Colloidal electrophoresis: Scaling analysis, Green-Kubo relation, and numerical results

TL;DR

This paper develops a theoretical framework for colloidal electrophoresis, deriving a Green-Kubo relation for mobility, and validates findings through simulations and finite-element solutions, highlighting the role of key parameters and salt effects.

Contribution

It introduces a systematic scaling analysis, derives a Green-Kubo formula for electrophoretic mobility, and compares theoretical predictions with numerical simulations.

Findings

Reduced electrophoretic mobility depends on a minimal set of parameters.

The Green-Kubo formula effectively predicts mobility from simulation data.

Salt-free and salt-containing systems can be mapped under certain conditions.

Abstract

We consider electrophoresis of a single charged colloidal particle in a finite box with periodic boundary conditions, where added counterions and salt ions ensure charge neutrality. A systematic rescaling of the electrokinetic equations allows us to identify a minimum set of suitable dimensionless parameters, which, within this theoretical framework, determine the reduced electrophoretic mobility. It turns out that the salt-free case can, on the Mean Field level, be described in terms of just three parameters. A fourth parameter, which had previously been identified on the basis of straightforward dimensional analysis, can only be important beyond Mean Field. More complicated behavior is expected to arise when further ionic species are added. However, for a certain parameter regime, we can demonstrate that the salt-free case can be mapped onto a corresponding system containing additional salt. The Green-Kubo formula for the electrophoretic mobility is derived, and its usefulness demonstrated by simulation data. Finally, we report on finite-element solutions of the electrokinetic equations, using the commercial software package COMSOL.