Utility and Importance of Poisson-Nernst-Planck Immittance-Spectroscopy Fitting Models

This feature article highlights work done by the author and others since 1953 on the Poisson-Nernst-Planck (PNP) continuum model for analyzing and fitting wide-range immittance-spectroscopy (IS) frequency-response data for unsupported materials with diffusing mobile charge species present. The small-signal PNP approach, one relevant for analyzing data involving ordinary or anomalous diffusion, is particularly important because it leads to estimates of the values of far more physically significant parameters than do other available IS fitting models. Unfortunately, its virtues were not well appreciated until recently, and it has thus not been used as widely as it should be. The present work aims at remedying this lack by providing a thorough description of the strengths and weaknesses of the model, its response possibilities, and its broad applicability. It deals with a neutral species that can fully or partially dissociate into positive and negative charged species of equal concentrations but arbitrary mobilities. The full model, including bulk, mobility, generation-recombination, electrode reaction, and adsorption parameters, is first described, and some of its simplified response functions are presented. It is also incorporated in the free LEVMW complex-nonlinear-least-squares fitting program, making all of its features available for analyzing experimental IS data sets. After a detailed review of relevant theoretical and experimental published work involving the PNP approach, exact graphical responses are presented of progressively more complicated PNP models mostly involving charge of only one sign mobile for all four IS immittance levels. Then it is shown to what degree the various PNP-model responses can be fitted within usual experimental error limits by other more common, but less physically germane continuum, discrete, and empirical models. The positive results of such fitting greatly expand the range of usefulness and applicability of the PNP models. Fits of exact and noisy IS Randles-circuit data sets involving a finite-length Warburg part are compared with those involving different PNP models, and the finite-length-Warburg complex-plane response is discussed and compared with that of the interface part of the PNP response. Finally, some other PNP full and interface response possibilities are discussed and illustrated, and results are presented that involve specific adsorption and adsorption-reaction electrode processes of physical interest to such fields as biology, corrosion, and energy storage. Since a composite PNP fitting model with charges of both signs mobile is shown to exactly fit both exact data sets derived from the ordinary Randles circuit and ones generalized to include additional low- or high-frequency relaxation behavior, its scope and utility for fitting and interpreting experimental data should make it the preferred alternative to most fitting circuits that involve both ordinary resistive and capacitive parameters as well as distributed elements such as finite-length Warburg ones.