Ab initio calculations of elastic constants and thermodynamic properties of γTiAl under high pressures

Intermetallics 18 (2010) 761–766

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Ab initio calculations of elastic constants and thermodynamic properties of gTiAl under high pressures Hongzhi Fu a, *, Zhiguo Zhao a, WenFang Liu b, Feng Peng a, Tao Gao c, Xinlu Cheng c a

Department of Physics, Luoyang Normal College, Luoyang 471022 People’s Republic of China College of Chemistry and Chemical Engineering, Luoyang Normal College, Luoyang 471022, People’s Republic of China c Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065 People’s Republic of China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 April 2008 Received in revised form 12 July 2009 Accepted 2 December 2009 Available online 23 December 2009

We have investigated the structural and elastic properties of gTiAl under high pressures using Vanderbilt-type ultrasoft pseudopotentials within the generalized gradient approximation correction (GGA) in the frame of density functional theory. The calculated pressure dependence of the normalized volume is in excellent agreement with the experimental results. The elastic constants and anisotropy as a function of applied pressure, the ratio of the normalized lattice parameters (a  a0)/a0, (c  c0)/c0 and the normalized volume (V  V0)/V0 with the applied pressure are presented. The variations of bulk modulus Ba, Bc, and the brittleness with the pressure are investigated. Through the quasi-harmonic Debye model, we also study the thermodynamic properties of gTiAl. The thermal expansion versus temperature and pressure, the thermodynamic parameters X (X: Debye temperature or specific heat) with pressure P, and the heat capacity of gTiAl at various pressures and temperatures are estimated. Ó 2009 Elsevier Ltd. All rights reserved.

PACS: 62.20.Dc 65.40.-b 62.50.þp 71.15.Ap Keywords: A. Titanium aluminides, based on TiAl B. Elastic properties E. Ab initio calculations

1. Introduction The gTiAl(space group P4/mmm,prototype AuCu) has been extensively investigated in the last 15 years because of its anomalous hardening behavior, the positive temperature dependence on yield stress, chemical stability, low density, and good oxidation resistance[1–4]. With close-packed L10 crystal structure, gTiAl is well established by experiments, which show anti-site atoms to compensate for the deviation of chemical composition from stoichiometry and thermal vacancy concentrations at melting temperature [5–8]. Up to now, there have been several theoretical methods applied to study the gTiAl. Badura and Scahaefer [9] studied theoretically the formation of thermal defects within the framework of a simple nearest neighbour bond model. In their treatment, bond energies of Ti–Ti and Al–Al were estimated from vacancy formation enthalpies of pure Ti and Al metals, and Ti–Al bond energy from ordering energy, respectively. A grand potential methods have been also applied to this problem using the defect energy with a full potential linearized augmented plane-wave method [10], the plane-wave * Corresponding author. Tel./fax: þ86 0379 65515016. E-mail address: [email protected] (H. Fu). 0966-9795/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2009.12.005

pseudopotential method [11] and the embedded-atom potential method [12], respectively. All the above models predict a general tendency of increasing vacancy concentration with the increase of Al content. With regard to computational investigations, many researchers concentrated on the effect of alloying elements, the phase stability in titanium-aluminum alloys[13–19]and the dislocation configurations [20,21]. In this letter, we focus on the thermodynamic properties and elastic properties of g-TiAl under high pressures in the atomic level. As known, elastic properties are important in many fields ranging from geophysics to materials research, and from chemistry to physics, because elastic properties are closely related to many fundamental solid-state properties, such as equation of state, specific heat, thermal expansion, Debye temperature, Gru¨neisen parameter, melting point, and so on. The knowledge of elastic constants is essential for many practical applications and relates to the mechanical properties of a solid, for example, load deflection, thermo-elastic stress, internal strain, sound velocities and fracture toughness [22]. From the elastic constants, one can obtain valuable information about the bonding characteristic between adjacent atomic planes and the anisotropic character of the bonding as well as the structural stability of a crystal.

