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Diffusion of hydrogen isotopes in 3C-SiC in HTR-PM: A first-principles study
Progress in Nuclear Energy xxx (xxxx) xxx
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Progress in Nuclear Energy journal homepage: http://www.elsevier.com/locate/pnucene
Diffusion of hydrogen isotopes in 3C-SiC in HTR-PM: A first-principles study Wenyi Wang a, b, c, d, Chuan Li a, b, c, Shun-Li Shang d, Jianzhu Cao a, b, c, Zi-Kui Liu d, Yi Wang d, Chao Fang a, b, c, e, * a
Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Advanced Nuclear Energy Technology, Beijing 100084, China The Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Beijing, 100084, China d Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA e Lab for High Technology, Tsinghua University, Beijing 100084, China b c
A R T I C L E I N F O
A B S T R A C T
Keywords: Tritium Silicon carbide Nuclear Diffusion Activation energy
Silicon carbide (SiC) is the main diffusion barrier of tri-isotropic particles in high-temperature gas-cooled reactor pebble-bed modules (HTR-PMs). When analyzing the source term of tritium in a HTR-PM primary circuit, it is essential to determine the amount of tritium released, which requires the diffusion coefficient of tritium in SiC. In the present work, the diffusion behavior of hydrogen in 3C-SiC is investigated using density functional theorybased first-principles calculations, which can be used to estimate the diffusivity of tritium while ignoring the neutron and mass effects. Different charge states of hydrogen (i.e., H0, H and Hþ) and all possible low-energy configurations and diffusion paths are also considered in the calculations. The results indicate that the charge state of hydrogen changes from negative to positive with the increase in Fermi energy. In addition, the most stable position of H in SiC is a tetrahedral site surrounded by four Si atoms. Furthermore, minimum diffusion barriers of 0.45, 0.46 and 1.52 eV are determined for H0, H and Hþ, respectively. The calculated diffusion coefficients from the present work agree well with those computed in the literature. In addition, the experimental results are closer to the negative hydrogen values computed in this study, indicating that the most likely charge state of hydrogen is negative. Our calculations provide a good reference for nuclear safety evaluation in HTR-PMs using the diffusivity of hydrogen in SiC.
1. Introduction Because of its excellent performance under high mechanical stress and temperature, silicon carbide (SiC) has been applied to advanced high-temperature nuclear fission reactors and future fusion reactors (Katoh et al., 2012; Malherbe et al., 2008; Olander, 2009). In high-temperature gas-cooled reactor pebble-bed modules (HTR-PMs), the most common fuel particles are tri-isotropic (TRISO)-coated parti cles (Fig. 1 (Malherbe, 2013)), which, from the inside to the outside, are composed as follows: (1) low-enriched UO2 kernels encapsulated by four successive layers, (2) porous pyrolytic carbon (buffer layer), (3) inner pyrolytic carbon, (4) SiC and (5) outer pyrolytic carbon. The main advantage of TRISO fuel particles is their excellent retention ability to various radioactive fission products under the operating conditions of small modular HTRs. As the main barrier for diffusion in fuel particles,
SiC plays an important role in preventing the release of radioactive fission products into the primary circuit under normal operation and accident conditions (Lee et al., 2007). Bulk SiC has six commonly used stacking configurations: 3C (zinc blende), 2H (wurtzite), 4H, 6H, 15R and 21R (Bekaroglu et al., 2010), with C, H and R representing cubic, hexagonal and rhombohedral crystal structures, respectively. The SiC used in TRISO fuel particles in HTR-PMs is 3C-SiC (or β-SiC). The interaction between adjacent atoms in SiC is very strong because each silicon is bonded with four carbon atoms, and vice versa for each carbon, which makes the structure of 3C-SiC especially stable (Wang et al., 2017). Tritium, as one of the hydrogen isotopes, has been strongly focused on in HTR-PMs because it emits harmful β-radiation (18.6 keV) and has a 12.3 years half-life. In HTR-PMs, there are two ways tritium can be generated: the ternary fission reaction in the fuel elements and the
* Corresponding author. Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China. E-mail address: [email protected] (C. Fang). https://doi.org/10.1016/j.pnucene.2019.103181 Received 2 July 2019; Received in revised form 9 September 2019; Accepted 12 October 2019 0149-1970/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Wenyi Wang, Progress in Nuclear Energy, https://doi.org/10.1016/j.pnucene.2019.103181
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should be emphasized that, although a few theoretical studies have been conducted on the interactions between H and 3C-SiC (Kim et al., 2009) (Aradi et al., 2001), a distinct diffusion pathway and the corresponding diffusion coefficient and diffusion mechanism are still unclear. These will be discussed in this paper.
neutron activation reaction. Most activation reaction of tritium is pro duced outside the TRISO fuel particles, and there are three main reaction sources: 3He (n, p) 3H, 6Li (n,α) 3H and 10B (n, 2α) 3H. It should be clarified that the tritium generated by activation reactions is released into the primary circuit, whereas for ternary fission reactions, a com plete SiC layer can be assumed to prevent all tritium from being released into the primary circuit. Therefore, only tritium in the defective particles will diffuse in the primary circuit, which is much less than that of the activation reaction. However, this assumption may not be so conserva tive, because the diffusion behavior of tritium in SiC is unclear. When analyzing the source term of tritium in the HTR-PM primary circuit, it is important and necessary to know the amount of tritium released from the fuel particles. In other words, the amount of tritium that will penetrate from the SiC layer into the primary circuit must be deter mined. If the results indicate that SiC is effective at blocking the diffu sion of tritium, all the tritium in the primary circuit is derived from the activation reaction. If the opposite is true, part of the tritium in the primary circuit may be considered to come from the ternary fission re action. The experiment can only provide a value of the diffusion coef ficient, and the value obtained by different experiments differs significantly (Causey et al., 1978; Esteban et al., 2002; Tam et al., 1995). In this case, the key point is to determine the diffusion coefficient of tritium in SiC. Furthermore, detailed mechanisms of diffusion are essential, which cannot directly be obtained from experimental analysis. To address this challenge, density functional theory (DFT)-based first-principles calculations (Kohn and Sham, 1965), as implemented in the Vienna Ab initio Simulation Package (VASP) (Kresse and Furth müller, 1996; Kresse and Hafner, 1993; Mehrer, 2007), are performed in the present work to provide a methodology at a level of accuracy close to or even exceeding that of the experimental data (Horn et al., 1988; Kohn and Sham, 1965; Lamble et al., 1988; Persson and Ishida, 1990). Several theoretical studies have been conducted on the behavior of H in SiC in terms of first-principles calculations. For example, the formation en ergies, defects and stable charge states of interstitial H in cubic and hexagonal SiC have been studied (Aradi et al., 2004; Chu and Estreicher, 1990; Kaukonen et al., 2003; Roberson and Estreicher, 1991). Aradi et al. (2004) carried out calculations using the FHI98MD code and the local density approximation to study the diffusion of H in perfect, p-type doped and radiation-damaged 4H-SiC. Chu and Estreicher (1990) calculated the similarities, differences and trends regarding the prop erties of interstitial H in cubic C, Si, BN, BP, AlP and SiC. In the present study, the first-principles approach is implemented to investigate the defect formation energy, the minimum energy diffusion path, the diffusion barrier and the diffusion coefficient of H in 3C-SiC. It
