The Effect of Alloying Elements on the Structural Stability, Mechanical Properties, and Debye Temperature of Al₃Li: A First-Principles Study

Abstract

The structural stability, mechanical properties, and Debye temperature of alloying elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) doped Al₃Li were systematically investigated by first-principles methods. A negative enthalpy of formation ΔH_f is predicted for all Al₃Li doped species which has consequences for its structural stability. The Sc, Ti, Zr, Nb, and Mo are preferentially occupying the Li sites in Al₃Li while the Co, Cu, and Zn prefer to occupy the Al sites. The Al–Li–X systems are mechanically stable at 0 K as elastic constants C_ij has satisfied the stability criteria. The values of bulk modulus B for Al–Li–X (X = Sc, Ti, Co, Cu, Zr, Nb, and Mo) alloys (excluding Al–Li–Zn) increase with the increase of doping concentration and are larger than that for pure Al₃Li. The Al₆LiSc has the highest shear modulus G and Young’s modulus E which indicates that it has stronger shear deformation resistance and stiffness. The predicted universal anisotropy index A^U for pure and doped Al₃Li is higher than 0, implying the anisotropy of Al–Li–X alloy. The Debye temperature Θ_D of Al₁₂Li₃Ti is highest among the Al–Li–X system which predicts the existence of strong covalent bonds and thermal conductivity compared to that of other systems.

1. Introduction

Lightweight structural materials such as the Al–Li based alloys have excellent comprehensive performance, such as low density, good corrosion resistance, and high elastic modulus [1,2], and it is the basic reason why Al–Li based alloys are so widely used in aviation and aerospace field. The metastable Al₃Li (δ′) precipitates has an important influence on the mechanical properties of Al–Li based alloys [3,4]. The δ′ phase is highly ordered with an L1₂ structure and forms as spheres possessing a cube–cube orientation relationship with matrix [5]. The lattice constant of δ′ phase (4.02 Å) and dilute Al–Li solid solutions (4.04 Å) are almost equal, and the corresponding precipitate-to-matrix misfit results in an interfacial strain of approximately 0.08 ± 0.02% [6]. Due to the small lattice misfit, strong orientational habit and low interfacial strains, the δ′ phase remain crystallographically coherent with the parent solid-solution matrix and the crystallographic orientation relationship is (111)_Al3Li//(111)_Al [7,8]. The δ′ precipitates are considered the most important strengthening phases of Al–Li alloys [9].

As far as the importance of δ′ precipitates is concerned, its structure and electronic properties, mechanical properties, nucleation and growth mechanism, and coarsening behavior have been widely studied both experimentally and theoretically [10,11,12,13]. The solubility and stability of δ′ phase in A1–Li alloy have been reported by Mao et al. [14], which suggest that vibrational entropy is essential for the simulation of solubility. Phase-field method is applied for the investigation of coarsening kinetics of δ′ precipitates in the binary Al–Li alloys [15]. The formation enthalpy, electronic structures, and vibrational and thermodynamic properties of the δ′ phase were systematically reported by employing the first-principles methods [16,17,18,19,20,21]. Yao et al. have reported that point defects play an important role in determining the physical properties of off-stoichiometric δ′ phase [22]. The δ′ phase in binary Al–Li alloys provide limited room temperature strength due to their inability to form in high volume fractions unlike precipitates in Al–Cu or Al–Zn–Mg alloys [23]. In this regard, it is important to improve the mechanical properties of δ′ phase. For strengthening phases, doping with the additional alloying elements is an effective way to improve their specific properties [24,25,26]. The alloying elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) are often adopted to improve the specific properties of alloys. However, no one has reported the influences of alloying elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) on the mechanical properties of δ′ phase.

In this paper, first-principles methods were employed to study the effect of alloying elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) and doping concentration on the structural stability, elastic properties, hardness, elastic anisotropy, and Debye temperature of Al₃Li phase.

