First-Principles Calculations on Relative Energetic Stability, Mechanical, and Thermal Properties of B2-AlRE (RE = Sc, Y, La-Lu) Phases

Abstract

The relative energetic stability, mechanical properties, and thermodynamic behavior of B2-AlRE (RE = Sc, Y, La-Lu) second phases in Al alloys have been investigated through the integration of first-principles calculations with the quasi-harmonic approximation (QHA) model. The results demonstrate a linear increase in the calculated equilibrium lattice constant a₀ with the ascending atomic number of RE, while the enthalpy of formation ΔH_f exhibits more fluctuating variations. The lattice mismatch δ between B2-AlRE and Al matrix is closely correlated with the transferred electron et occurring between Al and RE atoms. Furthermore, the mechanical properties of the B2-AlRE phases are determined. It is observed that the calculated elastic constants C_ij, bulk modulus B_H, shear modulus G_H, and Young’s modulus E_H initially decrease with increasing atomic number from Sc to Ce and then increase up to Lu. The calculated Cauchy pressure C₁₂-C₄₄, Pugh’s ratio B/G, and Poisson’s ratio ν for all AlRE particles exhibit a pronounced directional covalent characteristic as well as uniform deformation and ductility. With the rise in temperature, the calculated vibrational entropy (S_vib) and heat capacity (C_V) of AlRE compounds exhibit a consistent increasing trend, while the Gibbs free energy (F) shows a linear decrease across all temperature ranges. The expansion coefficient (α_T) sharply increases within the temperature range of 0~300 K, followed by a slight change, except for Al, AlHo, AlCe, and AlLu, which show a linear increase after 300 K. As the atomic number increases, both S_vib and C_V increase from Sc to La before stabilizing; however, F initially decreases from Sc to Y before increasing up to La with subsequent stability. All thermodynamic parameters demonstrate similar trends at lower and higher temperatures. This study provides valuable insights for evaluating high-performance aluminum alloys.

1. Introduction

The utilization of Al-based alloys is prevalent in the automotive, rail transit, aerospace, and other industries owing to their low density, high specific strength, exceptional welding strength, and excellent thermal conductivity [1,2,3,4,5]. However, the long-term operation of Al-based alloys at temperatures exceeding 200 °C poses significant challenges [6,7]. The addition of micro-alloying elements, such as rare earth (RE) elements, to conventional alloys is widely recognized for its significant enhancement in their mechanical properties [8,9], as the second phase particles composed of Al and RE elements with high melting points can effectively impede dislocation motion and grain boundary sliding [10,11,12,13,14,15] to make long-time running possible at high temperatures. Thus, the comprehensive and systematic investigation of the fundamental mechanical and thermodynamic properties of the secondary phases is crucial in facilitating the design process of conventional Al-based alloys. Although the scientists have conducted a lot of experimental and theoretical research on the second phases, e.g., B₂-AlRE, L1₂-, D0₁₉-, D0₂₂-, and D0₂₃-Al₃RE [16,17,18,19,20,21], the potential correlation between the stability, mechanical properties, and thermal properties and the position of RE elements in the periodic table has not been adequately proposed.

Herein, we first investigate the stability and mechanical and thermal properties of B₂-AlRE because of the simplicity and high symmetry of B₂ structures. The utilization of experimental research methods is challenging due to the high costs involved and the complexity of creating suitable experimental environments, based on our current understanding [22,23,24,25]. Material simulation, e.g., first-principles calculations, has become an important measure due to the significant improvement in computing power [26,27,28,29,30,31,32,33,34,35]. Tao et al. [36] have calculated the thermodynamic properties of B₂ AlRE based on the Debye model. The results showed that the expansion coefficient of AlSc and AlY is three times larger than that of the rest of AlRE. Wang et al. [37] have investigated elastic properties of AlRE (RE = Y, Tb, Pr, Nd, Dy) with B2-type structures in the finite temperature of 0~1000 K from first principles based on quasi-harmonic approximation (QHA). Their results showed that the calculated elastic constants (C₁₁, C₁₂, and C₄₄) decrease linearly, and their predicted results and the experimental data for pure Al provide excellent agreements. Recently, Liu et al. [38] have systematically investigated the stability, mechanical, and thermodynamic properties of Al-RE (RE = Sc, Y, and La-Lu) intermetallics using first-principles calculations combined with the Debye model. The results demonstrate that the formation energy of all AlRE compounds is lower than −28 kJ/mol, indicating favorable stability of the B2-AlRE structures. With the exception of AlCe, AlPr, AlEu, and AlYb, the remaining AlRE compounds exhibit superior hardness and strength compared to pure Al. Despite their comprehensive investigation into the AlRE phases, it is found that the simple Debye model inaccurately predicts thermodynamic behavior when utilizing Poisson’s ratio as a parameter. Furthermore, their research lacks potential mechanisms.

