The Effect of Nb Doping on the Properties of Ti-Al Intermetallic Compounds Using First-Principles Calculations

The crystal structures, stability, mechanical properties and electronic structures of Nb-free and Nb-doped Ti-Al intermetallic compounds were investigated via first-principles calculations. Seven components and eleven crystal configurations were considered based on the phase diagram. The calculated results demonstrate that hP8-Ti3Al, tP4-TiAl, tP32-Ti3Al5, tI24-TiAl2, tI16-Ti5Al11, tI24-Ti2Al5, and tI32-TiAl3 are the most stable phases. Mechanical properties were estimated with the calculated elastic constants, as well as the bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio and Pugh’s ratio following the Voigt–Reuss–Hill scheme. As the Al content increases, the mechanical strength increases but the ductility decreases in the Ti-Al compounds. This results from the enhanced covalent bond formed by the continuously enhanced Al-sp hybrid orbitals and Ti-3d orbitals. Nb doping (~5 at.% in this study) keeps the thermodynamical and mechanical stability for the Ti-Al compounds, which exhibit slightly higher bulk modulus and better ductility. This is attributed to the fact that the Nb 4d orbitals locate near the Fermi level and interact with the Ti-3d and Al-3p orbitals, improving the metallic bonds based on the electronic structures.


Introduction
Ti-Al alloys possess significant potential for applications due to their high strength, stiffness, hardness, thermal stability, and corrosion resistance (e.g., in Boeing 787) [1][2][3][4].As the aviation industry continues to advance at a rapid pace, there arises a need to further enhance the strengthening and toughening capabilities of Ti-Al alloys [5][6][7].Considerable efforts have been dedicated to the development of techniques such as heat treatment, thermomechanical processing, and alloying to effectively control the microstructure of Ti-Al alloys [8][9][10].Notably, the introduction of specific solute elements, such as Nb, as dopants into Ti-Al binary intermetallic compounds, has been found to induce microstructural alterations and subsequent changes in the properties.
A total of seven components and eleven configurations of Ti-Al binary intermetallic compounds have been reported based on experimental and theoretical Ti-Al binary equilibrium phase diagrams [11,12].However, only Ti 3 Al and TiAl are presently employed as base materials for Ti-Al alloys.Other intermetallic compounds, such as TiAl 3 with a high aluminum content and low density, have limited usage due to poor ductility and fracture toughness, despite possessing superior resistance to high-temperature oxidation

Materials and Methods
The periodic density functional theory (DFT) calculations were performed using the plane-wave pseudo-potential Vienna ab initio simulation package (VASP.5.4.4,Vienna, Austria) [24].The generalized gradient approximation as formulated by Perdew, Burke and Ernzerhof (GGA-PBE) was employed for the exchange-correlation functional [25].The projection enhanced wave (PAW) method proposed by Blöchl and implemented by Kresse and Joubert was used with a cutoff energy of 420 eV [26].A uniform mesh grid with a spacing of 0.03 Å was used to sample the complete Brillouin zone and calculate the density of states [27].Brillouin zone integrations were carried out with the Methfessel-Paxton technique with a 0.1 eV smearing of the electron levels [28].The PAW pseudopotentials considered were Ti 3p 6 3d 2 4s 2 , Al 2s 2 3p 1 , and Nb 4p 6 4d 4 5s 1 .The full relaxation structure optimization method was used to obtain the ground-state crystal structure of each compound.The total energy convergence parameter during optimization was 2 × 10 −6 eV/atom, the Hermann-Feynman force convergence parameter was 0.01 eV/Å, the tolerance shift was less than 0.002 Å, and the stress deviation per atom was less than 0.1 GPa.In addition, the single-crystal elastic matrix constants of the compounds in the ground-state structures were calculated using the stress-strain method according to the generalized Hooke's law.
According to experimental and theoretical Ti-Al binary equilibrium phase diagrams [11,12], there are seven chemical compositions and 11 phases for Ti-Al intermetallic compounds, i.e., hP8-Ti 3 Al, tP4-TiAl, cP2-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , oC12-TiAl 2 , tI16-Ti 5 Al 11 , tP28-Ti 2 A l5 , tI32-TiAl 3 , tI8-TiAl 3 , and cP4-TiAl 3 , with the person symbols written in front of the formula (Figure 1).Here, cP2-TiAl, oC12-TiAl 2 , and tI32-TiAl 3 were evaluated using thermodynamics calculations [11].All of the structures were considered in our calculations.Furthermore, we investigated the influences of Nb doping on the mechanical properties of the Ti-Al intermetallic compounds.About 5 at.%Nb atoms were considered in order to avoid too large a change in the crystal structures.eV/atom, the Hermann-Feynman force convergence parameter was 0.01 eV/Å, the tolerance shift was less than 0.002 Å, and the stress deviation per atom was less than 0.1 GPa.
In addition, the single-crystal elastic matrix constants of the compounds in the groundstate structures were calculated using the stress-strain method according to the generalized Hooke's law.According to experimental and theoretical Ti-Al binary equilibrium phase diagrams [11,12], there are seven chemical compositions and 11 phases for Ti-Al intermetallic compounds, i.e., hP8-Ti3Al, tP4-TiAl, cP2-TiAl, tP32-Ti3Al5, tI24-TiAl2, oC12-TiAl2, tI16-Ti5Al11, tP28-Ti2Al5, tI32-TiAl3, tI8-TiAl3, and cP4-TiAl3, with the person symbols written in front of the formula (Figure 1).Here, cP2-TiAl, oC12-TiAl2, and tI32-TiAl3 were evaluated using thermodynamics calculations [11].All of the structures were considered in our calculations.Furthermore, we investigated the influences of Nb doping on the mechanical properties of the Ti-Al intermetallic compounds.About 5 at.%Nb atoms were considered in order to avoid too large a change in the crystal structures.

