*3.2. Dependence of Elastic Constants on Temperature*

The elastic constants and modulus calculated at 300 K are shown in Figure 2. As shown in Figure 2a, the dependence of *C*<sup>11</sup> on *CCr* and *CAl* is similar to the results obtained at 0 K, as shown in Figure 1a. For *C*12, comparing to the results shown in Figure 1b, for *CAl* < 4 wt.%, a similar dependence of *C*<sup>12</sup> on *CCr* can be observed, that is, *C*<sup>12</sup> increases with *CCr*. However, for *CAl* equal to 4 wt.% and 5 wt.%, different trends were obtained. *C*<sup>12</sup> is almost a constant as a function of *CCr* for *CAl* = 4 wt.%. While for *CAl* = 5 wt.%, *C*<sup>12</sup> decreases firstly with *CCr* up to around 7 wt.% and then remains almost constant, as shown in the Figure 2 b. Furthermore, for a fixed *CCr*, *C*<sup>12</sup> always increases with the increase of *CAl*. For *C*44, *C*<sup>44</sup> increases with the increase of both *CCr* and *CAl*, a behavior that is different from the *C*<sup>44</sup> at 0 K, as previously shown in Figure 1c. Compared to the results at 0 K, the bulk modulus shows a similar dependence on the concentration of Cr and Al, as shown in Figures 1d and 2d. The shear modulus also shows a similar trend, but the critical *CCr* concentration for *G* (the intersection point) has changed from ~8 to ~15 wt.%. From these results, especially from the bulk modulus and shear modulus, it can be estimated that from the mutual effects of Cr and Al concentration on elastic properties, the concentration of Al should be not higher than 3% and the related Cr concentration should be higher than 10% in the bulk FeCrAl alloys. However, it should also be noted that the above results were obtained only according to the effect of concentration of Cr and Al on elastic properties but without considering the other defects and related defect interactions. Furthermore, since the formation of Al2O3 on the material surface is the main reason to increase the high-temperature oxidation resistance of FeCrAl alloy and addition of Cr is necessary to increase the corrosion resistance, the Cr and Al concentrations are expected to be higher

to satisfy this requirement. In fact, many researchers have suggested to add the other elements to increase the performance of FeCrAl alloy. Therefore, the optimization of Cr and Al concentration needs to consider various conditions, which is beyond the topic of the present paper. The optimization of FeCrAl alloy can be found elsewhere [26,27].

**Figure 2.** (**a**–**e**) Dependence of elastic constants, *C*11, *C*12, *C*44, bulk modulus and shear modulus on Cr and Al concentration at 300 k. (**f**) Example of statistical uncertainty for these elastic constants based on 50 simulations.

In this work, as stated in the Methods section, the elastic constants and modulus at 450 K, 600 K and 750 K were also calculated. Detailed analysis indicates that the results obtained at different temperatures, including 0 K, 300 K, 450 K, 600 K and 750 K, show a similar dependence on the concentration of Cr and Al. One example for *CAl* = 1 wt.% is shown in Figure 3, which clearly indicates the same dependence of elastic constants on Cr and Al concentration at different temperatures. Therefore, based on these results, the dependence of elastic constants and modulus on temperature can be explored.

The dependence of elastic properties on temperature is shown in Figure 4. The results for alloys with 1 wt.% Al and 4 wt.%, 8 wt.% and 14 wt.% Cr, are shown to illustrate the temperature effect. From this figure, it is evident that all elastic constants investigated in this work, *C*11, *C*12, *C*44, bulk modulus and shear modulus of these three alloys, decrease with increasing temperature from 300 K to 750 K, which is also consistent with the results of a previous study [27]. It is worth noting that in a previous experiment [28], the decrease of elastic constants with the increase of temperature has also been observed, in which the bulk modulus decreased from about 200 GPa at 300 K to about 150 GPa at 750 K, and the shear modulus decreased from about 77 GPa at 300 K to about 65 GPa at 750 K, both of which decreased by about 15% [28]. In the present work, the bulk and shear moduli decrease by about 6% and 9%, respectively, less than in the experiment. The main reason is that in the experiment, the sample was polycrystalline, while a single crystal is employed in our calculations. In a polycrystalline material, the effect of grain boundaries could contribute to the decrease of the elastic properties.

**Figure 3.** (**a**–**e**) Dependence of elastic constants, *C*11, *C*12, *C*44, bulk modulus and shear modulus on the concentration of Cr in an FeCrAl alloy with 1 wt.% Al at temperatures 0~750 k, (**f**) example of statistical uncertainty for these elastic constants based on 50 simulations.

**Figure 4.** (**a**–**e**) Dependence of elastic constants, *C*11, *C*12, *C*44, bulk modulus and shear modulus on temperature for 4Cr1Al (wt.%), 8Cr1Al (wt.%) and 14Cr1Al (wt.%) compositions at 300–750 K.

As shown in Figure 4, the dependence of each elastic constant on temperature follows the same trend for different concentration of Cr and Al, which, thus, can be described by the same equation but with different parameter values. For *C*11, this dependence can be described as:

$$C\_{11}(T) = A - BT \tag{1}$$

where *A* and *B* are parameters that depend on the concentration Cr and Al. For example, for 4Cr1Al (wt.%), 8Cr1Al (wt.%) and 14Cr1Al (wt.%), *A* is 236.23, 235.63 and 236.65 GPa, respectively, and *B* is 0.045, 0.045 and 0.047 GPa/K, respectively. The dependence of the other elastic constants, *C*12, *C*44, K and G, on temperature can be described by the following equation:

$$C(T) = A + BT - CT^2 \tag{2}$$

The fitted values for *A*, *B* and *C* for different concentrations of Cr and Al are listed in Table 2. In Figure 4, the fitted equations for FeCrAl alloy with *CCr* ~ 4%(wt.%) and *CAl* ~ 1%(wt.%) are also included.

**Table 2.** Fitted parameter values in the models of elastic constants and modulus as a function of temperature for Fe4Cr1Al (wt.%), Fe8Cr1Al (wt.%) and Fe14Cr1Al (wt.%). The models are given in Equations (1) and (2).


The nature of the decrease in the elastic constant after the temperature rises is due to the change in the lattice constant caused by thermal expansion [13]. This phenomenon can also be understood from the elastic constant temperature dependence formula derived by Leibfried and Ludwig in 1961 [29].
