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Interaction of helium–vacancy clusters with edge dislocations in α-Fe
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NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 265 (2007) 541–546 www.elsevier.com/locate/nimb
Interaction of helium–vacancy clusters with edge dislocations in a-Fe L. Yang
a,b
, X.T. Zu
a,*
, Z.G. Wang a, F. Gao c, X.Y. Wang d, H.L. Heinisch c, R.J. Kurtz
c
a
d
Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, China b School of Physics and Electronics Information, China West Normal University, Nanchong 637002, China c Pacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA National Key Laboratory for Surface Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621907, China Received 14 July 2007; received in revised form 29 August 2007 Available online 22 October 2007
Abstract Molecular static calculations were performed to evaluate the formation energies and binding energies of helium–vacancy (He–V) clusters in and near the core of an a/2<1 1 1>{1 1 0} edge dislocation in a-Fe with empirical potentials. The formation energies of these He–V clusters and their binding energies to the dislocation depend on the helium-to-vacancy ratio of the clusters. For the ratio equal to or larger than 1, the helium–vacancy clusters have negative binding energies on the compression side of the dislocation and strong positive binding energy on the tension side. However, for the ratio less than 1, the He–V clusters have positive binding energy on the both sides near the dislocation core. On the slip plane, the binding energies of the He–V clusters to the dislocation depend on not only the heliumto-vacancy ratio, but also the cluster size. Ó 2007 Elsevier B.V. All rights reserved. PACS: 02.70.c; 61.72.Ji; 61.72.Lk; 81.05.Bx Keywords: Molecular static; Edge dislocation; Helium–vacancy cluster; a-Fe
1. Introduction The effect of helium on the first wall structural materials has been widely recognized as one of the most crucial materials issues in nuclear fusion reactors, because the synergistic interaction of large amounts of helium with the existing and radiation-induced defects and microstructures in materials can significantly degrade their mechanical properties [1–3]. Computer simulations provide an important means of obtaining insight into fundamental understanding of the complex atomic-level processes of the defects controlling microstructural evolution in materials. For example, the interactions of helium atoms with the defects and microstructures in a-Fe (which is taken to be a first-order model for the ferritic steels that are promising materials for fusion reactors) have been systematically investigated *
Corresponding author. Tel./fax: +86 28 83201939. E-mail address: [email protected] (X.T. Zu).
0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.10.007
using molecular dynamics, molecular statics and Monte Carlo methods [4,5]. Atomistic calculations demonstrated the strong binding of He to grain boundaries (GBs) in aFe, finding that both substitutional and interstitial He atoms are trapped at GBs [6]. Recently, the interactions of helium atoms with edge dislocations in a-Fe were studied [7]. It is found that interstitial He atoms are either repelled from or trapped at edge dislocations in a-Fe, depending on the direction of approach. Near the dislocation core on the tension side He is strongly trapped as a crowdion with 1–2 eV greater binding energy than that of an octahedral interstitial in perfect iron, and in this form He atoms can migrate along the dislocation with a migration energy of 0.4–0.5 eV. Similar studies of He atoms in and near screw dislocations in a-Fe [8] determined that the binding energy of He to screw dislocations is about half that to edge dislocations, and that interstitial He atoms migrate along the screw with about the same migration energies as along the edge.
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In irradiated materials, in addition to single helium atoms, a large number of helium–vacancy clusters or bubbles are formed due to the extremely low solubility of helium in metals [9–12]. In the present work, molecular statics calculations are utilized to study the binding energy of small helium––vacancy (He–V) clusters in and around the core of the a/2 [1 1 1]{1 1 0} edge dislocation in a-Fe. Recent computer simulations have shown that the stability of He–V clusters in a perfect Fe lattice does not depend much on cluster size, but rather on the helium-to-vacancy (He/V) ratio in a-Fe [4,13]. However, the stability of He– V clusters in the ‘dislocated’ lattice has not been reported before. In this work, the binding energies of a large number of small He–V clusters to an edge dislocation were studied in detail. 2. Method and geometry The modified version of the MOLDY computer code [14] was used in this work. The interatomic potentials of Ackland [15], Wilson–Johnson [16] and Beck [17] were used to describe the interactions of Fe–Fe, Fe–He and He–He, respectively. The same set of interatomic potentials was also employed in the earlier calculations of the binding energies and atomic configurations of He–V clusters in perfect a-Fe, and to simulate the migration of He interstitials in grain boundaries and dislocations [4,7,18,19]. To be consistent with the body of earlier work, we have used this set of potentials in the present simulations. The cohesive energies calculated with these potentials are 4.316 and 0.00568 eV/atom for a perfect bcc Fe crystal and a perfect fcc He crystal, respectively. The formation energies of a substitutional He atom and an interstitial He atom using these potentials are calculated to be 3.25 and 5.25 eV, respectively. The computational model consists of a cubic cell of Fe atoms. The x, y and z axes of the simulated crystal were oriented along [1 1 1], ½ 1 1 2 and ½1 1 0 directions. An a/ 2h1 1 1i{1 1 0} edge dislocation was created according to the method developed by Osetsky and Bacon [20]. The following procedure has been adopted. The initial unrelaxed structure is presented as two half-crystals strained to have different lattice parameters in the direction b and joined along the dislocation slip plane. The two half-crystals are strained to have smaller and larger lattice repeat distance, respectively, in the b direction than their natural value b. The total size difference equals to b. Periodic boundary conditions were employed in the x and y directions, corresponding to the direction of the Burgers vector b, and the dislocation line direction, respectively, and fixed conditions were used across the z boundaries. The size of the calculated region is about 44 a0 (a0 is the lattice constant of Fe) for the [1 1 1] direction, 32 a0 for the ½ 1 1 2 direction and 45 a0 for the ½1 1 0 direction. The mobile region contains 10,6400 atoms. Thermal effects are not considered here and so simulation was carried out at 0 K by molecular static (MS) relaxation. The simulation cell containing an a/2h1 1 1i{1 1 0} edge dislo-
