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Effect of hydrogen atom concentration on hydrogen migration and bubble evolution in bcc iron
Nuclear Inst. and Methods in Physics Research B 461 (2019) 83–87
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Effect of hydrogen atom concentration on hydrogen migration and bubble evolution in bcc iron
T
Y.X. Weia, N. Gaob,c, , D. Wangd, C. Chena, L.P. Guoa, ⁎
⁎
a
Hubei Nuclear Solid Physic Key Laboratory, Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology, Wuhan University, 430072 Hubei, Wuhan, PR China b Institute of Modern Physics, Chinese Academy of Sciences, 73000 Lan Zhou, PR China c School of Nuclear Science and Technology, University of Chinese Academy of Sciences, 100049 Beijing, PR China d School of Materials Science and Engineering, Tsinghua University, 100084 Beijing, PR China
ARTICLE INFO
ABSTRACT
Keywords: Hydrogen High concentration Diffusion Molecular dynamics (MD)
The effect of the concentration of hydrogen (H) atoms on the hydrogen diffusion and evolution of hydrogen bubbles in body-centered cubic iron was investigated using molecular statics and molecular dynamics methods. The simulation results indicate that the local accumulation of H atoms can significantly influence their migration. Compared to the isolated H in a perfect lattice, the H atoms in a high-concentration H-cluster are trapped. After collecting a sufficient amount of hydrogen atoms, the H-vacancy cluster will push out one hydrogen atom instead of a self-interstitial atom, which indicates that loop-punching will not dominate the H-bubble growth in bcc iron. These results will provide a new understanding for the hydrogen-induced radiation damages in nuclear materials.
1. Introduction A high concentration of hydrogen (H) atoms generated in fusion reactors and spallation neutron sources [1,2], or stored from the environment in fission reactors [3], has been proven to seriously influence the properties of nuclear materials, such as the formation of new structures in surface layer [4], or appearance of embrittlement which is of significance for industrial applications [5,6]. Therefore, to explore the damage evolution related with hydrogen in a material requires an understanding of the hydrogen behavior after its implantation into a material, i.e., hydrogen diffusion and the clustering mechanism of the H-cluster. Owing to the experimental limitation of time and space scales, atomistic simulation methods are generally applied as complementary methods to provide more details in understanding the effects of H in materials. For example, the ab initio calculations or molecular dynamics (MD) have been used to calculate (1) the H diffusion barriers with different paths to describe the migration of H atoms, and (2) the binding energies between H atoms with H atoms and vacancies to describe the formation of H bubbles [7–9]. The rate equation approach has also been applied to understand the bubble growth [10]; however, this requires reliable parameters as well as a full comprehension regarding the changes of different defects during a period of irradiation.
⁎
In previous studies, most simulations conducted have focused on microstructural evolutions with a small number of H atoms. Recently, Song et al. [11] proposed a new atomic mechanism for H embrittlement in which the aggregation of H atoms at a high concentration (with 0.6–0.8 H per iron atom) can influence the ductile-to-brittle transition by inhibiting the dislocation emission at the crack in bcc iron. Therefore, an investigation into the effects of hydrogen concentration on the properties of different defects is required. In the present study, bodycentered cubic (bcc) iron as the main composition of ferritic steel, which is an attractive candidate material for a fusion reactor [12], was chosen as the matrix for simulations about the change of H behavior at different H concentrations. The effects of a high H concentration on the hydrogen migration and the H-bubble evolution have been extensively studied. In addition, according to previous studies, the loop punching mechanism during H-bubble growth has not been found in bcc iron, although such a mechanism has been found in many other materials such as copper and aluminum [13]. In this study, based on the effects of a high H concentration in iron, the kinetic process of H bubble growth was followed to answer the above question. The paper is organized as follows. The simulation method applied in this work is described in Section 2. Results and a discussion are then provided in Section 3. Finally, the concluding remarks are given in the final section.
Corresponding authors. E-mail addresses: [email protected] (N. Gao), [email protected] (L.P. Guo).
https://doi.org/10.1016/j.nimb.2019.09.025 Received 15 August 2018; Received in revised form 1 May 2019; Accepted 16 September 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 461 (2019) 83–87
Y.X. Wei, et al.