2. Methodology 2.1. Charge state When studying the diffusion of hydrogen atoms in SiC, it is necessary to consider the defect (H atom) in different charge states (Aradi et al., 2001). The most stable positions for different charged hydrogen atoms are different. Kaukonen et al. (2003) found that only singly positive or negative charge states of hydrogen are thermodynamically stable in SiC. In order to determine the charge state of hydrogen in 3C-SiC, the defect formation energies of hydrogen should be calculated in negative (H 1), neutral (H0) and positive (Hþ1) charge states. The formation energy of charged hydrogen in SiC can be written as Van de Walle and Neugebauer (2004) Ef ðH q Þ ¼ EðH q Þ
EðSiCÞ
nH μH þ q½EF þ EVBM ðSiCÞ�
(1)
where EðSiCÞ and EðHq Þ are the total energies of a perfect supercell of SiC without and with a hydrogen defect in charged state q, respectively, nH is the number of hydrogen atoms added (ni > 0) or removed (ni < 0) from the perfect supercell and is equal to unity in this study, μH is the chemical potential of hydrogen, EF is the Fermi energy ranging from the valence band maximum (VBM) to the conduction band minimum, and EVBM ðSiCÞ is the VBM of the perfect SiC. For a given value of EF , hydrogen is stable at the charge state with the lowest formation energy. 2.2. Diffusion coefficient The diffusion coefficient is an important parameter to determine and quantify the transport of radionuclides out of the fuel elements. The diffusion coefficient and its temperature dependence is commonly expressed by the Arrhenius equation (Lu and Zhang, 2013), D ¼ D0 e
Q=kB T
(2)
where Q is the activation energy for diffusion, D0 is a pre-exponential factor that is usually assumed to be independent of temperature T and kB is the Boltzmann constant. The activation energy Q can be calculated by the VASP, whereas D0 cannot be obtained directly. In order to obtain
Fig. 1. Configuration of TRISO-coated particle. 2
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the value of D0, we must fit a curve of the diffusion coefficient as a function of temperature (D–T curve). According to Wert and Zener (1949), the diffusion coefficient D of an interstitial atom can be described by
performed by the VASP code (Kresse and Furthmüller, 1996). The electron–ion interaction is described by the projector-augmented wave method (Kresse and Hafner, 1994; Kresse and Joubert, 1999; Vanderbilt, 1990), and the correlation energy and electron exchange are treated with the generalized gradient approximation of the Per dew–Burke–Ernzerhof form (Perdew et al., 1996). In the VASP calcu lations, a 2 � 2 � 2 supercell of 3C-SiC containing 64 atoms is employed for the simulations (shown in Fig. 2). The neutron and mass effects are not involved, because the strength of the gravitational field is consid erably weaker than that of electromagnetic interaction on this scale. The jump barrier is determined by the electronic interaction and not by the mass of the atom, which makes the energy barriers of the two isotopes identical (Aradi et al., 2001; Kim et al., 2009). Therefore, when calcu lating the energies, the behavior of the tritium atom can be described in terms of a hydrogen (H) atom. The cutoff energy for the plane-wave bases is 400 eV, and the Brillouin-zone integrations are carried out using 4 � 4 � 4 k-point meshes (Kaukonen et al., 2003). The electronic iteration convergence is at least 10 4 eV/atom. The lattice constant of the 3C-SiC is 4.38 Å according to the calculations presented in this paper, and it compares well to the experimental and theoretical values of 4.359 Å (Sherwin and Drummond, 1991) to 4.39 Å (Kim et al., 2009) in the literature. One H atom is positioned at various interstitial sites of 3C-SiC (see details in Sec. 3.1), and the structure is relaxed until all forces acting on the atoms are smaller than 0.01 eV/Å. The site with the lowest energy is defined as the IS, and the nearby site with the second-lowest energy is selected as the final state. The minimum energy pathway (MEP) and the corresponding activation energy Q value are determined using the climbing image-nudged elastic band (CI-NEB) method (Henkelman et al., 2000). In the present work, three images are used to calculate the energy barrier as more images cannot improve the results based on our tests (see Fig. 3). Because the mass of hydrogen isotopes is very small, there is no distinct difference among the diffusion coefficients of hydrogen, deuterium and tritium by the Debye method. Thus, phonon calculations are performed using the force constant, which is predicted by the VASP 5.2 code, and the phonon properties, calculated by the Yphon code (Wang et al., 2012, 2010).
(3)
D ¼ nβd 2 Γ
where n is the number of nearest-neighbor stable interstitial sites, β is the probability an interstitial atom jumps to a nearest-neighbor site and d is the projected length onto the direction of diffusion. For the present interstitial diffusion in the diamond structure (Jones, 2008), n ¼ 4 and β ¼ 1=6. Γ is the jump rate between two adjacent sites of the diffusion particle at a temperature T (Heinola and Ahlgren, 2010; Wert and Zener, 1949; Wimmer et al., 2008), � � kB T ZTS ΔE Γ¼ exp (4) h ZIS kB T where ZIS and ZTS are the partition functions for the initial state (IS) and transition state (TS), respectively, h is Planck’s constant and ΔE is the energy difference between TS and IS. At high temperatures (hv= 2kT≪ 1), the jump rate of Eq. (4) reduces to the results found by Vineyard (1957), Q3N 3 IS � � ΔE i¼1 vi Γ ¼ Q3N exp (5) 4 TS kB T i¼1 vi TS where vIS i and vi are the vibrational frequencies at the center of the Brillouin zone for IS and TS, respectively, and N is the total number of atoms in the supercell. At low temperatures, the jump rate of Eq. (4) reduces to the results found by Eyring (1935): � � kB T ΔE þ ΔEZP exp Γ¼ (6) h kB T
where ΔEZP is the difference in zero-point vibrational energy between the IS and TS. In the present work, ΔE in Eq. (4) and ΔEþ ΔEZP in Eq. (5) will be replaced by the temperature-dependent Gibbs energy difference ΔG between the IS and TS. In the case of zero external pressure, ΔG is equal to the Helmholtz energy difference ΔF, which can be estimated by the quasiharmonic approach (Shang et al., 2010), FðV; TÞ ¼ EðVÞ þ Fvib ðV; TÞ þ Fel ðV; TÞ
3. Results and discussion
(7)
In the present work, the most stable interstitial site must be selected to determine the formation energy of hydrogen and the IS of diffusion. The total energies of the H atom at various interstitial sites from the
where E is the total static energy at T ¼ 0 K determined by the fourparameter Birch–Murnaghan equation of state fitting (Shang et al., 2010), EðVÞ ¼ a þ bV
2 3
þ cV
4 3
þ dV
2
(8)
where a, b, c and d are parameters fitted to the equilibrium properties at seven different volumes, including the equilibrium volume (V0 ), energy (E0 ), bulk modulus (B0 ) and its pressure derivative (B’0 ) (Shang et al., 2010). Fvib in Eq. (7) is the vibrational contribution that can be obtained by the Debye model or the phonon density of state (DOS), and is the thermal electronic contribution for metals because of the nonzero electronic DOS at the Fermi level. From the above, the pre-factor D0 and the activation energy Q in the Arrhenius equation (Eq. (2)), as constant values, can be determined once the curve of D–T is fitted by Eq. (3). The purpose of this research is to compare the diffusion of different H ions in SiC and obtain the diffusion coefficient under different charged states. Further, the chemical forms of H in SiC is clarified to understand the microscopic mechanism of its diffusion. 2.3. Computational details
Fig. 2. Supercell of 3C-SiC lattice with 2 � 2 � 2 primitive cells, where the blue atoms represent silicon and the brown atoms represent carbon.
All DFT-based first-principles calculations in the present work are 3
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Fig. 3. Diffusion energy barriers of Hþ in 3C-SiC with different images (3/5/7) on the diffusion pathway.
