Fraction of Condensed Counterions around a Charged Rod: Comparison of Poisson−Boltzmann Theory and Computer Simulations

Abstract

We investigate the phenomenon of counterion condensation in a solution of highly charged rigid polyelectrolytes within the cell model. A method is proposed whichbased on the charge distribution functionidentifies 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/ξ of condensed ions for a Manning parameter ξ > 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 ξ our computer simulations predict charge oscillations, which mean-field theory is unable to reproduce.