2. Computational Studies

The Al₃Li phase has a cubic structure with a space group of pm-3m (No. 221), which contains 3 Al atoms and 1 Li atom (see Figure 1). The Al and Li atom occupy 3c (0 0.5 0.5) and 1a (0 0 0) Wyckoff position, respectively. Based on the Al₃Li phase, the supercells of 1 × 1 × 4 and 1 × 1 × 2 were constructed to study the doping effects at various alloying concentrations of 6.25 and 12.5%, respectively. In the supercells, Al or Li sites can be substituted by single alloying element X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo), while the chemical formulas of doped species can be represented as Al₃Li, Al₆LiX, Al₅Li₂X, Al₁₂Li₃X, and Al₁₁Li₄X, respectively.

All the first-principles calculations were carried out with CASTEP package [27] based on the density functional theory (DFT) [28]. The ultrasoft pseudopotential [29] in reciprocal space was performed to describe the ion-electron interactions. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) function [30] was applied to describe the exchange-correlation potential. Al 3s²3p¹, Li 1s²2s¹, Sc 3s²3p⁶3d¹4s², Ti 3s²3p⁶3d²4s², Co 3d⁷4s², Cu 3d¹⁰4s¹, Zn 3d¹⁰4s², Zr 4s²4p⁶4d²5s², Nb 4s²4p⁶4d⁴5s¹, and Mo 4s²4p⁶4d⁵5s¹ were treated as valence electrons. The plane-wave energy cutoff of 500 eV was selected for all calculations. The 21 × 21 × 21 k-points mesh was set for Al₃Li, and 21 × 21 × 11 and 21 × 21 × 5 k-points meshes were adopted for sampling the 1 × 1 × 4 and 1 × 1 × 2 supercells of Al₃Li, respectively. The convergence threshold of 5.0 × 10⁻⁶ eV/atom was chosen for maximum energy change.

3. Results and Discussion

3.1. Site Preference and Phase Stability

The predicted lattice constants, volume, mass density, and formation enthalpy of pure Al₃Li phase are listed in

Table 1. Simulated lattice constant a (Å), volume V (ų), mass density (Kg/m³), and formation enthalpy H_form (eV) of pure Al₃Li phase at 0 K.

Species	a	V	Mass Density	Hform
Present	4.034	65.65	2.223	−0.097
Cal. [17]	4.030	65.45	2.221	-
Cal. [14]	4.029	65.40	-	−0.100
Exp. [32]	4.01	64.48	2.260	-

. The DFT is one of the calculation methods to simplify the solution of the Schrodinger equation [31]. Thus, the discrepancies between the calculated and experimental values are inevitable. In Table 1, the obtained lattice constant and mass density is consistent with the experimental values, illustrating the reliability of the present computational model.

The structural stability of doped Al₃Li phase can be predicted by the simulated value of enthalpy of formation (ΔH_f). The ΔH_f can be calculated with the help of Formula (1) [33].

$$\Delta H_f=\frac{1}{n}({\mathrm{E}}_{\mathrm{tot}}-\mathrm{a}{\mathrm{E}}_{\mathrm{solid}}^{\mathrm{Al}}-\mathrm{b}{\mathrm{E}}_{\mathrm{solid}}^{\mathrm{Li}}-{\mathrm{E}}_{\mathrm{solid}}^{\mathrm{X}})$$

(1)

where E_tot represents the total energy of doped Al₃Li phase, $E_{\mathrm{solid}}^{\mathrm{Al}}$, $E_{\mathrm{solid}}^{\mathrm{Li}}$, and $E_{\mathrm{solid}}^{\mathrm{X}}$ denote the energies per atom of Al, Li, and X in solid states, n stands for the total number of atoms in the Al–Li–X system while a and b are the number of Al and Li atoms. The predicted ΔH_f and site occupancy behaviors of doping elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) in Al₃Li are listed in

Table 2. Simulated enthalpy of formation ΔH_f (eV) and site occupancy behaviors of doping elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) in Al₃Li at 0 K.

