Renormalization of Hard-Core Guest Charges Immersed in Two-Dimensional Electrolyte

Abstract

This paper is a continuation of a previous one [L. {Š}amaj, {\it J. Stat. Phys.} {\bf 120}:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike $\pm$ unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures $0\le β<2$. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the {\em pointlike} guest charge $Q$, in its stability regime $β| Q| < 2$. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius $σ$, which allows us to go beyond the stability border $β| Q| = 2$. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region $β| Q| >2$, the renormalized charge $Q_{\rm ren}$ turns out to be a periodic function of the bare charge $Q$ with period 1. The renormalized charge therefore does not saturate at a specific finite value as $| Q| \to\infty$, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits $β\to 0$ and $Q\to\infty$ with the product $βQ$ being finite, one reproduces correctly the monotonic dependence of $βQ_{\rm ren}$ on $βQ$ in the guest-charge stability region $β| Q| <2$ and the Manning-Oosawa type of counterion condensation with the uniform saturation of $βQ_{\rm ren}$ at the value $4/π$ in the region $β| Q|\ge 2$.