Fluctuating viscoelasticity based on a finite number of dumbbells

TL;DR

This paper derives viscoelastic properties from a finite number of dumbbells using stochastic calculus and thermodynamics, revealing finite-size effects on the Helmholtz free energy that influence the conformation tensor fluctuations.

Contribution

It introduces a thermodynamic approach to derive viscoelasticity with finite-size corrections and explicitly calculates the finite-size Helmholtz free energy contribution.

Findings

Finite-size effects modify the Helmholtz free energy.

Two derivation routes agree only with finite-size correction.

The generalized relaxation tensor remains unaffected by finite-size effects.

Abstract

Two alternative routes are taken to derive, on the basis of the dynamics of a finite number of dumbbells, viscoelasticity in terms of a conformation tensor with fluctuations. The first route is a direct approach using stochastic calculus only, and it serves as a benchmark for the second route, which is guided by thermodynamic principles. In the latter, the Helmholtz free energy and a generalized relaxation tensor play a key role. It is shown that the results of the two routes agree only if a finite-size contribution to the Helmholtz free energy of the conformation tensor is taken into account. Using statistical mechanics, this finite-size contribution is derived explicitly in this paper for a large class of models; this contribution is non-zero whenever the number of dumbbells in the volume of observation is finite. It is noted that the generalized relaxation tensor for the conformation tensor does not need any finite-size correction.