The Dynamics of Networks BY S. F. EDWARDS* Cavendish Laboratory, Cambridge Received 23rd January, 1974 A review is presented of recent work calculating the dynamic properties of polymers in solution, where an attempt is made to see how changes in frequency, density and flexibility take one through various regimes of viscosity to elastic behaviour and how the number of entanglements and cross links affect this behaviour. It is argued that a series of lengths and frequencies naturally present themselves, and most of these regimes are capable of theoretical resolution. The problem of considering the dynamics of networks is complicated by the pre- sence of many different parameters, and the fact that the macroscopic behaviour is a consequence of complex microscopic behaviour. Consider an assembly of JV chains each (for simplicity) of length L, randomly coiled in structure of effective step length 1.Suppose they are immersed in a liquid of viscosity v, at a temperature T, and that the natural period for the chain to flip from one local configuration is z (to be fixed more precisely later). External disturbances, e.g., shear waves have a frequency co. For this paper we consider 8 conditions, but the work is easily extended over most of the range. Consider the changes of circum- stance as parameters change. If V is the volume of the solution, the spacing of the chains can be described by The size of a chain by Ro = (V/N)*. R, = (L1)j and a further parameter introduced to characterize the mean freedom of a point on a chain in dense conditions, an entanglement radius Re = R$PR;*.(To see the significance of this variable, consider a heavily entangled chain which is diffusing very slowly, so that one can refer to an initial point on the chain r(;) and a subsequent position r,. One can expect r, to be found near r(:) and a reasonable representation would be ~ X P ( - Y 1 ( r n - 4°')2) n for the whole set of rn which make up the chain. Adding another chain JV 3 M+ 1 can be expected to increase y in the proportion 1 / M , i.e., y oc NL/V. One then expects the diffusion equation for the single entangled chain, which for a free chain is (;-bv")P(r, s) = 0 * at present at the Science Research Council, State House, High Holborn, London WClR 4TA. 4748 THE DYNAMICS OF NETWORKS to become where we have made the transition from the discrete set of points .. . r,, . . ., to the continuous curve ~(s). It is well known that the solution to (1.2) is exp { +(r - r(o))2} or i.e., or The significance of these parameters appears when presented in this way. When Ro % R, the solution is dilute, each chain is on average remote from the rest and the physical properties are dominated by the hydrodynamic behaviour of a single chain. The chains induce a change in viscosity proportional to M* where A4 = L/Z is a dimensionless version of the molecular weight. As the chains get closer to one another, the coherent hydrodynamical effects along any one chain are weakened by the screening effects of the (incoherent) inter- action with other chains. When Ro - R, and then R, > Ro, the hydrodynamic interaction is screened out and the viscosity correction varies like M.As the density or length of the chains further increases there comes a point where the diffusion of the chains is radically reduced and when Re - R, the viscosity jumps to M3m4 experimentally, but the subsequent discussion here will obtain M3. The material now is heavy entangled, but still creeps with a high viscosity. If it is cross linked one can introduce an analogue of the Re, which can be called R,, by R, = Z/n, n being the number of cross links per unit length. The final characteristic length associated in the gel with both entanglements and cross links we call R,, and so that in the case of no cross links R, = Re, whereas in the (phantom) case of no entanglements R, = R,.When R, is the dominant length the material is elastic and the elastic modulus varies as Rg2. Now consider time scales. The external time scale is w-l. This can be compared with the time it takes for hydrodynamic disturbance to traverse a polymer. From the Navier Stokes equation one can see that this time w - l is given by so that the hydrodynamic screening which alters the naive viscosity dependence from M to M* will disappear as o + > wh. Both this and the screening effect above will tend to return the viscosity to the simple M law, the more important effect being determined by the relative magnitudes of (v/c@ and Ro relative to R,, indeed (v/wh)* can be referred to as a hydrodynamic length Rh. In this discussion the polymer has been considered simply as a loose chain drifting in and with the fluid.In fact it will have internal barriers and these will be jumped in a manner just described for aS . F . EDWARDS 49 different problem by Kramers, and extended to polymer problems by Stockmayer and Verdier and by Edwards and Goodyear.2 There is a characteristic time associ- ated with the jumps of the molecule, (z/kT) or frequency oj. Lf o > wj the polymer appears as rigid and the