Counterions between walls with surface charge modulations: Exact Poisson-Boltzmann solutions

Abstract

We study the thermal equilibrium of a classical system of identical point charges moving between two walls with parallel surfaces at a distance $d$, charged symmetrically with position-dependent surface charge densities. Specifically, we study the effect of surface charge modulation on the effective force (pressure) between the walls in the Poisson-Boltzmann limit. Based on Monte Carlo simulations at high temperatures, it is predicted that surface charge modulation reduces the pressure between the walls compared to uniformly charged wall surfaces with the same average surface charge density. We restrict ourselves to surface charge modulations in only one direction and, using a general solution of the two-dimensional Liouville equation, we construct exact solutions for the electrostatic potential. The surface charge density on the walls is generated inversely from this potential, which means that our exact results apply to a limited set of models with a specific variation of the surface charge density with distance $d$. Explicit analytical results show that surface charge modulation can both increase (small distances $d$) and decrease (large $d$) the pressure between the walls.