State-of-Charge Implications of Thermal Runaway in Li-ion Cells and Modules

Four 3.25 Ah 18650 cylindrical Li-ion cells were electrochemically cycled two times using a standard protocol (described below) and then discharged to different SOCs. In each cycle, the cell was charged to 4.2 V under a constant-current (CC) mode at 0.5 C (1.625 A), followed by a constant-voltage (CV) charge phase at 4.2 V until the current declined to 0.02 C (0.065 A). After a 30-minute rest period, the cell was CC discharged to 2.5 V at 0.2 C (0.65 A). A discharge current pulse of 1.5 C (4.875 A) for 0.1 s was applied at 50% SOC in the discharge phase of the second cycle to determine the internal resistance of the cell. A T-type thermocouple was attached near the positive terminal of each cell to detect the temperature variation. Figure SF1a in the Supplementary Information shows the typical cell voltage and temperature response during this protocol. All four cells were recharged to 100% SOC using the same charging protocol as the standard test. After a 30 min rest phase, three of these cells were discharged at 0.2 C (0.65 A) to 3%, 33%, and 66% SOC, as shown in Fig. SF1b in Supplementary Information. The open circuit voltage (OCV) of these cells was allowed to stabilize before conducting the ARC experiments.

The NCA-based Li-ion cells at different SOCs were subject to an ARC test using the heat-wait-seek (HWS) protocol. The parameters of the ARC experimental setup are summarized in Table I. The OCV of each cell was monitored in-operando to understand the correlation between the occurrence of internal short circuit (ISC), thermal instability of the separator, and the onset of potential TR in the cell.

The cell-level TR process can be demarcated into different stages, with each stage denoting a distinct exothermic response. In this study, the TR phenomenon is segmented into four stages depending on the rate of temperature rise observed in the ARC experiments. Three characteristic temperatures, T₁, T₂, and T₃, are identified where T₁ denotes the onset of self-heating when the rate of temperature increase exceeds 0.02 °C min⁻¹, T₂ represents the beginning of TR when the rate of temperature rise exceeds 100 °C min⁻¹, and T₃ denotes the peak temperature encountered during the adiabatic TR. In the temperature range between T₁ and T₂, the temperature rate profile is divided into three stages (i.e., denoted by Stages 1, 2, and 3). Stage 1 denotes self-heating onset and comprises slow exothermic interactions, primarily driven by the SEI decomposition reaction. ⁴⁶ During Stage 2, a combination of exothermic reactions such as the formation of secondary SEI (due to the interaction between intercalated lithium in the anode and electrolyte) and its subsequent decomposition reaction occur, ^42,45 causing a larger rate of temperature increase than Stage 1. In addition, the exothermic reactions in Stage 2 release a large amount of gas, which increases the cell's internal pressure and causes significant swelling. Upon reaching a threshold pressure value due to gas accumulation, the safety vent of the cell opens to release a large amount of gas that causes a sudden decrease in cell temperature. Moreover, a series of chemical reactions such as cathode decomposition, phase transition, and oxygen liberation will occur inside the cell during Stage 3, resulting in an enormous heat generation and a large rate of temperature rise around 200 °C. ^38,47 Concomitantly, the separator melts, shrinks, and vaporizes to cause an ISC. ¹⁷ Once the cell temperature rate reaches T₂, the electrical energy released from ISC can cause electrolyte decomposition and combustion, and decomposition reactions of the binder and additives (Stage 4), releasing a significant amount of heat and posing potential TR hazards like fire and explosion.

The reaction rate corresponding to the exothermic interactions during various stages of the TR process can be described using the Arrhenius equation:

$$\begin{eqnarray}&&\frac{d\left[{c}_{i}\left(t\right)\right]}{{dt}}={A}_{m,i}\exp \left[-\frac{{E}_{a,i}}{{RT}\left(t\right)}\right]{\left[{c}_{i}\left(t\right)\right]}^{n}\end{eqnarray} \tag{ 1 }$$