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2. Theoretical method

AVib ðq; TÞ ¼ nKT 

2.1. Total energy electronic structure calculations In the electronic investigation, all calculations are performed based on the plane-wave pseudopotential density function theory (DFT) [23,24]. Vanderbilt-type ultrasoft pseudopotentials (USPP) [25] are employed to describe the electron–ion interactions. The effects of exchange correlation interaction are treated with the generalized gradient approximation (GGA) of Perdew–Burke–Eruzerhof(PBE) [26]. In the structure calculation, a plane-wave basis set with energy cut-off 550.00 eV is used. Pseudo-atomic calculations are performed for Al3s23p1and Ti3d24s2. For the Brillouinzone sampling, we adopt the 12  12  12Monkhorst–Pack mesh [27], where the self-consistent convergence of the total energy is at 107 eV/atom and the maximum force on the atom is below 105 eV/Å. All the total energy electronic structure calculations are implemented through the CASTEP code [28,29]. 2.2. Elastic properties The elastic constants are defined by means of a Taylor expansion of the total energy, E(V,d) for the system with respect to a small strain d of the lattice primitive cell volume V. The energy of a strained system is expressed as follows [30,31]:

2 EðV; dÞ ¼ EðV0 ; 0Þ þ V0 4

X

3

si xi di þ

i

1X Cij di xi dj xj 5; 2

(1)

ij



   9 q þ 3ln 1  eq=T  Dðq=TÞ ; 8T

(3)

where D(q/T) represents the Debye integral, K is the Boltzmann constant and n is the number of atoms per formula unit. For an isotropic solid, q is expressed by [35],

q ¼

rffiffiffiffiffi Z h 2 1=2 i1=3 BS 6p V ; f ðsÞ K M

(4)

where M is the molecular mass per formula unit, s the Poisson’s ratio and BS the adiabatic bulk modulus approximated by the static compressibility [32]

! d2 EðVÞ BS zBðVÞ ¼ V dV 2

(5)

and f(s) is given by Refs. [37,38]. Therefore, the non-equilibrium Gibbs function G*(V; P, T) as a function of (V; P, T) can be minimized with respect to volume V as

! vG* ðV; P; TÞ vV

¼ 0:

(6)

P; T

By solving Eq. (6), one can get the thermal equation of state (EOS) V(P,T). The isothermal bulk modulus BT is given by [32]

BT ðP; TÞ ¼ V

! v2 G* ðV; P; TÞ vV 2

;

(7)

P;T

where E(V0,0) is the energy of the unstrained system with equilibrium volume V0, si is an element in the stress tensor and xi, xj are factors of Voigt index. There are six independent components of the elastic tensor for gTiAl, i.e., C11, C12, C13, C33,C44 andC66. To obtain all elastic constants, we at least need six independent strains listed in Table 1. For each strain, a number of small values of d are taken to calculate the total energies for the strained crystal structure of gTiAl. According Eq.(1) and Table 1 the C11, C12, C13, C33,C44 andC66 are easily obtained. 2.3. Thermodynamic properties

The heat capacity CV and the thermal expansion (a) are expressed as

  3q=T CV ¼ 3nK 4Dðq=TÞ  q=T ; 1 e

gC V ; BT V

a ¼ 

where g is the Gru¨neisen parameter defined as

g¼ 

In our previous works, the quasi-harmonic Debye model [32] has been successfully applied to investigate the thermodynamic properties of PtC[33] and TiB2 [34]. In the quasi-harmonic Debye model, the non-equilibrium Gibbs functionG*(V;P,T) takes the form of

G* ðV; P; TÞ ¼ EðVÞ þ PV þ AVib ðqðVÞ; TÞ;

(2)

where E(V) is the total energy per unit cell for gTiAl, q(V) is the Debye temperature, and the vibrational Helmholtz free energy AVib can be written by [35,36] Table 1 Strains used to calculate the elastic constants of TiAl at zero pressure. 1 V0

v2 EðV;dÞ jd¼0 2 vd

Strain

Parameters(unlisted ei ¼ 0)

1

e1 ¼ d

2

e3 ¼ d

3

e4 ¼ 2 d

2C44

4

e1 ¼ 2d; e2 ¼ e3 ¼ 2d

1 ð5C11  4C12  2C13 þ C33 Þ 2

5 6

e1 ¼ e2 ¼ d; e3 ¼ 2d e1 ¼ e2 ¼ d; e3 ¼ 2d; e6 ¼ 2d

ðC11 þ C12  4C13 þ 2C33 Þ ðC11 þ C12  4C13 þ 2C33 þ 2C66 Þ

1 C 2 11 1 C 2 33

(8)

dlnqðVÞ dlnV

(9)

