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Effect of hydrogen concentration on various properties of gamma TiAl Ji Wei Wang, H.R. Gong* State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, China

article info

abstract

Article history:

First principles calculation reveals that the TiAlH phases become relatively less stable after

Received 16 September 2013

the addition of more H atoms, and that the H migration from tetrahedral interstitial site to

Received in revised form

octahedral site in gamma TiAl should be much easier than that between octahedral sites.

5 November 2013

Calculation also shows that H concentration has an important effect on mechanical

Accepted 12 November 2013

properties of TiAlH phases, and that the energetically favorable TiAlH phase should

Available online 15 December 2013

possess bigger E, G, and G/B values as well as lower elastic anisotropy. Moreover, it is found that the heat capacities of TiAlH phases increase with the increase of H concentration, and

Keywords:

that the coefficients of thermal expansion of TiAl and TiAlH phases decrease with the

TiAl and TiAlH

increase of pressure.

Phase stability

Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

Elastic property

reserved.

Thermodynamic property First principles calculation

1.

Introduction

During the past decades, the interaction between hydrogen (H) and gamma TiAl phase has induced considerable interests among researchers [1e7]. On the one hand, with high specific strength, good oxidation resistance at high temperature, and low density, gamma TiAl is widely considered as promising high-temperature structural materials such as aerospace, marine and automotive engine components [8e13], etc., which are apt to hydrogen attack or have close contacts with the fuel of H2 [14,15]. On the other hand, gamma TiAl is commonly regarded as a kind of promising materials for hydrogen storage, as it has a light weight and possesses a hydride forming element of Ti and a non-hydride forming element of Al, which are beneficial for both hydrogenation and dehydrogenation [16]. Furthermore, the gamma TiAl with

the porous structure could be utilized as a potential support of Pd membranes for the purpose of hydrogen separation and purification [17]. Regarding various properties of TiAlH phases, there are already a lot of investigations in the literature [1e7,16,18e24]. For instance, several experimental techniques were performed to find out diffusion coefficients of H [1,2], hydride formation [1,3,4,19], H adsorption and desorption [1,3,4,16,20], as well as effects of H on mechanical properties of TiAl [5e7,18e20], etc. Very recently, first principles calculation was conducted to reveal the effect of Al composition on structural stability, mechanical properties, elastic anisotropy, and electronic structures of various TiAl phases with a low H composition [24]. Unfortunately, it should be pointed out that there is no any study regarding the effect of H composition on structural stability and thermodynamic properties of gamma TiAl in the literature.

* Corresponding author. Tel.: þ86 731 88877387; fax: þ86 731 88710855. E-mail address: [email protected] (H.R. Gong). 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.11.045

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By means of highly accurate total energy calculations based on density function theory, the present study is therefore dedicated to have a systematic investigation of structural, mechanical, and thermodynamic properties of gamma TiAl phases as a function of H composition. The derived results will be compared with experimental evidence in the literature and the fundamental mechanism will be revealed through discussion, which could not only provide useful guidance for experimentalists, but also give a deep understanding of the underlying relationship between the structure and property of TiAlH phases.

2.

Calculation method

The present first principles calculation is based on the wellestablished Vienna ab initio simulation package (VASP) within the density functional theory [25]. The interactions between electrons and ions are described by the projectoraugmented wave (PAW) method [26]. The exchange and correlation items are treated by generalized gradient approximation (GGA) of Perdew et al. [27] and the cutoff energies are 400 eV for plane-wave basis and augmentation charge. For the Brillouin zone sampling, the temperature smearing method of Methfessel-Paxton [28] is used for dynamical calculation and the modified tetrahedron method of Blo¨chl-Jepsen-Andersen [29] is performed for static calculation. In the present study, six H concentrations are chosen, i.e., Ti1Al1H, Ti2Al2H, Ti4Al4H, Ti8Al8H, Ti16Al16H, and pure TiAl phases with the supercell models of 2, 4, 8, 16, 32, and 4 metal atoms, respectively. At each concentration, the ordered L10 (space group P4/mmm) is selected, and only one H atom is added at the octahedral (O) and tetrahedral (T) interstitial sites of each phase, respectively. It should be noted that for each phase, there is only one T site, while two O sites, i.e., the site surrounded by four Ti and two Al atoms is named O1, and the other by four Al and two Ti atoms is called O2. For simplicity, the H locations are symbolized by TimAlmH(O1), TimAlmH(O2), and TimAlmH(T) in the following text, tables, and figures, e.g., Ti4Al4H(O1) signifies the Ti4Al4H phase with H at the O1 site. After the test calculation, for Ti1Al1H, Ti2Al2H, Ti4Al4H, Ti8Al8H, Ti16Al16H, and TiAl phases, the k-meshes of 21 21 13, 13 13 13, 13 13 7, 7 7 13, 7 7 7, and 13 13 13 are adopted for relaxation calculations, while 25 25 17, 17 17 17, 17 17 9, 9 9 17, 9 9 9, and 17 17 17 for static calculations, respectively. In each calculation, periodic boundary conditions are added in three directions, and the unit cell is allowed to fully relax. The energy criteria are 0.1 and 0.01 meV for ionic and electronic relaxations, respectively, while 0.001 meV for static calculation.

3.

Results and discussion

3.1.

