Dynamical Considerations of Network Properties S. F. Edwards

Cavendish Laboratory, Cambridge CB3 OHE, UK

Elasticity is discussed as an aspect of viscoelasticity, which is described by the tube model. The effects of both crosslinks and entanglements contribute to this model and a discussion of how these effects can be quantified is given. At high enough concentration, entanglements ensure the existence of elastic effects even without crosslinks, and a theory is presented on how this dynamical phase change comes about.

Keywords networks,

entanglements, ruhher,

viscoelasticit y

1 INTRODUCTION

The phenomenon of the elastic behaviour of macromolecular networks was one of the first great successes of polymer theory. But after the first great result, that the simplest form of the free energy is

1

further advances are rather complex. Empirical additions to Fo obviously can improve the fit, but there is no convincing way in which a simple Mooney-Rivlin term like

comes out of a model. Models usually provide complex func- tions of the A’s which need some torturing before an Fl will emerge. Moreover, the kind of arguments based on non- dynamical studies, such as the original Kuhn and Griin, the James and Guth theory, and the simple and direct theory of Flory, say nothing about viscosity. However, it is well known that at a sufficiently (but not extemely) high frequency, melts and concentrated solutions have clear elastic properties in the absence of any permanent crosslinks, a ‘plateau modulus’, and that the modulus is a comparatively straightforward function of the concentration c:

(3) This behaviour is manifested for systems above the entangle- ment crisis in the zero frequency viscosity and a ready interpretation is that for large enough concentration (c ) and molecular weight M, the polymers are so entangled that within the time w1 (with u the applied frequency) they can slide around through their neighbours so that, if one considers ‘entanglements’ as some kind of temporary crosslink, at this frequency the melt behaves as a rubber with the entanglement density (whatever that may be) taking the place of the crosslink density.

Therefore it seems fruitful to approach the simple rubber network problem by a new route, studying viscoelasticity and seeing why one obtains the plateau modulus, and from that calculating the static modulus.

This paper will introduce the tube model, study the dynamical phase change which gives birth to it, and end with expressions for the static modulus.

2 THE TUBE MODEL

Although at first sight the picture of vast numbers of long entangled molecules in high thermal agitation is daunting, some general principles of theoretical physics come to one’s

aid. To do any effective mathematics one must have some picture which under certain conditions becomes rigorously correct, and then find that the parameter, which describes how far away one is from this picture, is small. The picture of very long molecules at high density is a good starting point, for clearly such a molecule can only move very slowly, and in a first approximation can only wriggle backwards and forwards mak- ing little real progress. Any one long chain interacts with a large number of other chains so that the effect of any one chain on the chain we are studying will be small. Thus, one is led to the picture of a single chain surrounded by some effective medium, the other chains, an effective medium which resists any attempt of the chain being studied to penetrate it. An exception to this remark is that the ends of the chain are free, so a slow motion along the length of the chain is possible. The ‘reptation’ is a slow motion relative to all others, and provides the small parameter on which a theory of polymer diffusion, and hence viscosity, etc., is based. Some of this is just hypothesis, but de Gennes showed more than a decade ago that reasonably convincing quantification can be made to con- firm this argument. This is not to say that the picture is proved, since there may be complex collective behaviour ignored by the single polymer picture. (Compare, for example, the theory of electrolytes. This screens the Coulomb potential to the Debye-Huckel potential, but the other effect is to bring into existence plasma excitations which are collective oscillations of the charge, contributing half the free energy. A naive theory would consider a gas of particles with screened interactions, and miss the collective oscillations.) But it is adopted here, since it can be made the basis of a respectable theory of viscoelasticity . Thus, if we imagine, in order to draw a diagram, that our chain is in the plane and the others are represented by dots, we have something like Fig. 1 where the locus of the polymer at time t=O is the full curve, at a later time the dashed curve, and the dotted lines represent the rough boundaries of the ‘tube’ that is the limit of the ‘excursion’ of the chain, Another way to look at it is to imagine a demon hauling one end of the chain in white keeping the other fixed. If the original chain had a step length

Fig. 1

BRITISH POLYMER JOURNAL, VOL. 17, NO. 2 1985 122

b, the topological skeleton (the 'primitive path') has a step length a, and indeed it is possible to show that the width of the tube is a also.

