First principles computation of composition dependent elastic constants of omega in titanium alloys: implications on mechanical behavior

For decades the poor mechanical properties of Ti alloys were attributed to the intrinsic brittleness of the hexagonal ω-phase that has fewer than 5-independent slip systems. We contradict this conventional wisdom by coupling first-principles and cluster expansion calculations with experiments. We show that the elastic properties of the ω-phase can be systematically varied as a function of its composition to enhance both the ductility and strength of the Ti-alloy. Studies with five prototypical β-stabilizer solutes (Nb, Ta, V, Mo, and W) show that increasing β-stabilizer concentration destabilizes the ω-phase, in agreement with experiments. The Young’s modulus of ω-phase also decreased at larger concentration of β-stabilizers. Within the region of ω-phase stability, addition of Nb, Ta, and V (Group-V elements) decreased Young’s modulus more steeply compared to Mo and W (Group-VI elements) additions. The higher values of Young’s modulus of Ti–W and Ti–Mo binaries is related to the stronger stabilization of ω-phase due to the higher number of valence electrons. Density of states (DOS) calculations also revealed a stronger covalent bonding in the ω-phase compared to a metallic bonding in β-phase, and indicate that alloying is a promising route to enhance the ω-phase’s ductility. Overall, the mechanical properties of ω-phase predicted by our calculations agree well with the available experiments. Importantly, our study reveals that ω precipitates are not intrinsically embrittling and detrimental, and that we can create Ti-alloys with both good ductility and strength by tailoring ω precipitates' composition instead of completely eliminating them.

That the presence of the w-phase within the β (body centered cubic or BCC) matrix causes embrittlement of Ti alloys is considered a fait accompli. However, what is not often recognized is that the type and nature of the w precipitates, plays a dominant role in determining the overall deformation behavior of the alloy. Metastable β titanium alloys, quenched from the high temperature single β phase-field, often exhibit athermal or quenched-in w precipitates that inherit the composition of the parent β matrix 1,2 . These precipitates often co-exist with α, αʹ, and α″ phases. These athermal-ω precipitates, within the β matrix, are typically not detrimental to plasticity if their volume fraction is not too high, and results in alloys that are ductile and often exhibit good strain-hardenability. Annealing such a quenched system, however, easily causes a compositional partitioning between the β and ω phases 3 . The resulting precipitates are now called isothermal ω and they correlate with a larger yield strength and a severe loss of ductility. The underlying causes of such dramatic variations in properties are not well understood.
The attractiveness of metastable β-Ti alloys is in its versatility and the wide range of mechanical properties that can be adjusted via alloying, and thermo-mechanical treatments 4,5 . In addition to superelastic and shape memory effects, Ti alloys can exhibit high ductility, transformation induced plasticity (TRIP), and twining induced plasticity (TWIP) effects, which can be achieved through the coexistence and reversable transformation between β and α″ phases 6,7 . Also, low elastic modulus values can be obtained in Ti alloys through tuning β-stabilizers concentration to near the lower limit of β-phase stability. Values corresponding to an average number of valence electrons per atom (e/a) of ~ 4.25 creates a Ti alloy with Young's modulus close to human bone 8,9 . However, Ti

Simulation methodology
The Vienna Ab Initio Simulation Package (VASP) 26,27 was used to perform the density functional theory (DFT) calculations. The Kohn-Sham equations of DFT were solved with projector-augmented-wave (PAW) potentials and wave functions 26 . Electron-electron exchange and correlation effects were described by the generalized gradient approximation (GGA) with the functional form proposed by Perdew-Burke-Ernzerhof (PBE) 27 . Careful convergence calculations yielded a plane-wave kinetic energy cutoff of 500 eV. A convergence accuracy of 10 -5 eV was used for the self-consistent electronic minimization. Monkhorst-Pack k-point meshes with a spacing of 15 × 15 × 15 Å −1 were used in conjunction with Methfessel-Paxton integration scheme using a smearing width of 0.2 eV. Applying these settings give energy accuracy of 1 me V/atom for our calculations.