Here $c\left(t\right)$ denotes the reactant concentration, $T\left(t\right)$ denotes the temperature at time $t,$ ${A}_{m}$ is frequency factor, ${E}_{a}$ is the activation energy, $n$ denotes the reaction order, and the subscript $i$ denotes the TR stage. During an ARC test which represents an adiabatic environment for the cell, the heat generation rate is proportional to the consumption of the reactant. Therefore, for each stage in the TR process,

$$\begin{eqnarray}&&\frac{{dT}\left(t\right)}{{dt}}=\frac{{m}_{i}{\unicode{x02206}H}_{i}}{M{C}_{p}}\left|\frac{d\left[{c}_{i}\left(t\right)\right]}{{dt}}\right|\end{eqnarray} \tag{ 2 }$$

Here $m$ denotes the reactant mass, and $\unicode{x02206}H$ denotes the exothermic reaction enthalpy. The total mass of the cell is $M$ and the specific heat capacity is denoted by ${C}_{p}.$

For any TR stage, integrating Eq. 2 from ${t}_{0}$ (start time) to $t,$ where the temperature changes from ${T}_{0,i}$ to $T(t),$ and the reactant concentration changes from ${c}_{i}\left[{t}_{0}\right]$ to ${c}_{i}(t),$ we have:

$$\begin{eqnarray}&&T(t)-{T}_{0,i}=\frac{{m}_{i}{\rm{\Delta }}{H}_{i}}{M{C}_{p}}\left[{c}_{i}\left({t}_{0}\right)-{c}_{i}\left(t\right)\right]\end{eqnarray} \tag{ 3 }$$

At the end of the heat generation for any specific stage, the cell temperature rises to ${T}_{e,i},$ and the reactant concentration is ${c}_{i}\left({t}_{0}\right)$ = 0. Therefore,

$$\begin{eqnarray}&&{T}_{e,i}-{T}_{0,i}=\frac{{m}_{i}{\rm{\Delta }}{H}_{i}}{M{C}_{p}}\left[{c}_{i}\left({t}_{0}\right)\right]\end{eqnarray} \tag{ 4 }$$

Dividing Eq. 3 by Eq. 4,

$$\begin{eqnarray}&&{c}_{i}\left(t\right)=\frac{{T}_{e,i}-T(t)}{{T}_{e,i}-{T}_{0,i}}{c}_{i}\left({t}_{0}\right)\end{eqnarray} \tag{ 5 }$$

Differentiating Eq. 5 with respect to time $t,$

$$\begin{eqnarray}&&\frac{{dT}(t)}{{dt}}=-\frac{{T}_{e,i}-{T}_{0,i}}{{c}_{i}\left({t}_{0}\right)}\frac{d\left[{c}_{i}\left(t\right)\right]}{{dt}}\end{eqnarray} \tag{ 6 }$$

Incorporating Eqs. 2 and 5 into Eq. 6 and taking the natural logarithm, we have:

$$\begin{eqnarray}&&\frac{{dT}(t)}{{dt}}={A}_{m,i}{c}_{i}{\left({t}_{0}\right)}^{n-1}\left({T}_{e,i}-{T}_{0,i}\right){\left(\frac{{T}_{e,i}-T\left(t\right)}{{T}_{e,i}-{T}_{0,i}}\right)}^{n}\exp \left[-\frac{{E}_{a,i}}{{RT}\left(t\right)}\right]\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\mathrm{ln}\left[\frac{\frac{{dT}(t)}{{dt}}}{{\left({T}_{e,i}-T(t)\right)}^{n}}\right]=\mathrm{ln}\left[\frac{{A}_{m,i}{c}_{i}{\left({t}_{0}\right)}^{n-1}}{{\left({T}_{e,i}-{T}_{0,i}\right)}^{n-1}}\right]-\frac{{E}_{a,i}}{{RT}\left(t\right)}\end{eqnarray} \tag{ 8 }$$

Assuming a first-order reaction, Eq. 8 is simplified to

$$\begin{eqnarray}&&\left[\frac{\frac{{dT}(t)}{{dt}}}{\left({T}_{e,i}-T(t)\right)}\right]=\mathrm{ln}\left[{A}_{m,i}\right]-\frac{{E}_{a,i}}{{RT}\left(t\right)}\end{eqnarray} \tag{ 9 }$$

The pre-exponential factor, ${A}_{m,i},$ and activation energy ${E}_{a,i}$ of the exothermic reaction for each stage can be determined by calculating the slope and intercept of the linear relationship between $\mathrm{ln}$ $\left[\displaystyle \frac{dT}{dt}/\left({T}_{e,i}-T\left(t\right)\right)\right]$ and $1/T(t).$ It is noted that ${T}_{e}$ is different at various stages of the TR process.

