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Ab initio contribution to the study of complexes formed during dilute FeCu alloys radiation
Nuclear Instruments and Methods in Physics Research B 202 (2003) 44–50 www.elsevier.com/locate/nimb
Ab initio contribution to the study of complexes formed during dilute FeCu alloys radiation C.S. Becquart a
a,*
, C. Domain
b
Laboratoire de M etallurgie Physique et G enie des Mat eriaux, UMR8517, Universit e de Lille 1, Bat. C6, F-59655 Villeneuve dÕAscq c edex, France b EDF-R&D D epartement MMC, Les Renardi eres, F-77818 Moret sur Loing c edex, France
Abstract Cu plays an important role in the embrittlement of pressure vessel steels under radiation and entities containing both Cu atoms and vacancies seem to appear as a consequence of displacement cascades. The characterisation of the stability as well as the migration of small Cu–vacancy complexes is thus necessary to understand and simulate the formation of these entities. For instance, cascade ageing studied by kinetic Monte Carlo or by rate theory models requires a good characterisation of such complexes which are parameters for these methods. We have investigated, by ab initio calculations based on the density functional theory, point defects and small defects in dilute FeCu alloys. The structure of small Cu clusters and Cu–vacancy complexes has been determined, as well as their formation and binding energies. Their relative stability is discussed. Vacancy migration energies in the presence of Cu atoms have been calculated and analysed. All the results are compared to the figures obtained with empirical interatomic potentials. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72.Ji; 71.15.Mb; 71.20.Be Keywords: Ab initio calculations; Radiation damage; FeCu; Computer simulation
1. Introduction The pressure vessels of light water reactors are made with low alloyed steels. Under irradiation, and depending upon temperature, dose and flux conditions, many defects are detected experimentally to form: self-interstitial atom (SIA) loops or
* Corresponding author. Tel.: +33-320-43-6927; fax: +33320-43-4040. E-mail address: [email protected] (C.S. Becquart).
microvoids as well as dilute solute precipitates or clouds. The latterÕs origin, structure and stability is still under investigation. It is now clearly established that neutron interaction with matter leads to the formation of displacement cascades, which produce the so-called primary damage: small regions containing a core of vacancies, either isolated or grouped in small clusters, surrounded by SIAs, themselves either isolated or in clusters. These primary defects are much too small to be resolved experimentally and molecular dynamics (MD) simulations [1–3] have proven very useful to characterise them. To investigate the evolution of these
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)01828-1
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
defects towards experimentally resolvable entities other tools such as kinetic Monte Carlo (KMC), or rate equations are used. Different KMC models exist: for instance the ‘‘object KMC’’, where one describes the fate of mobile entities (‘‘objects’’): vacancies and interstitials, either isolated or in clusters; or the ‘‘vacancy KMC’’, in which the vacancy diffusion through first neighbour jumps is reproduced in detail and the migration barriers it has to overcome are determined using empirical potentials [4,5] or simple pair interaction models [6]. In all these models, the parameterisation is a very difficult task as experimental data at this scale are scarce [7]. Furthermore, the interatomic potentials used in simulating the initial displacement cascades or the vacancy KMC have a strong influence on the results [3] and it is necessary to validate them, in particular the point defect properties they predict. This article presents some of the results pertinent to KMC or rate equation models obtained by ab initio calculations. In the first part we present the results obtained for bcc Cu and compare it to bcc Fe. We then determine the formation and binding energies of small vacancy clusters. The third part deals with Cu in the a-Fe matrix, and we discuss its migration properties. The last paragraph presents some mixed V–Cu complexes and small Cu clusters calculations. The ab initio results are compared to results obtained with empirical interatomic potentials, of the embedded atom method (EAM) type. Furthermore, as ab initio methods can only treat a limited number of atoms, the empirical calculations are also used to assess the convergence of the results with the system size. In all what follows EAM-1 refers to the EAM potential derived by Ludwig et al. [8], while FS-1 is the potential of Ackland et al. [9]. This work is part of the REVE (Reactor for Virtual Experiments) project, which aims at simulating irradiation effects in structural materials [10].
