Counter-ions at Charged Walls: Two Dimensional Systems

TL;DR

This paper analyzes the behavior of classical counter-ions near charged lines in 2D, comparing weak and strong coupling theories, and provides exact solutions for certain coupling constants, revealing phase transitions and attraction regimes.

Contribution

It offers a comprehensive comparison of weak and strong coupling theories for 2D Coulomb systems near charged lines, including exact solutions at specific coupling constants.

Findings

Density profile decay changes at Gamma=6.

Like-charge attraction exists at Gamma=4 and 6.

Exact solutions are obtained for Gamma=2,4,6.

Abstract

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double-layer), or between two such lines (interacting double-layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant Gamma is small, is briefly recapitulated (the coupling constant is defined as Gamma = beta e^2 where beta is the inverse temperature, and e the counter-ion charge). The opposite strong-coupling limit (Gamma -> infinity) is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. The weak- and strong-coupling theories are compared at intermediary values of the coupling constant Gamma=2 gamma (gamma=1,2,3), to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at Gamma=2 and up to relatively large finite N at Gamma=4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at Gamma=6. The like-charge attraction regime, possible in the strong coupling limit but precluded at mean-field level, survives for Gamma=4 and 6, but disappears at Gamma=2.