Molecular Theory for Rubber Elasticity

Consider a strand of polymer chain between two cross-links. The vector R between the positions of the two cross-links changes with deformation. Any molecular theory on rubber elasticity is based on the probability distribution function for R. As seen in Chapter 1, if the number of segments N on the strand is large, the probability distribution (R, N) of the end-to-end vector R is a Gaussian function 

There are a large number of configurations, Q, for the chain strand with a large N, which can all lead to the same end-to-end vector R. For a freely 

Consider the application of a uniaxial extension to a rectangular block of rubber, as shown in Fig. 2.1. The lengths of the three sides of the block are denoted by L°, L° and L° before deformation, and by L, Ly and Lz after deformation. We may choose any chain strand in the piece of rubber and let its one cross-link end be fixed at the origin of the coordinate system and denote the position of the other end at Rq = Xo, Yo, Zo) before deformation and at R = X,Y,Z) after deformation (Fig. 2.2). If the 

Furthermore, we assume that the density of the rubber does not change with the deformation. Then we have the relation 

According to Eq. (2.9), the entropies before and after deformation, denoted by So a.nd s respectively, are given by 

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