## Methodology
### Diffusion/Reaction Model  
At the nanoscale, the thermite reaction is simulated using a diffusion-reaction scheme, which allows the calculation of the combustion kinetics of the various species. The main assumption is that the chemistry driving the thermite combustion is restricted to a purely condensed phase process in which only oxygen is considered as a migrating species: oxygen diffuses through the growing oxide layer (i.e., the product of reaction) to react spontaneously upon reaching the fuel. It is, thus, assumed that the starting reactants and reaction products are in the solid state and exhibit thermal homogeneity. As shown in Figure 1, upon oxygen diffusion from the oxidizer and further reaction at the growing oxide interaction (i.e., oxide reaction product) with the fuel, the initial interface layer between the fuel and oxidizer (associated \(S_{init}\), Figure 1) propagates throughout the fuel particle (associated surface noted \(S_{fuel}\), Figure 1). Similarly, a front of reduction propagates into the oxidizer particle (noted \(S_{ox}\), Figure 1). The one-dimensional oxygen flux (noted \(\phi\)) along the \(x\) direction obeys the general form:  
\[\phi(x)=-\mathrm{D}\times S(x)\times\frac{dC(x)}{dx} \tag{1}\]  
where D and \(C(x)\) are the temperature dependent diffusion coefficient and concentration (in mol.m\({}^{-3}\)) of the oxygen species, respectively. \(S(x)\) is the surface area of the interface,  
Figure 1: Geometry of the modelled thermite system and main model parameters: top figure, general view of the system, with the reaction front propagating from the left to the right. Lower figure, a magnified view of one fuel/oxidizer particle pair, with its multiple interfaces as the reaction propagates. assumed to be flat, at a distance \(x\) from the center of a sphere of radius \(R\), either oxidizer or fuel as shown in Figure 1, expressed as:  
\[S(x)=\pi\Big{(}\mathrm{R}^{2}-x^{2}\Big{)} \tag{2}\]  
Note that \(\phi\) represents here the total flux and not the flux density, as generally used in 1D problems with a constant surface. In the Deal and Grove scheme [51], the flux is considered as constant between S\({}_{fuel}\) and S\({}_{ox}\). Therefore, Equation (1) can be integrated to express the oxygen concentration C at the various interfaces. The integral of 1/S(x) is a logarithmic function and we found:  
\[\mathrm{C(x)}=-(\phi/2\pi\mathrm{DR}\ \times\ \mathrm{ln}((\mathrm{R+x})/( \mathrm{R-x}))+Constant \tag{3}\]  
Applying this relation in the two regions that include, respectively, \(IB_{ox}\) and \(IB_{f}\) segments (see Figure 1), i.e., at point \(I\), \(B_{ox}\), and \(B_{f}\), summing them and imposing \(\mathrm{C}(B_{ox})=\mathrm{Cs}\), \(\mathrm{Cs}\) being the solubility limit in the reduced oxide, we found:  
\[\mathrm{C}\Big{(}B_{f}\Big{)}\ /2\pi\left[\frac{1}{R_{ox}D_{ox}}\mathrm{ln}\ \bigg{(}\frac{(R_{ox}+IO_{ox})(R_{ox}-O_{ox}B_{ox})}{(R_{ox}-IO_{ox})(R_{ox} +O_{ox}B_{ox}}\bigg{)}+\frac{1}{R_{f}D_{f}}\ln\left(\frac{(R_{f}+IO_{f})(R_{ f}-O_{f}B_{f})}{(R_{f}-IO_{f})(R_{f}+O_{f}B_{f}}\right)\bigg{]}\right] \tag{4}\]  
Note that \(D_{f}\) is the diffusion coefficient of oxygen in the oxidized fuel, as for instance Al\({}_{2}\)O\({}_{3}\) when dealing with aluminum-based thermites. In addition, at the reactive interface between the growing oxide and the fuel, the oxygen flux oxidizes Al at a rate \(v\), such that:  
\[\phi=v\ \pi\ \mathrm{C}\Big{(}B_{f}\Big{)}\left[R_{f}^{2}-\Big{(}IB_{f}\Big{)}^{2}\right] \tag{5}\]  
Combining Equations (4) and (5), we found the oxygen flux as a function of \(\mathrm{C}_{s}\), the solubility limit of oxygen in the metal oxide oxidizer, whatever its reduction degree, such as Cu\({}_{2}\)O or pure Cu, when considering CuO oxidizer.