A crucial prerequisite for first principles calculations is a reliable configurational model where the distribution of solute atoms is near the most stable configuration. To achieve this goal for the binary Ti-X (W, Mo, V, Ta, and Nb) alloys, the Cluster Expansion (CE) method implemented in a UNiversal CLuster Expansion code (UNCLE) was used to systematically search for the optimal random configurations that give thermodynamically stable alloys at T = 0 K 25,28 . The CE calculations were performed for the solid solution β-phase of our binary Ti-X alloys across a wide composition range using a bcc supercell with 16 atomic sites. To establish the cluster interaction parameters, the DFT total energies were calculated for 167, 201, 152, 160, 190 different training-set structures for Ti-W, Ti-Mo, Ti-V, Ti-Ta, and Ti-Nb respectively. Our DFT simulations fully relaxed the atomic positions, supercell volume, and supercell shape to obtain suitable equilibrium configurations.
The minimum energy pathways (MEPs) for β → ω transformation was calculated using the generalized solid state nudged elastic band (SSNEB) as implemented in the VTST package 29,30 . The atomic positions and cell degree of freedom (DOF) were relaxed simultaneously. A linear interpolation of atomic positions and cell vectors was used to guess the initial β → ω transformation path. Five intermediate images were used and the end point β and ω structures were fully relaxed with respect to cell parameters and atomic positions. Also, the forces were optimized below 10 -3 eV Å −1 on each image along the MEPs. Note that a direct full optimization of the volume and internal degree of freedom for the end points structures with low content of β-stabilizer caused a large distortion to the lattice and its symmetry, which created a more stable low temperature phase or other metastable phases. Therefore, the relaxation for structures with low β-stabilizer content was performed over two stages. The first stage minimized the volume and internal degree of freedom with a force threshold of 0.2 eV/Å while the second stage used the same minimization scheme but with a lower force threshold of 0.02 eV/Å.
The six elastic constants of the single crystal hexagonal ω-phase (c 11, c 12 , c 13 , c 33 , c 44 , and c 66 ) were calculated by applying ± 0.01 strains and measuring the response stress tensors. The polycrystalline elastic moduli were calculated using the Voigt-Reuss-Hill approximation 31 . The isotropic polycrystalline elastic moduli can be taken as an average of the single-crystal elastic constants. The theoretical upper bounds of isotropic polycrystalline bulk modulus B and shear modulus G are given by Voigt in terms of single crystal elastic constants c ij as 32 : And the lower bounds are given by Reuss as follows 32 : Here s ij are the elastic compliance constants, which are the components of the inverse elastic constant matrix. The predicted isotropic polycrystalline bulk modulus and shear modulus can be obtained as B = (B R + B V )/2 and G = (G R + G V )/2, respectively. Young's modulus and Poisson's ratio are Y = 9BG/(3B + G) and υ = (3B/2 − G)/(3B + G) , respectively 33 .

