A Smoluchowski equation for a sheared suspension of frictionally interacting rods

TL;DR

This work addresses how frictional contacts in dense, sheared suspensions of rod-like particles modify orientational dynamics and rheology. It extends Onsager–Doi theory using a Rayleighian variational framework to derive a self-consistent Smoluchowski equation and a friction-augmented stress tensor, incorporating both boundary-lubricated and solid Coulomb friction via a mean-field contact approach. A key contribution is a generalized mean-field count of interparticle contacts $\langle c\rangle$ as a function of volume fraction $\phi$, aspect ratio $L/D$, and nematic order $S$, which feeds the frictional dissipation into the dynamics. The framework recovers a Doi–Edwards-like description in the lubricated limit with renormalized drag and yields self-consistent corrections to the $\mathbf{Q}$-tensor dynamics and stress due to friction, providing a principled route to study DST and shear jamming in dense rod suspensions. Limitations include neglect of static friction sticking, potential spatial inhomogeneities, and challenges in closing the frictional terms without further closure approximations.

Abstract

In this work we develop constitutive equations for a dense, sheared suspension of frictionally interacting rods by applying Onsager's variational method as formulated by Doi. We treat both solid friction, of the Amontons-Coulomb form; and lubricated friction, which scales with relative tangential velocity at the contact point. Dissipation functions in terms of the rod angular velocity are derived via a mean field approach for each form of friction, and from these, a Rayleighian for dense suspensions of rigid rods under shear constructed. Derivatives of this Rayleighian with respect to rod angular velocity and velocity gradient give a Smoluchowski equation and stress tensor, respectively. We show that these are representable as perturbations to Doi's model for a sheared liquid crystal. We also suggest a form for the average number of contacts between rods as a function of volume fraction, aspect ratio, and nematic order parameter, generalizing Philipse's random contact equation for disordered packings.

Figures (3)

• Figure 1: Various concentration regimes of rod suspensions: a) dilute, b) semi-dilute, c) concentrated, for $S = 0$, d) concentrated, for $S >0$doiedwards
• Figure 2: Contact area for two rods. Each of the dotted lines has length $D/|\hat{\boldsymbol{u}}^{i} \times \hat{\boldsymbol{u}}^{j}|$, so the area is $D^{2}/|\hat{\boldsymbol{u}}^{i} \times \hat{\boldsymbol{u}}^{j}|$.
• Figure 3: Two rods in contact, the black dots are their centers of mass, the black triangle is their contact point. $\epsilon^{i}$ and $\epsilon^{j}$ represent the distance along rod centerlines to this point.