Generalized Equivalent Circuits

In the framework of linear irreversible thermodynamics it is possible to handle responses to electrical and/or chemical driving forces, as could be seen from the previous sections, by what is called Nernst-Planck- 

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FIGURE 1.5. a Three- electrode electrochemical cell, b General equivalent circuit, c equivalent circuit of the cell + potentiostat and current measurer (the symbols are defined in the text). 

Figure 2. Assumed generalized equivalent circuit of the semiconductor—electrolyte interface. Reduced equivalent circuit at high frequencies and the expression for the impedance at low and high frequencies.

Consider the situation under potentiostatic conditions. Here, the potential control takes care that the sum of the potential drop across the double layer, DL, and through the electrolyte up to the position of the RE (and possibly additional external series resistances) is constant, i.e. that U = DL + I Rn or / = (U - DL)/Rn. Rn is the sum of the uncompensated cell resistance and possible external resistances and I the total current through the cell. Hence, a perturbation of a state on the NDR branch towards larger values of Dl causes, on the one hand, a decrease of the faradaic current If, and, on the other hand, a decrease of the current through the electrolyte, I. The charge balance through the cell, which can be readily obtained from the general equivalent circuit of an electrochemical cell (Fig. 8), tells us whether the fluctuation is enhanced or decays ... 

Fig. 8. General equivalent circuit of an electrochemical cell. C double layer capacitance qSDL potential drop across the double layer Zp faradaic impedance Rij series resistance (comprising the uncompensated ohmic cell resistance and all external resistances). V is a potentiostatically fixed voltage drop. (It differs from the potentiostatically applied voltage by the constant potential drop across the RE see footnote 3).

Equivalent circuit analysis is well suited for analysis of EIS measurements of conversion coatings and is the primary method for interpreting EIS spectra from conversion coated metal surfaces. A widely accepted generalized equivalent circuit model for the EIS response of pitted conversion coatings is shown in Fig. 22a (66,67). Several related models discussed below are also shown. In the gener-... 

Using the generalized equivalent circuit model for conversion coated surfaces shown in Fig. 22, it is possible to track the time-dependent changes in the resistances and capacitances of the intact coating and evolving pits. Figure 25 shows representative Bode plots for CeCl3-passivated and bare A1 7075-T6 immersed in 0.5 M NaCl solution (81). Spectra like these were collected over 35... 

Figure 25 (a) Bode plot for CeCl3-treated 7075-T6 during immersion in 0.5 M NaCl solution, (b) Results of the equivalent circuit modeling of the EIS data from Fig. 25 using the generalized equivalent circuit model in Figure 22a. (From F. Mansfeld, S. Lin. S. Kim. H. Shih. Corrosion 45, 615 (1989).)... 

To separate the contribution of the anode from that of the cathode, Wagner et al. provided a more general equivalent circuit, as shown in Figure 6.33. The anode, the cathode, and the membrane resistance were in series. A parasitic inductance due to the mutual induction effect was also included. The cathode impedance was given by charge-transfer resistance (Rct0i) and constant phase element (CPE) in... 

The three experiments do not only introduce decisive mass and charge transport parameters, they also permit their determination. Some points relevant in this context will be investigated in the following. (Note that electrochemical measurement techniques are covered by Part II.1) At the end of this section we will have seen that—close to equilibrium—not only all the D s and the k s can be expressed as the inverse of a product of generalized resistances and capacitances, but that these elements can be implemented into a generalized equivalent circuit with the help of which one can study the response of a material on electrical and/or chemical driving forces. 

Fig. IIG General equivalent circuit with an unspecified element Z in the faradaic resistance branch, showing that the double layer capacitance can be obtained by extrapolation to sufficiently high frequency.

Figure 2.2.10. General equivalent circuit showing hierarchical structure and involving general distributed elements.

FIGURE 1.72. (a) General equivalent circuit representing an electroactive polymer film exhibiting redox conduction. Each individual circuit element corresponds to a particular kinetic/transport/structural property of the polymer, (b) Approximate equivalent circuits and associated Nyquist plots for various simplified limiting situations encountered in practice. Approximation (i) is valid when / ct is very low, / ps = = 0, the redox site... 

The general equivalent circuit incorporating RC approximations to the general impedance elements x, y, and z is illustrated in Fig. 1.85. The... 

FIGURE 6.20 Example of a generalized equivalent circuit model for a single-cell fuel cell. Models regarding real fuel cells often contain constant phase elements (CPE). (Adapted from Scribner Associates, 2010. http //www. scribner.com/technical-papers.html)... 

The electrochemical impedance Z (j co) obtained in this way can be described by the general equivalent circuit shown in Fig. 7-3. It consists of the double layer capacity Qi in parallel to the Faraday impedance Zp, which contains the most interesting information on the electrochemical reactions taking place at the electrode surface. The Faraday impedance Zp in general is a nonohmic complex function of the frequency co. In Fig. 7-3 the ohmic resistance Rq in series to Z and dl corresponds to the resistance of the electrolyte solution between the tip of the reference electrode RE and the working electrode WE. 

Figures. The generalized equivalent circuit of a single interface. Rpisthe resistance associated with the Faradaic current flow, is a generalized impedance associated with disorder either in the structure or in the dynamics (diffusion), C, and R, are associated with parallel charge accumulation modes with different relaxation times than the majority carriers such as surface states or minority carriers, is the space charge...

Recently, Fletcher et al. has proposed for carbon-based supercapacitors, a general equivalent circuit through the comprehensive and reasonably dependable explanation through open circuit voltage decay, capacitance loss at high frequency, and voltammetric distortions at high scan rate [94]. Table 4 shows the few selected conducting polymer nanocomposite supercapacitor performances. 

The typical IL system could be considered as a solvent-free system, in which it can simplify the EIS analysis significantly which spurs its wide use in the characterization of the IL-electrode interface. However, due to low mobility of ions in an IL and multiple molecular interactions present in an IL, more time is needed to reach to a steady state of IL-electrode interface structure and arrangement, when a potential is applied. Furthermore, the electron-transfer process in ILs is different from that in traditional solvents containing electrolytes. Thus, the interfacial structures of IL are more complex than other systems. Even the electrode geometry could affect the EIS results of IL systems. It is noted that the bulk ILs could not be simply described by a resistor (R ) as in classic electrochemical systems. And the electrode double layer in IL electrolyte couldn t be simply depicted as a capacitor. So the Randle equivalent circuit is not sufficient to describe an IL system. Significant efforts have been made to illustrate the properties of diffusion layer and the bulk ILs with equivalent circuits. However, currently there is no general equivalent circuit model to describe the interface of an IL system. 

FIGURE 9-2 General equivalent circuit and a typical complex impedance plot representing the electrode/electroactivefilm/solution system... 

In this context the construction of generalized equivalent circuits which include chemical and electrical effects is helpful [473]. 

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