Homo-interaction between parallel plates at constant charge

Various approximate solutions to the Poisson-Boltzmann (PB) equation have been derived to describe the interaction of electric diffuse double layers adjacent to charged surfaces. However, all these expressions are case-specific and accurate only in limited ranges of particle separations. None can cover the entire range of plate separations and/or surface charge densities generally found in real systems. In this paper, we derive an approximate expression for the force between two parallel similar plates with constant surface charge densities in a symmetrical electrolyte solution, which agrees well with the rather complex exact analytical solution over a wide range of plate separations. The method used is based on the so-called “compression” approach developed previously for the case of low surface charge densities. The results are also in good agreement with “exact” numerical solutions over a wide range where no restriction is actually required on the magnitudes of the surface charge densities, surface potentials or the distance between the plates. Furthermore, an expression for the derivative of the force is also given, which is fairly simple and is very useful in modelling, e.g. colloidal transport problems based on a force balance on particles in a colloidal system. In such cases it is very impractical to use either the exact analytical or numerical solution to the PB equation.