# Exact solution for the force-extension relation of a semiflexible polymer under compression

## TL;DR

This paper provides an exact analytical solution for the force-extension behavior of a semiflexible polymer under compression, revealing how thermal fluctuations influence buckling and related thermodynamic properties.

## Contribution

It introduces an exact solution using Mathieu functions for the wormlike chain model under compression, detailing the impact of thermal fluctuations on buckling and thermodynamics.

## Key findings

- Thermal fluctuations cause larger elongations than elastic rods at strong compression.
- Susceptibility peaks near the Euler buckling force, increasing with persistence length.
- Entropy and heat capacity vary non-monotonically with applied load.

## Abstract

Exact solutions for the elastic and thermodynamic properties for the wormlike chain model are elaborated in terms of Mathieu functions. The smearing of the classical Euler buckling instability for clamped polymers is analyzed for the force-extension relation. Interestingly, at strong compression forces the thermal fluctuations lead to larger elongations than for the elastic rod. The susceptibility defined as the derivative of the force-extension relation displays a prominent maximum at a force that approaches the critical Euler buckling force as the persistence length is increased. We also evaluate the excess entropy and heat capacity induced by the compresssion and find that they vary non-monotonically with the load. These findings are corroborated by pseudo-Brownian simulations.