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Solution and diffusion of hydrogen isotopes in tungsten-rhenium alloy
Journal of Nuclear Materials 491 (2017) 206e212
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Solution and diffusion of hydrogen isotopes in tungsten-rhenium alloy Fei Ren a, b, Wen Yin b, c, *, Quanzhi Yu c, Xuejun Jia c, Zongfang Zhao a, b, Baotian Wang b a
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, People's Republic of China Dongguan Branch, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China c Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China b
h i g h l i g h t s With or without ZPE corrections, the properties of H solution and diffusion in W and W-Re alloy are investigated. The presence of Re in W increases the solution energy and the real normal modes of vibration, compared to H in pure W. The concentration of Re does not influence noticeably the properties of H solution and diffusion in W-Re alloy. For H in perfect W and W-Re crystals, the presence of Re would decrease the solubility and increase the diffusivity.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2016 Accepted 28 April 2017 Available online 3 May 2017
Rhenium is one of the main transmutation elements forming in tungsten under neutron irradiation. Therefore, it is essential to understand the influence of rhenium impurity on hydrogen isotopes retention in tungsten. First-principle calculations were used to study the properties of hydrogen solution and diffusion in perfect tungsten-rhenium lattice. The interstitial hydrogen still prefers the tetrahedral site in presence of rhenium, and rhenium atom cannot act directly as a trapping site of hydrogen. The presence of rhenium in tungsten raises the solution energy and the real normal modes of vibration on the ground state and the transition state, compared to hydrogen in pure tungsten. Without zero point energy corrections, the presence of rhenium decreases slightly the migration barrier. It is found that although the solution energy would tend to increase slightly with the rising of the concentration of rhenium, but which does not influence noticeably the solution energy of hydrogen in tungsten-rhenium alloy. The solubility and diffusion coefficient of hydrogen in perfect tungsten and tungsten-rhenium alloy have been estimated, according to Sievert's law and harmonic transition state theory. The results show the solubility of hydrogen in tungsten agrees well the experimental data, and the presence of Re would decrease the solubility and increase the diffusivity for the perfect crystals. © 2017 Elsevier B.V. All rights reserved.
Keywords: Hydrogen isotopes Tungsten-rhenium alloy Solution and diffusion First-principle calculations
1. Introduction Tungsten (W) and its alloys are the promising plasma-facing material (PFM) for fusion reactor because of their high melting point, low sputtering yield, good thermal conductivity and low tritium (T) retention [1e3]. The plasma-facing components must withstand high heat and high flux particle bombardment escaping from the plasma, and hydrogen (H) isotopes can congregate on the surfaces of those components or penetrate through the surface and diffuse deep into the bulk [4e6]. In particular, H isotopes can be
* Corresponding author. Dongguan Branch, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China. E-mail address: [email protected] (W. Yin). http://dx.doi.org/10.1016/j.jnucmat.2017.04.057 0022-3115/© 2017 Elsevier B.V. All rights reserved.
trapped by various defects, such as vacancies, grain boundaries and impurity atoms, thus it would affect utilization efficiency of fusion fuel [7]. On the other hand, H isotopes retention in W-based materials may lead to modification of the materials' physical and mechanical properties. For instance, H is known to have a critical influence on metal embrittlement [8,9], due to vacancy formation induced by H would play a key role in the process of embrittlement [10,11]. Besides, H in W could have an impact on the brittle to ductile transition temperature [12]. Therefore, for safety and efficiency reasons, it is very important to understand the solution and diffusion of H isotopes in W-based materials. There are many experimental and computational methods developed to study H isotopes retention in W-based materials [2,13]. Frauenfelder [14] and Mazayev [15] measured the solubility of H in W in the temperature range of 1100 Ke2400 K, which agrees
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
The binding energy between two entities A and B is obtained as:
Eb ¼ ðEA þ EB Þ EAB þ Eref ;
(2)
where EA and EB are the total energies of the supercell containing A or B, respectively, EAB is the total energy of the supercell containing both A and B in interaction, Eref is the energy of the supercell without A and B. All the supercells contain the same number of sites. Note that a positive binding energy means attraction between A and B. According to Sievert's law, the solubility can be given approximately by:
sffiffiffiffiffi P DST Esol exp ; S¼ P0 kT
(3)
where P and P0 represent the background pressure and the standard pressure, respectively, k is the Boltzmann constant and T is the absolute temperature, the solution entropy DS is equal to about 4:7k [35]. The diffusion coefficient of interstitial atoms in solid solutions can be evaluated by Arrhenius equation:
D ¼ D0 exp Em=kT ;
(4)
where D0 is usually assumed to be the temperature independent diffusion constant, Em is the activation energy which is equal to migration energy for the interstitial atomic diffusion. According to the interstitial atomic diffusion theory presented by Wert and Zener [36], D0 can be written as:
n 2 D0 ¼ l v; 6
(5)
where n ¼ 4 is a geometrical factor of the number of equivalent jump paths for H jumping along tetrahedral site to another tetrahedral site in bcc structure metal, l is the jump length which equals pffiffiffi to 2a 4, and a is the lattice constant. v is the vibration frequency, which can be approximately evaluated by: =
with the calculated result by the first-principle calculations [16]. The diffusion coefficient of H in W was evaluated using the harmonic transition state theory (hTST) [17], which is agreement with the experimental result by Frauenfelder [14] at temperature above 1500 K. However, the calculated result of diffusion coefficient by Heinola [17] are much higher than the experimental results below 1500 K, which had been measured by Zakharov [18], Benamati [19], Otsuka [20], Ikeda [21] and Hoshihira [22], respectively. Furthermore, there is a significant difference in migration barrier of H in W between Frauenfelder's [14] experimental value and Heinola's [17] calculated value. It is worthy of note that the migration barrier was calculated by Johnson's [6] along a higher barrier path via octahedral site, although the calculated value is in accordance with Frauenfelder's [14] experimental result. Nevertheless, if the Frauenfelder's [14] experimental data at temperatures above 1500 K is fitted, the migration barrier of H in W will decrease from 0.39 eV to 0.25 eV. Therefore, it is very possible that there is a significant influence of lattice defects on H diffusion even at temperature up to about 1500 K, and Kong [16], Fernandez [23] and Oda [24] have corroborated this by first-principle calculations and Kinetic Monte Carlo method, respectively. Thus Kong's [16] calculated results agree with those experimental results when the influence of temperature and defects on H diffusion in W are considered. Under neutron irradiation, many transmutation elements are produced in PFM. Among them, rhenium (Re) is one of the main products [25], and calculations by Cottrell [26] and Gilbert [27] suggest that after several years of exposure of fusion neutrons, the concentration of Re in W might reach several percent. According to the previous experimental and theoretical studies for WRe alloy, the concentration of Re can be increased to above 25% in a body-centered cubic (bcc) structure [28e30], and the bcc structure is dynamically stable based on Federer's [28] experimental study, as well as Ekman's [29] and Chouhan's [30] calculations. Although Golubeva's [31] experimental results show the Re concentration in WeRe alloy does not influence noticeably the deuterium (D) retention in the range of 1e10%, and Tyburska [32] investigated D retention in damaged W-Re alloy, but the influence of Re on solubility of H in W is still not clear. Moreover, the solubility of the extrapolation of Frauenfelder's [14] datas and Kong's [16] calculated result are much lower than Benamati's [19] data of D in W bel investigated Re doped with 5% Re at 850 Ke885 K. Recently, Wro precipitation in irradiated bcc W-Re alloy and the interactions of Re and vacancy using first-principle calculations and Monte Carlo simulations [33]. However, the interactions between H and vacancy in W-Re alloy were not studied. Therefore, in this paper, we pay attention to the influence of Re on solution and diffusion of H isotopes in W-Re alloy. We carry out a systematic first-principle calculations to investigate the solution energies, migration barriers and binding energies of Re-H for H in perfect bcc W and W-Re crystals. Furthermore, the solubility and diffusion coefficient of H in perfect W and W-Re alloy were evaluated using Sievert's law and hTST.
