Density functional theory insight into the role of Al and V in Ti–6Al–4V dental implants: structural, electronic, and mechanical properties

Abstract

Commercially pure titanium and Ti–6Al–4V are the most commonly used materials for dental implants owing to their balanced mechanical properties and biocompatibility. However, much of the related research has focused primarily on experimental synthesis, lacking theoretical guidance and a deeper understanding of the underlying mechanical differences. To address this, we employ density functional theory (DFT) and the special quasi-random structure (SQS) method to construct a 64-atom supercell model and systematically analyze the effects of Al (α-phase stabilizer) and V (β-phase stabilizer) on the structural, electronic, and mechanical properties of Ti–Al–V alloys with various compositions. The results show that Al stabilizes the α-phase by reducing the formation energy through significant charge transfer, whereas V promotes β-phase formation due to its inherent body-centered cubic (BCC) phase tendency. Electronic structure analysis revealed that Al enhances stability through s/p orbital hybridization at deep energy levels, whereas V’s d-electrons dominate interactions near the Fermi level, weakening the bond strength. The moderate elastic modulus of α + β Ti–6Al–4V, combined with its structural isotropy and enhanced stability, results in superior tensile and yield strengths. On the basis of mechanistic insights from Ti–6Al–4V, potential alternative alloys suitable for dental implant applications are proposed.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12903-025-06672-1.

Background

Dental implant restoration is a method of replacing missing teeth with bioinert materials, which is highly favorable for providing patients with functionality and aesthetics close to natural teeth [1]. The implant is a bone-anchored device designed to withstand chewing forces embedded into the alveolar bone at the site of the missing tooth. It connects to the prosthetic tooth via an abutment, providing stability and support [2]. As a result, the properties of the implant material play a decisive role in the success of the implant procedure and the long-term experience of the patient [3]. Commercially pure titanium (cp-Ti) has emerged as the mainstream material for dental implants because of its excellent corrosion resistance, good biocompatibility, and favorable strength‒weight ratio, outperforming many other materials historically used in dentistry [4]. However, emerging clinical challenges (e.g., peri-implantitis, suboptimal osseointegration in osteoporotic bone) necessitate advanced alloys and surface coating with enhanced biomechanical and biofunctional properties [5].

With the development of titanium alloy materials, recent advancements reveal a paradigm shift toward alloy-based dental implants [6–8]. Generally, Ti alloys have two phases, including the α structure (hexagonal close-packed, HCP) and the β structure (body-centered cubic, BCC), which possess different mechanical properties. Researchers have focused on incorporating various alloying elements to stabilize either the α or β phase in titanium alloys [9], with the aim of modifying their phase composition, microstructure and properties. Therefore, multicomponent alloys, including Ti–Mo, Ti–Nb, Ti–Al–V, Ti–Zr–Cr, and Ti–Zr–Nb–Fe [10–14], can enhance strength, biocompatibility, and durability compared to cp-Ti, making them ideal for dental implants. Among the commercially available titanium alloys, Ti–6Al–4V is the most widely used biomedical alloy [15, 16], accounting for 50% of the total Ti production in addition to cp-Ti [17]. However, the majority of existing research has predominantly emphasized experimental synthesis, with limited emphasis on theoretical frameworks and a thorough exploration of the core mechanical distinctions between cp-Ti and Ti–6Al–4V [18].

Computational materials science based on density functional theory (DFT) has become an effective method for studying the properties of titanium alloys [19, 20], and can provide important guidance for the experimental synthesis of ideal alloy materials. For example, Liu et al. systematically studied the effects of 33 alloying elements on the elastic properties and solid solution strengthening of α-phase Ti alloys via DFT [21]. Their research revealed that VIII-group elements (Ru, Rh, Pd, Os, Ir, and Pt) show strong potential for enhancing the overall mechanical properties of Ti alloys. Wan et al. conducted a detailed investigation of the mechanical and electronic properties of the second phases and solid solutions in Ti–xAl–yV alloys based on DFT calculations [22]. Their research identified Ti₃₅Al₅V₂ and Ti₃₀Al₄V₂ as potential compositions with increased stability.

