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

This paper is a continuation of a previous one [L. Šamaj, J. Stat. Phys. 120:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike ± unit charges, the latter system being in the stabilityagainst-collapse regime of reduced inverse temperatures 0 ≤ 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 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 Qren 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| → ∞, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits β → 0 and Q → ∞ with the product βQ being finite, one reproduces correctly the monotonic dependence of βQren on βQ in the guest-charge stability region β|Q| < 2 and the Manning-Oosawa type of counterion condensation with the uniform saturation of βQren at the value 4/π in the region β|Q| ≥ 2.