The impact of steric repulsion on the total free energy of electric double layer capacitors

We present an analysis of the total free energy of a supercapacitor modelled with a composite diffuse layer (CDL) formed by a steric repulsive potential. The steric potential is modelled with a simple approximation to the Bikerman steric potential, enabling derivation of an analytical expression for the total free energy of the supercapacitor in terms of the size and valency of the electrolyte counterions and electrode potentials. The analytical expression for the total free energy of the supercapacitor matches the exact numerical Bikerman calculation at high potential with relative error close to 1%. This provides an upper bound over the more accurate Carnahan-Starling model. A maximum upper bound for the energy is also provided in the limit where bulk concentrations approach the ion concentration cap. We analyze the relative contribution of the steric interaction to the total free energy. At large voltages, the steric free energy is comparable in magnitude to that of the electrostatic free energy, and introduces ion-size effects in the energy of the supercapacitor. Consequently at high potentials the total free energy exceeds (doubles) the classical energy [Formula presented], indicating that this formula does not correctly describe the available stored energy from the experimentally measured capacitance.