Phase Stability, Elastic Modulus and Elastic Anisotropy of X Doped (X = Zn, Zr and Ag) Al3Li: Insight from First-Principles Calculations

In present work, the effects of alloying elements X (X = Zn, Zr and Ag) doping on the phase stability, elastic properties, anisotropy and Debye temperature of Al3Li were studied by the first-principles method. Results showed that pure and doped Al3Li can exist and be stable at 0 K. Zn and Ag elements preferentially occupy the Al sites and Zr elements tend to occupy the Li sites. All the Cij obey the mechanical stability criteria, indicating the mechanical stability of these compounds. The overall anisotropy decreases in the following order: Al23Li8Ag > Al3Li > Al23Li8Zn > Al24Li7Zr, which shows that the addition of Zn and Zr has a positive effect on reducing the anisotropy of Al3Li. The shear anisotropic factors for Zn and Zr doped Al3Li are very close to one, meaning that elastic moduli do not strongly depend on different shear planes. For pure and doped Al3Li phase, the transverse sound velocities νt1 and νt2 among the three directions are smaller than the longitudinal sound velocity νl. Moreover, only the addition of Zn is beneficial to increasing the ΘD of Al3Li among the three elements.


Introduction
Al-Li alloys have gained widespread attention for their use as lightweight structural materials in the aerospace field due to their low density, high elastic modulus and specific stiffness compared with other alloys [1][2][3][4]. The mechanical behaviors of Al-Li alloys depend to a large extent on the structure and properties of precipitates [5,6]. Al 3 Li (δ ) phase plays a vital strengthening role in high lithium aluminum alloys, especially in binary Al-Li alloys [7,8]. However, Al-Li alloys invariably possess poor ductility due to the planar slip caused by the shear of δ precipitates [8,9]. Given this situation, microalloying can improve the mechanical properties of Al-Li alloys.
The microalloying elements may affect the structure and properties of precipitate [10,11], which can be researched by different methods [12][13][14]. For example, the existence of Mg in Al-Li-Mg alloys increases the lattice parameters of the matrix and δ , indicating that Mg has been incorporated into δ [15]. Gault et al. have demonstrated that the Mg atoms are partitioned to Li sublattices in δ'precipitation [16,17]. Therefore, other alloying elements may also occupy the sublattice of either Al or Li in δ precipitate. Hirosawa and Sato researched the atomistic behavior of various microalloying elements in Al-Li alloys at 273 K, which revealed that Ag, Pb and Pd are preferentially occupying Al sites and Mg, Zn and Cu tend to occupy the Li sites [18]. In addition, the influence of microalloying elements on the precipitation process of δ phase has been systematically studied through experimental analysis [19,20]. Our previous work studied the influence of alloying elements on the elastic properties of δ phase at higher doping concentrations [21]. However, there are few reports concerning the influence of microalloying elements on the elastic anisotropy of doped δ phase at the doping concentration of 3.125 at. %.
In the present study, the enthalpy of formation and transfer energy were employed to predict the relative stability of doped Al 3 Li and the site preferences of alloying elements X (X = Zn, Zr and Ag), respectively. We adopted the first-principles method to reveal the effects of alloying elements X on the elastic modulus, anisotropy and Debye temperature of Al 3 Li phase.

Computational Studies
Al 3 Li phase is a cubic structure with a = b = c = 4.010 Å and a 2 × 2 × 2 supercell with 32 atoms which was employed to simulate the influence of doping elements, as shown in Figure 1. Only one alloying atom was selected to substitute either one Al atom or one Li atom, and the doping concentration of alloying element is 3.125 at. %. Hence the chemical formulas of doped Al 3 Li phase can be labeled as Al 24 Li 7 X and Al 23 Li 8 X (X = Zn, Zr and Ag), respectively. elements on the elastic properties of δ′ phase at higher doping concentrations [21]. However, there are few reports concerning the influence of microalloying elements on the elastic anisotropy of doped δ′ phase at the doping concentration of 3.125 at. %. In the present study, the enthalpy of formation and transfer energy were employed to predict the relative stability of doped Al3Li and the site preferences of alloying elements X (X = Zn, Zr and Ag), respectively. We adopted the first-principles method to reveal the effects of alloying elements X on the elastic modulus, anisotropy and Debye temperature of Al3Li phase.