The elastic properties of materials can be investigated via DFT calculations under idealized assumptions, including single-crystal models, zero-temperature conditions, and defect-free structures, which inherently differ from experimental environments. Practical alloys exhibit grain boundaries, textures, defects, and temperature-dependent behaviors that remain unaccounted for in these simulations. Furthermore, DFT cannot model phenomena requiring extended timescales or elevated temperatures, such as long-term fatigue evolution, high-temperature phase transformations, or plastic deformation mechanisms. Nevertheless, DFT effectively establishes composition-structure-property correlations for fundamental material characteristics like phase stability and elastic moduli. de Jong et al. demonstrated this capability by calculating bulk and shear moduli for 104 inorganic crystals with < 15% error relative to experimental data [23]. Similarly, Raabe et al. demonstrated that DFT calculations can predict the phase stability and mechanical properties of Ti-Nb and Ti-Mo alloys, which were verified by experiments [24]. Gutiérrez Moreno et al. employed DFT to predict Ti-Nb alloy phase stability trends, that closely aligned with experimental results, establishing critical guidance for rational alloy design [25].

The Special Quasi-Random Structure (SQS) method [26] was used to construct the supercell models with short-range chemical disorder characteristics, which can effectively simulate the effects of random atomic distribution and local lattice distortion on elastic properties in alloys. For example, in the Al-Ti alloy system, SQS successfully captured lattice distortions caused by disordered Ti atom occupancy and predicted the trend of elastic modulus changes with composition [27]. In the study of Ti-X (X = Mo, Nb, Al, etc.) alloys, SQS revealed the regulatory mechanism of solute atom local stress fields on anisotropic elastic constants [28]. However, the SQS method has critical limitations: (1) randomness-induced elastic constant fluctuations necessitate multi-configuration averaging; (2) low solute concentrations require prohibitively large supercells, escalating computational costs; (3) supercell size constraints prevent continuous composition modeling, mandating discrete sampling that impedes efficient global composition scanning. Therefore, SQS is more suitable for fine-grained elastic and electronic structure analysis at specific composition points, while comprehensive composition-property mapping of the entire composition space requires integration with methods like Coherent Potential Approximation (CPA) [29], cluster expansion [30], or machine learning-based prediction models [31] for accuracy-efficiency balance.

Nevertheless, current computational studies on Ti–Al–V alloys have focused primarily on high-symmetry unit cell structures or approximation methods [32, 33], leaving precise simulations of the random solid solution structure of Ti–6Al–4V, particularly for dental implant applications, still insufficiently explored. The microscopic mechanisms of α and β phase formation in the alloy after doping with Al and V, as well as the specific reasons behind the changes in alloy properties remain unclear. To address these issues, this study designed a supercell model containing 64 atoms to construct Ti–Al alloys and Ti–Al–V alloys with different doping proportions. The special quasi-random structure (SQS) method was employed to reflect the random arrangement of atoms in the Ti–Al–V with the given alloy. On the basis of DFT calculation, the effects of Al as an α-phase stabilizer and V as a β-phase stabilizer on the formation energy of the Ti–Al–V alloy and the underlying electronic structures were systematically investigated. Furthermore, the effects of doping atoms on the mechanical properties of the alloys were analyzed. These studies provide theoretical guidance and scientific basis for the rational design of alloys that meet the requirements of dental implant materials.

