Phantom Networks of Finite Chains

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.