An amended approximation of the non-Gaussian probability distribution function

Abstract

Network models of rubber elasticity are based on the conformational entropy of an idealized chain and mostly motivated by the non-Gaussian statistical theory by Kuhn and Grün. However, the non-Gaussian probability distribution function cannot be expressed in a closed form and requires an approximation. All such approximations applied in the literature demonstrate pronounced inaccuracies in comparison to the analytical solution. The ideal choice of the approximation function depends on a variety of factors, such as the chain parameters or the desired application of the approximation (the probability distribution function itself, the corresponding entropic energy, or the force developed by the chain). In addition, when making a choice regarding the best approximation for a given application, the applied error measure plays a significant role since the approximation that grants, for example, the best maximal relative error is not necessarily the same that provides the best mean absolute error. In the literature, this application-specific evaluation of available approximations is commonly disregarded. In this paper, we evaluate previously proposed approximations on the application-specific basis and develop an approach to derive a family of approximations for the free energy of a polymer chain in a broader range of the number of its chain segments. The analytical method based on the Padé technique delivers an approximation of the non-Gaussian probability distribution function that can be easily tailored depending on the desired application. The proposed approach is capable to provide much stronger predictions in comparison to the Kuhn and Grün model in a wide range of chain segment numbers.