207
v¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Em 2ml2 ;
(6)
2. Computational methods The solution energies of H in W or W-Re alloy can be defined by:
Esol ¼ E½W=ðW þ ReÞ þ H E½W=ðW þ ReÞ 0:5EH2 ;
(1)
where E½W=ðW þ ReÞ þ H is the total energy of W or W-Re supercell with a single H atom, and E½W=ðW þ ReÞ is the reference energy of perfect W or W-Re supercell. EH2 is the total energy of H molecule, our calculated value is 6.744 eV. The binding energy of a H molecule is 4.516 eV, which agrees well with the experimental value 4.747 eV [34].
where m is the mass of a H atom. In addition, according to reference [37] the diffusion coefficient can also be calculated as:
1 2 D ¼ l G; 6
(7)
where G is the jump rate, based on Eyring's theory [38,39] and hTST [40], it can be calculated by:
208
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
0 B P3N1 @ i
! hvt i 2kT
exp
hvt i kT
1exp
kT G¼ h
0 B P3N i @
! exp
1exp
g i 2kT
hv
g hv i kT
1
!C A Em exp kT 1
(8)
!C A
where, vti , vgi are the real normal modes of vibration on the transition state and the ground state of the initial state, respectively, N is the number of the vibrating atoms. The first-principle calculations are carried out by Vienna Ab initio Simulation Package (VASP) with the projector augmented wave potential method [41] and generalized gradient approximation [42] of the Perdew Burke Ernzerhof functionals [43,44]. The six outermost electrons for W (5d56s1) atom are taken into account as valence electrons. The plane wave energy cutoff for calculations is 500 eV which is large enough for convergence of geometrical structures and total energies. During the all calculations, all the shape and size of the supercells can be changed, a quasi-Newton algorithm is used to relax the ions into their instantaneous groundstate, and the atomic coordinates are relaxed until the Hellmann-Feynman forces are less than 0.01 eV/Å. The all calculations are done using 4 4 4 supercells with 128 atoms. A 3 3 3 k-point mesh is sampled using the Monkhorst and Pack scheme [45] with a Methfessel-Paxton smearing using a broadening of 0.15 eV, which proves to be sufficient for energy convergence of less than 0.001 eV per atom. The migration barriers are calculated using the nudged elastic band (NEB) method [46,47]. The calculation convergence and parameters stay the same for the ground state calculations.
3. Results 3.1. Results of first-principle calculations There are several crystalline forms for W, and the most stable lattice structure is bcc structure. The calculated bcc lattice constant of 3.172 Å agrees well with the experimental value of 3.165 Å [28,48]. Three types of W-Re supercells, 127W-1Re with concentration of 0.078% Re, 120W-8Re with concentration of 6.25% Re, and 96W-32Re with concentration of 25% Re are considered. The bcc structure of W-Re alloy is dynamically stable when the atomic content of Re is less than 27% [28e30]. To obtain reliable results, the solution and diffusion of H in 127W-1Re and 120W-8Re lattices are mainly investigated by first-principle calculations, due to several percent of sigma-phase incorporation is possible for W-25%Re alloy [31]. As shown in Fig. 1, suppose Re atoms are distributed equably in W-Re lattice. The calculated result shows that the lattice constant of W-Re alloy would decrease from 3.172 Å to 3.155 Å with the increase of Re content from 0.78% to 25%, which in accordance with Federer's [28] experimental result. The experimental result shows the lattice constant of W-Re alloy decreases from 3.160 Å to 3.153 Å with the increase in Re content from 4% to 13% [28]. Furthermore, the presence of Re will decrease volume of supercell a little bit due to Re is the adjacent element of W. Hence, it would be reasonable to use the kind of lattice models to calculate the properties of H in perfect W-Re alloy. Fig. 2 shows the first, second, third and fourth nearest neighboring tetrahedral (Td) sites of a Re atom indicated as 1NN, 2NN, 3NN and 4NN, respectively, for H in W-Re lattice.
Similarly, the nearest neighboring octahedral (Oh) and diagonal (Dg) sites of a Re atom can also be represented by 1NN, 2NN, 3NN and 4NN, respectively. First of all, we calculated the basic properties of H solution and diffusion in pure W. As shown in Table 1, the energy difference between Oh and Td sites is 0.378 eV, thus the most stable interstitial site of H in W is the Td site, consistent with the other calculations [16,17,49]. The corresponding solution energy of H at Td site with Zero Point Energy (ZPE) corrections is 1.056 eV, which is in good agreement with the experimental value 1.04 eV [50], and the previous calculated values [6,23]. The migration barrier with ZPE corrections is 0.173 eV, in excellent agreement with Johnson's [6] Fernandez's [23] 0.17 eV. As can be seen from Table 2, the interstitial H at 1NN sites of 127W-1Re lattice still prefers the Td site. The solution energy of H at 1NN Td site of 127W-1Re lattice is 0.096 eV bigger than H in pure W. When ZPE corrections are taken into account, the solution energy of H at 1NN Td site of 127W-1Re lattice is 0.179 eV bigger than H in pure W, that is because the vibrational frequencies at the ground state of H in 127W-1Re lattice increase significantly compared to H in pure W. There are two possible paths for H from one Td site migrate to another adjacent Td site. According to the NEB calculations and the difference of solution energies between H at Td and Oh interstitial sites, the most probable migration path of H passes through the Dg interstitial site in W-Re lattice, which is similar to the migration path of H in pure W. The migration barrier that H via 1NN Td site to the adjacent 1NN Td site for H in 127W1Re lattice is smaller than H in pure W. The binding energy between two H atoms in the nearest neighboring Td site is negative for H in pure W. Similarly, for H in 127W-1Re lattice, the binding energy of H-H pair which two H atoms are located at the adjacent 1NN Td sites is also negative, which means the two H atoms are repulsive each other. The binding energies of Re-H pairs which a H atom is located at 1NN, 2NN, 3NN and 4NN Td sites of a Re atom are 0.096 eV, 0.041 eV, 0.022 eV and 0.011 eV, respectively. Hence, the Re atom cannot act directly a trapping site of H, consistent with the Kong's calculated result [16]. For H in 127W-1Re and 120W-8Re lattices, the solution energies and migration barriers between the two nearest Td sites are shown in Fig. 3 and Fig. 4. The solution energy of H at Td site of 127W-1Re lattice decreases slowly with the increase in distance of Re-H. With considering ZPE corrections, the solution energy of H at 4NN Td site of 127W-1Re lattice is 0.072 eV bigger than H in pure W. For H in 120W-8Re lattice, the solution energies of H at Td sites are bigger than H in pure W, which is similar to H in 127W-1Re lattice. Moreover, if the distance of Re-H is same, the solution energies are almost equal for H in 127W-1Re and 120W-8Re lattices. With or without ZPE corrections, the solution energies of H at 1NN Td site of 96W-32Re are 1.042 eV and 1.243 eV, respectively, which are close to the solution energies of H at 1NN Td site of 120W-8Re. Thus, although the solution energy would tend to increase slightly with the rising of concentration of Re, but which does not influence noticeably the solution energy of H in W-Re alloy. Overall, in the presence of Re, the solution energy of H at Td sites will increase compared to H in pure W. Without considering ZPE corrections, the presence of Re would decrease the migration barrier between the two nearest Td sites slightly. With ZPE corrections, the migration barriers of H in 127W-1Re and 120W-8Re lattices are around 0.18 eV, 0.17 eV, respectively, which are close to the migration barrier of H in pure W. It can be seen from Fig. 5, for H in 127W-1Re and 120W-8Re lattices, the real normal modes of vibration on the ground state and the transition state increase significantly, compared to H in pure W, and if the distance of Re-H is same, the real normal modes of vibration are almost equal for H in 127W-1Re and 120W-8Re lattices.
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
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Fig. 1. The lattice models of 120W-8Re and 96W-32Re supercells. The gray and red balls represent W and Re atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The first, second, third and fourth nearest neighboring Td interstitial sites of a Re atom indicated as 1NN, 2NN, 3NN and 4NN, respectively, for H in W-Re lattice. The gray balls represent W atoms, the red ball represents Re atom. The green, blue, pink and yellow balls represent the H atoms at 1NN, 2NN, 3NN and 4NN Td interstitial sites, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Solubility and diffusion coefficient of hydrogen isotopes in tungsten and tungsten-rhenium alloy According to Sievert's law, the solubility can be estimated by
Table 1 Td (eV) and EOh (eV) are the The properties of H solution and diffusion in pure W. Esol sol solution energies of H at Td and Oh sites, respectively. Em (eV) is the migration barrier, vgi (THz) and vti (THz) are the real normal modes of vibration on the ground state and the transition state, respectively. EbHH (eV) is the binding energy between two H atoms in the nearest neighboring Td site. This work
With ZPE
0.924
1.056
0.95,a 0.89b
Oh Esol Em
1.302
1.324
1.33,a 1.29b
0.203 46.771
vti
64.737
EbHH a b
Ref. [17]. Ref. [16].