Experimental procedure
Ti-19 at.%V and Ti-6.5 at.% Mo alloys were prepared using an electric arc furnace. ~ 50-g ingots were prepared from pure Ti, V, and Mo pellets. The Ti-V ingot was flipped and remelted four times to ensure homogenous melting. The sample was cut to small specimens and β-solutionized at 900 °C for 30 min followed by water quenching. The β-solutionized samples were then annealed at 300 °C for 30 min to precipitate well developed ω-phase particles and stimulate compositional partitioning between β and ω phases. Similarly, the Ti-Mo sample was also β-solutionized at 900 °C for 30 min followed by water quenching. Unlike the Ti-V sample, the samples for Ti-Mo were annealed at 475 °C for 48 h. Transmission electron microscopy (TEM, FEI Tecnai F20-FEG TEM operating at 200 kV) was used for microstructure characterization. Compositional analysis at the atomic scale was conducted using 3D atom probe tomography (APT) in a local electrode atom probe (LEAP) 3000 × HR system from Cameca Inc. APT tips and TEM samples were prepared using Focused Ion Beam (FIB) technique with FEI Nova 200 NanoLab dual beam system. EDM machine was used to cut the tensile test dog bone shaped samples with a gauge length of 3 mm, width of 1 mm and thickness of 0.4-0.6 mm and used for mechanical testing. A (1) [c 11 + c 22 + c 33 + 2(c 12 + c 13 + c 23 )] (2) G V =

Results and discussion
Cluster expansion calculations and configurational models. The cluster expansion (CE) results are represented by a ground state phase diagram whose X and Y axes respectively represent composition and free energy as shown in Fig. 1 36 . This discrepancy may be attributed to using of 6 or fewer atoms per unit cell used by  www.nature.com/scientificreports/ Ravi et al. 35 for the ground state search, which reduces the number of atomic configurations that can be included within the training-set, hence affecting its output. Tables 1, 2, 3, 4 and 5 list the stable structures, symmetries, compositions, and schematic representations used for our binary Ti-X (X = W, Mo, V, Ta, and Nb) systems. Here we assume that the ordered structures are representative models for the β-phase. However, we do not imply that Ti forms intermetallic compounds with these β-stabilizing elements. For a specific composition there are many structures lying on the vertical line of the convex hull with small energy difference between them. For example, for Ti-12.5 at.% Mo, the structure closest to the convex hull has Im-3m symmetry as shown in Table 1, and the structure just vertically over it with the same composition has a symmetry of Pm. The energy difference between these two structures is 1 meV, which is sufficiently low, so both these structures are good configurational models for Ti-12.5 at.% Mo alloys. The computational equivalent of a real binary solid solution can be considered to be different arrangements of degenerate domains of different ordered structures. However, in this study we consistently use structures closest to the convex hull, and these structures may be considered as the best energetic models to represent our binary systems. β → ω transformation. After identifying the various ground state structures for the binary Ti-X β alloys using CE calculations, we investigated the effect of W, Mo, V, Ta, and Nb on the β → ω transformation, which is predicted to occur via an ordered displacement wave initiated by the softening of the q = (2/3,2/3,2/3) wave vector. This corresponds to the displacement of a pair of (111) atomic planes of β relative to each other causing their collapse to form a (0002) plane of ω but leaving the next plane unchanged. This process continues by collapsing the next pairs consecutively. There are four independent <111> directions within the bcc lattice triggering this transformation. However, our calculations chose a single <111> variant to simulate the transformation due to the difficulty in creating a supercell possessing the orientation and periodicity of all the four variants. This should, however, not affect the overall energetics of β → ω transformation.
The β and ω phases share a common trigonal symmetry respectively along their [111] and [0001] directions. This allows the creation of a common bcc supercell oriented along [111] direction to facilitate the manual displacement of (111) planes for the transformation simulation. The relaxed supercell model with this orientation for both β and ω phases are illustrated in Fig. 2, which also shows the collapse of {111} β planes during the β → ω transformation. The − → a , − → b , and − → c lattice vectors of β-phase supercell are oriented along [011], [211] and [0 11 ] respectively. The lattice constants for the relaxed β-phase with this orientation are 4.59, 7.95, and 2.82 Å for a, b, and c respectively. The (111) plane collapses along the − → c lattice vector during transformation, and the magnitude of the complete collapse is ± a β √ 3/12 that is half the spacing between two consecutive (111) planes. The atomic positions are (0, 0, 0), (2/3,1/3,1/2 + ȥ), and (2/3, 1/3, 1/2 -ȥ), where ȥ for bcc (β) and hexagonal (ω) are respectively 1/6 and 0. The next step ensures that each ground state structure obtained from CE is oriented as in Fig. 2. Lastly, the ground state structures were increased to 48 atom supercells and minimized by varying cell volume and ionic positions. Figure 3 shows the images along the MEP pathway β → ω transformation for pure Ti. There is no energy barrier for this transformation, which is in agreement with prior theoretical and experimental results predicting the higher stability of ω-phase for pure Ti at T = 0 K 22,37,38 . It should be noted that the absence of an energy barrier here does not mean that this transformation should occur because its driving force is high. The absence of vibrational entropy at T = 0 K may stabilize the β-phase. In addition, other martensitic phases such as αʹ and α″ may coexist with ω-phase which impose elastic constraints on β → ω transformation and raise the energy barrier. However, the MEPs in this study help us understand the β → ω transformation mechanism.
