Self-Similar Response of Electrode Polarization for Binary Electrolytes in Parallel Plate Capacitor Systems

Classical electrochemistry problem of polarization of an electrode immersed in a symmetric binary electrolyte and subjected to a small external ac voltage is revisited. The Nernst-Planck equations are simplified to the Debye-Falkenhagen equation, which is solved together with the Poisson equation, leading to analytical formulas for the space charge density and impedance of the system for two parallel plate electrodes. We then define a limit of thin electrical double layer and illustrate the emergence of the characteristic time scale, τ_c = λ_DL/D, a function of the Debye length, λ_D, the electrode separation distance, L, and the ionic diffusion coefficient D. Normalizing the impedance magnitude with the solution resistance and making the frequency dimensionless with the τ_c, we show that all analytical, numerical, and experimental data for different solution conductivities and electrode separation distances collapse onto a single curve. To account for the Stern layer effects, we conducted numerical simulations based on the modified Poisson-Nernst-Planck model and showed that the results agree with our analytical solution for a range of concentrations, with small discrepancies observed only above 0.1 M. Based on the proposed model, experimental impedance spectroscopy results at ac potentials can be used to obtain detailed knowledge of the corresponding surface (and space) charge densities on the electrodes.