Electrostatic Interaction Force between Planar Surfaces Due to 3-D Distribution of Sources of Potential (Charge)

Abstract

A closed form solution for the electrostatic double-layer force between two planar surfaces having sources of potential (charge) distributed along the x-, y-, and z-axes has been derived. The double-layer force is evaluated using the expression for the Lorentz force by combining the tangential and normal components of the electric field on and between the two approaching surfaces. The general analytical solutions are developed for two cases: The first case is for when the distribution of charges on the two interfaces and intermediate layers are given. The second case is for mixed boundary conditions where the potential distribution of the interacting surfaces and charge distributions originating from intermediate layers (protrusions) are provided. Provided is a numerical example of the interaction force between two planar surfaces, each having a periodic potential distribution as well as discretely placed charges located on protrusions away from the surface, demonstrating that the electrostatic interaction force is always attractive at smaller plate separations, whereas at intermediate separations the magnitude (and sign) of the interaction force is dictated by the relative voltage, i.e., charge-to-potential ratio, electrolyte concentration, and relative lateral displacement.