clustering in a perfect bcc iron, a region located at the center of the matrix was selected to include different numbers of H atoms for further relaxations, as shown in Fig. 1. Region I is a sphere with a radius of 10 Å, including H atoms, whereas region II, is the remainder of the cube outside of the sphere where no H atoms are initially included. Thus, the H atoms were inserted into region I to form a configuration with n H atoms, and the system was then relaxed using molecular statics (MS), followed by molecular dynamics (MD), at 300 K. The total MD simulation time reached up to 50 ps. In addition to the above case, the evolution of hydrogen–vacancy (H–V) clusters with a high concentration of H atoms was also simulated. A cluster of vacancies was first obtained by directly removing the Fe atoms at the center of the periodic calculation box. Then, H atoms were introduced one by one manually at the tetrahedral sites around the vacancy cluster. After each insertion of a H atom, a conjugate gradient relaxation was applied to minimize the system, followed by the MD at room temperature and zero pressure under the NPT ensemble. The time-step for the MD simulation was 1 fs, and the total MD simulation time reached up to 10 ps for each insertion. After the MD relaxation, a minimization was performed again for the next H insertion. The movement of H atoms can be calculated through the mean square displacement (MSD) method using the following equation:
Fig. 1. Schematic of computational box. The different concentrations of H atoms are inserted in Region I, and Region II is the remaining matrix of perfect bcc Fe atoms.
MSD = | S|2 = |S (ti )
2. Simulation method
(1)
S (t 0 )|2
where S is the displacement of H atoms from time ti to timet 0 , whereas S (ti ) and S (t 0 ) represent the position vectors of the H atoms at the corresponding time, respectively. The binding energy of a point defect i to a HnVm cluster was calculated as follows:
In this study, the Fe-H potential of Ramasubramaniam et al. [14] was applied to describe the interaction between Fe and H. The relaxations were conducted using a LAMMPS molecular dynamics simulator [15]. All simulations were performed with a simulation box of 20a0 × 20a0 × 20a0 (where a0 = 2.8553 Å is the lattice constant of αFe) containing 16,000 iron atoms. The X-, Y-, and Z-axes of the box were along [1 0 0], [0 1 0], and [0 0 1], respectively. The periodic boundary conditions were used for all three directions. Because the tetrahedral interstitial site has been proven to be a lower energy site for H atoms when no vacancies are available [14], all initial H atoms in this study were arranged at random tetrahedral interstitial sites, and the nearest distance between any two of these initially H atoms was restricted to more than 2 Å to prevent a strong repulsion between hydrogen atoms [8]. To efficiently simulate the concentration effect on hydrogen
Eb (i
Hn Vm) = Etot (i
Hn Vm)
Etot (Hn Vm + i )
(2)
where Etot(H ↔ HnVm) is the total energy when a point defect i is far from the HnVm cluster, whereas Etot(HnVm + i) is the total energy of the system after the HnVm cluster has trapped the point defect i. 3. Results and discussion 3.1. Concentration effect on the diffusion of H in perfect bulk The final distributions of H atoms with different numbers of H atoms Fig. 2. The distributions of H atoms after 50 ps relaxations as the number of H atoms (n) increases. The initial H-cluster configurations are shown in the inserts. The balls with different colors represent H atoms with different potential energies, as indicated by the color bar. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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(n) after 50 ps relaxations are shown in Fig. 2, in which only the H atoms are provided for clarity. According to the DFT data [8], when a single H atom is implanted in a perfect bcc iron lattice, it can diffuse quickly with a low energy barrier of 0.04 eV along the T-T and/or T-O-T paths, where T and O are the tetrahedral and octahedral sites, respectively. The binding energies between two H atoms are slightly positive with a H-H bound length of approximately 2.0–2.2 Å, and near zero or repulsive at other distances, indicating that H atoms do not combine strongly with each other. During the MD simulations, the H2 cluster maintained a dimer state for a short period and then decomposed into two isolated H atoms (Fig. 2a), followed by a quick diffusion of each H atom. Thus, the interaction between two H atoms was too weak to influence the hydrogen migration. A similar process has also been observed when the initial H-cluster contains ten H atoms. After a short period of relaxation, these H atoms also decomposed from the cluster and disperse along the random directions (Fig. 2b). However, when the number of H atoms, n, reached up to 50 or 100, the evolution of the Hcluster taken a different way. It is found that the H-cluster nearly maintained the cluster state and only limited H atoms were displaced away from the cluster, as shown in Fig. 2c and d. A similar phenomenon was more obvious with the number of H atoms reaching up to 200. Thus, under such a condition, the diffusions of H atoms were affected by their concentration. To understand the above results, the changes in the energetics, positions, and jumping rates of hydrogen were analyzed in the following section to demonstrate the concentration effect on the hydrogen behavior. The potential energy (Ep) of each H atom was expressed by a different color corresponding to the color bar in Fig. 2. The lowest and highest potential energies of a H atom are −0.034 eV and 0.96 eV, which were related with the H atoms inside the cluster and isolated H atoms away from the cluster, respectively. With an increase in the hydrogen concentration, an increasing number of H atoms became stable inside of the cluster with a low potential energy, which means that the high concentration can increase the stability of the H-cluster. This occurs because, when the concentration of H atoms is higher, the combination between H atoms becomes more powerful, thereby limiting their diffusions. Furthermore, the H atoms were found to move away from the initial tetrahedral positions to form a stable H-cluster. An example was shown in Fig. 3. During the simulations, even with a high concentration of H atoms, there were no Fe atoms pushed out by the Hcluster to form self-interstitial atoms (SIAs), which differs from the case
Fig. 4. Mean square displacement of H atoms in different states at 300 K.
