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David C. Silverman |
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The constant phase element arose during early studies in the 1940’s of dielectrics and was attributed to a combination of dispersion and absorption. The complex dielectric constant was found to be a function of the frequency raised to a fractional power. This analysis is directly applicable to electrochemical impedance spectroscopy. It offers a convenient method of accounting for the non-idealities alluded to in Analysis Using Simple Circuits. The impedance is affected by the interaction between frequency and all of the physical and chemical processes that respond to that frequency change within the electrochemical cell and across the corroding interface. The result is that one process may have multiple closely spaced dependencies on changing frequency. Or multiple processes may have similar but not the same responses to changing frequency. The result is often modeled as a phenomenological "RC" time constant the value of which may or may not be associated with one actual physical process. Some causes are:
The dispersion of relaxation across frequencies results in an alteration of the impedance spectrum. For example, the equation for the simple circuit presented in the section on Analysis Using Simple Circuits is:  (14)
This equation results in a semi-circle whose diameter is Rp. The curve would intersect the real axis at Rs as the frequency approaches infinity and at Rp+Rs when the frequency approaches zero. The angle of intersection with the abscissa would be 90o. When the time constant (RpC) is dispersed across a range of frequencies, the semicircle becomes shifted and appears as in this figure  (17)
where (jωτ)β accounts for depression below the real axis (e.g. dispersion) and τ is the phenomenological relaxation time constant loosely related to an RC time constant. The mathematical equations are generated by remembering that the term is expandable in terms of sines and cosines of the constant βπ/2. This figure shows the effect of a relatively small amount of dispersion on the impedance response to changing frequency in Nyquist format The following figures show how typical circuits might be written to take account of the constant phase element Some typical impedance spectra that at first glance might appear to be modeled by a simple RC time constants but which could only be fit by taking such dispersion into account are shown in these figures,  (500k).
The exponent is about 0.9. The second shows the impedance spectrum for steel
in a complex waste stream (D. C. Silverman, Electrochimica Acta, 38, 2075 (1993)).
In the latter case, the exponent was about 0.6 which lead to the large decrease in
the phase angle maximum. Both of these spectra were generated under turbulent
flow conditions in the rotating cylinder electrode to minimize mass transfer effects.
Inclusion of one constant phase element led to an excellent fit of the impedance
spectrum and reasonable agreement with corrosion rate estimates from mass loss of
the cylinder.
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David C. Silverman, Ph.D. - Primary Consultant |
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