Electrostatic interactions of colloid particles in symmetric electrolytes

Electrostatic interactions between two spherical particles and between a particle and a planar interface in a symmetric electrolyte have been quantitatively described. A numerical scheme has been developed for solving the nonlinear Poisson-Boltzmann equation expressed in the bispherical coordinate system. The method is based on the alternating direction over-relaxation procedure using the Newton-Raphson iteration. The novelty of the algorithm is the application of the grid transforming functions, enabling a more uniform distribution of mesh points in regions characterized by strong electric field. Using the numerical method, electric potential distribution (within and outside the sphere) have been calculated as well as the energy of interaction as a function of the particle interface separation. The boundary conditions in the form of the constant-potential, constant charge and the mixed case have been applied in these calculations. The energy profiles calculated for various screening lengths were compared with analytical approximations derived using the Hogg-Healy-Fuerstenau method and the Linear superposition approach (LSA). It was demonstrated that the LSA can be used as a good estimate of the potential distribution and interaction energy for a broad range of kappa a including kappa a < 1.