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Ab initio calculation of NbH phases with low H compositions
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 1 7 9 8 e1 1 8 0 6
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Ab initio calculation of NbH phases with low H compositions J.H. Long, H. Gong* School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang, Jiangxi 330063, China
article info
abstract
Article history:
Ab initio calculation shows that the BCC NbHx (0 x 0.5) structure with H at tetrahe-
Received 13 April 2014
dral(T) site is the most thermodynamically stable one among all the BCC, FCC, and HCP
Received in revised form
phases, and its negative heat of formation decreases linearly with the increase of H
13 May 2014
composition. Calculation also reveals that the elastic moduli of BCC(T) NbHx phases all
Accepted 15 May 2014
increase with the increase of H composition, and the BCC(T) NbHx phases remain ductile
Available online 14 June 2014
within the studied composition range (0 x 0.5). Moreover, it is found that the percentage anisotropy in shear (AG) and the universal anisotropic parameter (AU) are all
Keywords:
appropriate to describe the elastic anisotropy of NbH phases, and that H location should
NbeH phase
play an important role in elastic anisotropy. The fundamental mechanism of various
Heat of formation
properties is deeply understood by means of electronic structures, and the present results
Elastic properties
agree well with experimental investigations in the literature.
Electronic structure
Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Ab initio calculation
Introduction During the past years, the transition-metal niobium (Nb) and NbH phases have evoked a lot of investigations among researchers [1e13]. First of all, Nb is commonly believed as one of the most favorable hydrogen permeable candidates due to its highest hydrogen permeability among metals as well as its much lower price than palladium alloys [1e3]. On the other hand, with a very high melting point, excellent corrosion resistance, good mechanical properties, and small cross section of neutron absorption, Nb alloys have been widely utilized as high-temperature structural components such as nuclear fuel cladding and ITER divertor, which are sensitive to hydrogen or have close contacts with hydrogen [4e7]. Moreover, Nb and its alloys are also extensively used as superconducting materials, and hydrogen is believed to have an important effect on their superconducting properties [8e10].
It is well known that hydrogen could bring about considerable changes to the structure and stability of Nb [11,12], and the interaction between hydrogen and Nb with various structures (BCC, HCP, and FCC) should be deeply understood, in order to predict and control the properties of various NbH phases. In this regard, the published papers of NbHx with low hydrogen compositions in the literature are primarily concentrated on the BCC (a) structures [1e7,11,12], and the phase boundaries within low temperature ranges in the present Nb-H phase diagram are expressed by dashed lines to show the ambiguity of the compositions of these NbH phases [10,13]. More fundamental studies are therefore required to confirm the structural stability of various NbHx phases. As to the mechanical properties of Nb and NbH phases, some theoretical and experimental studies have been published in the literature [14e22]. In this respect, it is generally acknowledged that hydrogen embrittlement has become the main disadvantage of various Nb-based products, and that
* Corresponding author. Fax: þ86 791 86453201. E-mail address: [email protected] (H. Gong). http://dx.doi.org/10.1016/j.ijhydene.2014.05.097 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
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hydrogen composition should play a significant role on ductile/brittle behaviors of various NbH phases [3,5,10,19,21,22]. Nevertheless, the critical H composition to induce embrittlement of NbH phases is still unknown, and the fundamental mechanism of hydrogen embrittlement needs further investigation [3,21,22]. By means of highly accurate ab initio calculation, the present study is thus conducted to reveal structural stability, mechanical properties, elastic anisotropy, and electronic structures of NbHx phases with low hydrogen compositions (0 x 0.5). At each hydrogen composition, the BCC, FCC, and HCP structures are selected to compare with each other, and one hydrogen atom is added, respectively, at octahedral (O) and tetrahedral (T) sites of Nb. The derived results would reveal that hydrogen location and composition play important roles on various properties of NbH, which should bring about a profound understanding to the structures and properties of various NbH phases.
Theoretical methods The present calculation is based on the well-established Vienna ab initio simulation package (VASP) with the planewave basis and the projector-augmented wave method [23,24]. The exchange and correlation terms are expressed by generalized gradient approximation [25], and the cutoff energies are 350 eV for plane-wave basis. For k space integration, the first-order smearing [26] and modified tetrahedron methods [27] are chosen for dynamical and static calculations, respectively. To simulate NbeH phases with various H compositions, six H compositions are selected, i.e., Nb32H, Nb16H, Nb8H, Nb4H, Nb2H, and pure Nb phases with the supercell models of 32, 16, 8, 4, 2, and 2 Nb atoms, respectively. At each composition, the BCC, FCC, and HCP structures are chosen, and only one hydrogen atom is added at tetrahedral (T) and octahedral (O) sites of each structure, respectively. Specifically, the six hydrogen locations are expressed by the symbols of BCC(O),
BCC(T), FCC(O), FCC(T), HCP(O), and HCP(T), e.g., the BCC(T) means that the BCC NbH phase with hydrogen at the T site. After a series of test calculations, for pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with BCC structures, the kmeshes of 11 11 11, 11 11 11, 11 11 6, 6 6 11, 5 5 5, and 4 4 2 are adopted for relaxation calculations, respectively, while 13 13 13, 13 13 13, 13 13 7, 7 7 13, 7 7 7, and 6 6 4 for static calculations. For pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with FCC structures, the k-meshes of 11 11 11, 16 16 11, 11 11 11, 11 11 6, 6 6 11, and 5 5 5 are adopted for relaxation calculations, respectively, while 13 13 13, 19 19 13, 13 13 13, 13 13 7, 7 7 13, and 7 7 7 for static calculations. For pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with HCP structures, the k-mesh of 11 11 7, 11 11 7, 11 11 4, 7 7 7, 7 7 4, and 7 7 2 are selected for relaxation calculations, respectively, while 13 13 8, 13 13 8, 13 13 4, 9 9 9, 9 9 5, and 9 9 3 for static calculations. In addition, periodic boundary conditions are used in three directions and the unit cell is set for full relaxation. The energy criteria are 0.1 and 0.01 meV for ionic and electronic relaxations, respectively, while 0.001 meV for static calculation.
Results and discussion Structural stability We first find out the influence of hydrogen on crystal structure of NbH phases. After a series of calculation, the lattice constants of various BCC, FCC, and HCP NbH phases with the addition of hydrogen at T and O sites are calculated, respectively, and the results are listed in Table 1. For the sake of comparison, Table 1 also includes calculated and experimental lattice parameters of NbH phases and pure Nb in the literature [6,10,17,28]. It can be observed obviously from Table 1 that the obtained results from the present calculation match well with experimental data [10,28]. For instance, the present
Table 1 e Lattice constants of various NbH phases. BCC Exp.a,b
Present a(Å) Nb Nb32H Nb16H Nb8H Nb4H Nb2H a b c d
T site O site T site O site T site O site T site O site T site O site
3.321 3.328 3.307 3.334 3.303 3.346 3.274 3.367 3.275 3.413 3.255
c/a
Cal.c,d
a (Å)
a (Å)
a (Å)
3.305a 3.310b
3.309c
3.315b
3.2915d
4.233 4.239 4.236 4.248 4.242 4.266 4.251 4.304 4.265 4.362 4.596
1.019 1.029 3.326b 1.070 3.346b 1.083 3.386b 1.140
Ref. [28]. Extrapolated from experimental data in Refs. [28,10]. Ref. [17]. Ref. [6].