For Stages 1 to 3, the exothermic heat generation rate is given as follows:

$$\begin{eqnarray}&&{\dot{Q}}_{i}\left(t\right)=M{C}_{p}\cdot {A}_{m,i}\left({T}_{e,i}-T\left(t\right)\right)\exp \left[-\frac{{E}_{a,i}}{{RT}\left(t\right)}\right]\end{eqnarray} \tag{ 10 }$$

For Stage 4, the heat generation is calculated based on the total energy released from the cell during the TR process. ⁴⁸ While the mass and specific heat of the cell does not change significantly during the first three stages of the TR, there can be a significant mass loss and change in the specific heat capacity of the cell during Stage 4 (after the cell temperature reaches T₂) due to gas venting. During Stage 4, the change in enthalpy has been fitted to capture the temperature response during the ARC experiment. Therefore, for the last stage (Stage 4), the exothermic heat generation is given as follows:

$$\begin{eqnarray}&&{\dot{Q}}_{4}\left(t\right)=\frac{{\rm{\Delta }}H}{{\rm{\Delta }}t}\,\left(\frac{{dT}(t)}{{dt}}\geqslant 100\,^\circ {\rm{C}}/\min \right)\end{eqnarray} \tag{ 11 }$$

Here, $\unicode{x02206}H$ denotes the heat energy released during Stage 4, and $\unicode{x02206}t$ represents the time interval between the release of exothermic heat energy and the occurrence of TR.

Thus, the heat generated from the exothermic reactions in all the TR stages, denoted by ${Q}_{{gen}}\left(t\right),$ can be described as follows:

$$\begin{eqnarray}{\dot{Q}}_{{gen}}\left(t\right)=\left\{\begin{array}{cc}\displaystyle \sum _{i=1}^{4}\dot{{Q}_{i}}\left(t\right) & {if}\,T\left(t\right)\geqslant {T}_{2}\\ \displaystyle \sum _{i=1}^{3}\dot{{Q}_{i}}\left(t\right) & {otherwise}\end{array}\right.\end{eqnarray} \tag{ 12 }$$

The developed TR modeling framework captures the ARC response for an 18650 NCA-based Li-ion cell, following the HWS protocol. The cell temperature evolves based on the energy conservation equation:

$$\begin{eqnarray}&&M{C}_{p}\frac{{dT}\left(t\right)}{{dt}}={\dot{Q}}_{{gen}}\left(t\right)-{h}_{{ARC}}S\left[T\left(t\right)-{T}_{{ARC}}\left(t\right)\right]\end{eqnarray} \tag{ 13 }$$

Here, ${h}_{{ARC}}$ is the overall heat transfer coefficient (both convection and radiation) between the cell and the ARC chamber, $S$ is the surface area of the cell, and ${T}_{{ARC}}\left(t\right)$ denotes the temperature of the ARC chamber. The modeling parameters for the ARC simulations are summarized in Table II.

Table II. Model parameters for the simulated ARC signature and oven test.

Symbols	Descriptions	Values	References
$M$	Mass of the cell [g]	48.5	36
${C}_{p}$	Specific heat capacity of the cell [J/g-K]	0.83	36
$S$	Surface area of cell [cm2]	36.75	36
${h}_{{ARC}}$	Overall heat transfer coefficient in ARC test [W/m2-K]	30	Estimated
${h}_{{conv}}$	Convective heat transfer coefficient in oven test [W/m2-K]	5	26
$\varepsilon$	Cell emissivity [-]	1.0	26

For the oven test simulation of the 18650 Li-ion cells, an energy conservation equation in accordance with the formulation used by Hatchard and Dahn ²⁶ was employed in this study:

$$\begin{eqnarray}&&M{C}_{p}\frac{{dT}\left(t\right)}{{dt}}={\dot{Q}}_{{gen}}\left(t\right)-{h}_{{oven}}S\left[T\left(t\right)-{T}_{{oven}}\right]\end{eqnarray} \tag{ 14 }$$

Here, ${T}_{{oven}}$ denotes the surrounding temperature in the oven and ${h}_{{oven}}$ denotes the overall heat transfer coefficient, representing a combination of the convection and radiation effects. The convective heat transfer coefficient is assumed to be 5 W m⁻²-K, ^26,30 and depicts natural convection around the cell in a heated oven. The radiation heat transfer coefficient from the cell surface is based on the cell emissivity and oven temperature ($4\varepsilon \sigma {T}_{{oven}}^{3}$). Therefore,

$$\begin{eqnarray}&&{h}_{{oven}}={h}_{{conv}}+4\varepsilon \sigma {T}_{{oven}}^{3}\end{eqnarray} \tag{ 15 }$$

where $\varepsilon$ is the emissivity of the cell, and $\sigma$ is the Stefan-Boltzmann constant. The modeling parameters for the oven simulations are summarized in Table II.

