  Molecular chains of elastomer networks are modeled as ideal, finite, Freely
Jointed Chains (FJC). We first develop a compact, closed-form, mathematically
accurate representation of this model. We begin with the closed form of the
Pade approximations for the Inverse Langevin Function, modified by the Slater
(2003) method to map to the ideal FJC model. We fit the generalized form of
this expression to exact the series expression by Treloar (1975) for the
probability density function. The resulting fit yields the exact expression
with respect to the Treloar expression. We verify this exact fit from the
precise correlation of the moments from the fitted probability distribution
with the analytical FJC distribution moments. These expressions are
incorporated into the 8-chain geometry, to yield the affine network elasticity
expression. We extend the exact and compact expression of an ideal FJC to
determine the fluctuation distribution (about its mean position), of the
junction between two ideal FJCs. The fluctuation is the conditional probability
of the junction, given the separation of the distal points of the connected
FJCs. The fluctuation distribution corresponds to an ideal FJC, whose effective
number of segments, decreases linearly with the distal end separation. Finally,
we develop the elasticity relationship for such an ideal network whose
junctions can fluctuate to reduce the strain energy density. However, this
reduction itself decreases with deformation, which is incorporated in
developing resulting in the elasticity expression for a phantom network of
finite chains.
