Phase stability and anisotropic elastic properties of the Hf–Al intermetallics: A DFT calculation

Computational Materials Science 110 (2015) 10–19

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Phase stability and anisotropic elastic properties of the Hf–Al intermetallics: A DFT calculation Yong-Hua Duan a,⇑, Zhao-Yong Wu b, Bo Huang a, Shuai Chen a a b

School of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China Kunming Pei Si Engineering CO. Ltd., Kunming 650034, China

a r t i c l e

i n f o

Article history: Received 8 May 2015 Received in revised form 22 July 2015 Accepted 28 July 2015

Keywords: First-principles calculations Hf–Al intermetallics Stability Elastic anisotropy

a b s t r a c t The first-principles calculations based on density-functional theory were performed to investigate the structural properties, phase stability, and elastic properties of several selected Hf–Al intermetallics. The calculated formation enthalpies show that HfAl2 is the most stable in these intermetallics. The predicted elastic moduli results indicate that the order in the degree of elastic anisotropy is HfAl > Hf2Al > D022-HfAl3 > Hf3Al2 > Hf5Al3 > Hf2Al3 > D023-HfAl3 > L12-HfAl3 > Hf4Al3 > HfAl2. By using the procedure of Brugger, the calculated sound velocities in different directions for the Hf–Al intermetallics are anisotropic. The Debye temperatures were also estimated from the elastic constant evaluations of Hf–Al intermetallics. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Hafnium–aluminum (Hf–Al) alloys are of high-temperature structural applications due to their promising properties, including resistance to oxidation and corrosion, excellent elevatedtemperature strength, relatively low density and high points [1]. There are seven stable and three metastable intermetallic phases in Hf–Al, which are stable D023-HfAl3, D022-HfAl3, HfAl2, Hf2Al3, HfAl, Hf4Al3, Hf3Al2 and metastable L12-HfAl3, Hf5Al3, Hf2Al [2–5]. The existence of stable HfAl2, Hf2Al3, HfAl, Hf4Al3 and Hf3Al2 intermetallic phases has been confirmed [6–8]. An equilibrium transition between D023-HfAl3 and D022-HfAl3 around 650 °C has been found experimentally [9]. For the cubic L12-HfAl3 obtained by mechanical alloying, a transformation from L12-HfAl3 and D023-HfAl3 arises at 750 °C [4]. Due to impurity stabilization, the hexagonal Hf5Al3, similar to Zr5Al3, might be observed [9]. Also, the metastable Hf2Al can be stabilized by Si impurities [10,11]. Extensive works have been reported on Hf–Al intermetallics. The crystallographic details for these ten Hf–Al intermetallics have been investigated [12–16]. The phase stability for Hf–Al intermetallics has been intensively considered. An asymmetric convex hull for heat of formation of Hf–Al intermetallics indicates that HfAl2 is the most stable phase from an ab initio calculation [17].

⇑ Corresponding author at: School of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China. Tel./fax: +86 871 65136698. E-mail address: [email protected] (Y.-H. Duan). http://dx.doi.org/10.1016/j.commatsci.2015.07.053 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

The formation enthalpies of seven Hf–Al intermetallic compounds have been predicted from Miedema’s semi-empirical model and PARROT model [18], and the results are slightly different from those of Kaufman and Nesor [19]. The formation enthalpy of HfAl3 by the Knudsen cell-mass spectrometry in the temperature range 1280–1680 K is 44.7 kJ/(g atom) [20], which is more negative than the calorimetric value of 40.6 kJ/(g atom) [21]. The order of relative stabilities of L12, D022, and D023 structure in the HfAl3 intermetallics from their formation enthalpies is D023 > D022 > L12 [22]. The main contribution to the electric field gradient of Laves phase C14-HfAl2 comes from the p electrons [23]. In the aspect of elastic properties, theoretical studies are very limited. Only the isothermal bulk moduli of these ten Hf–Al intermetallics have been investigated and Hf3Al2 possesses the largest one [17]. Therefore, a systematical investigation on elastic constants and anisotropic properties is necessary. The first-principles calculations based upon electronic density-functional theory (DFT) have been proved to be a fundamental understanding on the mechanical properties and phase stability of solids. According to the results, a number of bulk properties including formation enthalpy, the relative stability of competing structures, elastic constants, and lattice parameters are derived. In present work, we performed the first-principles calculations to deeper clarify and understand the phase stability and elastic properties of Hf–Al intermetallics consisting of formation enthalpies, elastic constants, and polycrystalline mechanical properties: Young’s modulus E, shear modulus G, bulk modulus B, Poisson’s ratio m and anisotropy index A. Based on the calculated

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elastic constants and moduli, the Debye temperatures were also investigated. 2. Methods and computational details In present work, the first principles calculations were performed in Cambridge sequential total energy package (CASTEP) code [24]. The interactions between ionic core and valence electrons were indicated by ultra-soft pseudo-potentials (USPP), while the exchange correlation energy was describe by the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) parameterization [25]. Valence electrons for Al were 3s23p1 and for Hf were 5s25p65d26s2. The k points separation in the Brillouin zone of the reciprocal space were 20  20  20, 12  12  8, 12  12  4, 12  12  8, 8  4  12, 12  3  9, 9  9  12, 10  10  10, 16  16  16, 8  8  12, 10  10  12, and 15  15  10 for Al (fcc), D022-HfAl3, D023-HfAl3, HfAl2, Hf2Al3, HfAl, Hf4Al3, Hf3Al2, L12-HfAl3, Hf5Al3, Hf2Al, and a-Hf (hcp) respectively. After convergence tests, the cutoff energy for plane wave expansions was set as 500 eV. The self-consistent field (SCF) tolerance was set as 5  107 eV/atom. The crystallographic data of ten Hf–Al compounds, including the lattice parameters and atomic coordinates, are tabulated in Tables 1 and 2, respectively. 3. Results and discussion 3.1. Structural properties and stability The initial crystal structures have been built on the basis of the experimental crystallographic data of Al, a-Hf and ten Hf–Al intermetallics [10,12,14,22,26–31]. The optimized structural parameters are listed in Tables 1 and 2. It is found from Table 1 that the calculated lattice parameters agree very well with the available experimental data with the average deviation of 1%. The deviation is an intrinsic of the first-principles calculations using GGA. The

detailed comparison of atomic coordinates in Table 2 indicates a good agreement between experiment and theory. These agreements of optimized results with the experimental values in Tables 1 and 2 provide a confirmation that the computational method in present work is reliable and suitable. The cohesive energy (Ec) and formation enthalpy (DH) were calculated to investigate the phase stability of these Hf–Al intermetallics. Their expressions are:

Ec ¼ ½Etotal  ðxEHf þ yEAl Þ=ðx þ yÞ;

ð1Þ

h  i. bulk DH ¼ Etotal  xEbulk ðx þ yÞ; Hf þ yEAl

ð2Þ

where EHf and EAl are the total energies of isolated Hf and Al atoms, respectively. x, y are the number of atoms in the HfxAly formula. and Ebulk are the total energy Etotal is the total energy of HfxAly, Ebulk Hf Al of a bulk state Hf atom and Al atom, respectively. The cohesive energy is defined as the work when a compound is decomposed into several isolated atoms. Formation enthalpy is defined as the difference in total energy of the compound and the energies of its constituent elements in their stable states. A more negative cohesive energy indicates the larger binding force of the compound, while a more negative formation enthalpy corresponds to a better phase stability. The calculated cohesive energies and formation enthalpies of these ten intermetallics together with their available experimental and other theoretical calculated data [17–22,32,33] are tabulated in Table 3. From Table 3, the present calculated cohesive energies of Al, and a-Hf are 3.733 eV/atom and 6.845 eV/atom, respectively, they are in good agreement with the reported 3.691 eV/atom and 6.832 eV/atom for Al and a-Hf [32,33]. Fig. 1 plots the comparison of formation enthalpies between our results and other theoretical values. In Fig. 1, the present first-principles method calculated values are in the x-axis and the ab-initio approach calculated values are in the y-axis. The solid

Table 1 Calculated and experimental lattice parameters for Al, a-Hf and Hf–Al intermetallics. Phase

At% Hf

Space group

a Al

V (Å3)

Lattice parameters (Å) b

Experiment (Ref.)

c

0.0

Fm-3m

4.048 4.049

66.3 66.4

Present [26]

L12-HfAl3

25.0

Pm-3m

4.096 4.080

68.7 67.9

Present [22]

D022-HfAl3

25.0

I4/mmm

3.946 3.928

8.924 8.888

138.9 137.1

Present [12]

D023-HfAl3

25.0

I4/mmm

4.002 3.919

17.226 17.139

275.9 263.2

Present [22]

HfAl2

33.3

P63/mmc

5.257 5.230

8.704 8.651

208.3 204.9

Present [27]

Hf2Al3

40.0

Fdd2

9.557 9.523

13.800 13.763

5.529 5.522

729.2 723.7

Present [14]

HfAl

50.0

Cmcm

3.269 3.256

10.860 10.832

4.298 4.281

152.6 151.0

Present [12]

Hf4Al3

57.1

P-6

5.360 5.343

5.462 5.422

135.9 134.0

Present [27]

Hf3Al2

60.0

P42/mnm

7.582 7.549

6.898 6.906

396.5 393.6

Present [27]

Hf5Al3

62.5

P63mcm

8.114 8.066

5.725 5.678

326.4 319.9

Present [12]

Hf2Al

66.7

I4/mcm

6.853 6.776

5.245 5.372

246.3 246.6

Present [10]

a-Hf

100.0

P63/mmc

3.210 3.193

5.051 5.040

44.6 44.5

Present [28]

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Table 2 Calculated and experimental unit cell-internal parameters (Wyckoff positions) of Hf–Al intermetallics. Phase

Unit cell-internal parameters (Wyckoff positions) Present work (x, y, z)

Experiment (x, y, z) [Ref.]