Computational Studies
Al3Li phase is a cubic structure with a = b = c = 4.010 Å and a 2 × 2 × 2 supercell with 32 atoms which was employed to simulate the influence of doping elements, as shown in Figure 1. Only one alloying atom was selected to substitute either one Al atom or one Li atom, and the doping concentration of alloying element is 3.125 at. %. Hence the chemical formulas of doped Al3Li phase can be labeled as Al24Li7X and Al23Li8X (X = Zn, Zr and Ag), respectively. All the calculations in this work were carried out using the Cambridge Sequential Total Energy Package based on the density functional theory (DFT) [22]. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) function was employed [23]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) was selected to optimize the structure of pure and doped Al3Li, and the convergence thresholds for maximum displacement were set to 5.0 × 10 −4 Å. For all calculations, the plane-wave cut-off energy was set to 500.0 eV, and a Monkhorst-Pack mesh with 26 × 26 × 26 and 9 × 9 × 9 k-points was chosen to sample the Brillouin zone of pure Al3Li and alloying elements X doped Al3Li, respectively.

Phase Stability and Site Preference
In this work, the enthalpy of formation (ΔHf) is used to predict the relative stability of alloying elements X doped Al3Li [24] and the ΔHf can be calculated using the following formula [24]: where a and b correspond to the number of Al and Li atoms, Etot is the total energy of alloying elements X doped Al3Li phase, and E Al solid , E Li solid and E X solid represent the energies per atom of Al, Li, X in solid states, respectively. As shown in Table 1, the enthalpies of formation for doped samples are negative, which implies that they can exist and be stable [25,26]. Moreover, a phase with lower ΔHf is much easier to form than others. The ΔHf of All the calculations in this work were carried out using the Cambridge Sequential Total Energy Package based on the density functional theory (DFT) [22]. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) function was employed [23]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) was selected to optimize the structure of pure and doped Al 3 Li, and the convergence thresholds for maximum displacement were set to 5.0 × 10 −4 Å. For all calculations, the plane-wave cut-off energy was set to 500.0 eV, and a Monkhorst-Pack mesh with 26 × 26 × 26 and 9 × 9 × 9 k-points was chosen to sample the Brillouin zone of pure Al 3 Li and alloying elements X doped Al 3 Li, respectively.

Phase Stability and Site Preference
In this work, the enthalpy of formation (∆H f ) is used to predict the relative stability of alloying elements X doped Al 3 Li [24] and the ∆H f can be calculated using the following formula [24]: where a and b correspond to the number of Al and Li atoms, E tot is the total energy of alloying elements X doped Al 3 Li phase, and E Al solid , E Li solid and E X solid represent the energies per atom of Al, Li, X in solid states, respectively. As shown in Table 1, the enthalpies of formation for doped samples are negative, which implies that they can exist and be stable [25,26]. Moreover, a phase with lower ∆H f is much easier to form than others. The ∆H f of Al 24 Li 7 X (X = Zn and Ag) are larger than those of Al 23 Li 8 X (X = Zn and Ag), which indicates that Al 23 Li 8 X (X = Zn and Ag) are easier to form. For Zr doped Al 3 Li phase, the ∆H f of Al 24 Li 7 Zr is smaller than Al 23 Li 8 Zr, meaning that Al 23 Li 8 Zr is harder to form. Al 24 Li 7 Zr phase has the smallest value of ∆H f , suggesting that Al 24 Li 7 Zr is more stable than other doped samples. The site preference of a microalloying element in Al 3 Li phase was investigated by normalized transfer energy ( E X Li→Al ) and the formula of E X Li→Al is defined as follows [27,28]: Here, E X Li→Al and E X Al→Li represent the transfer energy of moving an alloying element X atom from a Li site to an Al site and from an Al site to a Li site, respectively. E X (Al) and E X (Li) stand for the energies of Al 3 Li with an alloying element X atom in an Al site and a Li site, respectively. E Al (Li) and E Li (Al) represent the energies of an Al antisite and a Li antisite of Al 3 Li. E Al3Li stands for the total energy of Al 3 Li. As listed in Table 1, the values of E X Li→Al for Zn and Ag doped Al 3 Li are negative, which means that Zn and Ag show strong Al site preference [27,28]. The value of E X Li→Al for Zr doped Al 3 Li is bigger than one, indicating that Zr elements tend to occupy the Li site. This result may be related to atomic radius difference between these elements. The atomic radius of Zn (1.39 Å) and Ag (1.44 Å) are closer to the atomic radius of Al (1.43 Å) and they tend to occupy the Al site. On the other hand, the atomic radius difference between Li (1.54 Å) and Zr (1.60 Å) is smaller than that of Al and Zr, which could be used to explain the site preference of Zr.

