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Critical concentration for hydrogen bubble formation in metals
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 395402 (http://iopscience.iop.org/0953-8984/26/39/395402) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 141.117.125.175 This content was downloaded on 19/05/2017 at 09:33 Please note that terms and conditions apply.
You may also be interested in: A review of modelling and simulation of hydrogen behaviour in tungsten at different scales Guang-Hong Lu, Hong-Bo Zhou and Charlotte S. Becquart Towards suppressing H blistering by investigating the physical origin of the H–He interaction in W Hong-Bo Zhou, Yue-Lin Liu, Shuo Jin et al. A comparative investigation of the behaviors of H in Au and Ag from first principles Quan-Fu Han, Zhen-Yu Zhou, Yuming Ma et al. Vacancy trapping mechanism for multiple hydrogen and helium in beryllium: a first-principles study Pengbo Zhang, Jijun Zhao and Bin Wen Review of hydrogen retention in tungsten T Tanabe Investigating behaviours of hydrogen in a tungsten grain boundary by first principles: from dissolution and diffusion to a trapping mechanism Hong-Bo Zhou, Yue-Lin Liu, Shuo Jin et al. The effect of irradiation-induced point defects on energetics and kinetics of hydrogen in 3C-SiC in a fusion environment Jingjing Sun, Yu-Wei You, Jie Hou et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395402 (9pp)
doi:10.1088/0953-8984/26/39/395402
Critical concentration for hydrogen bubble formation in metals Lu Sun1, Shuo Jin1, Hong-Bo Zhou1, Ying Zhang1, Wenqing Zhang2, Y Ueda3, H T Lee3 and Guang-Hong Lu1 1
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, People’s Republic of China 2 State Key Laboratory of High Performance Ceramics and Superfine Microstructures, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, People’s Republic of China 3 Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan E-mail: [email protected] Received 15 May 2014, revised 21 July 2014 Accepted for publication 11 August 2014 Published 10 September 2014 Abstract
Employing a thermodynamic model with previously calculated first-principle energetics as inputs, we determined the hydrogen (H) concentration at the interstitial and monovacancy as well as its dependence on temperature and pressure in tungsten and molybdenum. Based on this, we predicted the critical H concentration for H bubble formation at different temperatures. The critical concentration, defined as the value when the concentration of H at a certain mH-vacancy complex first became equal to that of H at the interstitial, was 24 ppm/7.3 GPa and 410 ppm/4.7 GPa at 600 K in tungsten and molybdenum in the case of a monovacancy. Beyond the critical H concentration, numerous H atoms accumulated in the monovacancy, leading to the formation and rapid growth of H-vacancy complexes, which was considered the preliminary stage of H bubble formation. We expect that the proposed approach will be generally used to determine the critical H concentration for H bubble formation in metals. Keywords: metal, critical hydrogen concentration, hydrogen bubble, thermodynamic model (Some figures may appear in colour only in the online journal)
1. Introduction
In a fusion reactor, metallic plasma-facing materials (PFMs) are exposed to high fluxes of H isotopes, which can lead to unwanted surface blistering as a result of H-PFM interaction [10]. When H is embedded in metals, it can accumulate in defects, leading to H bubble formation when the H concentration reaches a critical value. The local H concentration is believed to be directly associated with the formation process of bubbles [10], as well as influencing crack propagation [11] and thus plays a dominant role in defect formation and the degradation of the mechanical properties of metals. The high-Z metals tungsten (W) and molybdenum (Mo) are deemed to be the most promising candidates as PFMs due to their good thermal properties, low sputtering yields, and low H isotope retention [12–15]. It has been demonstrated that H isotopes have a strong interaction with defects in W and Mo [10, 13–15]. As a typical defect in metals, a vacancy has been demonstrated to be a trapping center for H, and H trapping behavior by a monovacancy in W and Mo has been
Impurity segregation (especially impurities with positive solution energies) with defects such as vacancy, grain boundary, and dislocation can cause significant changes of property, particularly the mechanical properties of metals [1, 2]. It is generally accepted that even a trace amount (ppm) of impurity can result in large variation in metal properties [3]. Hence, a quantitative analysis of the impurity concentration is quite useful in understanding the property degradation of metals induced by the impurity. Hydrogen (H) is a typical impurity in metals. The interaction between H and metals is one of the most important topics in material physics. The H-metal interaction is responsible for H-embrittlement in metals, which was a focus of study as early as the beginning of the last century [4–7]. The bonding of H with metals also determines the capacity of H storage materials and the desorption rate of H2 from metal hydrides [8, 9]. 0953-8984/14/395402+9$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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J. Phys.: Condens. Matter 26 (2014) 395402
determined via both the first-principle and the molecular dynamics method [16–19]. The first full and detailed description of H trapping and bubble formation in metals by first principles was by Liu et al [16]. They suggested that H tends to bind onto an isosurface of optimal charge density inside the monovacancy in W, and proposed a vacancy trapping mechanism for H bubble formation, which is considered to be generally applicable to other metals [16, 19]. More accurate energetics and configurations of multi-H at a monovacancy were determined for both W [17, 19] and Mo [18, 19], which shows one monovacancy can trap as many as 12 and 8 H with different trapping energies in W and Mo, irrespective of the zero point energy correction. On the other hand, the number of trapped H at the vacancy in W decreases with increasing temperature according to a molecular dynamics simulation [19] using a recently developed W-H potential [20], which suggests that temperature weakens the H trapping capacity of a vacancy. At 300 K, only ~6.5 H atoms can be trapped by a monovacancy, which decreases to ~5 H at 900 K. This confirms that the thermodynamic estimation of the number of trapped H atoms at the vacancy in W is basically correct and yields 5 H at 300 K [21]. Theoretically, using thermodynamic models with the firstprinciple calculated energetics of H as input parameters to the model, we were able to determine the H concentrations in metals such as W and Mo. For H concentrations in intrinsic metals without any defects, a thermodynamic model has been well-established as the well-known Sievert’s law. Using Sievert’s law, the temperature dependence of the concentration (solubility) of H in intrinsic W [22] and Mo [23] was examined systematically, and the H solubility was found to increase with increasing temperature. It is more practical, however, to determine the H concentrations in metals with defects, since a defect such as a vacancy exists in the metal-H system either intrinsically [24], or is induced by external factors such as an increase in temperature or irradiation. For example, a high temperature and high H gas pressure will induce the formation of superabundant vacancies in metals, resulting in the enhanced embrittlement of metals such as Pd, Ni, and Cu [24–26]. Taking the vacancy and H-vacancy complex in metals into consideration, the thermodynamic model for the intrinsic metal system can be further developed to be applicable for the defective metal system [27, 28]. Based on the developed thermodynamic model, the equilibrium H concentration as a function of temperature and pressure in metals with a certain concentration of monovacancy has been determined for Fe [27], Al [28, 29] and Mg [29]. The H concentration was shown to be strongly dependent on the H chemical potential, and the concentration of the H-vacancy complexes rapidly increases as the H chemical potential increases [27–29]. Unfortunately, the important, critical H concentration that is directly associated with H bubble formation and impacts on mechanical properties has not been explored so far, which significantly limits our understanding of H bubble formation and blistering phenomena. In this paper, we derive further formulations of the thermodynamic model that are more suitable for multiple H-vacancy complexes in metals. Based on these derived formulations, we
calculate the equilibrium H concentration and its dependence on the temperature and pressure in W and Mo. The necessary data, such as the H solution energies needed as inputs for the thermodynamic model, are directly derived from our previous first-principles calculations [19]. Most importantly, considering the monovacancy as a first step, we develop an approach to quantitatively determine the critical concentration of H at different temperatures, which marks a sharp increase in the number of H-vacancy complexes corresponding to the preliminary stage of H bubble formation and growth. This is expected to contribute to the evaluation of the H-induced failure of metallic PFMs in future fusion reactors. 2. Formulation of the thermodynamic model The equilibrium H concentration in a metal can be estimated based on basic thermodynamic principles. Here, only a monovacancy, the simplest case, is taken into account. Two types of H are considered for the dissolution of H into the metal, i.e. the interstitial H and mH-vacancy (mH-V) complexes that are distributed at a lattice site, where mH-V indicates a complex containing a monovacancy, and m indicates H atoms inside the monovacancy. In particular, there also are empty vacancies that trap no H atoms also, meaning m = 0. The Gibbs energy of the metal system changes due to the implantation of H, and the energy change reaches a minimal value with respect to the H concentration when the system reaches equilibrium. At a certain temperature and H pressure, the interstitial H and mH-V complexes inside the metal will be in equilibrium with the H gas. The equilibrium process of the interstitial H and mH-V complexes can be treated as being independent of each other. If the metal system containing the interstitial H and various mH-V complexes are considered as an ensemble, the total Gibbs-free energy can be defined as:
f m E fm − TS +pV , G = nHIEHI + ∑ nHV HV m
(1)
where S and V are the entropy and volume of the system (the change of the volume can be ignored), T and pare the temperm are the number of interstiature and pressure, and nHI and nHV tial H and the mH-V complexes. The corresponding formation fm energies, EHIf and EHV , can be defined as:
f EHI = EH−TIS − EBULK − μH , fm m −E EHV = EHV BULK +
1 EBULK − mμH , NM
(2) (3)
m are the total energies of a bulk where EBULK, EH−TIS, and EHV metal, a metal with a tetrahedral interstitial H, and an mH-V complex at a lattice site, respectively, and can be determined from density functional theory (DFT) calculations. NM is the number of total metal atoms and was 128 for this calculation. m is the H number at the monovacancy in the metal, m = 1 − 12 for W, and m = 1 − 8 for Mo according to our previous DFT calculations [19]. μH is the H chemical potential. It should be noted that the formation energies of mHvacancy complexes are correlated to the H number contained in the complexes as well as to the H chemical potential in the
2
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Table 1. The formation energies (eV) of mH-V complexes at a temperature of 0 K in W and Mo.