L12-HfAl3

Hf:1a Al:3c

0.00000 0.00000

0.00000 0.50000

0.00000 0.50000

D022-HfAl3

Hf:2a Al1:2b Al2:4d

0.00000 0.00000 0.00000

0.00000 0.00000 0.50000

0.00000 0.50000 0.25000

0.00000 0.00000 0.00000

0.00000 0.00000 0.50000

0.00000 [12] 0.50000 0.25000

D023-HfAl3

Hf:4e Al1:4c Al2:4d Al3:4e

0.00000 0.00000 0.00000 0.00000

0.00000 0.50000 0.50000 0.00000

0.11972 0.00000 0.25000 0.37523

0.00000 0.00000 0.00000 0.00000

0.00000 0.50000 0.50000 0.00000

0.11900 [15] 0.00000 0.25000 0.36600

HfAl2

Hf:4f Al1:2a Al2:6h

0.33333 0.00000 0.82919

0.66667 0.00000 0.65838

0.06372 0.00000 0.25000

0.33333 0.00000 0.82800

0.66667 0.00000 0.65600

0.06500 [29] 0.00000 0.25000

Hf2Al3

Hf:16b Al1:8a Al2:16b

0.18571 0.00000 0.18268

0.05298 0.00000 0.13619

0.00056 0.62676 0.49597

0.18500 0.00000 0.18500

0.05200 0.00000 0.12900

0.00000 [30] 0.63000 0.50000

HfAl

Hf:4c Al:4c

0.00000 0.00000

0.16304 0.42969

0.25000 0.25000

0.00000 0.00000

0.16700 0.42500

0.25000 [31] 0.25000

Hf4Al3

Hf1:1b Hf2:1f Hf3:2h Al:3j

0.00000 0.66667 0.33333 0.33331

0.00000 0.33333 0.66667 0.16669

0.50000 0.50000 0.25896 0.00000

Hf3Al2

Hf1:4f Hf2:4g Hf3:4d Al:8j

0.34441 0.20033 0.00000 0.12146

0.34441 0.79967 0.50000 0.12146

0.00000 0.00000 0.250000 0.21256

0.34000 0.20000 0.00000 0.12500

0.34000 0.80000 0.50000 0.12500

0.00000 [13] 0.00000 0.250000 0.21000

Hf5Al3

Hf1:4d Hf2:6g Al:6g

0.33333 0.23775 0.60308

0.66667 0.00000 0.00000

0.00000 0.25000 0.25000

0.33333 0.2300 0.59000

0.66667 0.00000 0.00000

0.00000 [13] 0.25000 0.25000

Hf2Al

Hf:8h Al:4a

0.15178 0.00000

0.65179 0.00000

0.00000 0.25000

0.15800 0.00000

0.65800 0.00000

0.00000 [10] 0.25000

line suggests a perfect agreement of these two methods, while two dashed lines are referred to an error bar of ±2.5 kJ/mol. For the formation enthalpies of these considered Hf–Al intermetallics in Fig. 1, the first-principles method agree well with the ab-initio approach and experiment. Fig.2 plots calculated cohesive energies and formation enthalpies as a function of the Hf-content. From Fig. 2(a), the cohesive energies have a declining trend with the increasing Hf-content. The smallest cohesive energy for Hf2Al suggests that the binding force of Hf2Al is the largest. For HfAl3, the cohesive energy of D023 structure is the most negative than D022 and L12 structures, indicating that the binding force for D023-HfAl3 is the largest in HfAl3 phases. The declining trend with the increasing Hf-content can be explained by the valency of Al and Hf. The d orbit in Hf has two unpaired electrons in terms of valence electrons of Hf. Al atom will provide valence electrons to form Hf–Al covalent bonds with Hf d-electrons in the process of Hf–Al bond formation. It leads to a reduction in the number of electrons to form the Hf–Al metal bonds. In Hf–Al system, the increasing Hf-content corresponds to the increasing number of Hf atoms in Hf–Al compound. It suggests that the valence electrons provided by Al atoms decrease, and the proportion of Hf–Al covalent bonds reduces in the Hf–Al compound with the increasing Hf-content. As a result, when the Hf-content increases, the binding force between Hf and Al decreases and the cohesive energy of Hf–Al compound reduces. Therefore, the highest Hf-content Hf2Al has the lowest cohesive energy. For the considered Hf–Al intermetallics, six intermetallics are stable at low temperature in the Hf–Al phase diagram, namely D023-HfAl3, HfAl2, Hf2Al3, HfAl, Hf4Al3 and Hf3Al2 [34]. Our calculations of formation enthalpies are illustrated a convex hull in Fig. 2(b). The convex hull is determined by D023-HfAl3, HfAl2 and

Fig. 1. Comparison of calculated formation enthalpies in this work for the Hf–Al intermetallics with the calculated values by ab initio calculations. The solid line shows unity (y = x) while the dotted lines present an error range of ±2.5 kJ/mol.

Hf4Al3. As shown in Fig. 2(b), the ground-state convex hull is asymmetric and skewed toward Al side with a maximum at HfAl2. The calculated formation enthalpies for the intermetallics located at the convex hull in Fig. 2(b) are 39.224 kJ/mol for D023-HfAl3, 43.860 kJ/mol for HfAl2, and 37.745 kJ/mol for Hf4Al3, respectively. The most negative formation enthalpy for HfAl2 suggests the best phase stability. The energies lying of Hf2Al3 and HfAl close to the convex hull (lie about 0.6 kJ/mol and 0.5 kJ/mol below the convex hull for Hf2Al3 and HfAl, respectively) indicate that they are stable at experimentally temperatures. As discussed by

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Fig. 2. Calculated cohesive energies (a) and formation enthalpies (b) plotted as a function of mole fraction of Hf for the Hf–Al intermetallics.

Table 3 The calculated cohesive energies and a comparison of formation enthalpies (DH) for Hf–Al intermetallic phases obtained by various methods: ab intio calculations (at 0 K), experiment (at different temperatures), and CALPHAD (at 298.15 K or the standard enthalpy of formation) modeling of Hf–Al phase diagram. Phase

Ec (eV/atom)

DH (kJ/mol) Present

ab initio

Experiment

CALPHAD

Al

3.733 3.691 [32]

0

L12-HfAl3

4.962

36.454

36.828 [17] 37.300 [22]

D022-HfAl3

5.040

38.991

38.649 [17] 38.900 [22]

D023-HfAl3

5.043

39.224

39.632 [17] 40.000 [22]

44.7 ± 2.4 [20] 40.6 ± 0.8 [21]

HfAl2

5.392

43.860

43.289 [17]

43.8 ± 1.3 [21]

41.673 [18]

Hf2Al3

5.621

42.767

41.796 [17]

40.8 ± 2.6 [20]

42.885 [18]

39.9 ± 2.0 [21]

38.077 [18] 41.818 [19] 41.077 [18]

HfAl

5.954

40.020

39.028 [17]

Hf4Al3

6.188

37.745

38.616 [17]

44.372 [19]

45.203 [18]

43.535 [19]

Hf3Al2

6.251

31.837

31.702 [17]

Hf5Al3

6.298

28.658

28.218 [17]

-41.023 [19]

Hf2Al

6.423

25.179

25.189 [17]

41.244 [18]

a-Hf

6.845 6.832 [33]

0

Ghosh et al. [17], entropic terms could very likely lead to the stabilization of Hf2Al3 and HfAl at experimentally accessible temperatures. Actually, Hf4Al3, Hf3Al2 and even Hf5Al3 intermetallics could have been formed both by disproportionation of HfAl and vaporization decomposition of Hf2Al3 [20]. For Hf3Al2, the formation enthalpy in the present work (31.837 kJ/mol) is in good agreement with the value by ab initio method (31.702 kJ/mol) [17]. The 3 kJ/mol energy lying above the convex hull for Hf3Al2 seems relatively large to be overcome by structural entropy differences at several hundred degrees Celsius. However, the negative enthalpy of formation for Hf3Al2 reveals its observed stability. The calculated formation enthalpies for D022-HfAl3 lie above the convex hull by 1 kJ/mol in Fig. 2(b), and this phase is experimentally observed to be stable at the high temperature. Experimentally an equilibrium transition from D023-HfAl3 to D022-HfAl3 has been established around 650 °C [9]. For the three considered metastable phases (L12-HfAl3, Hf5Al3 and Hf2Al), the calculated formation enthalpies lie about 3 kJ/mol, 5 kJ/mol and 4 kJ/mol above the convex hull, respectively. L12-HfAl3 is a metastable cubic structure. When L12-HfAl3 is heated, it will immediately transform to D022

structure or D023 structure depending on the temperature. In fact, the transformation of L12 in D023 at 750 °C was observed experimentally [4]. In Table 3, the formation enthalpies of L12-, D022-, and D023-HfAl3 are 36.454, 38.991 and 39.224 kJ/mol, respectively. It suggests that the order of phase stabilities of L12, D022, and D023 structure in our work is D023 > D022 > L12, which is in good agreement with the report [22]. Hf5Al3 and Hf2Al are observed experimentally that they were stabilized by interstitial impurities [2,9] and Si [10,11], respectively. Our calculated formation enthalpies, as listed in Table 3 and plotted in Fig. 2(b), are consistent with the explanation of these three metastable Hf–Al intermetallics in pure alloys. 3.2. Elastic constants and polycrystalline moduli The stiffness against an externally applied strain for a solid is determined by the elastic constant Cij. Using first-principles calculations by calculating the elastic strain energy as a function of appropriate lattice deformation through an introduced deformed cell, the elastic constants can be obtained. The elastic strain energy is given by follows [35]:

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Table 4 The calculated elastic constants for Hf–Al intermetallics. Phase

Elastic constants (GPa) C11

C12

Al

107.7 107.0a 176.6 193.6 205.6 248.1 232.3 177.2 256.7 207.3 194.7 162.5 191.4 190.1b

61.2 61.0 69.2 87.1 67.5 52.8 54.8 71.6 53.3 72.7 61.3 98.3 76.7 74.5

L12-HfAl3 D022-HfAl3 D023-HfAl3 HfAl2 Hf2Al3 HfAl Hf4Al3 Hf3Al2 Hf5Al3 Hf2Al a-Hf a b

C13

47.4 54.2 64.0 59.2 109.3 61.9 85.9 67.9 88.2 67.3 65.5

C22

C23.

219.8 232.3

74.1 55.5

C33

C44

217.8 200.0 220.2 213.7 199.3 215.8 188.6 183.3 185.3 212.6 204.4

28.9 28.0 68.5 92.4 82.5 86.1 76.3 63.6 81.4 95.4 93.3 77.8 62.8 60.0

C55

C66

123.3 100.5 78.9 125.1

56.3 71.7 78.1 66.5

Experiment, Ref. [36]. Experiment at 0 K, Ref. [37].

Table 5 The elastic compliance matrix of the Hf–Al intermetallics calculated from the elastic coefficients using the strain-stress method. Phase

Al L12-HfAl3 D022-HfAl3 D023-HfAl3 HfAl2 Hf2Al3 HfAl Hf4Al3 Hf3Al2 Hf5Al3 Hf2Al a-Hf

U ¼ DE=V 0 ¼ 1=2

Sij S11

S12

0.01579 0.00726 0.00662 0.00567 0.00446 0.00477 0.00914 0.00428 0.00619 0.00603 0.01061 0.00610

0.005720 0.00204 0.00278 0.00157 0.00066 0.00084 0.00174 0.00064 0.00124 0.00147 0.00495 0.00159

6 X 6 X C ij ei ej ; i

S13

0.00084 0.00111 0.00110 0.00103 0.00453 0.00104 0.00226 0.00169 0.00269 0.00143

S22

0.00530 0.00494

ð3Þ

j

DE ¼ Etotal ðV 0 ; dÞ  Etotal ðV 0 ; 0Þ;

S23

0.00161 0.00042

S44

0.00496 0.00560 0.00518 0.00552 0.00762 0.00523 0.00736 0.00637 0.00796 0.00568

0.03457 0.01459 0.01082 0.01211 0.01162 0.01310 0.01572 0.01229 0.01048 0.01072 0.01285 0.01593

S55

S66

0.00811 0.00995 0.01268 0.00800

0.01775 0.01394 0.01280 0.01503

Table 6 The calculated polycrystalline mechanical properties of Hf–Al intermetallics, including bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), B/G, Poisson’s ratio m.

ð4Þ

where Cij is the elastic constant. ei and ej are strains. DE is the energy difference. V0 is the volume of the original cell. We employed seven different strains d (0.0, ±0.01, ±0.02, ±0.03) for the deformed cell in the present work. The numbers of independent elastic constants for cubic, orthorhombic, tetragonal and hexagonal structures are three, nine, six and five, respectively. Tables 4 and 5 list the calculated elastic constants Cij and elastic compliance matrix Sij of the ten Hf–Al intermetallics, respectively. The good agreement of present results with experimental values for Al and a-Hf [36,37] suggests the reliability of our calculations. The mechanical stabilities can be discussed based on the generalized stability criteria: for cubic crystals, C11 > 0, C44 > 0, C11 > C12, C11 + 2C12 > 0 [38]; for hexagonal crystals, C11 > 0, C44 > 0, C11 > C12, (C11 + C12)C33  2C213 > 0 [39]; for tetragonal crystals, C11 > 0, C33 > 0, C44 > 0, C66 > 0, C11 > C12, C11 + C33  2C13 > 0, 2C11 + C33 + 2C12 + 4C13 > 0 [35]; for orthorhombic crystals, C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, C11 + C22  2C12 > 0, C11 + C33  2C13 > 0, C22 + C33  2C23 > 0, C11 + C22 + C33 + 2C12 + 2C13 + 2C23 > 0 [40]. In Table 4, the elastic constants of the cubic L12-HfAl3 are C11 = 176.6 > 0, C44 = 68.5 > 0, C11 = 176.6 > C12 = 69.2, and C11 + 2C12 = 315 > 0. They satisfy the cubic mechanical stability criteria. For the three hexagonal crystals (HfAl2, Hf4Al3 and Hf5Al3),

S33

Al L12-HfAl3 D022-HfAl3 D023-HfAl3 HfAl2 Hf2Al3 HfAl Hf4Al3 Hf3Al2 Hf5Al3 Hf2Al a-Hf a b

B

B [17]

G

E

B/G

m

76.7 105.0 107.6 106.9 119.7 115.8 120.2 120.1 121.4 107.4 117.7 114.1

74.2 103.8 105.4 105.3 118.3 114.3 117.4 116.1 123.0 109.9 112.1 110.3

26.5 62.2 86.6 81.5 89.4 73.3 71.6 89.1 75.8 66.7 56.8 63.1

71.3 155.8 204.8 195.0 214.7 181.6 179.2 214.3 188.2 165.8 146.8 159.8

2.89 1.69 1.24 1.31 1.34 1.58 1.68 1.35 1.60 1.61 2.07 1.81

0.35a, 0.349b 0.25 0.18 0.20 0.20 0.24 0.25 0.20 0.24 0.24 0.29 0.27a, 0.28b

This work. From Ref. [42].

