Role of the displacement current on Warburg-type behavior

We investigate the role of the displacement current in the analysis of the electric response of an electrolytic cell to an external stimulus. We show that several models proposed to interpret the spectra deduced by means of the impedance spectroscopy technique are questionable. In particular, we demonstrate that even in the frequency range below the Debye frequency the role of the displacement current is fundamental, and its omission leads to incorrect results for the impedance of the cell. In our analysis, the boundary conditions on the bulk current density are of Nernstian and of Ohmic type. The analysis is limited to a fully dissociated electrolyte, and for only one type of mobile ions, as discussed in several papers devoted to the subject. Particular attention is given to the spatial dependence of the current density. We show that Warburg-like behavior is never predicted in the framework of the Poisson-Nernst-Planck model, if the electric impedance of the cell is correctly evaluated. From this conclusion, valid for media with only one type of mobile ions, it follows that if Warburg-like behavior is experimentally observed the theoretical interpretation is still an open problem, and its origin is probably related to the boundary conditions.