Thermodynamic fluctuations in freely jointed chains under force

TL;DR

This paper investigates the fluctuations in the extension and angles of freely jointed chains under force, emphasizing the importance of considering variability rather than just average values in polymer physics models.

Contribution

It provides analytical and numerical analysis of thermodynamic fluctuations in freely jointed chains, highlighting their significance in modeling polymer stretching and deformation.

Findings

Fluctuations are significant at low forces and decrease with increasing chain length.

Link angles remain independent and fluctuate independently regardless of chain length.

Quantitative measures of fluctuations are provided for different forces and chain sizes.

Abstract

It is common to study polymer physics through the use of idealized single-chain models, and the most popular of these is the freely jointed chain model. In certain thermodynamic ensembles, statistical mechanical treatment of this model is analytically tractable or sometimes exactly solvable. This enables useful relations to be ascertained, like the expected chain end-to-end length as a function of an applied force. However, most of these relations return ensemble averages, which are values with inherent uncertainty, as opposed to deterministic values with no variance. This is an important distinction to understand and quantify, because the majority of studies to date involving single-chain models effectively treat these values as deterministic rather than fluctuating. To address this issue, thermodynamic fluctuations are examined in the freely jointed chain model. Specifically, the probability densities and standard deviations of the longitudinal, lateral, transverse, and radial portions of the chain extension, as well as the extension and link angles, are examined for different numbers of links and applied forces. Fluctuations in these quantities are shown to be considerable until the applied force becomes large. Increasing the number of links in the chain gradually reduces fluctuations in all quantities except for the link angles, since they are independent for freely jointed chains in the isotensional ensemble. Quantities are obtained analytically whenever possible and numerically otherwise. Overall, these results provide intuitive admonitions to consider when modeling the stretching of single polymer chains or the deformation of entire polymer networks.