Simple Description of the Capacitance of the Double Layer of a High Concentration Electrolyte

Recently, the electrochemical properties of an ionic liquid have received increasing attention. The simplest model of an ionic liquid is a fluid of charged hard spheres, all of which have the same diameter (the restricted primitive model or RPM), and the simplest theory of the double layer formed by an electrolyte is the Gouy-Chapman-Stern (GCS) theory. This theory is moderately useful for a low concentration electrolyte but is incorrect for a high concentration electrolyte, such as an ionic liquid. Among other things, the GCS theory predicts that the capacitance has a minimum at small electrode charge. In contrast, the capacitance of the double layer of an ionic liquid often has a maximum at small electrode charge. This is predicted by simulations and the modified Poisson-Boltzmann (MPB) theory. The mean spherical approximation (MSA) seems to agree with the simulations and MPB but is applicable only for small electrode charge. In this note we graft the MSA onto the GCS result and obtain qualitatively reasonable results for the capacitance that may be of use. This result is identical to the GCS theory at small concentrations and to the MSA at small electrode charge. A maximum is found for small electrode charge and high concentration.