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Nonlinear electrophoretic velocity of a spherical colloidal particle

Published online by Cambridge University Press:  31 July 2023

Richard Cobos Open the ORCID record for Richard Cobos [Opens in a new window]  and
Aditya S. Khair Open the ORCID record for Aditya S. Khair [Opens in a new window]
Richard Cobos
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Aditya S. Khair*
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
*
†Email address for correspondence: akhair@andrew.cmu.edu
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Abstract

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.

JFM classification

Complex Fluids: Colloids MHD and Electrohydrodynamics: Electrokinetic flows

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 968 , 10 August 2023 , A14
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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