First-principles calculations of hydrogen solution and diffusion in tungsten: Temperature and defect-trapping effects

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ScienceDirect Acta Materialia 84 (2015) 426–435 www.elsevier.com/locate/actamat

First-principles calculations of hydrogen solution and diffusion in tungsten: Temperature and defect-trapping effects ⇑

Xiang-Shan Kong,a Sheng Wang,a Xuebang Wu,a Yu-Wei You,a C.S. Liu,a, Q.F. Fang,a Jun-Ling Chenb and G.-N. Luob a

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, PO Box 1129, Hefei 230031, People’s Republic of China b Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, People’s Republic of China Received 17 June 2014; revised 23 September 2014; accepted 19 October 2014

Abstract—The solubility and diffusivity of hydrogen in tungsten are fundamental and essential factors in the application of this tungsten as a plasmafacing material, but data are scarce and largely scattered, indicating some important factors might be missed. We perform a series of first-principles calculations to predict the dissolution and diffusion properties of interstitial hydrogen in tungsten and the influence of temperature and the defecttrapping effect. The interstitial hydrogen always prefers the tetrahedral site over the temperature range 300–2700 K, and its migration path mainly advances via the nearest-neighboring tetrahedral sites. Our results reveal that both solution and activation energies are strongly temperature dependent. The predicted solubility and diffusivity show good agreement with the experimental data above 1500 K, but present a large difference below 1500 K, which can be bridged by the trapping effects of vacancies and natural trap sites. The present study reveals a dramatic effect of temperature and defect trapping on the hydrogen solution and diffusion properties in tungsten, and provides a sound explanation for the large scatter in the reported values of hydrogen diffusivity in tungsten. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Tungsten; Hydrogen solubility and diffusivity; Temperature and defect-trapping effects; First-principles calculations

1. Introduction Tungsten is the primary candidate for the plasma-facing material (PFM) of fusion reactors due to its low tritium retention, high melting point, high thermal conductivity and low sputtering yield for light elements [1–4]. In its role as a PFM, tungsten would be subject to high heat and particle flux of hydrogen escaping from the plasma, which would congregate on the tungsten surface or penetrate through it and diffuse deeper into the bulk. Hydrogen retention in tungsten leads to modification of the material’s physical and mechanical properties, such as assisting superabundant vacancy formation, inducing embrittlement, and reducing mechanical strength. More importantly, hydrogen retention in PFMs has been identified as a major safety and economic concern in fusion reactors [5]. Thus, the investigation of hydrogen retention in tungsten is an important task for fusion research. To study hydrogen retention in tungsten, knowledge of hydrogen solution and diffusion is fundamental and essential. The solubility and diffusivity play a key role in determining the recombination rate

⇑ Corresponding author. Tel.: +86 551 65591062; e-mail: csliu@issp. ac.cn

coefficient and should be directly associated with hydrogen trapping and bubble formation. In addition, they are also used as input in many experimental and computational analyses on recycling and retention of hydrogen in tungsten. Without reliable knowledge of the solubility and diffusivity, reliable predication of hydrogen recycling and retention in tungsten is hardly possible. Data for hydrogen solubility are scarce. Frauenfelder measured the solubility of hydrogen in 99.95% pure tungsten for the temperature range 1100–2400 K, which was given by S ¼ 9:3  103 expð1:04 eV=kTÞ H =ðW atm1=2 Þ (here, the solubility S is expressed in terms of the hydrogen/tungsten atomic fraction per square root of the hydrogen pressure on the tungsten in atmospheres) [6]. This agrees well with Mazayev’s measured data covering the temperature range 1900–2400 K [7]. The extrapolation of the data to lower temperatures is in fair agreement with Zakharov’s [8] (910 K < T < 1060 K) and Ikeda’s [9] (298 K < T < 353 K) data evaluated from their diffusivity and permeability data. Although the absolute values of the solubility among these reported data are within the same order of magnitude, their pre-exponential factors and activation energies are completely different. For example, the activation energies are 1.04, 0.03 and 0.19 eV for Frauenfelder [6], Zakharov [8] and Ikeda [9], respectively.

http://dx.doi.org/10.1016/j.actamat.2014.10.039 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

