Gee

The Present Status of the Theory of Rubber Elasticity* G. GE~ In this review a modified equation [or the lree energy o[ a single chain is proposed, based on a recent discussion of the excluded volume problem. A simple model o[ a network element is used to study departures from affine behaviour and the form of the stress/strain curve. Comparison with the Mooney equation suggests that the C~ term cannot be explained in terms of excluded volume, and it is suggested that the packing problem requires [urther study. Energy and volume changes on elongation are reviewed and the need for [urther work emphasized.

(1) Introduction IT IS not possible, in the course of a single lecture, to review all facets of the theory of rubber elasticity. Comments will therefore be centred on two topics of current interest: (a) the form of the stress/strain curve, and its dependence on chain statistics; (b) the volume and energy changes accompanying elongation. The quantitative interpretation of rubber elasticity 1 rests firmly on the use of Gaussian statistics to describe the behaviour of a single chain. The elastic network is then treated as an assembly of chains, with the assumption --which can be justified for Gaussian chains---of affine deformation of the junction points. From time to time the suggestion has been made that some of the observed discrepancies between theory and experiment may have their origin in departures from Gaussian statistics. It is well known that a free chain, in a neutral environment, is not Gaussian, due to the excluded volume effect. In a dilute solution the overall dimensions depend upon the nature of the solvent, and much use has been made of theta solvents, defined 2 as those in which the laws of ideal solutions hold at finite (low) concentrations. Under these circumstances the net excluded volume effect is zero, the expansion due to this cause being balanced out by the contraction occurring in a thermodynamically poor solvent. It is also generally accepted 3,4 that in a bulk polymer, the effective excluded volume must be zero, because the local structure is determined by interactions of segments, in which it is a matter of indifference whether these belong to the same or to different molecules. Quantitative analysis of the behaviour of networks of non-Gaussian chains raises two problems: (1) the description of the properties of a single chain, and (2) the study of possible departures from affine deformation. In this paper we make use of a recent treatment of the statistics of an infinite chain to define a parameter which can be used to describe departures from Gaussian statistics. The resulting free energy equation is then used in an accurate study of a 'network' of four chains, where departure from affine behaviour can be determined. *This review was vrevared during the tenure of a Fellowship o f the Michigan Foundat/on for Advanced Research, Midland, Michigan, U.S.A.. and was presented at the Great Lakes Conference on Polymer and Colloid Science, Detroit, Michigan, U.S.A., October 1965.

373

G. GEE

(2) The single molecule The simplest theories of rubber elasticityI treat a linear molecule as a random chain, made up of n links of length l, in which there are no restrictions on orientation. If one end of the chain is fixed, the probability P(r) of finding the other end in a specified volume element dv at a distance r is given by P(r) d r = (~Trn12)-3:2 exp ( - 3r2/2nP) dV (1) It is seen that P(r) has a maximum at r = 0, i.e. the most probable point at which to find the chain end is coincident with the beginning. The probability W(r) of finding the end at a distance between r and (r + dr) is then

W(r) dr = 4~'Pe(r) dr

(2).

which has a maximum at r~I. =(}nP)~. The mean square of r is easily shown to be o plays a central role in all theories of rubber elasticity and a reconsideration of its evaluation is one of the principal objectives of current work. Three factors have received attention, concerned with: (a) the recognition that the angle between consecutive links is fixed by valency considerations; (b) the fact that rotation about a single bond involves changes of energy, so that certain orientations are favoured; (c) the fact that molecules occupy a finite volume, so that conformations involving simultaneous occupation of a given volume element by two finks are impossible, Of these the first introduces no modifications which cannot be allowed for by redefining n and l, subject to the condition that nl is the outstretched length of the molecule. Energy barriers to rotation also leave the form of equation (3) unchanged, but , becomes a function of temperature. If we consider a chain in which each link has effectively a choice of two orientations, which differ in energy by AE, then 5 d In (r2)0/dT ~ A E / R T 2 (4) The experimental study of this equation has been one of the most fruitful of recent developments. It will be noted that to maintain equation (3), if AE @ 0, n and l must be temperature dependent. The effect of finite volume has proved difficult to treat quantitatively. Physically it is evident that exclusion of the more folded configurations must expand and broaden the distribution. Monte Carlo calculations on lattice models have led to the result s 0 --~ n x+~

