Another Way to View the CPE - ZARCs


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)

Nyquist plot for a ZARCZ = Ro / [ 1 + ( j ·omega·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 ·omega )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, RF ( 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 RF values perhaps more reasonable than a distribution of capacitances!

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(1) "Impedance Spectroscopy", JR Macdonald, ed., John Wiley, 1987. See Sec and Sec.
(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, RP, 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. B110 (2006) 20478. DOI: 10.1021/jp063055h

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