Methods

Calculation models

α-phase titanium (α-Ti) has a hexagonal close-packed (HCP) structure, which belongs to the P63/mmc space group and contains two basis atoms. In contrast, β-phase titanium (β-Ti) has a body-centered cubic (BCC) structure, belongs to the Im3̅m space group, and is defined by a basis atom located at the origin, as illustrated in Fig. S1. 4 × 4 × 2 α-Ti and 4 × 4 × 4 β-Ti supercell containing 64 atoms was subsequently constructed to serve as the base structure. The optimized geometric structure of the α-Ti₆₄ phase is shown in Fig. 1a. To handle the random arrangement of atoms in alloys, the disordered configurations within the supercell were modeled using the Special Quasi-Random Structure (SQS) method [26]. The SQS method is a computational approach that is widely employed to approximate disordered atomic configurations in solid solutions or alloys, which is particularly valuable for efficient and accurate modeling of complex alloys without the need to explicitly simulate every possible atomic arrangement. Two, four, and six Ti atoms in the supercell were replaced with Al atoms, resulting in the formation of a Ti₆₂Al₂, Ti₆₀Al₄, and Ti₅₈Al₆ (Ti = 94.5wt%; Al = 5.5wt%)​ binary alloys, as shown in Fig. 1b and S2. The Ti–6Al–4V alloy models were constructed by replacing eight Ti atoms in the supercell in the α-Ti and β-Ti supercells, where six Ti atoms were substituted by Al atoms and two Ti atoms were substituted by V atoms, as shown in Figs. 1c and d. This configuration formed a Ti₅₆Al₆V₂​ ternary alloy (Ti = 91 wt%; Al = 5.5 wt%; V = 3.5 wt%) whose mass ratio is close to that of Ti–6Al–4V (Ti = 90 wt%; Al = 6 wt%; V = 4 wt%). This supercell-based approach provides a tunable low-concentration alloy framework for investigating the effects of alloying elements (Al and V) on the structural, electronic, and mechanical properties of Ti–Al–V alloys.

Fig. 1.

Optimized geometry model, (a) α-Ti₆₄, (b) α-Ti₅₈Al₆, (c) α-Ti₅₆Al₆V₂, and (d) β-Ti₅₆Al₆V₂

DFT calculations

All density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP) [34, 35] with the projector-augmented wave (PAW) method [36, 37]. The employed PAW pseudopotentials included the following valence electron configurations: Ti (3s² 3p⁶ 4s² 3d²), V (3s² 3p⁶ 4s² 3d³) and Al (3s² 3p¹). The exchange-correlation functional was treated within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) parameterization [38]. The energy cutoff for the plane wave basis expansion was set to 400 eV. Partial occupancies of the Kohn − Sham orbitals were allowed using the Gaussian smearing method with a width of 0.1 eV. Γ-centered Monkhorst-Pack k-point meshes with grid densities corresponding to 2π × 0.03 Å⁻¹\AA\: spacing along reciprocal lattice vectors. For structural relaxation, the energy and force convergence criteria were set to 10⁻⁸ eV and 0.001 eV/Å respectively. Elastic constants were determined via the strain-stress approach, the elastic tensor C_ij was derived from the linear strain-stress response. The formation energy (E_form) of the alloy was calculated as follows:

1

where E_total(Ti_xAl_yV_z) refers to the total energy of the alloy unit cell; E_bulk(Ti), E_bulk(Al), E_bulk(V) are the energies per atom of the pure most stable bulk; and x, y, and z represent the numbers of Ti, Al, and V atoms in the supercell, respectively. Furthermore, density functional theory (DFT) calculations were employed to determine the crystal orbital Hamilton population (COHP), charge density differences (Bader charge analysis), and density of states (DOS).

Mechanical properties

The basic elastic properties, including elastic moduli, can be obtained from the elastic constants. The elastic response of an isotropic system is generally described by the bulk modulus (B) and shear modulus (G), which can be obtained by averaging single-crystal elastic constants. The most frequently used averaging methods are the Voigt [39], Reuss [40] and Hill [41] bounds. In Voigt’s and Reuss’s approximations, the equation takes the following forms:

2

3

4

5

where C_ij and S_ij denote the components of the elastic and compliance tensors respectively. The compliance constant matrix (S) and elastic constant matrix (C) are related by S = C ⁻¹. In addition, the arithmetic means of the Voigt and Reuss bounds, termed the Voigt-Reuss-Hill (VRH) average is also found to be a better approximation of the actual elastic behavior of a polycrystalline material,

6

7

The Young’s modulus (E), Poisson’s ratio (ν) andhardness (H) for an isotropic material are given by:

8

9

10

11

12

13

14

where the Debye temperature denoted as Θ_D, h and k_B represent Planck’s constant and Boltzmann’s constant, respectively, n signifies the number of atoms per unit cell, V_a is the atomic volume, ρ is mass density and v_m denotes the mean sound velocity derived from the longitudinal wave velocity (v_m) and transverse wave velocity (v_t).

