Determining the Limits of Fast Charging of a High-Energy Lithium-Ion NMC/Graphite Pouch Cell Through Combined Modeling and Experiments

We use a P3D model that was originally developed for studying aging in high-power lithium iron phosphate/graphite cylindrical cells. ⁴⁰ We reparameterized it for representing high-power lithium-ion pouch cell with graphite at the NE and NCA/LCO blend at the PE. ^39,43 Here, lithium plating was added. ^11,21 These studies demonstrate the versatility of P3D models with detailed chemistry. The transport scales are shown schematically in Fig. 2 and combine heat transport through the cell thickness and holder plates (in this paper referred to as macroscale or x scale), mass and charge transport inside the liquid electrolyte (mesoscale, y scale), and diffusive mass transport in the active materials (AM) particles (microscale, z scale), each solved in 1D and coupled via appropriate boundary conditions and upscaling relationships. Heat transport is described via conduction and uses heat source terms due to chemical reactions and ohmic losses. The electrodes are described in a continuum setting, that is, microstructure is not resolved. Transport of electrolyte-dissolved ionic species (Li⁺ and PF₆ ^–) is described as combined diffusion and migration in a Nernst-Planck setting with concentration-dependent diffusion coefficients. The lithium transport in the AM is modeled by Fickian diffusion with stoichiometry-dependent diffusion coefficient. The chemistry is described within a generalized framework allowing an arbitrary number and type of reactions at each of the involved interfaces. Thermodynamics and kinetics are evaluated with the open-source software Cantera, ⁴⁴ allowing a consistent definition and simple exchange of parameters. For details please see the Appendix and our previous works. ^11,21,39,40

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In the present work, we completely re-parameterized the model in order to represent the target cell and its aging behavior, then we validate the model by comparing simulations with experiments over a wide range of conditions.

In the present P3D model, the cell is represented by one single electrode pair on the macroscale. This means that only one single electrode-pair model (P2D model on the two lower scales) is used to describe electrical cell performance. The heat production calculated from the electrode pair is upscaled and evenly distributed over the macroscale. Spatial temperature gradients result from conduction along the cell thickness and convective heat losses at the surface. The average temperature on the macroscale is passed back to the P2D model. While predicted average temperature increase during cycling is notable (up to 1 °C at 2 C), predicted spatial temperature gradients are only small (<0.1 °C) for the thin (3.2 mm) pouch cell investigated here, thus justifying the use of only one electrode pair. The present modeling framework can also be applied to larger cells, where temporal and spatial temperature gradients can be considerably larger.

In the present model, the electrochemical reactions include two intercalation/de-intercalation charge-transfer reactions (NMC and graphite), the reversible plating and the two SEI formation reactions (with Li⁺ ions and Li metal as reactants, respectively). Required model parameters are rate coefficients and activation energies of all reactions as well as the double-layer capacitances of the two electrodes.

The reaction mechanism and rate coefficients are given in Table I. Intercalation/de-intercalation parameters (cf. Table I, Reactions 1, 5) were determined by fitting simulated EIS data to experimental values for individual temperatures, while activation energies were determined from Arrhenius plots (see Appendix). For the reversible plating reaction (cf. Table I, Reaction 2), we adopted the plating kinetics from our previous work ²¹ which had previously been derived by Ecker. ³⁵ The model includes SEI formation via two different pathways ¹¹ : one is "electrolyte-driven" (cf. Table I, Reaction 3) and forms SEI via the reaction of Li⁺ ions and electrons with the electrolyte. The rate expression includes accelerated kinetics from SEI cracking due to volume changes of the graphite particles during cycling ^8,19 . The other pathway, "plating-driven" (cf. Table I, Reaction 4), operates via a direct reaction of plated metallic lithium with electrolyte without charge transfer. Microscopically, this reaction takes place on the surface of plated lithium. In order to describe the positive feedback between plating and SEI formation, we set the plating-driven SEI formation rate proportional to the lithium volume fraction, that is, ${{\boldsymbol{k}}}_{{\bf{f}}}^{{\boldsymbol{0}}}\,$∼${{\boldsymbol{\varepsilon }}}_{{\bf{Li}}}.$ The SEI itself is assumed as ideally uniform in morphology and chemical composition, consisting of only lithium ethylene dicarbonate (LEDC), (CH₂OCO₂Li)

Table I. Interfacial chemical reactions and rate coefficients ("elyt-driven" = electrolyte-driven, "LP-driven" = lithium plating-driven).

No.	Electrode	Reaction	Label	Rate coefficient	Activation energy	Symmetry factor
(1)	Negative	Li+[elyt] + e– + V[C6] ⇄ Li[C6]	Intercalation	${{\boldsymbol{i}}}^{00}$ = 1.86·1013 A m−2	75.4 kJ mol−1. 39	0.5
(2)	Negative	Li+[elyt] + e– ⇄ Li[metal]	Plating	${{\boldsymbol{i}}}^{00}$ = 2.29·1013 A m−221,35	65.0 kJ mol−1. 35	0.492. 35
(3)	Negative	Li+[elyt] + C3H4O3[elyt] + e– ⇄ 0.5 LEDC[SEI] + 0.5 C2H4	SEI formation (elyt-driven)	${{\boldsymbol{k}}}_{{\bf{f}}}^{0}$ = $\left(\displaystyle \frac{1}{{{\boldsymbol{\delta }}}_{{\bf{S}}{\bf{E}}{\bf{I}}}}+{{\boldsymbol{k}}}_{{\bf{c}}}\cdot {{\boldsymbol{i}}}_{{\bf{c}}{\bf{h}}{\bf{g}}}^{{\boldsymbol{V}}}\cdot \displaystyle \frac{{\bf{d}}{{\boldsymbol{\sigma }}}_{{\bf{t}},{\bf{S}}{\bf{E}}{\bf{I}}}}{{\bf{d}}{{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}}\right)\cdot 8.65\cdot {10}^{-11}$ mol/(m2·s). 8 , a)	55.5 kJ mol−1. 8	0.5 8
(4)	Negative	Li[metal] + C3H4O3[elyt] ⇄ 0.5 LEDC[SEI] + 0.5 C2H4	SEI formation (LP-driven)	${{\boldsymbol{k}}}_{{\bf{f}}}^{0}\,$= ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}\cdot$4.65·10-7 mol/(m2·s) b)	5.0 kJ mol−1. b)	0.492 35
(5)	Positive	Li+[elyt] + e– + V[NMC] ⇄ Li[NMC]	Intercalation	${{\boldsymbol{i}}}^{00}$ = 1.17 ·1011 A m−2	61.4 kJ mol−1. 39	0.5

^a)Reaction rate set to represent both calendaric and cyclic SEI formation, see Ref.^8,11. ^b)Reaction rate set proportional to plated lithium volume fraction ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}.$ Values fitted to simulate experimental aging data.

All charge-transfer reactions are modeled with Butler-Volmer kinetics, while plating-driven SEI formation is a thermochemical reaction where the rate is described by standard mass-action kinetics (cf. Appendix). The kinetic parameters are area-specific, and we assume the reaction surface area to be constant and independent of lithium volume fraction. Hence, the present model does not consider effects like blocking of the graphite surface by metallic lithium or detachment of metallic lithium fragments.

The P3D model presented above is implemented in the in-house multiphysics software package DENIS (Detailed Electrochemistry and Numerical Impedance Simulation) ⁴⁰ and numerically solved using the implicit time-adaptive solver LIMEX. ^45,46 The spatial derivatives were discretized in a finite-volume scheme, using 20, 19 and 11 non-equidistant control volumes on the x, y and z scales, respectively.

The chemical thermodynamics and kinetics are evaluated with the open-source code Cantera ⁴⁴ (version 2.5.0a3), which is coupled to the DENIS transport models via the chemistry source terms. An in-depth explanation about modeling lithium-ion batteries with Cantera can be found in Mayur et al., ⁴⁷ further details are given in the Appendix.

MATLAB (version 2019a) is the interface for controlling all DENIS simulations, as well as for data evaluation and visualization. Electrochemical impedance simulations were performed using a current step/voltage relaxation protocol and subsequent Fourier transform. ⁴⁸

All experiments used 3.7 V 500 mAh high-energy lithium-ion pouch cells by the manufacturer Hunan CTS Technology Co., Ltd, China, cell model 333450. A total of 200 cells were purchased and 19 individual cells from the same batch were used for this study. The cell chemistry is graphite at the NE and NMC at the PE. The manufacturer specifies upper and lower cut-off voltages of 4.2 V and 3.0 V.

A pouch cell in pristine state was discharged down to the lower voltage limit of 3.0 V with a constant current of 0.012 C, then disassembled in an Argon-filled glove box under a monitored atmosphere with a water and oxygen content of less than 0.1 ppm. The electrodes and separator were rinsed with dimethyl carbonate (DMC) to remove all electrolyte residues. The total active electrode area was measured with a ruler, and the thicknesses of NE, PE, separator, and current collector were measured with a digital micrometer. Additionally, the mean particle size of both electrodes was determined optically with a Keyence Light Microscope VHX-7000. The measured values of the aforementioned parameters are listed in Table VI.

