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Ab initio study of interaction of helium with edge and screw dislocations in tungsten
Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Ab initio study of interaction of helium with edge and screw dislocations in tungsten Alexander Bakaev a,b,c,⇑, Dmitry Terentyev a, Petr Grigorev a,b,d, Matthias Posselt c, Evgeny E. Zhurkin b a
SCKCEN, Nuclear Materials Science Institute, Boeretang 200, Mol 2400, Belgium Department of Experimental Nuclear Physics K-89, Institute of Physics, Nanotechnology and Telecommunications, Peter the Great St. Petersburg Polytechnic University, 29 Polytekhnicheskaya str., 195251 St. Petersburg, Russia c Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400, 01328 Dresden, Germany d Ghent University, Applied Physics EA17 FUSION-DC, St. Pietersnieuwstraat, 41 B4, B-9000 Gent, Belgium b
a r t i c l e
i n f o
Article history: Received 31 August 2016 Received in revised form 28 November 2016 Accepted 28 November 2016 Available online xxxx Keywords: Dislocations Helium Tungsten Ab initio
a b s t r a c t The interaction of a single He atom with edge and screw dislocations in tungsten has been studied using ab initio calculations. It was revealed that He is strongly attracted to the core of both dislocations with the interaction energy of 1.3 and 3.0 eV for screw and edge dislocations, respectively, which corresponds to the detrapping temperature in thermal desorption spectroscopy experiments of about 500 K and 1050 K, respectively. The lowest energy positions for He around the dislocation cores are identified and the atomic structures are rationalized on the basis of elasticity theory considerations. Both types of dislocations exhibit a higher binding energy for He as compared to the He-He binding (known as self-trapping) and are weaker traps as compared to a single vacancy. It is, thus, concluded that the strong attraction to dislocation lines can contribute to the nucleation of He clusters in the temperature range which already excludes He self-trapping. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Tungsten is an important material for fusion applications, selected for use in International Thermonuclear Experimental Reactor (ITER) and DEMOnstration Power Station (DEMO) projects for divertor armor and first wall. During operation of thermonuclear fusion devices plasma-facing components undergo exposure to high energy fast neutrons and are bombarded by plasma ions such as H isotopes and He. This process inevitably leads to subsurface damage of the material due to accumulation (the socalled trapping) of H and He which reduces the resistance to a thermo-mechanical load and as the worst-case scenario can lead to a failure of the component. At present, significant efforts are dedicated to build up the theoretical and empirical models to assess the amount of gas atoms accumulated in the components during the operation and to estimate their storage for ITER-relevant conditions i.e. the high flux low energy plasma regime. Such a task requires deep understanding of a variety of processes including the very basic ones: the initial trapping of single H and He atoms in tungsten. ⇑ Corresponding author at: SCKCEN, Nuclear Materials Science Institute, Boeretang 200, Mol 2400, Belgium. E-mail address: [email protected] (A. Bakaev).
The experimental study of the problem of trapping and penetration of plasma components in tungsten-made components [1–3] is largely obstructed by the fact that the real material contains a certain density of lattice imperfections which include vacancies, dislocations and grain boundaries. The latter affect the penetration depth profile and may act as seeds for nucleation of high order clusters. In turn, the dislocations subdivide into edge and screw types which are likely traps for He (alike H [4]). Edge and screw dislocations experience different interaction with an interstitial He due to a difference in the elastic field around the core (being of larger size for edge dislocations [5]) and core structure itself. Since the real material contains not just pure straight edge and screw dislocations but rather a network of dislocations including mixed ones, it is rather difficult to deduce the actual trapping strength of particular types of dislocations using conventional experimental means (e.g. plasma exposure and thermal desorption spectroscopy), while this data is very important for numerical assessment of the H/He trapping and release. Therefore, in this work an alternative method – computational atomistic modelling – is chosen to determine the binding of He with the dislocations. From earlier theoretical studies [6] it is known that He interacts repulsively with any host atom in transition metals including W. Two He atoms bind strongly one to another (1 eV [7]) and this interaction vanishes at a distance of 3 Å [8]. He is strongly attracted
http://dx.doi.org/10.1016/j.nimb.2016.11.036 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
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A. Bakaev et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
to a vacancy – 4.5 eV [9] and it is positioned in the center of a vacancy, i.e. in the region with minimum of electron charge density. Also, the extended lattice defects such as grain boundaries and dislocations can play a role of trapping sites and diffusion channels thereby promoting deeper penetration of light interstitials into the material especially at a high temperature. In this work we apply ab initio calculations to deduce the interaction energy of a single He atom with ½h1 1 1i edge and screw dislocations and calculate the corresponding detrapping temperature. The energetically favourable positions around the dislocation core are identified. The contribution to the trapping of He in W from the dislocation of both screw and edge character is thus evaluated and compared with other typical trapping defects such as vacancy and grain boundaries.
