**3. Model**

The thermal runaway model is based on the kinetic equation parametrised by the temperature (*T*) and *α* (0 ≤ *α* ≤ 1), the fractional degree of conversion of the reactants:

$$\frac{\partial \alpha}{\partial t} = k(T)f(a) \tag{1}$$

The temperature dependence of the process rate is typically parametrised through the Arrhenius equation:

$$k(T) = A \exp^{-\frac{\nu}{R}} \tag{2}$$

where *A* and *Ea* are kinetic parameters. *A* is the pre-exponential factor, *Ea* the activation energy, and *R* the universal gas constant.

The fractional degree of conversion of the reactants *α* is determined experimentally as the fraction of the overall change in the physical properties that go with the process (ex: loss of mass). The overall transformation can generally involve more than a single reaction or, in other words, multiple steps, each of which has its specific extent of conversion. This is why some authors have proposed to model all the different steps of reactions. This is the case for Coman et al. [11] or Hatchard et al. [12]. Hatchard et al. [12] built their model by distinguishing the reaction of the SEI, the anode, and the cathode, and also introduced the state of charge (SOC). A couple (*A*, *Ea*) was identified for each reaction.

Generally, models consider only the total reaction. First, because the various reactions are quite differentiated during the first steps, each reaction has its own kinetics. For example, a reaction can be described by:

$$\frac{\partial \alpha}{\partial t} = \begin{cases} & k\_1(T)f\_1(a\_1) \text{ if } T \in [T\_{0\prime}, T\_1] \\ & k\_2(T)f\_2(a\_2) \text{ if } T \in [T\_{1\prime}, T\_2] \end{cases} \tag{3}$$

*α*<sup>1</sup> and *α*<sup>2</sup> are the specific conversion rates of the two individual reactions and their sum is *α* = *α*<sup>1</sup> + *α*<sup>2</sup> with:

$$\begin{cases} \quad \mathfrak{a}\_1 = 0 \text{ if } T \in \left[ T\_{1\prime}, T\_2 \right] \\ \quad \mathfrak{a}\_2 = 0 \text{ if } T \in \left[ T\_{0\prime}, T\_1 \right] \end{cases}$$

Second, even if the mechanism involves several steps, one of them can determine the overall kinetics. For instance, this would be the case for a mechanism composed of two consecutive reactions when the first reaction is significantly slower than the second. Then, the first process would determine the overall kinetics that would obey a single step, whereas the mechanism involves two steps.

In this study, ARC tests were stopped at 160 ◦C (for new cells) and 90 ◦C (for aged cells). The aged cells were subjected to post-abusive characterisation. The use of this temperature range guarantees keeping the cell intact. The principal reactions at this temperature are first the SEI and then anode reactions: consumption of Li in the anode when the tests stop at 90 ◦C, and the consumption of active materials in the anode for tests stopped at 160 ◦C. It seems reasonable to model the process as a single-step equation.

*f*(*α*) is the kinetic model. This function is based on physical-geometrical assumptions of regularly shaped bodies. We understand that these assumptions cannot describe heterogeneous systems. Therefore, the idea is to describe as easily as possible more complex reactions.

The first expression of *f*(*α*) comes from the experimental fitting of *α* through relation (1) under isothermal conditions:

$$f(\boldsymbol{a}) = \sum\_{k} a\_{k} \boldsymbol{a}^{k} \tag{4}$$

Sestak and Berggren [5] introduced a semi-empirical model (SB model) that estimates this:

$$f(\mathfrak{a}) = l\mathfrak{a}^m (1-\mathfrak{a})^n (-l n(1-\mathfrak{a}))^p \tag{5}$$

Several variations of this model exist. The three principal ones are as follows: The Avrami–Erofeev model:

$$f(a) = n(1 - a)(-
ln(1 - a))^{(n - 1)/n} \tag{6}$$

The Prout–Tompkins model:

$$f(\mathfrak{a}) = \mathfrak{a}^m (1 - \mathfrak{a})^n \tag{7}$$

and the Prout–Tompkins regular model, where (*p* = 0, *m* = *n* = 1).

The lack of information regarding the estimation of these parameters led us to propose a way to estimate them. In the following sections, a method to build a thermal runaway model is proposed. To model it some parameters are required:


The following sections describe how to obtain these parameters from ARC tests.