(5)

where, for long chains, 022 2> y > 0"18. Schatzki 7 has also tabulated distributions of end to end distances, from which it is possible to derive numerically 8 P(r) and W(r). These suffer from the fact that the particular lattice model chosen necessarily imposes a certain discontinuity, although Domb 9 has shown that all lattices give similar values of y (equation (5)). Recently Edwards 1° has succeeded in obtaining an asymptotic solution in closed form for the position of the nth link of an infinite chain, which may 374

THE PRESENT STATUS OF THE THEORY OF RUBBER ELASTICITY be written P(r) ~,~ exp [ - (1 "35/riP) (r - 0-87n°'6/a°'2) 2]

(6)

where the volume excluded by a link is vlo~.l = a P

(7)

so that a appears in equation (6) as a numerical parameter. In the limit of n > co, this leads to a mean square end to end distance (r2> o = 0" 7 5 5n1~Pa °'4

(8)

The exponent of n is here exactly 6/5, giving ~/ (equation (5))=0"20, in agreement with the lattice calculations. For the purpose of this paper, it has been a s s u m e d that an equation of the form (6) is valid also for finite values of n. Although this will not be strictly true, it should suffice to indicate semi-quantitatively the consequences of departure from Gaussian statistics. For application to the theory of elasticity, the distribution is conveniently expressed in terms of the configurational free energy of a chain whose ends are fixed at two specified points, a distance r apart. For a chain obeying Gaussian statistics, i.e. equation (1), this has the form G = const. + ( 3 k T / 2 n l S ) r ~

(9)

If such a chain is deformed from an initial length r . to a final length r, the increase of free energy is then A G = ( 3 k T / 2 r i P ) (r = - r~) = ( 3 k T / 2 n P ) ~ ( h 2 - 1)

(10)

where h is defined as r/r~. For non-Gaussian chains, G will no longer be linear in k=; numerical calculations have been reported 8 on the basis of Schatzki's work. If an equation of the form (6) holds, (9) and (10) must be replaced by G = const. + (1 "35/ni 2) (r - r*) 2

(11)

AG = (l . 3 5 k T / n P ) r ~ [()~ - b) 2 - (1 - b) 2]

(12)

If equation (6) held exactly we should have further r * = br~ = 0"87 Ia°2n °'~

(13)

and we m a y guess that this is the correct limit as n > c¢. The mean value of r 2 has been obtained as a function of b by numerical integration of ~ r a exp [ - ( 1 "35/nl 2) ( r - r*) 2] dr (r~>0 = ° ~ r ~ exp [ - ( 1 " 3 5 / n P ) ( r - r*) 2] dr

(14)

0

in terms of a further parameter B = 1.35r*2/nP. The limiting form for B-->- 0 is readily shown to be Y =
(15)

which may be compared with the known solution for small departures 375

G. GEE

from Gaussian statistics

o/nl2= 1 + 0"421 a ~/n

(16)

Equating (15) and (16) (ignoring the factor 1-111) leads to r*

>0"481an

(17)

A general equation for r*, and therefore for b, should have equations (13) and (17) as limiting forms. For the purpose of this paper we simply treat b as a parameter which is some sort of measure of the excluded volume, becoming zero for Gaussian chains.

L 4

~

. . b=O'8 /70"6 / ..-'~~0'4 /

..0

t~ ,¢1 ~

0

°o-"

I

1

I

a2

2

I

3

Figure /--Effect of parameter b on chain free energy Figure 1 gives a series of plots of AG against •2 for b between 0 and 0-8. The curves are qualitatively similar to those previously derived from Schatzki's data, but exact agreement is not expected, since in the latter elongation ratios were computed relative to r . . . . instead of to 0 (which is more difficult to estimate reliably). This has the effect of tilting the free energy curve; allowing for this, our calculations from Schatzki's data for n = 6 0 give results very similar to those of Figure I for b=0"6. (3) The network In a normal crosslinking procedure, we start with a system consisting of a close-packed assembly of long molecules, free to move and to change shape and position subject to the requirement of constant volume. Chemical reagents are introduced and are molecularly dispersed uniformly throughout the system. Chemical reaction occurs, in which the effective reagent is a very small part of a polymer molecule, so that each molecule may possess (say) 10" reactive units. A small fraction (say one per cent) of these react, so that the chemical reaction is in no way influenced by the chain character 376

