Mixed valency counterions between charged walls: An investigation using Monte Carlo simulation and comparison with Poisson-Boltzmann theory

Abstract

Systems consisting of parallel charged surfaces of infinite extent delimiting a dielectric medium, containing both monovalent and divalent counterions, have been investigated using canonical ensemble Monte Carlo simulation and a restricted primitive model. Singlet distributions, potential energies, and pressures have been obtained for counterion mixtures containing 0, 20, 40, 60, 80, and 100% of the charge as the monovalent species, with the surface charge either in the form of a regular lattice of discrete charges (the DSC model) or as a continuous sheet of charge (the CSC model). A solution of the Poisson-Boltzmann (PB) equation, which does not appear to have been obtained previously, has been derived for the CSC model and singlet density predictions are compared with the results of simulation. The PB equation provides a qualitative prediction of the singlet densities of all counterion species for the CSC system. In the DSC system the singlet charge density distributions show significant variations parallel to the surface which can be approximated by a Boltzmann distribution; the PB prediction for the singlet density normal to the surface is poorer than in the CSC system. Energies and pressures differ significantly between the DSC and CSC simulations, with the ion-wall part of the interaction being enhanced in the latter. The differences from PB predictions for all system properties increase with the divalent composition of the counterions.