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

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.