THE P R E S E N T STATUS OF THE T H E O R Y OF R U B B E R ELASTICITY

of the polymer. As a result of the chemical reaction the system now contains a large number of reacted points, and these will be randomly distributed in space. Each reacted point links together two of the original long molecules, and can now be thought of as a junction point from which four chains radiate. Most of these chains will terminate at some other junction point, and it is therefore convenient to speak of the length of molecule comprised between two junction points as a chain. A few chains will form a closed loop, returning to the same junction point. A number of chains equal to twice the original number of molecules will terminate in a free end. For most purposes, we can ignore both closed loops and free ends, and regard the structure as a perfect network. When the body is deformed by the application of forces, the whole network must respond, and will of course do so in such a way that the increase of free energy is a minimum. It is frequently assumed that the, network free energy is simply the sum of the free energies of the individual chains. If these are Gaussian, and a representative chain changes its end to end distance from ri to r, the resulting free energy increase will be given by equation (10). To obtain an expression for the network free energy, we have still to solve two problems: to evaluate ri; and to sum over all the chains. The simplest forms of theory put r~ =(rS>o =n/s, and thus obtain, for the representative chain, A G = 1.5 k T ()t s - 1) (18) The essential assumption involved in the summation is that each chain undergoes affine deformation, i.e. that its end to end distance changes proportionally to the bulk dimensions in the direction in which the chain lies. It then follows that if there are N chains in the network, and the total deformation is described by the three principal strains ~,1, h2 and h3, the total free energy of deformation is given by AG = 0"5 N k T (hl + h~ + h~ - 3 )

(19)

Wall and Flory11 have argued that an additional term is required when the deformation revolves a change of volume, to take account of the combination of the chains into a network; they modify (19) to •

8

AG = 0-5 N k T [hl + hl + hi - In (hi As h3) - 3]

(20)

The identification of ~ with 0 is based on the argument that any pair of segments in the linear polymer will on average be at this separation. When crosslinking occurs, the pair which form contiguous junction points become relatively fixed in position, and therefore retain this separation. This argument ignores a number of important considerations: (a) The value of o is temperature-independent, chains which are at their most probable lengths when formed will no longer be so in the undeformed test piece used in a normal mechanical test. (b) Unless all the crosslinks are formed simultaneously, the growing fragments of network will tend to contract, thereby changing the statistics 377

G. G E E

of the chains already incorporated. The reality of this tendency to contract is illustrated by the observation that crosslinking in dilute solution may be followed by syneresis of free liquid. The equilibrium here involves a balance between network contraction and the swelling (osmotic) pressure of the liquid, which will be operative even if all the crosslinks are formed at the same instant. Considerations such as these make it desirable to modify equation (19) [or (20)] by a factor <~)/o which Tobolsky et al. 12 have called the 'front factor'. Recent discussion has been concerned with the temperature and volume dependence of this factor. From the foregoing argument <~) is seen to represent an average of the chains as they exist in a test sample at the start of a deformation experiment. If we compare one experiment with another under different conditions (of temperature, pressure or state of dilution by a swelling agent) it is reasonable to assume <~> o= V20/swhere V0 is the volume of the undeformed sample in a particular experiment. It is clear that cannot change during an experiment, since it refers specifically to the initial conditions.
(21)

where ~b(T) accounts for the temperature dependence of
(22)

For most purposes, the expansion accompanying elongation of a solid elastomer at constant pressure and temperature is entirely negligible, and the stretching force is then given by 1=2 CTd/(T) [ L - V / L 2] •

(23)

where C is predicated to be independent of T and V. Assuming the validity of this equation, we can use it in conjunction with standard thermodynamic equations to draw conclusions regarding the change in energy at constant volume, and the change in volume at constant pressure. Without further physical assumption, we obtain s, is: (OU)

fe': " ~ and

_ _ , T d ln_~(T) , T d l n ( r 2 ) o "-dT ='~IT

r.r--

iov ~,0i--,~r. , =

k s - 1 = 2 jO CTqJ(lO

x_s

(24) (25)

where/3 is the coefficient of compressibility. The derivation of equations (21) to (25) has specificially assumed Gaussian statistics. If the representative chain is non-Gaussian, the analysis given is invalid. An attempt has been made s to investigate numerically the effect 378

T H E P R E S E N T STATUS OF T H E T H E O R Y O F R U B B E R ELASTICITY

of replacing equation (10) by a curve based on Schatzki's chain distri. bution data, retaining the crudest model of the network (three sets of mutually perpendicular chains). Uncertainties in the distribution curve made this inconclusive, but a similar analysis can now be made analytically by using equation (12). This leads straightforwardly to the result:

!

