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Solid State Sciences 12 (2010) 2120e2124

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Solid State Sciences

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Effects of Ta content on the phase stability and elastic properties of b TieTa alloysfrom first-principles calculations

C.Y. Wu, Y.H. Xin, X.F. Wang, J.G. Lin*

Faculty of Material and Optical-Electronic Physics, Key Laboratory of Low Dimensional Materials and Application Technology (Ministry of Education), Xiangtan University, Hunan,Xiangtan 411105, China

a r t i c l e i n f o

Article history:Received 24 June 2010Received in revised form6 September 2010Accepted 9 September 2010Available online 17 September 2010

Keywords:TieTa alloysElastic constantsElastic modulusFirst principles

* Corresponding author. Tel./fax: þ86 731 5829246E-mail address: lin_j_g@163.com (J.G. Lin).

1293-2558/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.solidstatesciences.2010.09.009

a b s t r a c t

The effects of Ta content on the phase stability, the elastic property and the electronic structure of b typeTieTa alloys were studied from first-principles calculations based on the density functional theory. It isfound that the phase stability, tetragonal shear constant C0 , bulk modulus, elastic modulus and shearmodulus of b type TieTa alloys increase with the Ta content increasing monotonously. The lowest elasticmodulus of the alloys is realized when the valence electron number (e/a) is around 4.25. Moreover, thephase stability of the alloys was discussed based on the calculated density of state (DOS).

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

The ideal biomedical implant materials should possess anexcellent biocompatibility, good corrosion resistance, high strengthand low elastic modulus (near to that of a human bone to thegreatest extent). Ti and its alloys can meet these requirements toa very high degree due to their low density, high strength and goodcorrosion resistance, etc. However, to be as the implant materials,the traditional a and aþ b Ti alloys may cause the stress shieldingresulting from biomechanical mismatch of the existing denseimplant materials, and moreover, the alloying elements, V and Al,may cause cytotoxic effects and adverse tissue reaction. It triggeredan extensive research to develop new Ti alloys in recent years,aiming at lowering the elastic moduli and promoting the safety ofthe alloys. It is well known that the high temperature b-phase of thetitanium has a cubic centered crystalline structure, while thea-phase has a hexagonal crystalline structure which providesb-alloys as compared to aþ b alloys, with an improved notchedfatigue resistance and a superior resistance to wear and abrasion[1e3]. The superior properties of b Ti alloys make them preferableto aþ b alloys for biomedical applications and promising alloys forto a wide range of implant applications. Hence, the new b Ti alloyscontaining nontoxic elements like Nb, Ta, Zr and so on are receiving

8.

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a great deal attention for biomedical application. In these alloys,TieTa alloys are expected to be promising biomaterials due to theirsuperior comprehensive properties, since Ta has the potential toenhance the strength, and to reduce the modulus of Ti alloys at thesame time. The experimental studies showed that the stability ofb phases and the mechanical properties of the Ti alloys are stronglydependent on the compositions, i.e., the contents of alloyingelements, and some nonequilibrium phases such as a0, a00 and uphases may precipitate in Ti alloys, which affect the modulus of thealloys [1e3]. Thus, it is of significance to research on the effects of Tacontent on the modulus and the strength of binary TieTa alloys.Recently, the computer-aided material design becomes a researchhotspot, and the first-principles methods based on electronicstructure theory are expected to provide a clue to understandingthe elastic properties and phase stability of b type Ti alloys. Ikehataet al [3] made a systematic study on Ti binary alloys by the first-principles calculations based on the density functional theory. Theresults indicated that rigid band model applied for TieX alloys(X¼Nb, Mo, Ta) and the related elastic properties of Ti alloys couldbe controlled by altering the valence electron number per atom ofTi alloys. However, the relations among the elastic properties, thephase stability and the electronic structure in the low elasticityTieTa alloys is not clear yet, and the effects of Ta content on b typeTieTa alloys have not been reported by the first principles calcu-lative method. Therefore, in this work, the effects of the Ta contenton the phase stability and elastic properties of TieTa alloyswith body-centre cubic (bcc) crystal structure were investigated by

Fig. 2. The change of the cohesive energy as a function of Ta content.

