Differential capacitance of ionic liquids according to lattice-gas mean-field model with nearest-neighbor interactions
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The Bragg-Williams free energy is used to incorporate nearest-neighbor interactions into the lattice gas model of a solvent-free ionic liquid near a planar electrode. We calculate the differential capacitance from solutions of the mean-field consistency relation, arriving at an explicit expression in the limit of a weakly charged electrode. The two additional material parameters that appear in the theory - the degree of nonideality and the resistance to concentration changes of each ion type - give rise to different regimes that we identify and discuss. As the nonideality parameter, which becomes more positive for stronger nearest-neighbor attraction between like-charged ions, increases and the electrode is weakly charged, the differential capacitance is predicted to transition through a divergence and subsequently adopt negative values just before the ionic liquid becomes structurally unstable. This is associated with the spontaneous charging of an electrode at vanishing potential. The physical origin of the divergence and the negative sign of the differential capacitance is a nonmonotonic relationship between the surface potential and surface charge density, which reflects the formation of layered domains alternatingly enriched in counterions and coions near the electrode. The decay length of this layered domain pattern, which can be many times larger than the ion size, is reminiscent of the recently introduced concept of underscreening.