**1. Introduction**

After the Fukushima Daiichi nuclear power disaster in 2011, development of accidenttolerant fuels (ATF) has been in great demand to enhance the safety of nuclear power plants, especially under accident conditions [1]. In addition to coated claddings, FeCrAl cladding is one of the possible technologies for use in ATF. In the past few years, based on this concept, FeCrAl alloys have been extensively studied and the advantages have been recognized, including improved high-temperature steam oxidation resistance [2], resistance to radiation damages [3], strength under both normal conditions and high-temperature accident conditions [4], and enhanced corrosion performance under normal conditions [5].

The roles of Cr and Al elements have been explored to understand the above properties. For example, Cr is added to improve the water corrosion resistance of these alloys at normal operating temperatures, and Al is added to improve the high-temperature oxidation resistance of the alloy through the formation of protective α-Al2O3 [6,7]. However, it should also be noted that various properties of FeCrAl alloys are strongly affected by the addition and concentrations of Cr and Al, including the mechanical properties. For example, the α phase in FeCrAl alloy will nucleate and grow at temperatures around 475 ◦C with a concentration of Cr, *CCr*, around 10~18 at.%, which leads to embrittlement of the alloy under normal conditions [8,9]. Meanwhile, under irradiation, the α phase could

**Citation:** Dai, H.; Yu, M.; Dong, Y.; Setyawan, W.; Gao, N.; Wang, X. Effect of Cr and Al on Elastic Constants of FeCrAl Alloys Investigated by Molecular Dynamics Method. *Metals* **2022**, *12*, 558. https://doi.org/10.3390/ met12040558

Academic Editor: Angelo Fernando Padilha

Received: 18 January 2022 Accepted: 22 March 2022 Published: 25 March 2022

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form even at 320 ◦C, with *CCr* around 10~18 at.% [10]. Hence, it is necessary to decrease the concentration of Cr to reduce the α phase formation and related embrittlement of the alloy under irradiation at medium temperatures [11]. For this purpose, Oak Ridge National Laboratory (ORNL) has prepared a variety of FeCrAl-Y alloys with different Cr concentrations to develop new low-Cr, high-strength FeCrAl-Y alloys [11]. They found that a FeCrAl alloy with *CCr* less than 13 wt.% can effectively avoid the formation of a brittle phase at 475 ◦C. Gussev et al. found that adding Al to the alloy can improve the oxidation resistance, but the higher Al concentration, *CAl*, increases, the greater the difficulty of welding and manufacturing the FeCrAl alloys [8,12].

In addition, previous studies have also reported that the concentration of Cr and Al could significantly affect the mechanical properties of FeCrAl alloys [12]. The elastic properties have been investigated for evaluating the survivability of cladding materials under energetic neutron irradiation [13]. In fact, mechanical properties, including the elastic modulus, yield stress, ultimate tensile stress, and so on, have been generally measured and estimated for selection of structural materials used in reactors before neutron irradiation experiments [14,15]. For example, before irradiation, it was reported by Liu et al. that the increase in Cr concentration will enhance the bulk modulus of the alloy. However, when the concentration of Al increases from 4.5 wt.% to 6 wt.%, the bulk modulus could also slightly decrease [14]. From experimental viewpoints, Speich et al. measured the shear modulus *G*, the ratio of the bulk modulus to the shear modulus K/G, and the Poisson's ratio *v* [16] using an ultrasonic pulse echo repeat method [17] at 298 K, indicating that the shear modulus and Poisson's ratio of the Fe-rich region of the Fe-Cr alloy showed a linear dependence with the increase of Cr concentration. Harmouche et al. prepared FeAl-B2 phase samples with Fe atomic fractions from 50.87% to 60.2%, and by measuring its Young's modulus at room temperature, the authors found that the dependence of Young's modulus on the concentration of Fe shows a quadratic relationship [18]. The dependence of elastic modulus, Poisson's ratio, and shear modulus on temperature has also been explored for FeCrAl alloy and almost no variation has been observed as a function of major alloy elements [12]. Based on the above literature review, it is clear that although there are many studies on the mechanical properties of binary alloys, research on the elastic properties of ternary alloys needs to be enriched, especially on the mutual effect of Cr and Al on the elastic constants of a ternary FeCrAl alloy. From the viewpoint of fundamental research, it would be necessary to investigate the Cr and Al concentration effect from the lower to higher values in order to present the possible dependence of elastic properties on these alloy elements.

Different from the effect of Cr and Al on mechanical properties before the irradiation, the effect of radiation-induced defects on the mechanical properties of FeCrAl alloy is another key factor in evaluating the performance of the selected material. From previous studies, it is well known that the elastic modulus is one of important parameters to be dynamically measured in order to estimate the state of structure materials during the performance of nuclear reactors [13]. Thus, understanding the elastic constant and modulus before and after the irradiation is important for the development and further optimization of FeCrAl alloys used in ATF design. Under irradiation, the super-saturated interstitials, vacancies, voids, and precipitates are generally observed, which are expected to affect the mechanical properties of FeCrAl alloys. For example, the formation of Cr-rich a phases in irradiated FeCrAl alloys has been investigated by small-angle neutron scattering (SANS) [19,20], atom probe tomography (APT) [21,22], high-efficiency STEM-EDS (energydispersive spectroscopy) [21] and a series of technical studies. Unfortunately, although the irradiation hardening and embrittlement of FeCrAl after irradiation have been investigated, the detailed dependence of elastic constants or modulus on irradiation defects in FeCrAl alloys is not well understood, so research comparing this to the dynamically measured elastic modulus of materials to increase the safe operation of nuclear reactors would be valuable.

Therefore, in order to shed light on above questions, in this work, molecular dynamics (MD) simulations have been performed to calculate the elastic constants of FeCrAl alloy with 1~15 wt.% Cr and 1~5 wt.% Al at different temperatures (0 K, 300 K, 450 K, 600 K and 750 K) to understand the concentration effect of alloy elements on elastic properties of FeCrAl alloy. The effects of radiation defects, including vacancies, interstitials and Cr precipitates on the elastic properties, are also simulated. Through these calculations, the elastic properties of FeCrAl alloy before and after the irradiation can be understood from an atomic scale. In the rest of this paper, the method of calculations is introduced in Section 2, results and discussion are provided in Section 3, and conclusion is given in the last section.