mH
1
2
3
4
5
6
7
8
9
10
11
12
W Mo
2.80 2.21
2.52 1.84
2.36 1.60
2.31 1.49
2.34 1.47
2.59 1.70
3.26 2.31
3.85 2.84
4.68 —
5.29 —
6.10 —
6.75 —
grand ensemble approach. The H chemical potential μH can be generally expressed as [30–32]: μH = μH (T = 0 K) + μH (T , p) ,
(4)
where μH (T = 0 K) is half of the energy of the H2 molecule at T = 0 K, which is −3.38 eV according to our previous DFT calculations [19]. μH (T , p) is the quantity depending on the temperature and H pressure, which has already been given by Fukai et al [31] and Hemmes et al [32] within a range of 0.1 MPa ≤ p ≤ 100 GPa and 100 K ≤ T ≤ 2000 K. Therefore, based on our previous DFT calculated energetics of H [19], combined with the given H chemical potential [31, 32], the formation energies of mH-V complexes can be derived at certain temperatures and H pressure as inputs for the thermodynamic model. The formation energies of mH-V complexes at a temperature of 0 K in W and Mo are listed in table 1. The configurational entropy Sc of the system is obtained by: Sc = kB lnΩHI ΩHV ,
Figure 1. The equilibrium H concentration as a function of temperature for H pressures of 0.17 MPa (red), 2 GPa (purple), 5 GPa (brown), 7 GPa (green), and 8 GPa (blue) in W.
(5)
where kB is the Boltzmann constant and ΩHI and ΩHV represent the number of configurations of the interstitial H and the H-V complex distributed in the interstitial and lattice sites in the metal, respectively. For a particular mH-V complex with m H atoms inside the complex, the distribution of the m H atoms in the complex Ω m can be defined as: Ω m=
m max ! , m ! (m max − m ) !
c HI =
(6)
c HV =
m max
∑ m
⎛ Ef ⎞ nHI N = I exp ⎜− HI ⎟ , NM NM ⎝ kBT ⎠
m mnHV = NM
m max
⎛ E fm ⎞ HV ⎟⎟ , ⎝ kBT ⎠
∑ mΩ m ⋅ exp ⎜⎜− m
c H = c HI + c HV,
(10)
In particular, in terms of m = 0, the monovacancy concentration dependent on temperature can be expressed as:
where m max is the maximum number of H atoms that can be trapped at a monovacancy to form an mH-V complex. Based on our DFT calculations, m max is 12 for W and 8 for Mo [19]. Based on equations (1)–(6) and the previous calculated formation and vacancy trapping energies of H as inputs [19], the concentration of H at the interstitial and the H-V complex can be expressed by:
⎛ E fm ⎞ m c HV = mΩ m ⋅ exp ⎜⎜− HV ⎟⎟ , ⎝ kBT ⎠
⎛ E f0 ⎞ c V = exp ⎜⎜− HV ⎟⎟ . ⎝ kBT ⎠
(11)
It is well-known that Sievert’s law can be employed to calculate the equilibrium H concentration in metals [22, 23, 31]. Sievert’s law is based on an ideal gas model, and is thus only applicable to the interstitial H at pressures lower than 0.1 Mpa [31]. Different from the Sievert’s law, the present thermodynamic model [27, 29] is not confined to an ideal gas model. Instead, the H chemical potential in the present model is derived from a modified van der Waals equation, in which the parameters are fitted using the experimental data, also including high H pressure data [31, 32]. Consequently, the present model can be employed to determine the H concentration at not only the interstitial, but also the H-V complex at even higher temperatures (e.g. >1000 K) and H pressures (e.g. >1 GPa).
(7)
(8) (9)
3. Results and discussion
where c HI and c HV represent the H concentration at the interstitial and total H-V complexes in the metal, respectively, and c H represents the total H concentration, i.e. the sum of the numbers of H at the interstitial and monovacancy in the metal. NI and NM are the number of interstitial sites and lattice sites in metals, respectively. Note that all the concentrations defined here are unitless. For a certain mH-V complex, the H concentration in the complex can be expressed as:
3.1. Equilibrium H concentration
Using the thermodynamic model derived above, we calculated the equilibrium H concentration and its dependence on temperature and H pressure in W. We first investigated the H concentration as a function of temperature from 300 K to 900 K for various H pressures based on equations (7)–(9), as 3
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Figure 2. (a) The equilibrium H concentration as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K
(green), and 900 K (blue) in W. The solid lines show the total H concentration and the dashed lines show the interstitial H concentration. (b) The critical H concentration as a function of temperature in W. The light yellow area represents the region of no H-V complex formation and the light cyan area represents the region of formation and rapid growth of H-V complexes in W.
shown in figure 1. The H concentration in W increases with increasing temperature for H pressures under ~7 GPa, indicating that high temperatures promote the dissolution of H in W. It was observed that beyond an H pressure of ~7 GPa, the H concentration at lower temperatures is greater than that at higher temperatures, presenting a trend contrary to rising temperature (figure 1). Next, we investigated the H concentration in W as a function of pressure at different fixed temperatures, as shown in figure 2(a). The H concentration is an increasing function of the H pressure because the chemical potential of H becomes larger with a higher H pressure, leading to a decrease in the formation of energy of H. Beyond a certain H pressure, the concentration of H exhibits a sharp increase along with an increase in H pressure. H can be located at either the interstitial or vacancy site. To further understand the sharp increase in H concentration with an increase in H pressure at a fixed temperature, we plotted the concentration of the interstitial H as a function of pressure using equation (7), with the formation energy given by equation (2), as shown by the dashed line in figure 2(a). Below a certain value of H pressure, the H concentration in W is mostly contributed by the interstitial H. Beyond this value, the concentration of interstitial H retains a similar increasing trend to below the pressure value. This implies that the sharp increase in H concentration originates from the increase in H at the H-V complexes, which is determined by equation (8), with the formation of energy determined by equation (3). Such a sharp increase in H at the H-V complexes suggests a critical pressure exists, beyond which an accumulation of numerous H atoms in the vacancy occurs. The existence of a critical H pressure may explain the different tendency of the H concentration to decrease as the temperature increases at a higher H pressure (beyond ~7 GPa) in comparison with a lower pressure, as shown in figure 1. At different temperatures, the critical pressure for the sharp increase of the H concentration is different, e.g. the pressure is ~ 6.5 GPa and ~ 7.7 GPa at 300 K and 700 K, respectively.
Figure 3. H trapping probability as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K (green), and 900 K (blue) in W.
When the pressure is lower than the critical pressure at the corresponding temperature, the H concentration increases along with an increase in temperature. In contrast, if the pressure is beyond the critical pressure, the H concentration increases to a much higher value, and the growth rate is higher at a lower temperature. This should result in a higher H concentration at lower temperatures beyond ~7 GPa in figure 1. In figure 3, we further plot the H trapping probability at the vacancy as a function of pressure, which is defined as the ratio of H concentration at the H-V complexes and the total H concentration in W, i.e. c HV c H in equation (9). The H trapping probability is almost zero below a certain pressure since H is mainly distributed at the interstitial sites. In contrast, beyond this pressure, the formation of H-V complexes drives H to segregate into the vacancy, leading to a sharp increase in H trapping probability. The probability finally reaches 1 at a certain high pressure, which primarily suggests the H concentration at the H-V complexes dominates in W. 4
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J. Phys.: Condens. Matter 26 (2014) 395402
concentration is thus mainly contributed to by H at the mH-V complex instead of the interstitial H. It can be noted that different mH-V complexes have different growth rates with increasing pressure. Namely, the H concentration at a different mH-V complex that is equal to that at the interstitial, varies according to different pressures. Once the H concentration at a certain mH-V complex exceeds that at the interstitial, the subsequent increase in mH-V complex leads to a sharp increase in the total H concentration at the vacancy. Therefore, we define the critical H concentration (c Hc) as the point at which the concentration of H at one certain mH-V complex first becomes equal to that at the interstitial, m corresponding to the intersection of the c HI curve and the c HV curve, e.g. 6 H-V at 300 K, as shown in figure 7. This is also the minimal value out of all the H concentrations at the mH-V complexes when they become equal to the interstitial complexes. Consequently, the H pressure corresponding to this H concentration is defined as the critical H pressure ( pHc). According to this definition, the values of the critical H concentration and the corresponding H pressure can be quantitatively determined from the following formulations. Under conditions where the concentration of H at the mH-V complex m ) and the interstitial (c ) are equal to each other: (c HV HI
Figure 4. The equilibrium H concentration as a function of temperature for H pressures of 0.17 MPa (red), 1 GPa (purple), 3 GPa (brown), 4 GPa (green), and 5 GPa (blue) in Mo.
