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

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 ΔHf 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 Cij 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 AU for pure and doped Al3Li is higher than 0, implying the anisotropy of Al–Li–X alloy. The Debye temperature ΘD 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.


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 Al 3 Li (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 L1 2 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 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.

Site Preference and Phase Stability
The predicted lattice constants, volume, mass density, and formation enthalpy of pure Al 3 Li phase are listed in Table 1. 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 Al 3 Li phase can be predicted by the simulated value of enthalpy of formation (∆H f ). The ∆H f can be calculated with the help of Formula (1) [33].
where E tot represents the total energy of doped Al 3 Li phase, E Al solid , E Li solid , and E X solid 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 ∆H f and site occupancy behaviors of doping elements X (X = Sc, Ti, Co, Cu, Zn, Zr, Nb, and Mo) in Al 3 Li are listed in Table 2. A negative ∆H f is predicted for all doped Al 3 Li, indicating its structural stability. Generally, a higher negative value of the ∆H f means the material is more stable. Thus, the Sc, Ti, Zr, Nb, and Mo are preferentially occupying the Li sites in Al 3 Li while the Co, Cu, and Zn prefer to occupy the Al sites. Moreover, the ∆H f of Al 6 LiZr 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.

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 Al 3 Li has three independent elastic constants, i.e., C 11 , C 12 , and C 44 . As shown in Figure 1, the doped Al 3 Li has tetragonal structure, which is slightly distorted from the pure Al 3 Li crystal [24,38]. Hence, there are six independent elastic constants (C 11 , C 12 , C 13 , C 33 , C 44 , and C 66 ) for doped Al 3 Li. Moreover, C ij can be used as an important criterion to judge the mechanical stability of pure and doped Al 3 Li. For the cubic crystals this criterion can be used as: For tetragonal crystals: The predicted C ij for pure and doped Al 3 Li at 0 K are summarized in Table 3. The simulated C ij of pure Al 3 Li 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 Al 3 Li 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 Al 3 Li, which indicate that the doping elements X (X =Sc, Ti, Zr, and Nb) can effectively improve the C ij of pure Al 3 Li. At high concentration (12.5 at %), the compression for Al 6 LiX (X = Sc, Zr, and Nb) and Al 5 Li 2 X (X = Co, Cu, and Zn) systems in the direction of the x-axis was more difficult than other axis owing the largest C 11 , and the deformation resistance for Al-Li-X (X =Ti and Mo) systems along z-axis are higher than others due to higher C 33 . When the doping concentration drops to 6.25 at %, the C 33 for Al 12 Li 3 X (X = Ti, Zr, Nb, and Mo) systems are higher than other C ij demonstrating that z-axis exhibits incompressibility. 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 Al 3 Li at 0 K and 0 GPa.

Phase
Species The elastic moduli (bulk modulus B, shear modulus G, and Young's modulus E) of pure and doped Al 3 Li 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 Al 3 Li. 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 Al 3 Li. On the other hand, the Al 12 Li 3 Nb 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 Al 3 Li. The Al 6 LiSc 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]: (1 2 ) 6(1 ) 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 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]: 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 Al 6 LiSc and Al 6 LiMo have maximum and minimum values of H, respectively. However, the difference between the hardness of Al 6 LiSc and Al 12 Li 3 Ti 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 Al 3 Li. 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 Al 3 Li 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. 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 ν 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]: where GV and GR represent the Voigt and Reuss shear modulus, BV and BR 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].

Debye Temperature
The Debye temperature ΘD 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 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 ν for Al 6 LiMo and Al 5 Li 2 Co are bigger than those of other Al-Li-X alloys, indicating that Al 6 LiMo and Al 5 Li 2 Co 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 Al 5 Li 2 Co and Al 6 LiMo) are close to 0.25, which shows that main chemical bonding is ionic bonding. The calculated ν for Al 6 LiMo (0.39) and Al 5 Li 2 Co (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 Al 6 LiMo and Al 5 Li 2 Co are different from that of other Al-Li-X alloys, and this may be the reason why Al 6 LiMo and Al 5 Li 2 Co 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 Al 3 Li can be predicted from the universal anisotropy index (A U ) and its formula is defined as follows [51]: 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 Al 3 Li 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 Al 5 Li 2 Co, Al 6 LiMo, and Al 12 Li 3 Mo) systems are near zero and smaller than that for pure Al 3 Li, which indicates that most doping elements can reduce the anisotropy of materials. The A U of Al 6 LiMo is much larger Materials 2018, 11, 1471 7 of 10 than other species. The reason behind this is that a large difference between C 44 (C 11 ) and C 66 (C 33 ) in Al 6 LiMo alloy [52].

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]: 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 Al 3 Li 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 Al 12 Li 3 Ti is higher among the Al-Li-X systems, which illustrates that the strength of covalent bonds and thermal conductivity of Al 12 Li 3 Ti are better than others. Furthermore, the predicted density of Al 12 Li 3 Ti (2.5 g/cm 3 ) is close to the density of Al 3 Li (2.2 g/cm 3 ), which demonstrates thatAl 12 Li 3 Ti may be a better reinforcement phase in aerospace materials.
Materials 2018, 11, x FOR PEER REVIEW 7 of 10 data. Thus, the ΘD was predicted from averaged sound velocity by employing the following formula [53]: where h, kB, n, NA, ρ, M, and vm stand for Planck's constant, Boltzmann's constant, total number of atoms, Avogadro's number, density, molecular weight, and average wave velocity, respectively. vs and vl represent the shear and longitudinal sound velocities of materials, respectively. As depicted in Figure 4, the predicted value of ΘD for pure Al3Li is 568.9 K at 0 K, which is in consistent with the already reported data (573 K) [19]. The ΘD decreases (increases) with the increase of Co, Cu, Zn, and Mo (Sc) concentration in the Al-Li-X systems. In general, a higher ΘD implies that the materials have higher thermal conductivity and stronger covalent bonds. The ΘD 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/cm 3 ) is close to the density of Al3Li (2.2 g/cm 3 ), which demonstrates thatAl12Li3Ti may be a better reinforcement phase in aerospace materials.

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:

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 Al 3 Li were systematically investigated through density functional theory. The main contents of this work can be summarized as follows: (1) All doped Al 3 Li systems are structural stability. The Sc, Ti, Zr, Nb, and Mo preferentially occupied the Li sites in Al 3 Li 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 Al 3 Li, which indicate that the doping elements X (X = Sc, Ti, Zr, and Nb) can effectively improve the C ij of pure Al 3 Li. 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 Al 3 Li. The Al 6 LiSc 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 Al 3 Li into ductility. Moreover, the Al 6 LiSc and Al 6 LiMo have maximum values of H and A U , respectively.
(4) A higher Θ D is observed for Al 12 Li 3 Ti, responsible for strong covalent bonds and higher thermal conductivity, compared to other Al-Li-X systems.