Half-cells measurements were carried out using PAT-Cell test cells from the company EL Cell, Germany. Harvested electrode rolls from the disassembled pouch cell were used for the assembly of experimental half-cells. Since the electrode rolls are coated with AM on both sides, we used N-Methyl-2-pyrrolidone (NMP) to remove the coating from one side. Next, the electrodes were punched into circular disks with 18 mm diameter, followed by a rinsing step in dimethyl carbonate (DMC) to remove any impurity. Pure lithium metal disks with the same diameter of 18 mm were prepared as counter electrodes. Apart from the electrodes, the test cell setup includes a PAT-Core, which comprises of a polypropylene insulation sleeve, a 220 μm thick polypropylene (PP) fibre/polyethylene (PE) membrane separator, a reed contact and a lithium metal reference ring. For electrolyte, we used 103 μl of 1 M LiPF₆ in EC:DMC 1:1 (v:v).

After the assembling of the experimental cells, a formation process was carried out with two constant current (CC with 4.4 mA) cycles and a third constant current-constant voltage (CC–CV) cycle. PE half-cells were cycled between 3.0 V and 4.3 V, whereas the voltage range for NE half-cells was 0.05 V to 0.8 V. After the cell formation, the OCV curves were obtained by applying a constant current of 0.088 mA in both charging and discharging directions. Both formation and OCV measurement were conducted using a battery cycler BaSyTec CTS LAB XL and inside a Friocell 55 climate chamber at 20 °C.

Electrochemical investigations of full cells were carried out with a battery cycler (BaSyTec CTS Lab XL) combined with an impedance analyzer (Zahner IM6 Electrochemical Workstation) in a climate chamber (MMM Group Fricocell 55). The investigated cell was placed between two aluminum plates, each with a thickness of 10 mm, to ensure symmetrical convection condition as the cell heats up under load. Due to the good thermal conductivity of aluminum, heat generated in the cell can be transferred to the surrounding air. Surface temperature of the cell was measured using an NTC sensor, which is embedded in the center of the base aluminum plate.

Electrochemical impedance spectroscopy (EIS) measurements were performed at different SOCs (20%, 35%, 50%, 65%, 80%, 100%) and at different temperatures (5 °C, 20 °C, 35 °C, 50 °C). The EIS measurements were recorded in galvanostatic mode by applying an excitation current of 20 mA within the frequency range of 10⁻² Hz to 10⁵ Hz. Before carrying out the EIS measurements, a discharge curve of the cell was obtained by discharging the cell with a small current of 50 mA (0.1 C) from a fully charged state at 20 °C (Step I). Based on the discharge curve, the quasi-open circuit voltages at the respective SOCs were obtained (Step II), with these values served as the cut-off voltages for the SOC adjustment in Step IV. For the first EIS measurement at 20 °C, the cell was charged to 100% SOC using a CCCV protocol (0.1 C, with cut-off current C/20), followed by a 1 h rest period (Step III). After the completion of the EIS measurement at 100% SOC, SOC adjustment (100% SOC to 80% SOC) was carried out by discharging the cell with a CCCV protocol (0.1 C, with cut-off current C/20), followed by a rest period of 1 h before carrying out the EIS measurement (Step IV). This SOC adjustment protocol was applied to the subsequent measurements, i.e. at 65%, 50%, 35% and 20% SOC (Step V). After the completion of the last EIS measurement at 20% SOC and 20 °C, the temperature was increased to 35 °C (Step VI). Step I to Step VI were repeated for the EIS measurements at 35 °C, 50 °C and 5 °C. At 5 °C, rest period of 1 h was insufficient for the cell to achieve a steady-state OCV after the adjustment to 100% SOC, thus, a total rest period of 5 h was applied.

Three different types of time-domain experiments were carried out in order to characterize the electrochemical and aging behavior, subsequently denoted as protocols A, B and C. In protocol A, the characteristics of the cell under non-aging conditions were investigated at different temperatures (5 °C, 20 °C, 35 °C, 50 °C) and at varying discharge current (0.05 C, 0.5 C, 1 C, 2 C, 3 C). The measurements were carried out by following a CCCV protocol in both charging and discharging directions with CV cut-off current C/20. To prevent possible aging effects caused by lithium plating, the charging current for every cycle was kept at a low current (0.2 C, with cut-off current C/20). For the same reason, we cycled the cells in such order: 20 °C, 35 °C, 50 °C, 5 °C, keeping measurements at 5 °C to the last, since a cell cycled at low temperature is more susceptible to lithium plating. Starting from a fully-discharged cell, successive cycles were performed for each temperature, with a rest period of 10 h allowed during the temperature change.

To investigate the cell behavior in the presence of aging, we carried two additional experiments. In protocol B, charge-discharge cycles were recorded at different temperatures (20 °C, 35 °C, 50 °C, 5 °C) and various C-rates (0.5 C, 1 C, 2 C, 5 C) in both charging and discharging directions. By applying charging currents greater than 1 C (thereby exceeding the manufacturer's recommended maximum charging current of 1 C), we induced aging due to lithium plating. Additionally, to capture the electrochemical behavior caused by calendar aging, we performed the same charge-discharge protocol at 20 °C with C-rates of 0.5 C, 1 C, 2 C, 5 C after a total storage time of 5 months at 20 °C.

Finally, for protocol C, cycles with discharge at a fixed C-rate (1 C) and varying charge rates were performed at 20 °C. Starting from a fully-discharged cell, successive cycles were performed with the following protocol: various CC charge (0.05 C, 0.5 C, 1 C, 2 C, 4 C) and fixed 1 C CCCV discharge, with cut-off current C/20.

Voltage relaxation ³³ and cell expansion ⁴⁹ were introduced as methods for detecting lithium plating. Both cell cycling and data acquisition of cell thickness were carried out with a BaSyTec CTS Lab XL to validate the simulation results.

The voltage relaxation measurement was performed on two cells, respectively at 5 °C and 20 °C, by charging with CC (0.5 C, 1 C and 2 C) from a discharged state to the upper voltage limit of 4.2 V, then followed by a 2 h rest period to monitor the voltage relaxation of the cells. At 20 °C, an evident voltage plateau was observed in the relaxation curve following 2 C CC charge, indicating the presence of plating from the charging step. For the measurements at 5 °C, no voltage plateau was detected at any of the charging rates.

A short cycling test was carried out at 5 °C on a third cell with a variation of charging current amplitudes (0.5 C, 1 C and 2 C) in sequence. For each charging C-rate, the cell was cycled for five times, following a CCCV charge—2 h rest—1 C CCCV discharge—2 h rest protocol. C/20 CV cut-off was used for (dis)charging. Irreversible cell expansion was observed at all three charging currents of 0.5 C (0.143 mm), 1 C (0.147 mm) and 2 C (0.213 mm).

An additional long cyclic aging test was performed as follows. Two cells were cycled at 20 °C for a total of 100 cycles, following a CCCV charge–30 min rest—CCCV discharge—30 min rest protocol. Apart from the charging current (1 C and 2 C, respectively), both cells had the same discharging current of 1 C and cut-off current of C/20. Minor capacity fade and smaller increase in cell thickness was detected at 1 C, with the capacity decreasing by 0.71% after 50 cycles and a total of 2.80% after 100 cycles. On the other hand, the cell charged at 2 C already experienced a capacity fade of 5.8% after only 5 cycles, with the decrease in capacity continued steadily to 13.47% after 100 cycles. The cell thickness was also measured at the end of 100 cycles, with the cells charged at 1 C and 2 C showing an increase by 0.09 mm and 0.17 mm, respectively.

In Fig. 3 simulated CCCV discharge curves are compared with experimental data at different temperatures and C-rates. Simulations were performed at different temperatures (20 °C, 35 °C, 50 °C, 5 °C) with protocol A described in "Electrochemical measurements of full cells" (CCCV charge at a fixed low C-rate of 0.2 C followed by variable CCCV discharge at 0.05 C, 0.5 C, 1 C, 2 C and 3 C, with cut-off current C/20). The CCCV charge and the rest period of 10 h are not shown in Fig. 3. The results of our simulations show a qualitatively good agreement with experiments. The model fidelity is generally better at lower C-rates compared to higher C-rates. In particular, the broadening and loss of features in the discharge curves as well as the increasing asymmetry between charge and discharge curves ⁵⁰ observed experimentally with increasing C-rate is not well reproduced by the model. It is noteworthy that our model reproduces very well the decrease in charge capacity at 5 °C (max charge throughput at 506 mAh at 0.05 C in Fig. 3d), but it is not able to reproduce the small increase observed at 35 °C and 50 °C (531 and 535 mAh at 0.05 C, respectively Figs. 3b and 3c).

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In Fig. 4 simulated CCCV charge-discharge cycles for protocol B (cf. "Electrochemical measurements of full cells") are compared with experimental data at different temperatures and various C-rates for both discharging and charging (0.5 C, 1 C, 2 C, 5 C). Starting from a fully discharged cell, successive cycles are performed for each temperature: 20 °C (a), 35 °C (b), 50 °C (c), 5°C (d) and then 20°C again (e), after a simulated break to replicate the experimental 5 months storage time (see "Electrochemical measurements of full cells"). Both simulations and experiments were carried out according to the following protocol: CCCV charging (cut-off current C/20) - CCCV discharging (cut-off current C/20), with a rest period of 10 h only allowed during the temperature change.