In the present study, we initially investigated the potential correlation between the stability of AlRE (RE = Sc, Y, La-Lu) compounds and the position of RE elements in the periodic table, which has been well proposed. Additionally, we have calculated and established the relationship between important mechanical parameters of AlRE phases, including elastic constants (bulk modulus, shear modulus), Young’s modulus, Cauchy pressure, Pugh’s ratio, Poisson’s ratio and Vicker’s hardness with respect to atomic number. Finally, based on thermodynamic property calculations for AlRE compounds and their dependence on atomic number, we have also discussed their thermodynamic properties.

2. Computation Detail

All calculations of physical properties for B2-AlRE structures were accomplished in the current work and implemented in VASP (Vienna ab initio Simulation Package) [39], which was employed in the theoretical density functional theory (DFT) according to the electronic wave-function expansion [40]. During all relaxation processes, we adopted a wave energy cut-off of 500 eV and considered the valence electron configurations of Al of 3s²3p¹, Sc of 3d¹4s², Y of 4d¹5s², La of 5d¹6s², Ce of 4f¹5d¹6s², (Pr-Eu) of 4f⁽³⁻⁷⁾6s², Gd of 4f⁷5d¹6s², (Tb-Yb) of 4f⁽⁹⁻¹⁴⁾6s² and Lu of 4f¹⁴5d¹6s². The generalized gradient approximation (GGA) functionals of Perdew, Burke and Ernzerhof (PBE) were used in this work, and the core–valence interactions were described as the projector augmented wave (PAW) [41] method. Here, for the B2 crystal, the red and violet balls in the center and corner denote RE and Al elements, respectively, as shown in Figure 1a, and a 15 × 15 × 15 k-point sampling girds was automatically carried out in the first Brillouin zone for the model, which adopted the Gamma-centered Monkhorst–Pack scheme [42]. A convergence criterion for total energy and Hellman–Feymann forces [43] force was established until the energy and forces tolerances reached 10⁻⁶ eV and 0.01 eV/Å, respectively, to stop calculation. Meanwhile, we used conjugate gradient (CG) minimization and Broyden–Fletcher–Goldfarb–Shannon (BFGS) schemes [44] to achieve the above convergence criteria.

Herein, the mechanical properties of B2-AlRE structures are investigated by the energy-strain method to calculate elastic constants implemented in the VASPKIT code [58]. The strains of −0.09~+0.09 were written in VPKIT.in files as input files of the VASPKIT code. Meanwhile, the thermodynamic performance is discussed using the Phonopy code [25] based on the density functional perturbation theory (DFPT) [59,60]. A 2 × 2 × 2 supercell (16 atoms) was adopted to obtain second-order force constants (2FC) and further calculate related thermodynamic performance in the pursuance of the Phonopy code.

3. Result and Discussion

3.1. The Relative Energetic Stability of B2-AlRE

To investigate the properties of B2-AlRE (RE = Sc, Y, La-Lu) particles in Al-based alloys, their relative energetic stability has first been discussed. A key indicator for evaluating the relative energetic stability of B2-AlRE called the enthalpy of formation (ΔH_f) is widely used, and for the binary phase, it is expressed as follows [61,62,63,64,65]:

$$\mathsf{\Delta }{H}_{\mathrm{f}}={\displaystyle \frac{\left(E_{\mathrm{tot}}-n_{\mathrm{Al}}{\mu }_{\mathrm{Al}}-n_{\mathrm{RE}}{\mu }_{\mathrm{RE}}\right)}{N}}$$

(1)

where E_tot and N (N = n_Al + n_RE) are the total energy and number of atoms per unit cell, respectively, and µ is the energy of a single atom of Al and RE elements from stable Al and RE bulks, respectively.

Meanwhile, we have investigated the effect of lattice mismatch δ of AlRE precipitates, which plays an important role as the larger lattice mismatch can significantly improve the whole strength of Al alloys by fixing the dislocation slips. It can be obtained by the difference in equilibrium lattice constants (ELCs) a₀ between Al matrix and AlRE second phase particles. It is defined as follows:

$$\delta =\left({\displaystyle \frac{a_0^{\mathrm{AlRE}}}{a_0^{\mathrm{Al}}}}-1\right)\times 100\%$$

(2)

