Unifying Chemical and Electrochemical Thermodynamics of Electrodes

Batteries are critical for electrified transportation and aviation, yet thermodynamic understanding of electrode materials remains lacking, as indicated by the often-seen violation of the second law of thermodynamics of open-circuit voltage (OCV) models. On the other hand, thermodynamic modeling rarely utilizes electrochemical data such as OCV, entropic heat (dOCV/dT), which contains rich thermodynamic information. This work introduces a framework of thermodynamic modeling of materials for electrochemical energy storage, using differentiable programming and gradient-based optimization of thermodynamic parameters. Using a modified Debye model that accounts for the phonon density of states, the thermodynamics of pure substances is modeled from experimental measurements of specific heat ($c_p$) as well as the phonon density of states $g(ω)$. Thermodynamics of mixing is modeled with measured entropic heat and OCV data. We demonstrate the differentiable thermodynamic modeling framework with forward and inverse problems. In the forward problem, i.e. determining phase diagram of graphite anode given thermochemical and electrochemical data, we show that in addition to accurate reproduction of phase diagram of LixC6, the fitted temperature-dependent OCV of graphite reaches 3.8 mV mean absolute error (MAE) for test set data measured at 10$^\circ$C, compared with 2.9-3.6 mV MAE for training set data measured at 25$^\circ$C - 57$^\circ$C. In the inverse problem, i.e. determining OCV of lithium iron phosphate (LFP) cathode from phase diagram constrained by thermochemical and electrochemical data, we demonstrate accurate reproduction of LFP OCV as well as phase diagram. This framework offers a unified treatment of chemical and electrochemical thermodynamic data for electrode materials.