Chee2940 Lecture 17 - Edl

CHEE2940: Particle Processing Lecture 17: Electrical Double Layers This Lecture Covers Origin of surface charge and structure of EDL Structure of EDL, conc. & potential distributions EDL interparticle force & energy Measuring surface potentials Chee 2940: electrical double layers

17.1

THE ELECTRICAL DOUBLE LAYER

• The electrical double layer or EDL occurs at the interface between a solid surface and its liquid medium. • How does this happen? Does it happen just between solid particles and water? • Understanding this is crucial to addressing the stability and aggregation of particles in liquid. Chee 3920: electrical double layers

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Particle-solution interface

Solid particle Solution

The electrical double layer (EDL) Chee 3920: electrical double layers

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17.2 ORIGIN OF SURFACE CHARGE • Most particles in an aqueous colloidal dispersion carry an electric charge. • There are many origins of this surface charge depending upon the nature of the particle and it's surrounding medium. • Three important mechanisms include: o Ionisation of surface groups o Differential loss of ions o Adsorption of charged species. Chee 3920: electrical double layers

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• Ionisation of Surface Groups Dissociation of any acidic groups on a particle surface will give a negatively charged surface. Dissociation of any basic groups on a particle surface will give a positively charged surface. The magnitude of the surface charge depends on the acidic or basic strengths of the surface groups and on the pH of the solution Chee 3920: electrical double layers

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In vacuum

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In solution

Examples: Oxides (SiO2)

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• Differential loss of ions from the crystal lattice If a crystal of Agl is placed in water, it starts to dissolve. +

-

If equal amounts of Ag and l ions were to dissolve, the surface would be uncharged. +

In fact, Ag ions dissolve preferentially leaving a negatively charged surface. Further examples: carbonates & phosphates. Chee 3920: electrical double layers

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Chee 3920: electrical double layers

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• Adsorption of charged species (ions and ionic surfactants) Surfactant ions may be specifically adsorbed onto the surface of a particle. Cationic surfactants would lead to a positively charged surface. Anionic surfactants would lead to a negatively charged surface Chee 3920: electrical double layers

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Chee 3920: electrical double layers

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17.3

STRUCTURE OF EDL

The double layer consists of two parts: • An inner region (Stern layer) where the counter-ions are strongly bound to the surface, and • An outer (diffuse) region where the counterand co-ions are less firmly associated with the surface. Chee 3920: electrical double layers

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Particle-solution interface Stern plane

Solid particle Solution

Diffuse layer Stern layer Chee 3920: electrical double layers

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Diffuse layer of EDL Distribution of ions in the diffuse layer is determined by the balance between the electrostatic (Coulomb) force and the Brownian force of thermal diffusion (First described by Boltzmann).

The Boltzmann distributions (Balance between electric potential energy and thermal energy)

 zi eψ ( x )  ni ( x ) = ni∞ exp −  k BT   Chee 3920: electrical double layers

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where x … distance from the solid surface ψ ( x ) … electric potential ni ( x ) … number concentration of ions i ni∞ … ni ( x ) in solution (at x = ∞ ) zi … valency of ions i -23 k B … Boltzmann constant (= 1.381×10 J/K) T … absolute temperature. Examples: For a solution containing CaCl2  2eψ   eψ  nCa ( x ) = nCa∞ exp − , nCl ( x ) = nCl∞ exp +   k BT   k BT  Chee 3920: electrical double layers

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n∞

Variation of co-ions and counter-ions at a charged surface Chee 3920: electrical double layers

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17.4

POTENTIAL DISTRIBUTION

The potential distribution, ψ ( x ), at the surface is required for quantifying the diffuse layer & EDL. It can be predicted using the Poisson equation (PE) from electrostatics. The PE states that the total electric flux through a closed surface is proportional to the total electric charge enclosed within the surface. Chee 3920: electrical double layers

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Mathematical description for the PE:

dψ εε 0 2 = − ρ ( x ) dx 2

ε0… permittivity of vacuum (8.854×10 C J m ) ε … dielectric constant of solution (=80 for water) ρ … charge density (of all ions) determined as -12

ρ ( x ) = ∑ ni ( x ) zi e i

-1

(= Sum of all ionic charges) -19

e… electronic charge (1.602×10 Chee 3920: electrical double layers

-1

C) 16

Inserting Boltzmann’s distribution for the ion concentration gives

 zi eψ  dψ εε 0 2 = −∑ zi eni∞ exp −  dx i  k BT  2

POISSON-BOLTZMANN EQUATION Example for NaCl salt 2  eψ   −eψ  dψ εε 0 2 = en∞ exp   − en∞ exp   dx  k BT   k BT  Chee 3920: electrical double layers

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Debye-Hückel linearisation (DHL) For small potential, exponential functions in the ±x PE can be linearised (e = 1 ± x + ⋅⋅⋅) to give (by Debye and Hückel)

d ψ  zi e ni∞  2 = ψ = κ ψ (DHL) 2 dx  εε 0 k BT  2

2 2

1/ 2

 e 2 ∑ zi ni∞  κ =   εε 0 k BT  2

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= Debye constant 18

Practical note: Salt concentration is given in molar concentration, ci [mol/L]. 3

Relationship between ci and ni [molecules/m ]:

ni = ci N A1000; NA …Avogadro #

Practical equations for κ: 1/ 2

1000 N Ae ∑ zi ci∞  κ =  εε 0 k BT   2

2

1/ 2

 2000 N Ae  =   εε 0 k BT  2

I

I = ∑ zi ci∞ / 2 = ionic strength [mol/L] 2

i

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Useful expressions o

κ [1/ m] = 3.283 × 10 I [mol/L] or κ [1/ nm] = 3.283 I [mol/L]

o

κ [1/ m] = 3.255 × 10 I [mol/L] or κ [1/ nm] = 3.255 I [mol/L]

