( 15.10.8) ] ~ r],

= BLo and so

( 15.10.9)

Although couched in terms of the shear rate rather than the shear stress this is clearly equivalent to eqn (4.10.15) or the Meter model of Table 15.1. The only drawbacks with this approach are (i) there is insufficient theoretical basis for one to be able to estimate what value the shear rate dependence parameter should have and (ii) there is no way in which one can introduce the energy involved in breaking links. Nor is it possible to incorporate any knowledge of the actual structure of the suspension into the analysis. All suspensions are assumed to consist of chains of particles (rather like linear polymers). It is clear from Fig. 15.2.3, however, that this type of equation can represent the viscosity behaviour of some pseudoplastic systems over a very wide range of shear rates; it has also been found to represent a very wide range of systems over a rather more limited range of shear rates. The relation between eqn (15.10.9) and the various models of Table 15.1, has been explored by Oka (1971) and that work is described by Goodwin (1975). Starting from the usual assumption that the number of bonds between particles decreases with increasing shear rate or shear stress, Oka shows that (15.10.10) where k’, a,and B are material constants. Integration of this expression for n < 1 and with = 0 when S = 0 gives the ‘generalized Casson equation’ (Exercise 15.10.2):

v

(S

+ a)1-”= ko + k’(y + p)’-”

(15.10.11)

which reduces to Casson’s equation:

d

i.

= ko

+ kl+

(15.10.12)

when a = 0, B = 0, and n = This equation has been used to describe the flow behaviour of blood and other materials like printing ink.

MICROSCOPIC BASIS OF RHEOLOGICAL MODELS

1753

Integration of eqn (15.10.10) with n = 1 generates the general power-law fluid (Exercise 15.10.2):

s=

.[(+].

Y+B

(15.10.13)

This will be indistinguishable from eqn (15.2.7) if a and /3 are not too large. For suspensions with a yield value (S = SBwhen Y = 0, and n = l), integration of eqn (15.10.10) produces the generalized Herschel and Bulkley equation (Exercise 15.10.3) (15.10.14)

A rather more elaborate version of the kinetic equation of flow (15.10.9) has been derived by Cooper et al. (1978). By analogy with the exponential effect of temperature on rate constant, they propose that the rate constant for bond breakage is a similar exponential function of the shear rate. This leads to a modified version of eqn (15.10.9). rl-rlcc VO

-

rlco

= [I

+ K F exp(-k/j)l-'

(15.10.15)

which they found gives a much better description of their experimental data than the Cross equation. It has, of course, an extra parameter and that will be expected to allow a better fit. There is, thus, no difficulty in providing a mathematical basis for the various expressions in Table 15.1 but providing an underlying physico-chemical model which relates the material parameters a, B, and n, to other known properties of the system is not so simple.

15.10.4 Flow of coagulated colloidal sols One system for which some success has been achieved in relating rheological behaviour to the underlying physico-chemical properties is the coagulated sub-micron sized, monodisperse sols studied by Hunter and his co-workers (1982, 1985). T o obtain reproducible flow behaviour, these systems were first subjected to a fairly high shear rate (- 3000 s-') for some minutes, until the applied stress required for that shear rate became constant. The shear rate was then decreased steadily and the corresponding shear stress recorded. The behaviour observed is very characteristic (Fig. 15.10.2), with a strictly linear regime at high shear rates and a pseudoplastic region at low shear rates. They may be described as plastic-pseudoplastic or non-ideal plastic systems with the transition from the curvilinear to linear behaviour occurring at a well defined critical shear rate (YJ. T o describe the high shear rate behaviour, Firth and Hunter (1976) used the Williamson-GoodeveGillespie procedure to separate the energy dissipation into a viscous and a 'structural' term. The viscous term was given by (compare with eqn ( 15.10.4)):

Ev = S V Y = VPLY. 2

(15.10.16)

754 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Shear rate (3 ) (s-l)

Fig. 15.1 0 -dsic shear diagram for a coagulated PMMA latex sol in 31.2% glycerol water. yc (=580 sP1) is the critical shear rate, above which the plastic (differential) viscosity, q p is ~ constant at 5.50 mPa s. The Bingham yield value, Se = 2.75 N m-2. (Particle radius = 125 nm, ionic strength = 0.01 mol LP1,5 = -21 mV.)

with the plastic viscosity, VPL being determined by the volume fraction ofJilocs (&) rather than that of the particles ($p). Thus, at low concentrations we might use the Einstein equation (4.10.9) with 4 = q5f but at higher particle concentrations a more appropriate expression would be (Krieger 1972):

VPL

[

= TO 1 --

(15.10.17)

where [q] = the intrinsic viscosity (defined in eqn (4.10.10) and equal to 2.5 for spheres). The packing parameter p % 0.6-0.7 and is chosen to match the asymptotically limiting v value when the particle concentration approaches close pack. This expression is used to evaluate & and hence to obtain the floc volume ratio C F (= ~ 4f/&) which is a measure of the degree of openness of the flocs. The other two parameters which characterize the high shear behaviour (Fig. 15.10.2) are the obtained by extrapolating critical shear rate, pc and the value of the shear stress (SB) the high shear behaviour down to p = 0. SBis called the Bingham yield value (even though the system is not an ‘ideal’ Bingham material) and Firth and Hunter (1976) have presented a model (the ‘elastic floc’ model) which makes it possible to describe these three parameters in terms of the shear history (in particular the initial high shear rate pmaxwhich determines the value of CFP)and the colloidal properties. The important colloidal properties of the system are the volume fraction of particles (4p),the particle radius ( r ) and the electrokinetic (or {-) potential of the particles. According to the model, this latter parameter measures the magnitude of the maximum attractive interaction force between two particles as a particleparticle ‘bond’ is stretched by the shear field. The relation between 5 and the force is derived directly

MICROSCOPIC BASIS OF RHEOLOGICAL MODELS

1755

from the DLVO theory of colloid stability (Chapter 12). Since the system is coagulated, there is no potential energy barrier separating the particles. All particleparticle interactions are attractive and range from their maximum value (when { = 0) to zero as I { I becomes sufficiently large to ensure dispersion (-45 mV for the poly(methylmethacrylate) latex particles used for most of the studies). The maximum (attractive) force between two particles occurs when they are in their minimum separation position (with surface separation dl) and is given by [eqns (11.3.15) and (12.5.2)]:

where B depends only on dl and the dielectric permittivity, E of the suspension medium but its exact form depends on the value of Ka. It is this force which determines when a particle-particle bond will break and this determines all aspects of the flow behaviour. The value of ( is thus crucial in determining the behaviour of the system for it determines all aspects of the structure and it is the only variable which is readily controllable. Low values of I {I imply a strong attraction between the particles. Flocs formed under such conditions have a large proportion of trapped solvent (CFP>> 1). As I ( I increases, the interparticle bonds become weaker and only more compact flocs can survive the initial high shear rate. Firth and Hunter (1976) show that CFP should decrease linearly with and this was shown to be so for a number of different systems. Above the critical shear rate, yc, the S - y behaviour is linear, indicating that the ‘structure’ in the system is in some sense constant. The model assumes that yc marks the point above which the shear field is sufficiently strong to separate pairs of flocs as fast as they are formed. Below this point, pairs of flocs can remain more or less permanently linked and as the shear rate falls towards zero the numbers of triplets, quadruplets, etc gradually increases. The ‘bond’ between two flocs is a particle-particle bond and its strength is again governed by the (-potential in the same way as the intrafloc bonds determine CFP.Not surprisingly, therefore, yc also decreases with increase in 5’. Although the structure of the flocs is not affected by the shearing process for yc < 9 < ymax,the size of the flocs does increase (by -25-5Oo/o) as the shear rate falls towards yc (-300- 900 s-’). The mean floc radius, a F , as determined in a Coulter counter (Hunter and Frayne 1980) is related to the shear rate by:

(’

aF

c(

y-0.4

(15.10.19)

and varies from -2 p m at ymaxto 4 p m at yC.Since the particles are about 0.1 p m in radius, each floc contains many hundreds of particles which makes it possible to describe the behaviour using rather crude averaging procedures. The model is not expected to work for much larger particles (>1 pm) because the shear forces then ensure that there are very few particles in each floc.

756 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

The theoretical estimate of SBwould appear to involve the calculation of the dissipated ) in overcoming the ‘structure’ in the system. additional energy (E, = S B ~ That is the way that SBhas been interpreted in the past (Fig. 15.10.1). It turns out, however, that when one examines the contributions to SBin more detail (van de Ven and Hunter 1977), the breaking of the particleparticle bonds between flocs is a relatively unimportant contributor. Even the energy involved in stretching particleparticle bonds inside the flocs (as they are rotated and distorted by the shear field) is unimportant. By far the largest contribution to E, comes from the viscous energy involved in moving liquid into and out of the space between particles in the floc as the floc is distorted, even though the scale of the distortion is very modest (-1Yo changes in floc shape for the systems under consideration here). It turns out that SBis a linear function of CFPand so it, too, decreases with (’. All three rheological parameters ( C F ~ , yc, and SB)thus show a similar dependence on the double layer potential of the particles. From that dependence it is possible to estimate the separation between the particles at which the attraction force is a maximum and, for the PMMA system, this turns out to be when the shear planes are separated by a distance of about 0.6 nm. The actual amount of interparticle stretching (6) which occurs inside the flocs is about half of that figure. The relative change in volume of the floc due to distortion is of the order 6 / a (van de Ven and Hunter 1977) and so, as remarked above, the degree of volume distortion of the flocs is only of order 1%. Nevertheless, the energy involved is a significant contribution to the overall dissipation process. The connection with the interparticle force comes about because it is that force which determines when the floc-floc bond will rupture and hence the extent of stretching. At low shear rates (P < Pc) as noted above, Ekdawi and Hunter (1983) have extended the model to describe the curvilinear (pseudoplastic) relation between S and 9. The treatment attempts to quantify the build-up of structure as flocs form doublets, triplets, etc as y decreases. The essential feature is that the flocs retain their integrity below yc and the suspension medium continues to flow between the flocs. The degree of distortion within a floc is smaller because the applied stress is smaller but the essential features of the high shear rate model are carried over into the low shear regime. The model also predicts that these systems will exhibit a small degree of viscoelasticity and this has been verified by van de Ven and Hunter (1979) and more recently by Hunter and Everett (1988). Again the relevant rheological parameter (the dynamic rigidity modulus, G’ (Section 15.4.4)) decreases with (’. More recently the relation between the primary yield value (or static shear yield stress) [So of Fig. 3.4.91 and the particle charge has been studied in great detail in the neighbourhood of the isoelectric point for a number of different mineral oxides (see, for example, Johnson et al. 1998). The primary yield value is measured using the vane technique of Nguyen and Boger (1983, 1985) in which one immerses a four bladed paddle into a suspension, allows the system to re-equilibrate, and then measures the torque required to turn the paddle or vane. This technique directly measures the interparticle forces at the junction where the material yields. Although the mechanism is quite different from that involved in the dynamic Bingham yield value described above, it should not be surprising that the dependence on zeta potential is identical. At the isoelectric (( = 0) point, the primary yield value must be a maximum and since { varies roughly linearly with pH in that region, a plot of yield value against pH is parabolic.

