Physical Properties Obtained from Measurement Model Analysis of Impedance Measurements
Authors: Liao Hangqi , Watson William , Dizon Arthur , Vivier Vincent , Tribollet Bernard , Orazem Mark E. 
Published: December 2020

This paper demonstrates the application of the Voigt measurement model to extract capacitance, ohmic resistance, and polarization resistance values from impedance data. The systems explored include a Randles circuit, films with exponential and power-law distributions of resistivity, systems exhibiting geometric capacitance, and systems showing geometry-induced frequency dispersion. The present work demonstrates that the measurement model serves as a useful means to provide quantitative estimates for parameters relevant to impedance spectroscopy data, including ohmic resistance, polarization resistance, and capacitance.
The ability to extract capacitance by use of the measurement model is important, because most impedance systems show distributed-time-constant behavior that is often fit by use of the CPE. The CPE parameters are related to the capacitance of the electrode, but extraction of a capacitance requires different approaches depending on the nature of the underlying distribution. For example, the Brug formula[1] is used in cases of a surface distribution of time constants, and the power-law model[2],[3] applies for a distribution through a film. Not all distributed-time-constant behaviors are represented by a CPE, and formulas like the Brug and Power-Law equations do not exist for such systems. Furthermore, the measurement model provides a unique ability to extract the high-frequency and low-frequency ohmic resistances for systems exhibiting an ohmic impedance.[4]

The application of the measurement model to extract parameters for experimental impedance data depends on the nature of the electrochemical system under investigation. For systems that have a uniform current and potential on the electrode surface, the ohmic resistance and capacitance may be extracted by regression of the measurement model to the part of the impedance spectrum that is found to be consistent with the Kramers-Kronig relations. The polarization resistance corresponding to the zero-frequency limit may also be estimated. For systems that have a nonuniform current and potential distribution, a characteristic frequency can be identified above which the impedance is influenced by the associated frequency dispersion. This characteristic frequency may be estimated for general systems using values of capacitance and ohmic resistance estimated by regression of the measurement model to the impedance spectrum. The estimate for capacitance may be updated be eliminating the frequencies above the characteristic frequency. This is an iterative process is completed when the estimated characteristic frequency is larger than the maximum frequency used in the regression. The iterative process will yield estimates for capacitance and low-frequency ohmic resistance. The iterative process may fail to yield a suitable convergence for cases in which the charge-transfer resistance is smaller than the ohmic resistance.
Used wisely, the measurement model may be a powerful tool for extracting the capacitance, ohmic resistance, and polarization resistance for impedance data. It can be applied both for systems exhibiting normal distributions of time constants and, if applied to frequencies sufficiently below the characteristic frequency for geometry-induced dispersion, for systems showing surface distributions of time-constants. Used in this way, the measurement model provides a powerful complement to the development of system-specific process models.

References

[1] G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, and J. H. Sluyters, “The Analysis of Electrode Impedances Complicated by the Presence of a Constant Phase Element,” Journal of Electroanalytical Chemistry, 176 (1984) 275-295.

[2] B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, “Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 1. Theory,” Journal of The Electrochemical Society, 157 (2010) C452-C457.

[3] B. Hirschorn, M. E. Orazem, B. Tribollet, V. Vivier, I. Frateur, and M. Musiani, “Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 2. Applications,” Journal of The Electrochemical Society, 157 (2010) C458-C463.

[4] O. Gharbi, A. Dizon, M. E. Orazem, M. T.T. Tran, B. Tribollet, and V. Vivier, “From Frequency Dispersion to Ohmic Impedance: A New Insight on The High-Frequency Impedance Analysis of Electrochemical Systems,” Electrochimica Acta, 320 (2019), 134609.