The equilibrium H concentration and H trapping probability as a function of temperature and H pressure are further examined for Mo, as seen in figures 4–6. Previous first-principles calculation [19] proposed that the atomic configuration and evolution of the optimal charge density isosurface for multi-H occupation at the vacancy in both W and Mo are the same because they have the same crystal structures and belong to the same subgroup VI with exactly the same number of valence electrons. Nonetheless, a similar trend of H concentration changing with temperature and H pressure in W and Mo was observed. However, because the H formation energy, as well as the H trapping behavior at the vacancy varies between W and Mo [19], the absolute values of the H concentration, the H trapping probability, and the H pressure required for a sharp increase in the H concentration in Mo are different from those in W.
m =c c HV HI ⎛ E fm ⎞ ⎛ Ef ⎞ N mΩm ⋅ exp ⎜⎜− HV ⎟⎟ = I exp ⎜− HI ⎟ , ⎝ kBT ⎠ ⎝ kBT ⎠ NM
(12)
and substituting the expressions of formation energies in equations (2) and (3) into equation (12), we can obtain the corresponding H chemical potential ( μHm) for the critical H concentration at which the H concentration at the mH-V complex is equal to that at the interstitial as:
μ Hm =
1 m −E EHV 6 kBT H−TIS + N EBULK ln + . (13) m − 1 mΩ m m−1
Note from equation (13) that the H chemical potential for the critical H concentration is a function of the H number m at the monovacancy as well as the temperature. Therefore, we can derive the H chemical potential for a different mH at the mH-V complex for a certain temperature. The corresponding m can be then obtained with a substitution H concentration c HV of the H chemical potential μHm into equation (10). As a result, m is also a function of the H number at the mH-V complex c HV m. According to the above definition, the critical H concentration c Hc is thus derived as the minimal value out of the H concentrations at the mH-V complexes when it becomes equal m (m ) = c (m ) . to that at the interstitial, i.e. c Hc = min [c HV ] HI Moreover, the corresponding critical H pressure ( pHc) can be derived by substituting equation (12) into equation (4). Based on the above definition, the critical H concentration as a function of temperature in W and Mo is plotted in figures 2(b) and 5(b), respectively, which obviously demonstrates that the critical H concentration increases along with an increase in temperature. Because the formation energy of the H-V complex increases and the diffusion of H is enhanced as the temperature rises, the H-V complex formation becomes more difficult as a result. Figures 2(b) and 5(b) can be treated
3.2. Critical H concentration for H-monovacancy complex formation
According to the dependence of the H concentration on the temperature and the H pressure in W and Mo as revealed above, we propose that the accumulation of numerous H atoms in the monovacancy actually occurs at a critical H pressure. This indeed corresponds to the critical H concentration associated with this critical pressure for the H-V complexes formation at a certain temperature in both W and Mo. The critical H concentration and pressure are defined as follows, as shown in figure 7, and the temperature of 300 K is used as an example. Below a critical H pressure at a certain temperature, the number of interstitial H, i.e. c HI in equation (9), increases steadily with increasing pressure, and the H concentration is now mainly contributed to by the interstitial H. As the H pressure continues to increase, for a certain m mH-V complex, the number of H at the mH-V complex, i.e. c HV in equation (10), rapidly increases and becomes equal to that at the interstitial once the critical pressure is reached. Beyond this critical pressure, the number of H at the mH-V complex continues to increase and exceeds that at the interstitial. The H 5
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J. Phys.: Condens. Matter 26 (2014) 395402
Figure 5. (a) The equilibrium H concentration as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K
(green), and 900 K (blue) in Mo. The solid lines show the total H concentration and the dashed lines show the interstitial H concentration. (b) The critical H concentration as a function of temperature in Mo. The light yellow area represents the region of no H-V complexes formation and the light cyan area represents the region of formation and rapid growth of H-V complexes in Mo.
Figure 7. The critical H concentration (c Hc) and pressure (p Hc) for the H-V complexes formation at a temperature of 300 K in W. The dashed line represents the H concentration at the interstitial (c HI). The solid lines represent the H concentration at different mH-V m ). complexes (c HV
Figure 6. The H trapping probability as a function of H pressure temperatures of 300 K (purple), 400 K (brown), 600 K (green), and 900 K (blue) in Mo.
as phase diagrams, divided into two phase regions. Under the critical H concentration, the interstitial H atoms dominate and almost no H-V complex forms, corresponding to the light yellow regions in figures 2(b) and 5(b). Beyond the critical H concentration, numerous H-V complexes form and grow rapidly, corresponding to the light cyan regions. These H-V complexes combine to form larger clusters, eventually leading to H bubble formation. The critical H concentration (atomic) and the critical H pressure for temperatures of 500 K to 900 K in W and Mo are presented in table 2. The critical H concentration in W is 24 ppm, and the corresponding critical H pressure is 7.3 GPa at a typical temperature of 600 K. For Mo, the critical H concentration is 410 ppm, and the corresponding critical H pressure is 4.7 GPa at 600 K. Here, the critical H concentration for the H-V complex formation in W is lower compared to that in Mo. The first-principles calculation [19] revealed that the trapping energy of H in W is ~0.2 eV lower than that in Mo, and the sequentially trapped number of H at the vacancy in
Table 2. The critical H concentration c Hc (ppm) and the
corresponding critical H pressure p Hc (GPa) within a temperature T (K) range of 500 K to 900 K in W and Mo. W
Mo
T (K)
c Hc
pHc
c Hc
p Hc
500 600 700 800 900
2.7 24 110 350 870
7.0 7.3 7.7 8.2 8.7
73 410 1380 3400 7000
4.3 4.7 5.3 5.8 6.4
W (12 H) is larger than that in Mo (8 H). The first-principles result implies that the vacancy in W has a stronger capability than that in Mo to trap H to form H-V complexes, consistent with the results of the critical H concentration. This suggests the H-V complex formation and thus the H bubble formation in W is easier than that in Mo. 6
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Consequently, although the H pressure generated from the H plasma is much lower than the critical one, such high pressure conditions inside the metal being required to form H bubbles is still reasonable based on the above discussion.
3.3. Discussions and comparison with experiments 3.3.1. H pressure. During the actual operation of a future
fusion reactor, W and Mo PFMs are exposed to H ions of low energy E (0–100 eV) and high fluxes n (1024–1025 m−2s−1) at the divertor [33]. The H ions undergo elastic collision with the PFMs, and the H pressure pHs on the PFMs can be estimated by pHs = 3 4 n 2mHE . Taking E = 100 eV, n = 1027 m−2s−1 (here we take the flux as even higher than that at the divertor) and mH = 1.67 × 10−27 kg, we obtain the H pressure on PFMs as pHs = 0.17 MPa. At this pressure, the H concentration and the H trapping probability in W (figures 1 and 3) and Mo (figures 4 and 6) also increase along with an increase in temperature. The absolute value of the equilibrium H concentration is extremely low at ~1.6 × 10−4 ppm for W and ~2.3 × 10−2 ppm for Mo at 600 K (figures 1 and 3). Most H atoms are distributed at the interstitial sites, and the untrapped H atoms diffuse to the surface and then enter the plasma through a recombination process [34]. From table 2, it is clear that the critical pressure corresponds to the critical H concentration predicted as the order of GPa is much higher than the pressure on the PFMs generated from the H plasma (0.17 MPa) as in the actual case estimated above. Namely, to form an H bubble, a high H fugacity (pressure) is actually required [10], though such a high pressure might not be achieved during the operation of a fusion reactor. Inside the H bubble, however, a high internal H pressure of up to several to several tens of GPa has been demonstrated [35]. First, we can roughly estimate the H pressure pHv inside the mH-V complex in this calculation. We employ the ideal gas state equation pHv Vv = mRT , taking the temperature as being 300 K and the ⎛ ⎞3 volume of the monovacancy as Vv = 4 3 π ⎜ 3 4 a⎟ , where a is ⎝ ⎠
