First principles study of Ti-Zr-Ta alloy phase stability and elastic properties

The effects of Ta and Zr content on the stability, elastic properties and electronic structure of Ti-Zr-Ta alloy phase were studied by first principles calculation method based on density functional theory. Moreover, Ti-Zr-Ta alloy was fabricated by spark plasma sintering (SPS) using spherical Ti powder,Ta powder, and Ti-Zr-Ta alloy powder produced by the plasma rotation electrode process (PREP). Afterward, the effects of sintering temperature, Zr and Ta content on the microstructure and mechanical properties of the alloy samples were investigated. The results showed that sintering temperature, Ta and Zr content were the key factors which affected the densification. When the sintering temperature was raised, the relative density and mechanical properties of Ti-Zr-Ta alloy were significantly increased. The first- principles calculation also indicated that Ti-1Zr-Ta possesses the lowest Young’s modulus and the best ductility, showing great potential of biomedical applications which agrees with the results of experimental results of alloy preparation.


Introduction
Due to the low density, excellent mechanical properties, good corrosion resistance, good biocompatibility, and low elastic modulus, titanium alloys are widely used in biomedical applications, such as artificial joints, trauma bone plates, cardiovascular scaffolds, dental implants and other medical instruments. There are two different crystal structures of Ti alloys, which are closely packed hexagonal α phase (hcp) and body centered cubic β phase (bcc) [1]. Because the stacking density of bcc is higher than that of hcp, so the atom diffuses rapidly in β phase, resulting in good plasticity and processability. In addition, high mechanical properties at room temperature also can be obtained by strengthening heat treatment. Moreover, with the continuous development and research of bio-titanium alloys, the new β-titanium alloys draw quite attention due to the good biocompatibility and similar elastic modulus compared with human bone, which have become a hot spot in the field of bio-titanium alloys [2,3]. The β-titanium alloys should be controlled in the β phase as much as possible, that is, the elastic modulus is 40-100 GPa. Therefore, the mechanical properties of the alloy could match the 5-30 GPa of human bone. So far, the new β-titanium alloys which have been developed mainly include Ti-13Nb-13Zr, Ti-15Mo-5Zr-3A1, Ti-Nb-Ta-Zr-Fe, Ti-12Mo-6Zr-2Fe, Ti-35Nb-7Zr-5Ta, Ti-Fe-Ta, Ti-Fe-Ta-Zr and other alloys [4][5][6][7][8][9][10]. Among them, Ti-Zr-Nb, Ti-Zr-Ni and Ti-Zr-Ta in Ti-Zr-based system still have huge components to be explored and developed, and further heat treatment and surface technology are effective means to improve the performance of biomaterials. At present, the Ti-Zr-based alloy is mainly prepared by vacuum melting method. To avoid the possible composition segregation,uneven structure and holes in the process of alloy preparation, it is often necessary to remelt the material for 2 ∼3 times due to the addition of Nb, Mo, Ta refractory metal elements in the alloy. However, the cost is the increase of material cost and the complexity of the preparation process. Spark plasma sintering (SPS) is a new type of special powder sintering forming technology. It not only has the advantages of low sintering temperature, fast cooling rate, short sintering time and high density, but also effectively suppresses the growth of material grain during the sintering process. Through controlling and optimizing the microstructure, it could improve the performance of the material [11].
