Nonlinear electrophoretic velocity of a spherical colloidal particle

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength $\beta =a^*e^*E_\infty ^*/k_B^*T^*$ , defined as the ratio of the product of the applied electric field magnitude $E_\infty ^*$ and particle radius $a^*$ , to the thermal voltage $k_B^*T^*/e^*$ , where $k_B^*$ is Boltzmann's constant, $T^*$ is the absolute temperature, and $e^*$ is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density $\sigma$ over a wide range of $\beta$ . Here, $\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$ , where $\sigma ^*$ is the dimensional surface charge density, and $\epsilon ^*$ is the permittivity of the electrolyte. For moderately charged particles ( $\sigma ={O}(1)$ ), the electrophoretic velocity is linear in $\beta$ when $\beta \ll 1$ , and its dependence on the ratio of the Debye length ( $1/\kappa ^*$ ) to particle radius (denoted by $\delta =1/(\kappa ^*a^*)$ ) agrees with Henry's formula. As $\beta$ increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is $\delta$ -dependent. For $\beta \gg 1$ , the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all $\delta$ . For highly charged particles ( $\sigma \gg 1$ ) in the thin-Debye-layer limit ( $\delta \ll 1$ ), our computations are in good agreement with recent experimental and asymptotic results.