of a helium cluster with a high concentration of helium atoms. The nearest distance between these H atoms after the relaxations was approximately 2.2 Å. The bcc iron lattice within and around the H-cluster was slightly distorted. When the H atoms were located at the positions between the Fe atoms, some of the Fe atoms (several of them were marked with a green point as a reference) departed from the initial positions, and the interatomic spacing between them became wider from 2.8 Å to approximately 3.2–3.7 Å after relaxation. A careful inspection of the positions of H atoms indicated that these H atoms were located at the octahedral site instead of the tetrahedral site, which led to a slightly distorted local structure. The migration of these H atoms can be described through the calculations of the mean square displacement (MSD). Based on the above results, H atoms can be divided into two different groups: the first contains H atoms at the center of the H-cluster (marked as Hg), and the second contains H atoms displaced far from the H-cluster (marked as Hd). The MSD results for two groups can be calculated and compared to that of an isolated H atom (marked as Hiso) in a perfect lattice. The results for the system with 200 H atoms were shown in Fig. 4. The Hiso atoms can diffuse quickly in metals and the slope of its curve representing its diffusion coefficient was approximately 20.84 Å2/ps in a perfect bcc crystal structure. However, as for the Hd atoms, the diffusion process has two stages. In the first stage with a simulation time from zero to approximately 43 ps, the movement of Hd atoms was limited, and the slope of this curve was approximately 3.01 Å2/ps. After the first stage, the Hd atoms started to diffuse quickly, resulting in an increase in slope to 20.86 Å2/ps, which was as fast as the Hiso atoms. Different from the above two cases, the slope of the MSD data of the Hg atoms was close to zero, indicating that the Hg atoms became almost immobile. Therefore, the high concentration can limit the hydrogen diffusion. 3.2. Concentration effect on the evolution of H-bubbles Different from the above case, the effects of the concentration of H atoms on the HnVm clusters were simulated with m from 9 to 138. After a careful investigation, the effects of H concentration on the HnVm clusters were similar. Thus, in the following, we taken HnV50 as an example (Fig. 5) to show the effect of the concentration of H atoms. From Fig. 5, the V50 cluster can trap the surrounding H atoms when their number was limited to less than 72. When the number of H atoms trapped by the cluster of vacancies reaches up to 72, the pressure in the cluster increased, and some H atoms would be pushed out of the cluster (Fig. 5c). The range of the cluster and the pushed H atoms were marked by a circle and green ball, respectively. Increasing the number of H
Fig. 3. The monoatomic layer sliced along the [1 0 0] axis. The small blue balls represent H atoms and the big red balls represent iron atoms. The two dotted lines of the Fe atomic centers along the 〈0 0 1〉 direction before the insertion of H are plotted as references to observe the local lattice distortion after H implantation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 85
Nuclear Inst. and Methods in Physics Research B 461 (2019) 83–87
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Fig. 7. The dependence of the critical number of H atoms (ns) trapped by the Vm cluster on the number (m) of vacancies in the cluster.
and a H atom to the HnV50 cluster were shown in Fig. 6. The results indicate that H atoms are less bound than SIAs to the H-cluster. After the number of H atoms trapped by the cluster of vacancies reached 72 (the critical value), the binding energies of a H atom and a SIA to the cluster decrease, which indicates that a saturated H bubble will push its surrounding atoms. The binding energy of a H atom to the cluster decreases to nearly zero (~0.05 eV). Therefore, the above kinetic process and an energy calculation can be used together to describe the evolution of an H–V cluster, that is, the development of pressure within the cluster as result of the increasing number of trapped H atoms provides a driving force to push the surrounding atoms (H atoms and iron atoms). Once a sufficient driving force is reached, the excess H atoms instead of the surrounding iron atoms will be pushed out of the cluster because of the weaker binding energies between the H atom and H-cluster. For other cases with a different number of vacancies, the critical number of H atoms before the H-cluster pushes out H atoms has also been obtained, as shown in Fig. 7. However, these results indicated that H atoms instead of SIAs are first pushed out from the H-cluster, which is different from the case of a He-bubble. Thus, it would be difficult for hydrogen bubbles to grow through the loop-punching mechanism in bcc iron.
