  Electric double layer supercapacitors are a fast-rising class of high-power
energy storage devices based on porous electrodes immersed in a concentrated
electrolyte or ionic liquid. As of yet there is no microscopic theory to
describe their surprisingly large capacitance per unit volume (volumetric
capacitance) of ~ 100 F/cm^3, nor is there a good understanding of the
fundamental limits on volumetric capacitance. In this paper we present a
non-mean-field theory of the volumetric capacitance of a supercapacitor that
captures the discrete nature of the ions and the exponential screening of their
repulsive interaction by the electrode. We consider analytically and via
Monte-Carlo simulations the case of an electrode made from a good metal and
show that in this case the volumetric capacitance can reach the record values.
We also study how the capacitance is reduced when the electrode is an imperfect
metal characterized by some finite screening radius. Finally, we argue that a
carbon electrode, despite its relatively large linear screening radius, can be
approximated as a perfect metal because of its strong nonlinear screening. In
this way the experimentally-measured capacitance values of ~ 100 F/cm^3 may be
understood.
