Dynamic density functional theory for the charging of electric double layer capacitors

We consider the charging of a model capacitor comprised of two planar electrodes and an electrolyte. Upon switching on a voltage difference, electric double layers build up in this setup, which we characterize with a classical dynamic density functional theory (DDFT) that accounts for electrostatic correlations and for molecular excluded volume of finite-sized ions and solvent molecules. Our DDFT predicts the electrode charge Q(t) to form exponentially with two timescales: at early times, the system relaxes on the RC time, namely, lambda L-D/[D(2 + sigma/lambda(D))], with lambda(D) being the Debye length, L being the electrode separation, sigma being the ion diameter, and D being the ionic diffusivity. Contrasting an earlier DDFT study, this early-time response does not depend on the applied potential. At late times, the capacitor relaxes with a relaxation time proportional to the diffusion time L-2/D.