Fig. 1. The change of the lattice constant of TieTa alloys as a function of Ta content.

C.Y. Wu et al. / Solid State Sciences 12 (2010) 2120e2124 2121

first-principles calculations. The Young’s modulus, the bulkmodulus and the shear modulus of the polycrystal from the elasticconstants of a single crystal were evaluated by theVoigteReusseHill (VRH) methods [4e6]. The change of the latticeconstant, cohesive energy, elastic properties, density of states (DOS)with Ta content were discussed in TieTa alloys, and the relationshipbetween the tetragonal shear constant C0 or C44 and the number ofvalence electrons (e/a) was established, aiming at tailoring thecomposition of TieTa alloys having the low elastic modulus.

2. Method

In the present study, density functional calculations within thegeneralized gradient approximation (GGA) [7] were performed inVienna Ab-initio Simulation Package (VASP) [8], PerdeweWang(PW91) version of the Projector augmentedwave (PAW)method forthe exchange-correlation potential was used for all calculations [9],in which, 4s, and 3d valence states for Ti and 5p, 6s and 5d valencestates for Ta were considered, respectively. In order to calculate thetotal energies, elastic properties and electronic structures of theb-type TieTa alloys with different alloying contents, a 16-atomsupercell with DO3 structure (cF16) containing eight bcc unit cellswas constructed, in which the lattice points were divided intofour kinds of the equivalent positions, marked with a (0,0,0),b (0.5,0.5,0.5), c (0.25,0.25,0.25) and d (0.75,0.75,0.75). The chemicalformulawas assumed as Ti1�xTax (x¼ 0, 0.0625, 0.1875, 0.25, 0.3125,0.5, 0.75, 1). All lattice points are occupied by Ti if x¼ 0, or by Ta ifx¼ 1, respectively. For theTi0.75Ta0.25 alloy, theatomicpositionsof4aare occupied by the four Ta atoms, the remaining positions, 4b, 4cand 4d, are occupied by the 12Ti atoms. To obtain the equilibriumgeometry for the constructed supercells, we performed a fullrelaxation both for atoms (to reduce the force acting on the atoms tozero), and for the size and the shape of the supercell (to make thesystem strain free). The method has been described in details inRef.[10].. For Ti0.75Ta0.25, the atoms were in 4b, 4c and 4d positionsfor the four Ta atoms, respectively, corresponding to the remainingpositions were occupied by 12 Ti atoms. The energies of theremaining three kinds of structures were obtained by the samecalculated method. Based on the calculated results and the lowestenergy principles, the equilibrium structure with the lowest energywas adopted for the calculation. The structure models for Ti0.5Ta0.5and Ti0.25Ta0.75 were obtained by the same method. For Ti0.9375Ta0.625 and Ti0.8125Ta0.1875, various local atom arrangements wereconsidered, and the structural models with the lowest energy weredetermined by the calculations in the different cases of the localatom arrangements according to the previous method. The cut-offenergy was chosen to be 400 eV for TieTa alloy system, and theBrillouin Zone (BZ) was sampled with a MonkhorstePack k-pointgrid [11]. For the structural relaxation and density of states (DOS)calculations, an 8� 8� 8 k-point mesh and a 12�12�12 k-pointmesh was used, respectively. The first-order MethfesselePaxton[12] method with a width of 0.2 eV was used for the structuralrelaxation, and the position of atoms were fully relaxed until thetotal forces on each ion were less than 0.001 eV/A. Then the totalenergy and the density of states (DOS) calculations were performedusing the tetrahedron method with Blöchel correction [13].