2. Computational details The ab initio calculations were performed using the density functional theory code Vienna Ab Initio Simulation Package (VASP) [10,11]. The projector-augmented wave (PAW) potentials [12,13] were applied. The electron exchange-correlation functional was described within the generalized gradient approximation using PW91 functionals [14], with a Vosko-Wilk-Nusair interpolation [15]. Ionic relaxation was carried out using the conjugate gradient algorithm with a force convergence criterion of 0.03 eV/Å. During the relaxation the cell shape and volume (equal to the equilibrium volume of bulk tungsten of 3.1781 Å) were kept constant. The energy cutoff equal to 450 eV was applied to get the converged values for total energy and interaction energy. The screw and edge dislocations were studied by two different crystals and approaches. The screw dislocation was simulated in the model box with a square-like periodic array of dislocation quadrupoles as described in Ref. [16]. The supercell vectors {C1, C2, C3} are not orthogonal and are given by C1 = n a1; C2 = ½n a1 + m a2 + ½a3; C3 = a3, where a1 = 1/3[1 1 2]; a2 = 1/2 [1 1 0]; a3 = [1 1 1] and (n, m) = (15, 9). Two components were added to C1 and C2 along a3 (1/(3m) and 1/(6m), respectively) to account for the shift along [1 1 0] direction of the centers of gravity of the upward- and downward-pointing triangles, on which the two dislocations constituting the dipole are centered. The resulting box had 270 atoms and the dimensions of 41.01 38.82 5.50 Å3. The k-point mesh of 1 3 11 was used. The length of the dislocation line was 2 b, where b = ½h1 1 1i. Such a box size along the (z) direction was selected intentionally to minimize the self-interaction for He atoms (which is strongest at the distance of 1.5 Å) which was described in Refs. [7,8]. The
edge dislocation was simulated by introducing the single edge dislocation with b = ½h1 1 1i in the center of the crystal with rigid boundary conditions (periodic along the dislocation line only) as described in Ref. [17]. The box with 246 atoms was dimensioned as 24.82 7.78 22.51 Å3 along [1 1 1] (x), [1 1 0] (y) and [1 1 2] (z) axes. A k-point mesh of 1 7 1 was applied. In order to find the lowest energy positions for a single He atom next to the dislocation, we have identified the tetrahedral positions (the lowest energy interstitial positions for He in bulk W [8]) around the dislocation core as is visualized in the Fig 1a and b for screw and edge dislocations, respectively. Such positions were used as initial configuration for the further relaxation. The interaction energy between an interstitial He atom and the dislocation EI was assessed following the standard definition applied in other DFT works [18–20]:
EI ¼ ½EdislHe þ Ebulk ½Edisl þ EHe ;
ð1Þ
where Edisl and EHe are the total energies of the box with the dislocation or He atom in the tetrahedral position only, respectively, Edisl-He is the energy of the system with He and dislocation and Ebulk refers to the total energy of the bulk tungsten. Following this notation, a negative value implies an attractive interaction and vice versa. 3. Results and discussion 3.1. Interaction of He with screw dislocations The results of the calculations show (see Fig. 2a) that the screw dislocation attracts strongly a single He atom with the maximum interaction energy of 1.3 eV, occurring in the vicinity of the dislocation core. Note that this value is by 0.3 eV lower that the interaction energy for a He-He pair equal to 1.0 eV. Such a value for the He-screw dislocation interaction implies that the screw dislocation can further enhance clustering of He atoms and facilitate the formation of He clusters on dislocation lines even at temperatures which already exclude He self-trapping. Unlike the case of hydrogen [4], the equilibrium positions (corresponding to the lowest energy state configuration) for He near the screw dislocation core do not coincide with the charge density depleted zones. On the contrary, the interaction between He and neighboring tungsten atoms is purely elastic and the formation of bonding is energetically unfavorable [6]. In Fig 2a one can easily identify a set of three zones extended along non-equivalent h1 1 0i directions (contained in non-equivalent {1 1 1} planes) with the lowest energy positions (i.e. strong attraction) for He. Whereas,
Fig. 1. The initial tetrahedral positions for He (blue) and for W atoms (black) next to screw (a) and edge (b) dislocations. The dislocation is oriented perpendicular to the image plane and its location can be identified by arrows (a) and by the sign \ (b). The octahedral interstitial positions (gray) are also shown in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
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A. Bakaev et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
(a) 4
(b) 4 3 -1.32
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<110>, A
<110>, Å
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0.75 1.16
-2
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-4
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0
1
2
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<112>, Å
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-3
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0
1
2