MICROSCOPIC BASIS OF RHEOLOGICAL MODELS

1757

500

400

300

200

100

0

9

8

7

10

11

12

10

11

12

P" 1.2

1.0

a

0.8 0

0.6

0.4

B 0.2

0.0

7

8

9

PH Fig. 15.10.3 The (a) measured and (b) normalized static shear yield stress (ty= So) of AKP-30 alumina suspensions as a function of both volume fraction and pH. 0 ,#J = 0.200; 0,#J = 0.225; A, #J = 0.250; A, = 0.275; #J = 0.300. (From Johnson et al. 1998, with permission.)

+,

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1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

1.2

1.o

0.8

0.6

0.4

0.2

0.0 0

500

1500

lo00

2000

2500

c2b V 2 ) <*

Fig. 15.1 0.4 The normalized static shear yield stress (ty= So)as a function of for several A D 30 alumina suspensions. The two central line plots correspond to d1 = 2.3 nm (- - --) and dl = 2.4 nm ( ) respectively. Line plots correspondingto dl = 1.4 and 3.4 nm are also given to show the sensitivity of the gradient to the choice of interparticle separation. (4 values as for Fig. 15.10.3.) (From Johnson et al. 1998, with permission.) ~

As the pH is varied around the i.e.p. the data for a variety of volume fractions can be placed on a master curve using a model which was developed by Kapur et al. (1997) (Fig. 15.10.3).The relative value of the yield stress, compared to the maximum value obtained at the i.e.p., can then be plotted against and the result is reasonably linear (Fig. 15.10.4). For alumina the approach distance turns out to be about 2.4 nm compared with the value of 0.6 nm found for PMMA, possibly an indication of the anisometric nature of the particles. The same group has also shown that adsorption of bulky specifically adsorbed molecules on the oxide surface not only shifts the i.e.p. but also pushes out the distance of approach. The whole yield stress versus pH curve is therefore shifted to the new i.e.p. and the maximum in the stress is reduced (Fig. 15.10.5.)

c2

15.10.5 Time-dependent systems: kinetic interpretation of thixotropy Denny and Brodkey (1962) applied a reaction kinetics approach to flow behaviour and obtained a result similar to eqn (15.10.6). Using the simple reaction scheme: unbroken bonds

k1 2

kz

broken bonds

(15.10.20)

MICROSCOPIC BASIS OF RHEOLOGICAL MODELS

500

I

I

I

1759

I

v 400

300 cd

@ r" 200

100

0 2

6

4

8

10

12

PH Fig. 15.10.5 The static yield stress (ty= SO) as a function of pH for a system with various additions of phosphate. The phosphate is specificallyadsorbed to the surface and it moves the i.e.p and at the same time increases the separation distance between the surfaces and so lowers the interparticle force and energy. (From Leong et al. 1993, with permission.)

to represent the structural effects, they write for the rate of structural breakdown: -

d(unbroken) = k',(unbroken)" dt

-

kz(broken)"

and so (15.10.2 1) VO

- Vco

VO

- Vco

[Thus the viscosity is introduced by assuming that it is directly proportional to the . ] amount of 'unbroken structure' which is a maximum at ~0 and a minimum at T ] ~ The rate constant 62 is assumed to be independent of shear (i.e. only Brownian motion leads to restructuring) whilst the breakdown rate constant depends on shear rate:

k', = kip

(15.10.22)

760 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

wherep is a constant which reflects the 'shear sensitivity' of the material. Under steady state conditions, dy/dt = 0 and, for n = m = 1, the Cross (1965) version of the Meter model is generated (Exercise 15.10.4):

(15.10.23) The novelty of the Denny and Brodkey approach lies in the solution for non-steady state conditions. Choosing possible values for m and n (e.g. m = 2 and n = 1) they obtain analytical solutions for the integrated form of eqn (15.10.21).Given the values of the parameters m,n, y, qo, roo,and p derived from steady state measurements it is then possible to evaluate k1 and k2 for a thixotropic material. For the heavy thixotropic mineral oil used in their studies they found kl = 1.21 x s-' and k2 = 0.94 x lop3s-' so that the 'equilibrium constant' for eqn (15.10.20), K = k l / k 2 = 1.29 x lo-". The very low value of kl indicates why the material is thixotropic: the rate of breakdown is so slow that it requires a considerable time to establish the steady state structure at a given shear rate. The hopeful aspect of the exercise was the agreement observed between the estimate of K from the time-dependent (thixotropic) data and the value obtained in the steady-state (pseudoplastic) regime (0.89 x lo-"). Unfortunately, the analysis is rather lengthy and requires a large amount of good data to enable reasonable values of the parameters to be extracted. The importance of the exercise lies in the demonstration of the link between thixotropy and pseudoplasticity. If the time for the establishment of the steady-state structure at a given shear rate is long compared to the measuring time, then the system will exhibit thixotropy. Pseudoplastic behaviour then appears as the limiting form of thixotropy when the time between successive measurements is long compared to the relaxation time for the structure in the system, which can then always exhibit its steady-state behaviour.

15.10.6 Elastic behaviour of concentrated sols We noted above (Section 15.10.4) that coagulated sols exhibit some degree of elastic behaviour, even at low particle concentrations (-7-10%). Those effects are, however, overwhelmed by the non-Newtonian viscous effects (van de Ven and Hunter 1979). For more concentrated sols the elastic properties can become extremely important, especially under conditions where the sol forms an ordered array of particles. Such a situation occurs when the electrolyte concentration is so low that double layer repulsion is the dominant interparticle force (see Section 14.2).The resulting lattice of particles often has a centre-to-centre distance R, of the order of the wavelength of visible light, so the system exhibits a characteristic multicoloured effect when viewed in white light. The colours are produced by Bragg diffraction from the various planes of particles, in much the same way as an atomic lattice diffracts the much smaller wavelength X-rays used in crystallography. A theoretical description of the high frequency limit of the dynamic rigidity or shear modulus, Gb, (see Sections 3.4 and 15.4.4)has been developed by Buscall e t al. (19826)

MICROSCOPIC BASIS OF R H E O L O G I C A LM O D E L S

I761

based on pair-wise additivity of the double layer repulsion forces. They show that G', is related to the potential energy of interaction, VT by:

(1 5.10.24)

+

where a! is a constant determined by the packing arrangement and R (= 2a H) is the centre-to-centre separation. Assuming that VT e VR,the repulsion potential energy and using, for VR(compare with eqn (12.5.5)) the expression valid for small KU:

VR= [4mocr&a2/R] exp(-KH)

(1 5.10.25)

they obtain (Exercise 15.10.5):

(15.10.26) Values of the theoretically estimated G',/I+$ can be plotted against the measured G', assuming a particular sort of packing (and, hence, a). The result is a good linear correlation (Fig. 15.10.6) from the slope of which one can estimate I $d I e 50 mV. This is a very reasonable value, in good agreement with (-potentials measured on comparable sols. Some further experimental data is examined by Goodwin et al. (1982) with similar very encouraging results.

~ o - ~ G(N: m-' ~

v2)

lvdz Fig. 15.10.6 Plot of the experimental limiting shear modulus C6, against G&(theor)/& for various volume fractions, 4, of a polymer latex in 5 x lop4 M NaCl solution. (&,(theor)/& values calculated for Ka = 2.48, (11= 0.833, and R = 2a (0.7443. The slope correspondsto I @d I 50 mV. (From Buscall et al. 19828, with permission, but corrected following recalculation by R.Williams [1991 personal communication.].)

762 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

The experimental determination of Gb, in these cases was done with a pulse shearometer (Rank Bros), the operation of which is described in an accompanying paper (Buscall et al. 1982~).This instrument effectively measures the velocity, v , of a shear wave (of frequency about 200 Hz) through the sample; the G’ value is then given by G‘ = p v2 where p is the density. The same procedure was used by Hunter and Everett (1988) in the experiments described at the end of Section 15.10.4. The ordered latices used in the studies of Buscall et al. show quite pronounced rigidity (G’ values of order 102-103 N m-’) and extremely high zero-shear viscosities (- lo6 times higher than that of water). The rigidity can also be studied as a function of compressive stress, as Barclay et al. (1972) and Buscall have done. Buscall et al. (19826) show that the shear modulus and the bulk (compressibility) modulus, K , are closely related to one another:

G‘,/K

= 9n/[32$k]

(15.10.27)

where $m is a packing constant. $, = 0.74 for hexagonal or face-centred cubic (f.c.c.) arrays and 0.68 for body-centred cubic (b.c.c) arrays. We, therefore, have G’,/K = 1.32 for f.c.c. and G’,/K = 1.478 for b.c.c. arrays. The experimental values for this ratio are found to be close to unity. For further information on the deformation and flow of concentrated systems see Adams et al. (1993).

Exercises 15.10.1 Show that the time taken for a particle of radius a to undergo Brownian diffusion through a distance equal to its radius is about 37cyoa3/kT (refer to Chapter 1.) 15.10.2 Integrate eqn (15.10.10) with S = 0 when p = 0 to establish eqn (15.10.11) and show how it reduces to eqn (15.2.7).Also show that for substances with a yield value (S= SBwhen p = 0) eqn (15.10.10) with n = 0 integrates to give the equation for Bingham flow. 15.10.3 Establish eqn (15.10.14) by the appropriate integration procedure and determine the conditions under which it yields the equation for the HerschelBulkley model. 15.10.4 Use eqns (15.10.21) and (15.10.22) for the Denny and Brodkey model to establish eqn (15.10.23) when dy/dt = 0 and n = m = 1. 15.10.5 Establish eqn (15.10.26) for the shear modulus. (Note that +I$ is rendered as @d in the original paper.)

References Abdel-Khalik, S.I., Hassager, O., and Bird, R.B. (1974). Polym. Engr. and Sci. 14, 859. Adams, M.J., Briscoe, B.J., and Kanjab, M. (1993).A h . Colloid Interface Sci. 44, 143-82.

