## 3 Results and Discussion
### Effective Thermal Conductivity Formulation
Note that when this effective conductivity is used, the particle size distribution is exclusively accounted for in the conduction, while the particle sizes considered for all mass transfer mechanisms remain constant. Table 2 compares the effect of the four above expressions of the conductivity. It assembles calculated initiation and combustion performances in terms of the initiation temperature, initiation delay, and burn rate relative to the proposed effective thermal conduction coefficient. We make use of a typical model set up of 80 nm Al: 99 nm CuO system (\(\xi\) = 1.0) at 60 %TMD unless otherwise mentioned. A bulk-like term (Equation (5)) accounting for a fully dense, 100%TMD packing, shows the highest conduction coefficient (83 W/mK) leading to a high burn rate value, at 22 m\(\cdot\)s\({}^{-1}\). The high conductivity is a slightly penalizing initiation because of the energy loss from the externally heated first cell into the broader apparatus (higher initiation delay, 85 ms). This also results in a slightly higher initiation temperature (1003 K). Next, using the formulation given in Equation (6), which introduces compaction, all values are reduced. The burn rate is divided by two, because the conductivity drops down to 50 W-mK\({}^{-1}\). This lower conductivity limits losses into the system that leads to a slightly lower initiation temperature, lowering the initiation delay down to 69 ms. In contrast to bulk like formulations, shifting to chain like structured materials, more in the scope of the considered nanostructure (Equations (4) and (7)), reduces the conductivity by an order of magnitude. The conductivity through the interfaces between oxide particles that traditionally have a smaller bulk thermal conductivity can bottleneck the heat propagation, slowing the overall burn rate issued from the self-sustained reaction. This modification greatly affects performances in comparison to homogeneous and dense materials. The burn rates drop into the cm\(\cdot\)s\({}^{-1}\) range. We saw previously that the homogeneous alternated chain has a 59 ms initiation and that both of these values fall within recently published data [36]. Equation (7) accounts for a distribution of different sized particles in the chain, which gives a more realistic picture of the experimental system in actual fabricated powders, offering an opportunity to quantify its impact on performances. Additionally, the numerous studies of in situ observation of reactive sintering has shown that the formation of microscale components due to agglomeration of nanoscale individuals occur on a faster timescale than that of the overall reaction, meaning some nanoscale material coalesces before initiation [36]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Formulation/Figure 6**} & **Effective Conduction** & **Initiation** & **Initiation** & **Burn Rate** \\  & **Coefficient** & **Temperature** & **Delay (ms)** & **(cm\(\cdot\)s\({}^{-1}\))** \\  & **(W\(\cdot\)mK\({}^{-1}\))** & **(K)** & & \\ \hline Equation (5)*/a) & 83.3 & 1003 & 85 & 22 \\ Equation (6)/b) & 50.2 & 997 & 69 & 11 \\ Equation (4)/c) & 2.9 & 944 & 59 & 0.6 \\ Equation (7)/d) & 0.6–1.2 & 943 & 23–42 & 0.2–0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated initiation and combustion performances for a thermite system of 80 nm Al: 99 nm CuO (\(\xi\) = 1.0) at 60 %TMD (except for * which is 100 %TMD) considering the different effective conductivity formulas. Multiple size dispersions of either just the fuel particles, just the oxide particles, or both were tested (see Supplementary Information file, Table S5). The dispersion was set up as a half-Gaussian distribution with the mean set to the average particle size utilized in these studies (i.e., 80 nm for aluminum and 100 nm for cupric oxide). Each effective conductivity reported in Table S5, corresponding to a certain variance, was defined as the average of three calculations with a different seed value utilized to generate the random diameter samples.