their elastic constants are consistent with the restrictions to hexagonal crystals. For the four tetragonal crystals (D022-HfAl3, D023-HfAl3, Hf3Al2 and Hf2Al) and two orthorhombic crystals (Hf2Al3 and HfAl), all the elastic constants meet the corresponding crystal’s mechanical stability criteria. The results reveal that all the ten Hf–Al intermetallics are mechanically stable. The elastic constant C11 characterizes the x direction resistance to linear compression, while the elastic constant C33 denotes the z direction resistance to linear compression. From Table 4, it can be

Y.-H. Duan et al. / Computational Materials Science 110 (2015) 10–19

15

Fig. 3. Comparison of calculated bulk modulus in this work for the Hf–Al intermetallics with the calculated values by ab initio calculations (a) and the bulk modulus as a function of mole fraction of Hf. In (a), the solid line shows unity (y = x) while the dotted lines present an error range of ±5 GPa, and in (b) the light dotted line is the ideal behavior of bulk modulus.

concluded that the calculated large C11 suggests the incompressibility of Hf–Al intermetallics under the x direction uniaxial stress. C33 larger than C11 for HfAl, Hf2Al and D022-HfAl3 indicates that it is more compressible along x direction than that along z direction for these three Hf–Al intermetallics. For the remaining Hf–Al intermetallics, their C11 larger than C33 implies that the z direction is more compressible than the x direction. Furthermore, it is well known that the elastic constant C44 indirectly determines the indentation hardness of a solid, and a large C44 implies a strong resistance to shear in the (1 0 0) plane [41]. The largest C44 for Hf3Al2 suggests that Hf3Al2 has the strongest resistance to shear in the (1 0 0) plane, as a result, Hf3Al2 should have the highest hardness. The polycrystalline moduli are more important than single-crystal elastic constants for materials. The polycrystalline bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio m have been obtained in terms of their elastic constants, and the results compared with the other theoretical values [17,42] are tabulated in Table 6. Bulk modulus is a measurement of the average bond strength of atoms for a solid [43]. Fig. 3 plots the comparison of bulk moduli by first-principles calculations (in the x-axis) with the ab-initio calculations (in the y-axis) [17] of the ten intermetallics and the variation of bulk modulus as a function of Hf content. With regard to all the Hf–Al intermetallics, the present calculated bulk moduli agree well with those from the ab-initio approach with differences less than 5 GPa in Fig. 3(a). The bulk moduli show positive deviation from the ideal behavior shown by the light dotted line in Fig. 3(b). As listed in Table 6 and plotted in Fig. 3, Hf3Al2 has the highest bulk modulus (121.4 GPa), while L12-HfAl3 possesses the lowest one (105.0 GPa). As a result, Hf3Al2 has the strongest average bond strength of atoms and L12-HfAl3 has the lowest one. Shear modulus reflects the resistance to reversible deformations under the shear stress [44,45]. A larger shear modulus G for HfAl2 (89.4 GPa) means its resistance to reversible deformations is higher. Young’s modulus (the ratio of stress and strain) is a measure of the stiffness of a solid. The material with a larger Young’s modulus is stiffer. As shown in Table 6, HfAl2 is stiffer than the other considered Hf–Al intermetallics owing to its higher Young’s modulus (214.7 GPa). The values of B/G and Poisson’s ratio m determine the brittleness and plasticity of a solid, that is, a solid with B/G < 1.75 or m < 0.26 usually is brittle, otherwise it is ductile [46,47]. Fig. 4 depicts the

Fig. 4. The ductile/brittle properties of the Hf–Al intermetallics. The dotted line denotes the dividing line between ductile and brittle solids. Solids with B/G > 1.75 and m > 0.26 are ductile, the opposite are brittle.

relationship among ductile or brittle properties, B/G and m. B/G > 1.75 and m > 0.26 for Hf2Al suggests that only Hf2Al is ductile. The remaining Hf–Al intermetallics are brittle with their B/G < 1.75 and m < 0.26. It is noted that, for three structures HfAl3 they are brittle, however, L12 structure is more ductile than D023 and D022 structures due to the order of their B/G and m in L12 > D023 > D022. Experiments on many face-centered-cubic metallic alloys also reveal that L12 structure is significantly more ductile than D022 and D023 structures owing to the lack of a sufficient number of slip systems in D022 and D023 structures [48]. 3.3. Anisotropy of elastic moduli The degree of elastic anisotropy is associated with the crystal symmetry. The universal anisotropic index AU is a universal measure to quantify the single crystal elastic anisotropy due to the consideration of both the shear and the bulk contributions unlike all other existing anisotropy measures [49]. We applied the universal anisotropic index AU and the compression and shear percent anisotropy (Acomp and Ashear) to describe the elastic anisotropy. The universal elastic anisotropy index AU and the percent anisotropy for a crystal [50,51] with any symmetry are expressed as follows:

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Y.-H. Duan et al. / Computational Materials Science 110 (2015) 10–19

Table 7 The calculated the ratio of bulk modulus (BV/BR) and shear modulus (GV/GR) in the Voigt and Reuss approximations, universal anisotropic index (AU), percent anisotropy (Ashear and Acomp) and shear anisotropic factors (A1, A2 and A3) of the Hf–Al intermetallics. Phase

BV/BR

GV/GR

AU

Ashear

Acomp

A1

A2

A3

Al L12-HfAl3 D022-HfAl3 D023-HfAl3 HfAl2 Hf2Al3 HfAl Hf4Al3 Hf3Al2 Hf5Al3 Hf2Al a-Hf

1.000 1.000 1.001 1.002 1.002 1.000 1.001 1.004 1.000 1.000 1.001 1.001

1.011 1.015 1.077 1.015 1.007 1.023 1.162 1.012 1.053 1.042 1.123 1.003

0.055 0.075 0.386 0.077 0.037 0.115 0.810 0.064 0.265 0.210 0.616 0.016

0.0057 0.0072 0.0370 0.0074 0.0034 0.0116 0.0747 0.0062 0.0257 0.0202 0.0581 0.0016