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Figure 4 shows severe aging during cycling with an evident decrease in charge capacity at C-rates > 1 C as early as 20 °C (Panel a): it can be concluded that the cell is strongly susceptible to plating and SEI formation. In Panels b) and c), respectively 35 °C and 50 °C, the charge capacity does not decrease with cycling but at 5 °C (Panel d) we observe quite a capacity drop, with the charge capacity at 2 C being as low as ∼380 mAh (at 5 C simulations underpredict the decrease observed experimentally - 370 mAh vs 336 mAh). It is possible to deduce the presence of plating by observing some peculiarities in the simulated discharge cycles at 5 C 20°C (Panel a) and for all C-rates at 5°C (Panel d): these "bumps" below 0.1 Ah correspond to voltage plateaus, due to the oxidation and eventual re-intercalation of the plated lithium formed during CCCV charging ^26–36 . In Fig. 4e we observe an increase again in the charge capacity, which is likely a matter of cycling at higher temperature (20°C for the aged cell vs the previous at 5 °C). Simulations here overpredict the capacity increase observed experimentally (for all C-rates, ∼390 mAh vs ∼360 mAh). To sum up the discussion of Fig. 4, the results of our simulations show generally a good agreement with the experiments for the wide range of C-rates and temperatures applied here.

The physicochemical model allows an insight into internal battery states. As the present model includes a continuity equation for both SEI and metallic lithium, ²¹ we can access the volume fractions during cycling. The data series shown in Fig. 4 is resumed in Fig. 5 by plotting selected observables and internal states over the C-rates for all temperatures. Each data point represents a single cycle, with overall experimental time progressing from left to right (i.e., the experiments started at 20 °C and 0.5 C, continued with 20 °C and 1 C, etc.). Panel (a) (simulations and experiments compared) shows the charge capacity for each individual charge. These data illustrate nicely how certain experimental conditions (in particular, low temperatures and high C-rates) lead to a decrease of capacity which irreversibly progresses into the following cycle. The data also illustrate the capability of the model to reproduce the experimental observations, that is, the model ages in a similar way than the experiments.

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Panel (b) and Panel (c) of Fig. 5 (both simulations only) show respectively the increasing SEI thickness and the maximum plated lithium volume fraction during each cycle. Panel (b) displays how the SEI layer thickens during cycling at 20 °C and, even more so, at 5 °C; at 35 °C, 50 °C and 20 °C (aged) no increase happens, consequently we can explain the capacity fade in Figs. 4a, 4d with the lithium-consuming SEI formation. On the other hand, plating in Panel (c) does not follow a constant trend but, being reversible, is caused by a combination of thermodynamics and kinetics according to cycling temperature. The highest plated lithium volume fraction is predicted at 5 °C and high C-rates, while at 50 °C, no plating occurs at all. As the plating-induced SEI formation rate is assumed proportional to the lithium volume fraction, large lithium volume fractions cause a fast increase of SEI thickness. It is interesting to note that the two 20 °C data sets show a different experimental behavior. Capacity loss after 2 C and 5 C rates is observed only for the initial data set, but not for the final one. The simulations are able to reproduce this finding.

According to protocol C (cf. "Electrochemical measurements of full cells"), experiments and simulations of consecutive cycles with varying CC charge rate (0.05, 0.5 C, 1 C, 2 C, 4 C) and CCCV discharge at a fixed C-rate (1 C, cut-off current C/20) were performed at 20 °C. In Fig. 6 the results are compared with the experiments (cf. "Experimental Methodology"). The charge throughput values of the curves in Fig. 6 are normalized at the end of CC charge (i.e., all cycles begin at 0 Ah to better show the progressive capacity loss due to fast charging). The results of our simulations show generally good agreement with the experiments, in particular including at 2 C and 4 C: the specific appearance below 0.1 Ah of the discharge curves at higher C-rates, which can be seen on the left side of Fig. 6, is again due to the discharge plateau, ^26–36 a plating hint associated with the simultaneous oxidation of deposited lithium and re-intercalation into the NE AM. Note the simulations show a shorter plateau as compared to the experiments.

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Overall, the macroscopic electrochemical behavior of the simulated cell agrees well with the experimental data for all the performed protocols, over the entire range of SOCs, C-rates, and temperatures studied. Worth noting, this is achieved with a single set of model parameters. We therefore conclude the general ability of the model to describe the behavior of the present experiments and next use it for predicting aging under additional operating conditions. It is worth noting that the aging model is not necessarily unique, because additional aging mechanisms not included model may also cause or at least contribute to the observed behavior. For example, the aging reactions used in the present model (SEI formation and lithium plating) lead to LLI as only degradation mode, while mechanisms leading to LAM (e.g., decomposition of active material or electrode dry-out) are not included. Still, irreversible lithium plating is typically considered the dominating mechanism for the conditions studied here (charging at fast rates and/or low temperatures). ^6,51,52 It is therefore a reasonable simplification not to include additional aging mechanisms.

In order to visualize the influence of operation conditions on cell aging, we have introduced before the concept of a aging colormap. ^11,21,22 These maps use a color look-up table to show aging (or other cell properties) as function of both temperature and charging current. Often they show a relatively sharp transition between harming and non-harming conditions.

We have carried out systematic simulations over a wide range of temperatures (−20 °C...60 °C in increments of 5 °C) and C-rates (0.05...2 C in increments of 0.5 C). The cell is pre-charged up to 100% SOC with a constant current of 0.02 C, hence simulations start with a CCCV discharge to 3.0 V followed by 30 min rest, a CCCV charge to 4.2 V (cut-off current C/20) and again final 30 min rest. In Fig. 7a the temperature increase ${{T}}_{max}\,-\,{{T}}_{amb}$ happening during cycling is shown as a function of temperature and C-rate: a slight warming can be observed for the high C-rates, with the cell reaching a +1 °C difference at 2 C and 5 °C. The temperature values represent averages over the cell thickness (x scale in Fig. 2): it should be noted that the predicted temperature gradient along the cell thickness is small (<0.1 °C for all conditions, results not shown) which is due to the small dimension of the investigated cell (3.2 mm thickness). The rather small temporal and spatial temperature gradients would likely allow to omit the thermal scale and use an isothermal aging-sensitive model to get similar results; yet this was not attempted here in order to keep the generalized P3D framework that we have applied before to other, larger cells with higher thermal gradients (e.g., a 26650 cylindrical cell with >10 °C temperature increase at 10 C ⁴⁰ ).

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Figure 7b shows the maximum lithium volume fraction ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}^{{\bf{m}}{\bf{a}}{\bf{x}}}$ reached during cycling as a function of temperature and C-rate. There is a clear transition between no-plating conditions (${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}^{{\bf{m}}{\bf{a}}{\bf{x}}}=0$) and plating conditions, for which ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}^{{\bf{m}}{\bf{a}}{\bf{x}}}\approx 0.035$ over a wide range of conditions. For constant C-rate, the transition is relatively sharp for decreasing temperature; at 1 C, it occurs between ca. 5 °C and 15 °C. For constant temperature, the transition is smooth for increasing C-rate; at 20 °C, plating starts above ca. 1.5 C (simulated ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}^{{\bf{m}}{\bf{a}}{\bf{x}}}=0.01$ at 20 °C 2 C); at 5 °C, plating starts above ca. 0.2 C.

Figure 7b correlates qualitatively with the experimental results discussed in "Detection of lithium plating", in which an evident voltage plateau was observed in the relaxation curve at 20 °C at 2 C, but not at 1 C or 0.5 C. Also, the long cycling test at 20 °C showed little aging at 1 C, but considerable aging at 2 C. At 5 °C, no voltage plateau was experimentally detected at any of the charging rates (0.5 C, 1 C and 2 C) during the relaxation measurements, while in the simulations plating is clearly predicted (simulated ${\varepsilon }_{{\rm{Li}}}^{{\rm{\max }}}=0.03$ at 5 °C 2 C): the absence of an experimental plateau could be due to a possible plating irreversibility at this temperature, and/or slow kinetics. However, the increase in cell thickness observed during the short cycling measurements at 5 °C at all investigated C-rates (0.5 C, 1 C, 2 C) ("Detection of lithium plating") does indicate the presence plating, consistent with the simulations.

In Figure 7c the SEI volume fraction ${\varepsilon }_{{\rm{SEI}}}^{{\rm{\max }}}$ formed within one CCCV cycle is shown: the highest value (∼0.12) is here found at −20 °C for C-rates ≥ 0.5 C, with the SEI produced mainly at low temperatures via the plating-driven reaction (cf. Table I, Reaction 4). This observation correlates well with the previous discussion on the comparison between simulations and cell expansion measurements from "Detection of lithium plating".

