Rubbers affine deformation model

To illustrate the usefulness of birefringence measurements in orientation studies, we now briefly discuss two simple models of orientation leading to different expressions of the second moment of the orientation function the affine deformation model for rubbers and the pseudo-affine model more frequently used for semi-crystalline polymers. 

One of the best-known of such schemes is the AFFINE deformation model for rubbers. The rubber is considered to be a network of flexible chains, and the macroscopic strain is imagined to be transmitted to the network such that lines joining the network junction points rotate and translate exactly as lines joining corresponding points marked on the bulk material. If we assume that the flexible chains consist of rotatable segments called random links , and that some statistical model can describe the configurational situation, it is then possible to obtain explicit expressions which relate the segmental orientation to the macroscopic deformation. 

The classical models of rubber elasticity reduce the elastic properties to a study of entropic springs. In the phantom model the EV interaction only gives the volume conservation, while in the affine deformation model it also is responsible for the affine position transformation of the crosslinks. Otherwise the entropic forces and the excluded volume interactions and consequently the entanglements, as they are a result of the EV interaction, completely decouple from the chain elasticity. The elasticity is entirely determined by the strand entropy. It is obvious that this is a... 

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. 

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 assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

The most important fact that you should grasp from this discussion is the entro-pic nature of rubber elasticity. Although the agreement between the simple model described here and experiment is not that great you have to keep in mind that there are both theoretical assumptions (e.g., affine deformation) and mathematical approximations (Gaussian chain statistics) that have... 

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 ... 

A pseudo-affine model predicts the variation of 2 with the deformation of a semi-crystalline polymer. It assumes that the distribution of crystal c axes is the same as the distribution of network chain end-to-end vectors r, in a rubber that has undergone the same macroscopic strain. Figure 3.12 showed the affine deformation of an r vector with that of a rubber block. 

The simplest explanation is that there is a rubber-like network present and that this has a maximum extensibility due to the degree of entanglement, which is constant for a given grade of polymer and depends on its molar mass and method of polymerisation. This limiting extensibility is not to be confused with the limit of applicability of the affine rubber model for predicting orientation distributions discussed in section 11.2.1 because the limiting extension can involve non-affine deformation. 

The simplest version of the rubber model makes the assumption of an affine deformation when the polymer is stretched the cross-link points move exactly as they would if they were points in a completely homogeneous medium deformed to the same macroscopic deformation (see section 6.4.4, fig. 6.12). The following additional assumptions are also made in the simplest form of the theory. 

Affine deformation This model assumes that the deformation of each configuration of the chains is affine in the macroscopic deformation. It is not compatible with known classical theories of rubber elasticity. 

Fig. 7. Co O and co O plotted against draw ratio, /.. for chain a.xes in uniaxially oriented tapes of polyethylene terephthalate.. cos O and O. cos determined from Raman intensity measurements on the 1616 (vn line. Curves (a) pseudo-affine cyg regate deformation model, atrves (b) affine rubber elasticity model.

Fig, 9. Comparison of cosf A and cos A obtained for oriented PVC % and oriented PMMA O Lower and upper curves show predicted relationship according to the affine rubber elasticity model and the pseudo-affine deformation scheme respectively. 

The Theory of Kuhn and Grun. The theory of birefringence of deformed elastomeric networks was developed by Kuhn and Griin and by Treloar on the basis of the same procedure as that used for the development of the classical theories of rubber-like elasticity (48,49). The pioneering theory of Kuhn and Griin is based on the affine network model that is, upon the application of a macroscopic deformation the components of the end-to-end vector for each network chain are assumed to change in the same ratio as that of the corresponding dimensions of the macroscopic sample. 

The affine network model assumes that the end-to-end vectors transform affinely with maaoscopic deformation, which amounts to saying that the chain ends are rigidly embedded in the medium. An improvement in this approximation is made by assuming that the junctions may fluctuate in the network. In a sense, this is a more realistic picture of rubbers since the chains are modeled as flexible. The instantaneous vector r joining two junctions at the ends of a network chain may be written as the sum of a time-averaged mean f and the instantaneous fluctuation Ar from this mean, that is. 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

In the Doi-Edwards theory the environment of a chain is modeled as a tube with a diameter which is constant over the tube length. Each subchain, which is the part of the polymer chain between localized entanglements, resides in a tube segment. The subchains react to an instantaneous strain affine deformation, as in the classical theory of rubber elasticity. Therefore, immediately after the imposition of a simple elongational strain at t 0, the stress is given by Eq. (6), with G 0 Se(X) is given by Eq. (8), and, in analogy with Eq. (9), the relaxational part of the modulus at t = 0 equals ... 

Small-angle neutron scattering (SANS) of labelled (deuterated) amorphous samples and rubber samples detects the size of the coiled molecules and the response of individual molecules to macroscopic deformation and swelling. It has been shown that uncrosslinked bulk amorphous polymers consist of molecules with dimensions similar to those of theta solvents in accordance with the Flory theorem (Chapter 2). Fernandez et al (1984) showed that chemical crosslinking does not appreciably change the dimensions of the molecules. Data on various deformed network polymers indicate that the individual chain segments deform much less than the affine network model predicts and that most of the data are in accordance with the phantom network model. However, defmite SANS data that will tell which of the affine network model and the phantom network model is correct are still not available. 

---