Based on a lumped thermal model, the temperature $T(t)$ of each cylindrical Li-ion cell in a 3 × 3 battery module can be expressed as:

$$\begin{eqnarray}&&M{C}_{p}\frac{{dT}\left(t\right)}{{dt}}=\dot{Q}{\left(t\right)}_{{gen}}+\dot{Q}{\left(t\right)}_{{diss}}+\dot{Q}{\left(t\right)}_{{heater}}\end{eqnarray} \tag{ 16 }$$

Here, $\dot{Q}{\left(t\right)}_{{gen}}$ represents the rate of heat generation in the cell due to TR, $\dot{Q}{\left(t\right)}_{{diss}}$ denotes the heat dissipation rate and $\dot{Q}{\left(t\right)}_{{heater}}$ denotes the heating power or heat input rate to the cell from the heater to trigger TR in the module. The value of $\dot{Q}{\left(t\right)}_{{heater}}$ = constant for the trigger cell and $\dot{Q}{\left(t\right)}_{{heater}}$ = 0 for the rest of the Li-ion cells in the module. The heat input to the trigger cell to cause TR is considered to be provided by a heating tape (wrapping the trigger cell) of constant heating power. The mass and the specific heat of the cell are assumed to be constant, and due to the low Biot number (${Bi}$ = 0.056) corresponding to the cylindrical cell configuration in this study, ^26,49 the temperature gradients within the cell have not been considered.

The heat dissipation from each cell in the Li-ion module is given by:

$$\begin{eqnarray*}&&\dot{Q}{\left(t\right)}_{{diss}}={\dot{Q}\left(t\right)}_{{cond},{tab}}+{\dot{Q}\left(t\right)}_{{cond},{air}}\end{eqnarray*}$$

$$\begin{eqnarray}&&+\,\dot{Q}{\left(t\right)}_{{rad},{cell}}+{\dot{Q}\left(t\right)}_{{conv},{surr}}+{\dot{Q}\left(t\right)}_{{rad},{surr}}\end{eqnarray} \tag{ 17 }$$

Here, $\dot{Q}{\left(t\right)}_{{cond},{air}}$ represents the conduction heat transfer between the cells (through stagnant air), $\dot{Q}{\left(t\right)}_{{rad},{cell}}$ is the radiation heat transfer between cells, $\dot{Q}{\left(t\right)}_{{cond},{tab}}$ is the conduction heat transfer through the connecting tabs, $\dot{Q}{\left(t\right)}_{{conv},{surr}}$ is the convection heat transfer from the cell to the air surrounding the cells, and $\dot{Q}{\left(t\right)}_{{rad},{surr}}$ is the radiation heat transfer from the cell to the surrounding air. ⁵⁰ A detailed description of the modeling approach to capture these different heat transfer modes has been presented in the Supplementary Information.