3.3.2. Critical concentration of H. Experimentally, the H concentration and bubble formation in metals are directly associated with the diffusivity of H, which was investigated as early as the 1960s. Frauenfelder [37] examined the diffusion behavior of H in W at temperatures of between 1100 K and 2400 K by measuring the degassing rates of a solid cylinder preloaded with H at a certain pressure, and gave the expression of H diffusivity as: D = 4.1 × 10−7exp(−0.39 eV / kT ) m2s−1. Zakharov et al [38] then determined the H diffusivity as D = 6.0 × 10−4exp(−1.07 eV / kT ) m2s−1 within a lower temperature range of 673 K to 1473 K by measuring the diffused H gas from the walls of a W tube. The diffusivity difference between the two works mainly originates from the different experimental temperature ranges [35]. First, several experimental studies have reported a retained D concentration under different radiation conditions in W. The D concentration was observed to be 550 ppm at a temperature of 700 K [39] and 400 to 500 ppm within a temperature range of 423 K to 973 K [40]. Furthermore, in a review by Causey and Venhaus [35], the D concentration was found to exceed from a few to a few tens of atomic ppm at temperatures below 1000 K with the D bubble formation. These experimentally observed H concentration values are comparable to our predicted critical H concentration results (see table 2). In the recent experimental work by Peng et al, the H isotope ion-driven permeation in W was investigated using a high flux ion beam test device coupled with a permeation device [41]. The W specimen with a thickness of 30 μm was exposed to deuterium (D) ions with an energy of 1 keV and a flux of 1019 ~ 1020 m−2s−1, and the permeation flux of D was measured at different irradiation temperatures. The atomic concentration of D was further derived according to the measured D permeation rate by using diffusivity measured by Zakharov and Frauenfelder [37, 38]. It is important to point out that the concentrations were indirectly determined from the measured experimental permeation rates. Therefore, the choice of diffusivity values plays a critical role in the calculated concentration values. It is suggested that an H bubble will form in W at certain temperatures when the D concentration is greater than the precipitation limit concentration, which is a function of D solubility and fugacity (SH p ), consistent with the previous experimental work on H bubble formation observed within the same temperature range [42]. Peng’s work provided clear information on the H bubble forms for those D concentration points at certain temperatures based on Zakharov’s diffusivity, which is determined within a temperature range of 673 K to 1473 K close to that for H bubble formation. As shown in figure 8, the experimental D concentration values with significant blistering and bubble formation observation for similar irradiation conditions [42, 43] (red filled triangles) are larger than the presently predicted critical values with one exception. Those D concentration points at which no bubble formation was experimentally
the metal lattice constant. The number of H atoms m at the monovacancy is set at 6 as an example. According to this estimation, the H pressure inside the mH-V complex is as high as 11.9 GPa. On the other hand, it is known that the surface of the metal is first exposed to a high flux of H plasma. Here, the incident flux of H at the surface is defined as Φs. A fraction of the H incident flux at the surface further permeates to the bulk, and such a permeation flux of H inside the metal is defined as Φb. Here, Φb = αΦs, where α is the fractional constant. The H permeation flux is known to be the function of the H solubility (SH), diffusivity (D H), and pressure. Thus, the resulting equa-
tion is: D Hb SH pHb = αD Hs SH pHs . Moreover, the diffusivity can be expressed as: D = D0 exp − Ea kBT where D0 is the preexponential factor and Ea is the activation energy of H, i.e. the H diffusion barrier. According to the above equations, we can roughly obtain the relationship between the H pressure at the bulk ( pHb) and the surface ( pHs ) as pHb = α 2⋅pHs ⋅exp 2Ebs kBT . Obviously, the migration barrier of H from the bulk to the surface (Ebs) should mainly be responsible for the large pressure difference. The derived H pressure in the bulk is then roughly estimated to be ~104 order of magnitude higher than that at the surface at a temperature of 600 K, based on the migration barrier (Ebs) of 0.27 eV reported in the previous work [36].
(
)
(
)
7
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J. Phys.: Condens. Matter 26 (2014) 395402
reduce the thermal conductivity of metals. The high H pressure inside bubbles is also responsible for the crack extension that is a result of H embrittlement [5, 11]. Therefore, the H bubbles originating from the accumulation of large quantities of H into defects are associated with the failure of metallic PFMs in future fusion reactors. In other words, the H-induced failure or property degradation (both thermal and mechanical) of metallic PFMs can be characterized by H bubble formation in PFMs, which can be considered the starting point of the failure of metallic PFMs under H plasma irradiation. We expect the approach used here to predict the critical H concentration in metals will contribute to the evaluation of the H-induced failure of metallic PFMs under H plasma irradiation in future fusion reactors. 4. Conclusions
Figure 8. The critical H concentration for H bubble formation in
comparison with the experimental observations [41] employing Zakharov’s diffusivity. Red-filled and dark hollow triangles represent the H concentration points with and without H bubble formation estimated by the experimental works, respectively.
We calculated the equilibrium H concentration at the interstitial and monovacancy, as well as its dependence on temperature and H pressure in W and Mo using a thermodynamic model with previously calculated first-principle energetics of H as the necessary input data for the model. At a certain temperature, the H concentration exhibits a sharp increase beyond the critical H pressure, corresponding to a critical H concentration for H-V complex formation in W and Mo. Such critical concentrations and pressures are defined as the values when the concentration of H at one certain mH-V complex first becomes equal to that of H at the interstitial, which are 24 ppm/7.3 GPa and 410 ppm/4.7 GPa at 600 K in W and Mo, respectively. Beyond such critical H concentrations, numerous H atoms accumulate in the monovacancy, leading to the formation and rapid growth of H-V complexes, which is considered the preliminary stage of H bubble formation. We expect the proposed approach to determine the critical H concentration for H bubble formation in metals will be applied to more complicated cases such as vacancy clusters and grain boundaries.
observed (the dark hollow triangles) are within our predicted phase region without H bubble formation. This implies that an H bubble will form when the H concentration reaches a high value at high temperatures (because the predicted critical H concentration is high), but the experimentally measured H concentration is far lower than the critical values, resulting in no bubble formation. Unfortunately, the experimental results provide little information on the critical H concentration for bubble formation. However, at the two temperatures of 610 K and 710 K, we can estimate the critical concentration ranges, since both concentration points with and without H bubble formation are present at the same temperature. One (710 K) is consistent with the present prediction, while the other is inconsistent (610 K). Nevertheless, we have only considered the simplest case of the monovacancy in the present thermodynamic calculation. This means the present calculations cannot be directly compared with the experimental results. In actuality, different kinds of vacancy exist in metals including di-vacancy, tri-vacancy, and even large vacancy clusters and voids. Furthermore, other types of defect such as grain boundaries and dislocations also play similar roles as vacancies for H bubble formation [19]. However, the main purpose of the present paper is to propose an approach to predict the H concentration based on the derived formulations of thermodynamic models, including a clear quantitative definition of the critical H concentration. Such an approach can be further generalized to multi-vacancy cases, as well as the cases of other types of defect such as grain boundaries and dislocations. As a next step, using molecular dynamics simulations, we will further calculate the critical H concentration taking the vacancy clusters with different sizes in W into consideration.
Acknowledgments This work is supported by the National Magnetic Confinement Fusion Program through Grant No. 2013GB109002 and the National Natural Science Foundation of China (NSFC) through Grant No. 51371019. G H Lu wishes to acknowledge support from The National Science Fund for Distinguished Young Scholars through Grant No. 51325103. References [1] Janisch R and Elsasser C 2003 Phys. Rev. B 67 224101 [2] Domain C, Becquart C S and Foct J 2004 Phys. Rev. B 69 144112 [3] Lu G H, Zhang Y, Deng S H, Wang T M, Kohyama M, Yamamoto R, Liu F, Horikawa K and Kanno M 2006 Phys. Rev. B 73 224115 [4] Troiano A R 1960 Trans. Am. Soc. Met. 52 54 [5] Hirth J P 1980 Metall. Mater. Trans. 11A 861