Element X	ΔHf		Site Preference	Element X	ΔHf		Site Preference
	Al6LiX	Al5Li2X			Al12Li3X	Al11Li4X
Sc	−0.267	−0.154	Li	Sc	−0.180	−0.127	Li
Ti	−0.232	−0.127	Li	Ti	−0.169	−0.108	Li
Co	−0.185	−0.202	Al	Co	−0.140	−0.152	Al
Cu	−0.057	−0.128	Al	Cu	−0.075	−0.111	Al
Zn	−0.017	−0.099	Al	Zn	−0.057	−0.098	Al
Zr	−0.276	−0.148	Li	Zr	−0.191	−0.118	Li
Nb	−0.188	−0.098	Li	Nb	−0.150	−0.094	Li
Mo	−0.100	−0.059	Li	Mo	−0.107	−0.075	Li

. A negative ΔH_f is predicted for all doped Al₃Li, indicating its structural stability. Generally, a higher negative value of the ΔH_f means the material is more stable. Thus, the Sc, Ti, Zr, Nb, and Mo are preferentially occupying the Li sites in Al₃Li while the Co, Cu, and Zn prefer to occupy the Al sites. Moreover, the ΔH_f of Al₆LiZr phase is smaller than other systems, which led us to predict that the occupancy of alloying elements Zr in Li site can substantially improve the stability of alloy.

3.2. Mechanical Properties

Elastic constants (C_ij) is an important parameter which can be used to predict the physical properties and mechanical stability of materials [34,35]. In this paper, the C_ij is obtained by employing the strain–stress method, based on the general Hooke’s law [36,37]. The pure cubic crystal of Al₃Li has three independent elastic constants, i.e., C₁₁, C₁₂, and C₄₄. As shown in Figure 1, the doped Al₃Li has tetragonal structure, which is slightly distorted from the pure Al₃Li crystal [24,38]. Hence, there are six independent elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆) for doped Al₃Li. Moreover, C_ij can be used as an important criterion to judge the mechanical stability of pure and doped Al₃Li. For the cubic crystals this criterion can be used as:

C₁₁ − C₁₂ > 0, C₁₁ > 0, C₄₄ > 0, C₁₁ + 2C₁₂ > 0

For tetragonal crystals:

$$C_{11}>\left|C_{12}\right|,C_{44}>0,C_{66}>0,C_{33}(C_{11}+C_{12})>2C_{13}^2$$

The predicted C_ij for pure and doped Al₃Li at 0 K are summarized in

Table 3. Simulated elastic constants C_ij (GPa), elastic moduli B, G, and E (GPa), hardness H and universal anisotropy index A^U for pure and doped Al₃Li at 0 K and 0 GPa.

