Statistical Mechanics of Semiflexible Polymer Chains from a new Generating Function

TL;DR

This paper introduces a new generating function approach to the Kratky-Porod model, providing exact and approximate expressions for semiflexible polymer properties across flexibility regimes, including a simple form for the polymer propagator.

Contribution

A novel generating function method for the Kratky-Porod model that yields exact and approximate expressions for polymer statistics, especially a simple form for the propagator.

Findings

Derived expressions for characteristic function, propagator, and mean square end-to-end distance.

Captures limits of flexible and stiff polymers exactly.

Predicts smooth crossover behavior between flexibility regimes.

Abstract

In this paper, we present a new approach to the Kratky-Porod Model (KP) of semiflexible polymers. Our solution to the model is based on the definition of a generating function which we use to study the statistical mechanics of semiflexible polymer chains. Specifically, we derive mathematical expressions for the characteristic function, the polymer propagator and the mean square end-to-end distance from the generating function. These expressions are valid for polymers with any number of segments and degree of rigidity. Furthermore, they capture the limits of fully flexible and stiff polymers exactly. In between, a smooth and approximate crossover behavior is predicted. The most important contribution of this paper is the expression of the polymer propagator which is written in a very simple and insightful way. It is given in terms of the exact polymer propagator of the Random Flight Model multiplied by an exponential that takes into account the stiffness of the polymer backbone.