0 0 0.0035 0.0009 0 0 0.0084 0.0021 0 0 0.0044 0.0003

1.043 1.076 1.267 1.110 1.012 0.932 1.611 0.934 1.113 1.111 1.516 0.932

1.043 1.076 1.267 1.110 1.012 1.106 1.561 0.934 1.113 1.111 1.516 0.932

1.043 1.076 1.315 1.455 1.001 0.658 1.077 1.000 1.160 1.000 1.072 0.868

AU ¼ 5

GV BV þ  6 P 0; GR BR

Acomp ¼

BV  BR ; BV þ BR

ð5Þ

Ashear ¼

GV  GR ; GV þ GR

ð6Þ

where BV (GV) and BR (GR) are the bulk modulus (shear modulus) in the Voigt and Reuss approximations, respectively. AU = 0 for a crystal means the crystal is isotropic. A larger deviation of AU from zero suggests a larger extent of anisotropy. The values of Acomp and Ashear can range from zero (isotropic) to 1 (the maximum anisotropic). Since the investigated crystal systems in the present work include several crystal structures, the shear anisotropic factors provide a measure of the degrees of anisotropy in atomic bonding in different planes. Therefore, the shear anisotropic factors A1, A2, A3 are calculated and discussed as follows [51]:

A1 ¼

4C 44 C 11 þ C 33  2C 13

for the f1 0 0g plane;

ð7Þ

A2 ¼

4C 55 C 22 þ C 33  2C 23

for the f0 1 0g plane;

ð8Þ

4C 66 A3 ¼ C 11 þ C 22  2C 12

for the f0 0 1g plane:

ð9Þ

A1, A2 and A3 = 1 means isotropic for a crystal. Otherwise, the crystal is anisotropic. Table 7 lists the results of elastic anisotropies. Although HfAl2, Hf5Al3, Hf3Al2 and Hf2Al3 are not cubic structure, their calculated Acomp are zero because of the same predicted bulk moduli in Voigt and Reuss approximations. According to Voigt and Reuss approximations, BV and BR are mainly determined by the compression moduli (C11, C22 and C33). Notably, for these four intermetallics the computed compression moduli C11, C22 and C33 in Table 4 exhibit small deviations from each other, implying that the anisotropies in bulk moduli in these crystals are relatively weak. It results in the non-directional dependence of the bulk moduli. The largest universal elastic anisotropic index for the Hf–Al intermetallics is orthorhombic HfAl due to its largest GV/GR. It suggests that the elastic moduli of HfAl are strongly dependent on different directions, and the calculated Acomp, Ashear and shear anisotropic factors (A1, A2 and A3) values for HfAl support this conclusion. Although HfAl2 belongs to hexagonal structure, it is elastically similar to the cubic crystal with the BV/BR and GV/GR close to 1 in Table 7. As a result, HfAl2 has the smallest universal elastic anisotropic index among these ten Hf–Al intermetallics. Moreover, HfAl2 possesses the percent anisotropy close to zero and shear anisotropic factors close to 1, indicating that the dependent of the elastic moduli on different directions for HfAl2 is slight. One can be concluded from the calculated elastic anisotropic indexes that the

order in anisotropy for Hf–Al intermetallics is HfAl > Hf2Al > D0 22-HfAl3 > Hf3 Al2 > Hf5 Al 3 > Hf 2Al 3 > D0 23-HfAl 3 > L12 -HfAl3 > Hf4 Al 3 > HfAl 2. As more direct and effective method to describe the elastic anisotropic behavior completely than elastic anisotropic indexes, the three-dimensional (3D) surface constructions of the directional dependence of reciprocal of Young‘s modulus are also investigated by the following expressions [38]:

  1 S44  2 2 2 2 2 2  l1 l2 þ l2 l3 þ l1 l3 ¼ S11  2 S11  S12  E 2

for cubic system; ð10Þ

1 2 2 4 2 2 ¼ ð1  l3 Þ S11 þ l3 S33 þ l3 ð1  l3 Þð2S13 þ S44 Þ for hexagonal system; E ð11Þ 1 4 4 2 2 2 2 4 2 2 ¼ S11 ðl1 þ l2 Þ þ ð2S13 þ S44 Þðl1 l3 þ l2 l3 Þ þ S33 l3 þ ð2S12 þ S66 Þl1 l2 E for tetragonal system; ð12Þ 1 4 4 4 2 2 2 2 2 2 2 2 ¼ l1 S11 þ l2 S22 þ l3 S33 þ 2l1 l2 S12 þ 2l1 l3 S13 þ 2l2 l3 S23 þ l2 l3 S44 E 2 2 2 2 þ l1 l3 S55 þ l1 l2 S66 for orthorhombic system; ð13Þ where Sij is the usual elastic compliance constant listed in Table 5. l1, l2 and l3 are the direction cosines. Fig. 5 plots the 3D surface constructions of the directional dependence of reciprocal of Young‘s modulus E. The magnitude of E along different orientations is represented by the surface in each graph. A spherical shape in the 3D surface construction denotes an isotropic system. The content of anisotropy is reflected by the deviation degree from the spherical shape. Moreover, the Young’s modulus in the three principal directions are also investigated and the expressions are given as E[100] = 1/S11, E[010] = 1/S22 and E[001] = 1/S33. For the cubic L12-HfAl3, a small deviation in 3D surface construction from the sphere indicates that the Young’s modulus for this cubic structure show a little anisotropy. It should be noted that, although C11 of L12-HfAl3 equals to C33, the Young’s moludi of L12-HfAl3 in three principal axes ([1 0 0], [0 1 0] and [0 0 1]), [1 1 0] and [1 1 1] directions by using the expressions introduce in the literature [52] are 137.7, 147.8 and 168.8 GPa, respectively. The difference in the directional Young’s moduli between principal axes and [1 1 1] direction for L12-HfAl3 is the largest one, indicating that the Young’s modulus of cubic L12-HfAl3 shows a little anisotropic property. Young’s modulus of tetragonal D022-HfAl3 shows strongly anisotropic: the 3D surface construction is visible deviate in shape from the sphere along x direction. The reason is C33 larger than C11 in D022-HfAl3, resulting in x direction is more compressible than z direction, and the values of E[100] (151.0 GPa), E[010]