Results and discussion

Lattice parameters and formation energy

The primitive α-Ti structure was first optimized, yielding lattice parameters of a = b = 0.293 nm and c = 0.465 nm, which is consistent with the HCP structure (space group P63/mmc), and agrees well with the experimental data (a = b = 0.295 nm, c = 0.468 nm) [42], validating the reliability of the computational methods and settings. For the alloy systems, the SQS method was used to model substitutional doping of Al and V in α-Ti₆₄. The lattice parameters, formation energies, and densities of the optimized structures were calculated and analyzed, as shown in Table S1 and Fig. 2. Upon doping with Al and V, the lattice constants decrease slightly, and the crystal system transitions from hexagonal to triclinic. This transformation is attributed to the smaller atomic radii of Al (1.43 Å) and V (1.34 Å) than that of Ti (1.47 Å), which induces symmetry changes and lattice distortion.

Fig. 2.

(a) E_form values and (b) densities of given alloys

In the case of the α-Ti–Al alloys, E_form and density decrease with increasing Al content, indicating that Al doping enhances the stability and contributes to light weight of the alloys. With the addition of a small amount of V, the E_form and density of α + β Ti₅₆Al₆V₂ remained lower than those of α-Ti₆₄ (Fig. 2). The underlying reason for the E_form transformational pattern is that although the most stable phase of Al is the face-centered cubic (FCC) structure with space group Fm3̅m, its metastable HCP structure with space group P63/mmc is only higher than 0.031 eV/atom of the FCC structure, as shown in Table S2. This small energy difference allows Al to readily transform from the FCC structure into the P63/mmc structure and combine with Ti to form a more stable α-Ti–Al alloy.

By the same principle, the inherent HCP ground-state structures of Mg and Zn may similarly contribute to α-Ti stabilization. Therefore, the E_form of α- and β-Ti doped with Mg and Zn were calculated, as shown in Fig. S3. For Mg-doped systems, the positive and progressively increasing E_form values in α-Ti-Mg demonstrate limited miscibility, while β-Ti-Mg shows even higher E_form values (Fig. S3a) restricting synthesis to low Mg concentrations [43]. Despite this constraint, Liu et al. successfully synthesized Ti-Mg alloys exhibiting low compression modulus (36–50 GPa), high strength (1500–1800 MPa), and bioactivity [44], indicating great potential in dental implants. For the Zn doped Ti, the α-Ti-Zn progressively decreasing negative E_form values with increasing Zn content (Fig. S3b), favoring spontaneous formation of solid solutions and intermetallic compounds over β-Ti-Zn’s positive and increasing trend, confirming Zn’s α-phase stabilization role consistent with Liang et al. [45]. Similarly, the doped Zn can lead to the high compressive strength (up to 1906 MPa) and low elastic modulus (16.6–26.8 GPa) of Ti-Zn alloys [46]. Since Zn is a biocompatible element and Zn²⁺ has the effect of inhibiting oral pathogenic bacteria, it has shown significant potential in experiments to replace Al for stabilizing α-Ti in future dental applications.

In the case of α-Ti–Al–V alloys, the E_form of α-Ti₅₆Al₆V₂ increases compared with that of the α-Ti₅₈Al₆ alloy, indicating that V acts as a destabilizer for α-Ti–Al–V alloys. The main difference between V and Al lies in the fact that the most stable phase of V is the BCC structure with space group Fm3̅m, but its metastable HCP structure with space group P63/mmc is significantly higher in energy by 0.254 eV/atom (Table S2). This large energy difference suggests that V has a strong tendency to maintain its BCC structure, destabilizing α-Ti alloys and promoting the formation of β-Ti alloys. This is an important reason why Nb, Ta, and Mo atoms act as stabilizers for β-Ti alloys [10, 25, 47]. To quantitatively probe V’s β-stabilizing role, we generated β-Ti₅₆Al₆V₂ by constructing a β-Ti₆₄ supercell and performing substitutional doping of Al/V atoms via the SQS method, maintaining identical composition to the α-Ti₅₆Al₆V₂ reference system. After optimization, the calculated E_form of β-Ti₅₆Al₆V₂ (−0.069 eV/atom) is lower than that of α-Ti₅₆Al₆V₂ (−0.064 eV/atom), confirming that V acts as a β-phase stabilizer. This result demonstrates that even a small amount of V doping can promote the formation of β-Ti alloys, which may lead to the formation of mixed-phase (α + β) Ti–Al–V alloys during the experimental synthesis process. Overall, α + β Ti₅₆Al₆V₂ still has a lower E_form than pure α-Ti, and its enhanced structural stability results in the improved tensile strength and yield strength [48]. Building on the mechanistic insights gained from the use of Al and V in regulating Ti alloys, we propose exploring alternative elements to design next-generation dental implant materials. For example, Mg or Zn, with their HCP structure and low-toxicity nature, can enhance the stability and biocompatibility of Ti alloys, while nontoxic β-phase stabilizers such as Nb, Mo, or Ta can be employed to fine-tune the formation of the β-phase, optimizing the mechanical properties and reduce stress shielding.