There have been some studies on the preparation of Ti-30Zr-based alloy [12], Ti-xZr-yTa [13], Ti-xZr-20Ta [14]. They have been indicated that Zr has unlimited solubility in Ti at all temperatures, which is able to impart solid solution strengthening, raising its transformation temperature and impart higher temperature strength onto Ti alloys. In addition, Ta acts as a β-stabilizing element, which can impart high ductility and room temperature malleability. However, none of those papers systematically studied the influence of Ta and Zr element content change on performance. Actually, the effects of alloying elements on the elastic constants, electronic structure and energy characteristics of the alloy could be studied by first principles calculation, providing a theoretical basis for the design of titanium alloys. Chun-Xia Li studied binary Ti-M(M = Nb, Mo, Ta) titanium alloy by first principles method, and explained the structural properties, lattice vibration properties and elastic properties of β, α 'and ω phase structures [15]. At present, most of the first principles method studies on titanium alloys are binary systems, and there are few studies on the theoretical calculation of ternary titanium alloys and the combination of theoretical calculation and experimental analysis. In this paper, the Ti-Zr-Ta ternary alloy model is established by first principles method. The relationship between the stability of Ti-xZr-xTa (x = 0 ∼3, 0-18.75at%) ternary alloy and the elements of Zr and Ta are analyzed by calculating the lattice constant and binding energy. The influence mechanism of Ta and Zr elements on the structural stability of Ti-xZr-xTa alloy was also analyzed. By calculating the elastic constant, the effect of Ta and Zr elements on the elastic modulus of the alloy was also studied. Afterwards, the mechanical properties of the ternary alloy prepared by SPS were tested. The relationship between alloy elements and properties is revealed from the electronic structure of the alloy, which provides a reference for the design of low elasticity and high strength Ti-Zr-Ta alloy with by theoretical calculation method.

Experiment method
The geometry optimization, elastic and electronic properties of Ti-xZr-Ta (x = 0 ∼3) and Ti-2Zr-xTa (x = 0 ∼3) alloy were calculated using the Vienna Ab-initio simulation package (VASP) based on density functional theory (DFT). The projected augmented wave (PAW) method was applied to describe the ion-electron interaction, the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) was used as the exchangecorrelation functional. The energy cutoff and convergence criteria for energy and force was set to be 450 eV, 10 −4 eV, and 0.01 eV/Å, respectively. Monkhorst-Pack special k-point mesh of 3×3×3 was proposed for the geometry optimization, while 9×9×9 was adopted for static calculations. For the Ti-2Zr-xTa systems, 16-atoms supercell with a body-centered cubic (bcc) structure was used to simulate the β-phase. The concentration of Zr element was kept at 12.5% (2 atom), while Ti atom was gradually replaced by Ta atom, and the concentrations vary from Ti-2Zr-0Ta (corresponding to 0) up to Ti-2Zr-3Ta (corresponding to 18.75%); the minimum compositional change of Ta is 0, and the supercell models of Ti-2Zr-xTa alloys are shown in figure 1(a). For the Ti-xZr-Ta systems, a 16-atoms supercell with the body-centered cubic (bcc) structure was used to simulate the β-phase. When the concentration of Ta element was kept at 6.25% (1 atom), Ti atom was gradually substituted Ti-3Zr-Ta (Blue for Ti, green for Zr, and red for Ta).
by Zr atom, and the concentrations change from Ti-0Zr-Ta (corresponding to 0) up to Ti-3Zr-Ta (corresponding to 18.75%); the minimum composition variation of Zr is 0, and the supercell models of Ti-xZr-Ta alloys are shown in figure 1(b). The configuration of supercell with different atomic positions has been tested to ensure the rationality and accuracy of Ti-2Zr-xTa and Ti-xZr-Ta (x = 0 ∼3) alloys. All concentrations in this paper were defined by atomic percentage. For each case, the geometrical optimization was always made first.
Pure Ti (53-150 μm), pure Ta (15-53 μm) and Ti-2Zr-2Ta alloy powder (53-150 μm) prepared by plasma rotating electrode process(PREP) method were sintered by SPS process. The sintering temperature was 1050°C-1250°C, the holding time was 10 min, and the pressure was 50 MPa. During the sintering process, the heating rate of was 100°C min −1 . Then the furnace is cooled to room temperature after holding for 10 min. The sintering process is shown in figure 2. The sample densities were determined by the Archimedes method and the microhardness values were obtained by using an HDI-1875 tester under a load of 0.49 N for 15 s. The XRD was performed with the XRD-7000 x-ray diffractometer (Shimadzu Manufacturing Institute, Japan). The test conditions were Cu Kα radiation, the maximum tube voltage was 60 kV, the maximum tube flow was 80 mA, the scanning speed was 4°min −1 , and the scanning range was 30°-80°. The scanning electron microscopy (JSM-6700F, SEM) was used to investigate the morphology. The compressive mechanical properties were determined using a universal testing machine (WDW-200, Changchun Kexin Test Instrument Co., Ltd, China) at a strain rate of 1.1×10 −3 s −1 .