3.3931d
FCC
HCP
Present
Present c/a
a (Å)
c/a
0.810
2.880 2.884 2.882 2.890 2.886 2.904 2.895 2.915 2.911 2.966 2.938
1.824 1.828 1.827 1.828 1.825 1.822 1.820 1.844 1.815 1.848 1.806
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lattice constants of 3.321, 3.334 and 3.413 Å for BCC Nb, BCC(T) Nb16H, and BCC(T) Nb2H phases are in good agreement with corresponding experimental values of 3.305, 3.332, and 3.385 Å, respectively [10,28], with an error of less than 1%. Moreover, Fig. 1 shows the derived atomic volumes of various NbHx phases as a function of H composition. It could be seen that compared with pure Nb, all the NbHx phases possess volume expansion with hydrogen addition, and the atomic volumes of NbHx phases increase almost linearly with the H composition. It should be pointed out that the linear volume expansion with the addition of H from the present calculation is consistent with similar experimental results of BCC NbHx phases in the literature [10]. One could also discern from Fig. 1 that H location has an important effect on volume expansion of NbHx, i.e., for the HCP and FCC structures, the volume expansion of T site seems much higher than the corresponding O site, while for BCC the volume of T site is slightly bigger when x > 0.19. To investigate the effect of hydrogen on structural stability of NbHx, the heat of formation, DHf, is derived as follows [29]: DHf ¼
ENbn H nENb 12 EH2 ; nþ1
(1)
where ENbn H , ENb , and EH2 are total energies of NbnH, pure BCC Nb (ground state), and H2 molecule, respectively. As a result, the derived DHf values of various NbHx phases are shown in Fig. 2 as a function of H composition. Several characteristics could be deduced from this figure. Firstly, it can be observed from Fig. 2 that the BCC phases are energetically more stable with lower heats of formation than FCC and HCP phases within the entire composition range. Specifically, at each H composition, the BCC(T) phase has the lowest and negative DHf value among six structures, indicating that the BCC(T) phases could be formed in real situation, which agrees well with the BCC NbH structures observed experimentally in the literature [10,13]. Secondly, it could be seen that the DHf values of all the six structures generally decrease with the increase of H composition. Especially, the DHf of BCC(T) NbHx phases decrease
21.0
HCP(T) HCP(O) BCC(T) BCC(O) FCC(T) FCC(O)
20.0
3
Volume(angstrom /atom)
20.5
19.5
19.0
18.5
18.0 0.0
0.1
Fig. 2 e Heats of formation of NbHx phases as a function of hydrogen composition.
0.2
0.3
0.4
0.5
H/Nb atomic ratio
Fig. 1 e Atomic volumes of NbHx phases as a function of hydrogen composition.
almost linearly with the increase of H composition, and such a linear change of DHf with H composition is in good agreement with similar experimental observations [10,13,30]. As shown in Fig. 2, the slope of the DHf curve of BCC(T) phases is calculated, through fitting, to be 25.94 kJ/(mol H), which is consistent with the corresponding experimental value of 26.74 kJ/(mol H) [13]. Thirdly, the location of H atom has an important effect on the DHf values of NbeH phases. At a certain H composition of the BCC structure, the DHf value of T site is much lower than that of O site, indicating that hydrogen thermodynamically prefers the T site of BCC NbH phase, which agrees well with experimental evidence in the literature [10,30]. As to HCP and FCC structures, however, it can be seen from Fig. 2 that at each H composition, the H atom would occupy the O site due to its lower DHf value. Fourthly, the H composition ranges of NbHx with negative DHf at 0 K are dissimilar for BCC, FCC, and HCP, i.e., the BCC(T) phases become energetically favorable with negative DHf values within the entire H composition range, whereas the FCC(O) phases are thermodynamically stable with negative DHf only when x is bigger than about 0.425. In addition, the DHf values of FCC(T), HCP(T), and HCP(O) NbHx phases are all positive, suggesting that they are all energetically unfavorable at 0 K when 0 x 0.5. We now find out the influence of hydrogen on structural energy differences of NbHx phases. Consequently, Fig. 3 shows the structural energy differences (DE) of NbHx phases as a reference of corresponding BCC(T) phases. Moreover, the structural energy differences of pure Nb are also derived and included in Fig. 3 for the sake of comparison. It could be seen that for pure Nb the present DE values of the FCC and HCP structures with respect to BCC are 31.45 and 28.53 meV/atom, respectively. It could be also discerned from Fig. 3 that the DE values of BCC(O) increase with the H composition, implying that the BCC(O) phases are energetically more unstable than BCC(T). For the FCC(T), FCC(O), and HCP(O) phases, however,
Structral energy differences(kJ/molH)
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bring about a deep understanding to the lower DHf value of the BCC(T) Nb4H phase shown in Fig. 2.
30
Mechanical properties BCC(O)-BCC(T) FCC(T)-BCC(T) FCC(O)-BCC(T) HCP(T)-BCC(T) HCP(O)-BCC(T)
20
10
0
0.0
0.1
0.2
0.3
0.4
0.5
H/Nb atomic ratio Fig. 3 e Structural energy differences (DE) of various NbHx phases with respect to corresponding BCC(T) phase as a function of hydrogen composition.
the DE values decrease with the H composition, which suggests that the relative stability of these phases as a reference of BCC(T) increases with more H compositions. It is of interest to get a deep understanding of the effect of H on structural stability of Nb at an electronic scale, and the electronic structures of various NbH phases are thus derived and compared with each other. As a typical example, Fig. 4 shows total density of states (DOSs) of Nb atom in BCC(T) and BCC(O) Nb4H phases. It can be discerned from Fig. 4 that the bandwidth of the DOSs of Nb in BCC(T) seems smaller than that in BCC(O). In addition, the DOS peaks of Nb in the BCC(T) phase at about 6.5 eV below the Fermi level (Ef) as well as 1.5 eV above Ef seem much higher than those in BCC(O). The above features regarding electronic structures indicates that the BCC(T) Nb4H phase should possess stronger chemical bonding than its BCC(O) counterpart, which would therefore
Fig. 4 e Comparison of total density of states of Nb atom in BCC(O) and BCC(T) Nb4H phases.