The dynamic instability of β-phase for group IV transition metals such as Ti, Zr, and Hf at T = 0 is attributed to the presence of imaginary frequencies in their phonon spectra corresponding to the lattice instability against specific atomic movement that causes phase transformation to more stable phases 39 . The transition is also related to an elastic instability that is confirmed by negative values of the tetragonal shear modulus Cʹ = (C 11 -C 12 ) 8,40 . However, the stability of bcc β-Ti increases through transfer from the s and p electrons, and this charge transfer increases the number of d-electrons by applying pressure 41 , or by alloying with transition metals such as W, Mo, V, Ta, and Nb 8,42 . A valence electrons per atom ratio (e/a) of 4.24 was also proposed as a critical number to stabilize the β-phase 8 . Figure 4 shows the MEP for the β → ω transformation in various Ti-X binary systems (X = W, Mo, V, Ta, Nb) and are compared with pure Ti. For all systems it can be seen that the stability of β-phase increases with increasing the solute content, which means that the omega transformation is progressively suppressed, and is in agreement with other theoretical and experimental results on the β stabilization by W, Mo, V, Ta, and Nb 17,42 . However, the stabilization rate of β-phase with increasing solute content is not the same for these elements. The stabilization rate of W and Mo is higher than V, Ta, and Nb. For example, 12.5 at.% Mo is required to suppress ω-phase formation in Ti-Mo system, while a concentration of more than 25 at.% Nb is required to suppress the transformation in Ti-Nb system. The higher stabilization rate of β-phase for W and Mo can be attributed to the high stability of their bcc structures in the pure form as suggested in a previous study 17 . Also, W and Mo belong to group VIa family and have more valence electrons, which increase the e/a ratio more than group Va elements V, Ta, and Nb. Clearly, the electronic structure effects, especially bonding, between Ti and β-stabilizing elements have implications for the phase stability in Ti alloys as will be shown later in this work.
Calculations of ω-phase elastic constants. Table 6 shows the calculated elastic constants of ω-phase for pure Ti and Ti-X binary alloys (X = W, Mo, V, Ta, and Nb). These single crystal elastic constants for pure Ti agrees well with previous calculations 43, 44 and finite temperature experiments 21 . Also, the calculated polycrystalline bulk modulus B, shear modulus G, and Young's modulus E of pure Ti ω-phase agree well with literature values 21,43,44 . Unlike the β-phase in pure Ti whose c 11 -c 12 < 0 indicating instability 45 13 , and c 44 > 0. This confirms the possibility of retaining metastable ω-phase at low temperatures.
The elastic constants (c 11, c 12 , c 13 , and c 33 ) of binary Ti-X alloys show no clear trend with increasing β-stabilizer content. However, both c 44 and c 66 decreases with increasing β-stabilizer content. It should be noted that in addition to calculating the elastic constants of the concentrations described in β → ω transformation section, we included the 25 at.% concentration for W and Mo, and 8.33 at.%, 16.6 at.% for V, Ta, and Nb using the same approach of getting the ground state structures obtained from the CE results. The study of these additional concentrations is for consistency and the completeness of the current work.