So if all the chains are long, there are many chains making up the tube and if one end of one of them passes our particular chain it hardly depletes the tube. Morover, for each dot vanish- ing in Fig. 1, on average another appears. So the tube will change only very slowly due to sideways motion. But the chain can always create and destroy the tube at its ends in the reptation process.

This is the high concentration/high molecular weight picture. How can it be checked? By many viscoelastic consequences' where good, although by no means perfect, results emerge. However, these results can take a, the primitive path step length, as a parameter, and it is interesting to ask if there is a direct measurement of a. This can be provided by the plateau modulus.

3 THE PLATEAU MODULUS

There are several ways of viewing the primitive path, i.e. the axis of the tube. A good one for this section is to think of a series of slip links, spaced a apart, and imagine that this is a representation of the complicated environment that really exists.

At a high frequency these slip links can be regarded as crosslinks, so that

where now

With a being a(c) the prediction is that the modulus is indepen- dent of molecular weight and a function of c alone. Experimen- tal plots for the modulus give a power

aw? a-1.2 ( 5 ) Can this power be derived from theory? It can if c is small but M so long, so that L h (where M has been converted into a length L). Then the theory is analogous to a critical phenomenon and there is only one length scale. This length scale is the screening length and it affords an effective ab2 which gives a=1.25. This fits, but is not at all the correct physical situation, since we are dealing with high densities in melts or near melts, and the chains are known to have almost unswollen dimensions. In these conditions chains are effec- tively 8 like. So what does theory predict there? There are three predictions.

(1) Gaussian chain behaviour is the only phenomenon present and the only length scale entering comes from the relation

Fig. 2

R2= Lb. Doi2 followed this argument through and it pre- dicts a=2. This is a purely dimensional argument (or scal- ing argument, but note it is much simpler than the scaling theory of the semi-dilute solution) and since it does not agree with experiment, it suggests that there is something wrong with the idea that entanglements can be completely discussed by using Gaussian chains. One can argue that as the chain experiences the tube, the individual chains it meets can be adequately modelled as straight, i.e. in Fig. 1 the chains can be thought of as a child's climbing frame of straight rods, i.e. simply to model the rest of the chains. This gives CT=I.O.~ A simple hypothesis is to say that the number of contacts is sufficient to characaerise the result, which also gives a=l.0.4 If one works directly from the Gaussian invariant formal- ism, one can argue that the tube can be directly modelled from the winding numbers, but since the integrals are rather singular at short distances (as indeed might be the case with two encountering random walks), one cuts all integrals off at b. Crude as this is, it was the first a p p r o a ~ h , ~ ~ ~ and since it gives a=0.5 might not be con- sidered worth mentioning if it were not for the fact that Donald and Evans' and Kramers,s using the results of Donald found just such an exponent in their study of crazes.

So theories give the plateau modulus between c3 and c3I2, and experiment c2.2 to C I . ~ .

This is a central problem in understanding the entangled state of polymerized matter, and clearly much work remains to be done. As long as we are only this close, all other studies of the condensed phase, including all rubber aspects, are partial.

4 THE DYNAMICAL PHASE CHANGE

Having had some trouble looking at our polymers as M+m, let us now look at the problem of fixed c and change the molecular weight. The spectacular phenomenon is that the viscosity which for dilute solutions have the Zimm value Milz rises in more concentrated solutions through M and then suddenly takes off with a high power M3.4. (For simplicity all our discus- sions will be as if the polymers were in 8 conditions, i.e. have hydrodynamic interactions, and cannot pass through one another, but have no potential, namely 2nd and higher virial interactions.)