To reveal the influence of hydrogen on mechanical properties of NbH phases, the elastic constants of NbH phases are derived by means of the following method [31,32]: a series of small strains are added to the unit cell to calculate the changes of total energy, and the elastic constants of single crystals are derived through quadratic polynomial fittings. The obtained elastic constants are then used to calculate the G of polycrystalline NbH phases by means of the Hill's approximation [33]. Moreover, bulk modulus (B) is determined through the Murnaghan equation of state [34], and Young's modulus (E) is obtained by E ¼ 9BG/(3B þ G) [33]. Accordingly, the derived elastic constants as well as B, G, and E values of Nb and BCC NbH phases are summarized in Table 2. The experimental and calculated data in the literature [6,14,18,20] are also listed in Table 2 for comparison. It can be observed from Table 2 that the present elastic properties are compatible with experimental values [18,20]. For instance, the present C11, C12, and B of BCC Nb are 242.5, 140.8, and 171.1 GPa, respectively, which agree well with corresponding values of 246, 133, and 171 GPa from experiments [18]. It should be pointed out that the present calculated C44 of BCC Nb is lower than the corresponding experimental value, similar to other calculated results [10,14]. Such theoretical underestimation of C44 of BCC Nb at ambient pressure would be probably attributed to the nesting properties of the Fermi surface with a van Hove singularity close to Ef [14,16]. We now investigate mechanical stability of NbHx. 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 [35]. This theory could be specified for the cubic structure as: C11 > 0, C211 > C212, and C44 > 0, for the HCP structure as: C11 > 0, C211 > C212, C33(C11 þ C12) > 2C213, and C11C33 > C213, and for the tetragonal structure as: (C11 C12) > 0, (C11 þ C33 2C13) > 0, (2C11 þ 2C12 þ C33 þ 4C13) > 0, C11 > 0, C33 > 0, C44 > 0, and C66 > 0 [36]. After a series of calculation, it could be discerned that the present C44 values of FCC and HCP NbHx phases are all negative (results not shown), suggesting that these FCC and HCP NbHx phases do not obey the above strain energy theory, and are therefore mechanically unstable at 0 K. Moreover, one could also observe from Table 2 that the BCC(O) Nb16H, Nb8H, and Nb4H phases possess mechanical instability due to their negative C66 values. On the contrary, the BCC(T) NbHx phases are all mechanically stable as they comply with the strain energy theory. It is of interest to explore the influence of hydrogen on elastic moduli of BCC(T) NbHx phases. As shown in Table 2, the present B values of BCC(T) Nb16H, Nb8H, Nb4H, and Nb2H phases are 172.9, 173.0, 173.95, 177.95 GPa, respectively, which match well with the corresponding data of 168.6, 169, 169.8, and 171.3 GPa measured experimentally at 528 K [20]. To have a clear comparison of elastic moduli, the calculated B, E, and G values of BCC(T) NbHx as well as BCC Nb are displayed in Fig. 5 as a function of H composition. It can be seen from Fig. 5 that the present calculated B values of BCC(T) NbHx phases have a
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Table 2 e Elastic constants and elastic moduli of BCC NbHx phases. B is bulk modulus, G is shear modulus, and E is Young's modulus. All values, except G/B, are in the unit of GPa. Phases
Structure
Nb
Nb32H Nb16H
T site O site T site
Nb8H
O site T site
Nb4H
O site T site
Nb2H
O site T site O site
a b c d
Type
C11
C12
Present Exp.a Cal.b Cal.c Present Present Present Exp.d Present Present Exp.d Present Present Exp.d Present Present Exp.d Present
242.5 246 247.8 247 246.6 248.1 249.3 239.3 236.6 254.5 237.5 228.1 271.3 233.5 228.8 312.6 228 304.6
140.8 133 133.1 138 136.5 142.6 134.7
C13
C33
132.3
252.2
149.6 132.3
130.8
265.5
170.0 125.3
121.0
271.3
175.2 110.4
118.2
313.5
118.0
104.4
375.1
C44 12.8 28 17.2 10.3 13.2 13.2 14.2 31.7 12.0 17.7 32.9 12.8 29.6 34.8 21.2 45.0 36.5 36.0
C66
9.1
8.1
0.7
8.2
32.9
B
G
E
G/B
171.1 171 171.3 174.3 173.2 174.4 172.9 168.6 173.2 173.0 169 171.7 173.95 169.8 176.11 177.79 171.3 180.83
23.13 37.24 28.58 21.61 24.43 23.35 25.91
66.40 104.16 81.21 62.27 70.00 67.06 70.03
0.135 0.218 0.167 0.124 0.141 0.134 0.150
43.49 29.86
120.39 84.71
0.251 0.171
10.87 42.94
31.92 119.03
0.063 0.247
140.73 62.62
333.39 168.12
0.799 0.352
56.30
153.02
0.311
Ref. [18]. Ref. [6]. Ref. [14]. Ref. [20]. Experimental values at 528 K.
slight and almost linear increase with the increase of H composition, which is in good agreement with similar experimental observations [18,20]. Interestingly, one could discern clearly from Fig. 5 that the G and E values of BCC(T) NbHx also increase with the increase of H composition, while the slopes of these two curves are much bigger than that of the curve of B, suggesting that the magnitude of the increase of elastic moduli of BCC(T) NbHx due to H addition should be as follows: E > G > B. Such an increase of B, G, and E values of BCC(T) NbHx as a function of H composition would be compatible with the decrease of DHf values shown in Fig. 2. Furthermore, the G/B values of BCC(T) NbHx phases are calculated and shown in Table 2 as well as Fig. 6. It should be
noted that the G/B index has been widely used as an empirical parameter to predict the ductility/brittleness of materials [37], i.e., the critical G/B value of 0.57 separates the ductile/brittle behavior; a smaller G/B value indicates more ductility, and vice versa. It could be detected from Fig. 6 that the G/B values of BCC(T) NbHx phases increase with the increase of the H composition, suggesting that the ductile behavior of BCC(T) NbHx should be increased with the decrease of H composition, which is in excellent agreement with similar experimental evidence [22]. Moreover, one can also see clearly from Fig. 6 that the BCC(T) NbHx phases are all ductile within the studied composition range, as their G/B values seem much less than the critical point of 0.57. Such an observation regarding
Fig. 5 e Elastic moduli of polycrystalline BCC(T) NbHx phases as a function of H composition.
Fig. 6 e G/B values of BCC(T) NbHx phases as a function of H composition.
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the ductility of NbHx phases from the present study is in excellent agreement with the non-brittle feature of NbH phases with low H compositions from experimental evidence [19]. It therefore implies that the experimentally observed hydrogen embrittlement in the literature would be probably attributed to NbHx phases with a high H composition, and further studies are needed to find out the exact H composition which could trigger the embrittlement of NbH phases. It is of further importance to find out the fundamental mechanism of the influence of hydrogen on ductility/brittleness of NbHx phases, and the electronic structures of NbHx phases are thus calculated and compared with each other. For instance, Fig. 7 displays total DOSs of Nb atom in BCC Nb and BCC(T) Nb2H phases. It can be seen that the bandwidth of Nb in BCC(T) Nb2H near Ef is much smaller than that of pure Nb, and a DOS peak at about 6.7 eV below Ef appears in Nb atom of BCC(T) Nb2H to indicate the strong hybridization between Nb and H. Moreover, the DOS value at Ef of Nb atom in BCC(T) Nb2H is 0.69 states/eV/atom, which seems much lower than the value of 1.25 states/eV/atom for pure Nb. These characteristics of DOSs indicate that the BCC(T) Nb2H phase should have more directional and stronger chemical bonding than BCC Nb, which could therefore provide a deep understanding to the brittle/ductile behavior of Nb and NbHx shown in Fig. 6.