cation, and the corresponding relationship of the directions is shown schematically in Fig. 1. A He–V cluster was introduced into the relaxed, dislocated model, and the new configuration was then relaxed again to obtain the minimum energy configuration. Binding energy calculations were performed for He–V clusters in and around the core of the dislocation. The binding energy of a He–V cluster to the dislocation is defined as the difference between the formation energy of the He–V cluster in a perfect Fe lattice and that in the ‘dislocated’ lattice. The formation energy of a He–V cluster in the dislocated lattice, Efd , is defined as tot fe he Efd ¼ U tot dþHen Vm U d þ m E n E ;
ð1Þ
tot where U tot dþHen Vm and U d are the total energy of the crystal which includes the edge dislocation, with and without the HenVm cluster, respectively. Efe =4.316 eV and Ehe =0.00567758 are the cohesive energies of bcc Fe and fcc He.
3. Results and discussion The previous atomistic simulations in perfect Fe indicate that the properties of He–V clusters mainly depend on the He/V ratio of clusters [4]. In this work, the binding energies of some small He–V clusters, i.e. HenVm (n = 1, 2, 3, 4 and m = 1, 2, 3, 4), with He/V ratios ranging from 0.25 to 4, were investigated in and around the core of the dislocation. 3.1. HenVm (n P m) As described above, the formation energies of He–V clusters within the dislocation were calculated using Eq. (1) after relaxing the simulated system to the minimum potential configuration. In the simulations, the vacancies of HenV2 (n = 1, 2, 3, 4) clusters lie along the <1 1 1> direction, and the vacancies of the HenV3 (n = 1, 2, 3, 4) and HenV4 (n = 1, 2, 3, 4) locate at the {1 1 0} plane, as shown in Fig. 2. In each case the He atoms were initially placed within the vacancy clusters, and the He–V clusters placed in the ‘dislocated’ lattices rapidly relaxed to the lowest
Fig. 1. Schematic view of the simulation cell with the corresponding crystallographic orientations.
L. Yang et al. / Nucl. Instr. and Meth. in Phys. Res. B 265 (2007) 541–546
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Fig. 2. Configurations of vacancies in HenVm clusters: (a) the divacancy in the HenV2 (n = 1, 2, 3, 4) clusters, (b) the vacancies of the HenV3 (n = 1, 2, 3, 4) clusters and (c) the vacancies of the HenV4 (n = 1, 2, 3, 4) clusters.
Fig. 3. Binding energies of HenVm (n P m) clusters to an a/2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the glide plane along a line through the center of the dislocation normal to the glide plane: (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1).
energy configurations. Fig. 3 shows the binding energy of a HenVm (n P m) cluster to the dislocation as a function of the distance of the cluster from the slip plane along a line running through the center of the dislocation from the ten-
sion to the compression side (along ½1 1 0 in Fig. 1): (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1). It can be seen that the binding energies of all the HenVm (n P m) clusters are positive on the tension side of the dislocation (negative distance from the glide plane) and negative on the compression side, which illustrates that the HenVm (n P m) clusters are strongly trapped on the tension side of edge dislocations and repelled from the compression side of edge dislocations in a-Fe. Using conjugate gradient relaxation, Heinisch et al. [7] investigated the interaction of a He interstitial with an edge dislocation, and the results showed that interstitial He atoms have negative binding energy on the compression side of the dislocation and strong positive binding energy on the tension side. The behavior of the He–V clusters near the dislocation is generally similar to that observed for single He interstitial. However, the binding energies for the He–V clusters are larger than that for a single interstitial or a substitutional He atom. Furthermore, at the same distance from the slip plane on the tension side of the dislocation, the binding energy increases with increasing He/V ratio, but decreases on the compression side. For the same He/V ratio, the binding energy increases with increasing He–V cluster size, as shown in Fig. 3(b). However, these results illustrate that the binding energies of the He–V clusters to the edge dislocation are less sensitive to cluster size than to the He/V ratio. All above may be attributed to the relaxed configurations of the clusters, which depend on the effects of the dislocation field. For example, near the dislocation core on the tension side, the relaxed He2V cluster lies along a <1 1 1> direction, forming a dumbbell configuration and has a maximum binding energy of 1.9 eV within the dislocation core. However, with the increase in the distance from the core on the compression side, the He2V cluster relaxes to locate in different directions, as shown in Fig. 4(a). It can be seen that within about 0.5 nm from the core the relaxed He2V cluster lies almost along the <1 1 1> direction on the tension side (negative distance from the glide plane). With increase in the distance on the tension side, the He2V relaxes to form a dumbbell configuration along the <1 0 0> direction, similar to that found in pure a-Fe [18]. However, at the first atom plane of the
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Fig. 4. The relaxed atomic configurations of (a) the He2V cluster and (b) the He3V cluster placed on the different atom positions from the slip plane along a line running through the center of the dislocation from the tension to the compression side, where the black and the gray spheres represent vacancies and He atoms, respectively.