Crystal Structure and Phase Stability
Table 1 lists the calculated lattice parameters and zero-temperature formation energy (ΔHr) of all Ti-Al intermetallic compounds including the experimental and other theoretical results for comparison.The formation energy was calculated using where Etol(TixAly) is the total energy of TixAly (f.u.), and Ecoh(Ti) and Ecoh(Al) are the cohesive energy of the Ti and Al crystals, respectively, which is the difference between the total energy of the Ti/Al crystal and the energy of a single Ti/Al atom [29].As shown in Table 1, the calculated lattice parameters are quite consistent with the experimental values, with the largest difference of 2.9% for tI8-TiAl3.Compared to the available Δ in our experiments (hP8-Ti3Al, tP4-TiAl, and cP4-TiAl3), our results are still consistent with the maximum error of 8% for hP8-Ti3Al.In addition, our results perfectly match other theoretical

Crystal Structure and Phase Stability
Table 1 lists the calculated lattice parameters and zero-temperature formation energy (∆H r ) of all Ti-Al intermetallic compounds including the experimental and other theoretical results for comparison.The formation energy was calculated using where E tol (Ti x Al y ) is the total energy of Ti x Al y (f.u.), and E coh (Ti) and E coh (Al) are the cohesive energy of the Ti and Al crystals, respectively, which is the difference between the total energy of the Ti/Al crystal and the energy of a single Ti/Al atom [29].As shown in Table 1, the calculated lattice parameters are quite consistent with the experimental values, with the largest difference of 2.9% for tI8-TiAl 3 .Compared to the available ∆H r in our experiments (hP8-Ti 3 Al, tP4-TiAl, and cP4-TiAl 3 ), our results are still consistent with the maximum error of 8% for hP8-Ti 3 Al.In addition, our results perfectly match other theoretical values using the same PAW-PBE method, showing only a slight difference with other theoretical methods.Experiment [31] Table 1 displays that the most stable structures are tP4 (P4/mmm), tI24 (I41/amd), and tI32 (I4/mmm), for the multiphase TiAl, TiAl 2 , and TiAl 3 , respectively.The cP2-TiAl is less stable than the tP4 phase with a formation energy of 13 kJ/mol higher, which is the B2 phase at high temperatures [12].The oC12-TiAl 2 , which was reported to be metastable [39], has a formation energy 0.4 kJ/mol higher than the tI24 structure.For TiAl 3 , cP4 is the most unstable phase, with a much higher formation energy, while tI8 has a formation energy very close to the isomorphic tI32 phase.The most stable phases of tP4-TiAl, tI24-TiAl 2 , and tI32-TiAl 3 were chosen for the subsequent calculations of the mechanical properties and Nb doping.
Furthermore, we performed the Nb-doping calculations for the Ti-Al intermetallic compounds in the most stable phases, including hp8-Ti 3 Al, tP4-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , tI16-Ti 5 Al 11 , tP28-Ti 2 Al 5 , and tI32-TiAl 3 .A 5% Nb atomic content was considered in order to keep the crystal structures nearly unchanged.Nb atoms were set to occupy Ti sites based on previous reports [17,18].There were two kinds of components considered for Nb-doped hp8-Ti 3 Al, i.e., unit-cell components of Ti 11 Al 4 N b (6.25 at.%Nb, 1 × 1 × 2 supercell of hp8-Ti 3 Al) and Ti 23 Al 8 N b (3.125 at.%Nb, 2 × 2 × 1 supercell hp8-Ti 3 Al).For tP4-TiAl and tI24-TiAl 2 , there was only one kind of Nb doping, with the unit-cell components of Ti Table 2 presents the calculated lattice constants and formation energies for all the Nb-doped Ti-Al intermetallic compounds, including the percentage change in the crystal structure parameters relative to the non-doped ones.We found that ~5 at.%Nb doping does not change the crystal shape, and the volume change remains within a very small range (<0.35%).The negative formation energies indicate that the Nb-doped Ti-Al compounds are thermodynamically stable.However, in all systems, only Nb-doped TiAl 3 (tI32-TiAl 3 -Nb-2) has lower formation energies than the non-doped ones by ~0.13 kJ/mol.The formation energy of hp8-Ti 3 Al-Nb-2 with 3.125 at.%Nb are lower than that with 6.250 at.%Nb.Clearly, Nb doping is not conducive to the thermodynamical stability of the Ti-Al intermetallic compounds.For the same component, such as tI24-Ti 2