X.-S. Kong et al. / Acta Materialia 84 (2015) 426–435

In addition, Benamati et al. [10] used their permeation data for hydrogen through tungsten with 5% rhenium to approximate the solubility and obtained a constant value (6  107 H =ðM atm1=2 Þ) over a limited temperature range of 850–885 K, which is much higher than the extrapolation of Frauenfelder’s data [6] to this temperature range. As with the solubility of hydrogen in tungsten, data on the diffusion coefficient of hydrogen in tungsten are also very limited. To this end, some experimental studies have been conducted to measure the diffusion coefficient of hydrogen in tungsten. Some earlier investigations on the hydrogen diffusivity at higher temperatures have been carried out by Moore et al. [11] (D ¼ 7:25  108 exp ð1:80 eV=kTÞ m2/s, 1200 K < T < 2500 K), Ryabchikov [12] (D ¼ 8:1  106 expð0:86 eV=kTÞ m2 =s, 1900 K < T < 2400 K), Frauenfelder [6] (D ¼ 4:1  107 expð0:39 eV =kTÞ m2/s, 1100 K < T < 2400 K), Zakharov [8] (D ¼ 6 104 expð1:07 eV=kTÞ m2/s, 910 K < T < 1060 K) and Benamati [10] (about 1:5  1010 m2/s, 850 K < T < 885 K) based on hydrogen degassing and permeation experiments, which are indirect measurements of hydrogen concentrations. Recently, the hydrogen diffusivity at lower temperatures has been measured by the tritium tracer technique in some experimental works, e.g. Otsuka et al. [13] (D ¼ 3  107 expð0:39 eV=kTÞ m2/s, 473 K < T < 673 K), lkeda et al. [9] (D ¼ 3:42  109 expð0:39 eV=kTÞ m2/s, 298 K < T < 353 K) and Hoshihira et al. [14] (D ¼ 4:3 109 expð0:39 eV=kTÞ m2/s, 293 K < T < 323 K). Taking literature data at higher temperatures and near room temperature into account, Ikeda and Otsuka [15] proposed a new diffusion coefficient (D ¼ 3:8  107 expð0:41 eV =kTÞ m2/s) as the most reliable hydrogen diffusivity, which is valuable for a wide temperature range of 250–2500 K. As summarized above, although various parameters for hydrogen solubility and diffusivity in tungsten have been recommended, few experimental values have been reported and moreover these parameters show large discrepancies. This is particularly true for the values of solution and activation energies. These discrepancies arise from the use of different techniques to obtain the data. In addition, the extremely low solubility of hydrogen in tungsten and the significant surface and trapping effects make the experimental measurements difficult and complex, particularly at low temperatures. At present, it is very hard to give a reliable value for the solubility and diffusivity in tungsten, and data for this is urgently required. As a result of progress in first-principles computational methods, it is now possible to calculate the solubility and diffusivity of foreign interstitial atoms in metals at a level of accuracy close to and sometimes better than available from experiments; examples include hydrogen and oxygen in nickel [16–18], oxygen in body-centered cubic (bcc) iron [19], carbon in palladium and palladium alloys [20], hydrogen in austenitic high-manganese steels [21], and so on. For these reasons, several researchers have used atomic calculations to investigate the solution and diffusion properties of hydrogen in tungsten. For example, Heinola et al. [22] and Liu et al. [23] have predicted the hydrogen diffusivity in tungsten as D ¼ 5:2  108 expð0:21 eV=kTÞ m2/s and D ¼ 1:57  107 expð0:19 eV=kTÞ m2/s, respectively. These two results are roughly comparable with each other. The slight difference between them may result from the different methods adopted for the jump rate calculation. Unfortunately, both of these diffusivities show large differences from the experimental values, particularly at

427

low temperatures, suggesting that some important factors might have been missed in these two first-principles calculations. The temperature effect may be one such omission, as this has been demonstrated to be an important factor for an accurate interpretation of experimental solution and diffusion data using first-principles computational methods [16,18–20], but was not considered in either Heinola et al.’s [22] or Liu et al.’s [23] works. According to the result reported by Dubrovinsky et al. [24], the thermal expansion of tungsten is calculated to be about 1.21%, 2.66% and 4.50% for temperatures of 298, 1205 and 2774 K, respectively. Recent results for the hydrogen behavior in tungsten under hydrostatic strain show that the volume expansion can change the solution and diffusion properties of hydrogen in tungsten [25,26]. Therefore, it can be expected that the temperature effect would affect hydrogen solution and diffusion properties in tungsten. Besides the temperature effect, the defect-trapping effect may also contributes to the large difference of the reported hydrogen solubility and diffusivity data, which has been found to significantly affect the hydrogen solution and diffusion behaviors in metals [27–30]. In tungsten, much work has demonstrated that lattice defects, e.g. vacancies, dislocations, impurities and grain boundaries, can trap multi-hydrogen atoms and impede hydrogen diffusion [31–40]. Until now, the defect-trapping effect has also been used by some researchers to explain discrepancies between the hydrogen diffusivity data in tungsten reported by themselves and by others [10,22]. However, little work has been done to quantify the effect of microstructural defect-trapping on hydrogen transport in tungsten and tungsten alloys. In this paper, we carry out a systematic first-principles calculation to investigate the temperature and defecttrapping effects on the dissolution and diffusion properties of interstitial hydrogen in tungsten. Here, the temperature effect is taken into account by the thermal expansion and vibration free-energy contribution, and the defect-trapping effect is described through the modified Fick’s equation according to Mac-Nabb and Foster [41]. Using the first-principles calculated results, reliable solubility and diffusivity data are predicted according to Sievert’s law and transition state theory, respectively. Together with the former reported hydrogen solution and diffusion results, our results reveal a dramatic effect of temperature and defect-trapping on the predicted hydrogen solution and diffusion properties in tungsten, and provide a sound explanation for the large scatter of the above-mentioned data on hydrogen diffusion in tungsten. 2. Computation method The solution energy is the energy required to place a hydrogen atom at a certain interstitial site f in the tungsten. It is defined by: WH f ESol  ðEW þ EH Þ; f ¼ E WH f

ð1Þ

where E is the total energy of the tungsten lattice with a single hydrogen atom at site f. The reference is given by the total energy per supercell of the perfect tungsten lattice (EW ) and the energy of a single hydrogen atom outside the tungsten (EH ¼ 3:40 eV, half the energy of a hydrogen molecule, which corresponds to the binding energy of 4.56 eV for a hydrogen molecule, consistent with experimental findings [42]).

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Sievert’s law states that, for low concentrations, the hydrogen solubility S f for a certain type of site f is, to a good approximation, given by: ! pffiffiffi DST  ESol P f S f ¼ pffiffiffiffiffi exp : ð2Þ kT P0 Here, P and P 0 are the background pressure and the reference pressure (we choose standard pressure in order to make a comparison with the experiment), respectively, k is the Boltzmann constant and T is the temperature. DS is the entropy of solution, equal to about 4:7k [43], which is almost negligible. Diffusion of interstitial solute atoms in solid solutions can be described as random jumps between interstitial positions. The diffusion coefficient can be written as [44]: 1 D ¼ k2 C; ð3Þ 6 where k is the jump length, and C denotes the jump rate between adjacent sites of the diffusing particle. In terms of the diffusion theory from Wert and Zener [45], the jump rate (denoted by WZ) can be written as: rffiffiffiffiffiffiffiffiffiffiffi   Eact Eact ; ð4Þ C¼n exp kT 2mk2 where n is the number of the nearest-neighbor interstitial positions, and m is the mass of a hydrogen atom. Eact is the activation energy, defined as the difference in the minimum energy of the transition state (ETS ) and of the initial state (EIS ), i.e. Eact ¼ ETS  EIS :