compression side, the He2V cluster is dissociated to form a substitutional He, with the second He atom migrating towards the slip plane, which results in significant displacements of its neighboring Fe atoms within the close-packed row, which was plotted in the rectangle in Fig. 4(a). At the second and further atom planes of the compression side, the directions of the relaxed He2V clusters change with different positions, and there is not a fixed direction that can be used to define their configurations. In the dislocation core, the relaxed He3V cluster lies in a {1 1 0} plane and forms a triangle configuration, where three interstitial He atoms locate at three corners of a triangle and the vacancy is at the center of three He atoms, as shown in Fig. 4(b). However, within about 0.4 nm away from the core, the He3V cluster is found to dissolve, forming a substitutional He at the vacancy site, with two He atoms relaxed into the slip plane, which results in significant displacements of their neighboring Fe atoms within the slip plane, which was not plotted in Fig. 4(b). The He3V clusters placed farther away from the dislocation core on the tension side spontaneously relax into the {1 0 0} plane, and they have smaller binding energies that decrease with increasing distance from the dislocation. Fig. 5 shows the binding energies of HenVm (n P m) clusters to the dislocation as a function of the distance of the clusters from the core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane; (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1). It is clear that the relaxed He–V clusters located near the dislocation core have the maximum binding energies. With increasing distance from the core, the binding energies rapidly decrease. When the distance is larger than 1.3 nm, the binding energies approach to zero, and the configurations of the He–V clusters on both sides of the dislocation core are almost the same. Therefore, within about 2.6 nm of the core, the HenVm (n P m) clusters are trapped
Fig. 5. Binding energies of the HenVm clusters to an a/2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of the distance from the dislocation core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane: (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1).
by the dislocation with a large binding energy on the slip plane. In the simulations, it is found that when the distance from the core is less than 0.5 nm or larger than 1.3 nm, the simulated system rapidly relaxes to the lowest energy configuration, but within the range from 0.5 nm to 1.3 nm, it achieves the lowest energy configuration with many relaxation time steps. This is attributed to the migration of He atoms within the clusters. However, for the distance beyond 1.3 nm, the configurations of the relaxed HenVm (n P m) clusters are similar to those in the perfect Fe lattice. For example, the He2V cluster lies along the <1 0 0> direction and the He3V cluster forms a triangle configuration in a {1 0 0} plane [18]. Also, Fig. 5 indicates that the binding energies of the HenVm (n P m) clusters to the dislocation along the slip plane depend on the He/V ratio. At the same distance
L. Yang et al. / Nucl. Instr. and Meth. in Phys. Res. B 265 (2007) 541–546
along the slip plane from the dislocation core, the binding energies of the clusters increase with increasing He/V ratio, which is similar to that observed in Fig. 3, where the binding energies of the HenVm (n P m) are investigated as a function of the distance normal to the slip plane. 3.2. HenVm (n < m) The formation of vacancy clusters created by displacement cascades in a-Fe containing different concentrations of substitutional He atoms has been studied using MD methods [18,12]. The results indicated that HenVm (n < m) clusters may be an important type of the He–V clusters produced in displacement cascades, and the studies of their interactions with microstructural features such as dislocations and grain boundaries are required to fundamentally understand the nucleation and formation of He bubbles at these microstructures. Therefore, the binding energies of small HenVm (n < m) clusters to an edge dislocation are investigated. In the simulations, it is found that most of HenVm (n P m) clusters placed in the ‘dislocated’ lattice rapidly relax to the lowest energy configurations. After relaxation, the initial atomic configurations change slightly, but the dissociation of He atoms from the clusters is not observed for any of the positions simulated, which may suggest that He atoms are strongly trapped by the vacancies in these clusters. Similar to the studies of the HenVm (n P m) clusters shown in Fig. 3, the binding energies of HenVm (n < m) clusters as a function of the distance from the slip plane along a line running through the center of the dislocation from the tension to the compression side (along the ½1 1 0 direction) are shown in Fig. 6. The maximum binding energies are about 1.1, 1.9 and 2.1 eV for the HeV2, HeV3 and HeV4 at the first atom plane of the compression side, respectively. Within about 2 Burgers vectors
Fig. 6. Binding energies of the HenVm (n < m) clusters to an a/ 2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the glide plane along a line through the center of the dislocation normal to the glide plane.
