Ion Diffusion in the Time-Dependent Potential of the Dynamic Electric Double Layer

The description of ion diffusion in the electric field set up by the electric double layer (EDL) is an important issue in many scientific fields because of its close relevance to diffusion-controlled chemical kinetics and ion transport occurring in the environmental, biological, and other systems with multiphase chemical reactions and charged particle transports. When considering ion diffusion in the EDL, the change of ion density in space leads to the change of potential with time. Currently, the static distribution of electric potential in the EDL at equilibrium is described by the nonlinear Poisson-Boltzmann equation. However, describing a time-dependent potential during a diffusion process in the EDL still remains a challenge. In this study, a dynamic Poisson-Boltzmann equation that describes the time-dependent potential was suggested for a slow diffusion problem. By combining the generalized nonsteady state diffusion equation (the linearized Fokker-Planck equation) with the time-dependent potential (the dynamic Poisson-Boltzmann equation), analytic solutions that can be expressed in algebraic form for describing the dynamic ion distribution with time-dependent potential and dynamic potential distribution in the EDL were obtained for the reflecting and adsorbing boundary conditions, respectively. The effective and simple approach for analytically solving the generalized linear equation of the complexity nonlinear diffusion in time-dependent potential advanced in the research could be potentially applied to solve other similar systems.