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Phase stability, mechanical properties and electronic structure of TiAl alloying with W, Mo, Sc and Yb: First-principles study
Journal of Alloys and Compounds 658 (2016) 689e696
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Phase stability, mechanical properties and electronic structure of TiAl alloying with W, Mo, Sc and Yb: First-principles study Hai Hu a, Xiaozhi Wu a, b, Rui Wang a, *, Weiguo Li c, Qing Liu b a
Institute for Structure and Function, Chongqing University, 401331, China College of Materials Science and Engineering, Chongqing University, 400044, China c College of Aerospace Engineering, Chongqing University, 400044, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 August 2015 Received in revised form 28 October 2015 Accepted 29 October 2015 Available online 9 November 2015
The phase stability and site preference of alloying elements W, Mo, Sc and Yb in L10 and B2 TiAl alloys were investigated using first-principles calculations. It is found that there will be a phase transition from L10 to B2 structure when the concentrations of W and Mo are about 10.50 at.% and 11.50 at.%, respectively. However, there is no phase transition for L10 TiAl with Sc and Yb. The elastic constants, bulk modulus, shear modulus, Poisson's ratio and hardness of TiAl alloys are also systematically presented. Based on Pugh ratio, the brittle/ductile transition are predicted for TiAl with W, Mo, Sc and Yb. It is found that W and Mo can effectively improve the ductility of TiAl due to the phase transition. Finally, the density of states is also used to analysis the mechanical properties, which is consistent with brittle/ ductile transition map of TiAl based alloys. © 2015 Elsevier B.V. All rights reserved.
Keywords: Phase stability Elastic constants TiAl First principles
1. Introduction TiAl has attracted significantly interesting due to its high melting temperature, adequate high temperature strength, excellent oxidation and corrosion resistance, good strength retention ability and low density is hopeful to be the new promising lightweight and heat-resistant functional material [1e3]. The inherent brittleness at low temperature, is by far the greatest hurdle that prevents their widely used in industrial applications [4e6]. TiAl has an ordered g-phase (ordered face-centered tetragonal L10 AuCutype) in which Ti and Al atoms alternatively occupy the (002) planes, and lattice constant ratio c/a is about 1.02 to show a slightly tetragonal (see Fig. 1). The low symmetric L10 phase cannot provide enough sliding systems to satisfy von Mises' criterion and exhibits poor ductility [7]. For handling this shortcoming, a lot of research work has been carried out to improve its ductility, such as heat treatment and elements alloying, as well as the introduction of a second reinforcing phase, etc [8e11]. Among them alloying is one of mature methods. Shili et al. suggest that Fe and Co atoms preferentially occupy the Al sites and can improve the electronic structures and elastic properties of TiAl, leading to the
* Corresponding author. E-mail addresses: [email protected] (R. Wang).
(X.
http://dx.doi.org/10.1016/j.jallcom.2015.10.270 0925-8388/© 2015 Elsevier B.V. All rights reserved.
Wu),
[email protected]
improvement of the ductility of TiAl [12]. V, Cr or Mn can also improve the ductility of TiAl alloys buy using a discrete variational (DV) Xa cluster method [6,13,14]. An alloy design strategy to improve the ductility of TiAl is to exploit the disordered b-phase (disordered BCC B2 AuCu-type) with alloying elements at elevated temperatures [15e18]. The B2 phase, which provides a sufficient number of independent sliding systems. Thus, it may improve the deformability at elevated temperature. Kimura et al. [19] have pointed out that V, Cr, Mn, Nb or Mo in L10 TiAl could lead to the change of phase equilibria and the B2 phase is stabilized. Cheng et al. [20] have suggested that the addition of B appears to be beneficial in TiAl that both strength and ductility are improved. Studies have shown that the addition of transition metal W can stabilize the high symmetry B2 phase nucleation preferably [21]. Moreover, the presence of 3 at.% Cr in L10 TiAl is expected to produce B2 phase and thus improves the tensile ductility [22]. In another study, B2 phase was produced to the TiAl alloy by adding 2 at.% Mo, hence improving room-temperature tensile ductility and strength [23]. Moreover, the addition of those transition metal elements into TiAl which weaken ped interactions but enhance ded interactions is most effective in improving the ductility in theory suggested by Morinaga et al. [6]. Thus, the stabilization of B2 phase is one of most interests to increase the ductility due to the enough independent slip systems. In addition, Hadi et al. [24] have suggested that the rare earth elements La and Er can stabilize the b
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3. Results and discussion 3.1. Site preference The formation enthalpy was calculated because it can directly reflect the site occupancy behaviors of the additive elements W, Mo, Sc and Yb in L10 and B2 TiAl. The formation enthalpy is defined as [29].
DH ¼ Etotal ðTi0:5 Al0:5x Mx Þ 0:5Esolid ðTiÞ ð0:5 xÞEsolid ðAlÞ xEsolid ðMÞ Fig. 1. The unit cells of (a) L10 and (b) B2 TiAl.
for the ternary elements on Al site. While, the ternary elements on Ti site can be written as
phase of TiAl and the B2 structure could be processed with the addition of rare earth elements La and Er investigated by scanning electron microscopy. Besides, Guo et al. [25] found that rare earth elements Y, Ce, Nd and Dy can improve the compressive ductility and yield strength of AlNi based eutectic alloy at room temperature. In this paper, we focus on the site occupancy, phase stability and transition, elastic properties of TiAl with ternary alloying elements W, Mo, Sc, and Yb using the first-principles density functional theory. Pugh's criteria and Poisson's ratio are used to reveal the brittle vs ductile behavior. Some basic physical phenomena, such as Young's modulus and hardness are also presented. The density of states (DOS) are used to establish the direct connection between the changes in interatomic bonding and the mechanical properties of the ternary alloying elements in TiAl. Finally, some conclusions are presented.
DH ¼ Etotal ðTi0:5x Al0:5 Mx Þ ð0:5 xÞEsolid ðTiÞ 0:5Esolid ðAlÞ xEsolid ðMÞ where Etotal(Ti0.5Al0.5xMx) and Etotal(Ti0.5xAl0.5Mx are the average total energy of each atom under equilibrium lattice parameter, Esolid(Ti), Esolid(Al) and Esolid(M) are the average total energy of each atom under bulk states, respectively. The site preference behaviors are given in Table 1. It can be seen that rare earth elements Sc and Yb always preferentially occupy the Ti sites both for L10 and B2 phase, and transition metals W and Mo have a weak site preference for the Ti site in L10 phase, while they have a strong site preference for the Al site in B2 phase. The results are in accordance with previous experimental and theoretical results [21,30,31].
3.2. Phase stability and transition
2. Computational methodology The first-principles calculations presented here are based on electronic density-functional theory (DFT) framework implemented in the VASP (Vienna ab-initio simulation) package [26,27]. All of our calculations were performed with PAW pseudo-potentials in generalized gradient approximation (GGA) refined by Perdew, Burke and Ernzerhof (PBE) [28]. VASP is a plane-wave code, and uses iterative strategies based on residual minimization and preconditioned conjugate-gradient techniques for the diagonalization of the KohneSham Hamiltonian. In the plane-wave soft-pseudopotential method, the electronic wave functions are expanded in plane waves. A plane-wave cutoff energy of 450 eV was employed throughout the calculations. Convergency of the total energies with respect to basis size and numbers of 15 15 15, 15 15 7, 7 7 15, 7 7 7 k-points for the Brillouin zone integration were used for geometry optimizations for the four types supercells (1 1 1, 1 1 2, 2 2 1, 2 2 2) respectively which are sufficiently large. The spin polarization was considered for the calculation of DOS and the elastic constants were derived from the total energy calculations correspond to single crystal elastic properties.