The c 11 and c 12 elastic constants for Ti-W and Ti-Mo systems increase while c 13 decreases with increasing concentration. However, this trend could not be observed for Ti-V, Ti-Ta, and Ti-Nb systems. Interestingly, for all systems studied here, the c 44 elastic constant decreases with increasing concentration indicating the mechanical stability of ω-phase decreases with increasing alloying content, which agrees with our predicted MEPs in Fig. 4. Also, c 66 first decreases then increases with concentration for Ti-W, Ti-Mo, and Ti-V systems, while it monotonically decreases with increasing concentration for Ti-Ta, and Ti-Nb systems. Figure 5 shows c 44 elastic constants of binary Ti-X alloy studied here. The sharp decrease in c 44 values for Ti-Mo, Ti-W, and Ti-V with increasing alloy concentration to some specific point is related to the instability of ω-phase at that concentration. We can relate this response to MEPs in Fig. 4 where the energy of ω-phase is higher than β-phase energy. The same is true for Ti-Ta and Ti-Nb systems. Also, this elastic instability sometimes manifests itself with negative values of elastic constants such as in the c 44 of Ti-Mo and Ti-V systems. Therefore, realistic elastic constants values should be considered within the concentration range of ω-phase stability, i.e. less than 12.5 at.% for (Mo, W), and 25 at% or less for (V, Ta, and Nb). This was confirmed experimentally before 38,46 .
To the best of our knowledge the only available calculated elastic constants for ω-phase, to compare our results with, are for Ti-25 at.% Nb reported by Lazer et al. 4 and Sun et al. 22 , and these values are listed in Table 6. Both studies used different structural models and some differences between calculated results for these two studies are expected. Our structural model for ω-phase with 25 at.%Nb is based on the cluster expansion, which is different from the previously used structural models, so some difference between our results and previously reported data is also expected. Figure 6 summarizes polycrystalline Young's moduli we calculated for Ti-X alloys. All β-stabilizers explored here decrease the Young's moduli of ω-phase with increasing concentration. However, if we consider the stable concentration range for ω-phase, which is below 12.5 at.% for W and Mo and 25 at.% for V, Ta, and Nb, we note that at 12.5 at.% concentration of W and Mo, the Young's modulus decreases by 17% and 5% respectively. However, a decrease of ~ 28% can be observed for V, Ta, and Nb with 25 at.% concentrations. Also, the shear modulus G decreases by ~ 15% for W and Mo, while a significant 32% decrease is seen for V, Ta, and Mo. The present calculations confirm that ω-phase has higher elastic moduli compared to metastable β and α″ phases 8,47 .
The composition of as-quenched (athermal) ω-phase is different from that of isothermal ω. The isothermal ω is obtained after aging below β-phase solvus and results from elemental partitioning of β-stabilizers during aging heat treatment 15,48 . Although it has not been shown before that changing the chemical composition of ω-phase directly affects its mechanical properties, the compositional partitioning between β and ω phases is known to    www.nature.com/scientificreports/ affect the deformation mode in Ti-Mo alloys 13 . Also, another study found that the high difference between the shear modulus of β and ω phases can suppress the TRIP and TWIP effects 12 . In fact, ω-phase with its large shear modulus obstructs the {1 1 0} β < 110 > β shuffle movement required to facilitate the β → α̋ martensitic transformation, which limits the TRIP effect. Our calculations suggest that the as quenched (athermal) ω-phase with high concentration of β-stabilizers can better enhance the elastic properties of Ti alloys than the solute-depleted ω-phase whose shear modulus is higher. Therefore, we argue that a thermomechanical treatment procedure that www.nature.com/scientificreports/ limits partitioning of β-stabilizers within the ω-phase should be considered when designing a high strength Ti alloys with good ductility. Such an insight has not been envisaged before because the ω-phase was considered to be intrinsically brittle with no recourse. Two prototype/model alloys were selected for complementary experimental investigations. These two are simple binary alloys involving two of the most commonly used β stabilizers in Ti alloys, V and Mo. The results are summarized in Figs. 7 and 8. While Fig. 7 shows the experimental results from the Ti-20 wt.% V alloy, Fig. 8 focuses on Ti-12 wt.% Mo alloy. Both these alloys form quenched-in athermal ω precipitates as well as isothermal ω precipitates.