Y.-H. Duan et al. / Computational Materials Science 110 (2015) 10–19

(151.0 GPa), and E[001] (201.8 GPa) also confirm the conclusion. The 3D surface construction of the Young’s modulus for the tetragonal D023-HfAl3 is characterized by incompressible along x direction than along z direction. For the hexagonal HfAl2, the 3D surface construction closed to a sphere and E[100] = E[010] = 224.2 GPa > E[001] = 193.1 GPa suggests its smallest anisotropy. As for the orthorhombic Hf2Al3 and HfAl, the deviate in shape from the sphere in HfAl is significantly larger than that in Hf2Al3. It also can be derived from their Young’s moduli in the three principal directions, which E[100], E[010], and E[001] for HfAl are 109.4, 202.4 and 131.2 GPa, respectively and E[100], E[010], and E[001] for Hf2Al3 are 214.6, 190.9 and 182.1 GPa, respectively. For the hexagonal Hf4Al3 with E[100] = E[010] = 233.6 GPa and E[001] = 191.0 GPa, similar to HfAl2, the 3D surface construction appears to be almost spherical in shape. However, the value of E[001]/E[100] for Hf4Al3 is smaller than that for HfAl2, suggesting that the anisotropy for Hf4Al3 is larger than that for HfAl2. Owing to the difference between C33 and C11 in Hf3Al2 (18.7 GPa) slightly smaller than that in Hf2Al (22.8 GPa), the deviate in shape from the sphere in Hf3Al2 is somewhat smaller than that in Hf2Al. It can be concluded, from the 3D surface construction of the Young’s modulus, that the degree of the elastic anisotropy for the considered Hf–Al intermetallics follows the order of HfAl > Hf2Al > D022-HfAl3 > Hf3Al2 > Hf5Al3 > Hf2Al3 > D023-HfAl3 > L12-HfAl3 > Hf4Al3 > HfAl2. This result is consistent with the results from the analysis of the universal elastic anisotropy indexes and the percent anisotropy.

17

3.4. Anisotropy of acoustic velocities The crystal symmetry and propagation direction determine the sound velocities. In an illustrative cubic crystal, the pure transverse and longitudinal modes only can be found in [0 0 1], [1 1 0] and [1 1 1] directions, while the sound propagating modes in other directions are the quasi-transverse or quasi-longitudinal waves. By the procedure of Brugger [53], the sound velocities in pure transverse and longitudinal modes of the Hf–Al intermetallics can be calculated from the single crystal elastic constants by expressions:

Cubic

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi C 11 =q; ½0 1 0mt1 ¼ ½0 0 1mt2 ¼ C 44 =q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 1 0ml ¼ ðC 11 þ C 12 þ 2C 44 Þ=2q; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ½1 1 0mt1 ¼ ðC 11  C 12 Þ=q; ½0 0 1mt2 ¼ C 44 =q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 1 1ml ¼ ðC 11 þ 2C 12 þ 4C 44 Þ=3q; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 1 2mt1 ¼ mt2 ¼ ðC 11  C 12 þ C 44 Þ=3q

ð14Þ

Hexagonal pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0ml ¼ ðC 11  C 12 Þ=2q; ½0 1 0mt1 ¼ C 11 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ½0 0 1mt2 ¼ C 44 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 0 1ml ¼ C 33 =q; ½1 0 0mt1 ¼ ½0 1 0mt2 ¼ C 44 =q

ð15Þ

½1 0 0ml ¼

Fig. 5. The directional dependence of the Young’s modulus for the Hf–Al intermetallics. The magnitude of Young’s modulus at different directions is represented by the contour. The units are in GPa.

18

Y.-H. Duan et al. / Computational Materials Science 110 (2015) 10–19

Table 8 The density (in g/cm3) and anisotropic sound velocities (in km/s) of the Hf–Al intermetallics. Phase

L12-HfAl3

D022-HfAl3

D023-HfAl3

HfAl2

Hf2Al3

HfAl

Hf4Al3

Hf3Al2

Hf5Al3

Hf2Al

6.48 5.220 3.251 3.251

6.29 5.548 4.427 3.833

6.31 5.708 3.991 3.616

7.53 3.601 5.740 3.381

8.04 5.375 2.646 3.133

9.06 4.422 2.813 3.716

9.91 3.203 5.090 2.866

9.99 4.555 2.796 3.090

10.10 2.570 4.391 3.039

10.34 3.964 2.536 2.743

5.435 4.071

6.474 2.910

6.129 3.308

4.672 2.596

4.364 1.762

[0 0 1]mt2

3.251

3.833

3.616

3.090

2.743

[1 1 1]ml

5.171 3.008

q [1 0 0]

[1 0 0]ml [0 1 0]mt1 [0 0 1]mt2

[1 1 0]

[1 1 0]ml ½1 1 0v t1

[1 1 1]

½1 1 2v t1;2 [0 1 0]

[0 1 0]ml [1 0 0]mt1 [0 0 1]mt2

[0 0 1]

[0 0 1]ml [1 0 0]mt1 [0 1 0]mt2

5.884 4.427 4.427

5.630 3.991 3.991

5.408 3.381 3.381

Table 9 The elastic wave velocity (in km/s) and the Debye temperature (in K) for the Hf–Al intermetallics.

Al L12-HfAl3 D022-HfAl3 D023-HfAl3 HfAl2 Hf2Al3 HfAl Hf4Al3 Hf3Al2 Hf5Al3 Hf2Al a-Hf

ml

mt

mm

HD

6.442

3.133

3.520

409 400 [42] 400 470 457 438 378 348 368 336 311 283 255 252 [56]

5.385 5.955 5.845 5.633 5.154 4.879 4.910 4.719 4.409 4.325 3.858

Orthorhombic pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0ml ¼ C 11 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 1 0ml ¼ C 22 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 0 1mt ¼ C 33 =q;

3.098 3.711 3.594 3.446 3.019 2.811 2.998 2.755 2.570 2.344 2.177

pffiffiffiffiffiffiffiffiffiffiffiffiffi C 66 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0mt1 ¼ C 66 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0mt1 ¼ C 55 =q; ½0 1 0mt1 ¼

3.441 4.089 3.966 3.804 3.348 3.122 3.311 3.055 2.851 2.615 2.421

pffiffiffiffiffiffiffiffiffiffiffiffiffi C 55 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ½0 0 1mt2 ¼ C 44 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 1 0mt2 ¼ C 44 =q ½0 0 1mt2 ¼

ð16Þ Tetragonal pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 0 0ml ¼ ½0 1 0ml ¼ C 11 =q; ½0 0 1mt1 ¼ C 44 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 1 0mt2 ¼ C 66 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ½0 0 1ml ¼ C 33 =q; ½1 0 0mt1 ¼ ½0 1 0mt2 ¼ C 66 =q pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 1 0ml ¼ ðC 11 þ C 12 þ 2C 66 Þ=2q; ½0 0 1mt1 ¼ C 44 =q; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 1 0mt2 ¼ ðC 11  C 12 Þ=2q

; ð17Þ

where q is the density. vl is the longitudinal sound velocity. vt1 and vt2 refer the first transverse mode and the second transverse mode, respectively. Table 8 shows the calculated sound velocities in different directions. Generally, C11, C22 and C33 determine the longitudinal sound velocities along [1 0 0], [0 1 0] and [0 0 1] directions, respectively, while C44, C55 and C66 correspond to the transverse modes [52]. As a result, the longitudinal sound velocities and the first transverse sound velocities are quite different in [1 0 0], [1 1 0] and [1 1 1] directions for L12-HfAl3 in Table 8. The anisotropic properties of sound velocities also indicate the elastic anisotropy in these intermetallics.