REFERENCES

Arp, P.A. and Mason, S.G. (1977). 3.Colloid Interface Sci. 61, 21-43. Barclay, L., Harrington, A., and Ottewill, R.H. (1972). Kolloid 2. 2. Polymere 250, 655. Batchelor, G.K. and Green J.T. (1972).3 Fluid Mech. 56, 375,401. Bird, R.B. and Curtiss, C.F. (1984). Physics Today 37, 3 W 3 . Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960). Transport phenomena. John Wiley, New York. Bird, R.B., Armstrong, R.C., and Hassager, 0. (1977). Dynamics of polymeric liquids, Vol. I Fluid mechanics. John Wiley, New York. Boger, D.V. (1977). Nature 265, 126. Boger, D.V., Tiu, C., and Uhlherr, P.H.T. (1980). Introduction to theflow properties of polymers. R.A.C.I. and Brit. SOC. Rheology, Australia. Bueche, F. (1952).J. Chem. Phys. 20, 1959. Buscall, R., Goodwin, J.W., Hawkins, M.W., and Ottewill, R.H. (1982a, b). 3. Chem. SOL.Faraday Trans. I 78,2873-88; 78,2889-99. Calderbank, P.H. and Moo-Young, M.B. (1959). Trans. Inst. Ch. Eng. (London) 37, 26. Carreau, P.J. (1968). Ph.D. Thesis, University of Wisconsin. Casson, N. (1959). In Rheology of disperse systems (ed. C.C. Mills). Pergamon Press, London. Caw, W.A. and Wylie, R.G. (1961). Brit. 3. Appl. Phys. 12,94. Christiansen, E.B., Ryan, N.W., and Stevens, W.E. (1955). A.I.Ch.E.3 1, 544. Cooper, P.G., Rayner, J.G., and Nicol, S.K. (1978).3 Chem.SOL.Faraday Trans. I 74,785-94. Cross, M.M. (1965). 3. Colloid Interface Sci. 20, 417. Davis, R.H. (1993). Adv. Colloid Interface Sci. 43, 17-50. de Haven, E.S. (1959). Ind. Eng. Chem. 51,63A46A; ibid. 813-16. Denny, D.A. and Brodkey, R.S. (1962). 3. Appl. Phys. 33, 2269-74. Dougherty, T.J. (1959). Ph.D. Thesis, Case Institute of Technology. (See Krieger 1972 and 1985). Ekdawi, N. and Hunter, R.J. (1983). 3. Colloid Interface Sci. 94, 35541. Eyring, H.J. (1936). 3 Chem. Phys. 4, 283. Firth, B.A. and Hunter, R.J. (1976). J. Colloid Interface Sci. 57,266-75. Fredrickson, A.G. (1964). Principles and applications of rheology. Prentice-Hall, New Jersey. Gillespie, T. (1960). 3 Colloid Sci. 15, 219. Ginn, R.F. and Metzner, A.B. (1969). Trans. SOL.Rheol. 13, 48. Goddard, D. and Miller, C. (1966). Rheol. Acta 5, 177. Goldsmith, H.L. and Mason, S.G. (1967). The microrheology of dispersions. In Rheology (ed. F.R. Eirich) Vol. 4, Chapter 2, pp. 85-250. Academic Press, New York. Goldsmith, H.L. and Mason, S.G. (1962). 3. Colloid Sci. 17,448. Goodeve, C.F. (1939). Trans. Faraday SOL.35, 342. Goodwin, J.W. (1975). In Colloid science Vol. 2, Chapter 7, pp. 246-93. Chemical Society: London. Goodwin, J.W. (1982). In Colloidal dispersions (ed. J.W. Goodwin) Chapter 8, pp. 165-96. Royal Society of Chemistry, London. Goodwin, J.W., Gregory, T., and Stile, J.A. (1982). Adv. Colloid Interface Sci. 17, 185-95. Govier, G.W. and Aziz, K. (1972). Flow of complex mixtures inpipes. Van NostrandReinhold, New York.

I763

764 I

1 5 : RHEOLOGY OF COLLOIDAL DISPERSIONS

Green, H. (1949). Industrial rheology and rheological structures, pp. 5242. John Wiley, New York; Chapman and Hall, London. Halmos, A.L. and Tiu, C. (1981). J. Texture Studies 12, 39. Han, C.D. (1976). Rheology in polymer processing. Academic Press, New York. Han, C.D., Kim, K.U., Siskovic, N., and Huang, C.R. (1975). Rheol. Acta 14,533. Huilgol, R.R. (1975). Continuum mechanics of viscoelastic liquids. Halsted Press, John Wiley and Sons, New York. Hunter, R.J. (1982). Adv. Colloid Interface Sci. 17, 197-212. Hunter, R.J. (1985). In Modern trends of colloid science in chemistry and biology (ed. H.-F. Eicke) pp. 184-202. Association of Swiss Chemists, Birkhauser Verlag, Basel. Hunter, R.J. and Everett, D.W. (1988). Proc. Xth World Congress of Rheology. Sydney. Hunter, R.J. and Frayne, J. (1980).J. Colloid Interface Sci. 76, 107-15. Jeffery, G.B. (1922). Proc. Roy. SOL.A102, 161. Johnson, S.B., Russell, A S . , and Scales, P.J. (1998). Colloids and Surfaces Series A PhysicoChem. and Eng. Aspects 141, 119-30. Kapur, P.C., Scales, P.J., Boger, D.V., and Healy, T.W. (1997). AIChEJ. 43, 1171. Krieger, I.M. (1972). Adv. Colloid Interface Sci. 3, 111-36. Krieger, I.M. (1985). In Polymer colloids (ed. R. Buscall, T. Corner, and J.F. Stageman). Chapter 6, pp. 21946. Elsevier Applied Science, London. Krieger, I.M. and Dougherty, T.J. (1959). Trans. SOL.Rheol. 111, 137-52. Krieger, I.M. and Elrod, H. (1953). J. Appl. Phys. 24, 134. Krieger, I.M. and Maron, S.H. (1952).J. Appl. Phys. 23,147; (1954) 25,72; (1959)

30, 1705. Leong, Y.K., Scales, P.J., Healy, T.W., Boger, D.V., and Buscall, R. (1993). J. Chem. SOL.(Faraday Trans.) 89, 2473-8. Lindsley, C.H. and Fischer, E.K. (1947). J. Appl. Phys. 18,988. Lodge, A S . (1964). Elastic liquids. Academic Press, New York. Mason, S.G. (1977). 3. Colloid Interface Sci. 58, 275-85. Mason, S.G. and Bartok, W. (1959). In Rheology of disperse systems (ed. C.C. Mill) Chapter 2. Macmillan (Pergamon), New York. Meter, D.M. (1964). Thesis. University of Wisconsin, Madison, Wis. USA. Mooney, M. (1931). J. Rheol. 2, 210. Mooney, M. (1951). J. Colloid Sci. 6, 162. Mooney, M. and Ewart, R.H. (1934). Physics 5, 350. Nawab, M.A. and Mason, S.G. (1958). Trans. Faraday SOL.54, 1712. Nguyen, QD. and Boger, D.V. (1983). Acta Rheologica 27, 321; 29(1985), 335. Noll, W. (1958). Arch. Rat. Mech. Anal. 2, 197. Oka, S. (1960). Rheology: theory and application (ed. F.R. Eirich) Vol. 3. Academic Press, New York. Oka, S. (1971).Jap. J Appl. Phys. 10,287. Ostwald, W. (1926). Kolloid 2. 38, 261. Philippoff, W. (1942). Viscositat der Kolloide. In Handbuch der Kolloidwissenschaft 9. Theodor Steinkopff, Dresden. Prandtl, L. (1928). 2. Angew. Math. Mech. 8, 85. Rabinowitch, B. (1929). Z. physik. Chem. (Leipzig) 145A, 1. Rayner, J.G., Cooper, P.G., and Nicol, S.K. (1979). Rheol. Acta 18, 297-302. Reiner, M. (1949). Deformation andjow. Interscience, New York. Reiner, M. (1960). Deformation, strain, andjow. Interscience, New York.

REFERENCES

Reiner, M. and Riwlin, R.S. (1927). Kolloid Zeit. 43, 1. Russel, W.B., Saville, D.A., and Schowalter, W.R (1989). Colloidal dispersions, pp. 525. Cambridge University Press, Cambridge. Saal, R.N.I. and Koens, J. (1933). J. Inst. Petrol. Technologists 19, 176. Scott-Blair, G.W. (1969). Elementay rheology. Academic Press, London. Sherman, P. (1970). Industrial rheology, Chapter 3, pp. 97-184. Academic Press, London. Skelland, A.H.P. (1967). Non-newtonianflow and heat transfer. John Wiley, New York. Takamura, K., Goldsmith, H.L., and Mason, S.G. (1981a, b).J. Colloid Interface Sci. 82, 175-89; 82, 190-202. Taylor, G.I. (1923). Phil. Trans. Royal Soc. A223, 289. Taylor, G.I. (1932). Proc. Roy. SOC. A138,41. Vadas, E.B., Goldsmith, H.L., and Mason, S.G. (1973). J. Colloid Interface Sci. 43, 630-48. van de Ven, T.G.M. (1982). Adv. Colloid Interface Sci. 17, 105-27. van de Ven, T.G.M. and Hunter, R.J. (1977). Rheol. Actu 16, 534-43. van de Ven, T.G.M. and Hunter, R.J. (1979).J. Colloid Interface Sci. 68, 13543. van de Ven, T.G.M. (1989). Colloidal hydrodynamics, pp. 582. Academic Press, London. van Wazer, J.R,Lyons,J.W., Kim, K.Y., and Colwell, R.E. (1963). Viscosity and JEow measurement. Interscience (John Wiley), New York. Walters, K. (1975). Rheomety. Chapman and Hall, London. Williamson, R.V. (1929). Ind. Eng. Chem. 21, 1108.

I765

APPENDIX A1

Calculation of the allowed surface interaction modes in modern Dispersion Force Theory Refer to Figure 11.6.1 which represents the lowest energy vibrational mode which can satisfy Maxwell’s equations in the vacuum between the two half spaces. The electrical potential can be represented as a function of the form 4 (x,y, 2) $(t) = 4 (x,y, x)exp (-iot)and Maxwell’s equations in a charge-free region with c = 00 require that the spatial part of the wave function (4)must satisfy Laplace’s equation (see Appendix A3):

024 = 0.

(Al. 1)

A suitable solution of this equation is (Exercise Al.1):

4(x9Y, 2) =f ( x ) eXP[i(ku+ k3 .)I

(A1.2)

where k2 and k3 are the magnitudes of the wave vectors in they and x directions. (See eqn (3.3.1) for a definition of the wave vector which is inversely proportional to the wavelengths in those directions.) If we put P = ki k: thenf(x) must satisfy:

+

(A1.3) so that

4 will satisfy eqn (Al.1). The solution to eqn (A1.3) is:

f ( x ) = A exp(kx) for x < 0 and f ( x ) = Bexp(-kx)

for x > L.

and, between the plates: f ( x ) = Cexp(kx)

+ Dexp(-kx)

for 0 < x < L.

(These functions are chosen to ensure thatf(x) is well behaved for large x.) The boundary conditions at the respective surfaces require that both the potential, 4, and the dielectric displacement (Section 3.2)’ (= E(w)E = -ed$/dx), must be continuous. Hence (Exercise Al.2):

A-C-D=O exp(-kL)B - exp(+kL)C - exp(-kL)D = 0 C I A- €oC COD= 0 €1 exp(-kL)B €0 exp(kL)C - €0 exp(kL)D = 0.

+

+

767

(Al.4)

768 I

APPENDICES

These equations are soluble only if the determinant of the coefficients is zero and this is so only if (Exercise A1.2):

(A1.5) where Al, = (€1 - €O)/(€I

I

+ €0).

Exercises A1 . I Al.2

Verify that eqn (A1.2) is a solution of eqn (Al.l) iff(x) is defined by eqn (A1.3). Establish the equations (A1.4) using the stated boundary conditions and hence obtain the dispersion relation (Al.5) by evaluating the relevant determinant.

APPENDIX A 2

Evaluation of the sum of the roots of the dispersion relation Cauchy’s integral theorem states that the value of a functionf(w) of a complex variable at a point (say a = q Zt) can be obtained from its values on the curve C using the expression:

+

(A2.1) where C is any closed curve surrounding the point a, and f (w) is analytic in the region containing C. The solutions, wj of the dispersion equation (1 1.6.3) must satisfy:

a(@) = &(w

- Wj) = 0

(A2.2)

which may be written

In D(w) =

cjln(w

-

wj).

(A2.3)

Differentiating with respect to w gives:

D’(w)/D(w) = Xi(@- wj)-l.