Phase	Species	C11	C33	C44	C66	C12	C13	B	G	E	H	AU
Al3Li	Present	129.7	-	37.7	-	29.4	-	62.8	42.2	103.5	7.73	0.099
	Cal. [17]	128	-	39	-	30	-	63.3	42.8	116.8	-	-
	Exp. [39]	123.6	-	42.8	-	37.2	-	66	43	105.9	-	-
Al12Li3Sc	Present	136.2	130.4	46.1	51.7	31.0	38.3	68.7	48.4	117.6	9.20	0.019
Al12Li3Ti	Present	141.3	147.7	52.4	54.4	35.2	31.2	69.5	54.0	128.6	11.10	0.007
Al11Li4Co	Present	138.0	118.9	39.4	48.1	29.0	39.2	67.7	44.2	108.9	7.90	0.085
Al11Li4Cu	Present	126.7	116.2	37.0	37.5	37.7	37.0	65.8	39.4	98.5	6.55	0.032
Al11Li4Zn	Present	124.3	121.8	36.9	38.9	33.8	31.2	62.5	40.6	100.1	7.23	0.049
Al12Li3Zr	Present	143.4	147.5	53.1	55.9	42.2	35.2	73.3	53.8	129.7	10.58	0.008
Al12Li3Nb	Present	147.8	163.5	53.1	54.9	50.8	35.5	78.1	54.5	132.6	10.28	0.041
Al12Li3Mo	Present	118.7	147.0	52.0	51.5	66.9	36.6	73.8	45.4	113.3	7.77	0.426
Al6LiSc	Present	146.6	136.8	61.3	62.0	35.2	41.3	74.3	57.7	137.6	11.88	0.040
Al6LiTi	Present	153.7	157.5	55.0	62.0	51.8	48.2	84.6	55.4	136.4	9.93	0.020
Al5Li2Co	Present	144.9	76.5	35.9	34.0	15.6	58.2	69.1	32.5	84.3	4.41	1.534
Al5Li2Cu	Present	134.6	122.0	37.3	41.8	39.3	36.6	68.4	41.6	103.8	7.03	0.055
Al5Li2Zn	Present	123.2	116.4	33.5	36.4	37.0	30.9	62.2	38.0	94.7	6.42	0.082
Al6LiZr	Present	153.9	143.0	55.7	60.5	51.6	47.2	82.4	54.6	134.2	9.88	0.026
Al6LiNb	Present	162.0	156.2	54.2	64.0	56.4	61.9	93.4	54.2	136.3	8.88	0.047
Al6LiMo	Present	90.3	146.3	39.6	50.9	86.8	67.9	85.0	20.2	56.1	1.48	17.33

. The simulated C_ij of pure Al₃Li has a strong correlation with the already reported experimental and calculated results [17,39], implying the accuracy of our simulations. As shown in Table 3, all the predicted C_ij satisfied the stability criteria, indicating that the pure and doped Al₃Li are mechanically stable at 0 K. All the C_ij of Al–Li–X (X =Sc, Ti, Zr, and Nb) systems are higher than those of pure Al₃Li, which indicate that the doping elements X (X =Sc, Ti, Zr, and Nb) can effectively improve the C_ij of pure Al₃Li. At high concentration (12.5 at %), the compression for Al₆LiX (X = Sc, Zr, and Nb) and Al₅Li₂X (X = Co, Cu, and Zn) systems in the direction of the x-axis was more difficult than other axis owing the largest C₁₁, and the deformation resistance for Al–Li–X (X =Ti and Mo) systems along z-axis are higher than others due to higher C₃₃. When the doping concentration drops to 6.25 at %, the C₃₃ for Al₁₂Li₃X (X = Ti, Zr, Nb, and Mo) systems are higher than other C_ij demonstrating that z-axis exhibits incompressibility.

The elastic moduli (bulk modulus B, shear modulus G, and Young’s modulus E) of pure and doped Al₃Li were estimated by Voigt–Reuss–Hill method [40,41]. Generally, B describes the resistance to volume change. As shown in Table 3 and Figure 2, the values of B for Al–Li–X (X = Sc, Ti, Co, Cu, Zr, Nb, and Mo) alloys increase with the increase of doping concentration which is higher than that of pure Al₃Li. The greater B is, the better the ability to resist to volume change is. When the doping concentration is constant, the values of B for Al–Li–Nb are greater than other counterpart species, indicating that Al–Li–Nb has the stronger resistance to volume change. Thus, we inferred that the addition of Nb can effectively improve the resistance to volume change in Al–Li–X. The G and E measure the resistance to shape change and stiffness of Al–Li–X system, respectively. At low concentration (6.25 at %), the addition of Cu and Zn elements can reduce the G and E of Al₃Li. On the other hand, the Al₁₂Li₃Nb has higher values of G and E. when the doping concentration up to 12.5 at %, the Sc, Ti, Zr, and Nb elements can play an important role in enhancing the G and E of Al₃Li. The Al₆LiSc has higher G and E which consequently gives stronger shear deformation resistance and stiffness. Comparative analysis of these parameters led us to conclude that the values of E (G) decrease with the higher concentration (from 6.25 to 12.5%) of Co, Zn, and Mo (Co, Zn, Nb, and Mo). In short, this high doping concentration may decrease the overall performance of the material.