0.461
0.173
Table 2 The properties of solution and diffusion of H at 1NN interstitial sites for 127W-1Re Td (eV), E Oh (eV) and E Dg (eV) are the solution energies of H at Td, Oh and Dg lattice. Esol sol sol g sites, respectively. Em (eV) is the migration barrier, vi (THz) and vti (THz) are the real normal modes of vibration on the ground state and the transition state, respectively. EbHH (eV) is the binding energy between two H atoms in the nearest neighboring Td site, and EbReH (eV) is the binding energy between Re and H at the 1NN Td site. This work
With ZPE
Td Esol Oh Esol Dg Esol
1.020
1.235
1.242
1.394
1.138
1.466
Em
0.168
vgi
0.189 62.223
vti
70.081
Other
Td Esol
vgi
combined with Eq. (3) and the above solution energies. To make a better comparison with the experiment, the background pressure of Frauenfelder's [14] experiment 79993 Pa (600 Torr) is used in the calculation. As shown in Fig. 6, the solubility of H in pure W is in excellent agreement with Frauenfelder's [14] experimental data, especially when ZPE corrections are considered. This is due to the solution energy of H in pure W with ZPE corrections agrees well with the experimental value. Besides, the defects cannot effectively trap H at the high temperature range [16]. Benamati [19] measured the D inventory parameters in W-5%Re, the experimental background pressure is 75600 Pa. As can be seen from Fig. 6, the Benamati's [19] experimental data is much higher than the calculated results, despite the presence of Re would increase the solution energy and Re cannot trap H. However, for the perfect lattice, it can be deducted from the calculated results that the presence of Re would decrease the solubility compared to H in pure W due to the increase in solution energy. Based on hTST, the diffusion coefficient can be evaluated by Eqs. (7) and (8) combined with the migration barrier and the real modes of vibration on the ground state and the transition state. Here, in order to compare the influence of Re on diffusivity, the diffusion coefficient of H in pure W is also calculated. As shown in Fig. 7, for the diffusion coefficient of H in pure W, the calculated result from this paper agrees Heinola's [17] calculation. There is discrepancy between the calculated results with or without ZPE corrections and the experimental datas. It has been proved that the temperature and defect trapping effects play an important role in diffusion of H in pure W [16]. As mentioned, the presence of Re would decrease
a
b
0.2, 0.2 46.7,a 46.0b 63.3,a 62.3b
EbHH EbReH
0.313
0.229
0.096
0.179
210
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Fig. 3. The solution energy as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices. Fig. 5. The real normal modes of vibration on the ground state and the transition state as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices.
Fig. 4. The migration barrier as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices.
the migration barrier without ZPE corrections slightly, and significantly increase the real normal modes of vibration on the ground state and the transition state. The concentration of Re for 120W-8Re lattice is 6.25%, which is close to the concentration of Re of Benamati's [19] experimental sample. Therefore, the diffusion parameters of H in 120W-8Re lattice are used to evaluate the diffusion coefficient of H in perfect W-Re alloy. It can be seen from Fig. 7, although Benamati's [19] experimental data is much smaller than the calculated results, but for the perfect crystal, the calculated results show that the presence of Re increases the diffusivity slightly. 4. Discussion The solubility and diffusion coefficient are crucial parameters for W-based PFMs, which are related to H isotopes transport. Hence, there is need to study the solution and diffusion of H isotopes in W and W-Re alloy. The calculated solubility of H in pure W agrees very well with Frauenfelder's [14] experimental data. The solution energies of D and T in pure W are calculated to be 1.095 eV and 1.113 eV, respectively. Similarly, the solution energies of D, T at 1NN Td sites of 127W-1Re lattice are 1.274 eV, 1.292 eV, respectively.
Thus according to Eq. (3), the solubility of H is maximum for H isotopes in W, which is in accordance with Esteban's [51] experimental result. The concentration of Re does not influence noticeably the solubility of H in W-Re alloy, due to the change of solution energy with the increase in concentration of Re is quite small, which is in accordance with Golubeva's [31] experimental result. However, the calculated solubility of H in 120W-8Re lattice is much lower than Benamati's [19] experimental data. The data was obtained by using the W-5%Re sample in the temperature range 850 Ke885 K. In addition, Golubeva's [31] experimental results show the thermal desorption spectroscopy (TDS) of W-Re alloy contains an additional peak at 850 K, which does not appear in TDS of pure W. The experimental W-Re samples contain some defects, such as there are voids exist in the bulk for all W-Re alloys [31]. Furthermore, the recent calculations show that the vacancy-Re clusters remain stable at temperature above 800 K, and solute segregation is found to occur in the form of voids decorated by Re atoms in the low Re concentration range [33]. A single Re atom can raise the total binding energies between two or three vacancies obviously [52]. However, the two vacancies repulse each other even where they are situated as far as in fifth nearest neighbor position in pure W [53]. Therefore, although Re atom cannot act directly as a trapping site of H, it is very possible that the presence of Re can increase the solubility of H in W-Re alloy by the interactions of Re with defects. Consequently, for the solubility of H isotopes in W-Re alloy, the Benamati's [19] experimental data is much higher than the calculated result of H in perfect W-Re alloy. The diffusion coefficient also can be evaluated by Wert and Zener model of Eqs. (4)e(6), which suggests the diffusion coefficient is proportional to the reciprocal square root of mass. Thus the diffusivity decreases with the increase in the mass of H isotopes, which has been proved by the experiment [51]. According to hTST, the diffusion coefficient of H in perfect crystal is calculated. That is, for H in perfect W or W-Re lattice, it is decided by the diffusion of a single H atom from a Td site to another site. Nevertheless, usually there are some defects exist in the experimental samples. Therefore, to compare the experiment and calculation, the influence of defects on the diffusion of H isotopes should be taken into account. For instance, the effects of defect trapping play a key role for diffusivity of H in pure W in the low temperature range, below about 1500 K, which have been verified by Kong [16], Fernandez
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the ground state and the transition state. Although the solution energy would tend to increase slightly with the rising of concentration of Re, but which does not influence noticeably the solution energy of H in W-Re alloy. Overall, in the presence of Re, the solution energies of H at Td sites will increase compared to H in pure W. Without ZPE corrections, the presence of Re in W would decrease slightly the migration barrier. It is found that the binding energy of H-H pair with or without ZPE corrections is also negative in presence of Re, and Re atom cannot act directly as a trapping site of H. According to Sievert's law and hTST, the solubility and diffusion coefficient of H in perfect W and W-Re alloy have been estimated. The solubility of H in W agrees well the experimental result, and the diffusivity of H in W is in accordance with the experimental data at high temperature. For H in perfect W and W-Re crystals, the results show the presence of Re would decrease the solubility and increase the diffusivity. The discrepancy between the calculated and experimental results has been discussed, and the possible reasons have been analyzed in this paper. Acknowledgments Fig. 6. The solubility as a function of reciprocal temperature. Datas are from Frauenfelder [14], Benamati [19], and Kong [16].
This work is supported by the National Natural Science Foundation of China (Grant Nos. 51371195, 11575289). References
Fig. 7. The diffusion coefficient as a function of reciprocal temperature. Datas are from Frauenfelder [14], Benamati [19], Zakharov [18], Kong [16] and Heinola [17].
