Structural stability and mechanical properties of Ti2AlX (X = Hf, Re) intermetallics: A DFT study

Accepted Manuscript Structural stability and mechanical properties of Ti2AlX (X = Hf, Re) intermetallics: A DFT study Ashish Pathak PII:

S0925-8388(16)33303-5

DOI:

10.1016/j.jallcom.2016.10.180

Reference:

JALCOM 39347

To appear in:

Journal of Alloys and Compounds

Received Date: 27 November 2015 Revised Date:

17 October 2016

Accepted Date: 18 October 2016

Please cite this article as: A. Pathak, Structural stability and mechanical properties of Ti2AlX (X = Hf, Re) intermetallics: A DFT study, Journal of Alloys and Compounds (2016), doi: 10.1016/ j.jallcom.2016.10.180. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Structural Stability and Mechanical Properties of Ti2AlX (X=Hf, Re) Intermetallics: A DFT Study AshishPathak Defence Metallurgical Research Laboratory, Kanchanbagh, Hyderabad-500 058, India

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Abstract Present work describes the structural stability and mechanical properties of B2, D019 and O phases of Ti2AlX (X= Hf, Re) intermetallics calculated using density functional theory (DFT) within generalized gradient approximation (GGA). The lattice constant values for all the

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three phases are computed. For Ti2AlHf, O phase is the most stable followed by D019 and B2 whereas for Ti2AlRe, O phase is the most stable followed by B2 and D019 phases. All the three phases in Ti2AlHf and Ti2AlRe have metallic bonding inferred from the positive

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Cauchy pressure and they satisfy the Born stability criteria in terms of elastic constants. Ductile deformation behaviour is expected based on the G/B ratios for these systems except O phase. The B2 phase exhibits very high anisotropy in comparison to those of D019 and O in both alloys and the anisotropy is higher in Ti2AlHf.

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KEYWORDS: A. intermetallics; E. ab-initio calculations, electronic structure calculations,

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mechanical properties of Aluminides.

Corresponding author. Tel.:+91-40-24586385; fax: +91-40-24340683 E-mail address: [email protected], [email protected] (AshishPathak)

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ACCEPTED MANUSCRIPT 1. Introduction Titanium-based alloys are used for high temperature applications because of their properties such as high melting temperature, high thermal conductivity and low density [1–3]. However, their practical applications are limited due to the lack of room temperature ductility and toughness. Strong directional bonding (SDB) and restricted number of slip systems in the

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ordered structure are the main governing factors that limit/reduce the room temperature ductility. These factors are sensitive to the structure and can be manipulated by controlling the ordering of the structure. Therefore, fractional replacement of the alloying elements by suitable ternary substitutions can be utilised to achieve a desired bonding behaviour. The Ti-

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based intermetallics can be classified in three classes of alloys, namely Ti3Al (α2 + B2), O phase (Ti2AlNb +B2) and TiAl (γ +β).

The B2 phases with X2AlM stoichiometry are formed by three groups of metallic

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elements namely, X (Ti, Zr and Hf), Al and M (V,Nb, Ta, Cr, Mo and W). At room temperature, the elements in the groups X and M crystallise with hexagonal close packed (hcp) structures and body centered cubic (bcc) structures, respectively. Both X and M may correspond to either an individual element or a combination of elements of each group [4]. The structural and mechanical properties of Ti2AlNb and Ti2AlZr have already been reported

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[5, 6].

The structural details of the B2 and the D019 phases in both Ti2AlHf and Ti2AlRe intermetallics are given in Tables 1 and 2. The O phase forms two alternative packings, i.e. O1 and O2 with elemental ordering over three crystallographically distinct sites (4(c1), 4(c2)

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and 8(g)) [3]. In the O1 phase, Ti and X atoms randomly occupy the 4(c2) and 8(g) sites while in the case of the O2 phase, 4(c2) and 8(g) sites are predominantly occupied by X and

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Ti atoms, respectively. In both phases, 4(c1) sites are occupied by Al atoms. The crystallographic structural details of the O2 (called O phase hereinafter throughout the paper) phase with Ti2AlX (X= Hf, Re) stoichiometry are given in Tables 1 and 2. Ravi et al. [7] have investigated electronic structure, phase stability and cohesive

properties of Ti2AlX (X = Nb, V, Zr) alloys for three phases, namely for B2, D019 and O based on tight binding (TB) approximation. The present work is concerned with a comprehensive study to investigate structural, electronic and mechanical properties of these three phases in the intermetallics Ti2AlX (X= Hf, Re) using first principles studies.

