Study on Modifying Mechanical Properties and Electronic Structure of Aerospace Material γ-TiAl Alloy

Abstract

γ-TiAl alloy is a lightweight high-temperature structural material, featuring low density, excellent high-temperature strength, creep resistance, etc. It is a key material in the aerospace field. However, the essential defects of γ-TiAl alloys, such as poor room-temperature plasticity and low fracture toughness, have become the biggest obstacles to their practical application. Therefore, in this paper, the physical mechanism of modification of the mechanical properties and electronic structure of γ-TiAl alloys by doping with Sc, V, and Si was investigated by using the first-principles pseudopotential plane wave method. This paper specifically calculates the geometric structure, phonon spectrum, mechanical properties, electron density of states, Mulliken population analysis, and differential charge density of γ-TiAl alloys before and after doping. The results show that after doping, the structural parameters of γ-TiAl have changed significantly, and the doping models all have thermodynamic stability. The B, G, and E values of the doped system are, respectively, within the range of 94–112, 57–69, and 143–170 GPa, indicating that the material’s ability to resist compressive deformation is weakened. Moreover, the B/G values change from 1.5287 to 1.6350, 1.7279, and 1.6327, respectively, and a transformation from brittleness to plasticity occurs. However, it is still lower than the critical value of 1.75, indicating that the doped γ-TiAl alloy material retains its high-strength characteristics while also exhibiting a certain degree of toughness. The total elastic anisotropy index of the doped system increases, and the degree of anisotropy of mechanical behavior significantly increases. The total electron density of states diagram indicates that γ-TiAl alloys possess conductive properties. The covalent interactions between doped atoms and adjacent atoms have been weakened to varying degrees, which is manifested as a significant change in the charge distribution around each atom. The above results indicate that the doping of Sc, V, and Si can effectively tune the mechanical properties and electronic structure of γ-TiAl alloys.

1. Introduction

Benefiting from the world’s industrial structure adjustment and the rapid rise of high-end manufacturing, the demand for lightweight and high-temperature-resistant structural materials in fields such as aerospace and petrochemicals is constantly increasing. The γ-TiAl alloy features light weight, high specific strength, excellent high-temperature oxidation resistance, and creep resistance. Within a certain service temperature range, it is one of the superior lightweight structural materials and can be applied to critical components such as aero-engine blades and thermal protection systems of spacecraft [1]. As a typical intermetallic compound material, γ-TiAl alloy inherently exhibits poor room-temperature ductility and low fracture toughness, posing substantial limitations for its implementation in practical engineering scenarios [2]. Additionally, the presence of numerous strongly directional Ti-Al and Al-Al bonds in γ-TiAl leads to non-uniform charge distribution and pronounced anisotropy of bond strength during their bonding process. It restricts dislocation motion at the atomic level and reduces plasticity in such materials, thereby limiting the widespread applicability of γ-TiAl alloys [3,4,5]. The low thermal conductivity, poor machinability, and high cutting temperatures of γ-TiAl alloys also often make it difficult to ensure the geometric accuracy and surface integrity of parts [6]. How to overcome these inherent problems and thereby fully leverage the advantages of such materials and enhance their comprehensive mechanical properties has also been one of the important research topics in the field of materials science for a long time.

To fully explore and enhance its potential application value, researchers have conducted systematic research around synthetic preparation technology and performance characterization. Through in-depth analysis and optimization, they strive to achieve greater breakthroughs. For instance, Song et al. [7] conducted conventional grinding and ultrasonic vibration-assisted grinding on γ-TiAl intermetallic compounds, respectively, comprehensively revealing the wear mechanism of abrasive grains. Luo et al. [8] prepared Ti-44Al-3Ta-0.3(Cr, W) (at%) rods using a cladding hot extrusion process and tested the tensile properties of the material at room temperature to 800 °C. Wang et al. [9] used first-principles and phase diagram calculation methods and found that the α₂ phase is more likely to transform into the O₂ phase and, under certain conditions, can generate the ω_o phase through a phase transition reaction, while the γ phase is more likely to transform into the ω_o phase but cannot generate the O₂ phase. Huang et al. [10] applied the EHT tight-bound band calculation method and found that the brittleness of γ-TiAl alloys is mainly related to the anisotropy of bond strength around Ti atoms and the composition or electron distribution of the bonds. A large difference in bond strength around the atoms leads to obvious brittleness.