Mechanical Properties
Table 3 shows the elastic matrix constants of hp8-Ti 3 Al, tP4-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , tI16-Ti 5 Al 11 , tP28-Ti 2 Al 5 and tI32-TiAl 3 , and the most stable Nb-doped phases of hp8-Ti 3 Al-Nb-2, tP4-TiAl-Nb-1, tP32-Ti 3 Al 5 -Nb-1, tI24-TiAl 2 -Nb, tI16-Ti 5 Al 11 -Nb-3, tP28-Ti 2 Al 5 -Nb-1 and tI32-TiAl 3 -Nb-2.The stiffness-related elastic constants directly reflect the mechanical stability [29], and the elastic matrix constants in Table 3 meet the mechanical stability criteria [45][46][47].Thus, the Ti-Al compounds and Nb-doped ones are mechanically stable.The tP4-TiAl values in Table 3 are consistent with the experimental report, with a difference of ~10%.Theoretically, the elastic matrix constants are sensitive to the initial calculation parameters in the first-principles calculations, such as the cutoff energy and K-points.For tP4-TiAl, our results, calculated with a cutoff energy of 420 eV, are closer to the values from the same method (PAW-GGA) with a cutoff energy of 450 eV [48]; however, they are somewhat higher than that those with a cutoff energy of 400 eV [49].In general, our results are consistent with previous theoretical results.As shown in Table 3, when Nb atoms are introduced into the Ti-Al intermetallic compounds, there are some changes on the elastic matrix constants: (1) hP8-Ti 3 Al-Nb-2 exhibits smaller C 11 , C 22 , C 44 , C 55 , C 66 values, but a larger C 33 , implying enhanced anisotropy; (2) tI32-TiAl 3 -Nb-2 possess increased elastic constants, in which C 11 , C 22 and C 33 increase over 5 GPa; (3) for all compounds, C 33 increases, and even more than 5 GPa for tP28-Ti 2 Al 5 -Nb-1 and tI32-TiAl 3 -Nb-2.Based on the elastic constants, the bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), and Pugh's ratio (K, B/G) were calculated using the Voigt-Reuss-Hill (VRH) scheme [52][53][54].With the calculated bulk modulus and shear modulus, Vickers hardness (H v ) can be calculated according to the empirical formula proposed by Chen et al. [54].The VRH approximation is known as the best method for the evaluation of the theoretical mechanical properties of polycrystalline materials, taking the value from the average of the Voigt and Reuss approximations [47,53].In addition, the Debye temperature (Θ D ) was evaluated in terms of the sound velocity [55,56].All of the calculated results are shown in Table 4.It is known that B reflects the compressibility of a solid under hydrostatic pressure, while G generally indicates the relationship between reversible deformation resistance and shear stress.E is defined as the ratio of stress to strain and is used to measure the stiffness of a material.A larger E means a higher stiffness with more covalent bond characteristics [57,58].For Ti-Al compounds (Table 4), B decreases as the Al content increases, with the highest value of 116.09GPa for Ti 3 Al and the lowest value of 106.86 GPa for TiAl 3 .G and E continually increase with increasing Al content.The Pugh's ratio K (B/G) is normally used to reflect the ductility of a compound, with a critical value of 1.75, i.e., being brittle when K < 1.75 and ductile when K > 1.75 [59].Likewise, Poisson's ratio ν reflects the chemical bonding