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ACCEPTED MANUSCRIPT 2. Methodology The present study has been carried out using a first principles pseudopotential method in the framework of the density functional theory (DFT) [8, 9]. ABINIT software has been employed for the same [10–13]. Exchange-correlation effects have been treated within the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE)

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formulation [14]. The valence electrons considered in the norm-conserving pseudopotential are 4 electrons (3d2 4s2) for Ti, 3 electrons (3s2 3p) for Al, 4 electrons (5d2 6s2) for Hf and 7 electrons (5d5 6s2) for Re atoms. A fast Fourier transform algorithm [15] has been utilized to convert the wavefunctions between the real and reciprocal lattices. Consequently, a conjugate

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gradient algorithm [16, 17] has been used to determine wave functions within the frame work of self-consistency. Convergence with respect to the plane wave cut-off energy and k-points has been verified and accordingly a plane wave cut-off energy of 80 Ry and k-mesh of 8×8×8

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have been used. The integration over the Brillouin zone (BZ) has been done with the Monkhorst–Pack scheme [18]. Convergence is assumed when the differences between energies or forces in two consecutive steps are less than 27×10-3 meV or 2.6 meV/Å, respectively.

The numbers of atoms considered in the present study for the B2, D019 and O phases been defined as 

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are 8, 8, 16 atoms, respectively in both the alloys. The energy of the formation per atom has 





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 =   
 − 
 − 
 − 
    

(1)

 where,
 is the total system energy of TilAlmXn (X= Hf, Re) alloy, TilAlmXn having     





are the total energy

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‘l’ Ti-atoms, ‘m’ Al-atoms and ‘n’ Hf or Re-atoms,
 ,
 and


per atom in their reference (ground) states for bulk Ti, Al and X (Hf / Re), respectively, and (l+m+n) denotes the total number of atoms considered in the supercell. The crystal structures for Ti, Hf and Re in their ground state are considered as hexagonal close packed (hcp) structure, and Al as face centered cubic (fcc) structure. The formation energy per atom for Ti2AlX (X= Hf, Re) for the three phases has been calculated and given in Table 3. The linear-response method implemented in the ABINIT software has been employed for the calculation of elastic constants. This method is used to calculate the second derivative of the total energy with respect to the strain. Xcrysden [19] software has been utilized for the visualization of the crystal structure and 2D charge density. 3

ACCEPTED MANUSCRIPT 3. Results First principles based DFT calculations have been performed to study the relative stability, electronic structure and elastic properties of B2, D019 and O phases in both the alloys. The crystal structures of these three phases for Ti2AlHf are given in Figs. 1–3. The

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equilibrium lattice constants of these phases have been obtained by varying the lattice constant and finding the one that gives the minimum energy. The calculated lattice constant values are given in Table 3. The lattice constant values for Ti2AlHf are higher than Ti2AlRe in all three phases. This effect is mainly due to the high atomic radius of Hf (1.58 Å)

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compared to that of Re (1.38 Å).

The formation energy / atom values of the B2, D019 and O phases obtained in the present study are presented in Table 3. It can be observed that the O phase has the lowest

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energy among all the three phases, meaning that the O phase is the most stable. However, the stability trends for all phases in both the alloys are different. In Ti2AlHf, the O phase is the most stable phase, followed by D019 and B2, whereas in Ti2AlRe, the O phase is most stable, followed by the B2 and D019 phases.