Fig. 5. HnV50 evolution with different n. The small blue balls represent H atoms, and the big red balls represent iron atoms. The yellow arrows indicate the displacement vectors of iron atoms. (a–e) The evolution process with an increasing number n of H atoms. Only those iron atoms that are initially the first-nearest neighbor to the cluster are shown indicated. (f) A monoatomic layer is shown after 200 H atoms are introduced, in which the two dotted lines mark the positions of Fe along the 〈0 1 0〉 direction before H implantation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusions Under a high concentration of H atoms, the migration and the evolution of H-bubbles in bcc iron have been studied using the MS and MD methods. In perfect iron bulk, the status of H atoms can be limited with an increase in the hydrogen concentration. With an increase in the H concentration, H atoms inside the H-cluster are almost immobile and prefer to locate at O-sites. These results suggest that the self-trapping state of H is available with a high hydrogen concentration, which provides a new understanding of hydrogen retention in such a high H production or permeation systems, including accelerator-driven subcritical reactor systems (ADS) and fusion reactor systems. In addition, the changes in H atom diffusion and aggregation under different concentrations also provide new insight into the differences between proton and neutron irradiation. When vacancies are present, there is a critical number of H atoms with an increase in the H concentration. After such a critical number, some H atoms instead of self-interstitial atoms will be pushed out from the H-vacancy cluster because the binding energy of H atoms to the Hbubbles is much lower than that of the SIAs, which explains why the loop-punching mechanism has an extremely low probability to occur
Fig. 6. The binding energy of point defects (H atoms and SIAs) to a HnV50 cluster as a function of the number of H atoms trapped in the cluster (n). The curves are calculated from Eq. (2).
atoms to 100 and 200, the H atoms pushed out from the H-cluster were observed to move around the nearby interstitial sites rather than being displaced away from the H-vacancy cluster. With more H atoms, more iron atoms within and around the H-clusters were found to be displaced, as marked by the yellow arrows (Fig. 5d). The H atoms were trapped at the octahedral sites located close to the inner surface of the cluster, leaving a vacancy cluster inside of the H-vacancy cluster, as shown in Fig. 5d and e. The binding energies of a self-interstitial atom 86
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for H-bubbles in bcc iron. Therefore, all these results indicate the reason why the enhancement of H atoms is weaker for loop growth than He atoms in experimental results [16,17].
steels, J. Alloys Compd. 293 (1999) 356–360. [5] L. Marchetti, E. Herms, P. Laghoutaris, J. Chêne, Hydrogen embrittlement susceptibility of tempered 9%Cr-1%Mo steel, Int. J. Hydrogen Energy 36 (2011) 15880–15887. [6] R.A. Oriani, P.H. Josephic, Equilibrium and kinetic studies of the hydrogen-assisted cracking of steel, Acta Metall. 25 (1977) 979–988. [7] K. Ohsawa, K. Eguchi, H. Watanabe, M. Yamaguchi, M. Yagi, Configuration and binding energy of multiple hydrogen atoms trapped in monovacancy in bcc transition metals, Phys. Rev. B – Condens. Matter Mater. Phys. 85 (2012) 1–8. [8] E. Hayward, C.C. Fu, Interplay between hydrogen and vacancies in α-Fe, Phys. Rev. B – Condens. Matter Mater. Phys. 87 (2013) 174103. [9] E. Hayward, C. Deo, Energetics of small hydrogen-vacancy clusters in bcc iron, J. Phys.: Condens. Matter 23 (2011) 425402. [10] S. Li, Y. Li, Y.C. Lo, T. Neeraj, R. Srinivasan, X. Ding, J. Sun, L. Qi, P. Gumbsch, J. Li, The interaction of dislocations and hydrogen-vacancy complexes and its importance for deformation-induced proto nano-voids formation in α-Fe, Int. J. Plast. 74 (2015) 175–191. [11] J. Song, W.A. Curtin, Atomic mechanism and prediction of hydrogen embrittlement in iron, Nat. Mater. 12 (2013) 145–151. [12] Q.Y. Huang, Y.C. Wu, J.G. Li, F.R. Wan, J.L. Chen, G.N. Luo, X. Liu, J.M. Chen, Z.Y. Xu, X.G. Zhou, X. Ju, Y.Y. Shan, J.N. Yu, S.Y. Zhu, P.Y. Zhang, J.F. Yang, X.J. Chen, S.M. Dong, Status and strategy of fusion materials development in China, J. Nucl. Mater. 386–388 (2009) 400–404. [13] J.B. Condon, T. Schober, Hydrogen bubbles in metals, J. Nucl. Mater. 207 (1993) 1–24. [14] A. Ramasubramaniam, M. Itakura, E.A. Carter, Interatomic potentials for hydrogen in α-iron based on density functional theory, Phys. Rev. B 79 (2009) 174101. [15] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19. [16] W. Zhang, Z. Shen, R. Tang, S. Jin, Y. Song, Y. Long, Proton-irradiation induced defects in modified 310S steels characterized with positron annihilation spectroscopy and transmission electron microscopy, Nucl. Instr. Meth. Phys. Res. B 427 (2018) 1–8. [17] W. Zhang, L. Guo, Z. Shen, J. Xin, Q. Huang, Evolution of dislocation loops induced by different hydrogen irradiation conditions, Materials 11 (2018) 2276.
