A simple approximation for the distribution of ions between charged plates in the weak coupling regime

TL;DR

This paper presents a simple analytical approximation for ion distribution between charged plates in the weak coupling regime, matching numerical and simulation results within 5% without added salt.

Contribution

It introduces a new, straightforward approximation for ion density in the weak coupling regime, improving understanding of electrostatic interactions in confined systems.

Findings

Analytic approximation predicts ion density within 5% accuracy

Comparison with numerical and simulation data validates the approximation

Higher order terms can enhance accuracy at the cost of simplicity

Abstract

The solution of the Poisson--Boltzmann equation for counterions confined between two charged plates is known analytically up to a constant, namely, the ion density in the middle of the channel. This quantity is relevant also because it gives access, through the contact theorem, to the osmotic pressure of the system. Here we compare the values of the ion density obtained by numerical and simulation approaches, and report a useful analytic approximation for the weak coupling regime in the absence of added salt, which predicts the value of the ion density in the worst case within 5%. The inclusion of higher order terms in a Laurent expansion can further improve the accuracy, at the expense of simplicity.