Guest Charges in an Electrolyte: Renormalized Charge, Long- and Short-Distance Behavior of the Electric Potential and Density Profiles

Abstract

We complement a recent exact study by L.Šamaj 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. InŠamaj's previous work, exact results for arbitrary coulombic coupling β were obtained for a system where all the charges are points, provided βQ < 2 and β < 2. Here, we first focus on the mean field situation which we believe describes correctly the limit β → 0 but βQ 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 βQ > 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 byŠamaj for the renormalized charge are not valid for βQ > 2, contrary to a hypothesis put forward byŠamaj. 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 (βQ ≥ 2 − β). 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 βQ = 2.

A Generalization of the Stillinger–Lovett Sum Rules for the Two-Dimensional Jellium

“…In this paper, we study a typical situation in the theory of colloidal mixtures [11,12]: a "guest" particle with charge Zq is immersed into the bulk interior of a Coulomb system, in our case the jellium. Possible values of the parameter Z are restricted as follows.…”

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

“…18 Notice that this is the opposite behavior than the one found in the symmetric electrolyte. 18 Notice that this is the opposite behavior than the one found in the symmetric electrolyte.…”

“…18 Simultaneously, according to the inflection point criterion, a new inflection point at r m Ͻ −1 can be identified. 18 Simultaneously, according to the inflection point criterion, a new inflection point at r m Ͻ −1 can be identified.…”

“…18 When Z⌫ increases above 2 − ⌫ ͑Z⌫Ͻ2,⌫Ͻ2͒, the coion cloud n + ͑r͒ = n + g 0+ ͑r͒ shows a change of behavior near the guest particle. 18 When Z⌫ increases above 2 − ⌫ ͑Z⌫Ͻ2,⌫Ͻ2͒, the coion cloud n + ͑r͒ = n + g 0+ ͑r͒ shows a change of behavior near the guest particle.…”

“…18 Figure 3 shows the values of ␣ 0s evaluated from the short-range values of g 0s ͑r͒. 18 Figure 3 shows the values of ␣ 0s evaluated from the short-range values of g 0s ͑r͒.…”