Impedance data modeling

In all real systems, some deviation from ideal behavior can be observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or if one or more of the dielectric materials in the system is inhomogeneous, many of these microscopic current filaments would be different. In a response to a small-amplitude excitation signal this would lead to frequency-dependent effects, which can often be modeled with simple distributed circuit elements. For example, many capacitors in EIS experiments, most prominently the double-layer capacitor often do not behave ideally due to the distribution of currents and electroactive species. Instead, these capacitors often act like a constant phase element (CPE), an element that has found widespread use in impedance data modeling. 

Hitzig et al. have produced a simplified model of the aluminium oxide layer(s) to explain impedance data of specimens prepared under different layer formation and sealing conditionsThe model also gives consideration to the formation of active and passive pits in the oxide layer. Shaw et al. have shown that it is possible to electrochemically incorporate molybdenum into the passive film which, as previously noted, improves the pitting resistance. 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

The technique of constructing an equivalent circuit for impedance analysis represents the exception to the general rule that a chosen model can be almost certain to be correct. It is all too easy to compile an equivalent circuit which fits the impedance data, but is altogether wrong. In fact, many practitioners would say that impedance studies are so susceptible to this fitting to a bogus model that another technique should always be applied as a form of validation . It is much more unlikely for two techniques to fit a particular model, and the latter still be wrong ... 

Inconsistency of performance with a bulk path at low vacancy concentration. A quantitative comparison between predictions of the Adler model and impedance data for LSC shows the poorest agreement (underprediction of performance) at low temperatures, high F02. and/or low Sr content. These are the conditions under which the bulk vacancy concentration (and thus also the ionic conductivity and surface exchange rate of oxygen with the bulk) are the lowest. These are exactly the conditions under which we would expect a parallel surface path (if it existed) to manifest itself, raising performance above that predicted for the bulk path alone. Indeed, as discussed more fully in section 5, the Adler model breaks down completely for LSM (a poor ionic conductor at open-circuit conditions), predicting an... 

AC impedance measurements were also made in bulk paints. A Model 1174 Solartron Frequency Response Analyser (FRA) with a Thompson potentiostat developed ac impedance data between 10 KHz and 0.1 Hz at the controlled corrosion potential The circuit has been described in the literature( ). 

Partial blocking effect was first identified for pure iron in contact with aerated sulphuric acid medium [55]. Corrosion of carbon steel in sodium chloride media clearly showed the porous layer effect (see Section 5.2) [74]. The same effect was found for zinc corrosion in sodium sulphate [75] and the properties of the layer which was demonstrated to be formed of an oxide/hydroxide mixture were further used for building a general kinetic model of anodic dissolution [76], usable for measurement of the corrosion rate from impedance data. 

An alternative approach between the basic RSC series circuit and the De Levie model (TLM) is to consider a porous electrode as a whole capacitance by simply using the impedance data [20] ... 

Two types of electrical analogy model for the interpretation of impedance data can be used based on combinations of resistances and capacitances, or based on transmission lines. These possibilities are now described. 

Figure 40 Equivalent circuit models for analyzing impedance data from degraded polymer coated metals, (a) General model, (b) Model I for coatings with defects corroding under activation control, (c) Model II for coatings with defects corroding under diffusion control. (From F. Mansfeld, M. W. Kendig, S. Tsai. Corrosion 38, 478 (1982) and M. Kendig, J. Scully. Corrosion 46, 22 (1990).)...

The interpretation of measured data for Z(oi) is carried out by their comparison with predictions of a theoretical model based either on the (analytical or numerical) integration of coupled charge-transport equations in bulk phases, relations for the interfacial charging and the charge transfer across interfaces, balance equations, etc. Another way of interpretation is to use an -> equivalent circuit, whose choice is mostly heuristic. Then, its parameters are determined from the best fitting of theoretically calculated impedance plots to experimental ones and the results of this analysis are accepted if the deviation is sufficiently small. This analysis is performed for each set of impedance data, Z(co), measured for different values of external parameters of the system bias potentials, bulk concentrations, temperature... The equivalent circuit is considered as appropriate for this system if the parameters of the elements of the circuit show the expected dependencies on the external parameters. 

However, although powerful numerical analysis software, e.g., Zview, is available to fit the spectra and give the best values for equivalent circuit parameters, analysis of the impedance data can still be troublesome, because specialized electrochemical processes such as Warburg diffusion or adsorption also contribute to the impedance, further complicating the situation. To set up a suitable model, one requires a basic knowledge of the cell being studied and a fundamental understanding of the behaviour of cell elements. 