literature (Kaukonen et al., 2003; Roberson and Estreicher, 1991) indicate that the H atom is stable at the tetrahedral (T) sites. Fig. 4 il lustrates two different Wyckoff positions as the tetrahedral interstitial sites (TSi site and TC site) and two other sites in 3C-SiC (C1 site and C2 site). The Tc and TSi sites are surrounded by four C atoms and four Si atoms, respectively, where the C1 site is the anti-bond site of the C atom with a distance of 1.132 Å, and the C2 site is close to the C atom with a
distance of 1.147 Å on the diagonal of the octahedral composed of six C atoms. It can be seen in Table 1 that the most stable site for H0 and Hþ is the C1 site, while H prefers TSi . For H0 , among all the stable sites, the energy of the C1 site is lower than that of the C2 site by 0.10 eV and that of the TSi site by 0.34 eV. Thus, the charged hydrogen atom at C1 site will be treated as the IS, while those at the C2 site and TSi site will be the potential final states for the diffusion of H0 . Because the energy differ ence between the C1 and TC sites is too high, the TC site will not be considered a potential final state for diffusion. Similarly, for Hþ , the possible MEP is from the C1 to C2 sites; for H , the possible MEP is from the (i) TSi to C1 site and (ii) TSi to C2 site. The most stable sites of H obtained in this study are consistent with other data in the literature (Kaukonen et al., 2003; Zhang and Zhang, 2009). The predicted length of the C–H bond for H at C1 in the present work is 1.13 Å, which is similar to those obtained by Aradi (1.14 Å) (Aradi et al., 2001) and Zhang and Zhang (1.13 Å) (Zhang and Zhang, 2009). The experimental bandgap value of 2.417 eV (Gary Lynn Harris, n.d.) is used in this work, because the common practice in the literature (Bockstedte et al., 2003) is to choose an experimental bandgap value for determining the charge states. The relative stabilities of the three charge states based on the Fermi level are shown in Fig. 5. The cross point of the Hþ and H states is EVBM þ 1:15 eV, so the possible charge states of hydrogen are Hþ and H . In this study, in order to compare the diffu sivities of different charge states, all three charged hydrogen atoms are considered. For H0 , C1 has the lowest energy of formation, as shown in Table 1, and is thus set as the IS for diffusion with three probable MEPs: (1) a direct path from C1 to C10 (C10 is the nearest site equivalent to C1), (2) from C1 to TSi and (3) a zigzag path from C1 to C2 and then to TSi . As illustrated in Table 2, path 1 is the most likely trajectory for H0 by comparing the TS energy. For Hþ , the energy barrier of path 1 is 0.46 eV, which is lower than the energy difference between any two other stable Table 1 Total relaxation energies of H atom located at the four different interstitial sites in crystalline 3C-SiC. The four sites are defined in Fig. 4. Interstitial sites
Fig. 4. Four different interstitial sites of H in β-SiC with C atoms at 4a(0,0,0) and Si atoms at 4c(0.25,0.25,0.25). (a) Tc site at 4d(0.75,0.75,0.75) surrounded by four C atoms, (b) TSi site at 4b(0.5,0.5,0.5) surrounded by four Si atoms, (c) C1 site (close to C atom with distance of 1.132 Å in the carbon-filled TSi site, which can be called the anti-bond site of C) and (d) C2 site (close to C atom with distance of 1.147 Å on the diagonal of the octahedral, which is composed of six C atoms).
Eo (eV) H0
H
Tc
485.05
494.57
475.29
485.71
493.69
477.50
C1 C2
486.05 485.95
495.60 495.43
476.25 476.14
TSi
4
Hþ
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this study indicate that the energy barriers for the aforementioned zigzag path (C1-C2-TSi -C20 -C10 ) and the direct path (C1-TSi ) are identical (0.51 eV, see Table 2). When the H atom diffuses from the IS to the final site, the energy profile can be described by a sinusoidal curve, which was first suggested by Wert and Zener (1949). Fig. 7 shows the energy barrier for H diffu sion in 3C-SiC with different charge states in comparison to the previous DFT and experimental results (Causey et al., 1978; Esteban et al., 2002; Kaukonen et al., 2003; Tam et al., 1995). It can be seen that the calcu lated results for H0 and Hþ in the present work and those in the literature are very close; however, the experimental values differ significantly. To determine why, a Si- or C-vacancy is added during the diffusion calcu lations of H0 . Shrader at al. studied the diffusion of silver in 3C-SiC (Shrader et al., 2011) and found that vacancy could cause the migra tion barrier at the interstitial position lower than that of silver substituting a Si or C atom. Therefore, a vacancy may play a key role in regulating the energy barrier of H in SiC. Fig. 7 depicts that the existence of vacancies raises the energy bar rier. The energy barrier for H0 in defect-free 3C-SiC, with a Si vacancy and with a C vacancy, is 0.52, 1.09 and 1.71 eV, respectively. This in dicates that the C vacancy may cause a larger increase in diffusion activation energy than the Si vacancy. Because a complete diffusion path should be diffused from one stable position to another, it is desirable that the impurity atoms can diffuse out rather than oscillate inside the crystal lattice. Therefore, the initial state and final state of a complete diffusion path are usually two physically equivalent locations. Therefore, the energies of initial state and final state for Hþ and H0 are the same and set to 0 eV. For H , the energies of initial state and final state are different,
Fig. 5. Relative stability of the interstitial charged hydrogen atom in 3C-SiC as a function of the Fermi energy.
sites. Thus, there is no need to calculate the barriers for the other paths. For H , the most stable site, TSi , is treated as the IS. The potential final states for the migration of H are the C1 and C2 sites. The CI-NEB calculation for path 4 does not converge with the energy barrier of roughly 2.36 eV because of the difficulty of computation (see Table 2). Path 5 gives a MEP value of 1.52 eV (see Table 2). It should be noted that Kaukonen et al. (2003) calculated the diffusion energy barriers for H0 (0.5 eV) and Hþ (0.5 eV), which are close to our values of 0.45 and 0.46 eV, respectively (see Table 2). Similar to the present case, a value more than 2 eV was estimated for H because of computational diffi culties. Note that the calculations by Kaukonen et al. (2003) were per formed by DFT with two complementary programs, i.e., AIMPRO and FINGER, and their supercell of 3C-SiC contained 128 atoms and the k-point was 2 � 2 � 2. Fig. 6 shows a schematic of different interstitial diffusion paths in 3CSiC. Path 1 is the MEP for H0 and Hþ , in which the hydrogen atom first starts from the C1 site and then jumps from C1 to C1’ the next-nearest neighbor of the carbon atom, identical to the diffusion path in the literature (Kaukonen et al., 2003). Path 2 starts from the C1 site via C2 to TSi , where the H atom at TSi has six equivalent diffusion directions. Finally, the complete diffusion path is expressed as C1-C2-TSi -C20 -C10 , and the energy barrier for the TSi -C20 -C10 path is the reverse of that of C1-C2-TSi . Path 3 is a direct jump from C1 to TSi . The CI-NEB results in
Table 2 Energy barriers (Q) for H with different charges along possible diffusion path ways, as shown in Fig. 6. Paths No.
1 2 3 4 5
Paths
C1 to C10 C1 to C2 C2 to TSi C1 to TSi
Q (eV) H0
Hþ
0.45 0.1 0.41
0.46
H
0.51
TSi to C1
2.36 (not converged)
TSi to C2
1.52
Fig. 6. (a) Schematic of the different diffusion paths for the hydrogen atom in 3C-SiC. Path 1: red line from C1 to C1’ the next-nearest neighbor of the carbon atom. Path 2: black line from C1 via C2 to TSi , i.e., C1-C2-TSi -C20 -C10 . Path 3: green line from C1 to TSi . The specific MEP for different charge states is shown in (b), (c) and (d). 5
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Fig. 7. Diffusion energy barriers of H/D/T in 3C-SiC from the calculations in this study and experiments in the literature denoted by H0&Hþ_calc. (Kaukonen et al., 2003), Expt.1 (Esteban et al., 2002), Expt.2 [40] and Expt.3 (Tam et al., 1995). It should be clarified that the reaction coordinates of the saddle point from the literature (Causey et al., 1978; Esteban et al., 2002; Kaukonen et al., 2003; Tam et al., 1995) are set by the author, in order to visually compare the literature data to the calculation results of this study.
Fig. 8. Diffusion coefficient of different charge states of H in 3C-SiC from the calculations in this study and experiments in the literature denoted by Expt.1 (Esteban et al., 2002), Expt.2 (Causey et al., 1978) and Expt.3 (Tam et al., 1995). Because the difference in diffusion coefficient for H, D and T is too small (see Table 3), only the results of H are presented in this figure. The X-axis is the reciprocal of temperature multiplied by 1000 and the unit of temperature(T) is Kelvin(K).