Similarly to our previous work, ¹¹ we can also describe aging in a practically meaningful way by converting the predicted SEI formation during the simulated cycle to capacity loss in % per day CL_D according to

$$\begin{eqnarray}&&{{\rm{CL}}}_{{\rm{D}}}\left({T}{,}{I}\right)=\left({F}\cdot {{z}}_{{\rm{SEI}}}\cdot {\rm{d}}{{n}}_{{\rm{SEI}}}/{{C}}_{{\bf{N}}}\right)\cdot {{\boldsymbol{N}}}_{{\rm{D}}}\cdot 100 \% .\end{eqnarray} \tag{ 1 }$$

Here, ${\bf{d}}{{\boldsymbol{n}}}_{{\bf{S}}{\bf{E}}{\bf{I}}}$ are the SEI moles formed during one cycle (summed over the complete NE thickness), ${{\boldsymbol{z}}}_{{\bf{S}}{\bf{E}}{\bf{I}}}$ is the moles of lithium ions bound in one mole of SEI (for LEDC, ${\left({\bf{C}}{{\bf{H}}}_{2}{\bf{O}}{\bf{C}}{{\bf{O}}}_{2}{\bf{L}}{\bf{i}}\right)}_{2},$ ${{\boldsymbol{z}}}_{{\bf{S}}{\bf{E}}{\bf{I}}}\,$ = 2), ${{\boldsymbol{C}}}_{{\bf{N}}}$ is the nominal capacity of the cell and N_D is the calculated number of cycles in one day (assuming continuous cycling of the battery, as per protocol). ${\bf{C}}{{\bf{L}}}_{{\bf{D}}}$ is directly proportional to the volume fraction of SEI formed during the cycle and, by referring to a fixed amount of cycling time (one day), includes the difference in cycling time for the compared C-rates. Figure 7d shows CL_D as a function of temperature (−20 °C...60 °C) and C-rate (0.05...2 C). It shows how easily this particular cell is prone to aging, since almost 100% capacity loss can be already observed after one day of continuous cycling at 1 C at −5 °C (${\bf{C}}{{\bf{L}}}_{{\bf{D}}}\,$ = 97%) and a full 100% capacity loss can be found at 2 C for −10 °C ≤ T ≤ 10 °C. As expected from the plating volume fraction ${{\boldsymbol{\varepsilon }}}_{{\bf{L}}{\bf{i}}}^{{\bf{m}}{\bf{a}}{\bf{x}}}$ (Panel a), the limit between aging/no aging zones tends to move towards higher temperatures with increasing C-rates. Different to ${{\boldsymbol{\varepsilon }}}_{{\bf{S}}{\bf{E}}{\bf{I}}}^{{\bf{m}}{\bf{a}}{\bf{x}}},$ CL_D shows decreasing values when further cooling down (∼64% at T = −20 °C for C-rates ≥ 1 C). This is due to the N_D factor, which includes in Eq. 1 the time length of CCCV cycles, hence the lower number of cycles per day at low C-rates and at cold temperatures for C-rates ≥ 1 C.

We can now compare qualitatively the long cyclic aging test results presented in "Detection of lithium plating" with simulated values calculated using ${\bf{C}}{{\bf{L}}}_{{\bf{D}}}$ from Fig. 7d. At 1 C the experimental capacity was seen decreasing by 0.71% after 50 cycles and a total of 2.80% after 100 cycles, while the simulated cell, when extrapolated from Fig. 7d, would show a higher capacity loss both at 50 cycles (2.6%) and 100 cycles (5.19%). At 2 C the experimental cell experienced a capacity fade of 5.8% after 5 cycles, while the simulated capacity loss would be 12.5%. After 100 cycles the simulation predicts a total failure of the cell (>100% capacity loss), while the experimental capacity fade was observed to be no more than 13.5%. While the simulations follow qualitatively the same trends as the experiments, the quantitative mismatch is obvious. This might be explained as follows. The parameterization of the model has been carried out using accelerated aging experiments down to ca. 30% capacity loss. As shown in Fig. 5, the experimental cell used for those experiments exhibited considerable capacity loss even at 20 °C. This is inconsistent with the experimental long cycling tests. This self-inconsistency of the different experimental data sets leads to the inconsistency of the simulations.

During experimental aging, SEI formation is known to slow down, with different SEI self-passivation mechanisms expected to dominate at different time scales, ^53,54 including diffusion-limited film growth and electrolyte consumption (EC is a main reactant in the SEI formation reaction) with increasing cycling time. ⁸ This effect is not captured in the simulations leading to Fig. 7d. Therefore, the results shown in Fig. 7d represent a cell close to beginning-of-life, where the longer-term cycling effects, including the aforementioned SEI self-passivation, are not captured.

Lithium-ion batteries are also known to exhibit nonlinear aging and develop "knees" towards end-of-life, as has been reviewed recently by Attia et al. ⁵¹ This is usually a result of the interaction between several aging mechanisms. The present model would in principle be able to describe such interaction (e.g., the impact of internal resistance increase due to SEI formation on half-cell potential and therefore the plating kinetics), but this would require long-term simulations ⁸ which is out of scope of the present study. Instead, the capacity loss shown in Fig. 7d is extrapolated from two consecutive simulated cycles of a fresh cell. Therefore, it must be noted again that the simulation results represent the aging behavior of the investigated cell at beginning-of-life.

Serena Carelli: Conceptualization, Methodology, Validation, Writing - Original draft preparation. Yan Ying Lee: Investigation, Writing - Original draft preparation. André Weber: Writing—Review and Editing, Supervision, Funding acquisition. Wolfgang G. Bessler: Conceptualization, Software, Writing—Review and Editing, Supervision, Funding acquisition.

The model features a pseudo-3D domain (cell scale, electrode pair scale, particle scale), see Fig. 2. All model equations are given in Table III. A list of symbols is provided . On the cell level (x scale in Fig. 2), heat transport is modeled in one dimension as conduction along the cell thickness and holder plates, using convective and radiative heat transfer as boundary conditions. This dimension runs perpendicular to the electrode sheets of the pouch cell. On the electrode pair level (y scale in Fig. 2), mass and charge transport of Li⁺ and PF₆ ^– ions in the liquid electrolyte and electrons in the solid electrode components is modeled in one dimension along the thickness of the electrode pair. Again, this is perpendicular to the electrode sheet area. For ion transport, we describe species fluxes due to migration and diffusion with a Nernst-Planck approach with concentration- and temperature-dependent diffusion coefficients. Electronic conductivity within the electrodes is assumed high and not rate-limiting. On the particle level (z scale in Fig. 2), the diffusion of intercalated lithium atoms in the bulk of the AM particles is modeled using a simple Fickian diffusion approach. The diffusion coefficients are assumed concentration and temperature dependent.

Table III. Summary of all model equations.

Macroscale (x direction): Heat transport in cell
Energy conservation	${\boldsymbol{\rho }}{{\boldsymbol{c}}}_{{\bf{p}}}\displaystyle \frac{\partial {\boldsymbol{T}}}{\partial {\boldsymbol{t}}}=\displaystyle \frac{\,\partial }{\partial {\boldsymbol{x}}}\,\left({\boldsymbol{\lambda }}\displaystyle \frac{\partial {\boldsymbol{T}}}{\partial {\boldsymbol{x}}}\right)+{\dot{{\boldsymbol{q}}}}^{{\bf{V}}}$
Heat flux at cell surface	${{\boldsymbol{J}}}_{{\boldsymbol{q}}}={\boldsymbol{\alpha }}\left({\boldsymbol{T}}-{{\boldsymbol{T}}}_{{\bf{a}}{\bf{m}}{\bf{b}}}\right)+\epsilon {{\boldsymbol{\sigma }}}_{{\bf{S}}{\bf{B}}}({{\boldsymbol{T}}}^{4}-{{\boldsymbol{T}}}_{{\bf{a}}{\bf{m}}{\bf{b}}}^{4})$
Total heat sources	${\dot{{\boldsymbol{q}}}}^{{\boldsymbol{V}}}=\displaystyle \frac{{{\boldsymbol{A}}}_{{\bf{e}}}}{{{\boldsymbol{V}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}}\left(\underset{{{\boldsymbol{L}}}_{{\bf{E}}{\bf{P}}}}{\overset{0}{\int }}\left({\dot{{\boldsymbol{q}}}}_{{\bf{c}}{\bf{h}}{\bf{e}}{\bf{m}}}\left({\boldsymbol{y}}\right)+{\dot{{\boldsymbol{q}}}}_{{\bf{o}}{\bf{h}}{\bf{m}}}\left({\boldsymbol{y}}\right)\right){\bf{d}}{\boldsymbol{y}}+{{\boldsymbol{R}}}_{{\bf{c}}{\bf{c}}}{{\boldsymbol{i}}}^{2}\right)$
Chemistry heat source	${\dot{{\boldsymbol{q}}}}_{{\bf{c}}{\bf{h}}{\bf{e}}{\bf{m}}}=\displaystyle \sum _{{{\boldsymbol{N}}}_{{\bf{r}}}}^{{\boldsymbol{n}}=1}({{\boldsymbol{r}}}_{{\boldsymbol{n}}}{{\boldsymbol{A}}}_{{\boldsymbol{n}}}^{{\bf{V}}}(-{\boldsymbol{\Delta }}{{\boldsymbol{H}}}_{{\boldsymbol{n}}}+{\boldsymbol{F}}{{\boldsymbol{\nu }}}_{{\bf{e}},{\boldsymbol{n}}}{\boldsymbol{\Delta }}{{\boldsymbol{\Phi }}}_{{\boldsymbol{n}}}))$
Ohmic heating	${\dot{{\boldsymbol{q}}}}_{{\bf{o}}{\bf{h}}{\bf{m}}}={\sigma }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}\cdot {\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{\partial {\boldsymbol{y}}}\right)}^{2}$