The calculated results containing ELCs a₀ and formation enthalpies ΔH_f on B2-AlRE (RE = Sc, Y, La-Lu) systems have been compared with earlier theoretical and experimental literature [45,46,47,48,51,52,66,67,68,69,70,71,72,73,74,75]. They are summarized in Table 1. It can be seen that many values of a₀ and ΔH_f deviate from previous work, especially for ΔH_f of AlRE (RE = Y, La-Lu) compared to the latest work of Liu et al. We can see that ΔH_f of Y, La-Sm, Gd-Tm, and Lu are lower than those in Liu et al., while the ΔH_f of AlRE (RE = Eu, Lu) shows the opposite. The main reasons are as follows:

(1)

The difference in ΔH_f between the current work and Liu et al.’s work first comes with the different selected ground states of Y and La-Lu. In this work, we choose −3.738, −6.202, −6.426, −4.873, −4.734, −4.727, −4.708, −4.673, −4.633, −4.601, −4.577, −4.545, −4.515, −4.491, −4.474, −4.457, −4.440, −4.433 eV as the energy of individual atom referring G. K. et al. [76] for Al, Sc, Y and the La-Lu elements, respectively.

(2)

The reason may be that the Eu_2 and Lu_2 potentials are adopted in previous work. We use PAW_GGA pseudopotentials of Sc, Y_sv, La and (Ce-Lu)_3 in the work.

Figure 1b demonstrates the calculated a₀ of AlRE compounds with the changing trend of RE atomic number. Clearly, the a₀ increases linearly from 3.372 Å for AlSc to 3.800 Å for AlLa and then decreases linearly to 3.523 Å for AlLu. All compounds have a lower a₀ value than the a₀ of Al of 4.044 Å. From Figure 1c, we show the ΔH_f of AlRE particles as a function of atomic number; the linear increase in ΔH_f is from −43.68 kJ/mol for AlSc to −31.08 kJ/mol for AlCe, and then it decreases to −39.07 kJ/mol for AlEr; finally, it increases to −31.73 kJ/mol for AlLu. All cases of ΔH_f of <30 kJ/mol in B2 crystals indicate AlRE structures have better stability than corresponding to the Al matrix and RE bulks according to Equation (1).

Figure 1d shows the δ of AlRE precipitates as a function of RE atomic number. A clear trend can be seen that the δ decreases linearly from 16.63% for AlSc to 7.86% for AlNd and then increases linearly to 12.87% for AlLu. The δ of all particles is higher than 4%, indicating all AlRE compounds will have a good precipitate strengthening effect (PSE), and the δ of AlSc, AlY, Al (Gd-Lu) has a larger PSE as their δ is higher than 10%. Further, the transferred electron e_t and RE atom radius R_p have been analyzed to explore the potential mechanisms, as insets in Figure 1d, with the variation in atomic number. One can be found that the e_t has an opposite trend of change for the δ, meaning that when transferring more electrons from Al to RE atoms, the δ will increase. Whereas the δ is not directly related to the R_p.

Table 1. The calculated equilibrium lattice constant a₀ (Å), formation enthalpy ΔH_f (kJ/mol), lattice mismatch δ (%), atom radius R_p (Å), transferred electron e_t (n) of B2-AlRE systems.

System	a0		Hf		δ	Rp	et
	This Work	Literature	This Work	Literature
Al	4.044	4.039 [77] 4.032; 3.982 [78];	-	-	-	1.856	-
AlSc	3.372	3.371 [66]; 3.315 [47] 3.388 [46]; 3.450 [45]	−43.68	−43.40 [22] −46.00 [79]	16.63	2.461	−2.030
AlY	3.601	3.522 [80]; 3.606 [81]	−41.14	−45.20 [22]	10.94	2.280	−1.189
AlLa	3.800	3.696 [82]	−34.67	−40.40 [83] −45.90 [84]	6.04	2.156	−0.962
AlCe	3.794	3.575 [85]; 3.675 [86]	−31.08	−41.40 [87] −46.00 [88]	6.19	2.334	−0.982
AlPr	3.756	3.533 [85]; 3.760 [86]	−32.89	−46.90 [22] −47.00 [86]	7.11	2.395	−0.986
AlNd	3.726	3.729 [89]	−34.40	−47.90 [90] −50.00 [90]	7.86	2.433	−1.000
AlPm	3.698	3.500 [91]	−36.18	−43.50 [90]	8.55	2.429	−1.029
AlSm	3.677	3.490 [49]	−36.82	−46.50 [22] −49.00 [53]	9.07	2.409	−1.058
AlEu	3.655	3.480 [49]	−38.08	−28.70 [22]	9.62	2.691	−1.120
AlGd	3.631	3.470 [49]	−38.95	−42.40 [22] −42.90 [54]	10.22	2.525	−1.142
AlTb	3.613	3.614 [50]; 3.470 [49]	−39.39	−42.50 [22]	10.65	2.493	−1.157
AlDy	3.595	3.597 [50]; 3.470 [49]	−39.61	−42.40 [22]	11.10	2.472	−1.170
AlHo	3.580	-	−39.43	−42.80 [22]	11.48	2.465	−1.188
AlEr	3.564	-	−39.07	−41.60 [22]	11.86	2.456	−1.204
AlTm	3.551	-	−38.38	−43.90 [22]	12.19	2.465	−1.220
AlYb	3.538	-	−38.36	−30.30 [22]	12.51	2.473	−1.264
AlLu	3.523	3.413 [45]	−37.13	−42.60 [22]	12.87	2.396	−1.256