In recent years, the doping of trace alloying elements or transition metal elements has been proven to improve the properties of γ-TiAl alloys [11,12], which has become a research hotspot among numerous researchers. For instance, Shao et al. [13] found that compared with traditional γ-TiAl alloys, introducing Nb for alloying treatment can significantly enhance the material’s ductility at room temperature and oxidation resistance in high-temperature environments, thereby optimizing its comprehensive performance. Han et al. [14] added 1.4% B to the Ti-46Al-8Nb alloy, and the secondary phase produced in the alloy was the TiB phase, indicating that the introduction of B can significantly refine the alloy structure. Imayev V.M. et al. [15] found that compared with NB-containing alloys, Ti-44Al-0.2b-based alloys doped with Zr and Zr + Hf exhibited higher creep resistance. He et al. [16] investigated the effect of La doping on the interface properties of γ-TiAl/α2-Ti₃Al and found that the energy of the interface model reached its minimum value when Al atoms at specific doping sites were replaced by La atoms. Lin [17] prepared Mm(NiCoMnZrTiAl)5.15 hydrogen storage alloy doped with zirconium, yttrium, and titanium elements and found that the doping of yttrium element enhanced the corrosion resistance of the alloy particles. Li et al. [18] investigated the mechanism of alloying on the surface oxygen adsorption of γ-TiAl alloys by using the first principle method of pseudopotential plane waves based on DFT. They found that the co-doping of Nb and Si gave them higher oxidation resistance than their individual addition. Song et al. [19] used the first-principles DFT method and found that the incorporation of Mn atoms weakened the anisotropy of covalent bonds in the system, explaining the mechanism of improved ductility in the doped system from the perspective of chemical bonds. Liu et al. [20] adopted the first-principles plane-wave pseudopotential method based on density functional theory to analyze the occupation rules of different transition metal elements in TiAl alloys and explain the influence of 5d transition metal doping on the structure and properties of TiAl alloy systems. Yu et al. [21] investigated the intrinsic properties of the γ’ phase in the (FeCoNi)₈₆Ti₇Al₇ high-entropy alloy and its alloying strategy by using the first-principles calculation method. They found that Ta and Nb are the two most ideal alloying elements, which can simultaneously enhance the high-temperature thermal stability, modulus, hardness, plasticity, and fault energy of the γ’ phase. Song et al. [22] investigated the segregation behavior of Ce at the TiAl₃/α-Al interface using first-principles calculations. Their study revealed that Ce spontaneously segregates at the nucleation interface between TiAl₃ and Al, enhancing interfacial stability and promoting surface nucleation. Chen et al. [23] studied the occupation of L1₀-TiAl doped with Sc and found that Sc preferentially occupied Ti positions. Yin et al. [24] studied trace Sc-doped TiAl-based alloys and found that the high-temperature yield strength and compressive strength of the doped system were significantly improved. Song et al. [25] studied Ce and V double-doped γ-TiAl-based alloys and found that the doping system had improved stability and plasticity, as well as weakened covalent bonding between certain orbital bonds. Song et al. [26] studied V-doped two-phase γ-TiAl/α₂-Ti₃Al and found that its ductility was improved. Song et al. [27] studied Si and Y double-doped γ-TiAl-based alloys and found that the doping system had stability and significantly improved oxidation effects. Dong et al. [28] found that the lower the Nb content in TiAl alloys, the stronger the effect of Si on improving the oxidation resistance of the alloys.

Based on the existing reports, the doping of Sc has been found to improve the high-temperature yield strength and compressive strength of TiAl-based alloys; dual doping involving V enhances the stability and plasticity of γ/α₂-TiAl-based alloys; and single or dual doping incorporating Si significantly improves the stability and oxidation resistance of TiAl/γ-TiAl-based alloy systems. The majority of the above research findings demonstrate that doping with Sc, V, and Si can effectively enhance the stability and oxidative properties of TiAl-based alloys. However, there are few reports on the modification studies of the mechanical properties and electronic structure of Sc, V, and Si-doped γ-TiAl alloys. In light of this, theoretically investigating the underlying mechanisms by which doping modulates the mechanical properties and electronic structure of γ-TiAl holds significant scientific guidance value. Therefore, in this paper, taking the aviation material γ-TiAl alloy as the matrix, Sc, V, and Si are selected as doping elements. The first-principles pseudopotential plane wave method, which is widely used in material structure design and performance calculation, is adopted to study the mechanical properties and electronic structure of Sc-, V-, and Si-doped γ-TiAl alloys. The physical mechanism of the mechanical properties and electronic structure of doped modulated γ-TiAl alloys is preliminarily revealed, providing a theoretical basis for the experimental research of this alloy and filling the gap in this field.

2. Model Construction and Computational Methods

The crystal structure of the intermetallic compound γ-TiAl alloy belongs to the L1₀-type ordered face-centered tetragonal crystal system with a space group of P4/mmm (#83) with lattice constants a = b = 0.398 nm and c = 0.404 nm [29]. The unit cell contains a total of 4 atoms, including 2 Ti atoms and 2 Al atoms. Their coordinates are Ti (0, 0, 0), (0.5, 0.5, 0), and Al (0, 0.5, 0.5). In this paper, 2 × 1 × 1 and 1 × 1 × 2 γ-TiAl supercells are constructed, respectively. To explore the influence of doping on the mechanical properties and electronic structure of γ-TiAl, the doping position is selected at the center of the supercell (0.5, 0.5, 0.5). Therefore, on the basis of the two supercells, Sc, V, and Si are, respectively, used to dope and replace the Al atom or Ti atom at the coordinate of (0.5, 0.5, 0.5). The doping models are shown in Figure 1 and Figure 2.

The computational method adopted in this paper is the first-principles pseudopotential plane wave method based on density functional theory (DFT), and the main computational work is accomplished by the Castep 8.0 software package [30]. The interaction between the ionic cores and electrons is treated using the ultrasoft pseudopotential (USPP) [31]. The generalized gradient approximation (GGA) is adopted to correct the exchange-correlation energy during the calculation, and the Perdew–Burke–Ernzerhof (PBE) functional is selected for the electron exchange-correlation [32]. The valence electron configurations selected for the calculations are Ti 3s²3p⁶3d²4s², Al 3s²3p¹, Sc 3s²3p⁶3d¹4s², V 3s²3p⁶3d³4s², and Si 3s²3p², and the other orbital electrons are treated as core electrons for the calculations. Set the plane wave cutoff energy to 310 eV, the self-consistent field (SCF) for each atom to 1.0 × 10⁻⁶ eV, the maximum atomic force to 0.05 eV/A, the maximum stress to 0.1 GPa, the maximum atomic displacement to 2 × 10⁻² A, and the k-point grid in the Brillouin zone to 3 × 6 × 6.

