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Corrosion Calculator
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TUTORIAL ON ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY
David C. Silverman
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Table of Contents
Diffusion Impedance
One type of impedance spectrum that is sometimes observed when examining such
phenomena as corrosion in soils, concrete, or stagnant water is shown in this figure
.
The supposition is that the linear portion of the response continues on to infinity
in real and imaginary coordinates as the frequency approaches zero. This behavior
is often misnamed the Warburg impedance. This structure is actually a subset of
diffusion influenced impedance. It should be called the infinite length Warburg
impedance to differentiate it from the more general finite-length Warburg solution.
The infinite length Warburg impedance is derived from a solution of the diffusion
equation in which migration is into a semi-infinite medium. This situation is
somewhat rare because in many practical systems diffusion occurs over finite distances.
In addition, note that one of the criteria for the frequency spectrum to be called
an impedance is that the spectrum must be finite valued at infinity. One way for
this criterion to be fulfilled is for the equivalent circuit to be an open circuit
or a capacitor at infinite distance. One should exhaust other possibilities before
stating that the situation is an infinite length Warburg impedance.
If one assumes that the Nernst diffusion-layer thickness is comparable to the distance
traveled by diffusing species while subjected to low frequency oscillating perturbations
then the following equation can be derived for the Warburg diffusion contribution to
the impedance:
 (20)
This solution is often appropriate for a stirred electrolyte or a rotating electrode,
environments often encountered in practical corrosion testing. Incorporating this
finite length Warburg impedance along with a relaxation "RC" time constant results
in the following analogous circuit
.
This circuit was simulated by using a time constant of 1 second, a solution resistance
of 10 ohm-cm2, a charge transfer resistance of 104 ohm-cm2
and a CPE exponent of 0.8. The result is shown in this figure
.
The solid line corresponds to the portion of the spectrum terminated at a low
measurable frequency of between 10-4 and 10-3 radian/s. This
portion of the curve has the appearance of an infinite Warburg impedance
.
In practice, before concluding that a spectrum corresponds to an infinite length
Warburg impedance, the corrosion practitioner should ensure that the structure is not
caused by terminating the spectrum at too high of a frequency. Addressing that question
may be difficult because sometimes the measurement cannot proceed to low enough
frequencies to extract the entire spectrum. This point can be seen by transforming
the above simulation to Bode format
.
As shown in this figure, most experimenters do not or cannot obtain spectra below
about 10-4 to 10-3 radian/s. For example, a frequency of 10-7
radian/s would take the better part of a year to complete. The key point is that
before one assumes an infinite Warburg impedance, one must fully examine all
other possible diffusion analyses. Further information can be found in
J. R. Macdonald, Impedance Spectroscopy, John Wiley & Sons, NY, 1987.
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Next Page: Pseudoinductance
Previous Page: Critical Criteria for Proper Spectra
Return to Table of Contents
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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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