The binary model alloy, Ti-19 at.%V (Ti-20 wt.%) was subjected to a β-solutionizing treatment, followed by water-quenching, and subsequent aging at 300 °C/30 min, below the ω-solvus temperature. The dark-field TEM (DFTEM) images of ω precipitates and tensile test results for the β-solutionized and annealed conditions are shown in Fig. 7a. The dark-field images for both conditions were recorded along the [110] β zone axis, and the corresponding selected-area electron diffraction (SAED) patterns are shown as insets. Highly refined scale ω precipitates (~ 5-10 nm) were observed in the β-solutionized microstructure, and the size scale of these ω precipitates remained nearly the same (~ 5-10 nm) after annealing at 300 °C/30 min. The presence of intensity maxima at 1/3 and 2/3   www.nature.com/scientificreports/ Figure 7b shows the atom probe tomography (APT) results from the β-solutionized condition. The binomial distribution plots were obtained by measuring every 100 atom block within the 3D atom probe reconstruction 49 . The V ion distribution plot (red dotted line) clearly follows the expected distribution from a random solid solution (black dotted line). This means that the V partitioning between β and ω phases was not perceptible within the limitations of this analysis technique using binomial distribution. This indicates the formation of athermal congruent ω precipitates within the β matrix. Therefore, both phases have the same composition in www.nature.com/scientificreports/ the solutionized condition, i.e. ω phase has a composition of Ti-19 at.% V. On the other hand, a significant partitioning of V between ω and β phases could be observed in the 300 °C/30 min annealed specimen as shown in Fig. 7c. A depletion of V is observed for the isothermal ω precipitates (~ 5 at.% V), while the surrounding β matrix contains ~ 24 at.% V. These compositions were determined from APT reconstructions, based on V profiles calculated across isoconcentration surfaces (or isosurfaces) used for delineating the ω/β interfaces. The engineering tensile stress vs engineering strain curves for both β-solutionized, and β-solutionized and 300 °C/30 min annealed conditions are shown in Fig. 7a. A significant strain hardening with an ultimate tensile strength (UTS) of ~ 800 MPa, and a ductility of 20% could be observed for the β-solutionized condition (red curve), while the β-solutionized and annealed specimen showed much higher UTS of ~ 1000 MPa with a limited ductility of < 2% (blue curve). This substantial change in the mechanical properties can be attributed to the dependence of the ω-phase mechanical properties on its β-stabilizer content. The trends in the variation of DFT-computed Young's modulus as a function of V content, can be inferred from Fig. 6. Based on these trends, the computed Young's modulus for the Ti-19 at.% V and Ti-5 at.% V can be approximated to be 125 GPa and 150 GPa respectively. This large increase in the modulus of the isothermal ω precipitates (due to the rejection of V) as compared to the quenched-in athermal congruent ω precipitates, leads to the higher yield strength in case of isothermally annealed condition. This change is also accompanied by the loss of deformability of the isothermal ω precipitates, leading to loss of ductility, as experimentally observed (Fig. 7a).
Similar experimental results from the Ti-12 wt% Mo alloy, equivalent to approximately Ti-6.5 at.% Mo, have been summarized in Fig. 8. The dark-field TEM (DFTEM) images of ω precipitates and tensile test results for the β-solutionized and annealed conditions are shown in Fig. 8a. The dark-field images for both conditions were recorded along the [110] β zone axis, and the corresponding selected-area electron diffraction (SAED) patterns are shown as insets. Very fine scale ω precipitates (~ 5-10 nm) were observed in the β-solutionized microstructure, Table 6. The elastic constants of ω-phase in Ti-X binary systems (X = W, Mo, V, Ta, and Nb) at 0 K obtained using first-principles calculated. The single crystal elastic constants c ij , bulk modulus B, shear modulus G, and Young's modulus E are all in GPa.  Ti  193.7 77  51  248  54  58  113  62  159   Ti  192.3 77  54.6 248  54.3 57  115  61  155  44 ab initio   Ti  194  81  54  245  54  56  118  63  158  43 ab initio   Ti  179  90  60  228  70  44  111  60  152  21 Expt   Ti-8.33 W  202  58  100  212  42  41 119. 2 Figure 8b shows the atom probe tomography (APT) results of the β-solutionized alloy. The binomial distribution plots were obtained by measuring every 100 atom blocks within the 3D atom probe construction 49 . Mo ion distribution plot (blue line) clearly follows the expected distribution from a random solid solution (red line). This means that the Mo partitioning between β and ω phases was not perceptible within the limitations of this analysis technique using binomial distribution. This indicates the congruent formation of ω precipitates within the β matrix. In other words, both phases have the same composition in the solutionized condition. On the other hand, a significant partitioning of Mo between ω/β could be observed in the annealed specimen as shown in Fig. 8c. A depletion of Mo is observed for the coarse ω precipitates (~ 1.5 at.