Constant phase elements

The impedance of ideally polarizable liquid electrodes (e.g., mercury, amalgams, indium-gallium) may be modeled by an R-C circuit (Fig. 4.1a). However, most impedance studies are now carried out at solid electrodes. At these electrodes the double-layer capacitance is not purely capacitive and often displays a certain frequency dispersion. Such behavior cannot be modeled by a simple circuit consisting of R, L, and C elements. To explain such behavior, a constant phase element (CPE) is usually used. 

In general, bulk dielectric dispersion in solids and liquids is well known and described in the literature [24, 297, 298]. In this chapter, the dispersion of capacitances at electrode surfaces in solutions will be discussed. The complex dielectric constant is described as 

Cole and Cole [297] described the observed dispersion by the function 

In such a case, the impedance of the ideal capacitor, HijcoC), must be replaced by the impedance of a CPE  

To more easily identify the presence of the CPE, Orazem et al. [303] proposed using Bode plots of impedances corrected for the solution resistance, loglZ versus log/, and effective capacitance plots Teff versus log/  

The two-component equivalent circuit models presented in the last section are of course too simple to mimic the admittance found with real biomaterials at all 

Let us calculate the characteristic properties of a general CPE as a conductor and a susceptor in parallel, both frequency dependent. The frequency dependence of the admittance Y = G + jB is sought so that die phase angle ((pq,e = arctan B/G) becomes frequency independent  

From Eq. 9.17, it is clear that to keep the phase angle constant, both G and B must be dependent on the frequency in the same way  

Here the introduction of a parameter x with the dimension of time and forming the product 

The introduction of x has the interesting consequence that it changes the dimension of Gi and Bi to also be that of an ideal conductance siemens. 

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Recently, a constant-phase element has been found607 to be present at pc-Pb/KF + HaO interfaces by impedance measurements. The Pb electrode was cathodically reduced before use. The assumption has been made that the CPE is due to the inhomogeneity of the metal surface. Frequency-... 

Surface roughness is also expected to result in depression of the capacitance semi-circle. This phenomenon, which is indeed apparent in both Figures 1 and 2, is, however, unrelated to surface area. Rather, it is attributable to surface heterogeneity, i.e. the surface is characterized by a distribution of properties. Macdonald (16) recently reviewed techniques for representing distributed processes. A transmission line model containing an array of parallel R/C units with a distribution of values is physically attractive, but not practical. An alternative solution is introduction of an element which by its very nature is distributed. The Constant Phase Element (CPE) meets such a requirement. It has the form P = Y0 wn... 

We found an equivalent electrical circuit that fits best the LixC6 electrode behavior at high frequency. The circuit consists of a resistor R in parallel with a constant phase element (CPE). The latter is defined with a pseudo-capacitance Q and a parameter a with 0< a <1 [6], The impedance of... 

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows... 

Sweep rate (mV s 1) Electrolyte Resistance (Q cm2) Surface/species Parameter (106fi Van 2) Constant.phase/ element exponent Oxide filmResist ance (k 2 cm2) Ox. film thickness (nm)... 

Andrade and Molina [46] have performed electrochemical impedance studies of mercury electrodes with hematite particles adhered at different electrode potentials. Adhesion of such particles was strong and the decrease in the impedance was accompanied by an increase in the number of attached particles. Experimental results were analyzed in terms of an equivalent circuit including the constant phase element (CPE), the magnitude of which appeared to be directly related to the electrode coverage. A pore model for the metal/hematite particles interface has been proposed. 

This is the phenomenon of the so-called constant phase element (CPE) as it follows from eqn. (115) that Yc is composed of an imaginary and a real component, with a frequency-independent phase angle cor/2. Though the phenomenon is most clearly discovered in impedance or admittance analyses, its effect in time-domain methods should not be ignored. 

Constant-phase elements were first used to explain dielectrical properties of polar liquids and solids, and were attributed to the presence of the investigated material properties as a partitioning between extreme conditions, rather than as constant or uniform parameters40. Furthermore in the... 

In this book, an explanation of capacitive behaviour in similar and comparable systems is not directly possible with constant-phase elements because such a comparison is only possible if n values are equal, particularly in the study of surfaces covered with polymer coatings where a unification of the envisaged parameters is necessary. The impedances measured match with a relatively large amount of samples, of which the structure can be complex, showing many sources of non-idealities (e.g. variations in thickness of the membrane, pore size and pore density42 7). A good indication if such non-idealities occur can be found in the values of n. If they are not comparable, non-idealities occur. 

