Numerical Solution of the Poisson–Boltzmann Equation for a Spherical Cavity

Abstract

The Poisson–Boltzmann equation is numerically solved for a spherical cavity filled with a charged electrolyte solution. The network method used makes it possible to solve the problem in the most general case: the electrolyte solution can have any number of ion types with valences having any value. Furthermore, no a priori assumption concerning electroneutrality at the center of the cavity is required. Electric potential and ion concentration profiles, as well as the total potential drop in the cavity, are calculated for different system parameter values. These results are discussed and compared to the corresponding results obtained for suspended particles. Important differences arise, except for very thin double layers. For instance, the usual definition of the Debye length can no longer be used, since the electrolyte solution is nonneutral in the whole volume of the cavity. Furthermore, the charge density at the center of the cavity cannot be assigned any arbitrary value, since the charge density and the ion densities are no longer independent quantities.