Phase Stability and Mechanical Properties of Al8Fe4RE via First-Principle Calculations

We report on the phase stability, elastic, electronic, and lattice dynamic properties of 17 Al8Fe4RE (RE = Sc, Y, La–Lu) intermetallic compounds (IMCs) using first-principle calculations. The calculated lattice constants coincided with the experimental results. The calculated enthalpy formation indicated that all the 17 IMCs are stable. The elastic constants and various moduli indicated that Al8Fe4RE can be used as a strengthening phase due to its high Young’s modulus and shear modulus. The 3D surfaces of Young’s modulus for Al8Fe4RE showed anisotropic behavior, and the values of hardness for the IMCs were high (about 14 GPa). The phonon spectra showed that only Al8Fe4Y had a soft mode, which means the other IMCs are all dynamically stable.


Introduction
Due to their low density, low thermal conductivity, relative high strength, and low material cost, Al-Fe-based alloys have been studied extensively over the last few decades [1][2][3][4]. Al-Fe-based alloys are promising, high-temperature structural materials; however, their limited ductility at room temperature and the reduction in strength above 600 • C obstruct their application as high-temperature structural materials. Nevertheless, some recent investigations have shown that the mechanical properties can be effectively improved by controlling the microstructure, composition, and alloying elements [5][6][7][8].
As we known, rare-earth (RE) elements are special modifiers that are commonly used in Al-based and Fe-based alloys. Thus, the addition of RE elements in Al-Fe-based alloys may affect the microstructure and improve the mechanical properties of these alloys. When RE elements are added, Al-Fe-RE intermetallic compounds (IMCs) form, which affects the phase relationship and microstructure of Al-Fe-based alloys. The mechanical properties of Al-Fe-based alloys are consequently improved due to the changes in composition and microstructure. In previous works, Al-Fe-RE (RE = Y, Ce, Nd, Gd, Er) ternary phase diagrams have been experimentally investigated [9][10][11][12][13], and the ternary IMCs have been determined. The Al 8 Fe 4 RE IMCs are observed at the Al-rich corner, and they have a tetragonal crystal structure. The 17 Al 8 Fe 4 RE (RE = Sc, Y, La, Ce, Nd, Eu-Er, and Lu) IMCs have also been previously determined in experiments [14][15][16][17][18][19][20][21][22][23][24][25]. Using the empirical electron theory (EET), Al 8 Fe 4 Ce was found to be favorable for the stability of the Al-based alloy as a strengthening phase [15]. The magnetic properties of Al 8 Fe 4 RE have also been investigated [16][17][18][19][20][21], and the electronic conductivity [22] and the negative magnetoresistivity [23] of Al 8 Fe 4 RE have also been studied. Using the lattice inversion method, the lattice constants and lattice vibration spectra of Al 8 Fe 4 RE (RE = Sc, Ce, Nd, Sm) have been reported [24,25]. As a potential strengthening phase and as magnetic materials, the structural stability and electronic and elastic properties of Al 8 Fe 4 RE are very important for material design and for further development. However, few studies have focused on the electronic and elastic properties of Al 8 Fe 4 RE IMCs. Thus, the aim of this work was to study the physical properties of 17 Al 8 Fe 4 RE (RE = Sc, Y, and La-Lu) IMCs using first-principle (FP) calculations.
The formation enthalpy and cohesive energy of the Al 8 Fe 4 RE alloys can be estimated from the following equations: where E(Al 8 Fe 4 RE), E(Al), E(Fe), and E(RE) are the equilibrium first-principles-calculated total energies of the Al 8 Fe 4 RE IMCs, Al, Fe, and rare earth element, respectively. In the calculation, the Al, Ce, and Yb have the face-centered cubic (FCC) structure, Fe and Eu have the body-centered cubic (BCC) structure, and the others have the hexagonal close packed (HCP) structure. The E single (Al), E single (Fe), and E single (RE) are the total energies of the isolated atoms. For a tetragonal structure, there are six independent single-crystal elastic constants: C 11 , C 12 , C 33 , C 13 , C 44 , and C 66 . The calculated details can be found in [33] and are not recalled here. The effective elastic moduli can be estimated with Voigt [34], Reuss [35], and Hill [36] methods. Usually, the Voigt-Reuss-Hill (VRH) value is used as an effective data [37].