2. Model Ab initio calculations based on the density functional theory have now demonstrated their
45
capability to treat enough atoms for investigating a large field in materials science. Our calculations have been performed using the Vienna Ab initio Simulation Package VASP, see [11,12] and references therein. The calculations were done in a plane-wave basis, using fully non-local Vanderbilttype ultrasoft pseudopotentials to describe the electron–ion interaction. Exchange and correlation were described by the Perdew–Zunger functional, adding a non-local correction in the form of the generalised gradient approximation (GGA) of Perdew and Wang. All the calculations except when explicitly mentioned were done in the spin polarised GGA. The pseudopotentials were taken from the VASP library. Brillouin zone (BZ) sampling was performed using the Monkhorst-Pack scheme. The supercell approach with periodic boundary conditions was used to simulate point defects as well as pure phases energetics. The defect calculations were performed at constant volume thus relaxing only the atomic position in a supercell dimensioned with the equilibrium lattice parameter for Fe. This allows one to use a smaller plane wave cut-off energy (240 eV in this work). Calculations with 54 (respectively 128) atom supercells were done with a BZ sampling of 125 (respectively 27) k points. More details on the method and in particular a comparison of full relaxation versus constant volume calculations for defects in Fe can be found in our previous work [13]. The binding energies between two entities in a bcc iron matrix should be calculated as follows. The binding energy EbA–B of an A–B pair is defined as the difference of the two system energies, system 1 where A and B do not interact, i.e. the distance between A and B is greater than the potential cutoff, and system 2 where A and B interact. The distance between A and B may be first nearest neighbor distance, second neighbor distance and so on. However, because of the limited supercell size which can be used in ab initio calculations, we determined the binding energies in an indirect manner. The energy EðN 1 þ AÞ of a supercell containing only defect A is added to that (EðN 1 þ BÞ) of the supercell containing only defect B. To that sum, one subtracts the energy EðN 2 þ A þ BÞ of the same supercell containing A and B
46
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
interacting added to that (EðN Þ) of the supercell containing no defect. Thus, EbAB ¼ ½EðN 1 þ AÞ þ EðN 1 þ BÞ ½EðN 2 þ A þ BÞ þ EðN Þ:
ð1Þ
Russell–Brown dispersion strengthening model which appears to model rather well the low workhardening rates observed in the FeCu system [15]. A bcc Cu bulk modulus lower than that of Fe is also predicted by EAM-1, but not by FS-1.
2.1. Preliminary calculations
2.2. Stability of vacancy and vacancy clusters
In the first stage of precipitation of Cu in Fe, the Cu precipitates are coherent with the matrix and therefore have a bcc structure [14]. We thus investigated bcc Cu. As can be seen in Table 1 even though the volume difference between the fcc and bcc phases of Cu seems underestimated, the volume difference between the bcc phase of Fe and the bcc phase of Cu compares well with the experimental data available. All the calculations predict that the lattice parameter of a-Fe is smaller than that of bcc Cu and that a small Cu precipitate should thus induce compressive stresses in the lattice as observed by Phythian et al. [14]. Furthermore, the bulk modulus of bcc Cu was found to be close to 140 GPa. It is lower than that of bcc Fe (160 GPa) [13]. This fact is consistent with the well-known
The vacancy formation energy has been determined in a previous work [13]. The value obtained of 1.95 eV (resp. 2.02 eV) for a 54 (resp. 128) atom supercell falls in the experimental range (1.53 eV [16]–2 eV [17]). The formation energies for small vacancy clusters are presented in Fig. 1 and the corresponding binding energies, obtained using Eq. (1) are displayed in Table 2. In general there is a good agreement with the few available experimental data as well as with the data predicted by the empirical interatomic potentials. 2.3. Cu migration and diffusion coefficient in Fe The cohesive energy for an AB alloy can be approximated using the following equation:
Table 1 Energy and volume difference between the bcc and fcc structures for Cu as well as the volume difference between a-Fe and bcc Cu. The calculations were done in the GGA as well as in the local density approximation (LDA) and compared to other ab initio calculations
Cu (GGA) Cu (LDA) LAPWa Exptb a b
DEfcc-bcc (eV/atom)
Xfcc Xbcc (%) Xfcc
XFe-bcc XCu-bcc (%) XFe-bcc
0.036 0.041 0.049 –
)0.41 )0.45 )0.8 )3.5
)4.03 )5.03 – )3.2
Ref. [24]. Ref. [25]: bcc Cu precipitates found to form under electron irradiation with a fluence of 0.6 C cm2 at 300 °C in Fe 1.5 wt.% Cu.
Fig. 1. Small vacancy clusters formation energies (128 atom supercells). The empty squares are vacancies, the empty circles are Fe atoms. Only the most stable configurations among the different cases studied are represented.
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
47
Table 2 Vacancy cluster binding energies (eV). Comparison with EAM-1 of Ludwig et al. [8], another EAM potential by [26] (data taken from [27]) and FS-1 of Ackland et al. [9] V–V V2 –V V3 –V a
Ab initio (128 atoms)
EAM-1
EAM-2a
FS-1
0.28 0.36 0.70
0.21 0.36 0.63
0.23 0.37 0.66
0.19 0.34 0.63
Ref. [26] (data calculated in [27]).
Ac%B B A c Ecoh ¼ cEcoh þ ð1 cÞEcoh þ DEm ;
ð2Þ
A B where c is the B content, Ecoh and Ecoh are the cohesive energies of pure A and pure B respectively c and DEm is the mixing energy. For a regular soc c lution, DEm is given by DEm ¼ xcð1 cÞ where x is an energy parameter (corresponding to the partial molar energy of B at infinite dilution or the heat of solution). The partial molar energy of Cu at infinite dilution was found to be 0.54 eV (resp. 0.55 eV) for a 54 (resp. 128) atom supercell [13]. This result is in good agreement with the experimental result of Mathon [18] as well as with the predictions of EAM-1. The vacancy migration energies for a supercell containing one Cu atom has been determined for different jumps. The results are displayed in Fig. 2(a) and compared with the predictions of EAM-1 and FS-1. The anisotropy in the different jumps for the ab initio calculations is of the order of 0.1 eV (the highest migration barrier being 0.67 eV, the lowest being 0.56 eV). This anisotropy is clearly overestimated with EAM-1, while FS-1 is in better agreement with the ab initio data.
