The Poisson-Boltzmann Theory for Two Parallel Uniformly Charged Plates

TL;DR

This paper provides exact solutions to the nonlinear Poisson-Boltzmann equation for two parallel charged plates in symmetric and asymmetric electrolytes, deriving relations between surface charge, separation, and pressure, with new asymptotic results.

Contribution

It introduces exact solutions using Weierstrass elliptic functions for the two-plate Poisson-Boltzmann problem, including new asymptotic results in different regimes.

Findings

Exact expressions for electrostatic potential and surface charge density for one plate.

Derived functional relations between charge, separation, and pressure for two plates.

New asymptotic results for various regimes of plate separation and electrolyte types.

Abstract

We solve the nonlinear Poisson-Boltzmann equation for two parallel and likely charged plates both inside a symmetric elecrolyte, and inside a 2 : 1 asymmetric electrolyte, in terms of Weierstrass elliptic functions. From these solutions we derive the functional relation between the surface charge density, the plate separation, and the pressure between plates. For the one plate problem, we obtain exact expressions for the electrostatic potential and for the renormalized surface charge density, both in symmetric and in asymmetric electrolytes. For the two plate problems, we obtain new exact asymptotic results in various regimes.