It can be inferred from Tables 1 and 2 that the atomic positions of the B2 structure are

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fixed, while 6(h) of the D019 as well as the 4(c) and 8(g) positions of the O phases are variables. The different atomic potentials resulting from different site occupancy could be the reason for this variation. Banumathy et al. [20] have demonstrated that the Ti atoms (6(h)) in the D019 structure are slightly shifted from their regular hexagonal close packed (hcp) atomic

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sites. They have related the shift to the presence of symmetry elements in the D019 structure, which allows atomic displacements along the mirrors within the P63/mmc space group. It is

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to be noted that the atomic shift of 6(h) positions has also been observed in the present calculations.

Similarly, as mentioned above, the Wyckoff positions of the O phase consist of

variable parameters and these {4(c1), (4c2) and 8(g)} positions in the O phase vary within the framework of the Cmcm space group depending upon the alloy composition. Consequently, the corresponding natures of the atomic potentials vary, and thus result in different values of atomic shift. Accordingly, the bond lengths in the respective neighbours change. The density of states (DOS) and projected density of states (PDOS) for constituent atoms of the B2, D019 and O phases in both the alloys are shown in Figs. 4–9. The DOS of the O phase is quite smooth in comparison to other phases. The value of DOS at the Fermi level (EF) is the lowest for the O phase compared to other phases in both alloys. This 4

ACCEPTED MANUSCRIPT indicates that the O phase is relatively stable among all the three phases, which is in agreement with the analysis of the formation energy/atom values (Table 3). It can be observed from the PDOS (Figs. 4−9) that the bonding behaviour of these alloys is due to peaks below EF exhibiting from the Ti(d)-X(d) orbitals. Further, the charge density distribution has been investigated to understand the bonding behaviour in these alloys. The

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2D charge density of the B2 phase in the (010) and (020) planes is shown in Fig. 10, and for the D019 and O phases in the (001) plane for both alloys, it is presented in Figs. 11−12. All three phases display strong electronic interactions for Ti-Hf / Ti-Re, but weak interactions for Ti-Al bonds. The Ti-Hf / Ti-Re atoms display strong bonding in the D019 and O phases

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although the extent of bonding is greater in the latter case. On the other hand, less interaction is seen in the B2 phase.

The elastic properties of single crystals of B2, D019 and O phases are described by

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C11, C12, C44; C11, C12, C33, C13, C44 and C11, C12, C22, C33, C13, C23, C44, C55, C66, respectively and given in Table 4. The nature of the metallic bonding of B2, D019 and O phases can be predicted based on Cauchy pressures [21]. These are defined as C1: C12 – C44 < 0

C1: C13 – C44 < 0, C2: C12 – C66 < 0

for cubic for hexagonal

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C1: C23 – C44 < 0, C2: C13 – C55 < 0, C3: C12 – C66 < 0 for orthorhombic along 〈100〉 〈010〉 and 〈001〉 directions.

The negative Cauchy pressure indicates more directional bonding, whereas a positive value denotes predominant metallic bonding. The calculated values of Cauchy pressures of

these phases.

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all the phases are positive (Table 5), which indicates the presence of major metallic bonds in

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The values of elastic constants can be used as a significant parameter for the material to provide the information about the stability of phases. The Born stability criteria for cubic and hexagonal crystals are given in equations (2) and (3), respectively [22, 23]: Cubic materials

S1: C44 > 0, S2: C11 > |C12| and S3: C11 + 2C12 > 0

(2)

Hexagonal materials S1: C12 > 0, S2: C33 > 0, S3: C44 > 0, S4: (C11-C12)/2 > 0 and S5: (C11 + C12) C33 - 2C132 > 0

(3)

The stability conditions for an orthorhombic crystal are given by equations (4)−(7) [22] S1: C11 + C22 + C33 + 2(C12 + C13 + C23) > 0

(4) 5

ACCEPTED MANUSCRIPT S2: C11 + C22 – 2C12 > 0

(5)

S3: C11 + C33 – 2C13 > 0

(6)

S4: C22 + C33 – 2C23 > 0

(7)

The values of elastic constants have been used to calculate the stability of the B2, D019 and O phases and suggest that these phases are mechanically stable (Table 5).