One can see quantitatively through this. The Rouse Chain has M 1 viscosity. Hydrodynamics changes this to MI", but the hydrodynamics screens out as the chains start to overlap and one obtains a Rouse form MI. Eventually there is an entangle- ment crisis where chains cannot get out of one another's way, and hence slow up to cause the fact that they cannot get out of one another's way. Can we put this into mathematics? This problem has been studied by Klein'o and here we present a more elaborate model of the motion.

The basic idea is that in low concentration the motion of the chain is uniform in all directions, subject only to its connec- tivity, but in high concentration the chain cannot easily move perpendicular to the tube, although it is free to diffuse along the tube. The route to an algebraic realisation of the magni- tudes governing this situation is to consider an infinitely long tube. The real tube is, of course, only as long as the polymer itself (or rather as long as its primitive path), but consider for the moment diffusion in an infinitely long tube. Simplify this further by studying a single particle. The tube is taken to be a random walk of step length a, so that if ss' denote arc lengths along the tube and r(s) are the coordinates of a point of the

BRITISH POLYMER JOURNAL, VOL. 17, NO. 2 1985 123

tube

({r(s)-r(s‘)}2)= /s-s’ / a (6) If the particle has a normal diffusion behaviour along the tube, it can be assumed that this is diffusion of the variable S(t) which gives the position along the tube and

({ S( t ) - S( t ’ ) }2)= D 1 t- t’ I

R(t) = r{ S ( t ) 1

(7) If R(r) are the coordinates of the particle

so that I

({R(t)-R(t’)}’)K b ‘ E dt-t’a

This means that compared to the standard diffusion law

({R(t)-R(t’)}’)=D 1 t-t‘ 1 (9) the particle does not diffuse if confined to a random walk tube.

Now consider the case of a particle which can diffuse with DII along the tube, and DI perpendicular to it. This problem is a physically realistic model of the diffusion of a changed particle in a strong magnetic field where the field lines are random walks, as for example could be found in a turbulent magnetic fluid. Suppose the field line is characterised by a correlation of its tangent

(r’(s) .r‘(s’))=exp ( - y l s - s ’ i ) (10) i.e.

1 1 ({r(s)-r(s’)}2)=- Is-s’I +- {l-(exp -yls-s’I)} (11)

(in polymer language a ‘wormlike chain’, the stiffness at small distances being needed to give good behaviour at short dis- tances). Then in two dimensions it is found that

Y Y 2

( ( A R)z)- ({ R(t) - R( t ’ ) } ’) = V‘DIID,t +\j%qiD, (@Tqi Jrzt+oii‘

(12)

Dl,=D =D ((AR)’)=Dr (13)

D,=O ( ( ~ ~ ) 2 ) = y \ j F t (14)

,- - \i D i l D i l )

This correctly gives the limits

As t-+m the limiting value is simple

( ( A R ) ~ ~ J D , , D ~ t =dt i.e. the effective diffusion is the geometric mean.

In three dimensions it is more complicated and the limit is found to be

which again, for small D, gives

(whereas for q>=DL it gives D=D).

To apply this result to a polymer, we argue that reptation is a small residual effect and the first approximation is to ignore it. Then one can expect at low molecular weight that

(18) D=D-D - 11- 1 -D where D is the free diffusion value, applicable when there are no entanglements, but anticipate that at a certain molecular weight (or concentration (Dl=O and D = O . This will corres- pond to the viscosity becoming infinite at a certain molecular

L,\ W

n

Primitive path

Fig. 3

u

fi Fig. 4

weight, whereas in reality reputation, will remain, so in fact a small d will persist and a high viscosity, given by reptation theory as M ’ , in fact M3.4. The diffusion of the polymer can be related to the diffusion of a particle by synthesizing the polymer diffusion by hops along the The picture one has is that the ‘wall’ of the tube consists of other polymers which are drifting by with D . When an end leaves the wall the confined polymer can take up that position. However, the position can only be confirmed if another polymer slides in behind it. This process of ‘tube renewal’ was shown by de Gennes to be small compared to reptation in sufficiently high M or c, but clearly is dominant in the situation where there are so few neighbours that the ‘tube’ hardly matters. Figures 3 and 4 show the idealised picture.

