Effects of discreteness of surface charges on the effective electrostatic interactions

In the present work, we study effects of relative arrangement of discrete surface charges on electrostatic potential of mean force (EPMF) between two surfaces immersed in a primitive model electrolyte solution. To help in exploring and ascertaining the relevant mechanism, different combinations of values for electrostatic parameters (such as ion concentration, diameter, and valence, solution temperature and relative permittivity) are investigated by solving classical density functional theory in three dimensional space, in combination with different discrete surface charge configurations and surface charge homogeneity configuration as the effects may alter along with change of the surface charge arrangement. Our findings include: (i) if counter-ion is univalent, the EPMF of charge homogeneity surface is always more repulsive than those of charge heterogeneity surfaces whether the discrete surface charges are distributed on the two face-face surfaces commensurately or non-commensurately; (ii) in presence of high valence counter-ion well depth of like charge attraction between two charge homogeneity surfaces approximately takes an averaged value of those for various charge heterogeneity surfaces whether the discrete surface charges are distributed on the two face-face surfaces commensurately or non-commensurately; (iii) the EPMF becomes ever more attractive or less repulsive with a centralization of the discrete surface charges whether the counter-ion valence is univalent, divalent or trivalent; and (iv) the EPMF in the case of charge heterogeneity surfaces is influenced by bulk electrolyte concentration, salt ion diameter and valence, solution temperature, and relative permittivity in the same manner as in the case of charge homogeneity surfaces. All of these findings can be explained self-consistently by a recently proposed hydrogen-bonding style mechanism (for dealing with the EPMF between two charge homogeneity surfaces) reinforced by two new concepts: counter-ion effective adsorption spaces belonging to different localized surface charges and hydrogen bond directivity.