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The pseudo-two-dimensional (P2D) model of lithium-ion batteries couples a volume-averaged treatment of transport, reaction, and thermodynamics to solid-state lithium diffusion in electrode particles. Here we harness the linear and nonlinear physics of the P2D model to evaluate the fundamental (linear) and higher harmonic (nonlinear) response of a LiCoO2|LiC6 cell subject to moderate-amplitude sinusoidal current modulations. An analytic-numeric approach allows the evaluation of the linearized frequency dispersion function that represents electrochemical impedance spectroscopy (EIS) and the higher harmonic dispersion functions we call nonlinear electrochemical impedance spectroscopy (NLEIS). Base case simulations show, for the first time, the full spectrum second and third harmonic NLEIS response. The effect of kinetic, mass-transport, and thermodynamic parameters are explored. The nonlinear interactions that drive the harmonic response break some of the degeneracy found in linearized models. We show that the second harmonic is sensitive to the symmetry of the charge transfer reactions in the electrodes, whereas EIS is not. At low frequencies, NLEIS probes aspects of the cell thermodynamics that are not accessible with EIS. In short, NLEIS has the potential to increase the number of physicochemical parameters that can be assessed in experiments similar in complexity to standard EIS measurements.

Nonlinear electrochemical impedance spectroscopy (NLEIS) is a moderateamplitude extension to linear EIS that provides a sensitive and complementary whole-battery diagnostic for charge transfer kinetics, mass transport, and thermodynamics. We present the first full-frequency, second harmonic NLEIS spectra for lithium-ion batteries using commercially available, 1.5 Ah LiNMC|C cells. The mathematical framework for NLEIS shows, and experiments confirm, that moderate-amplitude input modulations can generate a second harmonic output that does not intrinsically corrupt the linear EIS response. Experimental measurements at varied states-of-charge (SoC) and states-of-health (SoH) are used to illustrate and compare NLEIS and EIS data. At low frequencies, the second harmonic NLEIS spectrum is shown to produce a much more distinct response to SoC dependent thermodynamic and diffusion processes than linear EIS. By combining NLEIS and EIS, we are able to characterize degradation in early cell cycling (where cells lost <1% of initial capacity). We also show that NLEIS complements the characterization of charge transfer kinetics of linear EIS through the second harmonic sensitivity to symmetry. For example, NLEIS shows that fresh cells have high symmetry charge transfer (α a = α c = 0.5) on both electrodes, whereas early in the cycling there is a shift toward kinetics that favor oxidation on the positive electrode (α a,pos > 0.5, α c,pos < 0.5). Combined analysis of EIS and NLEIS spectra shows promise for improved parameter estimation and model validation. All experimental data and analysis code for this manuscript can be found on ECSarXiv.

No abstract

The quantitative analysis of electrochemical impedance spectroscopy (EIS) data is important for both characterization and prognostic applications in many electrochemical systems. Here we describe an open-source platform, the ImpedanceAnalyzer, for easy-to-use physics-based analysis of experimental EIS spectra. To demonstrate the use of the platform, we explore the basic capabilities of the pseudo two-dimensional (P2D) battery model to predict publicly available experimental EIS data from a 1500 mAh commercial lithium-ion (LiCoO 2 /graphite) cell. An a priori computed dataset of 38,800 P2D-based impedance spectra simulations, covering a wide range of frequencies (1 mHz to 100 kHz) and model parameters, enables a straightforward least squares matching approach for analyzing experimental spectra. We find an average error of 1.73% between the best-matching computed spectrum from the 38,800 member library and the experimental spectrum being analyzed. Our analysis shows there is significant opportunity to improve the fit between experimental data and physics-based impedance simulations by a combination of a larger computed dataset, local optimization, and further additions to the model physics. Electrochemical impedance spectroscopy (EIS) is a powerful tool for investigating a wide variety of electrochemical systems. 1-3 EIS spectra separate individual electrochemical processes by their characteristic timescales, enabling both qualitative and quantitative analysis of electron transport, 4,5 reaction rates and mechanisms, 6,7 intercalation processes, 8 mass transport, 9,10 and electrode structure. 11,12 The noninvasive nature of EIS also makes impedance measurements useful in prognostic applications such as fuel cell health estimations 13,14 or prediction of remaining useful lifetime in batteries. 15,16 Qualitative analysis of EIS spectra generally involves assessing the shape of Nyquist plot features to determine the relative importance of different physicochemical processes. 17,18 In contrast, quantitative analysis relies on fitting a model to the data in order extract values for specific thermodynamic, transport, and/or kinetic parameters. Most experimental datasets are analyzed quantitatively using an equivalent circuit analog. Fitting an equivalent circuit to EIS data is straightforward using standard least squares regression techniques. 19,20 A good fit can often be found with a relatively simple equivalent circuit, particularly if non-ideal elements like the constant phase element are used. Moreover, many simple equivalent circuits, like the Randles' circuit, 21 have physically interpretable parameters based on linearized electrochemical processes. However, as more complex equivalent circuits are derived and utilized, the lumped parameters can lose their direct physical interpretability and the structure of the equivalent circuit analogs themselves can be degenerate. 22 An alternative to equivalent circuits for quantitative analysis of EIS data is to directly fit the data with a physics-based mathematical model ...