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Fig. 2. The interaction energy for He atoms with the core of a screw (a) and edge (b) dislocation. Negative values imply attractive interaction. The lower the energy – the stronger the interaction. The dislocation line is oriented perpendicular to the image plane and the core center can be identified by arrows (a) and by the sign \ (b). The colormapped points represent the relaxed atomic positions of He atoms and the color code reflects the interaction energy. Note that in both figures W atoms (black circles) are visualized in the relaxed positions before introduction of He atoms. The dotted arrows shown in (a) represent two sets of non-equivalent h1 1 0i directions with more and less favourable positions for He atoms. The isolines of equal isotropic stress (in kBar) are shown in (b).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
other three non-equivalent h1 1 0i directions rotated by 60 degrees correspond to the zones with less favourable locations (interaction energy lower by two times). Such an interaction energy map follows well the displacement field map of the screw dislocation in transition metals as described in Ref. [21]: the zones with larger displacement allocate lower energy positions for He atoms. This information gives a hint for the identification of the migration path of a He atom approaching the dislocation core, as well as for the in-core migration. One can also note that the distance from the core, where the strong interaction energy (lower than 1 eV) is observed, is around 4 Å, which is fairly large given the relatively small strain induced by He atom.
release of He atoms in polycrystalline tungsten. Fig. 3 summarizes the energy for He-He self-interaction [7], He-vacancy interaction [9], interaction with symmetric tilted grain boundary of [1 0 0] P {0 1 3} 5 type [25] and for the edge and screw dislocations, as obtained here. All the calculations listed above were obtained using the same ab initio package and by applying projectoraugmented wave (PAW) potentials. There is also additional information in Fig. 3 on the right hand side (Y axis) where the detrapping temperature is visualized for each of the defect considered. It was estimated from the thermal desorption spectra (TDS) obtained by means of integrating of the release equation:
3.2. Interaction of He with edge dislocations
dCðtÞ Ei þ Em ¼ CðtÞm exp dt kT
The edge dislocations are also offering sites with a strong attraction for He atoms. In the immediate vicinity of the dislocation core, the interaction energy reaches 3.0 eV. This value is significantly larger than the He-He self-trapping energy (1 eV [7]) and is close to the He-vacancy interaction energy, being 4.5 eV [9]. The lowest energy positions for He atoms around the edge dislocation core do not belong to the zones with depleted electron charge density but follow well the hydrostatic stress distribution, as shown in Fig. 2b. Such a map was created after the relaxation (using a conjugate gradient algorithm with the energy convergence between two successive iterations of 1010 eV as a stopping criterion) of the same edge dislocation using a molecular statics algorithm and the embedded atom method (EAM) potential [22]. The stress map was computed using a virial calculation procedure embedded in LAMMPS [23,24]. Following such a procedure the stress tensor per atom is evaluated, then the pressure is calculated by summation of diagonal components of the stress tensor after dividing it by 3 O, where O is an atomic volume calculated as Voronoi tessellation for each atom. 3.3. Hierarchy of trapping defects for He Given the above obtained results and using additional data available in the literature, we constructed a ranking of different microstructural defects that may play a role in the trapping and
ð2Þ
where Ei – the interaction energy between He and the defect considered, Em – He bulk migration energy (equal to 0.06 eV as reported in the ab initio study [7]), k – Boltzmann factor, m – Debye frequency (typical lattice vibration frequency equal to 1013 s1 was used following [26]), T – temperature (varying in time as T(t) = 0.5t in order to mimic the TDS measurement done at a typical heating ramp rate of 0.5 K/s). The detrapping temperature reported in Fig. 3 corresponds to the peak in the TDS, i.e. when the release rate from the trap reaches the maximum. The ranking of the interaction strength shows that He selftrapping (1 eV) is likely to be the prevailing mechanism for the He accumulation under high flux exposures up to a temperature of 400 K. Screw dislocations and grain boundary interfaces are stronger traps with the interaction energy lower by 0.3–0.4 eV. Detrapping from these types of defects is expected to occur at temperature of about 500 K. Indeed, the TDS measurements done on polycrystalline tungsten irradiated by H+ ions at room temperature, including 8 keV He+ irradiation to a fluence 3 1022 He+/m2 [27], 500 eV He + irradiation to fluences 1021–1024 He+/m2 [28] and 250 eV He+ irradiation to a fluence 2.4 1017 He+/m2 (done for single crystal W [29]), show the He release with a peak around 500 K. This is attributed to the detrapping from grain boundaries and possibly dislocation loops near HenVm complexes [28]. The current
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
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282 376 470 564 658 752 846 940 1034 1128 1222 1316 1410 1504 1598 1692
-1.0
Interaction energy (eV)
-1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0 He-He
He-SD
He-GB
He4V+He He5V+He
He-ED
HeV+He He3V+He He2V+He
Detrapping temperature (K)
-0.5
He-Vac
P Fig. 3. The interaction energy of He to different microstructural defects such as interstitial He (self-trapping) [7], symmetrical tilted grain boundary (GB) [1 0 0]{0 1 3} 5 [25], HeN=05-vacancy [9], edge (ED) and screw (SD) dislocation (this work). The lower energy implies the stronger interaction. The right Y axis represents the detrapping temperature of He from the considered defect, calculated as described in the text.
