The Poisson-Boltzmann Theory for The Two-Plates Problem: Some Exact Results

TL;DR

This paper derives exact solutions to the nonlinear Poisson-Boltzmann equation for two parallel charged plates using elliptic functions, providing precise asymptotic interaction results and surface charge density renormalization.

Contribution

It presents the first exact solutions for the two-plate problem in nonlinear Poisson-Boltzmann theory using Weierstrass elliptic functions, including asymptotic behaviors.

Findings

Exact asymptotic interaction formulas

Explicit form of renormalized surface charge density

Solutions applicable to symmetric and asymmetric electrolytes

Abstract

The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric elecrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.