Derivation of a constitutive model for the rheology of jammed soft suspensions from particle dynamics

TL;DR

This paper derives a tensorial constitutive model for the rheology of jammed soft suspensions from microscopic particle dynamics, linking macroscopic stress evolution to particle correlations and packing fraction.

Contribution

It introduces a novel, microscopically-based tensorial constitutive model for jammed soft suspensions that incorporates particle correlations and microscopic parameters.

Findings

Derived a closed-form evolution equation for stress tensor.

Expressed model coefficients explicitly in terms of packing fraction.

Provides a non-linear model applicable to arbitrary uniform deformations.

Abstract

Considering the rheology of two-dimensional soft suspensions above the jamming density, we derive a tensorial constitutive model from the microscopic particle dynamics. Starting from the equation governing the $N$-particle distribution, we derive an evolution equation for the stress tensor. This evolution equation is not closed, as it involves the pair and three-particle correlation functions. To close this equation, we first employ the standard Kirkwood closure relation to express the three-particle correlation function in terms of the pair correlation function. Then we use a simple and physically motivated parametrization of the pair correlation function to obtain a closed evolution equation for the stress tensor. The latter is naturally expressed as separate evolution equations for the pressure and for the deviatoric part of the stress tensor. These evolution equations provide us with a non-linear tensorial constitutive model describing the rheological response of a jammed soft suspension to an arbitrary uniform deformation. One of the advantages of this microscopically-rooted description is that the coefficients appearing in the constitutive model are known in terms of packing fraction and microscopic parameters.