Elastic Properties
The elastic behavior, which when considered as a basic physical property, can be described by elastic constants (C ij ). Pure and Zr doped Al 3 Li have three independent elastic constants, i.e., C 11 , C 12 and C 44 . However, with the addition of alloying elements X (X = Zn and Ag), the lattice symmetry of Al 3 Li decreases, thus, increasing the independent C ij [29]. Hence, the Al 23 Li 8 X (X = Zn and Ag) phases have a tetragonal structure, and the average elastic constants were not considered. As shown in Table 2, the predicted C ij of Al 3 Li are in good agreement with other reported values [30,31]. In addition, all the obtained C ij obey the mechanical stability criteria [32,33], implying the mechanical stability of pure and doped Al 3 Li. The bulk modulus B, shear modulus G and Young's modulus E of pure and doped Al 3 Li phase were calculated by adopting Voigt-Reuss-Hill approximation [34]. As displayed in Table 2 3 Li, indicating that Al 23 Li 8 Zn has the highest stiffness among these phases. Furthermore, the B/G ratios for all considered compounds are less than 1.75, implying that these compounds tend to exhibit brittle behavior based on the Pugh formulation [35]. The B/G ratios of Zr doped Al 3 Li is larger than others, which suggests that Zr slightly decreases the brittleness of Al 3 Li. As a whole, the Zn element plays a prominent role in enhancing the stiffness and shear deformation resistance compared with Ag and Zr elements. These three alloying elements have different effects on the elastic properties of Al 3 Li, which may be due to the different strength of chemical bonds between different alloying elements and Al or Li.

Elastic Anisotropy
In order to comprehensively investigate the elastic anisotropy performance of Al 3 Li with doping elements, a variety of different elastic anisotropy indices were selected. The universal anisotropy index A U is applied to estimate the overall anisotropy of compounds and the calculation formula is as follows [36]: where G V (B V ) and G R (B R ) represent the Voigt and the Reuss shear (bulk) modulus, respectively. When the value of A U is zero, the crystal tends to be isotropic. Significant deviations from zero indicate high anisotropic characteristics. As displayed in Table 3, the A U decreases in the following order: Al 23   The shear anisotropic factors (A 1 , A 2 and A 3 ) can be used to describe the shear anisotropies of crystal and are defined as follows [37]: For cubic symmetry: For tetragonal symmetry: for the (010) or (100) plane (8) for the (110) plane (9) for the (001) plane For an isotropic crystal, the values of A 1 , A 2 and A 3 are equal to one, otherwise, the crystal presented anisotropy. As listed in Table 3, the predicted A 1 , A 2 and A 3 are less than one, which reveals that they show anisotropic characteristics in different shear planes. The shear anisotropy of Al 23 Li 8 Ag along the (010), (100) and (110) plane is much higher than others, while the degree of anisotropy of Al 23 Li 8 Zn is the smallest in the (001) plane. Compared with the other two microalloying elements, Ag slightly increased the shear anisotropy of Al 3 Li in the (010), (100) and (110) plane. In addition, the shear anisotropic factors for Zn and Zr doped Al 3 Li are close to one, meaning that their elastic moduli do not strongly depend on the (010), (100), (001) and (110) planes.
The three-dimensional (3D) surface construction of Young's modulus has become an effective method to display the elastic anisotropy of compounds visually and can be determined by the following expressions [38]: For cubic symmetry: For tetragonal symmetry: 1 E = S 11 l 4 1 + l 4 2 + (2S 13 + S 44 ) l 2 1 l 2 3 + l 2 2 l 2 3 + S 33 l 4 3 + (2S 12 + S 66 )l 2 1 l 2 2 (13) where, S ij represents the elastic compliance coefficients of pure and doped Al 3 Li, and l 1 , l 2 and l 3 stand for the directional cosines corresponding to the x, y and z axes. If the 3D surface diagram is spherical, it indicates that the phase shows isotropic behavior. Otherwise, the phase exhibits anisotropic behavior. As plotted in Figure 2, the 3D surface diagrams of pure and doped Al 3 Li deviate from the sphere, which match well with the results of A U . The 3D surface diagram of Al 24 Li 7 Zr and Al 23 Li 8 Zn are close to the sphere, especially Al 24 Li 7 Zr.

