Semianalytical approximation of Ion Adsorption Layers and Capacitance in Carnahan-Starling-like steric models

The Carnahan–Starling (CS) steric model is the best description of hard-sphere fluids within the mean-field theory. Here we introduce an approximation of the near-linear adsorption concentration profile of a counterion near an electrode for a CS model and derive the subsequent electric field and electrostatic potential profile in a double layer. This enables the derivation of a semianalytical approximation of the electrode charge density, differential capacitance, and total energies (grand potentials) of an electric double-layer capacitor. These semianalytical equations are valid for electrode potentials between 0.2–4 V and converge to the full numerical solutions of the CS model at high potentials of 1V and bulk concentration of 1M with relative errors less than 2% for the electrode charge densities, and less than 5% for the capacitance and total energies. We find the steric contribution comprises approximately one-quarter of the total energy at high electrode potentials, while the contribution from ideal ion entropies becomes insignificant. The model shows very good agreement with experimental measurements of an aqueous electrolyte, and good agreement at high potentials with computer simulations of an ionic liquid. These semianalytical approximations are effective for applications with concentrated solutions or ionic liquids at high applied voltages where the full numerical solution is computationally expensive or in some cases impossible.