3.3.3. Evaluation of the H-induced failure of PFMs.
The formation of H bubbles in metallic PFMs is quite harmful because H bubbles degrade the mechanical properties and 8
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[25] Fukai Y 2003 J. Alloys Compounds 356–7 263 [26] Fukai Y and Okuma N 1994 Phys. Rev. Lett. 73 1640 [27] Nazarov R, Hickel T and Neugebauer J 2010 Phys. Rev. B 82 224104 [28] Ji M, Wang C Z, Ho K M, Adhikari S and Hebert K R 2010 Phys. Rev. B 81 024105 [29] Ismer L, Park M S, Janotti A and van de Walle C G 2009 Phys. Rev. B 80 184110 [30] van de Walle C G and Neugebauer J 2004 J. Appl. Phys. 95 3851 [31] Sugimoto H and Fukai Y 1992 Acta. Metall. Mater. 40 2327 [32] Hemmes H, Driessen A and Griessen R 1986 J. Phys. C 19 3571 [33] Haasz A A, Davis J W, Poon M and Macaulay-Newcombe R G 1998 J. Nucl. Mater. 258–63 889 [34] Causey R A 2002 J. Nucl. Mater. 300 91 [35] Causey R A and Venhaus T J 2001 Phys. Scr. T94 9 [36] Heinola K and Ahlgren T 2010 Phys. Rev. B 81 073409 [37] Frauenfelder R 1968 J. Vac. Sci. Technol. 6 388 [38] Zakharov A P, Sharapov V M and Evko É I 1973 Fiz. Khim. Mekh. Mater. 9 149 [39] Macaulay-Newcombe R G and Thompson D A 1998 J. Nucl. Mater. 258–63 1109 [40] Causey R A, Wilson K, Venhaus T and Wampler W R 1999 J. Nucl. Mater. 266–9 467 [41] Peng H Y, Lee H T, Ohtsuka Y and Ueda Y 2013 J. Nucl. Mater. 438 S1063 [42] Shimada T, Kikuchi H, Ueda Y, Sagara A and Nishikawa M 2003 J. Nucl. Mater. 313–6 204 [43] Peng H Y, Lee H T, Ohtsuka Y and Ueda Y 2011 Phys. Scr. T145 014046
[6] Robertson I M and Birnbaum H K 1986 Acta Metall. 34 353 [7] Shih D S, Robertson I M and Birnbaum H K 1988 Acta Metall. 36 111 [8] Chen P, Xiong Z T, Luo J Z, Lin J Y and Tan K L 2002 Nature 420 302 [9] Li S, Jena P and Ahuja R 2006 Phys. Rev. B 73 214107 [10] Condon J B and Schober T J 1993 J. Nucl. Mater. 207 1 [11] Zielinski A 2001 J. Mater. Process. Technol. 109 206 [12] Janeschitz G 2001 J. Nucl. Mater. 290–3 1 [13] Zhou H B, Liu Y L, Jin S, Zhang Y, Luo G N and Lu G H 2010 Nucl. Fusion 50 025016 [14] Ogorodnikova O V 2009 J. Nucl. Mater. 390–1 651 [15] Sharpe J P, Kolasinski R D, Shimada M, Calderoni P and Causey R A 2009 J. Nucl. Mater. 390–1 709 [16] Liu Y L, Zhang Y, Zhou H B, Lu G H, Liu F and Luo G N 2009 Phys. Rev. B 79 172103 [17] Ohsawa K, Goto J, Yamakami M, Yamaguchi M and Yagi M 2010 Phys. Rev. B 82 184117 [18] Ohsawa K, Eguchi K, Watanabe H, Yamaguchi M and Yagi M 2012 Phys. Rev. B 85 094102 [19] Sun L, Jin S, Li X C, Zhang Y and Lu G H 2013 J. Nucl. Mater. 434 395 [20] Li X C, Shu X L, Liu Y N, Gao F and Lu G H 2011 J. Nucl. Mater. 408 12 [21] Heinola K, Ahlgren T, Nordlund K and Keinonen J 2010 Phys. Rev. B 82 094102 [22] Liu Y L, Zhou H B and Zhang Y 2011 J. Alloys Compounds 509 8277 [23] Duan C, Liu Y L, Zhou H B, Zhang Y, Jin S, Lu G H and Luo G N 2010 J. Nucl. Mater. 404 109 [24] Fukai Y 1995 J. Alloys Compounds 231 35
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Critical concentration for hydrogen bubble formation in metals
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 395402 (http://iopscience.iop.org/0953-8984/26/39/395402) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 141.117.125.175 This content was downloaded on 19/05/2017 at 09:33 Please note that terms and conditions apply.
You may also be interested in: A review of modelling and simulation of hydrogen behaviour in tungsten at different scales Guang-Hong Lu, Hong-Bo Zhou and Charlotte S. Becquart Towards suppressing H blistering by investigating the physical origin of the H–He interaction in W Hong-Bo Zhou, Yue-Lin Liu, Shuo Jin et al. A comparative investigation of the behaviors of H in Au and Ag from first principles Quan-Fu Han, Zhen-Yu Zhou, Yuming Ma et al. Vacancy trapping mechanism for multiple hydrogen and helium in beryllium: a first-principles study Pengbo Zhang, Jijun Zhao and Bin Wen Review of hydrogen retention in tungsten T Tanabe Investigating behaviours of hydrogen in a tungsten grain boundary by first principles: from dissolution and diffusion to a trapping mechanism Hong-Bo Zhou, Yue-Lin Liu, Shuo Jin et al. The effect of irradiation-induced point defects on energetics and kinetics of hydrogen in 3C-SiC in a fusion environment Jingjing Sun, Yu-Wei You, Jie Hou et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395402 (9pp)
doi:10.1088/0953-8984/26/39/395402
Critical concentration for hydrogen bubble formation in metals Lu Sun1, Shuo Jin1, Hong-Bo Zhou1, Ying Zhang1, Wenqing Zhang2, Y Ueda3, H T Lee3 and Guang-Hong Lu1 1
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, People’s Republic of China 2 State Key Laboratory of High Performance Ceramics and Superfine Microstructures, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, People’s Republic of China 3 Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan E-mail: [email protected] Received 15 May 2014, revised 21 July 2014 Accepted for publication 11 August 2014 Published 10 September 2014 Abstract
Employing a thermodynamic model with previously calculated first-principle energetics as inputs, we determined the hydrogen (H) concentration at the interstitial and monovacancy as well as its dependence on temperature and pressure in tungsten and molybdenum. Based on this, we predicted the critical H concentration for H bubble formation at different temperatures. The critical concentration, defined as the value when the concentration of H at a certain mH-vacancy complex first became equal to that of H at the interstitial, was 24 ppm/7.3 GPa and 410 ppm/4.7 GPa at 600 K in tungsten and molybdenum in the case of a monovacancy. Beyond the critical H concentration, numerous H atoms accumulated in the monovacancy, leading to the formation and rapid growth of H-vacancy complexes, which was considered the preliminary stage of H bubble formation. We expect that the proposed approach will be generally used to determine the critical H concentration for H bubble formation in metals. Keywords: metal, critical hydrogen concentration, hydrogen bubble, thermodynamic model (Some figures may appear in colour only in the online journal)
1. Introduction
In a fusion reactor, metallic plasma-facing materials (PFMs) are exposed to high fluxes of H isotopes, which can lead to unwanted surface blistering as a result of H-PFM interaction [10]. When H is embedded in metals, it can accumulate in defects, leading to H bubble formation when the H concentration reaches a critical value. The local H concentration is believed to be directly associated with the formation process of bubbles [10], as well as influencing crack propagation [11] and thus plays a dominant role in defect formation and the degradation of the mechanical properties of metals. The high-Z metals tungsten (W) and molybdenum (Mo) are deemed to be the most promising candidates as PFMs due to their good thermal properties, low sputtering yields, and low H isotope retention [12–15]. It has been demonstrated that H isotopes have a strong interaction with defects in W and Mo [10, 13–15]. As a typical defect in metals, a vacancy has been demonstrated to be a trapping center for H, and H trapping behavior by a monovacancy in W and Mo has been
Impurity segregation (especially impurities with positive solution energies) with defects such as vacancy, grain boundary, and dislocation can cause significant changes of property, particularly the mechanical properties of metals [1, 2]. It is generally accepted that even a trace amount (ppm) of impurity can result in large variation in metal properties [3]. Hence, a quantitative analysis of the impurity concentration is quite useful in understanding the property degradation of metals induced by the impurity. Hydrogen (H) is a typical impurity in metals. The interaction between H and metals is one of the most important topics in material physics. The H-metal interaction is responsible for H-embrittlement in metals, which was a focus of study as early as the beginning of the last century [4–7]. The bonding of H with metals also determines the capacity of H storage materials and the desorption rate of H2 from metal hydrides [8, 9]. 0953-8984/14/395402+9$33.00
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J. Phys.: Condens. Matter 26 (2014) 395402
determined via both the first-principle and the molecular dynamics method [16–19]. The first full and detailed description of H trapping and bubble formation in metals by first principles was by Liu et al [16]. They suggested that H tends to bind onto an isosurface of optimal charge density inside the monovacancy in W, and proposed a vacancy trapping mechanism for H bubble formation, which is considered to be generally applicable to other metals [16, 19]. More accurate energetics and configurations of multi-H at a monovacancy were determined for both W [17, 19] and Mo [18, 19], which shows one monovacancy can trap as many as 12 and 8 H with different trapping energies in W and Mo, irrespective of the zero point energy correction. On the other hand, the number of trapped H at the vacancy in W decreases with increasing temperature according to a molecular dynamics simulation [19] using a recently developed W-H potential [20], which suggests that temperature weakens the H trapping capacity of a vacancy. At 300 K, only ~6.5 H atoms can be trapped by a monovacancy, which decreases to ~5 H at 900 K. This confirms that the thermodynamic estimation of the number of trapped H atoms at the vacancy in W is basically correct and yields 5 H at 300 K [21]. Theoretically, using thermodynamic models with the firstprinciple calculated energetics of H as input parameters to the model, we were able to determine the H concentrations in metals such as W and Mo. For H concentrations in intrinsic metals without any defects, a thermodynamic model has been well-established as the well-known Sievert’s law. Using Sievert’s law, the temperature dependence of the concentration (solubility) of H in intrinsic W [22] and Mo [23] was examined systematically, and the H solubility was found to increase with increasing temperature. It is more practical, however, to determine the H concentrations in metals with defects, since a defect such as a vacancy exists in the metal-H system either intrinsically [24], or is induced by external factors such as an increase in temperature or irradiation. For example, a high temperature and high H gas pressure will induce the formation of superabundant vacancies in metals, resulting in the enhanced embrittlement of metals such as Pd, Ni, and Cu [24–26]. Taking the vacancy and H-vacancy complex in metals into consideration, the thermodynamic model for the intrinsic metal system can be further developed to be applicable for the defective metal system [27, 28]. Based on the developed thermodynamic model, the equilibrium H concentration as a function of temperature and pressure in metals with a certain concentration of monovacancy has been determined for Fe [27], Al [28, 29] and Mg [29]. The H concentration was shown to be strongly dependent on the H chemical potential, and the concentration of the H-vacancy complexes rapidly increases as the H chemical potential increases [27–29]. Unfortunately, the important, critical H concentration that is directly associated with H bubble formation and impacts on mechanical properties has not been explored so far, which significantly limits our understanding of H bubble formation and blistering phenomena. In this paper, we derive further formulations of the thermodynamic model that are more suitable for multiple H-vacancy complexes in metals. Based on these derived formulations, we