Calculated formation energies and stability
The physical properties of materials were almost determined by the lattice structure and atomic occupancy. Due to the equivalent lattice the constants Ti-2Zr-xTa and Ti-xZr-Ta alloys maintain the body-centered cubic structure after full geometric optimization. The calculation results are shown in figure 3. It displayed that the lattice parameter a and unit cell volume V of Ti-2Zr-xTa and Ti-xZr-Ta alloys gradually increased with the Ta or Zr concentration, which is due to the atomic radius of Ta (0.148 nm) and Zr (0.16 nm) are both larger than that of Ti (0.147 nm). The larger atomic radius causes the lattice expansion around the Ta or Zr atoms and increases the distortion of crystal lattice.
The type of crystal lattice also affects the Young's modulus which is an intrinsic characteristic of the materials. The cohesive energy(E coh ) and formation energy (E f ) were considered to determine the β phase stability of Ti-2Zr-xTa alloys with increasing the Ta concentration and Ti-xZr-Ta with increasing the Zr concentration. It is well known that the thermodynamic stability of the β phase is determined by E coh , while the metallurgical tendency to form a solid solution depends on E f . The larger E coh and E f suggest that it is easier to form Ti-2Zr-xTa and Ti-xZr-Ta alloys with β phase solid solution [16]. E coh and E f are described by equations (1) and (2):  lattice parameter, respectively; N total , N Ti , N Zr and N Ta are total numbers of atoms, Ti atoms, Zr atoms and Ta atom in supercell, respectively. Table 1 presents the calculated E coh , E f and average valence electron concentration e/a for Ti-2Zr-xTa and Ti-xZr-Ta alloys. It could be seen that the negative E coh and E f indicate that all these alloys can maintain the thermodynamic stability of β phase solid solution at 0 K in the absence of Ti-0Zr-Ta. The decrease of E coh indicates that the β phase of Ti-2Zr-xTa and Ti-xZr-Ta alloys becomes more stable with the addition of Zr or Ta. The average valence electron concentration e/a gives an empirical criterion for determining the phase composition of Ti alloys. The transition phases in Ti-2Zr-0Ta, Ti-2Zr-1Ta and Ti-0Zr-Ta alloys are predictable when e/a is less than 4.15, while β phase dominates when e/a 4.2 [17]. As shown in table 1 the average valence electron concentration e/a of Ti-2Zr-xTa and Ti-xZr-Ta alloys is in the range of 4.21 ∼4.38. Compared with the calculated results, it is found that the addition of Ta can improve the β phase stability more than the addition of Zr. Moreover, the calculation result is also consistent with Zr as a β phase neutral element.

Calculated elastic properties
The elastic constants reflect the ability of the crystal to resist elastic deformation. The elastic constants are related to many properties of crystal materials. Different crystal systems have different numbers of independent elastic constants. The elastic moduli (bulk modulus B, shear modulus G and Young's modulus E), Poisson's ratio ν and effects of anisotropy factor A of Ti-2Zr-xTa and Ti-xZr-Ta alloys on the mechanical properties with Ta and Zr concentration are discussed. Table 2 shows the calculated elastic constants Cij. For the bcc lattice, there are three independent elastic constants C 11 , C 12 and C 44 . As displayed in table 2 all alloys except Ti-0Zr-Ta meet the Born stability criteria for mechanical stability of the bcc structure: C 11 > 0, C 44 > 0, C 11 -C 12 > 0, C 11 +2C 12 > 0, which mean that all the alloys except Ti-0Zr-Ta could maintain the mechanical stability of β phase. The Cauchy pressure (C 12 -C 44 ) related to the bonding characteristic in the crystal lattice is also used to estimate the ductile or brittle behavior for material [18,19]. The positive Cauchy pressure indicates more metallic character, while the negative C 12 -C 44 represents for more covalent bonding characteristic. From table 2 the value of C 12 -C 44 indicates that Ti-2Zr-xTa and Ti-xZr-Ta alloys exhibit more metallic characteristics with increasing the Zr and Ta concentration, which also has a positive effect on improving ductility. The bulk modulus B, shear modulus G and Young's modulus E can be estimated from elastic constants by the Voigt-Reuss-Hill method, described as formula (3)-(8).
