Lattice-based mean-field models of ionic liquids neglect charge discreteness and ion correlations. To address these limitations, we propose separating the short-range and long-range parts of the electrostatic interaction by truncating the Coulomb potential below a fixed distance that is equal to or slightly larger than that between neighboring ions. Interactions and correlations between adjacent ions can then be modeled explicitly, whereas longer-ranged electrostatic interactions are captured on the mean-field level. We implement this approximation into the framework of modeling a compact, solvent-free ionic liquid by, first, considering terms up to the fourth order of the operator that represents the truncated Coulomb potential and, second, by accounting for electrostatic correlations between pairs of neighboring ions on the level of the quasi-chemical approach. A set of boundary conditions for the resulting self-consistent fourth-order differential equation follows from functional minimization of the free energy. The differential capacitance of an ionic liquid in contact with a planar electrode is calculated analytically up to quadratic order in the electrode's surface charge density by solving the linearized model and applying a perturbation approach valid beyond the linear regime. We demonstrate that charge discreteness enhances the differential capacitance, whereas electrostatic correlations between ion-ion pairs drive the transition from a bell-shaped to a camel-shaped profile of differential capacitance. Our approach offers a systematic way to further improve the treatment of charge discreteness, account for short-range electrostatic and non-electrostatic interactions, and include higher-order ion-ion correlations.
Keywords: boundary conditions; charge discreteness; electrostatics; ion correlations; ionic liquid; mean-field; quasi-chemical approximation; random mixing approximation.
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