Equivalent circuit fitting

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. 

About 30 individual grain boundaries, i.e. 30 different electrode configurations were investigated by microcontact impedance spectroscopy. The resulting histograms of the resistance, capacitance and relaxation frequency obtained from an equivalent circuit fit of the low-frequency arc are shown in Fig. 38. For comparison, a conventional (macroscopic) impedance measurement was performed on an identically prepared sample. The relaxation frequency of the grain boundary semicircle is indicated in Fig. 38c by a solid line. 

Table 7.1 Electrical component values of the equivalent circuit fitted to the measured impedance data during the anodic deposition process at various temperatures. Data are obtained from Ref [64],...

While it is sometimes difficult to get quantitative information about overpotentials from individual processes out of the Bode plots, they do provide a very good opportunity to identify important processes that govern the electrochemical response. This is helpful especially in selecting an appropriate biophysicochemical model for the system being studied (discussed in Section 8.4) that can be applied to such methods of data analysis as the equivalent circuits fitting described later. In addition, the Bode plot helps to understand the fundamental physical response of the system with respect to the perturbation in amplitude as a function of frequency. 

The equivalent circuit fitting for these experimental data is a Randles circuit. The Randles equivalent circuit is one of the simplest and most common circuit models of electrochemical impedance. It includes a solution resistance, Rs in series to a parallel combination of resistor. Ret, representing the charge transfer (corrosion) resistance and a double layer capacitor, Cai, representing the electrode capacitance (Badawy et al, 1999). In this case, the value of Rs can be neglected because the value is too small as compared to that of the value of Ret. The equivalent circuit for the Randles cell is shown in Figure 2. ... 

Normally, the impedance plots are fitted to an often-complex equivalent circuit. Mathematically, this means searching for a global solution in R". However, problems arise if a complicated equivalent circuit is found which does not allow physical interpretation. Therefore, it is preferable to run a wide variety of experiments with different samples rather than trying to fit in detail the results of a single measurement in order to analyze the resulting impedance plots. 

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 ... 

Cases in which Impedance Spectroscopy Becomes Limited. One might say that if one understands an interface well, the results of Z-to measurements can be readily understood. Of course, the interest is in the other direction, in using Z-to plots when one does not understand the interlace. Then the task is to find an interfacial structure and mechanism (and its resulting equivalent circuit) that provides a Z that is consistent in its dependence on to with the experimental results of the impedance measurement. This requires finding reasonable parameters to fit the value of the C s and R s as a function of to for the individual elements in the various equivalent circuits. If the shape of the calculated Z-to plot can only be made to match experiment by using C s and R s that are physically unreasonable, the proposed structure and the equivalent circuit to match it are not acceptable and another must be tried. 

One of the techniques that can be used here is computer simulation. A computer can be programmed to find values of the parameters in the elements of the competing equivalent circuit that maximize the fit over a large frequency range. As already mentioned, the extension of the plots to very low frequencies is desirable to cover a range that may be very information-bearing. But what of the stability of the electrode surface after, say, 1 hr in the solution ... 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given frequency. 

Nyquist plots (imaginary vs. real impedance) were matched against a variety of equivalent circuits (Miller, 1992). The best fit was with equivalent circuit 7 in... 

Fig. 13. (a) Sketch of the microelectrode configuration used to investigate the distribution of grain boundary properties, (b) Typical impedance spectrum calculated for a model sample (inset) consisting of 24 cubic grains and two microelectrodes on adjacent grains. An equivalent circuit consisting of two serial RC-elements (inset) can be used to fit the spectrum. 

Fig. 42. (a) Impedance spectrum obtained on a 30 pm LSM microelectrode at a temperature of approx. 800 °C. The inset shows the equivalent circuit used to fit the data, (b) Electrode resistance Rti as a function of the microelectrode diameter at a temperature of approx. 800 °C. The solid line is a linear regression of the resistance data and shows the proportionality of Rd to the inverse of the square of the microelectrode diameter. 

Figure 28 A typical Nyquist plot obtained from a nickel electrode polarized to low potentials (0.2 V versus Li/Li+) in PC solutions (1 M LiBF4 in this case). The equivalent circuit analog of 4 R C circuits in series and their separate Nyquist plots (four semicircles) are also shown. The frame in the lower right represents a typical fitting between the experimental data and this equivalent circuit analog [34]. (With copyright from The Electrochemical Society Inc.)...

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 18, taken from Ref. 77, describes several models proposed for the Li electrodes in solutions, their equivalent circuit analogs, and the expected impedance spectra (presented as Nyquist plots). Assuming parallel plate geometry for the solid electrolyte interface, as well as knowledge of the surface species involved from spectroscopy (and thus their dielectric constant, which is around 5 for many surface species formed on Li, including R0C02Li, Li2C03, LiF, ROLi, etc. [186]), it is possible to estimate the surface film s thickness from the electrode s capacitance (calculated from the model fitted to the spectra) ...

Figure 20 Scheme of the multilayer model of the Li-solution interphase, the division of the various layers, and the corresponding equivalent circuit analog, which can be fitted very well to the experimental data [49]. (With copyrights from The American Chemical Society, 1998.)... 

Figure 19 Schematic Bode plots from EIS measurements and equivalent circuits that could be used to fit them for various possible corrosion product deposit structures (A) nonporous deposit (passive film) (B) deposit with minor narrow faults such as grain boundaries or minor fractures (C) deposit with discrete narrow pores (D) deposit with discrete pores wide enough to support a diffusive response (to the a.c. perturbation) within the deposit (E) deposit with partial pore blockage by a hydrated deposit (1) oxide capacitance (2) oxide resistance (3) bulk solution resistance (4) interfacial capacitance (5) polarization resistance (6) pore resistance (7) Warburg impedance (8) capacitance of a hydrated deposit.

For these reasons, EIS has been explored as an alternative proof test for evaluation of conversion coatings. In these tests, conversion coated surfaces are exposed to an aggressive electrolyte for some period of time during which coating damage will accumulate. An impedance spectrum is collected and evaluated using a suitable equivalent circuit model and complex nonlinear last-squares fitting. 

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. 

When investigating an electrochemical system using EIS, the equivalent circuit model that has been constructed must be verified. An effective way to do so is to alter a single cell component and see if the expected changes in the impedance spectrum occur, or to keep adding components to the circuit to see if a suitable circuit can be achieved, until reaching a perfect fit. Nevertheless, empirical models should use as few components as possible. 

It should also be pointed out that an equivalent circuit is not unique. In describing the same AC impedance spectrum, several circuits may exhibit the same result. For example, a model that includes elements without any chemical basis and practical meaning can demonstrate a perfect fit. Various equivalent circuit models used in PEM fuel cells will be discussed in detail in Chapter 4. 

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