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It is well known that some anisotropic parameters have been introduced to describe elastic anisotropy quantitatively in the literature [43e50]. In the present calculation, two frequently used anisotropic parameters, i.e., the percentage anisotropy in shear (AG) [43] and the universal anisotropic parameter (AU) [47], are chosen to describe elastic anisotropy of pure Nb and various NbHx phases with the BCC structure. It should be emphasized that the elastic anisotropy is indicated by these two parameters in terms of elastic behavior from different perspectives, and it is of importance to compare one kind of results with the other. The AG and AU values are derived from the following formulas [43,47]: AG ¼
GV GR 100%; GV þ GR
(2)
GV BV þ 6; GR BR
(3)
AU ¼ 5
Elastic anisotropy has a significant effect on many mechanical-physical qualities, e.g., phase transformation [38], elastic instability [39], mechanical stability [40], etc. Basically, almost all kinds of materials are elastically anisotropic and a suitable definition of the anisotropic behavior seems very important in materials physics and engineering science. In this regard, the elastic anisotropy and related properties of pure Nb and Nb alloys have been studied by several groups [16,41,42]. To our knowledge, however, there is no any report on elastic anisotropy of NbHx phases in the literature. Therefore, it is of interest to find out the effect of hydrogen on elastic anisotropy of the NbH phases.
where B and G denote the bulk and shear moduli, respectively; the subscripts R and V mean Reuss and Voigt approximations, respectively. Fundamentally, AG is a single-valued parameter to express elastic anisotropy by means of shear modulus, i.e., zero refers to elastic isotropy and 100% indicates the maximum anisotropy [43]. AU is considered as a universal anisotropic parameter which describes all the stiffness coefficients from both bulk and shear contributions, i.e., a AU value of zero means isotropic materials and the deviation of AU from zero expresses the magnitude of elastic anisotropy [47]. Consequently, the derived AU and AG values of BCC(T) NbHx phases are shown in Fig. 8 as a function of H composition. It could be discovered from Fig. 8 that the shapes of two curves seem very similar, and that at a certain H composition, the characteristics of elastic anisotropy expressed by AG and AU are very close to each other, although their magnitude is quite different. Such a similar characteristic implies that both anisotropy indexes of AG and AU could be realistic to describe the elastic anisotropy of NbH phases.
Fig. 7 e Total density of states of Nb atom in BCC Nb (dotted line) and BCC(T) Nb2H (solid line) phases.
Fig. 8 e AG and AU values of BCC(T) NbHx phases as a function of H composition.
Elastic anisotropy
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Conclusions Ab initio calculation has been used to find out the structural stability, mechanical property, elastic anisotropy, and electronic structures of various NbHx phases with low H compositions. It is revealed that the BCC(T) phase is the most thermodynamically stable structure with the lowest DHf value when 0 < x 0.5. It is also shown that the magnitude of the increase of elastic moduli of BCC(T) NbHx due to H addition should be as follows: E > G > B, and that the G/B values of BCC(T) NbHx are all less than the critical point of 0.57. Moreover, two anisotropic parameters are chosen to describe the elastic anisotropy of various NbHx phases, and a strong
(a)
3
0.20
0.450.25
1
0.25 0.50
0.450.50 0.45
0.20 0.20 0.25
0.40
0.25 0.20
0.45 0.50 0.25
0.20
0.25
0.45
0.40
0.20
Nb
0.35
0.50
-2
-3
0.50 0.30 0.45
0.40
0
-1
0.35
0.20
2
Y (Angstrom)
In addition, Fig. 9 displays the AG values of BCC(T) and BCC(O) NbHx phases as a function of H composition. It can be seen from Fig. 9 that at a certain H composition, the AG value of the BCC(O) NbHx phase is bigger than its corresponding BCC(T) counterpart, suggesting that the BCC(O) NbHx phase should have higher elastic anisotropy than BCC(T). Considering that the BCC(T) NbHx phase is energetically more favorable than BCC(O) as shown in Fig. 2, one would probably conclude that at a certain composition the thermodynamically more stable BCC NbHx structure should possess lower elastic anisotropy, and vice versa. It should be pointed out that the above conclusion regarding the strong correlation between elastic anisotropy and structural stability of BCC NbHx phases matches well with similar statement by Zener [43]. To get a deeper understanding of elastic anisotropy of BCC NbHx phases, the charge density plots of (100) planes of BCC(T) and BCC(O) Nb4H are shown in Fig. 10 as typical examples. It could be seen that the charge densities of both Nb and H atoms in the BCC(T) phase are more close to the circular shape than those in BCC(O), signifying that the directional bonding of the BCC(T) Nb4H phase would be less than that of BCC(O). This difference of bonding directionality could probably bring about the smaller elastic anisotropy of the T site as revealed in Fig. 9.
0.45
0.30
0.35 0.50 0.30
0.40
0.50 0.20
H
0.35
0.45 0.50 0.30 0.20 -3
-2
0.25
0.45 0.50 -1
0
0.20 1
0.45 0.50 2
3
X (Angstrom)
(b)
3
2
1
Y (Angstrom)
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0.40 0.50
0.50 0.45
0.50 0.45
0.35
H
0.45 0.50 0.30
0.40 0.50
0.30
0.20
0.40 0
Nb
0.30 -1
0.350.25
-2
0.25 0.50 0.30
-3
-3
-2
0.25
Nb
0.45 0.20 0.35
0.40
0.45 0.50 0.35
0.45
0.50
0.30 0.45 0.50 0.35 0.25
0.40 -1
0.25
0
1
0.25 0.45 2
3
X (Angstrom) Fig. 10 e Charge density plots of (100) planes of (a) BCC(T) and (b) BCC(O) Nb4H. The charge density is in the unit of e/Å3.
correlation is revealed between structural stability and elastic anisotropy of the BCC NbHx phases. The derived results are compared with experimental observations in the literature and the agreements between them are very nice.
references
Fig. 9 e eThe AG values of BCC NbHx phases as a function of H composition.
