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≃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)JESOAN0013-465110.1149/1.2129195; P. Kornilovitch and Y. Jeon, J. Appl. Phys. 109, 064509 (2011)JAPIAU0021-897910.1063/1.3554445]. 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, λ_{D}L/D, where L is the width of the cell, D is the diffusivity of ions, and λ_{D} is the Debye length. Second, in the (experimentally relevant) thin-double-layer limit ε=λ_{D}/L≪1, there is a "weakly nonlinear" regime defined by V_{T}≲V≲V_{T}ln(1/ε), 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≳V_{T}ln(1/ε), 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, λ_{D}^{2}/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/ε). 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.

A mathematical model for electrical impedance spectroscopy of zwitterionic hydrogels.

We report a mathematical model for ion transport and electrical impedance in zwitterionic hydrogels, which possess acidic and basic functional groups that carry a net charge at a pH not equal to the isoelectric point. Such hydrogels can act as an electro-mechanical interface between a relatively hard biosensor and soft tissue in the body. For this application, the electrical impedance of the hydrogel must be characterized to ensure that ion transport to the biosensor is not significantly hindered. The electrical impedance is the ratio of the applied voltage to the measured current. We consider a simple model system, wherein an oscillating voltage is applied across a hydrogel immersed in electrolyte and sandwiched between parallel, blocking electrodes. We employ the Poisson-Nernst-Planck (PNP) equations coupled with acid-base dissociation reactions for the charge on the hydrogel backbone to model the ionic transport across the hydrogel. The electrical impedance is calculated from the numerical solution to the PNP equations and subsequently analyzed via an equivalent circuit model to extract the hydrogel capacitance, resistance, and the capacitance of electrical double layers at the electrode-hydrogel interface. For example, we predict that an increase in pH from the isoelectric point, pH = 6.4 for a model PCBMA hydrogel, to pH = 8 reduces the resistance of the hydrogel by ∼40% and increases the double layer capacitance by ∼250% at an electrolyte concentration of 0.1 mM. The significant impact of charged hydrogel functional groups to the impedance is damped at higher electrolyte concentration.