Electrophoresis of a charged soft particle in a charged cavity with arbitrary double-layer thickness

An analysis for the quasi-steady electrophoretic motion of a soft particle composed of a charged spherical rigid core and an adsorbed porous layer positioned at the center of a charged spherical cavity filled with an arbitrary electrolyte solution is presented. Within the porous layer, frictional segments with fixed charges are assumed to distribute uniformly. Through the use of the linearized Poisson-Boltzmann equation and the Laplace equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately determined. The modified Stokes and Brinkman equations governing the fluid flow fields outside and inside the porous layer, respectively, are solved subsequently. An explicit formula for the electrokinetic migration velocity of the soft particle in terms of the fixed charge densities on the rigid core surface, in the porous layer, and on the cavity wall is obtained from a balance between its electrostatic and hydrodynamic forces. This formula is valid for arbitrary values of κa, λa, r0/a, and a/b, where κ is the Debye screening parameter, λ is the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, a is the radius of the soft particle, r0 is the radius of the rigid core of the particle, and b is the radius of the cavity. In the limiting cases of r0 = a and r0 = 0, the migration velocity for the charged soft sphere reduces to that for a charged impermeable sphere and that for a charged porous sphere, respectively, in the charged cavity. The effect of the surface charge at the cavity wall on the particle migration can be significant, and the particle may reverse the direction of its migration.