3. Results and Discussion

3.1. Geometric Structure

This work calculates the structural parameters and binding energy of γ-TiAl supercells doped with Sc, V, and Si before and after to accurately predict their stable structures, providing a theoretical basis for the performance optimization and modification of γ-TiAl alloy materials.

Table 1. Structural parameters and binding energy of γ-TiAl before and after doping.

Sample		a (nm)	b (nm)	c (nm)	V (nm3)	E (eV)	Eb (eV)
2 × 1 × 1 γ-TiAl		0.7980	0.4003	0.4098	0.1309	−6641.3795	−5.4814
1 × 1 × 2 γ-TiAl		0.3994	0.3994	0.8213	0.1310	−6641.3145	−5.4732
Sc doped	Replace Al	0.8310	0.4068	0.4114	0.1391	−7861.0374	−5.4268
	Replace Ti	0.4087	0.4087	0.8204	0.1370	−6315.3163	−5.2140
V doped	Replace Al	0.8475	0.4113	0.3697	0.1289	−8560.5950	−5.8125
	Replace Ti	0.3951	0.3951	0.8122	0.1268	−7014.4108	−5.5418
Si doped	Replace Al	0.7916	0.3963	0.4035	0.1266	−6692.5999	−5.6896
	Replace Ti	0.3876	0.3876	0.8392	0.1260	−5145.4654	−5.3001

lists the structural parameters and binding energy of γ-TiAl before and after doping.

It can be concluded from Table 1 that the lattice constants a and b of the 2 × 1 × 1 and 1 × 1 × 2 γ-TiAl supercells after structural optimization have errors of no more than 0.6% compared with the results in reference [29] (0.398 nm), and the c/a ratio has an error of no more than 1.3% compared with that in reference [29]. It indicates that the current calculation method and model can accurately predict the basic structure of γ-TiAl, which supports the subsequent calculation of mechanical properties and electronic structure.

In Table 1, after the substitutional doping of Sc, V, and Si, the structural parameters of the γ-TiAl supercell have undergone significant changes. The atomic radii of each atom are as follows: Ti (0.140 nm), Al (0.125 nm), Sc (0.160 nm), V (0.135 nm), and Si (0.110 nm) [33]. Due to the significant difference in atomic radii between the doped Sc and Ti and Al and the size effect of atoms, the lattice expands. After geometric optimization of the doped unit cell, the overall lattice parameters change significantly through the stress release mechanism to release the local changes introduced by the size difference. However, the atomic radii of V are closest to Ti and Al, and the trends of the lattice constant c and volume V match the order of atomic size, but the trends of the lattice constants a and b do not match the order of atomic size. It is possible that the smaller V atoms free up the space of the Al sites, leading to an increase in a and b due to tensile strain in the a and b directions within the layer, while the c direction in the interlayer is due to the enhancement of bonding and the decrease in atomic size, leading to a decrease in c. When Si atoms are doped, the trend of the change in lattice constant matches the order of atomic size, while the trend of the change in volume V does not match the order of atomic size. This may be attributed to the greater contribution of c-axis contraction to the overall volume change, leading to a more pronounced volumetric variation compared to the average lattice constants.

The results of the binding energy E_b in Table 1 are given by the following expression [34]:

E_b = [E_tot(Ti_xA1_yM_m) − xE(Ti) − yE(Al) − mE(M)]/(x + y + m)

(1)

In the formula, E_tot (Ti_xAl_yM_m) represents the total energy of the doped system, where x is the number of Ti atoms in the system, y is the number of Al atoms in the system, and m is the number of dopant atoms M (such as Sc, V, or Si). E(Ti) is the total energy of a single isolated Ti dopant atom, E(Al) is the total energy of a single isolated Al dopant atom, and E(M) is the total energy of a single isolated M impurity atom.

By substituting the calculated relevant data into the above Formula (1), the binding energy data in Table 1 are obtained. The numerical values of the binding energy indicate that the variation trend of the binding energy E_b in the doped systems is E_b(V doped) < E_b(Si doped) < E_b(Sc doped), suggesting that the V-doped system is more stable, has stronger interatomic bonding, and has a higher thermodynamic driving force compared to the Sc-doped system. In other words, the V-doped system is easier to prepare and form than the Sc-doped system. Moreover, when the same atoms are doped, the binding energy of 2 × 1 × 1 γ-TiAl with the substitution position at Al is always lower than that of 1 × 1 × 2 γ-TiAl with the substitution position at Ti, indicating that the 2 × 1 × 1 γ-TiAl doping model is more likely to form and has more superiority than the 1 × 1 × 2 γ-TiAl doping model. Therefore, all subsequent calculations related to performance are carried out based on the 2 × 1 × 1 γ-TiAl doped model.

3.2. Phonon Spectrum

To evaluate the thermodynamic stability of the computational model, the phonon spectra of γ-TiAl before and after doping are calculated in this paper, as shown in Figure 3. As can be seen from Figure 3, all branches of the phonon spectra before and after doping are located above the zero value of the frequency axis, with no imaginary frequency phenomenon. This indicates that the crystal has no tendency for spontaneous structural instability in the ground state, and the material is structurally stable under static conditions with no risk of spontaneous phase transition.

