Interactions between Two Spherical Particles with Nonuniform Surface Potentials: The Linearized Poisson–Boltzmann Theory

Abstract

Using the linearized Poisson–Boltzmann theory, electrical double layer interactions are calculated between two nonuniform spherical colloidal particles with mean potential zero. Most results are for the case of surface potentials modeled by a single spherical harmonic and aligned relative to each other. As previously observed for flat surfaces, interactions decay more rapidly as a function of separation between spheres with such periodic, “single-mode” potentials than between spheres with uniform potentials. The Deryaguin approximation for single-mode spheres is tested and calculations are made of the force and torque that particles in a doublet exert on one another. Many of the concepts developed from models of flat plates with periodic aligned surface potentials are shown to be useful in this more general case. Attempts to explain recent differential electrophoresis experiments on the basis of nonuniform double layers fail in that the maximum restraining torques produced under plausible assumptions about the amplitude of nonuniformity are an order of magnitude smaller than those implied by the measurements. The main reason for this is that the torques are too small at the separations characteristic of a secondary minimum. The effect of misalignment of single-mode spheres is assessed by calculating the distribution of interaction energies over a set of relative orientations generated by quasi-random sampling. Finally, a method for generating spheres with “random” surface potentials is devised and potentials of mean force are calculated for pairs of spheres with such surface potentials. Comparison with the single-mode case is kept at a qualitative level, in the absence of detailed knowledge of how realistic such “random” surfaces are.