Applied Surface Science 255 (2008) 128–131
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Positron annihilation in vacancies at grain boundaries in metals J. Kuriplach a,*, O. Melikhova a,b, M. Hou b, S. Van Petegem c, E. Zhurkin d, M. Sˇob e,f a
Department of Low Temperature Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holesˇovicˇka´ch 2, CZ-180 00 Prague 8, Czech Republic Physique des Solides Irradie´s et des Nanostructures CP234, Universite´ Libre de Bruxelles, Bd du Triomphe, B-1050 Bruxelles, Belgium c Materials Science and Simulation, Paul Scherrer Institute, CH-5232 Villigen, Switzerland d Department of Experimental Nuclear Physics, St. Petersburg State Polytechnical University, Polytekhnicheskaya 29, 195251 St. Petersburg, Russia e Department of Chemistry, Faculty of Science, Masaryk University, Kotla´rˇska´ 2, CZ-611 37 Brno, Czech Republic f Institute of Physics of Materials, Academy of Sciences of the Czech Republic, v.v.i., Zˇizˇkova 22, CZ-616 62 Brno, Czech Republic b
A R T I C L E I N F O
A B S T R A C T
The behavior of vacancies in selected coherent grain boundaries (GBs) in Fe and Ni is studied by means of molecular dynamics simulations. Corresponding positron lifetimes are calculated using the atomic superposition method. There is a difference between the vacancy behavior in Fe and Ni in dependence on temperature. In Ni, vacancies at GBs appear to diminish substantially their free volume (and lifetime) with the increasing temperature, which can be attributed to ‘vacancy delocalization’. Contrary, GB vacancies remain stable up to apparently higher temperatures in Fe. ß 2008 Elsevier B.V. All rights reserved.
Available online 15 May 2008 PACS: 78.70.Bj 61.72.Mm 61.72.Ji 71.60.+z Keywords: Positron annihilation Positron trapping at grain boundaries Vacancies Positron lifetime Molecular dynamics Atomic superposition method
1. Introduction Positron annihilation (PA) at grain boundaries (GBs) and other internal interfaces is not yet well understood. A coherent/ideal GB represents a positron trap with a binding energy of about half of that for single vacancies and exhibits a lifetime that exceeds only slightly the bulk one [1,2]. The free volume associated with such GBs is rather small and positron trapping can be viewed as trapping in a two-dimensional area with a lowered atomic (or average electron) density, where positrons are localized just in one direction. There is only a little experimental evidence  of such kind of trapping, probably due to the fact that producing well deﬁned GBs in macroscopic samples suitable for PA measurements is a difﬁcult task. On the other hand, nanocrystalline (nc) materials contain a large volume fraction of GBs and a positron lifetime (PLT) component that can be attributed to PA at GBs is also detected experimentally (see e.g. [4,5] where nc-Fe and nc-Ni are studied, respectively). This
* Corresponding author. Tel.: +420 22191 2769; fax: +420 22191 2567. E-mail address: [email protected]
(J. Kuriplach). 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.05.199
component (usually 150–200 ps) does not correspond to some particular GB, but it is an average signal originating from many different types of positron traps present at various GBs in a given nanocrystalline material. Furthermore, it is assumed that such a component corresponds to positrons annihilating at vacancy-sized defects at GBs and not to PA at 2D regions related to GBs. The analysis of free volumes (FVs) at GBs in simulated samples of nanostructured materials unveils a broad distribution of free volumes and corresponding PLTs originating from grain boundaries [5–7]. Some FVs really correspond to defects with a size of about one vacancy (or even larger), but most of them are small FVs that are inherent to GBs themselves, as explained above. In general, GB vacancies have been recognized to inﬂuence GB diffusion and mechanical properties of GBs [8–11], but positron related characteristics were not studied in detail very often. Vacancies at GBs are hardly experimentally accessible and as an example of a use of a non-positron technique we mention a recent measurement of formation enthalpies of vacancies located at GBs for several bcc metals using nuclear g-resonance spectroscopy . In this contribution, we continue our previous study , where we investigated several coherent GBs in Fe and Ni, and here we concentrate on the structure and positron characteristics of
