## 3 Results and Discussion
### Effective Thermal Conductivity Formulation  
The preceding results are based on a model system in which the thermal conductivity is expressed (see Equation (4)) as a function of the compaction (%TMD) in the case of a perfect alternation of metal/metal oxide nanoparticles organized as a linear chain of homogeneous nanosized spheres (see Figure 2). From the literature, another common way to treat thermal conductivity of thermites is to consider an averaged and homogeneous bulk associating both the fuel and oxidizers (see Figure 6). [PERSON] et al. suggested such an effective thermal conduction expression, considering a continual heat flow through the compressed powder, which corresponds to a bulk material with the combined thermal properties of the fuel and the oxide species (\(o\), oxidizer; \(f\), fuel), given by [46]:  
\[\lambda_{eff}^{b}=\varepsilon_{o}\lambda_{o}+(1-\varepsilon_{o})\lambda_{f} \tag{10}\]  
with \(\varepsilon_{o}\) and \((1-\varepsilon_{o})\) being the mass fraction of the oxide and fuel species, respectively, in a fully dense system. To modify this value to account for an actual experimental %TMD compaction, we have slightly adjusted the formula to:  
\[\lambda_{eff}^{b}=\varepsilon_{o}\lambda_{o}+(\%TMD-\varepsilon_{o})\lambda_{f} \tag{11}\]  
In this case, the material is still treated as a bulk heterogeneous material, but now the energetic mixture is having voids up to the preset %TMD (Figure 6b). Both of these approaches do not consider the geometric factors that form the basis of this model, the sintered particle pair, in addition to be best fit for simulating a %TMD that is rarely seen in nanothermites (>60%). Note also, that these effective conductivities ignore the interfacial regions formed by the wetted contact between the fuel-oxide pairing, whose bottlenecking aspect is important at the scale of the diffusion/reaction equations in our model. A more elaborate version of Equation (9) considers the realistic diversity of non-constant particle sizes, as seen in the size distribution of an actual nanopowder, as well as the assortment of different sized agglomerates formed after reactive sintering. This formulation as shown in Figure 6d is:  
\[\frac{1}{\lambda_{eff}^{d}}=\frac{1}{6\%TMD}\sum_{i}(\frac{1}{R_{i}\lambda_{i }}ln\Bigg{[}\frac{R_{i}+(IO)_{i}^{+}}{R_{i}-(IO)_{i}^{+}}\times\frac{R_{i}+( IO)_{i}^{-}}{R_{i}-(IO)_{i}^{-}}\Bigg{]}\times\frac{\sum\Big{[}d_{i}^{+} \Big{(}3R_{i}^{2}-(IO)_{i}^{+2}\Big{)}+d_{i}^{-}\Big{(}3R_{i}^{2}-(IO)_{i}^{- 2}\Big{)}\Big{]}}{\text{L}^{2}}) \tag{12}\]  
Figure 6: One-dimensional diagrams of the “effective” material structure given by: (**a**) bulk material with combined thermal properties, (**b**) bulk material with voids filling space to %TMD, (**c**) single 1D chain of identical alternating fuel and oxide particles, and (**d**) a single one-dimensional chain of various sized alternating fuel and oxide particles. In this formulation, each particle possesses two moving interfaces from both sides of the chain axis. \(R_{\text{i}}\) is the radius of the particle \(i\) (\(i\) being the oxidizer or fuel). \(d_{\text{i}}^{+}\) and \(d_{\text{i}}^{-}\) are the converted materials being oxidized or reduced depending on the chemical nature of particle \(i\), \(+\) and \(-\) indicating the two opposite directions along the linear chain axis, from either sides of the center of particle \(i\), as defined in the Supplementary Information file S4. L is the length of the considered chain motif. Equation (7) is the generalization of Equation (4) to systems with different nanoparticle sizes. Here, the heat flux is expressed as a function of the temperature difference along a chain of sintered nanoparticles. For this reason, Equation (7) contains a sum representing all discrete interfaces between nanoparticles, which can be modified depending on the actual structure considered.