characteristics of compounds.Covalent bonds become weaker and metallic bonds become stronger as ν increases, with a critical value of 0.26 [59,60].Obviously, for Ti-Al compounds, only Ti 3 Al has a K higher than 1.75 and ν larger than 0.26, showing good ductility and strong metallic bonds.As the Al content increases, ν/K reduces, indicating the presence of reinforcing covalent bonds.This is consistent with the results of H v and Θ D (Table 4), both of which increase with increasing Al content.
As shown in Table 4, all Nb-doped Ti-Al compounds possess a larger B, in which Ti 5 Al 11 has the largest D-value of 5 GPa.Obviously, Nb doping can strengthen the compressibility of Ti-Al compounds under hydrostatic pressure.After Nb doping, G and E show a non-monotonic change with increasing Al content, i.e., decreasing G and E for Ti 3 Al, a very small influence on TiAl, Ti 3 Al 5 and TiAl 2 , and 2 (G) and 6 (E) GPa increase for Ti 5 Al 11 .As the Al content further increases, this increment decreases.It can be seen that Nb doping can increase the ductility of Ti-Al compounds, which is reflected in the increased Pugh's ratio K and Poisson's ratio ν.Obviously, Nb doping weakens the covalent bonds and strengthens the metallic bonds; thus, the Θ D and H v of Nb-doped Ti-Al compounds become smaller than the non-doped ones.In addition, Nb doping has a greater influence on low-Al-content systems such as hP8-Ti 3 Al and tP4-TiAl, with the ν increment being about 0.01. Figure 3a shows the three-dimensional plots of the Young's modulus of hp8-Ti 3 Al, tP4-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , tI16-Ti 5 Al 11 , tP28-Ti 2 Al 5 and tI32-TiAl 3 .The plots of hp8-Ti 3 Al, tP4-TiAl, and tI32-TiAl 3 are quite similar to previous theoretical reports [14].The anisotropy of single-crystal structures usually originates from the directional properties of covalent bond.From the visual observation, hp8-Ti 3 Al seems to have a greater isotropic Young's modulus.As the Al content increases, the Ti-Al compounds display a greater anisotropic Young's modulus.As shown in Figure 3b, after Nb doping, the Young's modulus anisotropy of hP8-Ti 3 Al-Nb-2 has a considerable change along the [100] and [010] directions, i.e., its anisotropy increases.However, the Young's modulus of tI32-TiAl 3 -Nb-2 decreases slightly along the [100] and [010] directions, gently weakening its anisotropy.For other Ti-Al compounds, Nb doping has almost no influence on the anisotropy.
bonds and strengthens the metallic bonds; thus, the ΘD and Hv of Nb-doped Tipounds become smaller than the non-doped ones.In addition, Nb doping has a influence on low-Al-content systems such as hP8-Ti3Al and tP4-TiAl, with the ν in being about 0.01.
Figure 3a shows the three-dimensional plots of the Young's modulus of hp tP4-TiAl, tP32-Ti3Al5, tI24-TiAl2, tI16-Ti5Al11, tP28-Ti2Al5 and tI32-TiAl3.The plots Ti3Al, tP4-TiAl, and tI32-TiAl3 are quite similar to previous theoretical reports [ anisotropy of single-crystal structures usually originates from the directional pr of covalent bond.From the visual observation, hp8-Ti3Al seems to have a greater i Young's modulus.As the Al content increases, the Ti-Al compounds display a gre isotropic Young's modulus.As shown in Figure 3b