because the MEP is from TSi to C2 in my research. Actually, the complete path is TSi -C2-C2’-TSi , while the energy barrier can be shown only by plotting the half path TSi -C2. According to Fig. 7, the energy barrier for H is 1.52 eV, which is between the two experimental estimations of 1.12 (Esteban et al., 2002) and 2.82 eV (Causey et al., 1978), respec tively. This indicates that the most likely charge state of hydrogen atom is negative. All the energy barrier data are listed in Table 3. The diffusion coefficients of charged hydrogen atoms in this study and in the literature listed in Table 3 are plotted in Fig. 8. The diffusion coefficients of Hþ from the Debye and Yphon method from the present work are depicted as a red line and black line, respectively, which are very close to each other, with an energy barrier of 0.46 eV by the phonon method and 0.53 eV by the Debye method. The values of experiments 1 and 3 are almost the same. The green line represents the diffusion co efficient of H calculated by the Debye method, which is between the lines of experiment 1 and 2. As the diffusion coefficient differences for H, D and T are too small (see Table 3), we only present the results of H in Fig. 8. The diffusion coefficients for T can be obtained as follows: DTþ ¼ 3:03 � 10 9 e
0:46eV=kB T
(9)
DT ¼ 8:53 � 10 7 e
1:52eV=kB T
(10)
not be able to fully block the release of tritium. When analyzing the source term in HTR-PM, the release rate and release fraction of the fission product from the TRISO-coated particles and fuel elements can be computed by a diffusion model, where the diffusion activation energy of tritium in 3C-SiC is one of the key input parameters and is usually set to zero because of the lack of data, which makes the computed results too conservative. Consequently, our results can be used to analyze the source term in the HTR-PM primary circuit. 4. Summary and conclusions In this study, the diffusion behaviors of H isotopes in 3C-SiC were studied using DFT-based first-principles calculations. It was found that the most stable interstitial site for Hþ and H0 was C1, which is near C with a distance of about 1.13 Å. The MEP was determined after all possible diffusion paths were analyzed by the CI-NEB results, and the results indicate that the MEP was C1-C10 for the impurities Hþ and H0 , and TSi -C2 for H . Only positive or negative charge states of the hydrogen atoms were found to be thermodynamically stable in SiC. Therefore, the chemical form where H actually exists in SiC is positively or negatively charged, because the behavior of the tritium atom can be described in terms of a hydrogen (H) atom. The diffusion coefficient of tritium for the positive charge state is DTþ ¼ 3:03 � 10 9 e 0:46eV=kB T , and that for the negative charge state is DT ¼ 8:53 � 10 7 e 1:52eV=kB T .
In comparison to the activation energies of 7.9 and 5.14 eV for Ag and Cs in 3C-SiC (Malherbe, 2013), respectively, the low activation energy (0.46 and 1.52 eV) in the present work indicates that 3C-SiC may
Table 3 Diffusion energy barriers and pre-factors for H/D/T atoms with different charges in this study and in the literature. D0 (m2 s 1) þ H D T H H H H D T T T a
3C-SiC (Phonon) 3C-SiC (Phonon) 3C-SiC (Phonon) 3C-SiC (Debye) 3C-SiC (Vsi) 3C-SiC (Vc) 3C-SiC 3C-SiC 3C-SiC 3C-SiC 3C-SiC
4.94 � 10 3.61 � 10 3.03 � 10 3.34 � 10
9
1.10 � 10 1.72 � 10 3.41 � 10 1.69 � 10
4a
Q (eV) 0
– 6
2.99 � 10
7
1.32 � 10 9.97 � 10 8.53 � 10 4.22 � 10
2.80 � 10 1.58 � 10
3
4.55 � 10 7.30 � 10
4
9 9 7
2 4 4a
4
þ 7 7 7
5
0.46 0.46 0.46 0.53 0.5 1.12a 2.82a 3.19a 1.11a
0
0.52 1.09 1.71 0.5
Ref.
Temp. (K)
DFT DFT DFT DFT DFT DFT Calc. Expt.1 Expt.2
This work This work This work This work This work This work Kaukonen et al. (2003) Esteban et al. (2002) Causey et al. (1978)
300–2000 300–2000 300–2000 300–2000
Expt.3
Tam et al. (1995)
1373–1873
– 1.52 1.52 1.52 1.58 1.22 >2
The energy barrier and pre-factor in Expt.1 and Expt.3 do not refer to any particular charge state. 6
Method
675–1029 773–1573
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The calculated diffusion coefficients from this study agree well with the computations in previous studies, and the experimental results are closer to the computed values of negative hydrogen, indicating that the most likely charge state of hydrogen is negative. The results in this work could support the study of behavior of tritium in HTR-PM and the cor responding new diffusion and release model of tritium in fuel particles should be built in the next step.
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B 41, 7892–7895. https://doi.org/10.1103/ PhysRevB.41.7892. Vineyard, G.H., 1957. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids 3, 121–127. https://doi.org/10.1016/0022-3697(57)90059-8. Wang, W., Li, C., Cao, J., Fang, C., 2017. Adsorption behaviors of cobalt on the graphite and SiC surface: a first-principles study. Sci. Technol. Nucl. Install. 2017 (1–8) https://doi.org/10.1155/2017/8296387. Wang, Y., Shang, S., Liu, Z.-K., Chen, L.-Q., 2012. Mixed-space approach for calculation of vibration-induced dipole-dipole interactions. Phys. Rev. B 85, 224303. https:// doi.org/10.1103/PhysRevB.85.224303. Wang, Y., Wang, J.J., Wang, W.Y., Mei, Z.G., Shang, S.L., Chen, L.Q., Liu, Z.K., 2010. A mixed-space approach to first-principles calculations of phonon frequencies for polar materials. J. Phys. Condens. Matter 22, 202201. https://doi.org/10.1088/ 0953-8984/22/20/202201. Wert, C., Zener, C., 1949. Interstitial atomic diffusion coefficients. Phys. 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Declaration of competing interest The authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgements This work is supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China, China [grant number ZX06901]; Pennsylvania State University, United States; the Office of Science of the U.S. DOE, United States [grant number DEAC02-05CH11231]; and the National Science Foundation, United States [grant number ACI-1053575]. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pnucene.2019.103181. References Aradi, B., De� ak, P., Gali, A., Son, N.T., Janz� en, E., 2004. Diffusion of hydrogen in perfect, p-type doped, and radiation-damaged 4H SiC. Phys. Rev. B 69, 233202. https://doi. org/10.1103/PhysRevB.69.233202. Aradi, B., Gali, A., De� ak, P., Lowther, J.E., Son, N.T., Janz� en, E., Choyke, W.J., 2001. Ab initio density-functional supercell calculations of hydrogen defects in cubic SiC. Phys. Rev. B 63, 245202. https://doi.org/10.1103/PhysRevB.63.245202. Bekaroglu, E., Topsakal, M., Cahangirov, S., Ciraci, S., 2010. First-principles study of defects and adatoms in silicon carbide honeycomb structures. Phys. Rev. B 81, 075433. https://doi.org/10.1103/PhysRevB.81.075433. Bockstedte, M., Mattausch, A., Pankratov, O., 2003. Ab initio study of the migration of intrinsic defects in 3 C SiC. Phys. Rev. B 68, 205201. https://doi.org/10.1103/ PhysRevB.68.205201. Causey, R.A., Fowler, J.D., Ravanbakht, C., Elleman, T.S., Verghese, K., 1978. Hydrogen diffusion and solubility in silicon carbide. J. Am. Ceram. Soc. 61, 221–225. https:// doi.org/10.1111/j.1151-2916.1978.tb09284.x. Chu, C.H., Estreicher, S.K., 1990. Similarities, differences, and trends in the properties of interstitial H in cubic C, Si, BN, BP, AlP, and SiC. Phys. Rev. B 42, 9486–9495. https://doi.org/10.1103/PhysRevB.42.9486. Esteban, G.A., Perujo, A., Legarda, F., Sedano, L.A., Riccardi, B., 2002. Deuterium transport in SiCf/SiC composites. J. Nucl. Mater. 307–311, 1430–1435. https://doi. org/10.1016/S0022-3115(02)01282-5. Eyring, H., 1935. The activated complex in chemical reactions. J. Chem. Phys. 3, 107–115. https://doi.org/10.1063/1.1749604. Gary Lynn Harris, n.d. Properties of Silicon Carbide. Heinola, K., Ahlgren, T., 2010. Diffusion of hydrogen in bcc tungsten studied with first principle calculations. J. Appl. Phys. 107, 113531. https://doi.org/10.1063/ 1.3386515. Henkelman, G., Uberuaga, B.P., J� onsson, H., 2000. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 113, 9901. https://doi.org/10.1063/1.1329672. Horn, K., Hohlfeld, A., Somers, J., Lindner, T., Hollins, P., Bradshaw, A.M., 1988. Identification of the s-derived valence-electron level in photoemission from alkalimetal adlayers on aluminum. Phys. Rev. Lett. 61, 2488–2491. https://doi.org/ 10.1103/PhysRevLett.61.2488. Jones, S.W., 2008. Diffusion in Silicon. Katoh, Y., Snead, L.L., Szlufarska, I., Weber, W.J., 2012. Radiation effects in SiC for nuclear structural applications. Curr. Opin. Solid State Mater. Sci. 16, 143–152. https://doi.org/10.1016/J.COSSMS.2012.03.005. Kaukonen, M., Fall, C.J., Lento, J., 2003. Interstitial H and H2 in SiC. Appl. Phys. Lett. 83, 923–925. https://doi.org/10.1063/1.1598646. Kim, J.H., Kwon, Y.D., Yonathan, P., Hidayat, I., Lee, J.G., Choi, J.-H., Lee, S.-C., 2009. The energetics of helium and hydrogen atoms in β-SiC: an ab initio approach. J. Mater. Sci. 44, 1828–1833. https://doi.org/10.1007/s10853-008-3180-2.