Table 1. The calculated equilibrium lattice constant a₀ (Å), formation enthalpy ΔH_f (kJ/mol), lattice mismatch δ (%), atom radius R_p (Å), transferred electron e_t (n) of B2-AlRE systems.

System	a0		Hf		δ	Rp	et
	This Work	Literature	This Work	Literature
Al	4.044	4.039 [77] 4.032; 3.982 [78];	-	-	-	1.856	-
AlSc	3.372	3.371 [66]; 3.315 [47] 3.388 [46]; 3.450 [45]	−43.68	−43.40 [22] −46.00 [79]	16.63	2.461	−2.030
AlY	3.601	3.522 [80]; 3.606 [81]	−41.14	−45.20 [22]	10.94	2.280	−1.189
AlLa	3.800	3.696 [82]	−34.67	−40.40 [83] −45.90 [84]	6.04	2.156	−0.962
AlCe	3.794	3.575 [85]; 3.675 [86]	−31.08	−41.40 [87] −46.00 [88]	6.19	2.334	−0.982
AlPr	3.756	3.533 [85]; 3.760 [86]	−32.89	−46.90 [22] −47.00 [86]	7.11	2.395	−0.986
AlNd	3.726	3.729 [89]	−34.40	−47.90 [90] −50.00 [90]	7.86	2.433	−1.000
AlPm	3.698	3.500 [91]	−36.18	−43.50 [90]	8.55	2.429	−1.029
AlSm	3.677	3.490 [49]	−36.82	−46.50 [22] −49.00 [53]	9.07	2.409	−1.058
AlEu	3.655	3.480 [49]	−38.08	−28.70 [22]	9.62	2.691	−1.120
AlGd	3.631	3.470 [49]	−38.95	−42.40 [22] −42.90 [54]	10.22	2.525	−1.142
AlTb	3.613	3.614 [50]; 3.470 [49]	−39.39	−42.50 [22]	10.65	2.493	−1.157
AlDy	3.595	3.597 [50]; 3.470 [49]	−39.61	−42.40 [22]	11.10	2.472	−1.170
AlHo	3.580	-	−39.43	−42.80 [22]	11.48	2.465	−1.188
AlEr	3.564	-	−39.07	−41.60 [22]	11.86	2.456	−1.204
AlTm	3.551	-	−38.38	−43.90 [22]	12.19	2.465	−1.220
AlYb	3.538	-	−38.36	−30.30 [22]	12.51	2.473	−1.264
AlLu	3.523	3.413 [45]	−37.13	−42.60 [22]	12.87	2.396	−1.256

3.2. The Mechanical Property

The research of mechanical properties of metal materials is fundamental and important research in the field of material science, and higher strength of second phases for Al alloys is a necessary condition for long-term operation at finite temperatures [92,93,94]. The elastic constants C_ij, bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio ν of a material often provide valuable information on the structural property. The C_ij is discussed in B2 systems, which depends on the symmetry of the crystal; in the cubic crystals, there are three independent elastic constants containing C₁₁, C₁₂ and C₄₄. They can be used to determine the mechanical stability in well-known Born stability criteria [22,95,96]:

$$C_{11}>0,C_{44}>0,C_{11}-C_{12}>0, C_{11}+2C_{12}>0$$

(3)

As can be seen from Table 1, all AlRE structures can meet the requirements of mechanical stability. For the B, it can be used to evaluate the ability to resist volume deformation and strength of the average atomic bond. Meanwhile, the G and E mainly demonstrate the change in the shape and tensile strength of materials. To calculate the values of B, G and E of AlRE monocrystals, Hill’s method adopted the arithmetic averages of polycrystalline B, G and E, which are obtained by Voigt–Reuss approximation. Here, based on Hill’s method, the values of B_H, G_H and E_H of AlRE can be obtained as follows:

$$B_{\mathrm{H}}=B_{\mathrm{V}}=B_{\mathrm{R}}={\displaystyle \frac{C_{11}+2C_{12}}{3}}$$