Declaration of Competing Interest There is no conflict of interest. Acknowledgements This research was supported by the National Natural Science Foundation of China (Project Nos. 11775162, 11675230, and 11975170). NG acknowledges the support of from the Youth Innovation Promotion Association CAS, China. The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University. References [1] N. Baluc, Materials for fusion power reactors, Plasma Phys. Control. Fusion. 48 (2006) B165. [2] F.A. Garner, B.M. Oliver, L.R. Greenwood, M.R. James, P.D. Ferguson, S.A. Maloy, W.F. Sommer, Determination of helium and hydrogen yield from measurements on pure metals and alloys irradiated by mixed high energy proton and spallation neutron spectra in LANSCE, J. Nucl. Mater. 296 (2001) 66–82. [3] F.A. Garner, E.P. Simonen, B.M. Oliver, L.R. Greenwood, M.L. Grossbeck, W.G. Wolfer, P.M. Scott, Retention of hydrogen in fcc metals irradiated at temperatures leading to high densities of bubbles or voids, J. Nucl. Mater. 356 (2006) 122–135. [4] A. Szummer, E. Jezierska, K. Lublińska, Hydrogen surface effects in ferritic stainless
87
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Effect of hydrogen atom concentration on hydrogen migration and bubble evolution in bcc iron
T
Y.X. Weia, N. Gaob,c, , D. Wangd, C. Chena, L.P. Guoa, ⁎
⁎
a
Hubei Nuclear Solid Physic Key Laboratory, Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology, Wuhan University, 430072 Hubei, Wuhan, PR China b Institute of Modern Physics, Chinese Academy of Sciences, 73000 Lan Zhou, PR China c School of Nuclear Science and Technology, University of Chinese Academy of Sciences, 100049 Beijing, PR China d School of Materials Science and Engineering, Tsinghua University, 100084 Beijing, PR China
ARTICLE INFO
ABSTRACT
Keywords: Hydrogen High concentration Diffusion Molecular dynamics (MD)
The effect of the concentration of hydrogen (H) atoms on the hydrogen diffusion and evolution of hydrogen bubbles in body-centered cubic iron was investigated using molecular statics and molecular dynamics methods. The simulation results indicate that the local accumulation of H atoms can significantly influence their migration. Compared to the isolated H in a perfect lattice, the H atoms in a high-concentration H-cluster are trapped. After collecting a sufficient amount of hydrogen atoms, the H-vacancy cluster will push out one hydrogen atom instead of a self-interstitial atom, which indicates that loop-punching will not dominate the H-bubble growth in bcc iron. These results will provide a new understanding for the hydrogen-induced radiation damages in nuclear materials.
1. Introduction A high concentration of hydrogen (H) atoms generated in fusion reactors and spallation neutron sources [1,2], or stored from the environment in fission reactors [3], has been proven to seriously influence the properties of nuclear materials, such as the formation of new structures in surface layer [4], or appearance of embrittlement which is of significance for industrial applications [5,6]. Therefore, to explore the damage evolution related with hydrogen in a material requires an understanding of the hydrogen behavior after its implantation into a material, i.e., hydrogen diffusion and the clustering mechanism of the H-cluster. Owing to the experimental limitation of time and space scales, atomistic simulation methods are generally applied as complementary methods to provide more details in understanding the effects of H in materials. For example, the ab initio calculations or molecular dynamics (MD) have been used to calculate (1) the H diffusion barriers with different paths to describe the migration of H atoms, and (2) the binding energies between H atoms with H atoms and vacancies to describe the formation of H bubbles [7–9]. The rate equation approach has also been applied to understand the bubble growth [10]; however, this requires reliable parameters as well as a full comprehension regarding the changes of different defects during a period of irradiation.
⁎
In previous studies, most simulations conducted have focused on microstructural evolutions with a small number of H atoms. Recently, Song et al. [11] proposed a new atomic mechanism for H embrittlement in which the aggregation of H atoms at a high concentration (with 0.6–0.8 H per iron atom) can influence the ductile-to-brittle transition by inhibiting the dislocation emission at the crack in bcc iron. Therefore, an investigation into the effects of hydrogen concentration on the properties of different defects is required. In the present study, bodycentered cubic (bcc) iron as the main composition of ferritic steel, which is an attractive candidate material for a fusion reactor [12], was chosen as the matrix for simulations about the change of H behavior at different H concentrations. The effects of a high H concentration on the hydrogen migration and the H-bubble evolution have been extensively studied. In addition, according to previous studies, the loop punching mechanism during H-bubble growth has not been found in bcc iron, although such a mechanism has been found in many other materials such as copper and aluminum [13]. In this study, based on the effects of a high H concentration in iron, the kinetic process of H bubble growth was followed to answer the above question. The paper is organized as follows. The simulation method applied in this work is described in Section 2. Results and a discussion are then provided in Section 3. Finally, the concluding remarks are given in the final section.
