Inequivalence of fixed-force and fixed-extension statistical ensembles for a flexible polymer tethered to a planar substrate

TL;DR

This paper demonstrates that for a tethered polymer, fixed-force and fixed-extension ensembles are not equivalent due to confinement effects, with analytical and simulation results highlighting significant differences especially under strong forces.

Contribution

It provides the first analytical demonstration of ensemble inequivalence caused by confinement in tethered polymers, supported by Monte Carlo simulations.

Findings

Ensemble inequivalence is caused by confinement to half space.

Strong compressional forces amplify the inequivalence.

Simulations of DNA-like chains confirm analytical predictions.

Abstract

Recent advances in single macromolecule experiments have sparked interest in the ensemble dependence of force-extension relations. The thermodynamic limit may not be attainable for such systems, that leads to inequivalence of the fixed-force and the fixed-extension ensemble. We consider an ideal Gaussian chain described by the Edwards Hamiltonian with one end tethered to a rigid planar substrate. We analytically calculate the force-extension relation in the two ensembles and we show their inequivalence which is caused by the confinement of the polymer to half space. The inequivalence is quite remarkable for strong compressional forces. We also perform Monte-Carlo simulations of a tethered wormlike chain with contour length 20 times its persistence length which corresponds to experiments measuring the conformations of DNA tethered to a wall. The simulations confirm the ensemble inequivalence and qualitatively agree with the analytical predictions of the Gaussian model. Our analysis shows that confinement due to tethering causes ensemble inequivalence, irrespective of the polymer model.