Analytical solutions of the Poisson-Boltzmann equation within an interstitial electrical double layer in various geometries

Abstract

Almost all research works conducted in the area of characterization of an electrical double layer (EDL) have been principally concerned with formation of the layer outside a charged particle, but there are cases in natural and artificial systems where the layer is generated within particles. In other words, it is basically assumed that charged particles are surrounded by an electrolyte solution; however, in reality, an aqueous phase may be bounded inside charged particles. In order to address such problems, it is essential to analytically solve the Poisson-Boltzmann (PB) equation to obtain the distribution of electric potential and concentration profiles of ions within an interior EDL. Despite that the governing PB equation remains the same, analytical solutions will be completely different from cases with an exterior EDL as boundary conditions are different. Solving the PB equation for an interstitial EDL has not been fully addressed yet except for parallel plate-like particles, whose outcomes are inconvenient-to-use due to implicit forms of solutions, involving complex integrals that are not expressible in a closed form and inclusion of parameters determined by iterative numerical techniques.This article is directed at deriving accurate analytical solutions for calculation of electric potential distribution within an interstitial EDL in various particle geometries. First, an exact analytical solution of the PB equation is obtained for slab-shaped particles containing an electrolyte solution. However, because of the inherent complexity of cylindrical and spherical operators resulting from the curvature of shells, we have to derive approximate analytical solutions in these geometries under some assumptions. To account for errors associated with such assumptions, correction factors are applied to the approximated terms. Then, the overall ranges of variation of the factors are evaluated by performing sensitivity analyses on key influential parameters and it is shown that the correction factors are independent of characteristics of a system. Eventually, the analytical solutions and corresponding numerical results are compared to prove the validity of the newly-derived formulas in all practical applications. The results from this research find implications in diverse chemical/physical/biological phenomena, technological processes, and medical and industrial applications.