Electronic structure analysis

In the α-Ti₆₄ system, the density of states (DOS) plot reveals that the electrons near the Fermi level are primarily contributed by the d-electrons of Ti, which dominate near the Fermi level, as shown in Fig. 3a. This finding indicates that the d-electrons of Ti play a critical role in the formation of metallic bonds, which is consistent with the significant contribution of d-electrons to bonding in transition metals. The charge density difference analysis showed strong metallic bonding between Ti atoms, with many electrons delocalizing between the atoms, exhibiting typical metallic bonding characteristics (Fig. 4a). The nondirectional nature of metallic bonds allows metal atoms to slide relative to each other under external forces, endowing α-Ti₆₄ with excellent electrical conductivity and ductility.

Fig. 3.

DOS and partial DOS of given alloys, (a) α-Ti₆₄, (b)‒(d) α-Ti₅₈Al₆, (e)‒(h) α-Ti₅₆Al₆V₂, and (i)‒(l) β-Ti₅₆Al₆V₂

Fig. 4.

Charge density difference images of given compounds with 3D view and cross- sectional view, (a) α-Ti₆₄, (b) α-Ti₅₈Al₆, (c) α-Ti₅₆Al₆V₂, and (d) β-Ti₅₆Al₆V₂. In the 3D view, the yellow and cyan areas represent charge accumulation and depletion respectively. In the cross-sectional view, the red and blue colors indicate the charge accumulation and depletion, respectively

After six Ti atoms were replaced with Al atoms to form the α-Ti₅₈Al₆, the s and p orbitals of Al overlapped with the s and p orbitals of Ti at deep energy levels, as shown in Figs. 3b‒d. However, the electronic structure near the Fermi level was still dominated by the d-electrons of Ti, with Al contributing almost nothing to the states near the Fermi level. This indicates that the electrons of Al primarily participate in bonding with Ti at deep energy levels, which is distinctly different from the d-electron interactions between the transition metals. This unique bonding behavior led to a significant reduction in the energy of the α-Ti–Al alloy system. The deep energy level bonding may also explain why C and O can act as α-phase stabilizers [8].

Furthermore, the charge density difference image reveals that Al loses only a small number of electrons, while accumulating a substantial number of electrons contributed by neighboring Ti atoms (Fig. 4b). This observation aligns with Bader charge analysis results, indicating that Al gains approximately one electron per atom through charge transfer from Ti. Such significant electron accumulation at Al sites likely drives the substantial reduction in system formation energy upon Al alloying. Similarly, Zn in α-Ti₅₈Zn₆ exhibits analogous behavior, acquiring 1.03 electrons per atom (Fig. S4a). Conversely, Mg in α-Ti₅₈Mg₆ lose 1.29 electrons per atom (Fig. S4b). This can be attributed to the higher electronegativity of Al (1.61) and Zn (1.65) than those of Ti (1.54), whereas Mg (1.31). This mechanistic divergence in charge transfer- extending beyond the shared HCP base structures of Mg, Zn, and Al- collectively governs the formation energy trends: Al and Zn doping decrease α-Ti formation energy through favorable electron acceptance, whereas Mg alloying elevates formation energy due to energetically unfavorable electron donation from Mg to Ti. These results can provide an important guidance for the experimental synthesis of stable α-Ti alloys for dental applications.