The physical state of the Li-ion cells and the unwound jelly roll after TR for the four SOC conditions is shown in Figs. SF3a–SF3d of the Supplementary Information. The severity of the TR process is higher as the SOC increases from 3% to 100%. The jelly roll for the 3% and 33% SOC cases remains unbroken and delaminated. The electrodes of the cells at 66% and 100% SOCs suffered significant structural damage with only charred carbon remaining, making it difficult to unfold the jelly roll. Figures 1a–1e depicts the ARC test results of the NCA-based Li-ion cells at 100%, 66%, 33%, and 3% SOC. The corresponding in-operando voltage profiles during the TR process are shown in Figs. SF4a–SF4d of the Supplementary Information. Amongst the four cases, the cell at 100% SOC shows the lowest temperature for self-heating onset (i.e., 72.67 °C), denoting a strong thermal instability signature at the anode-electrolyte interface and SEI decomposition interactions between the fully lithiated graphite and electrolyte. ¹⁴ The 100% SOC cell undergoes TR at the lowest temperature of 203.29 °C due to potential destabilization of the cathode, ¹⁹ and the cell temperature reaches a maximum value of 759.61 °C accompanied by a drop in the cell voltage to 0 V. In contrast, the cell at 66% SOC shows a higher onset temperature for self-heating (i.e., 86.93 °C) with TR occurring at 216.26 °C. It demonstrates a lower maximum temperature of the cell (i.e., 713.70 °C) compared to the 100% SOC case, indicating less energy release during the TR process. Interestingly, the cell at 33% SOC presents an onset temperature for self-heating at 86.46 °C, which is very close to that obtained for the cell at 66% SOC. This subsequently results in TR at 236.14 °C, demonstrating a higher T₂ than the cell at 66% SOC. The maximum temperature of the cell at 33% SOC is 566.52 °C, which suggests reduced energy release during TR than the previous cases. The cell at 3% SOC exhibits an exotherm at 87.69 °C, similar to the cell at 33% SOC. Temperature T₂ is not encountered for the lowest SOC case, and it reaches a maximum temperature of 300.75 °C. As expected, the cell at 3% SOC does not undergo TR and demonstrates the minimum thermal safety hazard, indicated by the highest onset temperature for self-heating and the least maximum temperature. The spider chart in Fig. 1c summarizes all the cell-level TR characteristics at different SOCs. A comparison of the heat generated during the TR process for the four SOC cases is shown in Fig. 1d, showing a non-linear variation from 8.5 kJ to 27.4 kJ as the SOC increases from 3% to 100%. The cell mass before and after TR for the four SOCs is shown in Fig. 1e. The 3% SOC case depicts the lowest mass loss (∼10%), while the 100% SOC shows the highest mass loss (∼35%) as a result of the TR process. The larger mass loss could be due to the significant amount of heat generated, which in turn leads to higher cell temperatures and venting during the TR process. ⁵¹

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Based on the ARC experiments, we infer that the onset temperature of exothermic reaction (T₁) in NCA-based Li-ion cells is not monotonically dependent on the SOC since the 3%, 33%, and 66% SOC cases have almost the same T₁. A reduced T₁ value is only detected for the cell with 100% SOC. In addition, the time at which complete TR occurs is not directly proportional to the SOC, as we observe a later TR onset for the 100% SOC cell than the 66% and 33% SOC cells. This is because the cells with 66% and 33% SOC experienced an extended heating period before T₁, leading to a faster TR onset than the cell at 100% SOC. We note that the in-operando voltage profile correlates with micro ISC and the opening of safety vents in the cells during the TR process. The initial drop in the cell voltage relates to the micro ISC within the cell, which can be driven by a combination of factors such as electrolyte depletion, porosity reduction in the electrodes, and separator shutdown. ¹⁷ These effects can increase the cell resistance, limiting the short-circuit current and generating a small amount of heat with negligible contribution to the temperature rate profile. The cell voltage experiences another sharp drop as the safety vent opens and recovers for a short period before dropping to 0 V when the TR occurs.

This section derives and analyzes the kinetic parameters of the exothermic reactions during the TR process, cognizant of the cell-level SOC. For each SOC, three sets of kinetic parameters that correspond to Stages 1, 2, and 3 of the ARC signatures are identified. For Stage 4, the energy released from the cell after TR occurs is calculated such that the maximum cell temperature is representative of the ARC experiment. While we focus majorly on the 100% SOC case here, further results for the other SOC cases (66%, 33%, and 3%) are presented in the Supplementary Information. Figure 2a depicts the rate of temperature change as a function of cell temperature for the 100% SOC case, where the different stages are demarcated during the cell-level TR process (see Fig. SF5 in Supplementary Information for cell temperature as a function of time). In between T₁ and T₂, two additional temperatures (i.e., T_a and T_b) are identified, where a significant change in the temperature increase rate is observed. These temperatures (T₁, T_a, T_b, and T₂) for the different SOCs are summarized in Table III. The temperature variation in Stage 1 is from T₁ to T_a, Stage 2 is between T_a and T_b, and Stage 3 is from T_b to T₂. The starting temperature of Stage 1 is identified as the onset of self-heating, while the end of Stage 3 is recognized as the point when TR occurs in the cell.

