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Atomic collision cascades in vanadium crystallites with grain boundaries
20
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
Atomic collision cascades in vanadium crystallites with grain boundaries S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich, A.V. Zheleznyakov and V.M. Chernov1 1
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia Bochvar All-Russia Research Institute for Inorganic Materials, Moscow, 123060, Russia
The effect of grain boundaries on the special features inherent to the evolution of atomic collision cascades and formation of radiationaffected regions in vanadium crystallites was investigated. The presence of grain boundaries in the material was found to have a considerable impact on the radiation damage pattern, specifically on the radiation-induced defect distribution and size. Grain boundaries were shown to serve as a barrier to propagation of atomic displacement cascades and to accumulate most of radiation-induced defects. Relaxed vanadium crystallites contained a relatively small number of clusters made up of point defects, i.e., vacancies and intrinsic interstitial atoms. Keywords: radiation-induced defects, molecular dynamics, grain boundaries
1. Introduction Investigations into structural damage to materials under irradiation are of particular interest for long-term predictions of changes in their mechanical properties. Primary evidence for the radiation effect is generation of atomic displacement cascades responsible for radiation-induced structural defects, formation of microstructures, and changes in the physical-mechanical properties of materials. Simulation of atomic displacement cascades, analysis of the resulting defects, investigation into their evolution and formation of relatively stable radiation-induced defects, as a rule, are performed for materials with ideal structure [1 4]. It is known, however, that extended interfaces may have a considerable impact on many of the properties of materials [58]. Experimental investigations into generation of atomic displacement cascades are extremely difficult to perform. Because of this, computer simulations appear to be the most suitable method for deriving information of primary radiation processes developing in materials under irradiation. We have performed a molecular-dynamics simulation of the effect of grain boundaries on the evolution pattern of atomic displacement cascades and formation of radiation
* Corresponding author Prof. Konstantin P. Zolnikov, e-mail: [email protected] Copyright © 2009 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2009.03.003
induced defects in vanadium crystallites. To this end, symmetrical tilt grain boundaries with different energies were chosen [5]. Vanadium crystallites were picked up because vanadium-base alloys exhibit the most promise for nuclear power engineering. For the purpose in hand, the influence of the distance of a primary knock-on atom from the grain boundary on the evolution pattern of displacement cascades in a vanadium crystallite was investigated. 2. Computational formalism Molecular-dynamics calculations were performed, using the Lammps code (http://lammps.sandia.gov) developed in National Sandia Laboratories, USA. Interatomic interactions in vanadium crystallites were described by means of an interatomic potential of the FinnisSinclair type placed at our disposal by M.I. Mendeleev (Ames Laboratory, USA). The potential used is of importance for a simulation of atomic collision cascades and enables such characteristics as the lattice parameter, elastic moduli, the energy of point defect formation and migration and so forth to be described with fairly high accuracy (Table 1). The simulated crystallites exhibited symmetrical tilt boundaries of two types: 613 (320)[001] and 6 17 (410)[001], allowing the use of periodic boundary conditions [11] employed for the simulation of atomic displacement cascades in this work. The choice of the simulated crystallite size
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
depended on the primary knock-on atom energy Åpka. In the calculations, the number of atoms in the computational grid was varied between 65 000 (for Epka < 500 eV) and 450 000 (for Epka > 500 eV). The minimum time it takes to initiate the evolution of a displacement cascade (for Epka < 50 keV) and formation of stable defects is usually 1015 ps. The minimum time increases moderately as Epka is further increased. By and large the computation time is determined by three parameters: primary knock-on atom energy, crystallite size and interatomic interaction potential. In the cascade evolution simulation, the integration time was changed three times. In the cascade nucleation stage, the integration time was 1017 s, whereas in the relaxation stage it was 1016 s. In the stable defect evolution stage, the integration time was 1015 s. Using special computer software for visualization of the structure and fields of the physical quantities involved, the evolution of damage zones was examined to reveal possible overlaps of different parts of the displacement cascade. The analysis was based on general assumptions about overlaps of parts of the cascade, because no unambiguous criterion for identifying a cascade overlap has hitherto been established. The crystallite damage analysis was performed with allowance made for the number of Frenkel pairs and population of clusters formed by a displacement cascade. Frenkel pair is taken to mean an interstitial atom and a closest stable vacancy. There are two ways of identifying positions of vacancies and interstitial atoms in the simulated crystallite [2]: (1) to look at the occupancy of spheres of a certain radius by displaced atoms and (2) to use WignerSeitz cells. The radius of the sphere (center of the sphere coincides with the lattice site) was chosen to be 0.3 of the vanadium crystal lattice parameter. If the atom leaves the sphere or WignerSeitz cell, it is considered to be a displaced atom. If there are no atoms inside the sphere or cell, a vacancy is believed to form there. If there are two or more atoms inside the sphere or cell, the situation corresponds to an interstitial atomic configuration. Note that both of the ways provide fairly close results for considering the number of structural defects at a certain instant of time [2]. In the simulation of atomic displacement cascades, point defects (vacancies and intrinsic interstitial atoms) may occur in the immediate vicinity of one another. If the distance between any closest point defects is smaller than a certain threshold value, they are said to belong to the same cluster. However, there is no clear criterion for determining the threshold distance. It is suggested here that the threshold distance be the radius of the second coordination sphere in the perfect vanadium lattice [2]. Then the sum of stable vacancies and interstitial atoms (intrinsic interstitials) in a cluster is calculated. Notably, the number of vacancies coincides with that of interstitial atoms.