3. Results and discussion 3.1. Elasticity To determine the elastic constants of gTiAl at high pressure, we apply the six strains[39] listed in Table 1.The atomic positions are optimized at all strains. For each strain, a number of infinitesimal parameter d are employed to calculate the total energy E, and the volume V is not kept as a constant (i.e.volume-non-conserving) when the lattice is distorted by eij. All the obtained energies are then fitted to a second-order polynomial in d, where the secondorder derivatives coefficients of the fitted E(V,d) with respect to d are used to calculate the elastic constants of TiAl. The zero pressure bulk modulus B0 and its first-order pressure derivative are determined by fitting the calculated total energy–volume data to the Birch–Murnaghan EOS[40].

"

DEðVÞ ¼ E  E0 ¼ B0 V0

# 1B0 Vn 1 Vn 0   þ þ B00 1  B00 B00 B00  1

(10)

where E0 is the equilibrium energy at zero pressure, Vn the volume 0 at pressure and B0 ¼ dB0/dP. The pressure P versus the normalized

H. Fu et al. / Intermetallics 18 (2010) 761–766

volume Vn is obtained through the following thermodynamic relationship:

i dE B h B0 P ¼  ¼ 00 Vn 0  1 dV B0

(11)

The obtained the elastic constants, Bulk modulus K, Young’s modulus E, Shear modulus G and Poisson’s ratio s at T ¼ 0 are presented in Table 2, together with the experimental data and other theoretical results [41–50]. The Ab initio computations have been also used to calculate the bulk modulus of the gTiAl, being equal to 110.69 GPa, which is close to Other’s literature value and experiment result [Table 1]. Furthermore, the agreements among them are also good. These elastic stiffness coefficients of tetragonal gTiAl are shown in Table 2 and satisfy the generalized elastic stability criteria for tetragonal crystals under hydrostatic pressures [51,52], making the tetragonal cell mechanically stable.

3.2. Structural behaviors We have investigated the total energy as a function of primitive cell volume for the L10 crystal structure of gTiAl. It is found that the most stable structure (i.e., the normalized volume Vn ¼ V/V0 ¼ 1.0, where V0 is the equilibrium volume at zero pressure) corresponds to the ratio c/a of about 1.012, and the equilibrium lattice parameters a and c are about 4.001 Å and 4.071 Å, respectively, which are consistent with experimental data [41] and other theoretical results [44,50]. The equilibrium ratio and the corresponding normalized lattice parameters (a  a0)/a0, (c  c0)/c0 and the normalized volume (V  V0)/V0 as a function of the applied pressure are plotted in Fig. 1, where a0, c0, and V0 are their values at T ¼ 0 and P ¼ 0, respectively. By fitting the calculated data to the fourth-order polynomial, we obtain their relationships at the temperature T ¼ 0 K, ða L a0 Þ=a0 [ L3:08 3 10L3 P D 5:896 3 10L5 P 2

C11  C12 > 0; ðC11 þ C33  2C13 Þ > 0

(12)

L 1:326 3 10L6 D 1:335 3 10L8 P 4

2C11 þ 2C12 þ C33 þ 4C13 > 0

(13)

ðc L c0 Þ=c0 [ L3:00 3 10L3 P D 6:70 3 10L5 P 2

C11 > 0;

C33 > 0;

C44 > 0;

C66 > 0

(14)

According to Ref[53], the ductile/brittle properties of metals could be related empirically to their elastic constants by the ratio RG/K of shear modulus G divided by bulk modulus K. If RG/K < 0.5, the material behaves in a ductile manner, otherwise the material behaves in a brittle manner. This was recently demonstrated in the study of brittle versus ductile transition in intermetallic compounds from first principles calculations [54,55]. For gTiAl, the calculated value of the RG/K is 0.619, showing the brittleness in nature, which hinders it from being used to important practical applications [56].

