Guest charges in an electrolyte: renormalized charge, long- and short-distance behavior of the electric potential and density profile

TL;DR

This paper investigates the electric potential and density profiles around a guest charge in a 2D electrolyte, focusing on the validity of existing formulas for renormalized charge and analyzing behavior at different coupling strengths.

Contribution

It extends previous exact results by studying the mean field limit for large guest charges and tests the validity of existing formulas for renormalized charge beyond their original assumptions.

Findings

Exact formulas for renormalized charge are invalid for 2Q>2.

Density profiles show a change in behavior at 2-2, indicating initial steps of counterion condensation.

The study clarifies the limits of previous theoretical predictions and provides new insights into charge behavior at different couplings.

Abstract

We complement a recent exact study by L. Samaj on the properties of a guest charge $Q$ immersed in a two-dimensional electrolyte with charges $+1/-1$. In particular, we are interested in the behavior of the density profiles and electric potential created by the charge and the electrolyte, and in the determination of the renormalized charge which is obtained from the long-distance asymptotics of the electric potential. In Samaj's previous work, exact results for arbitrary coulombic coupling $\beta$ were obtained for a system where all the charges are points, provided $\beta Q<2$ and $\beta < 2$. Here, we first focus on the mean field situation which we believe describes correctly the limit $\beta\to 0$ but $\beta Q$ large. In this limit we can study the case when the guest charge is a hard disk and its charge is above the collapse value $\beta Q>2$. We compare our results for the renormalized charge with the exact predictions and we test on a solid ground some conjectures of the previous study. Our study shows that the exact formulas obtained by Samaj for the renormalized charge are not valid for $\beta Q>2$, contrary to a hypothesis put forward by Samaj. We also determine the short-distance asymptotics of the density profiles of the coions and counterions near the guest charge, for arbitrary coulombic coupling. We show that the coion density profile exhibit a change of behavior if the guest charge becomes large enough ($\beta Q\geq 2-\beta$). This is interpreted as a first step of the counterion condensation (for large coulombic coupling), the second step taking place at the usual Manning--Oosawa threshold $\beta Q=2$.