% Mo), while a rich Mo content of ~ 16 at.% can be observed for the surrounding β matrix. These compositions were obtained by measuring the 1D composition profile across the ω/β interface. www.nature.com/scientificreports/ The engineering tensile stress vs engineering strain curves for both β-solutionized, and β-solutionized and annealed conditions are shown in Fig. 8a. A significant strain hardening with an ultimate tensile strength (UTS) of ~ 559 MPa, and a ductility of 35% could be observed for the β-solutionized condition (red curve), while the β-solutionized and annealed specimen showed much higher UTS of ~ 1070 MPa with a limited ductility of < 5% (blue curve). Similar to the previously discussed case of the Ti-19 at.% V alloy, in the case of the Ti-6 at.%Mo alloy, this significant change in mechanical properties can be attributed to changes in the Young's modulus of the ω-phase as a function of its β-stabilizer content. Figure 6 shows a that with increasing Mo content the Young's modulus of ω decreases. While the experimental values of compositions for the athermal congruent ω (8.33 at.% Mo) and the annealed isothermal ω (1.5 at.% Mo) may not exactly correspond to the discrete ω compositions shown in Fig. 6, it is important to note that the range of Mo compositions evaluated by the DFT computations, covers the entire experimental range of compositions. Furthermore, other additional factors such as the change Figure 6. Calculated Young's moduli of the ω-phase Ti-X binary systems (X = W, Mo, V, Ta, and Nb). Previous available calculated Young's moduli values for β-phase are also included for comparison 8,18,19 . www.nature.com/scientificreports/ in the deformation mode within the β matrix, from twinning to slip, and the size of ω-phase precipitates can also influence the mechanical response of the condition containing the isothermal ω precipitates 13 . A recent study by Lai et al. 16 revealed severe loss of ductility in the Ti-12 wt.% Mo alloy upon short term annealing for 10 min. at 400 °C. While this short aging treatment did not result in any discernable change in the volume fraction, size, and inter-particle spacing of ω precipitates, a significant Mo rejection from the isothermal ω precipitates was reported based on APT results 21  www.nature.com/scientificreports/ state at the fermi level coupled with the opening of a pseudogap near E F indicates a strong covalent bonding between atoms and implies a higher phase stability in metallic system 50 . The β-stabilizers elements considered in this study can be divided into two groups. The VIa group elements W and Mo show a high stabilization rate at increasing concentrations. On the contrary, Va family group elements V, Ta, and Nb has a lower stabilization rate at increasing composition. Therefore, for DOS calculations, we chose Mo, and Nb as representatives of these two groups. Thus, the DOS calculations were performed for the β and ω phases of Ti-Nb and Ti-Mo systems at their equilibrium geometries. The DOS calculation results are shown in Fig. 9.
The energy levels in Fig. 9 range from − 5 to 2 eV, and the zero-energy denoted by the vertical dashed line is the Fermi level. The DOS shapes for β and ω phases in Ti-Mo and Ti-Nb are similar. This confirms the previous finding that DOS spectrum shape depends on the crystal structure rather than the elements type 8,51 . A clear pseudogap can be seen between the low-energy bonding and high-energy anti-bonding regions for β-phase of Ti-Mo and Ti-Nb binary systems as shown in Fig. 9a, c respectively. The high value of the DOS at the Fermi level, which is located in the bonding region, especially in case of lower concentrations of Mo and Nb, indicates a metallic bonding between atoms and a reduction in β-phase stability. Considering the ω-phase, the Fermi level is located very close to the pseudogap, for lower alloying concentrations in both Ti-Mo and Ti-Nb systems as shown in Fig. 9b, d, which indicates a covalent bonding character and higher stability of ω-phase with respect www.nature.com/scientificreports/ to β-phase. However, with increasing Mo and Nb concentration, the valence d-electrons increase and the Fermi level shifts towards the pseudogap in β-phase, which creates a stronger metallic bonding and enhances stability. Furthermore, higher solute concentration shifts the Fermi level towards the antibonding region with higher values of DOS and destabilizes the ω-phase. These DOS observations of destabilization of ω-phase with increasing concentration is consistent with our predicted MEPs and elastic constants calculations.