(4)

$$G_{\mathrm{V}}={\displaystyle \frac{C_{11}-C_{12}+3C_{44}}{5}};{ \mathrm{G}}_{\mathrm{R}}={\displaystyle \frac{5\times \left(C_{11}-C_{12}\right)\times C_{44}}{4\times C_{44}+3\times \left(C_{11}-C_{12}\right)}};{ \mathrm{G}}_{\mathrm{H}}={\displaystyle \frac{G_{\mathrm{V}}+G_{\mathrm{R}}}{2}}$$

(5)

$$E_{\mathrm{H}}={\displaystyle \frac{9G_{\mathrm{H}}{B}_{\mathrm{H}}}{3B_{\mathrm{H}}+G_{\mathrm{H}}}}$$

(6)

where G_V, B_V, GR, and B_R are the minimum and maximum values of the polycrystalline, respectively.

Meanwhile, Vicker’s hardness H_V can be obtained by a simple empirical formula [97,98], and the uniform/nonuniform behaviors of materials can be evaluated by a value of Poisson’s ratio ν [99]; they can be given by:

$$H_{\mathrm{V}}=2{\left({\displaystyle \frac{G_{\mathrm{H}}^2}{B_{\mathrm{H}}}}\right)}^{0.585}-3$$

(7)

$$\nu ={\displaystyle \frac{3B_{\mathrm{H}}-2G_{\mathrm{H}}}{2\times \left(3B_{\mathrm{H}}+G_{\mathrm{H}}\right)}}$$

(8)

The import indicators that include elastic constants C_ij containing C₁₁, C₁₂, C₄₄, bulk modulus B_H, shear modulus G_H, Young’s modulus E_H, Cauchy pressure C₁₂-C₄₄, Pugh’s ratio B/G, Poisson’s ratio ν and Vicker’s hardness H_V listed in

Table 2. The calculated elastic constants C_ij containing C₁₁, C₁₂, C₄₄, bulk modulus B, shear modulus G, Young’s modulus E, Cauchy pressure C₁₂-C₄₄, Pugh’s ratio B/G, Poisson’s ratio ν and Vicker’s hardness H_V of B2-AlRE systems.