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(about 0.5 nm) of the core on both sides of the dislocation, the binding energies of the HenVm (n < m) clusters are positive, which is in contrast to those of the HenVm (n P m) clusters (in Fig. 3). It may suggest that the HenVm (n < m) clusters are trapped within about 1 nm of the dislocation core, and this is independent on which side the clusters are located. The binding energies rapidly decrease to about zero with increasing distance from the slip plane on both sides, which are different from those observed in the HenVm (n P m) clusters. However, the binding energies of the clusters located at distances of a few nm away from the dislocation core on the both sides of the slip plane also depend on the He/V ratio. The binding energy increases with increasing He/V ratio on the tension side, and decreases on the compression side. Fig. 7 shows the binding energies of HenVm (n < m) clusters to the dislocation as a function of the distance from the core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane. The binding energy decreases with increasing distance, which is similar to that of the HenVm (n P m) clusters, and the HenVm (n < m) clusters initially placed at all the positions along the [1 1 1] direction rapidly relax to the lowest energy configurations. When the distance is larger than 1.3 nm, the binding energies approach to zero. Similar to the HenVm (n P m) clusters, within about 2.6 nm of the dislocation core, the HenVm (n < m) clusters are trapped in the dislocation with a large binding energy. Furthermore, it is clear that the binding energies of the clusters depend on not only the He/V ratio, but also the cluster size at and near the dislocation core. For the same He/V ratio, such as HeV2 and He2V4, the binding energies increase with the cluster size. For the same number of vacancies in the He–V clusters, such as the HenV3 (n = 1, 2) and HenV4 clusters (n = 1, 2, 3), the binding energies increase with increasing He/V ratio. However, the binding energies of the clusters located farther away from the dislocation core on the slip
Fig. 7. Binding energies of the HenVm (n < m) clusters to an a/ 2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the core along the direction of the Burgers vector on the slip plane.
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plane depend on only the He/V ratio, and increase with increasing He/V ratio. This means that the He–V clusters with high He/V ratio are easier to be trapped by the dislocation than those with small He/V ratio, which may suggest that the sink strength for a dislocation to trap He–V clusters strongly depends on the He/V ratio. In general, the HenVm (n P m) clusters have larger binding energies than the HenVm (n < m) clusters (see Figs. 5 and 7), and it is likely that these clusters could not easily escape from the dislocation. Therefore, these clusters could significantly increase the probability of the nucleation and formation of helium bubbles within dislocation core. 4. Conclusions Atomistic simulations have been employed to study the interactions of small He–V clusters with an edge dislocation, for clusters having different He/V ratios. The results show that the HenVm (n P m) clusters are trapped in the dislocation on the tension side, but the HenVm (n < m) cluster are trapped within about 0.5 nm from the dislocation core on both the compression and tension sides. The binding energies of the HenVm (n P m) clusters to the dislocation do not depend much on cluster size, but rather on the He/V ratio. However, the binding energies of the HenVm (n < m) clusters to the dislocation depends on not only the He/V ratio, but also the cluster size at and near the dislocation core. The effects of dislocations on the formation of He–V clusters is very small for the distances more than about 1.3 nm from the dislocation core on the slip plane, and the binding energies of the clusters depends on only the He/V ratio at these positions. Acknowledgments X.T. Zu is grateful for the Program for Innovative Research Team in UESTC and the Program for New Century
Excellent Talents in University (NCET-04-0899). Other authors (F. Gao, R. Kurtz and H. Heinisch) are grateful to the support by the US Department of Fusion Energy Science under Contract DE-AC06-76RLO 1830. References [1] G.A. Cottrell, Fusion. Eng. Des. 66–68 (2003) 253. [2] N. Yamamoto, T. Chuto, Y. Murase, J. Nagakawa, J. Nucl. Mater. 329–333 (2004) 993. [3] I.I. Balachov, E.N. Shcherbakov, A.V. Kozlov, I.A. Portnykh, F.A. Garner, J. Nucl. Mater. 329–333 (2004) 617. [4] K. Morishita, R. Sugano, B.D. Wirth, T. Diaz de la Rubia, Nucl. Instr. and Meth. B 202 (2003) 76. [5] K. Morishita, R. Sugano, B.D. Wirth, J. Nucl. Mater. 323 (2003) 243. [6] R.J. Kurtz, H.L. Heinisch, J. Nucl. Mater. 329–333 (2004) 1199. [7] H.L. Heinisch, F. Gao, R.J. Kurtz, E.A. Le, J. Nucl. Mater. 