

The ZARC ElementAlthough 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 realaxis 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 ColeCole 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 

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The advantage of Eq. 1 and the ZARC or ColeCole 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 [ (1n)·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! More
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