Debye Temperature and Anisotropic Sound Velocities
The thermodynamic properties of crystals, such as specific heat and melting point, can be described by Debye temperature (Θ D ). The Θ D can be calculated by the following expression [39]:

Debye Temperature and Anisotropic Sound Velocities
The thermodynamic properties of crystals, such as specific heat and melting point, can be described by Debye temperature (ΘD). The ΘD can be calculated by the following expression [39]: Here, h, kB, n, NA, ρ, M and vm represent Planck's constant, Boltzmann's constant, total number of atoms, Avogadro's number, density, molecular weight and average wave velocity, respectively. vt and vl are the transverse and longitudinal sound velocities of materials, respectively.
The sound velocities, density and Debye temperature of pure and doped Al3Li phase are summarized in Table 4. The obtained ΘD of Al3Li is 570.9 K, which is consistent with the reported value in the literature [40]. The value of ΘD from high to low is as follows: Al23Li8Zn > Al3Li > Al23Li8Ag > Al24Li7Zr, which indicates that the ΘD of Al24Li7Zr is smaller than that of the others. Among the three elements, only the addition of Zn is beneficial to increase the ΘD of Al3Li at the doping concentration of 3.125 at. %.  Here, h, k B , n, N A , ρ, M and v m represent Planck's constant, Boltzmann's constant, total number of atoms, Avogadro's number, density, molecular weight and average wave velocity, respectively. v t and v l are the transverse and longitudinal sound velocities of materials, respectively.
The sound velocities, density and Debye temperature of pure and doped Al 3 Li phase are summarized in Table 4. The obtained Θ D of Al 3 Li is 570.9 K, which is consistent with the reported value in the literature [40]. The value of Θ D from high to low is as follows: Al 23 Li 8 Zn > Al 3 Li > Al 23 Li 8 Ag > Al 24 Li 7 Zr, which indicates that the Θ D of Al 24 Li 7 Zr is smaller than that of the others. Among the three elements, only the addition of Zn is beneficial to increase the Θ D of Al 3 Li at the doping concentration of 3.125 at. %. According to the Debye model, the sound velocity is one of the critical parameters for obtaining Debye temperature, which is closely related to the elastic properties of crystals. Longitudinal sound waves are related to the compressibility and density of crystals. The transverse sound velocity is considered to be the origin of shear deformation, which can be obtained by shear modulus and density [41]. In addition, the direction of sound propagation has an impact on the sound velocity of crystal. Thus, the sound velocity anisotropy of doped Al 3 Li phases were systemically studied and the formulas are as follows [41]: For tetragonal crystals: [ [110] : [111] : where v t1 and v t2 stand for the first and second mode transverse sound velocity. The directional sound velocities for pure and doped Al 3 Li phases are listed in Table 5.

Conclusions
We investigated the effects of alloying elements X (X = Zn, Zr and Ag) on the phase stability, elastic properties, anisotropy and Debye temperature of pure and doped Al 3 Li phase. The following conclusions are reached: (1) The enthalpies of formation for doped samples are negative, indicating that they can exist and be stable at 0 K. The Zn and Ag elements preferentially occupy the Al sites and Zr elements tend to occupy the Li sites.
(2) Pure and doped Al 3 Li are mechanically stable and tend to exhibit brittle behavior. The addition of Zr and Ag elements can improve the B. Moreover, the Zn element plays an obvious role in enhancing the stiffness and shear deformation resistance.
(3) The A U decreases in the following order: Al 23