Table 2 The elastic constants Cij (in GPa), Bulk modulus K, Young’s modulus E, Shear modulus G and Poisson’s ratio s, together with the experimental data and other theoretical results. TiAl

Experiment

Present work

Other calculations

a(Å) c/a C11(GPa) C12(GPa) C13(GPa) C33(GPa) C44(GPa) C66(GPa) Bulk modulus K (Gpa) Young’s modulus E (Gpa) Shear modulus G (Gpa) Poisson’s ratio s Debye temperature q (K)

3.997a 1.02a 186b, 183c 72b, 74.1c 74b, 74.4c 176b, 178c 101b, 105c 77b, 78.4c 109.78b, 113.29c

4.001 1.012 164 85.5 81.04 178.57 109.6 72.6 110.69

4.003d, 3.989k 1.014d, 1.011k 183e, 187g, 190h, 188i, 170k 74.1e, 74.8g, 105h, 98i, 79k 74.4e, 74.8g, 90h, 96i, 78k 178e, 182g, 185h, 190i, 178k 105e, 109g, 120h, 126i, 113k 78.4e, 81.2g, 50h, 100i, 73k 110e, 112d

182.9b, 160.32c

170.50

184.7e, 173j

a b c d e g h i j k

Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[41]. [42]. [43]. [44]. [45]. [46]. [47]. [48]. [49]. [50].

74.8b, 63.41c

68.57

0. 22b, 0.22c 584b

0.26 583

763

L 1:183 3 10L6 P 3 D 1:189 3 10L9 P 4

(15)

(16)

ðV L V 0 Þ=V 0 [ L9:22 3 10L3 P D 2:149 3 10L5 P 2 L 4:599 3 10L6 P 3 D 4:393 3 10L9 P 4

(17)

It is shown that, as pressure increases, the ratio of (a  a0)/a0, (c  c0)/c0 and (V  V0)/V0 decreases. The compression along c axis is smaller than that along the a axis, consistent with the comparatively stronger (Ti–Al) bonds that determine the c axis length. When pressure increases, the atoms in the interlayers become closer, and their interactions become stronger. On the other hand, the calculated elastic constants are shown as functions of pressure in Fig. 2. The present results agree well with Ref. [50]. In the calculated pressure range, it is surprising that C66, followed by C44 increases monotonically with increasing pressure, but their rates of increase are very moderate. However, the increasing rates of pressure dependence of the present C33, keeping in step with C12 and C13 over 5GPa is always larger than that of C11, in which the measured rapid increase under pressure is not clearly seen.

75.7e, 70j 0.220e, 0.234j 587e

Fig. 1. The variations about ratio of the normalized lattice parameters (a  a0)/a0, (c  c0)/c0 and the normalized volume (V  V0)/V0 with the applied pressure of gTiAl.

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Fig. 2. Elastic constants of gTiAl as functions of Pressure. Fig. 4. The thermal expansion versus temperature and pressure for gTiAl.

3.3. Anisotropy It is well known that microcracks are induced in alloys owing to the anisotropy of the coefficient of thermal expansion as well as elastic anisotropy [57]. Hence it is important to calculate elastic anisotropy in structural intermetallics in order to understand these properties and hopefully find mechanisms which will improve their durability. Essentially all the known crystals are elastically anisotropic, and a proper description of such an anisotropic behavior has, therefore, an important implication in engineering science as well as in crystal physics. The shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. To qualify the mechanical anisotropy of gTiAl, one can define the bulk modulus along the a axis (Ba) and the c axis (Bc) as follows[31]:

L dP Ba ¼ a ¼ da 2þb Bc ¼ c

dP Ba ¼ b dc

L ¼ 2ðC11 þ C12 Þ þ 4C13 b þ C33 b2

Fig. 3. The variation of anisotropy of Ba and Bc with pressures.