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Progress in Nuclear Energy journal homepage: http://www.elsevier.com/locate/pnucene
Diffusion of hydrogen isotopes in 3C-SiC in HTR-PM: A first-principles study Wenyi Wang a, b, c, d, Chuan Li a, b, c, Shun-Li Shang d, Jianzhu Cao a, b, c, Zi-Kui Liu d, Yi Wang d, Chao Fang a, b, c, e, * a
Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Advanced Nuclear Energy Technology, Beijing 100084, China The Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Beijing, 100084, China d Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA e Lab for High Technology, Tsinghua University, Beijing 100084, China b c
A R T I C L E I N F O
A B S T R A C T
Keywords: Tritium Silicon carbide Nuclear Diffusion Activation energy
Silicon carbide (SiC) is the main diffusion barrier of tri-isotropic particles in high-temperature gas-cooled reactor pebble-bed modules (HTR-PMs). When analyzing the source term of tritium in a HTR-PM primary circuit, it is essential to determine the amount of tritium released, which requires the diffusion coefficient of tritium in SiC. In the present work, the diffusion behavior of hydrogen in 3C-SiC is investigated using density functional theorybased first-principles calculations, which can be used to estimate the diffusivity of tritium while ignoring the neutron and mass effects. Different charge states of hydrogen (i.e., H0, H and Hþ) and all possible low-energy configurations and diffusion paths are also considered in the calculations. The results indicate that the charge state of hydrogen changes from negative to positive with the increase in Fermi energy. In addition, the most stable position of H in SiC is a tetrahedral site surrounded by four Si atoms. Furthermore, minimum diffusion barriers of 0.45, 0.46 and 1.52 eV are determined for H0, H and Hþ, respectively. The calculated diffusion coefficients from the present work agree well with those computed in the literature. In addition, the experimental results are closer to the negative hydrogen values computed in this study, indicating that the most likely charge state of hydrogen is negative. Our calculations provide a good reference for nuclear safety evaluation in HTR-PMs using the diffusivity of hydrogen in SiC.
1. Introduction Because of its excellent performance under high mechanical stress and temperature, silicon carbide (SiC) has been applied to advanced high-temperature nuclear fission reactors and future fusion reactors (Katoh et al., 2012; Malherbe et al., 2008; Olander, 2009). In high-temperature gas-cooled reactor pebble-bed modules (HTR-PMs), the most common fuel particles are tri-isotropic (TRISO)-coated parti cles (Fig. 1 (Malherbe, 2013)), which, from the inside to the outside, are composed as follows: (1) low-enriched UO2 kernels encapsulated by four successive layers, (2) porous pyrolytic carbon (buffer layer), (3) inner pyrolytic carbon, (4) SiC and (5) outer pyrolytic carbon. The main advantage of TRISO fuel particles is their excellent retention ability to various radioactive fission products under the operating conditions of small modular HTRs. As the main barrier for diffusion in fuel particles,
SiC plays an important role in preventing the release of radioactive fission products into the primary circuit under normal operation and accident conditions (Lee et al., 2007). Bulk SiC has six commonly used stacking configurations: 3C (zinc blende), 2H (wurtzite), 4H, 6H, 15R and 21R (Bekaroglu et al., 2010), with C, H and R representing cubic, hexagonal and rhombohedral crystal structures, respectively. The SiC used in TRISO fuel particles in HTR-PMs is 3C-SiC (or β-SiC). The interaction between adjacent atoms in SiC is very strong because each silicon is bonded with four carbon atoms, and vice versa for each carbon, which makes the structure of 3C-SiC especially stable (Wang et al., 2017). Tritium, as one of the hydrogen isotopes, has been strongly focused on in HTR-PMs because it emits harmful β-radiation (18.6 keV) and has a 12.3 years half-life. In HTR-PMs, there are two ways tritium can be generated: the ternary fission reaction in the fuel elements and the
* Corresponding author. Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China. E-mail address: [email protected] (C. Fang). https://doi.org/10.1016/j.pnucene.2019.103181 Received 2 July 2019; Received in revised form 9 September 2019; Accepted 12 October 2019 0149-1970/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Wenyi Wang, Progress in Nuclear Energy, https://doi.org/10.1016/j.pnucene.2019.103181
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should be emphasized that, although a few theoretical studies have been conducted on the interactions between H and 3C-SiC (Kim et al., 2009) (Aradi et al., 2001), a distinct diffusion pathway and the corresponding diffusion coefficient and diffusion mechanism are still unclear. These will be discussed in this paper.
neutron activation reaction. Most activation reaction of tritium is pro duced outside the TRISO fuel particles, and there are three main reaction sources: 3He (n, p) 3H, 6Li (n,α) 3H and 10B (n, 2α) 3H. It should be clarified that the tritium generated by activation reactions is released into the primary circuit, whereas for ternary fission reactions, a com plete SiC layer can be assumed to prevent all tritium from being released into the primary circuit. Therefore, only tritium in the defective particles will diffuse in the primary circuit, which is much less than that of the activation reaction. However, this assumption may not be so conserva tive, because the diffusion behavior of tritium in SiC is unclear. When analyzing the source term of tritium in the HTR-PM primary circuit, it is important and necessary to know the amount of tritium released from the fuel particles. In other words, the amount of tritium that will penetrate from the SiC layer into the primary circuit must be deter mined. If the results indicate that SiC is effective at blocking the diffu sion of tritium, all the tritium in the primary circuit is derived from the activation reaction. If the opposite is true, part of the tritium in the primary circuit may be considered to come from the ternary fission re action. The experiment can only provide a value of the diffusion coef ficient, and the value obtained by different experiments differs significantly (Causey et al., 1978; Esteban et al., 2002; Tam et al., 1995). In this case, the key point is to determine the diffusion coefficient of tritium in SiC. Furthermore, detailed mechanisms of diffusion are essential, which cannot directly be obtained from experimental analysis. To address this challenge, density functional theory (DFT)-based first-principles calculations (Kohn and Sham, 1965), as implemented in the Vienna Ab initio Simulation Package (VASP) (Kresse and Furth müller, 1996; Kresse and Hafner, 1993; Mehrer, 2007), are performed in the present work to provide a methodology at a level of accuracy close to or even exceeding that of the experimental data (Horn et al., 1988; Kohn and Sham, 1965; Lamble et al., 1988; Persson and Ishida, 1990). Several theoretical studies have been conducted on the behavior of H in SiC in terms of first-principles calculations. For example, the formation en ergies, defects and stable charge states of interstitial H in cubic and hexagonal SiC have been studied (Aradi et al., 2004; Chu and Estreicher, 1990; Kaukonen et al., 2003; Roberson and Estreicher, 1991). Aradi et al. (2004) carried out calculations using the FHI98MD code and the local density approximation to study the diffusion of H in perfect, p-type doped and radiation-damaged 4H-SiC. Chu and Estreicher (1990) calculated the similarities, differences and trends regarding the prop erties of interstitial H in cubic C, Si, BN, BP, AlP and SiC. In the present study, the first-principles approach is implemented to investigate the defect formation energy, the minimum energy diffusion path, the diffusion barrier and the diffusion coefficient of H in 3C-SiC. It
