Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T01:33:38.238Z Has data issue: false hasContentIssue false
  • English
  • Français

Electrophoresis of a colloidal sphere along the axis of a circular orifice or a circular disk

Published online by Cambridge University Press:  26 April 2006

Huan J. Keh  and
Liang C. Lien
Liang C. Lien
Affiliation:
Department of Chemical Engineering, National Taiwan University, Taipei 10764 Taiwan, ROC
Get access Rights & Permissions [Opens in a new window]

Abstract

The axisymmetric electrophoretic motion of a dielectric sphere along the axis of an orifice in a large conducting plane or of a conducting disk is considered. The radius of the orifice or the disk may be either larger or smaller than that of the sphere. The assumption of thin electrical double layers at the solid surfaces is employed. To solve the electrostatic and hydrodynamic governing equations both the electric and the flow fields are partitioned at the plane of the orifice or the disk. For each field, separate solutions are developed on both sides of the plane that satisfy the boundary conditions in each region and the unknown functions for the field at the fluid interface. The continuities of the electric current flux and the fluid stress tensor at the matching interface lead to sets of dual integral equations which are solved analytically to determine the unknown functions for the fields at the matching interface. Then, a boundary–collocation technique is used to satisfy the boundary conditions on the surface of the sphere.

The numerical solutions for the electrophoretic velocity of the colloidal sphere are presented for various values of a/b and a/d, where a is the particle radius, b is the radius of the orifice or the disk, and d is the distance of the particle centre from the plane of the wall. For the limiting case of electrophoresis of a sphere perpendicular to an infinite plane wall, our results for the boundary effects agree very well with the exact calculations using spherical bipolar coordinates. Interestingly, the electrophoretic velocity of a sphere approaching an orifice of a larger radius increases when the sphere is close to the orifice, and this velocity can be even larger than that for an identical sphere undergoing electrophoresis in an unbounded fluid. If the sphere has a radius larger than that of the orifice, or if the sphere has a smaller radius and is located sufficiently far from the orifice, its electrophoretic mobility will decrease with the increase of the spacing parameter a/d. For the electrophoretic motion of a sphere along the axis of and close to a disk of finite radius, the resistance to the particle movement can be stronger than that for an equal sphere undergoing electrophoresis normal to an infinite plane wall at the same value of a/d. As the particle approaches the disk wall, its mobility decreases steadily and vanishes at the limit a/d → 1. The boundary effects on the particle mobility and the fluid flow pattern in electrophoresis differ significantly from those of the corresponding sedimentation problem with which a comparison is made.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 305 - 333
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acrivos, A., Jeffrey, D. J. & Saville, D. A., 1990 Particle migration in suspensions by thermocapillary or electrophoretic motion. J. Fluid Mech. 212, 95.Google Scholar
Anderson, J. L.: 1981 Concentration dependence of electrophoretic mobility. J. Colloid Interface Sci. 82, 248.Google Scholar
Anderson, J. L.: 1989 Colloid transport by interfacial forces. Ann. Rev. Fluid Mech. 21, 61.Google Scholar
Batchelor, G. K.: 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245.Google Scholar
Chen, S. B. & Keh, H. J., 1988 Electrophoresis in a dilute dispersion of colloidal spheres. AIChE J. 34, 1075.Google Scholar
Dagan, Z., Pfeffer, R. & Weinbaum, S., 1982a Axisymmetric stagnation flow of a spherical particle near a finite planar surface at zero Reynolds number. J. Fluid Mech. 122, 273.Google Scholar
Dagan, Z., Weinbaum, S. & Pfeffer, R., 1982b An infinite-series solution for the creeping motion through an orifice of finite length. J. Fluid Mech. 115, 505.Google Scholar
Dagan, Z., Weinbaum, S. & Pfeffer, R., 1982c General theory for the creeping motion of a finite sphere along the axis of a circular orifice. J. Fluid Mech. 117, 143.Google Scholar
Davis, A. M. J., O'Neill, M. E. & Brenner, H. 1981 Axisymmetric Stokes flow due to a rotlet and Stokeslet near a hole in a plane wall: filtration flows. J. Fluid Mech. 103, 183.Google Scholar
Deblois, R. W. & Bean, C. P., 1970 Counting and sizing of submicron particles by the resistive pulse technique. Rev. Sci. Instrum. 41, 909.Google Scholar
Gluckman, M. J., Pfeffer, R. & Weinbaum, S., 1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705.Google Scholar
Happel, J. & Brenner, H., 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hunter, R. J.: 1987 Foundations of Colloid Science, Vol. I, p. 557. Clarendon.
Keh, H. J. & Anderson, J. L., 1985 Boundary effects on electrophoretic motion of colloidal spheres. J. Fluid Mech. 153, 417.Google Scholar
Keh, H. J. & Chen, S. B., 1988 Electrophoresis of a colloidal sphere parallel to a dielectric plane. J. Fluid Mech. 194, 377.Google Scholar
Keh, H. J. & Chen, S. B., 1989a Particle interactions in electrophoresis - I. Motion of two spheres along their line of centers. J. Colloid Interface Sci. 130, 542.Google Scholar
Keh, H. J. & Chen, S. B., 1989b Particle interactions in electrophoresis - II. Motion of two spheres normal to their line of centers. J. Colloid Interface Sci. 130, 556.Google Scholar
Keh, H. J. & Lien, L. C., 1989 Electrophoresis of a dielectric sphere normal to a large conducting plane. J. Chinese Inst. Chem. Engng 20, 283.Google Scholar
Keh, H. J. & Yang, F. R., 1990 Particle interactions in electrophoresis - III. Axisymmetric motion of multiple spheres. J. Colloid Interface Sci. (in press).Google Scholar
Kozak, M. W. & Davis, E. J., 1989 Electrokinetics of concentrated suspensions and porous media - I. Thin electrical double layers. J. Colloid Interface Sci. 127, 497.Google Scholar
Morrison, F. A.: 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34, 210.Google Scholar
Morrison, F. A. & Stukel, J. J., 1970 Electrophoresis of an insulating sphere normal to a conducting plane. J. Colloid Interface Sci. 33, 88.Google Scholar
O'Brien, V.: 1968 Form factors for deformed spheroids in Stokes flow. AIChE J. 14, 870.Google Scholar
Reed, C. C. & Anderson, J. L., 1980 Hindered settling of a suspension at low Reynolds number. AIChE J. 26, 816.Google Scholar
Reed, L. D. & Morrison, F. A., 1976 Hydrodynamic interactions in electrophoresis. J. Colloid Interface Sci. 54, 117.Google Scholar
Tranter, C. J.: 1951 On some dual integral equations. Q. J. Maths 2, 60.Google Scholar