3.3. Mechanical Properties

This work calculates the elastic constants of γ-TiAl alloys before and after doping with Sc, V, and Si, and systematically analyzes the regulation effect of these doped elements on the mechanical properties of γ-TiAl. The elastic constants of doped γ-TiAl are listed in

Table 2. Elastic constants of doped γ-TiAl.

Sample	C11	C12	C13	C33	C44	C66
γ-TiAl	191.3037	92.2742	81.2407	195.4126	121.6786	71.0021
Sc-doped γ-TiAl	139.0976	72.2648	68.4600	153.1293	85.6506	64.3171
V-doped γ-TiAl	186.9689	51.0905	92.5921	134.9768	112.3177	52.5427
Si-doped γ-TiAl	168.9596	90.8389	77.5810	172.6962	108.6766	64.0101

.

For γ-TiAl crystals with a tetragonal structure, the following four conditions (2)–(5) must be satisfied simultaneously according to the Bonn criterion [35], the criterion of mechanical stability:

C₁₁ > 0

(2)

C₄₄ > 0

(3)

C₁₁ > |C₁₂|

(4)

(C₁₁ + 2C₁₂) > 0

(5)

Data processing of the relevant elastic constants in Table 2 shows that the elastic constants of γ-TiAl alloys doped with Sc, V, and Si all meet the above four conditions, indicating that the system studied in this work has mechanical stability.

In the theory of crystal elasticity, there is a strong correlation between the magnitude of the elastic constant C₁₃ and the c/a ratio, with the smaller C₁₃, the larger c/a. It can be seen from Table 2 that only the elastic constant C₁₃ of V increases after doping, which is the result of the maximum compression of the c-axis lattice (0.4098 → 0.3697) and the maximum expansion of the a-axis lattice (0.7980 → 0.8475). However, the elastic constants of the other doped systems all decrease significantly.

Based on the Voigt–Ross–Russ–Hill method (VRH) [36,37,38], the elastic constants C_ij from Table 2, the bulk elastic modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio υ are deduced from Equations (6)–(15), as shown in

Table 3. Mechanical property parameters of doped γ-TiAl.

Sample	B (GPa)	G (GPa)	E (GPa)	υ	B/G
γ-TiAl	120.8216	79.0349	194.6594	0.2315	1.5287
Sc-doped γ-TiAl	94.3628	57.7156	143.8241	0.2460	1.6350
V-doped γ-TiAl	108.9627	63.06095	158.5890	0.2574	1.7279
Si-doped γ-TiAl	111.3678	68.2116	169.9394	0.2457	1.6327

.

B_V = (1/9)[2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃]

(6)

G_V = (1/30)(M + 3C₁₁−3C₁₂ + 12C₄₄ + 6C₆₆)

(7)

M = C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃

(8)

C² = (C₁₁ + C₁₂) C₃₃ − 2C₁₃²

(9)

B_R = C²/M

(10)

G_R = 15{(18 B_V/C²) + [6/(C₁₁ − C₁₂)] + (6/C₄₄) + (3/C₆₆)}⁻¹

(11)

B = (B_V + B_R)/2

(12)

G = (G_V + G_R)/2

(13)

E = 9BG/(3B + G)

(14)

υ = (3B − 2G)/[2(3B + G)]

(15)

Table 3 lists the mechanical property parameters of γ-TiAl before and after doping. As can be seen from Table 3, the higher bulk modulus B and shear modulus G indicate that γ-TiAl has good resistance to homogeneous compression and resistance to shear deformation, and its stiffness performance is good. The Young’s modulus E of γ-TiAl is about 195 GPa, which is higher than the benchmark value of conventional polycrystalline γ-TiAl (160~180 GPa). The calculated γ-TiAl has a relatively low Poisson’s ratio υ, indicating its weak plastic deformation ability. Together with the low B/G value, this reveals the inherent brittleness of the material.

When V is doped, the bulk elastic modulus B, shear modulus G, and Young’s modulus E decrease, but the B/G value (1.7279) is closer to the critical value of 1.75 for the ductile material characterization in Pugh’s criterion [39], which indicates the brittle-to-ductile transition of the material occurs after V doping. This may be due to the participation of the 3d electrons of V in metallic bond formation, which enhances the ability to slip between atomic layers, indicating a decrease in the stiffness properties of the material and an increase in toughness. When Si is doped, due to the fact that the atomic configuration of Si differs from that of Al by only one electron, the bulk modulus B and shear modulus G of the doped system are the closest to those of the undoped state, approximately 170 GPa Young’s modulus E indicates that the material still belongs to the high stiffness category, but the slightly higher Poisson’s ratio υ and B/G value of the doped system also indicate that the material has undergone a transformation from brittleness to toughness. Compared with the doping of V and Si, the doping of Sc has the greatest impact on the bulk modulus B, shear modulus G, and Young’s modulus E of the doped system. This is due to the atomic radius of Sc being larger than that of Ti and Al, which generates lattice defects and significantly alters mechanical properties. It is manifested as being prone to lattice distortion, more likely to form layer faults, a significant decrease in material stiffness performance, and a transformation from brittleness to toughness of the material. When Sc, V, and Si are doped, the bulk modulus B, shear modulus G, and Young’s modulus E of γ-TiAl all decrease significantly, indicating a decline in the stiffness performance of the doped system. In contrast, Poisson’s ratio υ and B/G value increase instead, suggesting an enhancement in the toughness of the doped system. The underlying mechanism for the enhanced toughness originates from Sc, V, and Si doping-induced atomic displacements within the unit cell. This modifies the lattice parameters of γ-TiAl, inducing asymmetric lattice distortion. Consequently, the resistance to dislocation glide is significantly reduced, thereby enhancing dislocation mobility and ultimately improving fracture toughness.

