Impedance data modeling Physicochemical models

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 ... 

These models provided a quantitative relationship between physicochemical parameters and impedance response, but the application of interpretation strategies did not keep pace with the model development. Interpretation was based on graphical examination of plotted data. In simple cases, plots could be used directly as described in Chapter 18. For more complicated cases, simulations could be compared graphically to data to reveal qualitative agreement. 

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. 

However, taking the impedance at each potential produces series of data values at different frequencies. Examples of complex plane impedance plots that is imaginary versus real part at various frequencies for different fuel cells are presented in Fig. 1.2. The polarization resistance is the only point corresponding to zero frequency, as indicated in the plots. One may observe that the impedance plots, besides R, produce much more information that is not available in steady-state measurements. Impedance plots display complex curves that are rich in information. Such information is contained in every point, not only in one value of R. However, one must know how to find this information on the system being studied. This is a more complex problem and can be solved by the proper physicochemical modeling. 

The experimental data that were checked by the Kramers-Kronig transforms may be used in modeling. First, usually, fit to an electrical equivalent model is carried out. It is important to use a proper weighting procedure and start with the simplest model. Then additional parameters can be added and their importance verified by the appropriate F- and r-tests. The number of adjustable parameters must be kept to a minimum. Additionally, comparison of the experimental and model impedances on complex plane and Bode plots should be carried out. Furthermore, plots of the residuals indicate the correctness of the model used. Next, on the basis of this fit, a physicochemical model might be constructed. One should check how the obtained parameters depend on the potential, concentration, gas pressiue, hydrodynamic conditions, etc. If a strange or unusual dependence is obtained, one should check whether the assumed model is physically correct in the studied case. This is the most difficult part of modeling. 

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