(18)

(19)

b¼

C11 þ C12  2C13 C33  C13

(21)

The anisotropy of Ba and Bc are also presented in Fig. 3. Unfortunately, there is no experimental data to check our calculated results at high pressure. Obviously, the contours of Ba and Bc appear different behavior for the whole pressure range, i.e. the contour of Ba nearly increasing lineally with pressure and the Bc increasing by trajectory. The value of Ba/Bc has a trend of gradual decline as the normalized volume V/V0 decreases. It is shown that the mechanical behavior of gTiAl under zero pressure is of large anisotropy. With the applied pressure increasing, the anisotropy Ba will gradually weaken. These results are similar to the hcp crystal interacting with central nearest-neighbor forces (CNNF) [58]. For this model the elastic anisotropy is independent of the interatomic potential to lowest order in P/C11, hence the anisotropy is dependent on the symmetry of the crystal only. In our foregoing work, we found that the ratio c/a changes with different pressure, that is, the structure is always varying with the applied pressure. Therefore, the elastic anisotropy may be different with pressure. The relationship

(20)

Fig. 5. Variations of thermodynamic parameters X (X: Debye temperature or specific heat) with pressure P. They are normalized by (X  X0)/X0, where X and X0 are the Debye temperature or specific heat under any pressure P and zero pressure at the temperatures of 300 K and 800 K, respectively.

H. Fu et al. / Intermetallics 18 (2010) 761–766

765

and the normalized volume (V  V0)/V0 for a given volume, as well as the pressure dependence of the normalized lattice parameters of gTiAl. The normalized volume at P ¼ 0 GPa is in excellent agreement with the experimental and other’s results. The elastic constants of gTiAl at high pressure from 0 GPa to 10 GPa are also calculated. In order to qualify the mechanical anisotropy, we have investigated the bulk modulus values Ba and Bc, and the pressure dependence of anisotropy. An analysis of the calculated parameters reveals the anisotropy of gTiAl, that is, in a certain range of applied pressure, the anisotropy is remarkable, but beyond this range the anisotropy Ba tends to reach its extremum. Finally, we investigate the thermodynamic properties, the relationships among the thermal expansion, temperature and pressure, as well as the variations of Debye temperature, and specific heat with pressure. Acknowledgment

Fig. 6. The heat capacity of gTiAl at various pressures and temperatures.

between the elastic anisotropy and the applied pressure also shows that Bc have no remarkable changes under high pressure. These behaviors may be correspond to the bonding situations in gTiAl, which is characterized as a strong cohesive bonding between Ti and Al layers and a weaker bonding in the pure Al layer. 3.4. Thermodynamic properties The obtained variations of the thermal expansion a with temperatures and pressures are shown in Fig. 4. We note that at zero pressure a increases exponentially with T at low temperatures and gradually approaches a linear increase at high temperatures. As the pressure increases, increase of a with temperature becomes smaller, especially at high temperature. At a given temperature, a decreases drastically with the increase of pressure. When the pressure is above 30 GPa, the thermal expansion a of 900 K is just a little larger than that of 600 K, which means that the temperature dependence of a is very small at high temperature and high pressure. In Fig. 5, we show the heat capacity CV and the Debye temperature q as a function of pressure P at the temperatures of 300K and 800K for gTiAl. It is shown that when the temperature is constant, the Debye temperature q increases almost linearly with applied pressures, indicating the change of the vibration frequency of particles in gTiAl under pressure. However, the heat capacity CV decreases with the applied pressures, which is because the effect of increasing pressure on gTiAl is the same as decreasing temperature of gTiAl. In Fig. 6, the heat capacity of gTiAl are plotted for several pressures. It is shown that when T < 1400K, the heat capacity CV is dependent on both the temperature T and the pressure P. This is due to the anharmonic approximations of the Debye model. However, at higher pressures and/or higher temperatures, the anharmonic effect on CV is suppressed, and CV is very close to the Dulong–Petit limit, as is similar to the case of MgO [59]. 4. Conclusions The elastic constants of gTiAl at high pressure are computed by the Vanderbilt-type ultrasoft pseudopotentials within the generalized gradient approximation correction (GGA) in the frame of density functional theory. We carry out total energy calculations over a wide range of volumes from 0.7 V0–1.2 V0, and obtain the equilibrium ratio of the lattice parameters (a  a0)/a0, (c  c0)/c0

This project was supported by the National Natural Science Foundation of China under grant No. 10376021, 10274055 and 40804034, and by the Research Fund for the Doctoral Program of High Education of China under grant No. 20020610001 and the Natural Science Foundation of the Education Department of Henan province of China under grant No. 2009B590001 and 2007140011, and by Henan Science and Technology Agency of china under grant No. 092102210314. 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]

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