Kinetic Monte Carlo simulation on influence of vacancy on hydrogen diffusivity in tungsten
Introduction
In fusion reactors, since tritium balance between production in breeding materials and consumption in plasma will be tight [1], [2], tritium retention in reactor materials needs to be minimized for establishing an efficient and sustainable tritium fuel cycle. Large tritium retention also causes radiation-safety concerns as tritium is radioactive. Therefore, understanding and controlling tritium behavior such as release and accumulation in reactor materials is an important subject in fusion engineering.
Among various materials to be used in fusion reactors, materials of plasma-facing components (PFCs) directly face to plasma and thus a large amount of tritium may be piled up in it. As a candidate material for PFCs, tungsten and tungsten-based alloys have received increasing attention due to its high melting point, low erosion rate, and low hydrogen solubility (1.0 eV as the solution energy by first-principles calculation [3] and 1.16 eV as the activation energy for solution by experiment [4], [5]). However, it has been shown that hydrogen retention is largely increased by irradiation defects [6], [7], [8]. For example, a mono-vacancy can trap multiple hydrogen atoms [9], [10]. Therefore, many researches have been performed for understanding the influence of radiation defects on tritium behavior in tungsten [11], [12], [13].
The diffusion coefficient of hydrogen is one of the fundamental physical quantities that govern the tritium behavior in tungsten. It is directly relevant with tritium release and permeation from/through tungsten. In addition, it is a vital input in analyzing experimental results of thermal desorption spectroscopy (TDS; or temperature programed desorption, TPD), which has been widely conducted to acquire information on hydrogen-defect interactions. Among several reported hydrogen diffusivities, the equation given by Frauenfelder [14] has been most utilized:where D (m2 s−1) is the diffusion coefficient of hydrogen at temperature T (K), and k (=8.62 × 10−5 eV K−1) is the Boltzmann constant. This formula was determined by degassing experiment of pre-loaded H2 gas in 1100–2400 K [14].
First-principles calculation based on density functional theory (DFT) has also been applied to hydrogen in tungsten. For achieving a diffusion coefficient, DFT calculation is often coupled with transition state theory (TST). For example in bcc-Fe, DFT + TST approach produced diffusion coefficients comparable with experimental results [15], [16]. Since bcc-Fe has similar characteristics to bcc-W in respect to hydrogen behavior, such as a positive solution energy and a low energy barrier for hydrogen diffusion, it is reasonable to expect a good result in tungsten as well. However, previous DFT calculations showed large disagreement with experiment in the case of tungsten: 0.20 eV as the migration barrier in DFT calculation [17] and 0.39 eV as the activation energy for diffusion in Frauenfelder's experiment [14] (Eq. (1)). Note that the migration barrier reported by Johnson and Carter [18], 0.42 eV and 0.39 eV with zero-point energy correction, was calculated along a higher-barrier path via octahedral site (O-site), not along the minimum-barrier path via trigonal site (Tri-site). Therefore, it is inappropriate to compare this calculation value with the experimental value.
Heinola and Ahlgren [19] examined the disagreement between DFT calculation and Frauenfelder's experiment by using DFT + TST approach. They showed that the diffusion coefficients obtained with DFT + TST approach are comparable with those of Frauenfelder's experiment, if experimental data at low temperatures (<1500 K) are excluded. The exclusion decreases the activation energy for hydrogen diffusion in Frauenfelder's experiment from 0.39 eV to 0.25 eV, as given in the following Arrhenius equation [19]:
This finding indicates significant influence of lattice defects which act as hydrogen traps up to high temperatures like 1500 K. Indeed, the influence of defects has been considered to be a main cause of scattering of hydrogen diffusion coefficients [20]. However, the influence is yet to be clearly quantified.
In the present study, we quantify the influence of traps in hydrogen diffusivity in bcc-W, in order to verify the Heinola's suggestion and Eq. (2). For this aim, we perform Kinetic Monte Carlo (KMC) simulation utilizing diffusion coefficients given by Eq. (2). Among several lattice defects which can act as hydrogen traps including grain boundary [21] and dislocation [22], we focus on vacancy in this study as a typical trap. Vacancy-hydrogen (V–H) interactions are modeled based on DFT calculation results reported by Ohsawa et al. [10]. Diffusion coefficients determined by KMC simulations in systems with and without vacancies are compared with those of Frauenfelder's experiment.
Section snippets
Kinetic models in KMC
KMC simulation pursues a system evolution by repeating some atomic-scale kinetic events that are relevant with a phenomenon of interest. Judgment whether an event attempt will succeed or fail is made using Monte Carlo simulation technique. For example, when the success probability of an event per attempt is p (0 < p < 1), this event will occur if a fraction, which is randomly generated between 0 and 1, is smaller than p. The frequency of event attempt is determined so that the expected
Results
Fig. 6 compares KMC simulation results with Frauenfelder's experimental data [14] and the formula given by Heinola and Ahlgren [19] (Eq. (2) in the present paper), which was fitted only to high-temperature data (>1500 K) of Frauenfelder's experiment. The KMC results clearly show that the effective hydrogen diffusivity is largely decreased at low temperatures as the vacancy concentration increases. The influence of vacancy is evident below 1600 K when the vacancy concentration is 10 ppm, and
Discussion
In this chapter, we discuss some possible errors in the simple model utilized in the present KMC simulation: influence of vacancy clustering, influence of grain boundary and dislocation, influence of impurity atoms, and influence of quantum effects. None of these influences is not involved in the present KMC simulation.
Conclusions
KMC simulations were performed to quantify the influence of trap in hydrogen diffusivity in bcc-W. As a typical trap, mono-vacancy is considered. In the KMC simulations, Frauenfelder's experimental results [14] were nicely reproduced when hydrogen and trap concentrations expected in the experiment were assumed in the simulation.
Those KMC results are consistent with the Heinola's interpretation [19] on the data of Frauenfelder's experiment: low-temperature data (<1500 K) are largely affected by
Acknowledgement
This research was supported by National Research Foundation of Korea under Nuclear Fusion Basic Research program. This research was also supported by BK 21 plus project and the Center for Advanced Research in Fusion Reactor Engineering (CARFRE). The calculations in this research were carried out using the HELIOS supercomputer system at the Computational Simulation Center of the International Fusion Research Center (IFERC-CSC) in Japan.
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