The ab initio calculations, and the empirical potential calculations done with FS-1 and EAM-1, all predict that the migration energy of the Cu atom towards the vacancy in an Fe matrix is lower than the migration energy of a vacancy in pure aFe. When the vacancy formation and migration energies are used to determine the diffusion coefficient of Fe (with the attempt frequency taken to be 3:65 1015 s1 after Soisson et al. [6] and assumed to be identical for Fe and Cu), we obtain a value of 7:6 1018 m2 s1 at 1000 K (DFe Fe ¼ 2:2 expð2:66 eV=kT Þ cm2 s1 ), which is very close to the experimental data [19]. When the migration barriers of Fig. 2(a) are introduced in the 9-frequencies model developed by Le Claire [20] for the hetero-diffusion in a bcc crystal (our results clearly show that the simplifying assumptions behind the so-called ‘‘model I’’ and ‘‘model II’’ [21] cannot apply for the FeCu system), we obtain a value of 0.50 for the Cu correlation factor at 1000 K. We thus find that the diffusion coefficient of Cu in Fe at 1000 K is 0.07 times bigger than that of Fe 2 1 (DCu Fe ¼ 2:2 expð2:44 eV=kT Þ cm s ), predicting a faster diffusion of Cu atoms in Fe than Fe atoms,
Fig. 2. Vacancy migration energy in a presence of one Cu atom: (a) ab initio, (b) potential FS-1, (c) potential EAM-1. The empty squares are vacancies, the empty circles are Fe atoms and the grey circles are Cu atoms. The bottom caption is the vacancy migration energy in pure Fe.
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C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
in agreement with experimental findings [19] and MD simulations with FS-1 [22]. 2.4. Cu and mixed V–Cu clusters formation and binding energies The formation energies of small Cu clusters are displayed in Fig. 3. Fig. 4 displays the formation energies of small Cu2 –V clusters. Different configurations have been tested and the most stable configuration is that of Fig. 4(d). In parenthesis
appears the value predicted by EAM-1 which disagrees on which configuration is the most stable. This discrepancy can be explained by the fact that the ab initio results favour the V–Cu pair in second nearest neighbour position, while EAM-1 predict the pair to be more stable when the Cu is first nearest neighbour to the vacancy (Table 3). For configurations (a)–(c), the vacancy is in first nearest neighbour position to the two Cu atoms, and the only difference among the configurations arises from the relative positions of these two Cu
Fig. 3. Small Cu clusters formation energies for 54 and 128 atom supercells. The empty squares are vacancies, the empty circles are Fe atoms and the grey circles are Cu atoms.
Fig. 4. Cu2 –V formation energies for 54 and 128 atom supercells. The number in parenthesis are the data obtained with the EAM-1 potential. The empty squares are vacancies, the empty circles Fe atoms and the grey circles are Cu atoms. In (a)–(c), the vacancy is in first nearest neighbour position to the two Cu atoms.
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
49
Table 3 Binding energies (eV) for different supercell sizes, evaluation for 128 atoms ab initio supercells of the binding energy considering only first nearest neighbour and second nearest neighbour pair interactions and binding energies (eV) for different supercell sizes evaluated with the EAM-1 potential of Ludwig et al. [8] System
Cu–V (1nn) Cu–V (2nn) Cu–V (3nn) Cu2 –V Cu3 –V V–V (1nn) V–V (2nn) V–V (3nn) V2 –V V3 –V Cu–Cu (1nn) Cu–Cu (2nn) Cu–Cu (3nn) Cu2 –Cu Cu3 –Cu
Ab initio 54 atoms (125 k points)
128 atoms (27 k points)
Evaluation from ab initio 1nn and 2nn interactions (128 atoms)
0.17 0.21 0.01 0.36 0.46 0.10 0.21 )0.05 0.32 0.62 0.15 0.04 )0.01 0.13 0.25
0.17 0.28 0.04 0.36 0.44 0.14 0.28 )0.02 0.36 0.70 0.14 0.03 )0.01 0.17 0.26
0.17 0.28 0 0.34 0.50 0.14 0.28 0 0.30 0.59 0.14 0.03 0 0.17 0.31