At 20 C: At 25 C:

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Example calculation for 0.01 mol/L solution of o CaCl2 at 20 C

cCa∞ = 0.01 mol/L , cCl∞ = 0.02 mol/L Chee 3920: electrical double layers

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mol mol 2 ( +2 ) × 0.01 + ( −1) × 0.02 mol L L I= = 0.03 2 L 2

κ = 3.283 0.03 = 0.569 nm

-1

1/ κ = 1.759 nm

1

κ

is the measure of the diffuse layer thickness

“The higher the ion concentration, the thinner the diffuse (EDL) layer” Chee 3920: electrical double layers

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Solution of the DHL for single planar surfaces

ψ ( x ) = ψ 0 exp ( −κ x ) where ψ0 is the potential at x = 0 (at the ‘surface’) Electric potential at the surface decays exponentially with the distance x. Electric potential is zero far from the surface (in the bulk). Chee 3920: electrical double layers

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Particle-solution interface Shear plane

Solid particle

Potential

Solution

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ζ = zeta potential = Potential at the shear plane

δ 1/κ

Distance

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17.5

INTERACTION BETWEEN EDL’s

• The double layer force occurs due to the interaction between two electrical double layers • The EDL force is due two contributions: The overlap of the electric potential distributions at the two surfaces (the Maxwell stress). The overlap of the ion concentration distributions at the two surfaces (the osmotic pressure). Chee 3920: electrical double layers

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ψ0

ψm

Potential profile between surfaces

Potential profile at single surfaces before interaction

Potential profiles at interacting EDL’s Chee 3920: electrical double layers

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• The EDL force can be attractive and/or repulsive • The EDL force depends on the charging mechanims (boundary conditions) occurring at the surfaces during the EDL interaction: - The constant surface potential (long contact time), or - The constant surface charge density (short contact time). • In this course: The EDL interaction between identical particles (the simplest case). Chee 3920: electrical double layers

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EDL repulsion between identical surfaces In this case, the EDL pressure, Π, only depends on the overlap of concentration (relative to the bulk)

  Π = k BT  ∑ ni ( D / 2 ) − ni∞   i  Boltzmann Eq. gives  zi eψ  ni ( x ) = ni∞ exp −   k BT    zi eψ m   Π = k BT ∑ ni∞ exp −  − 1 i  k BT    Chee 3920: electrical double layers

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ψm … potential at the middle plane For low potentials, the series expansion gives  z eψ    1 z e ψ Π = k BT ∑ ni∞ 1 − i m +  i m  + ⋅⋅⋅ − 1 k BT 2  k BT  i   2 eψ m ) ( 2 ni∞ zi + ⋅⋅⋅ Π = −eψ m ∑ ni∞ zi + ∑ 2k BT i i 2 eψ m ) ( 2 ni∞ zi Due to electroneutrality: Π = ∑ 2k BT i 2

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We have κ = 2

e

2

∑z

2

i

ni∞

εε 0 k BT

and the superposition

solution for planar surfaces at low potentials gives (see p. 22): ψ m = 2ψ 0 exp ( −κ D / 2 )

Π ( D ) = 2εε 0κ ψ 0 exp ( −κ D ) 2

2

EDL interaction energy per unit area of planar surfaces (Lecture 15, p. 19) ∞

E ( D ) = ∫ Π ( h ) dh = 2εε 0κψ 0 exp ( −κ D ) 2

D

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For the EDL interparticle force, the Derjaguin approximation (Lecture 15, p. 18) gives

F ( D ) = π RE ( D ) ; R … radius of the particles

F ( D ) = 2πεε 0κ Rψ 0 exp ( −κ D ) 2

The EDL interparticle energy, V ( D ) , is calculated as ∞

V ( D ) = ∫ FdD D

V ( D ) = 2πεε 0 Rψ 0 exp ( −κ D ) Chee 3920: electrical double layers

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17.6

MEASURING SURFACE POTENTIAL

The surface potential is required for determining the EDL interparticle force and energy. It can be measured by applying a voltage over the particles and move them relative to the liquid. Balancing the applied electric force and the drag force on a particle gives

E field Q = 6πµ RU Efield … applied electric field, Q … particle charge, µ … liquid viscosity, U… particle velocity Chee 3920: electrical double layers

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Electrophoretic mobility (measurable) is defined as

U m= E field Charge on the particle surface is calculated as

Q = 4πζεε 0 R ζ … zeta (surface) potential (at the shear plane)

3mµ ζ = (Hückel equation) 2εε 0 Chee 3920: electrical double layers

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Measuring the particle velocity allows the calculation of the zeta potential. Measuring techniques include: 1) Microscope for measuring distance and time travelled by a particle 2) Laser light scattering – Doppler shift 3) Electro acoustic method, etc.

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Example data for the zeta potential versus pH

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Isoelectric Point – IEP

• IEP is the pH at which the zeta potential is zero (The surface is net neutral). • It depends on the material through the surface charging mechanisms. Point of Zero Charge – PZC • Is usually the same as the IEP except when polyvalent specifically adsorbed ions are on the surface. Chee 3920: electrical double layers

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Chee 3920: electrical double layers

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Zeta potential of alumina vs pH and KNO3 concentration

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Zeta potential of alumina vs pH and LiNO3 concentration

IEP changes due to the specific adsorption of Li. Chee 3920: electrical double layers

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