J. Kuriplach et al. / Applied Surface Science 255 (2008) 128–131
vacancies introduced into coherent GBs, with the aim to help understanding the nature of vacancy-like trapping sites at GBs in general. 2. Computational methods In order to relax atomic conﬁgurations of selected GBs in Fe and Ni, we employed Parrinello–Rahman molecular dynamics (MD) . Structural simulations were performed using many-body potentials as follows: An embedded atom (EAM) potential described by Ludwig et al.  and a potential based on the second moment of the tight binding model in the form detailed in Ref.  were utilized, respectively, for Fe and Ni (these potentials are further labeled as P1). For the sake of comparison, in selected cases we also employed a ‘magnetic’, EAM-type potential introduced recently by Dudarev and Derlet  for Fe and the potential of Cleri and Rosato  for Ni (label P2). Box sizes used in structural simulations and positron calculations varied and boxes contained typically several thousands of atoms. Simulations were done at temperatures from 0 to 750 K (or 900 K) and at zero pressure. At non-zero temperatures, atomic coordinates were averaged over certain number of steps after reaching thermal equilibrium in order to avoid inﬂuence of thermal vibrations of atoms on positron characteristics. Before starting MD simulations atomic conﬁgurations (simulation boxes) containing selected GBs were prepared considering the coincident-site-lattice (CSL) principle. The distances between two adjacent GBs in all boxes were larger than 6 and 8 lattice constants for Fe and Ni, respectively. GB boxes with vacancies were created by removing an atom just from the GB plane. MD relaxed atomic conﬁgurations were employed in positron calculations carried out using the atomic superposition (ATSUP) method . In the course of positron calculations the electron–positron correlation potential and enhancement factor according to reference  were utilized. 3. Results and discussion According to the previous work  vacancies at some GBs behave in a different way at zero and elevated temperatures. Here we inspect this effect in more detail, including additional GBs. First, the results obtained in  are brieﬂy summarized. In the case of Ni, we investigated the twin GB and one tilt (S = 5 (2 1 0) [0 0 1]) and one twist (S = 5 (0 0 1)) GBs. In the case of the twin GB, the vacancy remains stable up to at least 600 K. For other two GBs the vacancies are observed to be ‘absorbed’ by the GB already at 300 K. Concerning Fe, we studied one twist GB (S = 5 (0 0 1)) only. This GB has very complex structure that changes with temperature, but the vacancy is stable at least up to 300 K. To proceed, we carried out detailed investigation of the S = 19 (3 3 1) [1 1 0] tilt GB in Ni. The results of our structure simulations and PLT calculations are summarized in Fig. 1, where dependencies of the PLT on the simulation temperature are plotted for the GB with and without the vacancy for both potentials used. We could detect several structural changes of the studied GB in the examined temperature range and corresponding PLT ranges are 122.0–126.5 and 126.4–131.1 ps for perfect GBs obtained with P1 and P2 potentials, respectively. PLT variations are partly due to structure changes and partly due to temperature effects. There are no apparent structural differences at 0 K for the two used potentials though there is a small difference in lifetimes. However, the structures start to somewhat deviate at non-zero temperatures. This is also the reason why we considered two different (non-equivalent) vacancy positions in the case of the P2 potential (see Fig. 1). For this potential it is apparent that except for 0 K PLTs corresponding to the vacancy in the GB drop to values that are
Fig. 1. Calculated lifetimes for the S = 19 (3 1 3) tilt GB in Ni in dependence on the temperature for the P1 and P2 potentials. Temperatures when a structural change occurred are indicated by arrows.