2. Methodology 2.1. Charge state When studying the diffusion of hydrogen atoms in SiC, it is necessary to consider the defect (H atom) in different charge states (Aradi et al., 2001). The most stable positions for different charged hydrogen atoms are different. Kaukonen et al. (2003) found that only singly positive or negative charge states of hydrogen are thermodynamically stable in SiC. In order to determine the charge state of hydrogen in 3C-SiC, the defect formation energies of hydrogen should be calculated in negative (H 1), neutral (H0) and positive (Hþ1) charge states. The formation energy of charged hydrogen in SiC can be written as Van de Walle and Neugebauer (2004) Ef ðH q Þ ¼ EðH q Þ
EðSiCÞ
nH μH þ q½EF þ EVBM ðSiCÞ�
(1)
where EðSiCÞ and EðHq Þ are the total energies of a perfect supercell of SiC without and with a hydrogen defect in charged state q, respectively, nH is the number of hydrogen atoms added (ni > 0) or removed (ni < 0) from the perfect supercell and is equal to unity in this study, μH is the chemical potential of hydrogen, EF is the Fermi energy ranging from the valence band maximum (VBM) to the conduction band minimum, and EVBM ðSiCÞ is the VBM of the perfect SiC. For a given value of EF , hydrogen is stable at the charge state with the lowest formation energy. 2.2. Diffusion coefficient The diffusion coefficient is an important parameter to determine and quantify the transport of radionuclides out of the fuel elements. The diffusion coefficient and its temperature dependence is commonly expressed by the Arrhenius equation (Lu and Zhang, 2013), D ¼ D0 e
Q=kB T
(2)
where Q is the activation energy for diffusion, D0 is a pre-exponential factor that is usually assumed to be independent of temperature T and kB is the Boltzmann constant. The activation energy Q can be calculated by the VASP, whereas D0 cannot be obtained directly. In order to obtain
Fig. 1. Configuration of TRISO-coated particle. 2
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the value of D0, we must fit a curve of the diffusion coefficient as a function of temperature (D–T curve). According to Wert and Zener (1949), the diffusion coefficient D of an interstitial atom can be described by
performed by the VASP code (Kresse and Furthmüller, 1996). The electron–ion interaction is described by the projector-augmented wave method (Kresse and Hafner, 1994; Kresse and Joubert, 1999; Vanderbilt, 1990), and the correlation energy and electron exchange are treated with the generalized gradient approximation of the Per dew–Burke–Ernzerhof form (Perdew et al., 1996). In the VASP calcu lations, a 2 � 2 � 2 supercell of 3C-SiC containing 64 atoms is employed for the simulations (shown in Fig. 2). The neutron and mass effects are not involved, because the strength of the gravitational field is consid erably weaker than that of electromagnetic interaction on this scale. The jump barrier is determined by the electronic interaction and not by the mass of the atom, which makes the energy barriers of the two isotopes identical (Aradi et al., 2001; Kim et al., 2009). Therefore, when calcu lating the energies, the behavior of the tritium atom can be described in terms of a hydrogen (H) atom. The cutoff energy for the plane-wave bases is 400 eV, and the Brillouin-zone integrations are carried out using 4 � 4 � 4 k-point meshes (Kaukonen et al., 2003). The electronic iteration convergence is at least 10 4 eV/atom. The lattice constant of the 3C-SiC is 4.38 Å according to the calculations presented in this paper, and it compares well to the experimental and theoretical values of 4.359 Å (Sherwin and Drummond, 1991) to 4.39 Å (Kim et al., 2009) in the literature. One H atom is positioned at various interstitial sites of 3C-SiC (see details in Sec. 3.1), and the structure is relaxed until all forces acting on the atoms are smaller than 0.01 eV/Å. The site with the lowest energy is defined as the IS, and the nearby site with the second-lowest energy is selected as the final state. The minimum energy pathway (MEP) and the corresponding activation energy Q value are determined using the climbing image-nudged elastic band (CI-NEB) method (Henkelman et al., 2000). In the present work, three images are used to calculate the energy barrier as more images cannot improve the results based on our tests (see Fig. 3). Because the mass of hydrogen isotopes is very small, there is no distinct difference among the diffusion coefficients of hydrogen, deuterium and tritium by the Debye method. Thus, phonon calculations are performed using the force constant, which is predicted by the VASP 5.2 code, and the phonon properties, calculated by the Yphon code (Wang et al., 2012, 2010).
(3)
D ¼ nβd 2 Γ
where n is the number of nearest-neighbor stable interstitial sites, β is the probability an interstitial atom jumps to a nearest-neighbor site and d is the projected length onto the direction of diffusion. For the present interstitial diffusion in the diamond structure (Jones, 2008), n ¼ 4 and β ¼ 1=6. Γ is the jump rate between two adjacent sites of the diffusion particle at a temperature T (Heinola and Ahlgren, 2010; Wert and Zener, 1949; Wimmer et al., 2008), � � kB T ZTS ΔE Γ¼ exp (4) h ZIS kB T where ZIS and ZTS are the partition functions for the initial state (IS) and transition state (TS), respectively, h is Planck’s constant and ΔE is the energy difference between TS and IS. At high temperatures (hv= 2kT≪ 1), the jump rate of Eq. (4) reduces to the results found by Vineyard (1957), Q3N 3 IS � � ΔE i¼1 vi Γ ¼ Q3N exp (5) 4 TS kB T i¼1 vi TS where vIS i and vi are the vibrational frequencies at the center of the Brillouin zone for IS and TS, respectively, and N is the total number of atoms in the supercell. At low temperatures, the jump rate of Eq. (4) reduces to the results found by Eyring (1935): � � kB T ΔE þ ΔEZP exp Γ¼ (6) h kB T
where ΔEZP is the difference in zero-point vibrational energy between the IS and TS. In the present work, ΔE in Eq. (4) and ΔEþ ΔEZP in Eq. (5) will be replaced by the temperature-dependent Gibbs energy difference ΔG between the IS and TS. In the case of zero external pressure, ΔG is equal to the Helmholtz energy difference ΔF, which can be estimated by the quasiharmonic approach (Shang et al., 2010), FðV; TÞ ¼ EðVÞ þ Fvib ðV; TÞ þ Fel ðV; TÞ
3. Results and discussion
(7)
In the present work, the most stable interstitial site must be selected to determine the formation energy of hydrogen and the IS of diffusion. The total energies of the H atom at various interstitial sites from the
where E is the total static energy at T ¼ 0 K determined by the fourparameter Birch–Murnaghan equation of state fitting (Shang et al., 2010), EðVÞ ¼ a þ bV
2 3
þ cV
4 3
þ dV
2
(8)
where a, b, c and d are parameters fitted to the equilibrium properties at seven different volumes, including the equilibrium volume (V0 ), energy (E0 ), bulk modulus (B0 ) and its pressure derivative (B’0 ) (Shang et al., 2010). Fvib in Eq. (7) is the vibrational contribution that can be obtained by the Debye model or the phonon density of state (DOS), and is the thermal electronic contribution for metals because of the nonzero electronic DOS at the Fermi level. From the above, the pre-factor D0 and the activation energy Q in the Arrhenius equation (Eq. (2)), as constant values, can be determined once the curve of D–T is fitted by Eq. (3). The purpose of this research is to compare the diffusion of different H ions in SiC and obtain the diffusion coefficient under different charged states. Further, the chemical forms of H in SiC is clarified to understand the microscopic mechanism of its diffusion. 2.3. Computational details
Fig. 2. Supercell of 3C-SiC lattice with 2 � 2 � 2 primitive cells, where the blue atoms represent silicon and the brown atoms represent carbon.
