The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle

Recent theoretical studies have suggested a significant enhancement in electro-osmotic flows over hydrodynamically slipping surfaces, and experiments have indeed measured $O(1)$ enhancements. In this paper, we investigate whether an equivalent effect occurs in the electrophoretic motion of a colloidal particle whose surface exhibits hydrodynamic slip. To this end, we compute the electrophoretic mobility of a uniformly charged spherical particle with slip length $λ$ as a function of the zeta (or surface) potential of the particle $ζ$ and diffuse-layer thickness $κ−1$⁠. In the case of a thick diffuse layer, $κa⪡1$ (where $a$ is the particle size), simple arguments show that slip does lead to an $O(1)$ enhancement in the mobility, owing to the reduced viscous drag on the particle. On the other hand, for a thin-diffuse layer $κa⪢1$⁠, the situation is more complicated. A detailed asymptotic analysis, following the method of O’Brien [J. Colloid Interface Sci. 92, 204 (1983)], reveals that an $O(κλ)$ increase in the mobility occurs at low-to-moderate zeta potentials (with $ζ$ measured on the scale of thermal voltage $kBT/e≈25mV$⁠). However, as $ζ$ is further increased, the mobility decreases and ultimately becomes independent of the slip length—the enhancement is lost—which is due to the importance of nonuniform surface conduction within the thin-diffuse layer, at large $ζ$ and large, but finite, $κa$⁠. Our asymptotic calculations for thick and thin-diffuse layers are corroborated and bridged by computation of the mobility from the numerical solution of the full electrokinetic equations (using the method of O’Brien and White [J. Chem. Soc., Faraday Trans. 2 74, 1607 (1978)]). In summary, then, we demonstrate that hydrodynamic slip can indeed produce an enhancement in the electrophoretic mobility; however, such enhancements will not be as dramatic as the previously studied $κa→∞$ limit would suggest. Importantly, this conclusion applies not only to electrophoresis but also to electro-osmosis over highly charged surfaces, wherein any inhomogeneities (e.g., due to curvature, roughness, charge patterning, or a variation in slip length) will drive nonuniform surface conduction, which prevents the significant slip-driven flow enhancements predicted for a uniform highly charged surface.