calculate the equilibrium H concentration and its dependence on the temperature and pressure in W and Mo. The necessary data, such as the H solution energies needed as inputs for the thermodynamic model, are directly derived from our previous first-principles calculations [19]. Most importantly, considering the monovacancy as a first step, we develop an approach to quantitatively determine the critical concentration of H at different temperatures, which marks a sharp increase in the number of H-vacancy complexes corresponding to the preliminary stage of H bubble formation and growth. This is expected to contribute to the evaluation of the H-induced failure of metallic PFMs in future fusion reactors. 2. Formulation of the thermodynamic model The equilibrium H concentration in a metal can be estimated based on basic thermodynamic principles. Here, only a monovacancy, the simplest case, is taken into account. Two types of H are considered for the dissolution of H into the metal, i.e. the interstitial H and mH-vacancy (mH-V) complexes that are distributed at a lattice site, where mH-V indicates a complex containing a monovacancy, and m indicates H atoms inside the monovacancy. In particular, there also are empty vacancies that trap no H atoms also, meaning m = 0. The Gibbs energy of the metal system changes due to the implantation of H, and the energy change reaches a minimal value with respect to the H concentration when the system reaches equilibrium. At a certain temperature and H pressure, the interstitial H and mH-V complexes inside the metal will be in equilibrium with the H gas. The equilibrium process of the interstitial H and mH-V complexes can be treated as being independent of each other. If the metal system containing the interstitial H and various mH-V complexes are considered as an ensemble, the total Gibbs-free energy can be defined as:
f m E fm − TS +pV , G = nHIEHI + ∑ nHV HV m
(1)
where S and V are the entropy and volume of the system (the change of the volume can be ignored), T and pare the temperm are the number of interstiature and pressure, and nHI and nHV tial H and the mH-V complexes. The corresponding formation fm energies, EHIf and EHV , can be defined as:
f EHI = EH−TIS − EBULK − μH , fm m −E EHV = EHV BULK +
1 EBULK − mμH , NM
(2) (3)
m are the total energies of a bulk where EBULK, EH−TIS, and EHV metal, a metal with a tetrahedral interstitial H, and an mH-V complex at a lattice site, respectively, and can be determined from density functional theory (DFT) calculations. NM is the number of total metal atoms and was 128 for this calculation. m is the H number at the monovacancy in the metal, m = 1 − 12 for W, and m = 1 − 8 for Mo according to our previous DFT calculations [19]. μH is the H chemical potential. It should be noted that the formation energies of mHvacancy complexes are correlated to the H number contained in the complexes as well as to the H chemical potential in the
2
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J. Phys.: Condens. Matter 26 (2014) 395402
Table 1. The formation energies (eV) of mH-V complexes at a temperature of 0 K in W and Mo.
mH
1
2
3
4
5
6
7
8
9
10
11
12
W Mo
2.80 2.21
2.52 1.84
2.36 1.60
2.31 1.49
2.34 1.47
2.59 1.70
3.26 2.31
3.85 2.84
4.68 —
5.29 —
6.10 —
6.75 —
grand ensemble approach. The H chemical potential μH can be generally expressed as [30–32]: μH = μH (T = 0 K) + μH (T , p) ,
(4)
where μH (T = 0 K) is half of the energy of the H2 molecule at T = 0 K, which is −3.38 eV according to our previous DFT calculations [19]. μH (T , p) is the quantity depending on the temperature and H pressure, which has already been given by Fukai et al [31] and Hemmes et al [32] within a range of 0.1 MPa ≤ p ≤ 100 GPa and 100 K ≤ T ≤ 2000 K. Therefore, based on our previous DFT calculated energetics of H [19], combined with the given H chemical potential [31, 32], the formation energies of mH-V complexes can be derived at certain temperatures and H pressure as inputs for the thermodynamic model. The formation energies of mH-V complexes at a temperature of 0 K in W and Mo are listed in table 1. The configurational entropy Sc of the system is obtained by: Sc = kB lnΩHI ΩHV ,
Figure 1. The equilibrium H concentration as a function of temperature for H pressures of 0.17 MPa (red), 2 GPa (purple), 5 GPa (brown), 7 GPa (green), and 8 GPa (blue) in W.
(5)
where kB is the Boltzmann constant and ΩHI and ΩHV represent the number of configurations of the interstitial H and the H-V complex distributed in the interstitial and lattice sites in the metal, respectively. For a particular mH-V complex with m H atoms inside the complex, the distribution of the m H atoms in the complex Ω m can be defined as: Ω m=
m max ! , m ! (m max − m ) !
c HI =
(6)
c HV =
m max
∑ m
⎛ Ef ⎞ nHI N = I exp ⎜− HI ⎟ , NM NM ⎝ kBT ⎠
m mnHV = NM
m max
⎛ E fm ⎞ HV ⎟⎟ , ⎝ kBT ⎠
∑ mΩ m ⋅ exp ⎜⎜− m
c H = c HI + c HV,
(10)
In particular, in terms of m = 0, the monovacancy concentration dependent on temperature can be expressed as:
where m max is the maximum number of H atoms that can be trapped at a monovacancy to form an mH-V complex. Based on our DFT calculations, m max is 12 for W and 8 for Mo [19]. Based on equations (1)–(6) and the previous calculated formation and vacancy trapping energies of H as inputs [19], the concentration of H at the interstitial and the H-V complex can be expressed by:
⎛ E fm ⎞ m c HV = mΩ m ⋅ exp ⎜⎜− HV ⎟⎟ , ⎝ kBT ⎠
⎛ E f0 ⎞ c V = exp ⎜⎜− HV ⎟⎟ . ⎝ kBT ⎠
(11)
It is well-known that Sievert’s law can be employed to calculate the equilibrium H concentration in metals [22, 23, 31]. Sievert’s law is based on an ideal gas model, and is thus only applicable to the interstitial H at pressures lower than 0.1 Mpa [31]. Different from the Sievert’s law, the present thermodynamic model [27, 29] is not confined to an ideal gas model. Instead, the H chemical potential in the present model is derived from a modified van der Waals equation, in which the parameters are fitted using the experimental data, also including high H pressure data [31, 32]. Consequently, the present model can be employed to determine the H concentration at not only the interstitial, but also the H-V complex at even higher temperatures (e.g. >1000 K) and H pressures (e.g. >1 GPa).
(7)
(8) (9)
3. Results and discussion
where c HI and c HV represent the H concentration at the interstitial and total H-V complexes in the metal, respectively, and c H represents the total H concentration, i.e. the sum of the numbers of H at the interstitial and monovacancy in the metal. NI and NM are the number of interstitial sites and lattice sites in metals, respectively. Note that all the concentrations defined here are unitless. For a certain mH-V complex, the H concentration in the complex can be expressed as:
3.1. Equilibrium H concentration
Using the thermodynamic model derived above, we calculated the equilibrium H concentration and its dependence on temperature and H pressure in W. We first investigated the H concentration as a function of temperature from 300 K to 900 K for various H pressures based on equations (7)–(9), as 3
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J. Phys.: Condens. Matter 26 (2014) 395402
Figure 2. (a) The equilibrium H concentration as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K
(green), and 900 K (blue) in W. The solid lines show the total H concentration and the dashed lines show the interstitial H concentration. (b) The critical H concentration as a function of temperature in W. The light yellow area represents the region of no H-V complex formation and the light cyan area represents the region of formation and rapid growth of H-V complexes in W.
shown in figure 1. The H concentration in W increases with increasing temperature for H pressures under ~7 GPa, indicating that high temperatures promote the dissolution of H in W. It was observed that beyond an H pressure of ~7 GPa, the H concentration at lower temperatures is greater than that at higher temperatures, presenting a trend contrary to rising temperature (figure 1). Next, we investigated the H concentration in W as a function of pressure at different fixed temperatures, as shown in figure 2(a). The H concentration is an increasing function of the H pressure because the chemical potential of H becomes larger with a higher H pressure, leading to a decrease in the formation of energy of H. Beyond a certain H pressure, the concentration of H exhibits a sharp increase along with an increase in H pressure. H can be located at either the interstitial or vacancy site. To further understand the sharp increase in H concentration with an increase in H pressure at a fixed temperature, we plotted the concentration of the interstitial H as a function of pressure using equation (7), with the formation energy given by equation (2), as shown by the dashed line in figure 2(a). Below a certain value of H pressure, the H concentration in W is mostly contributed by the interstitial H. Beyond this value, the concentration of interstitial H retains a similar increasing trend to below the pressure value. This implies that the sharp increase in H concentration originates from the increase in H at the H-V complexes, which is determined by equation (8), with the formation of energy determined by equation (3). Such a sharp increase in H at the H-V complexes suggests a critical pressure exists, beyond which an accumulation of numerous H atoms in the vacancy occurs. The existence of a critical H pressure may explain the different tendency of the H concentration to decrease as the temperature increases at a higher H pressure (beyond ~7 GPa) in comparison with a lower pressure, as shown in figure 1. At different temperatures, the critical pressure for the sharp increase of the H concentration is different, e.g. the pressure is ~ 6.5 GPa and ~ 7.7 GPa at 300 K and 700 K, respectively.