A solution to this problem is the use of electrical equivalent circuits without constant-phase elements (note that a good numerical simulation of the experiments can be obtained only by inserting constant-phase elements) only pure capacities are used. This method, although not convincing, results in comparable capacities. 

Another possibility is based on the fact that constant-phase elements represent processes that are not purely conservative (charging and discharging of a capacitor is a conservative process), first because of the fact that n is not equal to 1. From such an element, one can isolate the conservative contribution, which can then be further treated as a conservative process and simulated by a pure capacitor. The starting point for this separation is the existence of an angle frequency, to, for which48 ... 

Finally, it can be seen from Fig. 9.9a that the real impedance does not remain constant at low frequencies for the textile electrode, and this effect is more pronounced at higher electrolyte concentrations. Probably, Zr is influenced by other effects only occurring in the low-frequency range. This effect is frequently observed and described in the literature and is caused by non-uniformity of surfaces at the micro-scale, which in fact is the case for the textile electrodes. It is also not possible to explain this effect by a pure resistor or a pure capacitor in the electrical equivalent circuit. For this purpose, constant-phase elements are implemented as described in the theoretical discussion of electrochemical impedance spectroscopy (presented in Chapter 2, section 2.4). 

The electrochemical impedance of a real electrode is frequently represented by an equivalent circuit containing constant phase element (CPE) showing power-law frequency dependence as follows... 

The first resistance Rs is the resistance of the electrolyte outside the pores the R, elements are the electrolyte resistances inside the pores of the electrode and are the double layer capacitances along the pores. This model is called the Transmission Line Model (TLM) by De Levie. A careful selection of a set of Rv C values allows to calculate back the experimental plot such as the one presented in Figure 1.23 [28]. It can be noted that constant phase element (CPE) can be used to replace the capacitance C for better fitting, the CPE impedance ZCPE being ZCPE = l//(Cco) . 

Fig. 10. Equivalent circuits of electrode (a) general circuit of a thin-film electrode (b) the Randles circuit and (c) circuit with a constant phase element.

The characteristic frequency dependence shown in Fig. 12b implies the presence of a so-called constant phase element (CPE) in the electrode s equivalent circuit. The CPE impedance equals to [65]... 

Rugosity and porosity give rise to the so-called constant phase element (CPE), which can be described by groups of parallel or branched transmission lines. The CPE is manifested in real systems by an impedance spectrum altered from the expected shape, especially in the... 

For coatings with no pitting, the generalized model must be amended to account for that fact that all current must flow through the barrier coating. The coating resistivity, Rt, is on the order of 100 to 1000 MQ cm2 and behaves essentially as an open circuit under near-DC conditions (f = 0). The EIS response over the typically measured frequency domain is that of a constant phase element (CPE) in series with a solution resistance (Fig. 22b). 

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

Constant phase element — Figure. Theoretical impedance of a parallel connection of a CPE and a resistor, for < f = 0.5... 

Immittance — In alternating current (AC) measurements, the term immittance denotes the electric -> impedance and/or the electric admittance of any network of passive and active elements such as the resistors, capacitors, inductors, constant phase elements, transistors, etc. In electrochemical impedance spectroscopy, which utilizes equivalent electrical circuits to simulate the frequency dependence of a given elec-trodic process or electrical double-layer charging, the immittance analysis is applied. 

Electrochemical reactions consist of electron transfer at the electrode surface. These reactions mainly involve electrolyte resistance, adsorption of electroactive species, charge transfer at the electrode surface, and mass transfer from the bulk solution to the electrode surface. Each process can be considered as an electric component or a simple electric circuit. The whole reaction process can be represented by an electric circuit composed of resistance, capacitors, or constant phase elements combined in parallel or in series. The most popular electric circuit for a simple electrochemical reaction is the Randles-Ershler electric equivalent... 

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. 

As discussed in Chapter 3, the electrolyte/interface and associated electrochemical processes can be treated as an electric circuit consisting of electrical elements, including resistance, capacitors, constant phase elements, and so on. Although the commonly used electrical elements have already been described in Chapters 2 and 3, the following section provides a brief review to preface the ensuing discussion of EIS equivalent circuits and their related PEM fuel cell processes. 

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