Comp.	C11	C12	C44	BH	GH	EH	C12-C44	B/G	ν	HV
Al	107.12 109.98 [22]	64.75 60.11 [22]	37.67 31.30 [22]	78.88 76.73 [22]	29.90 28.59 [22]	79.64 76.30 [22]	27.08 28.81 [22]	2.638 2.684 [22]	0.331 0.334 [22]	5.28
AlSc	94.58 97.87 [22] 87.98 [66]	79.67 61.79 [22] 70.25 [66]	92.44 91.55 [22] 82.72 [66]	84.64 73.82 [22] 76.16 [66] 92.90 [47]	37.54 48.48 [22] 36.44 [66]	98.10 119.32 [22]	−12.77 - -	2.255 1.523 [22] 2.089 [66]	0.307 0.231 [22] -	7.36
AlY	79.37 81.23 [22] 81.26 [50]	59.65 54.17 [22] 54.58 [50]	66.68 70.30 [22] 62.79 [50]	66.22 63.19 [22] 79.50 [47] 63.47 [50]	32.06 36.92 [22]	82.82 92.70 [22]	−7.03 -	2.065 1.712 [22]	0.292 0.255 [22]	6.95
AlLa	67.38 66.28 [22]	44.87 41.35 [22]	48.43 48.97 [22]	52.37 49.66 [22] 56.30 [47]	27.21 28.46 [22]	69.59 71.69 [22]	−3.56 -	1.925 1.745 [22]	0.279 0.259 [22]	6.42
AlCe	64.58 62.15 [22] 62.66 [50]	50.03 46.38 [22] 52.64 [50]	45.08 40.95 [22] 42.53 [50]	54.88 51.64 [22] 69.30 [47] 55.98 [50]	22.30 21.51 [22]	58.92 56.66 [22]	4.95 -	2.461 2.401 [22]	0.321 0.317 [22]	4.26
AlPr	67.22 65.81 [22] 72.61 [50]	52.22 48.37 [22] 46.82 [50]	48.95 46.52 [22] 47.30 [50]	57.22 54.18 [22] 65.50 [47] 55.42 [50]	23.81 24.21 [22]	62.72 63.21 [22]	3.27 -	2.403 2.238 [22]	0.317 0.306 [22]	4.65
AlNd	69.70 68.88 [22] 74.25 [50]	53.89 51.32 [22] 48.71 [50]	52.41 49.83 [22] 50.24 [50]	59.16 57.17 [22] 57.22 [50]	25.36 25.39 [22]	66.56 66.34 [22]	1.49 -	2.333 2.252 [22]	0.312 0.307 [22]	5.08
AlPm	71.73 70.86 [22]	55.49 52.47 [22]	55.54 52.71 [22]	60.90 58.60 [22]	26.61 26.76 [22]	69.68 69.68 [22]	−0.06 -	2.289 2.190 [22]	0.309 0.302 [22]	5.40
AlSm	73.09 72.55 [22]	56.90 53.83 [22]	57.89 54.63 [22]	62.30 60.07 [22]	27.35 27.57 [22]	71.58 71.73 [22]	−0.99 -	2.278 2.179 [22]	0.309 0.301 [22]	5.56
AlEu	74.95 47.39 [22]	58.46 30.89 [22]	60.60 41.35 [22]	63.96 36.39 [22]	28.38 21.99 [22]	74.18 54.91 [22]	−2.13 -	2.253 1.655 [22]	0.307 0.248 [22]	5.80
AlGd	76.72 74.84 [22]	59.74 55.14 [22]	63.21 59.14 [22]	65.40 61.71 [22]	29.49 29.56 [22]	76.92 76.48 [22]	−3.47 -	2.218 2.087 [22]	0.304 0.293 [22]	6.09
AlTb	77.95 77.38 [22] 80.98 [50]	60.51 56.02 [22] 55.17 [50]	65.20 61.17 [22] 62.30 [50]	66.32 63.14 [22] 63.73 [50]	30.38 31.07 [22]	79.07 80.07 [22]	−4.69 -	2.183 2.032 [22]	0.301 0.289 [22]	6.33
AlDy	79.22 78.26 [22] 81.40 [50]	61.56 57.28 [22] 56.11 [50]	67.39 63.15 [22] 64.00 [50]	67.45 64.27 [22] 64.54 [50]	31.21 31.54 [22]	81.11 81.32 [22]	−5.83 -	2.161 2.038 [22]	0.300 0.289 [22]	6.54
AlHo	79.78 79.99 [22]	62.54 58.24 [22]	69.06 64.55 [22]	68.28 65.49 [22]	31.52 32.39 [22]	81.95 83.42 [22]	−6.52 -	2.166 2.022 [22]	0.300 0.288 [22]	6.58
AlEr	80.51 81.23 [22]	63.40 59.28 [22]	70.74 65.89 [22]	69.10 66.60 [22]	31.99 32.94 [22]	83.13 84.83 [22]	−7.34 -	2.160 2.022 [22]	0.299 0.288 [22]	6.68
AlTm	81.35 81.75 [22]	63.98 59.86 [22]	72.17 66.46 [22]	69.77 67.16 [22]	32.59 33.10 [22]	84.59 85.28 [22]	−8.19 -	2.141 2.029 [22]	0.298 0.288 [22]	6.83
AlYb	82.33 52.31 [22]	64.34 33.22 [22]	73.18 48.76 [22]	70.34 39.58 [22]	33.25 25.76 [22]	86.16 63.50 [22]	−8.84 -	2.116 1.537 [22]	0.296 0.233 [22]	7.02
AlLu	83.77 81.78 [22]	64.64 60.57 [22]	74.90 68.00 [22]	71.01 67.64 [22] 87.70 [47]	34.42 33.46 [22]	88.89 86.18 [22]	−10.27 -	2.063 2.021 [22]	0.291 0.288 [22]	7.38

are along with other calculated results. The calculated C_ij, B_H, G_H, E_H, and B/G reasonably concur with other results for all AlRE materials except for AlEu and AlYb. The reason may be that Eu_2 and Lu_2 potentials are used in previous work. We further plotted the relation between the C_ij, B_H, G_H, E_H, C₁₂-C₄₄, B/G, ν and atomic number in Figure 2. We can see from Figure 2a the C₁₁, C₄₄ and C₁₂ decrease linearly from 94.58, 92.44 and 64.75 GPa for C₁₁, C₄₄ and C₁₂ of AlSc to 64.58, 45.08 GPa for C₁₁, C₄₄ of AlCe and 44.87 GPa for C₁₂ of AlLa, respectively, and then increase linearly to 83.77, 74.90 GPa for C₁₁, C₄₄ and 64.64 GPa for C₁₂ of AlLu. From our calculated B_H, G_H and E_H, demonstrated in Figure 2b, it is clear that the G_H, E_H and B_H show the same trend for C₁₁, C₄₄ and C₁₂, respectively.