The equilibrium lattice and formation enthalpy for L10 and B2 structure are given in Table 2. The calculated lattice constant (a ¼ 3.999 Å and c/a ¼ 1.017) are in agreement with the experimental and theoretical values which suggest that a weak tetragonal of L10 TiAl. The predicted formation enthalpy of L10 TiAl, 38.92 kJ/ mol, is about 6% lower than experimental result (36.7 kJ/mol).
Table 2 The lattice constant (Å) and formation enthalpy DH (kJ/mol) of perfect stoichiometric L10 and B2 TiAl compared with previous calculations and available experimental measurements. TiAl L10
B2
a c/a DH a DH
This worka
Previous calculations
Experiment
3.999 1.017 38.92 3.187 25.42
3.995b,3.998c 1.02b,1.047c 39.2b, 38.96c, 41.47d 3.196b 25.5b, 26.04g,27.97h
3.99e 1.02e 36.7f 3.16i e
c,g
embedded atom method (EAM), Ref. [32]. LAPW/GGA, Ref. [32]. VASP-PAW/GGA. b full-potential linearized augmented plane-wave (FLAPW) method, Ref. [31]. e Ref. [36]. f Ref. [37]. i Annealing of the rolled specimens at 1000 C, Ref. [38].
d,h
a
Table 1 The formation enthalpy DH (kJ/mol) and site occupancy behaviors of the additive elements M (M ¼ Sc, Yb, W, Mo) in TiAl. TiAl
Sc
DH
ScAl 26.00 Ti 16.81 Ti
L10 site occupancy B2 site occupancy
Yb ScTi 35.83 25.18
YbAl 5.41 Ti 7.03 Ti
W YbTi 16.62 4.73
WAl 25.73 Ti 30.14 Al
Mo WTi 25.86 2.96
MoAl 28.94 Ti 32.83 Al
MoTi 31.17 10.86
H. Hu et al. / Journal of Alloys and Compounds 658 (2016) 689e696
Fig. 2. The formation enthalpy with various impurity concentrations. It is clear to see that there will be a structure phase transition from L10 type to a B2 type with increasing doping concentration when W ups to 10.50 at.% and Mo ups to 11.50 at.%. It is worth mentioning that when the doped Yb ups to about 20 at.% the TiAl could not been stably existed.
These deviations may result from the influence of temperature in experiment. Besides, about 6.5% error between our data and Rajendra's [32] calculation. The deviation may be caused by the selection of pseudopotential, k-points or cutoff energy. For B2 phase, the equilibrium lattice constants is 3.187 Å. The formation enthalpy of B2 structure, 25.42 kJ/mol, is significantly higher than L10 phase, indicating that the L10 phase is more stable than B2 phase (see Fig. 1).. The stabilization effects of transition metals (W, Mo) and rare earth elements (Sc, Yb) are shown in Fig. 2. The stability of L10 phase becomes worse and worse with the increasing of concentrations. However, it's worth mentioning that formation enthalpy of B2 phase has a slight decreases implying that the stability of B2 phase was stabilized effectively by W and Mo. In particular, it is found that there will be a phase transition from L10 type to a B2 type when the W ups to 10.50 at.% and Mo ups to 11.50 at.%. However, there is no phase transition for L10 TiAl with Sc and Yb. Therefore, we can predict that the ductility will be improved effectively due to the B2 phase with a number of independent sliding systems with W and Mo alloying elements. 3.3. Mechanical properties
Table 3 Calculated elastic constants (GPa) of L10 TiAl with previous calculations and available experimental measurements. Cij This work Exp.b Exp.c Cal.d Cal.e a b c d e
a
2 . All of crystal are C44 > 0, C66 > 0, C11 > jC12j, ðC11 þ C12 ÞC33 > 2C13 the Cij values for L10 TiAl satisfy these criteria [33]. In L10 TiAl which has the larger C11 and C33. This can be explained by the orientation and the nature of TieAl bonds in the crystal, but the difference between C11 and C33 is not obvious, which can be explained by the slight tetragonal symmetry of TiAl lattice. Following the convention of this work, the elastic constants of C11 and C33 are directly relate to sound propagation along the crystallographic. And the bonds parallel to the c-axis have a dominating effect on C11 while C33 is dominated the parallel to the a-axis [34]. Furthermore, the calculation results show that there are larger C33 or C11 than other Cij (isj) which makes the structure stiffer along the a-axis and c-axis. Which suggests that TiAl is stiffer along axial directions but less stiffer along non-axial directions. The elastic constants of TiAl with W, Mo, Sc and Yb are given in Fig. 3 (a), (b), (c) and (d), respectively. Based on the calculated elastic constants, the bulk modulus (B), shear modulus (G), Young's modulus (Y) and Poisson's ratio (n) can be evaluated in terms of the Voigt-Reuss-Hill (VRH) scheme [41]. The theoretical upper bounds Voigt approximation which assuming uniform strain throughout a polycrystal are given as follows
1 ð2C11 þ 2C12 þ 4C13 þ C33 Þ 9 1 ð2C11 C12 2C13 þ C33 þ 6C44 þ 3C66 Þ GV ¼ 15 BV ¼
and the theoretical lower bounds Reuss approximation which assuming uniform stress are written as
BR ¼ ½S11 þ S22 þ S33 þ 2ðS12 þ S13 þ S23 Þ1 GR ¼ ½4ðS11 þ S22 þ S33 Þ þ 4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ1 where Sij are elastic compliances and their values can be given by the inversion of elastic constants matrix ðSij ¼ Cij1 Þ. We estimated these bulk values by the Voigt-Reuss-Hill approximation:
1 ðB þ BV Þ 2 R 1 GH ¼ ðGR þ GV Þ 2 9BH GH Y¼ 3BH þ GH BH ¼
n¼
In order to analyze the alloying effect of transition metals (W, Mo) and rare earth elements (Sc, Yb) on the mechanical properties of TiAl, the elastic constants of TiAl are calculated. The elastic constants of TiAl are in consistent with previous results (see Table 3). The requirements of mechanical stability in a tetragonal
C11
C12
C13
C33
C44
C66
173 186 187 171 195
83 72 75 96 107
84 74 75 97 113
168 176 182 175 213
111 101 109 111 92
75 77 81 84 84
VASP PAW GGA. resonant ultrasonic spectroscopy (RUS) technique, see Ref. [39]. rectangular parallelepiped resonance (RPR)method, see Ref. [40]. plane-wave pseudopotential method, see Ref. [30]. embedded atom method, see Ref. [32].
691
3BH 2GH : 6BH þ 2GH
The bulk modulus, shear modulus are give in Fig. 3 (e). As we know, the bulk modulus reflects the resistance to bond-length change, and shear modulus directly correspond to the bond-angle change with applied stress. So, it is believed that the material is ductile if it has larger bulk modulus and smaller shear modulus. Both B and G increase with concentrations of W and Mo from 6.25% to 12.50%. But, the increment of B is larger than that of G. It means that the additional of ternary W and Mo can effectively improve the ductility of TiAl. Furthermore, Young's modulus is another important physical quantity which reflects the resistance of a material toward uniaxial tensions. Larger value indicating the stronger tensile strength. The calculated results are given in Fig. 3 (f). It is clearly that the low concentration of Sc and Yb has smaller Young's modulus. However, it has a sharp rise when the concentration exceeds a certain value (about 12.50 at.%) means that the high concentration of rare earth elements Sc and Yb will enhance the hardness of TiAl. But the opposite trends for W and Mo. The low concentration of W and Mo has smaller Young's modulus (L10) but high concentration corresponds to larger modulus (B2 phase).
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Fig. 3. The elastic constants, shear modulus and bulk modulus of TiAl with W, Mo, Sc and Yb.