study points to a possible contribution (not exclusive) from the screw dislocations as well. Whereas, the interstitial dislocation loops are expected to have much stronger binding as their structure is similar to that of the edge dislocation. The ½h1 1 1i{1 1 0} edge dislocation is a much stronger trap than a ½h1 1 1i screw dislocation and its interaction strength is comparable with the attraction of He by HeN=1;2;3-V clusters i.e. from 3 to 3.2 eV. The interaction energy of 3 eV corresponds to the detrapping temperature of about 1050 K. The TDS peak at this temperature was observed after the 500 eV He+ irradiation at temperature 300 K and 700 K to fluences of 1021–1024 He+/m2 [28] and 55 eV He+ irradiation of polycrystalline W at a surface temperature of 1300 K to a fluence of 1.5 1025 m2 [30]. Our study suggests that the edge dislocations (or dislocation loops) can contribute to the He trapping for the exposures done at high temperature or under neutron irradiation (as interstitial dislocation loops are directly produced in collision cascades), where the He selftrapping is no longer operating. A single vacancy is the strongest trap for He atoms (EI = 4.5 eV) with the detrapping temperature of about 1500 K. The release of He with several peaks around 1200–1500 K was registered in Ref. [31], where low energy He+ ion implantation (0.5–1.5 keV) was made so as to produce and immediately fill the vacancies. In that experimental work, the strongest trapping is assigned to the He-V, while sub-peaks below 1500 K were attributed to HeN-V clusters, as was later confirmed by the ab initio calculations.
ary. Whereas, the interaction strength for the edge dislocation lies in the range of the binding energies of He to the HeN-V cluster (for N = 4,5). The strongest trapping objects are therefore a single vacancy and di/tri He – vacancy clusters. Thus, we conclude that trapping of He to the screw dislocation operates in the temperature range below 500 K which is very close to the self-trapping temperature window (400 K and below), while trapping to the edge dislocation extends to a much higher temperature equal to 1050 K. This is an important result, since the dislocation loops (with pure edge or mixed character) are directly produced under neutron irradiation to which tungsten will also be subject to under fusion operational conditions. Therefore, neutron irradiation, generating both vacancy- and self-interstitial type defects will produce traps with comparable binding strength. However, since dislocations are line defects, one also needs to assess the diffusion of He along dislocation lines and He-He interaction in the vicinity of the dislocation core. This will help to clarify whether dislocation lines can also enhance He diffusion (via pipe diffusion [4,20,32]) and He self-trapping. Acknowledgements The work was supported by the funding received from the Euratom research and training programme 2014-2018 under grant agreement No 633053 (EUROfusion/Enabling Research programme). The views and opinions expressed herein do not necessarily reflect those of the European Commission. AB is grateful to HELIOS supercomputer cluster (Japan) for the computational resources provided.
