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Nonlinear electrophoresis of a tightly fitting sphere in a cylindrical tube

Published online by Cambridge University Press:  28 March 2018

J. D. Sherwood Open the ORCID record for J. D. Sherwood [Opens in a new window]  and
S. Ghosal
J. D. Sherwood*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
S. Ghosal
Affiliation:
Department of Mechanical Engineering, and Department of Engineering Sciencesand Applied Mathematics, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208, USA
*
†Email address for correspondence: jds60@cam.ac.uk
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Abstract

We investigate electrophoresis of a tightly fitting sphere of radius $R-h_{0}$ on the axis of a circular tube of radius $R$, using lubrication theory and ideas due to Schnitzer & Yariv (Phys. Fluids, vol. 26, 2014, 122002). The electrical charge clouds on both the cylindrical wall and the surface of the sphere are assumed thin compared to the gap between the sphere and cylinder, so that charge clouds do not overlap and ion exclusion effects are minimal. Nevertheless, non-uniform pumping of counter-ions within the charge clouds leads to a change in the ionic concentration outside the charge clouds in the narrow gap between sphere and cylinder. The electro-osmotic slip velocities at the two surfaces are modified, leading to a decrease in the electrophoretic velocity of the sphere at low Péclet numbers and an increase in the velocity at high Péclet numbers. When the field strength $E_{0}$ is low, it is known that the electrophoretic velocity $U_{0}$ is proportional to $E_{0}(\unicode[STIX]{x1D701}_{s}-\unicode[STIX]{x1D701}_{c})$ which is zero when the zeta potential $\unicode[STIX]{x1D701}_{s}$ on the sphere surface is equal to the zeta potential $\unicode[STIX]{x1D701}_{c}$ on the cylinder. The perturbation to the above low field strength electrophoretic velocity (at high Péclet number) is predicted to be proportional to $E_{0}^{3}(\unicode[STIX]{x1D701}_{s}+\unicode[STIX]{x1D701}_{c})^{2}(\unicode[STIX]{x1D70E}_{s}+\unicode[STIX]{x1D70E}_{c})$, where $\unicode[STIX]{x1D70E}_{s}$ and $\unicode[STIX]{x1D70E}_{c}$ are the surface charge densities on the sphere and cylinder. The choice of materials with similar or identical zeta potentials (and surface charge densities) for the cylinder and sphere should therefore facilitate the observation of velocities nonlinear in the field strength $E_{0}$, since the reference linear electrophoretic velocity will be small.

JFM classification

Complex Fluids: Colloids Low-Reynolds-number flows: Lubrication theory Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 843 , 25 May 2018 , pp. 847 - 871
Copyright
© 2018 Cambridge University Press 

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