  A parallel plate capacitor containing an electrolytic solution is the
simplest model of a supercapacitor, or electric double layer capacitor. Using
both analytical and numerical techniques, we solve the Poisson-Nernst-Planck
equations for such a system, describing the mean-field charging dynamics of the
capacitor, when a constant potential difference is abruptly applied to its
plates. Working at constant total number of ions, we focus on the physical
processes involved in the relaxation and, whenever possible, give its
functional shape and exact time constants. We first review and study the case
of a symmetric binary electrolyte, where we assume the two ionic species to
have the same charges and diffusivities. We then relax these assumptions and
present results for a generic strong (i.e. fully dissociated) binary
electrolyte. At low electrolyte concentration, the relaxation is simple to
understand, as the dynamics of positive and negative ions appear decoupled. At
higher electrolyte concentration, we distinguish several regimes. In the linear
regime (low voltages), relaxation is multi-exponential, it starts by the
build-up of the equilibrium charge profile and continues with neutral mass
diffusion, and the relevant time scales feature both the average and the
Nernst-Hartley diffusion coefficients. In the purely nonlinear regime
(intermediate voltages), the initial relaxation is slowed down exponentially
due to increased capacitance, while bulk effects become more and more evident.
In the fully nonlinear regime (high voltages), the dynamics of charge and mass
are completely entangled and, asymptotically, the relaxation is linear in time.