Phase stability

After a series of calculations, the lattice constants of gamma TiAl and various TimAlmH phases are derived and summarized in Table 1. For comparison, the experimental and calculated lattice constants of TiAl in the literature [30e34] are

also listed in Table 1. It can be seen that the present lattice constants a, c, and c/a of gamma TiAl are 3.990 A, 4.073 A, and 1.02, respectively, which match well with the corresponding experimental data (a ¼ 3.99 or 3.997 A; c ¼ 4.07 or 4.077 A; c/ a ¼ 1.02) [30,31]. In addition, it could be also deduced from Table 1 that both lattice constant c of O1 site and lattice constant a of O2 site increase with the increase of H concentration. Considering that there are two Al atoms in the c direction of O1 site as well as four Al atoms in the a directions of O2 site, the above increases of lattice constants imply that the interaction between Al and H atoms should be mainly repulsive, and the present statement would be compatible with the experimental observation that the absorption of hydrogen in TiAl decreases with the increase of Al composition [16]. To find out phase stability, the heats of formation (DHf) of TiAl and TimAlmH phases, are calculated by means of the following formulas, respectively: DHf ¼

ETiAl 2ðETi þ EAl Þ ; 4

DHf ¼

ETim Alm H mETiAl þ 12 EH2 ; 2m þ 1

(1)

(2)

where ETiAl, ETi, EAl, ETim Alm H , and EH2 are total energies of TiAl, HCP Ti atom, FCC Al atom, TimAlmH, and H2 molecule, respectively. As a result, Table 1 lists the derived DHf values of TiAl and various TimAlmH phases as well as corresponding DHf values of TiAl in the literature [33,35]. Moreover, Fig. 1 shows the comparison of DHf values of TimAlmH(O1), TimAlmH(O2), and TimAlmH(T) phases as a function of H concentration. Several features could be observed from Fig. 1 as well as Table 1. Firstly, one sees from Table 1 that the present DHf value of the gamma TiAl phases with the L10 structure is calculated to be 39.368 kJ/mol, which agrees well with the corresponding experimental value of 37.6 kJ/mol [35]. This Table 1 e Lattice constants (a and c) and heats of formation (DHf) of TiAl and TimAlmH phases with the L10 structure. Phase Type a ( A) c ( A) c/a DHf (kJ/mol H) TiAl

This work Exp. Exp. [31] Cal. [32] Cal. [33] Cal. [34] Ti16Al16H(O1) This work Ti16Al16H(O2) This work Ti16Al16H(T) This work This work Ti8Al8H(O1) This work Ti8Al8H(O2) Ti8Al8H(T) This work This work Ti4Al4H(O1) This work Ti4Al4H(O2) This work Ti4Al4H(T) This work Ti2Al2H(O1) This work Ti2Al2H(O2) This work Ti2Al2H(T) Ti1Al1H(O1) This work This work Ti1Al1H(O2) This work Ti1Al1H(T)

3.990 3.997 [30] 3.99 3.96 3.995 3.97 3.990 4.000 3.999 3.989 4.011 4.011 3.995 4.050 4.028 3.991 4.099 4.073 3.967 4.194 4.141

4.073 1.02 4.077 [30] 1.02 4.07 1.02 4.08 1.03 4.075 1.02 4.05 1.02 4.085 1.024 4.070 1.018 4.083 1.021 4.098 1.027 4.057 1.011 4.085 1.018 4.107 1.028 4.020 0.993 4.101 1.018 4.163 1.043 3.969 0.968 4.114 1.010 4.333 1.092 3.878 0.925 4.175 1.008

39.368 37.6 [35]

39.2 0.28 1.05 0.56 0.26 2.61 1.14 0.82 4.61 1.77 0.004 6.76 4.03 0.65 7.18 5.45

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obtained results are summarized in Fig. 2. It could be observed from this figure that the O1 site should be energetically more favorable with a lower energy of 0.350 eV than the O2 site, and an energy barrier of 0.911 eV should be overcome if H would migrate from O2 to O1. Such a big migration barrier implies that it should be difficult for H to diffuse from O2 to O1. On the other hand, one can see from Fig. 2 that the energy difference between O1 and T sites is 0.209 eV, and the energy barrier is 0.212 eV. It therefore follows that the H migration from T to O1 in gamma TiAl should be much easier than that from O2 to O1.

8

O1 O2 T

6 5 4 3 2 1

3.2.

0 -1 0.0

0.1

0.2

0.3

0.4

0.5

H/TiAl atomic ratio

Fig. 1 e Heat of formation of TimAlmH phases as a function of H composition.