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The anisotropy factors (A) for the B2, D019 and O phases are defined as For B2 phase A =



(8)

 !

For D019 phase 





, A = "" , A# = !

 !



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A = 

"

For orthorhombic phase $  ""

A =  A# = 

"

$%% !! "" !"

$&&  !!

!

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A = 

(9)

(10) (11) (12)

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where shear anisotropic factors A1, A2 and A3 for the O phase predict the bonding anisotropy among atoms in different planes. The shear anisotropic factor A1 denotes the anisotropy between the 〈011〉 and 〈010〉 directions on {100} shear planes while A2 and A3 describe the anisotropy between the 〈101〉 and 〈001〉 directions on {010} and between the 〈110〉 and 〈010〉

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directions on {001} shear planes, respectively.

The values of anisotropic factors should be 1 for an isotropic crystal and other than 1

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denotes the presence of anisotropy in elastic constants. The values of anisotropic factors obtained in the present study for the B2, D019 and O phases are other than 1, indicating the presence of anisotropy in these materials (Table 5). However, the extent of anisotropy is very high for the B2 phase in comparison to those of D019 and O phases. This observation is not surprising, since ordered B2 alloys are extensively elastically anisotropic [24]. The effective elastic modulus of isotropic polycrystalline cubic materials can be evaluated from the elastic constants by the following two approximations, namely the Voigt [25] and Reuss [26] which provide the information about the upper and lower limit. These are ' = '( = ') = +( =

* *!

* *! #* ,

#

(13) (14) 6

ACCEPTED MANUSCRIPT +) =

,* *! *

(15)

$* #* *!

where B is the bulk modulus and GV and GR are the shear modulus values obtained by the Voigt and Reuss approximations, respectively. For hexagonal materials, the elastic constants in the Voigt [25] and Reuss [26] approximations are given as (16)

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B. = / 2C + C + C## + 4C# 4/9

G. = /C + C## − 2C# + 6C$$ + 5C:: 4/15 B< =

 ! "" !"  ! "" $"

(17)

(18)

S + S = C## /C S − S = 1/C − C

(20) (21) (22)

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S# = −C# /C

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G< = 15/ 8S + 4S## − 4S − 8S# + 6S$$ + 3S:: (19)

S## = C + C /C

(23)

S$$ = 1/C$$

(24)

S:: = 1/C::

(25)

 C = C + C C## − 2C#

(26)

as 

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The bulk modulus and shear modulus for a polycrystalline orthorhombic alloy can be given

B. = @ /C + C + C## + 2C + C# + C# 4 

(27)

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G. = , /C + C + C## + 3C$$ + C,, + C:: 4 B< = /S + S + S## + 2S + S# + S# 4

(28)



(29) 

(30)

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G< = 15/4S + S + S## + 3S$$ + S,, + S:: − 4S + S# + S# 4

The average value of the two (Voigt and Reuss) estimates mentioned above is given by Hill’s [27] approximation for cubic, hexagonal and orthorhombic materials. The Voigt−Reuss−Hill (VRH) average values are given by B = BA =

BC BD

G = GA =

 EC ED

@BE

E = #B E υ =

#B E #B E



(31) (32) (33) (34)

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ACCEPTED MANUSCRIPT where B(=BH), G(=GH), E and ν are the bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio, respectively. The calculated values of B, G and E for the B2, D019 and O phases are given in Table 6 and are important for predicting the mechanical properties of these materials. Ti2AlHf has maximum values of B, G and E for D019, followed by the B2 and O phases, whereas in the

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case of Ti2AlRe, the O phase has the highest value of G and E, followed by the D019 and B2 phases.

The values of shear and bulk modulus can also be utilized in predicting the brittle and ductile behaviour of materials by calculating the G/B ratio [28]. The ratio

E B

> 0.57 is

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associated with brittle behaviour while other values are related with ductile behaviour. The present calculation clearly indicates that the B2 and D019 phases are ductile whereas the O

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phase shows brittle behaviour in both the alloys.