4. Concluding remarks References In this work we have studied the interaction of a single He atom with ½h1 1 1i{1 1 0} edge and ½h1 1 1i screw dislocations using the ab initio method. The energetically favourable positions for He in the vicinity of the dislocation cores were identified and compared with the local distributions of the electron charge density and hydrostatic pressure. The maximum interaction energy of He to the edge dislocation core is equal to 3.0 eV and it is more than double the interaction energy with the screw dislocation (1.3 eV). The interaction strength of several most typical microstructural defects was discussed in the light of the current results. It was revealed that the interaction of He with the screw dislocation, naturally available in the microstructure of the polycrystalline tungsten, is stronger than the He-He binding and is comparable with the interaction energy to the tilted grain bound-
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Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Ab initio study of interaction of helium with edge and screw dislocations in tungsten Alexander Bakaev a,b,c,⇑, Dmitry Terentyev a, Petr Grigorev a,b,d, Matthias Posselt c, Evgeny E. Zhurkin b a
SCKCEN, Nuclear Materials Science Institute, Boeretang 200, Mol 2400, Belgium Department of Experimental Nuclear Physics K-89, Institute of Physics, Nanotechnology and Telecommunications, Peter the Great St. Petersburg Polytechnic University, 29 Polytekhnicheskaya str., 195251 St. Petersburg, Russia c Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400, 01328 Dresden, Germany d Ghent University, Applied Physics EA17 FUSION-DC, St. Pietersnieuwstraat, 41 B4, B-9000 Gent, Belgium b
a r t i c l e
i n f o
Article history: Received 31 August 2016 Received in revised form 28 November 2016 Accepted 28 November 2016 Available online xxxx Keywords: Dislocations Helium Tungsten Ab initio
a b s t r a c t The interaction of a single He atom with edge and screw dislocations in tungsten has been studied using ab initio calculations. It was revealed that He is strongly attracted to the core of both dislocations with the interaction energy of 1.3 and 3.0 eV for screw and edge dislocations, respectively, which corresponds to the detrapping temperature in thermal desorption spectroscopy experiments of about 500 K and 1050 K, respectively. The lowest energy positions for He around the dislocation cores are identified and the atomic structures are rationalized on the basis of elasticity theory considerations. Both types of dislocations exhibit a higher binding energy for He as compared to the He-He binding (known as self-trapping) and are weaker traps as compared to a single vacancy. It is, thus, concluded that the strong attraction to dislocation lines can contribute to the nucleation of He clusters in the temperature range which already excludes He self-trapping. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Tungsten is an important material for fusion applications, selected for use in International Thermonuclear Experimental Reactor (ITER) and DEMOnstration Power Station (DEMO) projects for divertor armor and first wall. During operation of thermonuclear fusion devices plasma-facing components undergo exposure to high energy fast neutrons and are bombarded by plasma ions such as H isotopes and He. This process inevitably leads to subsurface damage of the material due to accumulation (the socalled trapping) of H and He which reduces the resistance to a thermo-mechanical load and as the worst-case scenario can lead to a failure of the component. At present, significant efforts are dedicated to build up the theoretical and empirical models to assess the amount of gas atoms accumulated in the components during the operation and to estimate their storage for ITER-relevant conditions i.e. the high flux low energy plasma regime. Such a task requires deep understanding of a variety of processes including the very basic ones: the initial trapping of single H and He atoms in tungsten. ⇑ Corresponding author at: SCKCEN, Nuclear Materials Science Institute, Boeretang 200, Mol 2400, Belgium. E-mail address: [email protected] (A. Bakaev).
The experimental study of the problem of trapping and penetration of plasma components in tungsten-made components [1–3] is largely obstructed by the fact that the real material contains a certain density of lattice imperfections which include vacancies, dislocations and grain boundaries. The latter affect the penetration depth profile and may act as seeds for nucleation of high order clusters. In turn, the dislocations subdivide into edge and screw types which are likely traps for He (alike H [4]). Edge and screw dislocations experience different interaction with an interstitial He due to a difference in the elastic field around the core (being of larger size for edge dislocations [5]) and core structure itself. Since the real material contains not just pure straight edge and screw dislocations but rather a network of dislocations including mixed ones, it is rather difficult to deduce the actual trapping strength of particular types of dislocations using conventional experimental means (e.g. plasma exposure and thermal desorption spectroscopy), while this data is very important for numerical assessment of the H/He trapping and release. Therefore, in this work an alternative method – computational atomistic modelling – is chosen to determine the binding of He with the dislocations. From earlier theoretical studies [6] it is known that He interacts repulsively with any host atom in transition metals including W. Two He atoms bind strongly one to another (1 eV [7]) and this interaction vanishes at a distance of 3 Å [8]. He is strongly attracted
http://dx.doi.org/10.1016/j.nimb.2016.11.036 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
2
A. Bakaev et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
to a vacancy – 4.5 eV [9] and it is positioned in the center of a vacancy, i.e. in the region with minimum of electron charge density. Also, the extended lattice defects such as grain boundaries and dislocations can play a role of trapping sites and diffusion channels thereby promoting deeper penetration of light interstitials into the material especially at a high temperature. In this work we apply ab initio calculations to deduce the interaction energy of a single He atom with ½h1 1 1i edge and screw dislocations and calculate the corresponding detrapping temperature. The energetically favourable positions around the dislocation core are identified. The contribution to the trapping of He in W from the dislocation of both screw and edge character is thus evaluated and compared with other typical trapping defects such as vacancy and grain boundaries.
