Discharging dynamics in an electrolytic cell

We analyze the dynamics of a discharging electrolytic cell comprised of a binary symmetric electrolyte between two planar, parallel blocking electrodes. When a voltage is initially applied, ions in the electrolyte migrate towards the electrodes, forming electrical double layers. After the system reaches steady state and the external current decays to zero, the applied voltage is switched off and the cell discharges, with the ions eventually returning to a uniform spatial concentration. At voltages on the order of the thermal voltage V-T = k(B)T/q similar or equal to 25 mV, where k(B) is Boltzmann's constant, T is temperature, and q is the charge of a proton, experiments on surfactant-doped nonpolar fluids observe that the temporal evolution of the external current during charging and discharging is not symmetric [V. Novotny and M. A. Hopper, J. Electrochem. Soc. 126, 925 (1979); P. Kornilovitch and Y. Jeon, J. Appl. Phys. 109, 064509 (2011)]. In fact, at sufficiently large voltages (several V-T), the current during discharging is no longer monotonic: it displays a "reverse peak" before decaying in magnitude to zero. We analyze the dynamics of discharging by solving the Poisson-Nernst-Planck equations governing ion transport via asymptotic and numerical techniques in three regimes. First, in the "linear regime" when the applied voltage V is formally much less than V-T, the charging and discharging currents are antisymmetric in time; however, the potential and charge density profiles during charging and discharging are asymmetric. The current evolution is on the RC timescale of the cell, lambda L-D/D, where L is the width of the cell, D is the diffusivity of ions, and lambda(D) is the Debye length. Second, in the (experimentally relevant) thin-double-layer limit epsilon = lambda(D)/L << 1, there is a "weakly nonlinear" regime defined by V-T less than or similar to V less than or similar to V-T ln(1/epsilon), where the bulk salt concentration is uniform; thus the RC timescale of the evolution of the current magnitude persists. However, nonlinear, voltage-dependent, capacitance of the double layer is responsible for a break in temporal antisymmetry of the charging and discharging currents. Third, the reverse peak in the discharging current develops in a "strongly nonlinear" regime V greater than or similar to V-T ln(1/epsilon), driven by neutral salt adsorption into the double layers and consequent bulk depletion during charging. The strongly nonlinear regime features current evolution over three timescales. The current decays in magnitude on the double layer relaxation timescale, lambda(2)(D)/D; then grows exponentially in time towards the reverse peak on the diffusion timescale, L-2/D, indicating that the reverse peak is the results of fast diffusion of ions from the double layer layer to the bulk. Following the reverse peak, the current decays exponentially to zero on the RC timescale. Notably, the current at the reverse peak and the time of the reverse peak saturate at large voltages V >> V-T ln(1/epsilon). We provide semi-analytic expressions for the saturated reverse peak time and current, which can be used to infer charge carrier diffusivity and concentration from experiments.