ð5Þ

Based on Eyring’s theory [46,47] of an activated complex within transition state theory [48,49], the jump rate (denoted by hTST) can also be calculated by:   kT Z TS Eact : ð6Þ C¼ exp h Z IS kT The Z TS and Z IS are partition functions for the transition state and the ground state of the initial state, respectively. Within the framework of the harmonic approximation, the quantum mechanical partition functions can be written as:  z hm Q3N 1 exp 2kTi   z i¼1 hm TS 1exp kT i Z ¼ : ð7Þ i Q3N expðht Z IS 2kT Þ i¼1 1exp hmi ð kT Þ where mzi and mi 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. It is noteworthy that the potential energy has negative curvature along the reaction coordinate at the saddle point yielding an imaginary vibration mode, m . Thus there is one real normal mode less at the saddle point than at the ground state of the initial state. The temperature effect is taken into account by the thermal expansion and vibration free-energy contributions. To investigate the thermal expansion effect, the temperaturedependent lattice constant is first defined. Lu et al. [50] have determined the thermal expansion coefficients of tungsten numerically, and these agree reasonably well with the

reported experimental data. Here, we use the analytical form proposed by Lu et al. to correlate the temperature with the lattice parameter value. However, these lattice constants (aLu T ) cannot be directly used to interpret our density functional theory (DFT) calculations because of the small difference between the DFT optimized lattice constant at 0 K and the experimental observation. Similarly to Ref. [20], we calculate the relevant lattice constant (aT ) to use in applying Lu our DFT data by multiplying the ratio aLu T =a0 with the DFT-optimized lattice constant (see Table 1). In the quasi-harmonic approximation, the total free energy of a system A with a given lattice constant aT at temperature T is calculated using the equation: F A ðT Þ ¼ EAc ðaT Þ þ F Am ðaT Þ;

ð8Þ

EAc ðaT Þ

where is the total static energy, obtained directly from first-principles calculations as the ground-state energy, and F Am ðaT Þ is the vibration free energy, which can be expressed as: F Am ðaT Þ ¼ Ri hmAi ðaT Þ=2 þ k B Tlnð1  expðhmAi ðaT Þ=kT ÞÞ;

ð9Þ

mAi ðaT Þ

where are the vibration modes of the system A. Thus, the temperature effect on the solution and activation energy can be evaluated by replacing the total energy in Eqs. (1) and (5) by the corresponding total free energy at a given temperature T. The energy in the above expressions is computed by DFT with the generalized gradient approximation [51] and projector augmented wave potentials [52], as implemented in the Vienna Ab initio Simulation Package (VASP). A supercell composed of 54 lattice points (3  3  3) is used. In all the calculations, the volume of the lattice is kept constant and all atoms are free to move during the relaxation. Following a series of test calculations a plane-wave cutoff of 500 eV and a k-point grid density of 5  5  5 are employed. The structural optimization is truncated when the force converges to less than 0.01 eV/ ˚ . Additionally, all transition states and activation energies A of diffusion paths are calculated by the nudged elastic band method associated with the climbing image (cNEB) [53,54]. The vibration frequencies of the system are calculated by assuming that localized vibrations of the interstitial hydrogen atom and its neighboring tungsten atoms are decoupled from vibrations of other metal atoms, i.e. only the interstitial hydrogen atom and its neighboring tungsten atoms are allowed to vibrate (marked by red large sphere in Fig. 1).

3. Results and discussion 3.1. Verification of the calculations To verify the accuracy of our calculations, we first calculate the basic properties of hydrogen solution and diffusion in tungsten at 0 K with an optimized lattice parameter of ˚ . Two possible interstitial sites, i.e. tetrahedral 3.177 A (Tet) and octahedral (Oct), are considered (see Fig. 1). As shown in Table 2, the hydrogen atom prefers to occupy the tetrahedral interstitial site rather than the octahedral interstitial site. The positive solution energy means that dissolution of hydrogen in tungsten is an endothermic process, consistent with the experimental observations [55,56]. The activation energy for hydrogen hopping between two

X.-S. Kong et al. / Acta Materialia 84 (2015) 426–435 Table 1. The relevant lattice constant (aT ) and their corresponding lattice expansion ratios (DaT =a0 ). ˚) Temperature (K) aT (A DaT =a0 ð%Þ 0 300 600 900 1200 1500 1800 2100 2400 2700

3.177 3.180 3.184 3.189 3.193 3.198 3.204 3.210 3.216 3.223

0.000 0.096 0.235 0.378 0.526 0.684 0.855 1.043 1.248 1.474

Fig. 1. The interstitial sites and their tungsten neighbor atoms. The tetrahedral and octahedral interstitial sites are represented by the small white and red spheres, respectively. The tungsten atoms are represented by the large sphere. Among these, the red large spheres denote the tungsten atoms, which are allowed to vibrate in the frequency calculation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

adjacent tetrahedral interstitial sites is 0.22 eV. In order to make a comparison with previous theoretical data [22], vibration frequencies of the ground and transition states are also calculated in the harmonic approximation by assuming that localized vibrations of hydrogen atoms are decoupled from vibrations of the metal atoms. The calculated three vibration modes of hydrogen at the ground state have two degenerate frequencies and one smaller frequency. At the transition state, there are two real normal modes and one imaginary mode referring to the negative curvature of the saddle point in the direction of the reaction path. It can be clearly seen from Table 1 that our calculated results are in good agreement with the previously reported theoretical data [22,57,58]. In addition, the small energy barrier suggests a shallow potential energy surface between the interstitial sites, which could lead to strong anharmonic effects given the significant value of zero point energy. Therefore, the energy hypersurface is explored in the form of constrained geometry optimizations, i.e. by placing the hydrogen atom at the potentially relevant interstitial sites and by relaxing the tungsten atoms, to verify the reasonableness of the harmonic approximation for the hydrogen atom in the interstitial sites. It can be clearly seen in Fig. 2a that the zero-point energies are mostly nearly zero and much lower than their corresponding static energies, indicating that the anharmonic effects are very weak.

