*3.1. Dependence of Elastic Constants on Concentration of Cr and Al*

The calculated elastic constants, *C*11, *C*12, *C*<sup>44</sup> and related bulk modulus and shear modulus, of FeCrAl alloy as a function of concentration of Cr and Al at 0 K are shown in Figure 1. The statistical uncertainties of these elastic constants and modulus were also calculated, which showed a similar dependence on concentration of Cr and Al, and thus just one example is presented in Figure 1f. Each panel in Figure 1 has five curves, corresponding to the five different concentrations of Al from 1 wt.% to 5 wt.%. For each curve, the data are plotted as a function Cr concentration from 1 wt.% to 15 wt.%. Some curves show an increasing trend, while others exhibit an opposite trend, indicating different dependencies of elastic constants on the concentrations of Al and Cr.

**Figure 1.** (**a**–**e**) Dependence of elastic constants, *C*11, *C*12, *C*44, bulk modulus and shear modulus on Cr and Al concentrations at 0 K. (**f**) Example of statistical uncertainties for these elastic constants based on 50 simulations.

As shown in Figure 1a, for the lowest concentration of Al at 1 wt.%, *C*<sup>11</sup> increases slightly with Cr concentration up to 15 wt.%. While for 2 to 5 wt.% Al, *C*<sup>11</sup> decreases accordingly. Furthermore, from Figure 1a, when *CCr* is in the range of 1–3 wt.%, *C*<sup>11</sup> increases with the increase of *CAl*, while the opposite is true for *CCr* in the range of 5–15 wt.%. Interestingly, all curves intersect at a common point at 4 wt.% Cr, indicating that at this Cr concentration, *C*<sup>11</sup> does not depend on *CAl*.

For *C*12, as shown in Figure 1b, when *CAl* is 1 wt.% or 2 wt.%, *C*<sup>12</sup> increases with *CCr* from 1 to 15 wt.%. When *CAl* is 3 wt.%, *C*<sup>12</sup> decreases firstly and then increases with *CCr*. When *CAl* is 4 wt.% or 5 wt.%, *C*<sup>12</sup> decreases with *CCr*. Furthermore, when *CCr* is fixed and *CCr* is lower than 5 wt.%, *C*<sup>12</sup> increases with the increase of *CAl*. When *CCr* is in the range of 5–13 wt.%, there is no clear dependence on *CAl* with a fixed *CCr*. For *C*44, as shown in Figure 1c, when *CAl* is fixed from 1 to 5 wt.%, *C*<sup>44</sup> increases firstly to a peak value and then decreases with the increase of *CCr*. It should also be noted that the peak is a function of *CAl*, that is, the higher *CAl* the higher the peak, and that the peak occurs at a smaller *CCr*. When *CCr* is fixed, *C*<sup>44</sup> increases with the increase of *CAl*.

In addition, the bulk modulus (*K*) and the shear modulus (*G*) have also been calculated according to the relationship between the elastic constants and K and G, derived by Voigt [25], as shown in Figure 1d,e. It is clear that when *CAl* is 1 wt.%, the bulk modulus increases with *CCr*, which is consistent with the results of previous studies [26], that is, the bulk modulus of the alloy increases with the increase of Cr content. When *CAl* is 2 wt.%, the bulk modulus decreases firstly and then increases with *CCr*. When *CAl* is higher than 2 wt.%, the bulk modulus decreases with the increase of *CCr*. When the concentration of Cr is fixed, the bulk modulus increases with the increase of *CAl* for *CCr* < 4 wt.%. For *CCr* in the range of 4–8 wt.%, the dependence of the bulk modulus on *CAl* is not linear. For *CCr* > 8 wt.%, the bulk modulus decreases with the increase of *CAl*. Regarding the shear modulus G, at 1 wt.% Al, the shear modulus is almost a constant with *CCr*. When the concentration of Al is higher than 1 wt.%, the shear modulus decreases with the increase of *CCr*. From this figure, there is also a common intersection of the shear modulus curves at a critical Cr concentration (8 wt.%), similar to the case of *C*<sup>11</sup> (at 4 wt.%). Thus, for *CCr* < 8 wt.%, the shear modulus increases with Al content, while the opposite is true for *CCr* > 8 wt.%.