Phase Stability
The lattice constants, formation enthalpies, cohesive energies of 17 Al 8 Fe 4 RE IMCs were calculated, and the obtained results are listed in Table 1 with experimental [14] and theoretical data [38]. It can be seen from Table 1 that the calculated lattice constants of Al 8 Fe 4 RE IMCs were all in coincident with the experimental data [14], and the lattice constants slightly reduced with the increase in atomic number, which is known as the "lanthanide contraction". This phenomenon occurs in RE pure elements and RE-bearing IMCs [39][40][41]. The formation enthalpies (∆H) and cohesive energies (E c ) of Al 8 Fe 4 RE IMCs were all negative, showing that all the Al 8 Fe 4 RE IMCs are stable. For Al 8 Fe 4 Gd, the formation energy of CALPHAD is −0.6114 eV/atom [38], and the calculated result was −0.4254 eV/atom. As we known, the CALPHAD data is estimated from some experimental phase and thermodynamic data, which is the reason for the difference in the two results. However, some further experiments are needed to validate the calculated ∆H and E c of Al 8 Fe 4 RE. The magnetic moments of Al 8 Fe 4 RE were also obtained. The magnetic moments changed from 1.4 to 1.6 µ B per Fe atom. Here, it should be noted that the RE_3 with f -electrons were kept frozen in core used in the present work.

Mechanical Properties
In order to shed some light on the mechanical properties of Al 8 Fe 4 RE IMCs, the elastic constants (C ij ) of Al 8 Fe 4 RE IMCs were calculated, and the results are listed in Table 2.
Obviously, the present elastic constants C ij of the 17 Al 8 Fe 4 RE IMCs met the requirement of stability conditions with C 11 > 0, C 33 > 0, C 44 > 0, C 66 > 0, (C 11 − C 12 ) > 0, (C 11 + C 33 − 2C 13 ) > 0, and [2(C 11 + C 12 ) + C 33  The bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (σ) of the 17 Al 8 Fe 4 RE IMCs were estimated, and the results are listed in Table 3. The bulk moduli (B) of the 17 Al 8 Fe 4 RE IMCs were larger than that of Al (72 GPa) [42], and the shear moduli (G) and Young's moduli (E) of the 17 Al 8 Fe 4 RE IMCs were three times that of Al (27 GPa and 71 GPa) [42]. In order to compare them clearly, the arithmetic average values of pure Al, Fe, and RE with the weight of composition were calculated for Al 8 Fe 4 RE IMCs, and the results are shown in Figure 1. Obviously, the presently calculated bulk moduli (B) were 1.2 times than that of the arithmetic average values, and the presently calculated G and E were close to two times the arithmetic average values. This indicates that the Al 8 Fe 4 RE IMCs may be used as a strengthening phase.  The bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (σ) of the 17 Al8Fe4RE IMCs were estimated, and the results are listed in Table 3. The bulk moduli (B) of the 17 Al8Fe4RE IMCs were larger than that of Al (72 GPa) [42], and the shear moduli (G) and Young's moduli (E) of the 17 Al8Fe4RE IMCs were three times that of Al (27 GPa and 71 GPa) [42]. In order to compare them clearly, the arithmetic average values of pure Al, Fe, and RE with the weight of composition were calculated for Al8Fe4RE IMCs, and the results are shown in Figure 1. Obviously, the presently calculated bulk moduli (B) were 1.2 times than that of the arithmetic average values, and the presently calculated G and E were close to two times the arithmetic average values. This indicates that the Al8Fe4RE IMCs may be used as a strengthening phase.  In order to illustrate the elastic anisotropy of Al 8 Fe 4 RE IMCs, the surfaces of Young's modulus for Al 8 Fe 4 RE IMCs are shown in Figure 2. The three-dimensional surface exhibited a spherical shape for an isotropic crystal. As can be seen in Figure 2, the isosurfaces of Young's modulus exhibited remarkable anisotropic behavior for all the Al 8 Fe 4 RE IMCs. In the light of the Pugh criterion [43], the B/G ratios for the 17 Al 8 Fe 4 RE IMCs were all smaller than 1.75, which reveals that the IMCs are prone to brittleness. A theoretical model [44] of linking Vickers hardness and moduli is via Hv = 2 × (k −2 G) 0.585 − 3, where Hv is Vickers hardness and k is the ratio B/G. In order to illustrate the elastic anisotropy of Al8Fe4RE IMCs, the surfaces of Young's modulus for Al8Fe4RE IMCs are shown in Figure 2. The three-dimensional surface exhibited a spherical shape for an isotropic crystal. As can be seen in Figure 2, the isosurfaces of Young's modulus exhibited remarkable anisotropic behavior for all the Al8Fe4RE IMCs. In the light of the Pugh criterion [43], the B/G ratios for the 17 Al8Fe4RE IMCs were all smaller than 1.75, which reveals that the IMCs are prone to brittleness. A theoretical model [44] of linking Vickers hardness and moduli is via Hv = 2 × (k −2 G) 0.585 − 3, where Hv is Vickers hardness and k is the ratio B/G. The calculated Vickers hardness of the 17 Al8Fe4RE IMCs were all about 14 GPa.