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Figures 2b–2d shows the linear regression results for $\mathrm{ln}$ $\left[\displaystyle \frac{dT}{dt}/\left({T}_{e}-T\left(t\right)\right.\right]$ and $1/T(t),$ in comparison with the experimental data for the three TR stages at 100% SOC. The value of T_e for different TR stages at SOC = 3%, 33%, 66%, and 100% are adopted from Table III, where T_a represents T_e for Stage 1, T_b represents T_e for Stage 2, and T₂ represents T_e for Stage 3 of the cell-level TR process. The linear regression relationship based on the ARC experimental data for the other three SOC cases has been presented in the Supplementary Information (see Figs. SF6–SF8). The kinetic parameters (pre-exponential factor, ${A}_{i}$ and activation energy, ${E}_{a,i}$) corresponding to the exothermic reaction in each stage are calculated from the slope and intercept given by Eq. 8. The calculated kinetic parameters for the different SOC cases are summarized in Table IV. For the exothermic interaction in Stage 4, the $\unicode{x02206}H$ values at different SOCs are summarized in Table V; it is observed that $\unicode{x02206}H$ exhibits a gradual reduction with a decrease in the SOC of the cell.

Table V. The enthalpy of the exothermic reaction in Stage 4 under different SOCs.

SOC	3%	33%	66%	100%
$\unicode{x02206}H$ (J)	0	13299.45	20024.45	22394.66

The developed TR model for the NCA-based Li-ion cell is validated with the experimentally measured TR response. Figures SF9a–SF9d in the Supplementary Information compares the computational and experimental results of the TR behavior at the four different SOCs. In addition, the computationally calculated TR temperatures are compared with the experimental results in Table ST1. The TR model accurately captures T₁ and T₂ associated with TR, and the trends of T₃ also show good agreement with the experimental results.

In this section, we use the developed TR model to investigate the cell-level thermal stability under various oven conditions. Based on the model, standard oven tests are simulated and analyzed across oven temperatures ranging from 100 to 200 °C at SOC = 3%, 33%, 66%, and 100%. For a 66% SOC case, the thermal response at oven temperatures of 100 °C, 125 °C, 150 °C, 175 °C, and 200 °C is shown in Fig. 3a. The Li-ion cell does not experience TR for oven temperatures of 100 °C and 125 °C. However, for a SOC of 66%, the cell undergoes TR at oven temperatures of 150 °C, 175 °C, and 200 °C. The initial rise in the cell temperature is attributed to convective heating, and the occurrence of TR in the cell brings about a steep increase in the cell temperature. At Stage 4, the cell reaches a peak temperature of 737.15 °C, 748.03 °C and 754.16 °C at oven temperatures of 150 °C, 175 °C, and 200 °C respectively. Figure 3b examines the effect of SOC on the TR behavior for an oven temperature of 150 °C. The cells with 66% and 100% SOC experience TR with peak temperatures of 748.10 °C and 704.84 °C, while TR does not occur for SOCs of 33% and 3% due to the lack of exothermic heat generation. With a reduction in the SOC, the onset of TR is delayed, and the peak temperature during TR decreases. Figure 3c depicts the thermal stability map for NCA-based Li-ion cells as a function of the oven temperature and SOC. By quantifying the peak cell temperature across these conditions, the occurrence of TR is demarcated based on the criteria that the maximum temperature is greater than the oven temperature. Depending on the cell SOC, the critical temperature of TR is identified to fall within the range of 130 °C to 250 °C. Regions of the regime map beyond this critical value are identified as unsafe zones. On the other hand, safe zones imply no occurrence of TR since the temperature rise is not significant enough to go beyond the operating oven temperature.