Electronic Structures
In order to gain an insight into the physical mechanisms, the calculations of the electronic structures were performed for the Ti-Al compounds of hp8-Ti 3 Al, tP4-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , tI16-Ti 5 Al 11 , tP28-Ti 2 Al 5 and tI32-TiAl 3 , and the most stable Nb-doped phases of hp8-Ti 3 Al-Nb-2, tP4-TiAl-Nb-1, tP32-Ti 3 Al 5 -Nb-1, tI24-TiAl 2 -Nb, tI16-Ti 5 Al 11 -Nb-3, tP28-Ti 2 Al 5 -Nb-1 and tI32-TiAl 3 -Nb-2.Figure 4 presents the total density of states (TDOSs) and the partial density of states (PDOSs).The TDOS displays a large distribution across the Fermi energy level (E F ), indicating that all compounds show a metallic behavior.In addition, a pseudo-energy gap (a pronounced valley near E F ) can be clearly observed from the TDOSs in Figure 4, which exists in the bonding and anti-bonding regions.The stability of a compound can be assessed based on the relative position of the E F and the pseudo-energy gap.When the E F lies to the right of the pseudo-energy gap, the electrons occupy the bonding region, indicating a stable structure.Conversely, when the E F lies to the left of the pseudo-energy gap, the electrons occupy the anti-bonding region, resulting in a less stable structure.The width of the pseudo-energy gap serves as an indicator of the strength of the covalent bond, and a wider gap suggests a stronger covalent interaction [61].
occupy the bonding region, indicating a stable structure.Conversely, when the EF lies to the left of the pseudo-energy gap, the electrons occupy the anti-bonding region, resulting in a less stable structure.The width of the pseudo-energy gap serves as an indicator of the strength of the covalent bond, and a wider gap suggests a stronger covalent interaction [61].As shown in Figure 4a, the pseudo-energy gap width increases with increasing Al content, implying an enhancement of the covalent bond.Thus, hP8-Ti3Al has the lowest pseudo-energy gap, in agreement with its best ductility and lowest Debye temperature of 496 K (Table 4).The PDOSs in Figure 4a show that the Al 3s and 3p orbitals are almost completely separated in hP8-Ti3Al.Near the Fermi level, Al-3p and Ti-3d form strong metallic bonds, accounting for the good ductility of hP8-Ti3Al.As the Al content increases, the Al 3s orbitals widen and hybridize with the Al-3p orbitals; moreover, this hybridization gradually strengthens.The enhanced Al-sp-hybridizing orbitals form strong covalent bonds with the Ti-3d orbitals, accounting for the enhancing mechanical strength and lower As shown in Figure 4a, the pseudo-energy gap width increases with increasing Al content, implying an enhancement of the covalent bond.Thus, hP8-Ti 3 Al has the lowest pseudo-energy gap, in agreement with its best ductility and lowest Debye temperature of 496 K (Table 4).The PDOSs in Figure 4a show that the Al 3s and 3p orbitals are almost completely separated in hP8-Ti 3 Al.Near the Fermi level, Al-3p and Ti-3d form strong metallic bonds, accounting for the good ductility of hP8-Ti 3 Al.As the Al content increases, the Al 3s orbitals widen and hybridize with the Al-3p orbitals; moreover, this hybridization gradually strengthens.The enhanced Al-sp-hybridizing orbitals form strong covalent bonds with the Ti-3d orbitals, accounting for the enhancing mechanical strength and lower ductility with increasing Al content.After Nb doping, the pseudo-energy gap width reduces (Figure 4b).This indicates that Nb doping weakens the character of the covalent bond, being consistent with the results of the Poisson's ratio and Debye temperature, as shown in Table 4.The electronic structures display that for hP8-Ti 3 Al-Nb-2 and tp4-TiAl-Nb-1, in which Al-sp hybridization is weak, the Nb 4d orbitals locate near the Fermi level (>−4 eV) and interact with the Ti-3d and Al-3p orbitals, strengthening the metallic bonds.This is consistent with the result that Nb doping increases the Poisson's ratio ν more significantly for hP8-Ti 3 Al and tP4-TiAl than for other Ti-Al intermetallic compounds.As the Al content increases, although some Nb 4d electrons participate in the formation of covalent bonds because of the enhanced Al-sp hybridization, the introduction of Nb 4d electrons improves the metallicity of the Ti-Al compounds.

