Statistical Mechanical Approach to Rubber Elasticity

The entropy changes that give rise to rubber elasticity may be modeled in terms of the chain statistics introduced in Section 14.1.2. For a chain whose end-to-end vector is fixed and equal to R, the number of conformations, il(R), that the chain can adopt is proportional to P(R, N). From Eq. (3), one thus obtains Eq. (20), where k is Boltzmann s constant and S is a constant. 

for constant U, the free energy is obtained from Eq. (21). 

The magnitude of the restoring force on a chain with end-to-end vector R is therefore given by Eq. (22). 

The total free energy change when a large deformation is applied to the chain, so that R = (x, y, z) changes to R = (Xix, X2y,X z), is given by Eq. (23). 

In an elastomer the forces are transferred to individual chains through the crosslinks, which constitute the nodes of a continuous network. If the chains linking these nodes are assumed to be freely jointed chains composed of N links of length b, Eq. (2) implies the spatial separation of the nodes, R, to be given by Eq. (24) in the undeformed state. 

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