atoms. The configurations (d)–(f) consist in a pair of Cu (A and B) atoms first nearest neighbours, with a vacancy situated in the different first nearest neighbour position of one of the Cu atom, for instance A. The distance between the vacancy and the second Cu atom thus changes for each configuration. The energy difference between these two sets of configurations depends thus only upon second and higher order interactions as well as on the geometry of the Cu2 –V cluster. One important result one can extract from these calculations is that in order to distinguish between the different configurations, one has to take into account at least second nearest neighbour interactions (as first nearest neighbour pair interactions are not sufficient to describe these differences). It can be noticed that configuration (e) has the highest formation energy (by ab initio as well as with EAM-1) and appears to be a transition configuration for the vacancy to leave the two Cu atoms. Table 3 summarises the binding energies for all the small clusters studied with two different supercell sizes. It appears that except for the Cu3 –Cu binding energy the agreement between supercells of different sizes is rather good. Our results indicate a strong vacancy affinity for copper atoms, in agreement with the positron annihilation experi-
EAM-1 54 atoms
128 atoms
2000 atoms
0.19 )0.04 0.00 0.19 0.32 0.16 0.18 )0.03 0.22 0.73 0.19 )0.03 )0.01 0.14 0.28
0.19 )0.03 )0.01 0.20 0.34 0.16 0.20 )0.04 0.36 0.62 0.19 )0.02 )0.01 0.15 0.30
0.19 )0.03 )0.01 0.20 0.34 0.16 0.21 )0.04 0.36 0.63 0.19 )0.02 )0.01 0.16 0.30
ments of Nagai et al. [23]. The main discrepancy between ab initio and EAM calculations concerns the binding energy of V with Cu atoms. The ab initio approach predicts generally a stronger interaction whose origin is the important V–Cu binding energy in second nearest neighbour positions. The binding energies have also been evaluated considering only the first and second nearest neighbour pair interactions (obtained for supercells containing 128 atoms) in order to quantify the contribution of these pair interactions to the local interactions. For these small V–Cu clusters the convergence of the binding energy with the supercell size is quite good, already with only 128 atoms, as confirmed by the EAM-1 results.
3. Conclusions The determination of the formation and binding energies of small copper–vacancy complexes is a challenging task because of the small energy differences involved. However, these data are crucial for modelling the microstructure evolution in Fe alloys under irradiation, and to assess the validity of empirical cohesive models. In these regards, ab initio calculations are valuable tools and
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C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
provide good insight and data for radiation damage studies. Our calculations indicate a strong vacancy affinity for copper atoms in the Fe matrix, and a slightly faster diffusion of a Cu impurity as compared to Fe, as expected from experiments. Furthermore, for the data we have compared, the ab initio results validate many of the trends of the results obtained with EAM-like potentials. Acknowledgements This research has been done using the CRI supercomputer of the USTL supported by the Fonds Europeens de Developpement Regional, as well as the CEA Grenoble supercomputers in the frame of an EDF-CEA contract. References [1] R.S. Averback, T. Diaz de la Rubia, Solid State Phys. 51 (1998) 281. [2] A.F. Calder, D.J. Bacon, J. Nucl. Mater. 207 (1993) 25. [3] C.S. Becquart, C. Domain, A. Legris, J.C. van Duysen, J. Nucl. Mater. 280 (2000) 73. [4] C. Domain, C.S. Becquart, J.C. van Duysen, in: MRS Symp. Proceedings: Multiscale Modelling of Materials, Boston, December 1998, Vol. 538, 1999, p. 217. [5] G.R. Odette, B.D. Wirth, J. Nucl. Mater. 251 (1997) 157. [6] F. Soisson, G. Martin, A. Barbu, Ann. Phys. 20 (1995) C313. [7] L. Malerba, C.S. Becquart, M. Hou, C. Domain, A. Grassi, these Proceedings. [8] M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder, Modell. Simul. Mater. Sci. Eng. 6 (1998) 19. [9] G.J. Ackland, D.J. Bacon, A.F. Calder, T. Harry, Philos. Mag. A 75 (1997) 713.
