Shape-dependence of electrophoretic mobility: an AI-assisted derivation
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.