Electrostatic Interactions between a Charged Sphere and a Charged Planar Surface in an Electrolyte Solution

Abstract

The electrostatic interaction energy for a charged sphere interacting with a low dielectric charged planar surface in an electrolyte solution is calculated. The calculations are based on a solution of the linearized Poisson–Boltzmann equation under the condition of constant charge density on both surfaces. The influence of sphere size, its dielectric constant, and net charge as well as planar surface charge density and electrolyte concentration on the interaction energy is demonstrated. A comparison is made between the interaction energies for a point charge interacting with a charged planar surface, calculated from the Gouy–Chapman theory, and our solution for a charged 2.5 Å sphere. The calculations show an asymmetric interaction energy for spheres of opposite sign of charge, i.e., when the surface and sphere are of the same (opposite) sign the interaction is stronger (weaker) than what is obtained from the Gouy–Chapman theory. On the other hand, the asymmetry is overestimated when the Deryaguin procedure is applied to calculate the interaction energy between a charged sphere and a charged planar surface. In the latter case the asymmetry is mainly due to overestimation of the repulsive energy when the sphere and the planar surface have the same sign of charge. It is shown that, as a first approximation, the interaction energy can be obtained by adding the contribution from three separate components: a point charge, located in the center of the sphere, interacting with the charged surface plus the interaction energy between an uncharged sphere and the charged surface plus the interaction energy between a charged sphere and the uncharged surface.