[10] S. Jumel et al., J. Test. Eval., JTEVA 30 (2002) 37. [11] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558; G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [12] G. Kresse, J. Furthm€ uller, Comput. Mater. Sci. 6 (1996) 15. [13] C. Domain, C.S. Becquart, Phys. Rev. B 65 (2002) 024103. [14] W.J. Phythian, A.J. Foreman, C.A. English, J.T. Buswell, M. Hetherington, K. Roberts, S. Pizzini, in: R.E. Stoller, A.S. Kumar, D.S. Gelles (Eds.), ASTM STP 1125, American Society for Testing and Materials, Philadelphia, 1992, p. 131. [15] K. Russell, L.M. Brown, Acta Metall. 20 (1972) 969. [16] H.E. Schaefer, K. Maier, M. Weller, D. Herlach, A. Seeger, J. Diehl, Scr. Metall. 11 (1977) 803. [17] L.D. Schepper, D. Segers, L. Dorikens-Vanpraet, M. Dorikens, G. Knuyt, L.M. Stals, P. Moser, Phys. Rev. B 27 (1983) 5257. [18] M.H. Mathon, Ph.D. Thesis, 1995. [19] H. Mehrer (Ed.), Numerical Data and Functional Relationships in Science and Technology, Landolt-B€ ornstein New Series III, Vol. 26, Springer-Verlag, Berlin, 1990, Chapter 3, p. 73, 179. [20] A.D. Le Claire, in: H. Eyring (Ed.), Physical Chemistry: An Advanced Treatise, Academic Press, New York, 1970, Vol. 10, Chapter 5. [21] J. Philibert, Atom Movements Diffusion and Mass Trans ditions de Physiques, 1991. port in Solids, Les E [22] J. Marian, B.D. Wirth, J.M. Perlado, G.R. Odette, T. Diaz de la Rubia, in: G.E. Lucas, L.L. Snead, M.A. Kirk, R.G. Elliman (Eds.), Microstructural Processes in Irradiated Materials, Mater. Res. Soc. Proc. 650, Material Research Society, PA, 2001, p. R6.9. [23] Y. Nagai, Z. Tang, M. Hasegawa, T. Kanai, M. Saneyasu, Phys. Rev. B 63 (2001) 134110. [24] Z.W. Lu, S.H. Wei, A. Zunger, Phys. Rev. B 41 (1990) 2699. [25] F. Maury, N. Lorenzelli, M.H. Mathon, C.H. de Novion, P. Lagarde, J. Phys.: Condens. Matter 6 (1994) 6. [26] R.A. Johnson, D.J. Oh, J. Mater. Res. 4 (1989) 4. [27] N. Soneda, T. Diaz de la Rubia, Philos. Mag. A 78 (1998) 995.
Ab initio contribution to the study of complexes formed during dilute FeCu alloys radiation C.S. Becquart a
a,*
, C. Domain
b
Laboratoire de M etallurgie Physique et G enie des Mat eriaux, UMR8517, Universit e de Lille 1, Bat. C6, F-59655 Villeneuve dÕAscq c edex, France b EDF-R&D D epartement MMC, Les Renardi eres, F-77818 Moret sur Loing c edex, France
Abstract Cu plays an important role in the embrittlement of pressure vessel steels under radiation and entities containing both Cu atoms and vacancies seem to appear as a consequence of displacement cascades. The characterisation of the stability as well as the migration of small Cu–vacancy complexes is thus necessary to understand and simulate the formation of these entities. For instance, cascade ageing studied by kinetic Monte Carlo or by rate theory models requires a good characterisation of such complexes which are parameters for these methods. We have investigated, by ab initio calculations based on the density functional theory, point defects and small defects in dilute FeCu alloys. The structure of small Cu clusters and Cu–vacancy complexes has been determined, as well as their formation and binding energies. Their relative stability is discussed. Vacancy migration energies in the presence of Cu atoms have been calculated and analysed. All the results are compared to the figures obtained with empirical interatomic potentials. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72.Ji; 71.15.Mb; 71.20.Be Keywords: Ab initio calculations; Radiation damage; FeCu; Computer simulation
1. Introduction The pressure vessels of light water reactors are made with low alloyed steels. Under irradiation, and depending upon temperature, dose and flux conditions, many defects are detected experimentally to form: self-interstitial atom (SIA) loops or
* Corresponding author. Tel.: +33-320-43-6927; fax: +33320-43-4040. E-mail address: [email protected] (C.S. Becquart).
microvoids as well as dilute solute precipitates or clouds. The latterÕs origin, structure and stability is still under investigation. It is now clearly established that neutron interaction with matter leads to the formation of displacement cascades, which produce the so-called primary damage: small regions containing a core of vacancies, either isolated or grouped in small clusters, surrounded by SIAs, themselves either isolated or in clusters. These primary defects are much too small to be resolved experimentally and molecular dynamics (MD) simulations [1–3] have proven very useful to characterise them. To investigate the evolution of these
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)01828-1
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
defects towards experimentally resolvable entities other tools such as kinetic Monte Carlo (KMC), or rate equations are used. Different KMC models exist: for instance the ‘‘object KMC’’, where one describes the fate of mobile entities (‘‘objects’’): vacancies and interstitials, either isolated or in clusters; or the ‘‘vacancy KMC’’, in which the vacancy diffusion through first neighbour jumps is reproduced in detail and the migration barriers it has to overcome are determined using empirical potentials [4,5] or simple pair interaction models [6]. In all these models, the parameterisation is a very difficult task as experimental data at this scale are scarce [7]. Furthermore, the interatomic potentials used in simulating the initial displacement cascades or the vacancy KMC have a strong influence on the results [3] and it is necessary to validate them, in particular the point defect properties they predict. This article presents some of the results pertinent to KMC or rate equation models obtained by ab initio calculations. In the first part we present the results obtained for bcc Cu and compare it to bcc Fe. We then determine the formation and binding energies of small vacancy clusters. The third part deals with Cu in the a-Fe matrix, and we discuss its migration properties. The last paragraph presents some mixed V–Cu complexes and small Cu clusters calculations. The ab initio results are compared to results obtained with empirical interatomic potentials, of the embedded atom method (EAM) type. Furthermore, as ab initio methods can only treat a limited number of atoms, the empirical calculations are also used to assess the convergence of the results with the system size. In all what follows EAM-1 refers to the EAM potential derived by Ludwig et al. [8], while FS-1 is the potential of Ackland et al. [9]. This work is part of the REVE (Reactor for Virtual Experiments) project, which aims at simulating irradiation effects in structural materials [10].