[23] and Oda [24]. Similarly, for the diffusivity of H in W-Re alloy, it can be deducted that defects would affect the diffusivity. Besides, recent calculations [33,52] show that the presence of Re can facilitate the formations of voids and vacancy clusters in W. Therefore, it is possible that the interactions of Re, H and vacancy would affect the diffusion coefficient of H in W-Re alloy significantly. 5. Conclusions In the present work, the properties of solution and diffusion of H in perfect W and W-Re alloy have been studied by first-principle calculations. The lattice constant of W-Re alloy decreases with the increase in concentration of Re, which agrees with the experimental result. In W-Re lattice, the interaction between Re atom and H atom is repulsive, and the interstitial H still prefers the Td site. The presence of Re in W increases the solution energy with or without ZPE corrections and the real normal modes of vibration on
[1] G. Federici, C.H. Skinner, J.N. Brooks, et al., Plasma-material interactions in current tokamaks and their implications for next step fusion reactors, Nucl. Fusion 41 (2001) 1967e2137. [2] S. Wurster, N. Baluc, M. Battabyal, et al., Recent progress in R&D on tungsten alloys for divertor structural and plasma facing materials, J. Nucl. Mater 442 (2013) S181eS189. [3] Y. Ueda, K. Tobita, Y. Katoh, PSI issues at plasma facing surfaces of blankets in fusion reactors, J. Nucl. Mater 313e316 (2003) 32e41. ^ne, S. Garcia-Argote, et al., Tritium absorption/desorp[4] A. El-Kharbachi, J. Che tion in ITER-like tungsten particles, Int. J. Hydrogen Energ 39 (2014) 10525e10536. [5] C. Grisolia, E. Hodille, J. Chene, et al., Tritium absorption and desorption in ITER relevant materials: comparative study of tungsten dust and massive samples, J. Nucl. Mater 463 (2015) 885e888. [6] D.F. Johnson, E.A. Carter, Hydrogen in tungsten: absorption, diffusion, vacancy trapping, and decohesion, J. Mater. Res. 25 (2010) 315e327. [7] J. Roth, E. Tsitrone, A. Loarte, et al., Recent analysis of key plasma wall interactions issues for ITER, J. Nucl. Mater 390e391 (2009) 1e9. [8] J.P. Hirth, Effects of hydrogen on the properties of iron and steel, Metall. Mater. Trans. A 11 (1980) 861e890. [9] N. Eliaz, A. Shachar, B. Tal, et al., Characteristics of hydrogen embrittlement, stress corrosion cracking and tempered martensite embrittlement in highstrength steels, Eng. Fail. Anal. 9 (2002) 167e184. [10] M. Nagumo, M. Nakamura, K. Takai, Hydrogen thermal desorption relevant to delayed-fracture susceptibility of high-strength steels, Metall. Mater. Trans. A 32 (2001) 339e347. [11] E. Hayward, C.C. Fu, Interplay between hydrogen and vacancies in a -Fe, Phys. Rev. B 87 (2014) 104103. [12] P. Gumbsch, Controlling factors for the brittle-to-ductile transition in tungsten single crystals, Science 282 (1998) 1293. [13] T. Tanabe, Review of hydrogen retention in tungsten, Phys. Scr. T159 (2014) 827e847. [14] R. Frauenfelder, Permeation of hydrogen through tungsten and molybdenum, J. Chem. Phys. 48 (1968) 3955e3965. [15] A.A. Mazayev, R.G. Avarbe, Y.N. Vilk, et al., Solubility of hydrogen in tungsten at high temperature and pressures 25 (1968) 267e270. [16] X.S. Kong, S. Wang, X. Wu, et al., First-principles calculations of hydrogen solution and diffusion in tungsten: temperature and defect-trapping effects, Acta Mater. 84 (2015) 426e435. [17] K. Heinola, T. Ahlgren, Diffusion of hydrogen in bcc tungsten studied with first principle calculations, J. Appl. Phys. 107 (2010) 113531, 113531-8. Evko, Hydrogen permeability of poly[18] A.P. Zakharov, V.M. Sharapov, E.I. crystalline and monocrystalline molybdenum and tungsten, Mater. Sci. 9 (1975) 149e153. [19] G. Benamati, E. Serra, C.H. Wu, Hydrogen and deuterium transport and inventory parameters through W and W-alloys for fusion reactor applications, J. Nucl. Mater. 283 (2000) 1033e1037. [20] T. Otsuka, T. Hoshihira, T. Tanabe, Visualization of hydrogen depth profile by means of tritium imaging plate technique: determination of hydrogen diffusion coefficient in pure tungsten, Phys. Scr. 138 (2009) 014052. [21] T. Ikeda, T. Otsuka, T. Tanabe, Determination of hydrogen diffusivity and
212
[22]
[23]
[24]
[25]
[26] [27] [28] [29] [30]
[31] [32] [33]
[34] [35] [36] [37]
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212 permeability in W near room temperature applying a tritium tracer technique, J. Nucl. Mater. 415 (2011) S684eS687. T. Hoshihira, T. Otsuka, R. Wakabayashi, et al., Hydrogen behavior near surface regions in Mo and W studied by tritium tracer technique, J. Nucl. Mater. 417 (2011) 559e563. N. Fernandez, Y. Ferro, D. Kato, Hydrogen diffusion and vacancies formation in tungsten: density Functional Theory calculations and statistical models, Acta Mater. 94 (2015) 307e318. T. Oda, D. Zhu, Y. Watanabe, Kinetic Monte Carlo simulation on influence of vacancy on hydrogen diffusivity in tungsten, J. Nucl. Mater. 467 (2015) 439e447. M.R. Gilbert, S.L. Dudarev, S. Zheng, et al., An integrated model for materials in a fusion power plant: transmutation, gas production, and helium embrittlement under neutron irradiation, Nucl. Fusion 52 (2012) 083019. G.A. Cottrell, Sigma phase formation in irradiated tungsten, tantalum and molybdenum in a fusion power plant, J. Nucl. Mater. 334 (2004) 166e168. M.R. Gilbert, J.C. Sublet, Neutron-induced transmutation effects in W and Walloys in a fusion environment, Nucl. Fusion 51 (2011) 043005. J.I. Federer, R.M. Steele, Identification of a beta-tungsten phase in tungstenrhenium alloys, Nat 205 (1965) 587e588. M. Ekman, K. Persson, G. Grimvall, Phase diagram and lattice instability in tungsten-rhenium alloys, J. Nucl. Mater. 278 (2000) 273e276. R.K. Chouhan, A. Alam, S. Ghosh, et al., Ab initio study of the phonon spectrum, entropy and lattice heat capacity of disordered Re-W alloys, J. Phys. Condens. Matter 24 (2012) 375401. A.V. Golubeva, M. Mayer, J. Roth, et al., Deuterium retention in rheniumdoped tungsten, J. Nucl. Mater. 363 (2007) 893e897. B. Tyburska Püschel, V.K. Alimov, On the reduction of deuterium retention in damaged Re-doped W, Nucl. Fusion 53 (2013) 8e10. J. Wrobel, D. Nguyen-Manh, S. Dudarev, et al., A first-principles model for anomalous segregation in dilute ternary tungsten-rhenium-vacancy alloys, J. Phys. Condens. Matter 29 (2017) 145403. K.P. Huber, G. Hertzberg, Molecular Spectra and Molecular Structure. IV: Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. Y. Fukai, The Metal-hydrogen System, Springer, Berlin, 1993. C. Wert, C. Zener, Interstitial atomic diffusion coefficients, Phys. Rev. 76 (1949) 1169e1175. J. Philibert, Atom movements: diffusion and mass transport in solids, France,
Les editions de Physique, 1991. [38] H. Eyring, The activated complex in chemical reactions, J. Chem. Phys. 3 (1934) 107e115. [39] H. Eyring, W.U. Dan, The theory of absolute reaction rates in solution, Trans. Faraday Soc. 34 (1938) 41e48. [40] E. Wigner, The transition state method, Trans. Faraday Soc. 34 (1938) 29e41. € chl, Projector augmented-wave method, Phys. Rev. B 50 (1994) [41] P.E. Blo 17953e17979. [42] J.P. Perdew, J.A. Chevary, S.H. Vosko, et al., Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B 48 (1993) 4978, 4978. [43] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 78 (1997) 1396. [44] Y. Zhang, W. Yang, Comment on “generalized gradient approximation made simple”, Phys. Rev. Lett. 4 (1997) 890. [45] H.J. Monkhorst, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188e5192. nsson H, A climbing image nudged elastic [46] G. Henkelman, B.P. Uberuaga, H. Jo band method for finding saddle points and minimum energy paths, J. Chem. Phys. 113 (2000) 9901e9904. nsson, Improved tangent estimate in the nudged elastic [47] G. Henkelman, H. Jo band method for finding minimum energy paths and saddle points, J. Chem. Phys. 113 (2000) 9978e9985. [48] E. Lassner, W.D. Schubert, Tungsten: Properties, Chemistry, Technology of the Element, Alloys, and Chemical Compounds, Springer, Berlin, 2011. [49] K. Ohsawa, F. Nakamori, Y. Hatano, et al., Thermodynamics of hydrogeninduced superabundant vacancy in tungsten, J. Nucl. Mater. 458 (2014) 187e197. [50] R. Frauenfelder, Solution and diffusion of hydrogen in tungsten, J. Vac. Sci. Technol. 6 (1969) 388e397. [51] G.A. Esteban, A. Perujo, L.A. Sedano, et al., Hydrogen isotope diffusive transport parameters in pure polycrystalline tungsten, J. Nucl. Mater. 295 (2001) 49e56. [52] Y.H. Li, H.B. Zhou, S. Jin, et al., Behaviors of transmutation elements Re and Os and their effects on energetics and clustering of vacancy and self-interstitial atoms in W, Nucl. Fusion 57 (2017) 046006. [53] C.S. Becquart, C. Domain, Ab initio calculations about intrinsic point defects and He in W, Nucl. Instrum. Meth. Phys. Res. B 225 (2007) 23e26.