4. Discussion

The negative formation energy values (Table 3) suggest the possible existence of B2, D019 and O phases in both alloys. The O phase is the most stable among the three phases. Similar behaviour has been obtained previously in Ti2AlNb and Ti2AlZr [5,6]. Therefore, this

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indicates the presence of all three phases in equilibrium. Furthermore, these three phases (B2, D019 and O) exist in experiments in the equilibrium phase diagram of Ti2AlNb and Ti2AlZr depending upon the compositions and heat treatments. Hence, a thorough experimental investigation of Ti2AlHf and Ti2AlRe systems will be interesting to validate the predictions

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made in this paper based on the energetics calculations. The DOS, PDOS and charge density reveal that the charge rearrangement is mainly

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among Ti and alloying X atoms in all three phases. The X (Hf, Re) atoms have more valence electrons compared to Ti atoms and therefore the Ti−X bond is stronger than the Ti−Ti bond. Hence, the alloying X atoms will attract the neighbouring Ti atoms towards them (Figs. 5−7). The perception of the stability of binary intermetallics based on the lower DOS values at the Fermi level (EF) has been reported by Xu et al. [29]. Though Ravi et al. [7] claimed that structural stability based on (DOS)min at EF is not always followed by metastable materials, it is clear from our calculations that the value of DOS at the Fermi level (EF) is the lowest for O phase in both alloys. This indicates that the O phase is more stable than the other phases and this observation is in agreement with the formation energy/atom values obtained in the present study (Table 3). However, similar trends of (DOS)min at EF and formation energy 8

ACCEPTED MANUSCRIPT /atom values are not seen in the B2 and D019 phases. Therefore, the concept of stability based on the DOS value at EF is not always true. Consequently, the relative stability of different phases of the intermetallics can be explained based on the association of DOS and formation energy. The ductility of Ti3Al is considerably enhanced by the addition of Nb [3]. The present

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study shows the beneficial effect of Re and Hf apart from Nb. Haydock [30] reported that an adequately large stress is required to expel the localised valence electrons from the bonding. The directional Al(p)-Ti(d) covalent bonding among Al and Ti atoms increases the shear strength and provides a large barrier to the shear deformation. This phenomenon is

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responsible for the embrittlement of these compounds [31]. The reduction of the p-d covalent bonding and increase in d-d bonding are believed to increase the ductility of intermetallic compounds [31]. Based on this argument, it can be suggested that, with the addition of Re

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and Hf, the ductility of Ti3Al may be further improved. This result can be further augmented with the Cauchy pressure and G/B ratio, calculated from the present analysis as shown in Tables 5 and 6. However, it should be noted that the O phase shows behaviour that is either brittle or on the ductile/brittle boundary based on the G/B ratio.

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5. Conclusions

1. First principles density functional theory (DFT) within generalized gradient approximation (GGA) has been utilized to investigate the structural stability and mechanical properties of the B2, D019 and O phases of Ti2AlX ( X= Hf, Re) intermetallics.

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2. In Ti2AlHf, the O phase is the most stable phase followed by D019 and B2, whereas in Ti2AlRe, the O phase is the most stable followed by the B2 and D019 phases.

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3. All three phases in Ti2AlHf and Ti2AlRe have metallic bonding inferred from the positive Cauchy pressure.

4. All three phases satisfy the Born stability criteria in terms of elastic constants and are associated with ductile behaviour (except the O phase) based on the G/B ratio. 5. The B2 phase exhibits very high anisotropy in comparison to the D019 and O phases.

Acknowledgements Author is grateful to Ministry of Defence, Government of India for financial support. The author is indebted to Director DMRL, Hyderabad for his encouragement. The author would like to place on record thanks to Director ANURAG, Hyderabad for the provision of

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ACCEPTED MANUSCRIPT computational facilities, Dr. R. Sankarasubramanian and Shri A. Mondal for the technical support.

References [1] C. T. Liu and H. Inouye, Metall. Trans. A 10A (1979) 1515.

