Impedance data modeling Equivalent circuit models

When we begin to investigate an electrochemical system, we normally know little about the processes or mechanisms within the system. Electrochemical impedance spectroscopy (EIS) can be a powerful approach to help us establish a hypothesis using equivalent circuit models. A data-fitted equivalent circuit model will suggest valuable chemical processes or mechanisms for the electrochemical system being studied. From Chapter 1, we know that a fuel cell is actually an electrochemical system involving electrode/electrolyte interfaces, electrode reactions, as well as mass transfer processes. Therefore, EIS can also be a powerful tool to diagnose fuel cell properties and performance. 

Passive oscillator mode Impedance analysis of the forced oscillation of the quartz plate provides valuable information about the coating even if the active mode is not applicable anymore. For impedance analysis, a frequency generator is used to excite the crystal to a constraint vibration near resonance while monitoring the complex electrical impedance and admittance, respectively, dependent on the applied frequency (Figure 2B). For low load situations near resonance, an equivalent circuit with lumped elements - the so-called Butterworth—van-Dyke (BVD) circuit — can be applied to model the impedance data. The BVD circuit combines a parallel and series (motional branch) resonance circuit. The motional branch consists of an inductance Lq, a capacitance Cq, and a resistance Rq. An additional parallel capacitance Co arises primarily from the presence of the dielectric quartz material between the two surface electrodes (parallel plate capacitor) also containing parasitic contributions of the wiring and the crystal holder (Figure 2B). 

The dc conductivities can be extracted from Nyquist plots of the complex impedance by fitting parameters of a model equivalent circuit to the data. The equivalent circuit always consists of a parallel connection of an Ohmic resistance and a constant phase element [40]. The same dc values are also obtained by identifying the conductivity values of the low-frequency plateau with the dc conductivity. This extraction of dc conductivities from the spectra of dried PEC is straightforward because electrode polarization effects are almost absent. 

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 ... 

EIS data is generally interpreted based on defining an appropriate equivalent circuit model that best fits the acquired data. The elements of the circuit model involve a specific arrangement of resistors, capacitors, and inductors that tacitly represent the physicochemical reality of the device under test. Under these circumstances the numerical value for chemical properties of the system can be extracted by fitting the data to the equivalent circuit model. Impedance measurements are typically described by one of two models ... 

Most often, the electrochemical impedance spectroscopy (EIS) measurements are undertaken with a potentiostat, which maintains the electrode at a precisely constant bias potential. A sinusoidal perturbation of 10 mV in a frequency range from 10 to 10 Hz is superimposed on the electrode, and the response is acquired by an impedance analyzer. In the case of semiconductor/electrolyte interfaces, the equivalent circuit fitting the experimental data is modeled as one and sometimes two loops involving a capacitance imaginary term in parallel with a purely ohmic resistance R. 

The last comment relates to the data analysis and the choice of appropriate models for impedance spectra. As shown by Orazem et al. [241], each single impedance spectrum can be fitted by a number of equivalent circuit analogs. Hence, the choice of a model has to be based on... 

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. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements 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. 

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

Refined models for mass transfer to a disk electrode are presented in Section 11.6. The equivalent circuit presented in Figure 20.12 was regressed to the impedance data. The mathematical formulation for the model is given as equation (17.1). Four models were considered for the convective-diffusion term Zd (/) ... 

While reaction parameters were not identified by regression to impedance data, the simulation presented by Roy et al. demonstrates that side reactions proposed in the literature can account for low-frequency inductive loops. Indeed, the results presented in Figures 23.4 and 23.5 show that both models can account for low-frequency inductive loops. Other models can also account for low-frequency inductive loops so long as they involve potential-dependent adsorbed intermediates. It is generally understood that equivalent circuit models are not unique and have therefore an ambiguous relationship to physical properties of the electrochemical cell. As shown by Roy et al., even models based on physical and chemical processes are ambiguous. In the present case, the ambiguity arises from uncertainty as to which reactions are responsible for the low-frequency inductive features. 

FRA systems are versatile, and they can be controlled to acquire and analyse the data required to construct Mott-Schottky plots, for example. Unfortunately, the ease of use of FRA-fitting software can lead to errors of interpretation that arise from a failure to relate fitting elements to the physical system. Several equivalent circuits may give the same frequency-dependent impedance response. No a priori distinction between degenerate circuits is possible, ft is necessary to study the system response as a function of additional experimental variables (DC voltage, concentration, mass transport conditions etc.) in order to establish whether the circuit elements are related in a predictable way to a model of the physical system. 

Usually an equivalent circuit is chosen and the fit to the experimental data is performed using the complex nonlinear least-squares technique. However, the model deduced from the reaction mechanism may have too many adjustable parameters, while the experimental impedance spectrum is simple. For example, a system with one adsorbed species (Section IV.2) may produce two semicircles in the complex plane plots, but experimentally, often only one semicircle is identified. In such a case, approximation to a full model introduces too many free parameters and a simpler model containing one time-constant should be used. Therefore, first the number and nature of parameters should be determined and then the process model should be constructed in consistency with the parameters found and the physicochemical properties of the process. 

We consider first the results on p-GaP. The impedance data for p-GaP has been a fruitful source of controversy, though not of comprehension. If a sample of p-GaP is held at a negative potential for a considerable period and then slowly ramped towards positive potentials, the a.c. impedance data cannot be analysed within the framework of the two-component model. Attempts to do so lead to Mott-Schottky plots whose slopes and intercepts are both frequency-dependent as shown in Fig. 25. If the data are analysed according to the more complex five-component equivalent circuit shown above, then a much better fit is obtained for the potential region more than about 0.6 V negative of the predicted flat-band potential. In this region, the Mott-Schottky plot is linear with a slope that corresponds reasonably well... 

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