2. Computational details The ab initio calculations were performed using the density functional theory code Vienna Ab Initio Simulation Package (VASP) [10,11]. The projector-augmented wave (PAW) potentials [12,13] were applied. The electron exchange-correlation functional was described within the generalized gradient approximation using PW91 functionals [14], with a Vosko-Wilk-Nusair interpolation [15]. Ionic relaxation was carried out using the conjugate gradient algorithm with a force convergence criterion of 0.03 eV/Å. During the relaxation the cell shape and volume (equal to the equilibrium volume of bulk tungsten of 3.1781 Å) were kept constant. The energy cutoff equal to 450 eV was applied to get the converged values for total energy and interaction energy. The screw and edge dislocations were studied by two different crystals and approaches. The screw dislocation was simulated in the model box with a square-like periodic array of dislocation quadrupoles as described in Ref. [16]. The supercell vectors {C1, C2, C3} are not orthogonal and are given by C1 = n a1; C2 = ½n a1 + m a2 + ½a3; C3 = a3, where a1 = 1/3[1 1 2]; a2 = 1/2 [1 1 0]; a3 = [1 1 1] and (n, m) = (15, 9). Two components were added to C1 and C2 along a3 (1/(3m) and 1/(6m), respectively) to account for the shift along [1 1 0] direction of the centers of gravity of the upward- and downward-pointing triangles, on which the two dislocations constituting the dipole are centered. The resulting box had 270 atoms and the dimensions of 41.01 38.82 5.50 Å3. The k-point mesh of 1 3 11 was used. The length of the dislocation line was 2 b, where b = ½h1 1 1i. Such a box size along the (z) direction was selected intentionally to minimize the self-interaction for He atoms (which is strongest at the distance of 1.5 Å) which was described in Refs. [7,8]. The
edge dislocation was simulated by introducing the single edge dislocation with b = ½h1 1 1i in the center of the crystal with rigid boundary conditions (periodic along the dislocation line only) as described in Ref. [17]. The box with 246 atoms was dimensioned as 24.82 7.78 22.51 Å3 along [1 1 1] (x), [1 1 0] (y) and [1 1 2] (z) axes. A k-point mesh of 1 7 1 was applied. In order to find the lowest energy positions for a single He atom next to the dislocation, we have identified the tetrahedral positions (the lowest energy interstitial positions for He in bulk W [8]) around the dislocation core as is visualized in the Fig 1a and b for screw and edge dislocations, respectively. Such positions were used as initial configuration for the further relaxation. The interaction energy between an interstitial He atom and the dislocation EI was assessed following the standard definition applied in other DFT works [18–20]:
EI ¼ ½EdislHe þ Ebulk ½Edisl þ EHe ;
ð1Þ
where Edisl and EHe are the total energies of the box with the dislocation or He atom in the tetrahedral position only, respectively, Edisl-He is the energy of the system with He and dislocation and Ebulk refers to the total energy of the bulk tungsten. Following this notation, a negative value implies an attractive interaction and vice versa. 3. Results and discussion 3.1. Interaction of He with screw dislocations The results of the calculations show (see Fig. 2a) that the screw dislocation attracts strongly a single He atom with the maximum interaction energy of 1.3 eV, occurring in the vicinity of the dislocation core. Note that this value is by 0.3 eV lower that the interaction energy for a He-He pair equal to 1.0 eV. Such a value for the He-screw dislocation interaction implies that the screw dislocation can further enhance clustering of He atoms and facilitate the formation of He clusters on dislocation lines even at temperatures which already exclude He self-trapping. Unlike the case of hydrogen [4], the equilibrium positions (corresponding to the lowest energy state configuration) for He near the screw dislocation core do not coincide with the charge density depleted zones. On the contrary, the interaction between He and neighboring tungsten atoms is purely elastic and the formation of bonding is energetically unfavorable [6]. In Fig 2a one can easily identify a set of three zones extended along non-equivalent h1 1 0i directions (contained in non-equivalent {1 1 1} planes) with the lowest energy positions (i.e. strong attraction) for He. Whereas,
Fig. 1. The initial tetrahedral positions for He (blue) and for W atoms (black) next to screw (a) and edge (b) dislocations. The dislocation is oriented perpendicular to the image plane and its location can be identified by arrows (a) and by the sign \ (b). The octahedral interstitial positions (gray) are also shown in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
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A. Bakaev et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
(a) 4
(b) 4 3 -1.32
2
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1
-0.81
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<110>, A
<110>, Å
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0
-0.49 -0.08 0.34
-1
0.75 1.16
-2
-2 -3
-4
-4
-4
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-2
-1
0
1
2
3
4
<112>, Å
-4
-3
-2
-1
0
1
2
3
4
<111>, Å
Fig. 2. The interaction energy for He atoms with the core of a screw (a) and edge (b) dislocation. Negative values imply attractive interaction. The lower the energy – the stronger the interaction. The dislocation line is oriented perpendicular to the image plane and the core center can be identified by arrows (a) and by the sign \ (b). The colormapped points represent the relaxed atomic positions of He atoms and the color code reflects the interaction energy. Note that in both figures W atoms (black circles) are visualized in the relaxed positions before introduction of He atoms. The dotted arrows shown in (a) represent two sets of non-equivalent h1 1 0i directions with more and less favourable positions for He atoms. The isolines of equal isotropic stress (in kBar) are shown in (b).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
other three non-equivalent h1 1 0i directions rotated by 60 degrees correspond to the zones with less favourable locations (interaction energy lower by two times). Such an interaction energy map follows well the displacement field map of the screw dislocation in transition metals as described in Ref. [21]: the zones with larger displacement allocate lower energy positions for He atoms. This information gives a hint for the identification of the migration path of a He atom approaching the dislocation core, as well as for the in-core migration. One can also note that the distance from the core, where the strong interaction energy (lower than 1 eV) is observed, is around 4 Å, which is fairly large given the relatively small strain induced by He atom.
