# Screened Coulomb Interactions With Non-uniform Surface Charge

## TL;DR

This paper derives a simple Fourier space expression for the screened Coulomb interaction between charged planes with non-uniform surface charge, applicable to periodic and random distributions, and analyzes the effect on disjoining pressure.

## Contribution

It introduces a new analytical framework for calculating disjoining pressure with non-uniform surface charge distributions in the small potential limit.

## Key findings

- Disjoining pressure can be expressed as a Fourier integral or sum over reciprocal lattice vectors.
- Charge overspill reduces the disjoining pressure compared to uniform charge models.
- The method applies to both periodic and random charge distributions.

## Abstract

The screened Coulomb interaction between a pair of infinite parallel planes with spatially varying surface charge is considered in the limit of small electrical potentials for arbitrary Debye lengths. A simple expression for the disjoining pressure is derived in terms of a two dimensional integral in Fourier space. The integral is evaluated for periodic and random charge distributions and the disjoining pressure is expressed as a sum over Fourier-Bloch reciprocal lattice vectors or in terms of an integral involving the autocorrelation function respectively. The force between planes with a finite area of uniform charge, a model for the DLVO interaction between finite surfaces, is also calculated. It is shown that the overspill of the charge cloud beyond the region immediately between the charged areas results in a reduction of the disjoining pressure, as reported by us recently in the long Debye length limit for planes of finite width.