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Solution and diffusion of hydrogen isotopes in tungsten-rhenium alloy Fei Ren a, b, Wen Yin b, c, *, Quanzhi Yu c, Xuejun Jia c, Zongfang Zhao a, b, Baotian Wang b a
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, People's Republic of China Dongguan Branch, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China c Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China b
h i g h l i g h t s With or without ZPE corrections, the properties of H solution and diffusion in W and W-Re alloy are investigated. The presence of Re in W increases the solution energy and the real normal modes of vibration, compared to H in pure W. The concentration of Re does not influence noticeably the properties of H solution and diffusion in W-Re alloy. For H in perfect W and W-Re crystals, the presence of Re would decrease the solubility and increase the diffusivity.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2016 Accepted 28 April 2017 Available online 3 May 2017
Rhenium is one of the main transmutation elements forming in tungsten under neutron irradiation. Therefore, it is essential to understand the influence of rhenium impurity on hydrogen isotopes retention in tungsten. First-principle calculations were used to study the properties of hydrogen solution and diffusion in perfect tungsten-rhenium lattice. The interstitial hydrogen still prefers the tetrahedral site in presence of rhenium, and rhenium atom cannot act directly as a trapping site of hydrogen. The presence of rhenium in tungsten raises the solution energy and the real normal modes of vibration on the ground state and the transition state, compared to hydrogen in pure tungsten. Without zero point energy corrections, the presence of rhenium decreases slightly the migration barrier. It is found that although the solution energy would tend to increase slightly with the rising of the concentration of rhenium, but which does not influence noticeably the solution energy of hydrogen in tungsten-rhenium alloy. The solubility and diffusion coefficient of hydrogen in perfect tungsten and tungsten-rhenium alloy have been estimated, according to Sievert's law and harmonic transition state theory. The results show the solubility of hydrogen in tungsten agrees well the experimental data, and the presence of Re would decrease the solubility and increase the diffusivity for the perfect crystals. © 2017 Elsevier B.V. All rights reserved.
Keywords: Hydrogen isotopes Tungsten-rhenium alloy Solution and diffusion First-principle calculations
1. Introduction Tungsten (W) and its alloys are the promising plasma-facing material (PFM) for fusion reactor because of their high melting point, low sputtering yield, good thermal conductivity and low tritium (T) retention [1e3]. The plasma-facing components must withstand high heat and high flux particle bombardment escaping from the plasma, and hydrogen (H) isotopes can congregate on the surfaces of those components or penetrate through the surface and diffuse deep into the bulk [4e6]. In particular, H isotopes can be
* Corresponding author. Dongguan Branch, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China. E-mail address: [email protected] (W. Yin). http://dx.doi.org/10.1016/j.jnucmat.2017.04.057 0022-3115/© 2017 Elsevier B.V. All rights reserved.
trapped by various defects, such as vacancies, grain boundaries and impurity atoms, thus it would affect utilization efficiency of fusion fuel [7]. On the other hand, H isotopes retention in W-based materials may lead to modification of the materials' physical and mechanical properties. For instance, H is known to have a critical influence on metal embrittlement [8,9], due to vacancy formation induced by H would play a key role in the process of embrittlement [10,11]. Besides, H in W could have an impact on the brittle to ductile transition temperature [12]. Therefore, for safety and efficiency reasons, it is very important to understand the solution and diffusion of H isotopes in W-based materials. There are many experimental and computational methods developed to study H isotopes retention in W-based materials [2,13]. Frauenfelder [14] and Mazayev [15] measured the solubility of H in W in the temperature range of 1100 Ke2400 K, which agrees
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
The binding energy between two entities A and B is obtained as:
Eb ¼ ðEA þ EB Þ EAB þ Eref ;
(2)
where EA and EB are the total energies of the supercell containing A or B, respectively, EAB is the total energy of the supercell containing both A and B in interaction, Eref is the energy of the supercell without A and B. All the supercells contain the same number of sites. Note that a positive binding energy means attraction between A and B. According to Sievert's law, the solubility can be given approximately by:
sffiffiffiffiffi P DST Esol exp ; S¼ P0 kT
(3)
where P and P0 represent the background pressure and the standard pressure, respectively, k is the Boltzmann constant and T is the absolute temperature, the solution entropy DS is equal to about 4:7k [35]. The diffusion coefficient of interstitial atoms in solid solutions can be evaluated by Arrhenius equation:
D ¼ D0 exp Em=kT ;
(4)
where D0 is usually assumed to be the temperature independent diffusion constant, Em is the activation energy which is equal to migration energy for the interstitial atomic diffusion. According to the interstitial atomic diffusion theory presented by Wert and Zener [36], D0 can be written as:
n 2 D0 ¼ l v; 6
(5)
where n ¼ 4 is a geometrical factor of the number of equivalent jump paths for H jumping along tetrahedral site to another tetrahedral site in bcc structure metal, l is the jump length which equals pffiffiffi to 2a 4, and a is the lattice constant. v is the vibration frequency, which can be approximately evaluated by: =
with the calculated result by the first-principle calculations [16]. The diffusion coefficient of H in W was evaluated using the harmonic transition state theory (hTST) [17], which is agreement with the experimental result by Frauenfelder [14] at temperature above 1500 K. However, the calculated result of diffusion coefficient by Heinola [17] are much higher than the experimental results below 1500 K, which had been measured by Zakharov [18], Benamati [19], Otsuka [20], Ikeda [21] and Hoshihira [22], respectively. Furthermore, there is a significant difference in migration barrier of H in W between Frauenfelder's [14] experimental value and Heinola's [17] calculated value. It is worthy of note that the migration barrier was calculated by Johnson's [6] along a higher barrier path via octahedral site, although the calculated value is in accordance with Frauenfelder's [14] experimental result. Nevertheless, if the Frauenfelder's [14] experimental data at temperatures above 1500 K is fitted, the migration barrier of H in W will decrease from 0.39 eV to 0.25 eV. Therefore, it is very possible that there is a significant influence of lattice defects on H diffusion even at temperature up to about 1500 K, and Kong [16], Fernandez [23] and Oda [24] have corroborated this by first-principle calculations and Kinetic Monte Carlo method, respectively. Thus Kong's [16] calculated results agree with those experimental results when the influence of temperature and defects on H diffusion in W are considered. Under neutron irradiation, many transmutation elements are produced in PFM. Among them, rhenium (Re) is one of the main products [25], and calculations by Cottrell [26] and Gilbert [27] suggest that after several years of exposure of fusion neutrons, the concentration of Re in W might reach several percent. According to the previous experimental and theoretical studies for WRe alloy, the concentration of Re can be increased to above 25% in a body-centered cubic (bcc) structure [28e30], and the bcc structure is dynamically stable based on Federer's [28] experimental study, as well as Ekman's [29] and Chouhan's [30] calculations. Although Golubeva's [31] experimental results show the Re concentration in WeRe alloy does not influence noticeably the deuterium (D) retention in the range of 1e10%, and Tyburska [32] investigated D retention in damaged W-Re alloy, but the influence of Re on solubility of H in W is still not clear. Moreover, the solubility of the extrapolation of Frauenfelder's [14] datas and Kong's [16] calculated result are much lower than Benamati's [19] data of D in W bel investigated Re doped with 5% Re at 850 Ke885 K. Recently, Wro precipitation in irradiated bcc W-Re alloy and the interactions of Re and vacancy using first-principle calculations and Monte Carlo simulations [33]. However, the interactions between H and vacancy in W-Re alloy were not studied. Therefore, in this paper, we pay attention to the influence of Re on solution and diffusion of H isotopes in W-Re alloy. We carry out a systematic first-principle calculations to investigate the solution energies, migration barriers and binding energies of Re-H for H in perfect bcc W and W-Re crystals. Furthermore, the solubility and diffusion coefficient of H in perfect W and W-Re alloy were evaluated using Sievert's law and hTST.
