Comparison of two generation-recombination terms in the Poisson-Nernst-Planck model

Two phenomenological forms proposed to take into account the generation-recombination phenomenon of ions are investigated. The first form models the phenomenon as a chemical reaction, containing two coefficients describing the dissociation of neutral particles in ions, and the recombination of ions to give neutral particles. The second form is based on the assumption that in thermodynamical equilibrium, a well-defined density of ions is stable. Any deviation from the equilibrium density gives rise to a source term proportional to the deviation, whose phenomenological coefficient plays the role of a life time. The analysis is performed by evaluating the electrical response of an electrolytic cell to an external stimulus for both forms. For simplicity we assume that the electrodes are blocking, that there is only a group of negative and positive ions, and that the negative ions are immobile. For the second form, two cases are considered: (i) the generation-recombination phenomenon is due to an intrinsic mechanism, and (ii) the production of ions is triggered by an external source of energy, as in a solar cell. We show that the predictions of the two models are different at the impedance as well as at the admittance level. In particular, the first model predicts the existence of two plateaux for the real part of the impedance, whereas the second one predicts just one. It follows that impedance spectroscopy measurements could give information on the model valid for the generation-recombination of ions.