429

Therefore, the harmonic approximation of the hydrogen atom in the interstitial sites is justified. Eyring’s transition state theory assumes that the diffusing atom in the transition state is in a quasi-thermodynamic equilibrium with the lattice. Due to the large difference in vibrational frequencies of the light hydrogen atom compared with the relatively low-frequency phonon modes of the heavy tungsten atoms, the approximation in the vibrational calculations with restricted hydrogen and nearest-neighbor tungsten atoms should be justified. Here, we calculated the energy barriers between two adjacent tetrahedral interstitial sites in the following two situations: (i) the hydrogen atom is free to move and all tungsten atoms are frozen; (ii) the hydrogen and nearest-neighbor tungsten atoms are free to move and other tungsten atoms are frozen. The calculated results are presented in Fig. 2b. The energy barrier (0.37 eV) of case (i) is much larger than the energy barrier (0.22 eV) with all tungsten atom displaced. This significant change in the effective diffusion barrier due to the position of the neighboring tungsten atoms suggests that the diffusion of hydrogen atoms would increase when the tungsten atoms at the saddle point open up a channel. The energy barrier of case (ii) is 0.22 eV, which is the same for the case with all the tungsten atoms displaced. This suggests that the relaxation of nearestneighbor tungsten atoms play a dominant role in the hydrogen diffusion, and the effects of other tungsten atoms can be neglected. Therefore, it is justified that the vibrational calculations are restricted to hydrogen and nearest-neighbor tungsten atoms. In addition, we also considered the situation of all-phonon calculations and found that the overall effect of the phonons is so small as to be negligible compared to the present calculations in our manuscript. 3.2. Solution of interstitial hydrogen in tungsten Fig. 3a presents the temperature-dependent solution energy. It can be clearly seen that the solution energies in both tetrahedral and octahedral geometry decrease with temperature, suggesting the required energy of hydrogen dissolution in tungsten is reduced. That is, the dissolution of hydrogen becomes easier with increasing temperature. Furthermore, the solution energies of hydrogen at both tetrahedral and octahedral sites are positive over the whole temperature range, indicating that the solution process of hydrogen in tungsten is always endothermic when the temperature increases. Note that the solution energy decreases much faster with the temperature in the tetrahedral geometry than in the octahedral geometry, i.e. its difference between the two geometries becomes larger with temperature. Generally, the lower the solution energy, the more stable the interstitial hydrogen geometry. Therefore, the interstitial site becomes more stable with the increase of temperature, and the stabilization is much more significant for the tetrahedral geometry than for the octahedral geometry when the temperature increases. Thus, only the tetrahedral interstitial site is considered below. To further understand the temperature-dependent behavior of the interstitial hydrogen solution energy in tungsten, the solution energy is decomposed into two parts according to Eqs. (1) and (7), i.e. the contribution of the static energy (ESol c ) and the contribution of the vibration Sol free energy (ESol m ). These are calculated as follows: E c ðT Þ W H Sol WH f W f ¼ EWH ða Þ  E ða Þ  E ; E ðT Þ ¼ F ða Þ  F T T T c m m ðaT Þ. c m As shown in Fig. 3b, the static energy contribution shows a

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Table 2. Comparison of properties of hydrogen solution and diffusion in tungsten at 0 K as obtained by DFT calculations: ESol Tet is the solution energy of hydrogen at the ground state, i.e. tetrahedral interstitial site, DESol Oct–Tet is the energy difference between the octahedral and tetrahedral solution sites, Eact is the activation energy of hydrogen jumping between two adjacent tetrahedral interstitial sites, mGS and mTS are the vibration frequencies at the ground state and at the transition state, respectively. Other ESol Tet (eV) DEOctTet Sol

a

This work b

c

0.90 , 0.92 , 0.95 0.40a, 0.38c 0.21c 46.7c, 46.7c, 34.7c 63.3c, 45.6c, i24.9c

(eV)

Eact (eV) mGS (THz) mTS (THz)

0.89 0.40 0.22 46.0, 46.0, 34.4 62.3, 46.3, i25.8

a

Ref. [57], VASP code, PAW-GGA-PBE. Ref. [58], VASP code, PAW-GGA-PW91. c Ref. [22],VASP code, PAW-GGA-PBE. b

(a) 1.0 0.8

Energy (eV)

0.6 0.4 0.2 0.0 -0.2

Static energy Zero-point energy

-0.4

(b)

0.6

Migration barrier (eV)

0.5

All W displaced Nearest-neighbor W displaced, frozen other W atoms Frozen all W atoms

0.4 0.3 0.2 0.1 0.0

0

5

10

15

20

25

Coordinate point

30

1

2

3

4

5

6

7

Reaction coordinate

Fig. 2. (a) The energy hypersurface of hydrogen in tungsten. The inset figure is a schematic diagram of coordinate points. The points 1, 15 and 10 are tetrahedral, octahedral and transition state interstitial sites, respectively. The zero point is the energy of a tetrahedral interstitial site. (b) The energy barriers between two first nearest-neighbor tetrahedral interstitial sites. The black square represents the situation that all atoms are free to move. The red circle represents the situation that the hydrogen and nearest-neighbor tungsten atoms are free to move and other tungsten atoms are frozen (case (ii)). The blue triangle represents the situation that the hydrogen atom is free to move and all tungsten atoms are frozen (case (ii)). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Strain (%) 0.10 0.24 0.38 0.53 0.68 0.86 1.04 1.25 1.47 1.3

(a)

1.4

(b)

1.2 1.1 1.0

1.2

0.9

Energy (eV)

