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

Abstract

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.

1. 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 Al₈Fe₄RE IMCs are observed at the Al-rich corner, and they have a tetragonal crystal structure. The 17 Al₈Fe₄RE (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), Al₈Fe₄Ce was found to be favorable for the stability of the Al-based alloy as a strengthening phase [15]. The magnetic properties of Al₈Fe₄RE have also been investigated [16,17,18,19,20,21], and the electronic conductivity [22] and the negative magnetoresistivity [23] of Al₈Fe₄RE have also been studied. Using the lattice inversion method, the lattice constants and lattice vibration spectra of Al₈Fe₄RE (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 Al₈Fe₄RE are very important for material design and for further development. However, few studies have focused on the electronic and elastic properties of Al₈Fe₄RE IMCs. Thus, the aim of this work was to study the physical properties of 17 Al₈Fe₄RE (RE = Sc, Y, and La–Lu) IMCs using first-principle (FP) calculations.

2. Computational Details

The FP calculations were performed with the VASP code [26,27] using the projector augmented wave (PAW) method [28,29] and the generalized gradient approximation (GGA) [30]. The GGA-PBE (Generalized Gradient Approximation-Perdew–Burke–Ernzerhof) potentials of Al, Fe, Sc, Y_sv, La_s, Yb_2, and RE_3 (others) were used in this work. The FP calculations were performed with cutoff energy of 500 eV, Monkhorst–Pack K-point meshes [31], and a 0.05 eV smearing parameter with the Methfessel–Paxton technique [32].

The formation enthalpy and cohesive energy of the Al₈Fe₄RE alloys can be estimated from the following equations:

$$\Delta H\left(A{l}_{8}F{e}_{4}RE\right)=E\left(A{l}_{8}F{e}_{4}RE\right)-8E\left(Al\right)-4E\left(Fe\right)-E\left(RE\right)$$

(1)

$$E_c\left(A{l}_{8}F{e}_{4}RE\right)=E\left(A{l}_{8}F{e}_{4}RE\right)-8E_{single}\left(Al\right)-4E_{single}\left(Fe\right)-E_{single}\left(RE\right)$$

(2)

where E(Al₈Fe₄RE), E(Al), E(Fe), and E(RE) are the equilibrium first-principles-calculated total energies of the Al₈Fe₄RE 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: C₁₁, C₁₂, C₃₃, C₁₃, C₄₄, and C₆₆. 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].

3. Results and Discussion

3.1. Phase Stability

The lattice constants, formation enthalpies, cohesive energies of 17 Al₈Fe₄RE IMCs were calculated, and the obtained results are listed in

Table 1. Lattice constants, formation enthalpy, cohesive energy, and magnetic moments of Al₈Fe₄RE IMCs.

Phases	Lattice Constants		△H (eV/atom)	Ec (eV/atom)	Magnetic (μB/Fe)	Ref.
	a (Å)	c (Å)
Al8Fe4Sc	8.597 8.70	5.001 4.81	−0.4238	−4.5211	1.426	Present [14]
Al8Fe4Y	8.696 8.750	5.024 5.060	−0.4322	−4.5221	1.500	Present [14]
Al8Fe4La	8.849 8.900	5.045 5.075	−0.3843	−4.4922	1.602	Present [14]
Al8Fe4Ce	8.829 8.793	5.046 5.047	−0.3843	−4.4963	1.599	Present [14]
Al8Fe4Pr	8.802 8.824	5.042 5.054	−0.3955	−4.5076	1.584	Present
Al8Fe4Nd	8.781 8.804 8.875	5.039 5.054 5.211	−0.4043	−4.5152	1.571	Present [14,24]
Al8Fe4Pm	8.762	5.035	−0.4122	−4.5173	1.559	Present
Al8Fe4Sm	8.748 8.770 8.863	5.032 5.053 5.188	−0.4162	−4.5223	1.549	Present [14,24]
Al8Fe4Eu	8.732 8.784	5.035 5.051	−0.4328	−4.5259	1.536	Present [14]
Al8Fe4Gd	8.719 8.743	5.028 5.052	−0.4254 −0.6114	−4.5283	1.524	Present [14,38]
Al8Fe4Tb	8.708 8.740	5.024 5.036	−0.4277	−4.5283	1.511	Present [14]
Al8Fe4Dy	8.697 8.728	5.022 5.050	−0.4291	−4.5275	1.499	Present [14]
Al8Fe4Ho	8.688 8.720	5.021 5.038	−0.4298	−4.5262	1.488	Present [14]
Al8Fe4Er	8.678 8.700	5.018 5.028	−0.4296	−4.5247	1.477	Present [14]
Al8Fe4Tm	8.669 8.688	5.016 5.037	−0.4288	−4.5224	1.466	Present [14]
Al8Fe4Yb	8.703 8.691	5.049 5.017	−0.3652	−4.2378	1.559	Present [14]
Al8Fe4Lu	8.653 8.687	5.012 5.030	−0.4256	−4.5174	1.450	Present [14]

with experimental [14] and theoretical data [38]. It can be seen from Table 1 that the calculated lattice constants of Al₈Fe₄RE 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 (ΔH) and cohesive energies (E_c) of Al₈Fe₄RE IMCs were all negative, showing that all the Al₈Fe₄RE IMCs are stable. For Al₈Fe₄Gd, 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 ΔH and E_c of Al₈Fe₄RE. The magnetic moments of Al₈Fe₄RE 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.