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[2] D. Banerjee, A.K. Gogia, T.K. Nandy, K. Muraleedharan and R.S. Misra, Structure Intermetallics ed R. Darolia, JJ Lewandowski, C.T. Liu, PL Martin and MV Nathal (Warrendale, PA: The Mineral, Metals and Materials Society) p 19. [3] D. Banerjee, Prog. Mater. Sci. 42 (1997) 135.

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[4] S. Naka, M. Thomas, M. Marty, G. Lapasset, T. Khan. In: R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle, M.V. Nathal, editors. Structural intermetallics, The Minerals, Metals and Materials Society; 1993.p. 647.

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[5] Ashish Pathak, A.K.Singh, Solid State Comm. 204 (2015) 9. [6] Ashish Pathak, A.K. Singh, Intermetallics 63 (2015) 37.

[7] C. Ravi, P. Vajeeston, S. Mathijaya, R. Asokamani, Phys. Rev. B. 60 (1999) 683. [8] P. Hohenberg, W. Kohn, Phys Rev. 136 (964) B864.

[9] W. Kohn and L. J. Sham, Phys Rev. 140 (1965) A1133.

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[10] http://www.abinit.org

[11] X. Gonze et. al., Computational Materials Science. 25 (2002) 478-492. [12] X. Gonze, Zeitschrift fuer Kristallographie. 220 (2005) 558-562 . [13] X. Gonze et. al., Computer Phys. Comm. 180(2009) 2582-2615.

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[14] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. lett. 77 (1996) 3685. [15] S. Goedecker, S. SIAM J. Sci. Comput. 18 (1997) 1605.

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[16] M.C. Payne, Rev. Mod. Phys. 64 (1992) 1045. [17] X. Gonze, Phys. Rev. B. 54 (1996) 4383. [18] H. J. Monkhorst , J.D. Pack, Phys. Rev. B. 13 (1976) 5188. [19] A. J. Kokalj, J. Mol Graphics Modelling. 17(1999)176-179; http://www.xcrysden.org [20] S. Banumathy, P. Ghosal and A. K. Singh, J. Alloys and Compounds. 394 (2005) 181. [21] H.Olijnyk and A.P. Jephcoat, J. Phys. Condens. Matter., 12 (2000) 10423. [22] M. Born, Proc Cambridge Philos. Soc. 36 (1940) 160. [23] F. I. Fedorov, Theory of elastic waves in crystals, New York: Plenum, 1968. [24] Yi-Shen Lin, M. Cak, V. Paidar and V. Vitek, Acta Materialia, 60 (2012) 881– 888 [25] W. Voigt, Lehrburch der Kristallphysik, Teubner, Leipzig 1928. 10

ACCEPTED MANUSCRIPT [26] A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49. [27] R. Hill, Proc. Phys. Soc. London A 65 (1952) 349-454. [28] S. F. Pugh, Philos. Mag. 45 (1954) 823-843. [29] J.-h. Xu, T. Oguchi, A. J. Freeman, Phys. Rev. B 35(1987) 6940. [30] R. Haydock, J. Phys. C. 14 (1981) 3807.

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[31] M. Morinaga, J. Asito, N. Yukawa, H. Adachi, Acta Metall. 38 (1990) 25.

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FIGURE CAPTIONS

Fig.1. Crystal structure of the B2 phase in Ti2AlHf alloy.

Fig. 2. Crystal structure of the D019 phase in Ti2AlHf alloy. Fig. 3. Crystal structure of the O phase in Ti2AlHf alloy.

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Fig. 4. DOS and PDOS for the B2 phase in Ti2AlHf alloy.

Fig. 5. DOS and PDOS for the B2 phase in Ti2AlRe alloy. Fig. 6. DOS and PDOS for the D019 phase in Ti2AlHf alloy. Fig. 7. DOS and PDOS for the D019 phase in Ti2AlRe alloy.

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Fig. 8. DOS and PDOS for the O phase in Ti2AlHf alloy. Fig. 9. DOS and PDOS for the O phase in Ti2AlRe alloy.