Energy (eV)

linear-like negative dependence on the temperature, and the value decreases about 0.20 eV when the temperature increases from 300 to 2700 K. The difference of the static energy contribution between tetrahedral and octahedral geometry is always about 0.40 eV, independent of the temperature. This behavior results from the volume expansion of the tungsten lattice and can be understand by the linear elasticity theory. Following linear elasticity theory, the hydrogen solution energy under hydrostatic strain (here, isotropic volume expansion) can be calculated as Sol ESol r ¼ E e¼0 þ 3reV e¼0 , where r is the hydrogen-induced lattice average stress in the equilibrium lattice, and e is the strain. Here, ESol e¼0 and V e¼0 are the hydrogen static solution energy and the equilibrium volume at 0 K, respectively. The hydrogen solution energy is a linear function of the strain e with a slope 3r. The r values are very close to each other when a single hydrogen atom is introduced at the tetrahedral or octahedral site [25]. Therefore, the static energy contribution in tetrahedral and octahedral geometry shows very similar behavior to the volume expansion and thus to the temperature increase.

1.0

0.8

0.8 0.7 0.3

(c)

0.2 0.1 0.0

0.6

-0.1 -0.2

0.4

-0.3

300 600 900 120015001800210024002700

300 600 900 120015001800210024002700

T (K)

T (K)

Fig. 3. The temperature-dependent solution energies (a) and their corresponding static energy (b) and vibration free-energy contribution (c). The square and circle represent the situation of the tetrahedral and octahedral geometry, respectively. The open square and circle denote the vibration free-energy contribution excluding the temperature effect.

Unlike the situation of the static energy contribution, the vibration free-energy contribution exhibits a different temperature-dependent behavior (see Fig. 3c). The vibration free-energy contribution in both tetrahedral and octahedral geometry decreases with temperature as it involves terms of the form expð1=kT Þ. The difference of the vibration free-energy contribution between the tetrahedral and octahedral geometry becomes larger with temperature. The vibration free-energy contribution decreases much faster with temperature in the tetrahedral geometry than in the octahedral geometry, which may be because there are two real normal modes fewer in the octahedral geometry than in the tetrahedral geometry. In particular, the changes of the vibration free-energy contribution in the tetrahedral and octahedral geometry are 0.07 and 0.5 eV, respectively, when the temperature increases from 300 to 2700 K, which are less than half and more than two times their respective corresponding static energy contributions, respectively. This suggests that the contribution of the static energy and vibration free energy plays a dominant role in the temperature-dependent behavior of the solution energy in the tetrahedral and octahedral geometry, respectively. In addition, an expansion of the lattice leads to a softening of vibration modes. However, the softening of vibration mode

X.-S. Kong et al. / Acta Materialia 84 (2015) 426–435

the hydrogen concentration in the intrinsic tungsten in the temperature range 600–800 K (e.g. only 2.07  1013, at 600 K) is very low, implying that the defect-free tungsten cannot effectively trap hydrogen at this temperature range. Consequently, hydrogen will be difficult to accumulate and form bubbles, different from the experimental observation of hydrogen blistering in tungsten at this temperature range [36]. The reason is that defects such as vacancies and grain boundaries have not been taken into account in the present work. This difference also indicates that defects (e.g. vacancies, dislocations and grain boundaries) would play a dominant role in the actual amount of hydrogen retention in tungsten in fusion reactors. 3.3. The diffusion of interstitial hydrogen in tungsten We now turn to describing the diffusion of interstitial hydrogen in tungsten. As mentioned above, the tetrahedral interstitial site is the most stable location and plays a dominant role for interstitial hydrogen. Therefore, we considered processes that allowed interstitial hydrogen to hop from a tetrahedral site to another adjacent tetrahedral site. Two possible hopping paths were chosen in cNEB calculations (see Fig. 1). The first one, named Path 1, connects two first nearest-neighboring tetrahedral sites directly, while the second one, named Path 2, connects two second nearestneighboring tetrahedral sites passing through an octahedral site. For Path 1, the hydrogen atom in the transition state is situated a little distance away from the center between two nearest-neighboring tetrahedral sites toward the neighboring octahedral site. For Path 2, the saddle point is the octahedral geometry, and so the activation energy is the energy difference between the tetrahedral and octahedral geometry. Fig. 5a presents the calculated temperature-dependent activation energies. The activation energies in both paths 1 and 2 increase with the temperature, from 0.18 to 0.40 eV and from 0.29 to 0.69 eV when the temperature increases from 300 to 2700 K for Paths 1 and 2, respectively. This implies much more energy is needed for hydrogen hopping between two adjacent tetrahedral sites. Similarly to the solution energy, the activation energy can be also decomposed into the contribution of the static IS energy (Emc ðT Þ ¼ ETS c ðaT Þ  E c ðaT Þ) and the contribution

Strain (%) 0.5

0.7

Energy (eV)

0.6

-4

1/2

Solubility (H/W-atm )

10

-7

10

-10

10

Path 1 Path 2

0.4 0.3

0.5

0.4 0.3

-13

10

0.2 0.1 0.3

(c)

0.2 0.1

0.2

This work Mazayev Frauenfelder Benamati Fit to this work

-16

10

-19

10

-22

10

0.0

0

5

10

15

20

-0.1

0.1

-0.2 300 600 900 1200 1500 1800 2100 2400 2700

T (K)

-25

10

0.10 0.24 0.38 0.53 0.68 0.86 1.04 1.25 1.47

(b)

(a)