[

bxl ]

~b:--X-X-~=O9ONkT-n--ff LI

1~-~7~-j

(26)

This equation is of questionable significance, for any such analysis ignores a very important consequence of departure from Gaussian statistics: it is no longer permissible to assume affine deformation. To investigate this problem, some numerical calculations have been made on a simple model. (4) A tetrahedral model We consider four chains meeting at a junction point, with their other ends initially at the corners of a regular tetrahedron of volume V0. A force (D is now applied along the direction of one chain, so that the tetrahedron is deformed to a triangular pyramid of volume, V = V0s~, and height As times that of the tetrahedron. Denote by ht the extension ratio of the axial f

Figure 2--Tetrahedral model

chain, and by h2 that of the other three chains (Figure 2). The problems to be examined are: (a) the position of the junction point, and (b) the magnitude of the force. The free energy is given by

AG/C = (Xl - b) 2+ 3(X2 - b) 2 - 4(1 - b) 2

(27)

and geometrical considerations require hl = (~AS- 2~1)~+ 8s2/9h

(28)

The position of the junction point will be such that OG/Oht =0, the solution to which is conveniently written in terms of y = AS- hi 379

G. GEE (4y+ b)2 = 9b2h (hs+ 33) 2 8s2+ X (hs+ 3y)2

(29)

Since y is always small, this is easily solved numerically. The stretching force f is then given by:

l=(oG I OXl)~

These equations have been used in two series of computations: (0 Taking s = 1, X= ~/6, AG has been plotted as a function of hi (Figure 3) for several values of b. Attine deformation requires the minimum to occur at h~=h; this is seen to apply for the Gaussian curve (b=0), but the other curves deviate substantially.

~ ~(~:2,45) Figure 3 - - D e p e n d e n c e

of AG on XI at fixed h = 2 " 4 5

tO

215 (ii) Load/elongation curves were evaluated for a range of values of b and s. These are represented in Figure 4 in the form of the function dp=fs-2(h-h-~) -1, while the departure from Gaussian behaviour is shown in Figure 5 by plotting y/hs. It is easily seen, from the form of the equations, that these functions do not depend on b and s independently, but only on b/s. Thus while b will increase with dilution, its effect on the elastic behaviour of a network is compensated by the increase in s. For h > 1, y / ~ shows a maximum at approximately 100 per cent elongation, representing a three per cent departure from affine deformation for b/s=0"2. For the same b/s, the increase in ~b from h = l is two per cent at h = 2 , increasing to five per cent at h=3, and eight percent at ~ = 4 . 38O

THE PRESENT STATUS OF THE THEORY OF RUBBER ELASTICITY --

2.6 ~

~

2-4

b/s=O.033

_....-- 0.1 0.2 0.25 0.333 0.4

~

(~22 2.0 I

I

,

1 Figure

1

I

I,

2

3

4

4--Dependenceof ~ onk

0'05 0,2

01 0033

0 ¢o

-0'05

-0"1 •

,

r

!

1

2

,

f

1

3

4

X Figure5--Departure from affine deformation

The Mooney equation--It is of interest to compare the results of these calculations with the observed behaviour of an elastomer. It has become customary to represent measurements made in simple elongation by means of the Mooney equation TM 1~ I/2(h-h-2)=CI +C2/X

(31)

which is generally found to hold fairly well over the range of elongation 1 < h < 2. If the polymer is swollen, the second term becomes less important, and the effect of swelling can be represented at least approximately by modifying equation (3 l) to the formiC: 381

G. GEE

= 2(x- x-')

x

(32)

where ~b=, the volume fraction of polymer in the swollen sample, is equivalent to 1/s s in our analysis. The si£mificance of the term C= has remained obscure, and some workers lr have dismissed it as an experimental artefact. It is not always sufficiently realized that the Mooney equation holds only over a very short range of h. Combining Treloar's observations 1' on bi-axial deformation with Mullins's measurements ~' on elongation, two typical curves of ~b versus 1/h are reproduced in Figure 6, in which the / / //Expt

Catc. b/s : 0.1

_ _ j / 2"5

Figure 6 - - D e p e n d e n c e

o f ~b o n

1/x

•

/

~" . ~ , , , _

Catc.