nice agreement implies that the present PAW method is capable of revealing the phase stability of TiAl and TimAlmH phases. Secondly, it could be clearly observed from Fig. 1 that at each H concentration, the descending sequence of DHf values of TimAlmH is as follows: O2 / T / O1. In other words, H location plays an important role in determining the phase stability of TimAlmH, and the O1 site should be energetically more favorable than corresponding O2 and T sites within the entire composition range. Thirdly, it can be seen from Fig. 1 that for each H site, the DHf values of TimAlmH phases generally become bigger when H concentration increases, with only an exception of the minimum DHf value of Ti4Al4H(O1), indicating that the TimAlmH phases should be relatively less stable after the addition of more H atoms. Such a feature of H concentration on the stability of TimAlmH seems quite different from that of TiH phases with an almost linear decrease of DHf with the increase of H concentration [36]. It should be pointed out that the above phase stability of TimAlmH as a function of H concentration would be beneficial to the processes of dehydrogenation and H permeability when TiAl is used as hydrogen storage and separation materials, respectively [16,24]. Fourthly, one could also discern from Fig. 1 that the O1 sites are energetically favorable with negative DHf at 0 K when the atomic ratio of H/TiAl is less than about 0.26, and the O2 as well as T sites are energetically unfavorable with positive DHf values within the entire composition range. As shown in Table 1, the negative DHf of Ti16Al16H(O1), Ti8Al8H(O1), and Ti4Al4H(O1) phase from the present study are very small values of 0.28, 0.26, and 0.82 kJ/(mol H), respectively. Such small negative DHf, together with positive DHf of other TimAlmH phases, suggest that it should be not easy for H atoms to dissolve in L10 TiAl, which matches well with the very low solubility of H in TiAl observed experimentally [16]. It is of further interest to investigate the H diffusion between O1, O2, and T sites in gamma TiAl. Accordingly, the nudgedelastic-band technique [37] is used in the present study to calculate the migration barrier of site diffusion of H, and the

Elastic properties

In the present study, the elastic constants of TiAl and TimAlmH phases are derived according to the method adopted by Wang and Ye [38] with the following main idea: small strains are first applied to the equilibrium lattice, the resulting change of total energy is determined, and from this information, elastic constants are deduced by means of quadratic polynomial fittings. The obtained single-crystal elastic constants are then used to derive shear modulus (G) of polycrystalline phases through the Hill’s approximation [39]. Moreover, bulk modulus (B) is determined by fitting the Murnaghan equation of state (EOS) [40], and Young’s modulus (E) is derived by the formula of E ¼ 9BG/(3B þ G) [39]. Accordingly, the derived elastic constants as well as B, G and E values of TiAl and various TimAlmH phases are summarized in Table 2. The corresponding experimental and calculated results available in the literature [41,42] are also listed in Table 2 for the sake of comparison. It could be seen from this table that the elastic properties from the present study are consistent with experimental and calculated data in the literature [41,42]. For instance, the present B, G, and E of TiAl are 112.8, 68.3, and 170.6 GPa, respectively, which agree with corresponding experimental values of 110, 74.8, and 183 GPa [41]. We now investigate mechanical stability of various TimAlmH phases. According to the strain energy theory, for a mechanically stable phase the strain energy should be positive, and the matrix of elastic constants should be positive, definite, and symmetric [43]. This theory could be expressed for the tetragonal structure as: (C11 C12) > 0, (C11 þ C33 2C13) > 0,

Relative energy (eV)

Heat of formation (KJ/mol H)

7

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8 0.6

0.6

O2 0.4

0.4 0.2

0.2

O1 T

0.0

0.0 -0.2

-0.2 0

1

2

3

4

5

6

7

8

9

Image number

Fig. 2 e Migration barrier of site diffusion of H from O2 and T to O1 in TiAl.

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Table 2 e Elastic constants (C11, C12, C13, C44, and C66) and elastic moduli (B, G, and E) of TiAl and TimAlmH phases. All values, except G/B, are in the unit of GPa.

Ti16Al16H(O1) Ti16Al16H(O2) Ti16Al16H(T) Ti8Al8H(O1) Ti8Al8H(O2) Ti8Al8H(T) Ti4Al4H(O1) Ti4Al4H(O2) Ti4Al4H(T) Ti2Al2H(O1) Ti2Al2H(O2) Ti2Al2H(T) Ti1Al1H(O1) Ti1Al1H(O2) Ti1Al1H(T)

Type

C11

C12

C13

C33

C44

C66

B

G

E

G/B

This work Exp. [41] Cal. [42] This work This work This work This work This work This work This work This work This work This work This work This work This work This work This work

171 186 170 175 172 171 179 170 171 192 168 172 204 151 159 220 134 135

88 72 79 87 88 87 89 90 88 88 88 93 86 99 97 102 113 104

88 74 78 85 84 83 83 87 84 82 87 78 79 85 83 66 79 87

173 176 178 173 178 172 176 177 173 180 176 186 172 179 170 150 197 165

110 101 113 109 109 107 107 109 105 104 107 99 94 104 88 60 102 106

69 77 73 70 69 70 76 70 72 83 66 82 91 73 90 106 56 93

112.8 110 109 113.1 112.7 112.5 113.5 113.4 112.1 114.2 111.9 111.3 114.8 113.4 109.5 108.3 113.7 103.8

68.3 74.8

170.6 183

0.606 0.68

69.7 69.5 68.9 71.5 68.3 68.1 75.2 66.6 70.3 75.9 61.9 62.2 66.8 49.1 56.2

173.4 172.9 171.6 177.3 170.7 169.8 185.1 166.7 174.2 186.6 157.2 156.8 166.1 128.8 142.7

0.616 0.617 0.612 0.630 0.603 0.607 0.659 0.595 0.632 0.661 0.546 0.568 0.616 0.432 0.541

