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 uses 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.Comment: 3 pages, 3 figure