Analytical solution of the Poisson-Nernst-Planck equations for an electrochemical system close to electroneutrality

摘要

Single charge densities and the potential are used to describe models of electrochemical systems. These quantities can be calculated by solving a system of time dependent nonlinear coupled partial differential equations, the Poisson-Nernst-Planck equations. Assuming small deviations from the electroneutral equilibrium, the linearized and decoupled equations are solved for a radial symmetric geometry, which represents the interface between a cell and a sensor device. The densities and the potential are expressed by Fourier-Bessels series. The system considered has a ratio between the Debye-length and its geometric dimension on the order of 10~(?4) so the Fourier-Bessel series can be approximated by elementary functions. The time development of the system is characterized by two time constants, τ_ c and τ_ g. The constant τ_ c describes the approach to the stationary state of the total charge and the potential. τ_ c is several orders of magnitude smaller than the geometry-dependent constant τ_ g, which is on the order of 10 ms characterizing the transition to the stationary state of the single ion densities.