Thermodynamic Effects on Nonlinear Electrochemical Impedance Spectra in Lithium NMC | Graphite Cells
Authors: Hu Victor Waiman , Schwartz Daniel T. 
Published: February 2020

Electrochemical impedance spectroscopy (EIS) is widely used to diagnose and characterize physicochemical processes in lithium-ion batteries.1 EIS relies on a low magnitude sinusoidal modulation of either current or voltage so that the battery has a linear response, allowing subsequent analyses to be done through either equivalent circuit methods or linearized physics-based models.2 The analysis can provide qualitative and quantitative information about charge-transfer, mass transport, and thermodynamics processes occurring in the cell.3,4 Linearization of an inherently nonlinear system, like a battery, intrinsically limits the information content of an impedance spectrum.
Nonlinear electrochemical impedance spectroscopy (NLEIS) provides all of the information of EIS—e.g. generates the familiar linear spectrum that satisfies Kramers-Kronig relationships—while also revealing additional physics. The key to NLEIS is modestly increasing the magnitude of the input sinusoidal modulation, thereby driving the system into the weakly nonlinear regime, while measuring output signal at twice the input modulation frequency (the second-harmonic). Volterra series analysis (common in the study of weak nonlinearity in electronics) produces a second-harmonic impedance (Z2(w)) spectrum that probes physicochemical processes in new ways, and that are often invisible to linearized EIS.5 From prior work, we know that EIS and NLEIS depend on different thermodynamic traits of a battery, locally, at different states of charge.
Samsung 18650 LiNMC | C batteries (1500 mAh) were cycled under aggressive conditions to promote aging and degradation. The open-circuit potentials of these batteries were determined through extremely low rate cycling with high-precision coulometry, and the first and second order derivatives of the open-circuit potential with respect to the capacity were calculated. These calculated derivatives from the full cells are related to the low-frequency harmonic responses in a nontrivial, but consequential manner. Prior works have shown that the first derivatives of each individual electrode affects both the linear and nonlinear impedance response, while the second derivatives only affects the nonlinear response.6

In this work, we have performed NLEIS at voltages of interest in the first and second order derivatives of the full cells to understand their impact on the linear and the second harmonic impedance Z2(w) spectra. Here, we look at how the magnitude of both full-cell derivatives and the sign (+/-) of the second derivative impact the magnitude and the direction of the low-frequency nonlinear response. Although the nonlinear responses are collected from full cells, we can use an extension of the pseudo-2D model of lithium-ion batteries, that also computes the higher order harmonic responses, to gain insight into thermodynamics of the individual electrodes.6 These measurements can be collected with more ease and in shorter time frames than traditional half-cell experiments and differential voltage analyses, which are time-intensive and can be difficult to properly control.7,8 Using both the linear and nonlinear impedances simultaneously can provide fundamental insight into the impacts of the system thermodynamics and can then be used to determine, in shorter measurement times and with more sensitivity, the state of health of lithium-ion batteries.

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