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[24] Kresse G, Joubert J. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B 1999;59:1758e75. [25] 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;46:6671e 87. [26] Methfessel M, Paxton AT. High-precision sampling for Brillouin-zone integration in metals. Phys Rev B 1989;40:3616e21. € chl PE, Jepsen O, Andersen OK. Improved tetrahedron [27] Blo method for Brillouin-zone integrations. Phys Rev B 1994;49:16223e33. € sser R, Bickmann K. Phase diagram of the NbeT system. J [28] La Nucl Mater 1985;132:244e8. [29] Liang CP, Gong HR. Fundamental influence of hydrogen on various properties of a-titanium. Int J Hydrogen Energy 2010;35:3812e6. [30] Plackowski T, Wlosewicz D, Sorokina NI. Specific heat of NbHx with high hydrogen concentration. Phys B 1995;212:119e24. [31] Wang SQ, Ye HQ. Ab initio elastic constants for the lonsdaleite phases of C, Si Ge. J Phys Condens Matter 2003;15:5307e14. [32] Liang CP, Gong HR. Phase stability, mechanical property, and electronic structure of MgeLi system. J Alloys Comp 2010;489:130e5. [33] Westbrook JH, Fleischer RL. Intermetallic compounds: principles practice, volume I: principles. London: John Wiley Sons; 1995. pp. 1e9. [34] Murnaghan FD. The compressibility of media under extreme pressures. Proc Natl Acad Sci U S A 1944;30:244. [35] Sadd MH. Elasticity: theory, applications, numerics. United States of America: Elsevier Inc.; 2005. pp. 291e2. [36] Beckstein O, Klepeis JE, Hart GLW, Pankratov O. Firstprinciples elastic constants and electronic structure of aPt2Si and PtSi. Phys Rev B 2001;63:134112. [37] Pugh SF. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Philos Mag 1954;45:823e43. [38] Man CS. On the correlation of elastic and plastic anisotropy in sheet metals. J Elast 1995;39:165e73. [39] Sukumaran S, Ranganath G. On some elastic instabilities in biaxial nematics. J Phys II 1997;7:583. [40] Wang JW, Gong HR. Effect of hydrogen concentration on various properties of gamma TiAl. Int J Hydrogen Energy 2014;39:1888e96. [41] Xu WW, Han JJ, Wang CP, Zhou Y, Wang Y, Kang YW, et al. Temperature-dependent mechanical properties of alpha-/ beta-Nb5Si3 phases from first-principles calculations. Intermetallics 2014;46:72e9. [42] Chihi T, Fatmi M, Ghebouli B. First-principles prediction of metastable niobium and tantalium nitrides M4N5 and M5N6 stoichiometry. Solid State Sci 2012;14:80e9. [43] Zener C. Elasticity and anelasticity of metals. Chicago: University of Chicago Press; 1948. [44] Chung DH, Buessem WR. The elastic anisotropy of crystals. J Appl Phys 1967;38:2010e2. [45] Ledbetter H, Migliori A. A general elastic-anisotropy measure. J Appl Phys 2006;100:063516. [46] Chung DH, Buessem WR. In: Vahldiek FM, Mersol SA, editors. Anisotropy in single crystal refractory compounds. New York: Plenum; 1968. pp. 217e45. [47] Ranganathan SI, Ostoja-Starzewski M. Universal elastic anisotropy index. Phys Rev Lett 2008;101:055504. [48] Ravindran P, Fast L, Korzhavyi PA, Johnnsson B. Density functional theory for calculation of elastic properties of
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Available online at www.sciencedirect.com
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Ab initio calculation of NbH phases with low H compositions J.H. Long, H. Gong* School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang, Jiangxi 330063, China
article info
abstract
Article history:
Ab initio calculation shows that the BCC NbHx (0 x 0.5) structure with H at tetrahe-
Received 13 April 2014
dral(T) site is the most thermodynamically stable one among all the BCC, FCC, and HCP
Received in revised form
phases, and its negative heat of formation decreases linearly with the increase of H
13 May 2014
composition. Calculation also reveals that the elastic moduli of BCC(T) NbHx phases all
Accepted 15 May 2014
increase with the increase of H composition, and the BCC(T) NbHx phases remain ductile
Available online 14 June 2014
within the studied composition range (0 x 0.5). Moreover, it is found that the percentage anisotropy in shear (AG) and the universal anisotropic parameter (AU) are all
Keywords:
appropriate to describe the elastic anisotropy of NbH phases, and that H location should
NbeH phase
play an important role in elastic anisotropy. The fundamental mechanism of various
Heat of formation
properties is deeply understood by means of electronic structures, and the present results
Elastic properties
agree well with experimental investigations in the literature.
Electronic structure
Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Ab initio calculation
Introduction During the past years, the transition-metal niobium (Nb) and NbH phases have evoked a lot of investigations among researchers [1e13]. First of all, Nb is commonly believed as one of the most favorable hydrogen permeable candidates due to its highest hydrogen permeability among metals as well as its much lower price than palladium alloys [1e3]. On the other hand, with a very high melting point, excellent corrosion resistance, good mechanical properties, and small cross section of neutron absorption, Nb alloys have been widely utilized as high-temperature structural components such as nuclear fuel cladding and ITER divertor, which are sensitive to hydrogen or have close contacts with hydrogen [4e7]. Moreover, Nb and its alloys are also extensively used as superconducting materials, and hydrogen is believed to have an important effect on their superconducting properties [8e10].
It is well known that hydrogen could bring about considerable changes to the structure and stability of Nb [11,12], and the interaction between hydrogen and Nb with various structures (BCC, HCP, and FCC) should be deeply understood, in order to predict and control the properties of various NbH phases. In this regard, the published papers of NbHx with low hydrogen compositions in the literature are primarily concentrated on the BCC (a) structures [1e7,11,12], and the phase boundaries within low temperature ranges in the present Nb-H phase diagram are expressed by dashed lines to show the ambiguity of the compositions of these NbH phases [10,13]. More fundamental studies are therefore required to confirm the structural stability of various NbHx phases. As to the mechanical properties of Nb and NbH phases, some theoretical and experimental studies have been published in the literature [14e22]. In this respect, it is generally acknowledged that hydrogen embrittlement has become the main disadvantage of various Nb-based products, and that
* Corresponding author. Fax: þ86 791 86453201. E-mail address: [email protected] (H. Gong). http://dx.doi.org/10.1016/j.ijhydene.2014.05.097 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
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hydrogen composition should play a significant role on ductile/brittle behaviors of various NbH phases [3,5,10,19,21,22]. Nevertheless, the critical H composition to induce embrittlement of NbH phases is still unknown, and the fundamental mechanism of hydrogen embrittlement needs further investigation [3,21,22]. By means of highly accurate ab initio calculation, the present study is thus conducted to reveal structural stability, mechanical properties, elastic anisotropy, and electronic structures of NbHx phases with low hydrogen compositions (0 x 0.5). At each hydrogen composition, the BCC, FCC, and HCP structures are selected to compare with each other, and one hydrogen atom is added, respectively, at octahedral (O) and tetrahedral (T) sites of Nb. The derived results would reveal that hydrogen location and composition play important roles on various properties of NbH, which should bring about a profound understanding to the structures and properties of various NbH phases.
Theoretical methods The present calculation is based on the well-established Vienna ab initio simulation package (VASP) with the planewave basis and the projector-augmented wave method [23,24]. The exchange and correlation terms are expressed by generalized gradient approximation [25], and the cutoff energies are 350 eV for plane-wave basis. For k space integration, the first-order smearing [26] and modified tetrahedron methods [27] are chosen for dynamical and static calculations, respectively. To simulate NbeH phases with various H compositions, six H compositions are selected, i.e., Nb32H, Nb16H, Nb8H, Nb4H, Nb2H, and pure Nb phases with the supercell models of 32, 16, 8, 4, 2, and 2 Nb atoms, respectively. At each composition, the BCC, FCC, and HCP structures are chosen, and only one hydrogen atom is added at tetrahedral (T) and octahedral (O) sites of each structure, respectively. Specifically, the six hydrogen locations are expressed by the symbols of BCC(O),
BCC(T), FCC(O), FCC(T), HCP(O), and HCP(T), e.g., the BCC(T) means that the BCC NbH phase with hydrogen at the T site. After a series of test calculations, for pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with BCC structures, the kmeshes of 11 11 11, 11 11 11, 11 11 6, 6 6 11, 5 5 5, and 4 4 2 are adopted for relaxation calculations, respectively, while 13 13 13, 13 13 13, 13 13 7, 7 7 13, 7 7 7, and 6 6 4 for static calculations. For pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with FCC structures, the k-meshes of 11 11 11, 16 16 11, 11 11 11, 11 11 6, 6 6 11, and 5 5 5 are adopted for relaxation calculations, respectively, while 13 13 13, 19 19 13, 13 13 13, 13 13 7, 7 7 13, and 7 7 7 for static calculations. For pure Nb, Nb2H, Nb4H, Nb8H, Nb16H, and Nb32H phases with HCP structures, the k-mesh of 11 11 7, 11 11 7, 11 11 4, 7 7 7, 7 7 4, and 7 7 2 are selected for relaxation calculations, respectively, while 13 13 8, 13 13 8, 13 13 4, 9 9 9, 9 9 5, and 9 9 3 for static calculations. In addition, periodic boundary conditions are used in three directions and the unit cell is set for full relaxation. The energy criteria are 0.1 and 0.01 meV for ionic and electronic relaxations, respectively, while 0.001 meV for static calculation.