3. Results and discussion

3.1. Lattice constants and phase stability

Fig. 1 shows the calculated lattice constants of b type Ti1�xTaxalloys as a function of Ta content. It also can be seen that the latticeconstants of TieTa alloys increase almost linearly with the increaseof Ta, which can be attributed to the fact that the atomic radius of Ta

is slightly bigger than that of Ti in the TieTa substitution solidsolution. The structural stability of crystal is correlated to itscohesive energy Ecoh, which is defined as the work to decomposethe crystal decomposes into the single atom. Hence, a low cohesiveenergy implies a high the stability. In the present work, the cohe-sive energies (Ecoh) of b-type Ti1�xTax crystal cells can be obtainedfrom the following equation:

EABcoh ¼�Ecoh � NAE

Aatom � NBE

Batom

�.ðNA þ NBÞ (1)

where Etot is the total energy of the alloy, EAatom and EBatom, are thetotal energies of the atoms A and B in the freedom states. NA and NBrefer to the number of A, B atoms in each unit cell. The results of thecohesive energy calculation are presented in Fig. 2. It can be seenthat the cohesive energy decreases with the content of Taincreasing in TieTa binary system, indicating that the structurestability of b phase increases with the content of Ta increasing inTieTa system. Mareci et al. [14] had also revealed that the micro-structures of the quenched TieTa alloys were sensitive to Tacontent, and the content of b phase increased with the increase ofthe Ta content in these alloys. This is consistent with our presentcalculated results.

Table 1The strains used to calculate the elastic constants of Ti1�xTax

Strain I Parameters (eijs0) DE/V0 to 0 (d2)

3(1) e11¼ e22¼d, e33¼ (1þ d)�2� 1 3(C11� C12) d2

3(2) e12¼ e21¼ d/2, e33¼ d2/(4� d2) 1/2VC44d2

3(3) e11¼ e22¼ e33¼ d 3/2V(C11þ 2C12)d2

C.Y. Wu et al. / Solid State Sciences 12 (2010) 2120e21242122

3.2. Elastic properties

3.2.1. Elastic constantsGenerally, the single crystal elastic constants can be obtained by

calculating the total energy as a function of the appropriate latticedeformations from first principles. The necessary number of strainsis determined by the crystal symmetry [15]. Either a particularelastic constant or a combination of elastic constants can bedetermined from the curvature of the curves of the total energyversus the strains. The internal energy (E(V,{ei})) of a crystal underan infinitesimal strain ei, with respect to the energy E(V0,0) of theunstrained geometry, can be written as

EðV ; feigÞ ¼ E�V0;0

�þ V0

2

Xij

Cijeiej þ.; (2)

where V0 is the volume of the unstrained systemwith E(V0,0) beingthe corresponding energy, Cij are the single crystal elastic constantsand the member of strain tensor 3¼ {ei, ej,.} are given in Voigtnotation. For the TieTa alloys having bcc structure, the independentelastic constants C11, C12 and C44 can be obtained by using thevolume conserving tetragonal strain (3(1)), the volume conservingmonoclinic strain (3(2)) and the simple hydrostatic strain (3(3)) asdescribed in detail in Table 1. For the total energy calculations, fivestrain steps were used. 3(1) and 3(3) start from �0.02 to 0.02 witha interval of 0.01, while for 3(2) which may cause the rather smallchanges in energy, the strain starts from �0.04 to 0.04 with a 0.02interval. The calculation results, the deformation energies DE¼ E(d)� E(0) versus the definite strain d and three independent elasticconstants are listed in Table 1. Normally, the mechanical stabilitycriteria for a cubic crystal are as follows [15]:

C44 > 0;C11 þ C12 > 0;C11 � C12 > 0

So, the tetragonal shear elastic constant C0 ¼ (C11� C12)/2 canrepresent the elastic stability of a cubic crystal. The elastic constantsof b type TieTa alloys with the different Ta contents are obtained onthe basis of the above theory, which are listed in Table 2. It can beseen that for b Ti and Ti0.9375Ta0.625, C0 is negative, indicating b phaseis unstable at 0 K. C0 andC44first decrease and then increasewith theTa content increasing. For comparison, the calculation results givenby Ikehata et al. [3]are also listed in Table 2. It is clear that the valuesof the elastic constants (C11, C12, C0 and C44), and their change trendwith the Ta content are in good agreement with those of Ikehata’s

Table 2Elastic properties of Ti1�xTax (unit: GPa).

