The worm-like chain model at small and large stretch

TL;DR

This paper analyzes the force-stretch relationship in the worm-like chain model, deriving asymptotic series solutions for small and large forces, and proposes a uniform approximation that improves existing models.

Contribution

It provides new asymptotic series solutions for the force-stretch relation and introduces a uniform approximation valid across all stretch values.

Findings

Asymptotic series solutions for small and large forces.

A proposed uniform approximation valid for all stretches.

Improved accuracy over existing models.

Abstract

The relation between force and stretch in the worm-like chain model of entropic elasticity is examined. Although no closed-form expression is valid for all values of forcing, solutions in the form of asymptotic series can be obtained under conditions of small and large applied force. The small and large stretch limits correspond to regular and boundary layer perturbation problems, respectively. The perturbation problems are solved and series solutions obtained for force as a function of stretch. The form of the asymptotic series suggest a uniform approximation valid for all stretch that is an improvement on existing approximations.