Results and discussion Structural stability We first find out the influence of hydrogen on crystal structure of NbH phases. After a series of calculation, the lattice constants of various BCC, FCC, and HCP NbH phases with the addition of hydrogen at T and O sites are calculated, respectively, and the results are listed in Table 1. For the sake of comparison, Table 1 also includes calculated and experimental lattice parameters of NbH phases and pure Nb in the literature [6,10,17,28]. It can be observed obviously from Table 1 that the obtained results from the present calculation match well with experimental data [10,28]. For instance, the present
Table 1 e Lattice constants of various NbH phases. BCC Exp.a,b
Present a(Å) Nb Nb32H Nb16H Nb8H Nb4H Nb2H a b c d
T site O site T site O site T site O site T site O site T site O site
3.321 3.328 3.307 3.334 3.303 3.346 3.274 3.367 3.275 3.413 3.255
c/a
Cal.c,d
a (Å)
a (Å)
a (Å)
3.305a 3.310b
3.309c
3.315b
3.2915d
4.233 4.239 4.236 4.248 4.242 4.266 4.251 4.304 4.265 4.362 4.596
1.019 1.029 3.326b 1.070 3.346b 1.083 3.386b 1.140
Ref. [28]. Extrapolated from experimental data in Refs. [28,10]. Ref. [17]. Ref. [6].
3.3931d
FCC
HCP
Present
Present c/a
a (Å)
c/a
0.810
2.880 2.884 2.882 2.890 2.886 2.904 2.895 2.915 2.911 2.966 2.938
1.824 1.828 1.827 1.828 1.825 1.822 1.820 1.844 1.815 1.848 1.806
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lattice constants of 3.321, 3.334 and 3.413 Å for BCC Nb, BCC(T) Nb16H, and BCC(T) Nb2H phases are in good agreement with corresponding experimental values of 3.305, 3.332, and 3.385 Å, respectively [10,28], with an error of less than 1%. Moreover, Fig. 1 shows the derived atomic volumes of various NbHx phases as a function of H composition. It could be seen that compared with pure Nb, all the NbHx phases possess volume expansion with hydrogen addition, and the atomic volumes of NbHx phases increase almost linearly with the H composition. It should be pointed out that the linear volume expansion with the addition of H from the present calculation is consistent with similar experimental results of BCC NbHx phases in the literature [10]. One could also discern from Fig. 1 that H location has an important effect on volume expansion of NbHx, i.e., for the HCP and FCC structures, the volume expansion of T site seems much higher than the corresponding O site, while for BCC the volume of T site is slightly bigger when x > 0.19. To investigate the effect of hydrogen on structural stability of NbHx, the heat of formation, DHf, is derived as follows [29]: DHf ¼
ENbn H nENb 12 EH2 ; nþ1
(1)
where ENbn H , ENb , and EH2 are total energies of NbnH, pure BCC Nb (ground state), and H2 molecule, respectively. As a result, the derived DHf values of various NbHx phases are shown in Fig. 2 as a function of H composition. Several characteristics could be deduced from this figure. Firstly, it can be observed from Fig. 2 that the BCC phases are energetically more stable with lower heats of formation than FCC and HCP phases within the entire composition range. Specifically, at each H composition, the BCC(T) phase has the lowest and negative DHf value among six structures, indicating that the BCC(T) phases could be formed in real situation, which agrees well with the BCC NbH structures observed experimentally in the literature [10,13]. Secondly, it could be seen that the DHf values of all the six structures generally decrease with the increase of H composition. Especially, the DHf of BCC(T) NbHx phases decrease
21.0
HCP(T) HCP(O) BCC(T) BCC(O) FCC(T) FCC(O)
20.0
3
Volume(angstrom /atom)
20.5
19.5
19.0
18.5
18.0 0.0
0.1
Fig. 2 e Heats of formation of NbHx phases as a function of hydrogen composition.
0.2
0.3
0.4
0.5
H/Nb atomic ratio
Fig. 1 e Atomic volumes of NbHx phases as a function of hydrogen composition.
almost linearly with the increase of H composition, and such a linear change of DHf with H composition is in good agreement with similar experimental observations [10,13,30]. As shown in Fig. 2, the slope of the DHf curve of BCC(T) phases is calculated, through fitting, to be 25.94 kJ/(mol H), which is consistent with the corresponding experimental value of 26.74 kJ/(mol H) [13]. Thirdly, the location of H atom has an important effect on the DHf values of NbeH phases. At a certain H composition of the BCC structure, the DHf value of T site is much lower than that of O site, indicating that hydrogen thermodynamically prefers the T site of BCC NbH phase, which agrees well with experimental evidence in the literature [10,30]. As to HCP and FCC structures, however, it can be seen from Fig. 2 that at each H composition, the H atom would occupy the O site due to its lower DHf value. Fourthly, the H composition ranges of NbHx with negative DHf at 0 K are dissimilar for BCC, FCC, and HCP, i.e., the BCC(T) phases become energetically favorable with negative DHf values within the entire H composition range, whereas the FCC(O) phases are thermodynamically stable with negative DHf only when x is bigger than about 0.425. In addition, the DHf values of FCC(T), HCP(T), and HCP(O) NbHx phases are all positive, suggesting that they are all energetically unfavorable at 0 K when 0 x 0.5. We now find out the influence of hydrogen on structural energy differences of NbHx phases. Consequently, Fig. 3 shows the structural energy differences (DE) of NbHx phases as a reference of corresponding BCC(T) phases. Moreover, the structural energy differences of pure Nb are also derived and included in Fig. 3 for the sake of comparison. It could be seen that for pure Nb the present DE values of the FCC and HCP structures with respect to BCC are 31.45 and 28.53 meV/atom, respectively. It could be also discerned from Fig. 3 that the DE values of BCC(O) increase with the H composition, implying that the BCC(O) phases are energetically more unstable than BCC(T). For the FCC(T), FCC(O), and HCP(O) phases, however,
Structral energy differences(kJ/molH)
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bring about a deep understanding to the lower DHf value of the BCC(T) Nb4H phase shown in Fig. 2.