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Fig. 10. 2D charge density for the B2 phase in (010) and (020) plane for (a) Ti2AlHf (b) Ti2AlRe alloys. Plane directions and charge density scales, units are also shown. Fig. 11. 2D charge density for the D019 phase in (001) plane for (a) Ti2AlHf (b) Ti2AlRe alloys. Plane directions and charge density scales, units are also shown. Fig. 12. 2D charge density for the O phase in (001) plane for (a) Ti2AlHf (b) Ti2AlRe alloys. Plane directions and charge density scales, units are also shown.

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ACCEPTED MANUSCRIPT Table 1. Crystallographic data of ordered Ti2AlHf intermetallic with B2, D019 and O phases.

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0 0 1/2 1/2 1/4 1/4 3/4 3/4

0 0 0 0 1/2 1/2 1/2 1/2

z 0 1/2 0 1/2 1/4 3/4 1/4 3/4

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Ti Ti Hf Hf Al Ti Ti Al

y

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B2: Cubic 8 atoms per cell ATOMIC POSITIONS: ATOMS x

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D019: Hexagonal P63/mmc or D46h; 8 atoms per unit cell hP8 ATOMIC POSITIONS: ATOMS WYCKOFF SYMMETRY x y z NOTATION Al 2(d) 6 m2 1/3 2/3 3/4 Ti / Hf 6(h) mm2 x 2x 1/4 Present Study 0.165 0.330 1/4 The sum of the addition of probability of the Al atoms (PAl) and Hf atoms (PHf), Ti atoms (PTi) on 2(d) and 6(h) sites, respectively is 1. The value of x is around 1/6 depending upon the alloy composition and corresponding site occupancies.

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O : Orthorhombic Cmcm or D17 2h ; 16 atoms per unit cell oC16 ATOMIC POSITIONS: ATOMS WYCKOFF SYMMETRY NOTATION Al 4(c1) m2m Present Study Hf 4(c2) m2m Present Study Ti 8(g) m2m Present Study

x

y

z

0 y1 1/4 0 0.1676 1/4 0 y2 1/4 0 0.6451 1/4 x3 y3 1/4 0.2254 0.9083 1/4

---------------------------------------------------------------------------------------The sum of the addition of probability of the Al atoms (PAl), Hf atoms (PHf) and of Ti atoms (PTi) on 4(c1), 4(c2) and 8(g) sites, respectively is 1. The values of y1, y2, x3 and y3 are close to 1/6, 2/3, 1/4 and 11/12 depending upon the alloy composition and corresponding site occupancies, respectively.

ACCEPTED MANUSCRIPT Table 2. Crystallographic data of ordered Ti2AlRe intermetallic with B2, D019 and O phases.

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0 0 1/2 1/2 1/4 1/4 3/4 3/4

0 0 0 0 1/2 1/2 1/2 1/2

z 0 1/2 0 1/2 1/4 3/4 1/4 3/4

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Ti Ti Re Re Al Ti Ti Al

y

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B2: Cubic 8 atoms per cell ATOMIC POSITIONS: ATOMS x

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D019: Hexagonal P63/mmc or D46h; 8 atoms per unit cell hP8 ATOMIC POSITIONS: ATOMS WYCKOFF SYMMETRY x y z NOTATION Al 2(d) 1/3 2/3 3/4 6 m2 Ti / Re 6(h) mm2 x 2x 1/4 Present Study 0.160 0.320 1/4 The sum of the addition of probability of the Al atoms (PAl) and Re atoms (PRe, Ti atoms (PTi) on 2(d) and 6(h) sites, respectively is 1. The value of x is around 1/6 depending upon the alloy composition and corresponding site occupancies.

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O : Orthorhombic Cmcm or D17 2h ; 16 atoms per unit cell oC16 ATOMIC POSITIONS: ATOMS WYCKOFF SYMMETRY NOTATION Al 4(c1) m2m Present Study Re 4(c2) m2m Present Study Ti 8(g) m2m Present Study

x

y

z

0 y1 1/4 0 0.1453 1/4 0 y2 1/4 0 0.6315 1/4 x3 y3 1/4 0.2502 0.8856 1/4

---------------------------------------------------------------------------------------The sum of the addition of probability of the Al atoms (PAl), Re atoms (PRe) and of Ti atoms (PTi) on 4(c1), 4(c2) and 8(g) sites, respectively is 1. The values of y1, y2, x3 and y3 are close to 1/6, 2/3, 1/4 and 11/12 depending upon the alloy composition and corresponding site occupancies, respectively.

