Differential Capacitance of Electrolytes at Weakly Curved Electrodes

The differential capacitance characterizes the ability of an electric double layer to store energy and is thus of fundamental importance for applications that employ electrostatic double-layer capacitors to store or rapidly convert energy. It is well-known that the differential capacitance depends not only on the chemical structure of the electrode and electrolyte but also on the electrode's geometry. However, attempts to describe how the curvature of an electrode affects the differential capacitance are sporadic and have focused on a few specific geometries. In the present work we carry out a systematic expansion of the differential capacitance up to second order in electrode curvature. The expansion, which applies to a large class of underlying theoretical models for the electric double layer, leads to three parameters that we calculate generally and then exemplify for a number of commonly used mean-field models: the classical Poisson-Boltzmann model, the lattice gas model, and a model that employs the Carnahan-Starling equation of state. The curvature of an electrode affects the differential capacitance in a rather complex manner depending on the electrode charge and concentration of ions in the bulk of the electrolyte. In most cases, spherical curvature tends to increase the differential capacitance whereas saddle curvature leaves it largely unaffected or decreases it slightly.