Corresponding authors. E-mail addresses: [email protected] (N. Gao), [email protected] (L.P. Guo).
https://doi.org/10.1016/j.nimb.2019.09.025 Received 15 August 2018; Received in revised form 1 May 2019; Accepted 16 September 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 461 (2019) 83–87
Y.X. Wei, et al.
clustering in a perfect bcc iron, a region located at the center of the matrix was selected to include different numbers of H atoms for further relaxations, as shown in Fig. 1. Region I is a sphere with a radius of 10 Å, including H atoms, whereas region II, is the remainder of the cube outside of the sphere where no H atoms are initially included. Thus, the H atoms were inserted into region I to form a configuration with n H atoms, and the system was then relaxed using molecular statics (MS), followed by molecular dynamics (MD), at 300 K. The total MD simulation time reached up to 50 ps. In addition to the above case, the evolution of hydrogen–vacancy (H–V) clusters with a high concentration of H atoms was also simulated. A cluster of vacancies was first obtained by directly removing the Fe atoms at the center of the periodic calculation box. Then, H atoms were introduced one by one manually at the tetrahedral sites around the vacancy cluster. After each insertion of a H atom, a conjugate gradient relaxation was applied to minimize the system, followed by the MD at room temperature and zero pressure under the NPT ensemble. The time-step for the MD simulation was 1 fs, and the total MD simulation time reached up to 10 ps for each insertion. After the MD relaxation, a minimization was performed again for the next H insertion. The movement of H atoms can be calculated through the mean square displacement (MSD) method using the following equation:
Fig. 1. Schematic of computational box. The different concentrations of H atoms are inserted in Region I, and Region II is the remaining matrix of perfect bcc Fe atoms.
MSD = | S|2 = |S (ti )
2. Simulation method
(1)
S (t 0 )|2
where S is the displacement of H atoms from time ti to timet 0 , whereas S (ti ) and S (t 0 ) represent the position vectors of the H atoms at the corresponding time, respectively. The binding energy of a point defect i to a HnVm cluster was calculated as follows:
In this study, the Fe-H potential of Ramasubramaniam et al. [14] was applied to describe the interaction between Fe and H. The relaxations were conducted using a LAMMPS molecular dynamics simulator [15]. All simulations were performed with a simulation box of 20a0 × 20a0 × 20a0 (where a0 = 2.8553 Å is the lattice constant of αFe) containing 16,000 iron atoms. The X-, Y-, and Z-axes of the box were along [1 0 0], [0 1 0], and [0 0 1], respectively. The periodic boundary conditions were used for all three directions. Because the tetrahedral interstitial site has been proven to be a lower energy site for H atoms when no vacancies are available [14], all initial H atoms in this study were arranged at random tetrahedral interstitial sites, and the nearest distance between any two of these initially H atoms was restricted to more than 2 Å to prevent a strong repulsion between hydrogen atoms [8]. To efficiently simulate the concentration effect on hydrogen
Eb (i
Hn Vm) = Etot (i
Hn Vm)
Etot (Hn Vm + i )
(2)
where Etot(H ↔ HnVm) is the total energy when a point defect i is far from the HnVm cluster, whereas Etot(HnVm + i) is the total energy of the system after the HnVm cluster has trapped the point defect i. 3. Results and discussion 3.1. Concentration effect on the diffusion of H in perfect bulk The final distributions of H atoms with different numbers of H atoms Fig. 2. The distributions of H atoms after 50 ps relaxations as the number of H atoms (n) increases. The initial H-cluster configurations are shown in the inserts. The balls with different colors represent H atoms with different potential energies, as indicated by the color bar. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
84
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(n) after 50 ps relaxations are shown in Fig. 2, in which only the H atoms are provided for clarity. According to the DFT data [8], when a single H atom is implanted in a perfect bcc iron lattice, it can diffuse quickly with a low energy barrier of 0.04 eV along the T-T and/or T-O-T paths, where T and O are the tetrahedral and octahedral sites, respectively. The binding energies between two H atoms are slightly positive with a H-H bound length of approximately 2.0–2.2 Å, and near zero or repulsive at other distances, indicating that H atoms do not combine strongly with each other. During the MD simulations, the H2 cluster maintained a dimer state for a short period and then decomposed into two isolated H atoms (Fig. 2a), followed by a quick diffusion of each H atom. Thus, the interaction between two H atoms was too weak to influence the hydrogen migration. A similar process has also been observed when the initial H-cluster contains ten H atoms. After a short period of relaxation, these H atoms also decomposed from the cluster and disperse along the random directions (Fig. 2b). However, when the number of H atoms, n, reached