Conclusion
The effect of five β-stabilizing elements W, Mo, V, Ta, and Nb on the phase stability and the elastic properties of ω-phase in binary Ti alloys were systematically studied using first-principles calculations. The CE method was utilized the find the stable atomic configurations for Ti-X alloys at various compositions. These atomic structures were later utilized to find the MEPs for β → ω martensitic transformation using the solid state nudged elastic band (SSNEB) method. A higher stabilization rate of β-phase with increasing composition was observed for Ti-W and Ti-Mo due to their higher number of valence electrons, while a lower stabilization rate was observed for Ti-V, Ti-Ta, and Ti-V systems. The compositions causing full stabilization of β-phase agree well with experimental studies. The main outcomes of the present study can be listed as follows: • The single crystal and the polycrystalline elastic properties of the ω-phase were predicted as a function of β-stabilizers' concentrations. In general, all β-stabilizing elements considered in this study decreased the elastic moduli of ω-phase with increasing concentration. A decrease of E by 17% and 5% could be observed for that Ti-W and Ti-Mo, respectively at 12.5 at.% concentration, while a decrease of ~ 28% could be observed for Ti-V, Ti-Ta, and Ti-Nb systems with 25 at.% concentration. Also, G decreases by ~ 15% for W and Mo and 32% for V, Ta, and Mo. An experimental Ti-6.5 at.% Mo alloy was prepared to study the mechanical properties with respect to Mo partitioning between the β and ω phases. A significant increase in tensile strength and a decrease in ductility was observed in the annealed condition due to the low content of Mo in the ω phase, while lower strength and high ductility was observed in the solutionized and quenched condition where Mo content in ω-phase was high and nearly equal to that of the β-phase. • The present study conclusively establishes that the elastic constants and shear modulus of quenched athermal ω significantly differ from those of aged isothermal ω, and this correlates well with the observed variations in the mechanical behavior of the Ti-alloys with β + ω microstructure. The lower shear modulus of athermal ω as compared to isothermal ω implies that the bonding is weaker in case of the former, and consequently athermal ω precipitates can be more easily sheared during plastic deformation. This can enhance deformability of the alloy, and correlates well with the higher ductility reported previously for β Ti alloys containing athermal ω www.nature.com/scientificreports/ precipitates. On ageing, the β stabilizing solutes are rejected by the ω precipitates leading to the formation of isothermal ω precipitates. Their significantly higher shear modulus makes isothermal ω precipitates less plastically deformable and more embrittling. • These results suggest that the negative effect of ω-phase on ductility can be remedied by following a careful heat treatment procedure, which promotes the precipitation of ω-phase with higher concentration of β-stabilizers and decreases the elastic constant of this phase. Such heat treatments could be similar to what was reported previously 14,52 . Therefore, the ω-phase may not always be detrimental like previously believed. In fact, β-stabilizers may even enhance the ductility of Ti-X alloys via tuning their elastic constants. Such insights were not available from experiments and point out that contrary to popular belief, the presence of the ω-phase in Ti alloys does not necessarily embrittle the alloy. • The present study revealed for the first time that the nature of bonding within the w phase changes substantially as a function of content of b stabilizing element. Thus, the DFT-based computed density of states (DOS) reveals that quenched athermal ω, with a higher b stabilizer content, exhibits more non-directional metallic type bonding. On the other hand, aged isothermal ω, with a lower b stabilizer content, exhibits more directional covalent type bonding.

Data availability
The data used to generate the results within the current study can be obtained from the corresponding author upon request.