viscosity does not decrease with w. But for w < w the poly- mer drifts freely. Of course the normal viscous drag and hydrodynamic effects are still present and complicate a complete analysis, but the general result is that a plateau in viscosity will appear as co > coj. Now proceed to the gel. In order to be a gel it must be sufficiently cross linked orland sufficiently entangled.The network of chains will drift through the fluid, but having some mean position, or an approximate mean position for uncross-linked gels, where the " mean position " will slowly drift. Obviously, there will be screening of the hydrodynamic effects, and one can envisage applying frequencies w > w j such that shear waves in the fluid have to force themselves past a virtually rigid network. At lower frequencies the network can sustain elastic vibrations which will be damped by the internal frictional effects within the network, but more important by the relative motion of the fluid to the gel. The elastic modulus will depend on Re and R,, but for high frequencies the effect of cross linking will weaken since the vibra- ting chain will not notice its reduction in degrees of freedom.This effect will appear as w reaches the natural period of the chain associated with R, i.e.? or the appropriate modification for mh and R,. The effects on entanglements will be two-fold; firstly the same effect as for cross links which is to weaken the effect, but also the effective number of entanglements will increase with w, for at a high fre- quency, configurations which are not true entanglements and from which a chain could extricate itself in time now appear as true entanglements, I have not yet produced a convincing criterion for this latter effect; it has been observed experi- mentally (Ferry, personal communication, 1974). Having catalogued the effects, the next section will give an outline of the mathe- matical structure which models them.In addition to well known results in the litera- ture this paper will amount to a review of work on polymer solutions by myself and K. F. Freed and on gel-liquid systems by myself and A. Miller.4 co = ~ T I V ~ R : 2. MATHEMATICAL MODELS Physical properties peculiar to polymerized systems have to do with the great length of the chain. It is therefore important to adopt a notation which clings to this essential point whilst casting all other characteristics of the molecules into various constants that appear. The study of these constants is of course important, but they do not help in getting those phenomena which are unique to polymer systems. A chain can therefore be simplified to a series of points R(,,) on an inextensible chain Such arrays appear in crystals and it is well known that the way to study long waves in a crystal is to consider fourier series of the unit cell coordinates.Likewise in polymer problems the key variable is the fourier sum R(,,- 1,1 = 1. The complex variable is more useful than the real. The finite number of units on the chain, N , implies that R, is not a perfect variable, in as much as its use implies the adequacy of a cyclic condition That this is not true does show up that the modes of motion of polymers are not perfectly separable in these fourier variables. Nevertheless, even when extensive machine calculations are made to =50 THE DYNAMICS OF NETWORKS find “ best possible ” Rouse-Zimm modes, these are extremely close to fourier variables.(It is possible to show that more accurate theories of dilute solutions than the Rouse-Zimm theory give even closer results to the fourier modes, and that for gels the fourier modes are indeed exact modes.) The probability distribution of the small q modes for Gaussian chains is well known to be with a consequent entropy in 6 conditions of 3kT 21 - - q21RqI2. Thus if we want the simplest theory of the Brownian motion of a chain, and postulate a viscous drag of [ and a random forcef, 3kT 5Rq++Rq = fq(t) for each q value. If the probability of finding the R4 is P(. . . rq . . . ; t ) then standard theory gives where (f,(t) f-q(t‘)> = 3ckT6(t - t’). (2.6) The equilibrium solution for P is (2.2). Eqn (2.5) amounts to the well known Rouse equation where only the q -+ 0 values are used.What we wish to do in this section is to put the effects of cross linking, hydro- dynamic interaction and entanglements into this kind of formalism. Before getting to gels proper, consider the effects of a surrounding liquid. The boundary condition between the polymer and the liquid can be taken to be that of no slip, i.e., &n> = u(R(n,, t ) (2.7) where U(E, t ) is the fluid velocity. (It doesn’t seem to matter what condition one takes as long as it is unambiguous, see Zwanzig 5 ) . The mathematical way to handle such a constraint is to use a Lagrange multiplier and to modify the equations of hydro- dynamics to Du = c ou,,6(r -R#) + F (2.8) 9 R P ) = -a,+ f (2.9) u,n where a labels the chain and n the monomer.where D denotes the operator of the linearized Navier Stokes equations and 9 the equation of motion of the polymer, and F, fare