up to 50 or 100, the evolution of the Hcluster taken a different way. It is found that the H-cluster nearly maintained the cluster state and only limited H atoms were displaced away from the cluster, as shown in Fig. 2c and d. A similar phenomenon was more obvious with the number of H atoms reaching up to 200. Thus, under such a condition, the diffusions of H atoms were affected by their concentration. To understand the above results, the changes in the energetics, positions, and jumping rates of hydrogen were analyzed in the following section to demonstrate the concentration effect on the hydrogen behavior. The potential energy (Ep) of each H atom was expressed by a different color corresponding to the color bar in Fig. 2. The lowest and highest potential energies of a H atom are −0.034 eV and 0.96 eV, which were related with the H atoms inside the cluster and isolated H atoms away from the cluster, respectively. With an increase in the hydrogen concentration, an increasing number of H atoms became stable inside of the cluster with a low potential energy, which means that the high concentration can increase the stability of the H-cluster. This occurs because, when the concentration of H atoms is higher, the combination between H atoms becomes more powerful, thereby limiting their diffusions. Furthermore, the H atoms were found to move away from the initial tetrahedral positions to form a stable H-cluster. An example was shown in Fig. 3. During the simulations, even with a high concentration of H atoms, there were no Fe atoms pushed out by the Hcluster to form self-interstitial atoms (SIAs), which differs from the case
Fig. 4. Mean square displacement of H atoms in different states at 300 K.
of a helium cluster with a high concentration of helium atoms. The nearest distance between these H atoms after the relaxations was approximately 2.2 Å. The bcc iron lattice within and around the H-cluster was slightly distorted. When the H atoms were located at the positions between the Fe atoms, some of the Fe atoms (several of them were marked with a green point as a reference) departed from the initial positions, and the interatomic spacing between them became wider from 2.8 Å to approximately 3.2–3.7 Å after relaxation. A careful inspection of the positions of H atoms indicated that these H atoms were located at the octahedral site instead of the tetrahedral site, which led to a slightly distorted local structure. The migration of these H atoms can be described through the calculations of the mean square displacement (MSD). Based on the above results, H atoms can be divided into two different groups: the first contains H atoms at the center of the H-cluster (marked as Hg), and the second contains H atoms displaced far from the H-cluster (marked as Hd). The MSD results for two groups can be calculated and compared to that of an isolated H atom (marked as Hiso) in a perfect lattice. The results for the system with 200 H atoms were shown in Fig. 4. The Hiso atoms can diffuse quickly in metals and the slope of its curve representing its diffusion coefficient was approximately 20.84 Å2/ps in a perfect bcc crystal structure. However, as for the Hd atoms, the diffusion process has two stages. In the first stage with a simulation time from zero to approximately 43 ps, the movement of Hd atoms was limited, and the slope of this curve was approximately 3.01 Å2/ps. After the first stage, the Hd atoms started to diffuse quickly, resulting in an increase in slope to 20.86 Å2/ps, which was as fast as the Hiso atoms. Different from the above two cases, the slope of the MSD data of the Hg atoms was close to zero, indicating that the Hg atoms became almost immobile. Therefore, the high concentration can limit the hydrogen diffusion. 3.2. Concentration effect on the evolution of H-bubbles Different from the above case, the effects of the concentration of H atoms on the HnVm clusters were simulated with m from 9 to 138. After a careful investigation, the effects of H concentration on the HnVm clusters were similar. Thus, in the following, we taken HnV50 as an example (Fig. 5) to show the effect of the concentration of H atoms. From Fig. 5, the V50 cluster can trap the surrounding H atoms when their number was limited to less than 72. When the number of H atoms trapped by the cluster of vacancies reaches up to 72, the pressure in the cluster increased, and some H atoms would be pushed out of the cluster (Fig. 5c). The range of the cluster and the pushed H atoms were marked by a circle and green ball, respectively. Increasing the number of H
Fig. 3. The monoatomic layer sliced along the [1 0 0] axis. The small blue balls represent H atoms and the big red balls represent iron atoms. The two dotted lines of the Fe atomic centers along the 〈0 0 1〉 direction before the insertion of H are plotted as references to observe the local lattice distortion after H implantation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 85
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Fig. 7. The dependence of the critical number of H atoms (ns) trapped by the Vm cluster on the number (m) of vacancies in the cluster.
