  There is a growing interest in the properties of ionic liquids (ILs) and
IL-solvent mixtures at metallic interfaces, particularly due to their
applications in energy storage. The main focus so far has been on electrical
double layers with ILs far from phase transitions. However, the systems in the
vicinity of their phase transformations are known to exhibit some remarkable
features, such as wetting transitions and capillary condensation. Herein, we
develop a mean-field model suitable for the IL-solvent mixtures close to
demixing, and combine it with the Carnahan-Starling (CS) and lattice-gas
expressions for the excluded volume interactions. This model is then solved
analytically, using perturbation expansion, and numerically. We demonstrate
that, besides the well-known camel and bell-shaped capacitances, there is a
bird-shaped capacitance, having three peaks as a function of voltage, which
emerges due to the proximity to demixing. In addition, we find that the
camel-shaped capacitance, which is a signature of dilute electrolytes, can
appear at high IL densities for ionophobic electrodes. We also discuss the
differences and implications arising from the CS and lattice-gas expressions in
the context of our model.
