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 [3] 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 [4] have been developed to obtain the single end-to-end distribution function [5, 6], achieving encouraging agreements with Monte Carlo simulations [7].
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 [10] 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 [17]. 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 [20].
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 [2], 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 [9].
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.