Hardness (H) is an important parameter of materials which can be estimated by the ability to resist localized deformation [42]. The defects (i.e., dislocations) and grain sizes of materials have great influence on the hardness [43] and it is very difficult to get the exact value of H through empirical method. In this paper, the H was roughly predicted by the following semi-empirical formulas [44]:

$$H=\frac{(1-2\nu )}{6(1+\nu )}E$$

(2)

As described in Figure 2, the H of Al–Li–Zn follow a descending trend with the increase of doping concentration, but it increases in case of Al-Li-Sc. For Al–Li–X (X = Ti, Co, Zr, Nb, and Mo), the H reached to the maximum when 6.25% doping concentration is considered. Besides, the Al₆LiSc and Al₆LiMo have maximum and minimum values of H, respectively. However, the difference between the hardness of Al₆LiSc and Al₁₂Li₃Ti is small (about 0.78 GPa). Hence, considering high cost of Sc, the addition of Ti may be the best choice to improve the hardness of pure Al₃Li.

The brittle or ductile behavior of Al–Li–X system can be roughly evaluated from the ratio of B/G. Hence, the materials tend to brittle (ductile) if the ratios of B/G is smaller (larger) than 1.75 [45]. As shown in Figure 3, all the Al–Li–X systems present brittle behavior with a doping concentration of 0 to 6.25 at.%. The Al–Li–X (X = Co and Mo) systems tend to be ductile due to the higher B/G ratios while other Al–Li–X systems still possess brittle behavior at high concentration (12.5 at.%). Comparative analysis of this behavior led us to suggest that the existence of Co and Mo can transform the intrinsic brittleness of Al₃Li into ductility. Moreover, the B/G ratios for the Al–Li–Sc system decrease with the increase of Sc concentration (from 0 to 12.5%), while an increasing trend of B/G ratios is found in the Al–Li–X (X = Co, Zn, and Mo) systems. In short, it is necessary to choose an appropriate doping element and its concentration for the desired ductility or brittleness of materials.

The Poisson’s ratio ν is defined as ν = (3B − 2G)/(6B + 2G) and adopted to reveal the stability of the crystal against shear stress. As shown in Figure 3, the values of $\nu$ for Al₆LiMo and Al₅Li₂Co are bigger than those of other Al–Li–X alloys, indicating that Al₆LiMo and Al₅Li₂Co have a higher structural plasticity [46,47]. Besides, the typical values of ν for ionic and metallic materials are 0.25 and 0.33, respectively [48,49]. The predicted ν for Al–Li–X systems (excluding Al₅Li₂Co and Al₆LiMo) are close to 0.25, which shows that main chemical bonding is ionic bonding. The calculated ν for Al₆LiMo (0.39) and Al₅Li₂Co (0.30) are closer to 0.33 than 0.25, implying that the metallic bonding plays the dominant position. Thus, the main chemical bonds of Al₆LiMo and Al₅Li₂Co are different from that of other Al–Li–X alloys, and this may be the reason why Al₆LiMo and Al₅Li₂Co tend to be ductile.

Furthermore, the elastic anisotropy plays a vital role in the mechanical/physical processes such as crack behavior and phase transformations [50]. The elastic anisotropy of pure and doped Al₃Li can be predicted from the universal anisotropy index (A^U) and its formula is defined as follows [51]:

$$A^U=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6$$

(3)

where G_V and G_R represent the Voigt and Reuss shear modulus, B_V and B_R are the Voigt and Reuss bulk modulus, respectively.

As listed in Table 3, the predicted A^U for pure and doped Al₃Li is higher than 0, implying the anisotropy of the Al–Li–X alloy. In case of Al–Li–X alloy, the A^U shows an upward trend with the increase of doping concentration (6.25 to 12.5%). The A^U for most of the Al–Li–X (except Al₅Li₂Co, Al₆LiMo, and Al₁₂Li₃Mo) systems are near zero and smaller than that for pure Al₃Li, which indicates that most doping elements can reduce the anisotropy of materials. The A^U of Al₆LiMo is much larger than other species. The reason behind this is that a large difference between C₄₄ (C₁₁) and C₆₆ (C₃₃) in Al₆LiMo alloy [52].