A hybrid analytical-collocation approach for fast simulation of the impedance response for a Li-ion battery using the pseudo-two dimensional model is presented. The impedance response of the spherical diffusion equations is solved analytically and collocation is performed on the resulting boundary value problem across the electrode and separator thickness using an orthogonal collocation scheme based on Gauss-Legendre points. The profiles for a frequency range from 0.5 mHz to 10 kHz are compared with the numerical solution obtained by solving the original model in COMSOL Multiphysics. The internal variable profiles across a wide range of frequencies are compared between the two methods and the accuracy, robustness, and computational superiority of the proposed hybrid analytical-collocation approach is presented. The limitations of the proposed approach are also discussed. A freeware for academic use that reads the various battery parameters and frequencies of interest as input, and predicts the battery impedance for a half cell and full cell, is also developed and a means to access it is reported in this paper.

Nonlinear electrochemical impedance spectroscopy (NLEIS) is a moderate-amplitude extension to linear EIS that provides a sensitive and complementary whole-battery diagnostic for charge transfer kinetics, mass transport, and thermodynamics. We present the first full-frequency, second harmonic NLEIS spectra for lithium-ion batteries using commercially available, 1.5 Ah LiNMC|C cells. The mathematical framework for NLEIS shows, and experiments confirm, that moderate-amplitude input modulations can generate a second harmonic output that does not intrinsically corrupt the linear EIS response. Experimental measurements at varied states-of-charge (SoC) and states-of-health (SoH) are used to illustrate and compare NLEIS and EIS data. At low frequencies, the second harmonic NLEIS spectrum is shown to produce a much more distinct response to SoC dependent thermodynamic and diffusion processes than linear EIS. By combining NLEIS and EIS, we are able to characterize degradation in early cell cycling (where cells lost less than 1% of initial capacity). We also show that NLEIS complements the characterization of charge transfer kinetics of linear EIS through the second harmonic sensitivity to symmetry. For example, NLEIS shows that fresh cells have high symmetry charge transfer (α_a = α_c = 0.5) on both electrodes, whereas early in the cycling there is a shift toward kinetics that favor oxidation on the positive electrode (α_(a,pos) greater than 0.5, α_(c,pos) less than 0.5). Combined analysis of EIS and NLEIS spectra shows promise for improved parameter estimation and model validation. All experimental data and analysis code for this manuscript can be found on ECSarXiv.

Extracting quantitative information from electrochemical experiments relies on mathematical models of the system under study. Over the past century, many simplified electrochemical systems have been analyzed mathematically, typically resulting in univariate plotting methods where physicochemical parameters are extracted from linear best-fits of experimental data. This approach includes famously successful methods such as Tafel plots, Nernst plots, Levich, and Koutecký-Levich plots, to name just a few. These graphical methods generally require a half-cell experiment that naturally has, or can be manipulated to have, just one or two dominant physicochemical processes governing the electrochemical response. There are a myriad of ways that voltammetric, amperometric, microelectrode, and hydrodynamic electrode half-cell experiments have been devised to fit this limiting-case paradigm. 1 Even in more complex electrochemical systems where coupled physicochemical processes obscure simple analysis, or in whole-cell systems where both electrodes contribute to the overall electrochemical response, thoughtful experimental design sometimes allows inference of the complex underlying phenomena from the experimental data. 2 In today's era of ubiquitous computing and data-rich experimentation, it is increasingly possible to use more sophisticated mathematical models to qualitatively and quantitatively interpret complex electrochemical experiments. [3][4][5][6][7][8] Across the broader scientific community, the culture around creating open source software and open datasets is a force for accelerating discovery and impact. [9][10][11][12] In this article, we primarily discuss one particular example of a communitydriven, open source software toolkit for analyzing electrochemical impedance spectroscopy (EIS) data. It is important to note that much of what we show here can be done today using commercial or closed-source software; however, by establishing an open framework for EIS analysis, and open datasets for testing models, we believe the convergence between complex physicochemical models and experimental data will be accelerated. A further benefit of open software and open data is that the reproducibility of the scientific literature can be improved.The toolkit we describe here is one of several projects that have come out of The Electrochemical Society's Hack Week and Data Science Sprint events; some of the other projects are briefly discussed in the Outlook section.