(2C11 þ 2C12 þ C33 þ 4C13) > 0, C11 > 0, C33 > 0, C44 > 0, and C66 > 0 [44]. It could be detected from Table 2 that TiAl and TimAlmH phases all obey the above strain energy theory, indicating that these phases should be mechanically stable. It is of importance to explore the effects of H on elastic moduli of polycrystalline TimAlmH. Consequently, the calculated B, G and E values of TiAl and various TimAlmH phases are summarized in Fig. 3 as a function of H composition. One sees clearly from Fig. 3 that at each H concentration, the descending sequence of E and G values is as follows: O1 / T / O2, signifying that the energetically favorable TimAlmH(O1) phase should possess better mechanical properties. It could be also observed that H concentration has an important effect on elastic properties of TimAlmH(O1) phases, i.e., the B, G and E values of Ti1Al1H(O1) first increase with the increase of H concentration and are then followed by a decrease. Such a convex curve of elastic properties of TimAlmH(O1) as a function of H concentration revealed from the present study is in excellent agreement with similar experimental observations in the literature [5]. In addition, the G/B values of TiAl and various TimAlmH phases are calculated and the results are shown in Fig. 4 and Table 2. It should be pointed out that the G/B value proposed by Pugh has been widely used as an empirical parameter to predict the brittleness/ductility of materials [45], i.e., the critical G/B value of 0.57 separates the brittle/ductile behavior; a bigger G/B value means more brittleness, and vice versa. One sees from Table 2 that the G/B ratio of TiAl from the present study is 0.606, which agrees well with the corresponding experimental value of 0.68 in the literature [41]. It could be also observed from Fig. 4 that at each H concentration, the TimAlmH(O1) phase should be more brittle with a bigger G/B value than corresponding TimAlmH(T) and TimAlmH(O2) as well as TiAl. Considering that TimAlmH(O1) is energetically more favorable revealed in Sec. 3.1, such a bigger G/B value of TimAlmH(O1) suggests that the addition of H should make TiAl more brittle, which matches well with similar experimental observations in the literature [5,6]. It is of interest to further find out the effects of H on elastic anisotropy of TimAlmH phases. In the present study, two

anisotropic indexes are selected to express the elastic anisotropy with the following formulas [46,47]: AG ¼

GV GR 100%; GV þ GR

(3)

190 180 170 160 150

E

140 130 120

Elastic modulus (GPa)

Phase TiAl

114 112 110

O1 O2 T

108 106

B

104

75 70 65 60 55

G

50 45 0.0

0.1

0.2

0.3

0.4

0.5

H/TiAl atomic ratio Fig. 3 e Young modulus (E), Bulk modulus (B), and shear modulus (G) of TiAl and TimAlmH phases as a function of H composition.

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Ti1Al1H(O1) should be less than that of Ti1Al1H(O2). Such a difference of bonding directionality would probably induce the lower elastic anisotropy of the O1 site than its corresponding O2 site shown in Fig. 6.

0.65

G/B

0.60

0.55

3.3. O1 O2 T

0.50

In addition to phase stability and mechanical property related before, thermodynamic property is also of vital importance for investigation, as various TiAl and TiAlH products are usually utilized under high temperature. Accordingly, the thermodynamic properties of TiAl and TiAlH phases are calculated through the quasi-harmonic Debye model and thermal electronic excitation. For a system at given volume (V) and temperature (T), the Helmholtz free energy F(V,T) is obtained through combining electronic Fel(V,T) and vibrational Fph(V,T) free energy [49,50]:

0.45

0.40 0.0

0.1

0.2

0.3

0.4

Thermodynamic properties

0.5

H/TiAl atomic ratio

Fig. 4 e G/B values of TiAl and TimAlmH phases as a function of H composition.

FðV; TÞ ¼ E0 ðVÞ þ Fph ðV; TÞ þ Fel ðV; TÞ;

AG is the percentage anisotropy in shear to express elastic anisotropy, i.e., a value of zero means elastic isotropy and a value of 100% denotes the maximum anisotropy [47]. AU is regarded as a universal anisotropic index which takes into account all the stiffness coefficients from both shear and bulk contributions, i.e., an AU value of zero represents isotropic crystals and the departure of AU from zero indicates the extent of elastic anisotropy [46]. Consequently, the values of AG and AU for TiAl and TimAlmH phases are derived, and the results are shown in Fig. 5. It can be seen from this figure that for each H site, the shapes of AG and AU curves as a function of H concentration are very close to each other, implying that both anisotropy indexes of AG and AU would be relevant to express the elastic anisotropy of TimAlmH phases. Moreover, one could discern from Fig. 5 that H location should have an important effect on elastic anisotropy of TimAlmH, i.e., at each H concentration, the descending sequence of elastic anisotropy is as follows: O2 / T / O1. Considering that the DHf values of TimAlmH have the same descending sequence of O2 / T / O1 revealed in Sec. 3.1, a conclusion would be probably drawn that generally speaking at a certain composition the energetically more favorable TimAlmH structure may have lower elastic anisotropy, and vice versa. It should be emphasized that the present statement regarding the strong correlation between structural stability and elastic anisotropy of various TimAlmH phases is in good agreement with similar observation regarding cubic crystals by Zener [48]. To have a deep understanding of elastic anisotropy of TimAlmH phases at an electronic scale, the charge density plots of (110) surfaces of Ti1Al1H(O1) and Ti1Al1H(O2) are shown in Fig. 6 as typical examples. It could be observed from this figure that compared with Ti1Al1H(O2), the Ti1Al1H(O1) phase has a charge density of Al atoms more close to the circular shape, and that the directionality of TieH interaction along the x axis has been considerably decreased, implying that the directional bonding of

where E0(V) is the static total energy at 0 K. Fph(V,T) is given by the quasi-harmonic Debye model: 9 Fph ðV;TÞ ¼ n kB QD ðVÞ þ 3kB T ln 1 eQD ðVÞ=T kB TDðQD ðVÞ=TÞ ; 8 (6) where D(y) is Debye integral defined as

