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The Poisson-Nernst-Planck equations describe the dynamics of charge transport in an electric field. Although they are relevant in many applications, a general solution is not known and several aspects are not well understood. In many situations nonlinear effects arise for which no analytical description is available. In this work, we investigate charge transport in a planar device on application of a voltage step. We derive analytical expressions for the dynamical behavior in four extreme cases. In the "geometry limited" regime, applicable at high voltages and low charge contents, we neglect diffusion and the electric field induced by the charges. This leads to a uniform movement of all charges until the bulk is completely depleted. In the "space charge limited" regime, for high voltages and high charge contents, diffusion is still neglected but the electric field is almost completely screened over transient space charge layers. Eventually, however, the bulk becomes depleted of charges and the field becomes homogeneous again. This regime is solved under the assumption of a homogeneous current density, and is characterized by a typical t -3/4 behavior. In the "diffusion limited" regime, valid for low voltages and low charge contents, diffusion is the dominant transport mechanism and prevents the charges from separating. This results in only very small deviations from a homogeneous charge distribution throughout the device. In the "double layer limited" regime, for low voltages and high charge contents, the combination of dominant diffusion and screening of the electric field results in large variations occurring only in thin double layers near the electrodes. Numerical simulations confirm the validity of the derived analytical expressions for each of the four regimes, and allow us to investigate the parameter values for which they are applicable. We present transient current measurements on a nonpolar liquid with surfactant and compare them with the external current predicted by the theoretical description. The agreement of the analytical expressions with the experiments allows us to obtain values for a number of properties of the charges in the liquid, which are consistent with results in other works. The confirmation by simulations and measurements of the derived theoretical expressions gives confidence about their usefulness to understand various aspects of the Poisson-Nernst-Planck equations and the effects they represent in the dynamics of charge transport.

The Poisson-Boltzmann (PB) equation is widely used to calculate the interaction between electric potential and the distribution of charged species. In the case of a symmetrical electrolyte in planar geometry, the Gouy-Chapman (GC) solution is generally presented as the analytical solution of the PB equation. However, we demonstrate here that this GC solution assumes the presence of a bulk region with zero electric field, which is not justified in microdevices. In order to extend the range of validity, we obtain here the complete numerical solution of the planar PB equation, supported with analytical approximations. For low applied voltages, it agrees with the GC solution. Here, the electric double layers fully absorb the applied voltage such that a region appears where the electric field is screened. For higher voltages (of order 1 V in microdevices), the solution of the PB equation shows a dramatically different behavior, in that the double layers can no longer absorb the complete applied voltage. Instead, a finite field remains throughout the device that leads to complete separation of the charged species. In this higher voltage regime, the double layer characteristics are no longer described by the usual Debye parameter kappa, and the ion concentration at the electrodes is intrinsically bound (even without assuming steric interactions). In addition, we have performed measurements of the electrode polarization current on a nonaqueous model electrolyte inside a microdevice. The experimental results are fully consistent with our calculations, for the complete concentration and voltage range of interest.

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