(A2.4)

I

INDEX Entries in bold face refer to major sections devoted to the topic. Entries in italics refer to minor sections. The letters e, f and t refer to exercises, figures and tables, respectively., s.a. and s.u. mean ‘see also’ and ‘see under’ respectively. Note that the same item may appear under different names in the index and an entry may not be exhaustive in its page listings.

Index Terms

Links

A absorption, optical measured in centrifuge

131

133

226

230

of sound, see attenuation spectrum, optical

557f

a.c. electrokinetics

416

acoustic impedance

253

acoustics

250

activation energy of nucleus formtn

96f

active sites, adsorption on

277

activity ionic

66

activity coefficient, at surface

290

correction for solubility

80

influence on sedimentation

l23

in micelle formation

451

additivity of forces

539

adhesion

101

between small particles

92e

of mica plates

614

adsorbate

259

adsorbed films, thickness of

605

adsorbent

284

487

782

265

482

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

adsorption

259

at gas-liquid interface

63

at gas-solid interface

277

at mercury-solution interface

309

482

at solid-liquid interface

287

356

694

709

336

499

503

290

300

344

356

485

40

293

chemical

482

521 competitive

287

discreteness of charge in

338

equilibrium

278

from dilute solution

288

inner (Stern) layer

490

mechanism

277

negative

349

of anions on mercury

336f

of complex adsorbates

298

of counterions, on oxides

503

of hydrolysed metal ions

513

of multivalent ions

509

of organics on mercury of p.d.i. of polymers (neutral)

341f

of surfactant

518

on porous solids

284

physical, energy of

277

positive, for area determination

348

potential, chemical

327

preferential, on crystals

203

specific, detection

329

337

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

adsorption isotherm

89f

BET

278

289f

280

Freundlich Gibbs

289f

291

63

Grahame

337

Langmuir

278

289f

290

Stern

326

337

509e

263f

266

526

635

Aerosil

10 s.a. silica

aerosols

3

AES

209

affinity of adsorbate

279

AFM

272

agent, dispersing, see surfactant aggregation by polymers aggregation

617 632 2

effect on flow

755

fractal character

625

number, in micelles

459t

472

461

prevention of, see stability reaction limited

625

alcohols, aliphatic surface tension in water viscosity of (ethyl) alkali metal ions, adsorption on Agl

67f 166t 353

alkanes (s.a. hydrocarbon) contact angle on PTFE

578

dielectric data

570t

Hamaker constant

572t

neutron scattering

690t

surface tension of

576

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

alkanoates, see carboxylates alkylammonium salts (s.a. CTAB) c.m.c. of

439

459t

alkyl chains, see hydrocarbon chains alkylsulphates

469t

(s.a. surfactant; SDS) alumina

523f

525

757f

364f

503

alumino-silicates, see clay minerals alunite

12f

amphipath, see surfactant amphoteric surface

359

anchoring in steric stabilization

628

angular velocity (s.u. rotation) annealing

83

approach distance, see separation approximation formulae Deryaguin for attractive forces for double-layer overlap

549 543f 591

between dissimilar particles

596

between spheres

598

s.a. Debye-Hückel theory aprotic solvents, micelles in

436

area, surface, determination of

348

argon, liquid

652

arsenious sulphide

37t

association (s.a. equilibrium constant) closed — model of

443

degree of

446

rate constant for micellization

443

association colloid

10

434

(s.a. surfactant) This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

atomic force microscope

272

(s.a. AFMJ) ATR-FTIR

275f

attapulgite

23 (s.a. clay minerals)

attenuation of sound

250

attraction force

533

electrostatic

634f

influence on surface tension

49

s.a. van der Waals force Auger electron spectroscopy, see AES average force Avogadro constant, estimation of axially symmetric surfaces

647 31

120

104

B barrier, repulsion

581

(s.a. potential energy) basal surface of clay minerals basic shear diagram Beer’s Law

20 154f

751f

238

beidellite

21

bentonite

421

723

(s.a. montmorillonite) benzene

48t

residence time in micelles

470t

ring, effect on c.m.c

439

spreading on water

104

Bessel functions

369

BET equation

281

betaines as permittivity probe

466

beta-radiation studies of adsorption

69

This page has been reformatted by Knovel to provide easier navigation.

754f

Index Terms

Links

bilayers

16f (s.a. vesicle)

bile salts, aggregation of

447

binding constants see equilibrium constants binding of ions

504

to micelles

459t

Bingham flow

153

717

733e

751f

biotite

730

21

biphenyl, residence time in micelles

470

block copolymers in stabilization

294

blood and blood substitutes Bohr theory of hydrogen

5 536

boiling chips

76

boiling point, influence of bubbl size

79t

Boltzmann constant, estimation of

31

Boltzmann equation

319

boundary condns.

767

in fluid mech.

179

120

398

666 Boyle point

635e

(s.a. theta-point) Bragg diffraction

672

Bragg-Williams equation

297

Bredig arc

7

bridge functions

664

bridging by polymers in flocn

632

Brij 35 micelles bromide ion (s.u. halides)

This page has been reformatted by Knovel to provide easier navigation.

403

Index Terms Brownian motion

Links 24

181

397

683f

751

620 dynamics

667

Langevin equation for

184

of spheroids

185

bubbles in a fluid pressure inside

75

shape of pendant and sessile

105

temperature at equilibrium

79t

vapour pressure in

76t

buoyancy force in sedimentation

116

228

512

517f

C cadmium carbonate calcite (calcium carbonate) Hamaker constant

11f 203f 572t

calcium fluoride Hamaker const.

570t

calcium oxalate monohydrate

356

calcium phosphate

356

calomel electrode

347

Camp number

620

capacitance

356

354

125f

differential

314

506

integral

323

493

496f

323

352

507 of double layer

315f 493

of mercury drops in presence of organics of parallel plates

314 342f 785

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

capacitance (Cont.) of Agl interface impedance of capacity of system to do work capillary condensation

352

493

132e 780 87

electrometer

313

flow in —

173

737

73f

84

hydrodynamic fractionation, see CHDF rise in powder

91

viscometer (s.u. Ostwald) wetting of

84f

capillary pressure

84

effect on melting point

78

captive bubble method

606f

carbon tetrachloride, viscosity of

166t

carboxylates c.m.c. of

439t

dimers of

447

Casson eqn

752

cation exchange capacity, see c.e.c. Cauchy plot

569

Cauchy’s theorem

560

768

c.c.c., see critical coagulation concn c.e.c.

24e (s.a. ion exchange)

cell,electrochemical models centrifugal forces in Couette in fractionation

316

347

423 172 249

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

centrifugation

120

229

(s.a. sedimentation) for studying stability ceramics

605 4

sintering of cetyl pyridinium bromide

83 348

cetyl trimethylammonium bromide (s.u. CTAB) CFP, CFT,CFV

629

chain length, influence on c.m.c.

438

characteristic frequency

127

shear stress

750

time, molecular

674

charcoal

472

281

charge density, electric

33

components of, at interfaces

482

effect on colloid stability

581

effect on inner layer

353f

494f

in diffuse double layer

388f

484f

on Agl

344

relative

349f

on colloids, generation

356

on micelle surface

459t

on oxides

361

on spheres

366

volume

319

charge neutralization in flocculation

365

634f

charge regltn. during interactn.

583

CHDF

246

590f

chemical equilibrium, influence of particle size

80f

This page has been reformatted by Knovel to provide easier navigation.

507

Index Terms

Links

chemical potential

780

of ions, in adsorption of small particles of surfactant, contributions to

327

345

491

78 452

chemical reactions, thermodynamics of

781

chemisorption

277

(s.a. free energy of adsorption) chi (χ) parameter

297

300

chi (χ)-potential

308

327

334

20f

22f

435

460

china, see ceramics chloride ion, see halides chord length measurement

212

chromophore absorbed in micelles

460

cigar-shaped particles, see spheroids circulation of a vector field

775

circumferential flow in viscometer

172

Clausius–Clapeyron equation

76

clay minerals

18 365

influence on soil water retention isomorphous substitution in swelling of

86 24e 608f

cleavage faces of mica

609

closed association model

443

closure relations

682

cloud point

701

cluster–cluster interaction formation in nucleation c.m.c.

626f 97 14

factors affecting

438

coagulated sols, flow of

753

This page has been reformatted by Knovel to provide easier navigation.

Index Terms coagulation

Links 34

by electrolyte

36

by potential control

35

fractals produced in

625

in shear

619

kinetics of

616

coal slurry, flow of

717

coalescence of emulsion drops

604

731

18

coefficient of variation

220

cohesion between surfaces

92e

101

(s.a. work of—) collision frequency

617

colloid vibration current (potential)

422

colloids

2

association

10

classification

2

5

examples

3t

6

monodisperse

8

preparation, chemical

6

significance of

4

size limits

2

stability

33

(see coagulation) structure

669

(s.a. dispersions) combinatorial entropy

297

common intersection point

349

compact double-layer

324

compensation plot for micellization complexes, formn. on oxides

510

456f 487

505

(s.a. equilibrium constant) compliance, creep

150

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

components influence on degrees of freedom composites compressibility factor

70 3 656

computer modelling of surface dissociation

503

concentration effect in electrokinetics

423

on surfactant props.

14

gradient, detection of

230

variation across surfaces

59f

condensation, capillary from vapour, effect of curvature in dispersion preparation

614

89f 74 7

condenser, see capacitance conductance, electrical of plugs cone and plate viscometer congruence test

379

420

427 175

739

493f

495f

conservation of ions, in electrokinetics

397

of matter in fluid mechanics

158

conservative field

776

constitutive equation

147

contact angle

728

111f

and wetting

100

determination of

112

on rough surfaces

108

theoretical estimates

576

time-dependence of

112

contact dissimilarity energy

297

contact value theorem

614

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

contamination, surface

260

continuity equation

159

contour integral, evaluation of

768

contrast matching

687

control plate

595

convolution theorem

657e

693 777

copolymers in stabilization

628

core-shell systems

693

correlation function

243f

applications

654

calculation

663

measurement of

657

pair

642

644f

660

time dependent

652

662

674f

correlation of fluctuations

553

co-surfactant

17f

Couette viscometer , theory of

170

673f 683f

717f

for non-Newtonian fluids

728

Coulombic effects in polymer adsorpn.

634

Coulomb’s law

779

Coulter counter

232f

counter-electrode, platinum

310f

counterions

504

effect on coagulation

37t

on micelles

439

valency, significance of

603

(s.a. DLVO theory, electrolyte) coverage in adsorption

278

of polymers

528

Cox-Mertz rule

727

677

c.i.p., see common intersection point This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

cratic contribution in micellization

451

creep

146

creep compliance

147

150f

critical approach distance

574

chain length and micelle size

475

coagulation concn

37

flocculation point

629t

micellization concn (see c.m.c)

435

radius for nucleation

97

cross-product of vectors

775

crystal, charge defects

356

equilibrium shape of

81

melting point

78

solubility of

80

structure of suspension vapour pressure of CTAB

37t

671f 78 464f

707f

(s.a. alkylammonium) in soap films layers on mica

607f 612

cumulants

245

cumulative distribution curve

222

curl of a vector field

775

curvature of surface

53f

effect on chemical potential

78

melting point of crystals

78

pressure

52

surface tension

97

This page has been reformatted by Knovel to provide easier navigation.