Therefore, we believe that the hardness will be reduced due to the phase transition. It is well known that the ductile material is malleable and can be easily adjusted to different shapes. However, the brittle materials is
difficult to machine. And on the other hand, the brittle materials are sensitive to the sudden failure under critical mechanical stress conditions. There is no uniform standard to evaluating the ductility or brittleness of a material directly based on elastic constants
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Fig. 4. The brittleeductile transition with various alloying concentrations. It is worthwhile to mention that the W and Mo can effectively improve the ductility of TiAl, or even transform its intrinsic brittleness into ductility. Moreover, it is shown that a small number of Sc (6.25 at.%) and Yb (<12.50 at.%) can also increase the ductility of TiAl-based alloy.
derived from linear elastic theory, but a reasonable representation of it can be realized by the combination of BH and GH. The ductility and brittleness can be judged according to Pugh's criterion (G/B) [35]. The G/B ratio separates ductile (<0.5) and brittle (>0.5) materials. Fig. 4 illustrates the phenomenological estimation of brittle vs ductile behavior of TiAl alloys with various of ternary W, Mo, Sc and Yb. First, we consider alloying of TiAl with rare earth elements Sc. The G/B ratio for TiAleSc alloys change their values only slightly. It is found that with the Sc concentration up to 6.25 at.%, but the ratio G/B decreases slightly. Which indicates that the low concentration of Sc can improve the ductility. While, there is an obvious rising trend of G/B from 6.25 at.% to 18.75 at.%. It means that the ductility will getting poor with higher concentration of Sc, but a steep decline from 18.75 at.% to 25.00 at.%. This undulating change may be caused by the complex interatomic bonding with the additional of rare earth elements Sc, and we will try to give a reasonable explanation by using of the electronic structure. However, it is found that the additive element of Yb can effectively
693
improve the ductility when the Yb concentration up to 6.25 at.% and the TiAleYb almost getting into the ductile area. But it was very regret that a steep rise of G/B with higher concentration of Yb contains from 6.25 at.% to 18.75 at.% means the ductility will deteriorate. The following is our most concern about the effects on TiAl of W and Mo. It's worth mentioning that it is not only improve the ductility of TiAl effectively but unexpectedly transform its intrinsic brittleness into ductility. We may conclude that the phase transition from L10 phase to a B2 phase is the essential reason for this phenomenon by the comparison with Sc and Yb. Besides the empirical Pugh's criterion, Poisson's ratio is an elastic coefficient that provides insight into chemical bonding of atoms regarding the variations of bond angle and bond length. In spite of the lack of a direct link between elastic properties and plastic deformation behavior, as evidenced by a large group of brittle ordered intermetallic compounds with high Poisson's ratio [42]. According to this point, the addition of W and Mo could improve the ductility in some extent. Based on the above analysis, we can conclude that the phase transition from L10 to B2 is the essential reason improving the ductility of TiAl. As we know, the atomic arrangements are not the same along different directions in lattice. The anisotropy makes the crystal has the different physical and chemical properties in different directions. And there is research showing that the influence of microcracks and lattice distortion on material are almost always caused by the elastic anisotropy [43]. The anisotropy index is a measure to quantify the anisotropy in real materials which can be traced back to the first time introduced into the cubic crystals by Zener [44]. In the recent years, a universal elastic anisotropy index AU has been introduced to describe the anisotropy of real materials by Ranganathan et al. [45]. The universal elastic anisotropy index AU can be written as AU ¼ 5 GGVR þ BBVR 6, which is single-value. The value of AU is equal to 0 respects the isotropic materials, otherwise it is anisotropy materials. The calculated universal elastic anisotropy index as a function of various doping concentration are presented in Fig. 5 (a). It is clearly shown that the smaller anisotropy index for TiAl with the ternary elements Sc and W under low concentration (<6.25 at.% for W, <18.75 at.% for Sc). This is why the smaller difference between C11 (C44) and C33 (C66) in the two doped regions (see Fig. 3 (a) and Fig. 3 (c)).
Fig. 5. Mechanical properties of TiAl with W, Mo, Sc and Yb: (a) Universal anisotropy index AU and Poisson's ratio n; (b) Vichers hardness Hv.
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Fig. 6. The calculated DOS of the TiAl and TiAl with W, Mo, Sc, Yb.
Besides ductile/brittle and anisotropy, hardness is also one of the most basic mechanical properties of the material, which is used to characterize the ability of solids resistance to elastic and permanent plastic deformation or brittle failure [46,47]. So far, it still
remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity. Results from elastic constants suggest that the hardness is closely associated with elastic properties. It is generally believed that there exists a necessary
H. Hu et al. / Journal of Alloys and Compounds 658 (2016) 689e696
695
Fig. 6. (continued).
inherent relationship between hardness and elastic modulus. Recently, a simplified formula of Vickers hardness has been proposed by Chen et al. [48,49] as summarized by Teter's empirical correlation [50]. The Vickers hardness can be expressed as HV ¼ 2ðk2 GÞ0:585 3, where k is the Pugh ratio (k ¼ G/B). It indicates that the hardness correlates with both shear modulus and Pugh's ratio. Fig. 5 (b) illustrates the estimation hardness of TiAl alloys with various concentrations of ternary W, Mo, Sc and Yb. It is shown a larger ratio of G/B means a higher hardness. The value of HV is sensitive to the kinds and concentrations of the additive elements. Overall, the transition metals W and Mo can reduce the hardness of TiAl. 3.4. Electronic structure The interactions in TiAl alloys are rather complex, because the compounds not only possess covalent bonding but also have significant metallic bonding. In order to understand the stability and bonding character in TiAl alloys, we calculated the electronic density of states (DOS) of TiAl with alloying elements (see Fig. 6). The spin polarization was considered for the calculation of DOS. But the DOS of spin up is almost symmetric to the DOS of spin down, and there are very subtle differences between the DOS of spin up and
spin down (see Fig. 6 (a) and (b)). In addition, the DOS of spin up is enough to give a reasonable explanation of the mechanical properties. Hence, we just give the DOS of spin up with various additional ternary elements contains. For the L10 and B2 TiAl phase the main contribution to the DOS around Fermi level (Ef) comes from d bands of Ti. The total DOS (TDOS) shows that there is a deep valley close to the Fermi level in L10 phase and this valley is termed as pseudogap. This pseudogap indicates the presence of covalent bonding in L10 TiAl. Therefore the L10 structure is stable [21,34]. The partial density of states (PDOS) suggest that the covalent mainly results from the strong interactions between Ti-3d and Al-3p orbitals (4 eV ~ 4 eV) since their PDOSs overlapping, and the metallic bonding between Ti and Al atoms is weak. The strong covalent but weak metallic interactions, causing the uneven distribution of these binding forces, results in cleavage fracture along the direction of metallic interactions. The structure stability of the materials is closely related to the position of Ef. The Fermi level is located in the valley of the overlapping regions between the bonding states and anti-bonding states then the system will be more stable [51,52]. This suggests that L10 phase is more stable than B2 type. The DOS structure of TiAl with W, Mo, Sc, Yb are also given in Fig. 6. The most interesting feature is that the Fermi level has an