Mesoscale (y direction): Mass and charge transport in electrode pair
Mass conservation of species i	$\displaystyle \frac{\partial \left({\varepsilon }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}\right)}{\partial {\boldsymbol{t}}}=-\displaystyle \frac{\partial {{\boldsymbol{J}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}+{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}}}^{{\bf{V}}}+{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}},{\bf{D}}{\bf{L}}}^{{\bf{V}}}$
Charge conservation	${{\boldsymbol{C}}}_{{\bf{D}}{\bf{L}}}^{{\boldsymbol{V}}}\displaystyle \frac{\partial \left({\rm{\Delta }}{\rm{\Phi }}\right)}{\partial {\boldsymbol{t}}}=\displaystyle {\sum }_{{\boldsymbol{i}}}{{z}}_{{\boldsymbol{i}}}{\boldsymbol{F}}\tfrac{\partial {{\boldsymbol{J}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}-{{\boldsymbol{i}}}_{{\boldsymbol{F}}}^{{\boldsymbol{V}}}$
Species fluxes (Nernst-Planck)	${{\boldsymbol{J}}}_{{\boldsymbol{i}}}=-{{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}-\displaystyle \frac{{{\boldsymbol{z}}}_{{\boldsymbol{i}}}{\boldsymbol{F}}}{{\boldsymbol{RT}}}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}{{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}\displaystyle \frac{\partial {{\boldsymbol{\Phi }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{\partial {\boldsymbol{y}}}$

Microscale (z direction): Mass transport in active materials particle

Mass conservation (Fick's 2nd law)

$\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}}{\partial {\boldsymbol{t}}}=\displaystyle \frac{1}{{{\boldsymbol{z}}}^{2}}\displaystyle \frac{\partial }{\partial {\boldsymbol{z}}}\left({{\boldsymbol{z}}}^{2}{{\boldsymbol{D}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}}{\partial {\boldsymbol{z}}}\right)$

Chemical kinetics and thermodynamics a)
Energy conservation	${\boldsymbol{\rho }}{{\boldsymbol{c}}}_{{\bf{p}}}\displaystyle \frac{\partial {\boldsymbol{T}}}{\partial {\boldsymbol{t}}}=\displaystyle \frac{\,\partial }{\partial {\boldsymbol{x}}}\,\left({\boldsymbol{\lambda }}\displaystyle \frac{\partial {\boldsymbol{T}}}{\partial {\boldsymbol{x}}}\right)+{\dot{{\boldsymbol{q}}}}^{{\bf{V}}}$
Heat flux at cell surface	${{\boldsymbol{J}}}_{{\boldsymbol{q}}}={\boldsymbol{\alpha }}\left({\boldsymbol{T}}-{{\boldsymbol{T}}}_{{\bf{a}}{\bf{m}}{\bf{b}}}\right)+\epsilon {{\boldsymbol{\sigma }}}_{{\bf{S}}{\bf{B}}}({{\boldsymbol{T}}}^{4}-{{\boldsymbol{T}}}_{{\bf{a}}{\bf{m}}{\bf{b}}}^{4})$
Total heat sources	${\dot{{\boldsymbol{q}}}}^{{\boldsymbol{V}}}=\displaystyle \frac{{{\boldsymbol{A}}}_{{\bf{e}}}}{{{\boldsymbol{V}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}}\left(\underset{{{\boldsymbol{L}}}_{{\bf{E}}{\bf{P}}}}{\overset{0}{\int }}\left({\dot{{\boldsymbol{q}}}}_{{\bf{c}}{\bf{h}}{\bf{e}}{\bf{m}}}\left({\boldsymbol{y}}\right)+{\dot{{\boldsymbol{q}}}}_{{\bf{o}}{\bf{h}}{\bf{m}}}\left({\boldsymbol{y}}\right)\right){\bf{d}}{\boldsymbol{y}}+{{\boldsymbol{R}}}_{{\bf{c}}{\bf{c}}}{{\boldsymbol{i}}}^{2}\right)$
Chemistry heat source	${\dot{{\boldsymbol{q}}}}_{{\bf{c}}{\bf{h}}{\bf{e}}{\bf{m}}}=\displaystyle \sum _{{{\boldsymbol{N}}}_{{\bf{r}}}}^{{\boldsymbol{n}}=1}({{\boldsymbol{r}}}_{{\boldsymbol{n}}}{{\boldsymbol{A}}}_{{\boldsymbol{n}}}^{{\bf{V}}}(-{\boldsymbol{\Delta }}{{\boldsymbol{H}}}_{{\boldsymbol{n}}}+{\boldsymbol{F}}{{\boldsymbol{\nu }}}_{{\bf{e}},{\boldsymbol{n}}}{\boldsymbol{\Delta }}{{\boldsymbol{\Phi }}}_{{\boldsymbol{n}}}))$
Ohmic heating	${\dot{{\boldsymbol{q}}}}_{{\bf{o}}{\bf{h}}{\bf{m}}}={\sigma }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}\cdot {\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{\partial {\boldsymbol{y}}}\right)}^{2}$
Mesoscale (y direction): Mass and charge transport in electrode pair
Mass conservation of species i	$\displaystyle \frac{\partial \left({\varepsilon }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}\right)}{\partial {\boldsymbol{t}}}=-\displaystyle \frac{\partial {{\boldsymbol{J}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}+{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}}}^{{\bf{V}}}+{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}},{\bf{D}}{\bf{L}}}^{{\bf{V}}}$
Charge conservation	${{\boldsymbol{C}}}_{{\bf{D}}{\bf{L}}}^{{\boldsymbol{V}}}\displaystyle \frac{\partial \left({\rm{\Delta }}{\rm{\Phi }}\right)}{\partial {\boldsymbol{t}}}=\displaystyle {\sum }_{{\boldsymbol{i}}}{{z}}_{{\boldsymbol{i}}}{\boldsymbol{F}}\tfrac{\partial {{\boldsymbol{J}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}-{{\boldsymbol{i}}}_{{\boldsymbol{F}}}^{{\boldsymbol{V}}}$
Species fluxes (Nernst-Planck)	${{\boldsymbol{J}}}_{{\boldsymbol{i}}}=-{{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\boldsymbol{i}}}}{\partial {\boldsymbol{y}}}-\displaystyle \frac{{{\boldsymbol{z}}}_{{\boldsymbol{i}}}{\boldsymbol{F}}}{{\boldsymbol{RT}}}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}{{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}\displaystyle \frac{\partial {{\boldsymbol{\Phi }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{\partial {\boldsymbol{y}}}$
Microscale (z direction): Mass transport in active materials particle
Mass conservation (Fick's 2nd law)	$\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}}{\partial {\boldsymbol{t}}}=\displaystyle \frac{1}{{{\boldsymbol{z}}}^{2}}\displaystyle \frac{\partial }{\partial {\boldsymbol{z}}}\left({{\boldsymbol{z}}}^{2}{{\boldsymbol{D}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}\displaystyle \frac{\partial {{\boldsymbol{c}}}_{{\bf{L}}{\bf{i}},{\bf{A}}{\bf{M}}}}{\partial {\boldsymbol{z}}}\right)$
Chemical kinetics and thermodynamics a)
Interfacial rate of electrochemical reaction n (Butler-Volmer)	${{\boldsymbol{r}}}_{{\boldsymbol{n}}}=\displaystyle \frac{{{{\boldsymbol{i}}}_{{\boldsymbol{n}}}}^{0}}{{\boldsymbol{F}}}\left[\exp \left(\displaystyle \frac{{{\boldsymbol{\alpha }}}_{{\bf{c}}}{\boldsymbol{zF}}}{{\boldsymbol{RT}}}{{\boldsymbol{\eta }}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}\right)-\exp \left(-\displaystyle \frac{\left(1-{{\boldsymbol{\alpha }}}_{{\bf{c}}}\right){\boldsymbol{zF}}}{{\boldsymbol{RT}}}{{\boldsymbol{\eta }}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}\right)\right]$
Exchange current density	${{\boldsymbol{i}}}_{{\boldsymbol{n}}}^{0}={{\boldsymbol{i}}}_{{\boldsymbol{n}}}^{00}\cdot \exp \left(-\displaystyle \frac{{{\boldsymbol{E}}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}}{{\boldsymbol{RT}}}\right)\cdot \displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{R}}}}^{{\boldsymbol{i}}=1}{\left(\displaystyle \frac{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}}{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{0}}\right)}^{\left(1-{{\boldsymbol{\alpha }}}_{{\boldsymbol{c}},{\boldsymbol{n}}}\right)}\cdot \displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{P}}}}^{{\boldsymbol{i}}=1}{\left(\displaystyle \frac{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}}{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{0}}\right)}^{{{\boldsymbol{\alpha }}}_{{\boldsymbol{c}},{\boldsymbol{n}}}}$
Overpotential	${{\boldsymbol{\eta }}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}={\boldsymbol{\Delta }}{\phi }^{{\bf{e}}{\bf{f}}{\bf{f}}}-{\boldsymbol{\Delta }}{\phi }_{{\boldsymbol{n}}}^{{\bf{e}}{\bf{q}}}={\boldsymbol{\Delta }}\phi -{{\boldsymbol{R}}}_{{\bf{S}}{\bf{E}}{\bf{I}}}^{{\bf{V}}}{{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}-{\boldsymbol{\Delta }}{\phi }_{{\boldsymbol{n}}}^{{\bf{e}}{\bf{q}}}$
Interfacial rate of thermochemical reaction n (mass-action kinetics)	${{\boldsymbol{r}}}_{{\boldsymbol{n}}}={{\boldsymbol{k}}}_{{\bf{f}},{\boldsymbol{n}}}\displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{R}}}}^{{\boldsymbol{i}}=1}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{| {{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}| }-{{\boldsymbol{k}}}_{{\bf{r}},{\boldsymbol{n}}}\displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{P}}}}^{{\boldsymbol{i}}=1}{{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{| {{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}| }$
Forward rate constant	${{\boldsymbol{k}}}_{{\bf{f}},{\boldsymbol{n}}}={{\boldsymbol{k}}}_{{\bf{f}},{\boldsymbol{n}}}^{0}\cdot \exp \left(-\displaystyle \frac{{{\boldsymbol{E}}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}}{{\boldsymbol{RT}}}\right)$
Reverse rate constant	${{\boldsymbol{k}}}_{{\bf{r}},{\boldsymbol{n}}}={{\boldsymbol{k}}}_{{\bf{f}},{\boldsymbol{n}}}^{0}\cdot \exp \left(-\displaystyle \frac{{{\boldsymbol{E}}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\boldsymbol{n}}}}{{\boldsymbol{RT}}}\right)\cdot \exp \left(\displaystyle \frac{{\boldsymbol{\Delta }}{{\boldsymbol{G}}}_{{\boldsymbol{n}}}^{0}}{{\boldsymbol{RT}}}\right)\cdot \displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{R}}},{{\boldsymbol{N}}}_{{\bf{P}}}}^{{\boldsymbol{i}}=1}{\left({{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{0}\right)}^{-{{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}}$
Species source terms	${\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}}}^{{\boldsymbol{V}}}=\displaystyle {\sum }_{{\boldsymbol{n}}=1}^{{{\boldsymbol{N}}}_{{\bf{r}}}}\left({{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}{{\boldsymbol{r}}}_{{\boldsymbol{n}}}{{\boldsymbol{A}}}_{{\boldsymbol{n}}}^{{\boldsymbol{V}}}\right)$
Equilibrium potential (Nernst equation)	${\boldsymbol{\Delta }}{\phi }_{{\boldsymbol{n}}}^{{\bf{e}}{\bf{q}}}=-\displaystyle \frac{{\boldsymbol{\Delta }}{{\boldsymbol{G}}}_{{\boldsymbol{n}}}^{0}}{{\boldsymbol{zF}}}-\displaystyle \frac{{\boldsymbol{RT}}}{{\boldsymbol{zF}}}\,\mathrm{ln}\left(\displaystyle \prod _{{{\boldsymbol{N}}}_{{\bf{R}}},{{\boldsymbol{N}}}_{{\bf{P}}}}^{{\boldsymbol{i}}=1}{\left(\displaystyle \frac{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}}{{{\boldsymbol{c}}}_{{\boldsymbol{i}}}^{0}}\right)}^{{{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}}\right)$
Gibbs reaction energy	${\boldsymbol{\Delta }}{{\boldsymbol{G}}}_{{\boldsymbol{n}}}^{0}=\displaystyle \sum _{{{\boldsymbol{N}}}_{{\bf{R}}},{{\boldsymbol{N}}}_{{\bf{P}}}}^{{\boldsymbol{i}}=1}{{\boldsymbol{\nu }}}_{{\boldsymbol{i}}}{{\boldsymbol{\mu }}}_{{\bf{i}}}^{0}$
Standard-state chemical potential	${{\boldsymbol{\mu }}}_{{\boldsymbol{i}}}^{0}={{\boldsymbol{h}}}_{{\boldsymbol{i}}}^{0}-{\boldsymbol{T}}{{\boldsymbol{s}}}_{{\boldsymbol{i}}}^{0}+\left({\boldsymbol{p}}-{{\boldsymbol{p}}}_{{\bf{r}}{\bf{e}}{\bf{f}}}\right){{\boldsymbol{\Omega }}}_{{\boldsymbol{i}}}$
Current, voltage, potentials
Cell voltage	${\boldsymbol{E}}={\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}},{\bf{c}}{\bf{a}}}-{\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}},{\bf{a}}{\bf{n}}}-{\boldsymbol{i}}\cdot {{\boldsymbol{R}}}_{{\bf{c}}{\bf{c}}}$
Cell current	${{\boldsymbol{I}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}=\displaystyle \frac{{{\boldsymbol{A}}}_{{\bf{e}}}}{{{\boldsymbol{V}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}}\cdot \displaystyle {\int }_{{\boldsymbol{y}}=0}^{{{\boldsymbol{L}}}_{{\bf{e}}{\bf{l}}{\bf{e}}{\bf{c}}{\bf{t}}{\bf{r}}{\bf{o}}{\bf{d}}{\bf{e}}}}\left({{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}+{{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\boldsymbol{V}}}\right){\bf{d}}{\boldsymbol{y}}$
Faradaic current density	${{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}={\boldsymbol{F}}{\dot{{\boldsymbol{s}}}}_{{\bf{e}}}^{{\boldsymbol{V}}}=\displaystyle \sum _{{{\boldsymbol{N}}}_{{\bf{r}}}}^{{\boldsymbol{n}}=1}{\boldsymbol{F}}({{\boldsymbol{\nu }}}_{{\bf{e}},{\boldsymbol{n}}}{{\boldsymbol{r}}}_{{\boldsymbol{n}}}{{\boldsymbol{A}}}_{{\boldsymbol{n}}}^{{\boldsymbol{V}}})$
Double layer current density	${{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}={{\boldsymbol{C}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}\displaystyle \frac{{\bf{d}}\left({\boldsymbol{\Delta }}{\boldsymbol{\Phi }}\right)}{{\bf{d}}{\boldsymbol{t}}}$
Species source term from double layer	${\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}},{\bf{D}}{\bf{L}}}^{{\bf{V}}}=\displaystyle \frac{{{\boldsymbol{z}}}_{{\boldsymbol{i}}}}{{\boldsymbol{F}}}{{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}$ with ${\boldsymbol{i}}={\bf{L}}{{\bf{i}}}^{+}$
Potential step (both electrodes)	${\boldsymbol{\Delta }}\phi ={\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}}}-{\phi }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}$
Multi-phase management
Volume fraction of phases	$\displaystyle \frac{\partial \left({{\boldsymbol{\rho }}}_{{\boldsymbol{j}}}{{\boldsymbol{\varepsilon }}}_{{\boldsymbol{j}}}\right)}{\partial {\boldsymbol{t}}}=\displaystyle {\sum }_{{\boldsymbol{i}}=1}^{{{\boldsymbol{N}}}_{{\bf{R}},{\boldsymbol{j}}},{{\boldsymbol{N}}}_{{\bf{P}},{\boldsymbol{j}}}}{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}}}^{{\bf{V}}}{\boldsymbol{Mi}}$
Feedback on transport coefficients (porous electrode theory)	${{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}=\tfrac{{{\boldsymbol{\varepsilon }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{{{{\boldsymbol{\tau }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}^{2}}{{\boldsymbol{D}}}_{{\boldsymbol{i}}}$

