Shape-dependence of electrophoretic mobility: an AI-assisted perturbation analysis
Journal Article
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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 modified through small axisymmetric spherical harmonic shape perturbations, at an arbitrary ratio of particle size to Debye length; ; ; ; kappa a; ; ; κ; a; ; $kappa a$; ; ; . Using a volume integral formulation combined with domain perturbation techniques, we derive a universal shape correction coefficient; ; ; ; sigma 2 left parenthesis kappa a right parenthesis; ; ; ; σ; 2; ; (; κ; a; ); ; $sigma _2(kappa a)$; ; ; such that the mobility along the axis of symmetry takes the compact form; ; ; ; upper C Subscript parallel to Baseline equals f Subscript upper H Baseline left parenthesis kappa a right parenthesis left bracket 1 plus epsilon c 2 sigma 2 left parenthesis kappa a right parenthesis right bracket; ; ; ; C; ∥; ; =; ; f; H; ; (; κ; a; ); ; [; 1; +; ε; ; c; 2; ; ; ; σ; 2; ; (; κ; a; ); ]; ; $C_parallel = f_H(kappa a),[1 + varepsilon c_2,sigma _2(kappa a)]$; ; ; , where; ; ; ; f Subscript upper H; ; ; ; f; H; ; ; $f_H$; ; ; is Henry’s function. We show that; ; ; ; sigma 2; ; ; ; σ; 2; ; ; $sigma _2$; ; ; interpolates between; ; ; ; plus 1 divided by 5; ; ; +; 1; ; /; ; 5; ; $+1/5$; ; ; in the thick-double-layer (Hückel) limit, governed solely by the Stokes drag correction, and zero in the thin-double-layer (Smoluchowski) limit. The perturbation theory agrees quantitatively with exact spheroidal solutions for both prolate and oblate orientations present in the literature. A key finding is that only the; ; ; ; upper P 2; ; ; ; P; 2; ; ; $P_2$; ; ; (quadrupolar) component of the particle shape affects the mobility at leading order; higher harmonics are electrophoretically silent. The physical formulation, the identification of the relevant asymptotic regime, and the interpretation of the results were carried out by the authors; the supporting perturbation algebra, numerical computations and figures were developed with AI assistance (Claude, Anthropic) under close author verification. The AI made mistakes, such as fabricating a coefficient and forcing an interpolation, which we flagged and corrected. We discuss the role and limitations of AI in theoretical research, with representative prompts provided in an appendix.;