Formula C11 C12 C0 C44

Ti 94.48 116.55 �11.04 38.06Ti0.9375Ta0.625 108.71 115.59 �3.44 39.61Ti0.8125Ta0.1875 129.06 120.27 4.40 33.28Ti0.75Ta0.25 134.57 127.12 3.73 28.08

129.9 121.6 4.1 38.6Ti0.6875Ta0.3125 150.85 125.10 12.87 36.58Ti0.5Ta0.5 181.80 138.86 21.47 45.18

163.4 132.8 15.6 39Ti0.25Ta0.75 221.41 153.20 34.11 53.18

207.0 145.3 30.85 55.6Ta 271.77 164.98 53.39 71.65

calculations basically, except for the values of C0, C44 for Ti0.5Ta0.5 andC44 of Ti0.75Ta0.25. The difference of these values may be caused bythe different forms of the exchange-correlation used in the presentwork. Thus, our calculations are reliable.

The polycrystalline shear modulus can be estimated from thesingle crystal elastic constants based on the arithmetic Hill averageas follows:

GHill ¼ 12ðGV þ GRÞ (3)

where GV and GR are the Voigt and Reuss averages, which can bedescribed as follow, respectively.

GV ¼ C11 � C12 þ 3C443

(4)

GR ¼ 54S11 � 4S12 þ 3S44

(5)

S11, S12 and S44 are the elastic compliances [16]. Therefore,Young’smodulus of the TieTa polycrystalline alloys can be obtainedby

E ¼ 9GBGþ 3B

(6)

The calculated bulk modulus B (B¼ 1/3(C11þ2C12)), the averageshear modulus G and the elastic modulus E of TieTa alloys with thedifferent compositions are listed in Table 2. It can be seen that G andB of the alloys increase monotonously with the increase of Tacontent. However, the Young’s modulus first decreases and thenincreases with the Ta content increasing. Our calculations agreewell with Ikehata’s. From the calculations, we can see that the alloywith Ta content of about 25 at.% exhibits the lowest elasticmodulus, which is about 37.90 GPa. Thus, the calculations fromfirst-principles provide a good guideline for tailoring the compo-sition of Ti alloys with low elastic constants.

3.2.2. Relationship between C44, C0 and the number of valenceelectrons

It is well documented that by addition of VB or VIB familyelements to pure Ti, we can obtain Ti binary alloys with the lowYoung’s moduli. To clarify the mechanisms of the low elasticmodulus in these alloys, Ikehata et al. [3] have done calculations ofthe elastic constants of some TieX (X¼V, Nb, Ta, Mo andW) binaryalloys by first-principles. They found that to realize the low elasticmoduli of the Ti binary alloys the valence electron number (e/a)should be controlled to around 4.20e4.24 to make the value of C0

nearly zero without changing the bcc structure of the alloys. In themean time, C44 of the alloys should be small. But their work did notgive the relationship between C44 and the number of valenceelectrons. To further clarify the mechanisms by which the low

C11eC12 B G E

�22.07 109.19 e e

�6.88 113.29 e e

8.79 123.20 5.45 44.497.45 129.60 13.06 37.90PW91 (present)8.2 PBE [3]

25.75 133.69 24.25 68.5942.93 153.18 33.51 93.71PW91 (present)30.6 PBE [3]68.21 175.94 44.5 139.81PW91 (present)61.7 PBE [3]

106.79 200.58 58.98 161.14

Fig. 3. Relationship between the (a) valence electronic number and C0; and (b) valence electronic number and C44.