207
v¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Em 2ml2 ;
(6)
2. Computational methods The solution energies of H in W or W-Re alloy can be defined by:
Esol ¼ E½W=ðW þ ReÞ þ H E½W=ðW þ ReÞ 0:5EH2 ;
(1)
where E½W=ðW þ ReÞ þ H is the total energy of W or W-Re supercell with a single H atom, and E½W=ðW þ ReÞ is the reference energy of perfect W or W-Re supercell. EH2 is the total energy of H molecule, our calculated value is 6.744 eV. The binding energy of a H molecule is 4.516 eV, which agrees well with the experimental value 4.747 eV [34].
where m is the mass of a H atom. In addition, according to reference [37] the diffusion coefficient can also be calculated as:
1 2 D ¼ l G; 6
(7)
where G is the jump rate, based on Eyring's theory [38,39] and hTST [40], it can be calculated by:
208
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
0 B P3N1 @ i
! hvt i 2kT
exp
hvt i kT
1exp
kT G¼ h
0 B P3N i @
! exp
1exp
g i 2kT
hv
g hv i kT
1
!C A Em exp kT 1
(8)
!C A
where, vti , vgi are the real normal modes of vibration on the transition state and the ground state of the initial state, respectively, N is the number of the vibrating atoms. The first-principle calculations are carried out by Vienna Ab initio Simulation Package (VASP) with the projector augmented wave potential method [41] and generalized gradient approximation [42] of the Perdew Burke Ernzerhof functionals [43,44]. The six outermost electrons for W (5d56s1) atom are taken into account as valence electrons. The plane wave energy cutoff for calculations is 500 eV which is large enough for convergence of geometrical structures and total energies. During the all calculations, all the shape and size of the supercells can be changed, a quasi-Newton algorithm is used to relax the ions into their instantaneous groundstate, and the atomic coordinates are relaxed until the Hellmann-Feynman forces are less than 0.01 eV/Å. The all calculations are done using 4 4 4 supercells with 128 atoms. A 3 3 3 k-point mesh is sampled using the Monkhorst and Pack scheme [45] with a Methfessel-Paxton smearing using a broadening of 0.15 eV, which proves to be sufficient for energy convergence of less than 0.001 eV per atom. The migration barriers are calculated using the nudged elastic band (NEB) method [46,47]. The calculation convergence and parameters stay the same for the ground state calculations.
3. Results 3.1. Results of first-principle calculations There are several crystalline forms for W, and the most stable lattice structure is bcc structure. The calculated bcc lattice constant of 3.172 Å agrees well with the experimental value of 3.165 Å [28,48]. Three types of W-Re supercells, 127W-1Re with concentration of 0.078% Re, 120W-8Re with concentration of 6.25% Re, and 96W-32Re with concentration of 25% Re are considered. The bcc structure of W-Re alloy is dynamically stable when the atomic content of Re is less than 27% [28e30]. To obtain reliable results, the solution and diffusion of H in 127W-1Re and 120W-8Re lattices are mainly investigated by first-principle calculations, due to several percent of sigma-phase incorporation is possible for W-25%Re alloy [31]. As shown in Fig. 1, suppose Re atoms are distributed equably in W-Re lattice. The calculated result shows that the lattice constant of W-Re alloy would decrease from 3.172 Å to 3.155 Å with the increase of Re content from 0.78% to 25%, which in accordance with Federer's [28] experimental result. The experimental result shows the lattice constant of W-Re alloy decreases from 3.160 Å to 3.153 Å with the increase in Re content from 4% to 13% [28]. Furthermore, the presence of Re will decrease volume of supercell a little bit due to Re is the adjacent element of W. Hence, it would be reasonable to use the kind of lattice models to calculate the properties of H in perfect W-Re alloy. Fig. 2 shows the first, second, third and fourth nearest neighboring tetrahedral (Td) sites of a Re atom indicated as 1NN, 2NN, 3NN and 4NN, respectively, for H in W-Re lattice.
Similarly, the nearest neighboring octahedral (Oh) and diagonal (Dg) sites of a Re atom can also be represented by 1NN, 2NN, 3NN and 4NN, respectively. First of all, we calculated the basic properties of H solution and diffusion in pure W. As shown in Table 1, the energy difference between Oh and Td sites is 0.378 eV, thus the most stable interstitial site of H in W is the Td site, consistent with the other calculations [16,17,49]. The corresponding solution energy of H at Td site with Zero Point Energy (ZPE) corrections is 1.056 eV, which is in good agreement with the experimental value 1.04 eV [50], and the previous calculated values [6,23]. The migration barrier with ZPE corrections is 0.173 eV, in excellent agreement with Johnson's [6] Fernandez's [23] 0.17 eV. As can be seen from Table 2, the interstitial H at 1NN sites of 127W-1Re lattice still prefers the Td site. The solution energy of H at 1NN Td site of 127W-1Re lattice is 0.096 eV bigger than H in pure W. When ZPE corrections are taken into account, the solution energy of H at 1NN Td site of 127W-1Re lattice is 0.179 eV bigger than H in pure W, that is because the vibrational frequencies at the ground state of H in 127W-1Re lattice increase significantly compared to H in pure W. There are two possible paths for H from one Td site migrate to another adjacent Td site. According to the NEB calculations and the difference of solution energies between H at Td and Oh interstitial sites, the most probable migration path of H passes through the Dg interstitial site in W-Re lattice, which is similar to the migration path of H in pure W. The migration barrier that H via 1NN Td site to the adjacent 1NN Td site for H in 127W1Re lattice is smaller than H in pure W. The binding energy between two H atoms in the nearest neighboring Td site is negative for H in pure W. Similarly, for H in 127W-1Re lattice, the binding energy of H-H pair which two H atoms are located at the adjacent 1NN Td sites is also negative, which means the two H atoms are repulsive each other. The binding energies of Re-H pairs which a H atom is located at 1NN, 2NN, 3NN and 4NN Td sites of a Re atom are 0.096 eV, 0.041 eV, 0.022 eV and 0.011 eV, respectively. Hence, the Re atom cannot act directly a trapping site of H, consistent with the Kong's calculated result [16]. For H in 127W-1Re and 120W-8Re lattices, the solution energies and migration barriers between the two nearest Td sites are shown in Fig. 3 and Fig. 4. The solution energy of H at Td site of 127W-1Re lattice decreases slowly with the increase in distance of Re-H. With considering ZPE corrections, the solution energy of H at 4NN Td site of 127W-1Re lattice is 0.072 eV bigger than H in pure W. For H in 120W-8Re lattice, the solution energies of H at Td sites are bigger than H in pure W, which is similar to H in 127W-1Re lattice. Moreover, if the distance of Re-H is same, the solution energies are almost equal for H in 127W-1Re and 120W-8Re lattices. With or without ZPE corrections, the solution energies of H at 1NN Td site of 96W-32Re are 1.042 eV and 1.243 eV, respectively, which are close to the solution energies of H at 1NN Td site of 120W-8Re. Thus, although the solution energy would tend to increase slightly with the rising of concentration of Re, but which does not influence noticeably the solution energy of H in W-Re alloy. Overall, in the presence of Re, the solution energy of H at Td sites will increase compared to H in pure W. Without considering ZPE corrections, the presence of Re would decrease the migration barrier between the two nearest Td sites slightly. With ZPE corrections, the migration barriers of H in 127W-1Re and 120W-8Re lattices are around 0.18 eV, 0.17 eV, respectively, which are close to the migration barrier of H in pure W. It can be seen from Fig. 5, for H in 127W-1Re and 120W-8Re lattices, the real normal modes of vibration on the ground state and the transition state increase significantly, compared to H in pure W, and if the distance of Re-H is same, the real normal modes of vibration are almost equal for H in 127W-1Re and 120W-8Re lattices.
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
209
Fig. 1. The lattice models of 120W-8Re and 96W-32Re supercells. The gray and red balls represent W and Re atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The first, second, third and fourth nearest neighboring Td interstitial sites of a Re atom indicated as 1NN, 2NN, 3NN and 4NN, respectively, for H in W-Re lattice. The gray balls represent W atoms, the red ball represents Re atom. The green, blue, pink and yellow balls represent the H atoms at 1NN, 2NN, 3NN and 4NN Td interstitial sites, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Solubility and diffusion coefficient of hydrogen isotopes in tungsten and tungsten-rhenium alloy According to Sievert's law, the solubility can be estimated by
Table 1 Td (eV) and EOh (eV) are the The properties of H solution and diffusion in pure W. Esol sol solution energies of H at Td and Oh sites, respectively. Em (eV) is the migration barrier, vgi (THz) and vti (THz) are the real normal modes of vibration on the ground state and the transition state, respectively. EbHH (eV) is the binding energy between two H atoms in the nearest neighboring Td site. This work
With ZPE
0.924
1.056
0.95,a 0.89b
Oh Esol Em
1.302
1.324
1.33,a 1.29b
0.203 46.771
vti
64.737
EbHH a b
Ref. [17]. Ref. [16].
