The rubber elasticity of a randomly cross-linked polymer network has been the subject of intensive research in the past [1, 2]. One significant feature of such a network is its finite extensibility (i.e. a strand of N monomers cannot extend to a length of more than N times the monomer length), an effect that the very much simplifying and indeed extremely successful Gaussian statistical theory does not account for. In order to capture such finitely extensible behaviour of the network, Kuhn and Grun  devised the so-called inverse Langevin approximation within the freely jointed chain model, and obtained the leading correction to the Gaussian single chain statistics. More recently, a powerful method based on moment calculations from the characteristic function  have been developed to obtain the single end-to-end distribution function [5, 6], achieving encouraging agreements with Monte Carlo simulations .
Under assumptions such as the affine deformation of junction points and the additivity of individual strand entropy, the end-to-end distribution fuction directly leads to the network elasticity through either a finite-chain average or a network average [1, 8, 9]. However, their agreement with experiments is incomplete, and many alternative theoretical models of elasticity have been proposed, ranging from the constrained fluctuation  and primitive path [11, 12] to sliplink [13, 14] and tube models [15, 16]. The relative merits of these approaches in fitting experimental data have been reviewed by Gottlieb and Gaylord . The message is somewhat unclear as none of the eight molecular models available were able to reproduce all of the experimental observations over the entire experimantal range. A particularly striking point was that a negative (the energy density derivative with respect to the second invariant of deformation tensor) was observed experimentally at small strain, but none of the models could predict it. Computer simulations [18, 19] have been helpful, and amongst other things they confirm the negative seen in experiments .
In this paper, we return to the picture of freely jointed chains cross-linked at junction points deforming under affine conditions. The end-to-end distribution function is derived as an expansion in inverse strand length, which ultimately approaches those obtained earlier [5, 6]. However, the expansion facilitated the full network average which follows to give the elastic free energy in a similar expansion. It is assumed that the fluctuation of the junction points will introduce a prefactor , but will not significantly alter the expansion otherwise. The method we use incorporates the saddle point integration and Wick's theorem, and the result is obtained in its full tensorial form which is not done before . This tensorial expression is then expressed as a function of the two invariants of the deformation tensor. The negative value of is then found. It appears that the affine deformation together with finite extensibility is the physical mechanism which gave rise to a negative . The gradual breakdown of the affine assumption with increasing strain accounts for the eventual failure of the affine model and the necessity of other alternative ones. The results presented here should be applicable to the networks with medium to long (for convergence) strand length at low deformations.
The error in the widely used inverse Langevin approximation is examined.