Battery modeling and phase transitions. On top (a), the lithium-ion battery schematic with LiFePO 4 positive electrode and graphite negative electrode. Electrodes are composed of multiple particles which differ in shape and size. The physics-based model is formulated approximating the battery positive and negative electrodes as two, spherical, particles (b), one for each electrode. x and r indicate the Cartesian and radial coordinates, respectively. The thicknesses of negative particle, separator, and positive particle domains are L n , L s , and L p , respectively. In (c), the phase transitions experienced by the positive particle during a discharge from 100% to 0% SOC are shown. The positive particle is initialized at a concentration θ bulk

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... term V cs depends on the OCP of the negative electrode U n , the OCP of positive electrode in charge (U ch p ) and the OCP of the positive electrode in discharge (U dis p ). These latter terms are then combined to produce the average positive electrode OCP defined as (U ch p + U dis p )/2 and shown in Figure 2 (c). Moreover, V cs depends on the positive and negative electrode overpotentials, η p and η n , and the electrolyte overpotential, ∆Φ e which are derived according to [35] as described in Note S1. ...

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... experimental data are made available to the public at the link provided at the end of the paper. The term V h captures the deviation of the battery cell voltage from the average OCV, V avg OCV , defined as (U avg p − U n ) and shown in Figure S2 in the supplemental material. The hysteresis assumes negative values over the 100% to 0% SOC discharge range (because the battery voltage trajectory is below the V avg OCV ) and positive over the 0% to 100% SOC charge range (because the battery voltage trajectory is above the V avg OCV ) . ...

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... Physics-based model: average core-shell ESPM As shown in Figure 2(a) and (b), the average core-shell ESPM approximates the battery positive and negative electrodes as two, spherical, single particles where transport of lithium ions in the solid (the single particle) and electrolyte phase is expressed by mass conservation equations, charge conservation is used in the electrolyte phase, and phase transitions in the positive particle are modeled with a mass balance equation and a moving boundary [30]. Figure 2(c) shows phase transitions experienced by the positive particle and the corresponding regions on the charge and discharge OCPs. ...

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... Physics-based model: average core-shell ESPM As shown in Figure 2(a) and (b), the average core-shell ESPM approximates the battery positive and negative electrodes as two, spherical, single particles where transport of lithium ions in the solid (the single particle) and electrolyte phase is expressed by mass conservation equations, charge conservation is used in the electrolyte phase, and phase transitions in the positive particle are modeled with a mass balance equation and a moving boundary [30]. Figure 2(c) shows phase transitions experienced by the positive particle and the corresponding regions on the charge and discharge OCPs. In line with the arguments of [21], at a given stoichiometry, the battery potential in discharge (lithiation) is lower than the one in charge (delithiation). ...

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... switching between the A and B scenarios in Figure 2(c) is required. Such a switching is implemented by means of the transition conditions described in Note S3, which ensure conservation of mass. ...

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... note applies to the positive particle core-shell model (the two-phase region in Figure 2(c)) and describes the mass balance equations used to transition between charge and discharge conditions and remap the core-shell solid phase concentration. In this note, both "charge to discharge" and "discharge to charge" transitions are analyzed. ...