To further characteristics of interatomic bonds, ductile and anisotropic behaviors of AlRE particles, we have plotted the Cauchy pressure C₁₂-C₄₄, Pugh’s ratio B/G and Poisson’s ratio ν as a function of atomic number in Figure 2c and d. Generally, when a negative C₁₂-C₄₄ of material occurs, indicating that its bonds present directional covalent characteristics. Whereas the B/G of >1.75 in B2 systems shows ductile behaviors [22], and a ν of higher/lower than 0.26 is a signal for the uniform/nonuniform behaviors of material corresponding stress responses [99]. It can be seen that the C12-C44 first increases linearly from −12.77 GPa for AlSc to 4.95 GPa for AlCe and then decreases linearly to −10.27 GPa for AlLu. The calculated C₁₂-C₄₄ indicates that all AlRE particles have a strong directional covalent characteristic as their C₁₂-C₄₄ is negative, except for AlCe of 4.95 GPa, AlPr of 3.27 GPa and AlNd of 1.49 GPa. The Al (CE-Nd) presents a stronger metallic characteristic between bonds. The B/G decreases from 2.638 for AlSc to 1.925 for AlLa and increases to 2.461 for AlCe; finally, it decreases linearly to 2.063 for Lu. Whereas the ν changes a little. All AlRE phases show good uniformity deformation and ductility as their B/G and ν are larger than 1.75 and 0.26, respectively.

3.3. The Thermodynamic Property

The thermal properties of solids play a key role in the mechanism phase transition, nucleation of the second phase and ability of materials to run at finite temperatures. In this article, a more accurate method called quasi-harmonic approximation (QHA) compared with a simple Debye model is adopted to compute the full vibrational spectra of a crystal on a grid of volumes [23,100] based on the DFPT scheme.

The vibrational entropy S describes an ordered state to a disordered state in a system, and the heat capacity C_V can be defined as the ability to absorb or release heat at unit temperature. The S and C_V of AlRE compounds can be calculated using the frequency of the phonon ${\omega }_{\mathrm{qj}}\left(V\right)$ at a constant volume as follows [23]:

$$S={\displaystyle \frac{1}{2\mathrm{T}}}{\displaystyle \sum }_{qj}\hslash {\mathsf{\omega }}_{qj}\coth \left[{\displaystyle \frac{\hslash {\mathsf{\omega }}_{qj}}{2{\kappa }_{\mathrm{B}}T}}\right]-{\kappa }_{\mathrm{B}}{\displaystyle \sum }_{qj}\mathrm{In}\left[2\sin \mathrm{h}\left({\displaystyle \frac{\hslash {\mathsf{\omega }}_{qj}}{{\kappa }_{\mathrm{B}}T}}\right)\right]$$

(9)

$$C_{\mathrm{V}}={\displaystyle \sum }_{qj}{\mathrm{C}}_{qj}={\displaystyle \sum }_{qj}{\kappa }_{\mathrm{B}}{\left({\displaystyle \frac{\hslash {\mathsf{\omega }}_{qj}}{{\kappa }_{\mathrm{B}}T}}\right)}^2{\displaystyle \frac{\exp \left(\frac{\hslash {\mathsf{\omega }}_{qj}}{{\kappa }_{\mathrm{B}}T}\right)}{{\left[\exp \left(\frac{\hslash {\mathsf{\omega }}_{qj}}{{\kappa }_{\mathrm{B}}\mathrm{T}}\right)-1\right]}^2}}$$

(10)

where q, j, T, k_B and ħ are the wave vector, j-th band, temperature, Boltzmann constant and reduced Planck constant, respectively.

We further calculate the coefficient of thermal expansion α_T and Gibbs free energy F, as following equations [23,100,101,102]:

$${\alpha }_{\mathrm{T}}={\displaystyle \frac{1}{3V}}·{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}T}}$$

(11)

$$F \left(V,T\right)=\min \left[U+F_{\mathrm{pho}.}\right]$$

(12)

where U and F_pho. are the sum of electronic internal energy and phonon-free energy of AlRE structures, respectively. And the F_pho. is derived as follows:

$$F={\displaystyle \frac{1}{2}}{\displaystyle \sum }_{qj}\hslash {\omega }_{qj}+{\kappa }_{B}T{\displaystyle \sum }_{qj}In\left[1-exp\left(-{\displaystyle \frac{\hslash {\omega }_{qj}}{{\kappa }_{B}T}}\right)\right]$$

(13)