6

O1 O2 T

5 4 U

(4)

A

GV B V þ 6: GR BR

3 2 1 0 40 35

O1 O2 T

30 25

AG(%)

AU ¼ 5

(5)

20 15 10 5 0.0

0.1

0.2

0.3

0.4

0.5

H/TiAl atomic ratio Fig. 5 e Anisotropic indexes AG and AU of TiAl and various TimAlmH phases as a function of H composition.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 8 8 8 e1 8 9 6

3 DðyÞ ¼ 3 y

Zy ex

x3 dx; 1

and CVph can be divided into the contributions from TiAl and H atoms,

(7)

0

and QD is Debye temperature expressed by rﬃﬃﬃﬃﬃ Z 2 1=2 1=2 B ; f ðnÞ 6p V n QD ðVÞ ¼ kB M

H CVph ¼ CTiAl Vph þ CVph ;

v¼

ð3B 2GÞ ; 2ð3B þ GÞ

(8)

Z gð3 ; VÞf ð3 ; V; TÞ3 d3

gð3 ; VÞ3 dε;

(12)

where f(ε,V,T) is Fermi distribution, and g(ε,V) is electronic density of states. Electronic entropy, Sel(V,T), is defined as: Z Sel ðV; TÞ ¼ kB

gðε; VÞff ðε; V; TÞln f ð3 ; V; TÞ þ ½1 f ð3 ; V; TÞln½1

f ð3 ; V; TÞgd3 : (13) After Helmholtz free energy is obtained at various volume, the volume as a function of temperature at ambient pressure, V(T)0, is determined by fitting F(V,T) into Vinet’s equation of states (EOS) [51]:

FðV; TÞ ¼ FðV; TÞ0 þ

4BðV;TÞ0 VðTÞ0

2 2 QE =T QE =T CH = e 1 ; Vph ¼ 3nH RðQE =TÞ e

(20)

2BðV;TÞ VðTÞ

0 0 2 2 ½B0 ðV;TÞ0 1 ½B0 ðV;TÞ0 1 VðTÞ 1=3 VðTÞ 1=3 0 3 1 þ 2 exp ðV; TÞ 1 1 ; B 3 B0 ðV; TÞ0 1 0 VðTÞ VðTÞ 2 0

1 vVðTÞ0 : VðTÞ0 vT

(15)

Total isochoric heat capacity, CV, is obtained by combining the contributions of electronic excitation CVel and phonon vibration CVph [52,53]: CV ¼ CVel þ CVph ;

CVel ¼ T

vSel ðV; TÞ ¼ ge T; vT

(14)

0

where B(V,T)0 and B0 (V,T) are bulk modulus and pressure derivatives of bulk modulus, respectively. Coefficient of thermal expansion, a(T), is obtained by aðTÞ ¼

(21)

where B is bulk modulus. After a series of calculations, the QD , CVel, CVph, CV, and CP values are derived for TiAl and TiAlH phases as a function of temperature. As typical examples, Fig. 7 shows the temperature-dependent CVel, CVph, CV, and CP values of TiAl, as well as and CP of several TiAlH(O1) phases at 1000 K. For comparison, the available experimental heat capacities of TiAl [54] are also included in Fig. 7. Several characteristics could be observed from these results. First, the calculated thermodynamic properties of TiAl from the present study are in good agreement with experimental data in the literature [54]. For instance, the present Cp value of TiAl at 1000 K is 58.64 J/K/mol, which matches well with the corresponding value of 61.13 J/K/mol from experiment [54]. In addition, the calculated Debye temperature (538.2 K) of TiAl phase at 300 K is also compatible with the corresponding experiment value of 584 K [41]. Such good

(11)

Z

(19)

CP ¼ CV þ a2 BVT;

where v is Poisson’s ratio. Electronic free energy, Fel(V,T), and thermal electronic energy, Eel(V,T), could be derived through:

Eel ðV; TÞ ¼

3QD =T ; CTiAl Vph ¼ 3nTiAl kB 4DðQD =TÞ QD =T e 1

where QE ¼ hyH =kB is Einstein temperature, R is gas constant, nTiAl and nH are the numbers of TiAl and H atoms per formula unit, respectively. Isobaric heat capacity, CP, is then calculated according to the following formula [52,53]:

(9)

(10)

Fel ðV; TÞ ¼ Eel ðV; TÞ TSel ðV; TÞ;

(18)

H where CTiAl Vph and CVph are calculated through Debye and Einstein models [52], respectively:

where n is the number of atoms per unit cell and M is the molecular mass per formula unit, B is the adiabatic bulk modulus, and f(n) is given by ( "