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obvious shift towards to the bonding area with the increasing of alloying concentrations for W and Mo contains (see Fig. 6 (a) and (c)) from 0 at.% to 6.25 at.%. Furthermore, the localized DOS is enhanced compared to the pure L10 phase mainly contributed by Ti-d and hybridized Al-p and W-d (Mo-d) orbitals. This also implies the increasing formation enthalpy with W and Mo contains, therefore the L10 phase's stability is getting worse and worse and there is phase transition to B2 phase. As mentioned above, the bonding properties can be reflected from the DOS structure. It can bee seen from the TDOS that the pure L10 has a wider pseudogap (>2 eV) and intrinsic brittleness. The bandwidth of pseudogap become narrow when the concentrations for W and Mo increasing ((see Fig. 6 (c), (d), (e) and (f)). It's worth mentioning that the bandwidth of pseudogap become more narrow when the concentrations of W and Mo is 12.50 at.% and the TiAl based alloys is B2 phase (see Fig. 6 (d)). Therefore, the transformation form L10 to B2 phase may be the predominant origin of the ductile/brittle transition of TiAl (see Fig. 4). The DOS of TiAl with Sc and Yb are shown in Fig. 6(g) (6.25 at.%), Fig. 6(h) (12.50 at.%), Fig. 6 (i) (18.75 at.%) and Fig. 6 (j) (25.00 at.%). As we can see, the lower concentration (6.25 at.%) of Sc and Yb has narrower pseudogap (<2 eV), but higher concentration (6.25 at.% ~ 18.75 at.% without phase transformation) has wider pseudogap (>2 eV). Thus, the metallic bond will be enhanced by lower concentrations of Sc and Yb (6.25 at.%). However, it's also worth mentioning that the pseudogap become very narrow (about 1.2 eV) at very high concentration (25.00 at.%) for Sc contains, which may explain the decreased G/B from the concentration of 18.75 at.% to 25.00 at.% in Fig. 4. Comparing with W and Mo, higher concentrations of Sc and Yb will not lead to phase transformation. Both metallic bond is also not enhanced, and ductility is also not improved. Furthermore, the PDOS show that one important factor to improve the ductility is due to the weakened ped interactions between Al and Ti atoms but enhanced ded interactions between W (Mo) and Ti atoms in TiAl based alloys [6]. Therefore, the analysis of DOS is consistent with the results of brittle/ductile map of TiAl based alloys (see Fig. 4). 4. Conclusions The phase stability and site preference of alloying elements W, Mo, Sc and Yb in L10 and B2 TiAl alloys were investigated using first-principles calculations. It is found that rare earth elements Sc and Yb always preferentially occupy the Ti sites both for L10 and B2 phase, and transition metals W and Mo have a site preference at Ti site in L10 phase, while they have a strong site preference for the Al site in B2 phase. It is interesting to find that there will be a phase transition from L10 to B2 structure when the concentrations of W and Mo are about 10.50 at.% and 11.50 at.%, respectively. This is in an agreement with previous theoretical and experimental results. However, there is no phase transition for TiAl alloyed with rare earth elements Sc and Yb. The elastic constants, bulk modulus, shear modulus, Poisson's ratio and hardness of TiAl alloys are also systematically presented. The Pugh's ratio was used to give a phenomenological estimation of brittle/ductile behaviors. The results show that the enhancement of ductility is due to the phase transition from the less slip systems L10 to a more slip systems B2 phase of TiAl based alloys. Finally, the DOS is also used to analysis the mechanical properties. The results show that the pseudogap
will become narrow for TiAl with W and Mo alloying elements, and the narrow pseudogap indicates the enhancement of metallic bond which can improve the ductility. These are in consistent with brittle/ductile transition map of TiAl based alloys. Acknowledgments The work is supported by the Natural Science Foundation of China (11104361) and Projects supported by the Fundamental Research Funds for the Central Universities (CDJZR14328801 and CQDXWL2014003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]
M. Yamaguchi, H. Inui, K. Ito, Acta Mater. 48 (2000) 307. F. Appel, M. Oehring, R. Wanger, Intermetallics 8 (2000) 1283. G. Ghosh, A. van de Walle, M. Asta, Acta Mater. 56 (2008) 3202. C.T. Liu, J.L. Wright, S.C. Deevi, Mater. Sci. Eng. A 329 (2002) 416. naff, A.L. Gloanec, Intermetallics 13 (2005) 543. G. He M. Morinaga, J. Saito, N. Yukawa, et al., Acta Metall. 38 (1990) 25. R.B. Schwarz, P.B. Desch, S. Srinivasan, et al., Nanostruct. Mater. 1 (1992) 37. G.A. Salishchev, R.M. Imayev, O.N. Senkov, et al., Mater. Sci. Eng. A 286 (2000) 236. M. Beschliesser, A. Chatterjee, A. Lorich, et al., Mater. Sci. Eng. A 329 (2002) 124. B.Y. Huang, Y.H. He, J.N. Wang, Intermetallics 7 (1999) 881. Z.C. Liu, J.P. Lin, S.J. Li, et al., Intermetallics 10 (2002) 653. S. Shu, F. Qiu, C. Tong, et al., J. Alloys Comp. 617 (2014) 302. S.C. Huang, E.L. Hall, Metall. Trans. A 22 (1991) 2619. S.C. Huang, E.L. Hall, Acta Metall. 39 (1991) 1053. T. Tetsui, K. Shindo, S. Kobayashi, et al., Scr. Mater. 47 (2002) 399. T. Tetsui, K. Shindo, S. Kobayashi, et al., Intermetallics 11 (2003) 299. F. Appel, M. Oehring, J.D.H. Paul, Adv. Eng. Mater. 8 (2006) 371. T. Tetsui, K. Shindo, S. Kaji, et al., Intermetallics 13 (2005) 971. M. Kimura, K. Hashimoto, H. Morikawa, Mater. Sci. Eng. A 152 (1992) 54. T.T. Cheng, M.R. Willis, I.P. Jones, Intermetallics 7 (1999) 89. R. Yu, L.L. He, Z.Y. Chen, et al., Intermetallics 10 (2002) 661. Y. Zheng, L. Zhao, K. Tangri, Mater. Sci. Eng. A 208 (1996) 80. Y.G. Li, M.H. Loretto, Acta Metall. Mater. 42 (1994) 2913. M. Haidi, M.A. Shafyei, M. Meratian, Mater. Sci. Eng. A 624 (2015) 1. J.T. Guo, K.W. Huai, Q. Gao, et al., Intermetallics 15 (2007) 727. G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. A. Zoroddu, F. Bernardini, P. Ruggerone, et al., Phys. Rev. B 64 (2001) 045208. C. Woodward, S. Kajihara, L.H. Yang, Phys. Rev. B 57 (1998) 21. R. Yu, L.L. He, H.Q. Ye, Phys. Rev. B 65 (2002) 184102. R.Z. Rajendra, Y. Mishin, Phys. Rev. B 68 (2003) 024102. J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985. P. Ravindran, L. Fast, P.A. Korzhavyi, et al., J. Appl. Phys. 84 (1998) 4891. S.F. Pugh, Philos. Mag. Ser. 45 (1954) 823. E.A. Brandes (Ed.), Smithells Metal Reference Book, sixth ed., Butterworth, , London, 1983. P.D. Desai, J. Phys. Chem. Ref. Data 16 (1987) 109. B.J. Inkson, H. Clemens, J. Marien, Scr. Mater. 38 (1998) 1377. Y. He, R.B. Schwarz, A. Migliori, S.H. Whang, J. Mater. Res. 10 (1995) 1187. K. Tanaka, T. Ichitsubu, H. Iniu, et al., Philos. Mag. Lett. 73 (1996) 71. O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909. K. Gschneidner, A. Russell, A. Pecharsky, et al., Nat. Mater. 2 (2003) 587. V. Tvergaard, J.W. Hutchinson, J. Am. Ceram. Soc. 71 (1988) 157. C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, 1948. S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. V.V. Brazhkin, A.G. Lyapin, R.J. Hemley, Philos. Mag. A 82 (2002) 231. A.S. Povarennykh, Press of AN USSR, Kiev, 1963. X.Q. Chen, H. Niu, D. Li, et al., Intermetallics 19 (2011) 1275. X.Q. Chen, H. Niu, C. Franchini, et al., Phys. Rev. B 84 (2011) 121405. D.M. Teter, MRS Bull. 23 (1998) 22. J.H. Xu, T. Oguchi, A.J. Freeman, Phys. Rev. B 35 (1987) 6940. T. Hong, A.J. Freeman, J. Mater. Res. 19916 (1991) 330.