3.2. Mechanical Properties

In order to shed some light on the mechanical properties of Al₈Fe₄RE IMCs, the elastic constants (C_ij) of Al₈Fe₄RE IMCs were calculated, and the results are listed in

Table 2. The calculated elastic constants of Al₈Fe₄RE intermetallic compounds (IMCs) (unit in GPa).

Phases	C11	C12	C13	C33	C44	C66
Al8Fe4Sc	266.15	49.09	53.62	268.07	68.43	76.57
Al8Fe4Y	264.94	46.05	52.36	259.71	70.02	77.63
Al8Fe4La	254.17	41.90	52.25	262.29	67.73	70.86
Al8Fe4Ce	252.33	42.85	51.22	255.08	68.08	71.60
Al8Fe4Pr	255.29	43.90	51.40	255.34	68.35	71.43
Al8Fe4Nd	257.70	44.72	51.42	256.22	68.77	71.96
Al8Fe4Pm	260.29	45.46	51.61	257.71	69.33	73.14
Al8Fe4Sm	261.44	45.87	51.76	258.24	69.50	73.91
Al8Fe4Eu	262.97	46.12	51.84	258.83	69.74	74.79
Al8Fe4Gd	263.60	46.01	51.87	258.92	69.75	76.04
Al8Fe4Tb	264.58	46.22	52.25	259.89	69.80	76.95
Al8Fe4Dy	265.29	46.15	52.38	260.28	70.09	77.59
Al8Fe4Ho	265.91	46.22	52.78	261.12	70.39	78.01
Al8Fe4Er	266.61	46.29	53.14	261.95	70.74	78.24
Al8Fe4Tm	267.35	46.46	53.64	262.82	70.90	77.65
Al8Fe4Yb	247.62	38.03	48.66	249.15	64.34	66.59
Al8Fe4Lu	254.53	44.55	51.63	250.65	67.35	73.88

.

Obviously, the present elastic constants C_ij of the 17 Al₈Fe₄RE IMCs met the requirement of stability conditions with C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, (C₁₁ − C₁₂) > 0, (C₁₁ + C₃₃ − 2C₁₃) > 0, and [2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃] > 0. For Al₈Fe₄RE (RE = Sc, La, Ce, Pr, Yb), C₁₁ < C₃₃ indicated that the bonding strength along the [100] and [010] directions was softer than that along the [001] direction. However, for the others, C11 > C33, the opposite tendency occurred. C44 < C66 meant the [100](001) shear was easier than the [100](010) shear for the 17 Al₈Fe₄RE IMCs.

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 calculated bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio v, B/G ratio, and hardness H of Al₈Fe₄RE IMCs.

Phases	B (GPa)	G (GPa)	E (GPa)	v	B/G	H
Al8Fe4Sc	123.66	83.85	205.17	0.224	1.475	13.94
Al8Fe4Y	121.23	84.56	205.82	0.217	1.434	14.59
Al8Fe4La	118.10	81.19	198.17	0.220	1.454	13.90
Al8Fe4Ce	116.68	81.01	197.36	0.218	1.440	14.07
Al8Fe4Pr	117.69	81.36	198.36	0.219	1.447	14.02
Al8Fe4Nd	118.52	81.93	199.76	0.219	1.447	14.09
Al8Fe4Pm	119.51	82.76	201.72	0.218	1.444	14.23
Al8Fe4Sm	119.99	83.12	202.59	0.219	1.444	14.28
Al8Fe4Eu	120.49	83.61	203.70	0.218	1.441	14.37
Al8Fe4Gd	120.63	83.96	204.44	0.218	1.437	14.48
Al8Fe4Tb	121.16	84.28	205.26	0.218	1.438	14.50
Al8Fe4Dy	121.41	84.65	206.05	0.217	1.434	14.69
Al8Fe4Ho	121.83	84.94	206.78	0.217	1.434	14.63
Al8Fe4Er	122.25	85.24	207.50	0.217	1.434	14.67
Al8Fe4Tm	122.78	85.25	207.68	0.218	1.440	14.58
Al8Fe4Yb	112.76	77.80	189.76	0.220	1.449	13.55
Al8Fe4Lu	117.26	81.02	197.57	0.219	1.447	13.97

. 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 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⁻²G)^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.

3.3. 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 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, demonstrating the hybridization of Fe-3d and Sc-3d. For the other Al₈Fe₄RE IMCs, their electronic structures were all similar to Al₈Fe₄Sc, so they are not shown here (see Supplementary data).

3.4. 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 C_v and entropy S of Al₈Fe₄RE IMCs are shown in Figure 5. The calculated C_v exhibited the expected T³ power law in the low temperature, and C_v 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 C_v of Al₈Fe₄RE could be found in the literatures. The present calculations should be a prediction, and further experiments are needed in the future.

4. 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.