Current, voltage, potentials
Cell voltage	${\boldsymbol{E}}={\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}},{\bf{c}}{\bf{a}}}-{\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}},{\bf{a}}{\bf{n}}}-{\boldsymbol{i}}\cdot {{\boldsymbol{R}}}_{{\bf{c}}{\bf{c}}}$
Cell current	${{\boldsymbol{I}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}=\displaystyle \frac{{{\boldsymbol{A}}}_{{\bf{e}}}}{{{\boldsymbol{V}}}_{{\bf{c}}{\bf{e}}{\bf{l}}{\bf{l}}}}\cdot \displaystyle {\int }_{{\boldsymbol{y}}=0}^{{{\boldsymbol{L}}}_{{\bf{e}}{\bf{l}}{\bf{e}}{\bf{c}}{\bf{t}}{\bf{r}}{\bf{o}}{\bf{d}}{\bf{e}}}}\left({{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}+{{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\boldsymbol{V}}}\right){\bf{d}}{\boldsymbol{y}}$
Faradaic current density	${{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}={\boldsymbol{F}}{\dot{{\boldsymbol{s}}}}_{{\bf{e}}}^{{\boldsymbol{V}}}=\displaystyle \sum _{{{\boldsymbol{N}}}_{{\bf{r}}}}^{{\boldsymbol{n}}=1}{\boldsymbol{F}}({{\boldsymbol{\nu }}}_{{\bf{e}},{\boldsymbol{n}}}{{\boldsymbol{r}}}_{{\boldsymbol{n}}}{{\boldsymbol{A}}}_{{\boldsymbol{n}}}^{{\boldsymbol{V}}})$
Double layer current density	${{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}={{\boldsymbol{C}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}\displaystyle \frac{{\bf{d}}\left({\boldsymbol{\Delta }}{\boldsymbol{\Phi }}\right)}{{\bf{d}}{\boldsymbol{t}}}$
Species source term from double layer	${\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}},{\bf{D}}{\bf{L}}}^{{\bf{V}}}=\displaystyle \frac{{{\boldsymbol{z}}}_{{\boldsymbol{i}}}}{{\boldsymbol{F}}}{{\boldsymbol{i}}}_{{\bf{D}}{\bf{L}}}^{{\bf{V}}}$ with ${\boldsymbol{i}}={\bf{L}}{{\bf{i}}}^{+}$
Potential step (both electrodes)	${\boldsymbol{\Delta }}\phi ={\phi }_{{\bf{e}}{\bf{l}}{\bf{d}}{\bf{e}}}-{\phi }_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}$

Multi-phase management
Volume fraction of phases	$\frac{\partial \left({{\boldsymbol{\rho }}}_{{\boldsymbol{j}}}{{\boldsymbol{\varepsilon }}}_{{\boldsymbol{j}}}\right)}{\partial {\boldsymbol{t}}}={\sum }_{{\boldsymbol{i}}=1}^{{{\boldsymbol{N}}}_{{\bf{R}},{\boldsymbol{j}}},{{\boldsymbol{N}}}_{{\bf{P}},{\boldsymbol{j}}}}{\dot{{\boldsymbol{s}}}}_{{\boldsymbol{i}}}^{{\bf{V}}}{{\boldsymbol{M}}}_{{\boldsymbol{i}}}$
Feedback on transport coefficients (porous electrode theory)	${{\boldsymbol{D}}}_{{\boldsymbol{i}}}^{{\bf{e}}{\bf{f}}{\bf{f}}}=\tfrac{{{\boldsymbol{\varepsilon }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}{{{{\boldsymbol{\tau }}}_{{\bf{e}}{\bf{l}}{\bf{y}}{\bf{t}}}}^{2}}{{\boldsymbol{D}}}_{{\boldsymbol{i}}}$

^a)as implemented in Cantera. ^44,47

The model framework includes continuity equations for all solid, liquid and gaseous phases present in the electrode, allowing to track formation and growth of new phases. In the continuum setting, all phases are characterized by their respective volume fractions. Lithium metal and LEDC are both included as solid phases at the NE, and their starting volume fractions are respectively set to an initial value of 10⁻¹¹ and 8·10⁻⁴. As SEI and metallic lithium grow, the electrode porosity is reduced accordingly.