2. Model Ab initio calculations based on the density functional theory have now demonstrated their
45
capability to treat enough atoms for investigating a large field in materials science. Our calculations have been performed using the Vienna Ab initio Simulation Package VASP, see [11,12] and references therein. The calculations were done in a plane-wave basis, using fully non-local Vanderbilttype ultrasoft pseudopotentials to describe the electron–ion interaction. Exchange and correlation were described by the Perdew–Zunger functional, adding a non-local correction in the form of the generalised gradient approximation (GGA) of Perdew and Wang. All the calculations except when explicitly mentioned were done in the spin polarised GGA. The pseudopotentials were taken from the VASP library. Brillouin zone (BZ) sampling was performed using the Monkhorst-Pack scheme. The supercell approach with periodic boundary conditions was used to simulate point defects as well as pure phases energetics. The defect calculations were performed at constant volume thus relaxing only the atomic position in a supercell dimensioned with the equilibrium lattice parameter for Fe. This allows one to use a smaller plane wave cut-off energy (240 eV in this work). Calculations with 54 (respectively 128) atom supercells were done with a BZ sampling of 125 (respectively 27) k points. More details on the method and in particular a comparison of full relaxation versus constant volume calculations for defects in Fe can be found in our previous work [13]. The binding energies between two entities in a bcc iron matrix should be calculated as follows. The binding energy EbA–B of an A–B pair is defined as the difference of the two system energies, system 1 where A and B do not interact, i.e. the distance between A and B is greater than the potential cutoff, and system 2 where A and B interact. The distance between A and B may be first nearest neighbor distance, second neighbor distance and so on. However, because of the limited supercell size which can be used in ab initio calculations, we determined the binding energies in an indirect manner. The energy EðN 1 þ AÞ of a supercell containing only defect A is added to that (EðN 1 þ BÞ) of the supercell containing only defect B. To that sum, one subtracts the energy EðN 2 þ A þ BÞ of the same supercell containing A and B
46
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
interacting added to that (EðN Þ) of the supercell containing no defect. Thus, EbAB ¼ ½EðN 1 þ AÞ þ EðN 1 þ BÞ ½EðN 2 þ A þ BÞ þ EðN Þ:
ð1Þ
Russell–Brown dispersion strengthening model which appears to model rather well the low workhardening rates observed in the FeCu system [15]. A bcc Cu bulk modulus lower than that of Fe is also predicted by EAM-1, but not by FS-1.
2.1. Preliminary calculations
2.2. Stability of vacancy and vacancy clusters
In the first stage of precipitation of Cu in Fe, the Cu precipitates are coherent with the matrix and therefore have a bcc structure [14]. We thus investigated bcc Cu. As can be seen in Table 1 even though the volume difference between the fcc and bcc phases of Cu seems underestimated, the volume difference between the bcc phase of Fe and the bcc phase of Cu compares well with the experimental data available. All the calculations predict that the lattice parameter of a-Fe is smaller than that of bcc Cu and that a small Cu precipitate should thus induce compressive stresses in the lattice as observed by Phythian et al. [14]. Furthermore, the bulk modulus of bcc Cu was found to be close to 140 GPa. It is lower than that of bcc Fe (160 GPa) [13]. This fact is consistent with the well-known
The vacancy formation energy has been determined in a previous work [13]. The value obtained of 1.95 eV (resp. 2.02 eV) for a 54 (resp. 128) atom supercell falls in the experimental range (1.53 eV [16]–2 eV [17]). The formation energies for small vacancy clusters are presented in Fig. 1 and the corresponding binding energies, obtained using Eq. (1) are displayed in Table 2. In general there is a good agreement with the few available experimental data as well as with the data predicted by the empirical interatomic potentials. 2.3. Cu migration and diffusion coefficient in Fe The cohesive energy for an AB alloy can be approximated using the following equation:
Table 1 Energy and volume difference between the bcc and fcc structures for Cu as well as the volume difference between a-Fe and bcc Cu. The calculations were done in the GGA as well as in the local density approximation (LDA) and compared to other ab initio calculations
Cu (GGA) Cu (LDA) LAPWa Exptb a b
DEfcc-bcc (eV/atom)
Xfcc Xbcc (%) Xfcc
XFe-bcc XCu-bcc (%) XFe-bcc
0.036 0.041 0.049 –
)0.41 )0.45 )0.8 )3.5
)4.03 )5.03 – )3.2
Ref. [24]. Ref. [25]: bcc Cu precipitates found to form under electron irradiation with a fluence of 0.6 C cm2 at 300 °C in Fe 1.5 wt.% Cu.
Fig. 1. Small vacancy clusters formation energies (128 atom supercells). The empty squares are vacancies, the empty circles are Fe atoms. Only the most stable configurations among the different cases studied are represented.