30
Mechanical properties BCC(O)-BCC(T) FCC(T)-BCC(T) FCC(O)-BCC(T) HCP(T)-BCC(T) HCP(O)-BCC(T)
20
10
0
0.0
0.1
0.2
0.3
0.4
0.5
H/Nb atomic ratio Fig. 3 e Structural energy differences (DE) of various NbHx phases with respect to corresponding BCC(T) phase as a function of hydrogen composition.
the DE values decrease with the H composition, which suggests that the relative stability of these phases as a reference of BCC(T) increases with more H compositions. It is of interest to get a deep understanding of the effect of H on structural stability of Nb at an electronic scale, and the electronic structures of various NbH phases are thus derived and compared with each other. As a typical example, Fig. 4 shows total density of states (DOSs) of Nb atom in BCC(T) and BCC(O) Nb4H phases. It can be discerned from Fig. 4 that the bandwidth of the DOSs of Nb in BCC(T) seems smaller than that in BCC(O). In addition, the DOS peaks of Nb in the BCC(T) phase at about 6.5 eV below the Fermi level (Ef) as well as 1.5 eV above Ef seem much higher than those in BCC(O). The above features regarding electronic structures indicates that the BCC(T) Nb4H phase should possess stronger chemical bonding than its BCC(O) counterpart, which would therefore
Fig. 4 e Comparison of total density of states of Nb atom in BCC(O) and BCC(T) Nb4H phases.
To reveal the influence of hydrogen on mechanical properties of NbH phases, the elastic constants of NbH phases are derived by means of the following method [31,32]: a series of small strains are added to the unit cell to calculate the changes of total energy, and the elastic constants of single crystals are derived through quadratic polynomial fittings. The obtained elastic constants are then used to calculate the G of polycrystalline NbH phases by means of the Hill's approximation [33]. Moreover, bulk modulus (B) is determined through the Murnaghan equation of state [34], and Young's modulus (E) is obtained by E ¼ 9BG/(3B þ G) [33]. Accordingly, the derived elastic constants as well as B, G, and E values of Nb and BCC NbH phases are summarized in Table 2. The experimental and calculated data in the literature [6,14,18,20] are also listed in Table 2 for comparison. It can be observed from Table 2 that the present elastic properties are compatible with experimental values [18,20]. For instance, the present C11, C12, and B of BCC Nb are 242.5, 140.8, and 171.1 GPa, respectively, which agree well with corresponding values of 246, 133, and 171 GPa from experiments [18]. It should be pointed out that the present calculated C44 of BCC Nb is lower than the corresponding experimental value, similar to other calculated results [10,14]. Such theoretical underestimation of C44 of BCC Nb at ambient pressure would be probably attributed to the nesting properties of the Fermi surface with a van Hove singularity close to Ef [14,16]. We now investigate mechanical stability of NbHx. 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 [35]. This theory could be specified for the cubic structure as: C11 > 0, C211 > C212, and C44 > 0, for the HCP structure as: C11 > 0, C211 > C212, C33(C11 þ C12) > 2C213, and C11C33 > C213, and for the tetragonal structure as: (C11 C12) > 0, (C11 þ C33 2C13) > 0, (2C11 þ 2C12 þ C33 þ 4C13) > 0, C11 > 0, C33 > 0, C44 > 0, and C66 > 0 [36]. After a series of calculation, it could be discerned that the present C44 values of FCC and HCP NbHx phases are all negative (results not shown), suggesting that these FCC and HCP NbHx phases do not obey the above strain energy theory, and are therefore mechanically unstable at 0 K. Moreover, one could also observe from Table 2 that the BCC(O) Nb16H, Nb8H, and Nb4H phases possess mechanical instability due to their negative C66 values. On the contrary, the BCC(T) NbHx phases are all mechanically stable as they comply with the strain energy theory. It is of interest to explore the influence of hydrogen on elastic moduli of BCC(T) NbHx phases. As shown in Table 2, the present B values of BCC(T) Nb16H, Nb8H, Nb4H, and Nb2H phases are 172.9, 173.0, 173.95, 177.95 GPa, respectively, which match well with the corresponding data of 168.6, 169, 169.8, and 171.3 GPa measured experimentally at 528 K [20]. To have a clear comparison of elastic moduli, the calculated B, E, and G values of BCC(T) NbHx as well as BCC Nb are displayed in Fig. 5 as a function of H composition. It can be seen from Fig. 5 that the present calculated B values of BCC(T) NbHx phases have a
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Table 2 e Elastic constants and elastic moduli of BCC NbHx phases. B is bulk modulus, G is shear modulus, and E is Young's modulus. All values, except G/B, are in the unit of GPa. Phases
Structure
Nb
Nb32H Nb16H
T site O site T site
Nb8H
O site T site
Nb4H
O site T site
Nb2H
O site T site O site
a b c d
Type
C11
C12
Present Exp.a Cal.b Cal.c Present Present Present Exp.d Present Present Exp.d Present Present Exp.d Present Present Exp.d Present
242.5 246 247.8 247 246.6 248.1 249.3 239.3 236.6 254.5 237.5 228.1 271.3 233.5 228.8 312.6 228 304.6
140.8 133 133.1 138 136.5 142.6 134.7
C13
C33
132.3
252.2
149.6 132.3
130.8
265.5
170.0 125.3
121.0
271.3
175.2 110.4
118.2
313.5
118.0
104.4
375.1
C44 12.8 28 17.2 10.3 13.2 13.2 14.2 31.7 12.0 17.7 32.9 12.8 29.6 34.8 21.2 45.0 36.5 36.0
C66
9.1
8.1
0.7
8.2
32.9
B
G
E
G/B
171.1 171 171.3 174.3 173.2 174.4 172.9 168.6 173.2 173.0 169 171.7 173.95 169.8 176.11 177.79 171.3 180.83
23.13 37.24 28.58 21.61 24.43 23.35 25.91
66.40 104.16 81.21 62.27 70.00 67.06 70.03
0.135 0.218 0.167 0.124 0.141 0.134 0.150
43.49 29.86
120.39 84.71
0.251 0.171
10.87 42.94
31.92 119.03
0.063 0.247
140.73 62.62
333.39 168.12
0.799 0.352
56.30
153.02
0.311
Ref. [18]. Ref. [6]. Ref. [14]. Ref. [20]. Experimental values at 528 K.
slight and almost linear increase with the increase of H composition, which is in good agreement with similar experimental observations [18,20]. Interestingly, one could discern clearly from Fig. 5 that the G and E values of BCC(T) NbHx also increase with the increase of H composition, while the slopes of these two curves are much bigger than that of the curve of B, suggesting that the magnitude of the increase of elastic moduli of BCC(T) NbHx due to H addition should be as follows: E > G > B. Such an increase of B, G, and E values of BCC(T) NbHx as a function of H composition would be compatible with the decrease of DHf values shown in Fig. 2. Furthermore, the G/B values of BCC(T) NbHx phases are calculated and shown in Table 2 as well as Fig. 6. It should be
noted that the G/B index has been widely used as an empirical parameter to predict the ductility/brittleness of materials [37], i.e., the critical G/B value of 0.57 separates the ductile/brittle behavior; a smaller G/B value indicates more ductility, and vice versa. It could be detected from Fig. 6 that the G/B values of BCC(T) NbHx phases increase with the increase of the H composition, suggesting that the ductile behavior of BCC(T) NbHx should be increased with the decrease of H composition, which is in excellent agreement with similar experimental evidence [22]. Moreover, one can also see clearly from Fig. 6 that the BCC(T) NbHx phases are all ductile within the studied composition range, as their G/B values seem much less than the critical point of 0.57. Such an observation regarding
Fig. 5 e Elastic moduli of polycrystalline BCC(T) NbHx phases as a function of H composition.