Where Bv and Gv are the bulk modulus and shear modulus calculated by Voigt approximation method, respectively; B R and G R are the bulk modulus and shear modulus calculated by Reuss approximation method, respectively. The bulk modulus B and shear modulus G is commonly used to determine the resistance of alloys against the volume deformation and shape deformation, respectively. Table 3 and figure 4 shows the predicted bulk modulus B, shear modulus G and Young's modulus E. It can be seen from figure 4(a) that Ta is more effective in enhancing the resistance to volume deformation of Ti-2Zr-xTa alloys. While the trend of shear modulus suggests that with increasing the Ta concentration, the characteristics of resisting shear deformation decrease after the initial increase. The shear modulus G is more susceptible to Ta concentration in comparison with bulk modulus B. The Young's modulus E can reveal the stiffness of Ti-2Zr-xTa alloys which tends to increase in the initial stage and then decrease with the addition of Ta. The trend of the elastic modulus indicates that the shear modulus is main factor to reduce the Young's modulus. However, as shown in figure 4(b), with the increase of Zr content, the change of bulk modulus B, shear modulus G and Young's modulus E, have no obvious regularity.

Electronic properties
In order to further understand the relationship between the phase stability and bonding properties between Ta and Ti, Zr elements, the total density of state (TDOS) of ternary alloys with different Ta and Zr contents was calculated and the results are shown in figure 5. The vertical black dashed line represents the Fermi level, which is the boundary between valence electrons and free electrons. The valence electrons of metallic materials are generally distributed below the Fermi level. Moreover, there are two distinct peaks on either sides of the Fermi energy level, called 'pseudo gap'. The wider the 'pseudo gap' is, the stronger the atomic bond and more stable the structure is [20,21]. For all cases, the total and partial of the DOS cross the Fermi level, suggesting that there is no Table 2. Calculated elastic constants C 11 , C 12 , C 44 , Cauchy pressure C 12 -C 44 of β phase in Ti-2Zr-xTa and Ti-xZr-Ta alloys alloys. Alloy Ti-2Zr-0Ta (0 Ta)  energy gap at the Fermi level. It is indicated that the bonding state is mainly in the energy range between 8 eV and 0 eV. Moreover, it is related that the stability of Ti-2Zr-xTa and Ti-xZr-Ta alloys alloys can be improved by increasing the Ta or Zr content, which is consistent with the previous cohesive energy results. According to the above first principle calculation, the elastic modulus of Ti-1Zr Ta (6.25at%) alloy is the lowest, and the elastic modulus is also affected by the phase composition and density of the material. Therefore, the alloy powders of Ti-Ta, Ti-1Zr -Ta (6.25at%) and Ti-1Zr-2Ta (12.5at%) were selected to prepare the alloys at different sintering temperatures. On the one hand, the calculated results are compared. On the other hand, the influence of the preparation process on the elastic modulus of the alloy is studied.   Figure 6 shows the SEM images of Ti,Ta and Ti-Ta-Zr powders prepared by PREP . It can be seen that the three kinds of powders are all regular spherical shape with smooth surface, and dendritic structures. Figure 7 shows the microstructure of alloys with different compositions and different sintering temperatures. For Ti-Ta alloy, as shown in figures 7(a) and (b), the structure is mainly composed of dark gray area, gray area and particles. Through energy spectrum analysis, it is found that the dark gray area is pure Ti phase, the gray area is Ti-Ta solid solution phase, and the particles are pure Ta. With the increase of sintering temperature, the size of the remaining pure Ta particles becomes smaller, indicating that the sintering temperature is one of the main reasons for promoting the rapid diffusion of Ti and Ta during SPS process. Ti-Zr-Ta alloy is obtained by sintering pure Ti powder and Ti-2Zr-2Ta alloy powder. It can be seen from figures 7(c)-(e) that with the increase of sintering temperature, the widmanstatten structure in the alloy gradually increases, that is, the Ti and Ti Zr Ta alloy powder are completely dissolved. Ti-Zr-Ta alloy is sintered from pure Ti powder, Ta powder and Ti-2Zr-2Ta alloy powder. It can be seen from figure 7(f) that when the sintering temperature is 1250°C, the particle structure could be found, and judged as pure Ta particle through energy spectrum analysis. Figure 8 shows the XRD patterns of Ti-Ta and Ti-Zr-xTa