3.3. Debye Temperature

The Debye temperature Θ_D is an important parameter of a solid and it is associated with thermodynamic properties of materials, such as entropy, thermal expansion, and vibrational internal energy. One of the standard methods of calculating the Debye temperature is from elastic constant data. Thus, the Θ_D was predicted from averaged sound velocity by employing the following formula [53]:

$${\Theta }_D=\frac{h}{k_B}{[\frac{3n}{4\pi }(\frac{N_A\rho }{M})]}^{1/3}{v}_m$$

(4)

$$v_m={[\frac{1}{3}(\frac{2}{{v_s}^3}+\frac{1}{{v_l}^3})]}^{-1/3}$$

(5)

$$v_s=\sqrt{\frac{G}{\rho }}$$

(6)

$$v_l=\sqrt{\frac{3B+4G}{3\rho }}$$

(7)

where h, k_B, n, N_A, ρ, M, and v_m stand for Planck’s constant, Boltzmann’s constant, total number of atoms, Avogadro’s number, density, molecular weight, and average wave velocity, respectively. v_s and v_l represent the shear and longitudinal sound velocities of materials, respectively.

As depicted in Figure 4, the predicted value of Θ_D for pure Al₃Li is 568.9 K at 0 K, which is in consistent with the already reported data (573 K) [19]. The Θ_D decreases (increases) with the increase of Co, Cu, Zn, and Mo (Sc) concentration in the Al–Li–X systems. In general, a higher Θ_D implies that the materials have higher thermal conductivity and stronger covalent bonds. The Θ_D of Al₁₂Li₃Ti is higher among the Al–Li–X systems, which illustrates that the strength of covalent bonds and thermal conductivity of Al₁₂Li₃Ti are better than others. Furthermore, the predicted density of Al₁₂Li₃Ti (2.5 g/cm³) is close to the density of Al₃Li (2.2 g/cm³), which demonstrates thatAl₁₂Li₃Ti may be a better reinforcement phase in aerospace materials.

4. Conclusions

The effect of alloying elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) and doping concentration on the structural stability, mechanical properties, and Debye temperature of Al₃Li were systematically investigated through density functional theory. The main contents of this work can be summarized as follows:

(1) All doped Al₃Li systems are structural stability. The Sc, Ti, Zr, Nb, and Mo preferentially occupied the Li sites in Al₃Li while the Co, Cu, and Zn prefer to occupy the Al sites rather than Li sites.

(2) All the C_ij of Al–Li–X (X = Sc, Ti, Zr, and Nb) systems are higher than those of pure Al₃Li, which indicate that the doping elements X (X = Sc, Ti, Zr, and Nb) can effectively improve the C_ij of pure Al₃Li. The values of B for Al–Li–X (X = Sc, Ti, Co, Cu, Zr, Nb, and Mo) alloys (excluding Al-Li-Zn) increase with the increase of doping concentration and are higher than that for pure Al₃Li. The Al₆LiSc has higher G and E, which consequences stronger shear deformation resistance and stiffness.

(3) All the Al–Li–X systems present brittle behavior with the increase in doping concentration (from 0 to 6.25 at %). The Al–Li–X (X = Co and Mo) systems tend to ductile while other Al–Li–X systems still possess brittle behavior at high concentration (12.5 at %), which suggests that the existence of Co and Mo can transform the intrinsic brittleness of Al₃Li into ductility. Moreover, the Al₆LiSc and Al₆LiMo have maximum values of H and A^U, respectively.

(4) A higher Θ_D is observed for Al₁₂Li₃Ti, responsible for strong covalent bonds and higher thermal conductivity, compared to other Al–Li–X systems.