In conclusion, when Sc, V, and Si are doped, the bulk elastic modulus B, shear modulus G, and Young’s modulus E of γ-TiAl all significantly decrease, while the Poisson’s ratio υ and B/G value increase instead. It indicates that the doped systems have all undergone a transformation from brittleness to toughness of the material. However, its B/G value is still below the critical value of 1.75, indicating that the doped γ-TiAl alloy material retains its high strength characteristics while also exhibiting a certain degree of toughness. It is worth mentioning that after V doping, the balance between the ductility and strength of the material is particularly prominent.

The γ-TiAl alloy, due to its L1₀-type ordered face-centered tetragonal structure, exhibits differential atomic arrangements along directions such as [001], [100], and [110], thereby inducing directional dependence in the elastic modulus. From the above results, it can be concluded that the doping and substitution of Sc, V, and Si play a regulatory role in the elastic properties of the γ-TiAl alloy system, influencing the anisotropy of the physical properties of γ-TiAl crystals in different directions. Elastic anisotropy can be quantified by elastic anisotropy indices (i.e., total elastic anisotropy index A^U, compressive anisotropy percentage A_comp, and shear anisotropy percentage A_shear). In addition, the anisotropic degree of the material on the (001), (010), and (100) planes can also be characterized by the shear anisotropic factors (A₁, A₂, and A₃). Based on the data in Table 2, the elastic anisotropy index and shear anisotropy factor of γ-TiAl alloys before and after doping are derived from the calculation Formulas (16)–(21) [40]. The calculated results are shown in

Table 4. Elastic anisotropy indices of doped γ-TiAl.

Sample	AU	Acomp	Ashear	A1	A2	A3
γ-TiAl	0.7318	0.0126%	6.8172%	2.1706	2.1403	1.3710
Sc doped γ-TiAl	0.7893	0.0505%	7.3073%	2.2060	1.9947	1.8290
V doped γ-TiAl	2.0109	0.0815%	16.7309%	3.2851	4.4167	0.8107
Si doped γ-TiAl	0.9128	0.0307%	8.3594%	2.3309	2.4966	1.6439

.

A^U = 5G_V/G_R + B_V/B_R − 6

(16)

A_comp = (B_V − B_R)/(B_V + B_R) × 100%

(17)

A_shear = (G_V − G_R)/(G_V + G_R) × 100%

(18)

A₁ = 4C₄₄/(C₁₁ + C₃₃−2C₁₃)

(19)

A₂ = 4C₅₅/(C₂₂ + C₃₃−2C₂₃)

(20)

A₃ = 4C₆₆/(C₁₁ + C₂₂−2C₁₂)

(21)

The degree of anisotropy of the material can be evaluated by the magnitude of the A^U value. When A^U takes the value of 0, it indicates that the material shows isotropic properties, while the greater the deviation of A^U from 0, the higher the degree of anisotropy of the material. It can be seen from Table 4 that when no doping and substitution are made, A^U = 0.7318, indicating that γ-TiAl is anisotropic. When Sc, V, and Si are doped, the A^U, A_comp, A_shear, and A₁ values of the system increase, indicating that the doping elements significantly enhance the anisotropy of the mechanical behavior of the material. Comparing the three doping elements. The various indices of V change most intensely, showing significant anisotropy. It might be that the 3d electron injection of V strengthens the metal bonds in the Ti-V-Al system. The increase in the values of C₁₃ and C₂₃ leads to a significant increase in A₁ and A₂. Moreover, the formation of directional charge accumulation in the (100) plane of the V-AL bond promotes the localization of electrons in the [100] direction, resulting in an increase in A₃. The significant decrease in C₁₂ further triggered the directional polarization of shear modulus, increasing A_shear and ultimately leading to a remarkable growth in A^U. After Sc doping, the larger atomic radius causes severe lattice distortion in the [100] crystal direction. At the same time, in the partial density of states diagram below, the delocalization of Sc’s 3d electrons weakens the bonding strength along the [100] direction, resulting in large changes in A₃. Furthermore, lattice distortion facilitates stacking fault formation, resulting in a moderate increase in the shear anisotropy index A_shear. Moreover, due to the significant reduction of C₂₃, A₂ decreases, ultimately resulting in the smallest change in the total elastic anisotropy index A^U. After doping with Si, the smaller Si atoms cause the bond lengths of Si-Al and Si-Ti to contract slightly; the increase in bond strength causes a slight increase in the percentage of compressive anisotropy A_comp. The minimal variation in structural parameters corresponds to the most homogeneous changes in elastic constants. Consequently, the anisotropy index exhibits a gradual and controlled increase post-calculation, indicating Si atoms’ distinctive bonding behavior and exceptional lattice compatibility.