351 (2006) 141. [8] H.L. Heinisch, F. Gao, R.J. Kurtz, J. Nucl. Mater. 367–370 (2007) 311. [9] E.H. Lee, J.D. Hunn, T.S. Byun, L.K. Mansur, J. Nucl. Mater. 280 (2000) 18. [10] A. van Veen, R.J.M. Konings, A.V. Fedorov, J. Nucl. Mater. 320 (2003) 77. [11] K. Ono, K. Arakawa, K. Hojou, J. Nucl. Mater. 307–311 (2002) 1507. [12] L. Yang, X.T. Zu, H.Y. Xiao, F. Gao, H.L. Heinisch, R.J. Kurtz, K.Z. Liu, Appl. Phys. Lett. 88 (2006) 091915. [13] L. Yang, X.T. Zu, H.Y. Xiao, F. Gao, H.L. Heinisch, R.J. Kurtz, Z.G. Wang, K.Z. Liu, Nucl. Instr. and Meth. B. 255 (2007) 63. [14] F. Gao, D.J. Bacon, P.E.J. Flewitt, T.A. Lewis, J. Nucl. Mater. 249 (1997) 77. [15] G.J. Ackland, D.J. Bacon, A.F. Calder, T. Harry, Philos. Mag. A 75 (1997) 713. [16] W.D. Wilson, R.D. Johnson, in: P.C. Gehlen, J.R. Beeler Jr., R.I. Jaffee (Eds.), Interatomic Potentials and simulation of Lattice Defects, Plenum, 1972, p. 375. [17] D.E. Beck, Mol. Phys. 14 (1968) 311. [18] L. Yang, X.T. Zu, H.Y. Xiao, F. Gao, H.L. Heinisch, R.J. Kurtz, Phys. B 391 (2007) 179. [19] F. Gao, H. Heinisch, R.J. Kurtz, J. Nucl. Mater. 351 (2006) 133. [20] Yu.N. Osetsky, D.J. Bacon, Model. Simul. Mater. Sci. Eng. 11 (2003) 427.
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 265 (2007) 541–546 www.elsevier.com/locate/nimb
Interaction of helium–vacancy clusters with edge dislocations in a-Fe L. Yang
a,b
, X.T. Zu
a,*
, Z.G. Wang a, F. Gao c, X.Y. Wang d, H.L. Heinisch c, R.J. Kurtz
c
a
d
Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, China b School of Physics and Electronics Information, China West Normal University, Nanchong 637002, China c Pacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA National Key Laboratory for Surface Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621907, China Received 14 July 2007; received in revised form 29 August 2007 Available online 22 October 2007
Abstract Molecular static calculations were performed to evaluate the formation energies and binding energies of helium–vacancy (He–V) clusters in and near the core of an a/2<1 1 1>{1 1 0} edge dislocation in a-Fe with empirical potentials. The formation energies of these He–V clusters and their binding energies to the dislocation depend on the helium-to-vacancy ratio of the clusters. For the ratio equal to or larger than 1, the helium–vacancy clusters have negative binding energies on the compression side of the dislocation and strong positive binding energy on the tension side. However, for the ratio less than 1, the He–V clusters have positive binding energy on the both sides near the dislocation core. On the slip plane, the binding energies of the He–V clusters to the dislocation depend on not only the heliumto-vacancy ratio, but also the cluster size. Ó 2007 Elsevier B.V. All rights reserved. PACS: 02.70.c; 61.72.Ji; 61.72.Lk; 81.05.Bx Keywords: Molecular static; Edge dislocation; Helium–vacancy cluster; a-Fe
1. Introduction The effect of helium on the first wall structural materials has been widely recognized as one of the most crucial materials issues in nuclear fusion reactors, because the synergistic interaction of large amounts of helium with the existing and radiation-induced defects and microstructures in materials can significantly degrade their mechanical properties [1–3]. Computer simulations provide an important means of obtaining insight into fundamental understanding of the complex atomic-level processes of the defects controlling microstructural evolution in materials. For example, the interactions of helium atoms with the defects and microstructures in a-Fe (which is taken to be a first-order model for the ferritic steels that are promising materials for fusion reactors) have been systematically investigated *
Corresponding author. Tel./fax: +86 28 83201939. E-mail address: [email protected] (X.T. Zu).
0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.10.007
using molecular dynamics, molecular statics and Monte Carlo methods [4,5]. Atomistic calculations demonstrated the strong binding of He to grain boundaries (GBs) in aFe, finding that both substitutional and interstitial He atoms are trapped at GBs [6]. Recently, the interactions of helium atoms with edge dislocations in a-Fe were studied [7]. It is found that interstitial He atoms are either repelled from or trapped at edge dislocations in a-Fe, depending on the direction of approach. Near the dislocation core on the tension side He is strongly trapped as a crowdion with 1–2 eV greater binding energy than that of an octahedral interstitial in perfect iron, and in this form He atoms can migrate along the dislocation with a migration energy of 0.4–0.5 eV. Similar studies of He atoms in and near screw dislocations in a-Fe [8] determined that the binding energy of He to screw dislocations is about half that to edge dislocations, and that interstitial He atoms migrate along the screw with about the same migration energies as along the edge.
