Analytical solution of the Poisson-Nernst-Planck equations in the linear regime at an applied dc-voltage

The analytical solution of the Poisson-Nernst-Planck equations is found in the linear regime as response to a dc-voltage. In deriving the results a new approach is suggested, which allows to fulfill all initial and boundary conditions and guarantees the absence of Faradaic processes explicitly. We obtain the spatiotemporal distribution of the electric field and the concentration of the charge carriers valid in the whole time interval and for an arbitrary initial concentration of ions. A different behavior in the short- and the long-time regime is observed. The crossover between these regimes is estimated.