3=2

3=2 #1 )1=3 2 1þn 11 þ n þ ; f ðnÞ ¼ 3 2 3 1 2n 31 n

1893

(16)

(17)

agreements imply that the theory implanted in the present calculation should be pertinent to reflect thermodynamic properties of TiAl phases. Second, at a given temperature, the CVph value of TiAl is much bigger than CVel, suggesting that lattice vibration should have a much more contribution to heat capacity than electrons within the entire temperature range. Third, one could observe clearly from Fig. 7(b) that both CV and CP of TiAlH(O1) phases are all bigger than corresponding values of TiAl, and increase with the increase of H concentration. It follows that more H atoms should bring about bigger heat capacity, which is consistent with the well-known law of Dulong and Petit [55]. The coefficient of volume thermal expansion (a) as a function of temperature is then calculated according to Eq. (15), and the coefficient of linear thermal expansion (aL) is

1894

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0.16 2

0.16

1

Y (Angstrom)

72

0.16

Heat capacities (J/mol K)

(a)

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 8 8 8 e1 8 9 6

Al 0.12 0.080 0.20 0.32 0.36 0.28 0.24 0.40

Ti

0

-1

0.32 H 0.40 0.28 0.360.20 0.24

0.32

Ti 0.40

(b) 64 60 56 52

0.36

0.0

0.16 0.16

-1

0

1

2

3

X (Angstrom)

(b)

3

0.12 2

Al 0.16 0.080

0.16

0.20 0.20

Y (Angstrom)

1

Ti

0

0.28

0.32

H

0.16

0.200.36 0.24 0.32 0.36

0.080 Al 0.12

-2

-1

0

0.5

0.16

1

(a)

40

CVel CVph CV CP CPExp.[54]

30 20 10

200

400

600

800

1000

Fig. 7 e Calculated heat capacities of (a) TiAl as a function of temperature and (b) various TiAlH(O1) phases at 1000 K. The dashed cubes are experimental values [54].

Ti

0.20

0.20 -2

0.4

Temperature(K)

-3 -3

0.3

50

0

0.160.20

0.16

0.2

0

0.24 0.28 0.360.32 0.20

-1

60

Heat capacities (J/mol K)

0.16

-3 -2

0.1

H/TiAl atomic ratio

0.12 0.080Al

-2

-3

CV CP

68

2

closer to the aL curve of Ti than the curve of Al, and that the present aL of TiAl is generally smaller than other calculated value in the literature [23]. Such a difference of aL between the present study and other calculations [23] would be probably due to different theoretical methods.

3

X (Angstrom) 5.0

Fig. 6 e Charge density plots of (110) planes of (a) Ti1Al1H(O1) and (b) Ti1Al1H(O2) phases. The charge density ˚ 3. is in the unit of e/A

Ti Al TiAl Al Exp.[57] Ti Exp.[56] TiAl Cal.[23]

4.5 4.0 3.5 3.0 -5

aL (10 /K)

obtained through the formula of aL ¼ a/3. Accordingly, the obtained aL values of TiAl as well as pure Ti and Al are summarized in Fig. 8. The available experimental and calculated results in the literature [23,56,57] are also listed for the sake of comparison. One could discern from this figure that the calculated results of Ti and Al are in good agreement with corresponding experiment values within the entire temperature range [56,57], and that the present aL value (1.587 105/ K) of TiAl at 750 K is consistent with the corresponding 1.267 105/K of Tie56Al alloy measured from experiments [58]. It is also of interest to see from Fig. 8 that at each temperature the aL of TiAl is lower than those of Ti and Al when the temperature is below 160 K, while located between those of Ti and Al above 160 K. Moreover, it could be observed that the aL curve of TiAl as a function of temperature is much

2.5 2.0 1.5 1.0 0.5 0.0 0

200

400

600

800

1000

Temperature (K)

Fig. 8 e Calculated coefficients of linear thermal expansion of Ti, Al and TiAl. The available experimental and calculated results in the literature [23,56,57] are also listed for comparison.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 8 8 8 e1 8 9 6

Furthermore, the temperature- and pressure-dependent aL values of TiAl and TiAlH(O1) are also derived and the results are shown in Fig. 9. It can be seen from Fig. 9(a) that the addition of H has an important effect on thermal expansion of TiAl, i.e., the aL of Ti1Al1H(O1) is bigger than that of TiAl, while the aL of Ti16Al16H(O1) and Ti2Al2H(O1) become smaller within the entire temperature range. One could also observe from Fig. 9(b) that the aL of TiAl and TiAlH(O1) decrease with the increase of pressure, and that the difference of aL between TiAl and TiAlH(O1) could be lowered under higher pressure. The above characteristics suggest that pressure could reduce the magnitude of H effect on thermal expansion of TiAl.

4.

