Electrochemical impedance spectroscopy (EIS) has been widely used to study the linear dynamics of many electrochemical systems, including Li-ion batteries.^1-4 Most physical and chemical processes in an electrochemical system possess distinct characteristic time constants, enabling EIS to distinguish these processes by their frequency response. The dynamics of the porous electrodes used inside most commercially available Li-ion batteries are governed by the conduction and mass transfer in the solid and the electrolyte phase, and electro-chemical kinetics at the solid-electrolyte interface. Because EIS is able to separate processes by their time-scales, it is often possible to see changes in each battery electrode as well as the degradation that leads to capacity fade of the batteries.^5-8

Equivalent circuits are the most commonly used models to study the impedance response of batteries; however, these models can suffer from a lack of physical interpretability and model degeneracy.⁹ Though much less used, physics-based models can provide more direct insight into the impedance response of a battery. Mathematically, physics-based impedance models are transformed from the time-domain to the frequency-domain by assuming a steady-periodic response, thereby eliminating time as an independent variable.¹⁰ Most rigorous physics-based models for the impedance of a Li-ion battery employ a numerical scheme to solve the resulting equations.^10-12 The resulting equations are computationally expensive to solve, and this reduces their usefulness for multi-parameter optimization and analysis of different mechanisms from experimental impedance data.

In this talk, we propose a hybrid analytical-collocation approach for simulating the impedance response of a Li-ion battery using the pseudo-two dimensional (P2D) model in real-time. 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 will also be discussed.

Acknowledgments

The authors would like to thank the Department of Energy (DOE) for providing partial financial support for this work, through the Advanced Research Projects Agency (ARPA-E) award number DE-AR0000275, along with the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the DOE through the Advanced Battery Material Research (BMR) Program (Battery500 consortium).

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