**2. Methods**

In this work, molecular dynamics (MD) simulations have been performed with LAMMPS software to calculate the elastic constants at different temperatures. The computational box was built with three directions along [100], [010] and [001]. There are 10 unit cells along each direction, thus, 2000 atoms in the matrix box. The interactions between atoms in FeCrAl are described by the F-S potential developed by Liao et al., which has been shown to be accurate to describe the characteristics of the elements in FeCrAl alloys [22]. The time step is 1 femtosecond (fs), and the total simulation time is up to 10 ps for each simulation.

The elastic tensor can be written as a 6 × 6 matrix using the Voigt concept, that is, *C*ij instead of *C*ijkl. According to this definition, during calculations, the elastic constants can be calculated by applying elementary deformations in the six distinct strain components and measuring the changes in the six stress components accordingly. In this work, the strain to induce the deformation of the simulation box was set as 10<sup>−</sup>6. Thus, according to the dependence of stress on strain, the elastic constants can be calculated. For temperatures higher than 0 K, thermal motion is included in the calculation of the stress. For the cubic lattice simulated in this work, the elastic tensor has three independent elastic constants, namely *C*11, *C*<sup>12</sup> and *C*44. With the above method, these elastic constants can be calculated with or without temperature effect. In fact, this method has been applied by different groups to investigate the elastic properties of materials. For example, Miller et al. studied the elastic constants of PbTe, SnTe and Ge0.08Sn0.92Te with this method [23]. In order to understand the influence of solute atoms on the elastic constants of FeCrAl alloys, 1~15 wt.% Cr and 1~5 wt.% Al atoms were introduced as substitutional solutes to form a solid solution, since FeCrAl alloys generally form a solid solution, as observed in previous experiments [10,24]. For convivence, the atomic concentration, at.%, is usually applied for atomic simulations. The conversion between wt.% and at.% can be obtained by including the atomic mass for Fe, Cr and Al. Examples of this unit conversion are listed in Table 1. It should also be noted that both Cr and Al atoms were introduced into the computational box in a randomly substitutional way by keeping the distance between each pair of substitutional atoms at least one lattice constant. As stated above, simulations at different temperatures (0 K, 300 K, 450 K, 600 K, 750 K) were performed to explore the effect of temperature on the elastic constants of the alloy. Furthermore, at each temperature, 50 simulations were performed to get reliable statistics and to estimate uncertainties in the results.

**Table 1.** Examples of conversion from wt.% to at.% for several FeCrAl alloys.


In addition, in order to consider the effect of vacancies, interstitials and Cr precipitates on elastic constants, a big simulation model with 36,000 atoms was built with three directions in the same orientation as before. The radiation-induced vacancies (and voids) are included in the computational box by changing, *n*, the number of vacancies in each vacancy

cluster or void and, *m*, the number of vacancy clusters or voids. For convenience, in this work, the total number of vacancies in the box was kept as 320, and thus *nm* = 320. The value of *m* is defined as 1, 2, 4, 5, 10, 20, 40, 80, 160 and 320, thus *n* is calculated accordingly. To investigate the effect of interstitials, different numbers of separated interstitials are were in the computational box. For a single interstitial, the <110> Fe-Fe or Fe-Cr dumbbell was built after the construction of FeCrAl alloy since the substitutional Al has lower formation energy, and thus, only the Fe-Fe and Fe-Cr dumbbells were built in the present work. As to the effect of Cr-rich α on elastic properties, similar to the model of vacancy clusters, the precipitates containing different numbers of Cr atoms were considered. The number of precipitates was 1 to 7 and two cases with a total number Cr atoms in the box up to 4300 (*CCr* ~ 11%) and 4850 (*CCr* ~ 13%), respectively, were considered. Following above method, simulations at 0 K were performed to explore the effects of vacancies, interstitials and Cr precipitates on the elastic constants of the alloys. In each case, 50 simulations were also performed to obtain reliable statistics and estimate the uncertainty in the results.