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Phase stability, mechanical properties and electronic structure of TiAl alloying with W, Mo, Sc and Yb: First-principles study Hai Hu a, Xiaozhi Wu a, b, Rui Wang a, *, Weiguo Li c, Qing Liu b a
Institute for Structure and Function, Chongqing University, 401331, China College of Materials Science and Engineering, Chongqing University, 400044, China c College of Aerospace Engineering, Chongqing University, 400044, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 August 2015 Received in revised form 28 October 2015 Accepted 29 October 2015 Available online 9 November 2015
The phase stability and site preference of alloying elements W, Mo, Sc and Yb in L10 and B2 TiAl alloys were investigated using first-principles calculations. It is found that there will be a phase transition from L10 to B2 structure when the concentrations of W and Mo are about 10.50 at.% and 11.50 at.%, respectively. However, there is no phase transition for L10 TiAl with Sc and Yb. The elastic constants, bulk modulus, shear modulus, Poisson's ratio and hardness of TiAl alloys are also systematically presented. Based on Pugh ratio, the brittle/ductile transition are predicted for TiAl with W, Mo, Sc and Yb. It is found that W and Mo can effectively improve the ductility of TiAl due to the phase transition. Finally, the density of states is also used to analysis the mechanical properties, which is consistent with brittle/ ductile transition map of TiAl based alloys. © 2015 Elsevier B.V. All rights reserved.
Keywords: Phase stability Elastic constants TiAl First principles
1. Introduction TiAl has attracted significantly interesting due to its high melting temperature, adequate high temperature strength, excellent oxidation and corrosion resistance, good strength retention ability and low density is hopeful to be the new promising lightweight and heat-resistant functional material [1e3]. The inherent brittleness at low temperature, is by far the greatest hurdle that prevents their widely used in industrial applications [4e6]. TiAl has an ordered g-phase (ordered face-centered tetragonal L10 AuCutype) in which Ti and Al atoms alternatively occupy the (002) planes, and lattice constant ratio c/a is about 1.02 to show a slightly tetragonal (see Fig. 1). The low symmetric L10 phase cannot provide enough sliding systems to satisfy von Mises' criterion and exhibits poor ductility [7]. For handling this shortcoming, a lot of research work has been carried out to improve its ductility, such as heat treatment and elements alloying, as well as the introduction of a second reinforcing phase, etc [8e11]. Among them alloying is one of mature methods. Shili et al. suggest that Fe and Co atoms preferentially occupy the Al sites and can improve the electronic structures and elastic properties of TiAl, leading to the
* Corresponding author. E-mail addresses: [email protected] (R. Wang).
(X.
http://dx.doi.org/10.1016/j.jallcom.2015.10.270 0925-8388/© 2015 Elsevier B.V. All rights reserved.
Wu),
[email protected]
improvement of the ductility of TiAl [12]. V, Cr or Mn can also improve the ductility of TiAl alloys buy using a discrete variational (DV) Xa cluster method [6,13,14]. An alloy design strategy to improve the ductility of TiAl is to exploit the disordered b-phase (disordered BCC B2 AuCu-type) with alloying elements at elevated temperatures [15e18]. The B2 phase, which provides a sufficient number of independent sliding systems. Thus, it may improve the deformability at elevated temperature. Kimura et al. [19] have pointed out that V, Cr, Mn, Nb or Mo in L10 TiAl could lead to the change of phase equilibria and the B2 phase is stabilized. Cheng et al. [20] have suggested that the addition of B appears to be beneficial in TiAl that both strength and ductility are improved. Studies have shown that the addition of transition metal W can stabilize the high symmetry B2 phase nucleation preferably [21]. Moreover, the presence of 3 at.% Cr in L10 TiAl is expected to produce B2 phase and thus improves the tensile ductility [22]. In another study, B2 phase was produced to the TiAl alloy by adding 2 at.% Mo, hence improving room-temperature tensile ductility and strength [23]. Moreover, the addition of those transition metal elements into TiAl which weaken ped interactions but enhance ded interactions is most effective in improving the ductility in theory suggested by Morinaga et al. [6]. Thus, the stabilization of B2 phase is one of most interests to increase the ductility due to the enough independent slip systems. In addition, Hadi et al. [24] have suggested that the rare earth elements La and Er can stabilize the b
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3. Results and discussion 3.1. Site preference The formation enthalpy was calculated because it can directly reflect the site occupancy behaviors of the additive elements W, Mo, Sc and Yb in L10 and B2 TiAl. The formation enthalpy is defined as [29].
DH ¼ Etotal ðTi0:5 Al0:5x Mx Þ 0:5Esolid ðTiÞ ð0:5 xÞEsolid ðAlÞ xEsolid ðMÞ Fig. 1. The unit cells of (a) L10 and (b) B2 TiAl.
for the ternary elements on Al site. While, the ternary elements on Ti site can be written as
phase of TiAl and the B2 structure could be processed with the addition of rare earth elements La and Er investigated by scanning electron microscopy. Besides, Guo et al. [25] found that rare earth elements Y, Ce, Nd and Dy can improve the compressive ductility and yield strength of AlNi based eutectic alloy at room temperature. In this paper, we focus on the site occupancy, phase stability and transition, elastic properties of TiAl with ternary alloying elements W, Mo, Sc, and Yb using the first-principles density functional theory. Pugh's criteria and Poisson's ratio are used to reveal the brittle vs ductile behavior. Some basic physical phenomena, such as Young's modulus and hardness are also presented. The density of states (DOS) are used to establish the direct connection between the changes in interatomic bonding and the mechanical properties of the ternary alloying elements in TiAl. Finally, some conclusions are presented.
DH ¼ Etotal ðTi0:5x Al0:5 Mx Þ ð0:5 xÞEsolid ðTiÞ 0:5Esolid ðAlÞ xEsolid ðMÞ where Etotal(Ti0.5Al0.5xMx) and Etotal(Ti0.5xAl0.5Mx are the average total energy of each atom under equilibrium lattice parameter, Esolid(Ti), Esolid(Al) and Esolid(M) are the average total energy of each atom under bulk states, respectively. The site preference behaviors are given in Table 1. It can be seen that rare earth elements Sc and Yb always preferentially occupy the Ti sites both for L10 and B2 phase, and transition metals W and Mo have a weak site preference for the Ti site in L10 phase, while they have a strong site preference for the Al site in B2 phase. The results are in accordance with previous experimental and theoretical results [21,30,31].
3.2. Phase stability and transition
2. Computational methodology The first-principles calculations presented here are based on electronic density-functional theory (DFT) framework implemented in the VASP (Vienna ab-initio simulation) package [26,27]. All of our calculations were performed with PAW pseudo-potentials in generalized gradient approximation (GGA) refined by Perdew, Burke and Ernzerhof (PBE) [28]. VASP is a plane-wave code, and uses iterative strategies based on residual minimization and preconditioned conjugate-gradient techniques for the diagonalization of the KohneSham Hamiltonian. In the plane-wave soft-pseudopotential method, the electronic wave functions are expanded in plane waves. A plane-wave cutoff energy of 450 eV was employed throughout the calculations. Convergency of the total energies with respect to basis size and numbers of 15 15 15, 15 15 7, 7 7 15, 7 7 7 k-points for the Brillouin zone integration were used for geometry optimizations for the four types supercells (1 1 1, 1 1 2, 2 2 1, 2 2 2) respectively which are sufficiently large. The spin polarization was considered for the calculation of DOS and the elastic constants were derived from the total energy calculations correspond to single crystal elastic properties.
The equilibrium lattice and formation enthalpy for L10 and B2 structure are given in Table 2. The calculated lattice constant (a ¼ 3.999 Å and c/a ¼ 1.017) are in agreement with the experimental and theoretical values which suggest that a weak tetragonal of L10 TiAl. The predicted formation enthalpy of L10 TiAl, 38.92 kJ/ mol, is about 6% lower than experimental result (36.7 kJ/mol).
