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TUTORIAL ON ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY
David C. Silverman
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Table of Contents
Two Capacitive Relaxation Time Constants
A large number of corroding systems demonstrate two relaxation time constants when
examined by electrochemical impedance spectroscopy. Among them are imperfect
coatings allowing internal ionic migration, surfaces protected by inhibitors,
and surfaces in which the structure varies. The impedance spectrum might be
modeled using a circuit shown in this figure
.
This circuit has been modified to take account of the fact that both of the capacitive
elements can suffer from dispersion and thus might be modeled using a constant
phase element in place of a true capacitance. This circuit is not the same as that
shown in this figure
because the capacitance is not a time constant. Care should be exercised
to ensure that the distinction is not overlooked.
Throughout the literature, impedance spectra have been reported in Nyquist
(real impedance vs. imaginary impedance) format with no attempt to examine the
same spectra in Bode format. This figure
shows a simulated impedance spectrum with two relaxation time constants plotted
in Nyquist format. The solution resistance is 10 ohm-cm2, the high frequency
resistance is 100 ohm-cm2, and the low frequency resistance is 105 ohm-cm2.
Two pairs of exponents were used for the constant phase elements, 1,1 and 0.95,0.85.
The main point is that in Nyquist format, the structure appears to exhibit only one
relaxation time constant independent of the value of the exponent. The two time
constants cannot be resolved in this format.
This figure
shows the same results when plotted in Bode format. Both relaxation time constants
are resolved in this format. Impedance spectra should always be examined in Bode format
especially when a complex structure is suspected. Also note the large effect that
changes in the exponent on the constant phase element can have on the resulting spectrum.
Neglecting the exponent on the constant phase element when performing a regression of
a model against the data can lead to wrong conclusions or poor values for the
extracted parameters (resistances and capacitances).
Following is an example of how this approach has been applied to examining steel
inhibited by low concentrations of hydroxyethylidene 1,1 diphosphonic acid (D. C.
Silverman, "Corrosion Prediction from Circuit Models - Application to Evaluation of
Corrosion Inhibitors", in Electrochemical Impedance: Analysis and Interpretation
(J. R. Scully, D. C. Silverman, and M. W. Kendig, ed.), ASTM STP 1188, p. 192,
American Society for Testing and Materials, Philadelphia, PA, 1993).
One of the impedance spectra generated in this study is shown in this figure
.
The exponents on the constant phase elements were 0.95 and 0.47. The latter value
is very close to that which might be expected for a diffusion process. The study
was performed in the rotating cylinder electrode under turbulent flow conditions.
Use of two CPE’s resulted in an excellent fit of the spectra. The following table
shows corrosion rates estimated from both mass loss and by assuming that the
resistance associated with the low frequency portion of the spectrum was the
polarization resistance. The table includes results for several phosphonic
inhibitors at several temperatures. Details are in the reference cited above.
The agreement supports both the model and that the designation of the resistance
associated with the low frequency portion of the spectrum reflects the corrosion
rate.
Inhibitor Study - Steel in HEDP/Water
Time-averaged Corrosion Rate for Samples (mm/y)
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| Impedance
|
Mass Loss
|
| 0.86 |
1.30 |
| 1.8 |
2.5 |
| 0.04 |
0.09 |
| 0.10 |
0.14 |
| 0.50 |
0.22 |
| 0.51 |
0.41 |
| 0.04 |
0.02 |
| 0.04 |
0.05 |
The capacitance values were estimated from the constant phase element parameters.
This figure .
shows the result. The question is "What do those capacitance values mean?"
For example capacitance values of 103 to 105
microfarad/cm2 have been hypothesized to be caused by the presence
of surface oxides. But, this statement should be considered an hypothesis until
other measurements or calculations confirm that these values can be tied to actual
physical processes compounds. This conclusion has general applicability.
A significant amount of verbiage is spent discussing the physical meaning of
circuit elements such as these in the published literature. But such elements
are models only. They often do not have a one to one correspondence with a
physical process especially since they may be averaged over more than one process.
Discussions of their relevance to actual physical processes should be treated as
conjecture unless their values can be directly calculated from the physical
processes in question.
As mentioned previously, two different analogous circuits can create the same
impedance because they mathematically simplify to the same equation. In addition,
two circuit models can create the same impedance because of numerical error
propagation during non-linear regression even if they do not mathematically
simplify into the same equation. Following is a case in point extracted from
D. C. Silverman, "On Ambiguities In Modeling Electrochemical Impedance Spectra
Using Circuit Analogues", Corrosion, Vol. 47, No. 2, p. 87 1991 1  (174k).
An impedance spectrum exhibiting two relaxation time constants was analyzed
by the following circuits,
and .
where Q=((Ajω)β)-1. The equations governing these
circuits are
 (18)
for the parallel circuit and
 (19)
for the series circuit. The fitted spectra are shown in these figures for
the nested parallel circuit
and the series circuit
.
The key point is that not only were the two curve-fits similar but the parameters
(resistances and constant phase elements) calculated from the curve fits were also almost
identical. The reason is that in this situation several orders of magnitude separated
the values of Q1 and Q2 with Q2 much less than Q1.
The ratio Q2/Q1 became much less than one and the quantity in
parentheses approximately equaled one. Under these circumstances equations (18)
and (19) became similar. Mathematical noise that normally occurs during non-linear
regression prevented a clear differentiation between the two models.
The key point is that even after one has an excellent fit of a model to the
spectrum, one still must analyze the mathematics to ensure that the solution is
not ambiguous because of relative values of the parameters.
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Next Page: Critical Criteria for Proper Spectra
Previous Page: Corrosion Rate Estimation
Return to Table of Contents
1 © NACE International publication and year shown in citation above. All rights reserved. Displayed with permission from NACE International, Houston, TX (http://www.nace.org). Published in Corrosion, in the month and year shown in the citation above.
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David C. Silverman, Ph.D. - Primary Consultant
E-Mail: dcsilverman@argentumsolutions.com
Phone: 314-576-3586
Fax: 314-754-9825
Address: The Argentum House
14314 Strawbridge Ct.
Chesterfield, MO 63017
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