C.Y. Wu et al. / Solid State Sciences 12 (2010) 2120e2124 2123

elastic moduli of Ti binary alloys is realized the change of the valuesof C44 or C0 as a function of the valence electrons is presented inFig. 3. It can be seen that C0 increases with the increase of e/a andthe value for C0 ¼ 0 is about 4.16 from Fig. 3(a). The result indicatesthat the TieTa alloys having bcc structure are stable when e/a�4.16. In addition, the change of the value C44 as a function of e/a isshown Fig. 3(b). It can be seen that C44 is the lowest when e/a isaround 4.20e4.25 in TieTa alloys having bcc structure. According tothe analysis above, the elastic modulus of the b Ti0.75Ta0.25 alloy is

Fig. 4. (a) Total density of state(DOS) and partial density of states(PDOS) of Ti75Ta25 alloyFermi level).

the lowest and the value of e/a is about 4.25. So the effect of C44 onthe elastic modulus is crucial. In addition, the experimentalinvestigations given by Kim et al. [17] indicate the low Young’smodulus in b Ti alloys are due to the low values of not only C0 butalso C44.Therefore, to realize a low elastic modulus TieTa binaryalloy having bcc structure, the influences of C0 on phase stabilityand C44 on the elastic modulus should be considered. From thispoint of view, the valence electronic number of the lowest elasticmodulus b TieTa alloys is around 4.25.

; (b)Total density of state(DOS) Ti75Ta25,Ti50Ta50 and Ti25Ta75 alloys(dash line note

C.Y. Wu et al. / Solid State Sciences 12 (2010) 2120e21242124

3.3. Density of states

The density of states (DOS) for the b type Ti1�xTax alloys wasfurther calculated at their equilibrium geometries. Fig. 4(a) depictsthe calculated DOS and partial density of states (PDOS) of b typeTi0.75Ta0.25 alloys, It can be seen that the DOS of the alloy originatesmainly from Ti 3d orbits and Ta 5d orbits, and the contribution of Ta5d orbits is predominant in the low-energy area below Fermi level(EF). It indicates the bonding DOS below Fermi level must increaseswith the increase of Ta content in TieTa alloys. To illustrate theeffects of Ta content on the phase stability of the TieTa alloys,the DOS of Ti0.75Ta0.25, Ti0.5Ta0.5 and Ti0.25Ta0.75 are calculated andthe results are shown in Fig. 4(b). EF is located in the anti-bondingregion of DOS, whichmeans that the phase stability of b structure isrelatively low in TieTa alloys at 0 K. Furthermore, EF of TieTa alloysDOS move slightly upward to the valley of DOS and the DOS of theFermi level of N(EF) decreases with the Ta content increase [18]. Itimplies that the stability of b type TieTa alloys is promoted grad-ually with the Ta content increasing.

4. Conclusions

In this work, the effects of the tantalum content on the phasestability and elastic properties effect of TieTa alloys with body-centre cubic (bcc) crystal structure were investigated from first-principle calculations by using the supercell and augmented planewaves plus local orbital method within the generalized gradientapproximation. The results indicate that the phase stability,tetragonal shear constant C0, bulk modulus, elastic modulus andshear modulus of b type TieTa alloys increase with the tantalumcontent increasing monotonously. When e/a of the TieTa alloys isaround 4.16, the tetragonal shear constant C0 is nearly zero. C44 isthe lowest when the value of e/a is around 4.25. The lowest elasticmodulus in the b type TieTa binary alloys can be realized by

controlling C44 to be the lowest and the tetragonal shear constant C0

is greater than or equal to zero. This work provides a guideline fortailoring the composition of TieTa alloys with low elastic constants.

Acknowledgements

The authors would like to acknowledge financial support fromthe National Natural Science Foundation of China (No. 10972190)and the Scientific Research Fund of Hunan Provincial EducationDepartment (09A089, 09C954).

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