Furthermore, when two V atoms substitute for Ti atoms in α-Ti₅₈Al₆, the formed α-Ti₅₆Al₆V₂ system has a relatively small impact on the DOS compared to the α-Ti₅₈Al₆ system, because of the similar valence electron distributions of the transition metals Ti and V, as shown in Fig. 3e‒h. Unlike Al, the d-electrons of V contribute significantly to the states near the Fermi level, overlapping extensively with the d-electrons of Ti. This indicates that the interaction between V and Ti is dominated primarily by their d-electrons. In contrast, the overlap between the DOSs of Al and V is minimal, suggesting weak interactions between them. Additionally, the introduction of V causes the s-orbital of Al to shift toward the Fermi level, resulting in an increase in the energy of the bonding orbitals and corresponding a weakening of the bond strength. The charge density difference analysis revealed that the charge distribution of V changed in manner similar to that of Ti, as shown in Fig. 4c. On average, Al gains approximately one electron through its interactions with both Ti and V. Correspondingly, the total Bader charge of V increases by 1.13 electrons. Despite this charge transfer similar to Al, inherent preference of V for maintaining the BCC ground-state structure did not promote further energy reduction in the system. These results indicate that the bonding orbitals between V and Ti exhibit higher energy levels, and that the interactions between Al and V are relatively weak. These factors reveal that doping V is unfavorable for lowering the overall energy of the system.

In addition, the electronic structure of β-Ti₅₆Al₆V₂ was determined, as shown in Fig. 4d. The total and partial DOSs for α-Ti₅₆Al₆V₂ and β-Ti₅₆Al₆V₂ are highly similar, indicating that the two phases have nearly identical electronic structures and properties. This suggests that the transformation from the α to the β phase in Ti–Al–V alloys is driven primarily by structural changes rather than significant electronic rearrangements. However, a small difference is observed in the deep energy levels of the Al s-orbital in β-Ti₅₆Al₆V₂, which exhibit a higher peak intensity than that of α-Ti₅₆Al₆V₂. This enhanced contribution from the Al s-orbital likely stabilizes the β-phase by slightly lowering its overall energy. Bader charge analysis reveals similar charge transfer behavior in both phases, with Al gaining an average of 1.05 electrons and V remaining almost unchanged by an average of 1.03 electrons. The stronger charge transfer of Al in the β-phase than in the α-phase may further lower its energy, contributing to the development of dual-phase (α + β) Ti–Al–V alloys.

We further conducted a detailed analysis of the changes in the bond strengths using Crystal Orbital Hamilton Population (COHP) for the α-Ti₅₈Al₆, α-Ti₅₆Al₆V₂, and β-Ti₅₆Al₆V₂ alloys, as shown in Table S3. Generally, a more negative integrated COHP value indicates a stronger bond strength. With the addition of V, the primary Ti–Ti bonds in both the α-Ti₅₆Al₆V₂ and β-Ti₅₆Al₆V₂ alloys weakened slightly compared to those in the α-Ti₅₈Al₆ alloy. In addition, the bond strength of V–Al is notably lower than the other bonds in the system, which leads to changes in the overall stability and structural characteristics of the alloys. For the α-Ti₅₆Al₆V₂ and β-Ti₅₆Al₆V₂ alloys, the bond strengths in are quite similar which may contribute to a marginal E_form change in the systems.

Elastic properties

Elastic modulus, poisson’s ratio and hardness

First, to determine the reliability of the calculated values of the elastic constants C11​, C12​, C13​, C33​, and C44​, the bulk modulus (B), shear modulus (G), and Young’s modulus (E) for α-Ti₆₄ are compared with the experimental results [49, 50], as shown in Fig. 5a; Table 1. The calculated values in this study are consistent with the reported values, indicating the effectiveness and accuracy of our method in predicting the elastic properties of alloys. As shown in Fig. 5b, all three α-type alloy materials exhibit similar bulk moduli (B), indicating comparable resistance to uniform compression. Ti₅₈Al₆ achieves the highest values of Young’s modulus (E), shear modulus (G), and hardness (H). These findings indicate that Al doping enhances the alloy’s overall mechanical strength and stiffness, aligning with experimental observations [51], thereby guiding experimental strategies for property control through aluminum content modulation. α-Ti56Al₆V₂ has intermediate values for E, G, B​/G ratio, H [52], and Poisson’s ratio (ν). These properties place α-Ti₅₆Al₆V₂ between α-Ti₆₄ and α-Ti₅₈Al₆, indicating a balanced combination of ductility and strength.

