Conveniently Revealing Lithium-Ion Battery Reaction Kinetics Using Nonlinear Frequency Response Analysis
Authors: Ulrich Julian , Lee Yan Ying , Popovic-Neuber Jelena , Weber Andre , Krewer Ulrike 
Published: September 2023

Nonlinear Frequency Response Analysis (NFRA) can be considered a nonlinear modification of galvanostatic EIS. Due to an increased AC excitation, higher harmonics are additionally present in the voltage response of the cell [1]. Here, a single particle model (SPM) [2] in the frequency domain is used for the simulation of the NFR of lithium-ion batteries (LiB). The time-dependent processes (double-layer charging, solid diffusion) are linear in currently available SPMs. Hence, they can be described algebraically as impedances in the frequency domain. From the nonlinear, time-independent equations (Butler-Volmer, Nernst), for the case of sinusoidal excitations, a system of algebraic equations for the fundamental and higher harmonics up to a freely selectable order N is derived. By combining these systems with the impedances, we obtain a still nonlinear SPM in the frequency domain, which can be solved in a purely algebraic way for the higher harmonics. The proposed model enables researchers to conveniently analyze the NFR of LiBs. This is especially beneficial for investigations of the reaction kinetics since charge transfer coefficients can be obtained directly by parameterizing the model with NFR measurements.
The abilities of the model are shown with two exemplary cell setups. At first, a symmetric cell built from metallic lithium is analyzed. The measured impedance is used to parameterize a simple equivalent circuit model (ECM), including transport through the solid electrolyte interphase (SEI). The resulting exchange current densities are used within the frequency domain SPM, which represents a nonlinear version of the ECM. Subsequently, the charge transfer coefficients of the plating/stripping reaction can be obtained conveniently by comparing the NFR measurements with the SPM response. Since the sum of obtained charge transfer coefficients is smaller than one, they are interpreted in the sense of Marcus-Hush-Chidsey kinetics [3], which is possible within a certain range of overpotential. This leads to a distinct value for the solvent reorganization energy. The calculated value is in coincidence with literature values, which are relying on high scan rate CVs with ultramicroelectrodes [4].
Secondly, a full pouch cell (NMC622/Graphite) was analyzed in a similar manner. Even though the SPM cannot capture microstructural effects of the porous electrode, a sufficient fit for the EIS measurement was found. After a subsequent parameterization of the charge transfer coefficients, the NFR measurement and simulation again agree well. The resulting values for intercalation reorganization energies at the graphite anode and NMC cathode are even lower than for the plating/stripping reaction. The reorganization energies could not be validated due to the lack of literature values. However, available values for LFP [5] and a value based on unpublished data for graphite [6], are also lower compared to the plating/stripping reaction.
The proposed method is very promising since it can directly capture the complete nonlinear kinetics of an electrode. Due to the frequency dependence of LiBs, the separation of kinetic and diffusive effects remains as convenient as it is for EIS.
If reorganization energies are considered as given, two persistent challenges in EIS analysis could be tackled: The separation of SEI resistance and charge transfer resistance since the latter is considered as nonlinear. In addition, the ionic transport resistance and charge transfer resistance could be separated for the same reason.
In the future, microstructural effects should be integrated into the model.

Vidaković-Koch, T.; Miličić, T.; Živković, L. A.; Chan, H. S.; Krewer, U.; Petkovska, M. Nonlinear Frequency Response Analysis: A Recent Review and Perspectives. Opin. Electrochem.
2021, 30, 100851. https://doi.org/10.1016/J.COELEC.2021.100851.

Ning, G.; Popov, B. N. Cycle Life Modeling of Lithium-Ion Batteries. Electrochem. Soc.
2004, 151 (10), A1584. https://doi.org/10.1149/1.1787631/XML.

Henstridge, M. C.; Laborda, E.; Rees, N. V.; Compton, R. G. Marcus–Hush–Chidsey Theory of Electron Transfer Applied to Voltammetry: A Review. Acta
2012, 84, 12–20. https://doi.org/10.1016/J.ELECTACTA.2011.10.026.

Boyle, D. T.; Kong, X.; Pei, A.; Rudnicki, P. E.; Shi, F.; Huang, W.; Bao, Z.; Qin, J.; Cui, Y. Transient Voltammetry with Ultramicroelectrodes Reveals the Electron Transfer Kinetics of Lithium Metal Anodes. ACS Energy Lett.
2020, 5 (3), 701–709. https://doi.org/10.1021/acsenergylett.0c00031.

Bai, P.; Bazant, M. Z. Charge Transfer Kinetics at the Solid-Solid Interface in Porous Electrodes. Commun.
2014, 5 (1), 1–7. https://doi.org/10.1038/ncomms4585.

Gao, T.; Han, Y.; Fraggedakis, D.; Das, S.; Zhou, T.; Yeh, C. N.; Xu, S.; Chueh, W. C.; Li, J.; Bazant, M. Z. Interplay of Lithium Intercalation and Plating on a Single Graphite Particle. Joule
2021, 5 (2), 393–414. https://doi.org/10.1016/J.JOULE.2020.12.020.

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