Analyzing Experimental Data Validity and Error Structure for Second-Harmonic Nonlinear Electrochemical Impedance Spectroscopy
Authors: Ji Yuefan , Shih An-Hung , Schwartz Daniel T. 
Published: December 2024

Second harmonic nonlinear electrochemical impedance spectroscopy (2nd-NLEIS) is a powerful and complementary technique to traditional EIS, often resolving model degeneracy issues in electroanalytical half-cell measurements and enabling better parameter assignment in two-electrode systems such as lithium-ion battery (LIB) experiments. Since the initial 2nd-NLEIS data for Li-ion batteries was reported by Murbach et al., [1] methods for testing the validity of 2nd-NLEIS datasets have been lacking. In EIS, the Kramers-Kronig (KK) relation is foundational for evaluating if experimental data meets key metrics for linearity, causality, and stationarity. [2] While the nonlinear nature of the 2nd-NLEIS response does not allow the direct application of KK relation, other approaches for data validation established in the EIS literature can be adapted to data validation and experimental error analysis in 2nd-NLEIS experiments.
In this work, we provide a pathway to quantify the quality of 2nd-NLEIS data, providing better confidence in the simultaneous analysis of EIS and 2nd-NLEIS. The Data validation for 2nd-NLEIS relies on the use of a series of nonlinear Voigt elements to describe a 2nd-NLEIS spectrum, enabling a foundational assessment of data quality and stochastic error analogous to the practice in linear EIS data quality assessments. [3,4] Our method builds on prior work showing the 2nd-NLEIS Randles circuit response is directly proportional to the square of the linear EIS response. [5] By feeding the KK-transformable EIS response into the 2nd-NLEIS governing equation, we obtained a KK equivalent nonlinear transformation for the 2nd-NLEIS. That is, the 2nd-NLEIS response can be represented by the sum of M nonlinear Randles in the absence of measurement noise. Similarly, the difference between the data and fit provides a measure of the experimental noise. Overall, this work leverages methods from linear EIS and basic circuit elements we’ve developed for the 2nd-NLEIS response to provide data quality quantification of the 2nd-NLEIS data.

References

[1] M.D. Murbach, V.W. Hu, D.T. Schwartz, Nonlinear Electrochemical Impedance Spectroscopy of Lithium-Ion Batteries: Experimental Approach, Analysis, and Initial Findings, J. Electrochem. Soc. 165 (2018) A2758–A2765. https://doi.org/10.1149/2.0711811jes.
[2] M. Urquidi-Macdonald, S. Real, D.D. Macdonald, Applications of Kramers—Kronig transforms in the analysis of electrochemical impedance data—III. Stability and linearity, Electrochimica Acta 35 (1990) 1559–1566. https://doi.org/10.1016/0013-4686(90)80010-L.
[3] B.A. Boukamp, A Linear Kronig‐Kramers Transform Test for Immittance Data Validation, J. Electrochem. Soc. 142 (1995) 1885–1894. https://doi.org/10.1149/1.2044210.
[4] M. Schönleber, D. Klotz, E. Ivers-Tiffée, A Method for Improving the Robustness of Linear Kramers-Kronig Validity Tests, Electrochimica Acta 131 (2014) 20–27. https://doi.org/10.1016/j.electacta.2014.01.034.
[5] Y. Ji, D.T. Schwartz, Second-Harmonic Nonlinear Electrochemical Impedance Spectroscopy: Part I. Analytical Theory and Equivalent Circuit Representations for Planar and Porous Electrodes, J. Electrochem. Soc. 170 (2023) 123511. https://doi.org/10.1149/1945-7111/ad15ca.