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In this section, we analyze the TRP behavior in a 3 × 3 battery module comprising NCA-based Li-ion cells at uniform SOCs (Figs. 4a–4b). The Li-ion cells in the 3 × 3 module are connected by M-type tabs, and the spacing between them is 0.1 mm, analogous to the earlier TRP studies ^23,50 for Li-ion battery modules. The central cell of the module (Cell 5) is heated with a power of 20 W to trigger TR and understand the potential pathways for TRP. Figures 5a–5d shows the temperature response of each cell in the module under convective conditions (${h}_{{conv}}$ = 5 W m⁻²-K) and an ambient temperature of 35 °C. For the Li-ion battery module with all the cells at SOC = 3%, the amount of heat generated due to the TR of the trigger cell (Cell 5) is not sufficient to propagate the TR to the neighboring cells. On the other hand, for a uniform SOC of 33%, a TR originating in the central cell (Cell 5) is observed to propagate through the battery module. The TR for the central cell occurs at t_TR = 30.92 min with a peak temperature of 532 °C. Subsequently, TR in the surrounding cells follows the following sequence: B5-B8-B2-{B4, B6}-{B1, B3, B7, B9}. Here, the cells within brackets simultaneously undergo TR. The TRP rate is defined as the ratio of the number of cells in the module that experience TR (excluding the trigger cell) to the time interval for TRP. ⁵² The time interval for TRP is calculated as the difference between the TR time of the trigger cell and the last cell undergoing TR in the module. Accordingly, the TRP rate in the module with SOC = 33% was calculated to be 1.82 min⁻¹. Similarly, for a uniform SOC = 66%, the TR of the central cell (Cell 5) occurs at t_TR = 19.33 min, which is much earlier than the onset time at SOC = 33%. For this case, the TRP sequence is B5-B8-B2-{B4, B6}-{B1, B3}-{B7, B9}, with a peak temperature of 723 °C occurring for Cells 1 and 3. The TRP rate for SOC = 66% was calculated to be 1.60 min⁻¹. Finally, for a uniform SOC = 100% in all the cells, the heat generation due to TR of the trigger cell (Cell 5) is insufficient to cause TR in the neighboring cells. With a smaller heating period of t_TR = 16.92 min for the 100% SOC case, the temperature of the cell adjacent (Cell 8) to the trigger cell at the onset of TR in the module is 93 °C. This temperature for the 100% SOC case is lower than the temperature of the same cell for the 66% SOC case (i.e., 102 °C) during TR onset. This is primarily because the frequency factor and activation energy for the exothermic reaction in Stages II and III at 100% SOC is higher than 66% SOC, leading to a much faster release of exothermic heat. During Stage IV of the exothermic reactions, the rate of temperature rise in the neighboring Cell 8 is 0.18 °C s⁻¹ at 100% SOC compared to 0.55 °C s⁻¹ at 66% SOC. Due to the rapid heat release from the cell at 100% SOC, there is insufficient time for the heat to be transferred to the neighboring cells. Thus, TRP does not occur for the battery module at 100% SOC. Such scenarios have also been reported in a prior experimental study by Wang et al. ⁵³ where the TRP in a 1 × 3 cylindrical Li-ion battery module under different SOC was analyzed. Overall, the proposed TR model developed using mechanistic insights from the cell-level ARC experiments, and subsequent thermo-kinetic analysis is a scalable framework that can capture the TRP behavior in a battery module under different thermal abuse conditions.

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Lastly, we study the TRP behavior in a battery module comprising cells with an imbalanced SOC distribution. To analyze this, we select N = 100 samples of 3 × 3 Li-ion battery modules that contain cells following a random normal distribution of the SOC. The mean SOC of the cells in the battery module is 66%, and the standard deviation (SD) is 5%,, ^54,55 as shown in Fig. 6a. At an ambient temperature of 35 °C and a cell spacing of 0.1 mm, the central cell of the module (Cell 5) was provided with a heating power of 20 W to trigger TR in each sample of the battery module. Figure 6b summarizes the simulated TRP outcomes for the 100 samples under the prescribed SOC distribution in terms of the following attributes: (i) the amount of critical thermal energy input required for Cell 5 to trigger TR in the module, (ii) exothermic heat generated due to TR, and (iii) the resulting TRP rate in the module. For the considered N = 100 samples (mean SOC = 66%, SD = 5%), the critical thermal energy input to trigger TR varies from 20.47 to 25.91 kJ with a mean value of 23.20 kJ. The heat generated from the exothermic reactions during TR ranges from 147.59 to 151.22 kJ with an average value of 149.40 kJ, and The TRP rate in the module varies between 1.33 to 1.72 min⁻¹ with a mean value of 1.52 min⁻¹. In summary, a hierarchical TR modeling framework was developed in this study to analyze the SOC-dependent TR response stemming at the cell-level and interrogate its critical implications for TRP within a typical Li-ion battery module. The development of such experimentally-informed virtual TR analytics platforms can enable the mechanistic design of optimal cooling strategies and intelligent battery thermal management systems to prevent cell-to-cell TRP in battery modules.

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