604

Index Terms

Links

cylinder double layer around

369

micellar

16f

cylinder-in-cylinder viscometer cylindrical metal analyser

170 267f

D damping of electron motion Deborah number de Broglie wavelength

132e 145

147

212e

685

Debye forces

536

Debye–Hückel parameter

320

715

(s.a. Debye length) Debye–Hückel theory

320

596

664 (s.a. approximation formulae) Debye length

670

Debye relaxation

131

Debye–Scherrer cone

673

decay curves for emission from micelles

463

deconvolution

243

defects in crystal lattice

356

deformation elastic, of solid

145

of fluid

153

degrees of freedom of association

70 446

del operator

772

delta function

565

density, gradient of, at surface

614

depletion interactions

634

depolarization of micelle fluorescence

471

459t

This page has been reformatted by Knovel to provide easier navigation.

644

Index Terms

Links

Deryaguin approxn

549

563

desorption

285

287e

from Hg-solution interface

342f

detergent, see surfactant deuterium oxide

687

690t

deviatoric stress tensor

164

715

diameter of particles

211

(s.u. size) dielectric constant

785

see dielectric permittivity; — response dielectric dispersion dielectric displacement

417f

420

126

774

complex — vector

129

dielectric permittivity

125

at interfaces

319

385

465t

515 average, in compact EDL

325

relative

125

785

128

131

570t

769

construction of

563

568f

to static electric field

124

516

dielectric response

die swell

556

725f

differential capacitance, see capacitance viscosity diffraction of light, Bragg

716 672

diffuse layer (s.u. electrical) diffusion

24

27

30f

28

241

393

401

662

404 coefficient

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

diffusion (Cont.) spheroids, rotational

185

translational

187

gradient in conc suspensions

617

181

diffusivity, see diffusion coefficient dilatant flow

153

dimension, fractal

626

dimensionless variables

720

322f

407

680

750

dimer formation by surfactants

447

dipole-dipole interaction

535

dipole moment, induced

536

412

dipole potential, see chi-potential dipole strength

189

disc centrifuge

229

discreteness of charge effect

338

193

421

discs (s.a. spheroids) light scattering from

141f

rotational diffusion

185

disperse phase

743f

2

dispersing agent, see surfactant dispersion force theory

533

552

767

(s.a. London, van der Waals) dispersion medium

542

dispersion relation

555

558

roots of

559

767

768

dispersions

2 181

193

254

423

698

760

concentrated mixed

704

model

8

preparation

6

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

dispersions (Cont.) stabilization

33

582

structure of

669

760

transport properties of

188

749

(s.a. colloids) displacement, mean square

31

displacement vector, see dielectric dissimilar surfaces, interaction of

594

dissipation factor

237

( s.a. damping, viscosity, friction) dissociation, degree of, see equilm. const. dissymetry ratio

140

141f

216

663

distance of approach, s.u critical sepn distribution function adsorbed polymer bimodal

299 224f

Gaudin-Schuman

224

ions near surfaces

350f

log-normal

217

Lorentz

241f

molecules at surface

614f

normal (Gaussian)

223

221

pair (s.u. correlation function) particle orientation quencher in micelles

185 472e

Schulz

225

size

213

disturbance velocity in sedimentation

179

div (divergence)

773

grad

324

of a tensor dividing surface between phases

742

774

777e 59f

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

DLA

625

DLVO theory

39f

581

747f

761 experimental confirmation

604

in suspension flow

755

DME

310f

capacitance Dodecyl compounds in micelles

332 521t

696f

441f

459t

domains of close approach

630

Donnan e.m.f.

354

Doppler line broadening

241f

dot product of vectors

772

double bonds, effect on c.m.c.

438

393

double layer, see electrical–– doublets of spheres in flow

746

Dougherty-Krieger relation

425f

drag (viscous), force

28

drainage of wetting films

606

drop of liquid, flow in

744

free energy of formation of motion in flow

121

95 744

on surface, shape see sessile & pendant vapour pressure of

74

dropping mercury electrode, see DME drop weight in surface tension measure

312

Dukhin number

407

dye, fluorescent, in micelles

470

dynamic light scattering

241

mobility

254

SIMS

268

392

This page has been reformatted by Knovel to provide easier navigation.

168

Index Terms

Links

dynamic light scattering (Cont.) structure factor

661

viscosity

726

dynamics of micelles

466

E eccentricity of shape

117

(s.a. spheroids) e.c.m.

313 (s.a. p.z.c.)

Esin and Markov effect on

331f

EDL, see electrical double layer effective medium theory Einstein’s eqn., for diffusion

197

425f

754

29

for viscosity of suspension

190

Einstein-Smoluchowski equation

31

elastic floc model

753

elastic response

145

760

electrical double layer

34f

304

365

compact region

324

335f

353f

484f

490

322f

484f

diffuse capacitance of compression

332 36

dynamics

400

free energy of

584

Gouy-Chapman model

317

Stern-Grahame version, see GCSG Helmholtz model

367

inner, see compact on oxide surfaces

361

overlap of

581

This page has been reformatted by Knovel to provide easier navigation.

494f

Index Terms

Links

electrical double layer (Cont.) statistical mechanics of

342

thickness, see Debye length electric field

770

alternating

416

dielectric response to

124

in double layer

516

electroacoustics

252

electrocapillarity

312

422

extracapillary maximum, see e.c.m. electrochemical potential in Agl system

307 345

electrochemistry

304

electrodes, polarized

309

reference reversible electrokinetics

310

331

346 373

395

489

493

equations of

375

397

in alternating fields

416

units

785e

(s.a. electroosmosis, electrophoresis, streaming potential, electroacoustics) electrokinetic sonic amplitude electrolyte, colloidal

252 2

(s.a. micelle, soap, surfactant) electrolyte conductivity

407

in porous plugs

378

effect on d.l. potential

36

on micellization

440

on e.c.m.

331

on surface potential of oxides

358

416

361

This page has been reformatted by Knovel to provide easier navigation.

426 415

Index Terms electrolytes, indifferent

Links 36

influence on attractive forces

571

simple, effect on coagulation

603

(s.a. ions) electromagnetic theory

124

electrometer, Lippmann

313

electron beam

262

binding energy

267

electrochemical potential of

307

response of-to a.c.field

263f

electron loss spectroscopy

263f

electron microscope

207

electron probe microanalysis

209

electro-osmosis

375

in plugs

377

electrophoresis

380

electrostatic patch interaction

633

potential of a phase

552

390 390

305

ellipsoidal particles (s.u. spheroid) elliptic integrals

587

Ellis model of shear

719t

e.m.f, Donnan

354

(s.a. potential, electrical) emulsification, spontaneous

18

emulsions

16

sedimentation of

119

viscosity

744

end correction in viscometry

173

end-to-end length

295

energy dissipation see viscosity interfacial, see surface energy

This page has been reformatted by Knovel to provide easier navigation.

412

Index Terms

Links

energy dissipation see viscosity (Cont.) internal levels of molecules

59 630

of micellization

456

of mixing of polymer

630

entropy, configurationl of gas

635e

correction in adsorption

337

of flocculation

630

of micellization

456

of mixing of polymer

630

equilibrium adsorptn. in approach

582 80f

constant, in micellization

443 27

liquid-vapour, influence of curvature

77

shape of crystals

81

unstable, of particle size

604

487

in a phase

theory of pseudoplastic flow

528

60

chemical, effect of particle size for adsorption

780

263

enthalpy of flocculation

surface

654

60

752 94

equipotential surfaces

77If

error function

222

ESA

252

ESCA, see XPS Esin and Markov coefficient

329

e.s.r. spectroscopy of micelles

472

ethanol (ethyl alcohol)

166

evanescent wave

338

273f

evaporation, work done in

56

excess Gibbs free energy

61

excess (surface) concn.

58

This page has been reformatted by Knovel to provide easier navigation.

349

Index Terms

Links

exchange reaction

488

excimers

460

excited states, molecular in interactn.

537

extensional flow

191

195

131

238

extension, of polymer, see RMS length extinction coefficient

F fabric conditioning

100

Faraday constant

320

fast coagulation

616

FECO fringes

270

Feret’s diameter

211

ferric oxide

12f

512f

525 coagulation of

37

Fick’s law of diffusion

28

field, conservative

776

electromagnetic, propagation time

547

external, thermodynamics of

782

flow —

188

flow fractionation

248

emission microscope ion microscope

29

264f 262

(s.a. electric field) films, soap, thickness of

606

wetting, on solids

605

flat double layers flocculation by polymer

586 41

floe volume ratio

754

floes

625

Flory-Huggins theory

295

628

This page has been reformatted by Knovel to provide easier navigation.

517f

Index Terms

Links

Flory point, (s.u. theta-point) flotation of minerals

105

flow field, in a suspension

188

357

582

157

714

(s.a. flux) in sedimentation

117f

in viscometers

717f

of dispersions

188

profile in capillaries flow behaviour

737f 153

(s.a. rheology, Newtonian liquid) fluctuations in Brownian motion in light scattering

184 243f

in soap films

607

fluid mechanics

157

fluids inelastic, time independent

715

time dependent

740

statistical mechanics of

638

viscoelastic

724

fluidity, microscopic, in micelles

470

fluorescence

276

probes in micelles fluoroscopy, total internal reflectn.

461

470

276f

fluoride ion, see halides fluorine, effect on c.m.c.

439

fluorocarbons in micelles

471

fluorodecanoic acid, contact angle on flux

577f 29f

diffusive

28

of fluid

159

of material

29f

of particles in coagulation

617

in force field

397

622

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

flux (Cont.) in sedimentation of vector field foam

182 773f 3

fog, formation of force between macrobodies

96 541

particles

38

spheres

646

two plates

39f

body, in sedimentation diffusion

179 28

image, of ions

339

long- and short-range

160

of adhesion, measurement

65

99

on fluid element

160

van der Waals

39f

(s.a. dispersion force) Viscous

28

117f

121

167 forces, additivity of

539

654

form factor

139

659

691

673

708 formamide, effect on aqueous micelles

442

micelles in

436

Fourier transform

655

660

777

708

fractal nature of aggregates

625

fractionation, see CHDF free energy, definition

780

elastic, in polymer mixing

631

electrical

584

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

free energy, definition (Cont.) Helmholtz of attraction of adsorption of droplet formation of formation of double layer of micellization of polymer mixing of surface of surfactant, contributions to

780 560 337

521t

96 584 449f 630 45 452

(s.a. Helmholtz – –, Gibbs – –) free energy change in double layer interaction between dissimilar plates in micelle formation

560 595 452

free polymer, effect on stability

634

free volume of polymer

296

freeze-thaw stability using polymers

628

frequency, characteristic, electron

127

of transition of e.m. radiation

585

297 536

557f 128

134

241

118

121

548 friction coefficient

29 182

(s.a. damping) fringes of equal chromatic order

270

fructose, effect on micelles

442

FTIR

275

functional, definition of

558

fur, animal, wetting of

105

fuzzy sets

203

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

G Galvani potential

305

gamma function

32

gap width, in Couette

171

gauche conformation of chains

476

Gaussian coils

139

Gaussian distribution

27f

GCSG model

325

gel layer on oxides

364

Gibbs adsorption equation Isotherm for charged surface

316 177e

731

483

63 63

67f

68

311

Gibbs convention

59f

Gibbs dividing surface

72e

312

Gibbs-Duhem eqn

63

74

783

Gibbs free energy

452

521

781

partial molal, see chemical potential surface – –

62e

(s.a. free energy) Gibbs surface excess glass

63 362

electrode

354

ruby

3

glycerol

166t

gold sol

5

Gouy-Chapman theory

317

in micellar solutions

454t

limitations of

342

gradient

37

771f

diffusion in cone suspension

181

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

gradient (Cont.) of chemical potential

28

of concentration

29

grad of a scalar field

770

of a vector

776

graft copolymers in stabilization

628

Grahame isotherm

337

Grahame model of double layer

326f

graphite

292

gravitational field

116

forces in soap films

606

in moving fluid

169

thermodynamic effects of group, hydrophilic or hydrophobic surface, dissociation of growth of crystals

782 10 356

501

93

Guinierp lot

140

692

gyration, radius of

140

693

H habit of crystals haematite

203f 12f

see ferric oxide Hagen-Poiseuille eqn.