release of He atoms in polycrystalline tungsten. Fig. 3 summarizes the energy for He-He self-interaction [7], He-vacancy interaction [9], interaction with symmetric tilted grain boundary of [1 0 0] P {0 1 3} 5 type [25] and for the edge and screw dislocations, as obtained here. All the calculations listed above were obtained using the same ab initio package and by applying projectoraugmented wave (PAW) potentials. There is also additional information in Fig. 3 on the right hand side (Y axis) where the detrapping temperature is visualized for each of the defect considered. It was estimated from the thermal desorption spectra (TDS) obtained by means of integrating of the release equation:
3.2. Interaction of He with edge dislocations
dCðtÞ Ei þ Em ¼ CðtÞm exp dt kT
The edge dislocations are also offering sites with a strong attraction for He atoms. In the immediate vicinity of the dislocation core, the interaction energy reaches 3.0 eV. This value is significantly larger than the He-He self-trapping energy (1 eV [7]) and is close to the He-vacancy interaction energy, being 4.5 eV [9]. The lowest energy positions for He atoms around the edge dislocation core do not belong to the zones with depleted electron charge density but follow well the hydrostatic stress distribution, as shown in Fig. 2b. Such a map was created after the relaxation (using a conjugate gradient algorithm with the energy convergence between two successive iterations of 1010 eV as a stopping criterion) of the same edge dislocation using a molecular statics algorithm and the embedded atom method (EAM) potential [22]. The stress map was computed using a virial calculation procedure embedded in LAMMPS [23,24]. Following such a procedure the stress tensor per atom is evaluated, then the pressure is calculated by summation of diagonal components of the stress tensor after dividing it by 3 O, where O is an atomic volume calculated as Voronoi tessellation for each atom. 3.3. Hierarchy of trapping defects for He Given the above obtained results and using additional data available in the literature, we constructed a ranking of different microstructural defects that may play a role in the trapping and
ð2Þ
where Ei – the interaction energy between He and the defect considered, Em – He bulk migration energy (equal to 0.06 eV as reported in the ab initio study [7]), k – Boltzmann factor, m – Debye frequency (typical lattice vibration frequency equal to 1013 s1 was used following [26]), T – temperature (varying in time as T(t) = 0.5t in order to mimic the TDS measurement done at a typical heating ramp rate of 0.5 K/s). The detrapping temperature reported in Fig. 3 corresponds to the peak in the TDS, i.e. when the release rate from the trap reaches the maximum. The ranking of the interaction strength shows that He selftrapping (1 eV) is likely to be the prevailing mechanism for the He accumulation under high flux exposures up to a temperature of 400 K. Screw dislocations and grain boundary interfaces are stronger traps with the interaction energy lower by 0.3–0.4 eV. Detrapping from these types of defects is expected to occur at temperature of about 500 K. Indeed, the TDS measurements done on polycrystalline tungsten irradiated by H+ ions at room temperature, including 8 keV He+ irradiation to a fluence 3 1022 He+/m2 [27], 500 eV He + irradiation to fluences 1021–1024 He+/m2 [28] and 250 eV He+ irradiation to a fluence 2.4 1017 He+/m2 (done for single crystal W [29]), show the He release with a peak around 500 K. This is attributed to the detrapping from grain boundaries and possibly dislocation loops near HenVm complexes [28]. The current
Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036
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A. Bakaev et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
282 376 470 564 658 752 846 940 1034 1128 1222 1316 1410 1504 1598 1692
-1.0
Interaction energy (eV)
-1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0 He-He
He-SD
He-GB
He4V+He He5V+He
He-ED
HeV+He He3V+He He2V+He
Detrapping temperature (K)
-0.5
He-Vac
P Fig. 3. The interaction energy of He to different microstructural defects such as interstitial He (self-trapping) [7], symmetrical tilted grain boundary (GB) [1 0 0]{0 1 3} 5 [25], HeN=05-vacancy [9], edge (ED) and screw (SD) dislocation (this work). The lower energy implies the stronger interaction. The right Y axis represents the detrapping temperature of He from the considered defect, calculated as described in the text.