The changing trends of various anisotropic indices caused by the above-mentioned doping are of great significance for evaluating the application potential of γ-TiAl alloys in the aerospace field. The weakening of the [001] direction caused by Sc doping makes the components susceptible to creep fracture and fatigue cracks, resulting in increased directional dependence of material properties. The drastic anisotropic changes caused by V doping lead to high residual stress and stress concentration at grain boundaries, exacerbating component fatigue life and damage tolerance. In contrast, the anisotropic increase after Si doping is relatively gentle, which is conducive to achieving a more uniform mechanical response in complex-shaped components or polycrystalline materials and may improve processing performance and service stability. This is more attractive for large-scale production and aerospace applications that require high reliability.

3.4. Electronic Density of States

To explore the influence mechanism of doping on the electronic structure of γ-TiAl alloy, the electronic density of states distribution of γ-TiAl before and after doping is calculated, as shown in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. As can be seen in Figure 4, the density of electronic states near the Fermi energy level is non-zero before and after doping, indicating that γ-TiAl possesses conductor properties. Moreover, the electronic state density at this location slightly increases after doping, indicating that doping can enhance the electrical conductivity of the γ-TiAl alloy. Figure 5, Figure 6, Figure 7 and Figure 8 show the fractional-wave density of states plots for each atom before and after doping of γ-TiAl alloys to dissect the intrinsic mechanism of how the doped atoms affect the electronic density of states.

As can be seen from Figure 4 and Figure 5, in the energy range of −58.6 eV to −55.63 eV, the partial density of states of Ti-3s orbitals dominates the contribution to the total density of states. Its orbital character exhibits high localization, forming a characteristic peak at −56.59 eV in the valence band. In the energy range of −33.96 eV to −31.65 eV, the density of states is mainly contributed by the Ti-3p orbital, which is highly localized in its orbital and forms a characteristic peak at the valence band of −32.92 eV, matching the sharp peak at the same energy position in the total density of states. The density of states in the energy range of −9.73 eV to −4.21 eV is mainly contributed by the Al-3s orbitals, and there is a characteristic wave peak in the valence band at −5.03 eV. In the energy interval from −4.21 eV to 5.00 eV, the density of states is mainly contributed by the Ti-3d orbitals and the Al-3p orbitals, both of which show signs of hybridization to form the characteristic peak shapes of density of states, and there are the characteristic waveforms of Al-3p and Ti-3d in both the valence band at −1.72 eV and the conduction band at 1.91 eV.

As can be seen from Figure 4 and Figure 6, after Sc doping, two eigenwave peaks dominated by Sc-4s and Sc-3p orbitals appear at −48.76 eV and −27.44 eV, and their orbital features show a high degree of localization. In the energy range of −9.73 eV to −0.89 eV, due to the introduction of a very small amount of Sc-3p state electrons, the Al-3p state is weakened, resulting in a reduction of the electron transition peak between Ti-3d and Al-3p, which corresponds to a decrease in the characteristic peak shape of the total density of states. Near the Fermi level, the slight introduction of Sc-3d electrons enhances the Ti-3d states, manifested as a slight increase in the characteristic peak of the density of states. This indicates that Sc doping strengthens the covalent interactions between Ti and Al. The localization peak of the deep energy level of Sc-4s causes distortion stress in the lattice, and the surrounding bonds are passively elongated, corresponding to the decrease in elastic modulus in the previous text and the increase in bond length around Sc in the following text.

As can be seen from Figure 4 and Figure 7, two characteristic wave peaks dominated by V-3s and V-3p orbitals with significant orbital localization appear at −63.62 eV and −37.47 eV after V doping. Near the Fermi level, the introduction of V-3d electrons weakens both Ti-3d and Al-3p states, while the V-3d orbital itself contributes new electronic states. The coupling of these orbitals leads to a slight increase in the characteristic peak of the density of states, indicating that V doping enhances the covalent interactions between Ti and Al. The Ti-V-Al metal bond in the system is strengthened, corresponding to a higher shear modulus.

As shown in Figure 4 and Figure 8, after Si doping, two characteristic peaks dominated by Si-3s and Si-3p orbitals appear at −8.53 eV and −2.58 eV in the partial density of states. In the energy range of −10.77 eV to −5.11 eV, the introduction of Si-3s electrons weakens both Al-3s and Al-3p states, while the Si-3s orbital itself contributes new electronic states. The hybridization of these orbitals results in a slight increase in the characteristic peak of the density of states, corresponding to the enhanced covalent property of the Al-Al bond in the following text. In the energy range of −5.11 eV to −1.14 eV, the introduction of Si-3p state electrons weakens the Al-3p state, reducing the electron transition peak between Ti-3d and Al-3p, which corresponds to the decrease in the characteristic peak shape of the density of states. In summary, the distribution of characteristic peaks in the total electronic density before and after Si doping does not change significantly. This is consistent with the fact mentioned earlier that the elastic modulus of Si after doping is closest to the value before doping.

3.5. Mulliken Population Analysis and Differential Charge Density

Table 5. Mulliken population analysis of doped atoms and their adjacent atoms.

Sample	Atom	s	p	d	Total	Charge (e)
γ-TiAl	Ti	2.31	6.72	2.84	11.88	0.12
	Al	0.98	2.14	0	3.12	−0.12
Sc-doped	Ti	2.39	6.82	2.78	11.99	0.01
	Al	1.01	2.13	0	3.14	−0.14
	Sc	2.32	6.77	1.79	10.87	0.13
V-doped	Ti	2.33	6.88	2.77	11.97	0.03
	Al	0.97	2.10	0	3.07	−0.07
	V	2.41	6.79	9.85	13.05	−0.05
Si-doped	Ti	2.34	6.69	2.86	11.89	0.11
	Al	0.98	2.10	0	3.07	−0.07
	Si	1.33	2.86	0	4.19	−0.19

lists the Mulliken charge population values of the dopant atoms (Sc, V, Si) and their adjacent atoms. Figure 9 shows the differential charge density of doped γ-TiAl. In the differential charge density of Figure 9, red indicates gained electrons, blue indicates lost electrons, and green indicates intermediate states.

