Statistics of Semiflexible Polymer Chains and the Generalized Borel Transform

TL;DR

This paper introduces the Generalized Borel Transform as a novel computational method to analyze the statistical mechanics of semiflexible polymers, providing exact and approximate expressions for key properties across different flexibility regimes.

Contribution

The paper presents the GBT as a new technique to evaluate the polymer propagator and structure factor for semiflexible chains, capturing flexible and stiff limits and crossover behavior.

Findings

Derived an expression for the polymer propagator valid for any segment number and flexibility.

Showed the structure factor decreases faster with increased semiflexibility.

Identified a power-law regime in the structure factor with exponents 2 and 1 for flexible and rigid chains.

Abstract

In this paper, we present a new approach to the discrete version of the Wormlike Chain Model (WCM) of semiflexible polymers. Our solution to the model is based on a new computational technique called the Generalized Borel Transform (GBT) which we use to study the statistical mechanics of semiflexible polymer chains. Specifically, we evaluate the characteristic function of the model approximately. Afterward, we compute the polymer propagator of the model using the GBT and find an expression valid for polymers with any number of segments and values of the semiflexibility parameter. This expression captures the limits of flexible and infinitely stiff polymers exactly. In between, a smooth and approximate crossover behavior is predicted. Another property of our propagator is that it fulfills the condition of finite extensibility of the polymer chain. We have also calculated the single chain structure factor. This property is a decreasing function of the wave vector, $k,$ until a plateau is reached. Our computations clearly show that the structure factor decreases faster with increasing wave vector when the semiflexibility parameter is increased. Furthermore, when the wave vector is large enough, there is a regime where the structure factor follows an approximate power law of the form $k^{-\theta}$ even for short polymer chains. $\theta$ is equal to two for flexible polymers and to one for rigid chains. We also compare our results to the predictions of other models.