forces acting, random and external. The equation of the polymer could be taken to be that of a chain without rigidity or mass, i.e., 3kT BRq = ----q2R z q (2.10)S . F . EDWARDS 51 i.e., if we think of -+ R(s, t ) s an arc label (2.11) But if we think of the polymer making random jumps of its own with a characteristic time z one has (2.12) One must solve for u in terms o f 0 and eliminate 0 from the equation by the condition I? = t, on the polymer. When this is done, one finds the form ( ~ k , + ~ q ' R ~ +.X(q, p)k, = 9+f 3kT ) (2.13) where% gives the effect of the liquid and 9 is the force stemmiiig from the forces F acting on the liquid, transmitted to the polymer.For example, in the case of a little (massless) sphere embedded in a liquid, X would be the Stokes term 6na9. For a dilute solution p - 0 or more precisely R, < Ro, 3- is independent of p but varies like 43, which is a well known result expressed in the present notation. This result comes from the fact that the motion of one part of the polymer forces the liquid to move, which then affects another part of the polymer cooperatively. However, a more accurate attack again puts in the time dependence of this hydrodynamic effect and replaces X in (2.13) by s.X(q, p, t - t')R(t') dt' i.e., one can fourier transform also on t, to get (2.14) The full form of X is complicated and is given by Edwards and Freed structure varies like but the according to which is the largest.Thus the hydrodynamic interaction is screened by p or o, i.e., by the intercession of other chains, or by the time it takes for the motion to get from one point on the polymer to another. The expression is further complicated by z, for the smallest q will be - L-l and if z is larger than this it implies that the chain responds so slowly to the forces acting on it that the hydrodynamic intervention is irrelevant. The details of these calculations we do not give here; the point made is that the variables q, o give a description of the various circumstances that arise. The static viscosity increment can be developed from (2.14) (and indeed the dyna- mic viscosity also), and has a simple form, being We shall not employ z further and consider chains freely moving.(2.16) (2.17) (2.18)52 THE DYNAMICS OF NETWORKS How can one extend this to situation in a gel where the chains are not just overlapping, so that hydrodynamic effects are screened out, but heavily entangled so that they can only move cooperatively ? When an oscillating shear wave is applied, elastic effects will dominate, but one framework so far should handle weak, steady shear and yield a viscosity. One can see how this might be handled by returning to (2.4) and (2.5). The effects of the random forcef, is countered by the fact that the chain is boxed in by its neighbours. Thus if one can expect where whereas interaction), h, Note the sign; the presence of the other chains to diminish this by h,(q) say, so that (2.20) ho is independent of q (there being screening of the hydrodynamic will be dependent on q and have the form for clearly the weakening effect of h, will be diminished over small h l h ) = h1(0)-h2q2. (2.2 1) distances and be at its strongest over large distances.Note also that q takes positive and negative values, and there cannot be terms linear in q since the resulting expression must be even. It could of course be that the expansion should be IqJ', but this re- presents a higher level of sophistication than is used here. We just expand it and keep the first term. Thus h = (ho-hl)+h2q2. (2.22) Now h, will depend on the number of polymers to give rise to entanglements, so for a fixed molecular weight, as p increases ho -hl decreases and at a critical density gives zero.Thereafter h = h2q2 and since at this point only the physical quantity Re can be involved, The viscosity in this regime is then (with 5 - p-I the screened term (2.15)) 6 V - OO .M3p3; PV R, > Ro > Re. (2.23) Experimentally the power of M is greater than 3, but the expansion (2.22) represents the simplest approach to the problem. An explicit realization of the structure of h l , h2 has been given by Edwards and Grant,6 but much more needs to be done to make this approach convincing. In particular, the transition has been treated by considering the behaviour past it, by the statement that only R, can appear. In fact, one can hold p constant and vary M so that through the transition hl will behave as h,(M, p ) taking the value ho as p + coy independent of M.One would like also to keep p fixed and study the transition as a function of M, which for technical algebraic reasons seems quite difficuIt in practice though not in principle. It will be noted that if one considers a( [ R,12)>lat one finds zero for q = 0, simply because ([R,12) is the equilibrium value. But in our fourier variables q = 0 corresponds to the motion of the centre of mass, R = -R,=, 1 N (2.24)S . F. EDWARDS 53 and directly by multiplying (2.20) by R* and integrating aW2 2h(0) - = - d t N (2.25) the normal Browman motion of the polymer as a whole. However, after