Electronic Properties
The density of states (DOS), electron localization function (ELF), and bonding charge density (BCD) for Al8Fe4Sc are plotted in Figure 3 as an example. It can be seen from Figure 3 that Al8Fe4Sc

Electronic Properties
The density of states (DOS), electron localization function (ELF), and bonding charge density (BCD) for Al 8 Fe 4 Sc are plotted in Figure 3 as an example. It can be seen from Figure 3 that Al 8 Fe 4 Sc showed metallic behavior, and the DOS at the Fermi level was mainly dominated by the Fe-3d state and Sc-3d states, evidencing the hybridization at the Fermi level. The ELF and BCD showed a depletion of the electronic density (ED) at the Al and Sc lattice sites, along with an increment of the ED at the Fe sites. This feature is consistent with the DOS plots in Figure 3a

Lattice Dynamical Properties
In order to check the dynamic stability, the phonon dispersion (PD) curves of Al8Fe4RE IMCs were calculated by combing VASP and PHONOPY codes [45]. For the PD calculation, we used 2 × 2 × 2 supercell containing 104 atoms for Al8Fe4RE and 5 × 5 × 5 k-point mesh. The calculated PD curves along Z-Γ-X-P-N-Γ directions and the phonon density of states (PDOS) are plotted in Figure 4. Among the 17 Al8Fe4RE IMCs, only Al8Fe4Y had the imaginary frequency, indicating that Al8Fe4Y is dynamically unstable. For the others, the calculated PD curves did not have any soft mode, confirming the dynamic stability of Al8Fe4RE (RE, La-Lu) IMCs. The heat capacity Cv and entropy S of Al8Fe4RE IMCs are shown in Figure 5. The calculated Cv exhibited the expected T 3 power law in the low temperature, and Cv reached a classic limit of 324.246 J·(K·mol) −1 at high temperature, which is consistent with the classic law of Dulong-Petit. However, no experimental data of Cv of Al8Fe4RE could be found in the literatures. The present calculations should be a prediction, and further experiments are needed in the future.

Lattice Dynamical Properties
In order to check the dynamic stability, the phonon dispersion (PD) curves of Al 8 Fe 4 RE IMCs were calculated by combing VASP and PHONOPY codes [45]. For the PD calculation, we used 2 × 2 × 2 supercell containing 104 atoms for Al 8 Fe 4 RE and 5 × 5 × 5 k-point mesh. The calculated PD curves along Z-Γ-X-P-N-Γ directions and the phonon density of states (PDOS) are plotted in

Summary
Using the first-principle calculations, the phase stability, elastic constants, various moduli, hardness, electronic, and lattice dynamical properties of Al8Fe4RE IMCs were investigated. The calculated lattice constants were all consistent with the experimental data. The formation enthalpies were all negative, meaning all the IMCs are stable from a thermodynamic point view. The calculated Young's and shear modulus were three times as large as that of Al and two times as large as that of arithmetic average values. The values of hardness of Al8Fe4RE IMCs were all about 14 GPa. All of abovementioned mechanical properties indicate that Al8Fe4RE IMCs may be a good strengthening phase for Al-Fe-based alloys. The calculated PD of Al8Fe4Y had a soft mode, and the others had no soft mode at any vectors, which means that Al8Fe4Y is dynamically unstable, while the others are all dynamically stable. The results are beneficial for the extensive application of Al8Fe4RE IMCs.

Summary
Using the first-principle calculations, the phase stability, elastic constants, various moduli, hardness, electronic, and lattice dynamical properties of Al 8 Fe 4 RE IMCs were investigated. The calculated lattice constants were all consistent with the experimental data. The formation enthalpies were all negative, meaning all the IMCs are stable from a thermodynamic point view. The calculated Young's and shear modulus were three times as large as that of Al and two times as large as that of arithmetic average values. The values of hardness of Al 8 Fe 4 RE IMCs were all about 14 GPa. All of abovementioned mechanical properties indicate that Al 8 Fe 4 RE IMCs may be a good strengthening phase for Al-Fe-based alloys. The calculated PD of Al 8 Fe 4 Y had a soft mode, and the others had no soft mode at any vectors, which means that Al 8 Fe 4 Y is dynamically unstable, while the others are all dynamically stable. The results are beneficial for the extensive application of Al 8 Fe 4 RE IMCs.

Conflicts of Interest:
The authors declare no conflict of interest.