alloys with different compositions and sintering temperatures. It shows that For Ti-Ta alloys α-Ti and β Phase composition, through further analysis of β phase, which includes β-Ta and β-Ti x Ta 1-x solid solution. With the increase of sintering temperature α-Ti phase gradually decreases, while the content of β phase gradually increases. When the Ta content in the alloy is further increased, the Ta content in β phase of the Ti-Zr-2Ta alloy dominants, with only a small amount of α-Ti phase. Moreover, the lattice constants of Ti-Ta, Ti-Zr-Ta alloys sintered at 1250°C are calculated by the XRD results, which are 0.636 nm and 0.649 nm, respectively. The error is less than 2.2% compared with 0.65 nm and 0.657 nm, which were calculated by first-principles. The comparison results of lattice constant calculation indicates that there is little difference between experimental results and theoretical calculations. Figure 9 shows the relative densities of Ti-Ta and Ti-Zr-xTa alloys with different compositions and sintering temperatures. It can be seen from the figure that the density of alloys with different compositions increases with the increase of sintering temperature; At the same sintering temperature, the density of Ti-Zr-Ta alloy is higher. Mainly because it is sintered from pure Ti powder and Ti-2Zr-2Ta alloy powder, and its melting point difference is small, while Ta powder is introduced into other alloys, and its melting point is 2900°C, which is not conducive to sintering densification.   Figure 10 shows the microhardness of Ti-Ta and Ti-Zr-xTa alloys with different compositions and sintering temperatures. It can be seen from the figure that the microhardness gradually increases with the increase of sintering temperature, which is related to the increase of density; For the alloys with different compositions, their solution strengthening and particle strengthening also have certain effects. Figure 11 shows the mechanical properties of Ti-Ta and Ti-Zr-xTa alloys with different compositions and different sintering temperatures. It can be seen from the figure that with the increase of sintering temperature the tensile strength and yield strength of alloys with different compositions gradually increase, which is related to the increase of density to a certain extent. The tensile strength and yield strength also increase significantly with the increase of Ta and Zr elements content. However, the change of elastic modulus shows the opposite trend. According to the XRD analysis results, with the increase of Ta and Zr element content and sintering temperature, the content of β phase for the alloys also increased which can effectively reduce the elastic modulus [12]. Therefore, with the increase of sintering temperature and the content of Ta and Zr elements, the elastic modulus decreases continuously, between 35-50 GPa, which is close to the elastic modulus of human bone of 5 ∼30 GPa. The above analysis shows that the experimental results are basically consistent with the trend of the first principle calculation results. However, the values of elastic modulus between experiment results and calculation had much difference. The reason for this difference may be that, during the alloy preparation process there are some defects in the alloy, moreover, the prepared alloy are not completely compact (low density) titanium alloy, which must have a certain gap with the theoretical value.The experiment results also shows that  the dynamics properties of Ti-Zr-Ta alloy are higher than that of Ti-Ta alloy, indicating that the bonding strength and cohesive energy of Ti-Zr-Ta alloy is more stable, which is consistent with the calculated result.

Conclusions
The binding energy, electronic structure and elastic constant of the Ti-xZr-xTa (x = 0 ∼3, 0-18.75at%) alloy were calculated based on the first principle. The effects of of Ta and Zr content on the phase stability and elastic properties of Ti-Zr-Ta alloy were discussed, and the calculation results were verified by experiments. The calculation results shows that the addition Ta could improve the β-phase stability of Ti-2Zr-xTa alloys. The bulk modulus B, shear modulus G, Young's modulus E of Ti-Zr-Ta alloy are lower. The experimental results show that the elastic modulus of Ti-1Zr-Ta alloy has the lowest elastic modulus among Ti-Ta, Ti-1Zr-Ta (6.25 at%) and Ti-1Zr-2Ta. Meanwhile, the elastic modulus of Ti-1Zr-Ta alloy increases slightly with the increase of sintering temperature, which is mainly affected by the alloy density. The results are consistent with the results calculated by the first principle.