Solution to the two-body Smoluchowski equation with shear flow for charge-stabilized colloids at low to moderate P\'eclet numbers

TL;DR

This paper presents an analytical method to determine the microstructure of sheared colloidal suspensions by solving the two-body Smoluchowski equation, revealing how shear flow affects particle correlations and structure factors.

Contribution

The authors develop a higher-order analytical solution to the two-body Smoluchowski equation under shear, applicable to various interaction potentials, and analyze the shear-induced microstructural distortions.

Findings

Shear causes a broadening and shift of the structure factor peak at low wavevectors.

Distortion of the structure factor scales as a power law with Péclet number.

Yukawa interactions show significantly larger shear-induced structural changes than hard spheres.

Abstract

We developed an analytical theoretical method to determine the microscopical structure of weakly to moderately sheared colloidal suspensions in dilute conditions. The microstructure is described by the static structure factor, obtained by solving the stationary two-body Smoluchowski advection-diffusion equation. The singularly perturbed PDE problem is solved by performing an angular averaging over the extensional and compressing sectors and by the rigorous application of boundary-layer theory (intermediate asymptotics). This allows us to expand the solution to a higher order in P\'eclet with respect to previous methods. The scheme is independent of the type of interaction potential. We apply it to the example of charge-stabilized colloidal particles interacting via the repulsive Yukawa potential and study the distortion of the structure factor. It is predicted that the distortion is larger at small wavevectors $k$ and its dependence on $Pe$ is a simple power law. At increasing $Pe$, the main peak of the structure factor displays a broadening and shift towards lower $k$ in the extensional sectors, which indicates shear-induced spreading out of particle correlations and neighbor particles locally being dragged away from the reference one. In the compressing sectors, instead, a narrowing and shift towards high $k$ is predicted, reflecting shear-induced ordering near contact and concomitant depletion in the medium-range. An overall narrowing of the peak is also predicted for the structure factor averaged over the whole solid angle. Calculations are also performed for hard spheres, showing good overall agreement with experimental data. It is also shown that the shear-induced structure factor distortion is orders of magnitude larger for the Yukawa repulsion than for the hard spheres.