One-dimensional counterion gas between charged surfaces: Exact results compared with weak- and strong-coupling analysis

TL;DR

This paper provides exact solutions for a one-dimensional counterion gas between charged surfaces and compares these results with weak- and strong-coupling approximations, revealing the regimes where each approximation is valid.

Contribution

It offers the first exact statistical integral for a 1D counterion system and compares it with classical approximations, highlighting their applicability.

Findings

Weak coupling approximation is accurate for large numbers of counterions.

Fluctuations are smaller in 1D than in 3D systems.

Strong coupling approximation better for few counterions.

Abstract

We evaluate exactly the statistical integral for an inhomogeneous one-dimensional counterion-only Coulomb gas between two charged boundaries and from this compute the effective interaction, or disjoining pressure, between the bounding surfaces. Our exact results are compared with the limiting cases of weak and strong coupling which are the same for 1D and 3D systems. For systems with a large number of counterions it is found that the weak coupling (mean-field) approximation for the disjoining pressure works perfectly and that fluctuations around the mean-field in 1D are much smaller than in 3D. In the case of few counterions it works less well and strong coupling approximation performs much better as it takes into account properly the discreteness of the counterion charges.