Fig. 5.

a Comparison of the calculated C11​, C12​, C13​, C33​, C44, B, E, and G values of α-Ti₆₄ with experiment results. b B, E, and G values of α-Ti₆₄, α-Ti₅₈Al₆, α-Ti₅₆Al₆V₂, and β-Ti₅₆Al₆V₂

Table 1.

B, E, G, B/G, ν,Cp, H, and Debye temperatures of α-Ti₆₄, α-Ti₅₈Al₆, α-Ti₅₆Al₆V₂, and β-Ti₅₆Al₆V₂

Compounds	B (GPa)	E (GPa)	G (GPa)	B/G	ν	Cp (GPa)	H (GPa)	Debye temperature (K)
α-Ti64	114.6	118.6	44.7	2.57	0.33	56.3	3.10	402.0
α-Ti58Al6	114.3	134.7	51.7	2.21	0.30	30.9	4.90	439.4
α-Ti56Al6V2	115.4	128.3	48.8	2.36	0.32	36.6	4.08	426.7
β-Ti56Al6V2	115.5	128.7	49.0	2.36	0.31	39.3	4.11	427.4

For β-Ti₅₆Al₆V₂, our results indicate that the elastic moduli are very similar to those of α-Ti₅₆Al₆V₂. This suggests that at low concentrations, V doping does not significantly influence the elastic moduli of the α and β phases, which is consistent with experimental findings [53]. As a result, the average Young’s modulus of Ti₅₆Al₆V₂ with the α + β phase aligns well with the experimental data, showing a slight increase compared with that of commercially pure titanium [54]. Furthermore, studies suggest that as the V concentration continues to increase, the elastic modulus of the β-phase further decreases [33].

According to Pugh’s criterion [55], materials are brittle if B/G < 1.75 and ductile if B​/G​>1.75. Poisson’s ratio (> 0.26) further confirmed its ductility, which was associated with the B​/G change trend. Besides, when the Cauchy pressure (Cp) [56] is greater than 0, it indicates that metallic bonding dominates as in the electron analysis above, and the material tends to exhibit ductile behavior. All four materials exhibited good ductility, with α-Ti₆₄ achieving the highest ductility. The Debye temperature reflects the strength of the interatomic forces. Materials with higher Debye temperatures have lower thermal expansion coefficients because of constrained atomic movement. Higher Debye temperatures lead to greater resistance to deformation and higher Young’s modulus. Ti₅₈Al₆ has the highest Debye temperature (439.4 K), Ti₆₄ has the lowest (402 K), and α-Ti₅₆Al₆V₂ and β-Ti₅₆Al₆V₂ have similar values. These results are consistent with trends in the E_form and elastic properties. Finally, all four materials satisfied the elastic stability criteria [57], ensuring their mechanical stability under elastic deformation.

Elastic anisotropy

The Elastic Anisotropy Index (A_u) [58] quantifies the directional dependence of a material’s stiffness, and is calculated using the following formula:

15

α-Ti₆₄ exhibited significant anisotropy (A_u = 0.3), indicating nonuniform stiffness across different directions. For Ti₅₈Al₆, the addition of Al significantly reduces the anisotropy to A_u = 0.03, demonstrating minimal directional stiffness variation. For α-Ti₅₆Al₆V₂ and β-Ti₅₆Al₆V₂, the incorporation of V slightly increased the anisotropy to A_u = 0.06 and A_u = 0.07 respectively, but directional stiffness variation remained low. Materials with high anisotropy are more prone to develop microcracks because the directional dependence can create weak points in the material, leading to crack initiation and propagation under stress. The introduction of Al and V induces a structural transformation from a hexagonal to a triclinic crystal system, which is associated with reduced anisotropy. This makes α + β Ti₅₆Al₆V₂ more reliable and stable, particularly in terms of tensile and yield strengths, for practical applications.

