Another Way to View the CPE - ZARCs

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The ZARC Element

Although we use the Constant Phase Element ( CPE ) as a 'fundamental' circuit element today, the earliest explanations of depressed semicircles modeled the semicircle directly as a system with a distribution of time constants.(ref 1)

Z = Ro / [ 1 + ( j ··T )ⁿ ]        (Eq. 1)

where Ro is the low frequency real-axis intercept, T is the mean time constant, and n is related to the depression angle, as shown in the sketch at the right. Macdonald (ref 1) calls this ZARC element. It is also referred to as the Cole-Cole circuit element (ref 1,5).

Jovic and Orazem (ref 2) point out that the same Z can also be obtained from the parallel combination of a resistor and a CPE.

Z = Ro / [ 1 + Ro·Q ( j · )ⁿ ]        (Eq. 2)

Most of the commercially available EIS fitting programs yield values of Ro and Q as defined by Eq. 2. Of course, Eq. 1 and Eq. 2 are interchangeable if

Ro·Q = T ⁿ       (Eq. 3)

The advantage of Eq. 1 and the ZARC or Cole-Cole circuit element is that it encourages us to think in terms of a distribution of time constants ( T ) rather than a distribution of capacitances! It is reasonable to think that an electrochemical activation energy ( E^* ) might not have the same value at all points on an electrode surface. If we assume that the probability of finding an activation energy follows Eq. 4 ( ref 3 )

P( E^* ) ~ exp [ (1-n)·E^* / k·T ]      (Eq. 4)

and that the time constant, T, follows Eq. 5 ( ref 3 )

 T( E^* ) = T°·exp [  E^* / k·T ]      (Eq. 5)

then Z will follow Eq. 1, and a depressed semicircle will be observed (Macdonald, ref 1).

If we think of  T( E^* ) in Eq. 5 as R(E^*)·C, then the faradaic resistance, R_F ( ref 4 ) depends upon the activation energy in a familiar way. 

A poly crystalline metal electrode may show CPE/ZARC behavior. Although the capacitance of different crystal faces may differ somewhat, I would not expect the difference to be large. Differences in rate constants at different crystal faces, or at corners and steps on a face, or at grain boundaries, may differ considerably, however. Varying states of oxidation on a carbon surface can influence electron transfer rate constants considerably, as the work of McCreery has shown. These facts make a distribution of R_F values perhaps more reasonable than a distribution of capacitances!

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REFERENCES
(1) "Impedance Spectroscopy", JR Macdonald, ed., John Wiley, 1987. See Sec 2.1.2.3 and Sec. 2.2.3.4
(2) "Extension of the measurement model approach for deconvolution of underlying distributions for impedance measurements," ME Orazem, P Shukla, MA Membrino, Electrochimica Acta, 47(2002) 2027. DOI: 10.1016/S0013-4686(02)00065-8
(3) My apologies. The T in kT is Temperature.  T is the time constant.
(4) You may know it as Polarization Resistance, R_P, or think of its related inverse as  Exchange Current, i°, or Standard Heterogeneous Rate Constant, k°_SH. Your preference depends on your background and electrochemical dialect!
(5) "Electrochemical Impedance Spectroscopy Investigations of a Microelectrode Behavior in a Thin-Layer Cell: Experimental and Theoretical Studies", C Gabrielli, M Keddam, N Portail, P Rousseau, H Takenouti, V Vivier,  J. Phys. Chem. B,  110 (2006) 20478. DOI: 10.1021/jp063055h

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