0.461
0.173
Table 2 The properties of solution and diffusion of H at 1NN interstitial sites for 127W-1Re Td (eV), E Oh (eV) and E Dg (eV) are the solution energies of H at Td, Oh and Dg lattice. Esol sol sol g sites, respectively. Em (eV) is the migration barrier, vi (THz) and vti (THz) are the real normal modes of vibration on the ground state and the transition state, respectively. EbHH (eV) is the binding energy between two H atoms in the nearest neighboring Td site, and EbReH (eV) is the binding energy between Re and H at the 1NN Td site. This work
With ZPE
Td Esol Oh Esol Dg Esol
1.020
1.235
1.242
1.394
1.138
1.466
Em
0.168
vgi
0.189 62.223
vti
70.081
Other
Td Esol
vgi
combined with Eq. (3) and the above solution energies. To make a better comparison with the experiment, the background pressure of Frauenfelder's [14] experiment 79993 Pa (600 Torr) is used in the calculation. As shown in Fig. 6, the solubility of H in pure W is in excellent agreement with Frauenfelder's [14] experimental data, especially when ZPE corrections are considered. This is due to the solution energy of H in pure W with ZPE corrections agrees well with the experimental value. Besides, the defects cannot effectively trap H at the high temperature range [16]. Benamati [19] measured the D inventory parameters in W-5%Re, the experimental background pressure is 75600 Pa. As can be seen from Fig. 6, the Benamati's [19] experimental data is much higher than the calculated results, despite the presence of Re would increase the solution energy and Re cannot trap H. However, for the perfect lattice, it can be deducted from the calculated results that the presence of Re would decrease the solubility compared to H in pure W due to the increase in solution energy. Based on hTST, the diffusion coefficient can be evaluated by Eqs. (7) and (8) combined with the migration barrier and the real modes of vibration on the ground state and the transition state. Here, in order to compare the influence of Re on diffusivity, the diffusion coefficient of H in pure W is also calculated. As shown in Fig. 7, for the diffusion coefficient of H in pure W, the calculated result from this paper agrees Heinola's [17] calculation. There is discrepancy between the calculated results with or without ZPE corrections and the experimental datas. It has been proved that the temperature and defect trapping effects play an important role in diffusion of H in pure W [16]. As mentioned, the presence of Re would decrease
a
b
0.2, 0.2 46.7,a 46.0b 63.3,a 62.3b
EbHH EbReH
0.313
0.229
0.096
0.179
210
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212
Fig. 3. The solution energy as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices. Fig. 5. The real normal modes of vibration on the ground state and the transition state as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices.
Fig. 4. The migration barrier as a function of the neighboring Td site of a Re atom, for H in 127W-1Re and 120W-8Re lattices.
the migration barrier without ZPE corrections slightly, and significantly increase the real normal modes of vibration on the ground state and the transition state. The concentration of Re for 120W-8Re lattice is 6.25%, which is close to the concentration of Re of Benamati's [19] experimental sample. Therefore, the diffusion parameters of H in 120W-8Re lattice are used to evaluate the diffusion coefficient of H in perfect W-Re alloy. It can be seen from Fig. 7, although Benamati's [19] experimental data is much smaller than the calculated results, but for the perfect crystal, the calculated results show that the presence of Re increases the diffusivity slightly. 4. Discussion The solubility and diffusion coefficient are crucial parameters for W-based PFMs, which are related to H isotopes transport. Hence, there is need to study the solution and diffusion of H isotopes in W and W-Re alloy. The calculated solubility of H in pure W agrees very well with Frauenfelder's [14] experimental data. The solution energies of D and T in pure W are calculated to be 1.095 eV and 1.113 eV, respectively. Similarly, the solution energies of D, T at 1NN Td sites of 127W-1Re lattice are 1.274 eV, 1.292 eV, respectively.
Thus according to Eq. (3), the solubility of H is maximum for H isotopes in W, which is in accordance with Esteban's [51] experimental result. The concentration of Re does not influence noticeably the solubility of H in W-Re alloy, due to the change of solution energy with the increase in concentration of Re is quite small, which is in accordance with Golubeva's [31] experimental result. However, the calculated solubility of H in 120W-8Re lattice is much lower than Benamati's [19] experimental data. The data was obtained by using the W-5%Re sample in the temperature range 850 Ke885 K. In addition, Golubeva's [31] experimental results show the thermal desorption spectroscopy (TDS) of W-Re alloy contains an additional peak at 850 K, which does not appear in TDS of pure W. The experimental W-Re samples contain some defects, such as there are voids exist in the bulk for all W-Re alloys [31]. Furthermore, the recent calculations show that the vacancy-Re clusters remain stable at temperature above 800 K, and solute segregation is found to occur in the form of voids decorated by Re atoms in the low Re concentration range [33]. A single Re atom can raise the total binding energies between two or three vacancies obviously [52]. However, the two vacancies repulse each other even where they are situated as far as in fifth nearest neighbor position in pure W [53]. Therefore, although Re atom cannot act directly as a trapping site of H, it is very possible that the presence of Re can increase the solubility of H in W-Re alloy by the interactions of Re with defects. Consequently, for the solubility of H isotopes in W-Re alloy, the Benamati's [19] experimental data is much higher than the calculated result of H in perfect W-Re alloy. The diffusion coefficient also can be evaluated by Wert and Zener model of Eqs. (4)e(6), which suggests the diffusion coefficient is proportional to the reciprocal square root of mass. Thus the diffusivity decreases with the increase in the mass of H isotopes, which has been proved by the experiment [51]. According to hTST, the diffusion coefficient of H in perfect crystal is calculated. That is, for H in perfect W or W-Re lattice, it is decided by the diffusion of a single H atom from a Td site to another site. Nevertheless, usually there are some defects exist in the experimental samples. Therefore, to compare the experiment and calculation, the influence of defects on the diffusion of H isotopes should be taken into account. For instance, the effects of defect trapping play a key role for diffusivity of H in pure W in the low temperature range, below about 1500 K, which have been verified by Kong [16], Fernandez
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the ground state and the transition state. Although the solution energy would tend to increase slightly with the rising of concentration of Re, but which does not influence noticeably the solution energy of H in W-Re alloy. Overall, in the presence of Re, the solution energies of H at Td sites will increase compared to H in pure W. Without ZPE corrections, the presence of Re in W would decrease slightly the migration barrier. It is found that the binding energy of H-H pair with or without ZPE corrections is also negative in presence of Re, and Re atom cannot act directly as a trapping site of H. According to Sievert's law and hTST, the solubility and diffusion coefficient of H in perfect W and W-Re alloy have been estimated. The solubility of H in W agrees well the experimental result, and the diffusivity of H in W is in accordance with the experimental data at high temperature. For H in perfect W and W-Re crystals, the results show the presence of Re would decrease the solubility and increase the diffusivity. The discrepancy between the calculated and experimental results has been discussed, and the possible reasons have been analyzed in this paper. Acknowledgments Fig. 6. The solubility as a function of reciprocal temperature. Datas are from Frauenfelder [14], Benamati [19], and Kong [16].
This work is supported by the National Natural Science Foundation of China (Grant Nos. 51371195, 11575289). References
Fig. 7. The diffusion coefficient as a function of reciprocal temperature. Datas are from Frauenfelder [14], Benamati [19], Zakharov [18], Kong [16] and Heinola [17].