21 Table 1
Properties of vanadium calculated with the use of MIM* potential Properties
Experiment
Lattice parameter, nm
Calculation (ÌIÌ)
0.3039 [9]
0.3030
Binding energy, eV/atom
5.31*
5.016
Elastic modulus C11 , GPa
229 [9]
228
Elastic modulus C12 , GPa
119 [9]
119
Elastic modulus C44 , GPa
43 [9]
42
Vacancy formation energy
Efv , eV
2.48*
2.49
Vacancy migration energy + vacancy formation energy Emv Efv , eV
3.19*
3.27
Interstitial formation energy Efi ( 100 ), eV
3.22*
3.21
Interstitial formation energy Efi ( 110 ), eV
3.09*
3.09
2.81*
2.81
2.81*
2.83
2190 [10]
3119
Interstitial formation energy Efi ( 111 ), eV Crowdion formation energy eV/atom
E fc,
Melting temperature Tm, K * M.I. Mendeleev, private communication
3. Atomic displacement cascades in the perfect vanadium lattice Before generation of atomic displacement cascades, the simulated vanadium crystallites were relaxed at 10 K. As soon as the primary knock-on atom was initiated, a chain of atomic displacements began to develop in the crystallite. The number of displaced atoms and the size of the radiationaffected zone increased simultaneously until the energy transferred by the primary knock-on atom to its nearest neighbors was distributed over the bulk of the simulated crystallite, so that no atoms with energy above that of the threshold displacement was left. This stage in the development of the cascade is called ballistic. Late in this stage, the
Fig. 1. Peak time versus primary knock-on atom energy Åpka
22
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
b
Fig. 2. Number n of point defects versus Å pka at the peak time (a) and in the cooling stage (b)
number of defects produced by atomic displacements is at its maximum. Then a recombination (relaxation) stage begins, where the number of defects reduces down to a certain relatively stable value after which only diffusion processes will change the number and distribution pattern of the defects in the crystallite. It is to be noted that the kinetic temperature in the recombination stage reaches high values (thermal peak) in the cascade region. Late in the ballistic stage, the essential characteristics of the displacement cascade are as follows: (1) the maximum number of the defects produced, (2) the peak time (time it
takes to generate a maximum number of defects), (3) the volume of the region where a cascade develops and (4) the cascade density (number of defects per unit volume of the cascade region at the peak time). These characteristics depend on the primary knock-on atom energy and, generally speaking, on the crystallite temperature. The latter is ignored here. The peak time as a function of cascade energy is presented in Fig. 1. As can be seen from the figure, the curve slope decreases as the energy increases. This behavior is likely to be due to splitting of the main cascade into subcas-
à
b
c
d
Fig. 3. Point defects in a simulated crystallite containing 450 000 atoms at different instants of time: 0.1 (à), 0.2 (b), 0.4 (c), 2 ps (d ). Gray and black circles correspond to vacancies and interstitials, respectively. The initial direction of the primary knock-on atom is along [320] and the primary knockon atom energy is 1 keV
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
23 b
)
)
[410]
[320]
[001]
[001]
[230]
[140]
Fig. 4. Projections of symmetrical tilt grain boundaries onto the plane (001) for the grain boundaries 613(à) and 617(b). Shaded are periodicity regions along a grain boundary. Black and gray circles show atoms lying in adjacent atomic planes
cades initiated by secondary knock-on atoms with energy below that of the primary knock-on atom. The secondary knock-on atoms generate cascades with shorter peak time compared to the main cascade and, on the whole, decrease the peak time. The maximum number of defects in the vicinity of an atomic displacement cascade as a function of energy of the primary knock-on atom is shown in Fig. 2(a). The relation is near-linear, with the number of defects in the cascade increasing with increase in the energy of the cascade. However, the number of defects produced in a relaxed crystallite is a nonlinear function of energy of the primary knock-on atom (Fig. 2(b)). The decrease in the rate of growth of the curve with increase in the primary knock-on atom energy (Fig. 2(b)) is attributable to the splitting of the main cascade into subcascades. The relative position of point defects in the simulated vanadium crystallite in different evolution stages of an atomic displacement cascade is shown in Fig. 3. The initial velocity of the primary knock-on atom was oriented along the crystallographic direction [320]. The primary knockon atom energy was 1 keV. Figure 3(a and b) corresponds to the ballistic stage of the evolution of the cascade. Figure 3 (c and d) illustrates the recombination stage and the cooling stage of the evolution of radiation damage, respectively.