Energy (eV)

with temperature has little effect on the temperaturedependent behavior of the vibration free-energy contribution. For example, the degenerate frequency of the hydrogen atom in the tetrahedral site is shifted from 46.0 to 43.8 THz if the volume is expanded by 3.162% (corresponding to a temperature increase from 0 to 2100 K), while the vibration free-energy contribution increases only slightly, about 0.02 eV, when the vibration modes of the ˚ are used system with the lattice constant a2100 ¼ 3:210 A instead of the system with the lattice constant ˚ . Therefore, it is reasonable to use the vibraa0 ¼ 3:177 A tion frequencies of the system with the DFT-optimized lattice constant at all temperatures. Similar results have also been found in other metal and metal alloying system [20]. In the quasi-harmonic approximation, lattice vibrations are evaluated only from the harmonic part of the interatomic potential. To some extent, anharmonic effects are taken into account by the volume dependence of the phonon frequencies. Therefore, according to the above results, it is reasonable to conjecture a very weak anharmonic effect for hydrogen in the interstitial sites, which is consistent with the above results of the energy hypersurface. According to Sievert’s law, the solubility can be estimated by Eq. (2) using the above solution energy. Fig. 4 shows the hydrogen solubility as a function of reciprocal temperature. Although the solution energy ESol depends on the temperature during our calculation process, it can be clearly seen from Fig. 4 that the calculated solubility still shows nearly Arrhenius-type behavior. Therefore, one can try to fit the calculated solubility constants with the Arrhe nius form (S ¼ S 0 expðESol =kTÞ). The fitting values of the pre-exponential factor and activation energy are 9:9  103 and 1.25 eV, respectively, which are quite close to Frauenfelder’s values [6]. However, our calculated solubility is about three orders of magnitude smaller than Benamati’s data [10], which was obtained in tungsten samples doped with 5% rhenium. Does alloying of rhenium in tungsten act as an additional trapping site and affect the hydrogen solution in tungsten? The interaction of rhenium with hydrogen has been further studied. The binding energies of the Re–H pair at the first and second nearest-neighbor position are 0.10 and 0.04 eV, respectively, and then rapidly drop to nearly zero with increasing distance, suggesting a very weak repulsive interaction between rhenium and hydrogen. Thus, rhenium cannot act as a trapping site of hydrogen, consistent with the recent experimental results. Therefore, it is reasonable to compare Benamati’s data with other data obtained in tungsten. In addition,

431

25

30

300 600 900 120015001800210024002700

T (K)

35

10000/T(1/K)

Fig. 4. Solubility of hydrogen in tungsten. Data are from Frauenfelder [6], Mazayev [7] and Benamati [10].

Fig. 5. The temperature-dependent activation energies (a) and their corresponding static energy (b) and vibration free-energy contribution (c). The square and triangle represent the situation of Paths 1 and 2, respectively.

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IS of the vibration free energy (Emm ¼ F TS m ðaT Þ  F m ðaT Þ). As shown in Fig. 5b and c, the static energy contributions are always about 0.2 and 0.4 eV for Paths 1 and 2, respectively, independent of the temperature, while the vibration free-energy contribution for both Paths 1 and 2 increases with the temperature. These temperature-dependent behaviors can be understood by a similar explanation to that applied above to solution energy. The straightforward correlation between the total activation energy and the corresponding vibration free-energy contribution suggests that the contribution of the vibration free energy plays a dominant role in the behavior of the activation energy with temperature. Note that the activation energy of Path 1 is always less than that of Path 2, and the difference between them becomes larger as the temperature increases. Hence, it can be deduced that the migration path of interstitial hydrogen mainly advances via the nearest-neighboring tetrahedral sites in tungsten. Using Eq. (3) and the methodologies described in Section 2, the diffusion coefficients of hydrogen in tungsten with and without the temperature effect can be predicted. Four methods are adopted to predict diffusion coefficients: the jump rate from the Wert and Zener model of Eq. (4) with and without the temperature effect (WZ-TE and WZ-UNTE); and the jump rate from the harmonic transition state theory of Eqs. (6) and (7) with and without the temperature effect (hTST-TE and hTST-UNTE). For the situation with the temperature effect, i.e. WZ-TE and hTST-TE, the above temperature-dependent activation energy and vibration modes are used. As shown in Fig. 6, the predicted diffusivity data from WZ-TE and hTST-TE are similar to each other and show nearly Arrhenius behavior, although both WZ-TE and hTST-TE provide temperature-dependent activation energy and pre-exponential

-7

10

-7

10

(a)

-8

10

-16

Deff(Etrap=1.3)

-10

10

-11

10

-12

10

-13

10

-14

10 10

10

500

2

-15

hTST+TE WZ+TE Frauenfelder Zakharov Benamati Otsuka Ikeda Hoshihira Fit-here Deff(Etrap=0.5)

-9

10

Diffusion coefficient (m /s)

2

Diffusion coefficient (m /s)

(b)

-8

10

-9

10

-10

10

-11

10

-12

10

-13

10

Heinola Liu Frauenfelder Ikeda-Otsuka

-14

10

-15

10

-16

10

1000 1500 2000 2500

0

T (K) 2500K

-7

10

5

10

15

20

25

30

35

10000/T (1/K) 1666K

1250K

factor. Here, their average values are considered as our predicted hydrogen diffusivity in tungsten and can be well fitted by the Arrhenius equation D ¼ D0 expðEact =kT Þ with D0 ¼ 7:44  108 m2/s and Eact ¼ 0:13 eV. The fitted values are consistent with our calculated results at high temperature and are slightly larger at room temperature. When the temperature effect is excluded, we only use the diffusion activation energy (0.22 eV) and the vibration modes of the system with the DFT-optimized lattice parameter a0 ¼ 3:177. The predicted diffusivity from WZ-UNTE and hTST-UNTE can be well fitted by the Arrhenius equation D ¼ 1:72  107 expð0:22 eV=kTÞ m2/s and D ¼ 5:70 108 expð0:22 eV=kTÞ m2/s, respectively, agreeing very well with the Liu et al.’s [23] and Heinola et al.’s [22] results, respectively. For clarity, we present Liu et al.’s and Heinola et al.’s results in Fig. 6b and c. Note that there are obvious differences between the predicted diffusivity with and without the temperature effect. It is well known that the agreement with experiment is an extremely important criterion to verify theoretical predictions. Therefore, the available experimental data are also shown in Fig. 6 to compare the estimated diffusivity with experimentally measured values. Among them, the diffusivity data at high temperatures, reported by Frauenfelder [6], has been usually believed to be the most reliable data, because it was obtained at elevated temperatures and is likely less influenced by both surface and trapping effects. The experimental diffusivity at the low-temperature range is inevitably disturbed by the defect-trapping effect, and the values usually fall under the extrapolated data from the high-temperature region. It has been suggested that hydrogen diffusion is strongly influenced by the defect-trapping effect even up to temperatures of about 1500 K [10,55]. As shown in Fig. 6, when the temperature effect is not considered, the predicted hydrogen diffusivity presents a slight difference with the high-temperature experimental data above 1500 K, but has a significant deviation at the temperature range below 1500 K. After including the temperature effect, our predicted diffusivity is in very good agreement with the experimental data at temperatures above 1500 K, but much higher than the experimental diffusivity at temperatures below 1500 K. This perfect consistency of diffusivity at the high-temperature range implies that our calculations accurately describe the diffusion of interstitial hydrogen in tungsten. It also suggests that the temperature effect must be taken into account for an accurate interpretation of experimental hydrogen diffusion data in tungsten. The significant deviation at temperatures below 1500 K can be explained by the defect-trapping phenomenon. 3.4. Defect-trapping effects on hydrogen diffusivity

