An Improved Non-Gaussian Statistical Theory of Rubber Elasticity for Short Chains

Abstract

The mechanical behavior of polymers has long been described by the non-Gaussian statistical model. Non-Gaussian models are generally based on the Kuhn-Grün (KG) distribution function, which itself is derived from the first order approximation of the complex Rayleigh’s exact Fourier integral distribution. The KG function has gained such a broad acceptance in the field of polymer physics that the non-Gaussian theory is often used to describe chains with various flexibility ratios. However, KG function is shown to be only relevant for long chains, with more than 40 segments. Here, we propose a new accurate approximation of the entropic force resulted from Rayleigh distribution function of non-Gaussian chains. The approximation provides an improved version of inverse Langevin function which has a limited error value with respect to the exact entropic force. The proposed function provides a significantly more accurate estimation of the distribution function than KG functions for small and medium-sized chains with less than 40 segments.