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Table 3. Equilibrium lattice constants and formation energy/atom (Efor/atom) of Ti2AlX (X= Hf, Re) are given for three phases namely B2, D019 and O. Lattice constants (Å)

Phase

Ti2AlRe

a

b

c

(eV)

B2

3.3499

3.3499

3.3422

-0.1764

D019

5.8986

5.8986

4.7778

-0.2227

O

6.3150

9.9245

4.8398

-0.3469

B2

3.2155

3.2155

3.2155

-0.2653

D019

5.6576

5.6576

4.6958

-0.2535

O

5.9837

9.4034

4.5856

-0.5128

RI PT

Ti2AlHf

Efor /atom

SC

System

Phase

M AN U

Table 4: Calculated single crystal elastic constants for B2, D019 and O phases of Ti2AlX (X= Hf, Re) intermetallic. Elastic constants (GPa)

C12

C22

B2

110

88

C11

D019

177

85

C11

O

133

72

156

119

D019

229

105

O

187

171

C23

C44

C55

C66

C11

C13 Ti2AlHf C12

C12

74

C44

C44

215

54

C13

43

C44

(C11 − C12)/2

63

38

28

52

61

AC C

119

177

Ti2AlRe

C11

C11

C12

C12

99

C44

C44

C11

253

66

C13

50

C44

(C11 − C12)/2

264

129

28

10

90

93

EP

B2

C33

TE D

C11

249

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Table 5: Cauchy pressures, Born stability criteria and anisotropic values for B2, D019 and O phases for Ti2AlX (X = Hf, Re) intermetallics

Phase

Cauchy pressures

Born Stability Criteria values

Anisotropic values

(GPa)

(GPa) C2

C3

S1

S2

S3

S4

S5

Ti2AlHf 14

D019

11

39

O

10

11

11

74

22

286

85

177

28

46

827

160

184

272

D019

16

43

O

18

39

26

99

37

394

105

253

50

62

1252

198

193

457

50498

0.9348

1.2147

1.5741

0.6087

0.7647

1.5250

5.3514

M AN U

20

A3

6.7272

Ti2AlRe B2

A2

SC

B2

A1

RI PT

C1

75790

0.8065

1.1048

1.5909

0.2073

0.7877

1.8788

Table 6: Calculated polycrystalline mechanical properties such as Bulk modulus (B), Shear Modulus (G),

Re) intermetallics. Phase

TE D

Young’s modulus (E), G/B ratio and Poisson’s ratio (ν) for B2, D019 and O phases for Ti2AlX (X= Hf,

Elastic constants (GPa) BR

B

GV

EP

BV

GR

G

E

G/B

υ

Ti2AlHf

95

95

95

49

23

36

95

0.3739

0.3338

D019

106

106

106

51

53

52

135

0.4921

0.2886

92

68

80

49

43

46

116

0.5738

0.2592

O

B2

AC C

B2

Ti2AlRe

131

131

131

67

36

52

137

0.3918

0.3267

D019

132

132

132

64

45

54

144

0.4136

0.3182

O

139

75

107

67

120

95

219

0.8851

0.1581

AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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AC C

EP

TE D

M AN U

SC

RI PT

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ACCEPTED MANUSCRIPT Research Highlights

Electronic structural, mechanical properties of Ti2AlX (X= Hf, Re) intermetallics.

•

All the three phases in Ti2AlHf and Ti2AlRe show metallic bonding.

•

Both the alloys satisfy the stability criteria in terms of elastic constants.

•

Ductile/brittle behavior has been explained based on G/B ratios.

•

B2 phase shows high anisotropy.

AC C

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