roughly the same as for the perfect GB. A closer inspection of atoms around the vacancy site reveals that some of these atoms are displaced signiﬁcantly from their positions in the prefect GB, which results in a substantial decrease of the free volume associated with the vacancy (compared to non-relaxed positions or 0 K situation) and in the related drop in the PLT. This effect, called ‘vacancy delocalization’, was also observed in  for the S = 5 (2 1 0) [0 0 1] tilt GB in Cu (fcc). In the case of the potential P1, the decrease of the PLT with temperature corresponding to the vacancy in the GB plane is rather gradual with an exception at 750 K where a small ‘peak’ appears due to a different way of atomic rearrangements around the vacancy. The PLT drop has a similar origin as in the previous case though atomic positions differ. In particular, the ‘vacancy instability’  manifests here, i.e. the vacancy is unstable at the original position where it was introduced and jumps to another position, but vacancy delocalization takes place even in this new position that is located at the second atomic plane next to the GB plane. Furthermore, we investigated various structure modiﬁcations of the S = 19 GB in Ni. In this particular study only the P1 potential was used. In addition, the distance between adjacent GBs in the simulation box was increased 1.5 times in order to check the inﬂuence of the size of the simulation box on results. The structure obtained after relaxing the CSL box will be denoted by the symbol S1. At 0 K this structure is identical to the structure investigated using a smaller box (see above). But as the temperature increases, corresponding structures start to differ. Further structural modiﬁcations were obtained by considering the translational degrees of freedom in directions parallel to the GB plane (several small translations in different directions were examined). In this way, we could ﬁnd other two structures (S2 and S3). The last one (S4) was obtained by quenching the high-temperature structure (600 K) originating from the structure S1.1 Corresponding GB energies are in all cases around 1 J/m2 (at 0 K), with the difference between the highest (S4) and
1 In the following, each of symbols S1, S2, S3 and S4 designates all different structures (at different temperatures) that evolved from the 0 K structure denoted originally by the corresponding symbol. Different structures exhibit different atomic arrangements, but fall into the same GB type (i.e. S = 19 (3 1 3) [1 1 0]) from the geometrical point of view.
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disappears/anneals out (the perfect bcc lattice can be observed only). The corresponding lifetime is 124 ps (0 K) that remains constant (except for small temperature effects) as the temperature increases and drops to 104 ps at 900 K. If a vacancy is introduced into the GB plane, it remains at its original position up to 750 K. At 900 K a vacancy in perfect bulk is detected instead. This is also reﬂected by the PLT which amounts to 200 ps at 0 K (with almost no change with temperature) and then jumps to 163 ps at 900 K. When the P2 potential is considered, the behavior of the perfect GB is identical except the GB remains stable even at 900 K. The positron lifetime is 120 ps at 0 K and varies only slightly with the increasing temperature. A vacancy introduced into the GB plane remains stable up to 750 K (the PLT varies from 192 to 196 ps) and at 900 K vacancy delocalization can be observed with the corresponding PLT drop to 127 ps. 4. Conclusions
Fig. 2. Calculated lifetimes for different structural modiﬁcations of S = 19 (3 1 3) tilt GB in Ni in dependence on the temperature for the P1 potential.
lowest (S2) GB energies about 0.2 J/m2. This result for the GB energy agrees well with the value 1.2 J/m2 given in  for the same GB. Fig. 2 summarizes results of structure simulations and PLT calculations for all GB structures with and without vacancies. Concerning perfect GBs, at low temperatures we have four structures whereas at higher temperatures only three are present (S4 transforms to S1 at 600 K; see above). According to the PLT we can distinguish two structural classes (at low temperatures S1, S2, and S3 constitute the ﬁrst class and S4 the second class; at higher temperatures S2 and S3 belong to the ﬁrst class whereas S1, which transforms to S4, belongs to the second class) that differ by about 7 and 11 ps at low and high temperatures, respectively. This difference can be related to different atomic arrangements of these two classes in the GB region and could, in principle, be used to detect some structural changes (e.g. S1 ! S4) using PLT measurements. When vacancies are considered, the situation becomes