Table 2 The lattice constant (Å) and formation enthalpy DH (kJ/mol) of perfect stoichiometric L10 and B2 TiAl compared with previous calculations and available experimental measurements. TiAl L10
B2
a c/a DH a DH
This worka
Previous calculations
Experiment
3.999 1.017 38.92 3.187 25.42
3.995b,3.998c 1.02b,1.047c 39.2b, 38.96c, 41.47d 3.196b 25.5b, 26.04g,27.97h
3.99e 1.02e 36.7f 3.16i e
c,g
embedded atom method (EAM), Ref. [32]. LAPW/GGA, Ref. [32]. VASP-PAW/GGA. b full-potential linearized augmented plane-wave (FLAPW) method, Ref. [31]. e Ref. [36]. f Ref. [37]. i Annealing of the rolled specimens at 1000 C, Ref. [38].
d,h
a
Table 1 The formation enthalpy DH (kJ/mol) and site occupancy behaviors of the additive elements M (M ¼ Sc, Yb, W, Mo) in TiAl. TiAl
Sc
DH
ScAl 26.00 Ti 16.81 Ti
L10 site occupancy B2 site occupancy
Yb ScTi 35.83 25.18
YbAl 5.41 Ti 7.03 Ti
W YbTi 16.62 4.73
WAl 25.73 Ti 30.14 Al
Mo WTi 25.86 2.96
MoAl 28.94 Ti 32.83 Al
MoTi 31.17 10.86
H. Hu et al. / Journal of Alloys and Compounds 658 (2016) 689e696
Fig. 2. The formation enthalpy with various impurity concentrations. It is clear to see that there will be a structure phase transition from L10 type to a B2 type with increasing doping concentration when W ups to 10.50 at.% and Mo ups to 11.50 at.%. It is worth mentioning that when the doped Yb ups to about 20 at.% the TiAl could not been stably existed.
These deviations may result from the influence of temperature in experiment. Besides, about 6.5% error between our data and Rajendra's [32] calculation. The deviation may be caused by the selection of pseudopotential, k-points or cutoff energy. For B2 phase, the equilibrium lattice constants is 3.187 Å. The formation enthalpy of B2 structure, 25.42 kJ/mol, is significantly higher than L10 phase, indicating that the L10 phase is more stable than B2 phase (see Fig. 1).. The stabilization effects of transition metals (W, Mo) and rare earth elements (Sc, Yb) are shown in Fig. 2. The stability of L10 phase becomes worse and worse with the increasing of concentrations. However, it's worth mentioning that formation enthalpy of B2 phase has a slight decreases implying that the stability of B2 phase was stabilized effectively by W and Mo. In particular, it is found that there will be a phase transition from L10 type to a B2 type when the W ups to 10.50 at.% and Mo ups to 11.50 at.%. However, there is no phase transition for L10 TiAl with Sc and Yb. Therefore, we can predict that the ductility will be improved effectively due to the B2 phase with a number of independent sliding systems with W and Mo alloying elements. 3.3. Mechanical properties
Table 3 Calculated elastic constants (GPa) of L10 TiAl with previous calculations and available experimental measurements. Cij This work Exp.b Exp.c Cal.d Cal.e a b c d e
a
2 . All of crystal are C44 > 0, C66 > 0, C11 > jC12j, ðC11 þ C12 ÞC33 > 2C13 the Cij values for L10 TiAl satisfy these criteria [33]. In L10 TiAl which has the larger C11 and C33. This can be explained by the orientation and the nature of TieAl bonds in the crystal, but the difference between C11 and C33 is not obvious, which can be explained by the slight tetragonal symmetry of TiAl lattice. Following the convention of this work, the elastic constants of C11 and C33 are directly relate to sound propagation along the crystallographic. And the bonds parallel to the c-axis have a dominating effect on C11 while C33 is dominated the parallel to the a-axis [34]. Furthermore, the calculation results show that there are larger C33 or C11 than other Cij (isj) which makes the structure stiffer along the a-axis and c-axis. Which suggests that TiAl is stiffer along axial directions but less stiffer along non-axial directions. The elastic constants of TiAl with W, Mo, Sc and Yb are given in Fig. 3 (a), (b), (c) and (d), respectively. Based on the calculated elastic constants, the bulk modulus (B), shear modulus (G), Young's modulus (Y) and Poisson's ratio (n) can be evaluated in terms of the Voigt-Reuss-Hill (VRH) scheme [41]. The theoretical upper bounds Voigt approximation which assuming uniform strain throughout a polycrystal are given as follows
1 ð2C11 þ 2C12 þ 4C13 þ C33 Þ 9 1 ð2C11 C12 2C13 þ C33 þ 6C44 þ 3C66 Þ GV ¼ 15 BV ¼
and the theoretical lower bounds Reuss approximation which assuming uniform stress are written as
BR ¼ ½S11 þ S22 þ S33 þ 2ðS12 þ S13 þ S23 Þ1 GR ¼ ½4ðS11 þ S22 þ S33 Þ þ 4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ1 where Sij are elastic compliances and their values can be given by the inversion of elastic constants matrix ðSij ¼ Cij1 Þ. We estimated these bulk values by the Voigt-Reuss-Hill approximation:
1 ðB þ BV Þ 2 R 1 GH ¼ ðGR þ GV Þ 2 9BH GH Y¼ 3BH þ GH BH ¼
n¼
In order to analyze the alloying effect of transition metals (W, Mo) and rare earth elements (Sc, Yb) on the mechanical properties of TiAl, the elastic constants of TiAl are calculated. The elastic constants of TiAl are in consistent with previous results (see Table 3). The requirements of mechanical stability in a tetragonal
C11
C12
C13
C33
C44
C66
173 186 187 171 195
83 72 75 96 107
84 74 75 97 113
168 176 182 175 213
111 101 109 111 92
75 77 81 84 84
VASP PAW GGA. resonant ultrasonic spectroscopy (RUS) technique, see Ref. [39]. rectangular parallelepiped resonance (RPR)method, see Ref. [40]. plane-wave pseudopotential method, see Ref. [30]. embedded atom method, see Ref. [32].
691
3BH 2GH : 6BH þ 2GH
The bulk modulus, shear modulus are give in Fig. 3 (e). As we know, the bulk modulus reflects the resistance to bond-length change, and shear modulus directly correspond to the bond-angle change with applied stress. So, it is believed that the material is ductile if it has larger bulk modulus and smaller shear modulus. Both B and G increase with concentrations of W and Mo from 6.25% to 12.50%. But, the increment of B is larger than that of G. It means that the additional of ternary W and Mo can effectively improve the ductility of TiAl. Furthermore, Young's modulus is another important physical quantity which reflects the resistance of a material toward uniaxial tensions. Larger value indicating the stronger tensile strength. The calculated results are given in Fig. 3 (f). It is clearly that the low concentration of Sc and Yb has smaller Young's modulus. However, it has a sharp rise when the concentration exceeds a certain value (about 12.50 at.%) means that the high concentration of rare earth elements Sc and Yb will enhance the hardness of TiAl. But the opposite trends for W and Mo. The low concentration of W and Mo has smaller Young's modulus (L10) but high concentration corresponds to larger modulus (B2 phase).
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Fig. 3. The elastic constants, shear modulus and bulk modulus of TiAl with W, Mo, Sc and Yb.