(c) 2

D (m /s)

Traps effectively increase the time to desorb a hydrogen atom from its lattice site and act to increase the apparent activation energy for diffusion. Commonly, according to the classic Mac-Nabb and Forester formula [41], the effective diffusivity in the defect field can be simply described by:

-8

10

4

6

8

10

10000/T (1/K)

Fig. 6. Diffusivity of hydrogen in tungsten: (a) diffusion coefficient vs. temperature; (b) diffusion coefficient vs. the reciprocal of temperature; (c) diffusion coefficient vs. the reciprocal of temperature for the hightemperature range. Data are from Frauenfelder [6], Zakharov [8], Benamati [10], Otsuka [13], Ikeda [9], Hoshihira [14], Heinola [22], Liu [23] and Ikeda-Ostuka [15].

Deff ¼

Dperf 1 þ ctrap exp



Etrap kT

;

ð10Þ

where Dperf is the diffusivity at the perfect system without traps, ctrap is the trap concentration and Etrap is the trapping energy of hydrogen in the trap site. Two types of typical trapping energies have been obtained by fitting the temper-

X.-S. Kong et al. / Acta Materialia 84 (2015) 426–435

ature position of the thermal desorption spectra of hydrogen in tungsten: one is 0.5 eV with a concentration of  102 and lower [59,60], the other is 1.3 eV with a concentration of  106 [60,61]. The first one may be associated with trapping at impurities, dislocations and grain boundaries, usually termed natural traps, while the second one may be associated with trapping at vacancies [59–61]. Substituting these values into Eq. (10), the effective diffusivity in the field of natural traps and vacancies can be calculated, respectively. Although our calculations are rather crude and simple, they still give some indication of the influence of the defect-trapping phenomenon on hydrogen diffusion in tungsten. As shown in Fig. 6, the predicted effective diffusivity considering natural traps shows fairly good consistency with the experimental values for the temperature range of 300–600 K (including Otsuka’s [13], Ikeda’s [9] and Hoshihira’s [14] data), while the predicted effective diffusivity considering vacancies is in very good agreement with the experimental values for the temperature range of 800–1200 K (including Benamati’s data [10], Zakharov’s data [8] and two of Frauenfelder’s data points [6]). Therefore, it can be speculated that the hydrogen diffusion in tungsten would be significantly influenced by natural traps, such as impurities, dislocations and grain boundaries, at the temperature range of 300–600 K, and is dominated by vacancies in the temperature range of 800–1200 K. This speculation can be understood as follows. In the temperature range of 300–600 K, the hydrogen atoms trapped by natural traps can be activated due to the lower trapping energy, while the hydrogen atoms in the vacancy are firmly captured and locked irremovably because of the higher trapping energy. Therefore, the natural traps, such as, impurities, dislocations and grain boundaries, have a dominant impact on the hydrogen effective diffusivity for the this low-temperature range. With the temperature increases, since the temperature is above the trapping energy of the natural traps, these exert little effect on the effective diffusivity and can be considered negligible. The hydrogen atoms trapped in the vacancy would be activated, and the vacancy would play a dominate role in this temperature range. It should be pointed out that this speculation should be carefully checked in further works with a more complex theoretical framework for hydrogen diffusion in the defect field, where systematic identification of the role of different trap states (i.e. impurities, dislocations, grain boundaries, vacancies, self-interstitial, defect clusters, etc.) and accurate quantification of their energetic levels and concentrations are needed.

433

Most theoretical works based on transition state theory inherently provide temperature-dependent activation energies and pre-exponential factors. This can be understood by the definition of the activation energy. As shown in Eq. (5), the activation energy is the difference between the free energies of the transition state and the initial state. The free energy of the system usually varies with the temperature. Therefore, it is natural to suppose a temperature-dependent activation energy. Due to the strong relation between the activation energy and pre-exponential factor, as shown in Eq. (4), the pre-exponential factor is also temperature dependent. Although the activation energy and pre-exponential factor are temperature-dependent, most theoretical data of the impurity diffusivity in the solid still show nearly Arrhenius-type behavior and can be approximately fitted using the Arrhenius equation with constant activation energy and pre-exponential factor [16–19,21]. The underlying reason might be connected with the strong relation between the constant activation energy and pre-exponential factor. Using the temperature-dependent activation energy and pre-exponential factor data calculated by first-principles calculation, a nearly linear relationship between the logarithm of the pre-exponential factor and the activation energy is observed (see Fig. 7a). This is a very similar manifestation of the Meyer–Neldel rule (usually referred to as the compensation effect, and applied to different physical and physicochemical processes, such as the diffusion of solute atoms and grain boundary in crystalline solids [63–65]). This relationship indicates that the alteration of the diffusivity resulting from the decrease or increase in the activation energy is moderated by the variation of the pre-exponential factor. Therefore, the temperature-dependent activation energy and pre-exponential factor are coupled so that the impurity diffusivity in the solid can be described reasonably well by the Arrhenius equation. For example, the predicted constant activation energy and pre-exponential factor of hydrogen in tungsten using the universal Arrhenius equation is 0.13 eV and 7:44  108 m2/s, respectively. We plot the ratio expðEact ðT Þ=kT Þ=expð0:13=kTÞ and D0 ðT Þ=ð7:44 108 Þ and their product in Fig. 7b. It can be clearly seen that the values of expðEact ðT Þ=kTÞ=expð0:13=kTÞ increase with temperature, D0 ðT Þ=ð7:44  108 Þ decreases with temperature, and their products are always nearly 1 -6.6