736

half cell, see electrode half-spaces, interaction between halides, adsorptn at Hg-soln interface

541

561

336f

(s.a. silver halides) Hamaker constants effect of suspn. medium

533

563

542

571

(s.a. van der Waals, forces)

This page has been reformatted by Knovel to provide easier navigation.

572t

Index Terms

Links

hard sphere model

650f

661

188

749

53

90

transport properties of harmonic mean HDC

246

head group

522

dissociation of

356

of surfactant, steric effects

472

significance in micelle formn

438

heat capacity, influence on micelles

440

heat of adsorption

282

of phase transition in small crystals Helmholtz model of double layer capacitance of

501 453 292

78 367f 323

(s.a. OHP, IHP) Helmholtz free energy, surface at electrified interface hematite

62 317e 12f

see ferric oxide hemimicelles

524

Henry eqn.

382

Herschel-Bulkley model

717

heterocoagulation

597

heterodisperse systems

213

753

(s.a. polydispersity) heterodyning

244

high frequency conductance

420

histogram of particle sizes

213f

HN Capproxn.

664

Hofmeister series

615

homodisperse sols, see monodisperse homodyning homogeneous nucleation

243 93

This page has been reformatted by Knovel to provide easier navigation.

669

Index Terms

Links

homologous series, alkanes

452

Hooke’s Law

145

HOTS

141

Hückel equation

381

hydration forces

613f

of head groups in micellization

438

(s.a. structural forces) hydrazine, micelles in

436

hydrocarbon chains in micelles

438

surface tension

48t

576

(s.a. alkanes; wax) hydrocolloid

269

hydrodynamic chromatography, see HDC hydrodynamic correctn. to stability ratio

623

focusing in particle counters

235

fractionation

246

interactions

179

hydrodynamics, vector theory of

157

hydrogen atom, Bohr theory of

536

hydrogen bonds

19

41

447

516 hydrolysis of metal ions

513

hydrophilic head groups

13

hydrophobic interaction

615

in micelle formation

437

457

hydrostatic pressure

173

379

hydroxyl groups on oxide surfaces

356

501

hypernetted chain approxn.

664

hysteresis

723f

453

740

I ideal gas eqn in 2 dimensions

103

This page has been reformatted by Knovel to provide easier navigation.

606

Index Terms

Links

ideal liquids and solids

145

i.e.p.

491 (s.a. p.z.r and p.z.c.)

IHP

326f

image force image formn. by electron microscope immersion lens impedance, electrical incompressibility of liquid in Kelvin and Laplace equations

339 207f 205 132e

416

158

170

97

indicators of pH in micelles

464

inelastic fluids

715

time dependent

655

721

740

inertia factor

422

ink-bottle effect in adsorption

89f

inner double-layer, see electrical – inner Helmholtz plane, see IHP inner potential, see Galvani– inner product

772

insecticide application

100

integral capacity (s.u. capacitance) intensity of fluorescence of scattering

461 133

236

657

669 interaction between molecules

535

639

between spheres

598

746f

hydrodynamic, in settling

179

in micelles

435

472

of macrobodies

539

541

of particles

581

591

of polymers

628

potential energy of

601

This page has been reformatted by Knovel to provide easier navigation.

554

Index Terms

Links

interaction between molecules (Cont.) surface conditions during

582

(s.a. van der Waals) intercalation in clay minerals interface, influence on sedimentatn.

23 119

liquid–gas

63

solid–gas

259

interfacial charge

482

potential, see under potential electrical structure

614f

651f

tension

48

574

interferometry

230

270

interior of micelles

460

472

intermediate scattering function

662

interpenetratn. domain of polymers

630

intrinsic binding constants

488

501

192

754

24e

488

(s.a. equilibrium constants) intrinsic viscosity iodide, see halides; silver iodide ion exchange (s.a. equilibrium constant) ionic strength

320

(s.a. electrolyte) ionization of micelles potential of H atom

466 536

ion pairs

516

ions, binding

504

to micelles

459t

discreteness correction

338

distribution near surfaces

350

indifferent metallic, hydrolysable

36 513

This page has been reformatted by Knovel to provide easier navigation.

502

Index Terms

Links

ions, binding (Cont.) multivalent, adsorption

509

potential-determining

33

specifically adsorbed

327

ion self-atmosphere effect

338

ion size, influence on adsorption

337

344

485

iron(III) oxide, see ferric oxide isodisperse sols, see monodisperse sols isoelectric point, see i.e.p. isomorphous substitution isosteric heat of adsorption

22 286e

292

isotherm, see adsorption —

J Jones-Ray effect

605

jump potential, see chi potential

K Kaolinite

20f

287e

(s.a. clay minerals ) Keesom forces

535

Kelvin equation

72

limits of applicability kidney stones kinetic pressure kinetics of coagulation

97 355 49 616

of flow

750

of micelle formation

467

of nucleation kinetic theory of pressure Kirchhoff’s law

96 654 81

This page has been reformatted by Knovel to provide easier navigation.

421

Index Terms

Links

Krafft point

440

Kramers-Kronig relation

557

Kronecker delta

165

Kuhn segment (of polymer)

295

kurtosis

215

L LALLS

140

659

Langevin equation

184

Langmuir isotherm

278

499

Laplace’s equation

411

774

Laplace operator

319

774

Laplace pressure

72

90

latex, polymer

198f

229

lecithins

497f (s.a. phosphatidyl compds.)

LEED

263f

Lennard-Jones fluid potential

653 640f

lens, oil, on water

102f

lifetime, emission

462

Lifshitz theory

552

of surface tension light scattering

574 133

angular dependence

142f

apparatus

240f

by large particles

239f

dynamic, see PCS

241

for particle sizing

141f

in stability studies

607

theory

133

Mie

661

143f

236

141

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

light scattering (Cont.) Rayleigh

134

RGD

139

vector

139

140f

linearization

147

407

of Navier-Stokes equation

178

398

of Poisson-Boltzmann equation

320

linear response theory

655

line integral

769

lines of force

771

lipids

775

497f

lipophile solubility in micelles

441

liposomes, see vesicles Lippman equation

312

liquid junction

331

liquid films

605

liquids, ideal viscous

146

in capillaries

737

in contact

71e

structure of

641

347

(s.a. fluids) London force

536

(s.a. van der Waals —) loops, polymer Lorentz function loss modulus

41f 133e

242

726

low-angle laser light scattering, see LALLS low energy electron diffraction, see LEED solids

576

low frequency electrokinetics

418

lubrication approximation

176

This page has been reformatted by Knovel to provide easier navigation.

241 407

Index Terms Ludox

Links 10

s.a. silica luminescence quenching

462

lyophilic solute

441

effect on surface tension lyotropic series

68f 615

M macromolecules, see polymers macropores

284

macropotential in adsorption

337

339

133e

559

magnification of microscope

205

208

Martin’s diameter

211

MASIF

271

mass area mean diameter

216t

magnetic permeability

spectrometry, see SIMS, TDMS matrix representation of light scattering

240

stress

161

tensors

776

Maxwell e.m. eqns

555

frequency

421

stress

587

mean

559

214

curvature

54

diameter

216t

geometric –

217

harmonic –

54

ionic activity

69

particle size

214

spherical approxn. (s.a. MSA)

664

This page has been reformatted by Knovel to provide easier navigation.

767

Index Terms

Links

mean value theorem

148

melting, effect of curvature memory function

78 128

meniscus, shape of

110e

and wetting

104

mercury, injection porosimeter surface tension of viscosity

147

92e 48t

575

166t

mercury drop, see DME mercury-solution interface

309

mesopores

284

metals, dielectric response

569

Meter model of shear

719t

mica sheets capillary condensation on

21 98

Hamaker constant

572t

interaction between

609

streaming potential of

610

mica, white (muscovite) micellar solutions thermodynamics of micelle

22f

270

10

435

450 15f

aggregation numbers

459t

dynamics of

466

formation of

14

457

shape and size

16f

24e

solubility in

17f

441

statistical mechanics

476

micellization

24e

(s.a. c.m.c) as phase transition

445

effect of molecular packing on

472

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

micellization (Cont.) in non-aqueous media

457

phenomenology of

438

microemulsions

17f

microfluidity in micelles

470

micropores

284

micropotential for adsorbed ions

339

microrheology

741

microscope

204

atomic force, see AFM electron

207

(s.a. SEM and STM) field emission

264f

field ion

262

optical

205

scanning, see SFM & STM total internal reflectance, see TIRM ultra-

206f

microtubule

15

microviscosity

470

microwaves

567

Mie scattering

141

mill, colloid

7

mineral, clay —, (s.u. clay) ores (s.u. flotation) oxides (s.u. oxides) MINEQL

503

mixtures, coagulation of

596

mobility, dynamic

254

electrophoretic

380

mode, vibrational, of e.m. radiation

422

560

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

models of double layer

305

Grahame

482

326f

site binding

487

triple layer

488

(s.u. Helmholtz; Gouy-Chapman; Stern) models of micellisation modifiers of crystal habit modulus, relaxation shear

443 203f 147 761

storage and loss Young’s

151

726

145

molar mass (sym. molecular weight) determ. of

138

of polymers

121

and RMS length molecular dynamics

138

294 666

molecule, amphipathic or amphiphilic

10

hydrophobic, sol’ty in micelles

441

interaction with a macrobody

539

moment, turning, see torque moments of a distribution monochromator for neutrons monodisperse sols

215 687f 11f

s.u. size monolayers, gas

284

spreading

103

576

monomer exchange in micelles

443

467

Monte Carlo calculations

666

montmorillonite

23f

24e

(s.a. clay minerals) motion, simple harmonic

126

This page has been reformatted by Knovel to provide easier navigation.

154

Index Terms

Links

moving boundary in electrophoresis

392

MSA

664

multiple equilm. models of micelles

445

multiple exposure in SEM

210

muscovite (syn. white mica)

22f

678

682

N Navier-Stokes eqn.

167

in Couette flow

172

in sedimentation

178

397

(s.a. linearisation) negative adsorption thixotropy Nernst equation

299f

349

722 345

486

breakdown

361

503

slope

355

490

782 492

506f neutron reflectivity scattering

705 669

amplitude

686t

by adsorbed films

695

by suspensions

684

Newtonian liquid

688f

146

165

161

167

397

715

728

734

suspensions

741

750

non-stick surfaces

576

Newton’s Laws in fluid mechanics n.m.r. spectroscopy non-interpenetrational domain non-ionic surfactants

471 631f 700

(s.u. surfactants) non-Newtonian fluids

This page has been reformatted by Knovel to provide easier navigation.

