Network affine deformation

Fig. 3a. P200 and P400 as a function of draw ration for the pseudo-affine deformation scheme (uniaxially oriented sample) b P20o and P400 as a function of draw ratio X for the rubber network affine deformation scheme (N = 6, uniaxially oriented sample). Reproduced from Journal of Polymer Science by permission of the publishers, John Wiley Sons Incs (C)...

Let us consider further the calculation of the same characteristics of a PCP network, stretched up to some fixed drawing ratio X, for example, X = 3. Assuming network affine deformation the value of R at A, = 3 () becomes [6] ... 

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

Such considerations appear to be very relevant to the deformation of polymethylmethacrylate (PMMA) in the glassy state. At first sight, the development of P200 with draw ratio appears to follow the pseudo-affine deformation scheme rather than the rubber network model. It is, however, not possible to reconcile this conclusion with the temperature dependence of the behaviour where the development of orientation reduces in absolute magnitude with increasing temperature of deformation. It was proposed by Raha and Bowden 25) that an alternative deformation scheme, which fits the data well, is to assume that the deformation is akin to a rubber network, where the number of cross-links systematically reduces as the draw ratio is increased. It is assumed that the reduction in the number of cross-links per unit volume N i.e. molecular entanglements is proportional to the degree of deformation. 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

Structures of Type (b) can neither be counted as a single elastically effective network chain, nor, if the chain lengths differ, as two chains, because affine deformation for the chain end-to-end vectors does not apply. 

Wang and Goth (178) have applied the JG-type derivation (see Section III-l) using a series expansion of qt (r ) valid at not too large extensions. In this way the artificial splitting up in three sets of network chains, all having the same chain-end distances, is avoided. The assumption of affine deformation. however, still has to be made moreover, the fluctuations in crosslink position were assumed to be independent of the imposed strain. 

Since affine deformation cannot be proven for the non-Gaussian network chain defined by Eq. (IV-30), Blokland uses Eq. (IV-5) to derive the elastic free energy of the network. This yields ... 

Since swelling introduces an affine deformation of the network, the above calculation holds for any swelling degree and the distance d between first neighbor crosslinks can be written as ... 

In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30) ), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between that of the continuum and that of the network strands. Wagner [33] showed that the Doi Edwards strain function... 

The components of the end-to-end distance vector of each chain change in the same ratio as the corresponding dimensions of the bulk network. This means that the network undergoes an affine deformation. 

Figure 3.8 shows a diagram of the deformation. The sample initially possesses dimensions Xq, yo, zq, and when deformed it takes on dimensions X, y, z. According to the affine deformation model, the end-to-end distance vector of the chains of the strained network will change its components in... 

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

Comparison of Eqs. (3.36a) and (3.33) indicates that the value of modulus G obtained from the affine deformation model is two times the value corresponding to the phantom network. This would mean that the latter model is more applicable in the region of moderate deformations and the affine model is more suitable in the region of low deformations. 

Assume that each network strand has A monomers. One network strand, shown in Fig. 7.2, has end-to-end vector Rq with projections along the x, y, and z directions of R o, RyQ, and R in the undeformed state. In the affine network model, the positions of the junction points (the ends of the strands) are always fixed at particular points in space by the deformation and not allowed to fluctuate. For affine deformation, the end-to-end vector of the same chain in the deformed state is R (see Fig. 7.2) with projections along the x, y, and z directions of... 

This equation is the Flory form of the elastic part of the free energy of a network. The mean-square end-to-end distance of network strands in their preparation state is 7 q. Assuming affine deformation on the length scales of a network strand, the mean-square end-to-end distance in the final state is = XRoY- Modern treatments of network swelling and elasticity utilize a more general form of Eq. (7.69) for the elastic energy of a swollen or deformed network strand, known as the Panyukov form... 

An early model based on crosslinked rubbers put forward by Flory and Rehner (1943) assumed that chain segments deform independently and in the same manner as the whole sample (affine deformation) where crosslinks were fixed in space. James and Guth (1943) then described a phantom-network model that allowed free motion of crosslinks about the average affine deformation. The stress (cr) described from these theories can be described in the following equations ... 

An interesting comparison has been drawn between the prediction of pseudo-affine deformation and affine deformation of a rubbery network [3] which showed that the orientation distribution functions (cos " /3) grow at a rate which increases with increasing draw ratio in contrast to the behavior of the pseudo-affine scheme. 

In the consideration of real networks, it is clear that this assumption is overly restrictive. For example, consider a particular chain that happens to be reasonably extended in the unstrained network state. In the neighborhood of a crosslink point for this chain, we will find many other crosslink points for other chains. At least some of these other chains are expected to be in less extended configurations than the chain under consideration. Upon deformation, the tendency for the already extended chain to further elongate would be expected to be less than the tendency for the more relaxed chains to elongate. This being the case, the positions of crosslink points would be expected to move past one another in a manner not strictly defined by the affine deformation. 

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