more complex. There are more PLT ‘bands’ that exhibit signiﬁcantly larger mutual differences compared to perfect structures, but they approach a level that is close to perfect GB lifetimes as the temperature increases. In some cases there are two non-equivalent sites at the GB plane that need to be examined for vacancy behavior, which explains the existence of two curves or curve splitting (joining) for some structures. The general behavior is that the variety of possible vacancy sites decreases with temperature due to the decreasing number of distinct structures and also vacancy instability and delocalization effects described above. For example, the S4 structure exhibits two different vacancy sites at lower temperatures, but at 600 K one of them becomes unstable and jumps to the second vacancy position which can only be observed. Finally, we discuss results obtained for the S = 5 (3 1 0) [0 0 1] tilt GB in Fe. This GB was studied using both potentials P1 and P2 for iron. The structure after the CSL construction (without any shifts) was considered here. In the case of the P1 potential, the GB structure remains the same up to 750 K and at 900 K the GB
The study of selected coherent GBs containing vacancies in Ni (fcc) and Fe (bcc) was performed. It turns out that vacancies at GBs in Ni and Fe exhibit different behaviors. In Ni, vacancies at GBs appear to diminish their free volume (and PLT) with the increasing temperature. When the temperature further increases, the corresponding free volume practically disappears and the corresponding PLT is roughly the same as for the perfect GB. These effects are due to vacancy delocalization and instability. On the other hand, vacancies at GBs in Fe are stable up to apparently higher temperatures (compared to Ni) and the corresponding PLT remains almost unchanged, being somewhat larger than that for vacancies in bulk. An instability of vacancies could be observed at very high temperatures only. Such behaviors are in some respect compatible with other observations found in literature. Namely, Eldrup and Singh  observed in neutron irradiated Cu (fcc) and Fe (bcc) that the damage accumulated during irradiation has substantially different character for Cu and Fe. Fe exhibits larger vacancy clusters, which does not happen for Cu where rather a larger amount of stacking fault tetrahedra (i.e. a defect with very low associated free volume) is found. Nevertheless, additional research is needed in order to understand in detail the difference between GB vacancy behaviors in fcc and bcc metals. From the methodological point of view, there are differences in GB structures (both perfect and with vacancies) obtained with different potentials. In this respect, a help from an experimental and/or ab initio study would be useful in order to select suitable simulation parameters. Further work is in progress to analyze ‘classes’ of possible vacancy sites at GBs considering vacancy instability and delocalization effects. In addition, more general vacancy conﬁgurations at GBs are also being investigated in order to consider more general GB structures that involve a higher level of imperfection, keeping simultaneously a well deﬁned GB geometry. Finally, Monte Carlo studies are planned to check the vacancy behavior in a longer time scale than allowed within the MD scheme. Acknowledgments We are grateful to M.J. Puska for providing us with his ATSUP code that served as a basis for further developments. This research is supported by the Grant Agency of the Czech Republic (CR) (Project No. 202/06/1509) and by the Grant Agency of the Academy of Sciences of the CR (Project No. IAA1041302). This work is part of the research plans MSM 0021620834 and MSM 0021622410 ﬁnanced by the Ministry of Education (ME) of the CR and of the research plan AV0Z20410507 supported by the Academy of
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Sciences of the CR. The presented research is also carried out within the framework of the COST Action P19 ‘‘Multiscale Modelling of Materials’’ and supported by the ME of the CR (Project No. COST OC 165). References  S. Van Petegem, J. Kuriplach, M. Hou, E.E. Zhurkin, D. Segers, A.L. Morales, S. Ettaoussi, C. Dauwe, W. Mondelaers, Mater. Sci. Forum 363–365 (2001) 210.  J. Kuriplach, O. Melikhova, M. Hou, S. Van Petegem, E. Zhurkin, M. Sˇob, Phys. Stat. Sol. C 4 (2007) 3461.  See e.g., X.Y. Qin, J.S. Zhu, L.D. Zhang, X.Y. Zhou, J. Phys.: Condens. Matter, 10 (1998) 3075.  H.E. Schaefer, R. Wu¨rschum, Phys. Lett. A 119 (1987) 370.  S. Van Petegem, F. Dalla Torre, D. Segers, H. Van Swygenhoven, Scripta Mater. 48 (2003) 17.  S. Van Petegem, J. Kuriplach, Acta Phys. Polonica A 107 (2005) 769.
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