*3.2. Estimation of Parameters* (*l*, *m*, *n*, *p*)

Once the onset temperature of each cell is determined, the parameters of the model can be identified.

The demonstration is realised under adiabatic conditions. The energy conservation equation for an adiabatic reaction guarantees:

$$\frac{\partial T}{\partial t} = \frac{E\_{\text{Tot}}}{\mathbb{C}\_{\text{tot}}} \frac{\partial \alpha}{\partial t} \tag{10}$$

where *ETot* (*J*) is the total heat that can be produced by the reaction and *Ctot* is the total capacity (*JK*<sup>−</sup>1).

The heating rate is denoted by:

$$
\beta(t) = \frac{\partial T}{\partial t} \tag{11}
$$

With this notation the thermal runaway model (1) becomes:

$$
\ln \left( \beta(t) \frac{C\_{tot}}{E\_{Tot}} \right) - \ln(f(a)) = \ln(A) - \frac{E\_a}{RT} \tag{12}
$$

Therefore:

$$\ln(f(a)) = \frac{E\_a}{RT} + \ln(\frac{\beta(t)}{A}\frac{\mathbb{C}\_{tot}}{E\_{Tot}}) \tag{13}$$

At the start of the reaction *t* = *t*0, *α* = 0. The reactants begin to transform and we can consider that the heat rate is constant:

$$
\beta(t) = \beta \tag{14}
$$

Therefore, the SB kinetic model can be approached by a limited development:

$$
\ln(f(\mathfrak{a})) \underset{\mathfrak{a} \to \mathfrak{0}}{\sim} \ln(l) + (m+p)\ln(\mathfrak{a}) - n\mathfrak{a} \tag{15}
$$

Therefore, when *α* → 0 Equation (13) is:

$$\frac{E\_d}{RT} + \ln(\frac{\beta(t)}{A}\frac{C\_{tot}}{E\_{Tot}}) = \ln(l) + (m+p)\ln(\alpha) - na \tag{16}$$

By deriving Equation (16) by *T*:

$$-\frac{E\_a \beta}{RT^2} = (m+p)\frac{\partial \alpha}{\partial t}\frac{1}{\alpha} - n\frac{\partial \alpha}{\partial t} \tag{17}$$

Due to the adiabatic condition *∂α <sup>∂</sup><sup>t</sup>* verifies (10) so:

$$-\frac{E\_a E\_{Tot}}{RC\_{tot}T^2} = (m+p)\frac{1}{a} - n \tag{18}$$

Three cases are possible at the beginning of the reaction (*α* → 0):

• *Ea* = 0. The reaction is instantaneous. Then:

$$\frac{m+p}{n} = 0$$

• *Ea* > 0 then:

$$\frac{m+p}{n} < 0$$

• *Ea* < 0. The phenomenon occurs in a chain reaction. Then:

$$\frac{m+p}{n} > 0$$

At the beginning of the reaction, it seems reasonable that *Ea* be positive or null. This implies: *<sup>m</sup>* <sup>+</sup> *<sup>p</sup>*

$$\frac{n+p}{n} \le 0 \tag{19}$$

Gorbatchev [13] has shown that no more than two kinetic exponents are necessary to describe any kinetic schema, so we choose *p* = 0.

*n* is considered as the order of the reaction, so *n* > 0, and therefore:

$$m \le 0\tag{20}$$

Let us study the Gibbs free energy of activation, defined as:

$$
\Delta G = \Delta H - T\Delta S = -RT\ln\left(\frac{kh}{k\_B T}\right) \tag{21}
$$

where *kB* is the Boltzmann constant and *h* the Planck constant and so *kB <sup>h</sup>* <sup>=</sup> 2.084 <sup>×</sup> 1010. *<sup>k</sup>* is the constant reaction.

Chemical thermodynamics explains that if


• Δ*G* > 0: additional energy must be input for the reaction to occur.

In ARC tests, the cell is heated for a few minutes and then there is a 35 min break: if the heat rate is not over 0.02 ◦C a new step is started. During these first steps additional energy is necessary to trigger the reactions. Therefore, Δ*G* must be positive. At the onset temperature: *<sup>T</sup>*0, the thermal runaway starts and reactions become spontaneous so <sup>Δ</sup>*<sup>G</sup>* < 0.

We consider that at *t*0, when *T* = *T*<sup>0</sup> the reaction is at its state of equilibrium. The free enthalpy is then equal to 0.

Therefore, at *t* = *t*0:

$$\ln\left(\frac{kh}{k\_B T\_0}\right) = 0 \iff k\_0 = \frac{k\_B T\_0}{h} \tag{22}$$

From (1) and (10):

$$\ln(k) = \ln\left(\frac{\beta(t)C\_{\text{tot}}}{E\_{\text{Tot}}}\right) - \ln(f(\alpha))\tag{2.3}$$

At *T*0:

$$\ln(k\_0) = \ln\left(\frac{\beta(t\_0)\mathbb{C}\_{tot}}{E\_{Tot}}\right) - \left(\ln(l) + m\ln(a\_0) + n\ln(1-a\_0)\right) \tag{24}$$

Then:

$$m\ln(a\_0) = \ln\left(\frac{\beta(t\_0)C\_{tot}}{E\_{Tot}}\right) - \ln(k\_0) - n\ln(1-a\_0) - \ln(l)\tag{25}$$

*<sup>m</sup>* <sup>≤</sup> 0 and *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; 1 so:

⎧ ⎪⎨ ⎪⎩

$$
\ln\left(\frac{\beta(t\_0)\mathbb{C}\_{tot}}{E\_{Tot}}\right) - \ln(k\_0) - n\ln(1 - a\_0) - \ln(l) \ge 0\tag{26}
$$

Therefore:

$$\begin{array}{l} \ln(l) \le \ln\left(\frac{\beta(t\_0)C\_{\text{tot}}}{E\_{\text{Tot}}}\right) - \ln(k\_0) - n\ln(1 - a\_0) = \ln(l\_{\text{max}})\\ m = \frac{\ln\left(\frac{\beta(t\_0)C\_{\text{tot}}}{E\_{\text{Tot}}}\right) - \ln(k\_0) - n\ln(1 - a\_0) - \ln(l)}{\ln(a\_0)} \end{array} \tag{27}$$

*l* will be chosen so that the Gibbs free energy is positive before *T*<sup>0</sup> and negative after, and so that ln(*l*) ≤ ln(*lmax*) is verified. Until these conditions are verified, ln(*l*) is reduced per step to 1.

For a fixed *n*, Algorithm 1 can be used to choose the parameter (*l*, *m*).

*3.3. Estimation of* (*Ea*, *A*)

Energy activation (*Ea*) and pre-exponential factor *A* are calculated thanks to Equation (13) and the calculated parameters (*l*, *m*) (Figure 2).

**Figure 2.** Estimation of *Ea* and *<sup>A</sup>* for fresh cell data: *ln*(*k*(*T*)) = *ln*(*A*) <sup>−</sup> *Ea RT* .

The pre-exponential is defined as:

$$
\ln(A\_0) = \ln(A) + \ln(l) \tag{28}
$$

## *3.4. Remarks about the Temperature Effects*

The rate constant *k* is defined by the Arrhenius law, where energy of activation is considered a constant parameter (independent of the temperature). This approximation proves to be a very good one, at least over a moderate temperature range. In this case, kinetic phenomena can be interpreted as a set of elementary reactions. This theory is totally adapted for gas phases where chemical transformations take place in a series of isolated binary collisions of molecules, but less for solid or liquid phases or when the reactions are blended [5,14].

There is another way of calculating the rate constant. The expression comes from the transition state theory defined by Equation (21). In this case the energy of activation *Ea* and pre-exponential factor *A* are dependent on the temperature. It is this theory that is used to determined the values of (*m*, *l*). It seems well-adapted because in the system the reactions are blended and are in different phases. Pulses of heat impact the progress of the reaction (*α*) and so the transitory state of equilibrium. The level of energy of activation required to realise the reaction will change.

#### *3.5. Remarks about the Precision of Measurements*

Algorithm 1 updates *lmax* until all the values of *G* are negative before *T*0. Sometimes, due to the precision of the measurement, few points are not under 0 during several iterations. For example, in Figure 3, 20 iterations more were performed because 3 points were still positive. *lmax* was then overvalued.

To avoid this kind of problem, instead of verifying *G*[0 : *idx*0] ≤ 0, a tolerance is added. In theory no value can be positive. In this case, we accepted a maximum of 10 positive values.

**Figure 3.** Number of negative *G* points as function of the number of iterations for a cell aged at 45 ◦C.

#### **4. Results**

All the parameters described were calculated on aged cells in order to appreciate the impact of the ageing on the kinetics model. The values were crossed with experimental results to verify the coherence of the values obtained.

#### *4.1. Post-Mortem Analysis*

The negative electrode is the site of the exchangeable lithium loss (SEI, Li-plating, ...). This is why the post-mortem analyses (GD-OES, Li-NMR and EIS) were focused on it. The results are presented in the next paragraph.

#### 4.1.1. GD-OES and Li-NMR

Figure 4a presents the composition of the negative electrode surface after the different ageing conditions obtained by GD-OES. GD-OES allows us to study the chemical composition of the surface in the first 1.5 μm (total thickness of electrode is 44 μm) and allows us to understand which ageing mechanism takes place for each ageing condition.

For all aged cells, GD-OES measurements show that Si particles, initially well distributed in the negative electrode depth, are concentrated on the electrode surface. This migration should be related to the formation of Li-silicates [15], which consume cyclable Li.

Compared to fresh cells, the quantity of *Li* increases whereas the quantities of *O* (oxygen) and *P* decrease for cells aged at −20 ◦C. This shows that SEI does not grow on the electrode surface and it reveals the presence of metallic Li on its surface. Li-NMR measurements presented in Figure 4b confirm this observation.

When cells are aged at 25 ◦C and 45 ◦C, *Li* increases while *O* decreases and *P* increases, which reveals SEI growth by salt degradation [16].

At 45 ◦C calendar ageing, the observations are the same for both CV and OCV conditions. The increase in *Li* coupled with the increase in *O* and the decrease in *P* reveals that SEI grows by solvent degradation [16]. The mechanism is more pronounced for CV than OCV conditions.

Cells aged at 0 ◦C have no preponderant ageing mechanism considering the very small change in *O* and *P* quantities. 7Li-NMR MAS (Figure 4b) shows that the observed Li is not in a metallic state. The Li detected here by the GD-OES analysis is not contained in the SEI layer and not in a metallic state.

**Figure 4.** GD-OES and Li-NMR measurements. (**a)** Elementary composition of negative electrode surface by GD-OES analyses after ageing. (**b**) Li-NMR measurements of negative electrodes after ageing.