and a H atom to the HnV50 cluster were shown in Fig. 6. The results indicate that H atoms are less bound than SIAs to the H-cluster. After the number of H atoms trapped by the cluster of vacancies reached 72 (the critical value), the binding energies of a H atom and a SIA to the cluster decrease, which indicates that a saturated H bubble will push its surrounding atoms. The binding energy of a H atom to the cluster decreases to nearly zero (~0.05 eV). Therefore, the above kinetic process and an energy calculation can be used together to describe the evolution of an H–V cluster, that is, the development of pressure within the cluster as result of the increasing number of trapped H atoms provides a driving force to push the surrounding atoms (H atoms and iron atoms). Once a sufficient driving force is reached, the excess H atoms instead of the surrounding iron atoms will be pushed out of the cluster because of the weaker binding energies between the H atom and H-cluster. For other cases with a different number of vacancies, the critical number of H atoms before the H-cluster pushes out H atoms has also been obtained, as shown in Fig. 7. However, these results indicated that H atoms instead of SIAs are first pushed out from the H-cluster, which is different from the case of a He-bubble. Thus, it would be difficult for hydrogen bubbles to grow through the loop-punching mechanism in bcc iron.
Fig. 5. HnV50 evolution with different n. The small blue balls represent H atoms, and the big red balls represent iron atoms. The yellow arrows indicate the displacement vectors of iron atoms. (a–e) The evolution process with an increasing number n of H atoms. Only those iron atoms that are initially the first-nearest neighbor to the cluster are shown indicated. (f) A monoatomic layer is shown after 200 H atoms are introduced, in which the two dotted lines mark the positions of Fe along the 〈0 1 0〉 direction before H implantation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusions Under a high concentration of H atoms, the migration and the evolution of H-bubbles in bcc iron have been studied using the MS and MD methods. In perfect iron bulk, the status of H atoms can be limited with an increase in the hydrogen concentration. With an increase in the H concentration, H atoms inside the H-cluster are almost immobile and prefer to locate at O-sites. These results suggest that the self-trapping state of H is available with a high hydrogen concentration, which provides a new understanding of hydrogen retention in such a high H production or permeation systems, including accelerator-driven subcritical reactor systems (ADS) and fusion reactor systems. In addition, the changes in H atom diffusion and aggregation under different concentrations also provide new insight into the differences between proton and neutron irradiation. When vacancies are present, there is a critical number of H atoms with an increase in the H concentration. After such a critical number, some H atoms instead of self-interstitial atoms will be pushed out from the H-vacancy cluster because the binding energy of H atoms to the Hbubbles is much lower than that of the SIAs, which explains why the loop-punching mechanism has an extremely low probability to occur
Fig. 6. The binding energy of point defects (H atoms and SIAs) to a HnV50 cluster as a function of the number of H atoms trapped in the cluster (n). The curves are calculated from Eq. (2).
atoms to 100 and 200, the H atoms pushed out from the H-cluster were observed to move around the nearby interstitial sites rather than being displaced away from the H-vacancy cluster. With more H atoms, more iron atoms within and around the H-clusters were found to be displaced, as marked by the yellow arrows (Fig. 5d). The H atoms were trapped at the octahedral sites located close to the inner surface of the cluster, leaving a vacancy cluster inside of the H-vacancy cluster, as shown in Fig. 5d and e. The binding energies of a self-interstitial atom 86
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for H-bubbles in bcc iron. Therefore, all these results indicate the reason why the enhancement of H atoms is weaker for loop growth than He atoms in experimental results [16,17].
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Declaration of Competing Interest There is no conflict of interest. Acknowledgements This research was supported by the National Natural Science Foundation of China (Project Nos. 11775162, 11675230, and 11975170). NG acknowledges the support of from the Youth Innovation Promotion Association CAS, China. The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University. References [1] N. Baluc, Materials for fusion power reactors, Plasma Phys. Control. Fusion. 48 (2006) B165. [2] F.A. Garner, B.M. Oliver, L.R. Greenwood, M.R. James, P.D. Ferguson, S.A. Maloy, W.F. Sommer, Determination of helium and hydrogen yield from measurements on pure metals and alloys irradiated by mixed high energy proton and spallation neutron spectra in LANSCE, J. Nucl. Mater. 296 (2001) 66–82. [3] F.A. Garner, E.P. Simonen, B.M. Oliver, L.R. Greenwood, M.L. Grossbeck, W.G. Wolfer, P.M. Scott, Retention of hydrogen in fcc metals irradiated at temperatures leading to high densities of bubbles or voids, J. Nucl. Mater. 356 (2006) 122–135. [4] A. Szummer, E. Jezierska, K. Lublińska, Hydrogen surface effects in ferritic stainless
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