Therefore, we believe that the hardness will be reduced due to the phase transition. It is well known that the ductile material is malleable and can be easily adjusted to different shapes. However, the brittle materials is
difficult to machine. And on the other hand, the brittle materials are sensitive to the sudden failure under critical mechanical stress conditions. There is no uniform standard to evaluating the ductility or brittleness of a material directly based on elastic constants
H. Hu et al. / Journal of Alloys and Compounds 658 (2016) 689e696
Fig. 4. The brittleeductile transition with various alloying concentrations. It is worthwhile to mention that the W and Mo can effectively improve the ductility of TiAl, or even transform its intrinsic brittleness into ductility. Moreover, it is shown that a small number of Sc (6.25 at.%) and Yb (<12.50 at.%) can also increase the ductility of TiAl-based alloy.
derived from linear elastic theory, but a reasonable representation of it can be realized by the combination of BH and GH. The ductility and brittleness can be judged according to Pugh's criterion (G/B) [35]. The G/B ratio separates ductile (<0.5) and brittle (>0.5) materials. Fig. 4 illustrates the phenomenological estimation of brittle vs ductile behavior of TiAl alloys with various of ternary W, Mo, Sc and Yb. First, we consider alloying of TiAl with rare earth elements Sc. The G/B ratio for TiAleSc alloys change their values only slightly. It is found that with the Sc concentration up to 6.25 at.%, but the ratio G/B decreases slightly. Which indicates that the low concentration of Sc can improve the ductility. While, there is an obvious rising trend of G/B from 6.25 at.% to 18.75 at.%. It means that the ductility will getting poor with higher concentration of Sc, but a steep decline from 18.75 at.% to 25.00 at.%. This undulating change may be caused by the complex interatomic bonding with the additional of rare earth elements Sc, and we will try to give a reasonable explanation by using of the electronic structure. However, it is found that the additive element of Yb can effectively
693
improve the ductility when the Yb concentration up to 6.25 at.% and the TiAleYb almost getting into the ductile area. But it was very regret that a steep rise of G/B with higher concentration of Yb contains from 6.25 at.% to 18.75 at.% means the ductility will deteriorate. The following is our most concern about the effects on TiAl of W and Mo. It's worth mentioning that it is not only improve the ductility of TiAl effectively but unexpectedly transform its intrinsic brittleness into ductility. We may conclude that the phase transition from L10 phase to a B2 phase is the essential reason for this phenomenon by the comparison with Sc and Yb. Besides the empirical Pugh's criterion, Poisson's ratio is an elastic coefficient that provides insight into chemical bonding of atoms regarding the variations of bond angle and bond length. In spite of the lack of a direct link between elastic properties and plastic deformation behavior, as evidenced by a large group of brittle ordered intermetallic compounds with high Poisson's ratio [42]. According to this point, the addition of W and Mo could improve the ductility in some extent. Based on the above analysis, we can conclude that the phase transition from L10 to B2 is the essential reason improving the ductility of TiAl. As we know, the atomic arrangements are not the same along different directions in lattice. The anisotropy makes the crystal has the different physical and chemical properties in different directions. And there is research showing that the influence of microcracks and lattice distortion on material are almost always caused by the elastic anisotropy [43]. The anisotropy index is a measure to quantify the anisotropy in real materials which can be traced back to the first time introduced into the cubic crystals by Zener [44]. In the recent years, a universal elastic anisotropy index AU has been introduced to describe the anisotropy of real materials by Ranganathan et al. [45]. The universal elastic anisotropy index AU can be written as AU ¼ 5 GGVR þ BBVR 6, which is single-value. The value of AU is equal to 0 respects the isotropic materials, otherwise it is anisotropy materials. The calculated universal elastic anisotropy index as a function of various doping concentration are presented in Fig. 5 (a). It is clearly shown that the smaller anisotropy index for TiAl with the ternary elements Sc and W under low concentration (<6.25 at.% for W, <18.75 at.% for Sc). This is why the smaller difference between C11 (C44) and C33 (C66) in the two doped regions (see Fig. 3 (a) and Fig. 3 (c)).
Fig. 5. Mechanical properties of TiAl with W, Mo, Sc and Yb: (a) Universal anisotropy index AU and Poisson's ratio n; (b) Vichers hardness Hv.
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Fig. 6. The calculated DOS of the TiAl and TiAl with W, Mo, Sc, Yb.
Besides ductile/brittle and anisotropy, hardness is also one of the most basic mechanical properties of the material, which is used to characterize the ability of solids resistance to elastic and permanent plastic deformation or brittle failure [46,47]. So far, it still
remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity. Results from elastic constants suggest that the hardness is closely associated with elastic properties. It is generally believed that there exists a necessary
H. Hu et al. / Journal of Alloys and Compounds 658 (2016) 689e696
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Fig. 6. (continued).
inherent relationship between hardness and elastic modulus. Recently, a simplified formula of Vickers hardness has been proposed by Chen et al. [48,49] as summarized by Teter's empirical correlation [50]. The Vickers hardness can be expressed as HV ¼ 2ðk2 GÞ0:585 3, where k is the Pugh ratio (k ¼ G/B). It indicates that the hardness correlates with both shear modulus and Pugh's ratio. Fig. 5 (b) illustrates the estimation hardness of TiAl alloys with various concentrations of ternary W, Mo, Sc and Yb. It is shown a larger ratio of G/B means a higher hardness. The value of HV is sensitive to the kinds and concentrations of the additive elements. Overall, the transition metals W and Mo can reduce the hardness of TiAl. 3.4. Electronic structure The interactions in TiAl alloys are rather complex, because the compounds not only possess covalent bonding but also have significant metallic bonding. In order to understand the stability and bonding character in TiAl alloys, we calculated the electronic density of states (DOS) of TiAl with alloying elements (see Fig. 6). The spin polarization was considered for the calculation of DOS. But the DOS of spin up is almost symmetric to the DOS of spin down, and there are very subtle differences between the DOS of spin up and
spin down (see Fig. 6 (a) and (b)). In addition, the DOS of spin up is enough to give a reasonable explanation of the mechanical properties. Hence, we just give the DOS of spin up with various additional ternary elements contains. For the L10 and B2 TiAl phase the main contribution to the DOS around Fermi level (Ef) comes from d bands of Ti. The total DOS (TDOS) shows that there is a deep valley close to the Fermi level in L10 phase and this valley is termed as pseudogap. This pseudogap indicates the presence of covalent bonding in L10 TiAl. Therefore the L10 structure is stable [21,34]. The partial density of states (PDOS) suggest that the covalent mainly results from the strong interactions between Ti-3d and Al-3p orbitals (4 eV ~ 4 eV) since their PDOSs overlapping, and the metallic bonding between Ti and Al atoms is weak. The strong covalent but weak metallic interactions, causing the uneven distribution of these binding forces, results in cleavage fracture along the direction of metallic interactions. The structure stability of the materials is closely related to the position of Ef. The Fermi level is located in the valley of the overlapping regions between the bonding states and anti-bonding states then the system will be more stable [51,52]. This suggests that L10 phase is more stable than B2 type. The DOS structure of TiAl with W, Mo, Sc, Yb are also given in Fig. 6. The most interesting feature is that the Fermi level has an