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In irradiated materials, in addition to single helium atoms, a large number of helium–vacancy clusters or bubbles are formed due to the extremely low solubility of helium in metals [9–12]. In the present work, molecular statics calculations are utilized to study the binding energy of small helium––vacancy (He–V) clusters in and around the core of the a/2 [1 1 1]{1 1 0} edge dislocation in a-Fe. Recent computer simulations have shown that the stability of He–V clusters in a perfect Fe lattice does not depend much on cluster size, but rather on the helium-to-vacancy (He/V) ratio in a-Fe [4,13]. However, the stability of He– V clusters in the ‘dislocated’ lattice has not been reported before. In this work, the binding energies of a large number of small He–V clusters to an edge dislocation were studied in detail. 2. Method and geometry The modified version of the MOLDY computer code [14] was used in this work. The interatomic potentials of Ackland [15], Wilson–Johnson [16] and Beck [17] were used to describe the interactions of Fe–Fe, Fe–He and He–He, respectively. The same set of interatomic potentials was also employed in the earlier calculations of the binding energies and atomic configurations of He–V clusters in perfect a-Fe, and to simulate the migration of He interstitials in grain boundaries and dislocations [4,7,18,19]. To be consistent with the body of earlier work, we have used this set of potentials in the present simulations. The cohesive energies calculated with these potentials are 4.316 and 0.00568 eV/atom for a perfect bcc Fe crystal and a perfect fcc He crystal, respectively. The formation energies of a substitutional He atom and an interstitial He atom using these potentials are calculated to be 3.25 and 5.25 eV, respectively. The computational model consists of a cubic cell of Fe atoms. The x, y and z axes of the simulated crystal were oriented along [1 1 1], ½ 1 1 2 and ½1 1 0 directions. An a/ 2h1 1 1i{1 1 0} edge dislocation was created according to the method developed by Osetsky and Bacon [20]. The following procedure has been adopted. The initial unrelaxed structure is presented as two half-crystals strained to have different lattice parameters in the direction b and joined along the dislocation slip plane. The two half-crystals are strained to have smaller and larger lattice repeat distance, respectively, in the b direction than their natural value b. The total size difference equals to b. Periodic boundary conditions were employed in the x and y directions, corresponding to the direction of the Burgers vector b, and the dislocation line direction, respectively, and fixed conditions were used across the z boundaries. The size of the calculated region is about 44 a0 (a0 is the lattice constant of Fe) for the [1 1 1] direction, 32 a0 for the ½ 1 1 2 direction and 45 a0 for the ½1 1 0 direction. The mobile region contains 10,6400 atoms. Thermal effects are not considered here and so simulation was carried out at 0 K by molecular static (MS) relaxation. The simulation cell containing an a/2h1 1 1i{1 1 0} edge dislo-
cation, and the corresponding relationship of the directions is shown schematically in Fig. 1. A He–V cluster was introduced into the relaxed, dislocated model, and the new configuration was then relaxed again to obtain the minimum energy configuration. Binding energy calculations were performed for He–V clusters in and around the core of the dislocation. The binding energy of a He–V cluster to the dislocation is defined as the difference between the formation energy of the He–V cluster in a perfect Fe lattice and that in the ‘dislocated’ lattice. The formation energy of a He–V cluster in the dislocated lattice, Efd , is defined as tot fe he Efd ¼ U tot dþHen Vm U d þ m E n E ;
ð1Þ
tot where U tot dþHen Vm and U d are the total energy of the crystal which includes the edge dislocation, with and without the HenVm cluster, respectively. Efe =4.316 eV and Ehe =0.00567758 are the cohesive energies of bcc Fe and fcc He.
3. Results and discussion The previous atomistic simulations in perfect Fe indicate that the properties of He–V clusters mainly depend on the He/V ratio of clusters [4]. In this work, the binding energies of some small He–V clusters, i.e. HenVm (n = 1, 2, 3, 4 and m = 1, 2, 3, 4), with He/V ratios ranging from 0.25 to 4, were investigated in and around the core of the dislocation. 3.1. HenVm (n P m) As described above, the formation energies of He–V clusters within the dislocation were calculated using Eq. (1) after relaxing the simulated system to the minimum potential configuration. In the simulations, the vacancies of HenV2 (n = 1, 2, 3, 4) clusters lie along the <1 1 1> direction, and the vacancies of the HenV3 (n = 1, 2, 3, 4) and HenV4 (n = 1, 2, 3, 4) locate at the {1 1 0} plane, as shown in Fig. 2. In each case the He atoms were initially placed within the vacancy clusters, and the He–V clusters placed in the ‘dislocated’ lattices rapidly relaxed to the lowest
Fig. 1. Schematic view of the simulation cell with the corresponding crystallographic orientations.
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Fig. 2. Configurations of vacancies in HenVm clusters: (a) the divacancy in the HenV2 (n = 1, 2, 3, 4) clusters, (b) the vacancies of the HenV3 (n = 1, 2, 3, 4) clusters and (c) the vacancies of the HenV4 (n = 1, 2, 3, 4) clusters.