Figure 3. H trapping probability as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K (green), and 900 K (blue) in W.
When the pressure is lower than the critical pressure at the corresponding temperature, the H concentration increases along with an increase in temperature. In contrast, if the pressure is beyond the critical pressure, the H concentration increases to a much higher value, and the growth rate is higher at a lower temperature. This should result in a higher H concentration at lower temperatures beyond ~7 GPa in figure 1. In figure 3, we further plot the H trapping probability at the vacancy as a function of pressure, which is defined as the ratio of H concentration at the H-V complexes and the total H concentration in W, i.e. c HV c H in equation (9). The H trapping probability is almost zero below a certain pressure since H is mainly distributed at the interstitial sites. In contrast, beyond this pressure, the formation of H-V complexes drives H to segregate into the vacancy, leading to a sharp increase in H trapping probability. The probability finally reaches 1 at a certain high pressure, which primarily suggests the H concentration at the H-V complexes dominates in W. 4
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concentration is thus mainly contributed to by H at the mH-V complex instead of the interstitial H. It can be noted that different mH-V complexes have different growth rates with increasing pressure. Namely, the H concentration at a different mH-V complex that is equal to that at the interstitial, varies according to different pressures. Once the H concentration at a certain mH-V complex exceeds that at the interstitial, the subsequent increase in mH-V complex leads to a sharp increase in the total H concentration at the vacancy. Therefore, we define the critical H concentration (c Hc) as the point at which the concentration of H at one certain mH-V complex first becomes equal to that at the interstitial, m corresponding to the intersection of the c HI curve and the c HV curve, e.g. 6 H-V at 300 K, as shown in figure 7. This is also the minimal value out of all the H concentrations at the mH-V complexes when they become equal to the interstitial complexes. Consequently, the H pressure corresponding to this H concentration is defined as the critical H pressure ( pHc). According to this definition, the values of the critical H concentration and the corresponding H pressure can be quantitatively determined from the following formulations. Under conditions where the concentration of H at the mH-V complex m ) and the interstitial (c ) are equal to each other: (c HV HI
Figure 4. The equilibrium H concentration as a function of temperature for H pressures of 0.17 MPa (red), 1 GPa (purple), 3 GPa (brown), 4 GPa (green), and 5 GPa (blue) in Mo.
The equilibrium H concentration and H trapping probability as a function of temperature and H pressure are further examined for Mo, as seen in figures 4–6. Previous first-principles calculation [19] proposed that the atomic configuration and evolution of the optimal charge density isosurface for multi-H occupation at the vacancy in both W and Mo are the same because they have the same crystal structures and belong to the same subgroup VI with exactly the same number of valence electrons. Nonetheless, a similar trend of H concentration changing with temperature and H pressure in W and Mo was observed. However, because the H formation energy, as well as the H trapping behavior at the vacancy varies between W and Mo [19], the absolute values of the H concentration, the H trapping probability, and the H pressure required for a sharp increase in the H concentration in Mo are different from those in W.
m =c c HV HI ⎛ E fm ⎞ ⎛ Ef ⎞ N mΩm ⋅ exp ⎜⎜− HV ⎟⎟ = I exp ⎜− HI ⎟ , ⎝ kBT ⎠ ⎝ kBT ⎠ NM
(12)
and substituting the expressions of formation energies in equations (2) and (3) into equation (12), we can obtain the corresponding H chemical potential ( μHm) for the critical H concentration at which the H concentration at the mH-V complex is equal to that at the interstitial as:
μ Hm =
1 m −E EHV 6 kBT H−TIS + N EBULK ln + . (13) m − 1 mΩ m m−1
Note from equation (13) that the H chemical potential for the critical H concentration is a function of the H number m at the monovacancy as well as the temperature. Therefore, we can derive the H chemical potential for a different mH at the mH-V complex for a certain temperature. The corresponding m can be then obtained with a substitution H concentration c HV of the H chemical potential μHm into equation (10). As a result, m is also a function of the H number at the mH-V complex c HV m. According to the above definition, the critical H concentration c Hc is thus derived as the minimal value out of the H concentrations at the mH-V complexes when it becomes equal m (m ) = c (m ) . to that at the interstitial, i.e. c Hc = min [c HV ] HI Moreover, the corresponding critical H pressure ( pHc) can be derived by substituting equation (12) into equation (4). Based on the above definition, the critical H concentration as a function of temperature in W and Mo is plotted in figures 2(b) and 5(b), respectively, which obviously demonstrates that the critical H concentration increases along with an increase in temperature. Because the formation energy of the H-V complex increases and the diffusion of H is enhanced as the temperature rises, the H-V complex formation becomes more difficult as a result. Figures 2(b) and 5(b) can be treated
3.2. Critical H concentration for H-monovacancy complex formation
According to the dependence of the H concentration on the temperature and the H pressure in W and Mo as revealed above, we propose that the accumulation of numerous H atoms in the monovacancy actually occurs at a critical H pressure. This indeed corresponds to the critical H concentration associated with this critical pressure for the H-V complexes formation at a certain temperature in both W and Mo. The critical H concentration and pressure are defined as follows, as shown in figure 7, and the temperature of 300 K is used as an example. Below a critical H pressure at a certain temperature, the number of interstitial H, i.e. c HI in equation (9), increases steadily with increasing pressure, and the H concentration is now mainly contributed to by the interstitial H. As the H pressure continues to increase, for a certain m mH-V complex, the number of H at the mH-V complex, i.e. c HV in equation (10), rapidly increases and becomes equal to that at the interstitial once the critical pressure is reached. Beyond this critical pressure, the number of H at the mH-V complex continues to increase and exceeds that at the interstitial. The H 5
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Figure 5. (a) The equilibrium H concentration as a function of H pressure for temperatures of 300 K (purple), 400 K (brown), 600 K
(green), and 900 K (blue) in Mo. The solid lines show the total H concentration and the dashed lines show the interstitial H concentration. (b) The critical H concentration as a function of temperature in Mo. The light yellow area represents the region of no H-V complexes formation and the light cyan area represents the region of formation and rapid growth of H-V complexes in Mo.
Figure 7. The critical H concentration (c Hc) and pressure (p Hc) for the H-V complexes formation at a temperature of 300 K in W. The dashed line represents the H concentration at the interstitial (c HI). The solid lines represent the H concentration at different mH-V m ). complexes (c HV
Figure 6. The H trapping probability as a function of H pressure temperatures of 300 K (purple), 400 K (brown), 600 K (green), and 900 K (blue) in Mo.
as phase diagrams, divided into two phase regions. Under the critical H concentration, the interstitial H atoms dominate and almost no H-V complex forms, corresponding to the light yellow regions in figures 2(b) and 5(b). Beyond the critical H concentration, numerous H-V complexes form and grow rapidly, corresponding to the light cyan regions. These H-V complexes combine to form larger clusters, eventually leading to H bubble formation. The critical H concentration (atomic) and the critical H pressure for temperatures of 500 K to 900 K in W and Mo are presented in table 2. The critical H concentration in W is 24 ppm, and the corresponding critical H pressure is 7.3 GPa at a typical temperature of 600 K. For Mo, the critical H concentration is 410 ppm, and the corresponding critical H pressure is 4.7 GPa at 600 K. Here, the critical H concentration for the H-V complex formation in W is lower compared to that in Mo. The first-principles calculation [19] revealed that the trapping energy of H in W is ~0.2 eV lower than that in Mo, and the sequentially trapped number of H at the vacancy in
Table 2. The critical H concentration c Hc (ppm) and the
corresponding critical H pressure p Hc (GPa) within a temperature T (K) range of 500 K to 900 K in W and Mo. W
Mo
T (K)
c Hc
pHc
c Hc
p Hc
500 600 700 800 900
2.7 24 110 350 870
7.0 7.3 7.7 8.2 8.7
73 410 1380 3400 7000
4.3 4.7 5.3 5.8 6.4
W (12 H) is larger than that in Mo (8 H). The first-principles result implies that the vacancy in W has a stronger capability than that in Mo to trap H to form H-V complexes, consistent with the results of the critical H concentration. This suggests the H-V complex formation and thus the H bubble formation in W is easier than that in Mo. 6
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Consequently, although the H pressure generated from the H plasma is much lower than the critical one, such high pressure conditions inside the metal being required to form H bubbles is still reasonable based on the above discussion.