All DFT-based first-principles calculations in the present work are 3
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Fig. 3. Diffusion energy barriers of Hþ in 3C-SiC with different images (3/5/7) on the diffusion pathway.
literature (Kaukonen et al., 2003; Roberson and Estreicher, 1991) indicate that the H atom is stable at the tetrahedral (T) sites. Fig. 4 il lustrates two different Wyckoff positions as the tetrahedral interstitial sites (TSi site and TC site) and two other sites in 3C-SiC (C1 site and C2 site). The Tc and TSi sites are surrounded by four C atoms and four Si atoms, respectively, where the C1 site is the anti-bond site of the C atom with a distance of 1.132 Å, and the C2 site is close to the C atom with a
distance of 1.147 Å on the diagonal of the octahedral composed of six C atoms. It can be seen in Table 1 that the most stable site for H0 and Hþ is the C1 site, while H prefers TSi . For H0 , among all the stable sites, the energy of the C1 site is lower than that of the C2 site by 0.10 eV and that of the TSi site by 0.34 eV. Thus, the charged hydrogen atom at C1 site will be treated as the IS, while those at the C2 site and TSi site will be the potential final states for the diffusion of H0 . Because the energy differ ence between the C1 and TC sites is too high, the TC site will not be considered a potential final state for diffusion. Similarly, for Hþ , the possible MEP is from the C1 to C2 sites; for H , the possible MEP is from the (i) TSi to C1 site and (ii) TSi to C2 site. The most stable sites of H obtained in this study are consistent with other data in the literature (Kaukonen et al., 2003; Zhang and Zhang, 2009). The predicted length of the C–H bond for H at C1 in the present work is 1.13 Å, which is similar to those obtained by Aradi (1.14 Å) (Aradi et al., 2001) and Zhang and Zhang (1.13 Å) (Zhang and Zhang, 2009). The experimental bandgap value of 2.417 eV (Gary Lynn Harris, n.d.) is used in this work, because the common practice in the literature (Bockstedte et al., 2003) is to choose an experimental bandgap value for determining the charge states. The relative stabilities of the three charge states based on the Fermi level are shown in Fig. 5. The cross point of the Hþ and H states is EVBM þ 1:15 eV, so the possible charge states of hydrogen are Hþ and H . In this study, in order to compare the diffu sivities of different charge states, all three charged hydrogen atoms are considered. For H0 , C1 has the lowest energy of formation, as shown in Table 1, and is thus set as the IS for diffusion with three probable MEPs: (1) a direct path from C1 to C10 (C10 is the nearest site equivalent to C1), (2) from C1 to TSi and (3) a zigzag path from C1 to C2 and then to TSi . As illustrated in Table 2, path 1 is the most likely trajectory for H0 by comparing the TS energy. For Hþ , the energy barrier of path 1 is 0.46 eV, which is lower than the energy difference between any two other stable Table 1 Total relaxation energies of H atom located at the four different interstitial sites in crystalline 3C-SiC. The four sites are defined in Fig. 4. Interstitial sites
Fig. 4. Four different interstitial sites of H in β-SiC with C atoms at 4a(0,0,0) and Si atoms at 4c(0.25,0.25,0.25). (a) Tc site at 4d(0.75,0.75,0.75) surrounded by four C atoms, (b) TSi site at 4b(0.5,0.5,0.5) surrounded by four Si atoms, (c) C1 site (close to C atom with distance of 1.132 Å in the carbon-filled TSi site, which can be called the anti-bond site of C) and (d) C2 site (close to C atom with distance of 1.147 Å on the diagonal of the octahedral, which is composed of six C atoms).
Eo (eV) H0
H
Tc
485.05
494.57
475.29
485.71
493.69
477.50
C1 C2
486.05 485.95
495.60 495.43
476.25 476.14
TSi
4
Hþ
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this study indicate that the energy barriers for the aforementioned zigzag path (C1-C2-TSi -C20 -C10 ) and the direct path (C1-TSi ) are identical (0.51 eV, see Table 2). When the H atom diffuses from the IS to the final site, the energy profile can be described by a sinusoidal curve, which was first suggested by Wert and Zener (1949). Fig. 7 shows the energy barrier for H diffu sion in 3C-SiC with different charge states in comparison to the previous DFT and experimental results (Causey et al., 1978; Esteban et al., 2002; Kaukonen et al., 2003; Tam et al., 1995). It can be seen that the calcu lated results for H0 and Hþ in the present work and those in the literature are very close; however, the experimental values differ significantly. To determine why, a Si- or C-vacancy is added during the diffusion calcu lations of H0 . Shrader at al. studied the diffusion of silver in 3C-SiC (Shrader et al., 2011) and found that vacancy could cause the migra tion barrier at the interstitial position lower than that of silver substituting a Si or C atom. Therefore, a vacancy may play a key role in regulating the energy barrier of H in SiC. Fig. 7 depicts that the existence of vacancies raises the energy bar rier. The energy barrier for H0 in defect-free 3C-SiC, with a Si vacancy and with a C vacancy, is 0.52, 1.09 and 1.71 eV, respectively. This in dicates that the C vacancy may cause a larger increase in diffusion activation energy than the Si vacancy. Because a complete diffusion path should be diffused from one stable position to another, it is desirable that the impurity atoms can diffuse out rather than oscillate inside the crystal lattice. Therefore, the initial state and final state of a complete diffusion path are usually two physically equivalent locations. Therefore, the energies of initial state and final state for Hþ and H0 are the same and set to 0 eV. For H , the energies of initial state and final state are different,
Fig. 5. Relative stability of the interstitial charged hydrogen atom in 3C-SiC as a function of the Fermi energy.
sites. Thus, there is no need to calculate the barriers for the other paths. For H , the most stable site, TSi , is treated as the IS. The potential final states for the migration of H are the C1 and C2 sites. The CI-NEB calculation for path 4 does not converge with the energy barrier of roughly 2.36 eV because of the difficulty of computation (see Table 2). Path 5 gives a MEP value of 1.52 eV (see Table 2). It should be noted that Kaukonen et al. (2003) calculated the diffusion energy barriers for H0 (0.5 eV) and Hþ (0.5 eV), which are close to our values of 0.45 and 0.46 eV, respectively (see Table 2). Similar to the present case, a value more than 2 eV was estimated for H because of computational diffi culties. Note that the calculations by Kaukonen et al. (2003) were per formed by DFT with two complementary programs, i.e., AIMPRO and FINGER, and their supercell of 3C-SiC contained 128 atoms and the k-point was 2 � 2 � 2. Fig. 6 shows a schematic of different interstitial diffusion paths in 3CSiC. Path 1 is the MEP for H0 and Hþ , in which the hydrogen atom first starts from the C1 site and then jumps from C1 to C1’ the next-nearest neighbor of the carbon atom, identical to the diffusion path in the literature (Kaukonen et al., 2003). Path 2 starts from the C1 site via C2 to TSi , where the H atom at TSi has six equivalent diffusion directions. Finally, the complete diffusion path is expressed as C1-C2-TSi -C20 -C10 , and the energy barrier for the TSi -C20 -C10 path is the reverse of that of C1-C2-TSi . Path 3 is a direct jump from C1 to TSi . The CI-NEB results in
Table 2 Energy barriers (Q) for H with different charges along possible diffusion path ways, as shown in Fig. 6. Paths No.
1 2 3 4 5
Paths
C1 to C10 C1 to C2 C2 to TSi C1 to TSi
Q (eV) H0
Hþ
0.45 0.1 0.41
0.46
H
0.51
TSi to C1
2.36 (not converged)
TSi to C2
1.52
Fig. 6. (a) Schematic of the different diffusion paths for the hydrogen atom in 3C-SiC. Path 1: red line from C1 to C1’ the next-nearest neighbor of the carbon atom. Path 2: black line from C1 via C2 to TSi , i.e., C1-C2-TSi -C20 -C10 . Path 3: green line from C1 to TSi . The specific MEP for different charge states is shown in (b), (c) and (d). 5
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Fig. 7. Diffusion energy barriers of H/D/T in 3C-SiC from the calculations in this study and experiments in the literature denoted by H0&Hþ_calc. (Kaukonen et al., 2003), Expt.1 (Esteban et al., 2002), Expt.2 [40] and Expt.3 (Tam et al., 1995). It should be clarified that the reaction coordinates of the saddle point from the literature (Causey et al., 1978; Esteban et al., 2002; Kaukonen et al., 2003; Tam et al., 1995) are set by the author, in order to visually compare the literature data to the calculation results of this study.