(b)

(a) 3

-6.7

Ratio

2

Accurate measurement of activation energy and preexponential factor is critical for understanding the thermal activation processes of the interstitial impurity diffusion in solids. To date, they are almost exclusively extracted using the universal Arrhenius equation D ¼ D0 expðEact =kT Þ. Inherently, the standard Arrhenius plot method assumes that the activation energy and pre-exponential factor are invariant constants with temperature. However, this is not the actual case. Some experimental and theoretical results show that both activation energy and pre-exponential factor are usually temperature dependent [16–21,62]. For example, the experimental data gives a large activation energy at high temperature and a low activation energy at low temperature for diffusion of carbon in palladium [62].

log(D0(T)) (m /s)

3.5. Discussion 2 1

-6.8

0 0.2

0.3

Eact(T) (eV)

0.4

0

600 1200180024003000

T (K)

Fig. 7. (a) The dependence of the logarithm of the pre-exponential factor on the activation energy. (b) The ratio expðEact ðT Þ=kTÞ=expð0:13=kTÞ (square) and D0 ðT Þ=ð7:44  108 Þ (triangle) and their product (circle).

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except at low temperatures, indicating a good couple relationship between the activation energy and the pre-exponential factor. Consequently the diffusivity can be described quite well by the Arrhenius equation. Note that the fitted constant activation energy and pre-exponential factor are different to our calculated value using firstprinciples calculations. Therefore, one should be very cautious in making a comparison between the experimentally measured and the first-principles calculated activation energies and pre-exponential factors. 4. Conclusions The temperature-dependent dissolution and diffusion characteristics of interstitial hydrogen in tungsten have been studied through first-principles calculations combined with the transition state theory. The temperature effect is taken into account by the thermal expansion and vibration free-energy contribution. Tetrahedral and octahedral interstitial sites are considered. An interstitial hydrogen atom always prefers to occupy a tetrahedral interstitial site rather than an octahedral interstitial site over the investigated temperature range of 300–2700 K. The solution energy in both tetrahedral and octahedral geometry decreases with temperature, suggesting the dissolution of hydrogen becomes easier with increasing temperature. The solution energy decreases much faster with temperature in the tetrahedral geometry than in the octahedral geometry, meaning the stabilization is much more significant for the tetrahedral site than for the octahedral site when the temperature increases. Two possible hopping paths, i.e. hydrogen hopping from a tetrahedral site to its first or second nearestneighboring tetrahedral site, are considered. The activation energies in these two paths increase with temperature, implying much more energy is needed for hydrogen hopping between two adjacent tetrahedral sites. The activation energy of Path 1 is always less than that of Path 2, and the difference between them becomes larger as the temperature increases. Hence, it can be deduced that the migration path of interstitial hydrogen mainly advances via the nearestneighboring tetrahedral sites in tungsten. To further understand the temperature-dependent behavior of the interstitial hydrogen solution and activation energy in tungsten, the solution and activation energy are decomposed into two parts, i.e. the contribution of the static energy and the contribution of the vibration free energy. The straightforward correlation between the total solution and activation energy behavior and their corresponding vibration free-energy contribution suggests that the contribution of the vibration free energy plays a dominant role in the behavior of the activation energy with temperature. In addition, the behavior of the static energy contribution with temperature can be understood in terms of linear elasticity theory. Based on the calculated solution and activation energy, the hydrogen solubility and diffusivity are predicted. Our predicted hydrogen solubility and diffusivity are in very good agreement with the experimental data at temperatures above 1500 K, but much smaller and higher than their corresponding experimental values at temperatures below 1500 K, respectively. The significant deviation at temperatures below 1500 K can be explained by the defect-trapping phenomenon. According to the classic Mac-Nabb and Forester formula, the effective hydrogen diffusivity in the defect field has been calculated. Based on these results, it can be

speculated that the hydrogen diffusion in tungsten would be significantly influenced by natural traps, such as impurities, dislocations and grain boundaries, in the temperature range of 300–600 K, and it is dominantly affected by vacancies in the temperature range of 800–1200 K. However, this speculation needs to be carefully checked in future works. Finally, we discussed the possibility of using the standard Arrhenius plot method to accurately predict the activation energy and pre-exponential factor of interstitial impurity diffusion in solids. Although both the activation energy and the pre-exponential factor are usually temperature dependent, the predicted diffusivity of interstitial impurities in the solid by first-principles calculations still shows nearly Arrhenius-type behavior with temperature. The underlying reason might be connected to the coupled relationship between the constant activation energy and preexponential factor, e.g. a nearly linear relationship between the logarithm of the pre-exponential factor and the activation energy. According to these results, we suggest that one should be very cautious in making a comparison between the experimentally measured and the first-principles calculated activation energies and pre-exponential factors. Acknowledgments This work is financially supported by the National Magnetic Confinement Fusion Program (Grant No. 2011GB108004), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA03010303) and the National Natural Science Foundation of China (Grant Nos. 91026002, 91126002 and 11375231). The Center for Computation Science, Hefei Institutes of Physical Sciences is acknowledged for computational support. This research project was part of the CRP carried out under the sponsorship of the IAEA.

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