Attraction of like-charged walls with counterions only: Exact results for the 2D cylinder geometry

TL;DR

This paper derives exact results for the effective interaction between like-charged walls with counterions in a 2D cylindrical geometry, revealing conditions under which the walls attract or repel depending on charge density and coupling strength.

Contribution

It provides exact solutions for the 2D Coulomb system at specific couplings, demonstrating how like-charged walls can attract due to counterion effects in a cylindrical geometry.

Findings

Pressure remains positive and decays monotonically with distance.

Numerical results for finite counterions closely match the thermodynamic limit.

Attraction occurs when both walls have nonzero charge density at larger distances.

Abstract

We study a 2D system of identical mobile particles on the surface of a cylinder of finite length $d$ and circumference $W$, immersed in a medium of dielectric constant $\varepsilon$. The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge $-e$ ($e$ being the elementary charge) are coined as ``counterions''; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions $N$. Our task is to derive an effective interaction between the end-circles mediated by the counterions in thermal equilibrium at the inverse temperature $\beta$. The exact solution of the system at the free-fermion coupling $\Gamma \equiv \beta e^2/\varepsilon =2$ is used to test the convergence of the pressure as the (even) number of particles increases from $N=2$ to $\infty$. The pressure as a function of distance $d$ is always positive (effective repulsion between the like-charged circles), decaying monotonously; the numerical results for $N=8$ counterions are very close to those in the thermodynamic limit $N\to\infty$. For the couplings $\Gamma=2\gamma$ with $\gamma=1,2,\ldots$, there exists a mapping of the continuous two-dimensional (2D) Coulomb system with $N$ particles onto the one-dimensional (1D) lattice model of $N$ sites with interacting sets of anticommuting variables. This allows one to treat exactly the density profile, two-body density and the pressure for the couplings $\Gamma=4$ and $6$, up to $N=8$ particles. Our main finding is that the pressure becomes negative at large enough distances $d$ if and only if both like-charged walls carry a nonzero charge density. This indicates a like-attraction in the thermodynamic limit $N\to\infty$ as well, starting from a relatively weak coupling constant $\Gamma$ in between 2 and 4.