(Keynote) Modeling of Impedance
Authors: Newman John 
Published: February 2020

Sophisticated models exist for many electrochemical systems, based on physical and chemical laws. These may be transient or steady. They generally involve many physical quantities, and a corresponding number of governing equations, distributed over a spatial region. Solution methods frequently involve linearization over nonlinearities and iteration to account for or remove any such approximations.1 It is straightforward to extend such models to alternating-current impedance spectroscopy by taking advantage of the solution method already used. Impedance modeling already involves linearization of the problem, and the solution method takes advantage of this. Instead of having a nonlinear steady problem, which needs to be iterated over, one now has a linear problem but with the frequency w of the perturbation as a parameter. w varies from very small to vary large to elucidate the various time constants of the original problem.
This approach has advantages over the use of equivalent circuits, where the circuit parameters often have an unclear relationship to the original nonlinear physical model. These parameters can include resistors, capacitors, and inductances as well as constant-phase-angle elements which are more fitting parameters rather than physically meaningful constructs. One advantage of equivalent circuits is that the results for the real and imaginary parts of any quantity, such as potential, current, or concentration, obey the Kramers-Kronig relations, thus providing a check on the results and providing a suitable fit for the experimental results, which must also necessarily obey the Kramers-Kronig relations. However, the approach suggested here is also based on physical laws, and the results will also obey the Kramers-Kronig relations.
Examples discussed in more detail in the presentation include
1. The effective resistance and capacitance of a disk electrode embedded in an insulating disk, where concentrations gradients are ignored but interfacial kinetics and capacitance can be treated.2 Frequency dispersion (where the effective capacitance and resistance depend on frequency) arise in a straightforward manner due to the geometry of the system.
2. A hydrodynamically modulated rotating disk, where the rotation speed is varied with small-amplitude sinusoidal modulation.3 This example is particularly instructive because the problem can be broken down into three parts: a. The nonlinear hydrodynamics, b. the concentration profiles in the solution, and c. the interfacial phenomena.
3. Porous electrodes, where all profiles vary with the distance through the porous electrode and where complex interfacial phenomena and transient behavior can be treated.4,5 However, to follow the rules of impedance analysis, the system needs to be treated at open-circuit, where a valid steady state exists (or where the modulation frequency is large compared to the time constants of the system). This limitation does not apply to a fuel cell, where a valid steady state is possible.
4. Kramers-Kronig analysis, which illustrates what is involved and demonstrates that numerical results obey the Kramers-Kronig relations.6

*The author of this work is an Electrochemical Society Honorary Member

References

1. John Newman, and Karen E. Thomas-Alyea, Electrochemical Systems, Third Edition (New York: Wiley, 2004), Appendix C.
2. John Newman, "Frequency Dispersion in Capacity measurements at a Disk Electrode," Journal of the Electrochemical Society, 117 (1970), 198-203.
3. Bernard Tribollet and John Newman, "The Modulated Flow at a Rotating Disk Electrode," Journal of the Electrochemical Society, 130 (1983), 2016-2026.
4. Marc Doyle, Jeremy P. Meyers, and John Newman, "Computer Simulations of the Impedance Response of Lithium Recharageable Batteries," Journal of the Electrochemical Society, 147 (2000), 99-110.
5. Jeremy P. Meyers, Marc Doyle, Robert M. Darling, and John Newman, "The Impedance Response of a Porous electrode Composed of Intercalation Particles," Journal of the Electrochemical Society, 147 (2000), 2930-2940.
6. Milan M. Jakšiæ and John Newman, "The Kramers-Kronig Relations and Evaluation of Impedance for a Disk Electrode," Journal of the Electrochemical Society, 133 (1986), 1097-1101.