I

0

0.5

1/~

his = 0.4

I

I,

1

l"b

broken lines represent equation (32). On the same figure are included two curves replotted from Figure 5. Comparison cannot of course be exact, since our model represents only a typical dement of the network, deformed in a particular way. Nevertheless, two conclusions seem justified: (I) a very large ratio b/s would be needed to produce any detectable change in ~b; (2) even with a large ratio b/s, the highly characteristic fall of ~b from h = 1 to h = 2 is not reproduced at all. In the light of these observations, the suggestion made previously8, that the C~ term might be due at least in part to the excluded volume, must be withdrawn. It is more difficult to make any positive contribntion to the interpretation of C2, more particularly since recent experimental work 19 has tended to contradict earlier evidence that C2 does not vary widely. It may, however, be worthwhile to call attention a g a i n 2°' 2t to the fact that the theoretical treatment ignores completely the problem of molecular packing in the solid state. This will greatly reduce the configurational entropy of the system, but will only contribute to the observed force if the packing free energy changes on deformation. DiMarzio ~2 has recently concluded, on the basis of an admittedly crude analysis, that while this will indeed contribute to C~, by itself it can account for no more than a small part of a typical value of 382

THE PRESENT STATUS OF THE THEORY OF RUBBER ELASTICITY C~. Before accepting this as a definitive result, it is pertinent to comment that the elastic free energy associated with the variation of ~b in Figure 6, from h = I to h = 2 , amounts only to 0.02 cal/emL This is only one tenth of the amount by which the entropy ( x T) of solid natural rubber has been estimated2~ (from solution thermodynamics) to fall short of the value for a random assembly. (5) Volume and energy changes on stretching For most purposes, it is a sufficiently good approximation to say that an elastomer elongates without change of volume. This cannot of course be strictly true, since the hydrostatic component of the tensile force must produce a dilation, whose magnitude is given, according to the network theory outlined above, by equation (25). Recently Tobolsky ~" has suggested that C in equation (23) may be volume dependent. The direct experimental determination of dilation is difficult, and equation (25) cannot yet be said to have been adequately tested. Strong, though indirect, evidence of its accuracy comes from a study of the thermoelastic behaviour of elastomers. Equation (24) provides a basis for the interpretation of stress/temperature measurements; in conjunction with equation (4), valuable information becomes accessible on the energy differences between different rotational states of a chain. As it stands, equation (24) calls for temperature coefficients at constant volume, which have hitherto not been available. However, the constant pressure coefficients are readily converted, if the volume change is known. Assuming the dilation calculated from equation (25) to be correct, it is easily shown~ that equation (24) is equivalent, to

(//T)]

d In o 0 In + x 3- 1 ~ (33) dT = ~ JP, L where ot is the coefficient of cubical expansion. Flory and his co-workers have made a series of very careful measurements on a range of polymers, and have interpreted them by means of this equation. Their work may be illustrated by quoting some results 24 for atactic poly(isopropyl acetate): )t 1"134 1 '214 1 "256 1 "305 1-364

_ 103 [0 In (//T)] ot OT ]p. r. 103 A~ ------'1 1"78 1-07 1-00 0"81 0"68

1"50 0"87 0"70 0"56 0"45

103 d In o dr 0"28 0"20 0"30 0-25 0"23

The importance of the correction term a/(h 3- 1) is obvious, but the fact that the final column shows no systematic dependence on ~ suggests that the dilation has been correctly estimated. Taken by itself, this evidence is far from compelling, as it is easily shown that a modification of equation (25) may change the figures in the last column without causing them to depend on h. More convincing is the cumulative effect of a series of such investigations2~-29in some of which an independent estimate of d In 0/dT 383

G. G E E

was obtained from dilute solution measurements. Moreover, the values obtained have several times been found consistent with calculations based on the energies of different chain conformations. Current work should shortly lead either to complete confirmation of these conclusions, or to the need for a reassessment. Several investigators are reexamining the experimental problem of dilation measurements, with the hope of obtaining more definitive results. The direct measurement of temperature coefficients at constant volume has also been undertaken, but the preliminary results~° which have been reported are not quite of the precision required. SUMMARY AND CONCLUSIONS

