Computer Simulations of the Electric Double Layer

Abstract

We describe the Lekner–Sperb summation technique used to calculate the Coulomb interaction in 2D periodic systems, and discuss in detail the methods used to perform computer simulations of counterions close to charged objects (electric double layer). We focus on three situations where the double layer is formed: (1) when a wall with smeared out surface charge is in the presence of its counterions, (2) when the surface charge at the wall is modulated (e.g. formed by discrete charges), and (3) when two simple double layers interact with each other (i.e., the counterions are confined between two walls). Using Monte Carlo (MC) simulations, we obtain counterion density profiles around the charged objects and use them to test both Poisson–Boltzmann (asymptotically exact in the limit of low surface charge and/or low counterion valence) and the strong coupling theory, which becomes exact in the opposite limit of high surface charge and/or high counterion valence. For the case of single wall with smeared out surface charge (system 1), we also study the the counterion pair correlation function, which indicates a behavioral change from a three-dimensional, weakly correlated counterion distribution (at low coupling) to a two-dimensional, strongly correlated counterion distribution (at high coupling), which is paralleled by the specific heat capacity which displays a rounded hump at intermediate coupling strengths. For the charge-modulated case (system 2), we show (for high surface charge and/or multivalent counterions) that the counterions tend to stay in the close vicinity of the surface and are laterally correlated with the surface charges when the minimum approach distance between the latter and counterions is smaller than the distance between surface charges. We obtain a set of parameters for which the average counterion density profiles are very different from the ones obtained with smeared out charged surfaces, and show that in the regime where the classical Poisson–Boltzmann theory is expected to fail, the strong coupling theory agrees very well with Monte Carlo simulations. Finally, for the case of counterions confined between two equally charged walls (system 3), we analyze the inter-wall pressure and establish the complete phase diagram, featuring attraction between the walls for large enough coupling strength and at intermediate wall separation.