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TUTORIAL ON ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

David C. Silverman


Table of Contents

Overview of Tutorial
Overview of EIS
Analysis Using Simple Circuits
Constant Phase Element
Corrosion Rate Estimation
         Two Capacitive Relaxation Time Constants
Critical Criteria for Proper Spectra
Diffusion Impedance
Pseudoinductance

Analysis Using Simple Circuits

Most corrosion practitioners have attempted to analyze impedance spectra using combinations of analogous circuit elements. The reasoning used to justify this approach might be summarized by the follow statements:
  • Corrosion of alloys and related conductive materials is an electrochemical degradation process governed by kinetics and thermodynamics.
  • This chemistry is often difficult to interpret in real-life complex systems normally encountered. Time and funding usually do not allow for extensive study.
  • Analogous circuit elements bridge gaps in knowledge to allow the corrosion practitioner to use electrochemical impedance spectroscopy to estimate corrosion rates and corrosion mechanisms in poorly characterized systems.
Circuit elements often used to model electrochemical impedance spectra are combinations of resistances, capacitances, and inductors. The impedance equations for the elements are:
                                                     Resistor:                           (10)

                                                    Capacitor                  (11)

                                                     Inductor                       (12)

In these equations, R is a resistor, C is a capacitor, and L is an inductor. These circuit elements are often joined in a variety of ways to model the circuit. The procedure employed is to build a circuit and curve fit that circuit to the measured spectra. A good fit is often judged to be a good model. Points often overlooked are:
  1. An analogous circuit element successfully used in a model may not have a 1 to 1 correspondence to an actual physical corrosion process.
  2. Redundancy of circuits means that two seemingly different circuits can result in identical impedance spectra. ( see S. Fletcher, J. Electrochem. Soc., 141, 1893 (1994) for a number of degenerate examples)
The following figure shows a simple circuit of a resistor in series with a parallel combination of a resistor and capacitor. This circuit is often invoked to model a simple corrosion process consisting of a corrosion reaction, a surface capacitance and the uncompensated solution resistance . This circuit may be modeled by the following equation

                                                                            (13)

where Rs is often designated the solution resistance, Rp is often designated as the polarization resistance, and C often designated as is an interfacial capacitance between the surface of the corroding material and the fluid. Sometimes C is said to be a double layer capacitance but this designation is often far too simple in corroding systems. Many practitioners do not realize that fact.

This circuit expands out to the equation often seen as a semicircle when plotted on real versus imaginary coordinates (Nyquist format):

                                                              (14)

Note that the frequency is not an explicit variable when the spectrum is plotted in real versus imaginary coordinates. A better way to plot this information is in Bode format in which the impedance magnitude and phase angle are plotted separately versus the frequency. The equations that govern this plot are:

                                                                   (15)



                                                                                     (16)

This figure shows a plot of equations (15) and (16) . for the case of Rs = 10 ohm-cm2, Rp = 10000 ohm-cm2, and C = 0.0001F/cm2. That this simple model can fit actual corrosion data is shown in this figure . for the case of titanium in a weak acid solution. Other examples can be found in the literature. (See D.C.Silverman, "Practical Corrosion Prediction Using Electrochemical Techniques", in Uhlig’s Corrosion Handbook, 2nd ed. (R. W. Revie, ed.), 1179, John Wiley, N.Y., 2000     (1800k)).

But, applicability of simple models such as those discussed here tends to be the exception, not the rule. Some of the contributors to this complexity are:
  1. Multiple relaxation time constants (multiple RC time constants which may have no 1:1 correspondence to circuit elements)
  2. Distributed processes (e.g. transmission line)
  3. Diffusion processes
  4. Additional surface reactions affected at low frequency (pseudoinductance)
  5. Cell geometry (electrode placement)
  6. Heterogeneous surface
  7. Surface Roughness
  8. Violation of requirements for the frequency response to be an impedance
Note that some of the above contributors are the result of poor experimental design. Others are part of the physics of the system and cannot be avoided. One of the methods of accounting for some of these influences is by using the constant phase element in place of the capacitance. This approach can be especially useful when the complex chemistry prevents a kinetic model from being developed or when time and money preclude an extensive study.



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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