Manning condensation in two dimensions

TL;DR

This paper provides exact analytical results for counterion condensation phenomena around a macroion in two dimensions, confirming mean-field predictions and revealing a series of single-ion condensation transitions.

Contribution

It presents the first exact solutions for two-dimensional counterion condensation, including transition thresholds and multiple condensation steps, aligning well with simulation data.

Findings

Counterion condensation threshold matches mean-field theory

Multiple single-ion condensation transitions occur with increasing macroion charge

Analytical results agree with Monte-Carlo simulations

Abstract

We consider a macroion confined to a cylindrical cell and neutralized by oppositely charged counterions. Exact results are obtained for the two-dimensional version of this problem, in which ion-ion and ion-macroion interactions are logarithmic. In particular, the threshold for counterion condensation is found to be the same as predicted by mean-field theory. With further increase of the macroion charge, a series of single-ion condensation transitions takes place. Our analytical results are expected to be exact in the vicinity of these transitions and are in very good agreement with recent Monte-Carlo simulation data.