Concluding remarks

First principle calculation has been conducted to investigate phase stability, elastic properties, and thermodynamic properties of TiAl and TimAlmH phases. It is demonstrated that it should be not easy for H atoms to dissolve in L10 TiAl due to small negative DHf, and positive DHf of TimAlmH phases, and that TimAlmH(O1) is energetically more favorable within the entire composition range. It is also shown that both H location and concentration have an important effect on mechanical properties, heat capacity, and coefficients of thermal expansion of TimAlmH phases. The calculated results are compared with available experimental results in the literature and the agreements between them are fairly good. 2.4

(b)

2.2

0GPa 3GPa 6GPa 9GPa

-5

aL (10 /K)

2.0 1.8 1.6 1.4 1.2 0.0

0.1

0.2

0.3

0.4

0.5

H/TiAl atomic ratio 2.4

(a)

2.0

-5

aL (10 /K)

1.6 1.2

TiAl Ti16Al16H(O1) Ti2Al2H(O1) Ti1Al1H(O1)

0.8 0.4 0.0 0

200

400

600

800

1000

Temperature (K) Fig. 9 e (a) Temperature-dependent (under 0 GPa) and (b) pressure-dependent (at 1000 K) coefficients of linear thermal expansion of TiAl and TiAlH(O1) phases.

1895

Acknowledgments This research work is supported by the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110162110045), and Key Project of Science and Technology of Hunan Province (Grant No. 2012FJ2003).

references

[1] Estupin˜an HA, Uribe I, Sundaram PA. Hydrogen permeation in gamma titanium aluminides. Corros Sci 2006;48:4216e22. [2] Sundaram PA, Wessel E, Ennis PJ, Quadakkers WJ, Singheiser L. Diffusion coefficient of hydrogen in a cast gamma titanium aluminide. Scr Mater 1999;41(1):75e80. [3] Takasaki A, Furuya Y. Hydride formation and thermal desorption spectra of hydrogen of cathodically charged single-phase gamma titanium aluminide. Scr Mater 1999;40(5):595e9. [4] Matejczyk DE, Rhodes GG. Second phase formation in gamma titanium aluminide during high pressure hydrogen charging. Scr Metall 1990;24(7):1369e73. [5] Ruales M, Martell D, Vazquez F, Just FA, Sundaram PA. Effect of hydrogen on the dynamic elastic modulus of gamma titanium aluminide. J Alloy Compd 2002;339:156e61. [6] Sundaram PA, Basu D, Steinbrech RW, Ennis PJ, Quadakkers WJ, Singheiser L. Effect of hydrogen on the elastic modulus and hardness of gamma titanium aluminides. Scr Mater 1999;41(8):839e45. [7] Liu XW, Su YQ, Luo LS, Liu JP, Guo JJ, Fu HZ. Effect of hydrogen on hot deformation behaviors of TiAl alloys. Int J Hydrogen Energy 2010;35:13322e8. [8] Kim YW. Intermetallics alloys based on gamma titanium aluminide. JOM 1989;41(7):24e30. [9] Stroosnijder MF, Zheng N, Quadakkers WJ, Hofman R, Gil A, Lanza F. The effect of niobium ion implantation on the oxidation behaviour of g-TiAl-based intermetallic. Oxid Met 1996;46(1e2):19e35. [10] Liu CT. Recent advances in ordered intermetallics. Mater Res Soc Symp Proc 1993;288:3e19. [11] Lipsitt HA. High temp. Order intermetallic alloys. Mater Res Soc Symp Proc 1985;39:351e64. [12] Fleischer RL, Dimiduk DM, Lipsitt HA. Intermetallic compounds for strong high-temperature materials: status and potential. Ann Rev Mater Sci 1989;19:231e63. [13] Nickel H, Zheng N, Elschner A, Quadakkers WJ. The oxidation behaviour of niobium containing g-TiAl based intermetallics in air and argon/oxygen. Microchim Acta 1995;119:23e9. [14] Haidar J, Gnanarajan S, Dunlop JB. Direct production of alloys based on titanium aluminides. Intermetallics 2009;17:651e6. [15] Hamzah E, Suardi K, Ourdjini A. Effect of microstructures on the hydrogen attack to gamma titanium aluminide at low temperature. Mater Sci Eng A 2005;397:41e9. [16] Hashi K, Ishikawa K, Suzuki K, Aoki K. Hydrogen absorption and desorption in the binary Ti-Al system. J Alloy Compd 2002;330/332:547e50. [17] Gong HR, He YH, Huang BY. Bond strength and interface energy between Pd membranes and TiAl supports. Appl Phys Lett 2008;93:101907. [18] Legzdina D, Robertson IM, Birnbaum HK. Redistribution of Ti and Al in deuterium charged TiAl. Scr Metall Mater 1992;26(11):1737e41. [19] Gao J, Wang YB, Chu WY, Hsiao CM. A study of hydride in TiAl after cathodic charging. Scr Metall Mater 1992;27:1219e22.

