Exact solution of the Poisson-Nernst-Planck equations in the linear regime

Based on the Poisson-Nernst-Planck equations (PNP), the spatiotemporal charge, concentration profile, and the electric field in polyelectrolytes are analyzed. The system is subjected to a dc applied voltage. Different to recent papers we obtain an exact analytical solution of the PNP in the linear regime, which is characterized by an inevitable coupling between the spatial and the temporal behavior. In the long time limit the systems tends in a nonexponential manner to the steady state predicted by the Debye-Hueckel theory, where the time scale for the crossover into the steady state is determined by the Debye screening length and the initial concentration. The higher the initial concentration is the faster the system evolves into the stationary state. The Debye screening length characterizes not only the asymptotic behavior but also the spatiotemporal evolution of the system at finite times. Using experimental data the concentration profile and the electric field is shown to be on a master curve parametrized by the screening length. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3223724]