The Constant Phase Element

When a charge transfer proceeds at the electrode, the equivalent circuit consists of Ca and the charge transfer resistance R in parallel. Therefore, the corresponding Nyquist impedance plot represents a semicircular arc. Analogous to 

A full discussion of the distribution of relaxation times as the origin of constant-phase elements is available in the literature [5]. 

In another report [4], an error in the interpretation of the CPE is pointed out. On one hand, the double-layer capacity is replaced by the CPE, i. e. the CPE is a property of the double layer itself. On the other, the CPE is discussed as originating from surface inhomogeneities. 

For an ideally polarized electrode, the impedance consists of the double-layer capacity Cd and the solution resistancein series. In the impedance plane plot, a straight vertical line results intersecting the Z -axis at Z =. At solid electrodes, especially 

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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... 

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. 

The constant phase element (CPE) is a non-intuitive circuit element that was discovered in the course of investigations into responses from real systems. In general, a Nyquist plot (also called a Cole-Cole plot or complex impedance plane plot) should be a semicircle with the centre on the x-axis. However, the observed plot for some real systems was indeed the arc of a circle but with the centre located somewhere below the x-axis. Figure 4.1 shows the impedance spectra of a circuit of a resistor and a constant phase element connected in parallel. The centre of the semicircle is located at (l-n)x90° below the real axis. 

Research Solutions and Resources LLC (2009) The constant phase element (CPE). http //www.consultrsr.com/resources/eis/cpel.htm. Accessed 19 May 2009. 

C. H. Hsu and F. Mansfeld, "Technical Note Concerning the Conversion of the Constant Phase Element Parameter Yq into a Capacitance," Corrosion, 57 (2001) 747-748. 

EIS for a LiMn2O4 electrode at 20°C is shown in Figure 10.7, whereas the equivalent circuit used for fitting experimental impedance data is depicted in Figure 10.8 (Sinha and Munichandraiah, 2008). The Nyquist plot can be described in terms of two superimposed semicircles. Because the semicircles are depressed with their centers below the real axis, constant phase elements are introduced in this circuit. Here, R, and gj represent the resistance of the surface film on LiMn2O4 and the constant phase element (CPE) representing the film... 

Figure 23.4B shows the eorresponding equivalent circuit. The equivalent circuit consists of a parallel combination of the resistance (Ri) and a constant phase element CPE ) in series with the electrolyte resistance (/fg). In general, the constant phase element CPE ) is used to replace the capacitance and is defined by ... 

In fact, the impedance spectroscopy of the WO3 photoanodes under strong illumination (ca. 0.3 W/cm ) reveals that the photoelectrode is essentially dominated, in the complex plane, by a kinetic loop the single semicircle with a maximum in the 1-10 Hz interval (depending on the potential) can be reasonably attributed to the charge transfer across the semiconductor-electrolyte interface (Fig. 57). It can be well fitted by a single R QR ) equivalent circuit [78], where R is the ohmic contribution, Q is the constant phase element which takes into account... 

S. Skale, V. Dolecek, M. Slemnik, Substitution of the constant phase element by Warburg impedance for protective coatings, Corros. Sci. 49 (2007) 1045-1055. 

In addition to capacitors and resistors, equivalent circuit models include elements that do not have electrical analogs, i.e., as the Warburg (W) element and the constant phase element (CPE). These elements can explain the deviations from theoretical predictions of the models. The Warburg element is frequency-dependent, and its impedance may be represented by following equation ... 

In the nineteenth century, the Maxwell equations appeared, based upon electromagnetic theory and juxtaposed with light. In the twentieth century came the Debye tradition of regarding biomolecules as polar materials with exponential relaxation. However, experimental findings showed that most dielectrics do not show exponential relaxation, but fractional power law relaxation. Fractional power law was the universal property. Then came the Cole models based upon a new component, the constant phase element (CPE). 

The problem when trying to make an electrical model of the physical or chemical processes in tissue is often that it is not possible to mimic the electrical behavior with ordinary lumped, physically realisable components such as resistors (R), capacitors (C), inductors, semiconductor components, and batteries. Let us mention three examples 1) The constant phase element (CPE), not realizable with a finite number of ideal resistors and capacitors. 2) The double layer in the electrolyte in contact with a metal surface. Such a layer has capacitive properties, but perhaps with a capacitance that is voltage or frequency dependent. 3) Diffusion-controlled processes (see Section 2.4). Distributed components such as a CPE can be considered composed of an infinite number of lumped components, even if the mathematical expression for a CPE is simple. 

There are several other, more complicated elements available to describe the various processes that can occur in a photoelectrochemical cell, such as the Warburg element (to model diffusion), the Constant Phase Element (CPE, used to describe processes that have a distribution of time constants or activation energies), and transmission lines (to model porous electrodes [47]). Porous electrodes and CPE elements that represent nonideal capacitive elements are briefly discussed below. For more detailed information, the reader is referred to the literature [48, 49]. 

Nonideal Space Charge Capacitance The Constant Phase Element... 

Hsu, C.H., Mansfeld, F. Technical note concerning the conversion of the constant phase element parameter Yq into a capacitance. Corrosion 57, 747-748 (2001)... 

The constant phase element (CPE) has been used to describe both the doublelayer capacitance and the low-frequency pseudocapacitance as well as the diffusion impedance [22,24,30,33,59,71,101,139,140] ... 

However, in most materials, the ideal behavior that the Debye model depicts is rarely observed and some more sophisticated models are currently proposed for taking into account deviations from ideality. One of the widely used ingredients is the constant phase element behavior, explained in Chapter 10 in case study G6 Constant Phase Elanent which can be adapted to spatially reduced properties. 

Fig. 8 Equivalent circuit for an electroactive SAM. Ru is the uncompensated resistance, CPE is the constant phase element, Ra is the adsorption resistance and Ca is the adsorption capacitance.

The experimental smdy of the solid-solid interface is complicated by a further problem. It is often (perhaps usually) observed that, instead of a purely capacitative behavior, the interface shows significant frequency dispersion. Several authors have found excellent agreement of this behavior with the dispersion shown by the constant-phase element (Bottelberghs and Broers [1976], Raistrick et al. [1977]). Although the amount of frequency dispersion is influenced by electrode roughness and other aspects of the quality of the interface (i.e. nonuniform current distribu-... 

The Constant Phase Element and Its Simple Combinations. Although Warburg and open-ended diffusion effects frequently appear in supported situations and sometimes in unsupported ones and exhibit characteristic 6 = 45° lines in the Z or plane, one often finds approximate straight-line behavior over a limited frequency range with 0 45° (e.g. McCann and Badwal [1982]). Then the frequency response of Z and Z is no longer proportional to but to some other power of (0. To describe such response it is convenient to write, as in Eq. (7) in Section 1.3, at the admittance level,... 

For supercapacitor development, special ECs have been proposed to fit the experimental results. For example. Figure 7.13 shows three proposed ECs cited in the literature. The first (a) is similar to that in Figure 7.10a, except that the parallel leakage resistance is connected in parallel to the pseudocapacitance Cp rather than to - Cp [10]. The second EC (b) and the third (c) are transmission line models to take care of the porous electrode layer [8,10,11]. Note that the third EC model uses the constant phase elements (Q,) rather than pure capacitances that mainly deal with the inclined Nyquist line at the low frequency range. In Figure 7.13b and c, the magnitudes of R, Q, and can be the same or different, depending on the real situation. The constant phase element (Qj) can be expressed as... 

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