transition - = o aR2 a t (2.26) since h(q) = 0 at q = 0. The transition corresponds then to the chains ceasing to drift except at the very slow level implied by the motion of the free ends which do not have entanglements.Under these conditions it is not the low viscosity which will appear as the dominant physical phenomenon, but the emergence of elastic moduli at all but the lowest frequencies. In order to study this aspect, a good start is to consider the effects of cross linking on the network. The constraint is now that certain points on certain chains are the same as certain points on other chains R(@)(SP, t ) = P ( s e , t). As a model for this we can consider an average position of one chain, say R(A) and another R(g). Then and R(2) will not stray far from R(A) and R(g). The kind of equation one might expect is then (2.27) where 402 represents effects restoring R4 to its mean position. Now 402 will be the result of the various cross links and entanglements.Suppose it is known, then one can estimate the effect of one link between chains 1 and 2 by putting in a Lagrange multiplier into equations of motion which are otherwise (2.27). The total effect of such terms must then be 4:. This programme is easily carried out, and yields (2.28) 402 = n/SG(q, w) dq where i.e. (2.29) (2.30) together with a change in structure of the equation, by which the many other chains interacting with the one particular one under consideration are reasonably represented by the mean displacement of the solid (2.31) (2.32) Physically, this means that the average effect of cross links is to tie down a chain relative to a mean position plus the average distortion of the materials as a whole. One can now go ahead and solve (2.32) for R in terms of u and hence derive equations54 THE DYNAMICS OF NETWORKS for u directly, which give the elastic equations of motion and conform in structure with classical work, but also give the clamping and frequency shifts.A detailed account is given in Edwards.2 Notice however that qg is frequency dependent, i.e., the effect of a cross link depends on frequency, in the limit of a very high frequency nq, = n2 as m-+O which then modifies the wave equation, which is now C02&+kTqOV2Uk = F k . (2.33) (2.34) (2.35) (2.36) Finally, one can add entanglements into (2.32) in the static sense very easily, for the additional constraint must be proportional to the density and the qo equation modifies to 402 = nqo+Cp. (2.37) But the full theory is more difficult, and it is speculated upon in the next section.FURTHER PROBLEMS An outline has been given of how increasing density leads to chains getting less and less mobile until they reach a state of very low mobility and high viscosity. From the other end of a cross linked elastic solid, the entanglement of the chains acts as an enhancement of the cross links and increases the elastic modulus in a well defined way (provided that the material is randomly cross linked these predictions can be checked, see for example Allen et al. 9 ’). There are two obvious areas which should be explored. First, when there are cross links the formula for their effect is frequency dependent. One can expect that for entanglements to be so also, but in two ways. One is that the response of the chain to forces is frequency dependent-the same effect as for cross links, but also that the number of entanglements increases.For example at low frequency the configurations / (I, are the same from the point of view of cal- culating the effect of entanglement on the en- tropy : one can deform one into the other. However at high frequency only a small segment of the chain notices the rest of the reduced. Another problem is the precise behaviour when there are no cross links. At a high enough frequency, the number of entanglements will increase so that the presenceS. F. EDWARDS 55 of cross links is unimportant, and one can set the problem up as an elasticity problem. But as frequency or density diminishes one reaches the region where a highly viscous liquid is a more appropriate description. How can one span these regions? The principal difficulty seems to be in the fact that the formal theory of entanglements (which is not touched on in this review) though quite easy to write down in a formal way, leads to very awkward mathematics. Yet the physical picture is clear enough, so I hope to be able to find an adequate bridge. I have benefited from discussions with Prof. Ferry and Stockmayer at the 1973 Les Houches Summer School, and as ever from G. Allen and K. F. Freed. W. F. Stockmayer and P. H. Verdier, J. Chem. Phys., 1962, 36,227. S . F. Edwards and A. G. Goodyear, J. Phys. A , 1972,5,965. S . F. Edwards and K. F. Freed, J. Chem. Phys., 1974, 61, 1189, 3626. S. F. Edwards and A. Miller, to be published. R. Zwanzig, private communication. S. F. Edwards and J. W. V. Grant, J. Phys. A , 1973,6, 1169, 1186. 'S. F. Edwards, J. Phys. A , 1974, 7, 318. G. Allen et al., Proc. Roy. SOC. A, 1973, 334,453,465, 477. G. Allen, Farday Disc. Chem. Soc., 1974,57, 19.