The main stream of progress depends on a network theory in which the chains show Gaussian behaviour. This has achieved notable successes, but some caution is needed in accepting present evidence as a complete justification of all the details of the theory. Perhaps the most striking recent advance is in the interpretation of the stress/temperature coefficient of a network in terms of the energy differences between rotational positions of a single chain. The good agreement found, both with measurements on dilute solutions and with theoretical values, is particularly convincing. It is clear, however, that this does no more than show the reality of this factor in determining the end to end distance of a chain; it is not by itself evidence concerning the distribution. An attempt has been made here to assess the effects of departures from Gaussian behaviour consequent upon an excluded volume. Without claiming quantitative validity for the analysis, it seems justifiable to conclude that this would not modify the elastic properties of a network in the way empirically described by the C~ term. In principle, the most critical experiments which could be performed involve the use of samples swollen before and/or after crosslinking. There is evidence~1, however, that this introduces a new source of uncertainty, which is particularly important when we consider a polymer swollen to less than its saturation value, since the (negative) free energy of mixing is then large compared with that involved in network deformation. Closely related to this problem is the suggestion reiterated here that the C2 term reflects, at least in part, changes of packing. It is clear that more detailed understanding of molecular arrangements in the solid polymer, both dry and swollen, is greatly to be desired.

I gratefully acknowledge discussion with S. F. Edwards of the use I have made of his work. Department o¢ Chemistry, University of Manchester (Received January 1966) REFERENCES 1 See, for example, TRr~LO~, L. R. G. The Physics of Rubber Elasticity, Chapters 3 and 4. Oxford University Press: London, 1958.

384

T H E P R E S E N T STATUS O F T H E T H E O R Y O F R U B B E R ELASTICITY 2 FLORY, P. J. Principles o/ Polymer Chemistry, p 425. Cornell University Press : Ithaca, 1953 3 FLORY, P. J. Trans. Faraday Soc. 1961, 57, 829 BtmCHE, F., KINZIG, B. J. and COVEN, C. J. Polymer Letters, 1965, 3, 399 5 FLORY, P. J., HOEVE, C. A. J. and CIFERRI, A. J. Polym. Sci. 1959, 34, 337 WALL, F. T. and ERPENBECK, J. J. J. chem. Phys. 1959, 30, 634 7 SCHATZKI, T. F. J. Polym. Sci. 1962, 57, 337 s GEE, G. Proceedings of the Natural Rubber Producers Research ,4ssociation Jubilee Con[erence, Cambridge 1964, p 125. Maclaren: Glasgow 9 DOMB, C. J. chem. Phys. 1963, 38, 2957 10EDWARDS, S. F. Proc. phys. Soc., Lond. 1965, 85, 613 11 WALL, F. T. and FLORY, P. J. J. chem. Phys. 1950, 18, 108; 1951, 19, 1435 12TOBOLSKY, A. V., CARLSON, D. W. and INDICTOR, N. J. Polym. Sci. 1961, 54, 175 13 KHASANOVICH, T. N. J. appl. Phys. 1959, 30, 948 14 MOONEY, M. J. appl. Phys. 1940, 11, 582 is RIVLIN, R. S. and SAUNOERS, D. W. Phil. Trans. ,4, 1951, 243, 251 16 MULLINS, L. J. appl. Polym. Sci. 1959, 2, 257 17C1FERRI, A. and FLORY, P. J. J. appl. Phys. 1959, 30, 1498 is TRELOAR, L. R. G. Proc. phys. Soc., Lond. 1948, 60, 135 19KRAUS, G. and MOCZVGEMBA,G. A. J. Polym. Sci. (A), 1964, 2, 277 ~ GEE, G. Trans. Faraday Soc. 1946, 42, 585 21 VOLKENSTEIN, i . V. Configurational Statistics o~ Polymeric Chains, p 545. Interscience : New York, 1963 -*"DtMARZIO, E. A. J. chem. Phys. 1961, 35, 658; 1962, 36, 1563 BOOTH, C., GEE, G., JONES, M. N. and TAYLOR, W. n . Polymer, Lond. 1964, 5, 353 2~TOBOLSKY, A. V. Private communication 2z MARK, J. E. and FLORY, P. J. J. ,4met. chem. Soc. 1965, 87, 1423 26 MARl<, J. E. and FLORY, P. J. J. Amer. chem. Soc. 1965, M, 1415 27CIFERRI, A. HOEVE, C. A. J. and FLORY, P. J. J. ,4mer. chem. Soc. 1961, 83, 1015 OROFINO, T. A. and CIFERRI, A. J. phys. Chem. 1964, 68, 3136 29 MARK, J. E. and FLORY, P. J. J. Amer. chem. Soc. 1964, 86~ 138 30 ALLEN, G., BIANCI~I,O. and PRICE, C. C. Trans. Faraday Soc. 1963,59, 2493 31 GEE, G., HERBERT, J. B. i . and ROBERTS, R. C. Polymer, Lond. 1965, 6, 541

385