study points to a possible contribution (not exclusive) from the screw dislocations as well. Whereas, the interstitial dislocation loops are expected to have much stronger binding as their structure is similar to that of the edge dislocation. The ½h1 1 1i{1 1 0} edge dislocation is a much stronger trap than a ½h1 1 1i screw dislocation and its interaction strength is comparable with the attraction of He by HeN=1;2;3-V clusters i.e. from 3 to 3.2 eV. The interaction energy of 3 eV corresponds to the detrapping temperature of about 1050 K. The TDS peak at this temperature was observed after the 500 eV He+ irradiation at temperature 300 K and 700 K to fluences of 1021–1024 He+/m2 [28] and 55 eV He+ irradiation of polycrystalline W at a surface temperature of 1300 K to a fluence of 1.5 1025 m2 [30]. Our study suggests that the edge dislocations (or dislocation loops) can contribute to the He trapping for the exposures done at high temperature or under neutron irradiation (as interstitial dislocation loops are directly produced in collision cascades), where the He selftrapping is no longer operating. A single vacancy is the strongest trap for He atoms (EI = 4.5 eV) with the detrapping temperature of about 1500 K. The release of He with several peaks around 1200–1500 K was registered in Ref. [31], where low energy He+ ion implantation (0.5–1.5 keV) was made so as to produce and immediately fill the vacancies. In that experimental work, the strongest trapping is assigned to the He-V, while sub-peaks below 1500 K were attributed to HeN-V clusters, as was later confirmed by the ab initio calculations.
ary. Whereas, the interaction strength for the edge dislocation lies in the range of the binding energies of He to the HeN-V cluster (for N = 4,5). The strongest trapping objects are therefore a single vacancy and di/tri He – vacancy clusters. Thus, we conclude that trapping of He to the screw dislocation operates in the temperature range below 500 K which is very close to the self-trapping temperature window (400 K and below), while trapping to the edge dislocation extends to a much higher temperature equal to 1050 K. This is an important result, since the dislocation loops (with pure edge or mixed character) are directly produced under neutron irradiation to which tungsten will also be subject to under fusion operational conditions. Therefore, neutron irradiation, generating both vacancy- and self-interstitial type defects will produce traps with comparable binding strength. However, since dislocations are line defects, one also needs to assess the diffusion of He along dislocation lines and He-He interaction in the vicinity of the dislocation core. This will help to clarify whether dislocation lines can also enhance He diffusion (via pipe diffusion [4,20,32]) and He self-trapping. Acknowledgements The work was supported by the funding received from the Euratom research and training programme 2014-2018 under grant agreement No 633053 (EUROfusion/Enabling Research programme). The views and opinions expressed herein do not necessarily reflect those of the European Commission. AB is grateful to HELIOS supercomputer cluster (Japan) for the computational resources provided.
4. Concluding remarks References In this work we have studied the interaction of a single He atom with ½h1 1 1i{1 1 0} edge and ½h1 1 1i screw dislocations using the ab initio method. The energetically favourable positions for He in the vicinity of the dislocation cores were identified and compared with the local distributions of the electron charge density and hydrostatic pressure. The maximum interaction energy of He to the edge dislocation core is equal to 3.0 eV and it is more than double the interaction energy with the screw dislocation (1.3 eV). The interaction strength of several most typical microstructural defects was discussed in the light of the current results. It was revealed that the interaction of He with the screw dislocation, naturally available in the microstructure of the polycrystalline tungsten, is stronger than the He-He binding and is comparable with the interaction energy to the tilted grain bound-
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Please cite this article in press as: A. Bakaev et al., Ab initio study of interaction of helium with edge and screw dislocations in tungsten, Nucl. Instr. Meth. B (2016), http://dx.doi.org/10.1016/j.nimb.2016.11.036