Visualization using 3D diagrams

We employed ElasticPOST software to generate three-dimensional diagrams that visually compare anisotropic behavior across different doping alloys [59]. These diagrams highlight the directional differences in stiffness for α-Ti₆₄ and its doped alloys, providing a clear and intuitive representation of the mechanical anisotropy as shown in Fig. 6; Table 2. The mechanical properties of α-Ti₆₄ exhibited a wide range of stiffness values, indicating its potential for varying the performance under different conditions. In contrast, Ti₅₈Al₆ exhibited more uniform mechanical properties, suggesting consistent performance. The α + β Ti₅₆Al₆V₂ alloy also achieved good uniformity, albeit slightly lower than that of α-Ti₅₈Al₆, making it a reliable option with balanced properties.

Fig. 6.

Three-dimensional images of B (a), E (b), G (c), and v (d) for α-Ti₆₄, α-Ti₅₈Al₆, α-Ti₅₆Al₆V₂, and β-Ti₅₆Al₆V₂

Table 2.

Variations in the elastic moduli B, E, G, ν, of α-Ti₆₄, α-Ti₅₈Al₆, α-Ti₅₆Al₆V₂, and β-Ti₅₆Al₆V₂

Compounds	B (GPa)	E (GPa)	G (GPa)	v
	Min–Max (Diff)	Min–Max (Diff)	Min–Max (Diff)	Min–Max (Diff)
α-Ti64	110.132–123.416 (13.284)	106.539–180.633 (74.094)	36.221–58.064 (21.843)	0.155–0.515 (0.360)
α-Ti58Al6	109.746–119.174 (9.428)	126.220–150.407 (24.187)	47.741–56.922 (9.181)	0.243–0.361 (0.118)
α-Ti56Al6V2	112.058–122.044 (9.986)	112.959–150.617 (37.658)	41.908–58.088 (16.180)	0.204–0.418 (0.214)
β-Ti56Al6V2	108.859–122.600 (13.741)	119.740–154.516 (34.776)	42.727–55.616 (12.889)	0.217–0.415 (0.198)

Conclusion

In summary, aluminum (Al) stabilized the α-phase in titanium alloys by reducing the formation energy through significant charge transfer and deep-level s/p orbital hybridization with titanium (Ti). In contrast, vanadium (V) destabilizes the α-phase and promotes the formation of the β-phase owing to its inherent body-centered cubic (BCC) structure preference, revealing strong d-electron interactions with Ti near the Fermi level as a transition metal. The Al and V preferences resulted in comparable energies for the α and β phases. This energy balance promoted the transition from α-Ti to β-Ti alloys. The α + β Ti₅₆Al₆V₂ alloy exhibited similar electronic structures in both the α and β phases, leading to comparable elastic properties that were intermediate between α-Ti₆₄ and α- Ti₅₈Al₆. This balance of the elastic modulus, combined with its better tensile strength and yield strength, good ductility and isotropy, made it highly suitable for dental implant applications. Leveraging these mechanistic insights, we suggest investigating low-toxicity alternatives like Mg or Zn to improve the stability of Ti alloys, alongside β-phase stabilizers such as Nb, Mo, or Ta, to develop next-generation dental implant materials with superior biocompatibility and performance.

Abbreviations

cp-Ti

Commercially pure titanium

HCP

Hexagonal close-packed

BCC

Body-centered cubic

SQS

Special Quasi-Random Structure

DFT

Density functional theory

VASP

Vienna Ab initio Simulation Package

PAW

Projector-augmented wave

COHP

Crystal Orbital Hamilton Population

DOS

Density of state

Authors’ contributions

Conceptualization and Methodology: Yang Yang, Jiu-Ning Wang, Xue-Cheng Liu. Data Curation and Formal Analysis: Li-Xia Hu, Jiu-Ning Wang. Investigation and Resources: Xue-Cheng Liu, Wei Xu. Original draft preparation: Yang Yang, Qasim. Writing, Reviewing, and Editing: Jiu-Ning Wang, Qasim, Wei Xu. Supervision and Project Administration: Wei Xu.

Funding

This research was Supported by Science and Technology Research Program of Chongqing Municipal Education Commission of China (Grant No. KJQN202400825).

Data availability

The data supporting this study’s findings are available from the corresponding author upon reasonable request.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.