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).
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
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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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