Table 6. Mulliken population analysis of bonds adjacent to doped atoms.

Sample	Bond	Population	Length (Å)
γ-TiAl	Al-Al	0.62	2.81
	Al-Ti	0.81	2.84
Sc-doped γ-TiAl	Sc-Al	0.31	2.95
	Sc-Ti	0.27	2.89
V-doped γ-TiAl	V-Al	0.25	2.91
	V-Ti	0.57	2.77
Si-doped γ-TiAl	Si-Al	0.57	2.80
	Si-Ti	0.78	2.83

presents the Mulliken population analysis of the bonds between the dopant atoms (Sc, V, Si) and their adjacent atoms.

Table 7. Mulliken population analysis of the peripheral bonds of doped atoms.

Sample	Bond	Population	Length (Å)
γ-TiAl	Al-Al	0.62	2.81
	Ti-Ti	−0.35	2.81
	Al-Ti	0.81	2.84
Sc-doped γ-TiAl	Al-Al	0.52	2.87
	Ti-Ti	0.08	2.99
	Al-Ti	1.01	2.89
V-doped γ-TiAl	Al-Al	0.26	3.00
	Ti-Ti	−0.69	2.93
	Al-Ti	1.04	2.77
Si-doped γ-TiAl	Al-Al	0.64	2.81
	Ti-Ti	−0.82	2.75
	Al-Ti	0.74	2.83

presents the Mulliken population analysis of the bonds between adjacent atoms (Ti, Al) and representative peripheral atoms.

It can be seen from Table 5 and Figure 9 that the electronic structures of the systems all change to varying degrees after the substitution of the doped atoms. Before doping, the Al atom gains electrons (−0.12e), corresponding to the red circular charge distribution around the Al atom in Figure 9; the Ti atom loses electrons (0.12e), corresponding to the blue area around the Ti atom in Figure 9. Combining Table 6 and Table 7, since the Ti-Ti bond forms an antibond, the charge distribution around the Ti atom is all in the shape of an “8”. After doping, the Sc atoms occupying the Al sites lose electrons. The red region transforms into a blue electron-deficient region, with most of the electrons being captured by the adjacent Ti atoms and a small portion by the adjacent Al atoms. This indicates that there is a sharing of electrons between Ti and Al atoms, which is manifested as the formation of a directional covalent bond between Ti and Al atoms, consistent with the increase in the bond population value of the Al-Ti bond (0.81 → 1.01) as mentioned below. The V atoms occupying the Al sites capture a small amount of electrons, causing the blue electron-deficient area around them to shrink and the red electron-rich area to expand. The adjacent Ti atoms also gain electrons, while the Al atoms lose a small amount of electrons. The red area around the former increases, and the red area around the latter slightly decreases. This also indicates that the covalent interaction between the Ti and Al atoms is enhanced. The Si atoms occupying the Al sites capture electrons from the neighboring Ti and Al atoms, and a smaller orange-colored region appears around them; the Ti and Al atoms lose a small amount of electrons, and the surrounding electron-gaining and electron-losing regions do not change much compared with those before doping.

The Mulliken population is centered, and the population value can reflect the strength of the bond: a positive value indicates a covalent bond interaction, and the larger the value, the stronger the covalent bond; a negative value indicates an antibond, and there is mutual repulsion between atoms. Bond length affects bond strength: generally, the shorter the bond length, the stronger the bond [29]. It can be seen from Table 6 and Table 7 that:

After Sc doping, the population values of Sc-Al and Sc-Ti bonds decrease, indicating that the introduction of Sc weakens the covalent interaction between Sc-Al and Sc-Ti atoms; the bond lengths of Sc-Al and Sc-Ti are both longer than those before doping, and the bond strength decreases after doping. Meanwhile, the population value of the peripheral Al-Al bond of the doped atoms decreases, while the Al-Ti bond increases (0.81 → 1.01), the strong covalent bond further strengthens, and the Ti-Ti bond changes from negative to positive (−0.35 → 0.08); the antibonding state is eliminated, and a weak covalent/metallic bond is formed. Overall, the covalent interaction is enhanced, and the bond strength decreases. The transition of the Ti-Ti bond population value from negative to positive improves the plasticity of the material, while the increase in the Al-Ti bond population value leads to strong covalency and thus increases brittleness. However, overall, the plasticity is enhanced.

After V doping, the population values of V-Al and V-Ti bonds decrease, weakening the covalent interaction; the V-Al bond length is longer than that before doping, and the bond strength decreases, while the V-Ti bond length is shorter than that before doping, and the bond strength increases. Meanwhile, the population values of the outer bonds Al-Al and Ti-Ti (negative values) decrease, and the bond lengths increase, indicating that the covalent interaction between Al-Al atoms weakens, and no chemical bond is formed between Ti-Ti atoms, with the bond strength weakening. As the Al-Ti bond population value increases, the bond length shortens, the covalent interaction between Al and Ti atoms intensifies, and the bond strength enhances. The weak V-Al and Al-Al bonds improve plasticity, while the strong covalent nature of the Al-Ti bond increases brittleness. However, overall plasticity is enhanced.

