(Invited) Parameter Identification in Electrochemical Impedance Spectroscopy Applications: Analysis of Sensitivity
Authors: Boinet Mickael , Condolf Cyril , Goulet Remi , Tribollet Bernard , Vivier Vincent 
Published: February 2020

EIS is a powerful non-destructive technique that can access essential information about interfacial and bulk material parameters through the use of low energy time varying electrical excitations. When applied to an electrochemical system, IS can provide information on reaction parameters, corrosion rates [1], oxide characteristics, porosity, coating integrity [2-3], mass transport, and many other electrode/interface characteristics [4].
Software packages for the analysis of impedance data have also emerged [5–7]. The typical goal of an EIS experiment is to gather significant information regarding the physicochemical phenomena taking place in an electrochemical system, and the experimental data are then compared to a model that describes the system.
The model can be purely empirical, being composed of resistors, capacitors, and generalized electrochemical elements, or otherwise it can be based on the physics of the system. Model parameters are evaluated through an inverse method. Minimization algorithms such as the Complex Non Linear Least Squares (CNLS) method, the Gauss-Newton method, the Levenberg-Marquardt method or the Downhill Simplex algorithm are used to determine the value of the parameters that minimize the sum of squared differences between the experimental and predicted data. Regardless of the method, the minimum value of the χ2 statistic is commonly used to assess the quality of the regression. χ2 is a non-weighted sum of squared differences:
(1)
In this equation, nf, Re, Im and |Z| represent the total number of frequencies constituting the impedance spectrum, the real and imaginary part and the modulus of the impedance, respectively. The subscripts e and c stand for the experimental and the calculated data, respectively. The standard deviation for the measurement is assumed to be Z= 0.01.|Ze| i.e.assuming a standard deviation for the impedance measurement of 1% of the modulus. For a complex regression, the fitting quality was determined by the standard deviation q.
(2)
Where n is the degree of freedom for the regression, i.e., the number of observations 2nf minus the number of adjustable parameters. The Smaller χ2 , the smaller the differences between the observed and modeled values and the better the fit. It is generally considered that a fit if satisfactory if the value of q is of the order of 1. A value smaller than 1 means that error was overestimated.
χ2 provides a useful single numerical value for assessing the quality of the regression, but χ2does not provide any information on the accuracy of the different parameters evaluated.
It is important to know if and how much the model, and subsequently the χ2or q are sensitive to the different parameters. Hence the sensitivity to the different parameters provides information on the accuracy of the identification technique.
In this communication, we will show the results of the sensitivity analysis performed on three model cases: a blocking electrode and the ferri/ferrocyanide system, silicon nitride thin layers. Confidence intervals (sensitivity) for the fitted parameters were estimated empirically from the change (of +/- 5%) in a single parameter value yielding an increase of the standard deviation q characterizing the fit quality, the other parameters staying fixed to their optimal value or unfixed in order to estimate interactions between parameters.
Basically, if a confidence interval is very wide, the data doesn’t define that parameter very well, or that the model proposed is not optimum. If a confidence interval is very narrow, this parameter becomes an important parameter in the overall model.
This approach is helpful in order to evaluate the robustness of the model and to validate the physical interpretation of model Parameters.

Keywords: electrochemical impedance spectroscopy, regression model, sensitivity, confidence interval

References

[1] Srisuwan, N., Ochoa, N., Pébère, N., Tribollet, B., Corrosion Science, Volume 50, Issue 5, May 2008, p 1245-1250
[2] NACE 2012 March 11 – Technology Exchange Group 097X, Electrochemical Measurements
[3] Introduction to Corrosion Science, E. McCafferty
[4] M.E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, The Electrochemical Society, Series, Ed. Wiley, 2008.
[5] B.A. Boukamp, Solid State Ionics 18–19 (1986) 136.
[6] J.R. Macdonald, L.D. Potter Jr., Solid State Ionics 24 (1987) 61.
[7] J.R. Macdonald, CNLS (Complex Nonlinear Least Squares) lmmittance Fitting, Program LEVM Manual, Houston, TX, Version 7.11 Edition, 1999.

Figure 1