Two main treatments within classical simulations for modeling a charged surface are using explicit, discrete charges and continuous, uniform charges. The computational cost can be substantially reduced if, instead of discrete surface charges, one use
s an electric field to represent continuous surface charges. In addition, many electrolyte theories, including the Poisson--Boltzmann theory, are developed on the assumption of uniform surface charge. However, recent simulations have demonstrated with discrete surface charges, one observes much stronger charge reversal, compared to the surfaces with continuous surface charges, when the lattice constant becomes notably larger than the ion diameter. These examples show that the two treatments for modeling a charged dielectric interface can lead to substantially different results. In this short note, we calculate the electrostatic force for a single point charge above an infinite plane, and compare the differences between discrete and continuous representations of surface charges. Our results show that while the continuous, uniform surface charge model gives a quite simple picture, the discrete surface charge model can offer several different cases even for such a simple problem, depending on the respective values of ion size versus lattice spacing and a self-image interaction parameter.