Figure 3a–d show the calculated S_vib, C_V, F and α_T of AlRE particles as functions of temperature. It can be found that the S_vib first increases sharply from 0 to ~66.81 J/(K/mol), and then the increasing rate slows down, indicating that the tendency of ordered systems in lower temperatures to turn into a disordered state is stronger. Meanwhile, more vibration modes are inspired by rising heat. From the picture, the C_V increases sharply from 0 to 23.49 J/(K/mol) with a rise in the range of ~300 K as vibrations of the atoms become energetic, and then attains a constant value Dulong–Petit limit of 25 J/(K/mol) at high temperatures. Clearly, the S_vib and C_V of Al and AlSc are larger than those of the rest of the structures at the limiting temperature. According to classical nucleation theory (CNT) [48,77,103], the F of bulks as a driving force of the nucleation process of the second phase is an important indicator. It can be found that the F of all cases decreases linearly with the whole temperature. One can clearly find that, in the whole range of temperature, all AlRE have a lower F than Al matrix of ~3.70 eV, showing they have a larger chemical driving force during the nucleation process of particles. The F of AlSc of −5.38 eV and AlY of −5.48 eV are lower than that of rest particles, indicating AlSc and AlY may first nucleate in the environment of trace RE elements. The α_T presents the ability of materials to expand or contract during the change in temperature. For the calculated α_T, we first compared the current work with the previous experiment [100]. Clearly, as can be seen, the errors between the current theoretical and the experiment values of α_T are increasing. The reason may be that the rise in temperature causes the quasi-harmonic movement of atoms to gradually become non-quasi-harmonic with the increase in temperature. The α_T of all compounds contacting pure Al increases sharply from 0 to 1.49 × 10⁻⁵ K⁻¹ in the range of 300 K, and then it changes a little, except for Al, AlHo, AlCe and AlLu. They would increase linearly to 2.48 × 10⁻⁵ K⁻¹ after 300 K.

To study the potential law of S_vib, and C_V, F and α_T of AlRE with the change in atomic number, we have plotted them in Figure 4a–d. First, at lower and higher points, we can see clearly that both the S_vib and C_V of AlRE compounds increase from AlSc to AlLa and then change a little. Whereas the F first decreases AlSc to AlY at 300 and 600 K and then increases to AlLa; finally, it changes a little. For the α_T, the trend of change is the same at lower and higher temperatures, except for AlTb. There are two lower peaks for AlLa and AlEu and one higher peak for AlHo at both temperatures. The α_T of All AlRE compounds are lower than that of pure Al at both temperatures, except for AlHo of 1.49 × 10⁻⁵ K⁻¹ at 300 K, indicating AlRE phases have good stability at higher temperatures.

4. Conclusions

In the current work, the structural stability, mechanical and thermodynamic properties of B2-AlRE (RE = Sc, Y, La-Lu) second phases in Al alloys have been investigated by combining first-principles calculations with the quasi-harmonic approximation (QHA) model. The main results are as follows:

(1)

The lattice constant a₀ exhibits a linear increase from AlSc to AlLa, followed by a linear decrease towards AlLu. In contrast, the ΔH_f demonstrates a linear increase from AlSc to AlCe, followed by a decrease until AlEr; finally, it increases again up to AlLu.

(2)

The mismatch δ with Al matrix for all particles is higher than 4%, and the δ decreases linearly from AlSc to AlNd and then increases linearly to AlLu. The transferred electron e_t have an opposite trend of change for the δ, meaning that the δ is closely related to the e_t between Al and RE atoms.

(3)

With the increase in atomic number, the elastic constants C₁₁, C₄₄ and C₁₂ decrease linearly first and then increase linearly to C₁₁, C₄₄ and C₁₂ of AlLu. The G_H, E_H and B_H show the same trend for C₁₁, C₄₄ and C₁₂, respectively.

(4)

The Cauchy pressure C₁₂-C₄₄ first increases linearly from AlSc to AlCe and then decreases linearly to AlLu. The Pugh’s ratio B/G decreases from AlSc to AlLa and increases to AlCe; finally, it decreases linearly to AlLu. Whereas Poisson’s ratio ν changes a little.

(5)

With the increase in temperature, the vibrational entropy S_vib and heat capacity C_V show a regularity rise feature, while the Gibbs free energy F of all cases decreases linearly in the whole temperature. The expansion coefficient α_T increases sharply in the temperature range of 0~300 K at first, and then it changes a little, except for those of Al, AlHo, AlCe and AlLu, which increase linearly after 300 K.

(6)

Both S_vib and C_V increase from AlSc to AlLa and then change a little, while the F first decreases AlSc to AlY and then increases to AlLa; finally, it changes a little. For the α_T, the trend of change is the same at lower and higher temperatures, except for AlTb. There are two lower peaks for AlLa and AlEu and one higher peak for AlHo at both temperatures.