On the Analysis of Non-stationary Impedance Spectra

Does it make sense to make impedance spectroscopy in non-stationary conditions? And if yes, how should one analyze the results? The main goal of this work is to give a satisfactory answer to these two questions. Regarding the first question and according to the usual interpretation of the concept of impedance, the answer should be no, because impedance is not defined as timedependent and, therefore, there should not exist an impedance out of stationary conditions. However, under some conditions, it is possible to show that time dependence can be conciliated into the concept of impedance. The mathematical formulation of this problem is presented in this article. In literature it is possible to find some works which deal with the concept of dynamic impedance. Ragoisha et al. introduced the idea of potentiodynamic electrochemical impedance spectroscopy (PDEIS) [1]. This technique is based on a subsequent set of impedance spectra measured on stationary conditions on a staircase potential profile. The impedance is collected in the standard fashion, a single frequency at the time. The literature is full of examples of series of impedance spectra used to study an electrochemical reaction in a range of potential. Some noticeable examples are the Fourier transform ac voltammetry employed especially by Bond and coworkers (for example reference [2–4]) and the Fourier transforms electrochemical impedance spectroscopy (FT-EIS) reported also by Pettit and Roy (some examples [5, 6]). In both cases, a multisine signal is superimposed to a step by step potential ramp and the impedance is calculated through Fourier analysis. Interpretation of the results in these cases is achieved though fitting of the spectra with an equivalent circuit which describes the physico-chemical behavior of the system or simply provides a rational series of parameters which are then studied as function of the potential. Darowicki achieved an instantaneous impedance spectrum using short-time Fourier transforms (STFT) [7–11]. In this technique a multisine is superimposed to a true linear potential ramp and the data are recorded in continuous. The impedance is then calculated breaking the recorded data in blocks and performing STFT on every block. The STFT have the advantage that can capture both the frequency and the time dependence of the signals. In this way it is possible to recover sets of impedance spectra in a range of frequencies as function of time. Sacci and Harrington used very similar approach in what they called dynamic electrochemical impedance spectroscopy (dEIS) [12–15]. They did not investigate the properties of the STFT, but provided valuable discussions regarding the problem of performing Fourier analysis on a dynamic system. They implemented a baseline correction to compensate for the evolution of the system and discussed which frequency constrain should be taken in superimposing a frequency signal on a linear voltammetry [15]. Abstract : In this work we study the possibility to analyze the non-stationary impedance spectra by employing standard equivalent circuits. For this purpose, the dynamic multi-frequency analysis (DMFA) is introduced and compared with a set of consecutive stationary impedance spectra. In order to prove the hypothesis, the data are obtained by the simulation in the time domain of the electron transfer process between an electrode and a free-diffusing redox couple in solution. During the simulation, the system is perturbed with a cyclic voltammetry superimposed to a small multisine perturbation (DMFA) or with a series of stationary impedance spectroscopies. Also, a new fitting algorithm, which takes into account the correlation between consecutive spectra, is proposed and tested. Although the Randles circuit can be used to fit both dynamic and stationary impedance spectra, the values of the fitting parameters are different and depend on the direction of the scan and on the rate. This is related to the influence of the diffusion profile on the fitting parameters.