Calculations of Double-Layer Electrostatic Interactions for the Sphere/Plane Geometry.

A numerical scheme for solving the nonlinear Poisson-Boltzmann equation for the sphere/sphere and plane/sphere geometries has been developed. The method is based on an alternating direction overrelaxation procedure using the Newton-Raphson iteration to solve the nonlinear equation stemming from finite-difference discretization. The novelty of the algorithm consists in using the grid transforming functions that allow a more uniform distribution of mesh points in the vicinity of the particle. The method was used to perform extensive calculations for opposite surface potentials of the interface and the particle immersed in a symmetric electrolyte solution. The electric potential distribution (within and outside the sphere) was calculated, as well as the force and energy of interaction (from the integral of the force over separation distance), for the constant potential, the constant charge, and the mixed cases. The energy profiles calculated for various kappaa were compared with the analytical approximations derived using the Hogg-Healy-Fuerstenau method and the linear superposition approach (LSA). These calculations enabled us to conclude that the LSA can be used as a good estimate of interaction energy for a broad range of kappaa values at distances greater than kappa-1, i.e., for problems pertinent to colloid particle adsorption.