C.S. Becquart, C. Domain / Nucl. Instr. and Meth. in Phys. Res. B 202 (2003) 44–50
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Table 2 Vacancy cluster binding energies (eV). Comparison with EAM-1 of Ludwig et al. [8], another EAM potential by [26] (data taken from [27]) and FS-1 of Ackland et al. [9] V–V V2 –V V3 –V a
Ab initio (128 atoms)
EAM-1
EAM-2a
FS-1
0.28 0.36 0.70
0.21 0.36 0.63
0.23 0.37 0.66
0.19 0.34 0.63
Ref. [26] (data calculated in [27]).
Ac%B B A c Ecoh ¼ cEcoh þ ð1 cÞEcoh þ DEm ;
ð2Þ
A B where c is the B content, Ecoh and Ecoh are the cohesive energies of pure A and pure B respectively c and DEm is the mixing energy. For a regular soc c lution, DEm is given by DEm ¼ xcð1 cÞ where x is an energy parameter (corresponding to the partial molar energy of B at infinite dilution or the heat of solution). The partial molar energy of Cu at infinite dilution was found to be 0.54 eV (resp. 0.55 eV) for a 54 (resp. 128) atom supercell [13]. This result is in good agreement with the experimental result of Mathon [18] as well as with the predictions of EAM-1. The vacancy migration energies for a supercell containing one Cu atom has been determined for different jumps. The results are displayed in Fig. 2(a) and compared with the predictions of EAM-1 and FS-1. The anisotropy in the different jumps for the ab initio calculations is of the order of 0.1 eV (the highest migration barrier being 0.67 eV, the lowest being 0.56 eV). This anisotropy is clearly overestimated with EAM-1, while FS-1 is in better agreement with the ab initio data.
The ab initio calculations, and the empirical potential calculations done with FS-1 and EAM-1, all predict that the migration energy of the Cu atom towards the vacancy in an Fe matrix is lower than the migration energy of a vacancy in pure aFe. When the vacancy formation and migration energies are used to determine the diffusion coefficient of Fe (with the attempt frequency taken to be 3:65 1015 s1 after Soisson et al. [6] and assumed to be identical for Fe and Cu), we obtain a value of 7:6 1018 m2 s1 at 1000 K (DFe Fe ¼ 2:2 expð2:66 eV=kT Þ cm2 s1 ), which is very close to the experimental data [19]. When the migration barriers of Fig. 2(a) are introduced in the 9-frequencies model developed by Le Claire [20] for the hetero-diffusion in a bcc crystal (our results clearly show that the simplifying assumptions behind the so-called ‘‘model I’’ and ‘‘model II’’ [21] cannot apply for the FeCu system), we obtain a value of 0.50 for the Cu correlation factor at 1000 K. We thus find that the diffusion coefficient of Cu in Fe at 1000 K is 0.07 times bigger than that of Fe 2 1 (DCu Fe ¼ 2:2 expð2:44 eV=kT Þ cm s ), predicting a faster diffusion of Cu atoms in Fe than Fe atoms,
Fig. 2. Vacancy migration energy in a presence of one Cu atom: (a) ab initio, (b) potential FS-1, (c) potential EAM-1. The empty squares are vacancies, the empty circles are Fe atoms and the grey circles are Cu atoms. The bottom caption is the vacancy migration energy in pure Fe.
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in agreement with experimental findings [19] and MD simulations with FS-1 [22]. 2.4. Cu and mixed V–Cu clusters formation and binding energies The formation energies of small Cu clusters are displayed in Fig. 3. Fig. 4 displays the formation energies of small Cu2 –V clusters. Different configurations have been tested and the most stable configuration is that of Fig. 4(d). In parenthesis
appears the value predicted by EAM-1 which disagrees on which configuration is the most stable. This discrepancy can be explained by the fact that the ab initio results favour the V–Cu pair in second nearest neighbour position, while EAM-1 predict the pair to be more stable when the Cu is first nearest neighbour to the vacancy (Table 3). For configurations (a)–(c), the vacancy is in first nearest neighbour position to the two Cu atoms, and the only difference among the configurations arises from the relative positions of these two Cu
Fig. 3. Small Cu clusters formation energies for 54 and 128 atom supercells. The empty squares are vacancies, the empty circles are Fe atoms and the grey circles are Cu atoms.
Fig. 4. Cu2 –V formation energies for 54 and 128 atom supercells. The number in parenthesis are the data obtained with the EAM-1 potential. The empty squares are vacancies, the empty circles Fe atoms and the grey circles are Cu atoms. In (a)–(c), the vacancy is in first nearest neighbour position to the two Cu atoms.