Conclusions
The first-principles density functional theory (DFT) was employed to study the crystal structures, stability, mechanical properties, anisotropy, and electronic structures of Nb-free and Nb-doped Ti-Al intermetallic compounds, including seven components and eleven crystal configurations based on the phase diagrams.The calculated total energies reveal that hP8-Ti 3 Al, tP4-TiAl, tP32-Ti 3 Al 5 , tI24-TiAl 2 , tI16-Ti 5 Al 11 , tI24-Ti 2 Al 5 , and tI32-TiAl 3 are the most stable phases.Mechanical properties were estimated using the calculated elastic constants, as well as the bulk modulus, shear modulus, Young's modulus, Poisson's ratio and Pugh's ratio following the Voigt-Reuss-Hill scheme.As the Al content increases, the bulk, shear and Young's modulus increase but the Poisson's ratio decreases for Ti-Al compounds, indicating the strengthened mechanical properties and weakened ductility.This is due to the enhanced covalent bonds, which are formed by the continuously enhanced Al-sp hybrid orbitals and Ti-3d orbitals.Nb doping (~5 at.% used in this study) maintains thermodynamic and mechanical stability for the Ti-Al compounds.Moreover, Nb-doped tI32-TiAl 3 has a lower formation enthalpy than the non-doped ones.The mechanical results show that Nb doping brings a slightly larger bulk modulus and better ductility for Ti-Al compounds.The electronic structures display that the Nb 4d orbitals locate near the Fermi level and interact with the Ti-3d and Al-3p orbitals, strengthening the metallic bonds in the Ti-Al compounds.Nb doping also increases the mechanical anisotropy of hP8-Ti 3 Al.
, after Nb doping, the Young's m anisotropy of hP8-Ti3Al-Nb-2 has a considerable change along the [100] and [010 tions, i.e., its anisotropy increases.However, the Young's modulus of tI32-TiAl3-N creases slightly along the [100] and [010] directions, gently weakening its anisotro other Ti-Al compounds, Nb doping has almost no influence on the anisotropy.

Figure 3 .
Figure 3.The 3D projection of the Young's modulus of (a) Ti-Al and (b) Nb-doped Ti-Al co pounds.

Figure 3 .
Figure 3.The 3D projection of the Young's modulus of (a) Ti-Al and (b) Nb-doped Ti-Al compounds.

Figure 4 .
Figure 4.The total density of states (TDOSs), partial density of states (PDOSs) and trend of the pseudo-energy gap width with increasing Al content of the (a) Ti-Al and (b) Nb-doped Ti-Al compounds.

Figure 4 .
Figure 4.The total density of states (TDOSs), partial density of states (PDOSs) and trend of the pseudoenergy gap width with increasing Al content of the (a) Ti-Al and (b) Nb-doped Ti-Al compounds.

Table 1 .
The calculated lattice parameters (Å) and formation enthalpy (kJ/mol) of the Ti-Al intermetallic compounds.

Table 2 .
The calculated lattice parameters (Å) and formation enthalpy (∆H r , kJ/mol) of the Nb-doped Ti-Al intermetallic compounds, and the percentage change in the structural parameters relative to those of the non-doped ones.x% is the atomic content of Nb in percentage.Boldface denotes the most stable structure in the same Ti-Al component compounds.

Table 3 .
The elastic constants (GPa) of the Ti-Al and Nb-doped Ti-Al intermetallic compounds.