Ahn et al. have developed fibre-based composite electrode structures suitable for oxygen reduction in fuel cell cathodes (containing high electrochemically active surface areas and high void volumes) [22], The impedance data obtained at -450 mV (vs. SCE), in the linear region of the polarization curves, are shown in Figure 6.22. Ohmic, kinetic, and mass transfer resistances were determined by fitting the impedance spectra with an appropriate equivalent circuit model. 

There were significant difficulties associated with fitting models to impedance data. The electrochemical systems frequently did not conform to the assumptions made in the models, especially those associated with electrode uniformity. Constant-phase elements (CPEs), described in Chapter 13, were introduced as a convenient general circuit element that was said to account for distributions of time constants. The meaning of the CPE for specific systems was often disputed. 

Example 3.5 Evaluation of Chi-Squared Statistics Consider that, for a given measurement, regression of a model to real and imaginan/ parts of impedance data yielded = 130. Measurements were conducted at 70 frequencies. The regressed parameters needed to model the data included the solution resistance and 9 Voigt elements, resulting in use of 19 parameters. Under assumption that the variances used in the evaluation ofxf were obtained independently, evaluate the hypothesis that the x value cannot be reduced by refinement of the model. 

There are, therefore, two primary concerns with the use of the CPE for modeling impedance data ... 

Graphical methods provide a first step toward interpretation and evaluation of impedance data. An outline of graphical methods is presented in Chapter 16 for simple reactive and blocking circuits. The same concepts are applied here for systems that are more typical of practical applications. The graphical techniques presented in this chapter do not depend on any specific model. The approaches, therefore, can provide a qualitative interpretation. Surprisingly, even in the absence of specific models, values of such physically meaningful parameters as the double-layer capacitance can be obtained from high- or low-frequency asymptotes. 

Figure 17.11 Traditional representation of impedance data obtained for the AZ91 alloy at the corrosion potential after different immersion times in 0.1 M NaCI a) complex-impedance-plane or Nyquist representation (the lines represent the measurement model fit to the complex data sets) b) Bode representation of the magnitude of the impedance as a function of frequency and c) Bode representation of the phase angle as a function of frequency. (Taken from Orazem et al. ° and reproduced with permission of The Electrochemical Society.)...

The graphical representations presented here are intended to enhance analysis and to provide guidance for the development of appropriate physical models. While visual inspection of data alone cannot provide all the nuance and detail that can, in principle, be extracted from impedance data, the graphical methods described in this chapter can provide both qualitative and quantitative evaluation of impedance data. 

Under certain limiting conditions, graphical methods for analysis of impedance data can be beised on the physics of the system under study. Use of such methods does not provide the detailed information that may be available from use of regression techniques, presented in Chapter 19. The graphical methods may, however, complement the development of detailed process models by identifying frequency ranges in which the process model must be improved. 

Equations (19.24) to (19.29) can be applied for complex noiUinear least-squares regression by concatenating the real and imaginary impedance data Z,- to form a data vector with length equal to twice the number of measured frequencies. A similar concatenation applies for the model values Z(Ti P). Press et al. provide a very approachable discussion of the least-squares methods and their implementation. 

Selection of an appropriate initial value for the time constant is critical for regression of the Voigt model (see equation (20.5)) to impedance data. Inductive loops can be modeled by the Voigt model by allowing the resistance values to be... 

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic... 

The quality of a regression can also be assessed by visual inspection of plots. Of course, some plots are more sensitive than others to the level of agreement between model and experiment. As will be demonstrated in this chapter, the plot types can be categorized as given in Table 20.1. The comparison of plot types is presented in the subsequent sections for regression of models to a specific impedance data set. 

Poor Sensitivity Modulus The Bode magnitude representation is singularly incapable of distinguishing between impedance models unless they provide extremely poor fits to impedance data. 

For the purpose of demonstrating the means of assessing regression quality, three models were applied for the aiialysis of the impedance data ... 

The equivalent circuit presented in Figure 20.1 was regressed to the impedance data. The mathematical formulation for the model is given as... 

Table 20.2 Estimated /v values for the regression of the model presented in Figure 20.1 to impedance data obtained for reduction of ferricyanide on a Pt rotating disk electrode.

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