The chemical thermodynamics and kinetics are calculated with the Cantera software suite (version 2.5.0a3). ⁴⁴ Details on the use of Cantera for lithium-ion batteries are given by Mayur et al. ⁴⁷ For the thermodynamics of the two AM (NMC, graphite) we use Cantera's BinarySolutionTabulatedThermo class. The electrolyte phase is described through the IdealSolidSolution class, with the standard concentration set to a unity value. Both Li[metal] and ${({\bf{C}}{{\bf{H}}}_{2}{\bf{O}}{\bf{C}}{{\bf{O}}}_{2}{\bf{L}}{\bf{i}})}_{2}\left[{\bf{S}}{\bf{E}}{\bf{I}}\right]$ species are described through Cantera's StoichSubstance class (stoichiometric_solid). For the kinetics, note Table I contains both electrochemical and thermochemical reactions, which are treated within the consistent framework provided by Cantera's interfaceKinetics class. Kinetics involving solid phases, such as metallic lithium, require special attention: thermodynamically, the activity of solid phases is unity regardless of the amount of solid. However, reaction kinetics must go to zero if the phase vanishes. This is not considered with standard mass-action kinetics. In the present model, we define for metallic lithium a limiting volume fraction of 10^–10, below which the decomposition rate is set to zero. For this we use Cantera's phaseExistence functionality implemented in the interfaceKinetics class. During right-hand side calculation, we set the phaseExistence flag to false if the volume fraction is below 10^–10, causing Cantera to set the reaction rate to zero. The Cantera input file is available from the authors upon request.

Approach

Cell at thermodynamic equilibrium

The base parameters needed for the lithium-ion battery model are those that are associated with the thermodynamic equilibrium, that is, that describe the open-circuit voltage ${{\boldsymbol{V}}}^{0}$ as function of charge throughput ${\boldsymbol{Q}}.$ Parameters related to nonequilibrium effects such as transport on multiple scales and finite reaction kinetics can only be identified reliably if the equilibrium behavior is modeled correctly.

We start by collecting molar thermodynamic properties (molar enthalpies ${{\boldsymbol{h}}}_{{\bf{L}}{\bf{i}}[{\bf{A}}{\bf{M}},{\boldsymbol{i}}]}^{0}$ and molar entropies ${{\boldsymbol{s}}}_{{\bf{L}}{\bf{i}}[{\bf{A}}{\bf{M}},{\boldsymbol{i}}]}^{0}$) of intercalated lithium in NMC and graphite. They are obtained from our own experiments of half-cell potential vs lithium metal ${{\boldsymbol{E}}}_{{\bf{A}}{\bf{M}},{\boldsymbol{i}}}^{{\bf{e}}{\bf{q}}}$ and from literature data of stoichiometry values (${{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}[{\bf{A}}{\bf{M}},{\boldsymbol{i}}]}$ vs ${{\boldsymbol{V}}}^{0}$ - shown in Fig. 10a) and temperature dependence ${{\boldsymbol{E}}}_{{\bf{A}}{\bf{M}},{\boldsymbol{i}}}^{{\bf{e}}{\bf{q}}}/{\bf{d}}{\boldsymbol{T}},$ including a correction for the entropy of the lithium metal counter electrode. ⁴⁷ Data were selected and processed for NMC ^58,59 and graphite ^60,61 and the resulting molar thermodynamic data are shown in Fig. 10b. We furthermore collected molar thermodynamic data for all other species present in the model, which are summarized in Table IV and form the thermochemical basis of the model.

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Table IV. Thermodynamic properties of all species included in the model.

Species	Molar enthalpy ${{\boldsymbol{h}}}_{{\boldsymbol{i}}}$/kJ·mol−1	Molar entropy ${{\boldsymbol{s}}}_{{\boldsymbol{i}}}$/J·mol−1·K−1	References
Li[NMC]	See Fig. 10b (left)	See Fig. 10b (left)	58, 59
V[NMC]	0	0	Reference value
Li[C6]	See Fig. 10b (right)	See Fig. 10b (right)	28, 29
V[C6]	0	0	Reference value
LEDC[SEI]	−1383	90.07	Calculated 63 assuming SEI formation potential of 0.8 V vs Li/Li+
C2H4	52 a)	219 a)	57
Li[metal]	0	29.12	21, 47
C3H4O3[elyt]	−578 a)	175 a)	57
C4H8O3[elyt]	0	0	Dummy value (not chemically active)
${{\rm{Li}}}^{+}$[elyt]	0	0	Reference value
${{\rm{PF}}}_{6}^{-}$[elyt]	0	0	Dummy value (not chemically active)

^a)Values are assumed T-dependent57, here given at 298 K.

Apart from the chemical thermodynamics, the ${{\boldsymbol{V}}}^{0}({\boldsymbol{Q}})$ behavior depends on the available electrode capacity and electrode balancing, which are governed by electrode volume, volume fraction of AM, density of AM, and the stoichiometry range the AM are cycled in. Here we apply the methodology developed by Mayur et al. ⁶² for self-consistent parameter identification: the resulting parameters are included in Table V and Fig. 10. Furthermore, Table V defines all phases and species assumed in both electrodes and separator.

Table V. Properties of all bulk phases included in the model.

Layer	Phase	Initial volume fraction $\varepsilon$	Density $\rho$/kg·m–3	Species (initial mole fraction X)
PE	NMC	0.456	2500	Li[NMC], V[NMC] (depends on SOC)
	Electrolyte	0.357	1270 39	C3H4O3 [elyt] (0.52), C4H8O3 [elyt] (0.34), ${{\rm{Li}}}^{+}$[elyt] (0.07), ${{\rm{PF}}}_{6}^{-}$[elyt] (0.07)
	Gas phase	0.080	From ideal gas law	N2 (1)
	Electron conductor	0.107	2000	No chemically active species
Separator	Separator	0.5 67	777
	Electrolyte	0.470	1270	same as at PE
	Gas phase	0.030	From ideal gas law	N2 (1)
NE	C6	0.308	2270	Li[C6],V[C6] (depends on SOC)
	Electrolyte	0.335	1270	same as at PE
	LEDC	0.0008	1300 68	(CH2OCO2Li)2
	Lithium carbonate	0.0092	2100	Li2CO3
	Lithium metal	10-11	534 69	Li
	Electron conductor	0.267	2000	No chemically active species
	Gas phase	0.080	1.14 70	N2 (0.99), C2H4 (0.01)

Geometry and transport parameters

All geometry and transport parameters of the modeling domain at macro-, meso- and microscale are summarized in Table VI.

Table VI. Geometry and transport parameters of the P3D modeling domain.

