Methods for Representing Impedance

Impedance data are presented in different formats to emphasize specific classes of behavior. The impedance format emphasizes the values at low frequency, which t5rpically are of greatest importance for electrochemical systems that are influenced by mass transfer and reaction kinetics. The admittance format, which emphasizes the capacitive behavior at high frequencies, is often employed for solid-state systems. The complex capacity format is used for dielectric systems in which the capacity is often the feature of greatest interest. 

The methods for representing impedance data are illustrated here for the two simple RC electrical circuits shown in Table 16.1. A review of Chapters 1 and 4 may be useful. The summary of relationships eunong complex impedance, real and imaginary parts of the impedance, and the phase eingle and magnitude fotmd in Tables 1.1,1.2, and 1.3 may also be useful. 

Complex impedance Zr 2 Time constant Re 1 none wCR 1+iujRC) RC 

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The effect of adsorption of electroactive species in ac methods is taken into account by modification of the equivalent circuit representing the electrode reaction (44, 64-68). This is usually accomplished by adding an adsorption impedance in parallel with the Warburg impedance and double-layer capacitance. Expressions for this impedance have been suggested for reversible (65, 66) and irreversible (67, 68) systems, but the complexity of the resulting analysis has limited the application of these techniques. 

The film grown on an electrode surface has a duplex structure with a thin, compact first layer that is directly on the electrode surface and a porous second layer contacting the electrolyte. An equivalent circuit can be used to represent the electrical properties of this film. The components of an equivalent circuit can be determined by impedance spectroscopy. Therefore, this method has become one of the key methods for the characterization of conducting polymers. 

Electrochemical methods are well adapted for characterizing the corrosion behavior of coated metals in solution. Because of the high resistance of organic coatings, ac methods are generally more suited than dc polarization methods. In electrochemical impedance spectroscopy (EIC) one measures the response of the coated electrode to a small amplitude ac perturbation as a function of frequency (Chapter 5). The interpretation of the measured frequency response, in principle, requires a physical model. However, for coated metals useful information is more easily obtained by representing the metal-coating-electrolyte interface by an electrical circuit (equivalent circuit). 

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Three common methods [cyclic voltammetry (CV), charging-discharging curve (CDC), and electrochemical impedance spectroscopy (EIS)] are briefly introduced. For fast screening of electrode materials, the conventional ex situ three-electrode cell is the choice test method. For in situ characterization of materials and supercapacitor performance, the two-electrode test cell can more closely represent the real conditions encountered during operation. 

The electrode layers formed using die physical loading method are usually relatively thicker (more than 10 pm in thickness), and the composite layers are composed of nanoparticles of the electrode material and the ionic polymer. These layers are both electronically and ionically conductive. The impedance for such electrodes is assumed to be similar to diat of porous electrodes. Levie (1963, 1964) was the first to develop a transmission line circuit (TLC) model of the porous electrode consisting of the electrolyte resistance and the double-layer capacitance. Subsequently, a number of authors proposed modified TLC models for the impedance of porous electrodes on the basis of Levie s model. Bisquert (2000) reviewed the various impedance models for porous electrodes. The composite electrode layers prepared by the physical loading method could be successfully represented by the impedance model for porous electrodes, as shown in Fig. 6d this model is composed of the double-layer capacitance, Cj, the Warburg diffusion capacitance, W and the electrolyte resistance, 7 (Liu et al. 2012 Cha and Porfiri 2013). 

The complications and sources of error associated with the polarization resistance method are more readily explained and understood after introducing electrical equivalent circuit parameters to represent and simulate the corroding electrochemical interface (1,16-20). The impedance method is a straightforward approach for analyzing such a circuit. The electrochemical impedance method is conducted in the frequency domain. However, insight is provided into complications with time domain methods given the duality of frequency and time domain phenomena. The simplest form of such a model is shown in Fig. 3a. The three parameters (Rp, Rs, and C d,) that approximate a corroding electrochemical inter-... 

Figure 6.21 shows the AC impedance spectra for the cathodic ORR of the cell electrodes prepared using the conventional method and the sputtering method. It can be seen that the spectra of electrodes 2 and 3 do not indicate mass transport limitation at either potentials. For electrode 1, a low-frequency arc develops, due to polarization caused by water transport in the membrane. It is also observable that the high-frequency arc for the porous electrode is significantly depressed from the typical semicircular shape. Nevertheless, the real-axis component of the arc roughly represents the effective charge-transfer resistance, which is a function of both the real surface area of the electrode and the surface concentrations of the species involved in the electrode reaction. 

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