Renormalization of a Hard-Core Guest Charge Immersed in a Two-Dimensional Electrolyte

Abstract

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 Q 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| → ∞, 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 βQ 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 ren at the value 4/π in the region β|Q| ≥ 2.

The double-layer structure of overscreened surfaces by smeared-out ions

“…This is in contrast to point charges, w(r) = δ(r), where charge inversion occurs when strong correlations give rise to lateral ordering of counterions within a double-layer. Such a correlation-based mechanism can only arise within an improved theory beyond the mean-field [3][4][5][6].…”

“…Such nonlinear renormalization of an effective charge carried out for point-ions does not lead to charge inversion but merely to charge saturation. To capture charge inversion for point-ions one needs to go beyond the mean-field level of description [2][3][4][5].…”

Renormalized charge in a two-dimensional model of colloidal suspension from hypernetted chain approach

“…2 are very close. 16,17 By increasing Z further, more counterions collapse and the previous description predicts a renormalized charge oscillating between two extremes. Naturally, this behavior can be ascribed to the correlations between the components of the system.…”

“…17, leading to an oscillatory behavior of the renormalized charge, applies only to point particles = 0 = 0. 17, leading to an oscillatory behavior of the renormalized charge, applies only to point particles = 0 = 0.…”

Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

“…The two-dimensional symmetric Coulomb gas of pointlike ±q charges, interacting via the logarithmic Coulomb potential, is integrable in the whole interval of couplings 0 ≤ Γ ≡ βq 2 < 2 where oppositely charged couples of charges do not collapse [14]. The concept of renormalized charge has been shown to be valid for a charged conductor wall [15], a pointlike guest charge [16] and for a guest charge with a small hard core [17,18] which permits one to go beyond the stability threshold. It is interesting that for a guest charge with a small hard core at a finite temperature [18], Q eff turns out to be a non-monotonous function of Q bare ; the same phenomenon was observed also in Monte-Carlo [19] and molecular-dynamics [20] simulations of the salt-free cell model.…”