Network Gaussian

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. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

From a theory of non-Gaussian network Hert-Smith34,3S derived dW/dlt = G exp [, (/, - 3)21 dW/dI2 =Gk2II2... 

Distribution functions for the end-to-end separation of polymeric sulfur and selenium are obtained from Monte-Carlo simulations which take into account the chains geometric characteristics and conformational preferences. Comparisons with the corresponding information on PE demonstrate the remarkable equilibrium flexibility or compactness of these two molecules. Use of the S and Se distribution functions in the three-chain model for rubberlike elasticity in the affine limit gives elastomeric properties very close to those of non-Gaussian networks, even though their distribution functions appear to be significantly non-Gaussian. 

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

Flory19> has shown that for Gaussian networks the connection between the temperature dependence of the force at constant volume and the temperature coefficient of molecular dimensions d In 0/dT holds for all types of distortions. According to Treloar 32), the equation of state for torsion of the Gaussian networks is... 

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

In Chapter III, section 1 the elasticity theory for ideal Gaussian networks is reviewed. Since in actual practice dry rubbery networks virtually never adhere to these theories — even in the range of moderate strains — experimental verification of the theory has been undertaken... 

It must be concluded that chain ordering may be a reality at least in a number of networks and should be taken into account as a possible source of deviations from Gaussian network behaviour (see (IV-3)). 

The Gaussian network theory can, therefore, predict phase separation in partially swollen systems. Phase separation can be induced by increasing the degree of crosslinking or increasing the interaction parameter %. Examples are given in Fig. 16. 

All dry networks and many swollen networks exhibit deviations from the Gaussian network behaviour discussed in the previous chapter. These deviations may have various causes ... 

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 ... 

The simplest of these approaches includes Gaussian Network Models (GNM) or Elastic Network Models (ENM) which assume that the native state represents the minimum energy configuration. A structure is represented as a network of beads connected by harmonic springs.12,13 One bead represents one residue and is usually centered on the position of the Ca carbon. Single parameter harmonic interactions are assigned to bead pairs which fall within a certain cutoff distance Rc. In case of proteins, Rc is usually around 8-10 A. The representation of the molecule in the... 

For dry (unswollen) Gaussian networks, the chemical contibution to the modulus is given by... 

A number of workers have treated non-Gaussian networks theoretically in terms of this finite extensibility problem. The surprising conclusion is that the effect on simple statistical theory is not as severe as might be expected. Even for chains as short as 5 statistical random links at strains of up to 0.25, the equilibrium rubbery modulus is increased by no more than 20-30 percent (typical epoxy elastomers rupture at much lower strains). Indeed, hterature reports of highly crosslinked epoxy M, calculated from equilibrium rubbery moduh are consistently reasonable, apparently confirming this mild finite extmsibiUty effect. 

A.S.Lodge, J.Meissner, On the use of instantaneous strains, superposed on shear and elongational flows of poljnneric liquids, to test Gaussian network hypothesis and to estimate the segment concentration and its variation during flow, Rheol. Acta 11 (1972), 351-352. 

For the same Gaussian networks, Treloar (1975) derived an equation for the shear modulus, G ... 

It is worthwhile to note that the equations describing the behavior of a Gaussian network [Eqs. (3.33) and (3.36)] depend on the structure of the network via a single parameter N, while the descriptions of non-Gaussian networks [Eq. (3.44)] include a second structural parameter, n. Therefore the number of chains, iV, of the network determines the behavior observed in... 

Thus, very shortly after a step deformation from rest (long enough for local relaxation of the chain) the response is the same as for a Gaussian network with affine junction displacements ... 

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