1896

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 8 8 8 e1 8 9 6

[20] Combres Y, Tsuyama S, Kishi T. Surface precipitation after cathodic charging of hydrogen and heat treatment in air for the TiAl intermetallic compounds. Scr Metall Mater 1992;27(4):509e14. [21] Liu XW, Su YQ, Luo LS, Li K, Dong FY, Guo JJ, Fu HZ. Effect of hydrogen treatment on solidification structures and mechanical properties of TiAl alloys. Int J Hydrogen Energy 2011;36:3260e7. [22] Su YQ, Liu XW, Luo LS, Zhao L, Guo JJ, Fu HZ. Hydrogen solubility in molten TiAl alloys. Int J Hydrogen Energy 2010;35:8008e13. [23] Fu HZ, Zhao ZG, Liu WF, Peng F, Gao T, Cheng XL. Ab initio calculations of elastic constants and thermodynamic properties of g TiAl under high pressures. Intermetallics 2010;18:761e6. [24] Chen S, Liang CP, Gong HR. Structure stability, mechanical property and elastic anisotropy of TiAl-H system. Int J Hydrogen Energy 2012;37:2676e84. [25] Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals. Phys Rev B 1993;47:558. [26] Kresse G, Joubert J. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B 1999;59:1758. [27] Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, et al. Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation. Phys Rev B 1992;46:6671. [28] Methfessel M, Paxton AT. High-precision sampling for Brillouin-zone integration in metals. Phys Rev B 1989;40:3616. [29] Blo¨chl PE, Jepsen O, Andersen OK. Improved tetrahedron method for Brillouin-zone integrations. Phys Rev B 1994;49:16223. [30] Person WB. A handbook of lattice spacing and structure of metals and alloys. Oxford: Pergamon; 1987. pp. 1e2. [31] Brandes EA. Smithells metals reference book. London: Butterworths Press; 1983. p. 275. [32] Benedek R, Seidman DN, Woodward C. The effect of misfit on heterophase interface energies. J Phys Condens Matter 2002;14:2877e900. [33] Yu R, He LL, Ye HQ. Effect of W on structural stability of TiAl intermetallics and the site preference of W. Phys Rev B 2002;65:184102. [34] Zhou LG, Dong L, He LL, Zhang CB. Ab initio pseudopotential calculations on the effect of Mn doped on lattice parameters of L10 TiAl. Intermetallics 2000;8:637e41. [35] Hultgren R, Desi PD, Hawkins DT, Gleiser M, Kelley KK. Selected values of the thermodynamic properties of binary alloys. Materials Park(OH): ASM; 1973. [36] Liang CP, Gong HR. Structural stability, mechanical property and phase transition of the Ti-H systerm. Int J Hydrogen Energy 2010;35:11378e86. [37] Henkelman G, Uberuaga BP, Jo´nsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 2000;113:9901e4; Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J Chem Phys 2000;113:9978e85.

[38] Wang SQ, Ye HQ. Ab initio elastic constants for the lonsdaleite phase of C, Si, Ge. J Phys Condens Matter 2003;15:5307e14. [39] Westbrook JH, Fleischer RL, editors. Principles. Intermetallic compounds: principles practice, vol. I. London: John Wiley Sons; 1995 [chapters 1 and 9]. [40] Murnaghan FD. Proc Natl Acad Sci U S A 1944;30:244. [41] He Y, Schwarz RB, Migliori A, Hang SH. Elastic constants of single crystal g-TiAl. J Mater Res 1995;10(5):1187e95. [42] Sot R, Kurzydowski KJ. Ab initio calculations of elastic properties of Ni3Al and TiAl under pressure. Mater Sci 2005;23(3):587e90. [43] Sadd MH. Elasticity: theory, applications, numerics. United States of America: Elsevier Inc.; 2005. pp. 291e2. [44] Beckstein O, Klepeis JE, Hart GLW, Pankratov O. Firstprinciples elastic constants and electronic structure of aPt2Si and PtSi. Phys Rev B 2001;63:134112. [45] Pugh SF. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Philos Mag 1954;45:823e43. [46] Ranganathan SI, Ostoja-Starzewski M. Universal elastic anisotropy index. Phys Rev Lett 2008;101. 055504e4. [47] Chung DH, Buessem WR. In: Vahldiek FM, Mersol SA, editors. Anisotropy in single crystal refractory compounds. New York: Plenum; 1968. pp. 217e45. [48] Zener C. Elasticity and anelasticity of metals. Chicago: University of Chicago Press; 1948. [49] Wang Y, Wang JJ, Zhang H, Manga VR, Shang SL, Chen LQ, et al. A first-principles approach to finite temperature elastic constants. J Phys Condens Matter 2010;22:225404e11. [50] Otero-de-la-Roza A, Abbasi-Pe´rez D, Luan˜a V. GIBBS2: a new version of the quasiharmonic model code. II. Models for solid-state thermodynamics, features and implementation. Comput Phys Commun 2011;182:2232e48. [51] Vinet P, Ferrante J, Smith JR, Rose JH. A universal equation of state for solids. J Phys C Solid State 1986;19:L467e73. [52] Setoyama D, Matsunaga J, Muta H, Uno M, Yamanaka S. Mechanical properties of titanium hydride. J Alloy Compd 2004;381:215e20. [53] Setoyama D, Matsunaga J, Ito M, Muta H, Kurosaki K, Uno M, et al. Thermal properties of titanium hydrides. J Nucl Mater 2005;344:298e300. [54] Barin I. Thermochemical data of pure substances. New York, Basel, Cambridge, Tokyo: Weinheim; 1995. [55] Petit AT, Dulong PL. Recherches sur quelques points importants de la the´orie de la chaleur. Ann Chim Phys 1819;10:395e413. [56] Hidnert P. Thermal expansion of titanium. J Res Natl Bur Stand 1943;30:101e5. [57] Gray DE. American Institute of Physics handbook. New York: McGraw-Hill; 1972. [58] He Y, Schwarz RB, Darling T, Hundley M, Whang SH, Wang ZM. Elastic constants and thermal expansion of single crystal g-TiAl from 300 to 750 K. Mater Sci Eng A 1997;239e240:157e63.