On the Graphical Analysis of the Impedance Response of Passive Electrodes
Authors: Gharbi Oumaima , Orazem Mark E. , Tran Mai T.T. , Tribollet Bernard , Turmine Mireille , Vivier Vincent 
Published: July 2022

The analysis of the capacitive behavior of a passive electrode is often a difficult subject because the response obtained by electrochemical impedance spectroscopy (EIS) is characterized by a non-ideal behavior that we try to model by using a constant phase element (CPE) or a Young impedance [1, 2]. In doing so, both the contribution of the double layer and the effects of the current and potential distributions due to the geometry of the electrode are usually neglected.
In the case of the contribution of the double layer, such an approximation can be verified a posteriori when data analysis is performed assuming that each of the contributions is a capacitance and that the double layer capacitance, the value of which is usually in the range 10 to 50 µF cm-2, is at least one order of magnitude larger than the value of the impedance of the passive film. However, this simplified approach is often invalid since both the impedance of passive film and the double layer relaxation usually exhibit a non-ideal behavior, thus resulting in the use of CPEs to account for the distribution of dielectric properties of the interface as well as experimental factors (position of the reference electrode, adsorption of impurities...).
We present in this work, a new method that uses the graphical analysis of the experimental data in the different frequency ranges (Figure 1), including the low-frequency domain, in order to determine the values of the capacitances without resorting to fitting procedures involving complex functions. This analysis, which is based on the electrical description of the interface, allows the determination of the thickness of a thin layer at the interface, independently of the use of a CPE to describe the distribution of time constants [3]. This will be first demonstrated on synthetic data and then used for the determination of thickness of the oxide film formed on Al electrode.
In a second part, we will show that this analysis can be extended to more complicated cases, including when frequency dispersion is observed in the high frequency domain [4]. In that case, the ohmic impedance needs to be properly estimated and then corrected on the experimental results before any data analysis.
References
[1] S.P. Harrington, T.M. Devine, Analysis of electrodes displaying frequency dispersion in Mott-Schottky tests, J. Electrochem. Soc., 155 (2008) C381-C386.
[2] B. Hirschorn, M.E. Orazem, B. Tribollet, V. Vivier, I. Frateur, M. Musiani, Determination of effective capacitance and film thickness from constant-phase-element parameters, Electrochim. Acta, 55 (2010) 6218-6227.
[3] O. Gharbi, M.T.T. Tran, M.E. Orazem, B. Tribollet, M. Turmine, V. Vivier, Impedance Response of a Thin Film on an Electrode: Deciphering the Influence of the Double Layer Capacitance, Chemphyschem, 22 (2021) 1371-1378.
[4] O. Gharbi, A. Dizon, M.E. Orazem, M.T.T. Tran, B. Tribollet, V. Vivier, From frequency dispersion to ohmic impedance: A new insight on the high-frequency impedance analysis of electrochemical systems, Electrochim. Acta, 320 (2019) 134609.
Figure 1: Nyquist representation of the impedance calculated for a passive film obeying to a power law model in series with a non-ideal double layer described by a CPE. The HF domain allows the thin oxide films properties to be obtained while the low frequency domain gives some information about the double layer.

Figure 1