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Table 3 Binding energies (eV) for different supercell sizes, evaluation for 128 atoms ab initio supercells of the binding energy considering only first nearest neighbour and second nearest neighbour pair interactions and binding energies (eV) for different supercell sizes evaluated with the EAM-1 potential of Ludwig et al. [8] System
Cu–V (1nn) Cu–V (2nn) Cu–V (3nn) Cu2 –V Cu3 –V V–V (1nn) V–V (2nn) V–V (3nn) V2 –V V3 –V Cu–Cu (1nn) Cu–Cu (2nn) Cu–Cu (3nn) Cu2 –Cu Cu3 –Cu
Ab initio 54 atoms (125 k points)
128 atoms (27 k points)
Evaluation from ab initio 1nn and 2nn interactions (128 atoms)
0.17 0.21 0.01 0.36 0.46 0.10 0.21 )0.05 0.32 0.62 0.15 0.04 )0.01 0.13 0.25
0.17 0.28 0.04 0.36 0.44 0.14 0.28 )0.02 0.36 0.70 0.14 0.03 )0.01 0.17 0.26
0.17 0.28 0 0.34 0.50 0.14 0.28 0 0.30 0.59 0.14 0.03 0 0.17 0.31
atoms. The configurations (d)–(f) consist in a pair of Cu (A and B) atoms first nearest neighbours, with a vacancy situated in the different first nearest neighbour position of one of the Cu atom, for instance A. The distance between the vacancy and the second Cu atom thus changes for each configuration. The energy difference between these two sets of configurations depends thus only upon second and higher order interactions as well as on the geometry of the Cu2 –V cluster. One important result one can extract from these calculations is that in order to distinguish between the different configurations, one has to take into account at least second nearest neighbour interactions (as first nearest neighbour pair interactions are not sufficient to describe these differences). It can be noticed that configuration (e) has the highest formation energy (by ab initio as well as with EAM-1) and appears to be a transition configuration for the vacancy to leave the two Cu atoms. Table 3 summarises the binding energies for all the small clusters studied with two different supercell sizes. It appears that except for the Cu3 –Cu binding energy the agreement between supercells of different sizes is rather good. Our results indicate a strong vacancy affinity for copper atoms, in agreement with the positron annihilation experi-
EAM-1 54 atoms
128 atoms
2000 atoms
0.19 )0.04 0.00 0.19 0.32 0.16 0.18 )0.03 0.22 0.73 0.19 )0.03 )0.01 0.14 0.28
0.19 )0.03 )0.01 0.20 0.34 0.16 0.20 )0.04 0.36 0.62 0.19 )0.02 )0.01 0.15 0.30
0.19 )0.03 )0.01 0.20 0.34 0.16 0.21 )0.04 0.36 0.63 0.19 )0.02 )0.01 0.16 0.30
ments of Nagai et al. [23]. The main discrepancy between ab initio and EAM calculations concerns the binding energy of V with Cu atoms. The ab initio approach predicts generally a stronger interaction whose origin is the important V–Cu binding energy in second nearest neighbour positions. The binding energies have also been evaluated considering only the first and second nearest neighbour pair interactions (obtained for supercells containing 128 atoms) in order to quantify the contribution of these pair interactions to the local interactions. For these small V–Cu clusters the convergence of the binding energy with the supercell size is quite good, already with only 128 atoms, as confirmed by the EAM-1 results.
3. Conclusions The determination of the formation and binding energies of small copper–vacancy complexes is a challenging task because of the small energy differences involved. However, these data are crucial for modelling the microstructure evolution in Fe alloys under irradiation, and to assess the validity of empirical cohesive models. In these regards, ab initio calculations are valuable tools and
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provide good insight and data for radiation damage studies. Our calculations indicate a strong vacancy affinity for copper atoms in the Fe matrix, and a slightly faster diffusion of a Cu impurity as compared to Fe, as expected from experiments. Furthermore, for the data we have compared, the ab initio results validate many of the trends of the results obtained with EAM-like potentials. Acknowledgements This research has been done using the CRI supercomputer of the USTL supported by the Fonds Europeens de Developpement Regional, as well as the CEA Grenoble supercomputers in the frame of an EDF-CEA contract. References [1] R.S. Averback, T. Diaz de la Rubia, Solid State Phys. 51 (1998) 281. [2] A.F. Calder, D.J. Bacon, J. Nucl. Mater. 207 (1993) 25. [3] C.S. Becquart, C. Domain, A. Legris, J.C. van Duysen, J. Nucl. Mater. 280 (2000) 73. [4] C. Domain, C.S. Becquart, J.C. van Duysen, in: MRS Symp. Proceedings: Multiscale Modelling of Materials, Boston, December 1998, Vol. 538, 1999, p. 217. [5] G.R. Odette, B.D. Wirth, J. Nucl. Mater. 251 (1997) 157. [6] F. Soisson, G. Martin, A. Barbu, Ann. Phys. 20 (1995) C313. [7] L. Malerba, C.S. Becquart, M. Hou, C. Domain, A. Grassi, these Proceedings. [8] M. Ludwig, D. Farkas, D. Pedraza, S. Schmauder, Modell. Simul. Mater. Sci. Eng. 6 (1998) 19. [9] G.J. Ackland, D.J. Bacon, A.F. Calder, T. Harry, Philos. Mag. A 75 (1997) 713.
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