3.3. Discussions and comparison with experiments 3.3.1. H pressure. During the actual operation of a future
fusion reactor, W and Mo PFMs are exposed to H ions of low energy E (0–100 eV) and high fluxes n (1024–1025 m−2s−1) at the divertor [33]. The H ions undergo elastic collision with the PFMs, and the H pressure pHs on the PFMs can be estimated by pHs = 3 4 n 2mHE . Taking E = 100 eV, n = 1027 m−2s−1 (here we take the flux as even higher than that at the divertor) and mH = 1.67 × 10−27 kg, we obtain the H pressure on PFMs as pHs = 0.17 MPa. At this pressure, the H concentration and the H trapping probability in W (figures 1 and 3) and Mo (figures 4 and 6) also increase along with an increase in temperature. The absolute value of the equilibrium H concentration is extremely low at ~1.6 × 10−4 ppm for W and ~2.3 × 10−2 ppm for Mo at 600 K (figures 1 and 3). Most H atoms are distributed at the interstitial sites, and the untrapped H atoms diffuse to the surface and then enter the plasma through a recombination process [34]. From table 2, it is clear that the critical pressure corresponds to the critical H concentration predicted as the order of GPa is much higher than the pressure on the PFMs generated from the H plasma (0.17 MPa) as in the actual case estimated above. Namely, to form an H bubble, a high H fugacity (pressure) is actually required [10], though such a high pressure might not be achieved during the operation of a fusion reactor. Inside the H bubble, however, a high internal H pressure of up to several to several tens of GPa has been demonstrated [35]. First, we can roughly estimate the H pressure pHv inside the mH-V complex in this calculation. We employ the ideal gas state equation pHv Vv = mRT , taking the temperature as being 300 K and the ⎛ ⎞3 volume of the monovacancy as Vv = 4 3 π ⎜ 3 4 a⎟ , where a is ⎝ ⎠
3.3.2. Critical concentration of H. Experimentally, the H concentration and bubble formation in metals are directly associated with the diffusivity of H, which was investigated as early as the 1960s. Frauenfelder [37] examined the diffusion behavior of H in W at temperatures of between 1100 K and 2400 K by measuring the degassing rates of a solid cylinder preloaded with H at a certain pressure, and gave the expression of H diffusivity as: D = 4.1 × 10−7exp(−0.39 eV / kT ) m2s−1. Zakharov et al [38] then determined the H diffusivity as D = 6.0 × 10−4exp(−1.07 eV / kT ) m2s−1 within a lower temperature range of 673 K to 1473 K by measuring the diffused H gas from the walls of a W tube. The diffusivity difference between the two works mainly originates from the different experimental temperature ranges [35]. First, several experimental studies have reported a retained D concentration under different radiation conditions in W. The D concentration was observed to be 550 ppm at a temperature of 700 K [39] and 400 to 500 ppm within a temperature range of 423 K to 973 K [40]. Furthermore, in a review by Causey and Venhaus [35], the D concentration was found to exceed from a few to a few tens of atomic ppm at temperatures below 1000 K with the D bubble formation. These experimentally observed H concentration values are comparable to our predicted critical H concentration results (see table 2). In the recent experimental work by Peng et al, the H isotope ion-driven permeation in W was investigated using a high flux ion beam test device coupled with a permeation device [41]. The W specimen with a thickness of 30 μm was exposed to deuterium (D) ions with an energy of 1 keV and a flux of 1019 ~ 1020 m−2s−1, and the permeation flux of D was measured at different irradiation temperatures. The atomic concentration of D was further derived according to the measured D permeation rate by using diffusivity measured by Zakharov and Frauenfelder [37, 38]. It is important to point out that the concentrations were indirectly determined from the measured experimental permeation rates. Therefore, the choice of diffusivity values plays a critical role in the calculated concentration values. It is suggested that an H bubble will form in W at certain temperatures when the D concentration is greater than the precipitation limit concentration, which is a function of D solubility and fugacity (SH p ), consistent with the previous experimental work on H bubble formation observed within the same temperature range [42]. Peng’s work provided clear information on the H bubble forms for those D concentration points at certain temperatures based on Zakharov’s diffusivity, which is determined within a temperature range of 673 K to 1473 K close to that for H bubble formation. As shown in figure 8, the experimental D concentration values with significant blistering and bubble formation observation for similar irradiation conditions [42, 43] (red filled triangles) are larger than the presently predicted critical values with one exception. Those D concentration points at which no bubble formation was experimentally
the metal lattice constant. The number of H atoms m at the monovacancy is set at 6 as an example. According to this estimation, the H pressure inside the mH-V complex is as high as 11.9 GPa. On the other hand, it is known that the surface of the metal is first exposed to a high flux of H plasma. Here, the incident flux of H at the surface is defined as Φs. A fraction of the H incident flux at the surface further permeates to the bulk, and such a permeation flux of H inside the metal is defined as Φb. Here, Φb = αΦs, where α is the fractional constant. The H permeation flux is known to be the function of the H solubility (SH), diffusivity (D H), and pressure. Thus, the resulting equa-
tion is: D Hb SH pHb = αD Hs SH pHs . Moreover, the diffusivity can be expressed as: D = D0 exp − Ea kBT where D0 is the preexponential factor and Ea is the activation energy of H, i.e. the H diffusion barrier. According to the above equations, we can roughly obtain the relationship between the H pressure at the bulk ( pHb) and the surface ( pHs ) as pHb = α 2⋅pHs ⋅exp 2Ebs kBT . Obviously, the migration barrier of H from the bulk to the surface (Ebs) should mainly be responsible for the large pressure difference. The derived H pressure in the bulk is then roughly estimated to be ~104 order of magnitude higher than that at the surface at a temperature of 600 K, based on the migration barrier (Ebs) of 0.27 eV reported in the previous work [36].
(
)
(
)
7
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reduce the thermal conductivity of metals. The high H pressure inside bubbles is also responsible for the crack extension that is a result of H embrittlement [5, 11]. Therefore, the H bubbles originating from the accumulation of large quantities of H into defects are associated with the failure of metallic PFMs in future fusion reactors. In other words, the H-induced failure or property degradation (both thermal and mechanical) of metallic PFMs can be characterized by H bubble formation in PFMs, which can be considered the starting point of the failure of metallic PFMs under H plasma irradiation. We expect the approach used here to predict the critical H concentration in metals will contribute to the evaluation of the H-induced failure of metallic PFMs under H plasma irradiation in future fusion reactors. 4. Conclusions
Figure 8. The critical H concentration for H bubble formation in
comparison with the experimental observations [41] employing Zakharov’s diffusivity. Red-filled and dark hollow triangles represent the H concentration points with and without H bubble formation estimated by the experimental works, respectively.
We calculated the equilibrium H concentration at the interstitial and monovacancy, as well as its dependence on temperature and H pressure in W and Mo using a thermodynamic model with previously calculated first-principle energetics of H as the necessary input data for the model. At a certain temperature, the H concentration exhibits a sharp increase beyond the critical H pressure, corresponding to a critical H concentration for H-V complex formation in W and Mo. Such critical concentrations and pressures are defined as the values when the concentration of H at one certain mH-V complex first becomes equal to that of H at the interstitial, which are 24 ppm/7.3 GPa and 410 ppm/4.7 GPa at 600 K in W and Mo, respectively. Beyond such critical H concentrations, numerous H atoms accumulate in the monovacancy, leading to the formation and rapid growth of H-V complexes, which is considered the preliminary stage of H bubble formation. We expect the proposed approach to determine the critical H concentration for H bubble formation in metals will be applied to more complicated cases such as vacancy clusters and grain boundaries.
observed (the dark hollow triangles) are within our predicted phase region without H bubble formation. This implies that an H bubble will form when the H concentration reaches a high value at high temperatures (because the predicted critical H concentration is high), but the experimentally measured H concentration is far lower than the critical values, resulting in no bubble formation. Unfortunately, the experimental results provide little information on the critical H concentration for bubble formation. However, at the two temperatures of 610 K and 710 K, we can estimate the critical concentration ranges, since both concentration points with and without H bubble formation are present at the same temperature. One (710 K) is consistent with the present prediction, while the other is inconsistent (610 K). Nevertheless, we have only considered the simplest case of the monovacancy in the present thermodynamic calculation. This means the present calculations cannot be directly compared with the experimental results. In actuality, different kinds of vacancy exist in metals including di-vacancy, tri-vacancy, and even large vacancy clusters and voids. Furthermore, other types of defect such as grain boundaries and dislocations also play similar roles as vacancies for H bubble formation [19]. However, the main purpose of the present paper is to propose an approach to predict the H concentration based on the derived formulations of thermodynamic models, including a clear quantitative definition of the critical H concentration. Such an approach can be further generalized to multi-vacancy cases, as well as the cases of other types of defect such as grain boundaries and dislocations. As a next step, using molecular dynamics simulations, we will further calculate the critical H concentration taking the vacancy clusters with different sizes in W into consideration.
Acknowledgments This work is supported by the National Magnetic Confinement Fusion Program through Grant No. 2013GB109002 and the National Natural Science Foundation of China (NSFC) through Grant No. 51371019. G H Lu wishes to acknowledge support from The National Science Fund for Distinguished Young Scholars through Grant No. 51325103. References [1] Janisch R and Elsasser C 2003 Phys. Rev. B 67 224101 [2] Domain C, Becquart C S and Foct J 2004 Phys. Rev. B 69 144112 [3] Lu G H, Zhang Y, Deng S H, Wang T M, Kohyama M, Yamamoto R, Liu F, Horikawa K and Kanno M 2006 Phys. Rev. B 73 224115 [4] Troiano A R 1960 Trans. Am. Soc. Met. 52 54 [5] Hirth J P 1980 Metall. Mater. Trans. 11A 861
3.3.3. Evaluation of the H-induced failure of PFMs.
The formation of H bubbles in metallic PFMs is quite harmful because H bubbles degrade the mechanical properties and 8
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