Index Terms normal distribution

Links 27f

(s.a.u. distribution) vector

160

771

normal stress

176

715

difference

726

734

no-slip condition

190

377

nuclear magnetic resonance

471

(s.a. pressure)

nucleation

740

9f

heterogeneous

107

homogeneous

93

number averages

216t

numerical methods

383f

nylon, electrokinetics of

489f

O oblate spheroids

202f

OCM model

679

Oden balance

227

OHP

326f

oil immersion technique

205

oil spreading on water

102

oil-water interface, adsorpn at

71e

oleophiles, sol’ty in micelles

441

olive oil, viscosity of

166t

one component macrofluid (s.u. OCM) Onsager reciprocity relations

384

opal, synthetic

10

operator, vector

772

774

ores (s.u. flotation, oxides) organics, influence on micelles

441

This page has been reformatted by Knovel to provide easier navigation.

776

Index Terms

Links

orientation of particles

117

185

190

741

Ornstein–Zernike eqn.

663

665

oscillating repulsive forces

613

in flow

oscillating shear

150f

oscillator, damped

132e

strength

725

537

565

osmotic compressibility

660

669

osmotic pressure experiments

605

in interaction theory

583

in sedimentation/diffusion

182

of polymer solutions

628

Ostwald ripening

38

viscometer

173

outer Helmholtz plane

586

734

326f

outer potential, see Volta — overlap of double layers (s.u. interaction) oxides, metallic congruence test on

501 495f

charge generation on

356

models of

361

P packing, in micelles

472

paint

144

pair correlation function

642

pair potential

639

(s.u. interaction) pairwise additivity

539

palisade structure in micelles

442

Pallman effect

354

677

parabolic flow (s.u. Poiseuille) This page has been reformatted by Knovel to provide easier navigation.

614

Index Terms

Links

paraffin; see alkane; hydrocarbon, wax. partial molal quantities

784

(s.a. chemical potential) particles in flow (s.u. orientation) shape of size of

202f 200

see size, particle, detn. of small

72

excess pressure in

78

solubility of

80

partition function of a vibration patch interaction

560 634f

P.B. equation (s.u. Poisson) PCS

241

in electrokinetics

661

392

p.d.i., see ions, potential-determining Péclet number

191

194

pendant drops

105

penetrating background

683

PEO

527

629t

Percus–Yevick approxn.

661

664

perfectly polarised electrode

309

periodic boundary conditions

666f

permittivity, see dielectric – persistence parameter (polymer)

295

perturbation methods

665

theory

537

phase rule

69

separation model, of micellization phase angle (difference)

445 658

in electrical signals

130

inrheology

152

This page has been reformatted by Knovel to provide easier navigation.

678

Index Terms

Links

phases, potential difference between

307

pH at interfaces

464

error

354

phlogopite

21

phosphate ion adsorption phosphatidyl cmpds.

275 497f

513

phospholipids, see vesicles photo-electrons

267

photon correlation spectroscopy, see PCS physical adsorption, energy of

277

pipette method of size determn.

226

Planck’s constant

212

536

plane of shear, see surface of shear plant roots, water uptake by plasma frequency plastic, contact angle on flow

86 569 577f 154

754

platelets, clay

20f

plate-like interaction

586

PMMA

570t

690t

175

717f

point of zero charge, see p.z.c of zeta reversal, see p.z.r. Poiseuille equation in HDC Poiseuille flow in capillaries in electrokinetics

378

Poisson-Boltzmann eqn

320

342

in electrokinetics

376

398

solution

317

for cylinders

369

for spheres

365

Poisson distrn. for quencher in micelles

472e

This page has been reformatted by Knovel to provide easier navigation.

754f

Index Terms

Links

Poisson equation

319

in electrokinetics

397

polar groups and c.m.c.

438

polarity, interfacial

465

polarizability

536

low, surfaces

576

molecular

137

774

784

472

143e

polarization of light in fluidity study

471

in scattering

135f

polarization vector

125

polarized electrode

309

313

double layer

406

409

pollution, colloid effects

5

poly(acrylamide)

632

polyacrylate

629t

poly(acrylonitrile)

690t

poly(dimethylsiloxane)

629t

polydispersity, degree of

219

of micelles

633

448

polyelectrolytes

632

poly(iso-butylene)

629t

polymers, organic

294

adsorption

41

298

632

748

bridging cationic, structure

632f

dielectric data

570t

flocculation by

416f

41

Hamaker constant

572t

in solution, stabilizing action

634

632

This page has been reformatted by Knovel to provide easier navigation.

628

Index Terms

Links

polymers, organic (Cont.) osmotic pressure of

628

solution thermodynamics of

295

630

229

690t

748f

143f

662f

687

696

699f

(s.a. latex, polymer) poly(methylmethacrylate) (s.u. PMMA) poly(oxyethylene) see PEO polystyrene scattering poly(tetrafluoroethylene) see PTFE pore size

92e

Porod region porosimeter, mercury injection

142f 87

porous layer on oxides

364

porous plug , electrokinetics in

377

potential, chemical adsorption

327

499

potential, electrical around cylinder

369

sphere

34f

365

at flat surface

317

322f

between dissimilar plates

594

influence on coagulation

603

interfacial

464

gradient of

772f

of a phase

305

of highly charged systems

424f

of liquid junction

354

of mean force

646

surface

308

theory of

305

potential-determining ions (s.u. ions)

This page has been reformatted by Knovel to provide easier navigation.

503

Index Terms

Links

potential energy barrier to coagulation

621

of interaction

585

approximation formulae

591

total

601

potential, gravito-chemical

119

potentiometer in electrochemistry

310

639f

pottery, see ceramics powders, determ. of contact angle of power, dissipation law model of shear

91 132e 720f

733e

736f

753 precision of distribution

27f

pressure average, in flow

165

difference across surface

52

drop in capillary

173

dynamic

51f

effect on micellization

440

effect on surface tension

49

excess in small solid particles

80

hydrostatic inside bubble or drop

189

173

379

736

313

607

52

molecular basis of

654

static

51f

stat. mech. calcn.

665

primary minimum

602f

610

yield value

718

756

primitive model principal radii of curvature

679 53

probability for particle pairs (s.u. correlation) paper

223 This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

probe molecules

461

profile of bubbles and drops

105

313

profile of concentration

230f

299

350

projected area diameter

211

propagation time of e.m. field

547

properties, physical, of surfactants

434

protection by polymers

628

pseudoplastic flow

153

718

749

pseudopotential

685

PTFE, contact angles on

578f

pulse counters, electrical

232

radiolysis in micelle studies

467

pyrene in micelles

460

pyridinium N-phenol betaine

466

pyrophyllite

22f

s.a. clay minerals p.z.c. determination

331

of silver iodide

347

pristine

511

s.a. e.c.m. p.z.r.

518f

521

Q Quadrupole–dipole interaction

536

quantum theory, perturbation

537

quartz, crystalline and fused

570t

Cauchy plot

569

Hamaker constant

572

710

quasi-elastic scattering (QELS) (s.u. PCS) quaternary ammonium ion, head grp

459t

quenching of fluorescence in micelles

461

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

R Rabinowitch-Mooney eqn.

738

radiotracers in adsorption studies

484

radial distribution function (s.u. correlation) radius, critical

96f

limiting, for Kelvin eqn

99

of curvature

53

76t

144e

295

of gyration rain-making, with Agl

353

random coil

294

light scattering from random walk

141f 25

295

(s.a. Brownian motion) rate constant for micelle formation

467

rate of coagulation

616

(s.a. kinetics) rate of nucleation

96

rate of shear, see shear rate rate of strain tensor

165

in flowing suspension

189

physical significance of

165

rationalization of units

715

785

Rayleigh-Gans-Debye scattering , see RGD Rayleigh light scattering

134

Rayleigh ratio

137

reaction quotient

782

reciprocity reins, in electrokinetics

384

reduced (electrical) potential

322f

reflection electron diffraction see RHEED reflection of light

273f

of neutrons

705

This page has been reformatted by Knovel to provide easier navigation.

299

Index Terms

Links

refractive index

684

and dielectric response

131

gradient of

230

of neutrons

686

Reiner-Riwlin eqn.

730

relative percentage frequency

217

relative permittivity, see dielectric — relaxation

129

Debye —, for dipoles

131

frequency

566

modulus

147

time

167

in micelles repeptization replication for TEM repulsive force between particles

406

469t 40 209 598e

rescaled MSA theory, see RMSA residence time in micelles

470t

resistivity in particle counters

234f

resolution of microscope, electron optical

207 205f

resonance frequency

128

response to mechanical stress

144

retardation of attractive forces

547

Reynolds number in sedimentn.

178

Internal

745

RF value in HDC

247

RGD scattering

139

RHEED

265

rheology of colloids W

144

microscopic models

749

rheopexy

154

556

750 692 713 722

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

rigid body motion of fluid

163

rigidity modulus

726

ripples in soap films

607

RLA

625

RMS displacement

31

in self-diffusion

183

RMS end-to-end length of polymer

295

RMSA theory

682

762

702

rods (s.a. spheroids) light scattering from

141f

micellar

16f

suspension of, viscosity

742

475

root mean square (s.u. RMS) rot (or curl) of a vector field

775

rotation

163

in Couette viscometer rotational diffusion

172 185

roughness (sym. rugosity) effect on capillary condensation

90f

effect on contact angle

111

of polished surfaces

270

ruthenium oxide tris-(bipyridyl) ion

356 462

S salt concentration, see electrolyte — SANS

659

camera

688f

sapphire, Hamaker constant

572

Saxen’s relation in electrokinetics

384

scalar field, gradient of

770

scalar product of vectors

772

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

scanning tunnelling microscopy

268

force microscope, see AFM probe microscope scattering, intensity of

273 658

amplitude (length) density

686

matrix

240

studies of structure

669

vector

139

241

392

(s.a.u. light— & neutron —) Schlieren optics Schultz-Hardy rule

230f 603

SDS (s.a. dodecyl compounds) c.m.c.

441f

ion binding in

459t

micelle properties

15f

461f 454t

469t

secondary ion mass spectrometry minimum

611

second virial coefficient

238

sedimentation

178

226

centrifugal

120

229

for stability study

605

coefficient

121

effect of particle interaction on

179

equilibrium

119

field flow fractionation

248

of cone., suspensions

179

under gravity

116

velocity

178 122

226

181f

seeding, see nucleation segment density distribution (polymer)

299f

self-assembly, s.a. aggregation; micellization self-atmosphere effect

338

This page has been reformatted by Knovel to provide easier navigation.

228

Index Terms

Links

self-diffusion in cone suspensions

183

Sellmeier damped oscillator

131

SEM

209

separation, interparticle

543f

sessile drops

105

settling radius, equiv

118

SFA

270

SFFF

248

SFM

272f

shadow casting for TEM

208f

shape, effect of impurity adsorption

203f

of crystals

81

of liquid drops

105

of meniscus

104

of micelles

472

shear behaviour

590f

609

201

719t

characteristics of suspensions

144

induced coagulation

619

modulus

146

oscillating

151

plane

377

385

rate

146

165

713 761f

(s.a. rate of strain) relaxation modulus stress

147 146f

thinning behaviour

716

749

viscosity, (s.u. viscosity) silica

709

gel layers on sol

758

364 362f

506

silicates, layer-lattice, see clay minerals silicon dioxide, see silica; oxides This page has been reformatted by Knovel to provide easier navigation.

