Statistical Mechanics of Systems with Internal Constraints: Rubber Elasticity

Abstract

The general formulation of the statistical mechanical description of systems with internal constraints is recast into a form in which the ensemble average is written as a summation over the initial and final microscopic states of the system. The functional integral formulation previously developed to treat polymer networks is generalized to include these permanent constraints. The statistical mechanics of rubber elasticity is considered using a simple model in which the cross linkages are taken to be permanent topological constraints on the polymer network; these linkages are taken to be at the same places along the chains in both the initial and the deformed states. Entanglements are ignored, and the elasticity of this ``phantom'' network is obtained by using a variational principle for the free energy. The analog of the average affine deformation of the network is obtained as a variational result of the calculation. The elastic modulus obtained is in agreement with Flory's theory; however, the present statistical mechanical formulation is capable of also considering the effects of interactions between different polymer chains, chain stiffness, etc., and closed circuits in the polymer network.