After Si doping, the population values of Si-Al and Si-Ti bonds are slightly lower than those before doping, the bond lengths shorten, and the bond strengths increase. Meanwhile, the population value of the peripheral Al-Al bond slightly increases, while the population values of the Ti-Ti bond (negative value) and the Al-Ti bond decrease, indicating that the overall covalent interaction weakens and the repulsive force between Ti-Ti atoms increases; the bond lengths of all bonds become shorter than before doping, and the bond strength increases. The weakening of the overall covalent interaction and the strengthening of the Ti-Ti antibonding indicate that the material is dominated by brittleness, which is consistent with the fact that the B/G value is the lowest among the three doped systems after Si doping, as mentioned earlier.

According to the Morrinaga theory [4], the fundamental reasons for the transformation of brittleness and toughness of γ-TiAl alloys are clearly presented in

Table 8. The correlation between the Mulliken Population and macroscopic mechanical behavior.

Bond Sample		Population Variation	Bond Length Variation	Bonding Evolution	Mechanical Behavior
Sc-doped	Al-Al	−0.1	+0.06	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	Ti-Ti	+0.43	+0.18	anti-bond state → weak covalent/metal bond	dislocation slip resistance ↓ → plasticity ↑
	Al-Ti	+0.2	+0.05	strong covalent bond enhancement	lattice resistance ↑ → plasticity ↓
	Sc-Al	−0.31	+0.14	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	Sc-Ti	−0.54	+0.05	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
V-doped	Al-Al	−0.36	+0.19	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	Ti-Ti	−0.34	+0.12	enhanced anti-bond repulsion	lattice resistance ↑ →plasticity ↓
	Al-Ti	+0.23	−0.07	strong covalent bond enhancement	lattice resistance ↑ →plasticity ↓
	V-Al	−0.37	+0.1	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	V-Ti	−0.24	−0.07	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
Si-doped	Al-Al	+0.02	0	covalent bond enhancement	lattice resistance ↑ →plasticity ↓
	Ti-Ti	−0.47	−0.06	enhanced anti-bond repulsion	lattice resistance ↑ →plasticity ↓
	Al-Ti	−0.07	−0.01	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	Si-Al	−0.05	−0.01	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑
	Si-Ti	−0.03	−0.01	covalent bond weakening	dislocation slip resistance ↓ → plasticity ↑

Note: In Table 8, “↑” indicates an increase, and “↓” indicates a decrease.

, and the correlation between the Mulliken population and macroscopic mechanical behavior is established.

4. Conclusions

In this paper, we used first-principles methods to construct supercell models of γ-TiAl doped with Sc, V, and Si, respectively, and systematically calculated the geometric structure, phonon spectrum, mechanical properties, electronic density of states, Mulliken population analysis, and differential charge density. The results show the following:

(1)

Structural stability: After the incorporation of Sc, V, and Si, due to the difference in atomic radius, the structural parameters of γ-TiAl change significantly. The calculation results of the phonon spectrum show that all branches of the phonon spectrum before and after doping are located above the zero value of the frequency axis, with no imaginary frequency phenomenon, indicating that the calculation model in this paper has specific thermodynamic stability. Meanwhile, according to the criterion of mechanical stability—the Bonn criterion, it is concluded that the structure of γ-TiAl is stable before and after doping.

(2)

Mechanical property transformation: After doping, the bulk elastic modulus B, shear modulus G, and Young’s modulus E of γ-TiAl all significantly decreased, while the Poisson’s ratio υ and B/G value increased. The doped system underwent a transformation from brittleness to toughness.

(3)

Anisotropic enhancement: The A^U, A_comp, A_shear, and A₁ values of the doped system all increased, indicating that doping can significantly enhance the anisotropy of the material’s mechanical behavior.

(4)

Conductor properties: The electron density of states near the Fermi level is not zero before and after doping, indicating that γ-TiAl possesses conductor properties. Moreover, the electronic density of states at this location slightly increases after doping, indicating that doping can enhance the electrical conductivity of γ-TiAl alloys.

(5)

Bonding characteristics: The internal mechanism by which doped atoms affect the electron density of states was reasonably analyzed through the partial wave density of states diagrams of each atom before and after doping. According to the Mulliken population analysis and the differential charge density diagram, incorporating Sc and V enhances the covalent interaction of the Al-Ti bond. However, the incorporation of Si weakens the covalent interaction of the Al-Ti bond while shortening the bond lengths and increasing the bond strength.

Therefore, based on the above research results and weighing mechanical properties, anisotropy levels, electronic properties, and bonding properties, we can conclude from practical applications that Sc doping enhances the toughness of γ-TiAl alloy materials, and the total elastic anisotropy index A^U changes minimally, indicating that under centrifugal loads, stresses in aircraft engine blades will not vary depending on direction, thereby significantly reducing the risk of crack initiation. V-doping, due to its significant enhancement of anisotropy and metal bond strength, as well as the targeted improvement of axial creep resistance by electron localization in the [100] direction, is suitable for high-directional load scenarios. Si doping, with its balanced toughness enhancement, controllable anisotropy, and bond-strengthening effect, is an ideal choice for fatigue-resistant polycrystalline components. The research results provide an important theoretical basis for the development of high-performance γ-TiAl-based alloys.