Effective CPE Coefficient

The slopes at the high-frequency asymptotes appear to be in good agreement for the three data sets, but a more detailed analysis reveals some trending. Values for the CPE exponent are provided in Table 17.2. A small increase in the value of a is evident as the immersion time increases. 

The graphical representations presented here are intended to enhance analysis and to provide guidance for the development of appropriate physical models. While visual inspection of data alone cannot provide all the nuance and detail that can, in principle, be extracted from impedance data, the graphical methods described in this chapter can provide both qualitative and quantitative evaluation of impedance data. 

Remember 17.4 The graphical methods described in this chapter are general and can be applied to both reactive and blocking systems. 

As shown in Section 13.3, logarithmic plots of the imaginary part of the impedance can be used to show the high-frequency pseudo-CPE behavior caused by nonimiform current distribution. A graphical analysis similar to that demonstrated in Example 17.1 was employed by Huang et al. ° to obtain values for parameters Rg, a, and Q associated with the impedance of a glassy carbon disk in KCl solutions and an oxide-covered stainless steel electrode in 0.05 M NaCl + 0.005 M Na2S04 electrolyte. 

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An effective capacitance, or, when a 1, an effective CPE coefficient may be obtained directly from the imaginary part of the impedance as... 

Figure 17.6 Effective CPE coefficient defined by equation (17.7), scaled by the input value of the double-layer CPE coefficient, as a function of frequency with a as a parameter for the Randles circuit presented as Figure 17.1(a). (Taken from Orazem et al. and reproduced with permission of The Electrochemical Society.)...

The ratio of effective CPE coefficient, calculated using equation (17.7), to the expected value is given in Figure 17.10. The effective CPE coefficient is seen to be independent of frequency. 

The effective CPE coefficient representation in Figures 17.6 and 17.13 yields, for a = 1, information concerning the high-frequency capacitance of the system. In the case that a < 1, Figures 17.6 and 17.13 yield an effective CPE coefficient Qeg that can be related to the film capacitance through a model of the distributed time constants following Brug et al. ° ... 

The term "constant phase element" stems from the fact that the phase angle of the portion of a circuit represented by such an element is AC frequency-independent. The impedance of a CPE has the form of dependency on an effective CPE coefficient Q [ohm s ] [3, p. 211] as ... 

Remember 17.3 Calculation of an effective capacitance or CPE coefficient according to equation (17.7) yields, in the high-frequency limit, properties associated with the electrode under study. 

The model is extended by a constant phase element (CPE), to include the possible effect of inhomogeneity of time constants (Bisquert and Compte 2001). The coefficient pis used... 

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