Fig. 3. Binding energies of HenVm (n P m) clusters to an a/2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the glide plane along a line through the center of the dislocation normal to the glide plane: (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1).
energy configurations. Fig. 3 shows the binding energy of a HenVm (n P m) cluster to the dislocation as a function of the distance of the cluster from the slip plane along a line running through the center of the dislocation from the ten-
sion to the compression side (along ½1 1 0 in Fig. 1): (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1). It can be seen that the binding energies of all the HenVm (n P m) clusters are positive on the tension side of the dislocation (negative distance from the glide plane) and negative on the compression side, which illustrates that the HenVm (n P m) clusters are strongly trapped on the tension side of edge dislocations and repelled from the compression side of edge dislocations in a-Fe. Using conjugate gradient relaxation, Heinisch et al. [7] investigated the interaction of a He interstitial with an edge dislocation, and the results showed that interstitial He atoms have negative binding energy on the compression side of the dislocation and strong positive binding energy on the tension side. The behavior of the He–V clusters near the dislocation is generally similar to that observed for single He interstitial. However, the binding energies for the He–V clusters are larger than that for a single interstitial or a substitutional He atom. Furthermore, at the same distance from the slip plane on the tension side of the dislocation, the binding energy increases with increasing He/V ratio, but decreases on the compression side. For the same He/V ratio, the binding energy increases with increasing He–V cluster size, as shown in Fig. 3(b). However, these results illustrate that the binding energies of the He–V clusters to the edge dislocation are less sensitive to cluster size than to the He/V ratio. All above may be attributed to the relaxed configurations of the clusters, which depend on the effects of the dislocation field. For example, near the dislocation core on the tension side, the relaxed He2V cluster lies along a <1 1 1> direction, forming a dumbbell configuration and has a maximum binding energy of 1.9 eV within the dislocation core. However, with the increase in the distance from the core on the compression side, the He2V cluster relaxes to locate in different directions, as shown in Fig. 4(a). It can be seen that within about 0.5 nm from the core the relaxed He2V cluster lies almost along the <1 1 1> direction on the tension side (negative distance from the glide plane). With increase in the distance on the tension side, the He2V relaxes to form a dumbbell configuration along the <1 0 0> direction, similar to that found in pure a-Fe [18]. However, at the first atom plane of the
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Fig. 4. The relaxed atomic configurations of (a) the He2V cluster and (b) the He3V cluster placed on the different atom positions from the slip plane along a line running through the center of the dislocation from the tension to the compression side, where the black and the gray spheres represent vacancies and He atoms, respectively.
compression side, the He2V cluster is dissociated to form a substitutional He, with the second He atom migrating towards the slip plane, which results in significant displacements of its neighboring Fe atoms within the close-packed row, which was plotted in the rectangle in Fig. 4(a). At the second and further atom planes of the compression side, the directions of the relaxed He2V clusters change with different positions, and there is not a fixed direction that can be used to define their configurations. In the dislocation core, the relaxed He3V cluster lies in a {1 1 0} plane and forms a triangle configuration, where three interstitial He atoms locate at three corners of a triangle and the vacancy is at the center of three He atoms, as shown in Fig. 4(b). However, within about 0.4 nm away from the core, the He3V cluster is found to dissolve, forming a substitutional He at the vacancy site, with two He atoms relaxed into the slip plane, which results in significant displacements of their neighboring Fe atoms within the slip plane, which was not plotted in Fig. 4(b). The He3V clusters placed farther away from the dislocation core on the tension side spontaneously relax into the {1 0 0} plane, and they have smaller binding energies that decrease with increasing distance from the dislocation. Fig. 5 shows the binding energies of HenVm (n P m) clusters to the dislocation as a function of the distance of the clusters from the core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane; (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1). It is clear that the relaxed He–V clusters located near the dislocation core have the maximum binding energies. With increasing distance from the core, the binding energies rapidly decrease. When the distance is larger than 1.3 nm, the binding energies approach to zero, and the configurations of the He–V clusters on both sides of the dislocation core are almost the same. Therefore, within about 2.6 nm of the core, the HenVm (n P m) clusters are trapped
Fig. 5. Binding energies of the HenVm clusters to an a/2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of the distance from the dislocation core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane: (a) HenVm (n P m, m = 1) and (b) HenVm (n P m, m > 1).