5.229 2.646 3.081

5.064 2.813 2.650

5.156 3.133 3.081

4.690 3.716 2.650

4.666 2.866 2.866

4.345 2.796 2.796

4.260 3.039 3.039

4.233 2.536 2.536

Debye temperature HD is a fundamental parameter for the thermodynamic properties of a solid. It is consistent with specific heat, thermal expansion and elastic constants. The Debye temperature can be estimated from the elastic constant evaluations [54,55]:

HD ¼

  1 h 3n NA q 3 tm ; k 4p M

tm ¼

  13 1 2 1 þ ; 3 t3t t3l

ð18Þ

tl ¼

  12 4G Bþ q ; 3

mt ¼

 12 G

q

; ð19Þ

where h is the Planck constant, respectively; NA is the Avogadro number, M is the molecular weight. q is the density. B and G are bulk modulus and shear modulus, respectively. tm is the average sound velocity. tl is the longitudinal velocity and tt is the transverse sound velocity. Table 9 lists the calculated Debye temperatures for Hf–Al intermetallics. The calculated Debye temperatures of Al and Hf agree well with the reported values [42,56]. As the Hf content increases, the cohesive energy and binding force of Hf–Al intermetallic decrease in terms of previous analysis of valency of Hf and Al in Section 3.1. The high binding force corresponds to a large Debye temperature. As a result, Debye temperatures also follow the decreasing trends with the increasing Hf content except L12-HfAl3 and Hf4Al3. It is well known that Debye temperature is related to the elastic moduli and density. The Debye temperatures of these intermetallics are relatively small due to their small mechanical moduli and large densities. The elastic wave velocities of L12-HfAl3 are smaller than D022-HfAl3 and D022-HfAl3 due to its smaller mechanical moduli and larger densities. Thereby, the Debye temperature of L12-HfAl3 is smaller than D022-HfAl3 and D022-HfAl3. For Hf4Al3, the cohesive energy is slightly smaller than that of HfAl with a difference of 0.234 eV/atom, the larger shear modulus determines its larger Debye temperature than HfAl. The largest HD is 457 K for D023-HfAl3 while the lowest one is 283 K for Hf2Al. However, there is no experimental and theoretical data available for the Debye temperatures of Hf–Al intermetallics so far. Therefore, our calculated results can provide support for future works on Hf–Al intermetallics. 4. Conclusions The phase stability, elastic moduli, and elastic anisotropy properties of Hf–Al intermetallics were investigated and discussed using the first-principles calculations. It is found that the equilibrium lattice parameters are slightly overestimated within the

Y.-H. Duan et al. / Computational Materials Science 110 (2015) 10–19

GGA. Results of formation enthalpies, which agree well with the other theoretical values, show that HfAl2 is more energetically stable than the other Hf–Al intermetallics. Based on the calculated single-crystal elastic constants, the mechanical moduli of these intermetallics were obtained. With regard to all the Hf–Al intermetallics, the first-principles calculated bulk moduli agree well with those from the ab-initio approach with differences less than 5 GPa. The elastic anisotropy was characterized by several different anisotropic indexes (AU, Acomp, Ashear, A1, A2, and A3) and three-dimensional surface construction. The results suggest that the selected Hf–Al intermetallics show certain anisotropy in mechanical properties, and HfAl has the most anisotropy while HfAl2 is the least anisotropic. Additionally, the calculated sound velocities in different directions also imply their anisotropic. Debye temperatures, estimated from the elastic constant evaluations, generally follow the decreasing trends with the increasing Hf content. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant no. 51201079. References [1] R.M. Cahn, Intermetallics 6 (1998) 563. [2] R. Marazza, R. Ferro, G. Rambaldi, D. Mazzone, J. Less-Common Met. 37 (1974) 285. [3] Q.Z. Hong, D.A. Lilienfeld, J.W. Mayer, J. Appl. Phys. 64 (1988) 4478. [4] S. Srinivasan, P.B. Desch, R.B. Schwarz, Scr. Metall. Mater. 25 (1991) 2513. [5] A.F. Norman, P. Tsakiropoulos, Mater. Sci. Eng., A 134 (1991) 1234. [6] B.B. Rath, G.P. Mohanty, L.F. Mondolfo, J. Inst. Met. 89 (1961) 248. [7] M. Potzschke, K. Schubert, Z. Metallkd. 53 (1962) 548. [8] I.A. Tsyganova, M.A. Tylkina, E.M. Savitskiy, Izv. Akad. Nauk SSSR, Met. 1 (1970) 160. [9] K. Schubert, H.G. Meissner, M. Pötschke, W. Rossteutscher, E. Stolz, Naturwissenschaften 49 (1962) 57. [10] H. Nowotny, O. Schob, F. Benesovsky, Monatsh. Chem. 92 (1961) 1301. [11] W. Rieger, H. Nowotny, F. Benesovsky, Monatsh. Chem. 95 (1964) 1417. [12] H. Boller, H. Nowotny, A. Wittmann, Monatsh. Chem. 91 (1960) 1174. [13] L.E. Edsammar, Acta Chem. Scand. 14 (1960) 1220. [14] L.E. Edsammar, Acta Chem. Scand. 14 (1960) 2244. [15] A.E. Dwight, J.W. Downey, R.A. Conner, Acta Crystallogr. 14 (1961) 75. [16] J.C. Schuster, H. Nowotny, Z. Metallkd. 71 (1980) 341. [17] G. Ghosh, M. Asta, Acta Mater. 53 (2005) 3225.

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