  We propose a Poisson-Bikerman (PBik) formula for calculating the differential
capacitance (DC) of electrical double layers (EDLs) in aqueous electrolytes or
ionic liquids. The PBik theory is a generalization of the classical
Poisson-Boltzmann theory to include different steric energies of
different-sized ions and water similar to different electrical energies for
different-charged ions. Water and ions with interstitial voids in this
molecular mean field theory have their physical volumes as they do in molecular
dynamics simulations. The PBik formula derived from Fermi distributions of ions
and water in arbitrary shape and volume reduces to the Bikerman-Freise formula
derived from the lattice model of equal-sized ions. The DC curves predicted by
the Gouy-Chapman formula are U-shaped (for point-like ions with zero volume and
very dilute solutions). The curves change from U shape to camel shape
(Bactrian) and then to bell shape (for finite size ions) as the volume fraction
of ions and water changes from zero to medium value then to large value. The
transition is characterized by critical and inflection voltages in terms of the
particle volume fraction. These voltages determine steric and electrical
energies that describe the space/charge competition and saturation properties
of ions and water packed in the condensed layer of EDLs under high field
conditions. Steric energy is as important as electrical energy in these
conditions. PBik computes symmetric DC curves from delicately balanced steric
interactions of asymmetric-size ions and water like the experimental data of
KPF6 in aqueous solution. It computes asymmetric curves and captures delicately
balanced steric or electrical interactions of ions having different volumes or
charges in ionic liquids.