by the dislocation with a large binding energy on the slip plane. In the simulations, it is found that when the distance from the core is less than 0.5 nm or larger than 1.3 nm, the simulated system rapidly relaxes to the lowest energy configuration, but within the range from 0.5 nm to 1.3 nm, it achieves the lowest energy configuration with many relaxation time steps. This is attributed to the migration of He atoms within the clusters. However, for the distance beyond 1.3 nm, the configurations of the relaxed HenVm (n P m) clusters are similar to those in the perfect Fe lattice. For example, the He2V cluster lies along the <1 0 0> direction and the He3V cluster forms a triangle configuration in a {1 0 0} plane [18]. Also, Fig. 5 indicates that the binding energies of the HenVm (n P m) clusters to the dislocation along the slip plane depend on the He/V ratio. At the same distance
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along the slip plane from the dislocation core, the binding energies of the clusters increase with increasing He/V ratio, which is similar to that observed in Fig. 3, where the binding energies of the HenVm (n P m) are investigated as a function of the distance normal to the slip plane. 3.2. HenVm (n < m) The formation of vacancy clusters created by displacement cascades in a-Fe containing different concentrations of substitutional He atoms has been studied using MD methods [18,12]. The results indicated that HenVm (n < m) clusters may be an important type of the He–V clusters produced in displacement cascades, and the studies of their interactions with microstructural features such as dislocations and grain boundaries are required to fundamentally understand the nucleation and formation of He bubbles at these microstructures. Therefore, the binding energies of small HenVm (n < m) clusters to an edge dislocation are investigated. In the simulations, it is found that most of HenVm (n P m) clusters placed in the ‘dislocated’ lattice rapidly relax to the lowest energy configurations. After relaxation, the initial atomic configurations change slightly, but the dissociation of He atoms from the clusters is not observed for any of the positions simulated, which may suggest that He atoms are strongly trapped by the vacancies in these clusters. Similar to the studies of the HenVm (n P m) clusters shown in Fig. 3, the binding energies of HenVm (n < m) clusters as a function of the distance from the slip plane along a line running through the center of the dislocation from the tension to the compression side (along the ½1 1 0 direction) are shown in Fig. 6. The maximum binding energies are about 1.1, 1.9 and 2.1 eV for the HeV2, HeV3 and HeV4 at the first atom plane of the compression side, respectively. Within about 2 Burgers vectors
Fig. 6. Binding energies of the HenVm (n < m) clusters to an a/ 2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the glide plane along a line through the center of the dislocation normal to the glide plane.
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(about 0.5 nm) of the core on both sides of the dislocation, the binding energies of the HenVm (n < m) clusters are positive, which is in contrast to those of the HenVm (n P m) clusters (in Fig. 3). It may suggest that the HenVm (n < m) clusters are trapped within about 1 nm of the dislocation core, and this is independent on which side the clusters are located. The binding energies rapidly decrease to about zero with increasing distance from the slip plane on both sides, which are different from those observed in the HenVm (n P m) clusters. However, the binding energies of the clusters located at distances of a few nm away from the dislocation core on the both sides of the slip plane also depend on the He/V ratio. The binding energy increases with increasing He/V ratio on the tension side, and decreases on the compression side. Fig. 7 shows the binding energies of HenVm (n < m) clusters to the dislocation as a function of the distance from the core along the direction of the Burgers vector (along the [1 1 1] direction in Fig. 1) on the slip plane. The binding energy decreases with increasing distance, which is similar to that of the HenVm (n P m) clusters, and the HenVm (n < m) clusters initially placed at all the positions along the [1 1 1] direction rapidly relax to the lowest energy configurations. When the distance is larger than 1.3 nm, the binding energies approach to zero. Similar to the HenVm (n P m) clusters, within about 2.6 nm of the dislocation core, the HenVm (n < m) clusters are trapped in the dislocation with a large binding energy. Furthermore, it is clear that the binding energies of the clusters depend on not only the He/V ratio, but also the cluster size at and near the dislocation core. For the same He/V ratio, such as HeV2 and He2V4, the binding energies increase with the cluster size. For the same number of vacancies in the He–V clusters, such as the HenV3 (n = 1, 2) and HenV4 clusters (n = 1, 2, 3), the binding energies increase with increasing He/V ratio. However, the binding energies of the clusters located farther away from the dislocation core on the slip
Fig. 7. Binding energies of the HenVm (n < m) clusters to an a/ 2h1 1 1i{1 1 0} edge dislocation in a-Fe as a function of their distance from the core along the direction of the Burgers vector on the slip plane.
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plane depend on only the He/V ratio, and increase with increasing He/V ratio. This means that the He–V clusters with high He/V ratio are easier to be trapped by the dislocation than those with small He/V ratio, which may suggest that the sink strength for a dislocation to trap He–V clusters strongly depends on the He/V ratio. In general, the HenVm (n P m) clusters have larger binding energies than the HenVm (n < m) clusters (see Figs. 5 and 7), and it is likely that these clusters could not easily escape from the dislocation. Therefore, these clusters could significantly increase the probability of the nucleation and formation of helium bubbles within dislocation core. 4. Conclusions Atomistic simulations have been employed to study the interactions of small He–V clusters with an edge dislocation, for clusters having different He/V ratios. The results show that the HenVm (n P m) clusters are trapped in the dislocation on the tension side, but the HenVm (n < m) cluster are trapped within about 0.5 nm from the dislocation core on both the compression and tension sides. The binding energies of the HenVm (n P m) clusters to the dislocation do not depend much on cluster size, but rather on the He/V ratio. However, the binding energies of the HenVm (n < m) clusters to the dislocation depends on not only the He/V ratio, but also the cluster size at and near the dislocation core. The effects of dislocations on the formation of He–V clusters is very small for the distances more than about 1.3 nm from the dislocation core on the slip plane, and the binding energies of the clusters depends on only the He/V ratio at these positions. Acknowledgments X.T. Zu is grateful for the Program for Innovative Research Team in UESTC and the Program for New Century
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