Fig. 6 e G/B values of BCC(T) NbHx 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 1 7 9 8 e1 1 8 0 6
the ductility of NbHx phases from the present study is in excellent agreement with the non-brittle feature of NbH phases with low H compositions from experimental evidence [19]. It therefore implies that the experimentally observed hydrogen embrittlement in the literature would be probably attributed to NbHx phases with a high H composition, and further studies are needed to find out the exact H composition which could trigger the embrittlement of NbH phases. It is of further importance to find out the fundamental mechanism of the influence of hydrogen on ductility/brittleness of NbHx phases, and the electronic structures of NbHx phases are thus calculated and compared with each other. For instance, Fig. 7 displays total DOSs of Nb atom in BCC Nb and BCC(T) Nb2H phases. It can be seen that the bandwidth of Nb in BCC(T) Nb2H near Ef is much smaller than that of pure Nb, and a DOS peak at about 6.7 eV below Ef appears in Nb atom of BCC(T) Nb2H to indicate the strong hybridization between Nb and H. Moreover, the DOS value at Ef of Nb atom in BCC(T) Nb2H is 0.69 states/eV/atom, which seems much lower than the value of 1.25 states/eV/atom for pure Nb. These characteristics of DOSs indicate that the BCC(T) Nb2H phase should have more directional and stronger chemical bonding than BCC Nb, which could therefore provide a deep understanding to the brittle/ductile behavior of Nb and NbHx shown in Fig. 6.
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It is well known that some anisotropic parameters have been introduced to describe elastic anisotropy quantitatively in the literature [43e50]. In the present calculation, two frequently used anisotropic parameters, i.e., the percentage anisotropy in shear (AG) [43] and the universal anisotropic parameter (AU) [47], are chosen to describe elastic anisotropy of pure Nb and various NbHx phases with the BCC structure. It should be emphasized that the elastic anisotropy is indicated by these two parameters in terms of elastic behavior from different perspectives, and it is of importance to compare one kind of results with the other. The AG and AU values are derived from the following formulas [43,47]: AG ¼
GV GR 100%; GV þ GR
(2)
GV BV þ 6; GR BR
(3)
AU ¼ 5
Elastic anisotropy has a significant effect on many mechanical-physical qualities, e.g., phase transformation [38], elastic instability [39], mechanical stability [40], etc. Basically, almost all kinds of materials are elastically anisotropic and a suitable definition of the anisotropic behavior seems very important in materials physics and engineering science. In this regard, the elastic anisotropy and related properties of pure Nb and Nb alloys have been studied by several groups [16,41,42]. To our knowledge, however, there is no any report on elastic anisotropy of NbHx phases in the literature. Therefore, it is of interest to find out the effect of hydrogen on elastic anisotropy of the NbH phases.
where B and G denote the bulk and shear moduli, respectively; the subscripts R and V mean Reuss and Voigt approximations, respectively. Fundamentally, AG is a single-valued parameter to express elastic anisotropy by means of shear modulus, i.e., zero refers to elastic isotropy and 100% indicates the maximum anisotropy [43]. AU is considered as a universal anisotropic parameter which describes all the stiffness coefficients from both bulk and shear contributions, i.e., a AU value of zero means isotropic materials and the deviation of AU from zero expresses the magnitude of elastic anisotropy [47]. Consequently, the derived AU and AG values of BCC(T) NbHx phases are shown in Fig. 8 as a function of H composition. It could be discovered from Fig. 8 that the shapes of two curves seem very similar, and that at a certain H composition, the characteristics of elastic anisotropy expressed by AG and AU are very close to each other, although their magnitude is quite different. Such a similar characteristic implies that both anisotropy indexes of AG and AU could be realistic to describe the elastic anisotropy of NbH phases.
Fig. 7 e Total density of states of Nb atom in BCC Nb (dotted line) and BCC(T) Nb2H (solid line) phases.
Fig. 8 e AG and AU values of BCC(T) NbHx phases as a function of H composition.
Elastic anisotropy
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 1 7 9 8 e1 1 8 0 6
Conclusions Ab initio calculation has been used to find out the structural stability, mechanical property, elastic anisotropy, and electronic structures of various NbHx phases with low H compositions. It is revealed that the BCC(T) phase is the most thermodynamically stable structure with the lowest DHf value when 0 < x 0.5. It is also shown that the magnitude of the increase of elastic moduli of BCC(T) NbHx due to H addition should be as follows: E > G > B, and that the G/B values of BCC(T) NbHx are all less than the critical point of 0.57. Moreover, two anisotropic parameters are chosen to describe the elastic anisotropy of various NbHx phases, and a strong
(a)
3
0.20
0.450.25
1
0.25 0.50
0.450.50 0.45
0.20 0.20 0.25
0.40
0.25 0.20
0.45 0.50 0.25
0.20
0.25
0.45
0.40
0.20
Nb
0.35
0.50
-2
-3
0.50 0.30 0.45
0.40
0
-1
0.35
0.20
2
Y (Angstrom)
In addition, Fig. 9 displays the AG values of BCC(T) and BCC(O) NbHx phases as a function of H composition. It can be seen from Fig. 9 that at a certain H composition, the AG value of the BCC(O) NbHx phase is bigger than its corresponding BCC(T) counterpart, suggesting that the BCC(O) NbHx phase should have higher elastic anisotropy than BCC(T). Considering that the BCC(T) NbHx phase is energetically more favorable than BCC(O) as shown in Fig. 2, one would probably conclude that at a certain composition the thermodynamically more stable BCC NbHx structure should possess lower elastic anisotropy, and vice versa. It should be pointed out that the above conclusion regarding the strong correlation between elastic anisotropy and structural stability of BCC NbHx phases matches well with similar statement by Zener [43]. To get a deeper understanding of elastic anisotropy of BCC NbHx phases, the charge density plots of (100) planes of BCC(T) and BCC(O) Nb4H are shown in Fig. 10 as typical examples. It could be seen that the charge densities of both Nb and H atoms in the BCC(T) phase are more close to the circular shape than those in BCC(O), signifying that the directional bonding of the BCC(T) Nb4H phase would be less than that of BCC(O). This difference of bonding directionality could probably bring about the smaller elastic anisotropy of the T site as revealed in Fig. 9.
0.45
0.30
0.35 0.50 0.30
0.40
0.50 0.20
H
0.35
0.45 0.50 0.30 0.20 -3
-2
0.25
0.45 0.50 -1
0
0.20 1
0.45 0.50 2
3
X (Angstrom)
(b)
3
2
1
Y (Angstrom)
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0.40 0.50
0.50 0.45
0.50 0.45
0.35
H
0.45 0.50 0.30
0.40 0.50
0.30
0.20
0.40 0
Nb
0.30 -1
0.350.25
-2
0.25 0.50 0.30
-3
-3
-2
0.25
Nb
0.45 0.20 0.35
0.40
0.45 0.50 0.35
0.45
0.50
0.30 0.45 0.50 0.35 0.25
0.40 -1
0.25
0
1
0.25 0.45 2
3
X (Angstrom) Fig. 10 e Charge density plots of (100) planes of (a) BCC(T) and (b) BCC(O) Nb4H. The charge density is in the unit of e/Å3.
correlation is revealed between structural stability and elastic anisotropy of the BCC NbHx phases. The derived results are compared with experimental observations in the literature and the agreements between them are very nice.
references
Fig. 9 e eThe AG values of BCC NbHx phases as a function of H composition.
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