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obvious shift towards to the bonding area with the increasing of alloying concentrations for W and Mo contains (see Fig. 6 (a) and (c)) from 0 at.% to 6.25 at.%. Furthermore, the localized DOS is enhanced compared to the pure L10 phase mainly contributed by Ti-d and hybridized Al-p and W-d (Mo-d) orbitals. This also implies the increasing formation enthalpy with W and Mo contains, therefore the L10 phase's stability is getting worse and worse and there is phase transition to B2 phase. As mentioned above, the bonding properties can be reflected from the DOS structure. It can bee seen from the TDOS that the pure L10 has a wider pseudogap (>2 eV) and intrinsic brittleness. The bandwidth of pseudogap become narrow when the concentrations for W and Mo increasing ((see Fig. 6 (c), (d), (e) and (f)). It's worth mentioning that the bandwidth of pseudogap become more narrow when the concentrations of W and Mo is 12.50 at.% and the TiAl based alloys is B2 phase (see Fig. 6 (d)). Therefore, the transformation form L10 to B2 phase may be the predominant origin of the ductile/brittle transition of TiAl (see Fig. 4). The DOS of TiAl with Sc and Yb are shown in Fig. 6(g) (6.25 at.%), Fig. 6(h) (12.50 at.%), Fig. 6 (i) (18.75 at.%) and Fig. 6 (j) (25.00 at.%). As we can see, the lower concentration (6.25 at.%) of Sc and Yb has narrower pseudogap (<2 eV), but higher concentration (6.25 at.% ~ 18.75 at.% without phase transformation) has wider pseudogap (>2 eV). Thus, the metallic bond will be enhanced by lower concentrations of Sc and Yb (6.25 at.%). However, it's also worth mentioning that the pseudogap become very narrow (about 1.2 eV) at very high concentration (25.00 at.%) for Sc contains, which may explain the decreased G/B from the concentration of 18.75 at.% to 25.00 at.% in Fig. 4. Comparing with W and Mo, higher concentrations of Sc and Yb will not lead to phase transformation. Both metallic bond is also not enhanced, and ductility is also not improved. Furthermore, the PDOS show that one important factor to improve the ductility is due to the weakened ped interactions between Al and Ti atoms but enhanced ded interactions between W (Mo) and Ti atoms in TiAl based alloys [6]. Therefore, the analysis of DOS is consistent with the results of brittle/ductile map of TiAl based alloys (see Fig. 4). 4. Conclusions The phase stability and site preference of alloying elements W, Mo, Sc and Yb in L10 and B2 TiAl alloys were investigated using first-principles calculations. It is found that rare earth elements Sc and Yb always preferentially occupy the Ti sites both for L10 and B2 phase, and transition metals W and Mo have a site preference at Ti site in L10 phase, while they have a strong site preference for the Al site in B2 phase. It is interesting to find that there will be a phase transition from L10 to B2 structure when the concentrations of W and Mo are about 10.50 at.% and 11.50 at.%, respectively. This is in an agreement with previous theoretical and experimental results. However, there is no phase transition for TiAl alloyed with rare earth elements Sc and Yb. The elastic constants, bulk modulus, shear modulus, Poisson's ratio and hardness of TiAl alloys are also systematically presented. The Pugh's ratio was used to give a phenomenological estimation of brittle/ductile behaviors. The results show that the enhancement of ductility is due to the phase transition from the less slip systems L10 to a more slip systems B2 phase of TiAl based alloys. Finally, the DOS is also used to analysis the mechanical properties. The results show that the pseudogap
will become narrow for TiAl with W and Mo alloying elements, and the narrow pseudogap indicates the enhancement of metallic bond which can improve the ductility. These are in consistent with brittle/ductile transition map of TiAl based alloys. Acknowledgments The work is supported by the Natural Science Foundation of China (11104361) and Projects supported by the Fundamental Research Funds for the Central Universities (CDJZR14328801 and CQDXWL2014003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]
M. Yamaguchi, H. Inui, K. Ito, Acta Mater. 48 (2000) 307. F. Appel, M. Oehring, R. Wanger, Intermetallics 8 (2000) 1283. G. Ghosh, A. van de Walle, M. Asta, Acta Mater. 56 (2008) 3202. C.T. Liu, J.L. Wright, S.C. Deevi, Mater. Sci. Eng. A 329 (2002) 416. naff, A.L. Gloanec, Intermetallics 13 (2005) 543. G. He M. Morinaga, J. Saito, N. Yukawa, et al., Acta Metall. 38 (1990) 25. R.B. Schwarz, P.B. Desch, S. Srinivasan, et al., Nanostruct. Mater. 1 (1992) 37. G.A. Salishchev, R.M. Imayev, O.N. Senkov, et al., Mater. Sci. Eng. A 286 (2000) 236. M. Beschliesser, A. Chatterjee, A. Lorich, et al., Mater. Sci. Eng. A 329 (2002) 124. B.Y. Huang, Y.H. He, J.N. Wang, Intermetallics 7 (1999) 881. Z.C. Liu, J.P. Lin, S.J. Li, et al., Intermetallics 10 (2002) 653. S. Shu, F. Qiu, C. Tong, et al., J. Alloys Comp. 617 (2014) 302. S.C. Huang, E.L. Hall, Metall. Trans. A 22 (1991) 2619. S.C. Huang, E.L. Hall, Acta Metall. 39 (1991) 1053. T. Tetsui, K. Shindo, S. Kobayashi, et al., Scr. Mater. 47 (2002) 399. T. Tetsui, K. Shindo, S. Kobayashi, et al., Intermetallics 11 (2003) 299. F. Appel, M. Oehring, J.D.H. Paul, Adv. Eng. Mater. 8 (2006) 371. T. Tetsui, K. Shindo, S. Kaji, et al., Intermetallics 13 (2005) 971. M. Kimura, K. Hashimoto, H. Morikawa, Mater. Sci. Eng. A 152 (1992) 54. T.T. Cheng, M.R. Willis, I.P. Jones, Intermetallics 7 (1999) 89. R. Yu, L.L. He, Z.Y. Chen, et al., Intermetallics 10 (2002) 661. Y. Zheng, L. Zhao, K. Tangri, Mater. Sci. Eng. A 208 (1996) 80. Y.G. Li, M.H. Loretto, Acta Metall. Mater. 42 (1994) 2913. M. Haidi, M.A. Shafyei, M. Meratian, Mater. Sci. Eng. A 624 (2015) 1. J.T. Guo, K.W. Huai, Q. Gao, et al., Intermetallics 15 (2007) 727. G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. A. Zoroddu, F. Bernardini, P. Ruggerone, et al., Phys. Rev. B 64 (2001) 045208. C. Woodward, S. Kajihara, L.H. Yang, Phys. Rev. B 57 (1998) 21. R. Yu, L.L. He, H.Q. Ye, Phys. Rev. B 65 (2002) 184102. R.Z. Rajendra, Y. Mishin, Phys. Rev. B 68 (2003) 024102. J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985. P. Ravindran, L. Fast, P.A. Korzhavyi, et al., J. Appl. Phys. 84 (1998) 4891. S.F. Pugh, Philos. Mag. Ser. 45 (1954) 823. E.A. Brandes (Ed.), Smithells Metal Reference Book, sixth ed., Butterworth, , London, 1983. P.D. Desai, J. Phys. Chem. Ref. Data 16 (1987) 109. B.J. Inkson, H. Clemens, J. Marien, Scr. Mater. 38 (1998) 1377. Y. He, R.B. Schwarz, A. Migliori, S.H. Whang, J. Mater. Res. 10 (1995) 1187. K. Tanaka, T. Ichitsubu, H. Iniu, et al., Philos. Mag. Lett. 73 (1996) 71. O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909. K. Gschneidner, A. Russell, A. Pecharsky, et al., Nat. Mater. 2 (2003) 587. V. Tvergaard, J.W. Hutchinson, J. Am. Ceram. Soc. 71 (1988) 157. C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, 1948. S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. V.V. Brazhkin, A.G. Lyapin, R.J. Hemley, Philos. Mag. A 82 (2002) 231. A.S. Povarennykh, Press of AN USSR, Kiev, 1963. X.Q. Chen, H. Niu, D. Li, et al., Intermetallics 19 (2011) 1275. X.Q. Chen, H. Niu, C. Franchini, et al., Phys. Rev. B 84 (2011) 121405. D.M. Teter, MRS Bull. 23 (1998) 22. J.H. Xu, T. Oguchi, A.J. Freeman, Phys. Rev. B 35 (1987) 6940. T. Hong, A.J. Freeman, J. Mater. Res. 19916 (1991) 330.