Shape-dependence of electrophoretic mobility

TL;DR

This paper develops a universal perturbation theory to quantify how nearly spherical particle shapes influence electrophoretic mobility across different Debye length regimes, revealing shape effects are dominated by quadrupolar components.

Contribution

It introduces a shape correction coefficient that interpolates between thick and thin double-layer limits, accounting for shape effects at arbitrary Debye lengths, validated against spheroid solutions.

Findings

Shape correction coefficient varies from +1/5 to 0 across regimes.

Only quadrupolar shape components significantly affect mobility at leading order.

Perturbation theory matches exact spheroid solutions for prolate and oblate particles.

Abstract

The electrophoretic mobility of a spherical particle is well understood, yet how particle shape modifies this mobility at arbitrary Debye length remains an open question. Here, we compute the electrophoretic mobility of a nearly spherical particle whose surface is described by $r_s(\theta) = a[1 + \varepsilon f(\theta)]$, with $\varepsilon \ll 1$, at arbitrary ratio of particle size to Debye length $\kappa a$. Using a volume-integral formulation combined with domain perturbation techniques, we derive a universal shape correction coefficient $\sigma_2(\kappa a)$ such that the mobility takes the compact form $C_\parallel = f_H(\kappa a)\,[1 + \varepsilon\,c_2\,\sigma_2(\kappa a)]$, where $f_H$ is Henry's function. We show that $\sigma_2$ interpolates between $+1/5$ in the thick-double-layer (H\"{u}ckel) limit, governed solely by the Stokes drag correction, and zero in the thin-double-layer (Smoluchowski) limit, recovering the classical shape-independence theorem. The perturbation theory agrees quantitatively with exact spheroid solutions for both prolate and oblate orientations. A key finding is that only the $P_2$ (quadrupolar) component of the particle shape affects the mobility at leading order; higher harmonics are electrophoretically silent due to angular selection rules governing the coupling between the dipolar applied field and the shape perturbation. The results in this paper were generated using Claude Code (Anthropic, Opus 4.6 model) with supervision from the authors. Our thoughts on the usage of AI for theoretical research, along with representative prompts from the development process, are provided in the manuscript and Appendix.