Fig. 8. Diffusion coefficient of different charge states of H in 3C-SiC from the calculations in this study and experiments in the literature denoted by Expt.1 (Esteban et al., 2002), Expt.2 (Causey et al., 1978) and Expt.3 (Tam et al., 1995). Because the difference in diffusion coefficient for H, D and T is too small (see Table 3), only the results of H are presented in this figure. The X-axis is the reciprocal of temperature multiplied by 1000 and the unit of temperature(T) is Kelvin(K).
because the MEP is from TSi to C2 in my research. Actually, the complete path is TSi -C2-C2’-TSi , while the energy barrier can be shown only by plotting the half path TSi -C2. According to Fig. 7, the energy barrier for H is 1.52 eV, which is between the two experimental estimations of 1.12 (Esteban et al., 2002) and 2.82 eV (Causey et al., 1978), respec tively. This indicates that the most likely charge state of hydrogen atom is negative. All the energy barrier data are listed in Table 3. The diffusion coefficients of charged hydrogen atoms in this study and in the literature listed in Table 3 are plotted in Fig. 8. The diffusion coefficients of Hþ from the Debye and Yphon method from the present work are depicted as a red line and black line, respectively, which are very close to each other, with an energy barrier of 0.46 eV by the phonon method and 0.53 eV by the Debye method. The values of experiments 1 and 3 are almost the same. The green line represents the diffusion co efficient of H calculated by the Debye method, which is between the lines of experiment 1 and 2. As the diffusion coefficient differences for H, D and T are too small (see Table 3), we only present the results of H in Fig. 8. The diffusion coefficients for T can be obtained as follows: DTþ ¼ 3:03 � 10 9 e
0:46eV=kB T
(9)
DT ¼ 8:53 � 10 7 e
1:52eV=kB T
(10)
not be able to fully block the release of tritium. When analyzing the source term in HTR-PM, the release rate and release fraction of the fission product from the TRISO-coated particles and fuel elements can be computed by a diffusion model, where the diffusion activation energy of tritium in 3C-SiC is one of the key input parameters and is usually set to zero because of the lack of data, which makes the computed results too conservative. Consequently, our results can be used to analyze the source term in the HTR-PM primary circuit. 4. Summary and conclusions In this study, the diffusion behaviors of H isotopes in 3C-SiC were studied using DFT-based first-principles calculations. It was found that the most stable interstitial site for Hþ and H0 was C1, which is near C with a distance of about 1.13 Å. The MEP was determined after all possible diffusion paths were analyzed by the CI-NEB results, and the results indicate that the MEP was C1-C10 for the impurities Hþ and H0 , and TSi -C2 for H . Only positive or negative charge states of the hydrogen atoms were found to be thermodynamically stable in SiC. Therefore, the chemical form where H actually exists in SiC is positively or negatively charged, because the behavior of the tritium atom can be described in terms of a hydrogen (H) atom. The diffusion coefficient of tritium for the positive charge state is DTþ ¼ 3:03 � 10 9 e 0:46eV=kB T , and that for the negative charge state is DT ¼ 8:53 � 10 7 e 1:52eV=kB T .
In comparison to the activation energies of 7.9 and 5.14 eV for Ag and Cs in 3C-SiC (Malherbe, 2013), respectively, the low activation energy (0.46 and 1.52 eV) in the present work indicates that 3C-SiC may
Table 3 Diffusion energy barriers and pre-factors for H/D/T atoms with different charges in this study and in the literature. D0 (m2 s 1) þ H D T H H H H D T T T a
3C-SiC (Phonon) 3C-SiC (Phonon) 3C-SiC (Phonon) 3C-SiC (Debye) 3C-SiC (Vsi) 3C-SiC (Vc) 3C-SiC 3C-SiC 3C-SiC 3C-SiC 3C-SiC
4.94 � 10 3.61 � 10 3.03 � 10 3.34 � 10
9
1.10 � 10 1.72 � 10 3.41 � 10 1.69 � 10
4a
Q (eV) 0
– 6
2.99 � 10
7
1.32 � 10 9.97 � 10 8.53 � 10 4.22 � 10
2.80 � 10 1.58 � 10
3
4.55 � 10 7.30 � 10
4
9 9 7
2 4 4a
4
þ 7 7 7
5
0.46 0.46 0.46 0.53 0.5 1.12a 2.82a 3.19a 1.11a
0
0.52 1.09 1.71 0.5
Ref.
Temp. (K)
DFT DFT DFT DFT DFT DFT Calc. Expt.1 Expt.2
This work This work This work This work This work This work Kaukonen et al. (2003) Esteban et al. (2002) Causey et al. (1978)
300–2000 300–2000 300–2000 300–2000
Expt.3
Tam et al. (1995)
1373–1873
– 1.52 1.52 1.52 1.58 1.22 >2
The energy barrier and pre-factor in Expt.1 and Expt.3 do not refer to any particular charge state. 6
Method
675–1029 773–1573
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The calculated diffusion coefficients from this study agree well with the computations in previous studies, and the experimental results are closer to the computed values of negative hydrogen, indicating that the most likely charge state of hydrogen is negative. The results in this work could support the study of behavior of tritium in HTR-PM and the cor responding new diffusion and release model of tritium in fuel particles should be built in the next step.
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Declaration of competing interest The authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgements This work is supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China, China [grant number ZX06901]; Pennsylvania State University, United States; the Office of Science of the U.S. DOE, United States [grant number DEAC02-05CH11231]; and the National Science Foundation, United States [grant number ACI-1053575]. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pnucene.2019.103181. References Aradi, B., De� ak, P., Gali, A., Son, N.T., Janz� en, E., 2004. Diffusion of hydrogen in perfect, p-type doped, and radiation-damaged 4H SiC. Phys. Rev. B 69, 233202. https://doi. org/10.1103/PhysRevB.69.233202. Aradi, B., Gali, A., De� ak, P., Lowther, J.E., Son, N.T., Janz� en, E., Choyke, W.J., 2001. Ab initio density-functional supercell calculations of hydrogen defects in cubic SiC. Phys. Rev. B 63, 245202. https://doi.org/10.1103/PhysRevB.63.245202. Bekaroglu, E., Topsakal, M., Cahangirov, S., Ciraci, S., 2010. First-principles study of defects and adatoms in silicon carbide honeycomb structures. Phys. Rev. B 81, 075433. https://doi.org/10.1103/PhysRevB.81.075433. Bockstedte, M., Mattausch, A., Pankratov, O., 2003. Ab initio study of the migration of intrinsic defects in 3 C SiC. Phys. Rev. B 68, 205201. https://doi.org/10.1103/ PhysRevB.68.205201. Causey, R.A., Fowler, J.D., Ravanbakht, C., Elleman, T.S., Verghese, K., 1978. Hydrogen diffusion and solubility in silicon carbide. J. Am. Ceram. Soc. 61, 221–225. https:// doi.org/10.1111/j.1151-2916.1978.tb09284.x. Chu, C.H., Estreicher, S.K., 1990. Similarities, differences, and trends in the properties of interstitial H in cubic C, Si, BN, BP, AlP, and SiC. Phys. Rev. B 42, 9486–9495. https://doi.org/10.1103/PhysRevB.42.9486. Esteban, G.A., Perujo, A., Legarda, F., Sedano, L.A., Riccardi, B., 2002. Deuterium transport in SiCf/SiC composites. J. Nucl. Mater. 307–311, 1430–1435. https://doi. org/10.1016/S0022-3115(02)01282-5. Eyring, H., 1935. The activated complex in chemical reactions. J. Chem. Phys. 3, 107–115. https://doi.org/10.1063/1.1749604. Gary Lynn Harris, n.d. Properties of Silicon Carbide. Heinola, K., Ahlgren, T., 2010. Diffusion of hydrogen in bcc tungsten studied with first principle calculations. J. Appl. Phys. 107, 113531. https://doi.org/10.1063/ 1.3386515. Henkelman, G., Uberuaga, B.P., J� onsson, H., 2000. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 113, 9901. https://doi.org/10.1063/1.1329672. Horn, K., Hohlfeld, A., Somers, J., Lindner, T., Hollins, P., Bradshaw, A.M., 1988. Identification of the s-derived valence-electron level in photoemission from alkalimetal adlayers on aluminum. Phys. Rev. Lett. 61, 2488–2491. https://doi.org/ 10.1103/PhysRevLett.61.2488. Jones, S.W., 2008. Diffusion in Silicon. Katoh, Y., Snead, L.L., Szlufarska, I., Weber, W.J., 2012. Radiation effects in SiC for nuclear structural applications. Curr. Opin. Solid State Mater. Sci. 16, 143–152. https://doi.org/10.1016/J.COSSMS.2012.03.005. Kaukonen, M., Fall, C.J., Lento, J., 2003. Interstitial H and H2 in SiC. Appl. Phys. Lett. 83, 923–925. https://doi.org/10.1063/1.1598646. Kim, J.H., Kwon, Y.D., Yonathan, P., Hidayat, I., Lee, J.G., Choi, J.-H., Lee, S.-C., 2009. The energetics of helium and hydrogen atoms in β-SiC: an ab initio approach. J. Mater. Sci. 44, 1828–1833. https://doi.org/10.1007/s10853-008-3180-2.
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