Parameter	Value	References
Cell thickness	3.2 mm	Measured
Aluminum plate (right/left) thickness	10 mm	Measured
Active electrode area ${A}_{{\rm{e}}}$	0.030767 m2	Measured
Cell thermal conductivity $\lambda$	0.9 W·m–1·K–1	39,71
Aluminum plate (right/left) thermal conductivity $\lambda$	237 W·m–1·K–1	39,72
Cell heat capacity $\rho {c}_{{\rm{P}}}$	0.95 J·g–1·K–1	39,73
Aluminum plate (right/left) heat capacity $\rho {c}_{{\rm{P}}}$	0.897 J·g–1·K–1	39,72
Heat transfer coefficient $\alpha$	157 W·m–2·K–1	Measured 39
Thickness of PE	84.0 μm	Measured
Thickness of separator	16.0 μm	Measured
Thickness of NE	92.0 μm	Measured
Tortuosity of PE $\tau$	1.29	Bruggeman relationship (calc.)
Tortuosity of separator $\tau$	1.21	Bruggeman relationship (calc.)
Tortuosity of NE $\tau$	1.31	Bruggeman relationship (calc.)
Diffusion coefficients ${D}_{{{\rm{Li}}}^{+}},$ ${D}_{{{\rm{PF}}}_{6}^{-}}$	See Eqs. 14 and 15	39,40,63
Specific surface area NMC/electrolyte ${A}_{{\rm{NMC}}}^{V}$	1.07 ·106 m2 m−3	$3{\varepsilon }_{{\rm{AM}}}/{r}_{AM}$
Specific surface area graphite/electrolyte ${A}_{{\rm{C}}6}^{V}$	6.94 · 105 m2 m−3	$3{\varepsilon }_{{\rm{AM}}}/{r}_{AM}$
NE double layer capacitance ${C}_{{\rm{DL}}}^{{\rm{V}}}$	3.0·10-2 F·m2	Fitted to EIS data
PE double layer capacitance ${C}_{{\rm{DL}}}^{{\rm{V}}}$	1.8·10-2 F·m2	Fitted to EIS data
Ohmic resistance of current collection system ${R}_{{\rm{cc}}}^{0}$	9.0·10-1 mΩ·m2	Fitted to EIS data
Electrical conductivity of the SEI layer ${\sigma }_{{\rm{SEI}}}$	1.0·10-5 S m−1	Assumed 74
Graphite stoichiometry range ${X}_{{\rm{Li}}\left[{{\rm{C}}}_{6}\right]}$ (0...100% SOC)	0.025...0.7584	Optimization
NMC stoichiometry range ${X}_{{\rm{Li}}\left[{\rm{NMC}}\right]}$ (0...100% SOC)	0.8647...0.2488	Optimization
Radius of PE particles ${r}_{{\rm{NMC}}}$	$2.8\cdot {10}^{-6}{\rm{m}}$	Measured
Diffusion coefficient of Li in NMC ${D}_{{\rm{Li}},\,{\rm{NMC}}}$	See Fig. 11a	Measured 64 + activation energy 65
Radius of NE particles ${r}_{{{\rm{C}}}_{6}}$	$4.32\cdot {10}^{-6}{\rm{m}}$	Measured
Diffusion coefficient of Li in graphite ${D}_{{\rm{Li}},{{\rm{C}}}_{6}}$	See Fig. 11b	Measured 66 + activation energy 60

We assume the electrolyte to be composed of EC, EMC, and LiPF₆; the exact composition is unknown. We use an electrolyte transport model based on dilute solution theory, ⁴⁰ but use concentration-dependent diffusivities (accounting for the interaction between the ions in the concentrated solution). The expressions for the ion diffusion coefficients are

$$\begin{eqnarray}&&\begin{array}{c}{{\boldsymbol{D}}}_{{\bf{L}}{{\bf{i}}}^{+}}=2.06\cdot {10}^{-10}{{\bf{m}}}^{2}{{\bf{s}}}^{-1}\,\cdot \,\exp \left(-\,\displaystyle \frac{{{\boldsymbol{c}}}_{{\bf{L}}{{\bf{i}}}^{+}}}{1000\,{\bf{m}}{\bf{o}}{\bf{l}}\,{{\bf{m}}}^{-3}}\right)\\ \,\cdot \,\exp \left(-\,\displaystyle \frac{32.0\,{\bf{k}}{\bf{J}}\,{\bf{m}}{\bf{o}}{{\bf{l}}}^{-1}\,}{{\boldsymbol{R}}}\left(\displaystyle \frac{1}{{\boldsymbol{T}}}-\displaystyle \frac{1}{298\,{\bf{K}}}\right)\right),\end{array}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\begin{array}{c}{{\boldsymbol{D}}}_{{\bf{P}}{{\bf{F}}}_{6}^{-}}=4.81\cdot {10}^{-10}{{\bf{m}}}^{2}{{\bf{s}}}^{-1}\,\cdot \,\exp \left(-\,\displaystyle \frac{{{\boldsymbol{c}}}_{{\bf{P}}{{\bf{F}}}_{6}^{-}}}{1000\,{\bf{m}}{\bf{o}}{\bf{l}}\,{{\bf{m}}}^{-3}}\right)\\ \,\cdot \,\exp \left(-\,\displaystyle \frac{32.0\,{\bf{k}}{\bf{J}}\,{\bf{m}}{\bf{o}}{{\bf{l}}}^{-1}\,}{{\boldsymbol{R}}}\left(\displaystyle \frac{1}{{\boldsymbol{T}}}-\displaystyle \frac{1}{298\,{\bf{K}}}\right)\right)\,.\end{array}\end{eqnarray} \tag{ 15 }$$

We refer the reader to our previous works ^39,40,63 for a detailed description of our electrolyte transport model. We also assume an Arrhenius-type dependence on temperature and an activation energy of ${{\boldsymbol{E}}}_{{\bf{a}}{\bf{c}}{\bf{t}}}$ = 32.0 kJ mol⁻¹.

Lithium diffusion inside the AM strongly influences cell behavior. Figure 11 shows diffusion coefficients of intercalated lithium as function of intercalation stoichiometry for the two AM. We applied a literature research for concentration- and temperature-dependent diffusion coefficients: for NMC, we chose the work from Noh et al., ⁶⁴ shown in Fig. 11a, where GITT experiments have been conducted over a nearly complete intercalation stoichiometry range. We included an Arrhenius temperature dependence with an activation energy of 45.45 kJ mol⁻¹ as found by Cui et al. ⁶⁵ For graphite, we took as reference the work from Levi, ⁶⁶ where the diffusion was investigated using both PITT and EIS techniques. The values are shown in Fig. 11b. They were also used in the models by Kupper ⁴⁰ and in our previous works. ^11,21,39 An activation energy of 44.0 kJ mol⁻¹ was used, as average between the two values experimentally measured in Ecker et al. ⁶⁰

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Electrochemical parameters and SEI model

The electrochemical parameters include the exchange current density factors ${{\boldsymbol{i}}}^{00}$ and activation energies ${{\boldsymbol{E}}}_{{\bf{a}}{\bf{c}}{\bf{t}},{\bf{f}}}$ of the two charge-transfer reactions (NMC and graphite) as well as double-layer capacitances of both electrodes. These parameters were obtained by fitting simulated EIS data to experimental values at two different SOC (20%, 50%) and three different temperatures (5 °C, 20 °C, 35 °C). The activation energies were obtained from Arrhenius plots. The resulting values of the double layer capacitances are included in Table VI, while the reaction mechanism and rate coefficients for all AM are reported in Table I.

A comparison of EIS simulations (after parameter fitting) and measurements is shown in Fig. 12. Simulations were carried out for frequencies from 10^-2 Hz up to 10⁵ Hz, SOCs of 20%, 35%, 50%, 65%, 80% and 100%, and temperatures of 5 °C, 20 °C, 35 °C and 50 °C. Both experiments and simulations show two features: (i) a small flattened semi-circle at high frequency (between 10 Hz and 100 Hz) which can be assigned to the PE charge-transfer reaction and double layer, overlapped within a larger semi-circle at medium frequency which can be assigned to the NE charge-transfer reaction and double-layer; (ii) a Warburg-type feature at low frequency (< 0.1 Hz) which can be assigned to lithium diffusion in the AM. There is qualitative agreement between model and experiment, except for 5 °C in Panel a) and generally SOC 100% at all temperatures.

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About the SEI model, several parameters are needed to simulate both calendaric and cyclic SEI formation, cf. Kupper et al. ⁸ The applied rate expression is given in Table I (Eq. 3). The SEI thickness is calculated as

$$\begin{eqnarray}&&{{\boldsymbol{\delta }}}_{{\bf{S}}{\bf{E}}{\bf{I}}}=\left(\sqrt[3]{\left(1+\displaystyle \frac{{\varepsilon }_{{\bf{L}}{\bf{E}}{\bf{D}}{\bf{C}}}}{{{\boldsymbol{\varepsilon }}}_{{{\bf{C}}}_{6}}}\right)}-1\right)\cdot {{\boldsymbol{r}}}_{{{\bf{C}}}_{6}}.\end{eqnarray} \tag{ 16 }$$

The current density during charge is

$$\begin{eqnarray}&&{{\boldsymbol{i}}}_{{\bf{c}}{\bf{h}}{\bf{g}}}^{{\boldsymbol{V}}}=\left\{\begin{array}{c}0\,for\,{{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}\gt 0\,\left({\rm{discharge}}\right)\\ -{{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}\,for\,{{\boldsymbol{i}}}_{{\bf{F}}}^{{\boldsymbol{V}}}\lt 0\left({\rm{charge}}\right)\end{array}\right..\end{eqnarray} \tag{ 17 }$$

The derivative of the tangential SEI stress with respect to stoichiometry, ${\bf{d}}{{\boldsymbol{\sigma }}}_{{\bf{t}},{\bf{S}}{\bf{E}}{\bf{I}}}/{\bf{d}}{{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]},$ is given by the polynomial

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{\bf{d}}{\sigma }_{{\bf{t}},{\bf{S}}{\bf{E}}{\bf{I}}}\left({{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}\right)}{{\bf{d}}{{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}}=\left(-931\cdot {{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}^{4}+1319.96\cdot {{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}^{3}\right.\\ \left.-201.684\cdot {{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}^{2}-240.56\cdot {{\boldsymbol{X}}}_{{\bf{L}}{\bf{i}}\left[{{\bf{C}}}_{6}\right]}+67.9\right){\bf{M}}{\bf{P}}{\bf{a}}\end{array}\end{eqnarray} \tag{ 18 }$$

The parameter describing the calendaric over cyclic aging rate is set to ${{\boldsymbol{k}}}_{{\boldsymbol{c}}}=27.5\tfrac{{{\bf{m}}}^{3}}{{\bf{A}}\cdot {\bf{M}}{\bf{P}}{\bf{a}}}.$