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The rapid and efficient exchange of ions between porous electrodes andaqueous solutions is important in many applications, such as electrical energystorage by super-capacitors, water desalination and purification by capacitivedeionization (or desalination), and capacitive extraction of renewable energyfrom a salinity difference. Here, we present a unified mean-field theory forcapacitive charging and desalination by ideally polarizable porous electrodes(without Faradaic reactions or specific adsorption of ions) in the limit ofthin double layers (compared to typical pore dimensions). We illustrate thetheory in the case of a dilute, symmetric, binary electrolyte using theGouy-Chapman-Stern (GCS) model of the double layer, for which simple formulaeare available for salt adsorption and capacitive charging of the diffuse partof the double layer. We solve the full GCS mean-field theory numerically forrealistic parameters in capacitive deionization, and we derive reduced modelsfor two limiting regimes with different time scales: (i) In the"super-capacitor regime" of small voltages and/or early times where the porouselectrode acts like a transmission line, governed by a linear diffusionequation for the electrostatic potential, scaled to the RC time of a singlepore. (ii) In the "desalination regime" of large voltages and long times, theporous electrode slowly adsorbs neutral salt, governed by coupled, nonlineardiffusion equations for the pore-averaged potential and salt concentration.

Nonlinear Dynamics of Capacitive Charging and Desalination by Porous Electrodes