The fraction of condensed counterions around a charged rod: Comparison of Poisson-Boltzmann theory and computer simulations

TL;DR

This study compares Poisson-Boltzmann theory with molecular dynamics simulations to analyze counterion condensation around charged rods, revealing limitations of mean-field theory at high salt concentrations and multivalent ions.

Contribution

It introduces a method to identify condensed ions and compares theoretical predictions with simulations, highlighting discrepancies at high charge and salt levels.

Findings

Poisson-Boltzmann theory predicts a fixed fraction of condensed ions for xi>1.

Simulations show stronger condensation than PB theory, especially with multivalent ions.

Charge oscillations are observed in simulations at high salt and charge, not captured by PB theory.

Abstract

We investigate the phenomenon of counterion condensation in a solution of highly charged rigid polyelectrolytes within the cell model. A method is proposed which -- based on the charge distribution function -- identifies both the fraction of condensed ions and the radial extension of the condensed layer. Within salt-free Poisson-Boltzmann (PB) theory it reproduces the well known fraction 1-1/xi of condensed ions for a Manning parameter xi>1. Furthermore, it predicts a weak salt dependence of this fraction and a breakdown of the concept of counterion condensation in the high salt limit. We complement our theoretical investigations with molecular dynamics simulations of a cell-like model, which constantly yield a stronger condensation than predicted by PB theory. While the agreement between theory and simulation is excellent in the monovalent, weakly charged case, it deteriorates with increasing electrostatic interaction strength and, in particular, increasing valence. For instance, at a high concentration of divalent salt and large xi our computer simulations predict charge oscillations, which mean-field theory is unable to reproduce.