Planar screening by charge polydisperse counterions

Abstract

We study how a neutralising cloud of counterions screens the electric field of a uniformly charged planar membrane (plate), when the counterions are characterised by a distribution of charges (or valence), . We work out analytically the one-plate and two-plate cases, at the level of non-linear Poisson–Boltzmann theory. The (essentially asymptotic) predictions are successfully compared to numerical solutions of the full Poisson–Boltzmann theory, but also to Monte Carlo simulations. The counterions with smallest valence control the long-distance features of interactions, and may qualitatively change the results pertaining to the classic monodisperse case where all counterions have the same charge. Emphasis is put on continuous distributions , for which new power-laws can be evidenced, be it for the ionic density or the pressure, in the one- and two-plates situations respectively. We show that for discrete distributions, more relevant for experiments, these scaling laws persist in an intermediate but yet observable range. Furthermore, it appears that from a practical point of view, hallmarks of the continuous behaviour are already featured by discrete mixtures with a relatively small number of constituents.