Thus, since when different processes must all be involved it is the time-scales which must be added, and since both the removal of one tube polymer and the insertion of the other taken together remove the square robt in Eqn (17), a careful analysis of the processes leaves us with

124 BRITISH POLYMER JOURNAL, VOL. 17, NO. 2 1985

where c = l is the maximum concentration possible, to is the free polymer time-scale, and L the length of the average polymer.

If we replace L by M and introduce M o as the critical value of M

D = O (20) or

1. or absorbing 1-c into Mo by M , = M o / v G

If reptation is now added one eventually reaches the final form for M<M,

(1- 1

1-( M/M,)* \ 1 + ( 2 M 2 D / A ~ ) ( 1-M21Mf) rl=

where AR is the constant which gives this expression meeting the reptation value

M 3 rl=-

A R

at M=M, and thereafter for M>M, Eqn (24) holds.

This discussion is much abbreviated and will be published in full elsewhere. The point of introducing it is to show that the concept of the tube implies a viscosity behaviour very much like that found, and indeed one which we believe can pin down further the specification of the tube.

5 THE SLIP LINK MODULUS

The final comment on the tube model is to note that it can be used in its slip link version to calculate an effective modulus due to entanglement. This has been done by Ball et al.I3 using the replica method of calculation which states that if e, a measure of the amount the chain in Fig. 2, is allowed to slip

through the link, then each slip link contributes

A f ( l + & ) , 1+&A?

Y2 Z-- + % C l o g ( l + & A f ) 1

so that if the concentration of crosslinks is c, and the number concentration of chains is c, one has

2F VuT-

If c, is zero then Fis zero as it must be, and if c, is large then one has -c,22; from crosslinks and c/ab2 (expression ( 2 5 ) ) from entanglements. It is possible to give an elementary derivation of this expression on the lines of the Flory single segment theory of elasticity, but in this paper it is simply quoted.

The point being made is that since the tube model gives a reasonable account of viscoelasticity of the melt, it cannot be ignored in rubber elasticity proper, and although other effects such as the detailed environment of the crosslink may also matter14 the present effect must also be found.

In fact quite good agreement with experiment has been found by Thirion and Weil.15

References 1 Graessley, W.W., Adv. Polym. Sci., 1982, 41, 67. 2 Doi, M., 1. Phys. A, 1975, 8, 959. 3 Edwards, S.F. & Evans, K.E., J. Chem. SOC. Farad. 11, 1981, 1891. 4 de Gennes, P.G., J . Phys. (Paris), 1974,35, L133. 5 Edwards, S.F., Proc. Phys. Soc., 1967, 92, 9. 6 Deam, R.T. & Edwards, S.F., Phil. Trans. A, 1976,280,317. 7 Donald, A.M. & Evans, K.E., Polymer, in press. 8 Donald, A.M. & Kramer, E.J., Phil. Mag. A, 1981,43, 857. 9 Donald, A.M. & Kramer, E.J., Polymer, 1982, 23, 461.

10 Kelin, J., Macromolecules, 1978, 11, 852. 11 Orwoll, R.A. & Stockmayer, W.M., Adv. Chem. Phys., 1969, 305. 12 Edwards, S.F. & Goodyear, A.G., J . Phys. A, 1972,5,965. 13 Ball, R.C., Doi, M., Edwards, S.F. & Warner, M., Polymer, 1981,22,

14 Flory, P.J., Br. Polym. J . , 1985, 17, 96. 15 Thirion, P. & Weil, T., Polymer, 1984, 25, 609.

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