716

Index Terms

Links

silver halide (iodide) sol

344

ion adsorption

362f

345

negative adsorption on

351f

Nernstian behaviour of

360

506

surfactant adsorption on

519

527

SIMS

494f

268

sintering

83

site dissociation

357

359f

501 SI units, rationalized

784

size distributions

213

(s.u. distribution) different averages

216t

of micelles

468f

size, particle, detn. of

200

acoustic methods

250

electroacoustics

252

hydrodynamic methods

246

light scattering

236

microscopic observation

204

sedimentation

226

pulse counting (zone sensing)

232

skewness

215

sky, blue colour of

137

slip between solid and liquid

732

velocity in electro-osmosis

391f

slow coagulation

621

slurry

146

255f

411

small angle neutron scattering (s.u. SANS ) particles, thermodynamics of

72

This page has been reformatted by Knovel to provide easier navigation.

487

Index Terms

Links

Smoluchowski eqn.

377

381

408

423 in coagulation soap films, thickness of

616 606

soap, ionic, see surfactant sodium soaps

13f

(s.a. SDS) cholate, aggregation of

447

fluoride, electrocapillarity

315

soil water, capillary forces in

86

solids ideal, elastic

145

low-energy

576

surface energy

49

solid-liquid-vapour interface

100

solubility of small particles

80

solubilization by micelles

441

solvation of particles

614

influence on sedimentation

460

118

solvent, good, for polymer

628

solvent structure

602

sound attenuation & velocity

250

spacial frequency

656

specific adsorption

327

detection of

328

651f

336f

spectroscopy attenuated total reflectance of micellar solutions

275f 460

471

spectrum, absorption (s.u. absorption) sphere, light scattering from spheres, electrical double layer on

141f 365

interaction between

598

rotation in fluid flow

745f

This page has been reformatted by Knovel to provide easier navigation.

504

Index Terms

Links

sphere-plate interaction

543f

spheroids, oblate and prolate

202f

Brownian motion of

185

rotational diffusion of

186

sedimentation of

117

viscosity of

192f

spreading coefficient

104

spreading pressure

103

stability

743f 750

38

of colloids stability ratio

616 616

624f

stabilization, by polymer, see stericby free polymer

634

electrostatic

621

stabilizers (s.u. polymers, surfactants) standard deviation

214

standard distribution (s.u. normal) standard state for micellization

455

stannic oxide

362f

static pressure

51f

structure factor statistical thermodynamics mechanics

657 560 342

of chain packing in micelles

476

of polymers

295

statistics of size distrn.

213

stearate, sodium

13f

steric stabilization

638

40

628

Stern isotherm

337

509e

Stern layer conductance

428

Stern model of double layer

325

on oxides

495

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Stirling approximation

Links 26

STM

268

Stokes equations

178

in electrokinetics

397

Stokes settling radius

116

strain

145f

complex

152

rate

151 see shear rate

streaming current potential streamlines orientation of particles along stress

389

420f

378

389

117 190

741

144

applied

146f

716

normal

168

715

oscillating

151

relaxation

148f

tensile

149f

86

stress in a moving fluid at equilibrium stress tensor

145f 162 161

macroscopic

166

in suspensions

188

structural effects in flow

751

forces in stability

602f

structure factor, static

188

657

makers & breakers

442

measurement of

669

suction pressure

160

86

sulphate ion as head group, see SDS negative adsorption on Agl

351

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

sulphates, c.m.c. of

454t

sulphonate head group

521

sulphur sol

469t

8

supercooling

77

superheating, of liquids

76

supernatant, see centrifugation Galvani potential of

346

superposition, linear of electric fields

344

of sedimentation and diffusion

179

supersaturation, degree of surface coverage, by polymer Nernstian surface area

96 299f 345

355

4e

determination

348

surface charge density

323

485

(s.a. charge density) constant, during interaction

590

generation of

356

surface complex formn.

361

surface composition, determ. of

262

surface (excess) concentration

63

486

407

422

anomalous

396

427t

surface energy

45

surface conductance effects

487

(s.a. surface tension) effect on m.p. of small crystals theoretical estimate surface excess quantities

78 574 57

surface force apparatus, see SFA and MASIF surface heterogeneity and hysteresis

111f

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524

Index Terms

Links

surface modes of vibration

553

surface of shear

377

of micelles

477

of tension

57

385

489

59f

65

475 surface phases surface potential

69 308

constancy, during interaction

585

surface props., theor. estimate of

574

surface sites

357

surface tension definition

47f 67

effect of dispersion forces on

574

effect of electrolyte

68f

from Lippman equation

315

lowering by adsorption

68f

measurement

313

of mercury-solution interface

312 49

table of

48t

topography surfactant adsorption

314

49

of solids surface thermodynamics

501

45

dependence on concentration

molecular origins

485

44

59

210f

269f

272

13f

10

434

68f

518

695

705 aggregation of cationic, in soap film

16f

17f

607f

c.m.c. of (see c.m.c.) influence on interfacial rigidity

744

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Index Terms

Links

surfactant (Cont.) non-ionic, adsorption scattering

526 700f

susceptibility

655

suspension effect

354

suspension medium, effect on van der Waals force

542

suspensions: see colloids; dispersions swelling of clays

608

symmetry, axial, of surfaces

104

of stress tensor syneresis

716 741

T tails, polymer talc

41f

298

21 s.a. clay minerals

TDMS

285

TEM

207

temperature, compensation

456f

effect on floccn: see CFT ; LCFT ; UCFT effect on micellization on van der Waals force on vapour pressure of droplets lower consolute — of homogeneous nucleation

440 560 76

79t

701 96

programmed desorption

285

theta–

628

tensile strength of liquids

86

tension, hydrostatic

86

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628

Index Terms

Links

tensor representation of grad v

776

stress

161

unit

162

thermal desorption mass spectrometry

285

thermodynamics in absence of surfaces

780

of adsorption

63

of charged systems

311

of irreversible processes

384

of micelle formation

442

of polymer stabilization

72

of surfaces

44

theta-(θ) point

628

thin double-layer approximation films

292

450

630f

of small systems

— temperature

188

629t 405 606f

in electrokinetics thixotropy

408 154

722f

758 Thomson equation application to solids time, coagulation effects in surface tension for diffusion step

76 78 617 112 27

propagation, for e.m. field

547

residence, in micelles

470t

tin oxide

362f

TIRF

276f

TIRM

273

titania (TiO2) charge and ζ-potential

364f

184

484f

504f

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740

Index Terms

Links

torque, in viscometry

171

729

185

742

171f

176

total internal reflection

273

274

TPD

285

tracer diffusion

183

traction

144

trains, polymer

41f

299

194f

745

on particles in shear torsion wire, use in viscometry

trajectories of particles of fluid

117f

transducers

250

transient behaviour

112

transition dipole

537

phase — in micellization

445

translation of fluid element

117

transmission of e.m. waves

556

transport, convective

397

transport properties

157

of suspensions Triton X-100

178 464f

turbidity

236

Tyndall effect

23e

Tyndall spectra, higher-order

141

528

U UCFT

629t

ultracentrifuge

230f

ultrahigh vacuum

261f

ultramicroscope

206f

ultrasonic absorption ultrasonication ultraviolet interpolation

250 7 568

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216

Index Terms

Links

units, electrical

784

unit normal

160f

tensor

162

vectors

160

universal potential

681

urea as H-bond breaker

442

UV interpolation of dielectric funcn

568

V vacuum

261f

valency, counterion influence on coagulation

37t 603

vanadium pentoxide

204

vanadyl ion, mobility on micelles

472

van der Waals equation

535

and Boyle temperature

321

628

van der Waals force

39f

533

calculation of

563

571

contribution to work of adhesion

574

effect of retardation

547

exptl study of

601

relation to hydrophobic force

611f

615f

van Hove correlation function

662

van Kampen method

552

vapour, definition

88

vapour pressure, effect of temperature

76

lowering, inside bubbles

75

of small particles

74

variance

221

vector calculus

770

vector field

770

vector, unit

160

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769

Index Terms

Links

vector cross-product of

775

dot (scalar) product of

772

velocimetry, sound (for sizing)

250

velocity, fluid

164

in sedimentation gradient in electro-osmosis in capillary flow

117

121

377

385

179

737f

of sound

250

terminal

123e

vermiculite

23

(s.a. clay minerals) swelling of vesicles

608f 498

molecular packing in

17f

vibration, surface-modes

553

virial coefficient

238

viscoelastic behaviour

147

512

517e

152

724

756 viscoelectric effect

386

viscometric flows

715

viscosity apparent

153

716

complex

151

726

differential

153

716

dynamic

726

effect of shear rate

154f

in the double layer

385

intrinsic, for spheroids

754

192f

measurement

170

modified, inside micelles

470

of dispersions

153

750

concentrated

193

196f

dilute spheres

189

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Index Terms

Links

viscosity apparent (Cont.) of liquids Volta potential of disperse phase volume, excluded

166t 306 308 296

volume fraction effect on electroacoustics

254

on sedimentation

180

on viscosity

195

754

175

384

121

143e

172

735

volume flow rate molar

W wall shear stress water anomalous capillary force

100f

Cauchy plot for

569

— organic mixtures, permittivity

465

displacement by organics at DME

341

Hamaker constant

572t

influence on stability

613

interfacial tension

575

liquid-vapour equilibrium

97e

movement in tall trees

86

neutron scattering by

690t

penetration into micelles

476

proofing

100

structure, influence on micelles

437

surface tension

48t

viscosity

146

wavelength, characteristic

547

wave motion of light, eqns for

134

166t 559

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735

Index Terms

Links

wave vector

135f

658

767

487

501

769 s.a. light scattering vector wax, paraffin, contact angle on

577f

weak acid head groups

357

weighting of a distribution

215

Weissenberg effect

725f

wetting and contact angle

100

of liquid in a capillary

84

of PTFE byalkanes

727

577

wetting agent, see surfactant Wilson cloud chamber work, capacity of system to do

77 780

electrical

782

done in charging double layer

584

integral

776

mechanical, in surface processes work of bringing ion into double layer

56 319

of co- and adhesion

101

of creating surface

47

X XPS

266

X-ray adsorption in sedimentation

226

X-ray photoelectron spectroscopy

266

xylose, effect on micelles

442

Y Young-Laplace equation

54

limits of applicability

97

thermodynamic derivation

60

105

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Index Terms

Links

Young’s equation

103

thermodynamic derivation

107

Young’s modulus

145

yield value

717

754

Z Zeiss-Endter particle sizer

212

zero point of charge, see p.z.c. and e.c.m. zeros of the dispersion relation

768

zeta potential (s.a. electrokinetics)

377

424f

489

494

518f

523

in coagulation

604

in flow

755

758

of lecithins

498

513

Zetasizer (Malvern)

392f

Zimm plot

238

zinc sulphide

11f

zone sensing

255f

see pulse counters zwitterionic surfaces

359

363

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488