[23] and Oda [24]. Similarly, for the diffusivity of H in W-Re alloy, it can be deducted that defects would affect the diffusivity. Besides, recent calculations [33,52] show that the presence of Re can facilitate the formations of voids and vacancy clusters in W. Therefore, it is possible that the interactions of Re, H and vacancy would affect the diffusion coefficient of H in W-Re alloy significantly. 5. Conclusions In the present work, the properties of solution and diffusion of H in perfect W and W-Re alloy have been studied by first-principle calculations. The lattice constant of W-Re alloy decreases with the increase in concentration of Re, which agrees with the experimental result. In W-Re lattice, the interaction between Re atom and H atom is repulsive, and the interstitial H still prefers the Td site. The presence of Re in W increases the solution energy with or without ZPE corrections and the real normal modes of vibration on
[1] G. Federici, C.H. Skinner, J.N. Brooks, et al., Plasma-material interactions in current tokamaks and their implications for next step fusion reactors, Nucl. Fusion 41 (2001) 1967e2137. [2] S. Wurster, N. Baluc, M. Battabyal, et al., Recent progress in R&D on tungsten alloys for divertor structural and plasma facing materials, J. Nucl. Mater 442 (2013) S181eS189. [3] Y. Ueda, K. Tobita, Y. Katoh, PSI issues at plasma facing surfaces of blankets in fusion reactors, J. Nucl. Mater 313e316 (2003) 32e41. ^ne, S. Garcia-Argote, et al., Tritium absorption/desorp[4] A. El-Kharbachi, J. Che tion in ITER-like tungsten particles, Int. J. Hydrogen Energ 39 (2014) 10525e10536. [5] C. Grisolia, E. Hodille, J. Chene, et al., Tritium absorption and desorption in ITER relevant materials: comparative study of tungsten dust and massive samples, J. Nucl. Mater 463 (2015) 885e888. [6] D.F. Johnson, E.A. Carter, Hydrogen in tungsten: absorption, diffusion, vacancy trapping, and decohesion, J. Mater. Res. 25 (2010) 315e327. [7] J. Roth, E. Tsitrone, A. Loarte, et al., Recent analysis of key plasma wall interactions issues for ITER, J. Nucl. Mater 390e391 (2009) 1e9. [8] J.P. Hirth, Effects of hydrogen on the properties of iron and steel, Metall. Mater. Trans. A 11 (1980) 861e890. [9] N. Eliaz, A. Shachar, B. Tal, et al., Characteristics of hydrogen embrittlement, stress corrosion cracking and tempered martensite embrittlement in highstrength steels, Eng. Fail. Anal. 9 (2002) 167e184. [10] M. Nagumo, M. Nakamura, K. Takai, Hydrogen thermal desorption relevant to delayed-fracture susceptibility of high-strength steels, Metall. Mater. Trans. A 32 (2001) 339e347. [11] E. Hayward, C.C. Fu, Interplay between hydrogen and vacancies in a -Fe, Phys. Rev. B 87 (2014) 104103. [12] P. Gumbsch, Controlling factors for the brittle-to-ductile transition in tungsten single crystals, Science 282 (1998) 1293. [13] T. Tanabe, Review of hydrogen retention in tungsten, Phys. Scr. T159 (2014) 827e847. [14] R. Frauenfelder, Permeation of hydrogen through tungsten and molybdenum, J. Chem. Phys. 48 (1968) 3955e3965. [15] A.A. Mazayev, R.G. Avarbe, Y.N. Vilk, et al., Solubility of hydrogen in tungsten at high temperature and pressures 25 (1968) 267e270. [16] X.S. Kong, S. Wang, X. Wu, et al., First-principles calculations of hydrogen solution and diffusion in tungsten: temperature and defect-trapping effects, Acta Mater. 84 (2015) 426e435. [17] K. Heinola, T. Ahlgren, Diffusion of hydrogen in bcc tungsten studied with first principle calculations, J. Appl. Phys. 107 (2010) 113531, 113531-8. Evko, Hydrogen permeability of poly[18] A.P. Zakharov, V.M. Sharapov, E.I. crystalline and monocrystalline molybdenum and tungsten, Mater. Sci. 9 (1975) 149e153. [19] G. Benamati, E. Serra, C.H. Wu, Hydrogen and deuterium transport and inventory parameters through W and W-alloys for fusion reactor applications, J. Nucl. Mater. 283 (2000) 1033e1037. [20] T. Otsuka, T. Hoshihira, T. Tanabe, Visualization of hydrogen depth profile by means of tritium imaging plate technique: determination of hydrogen diffusion coefficient in pure tungsten, Phys. Scr. 138 (2009) 014052. [21] T. Ikeda, T. Otsuka, T. Tanabe, Determination of hydrogen diffusivity and
212
[22]
[23]
[24]
[25]
[26] [27] [28] [29] [30]
[31] [32] [33]
[34] [35] [36] [37]
F. Ren et al. / Journal of Nuclear Materials 491 (2017) 206e212 permeability in W near room temperature applying a tritium tracer technique, J. Nucl. Mater. 415 (2011) S684eS687. T. Hoshihira, T. Otsuka, R. Wakabayashi, et al., Hydrogen behavior near surface regions in Mo and W studied by tritium tracer technique, J. Nucl. Mater. 417 (2011) 559e563. N. Fernandez, Y. Ferro, D. Kato, Hydrogen diffusion and vacancies formation in tungsten: density Functional Theory calculations and statistical models, Acta Mater. 94 (2015) 307e318. T. Oda, D. Zhu, Y. Watanabe, Kinetic Monte Carlo simulation on influence of vacancy on hydrogen diffusivity in tungsten, J. Nucl. Mater. 467 (2015) 439e447. M.R. Gilbert, S.L. Dudarev, S. Zheng, et al., An integrated model for materials in a fusion power plant: transmutation, gas production, and helium embrittlement under neutron irradiation, Nucl. Fusion 52 (2012) 083019. G.A. Cottrell, Sigma phase formation in irradiated tungsten, tantalum and molybdenum in a fusion power plant, J. Nucl. Mater. 334 (2004) 166e168. M.R. Gilbert, J.C. Sublet, Neutron-induced transmutation effects in W and Walloys in a fusion environment, Nucl. Fusion 51 (2011) 043005. J.I. Federer, R.M. Steele, Identification of a beta-tungsten phase in tungstenrhenium alloys, Nat 205 (1965) 587e588. M. Ekman, K. Persson, G. Grimvall, Phase diagram and lattice instability in tungsten-rhenium alloys, J. Nucl. Mater. 278 (2000) 273e276. R.K. Chouhan, A. Alam, S. Ghosh, et al., Ab initio study of the phonon spectrum, entropy and lattice heat capacity of disordered Re-W alloys, J. Phys. Condens. Matter 24 (2012) 375401. A.V. Golubeva, M. Mayer, J. Roth, et al., Deuterium retention in rheniumdoped tungsten, J. Nucl. Mater. 363 (2007) 893e897. B. Tyburska Püschel, V.K. Alimov, On the reduction of deuterium retention in damaged Re-doped W, Nucl. Fusion 53 (2013) 8e10. J. Wrobel, D. Nguyen-Manh, S. Dudarev, et al., A first-principles model for anomalous segregation in dilute ternary tungsten-rhenium-vacancy alloys, J. Phys. Condens. Matter 29 (2017) 145403. K.P. Huber, G. Hertzberg, Molecular Spectra and Molecular Structure. IV: Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. Y. Fukai, The Metal-hydrogen System, Springer, Berlin, 1993. C. Wert, C. Zener, Interstitial atomic diffusion coefficients, Phys. Rev. 76 (1949) 1169e1175. J. Philibert, Atom movements: diffusion and mass transport in solids, France,
Les editions de Physique, 1991. [38] H. Eyring, The activated complex in chemical reactions, J. Chem. Phys. 3 (1934) 107e115. [39] H. Eyring, W.U. Dan, The theory of absolute reaction rates in solution, Trans. Faraday Soc. 34 (1938) 41e48. [40] E. Wigner, The transition state method, Trans. Faraday Soc. 34 (1938) 29e41. € chl, Projector augmented-wave method, Phys. Rev. B 50 (1994) [41] P.E. Blo 17953e17979. [42] J.P. Perdew, J.A. Chevary, S.H. Vosko, et al., Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B 48 (1993) 4978, 4978. [43] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 78 (1997) 1396. [44] Y. Zhang, W. Yang, Comment on “generalized gradient approximation made simple”, Phys. Rev. Lett. 4 (1997) 890. [45] H.J. Monkhorst, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188e5192. nsson H, A climbing image nudged elastic [46] G. Henkelman, B.P. Uberuaga, H. Jo band method for finding saddle points and minimum energy paths, J. Chem. Phys. 113 (2000) 9901e9904. nsson, Improved tangent estimate in the nudged elastic [47] G. Henkelman, H. Jo band method for finding minimum energy paths and saddle points, J. Chem. Phys. 113 (2000) 9978e9985. [48] E. Lassner, W.D. Schubert, Tungsten: Properties, Chemistry, Technology of the Element, Alloys, and Chemical Compounds, Springer, Berlin, 2011. [49] K. Ohsawa, F. Nakamori, Y. Hatano, et al., Thermodynamics of hydrogeninduced superabundant vacancy in tungsten, J. Nucl. Mater. 458 (2014) 187e197. [50] R. Frauenfelder, Solution and diffusion of hydrogen in tungsten, J. Vac. Sci. Technol. 6 (1969) 388e397. [51] G.A. Esteban, A. Perujo, L.A. Sedano, et al., Hydrogen isotope diffusive transport parameters in pure polycrystalline tungsten, J. Nucl. Mater. 295 (2001) 49e56. [52] Y.H. Li, H.B. Zhou, S. Jin, et al., Behaviors of transmutation elements Re and Os and their effects on energetics and clustering of vacancy and self-interstitial atoms in W, Nucl. Fusion 57 (2017) 046006. [53] C.S. Becquart, C. Domain, Ab initio calculations about intrinsic point defects and He in W, Nucl. Instrum. Meth. Phys. Res. B 225 (2007) 23e26.