4. The influence of grain boundaries on the formation of atomic displacements in a cascade Symmetrical tilt or twin grain boundaries formed from two crystals of bilateral symmetrical structure are the simplest configuration of all grain boundaries. Table 2 gives parameters of 6 types of symmetrical tilt grain boundaries formed in bcc crystals: directions of rotation axes, angles )
Fig. 5. Energy of symmetrical tilt grain boundaries in a vanadium crystallite versus the tilt angle ) before (open circles) and after relaxation (full circles)
Table 2
Symmetrical tilt grain boundary parameters in vanadium crystallite Normals to planes (320)(230)
Rotation axis
(530)(350) (210)(120) (310)(130)
[001]
)
6
22.62q
13
28.07q
17
36.87q
5
53.13q
5
(410)(140)
61.93q
17
(510)(150)
67.38q
13
Fig. 6. Mean potential energy per atom Eð along the Y-direction normal to the 613 grain boundary plane
24
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 Table 3
The distance of the primary knock-on atom from the symmetrical tilt grain boundary Type of grain boundary Distance of the primary knock-on atom from the grain boundary, nm Direction of primary knock-on atom velocity
(320)[001] 613
(410)[001] 617
[320]
[410]
1.04, 1.81, 2.65, 3.16, 3.87, 4.42
of rotation of one grain relative to another, and the number 6 of coincidence lattice sites. Figure 4 shows projections of unrelaxed crystallite structures with the grain boundaries 613 and 617 onto the plane (001). During relaxation of the initial bicrystal, the atoms near the grain boundary are found to experience the largest displacements. It is from the atomic displacement that the grain-boundary width can be estimated. Atoms away from this region retain bilateral symmetrical order relative to the boundary. Bicrystals with different types of grain boundaries were relaxed at room temperature to produce stable structure. The grain-boundary energy ÅGB before and after relaxation is presented in Fig. 5. These calculations show no correlation between the grain-boundary energy and tilt angle. The grain boundary (410) [001] 617 exhibits the highest energy, whereas the lowest energy is obtained for the grain boundary (320) [001] 613. That is why these grain boundaries were chosen to investigate the effect of extended interfaces on the evolution pattern of atomic displacement cascades and subsequent formation of radiation-affected regions. The grain-boundary width is generally several atomic layers. Due to disordering of these layers, many of the physical properties (thermal expansion, electrical resistivity, elastic moduli and other characteristics) near the grain boundary may differ essentially from the bulk properties upon relaxation and exhibit a high degree of anisotropy. The grainboundary width can be determined from calculations of changes in the potential atomic binding energy Åð in the à
1.02, 1.84, 2.57, 3.24, 3.87, 4.53
direction normal to the grain-boundary plane (Fig. 6). The plot was constructed by breaking the simulated crystallite into thin layers normal to the grain-boundary plane and the atomic binding energy per atom inside a layer was averaged. The binding energy per atom for the perfect vanadium lattice at 10 K was 5.014 eV. The grain-boundary width was found from the condition that the binding energy inside the boundary differs from its value for the perfect lattice and as a rule is higher by more than 0.01 eV. An atomic displacement cascade begins to develop at the instant the primary knock-on atom is generated. To gain insight into the influence of grain boundaries on the evolution of displacement cascades, we examined the distance of the primary knock-on atom from the interface. The energy of the primary knock-on atom was 1 keV in all cases studied and the direction of the velocity of the primary knock-on atom was chosen to be normal to the grain-boundary plane. The distances of the primary knock-on atom from the grainboundary plane are listed in Table 3. The special features of the evolution of atomic displacement cascades in materials with grain boundaries are in many ways similar to the processes involved in materials with ideal structure. In particular, the largest number of defects in both cases is generated within half a picosecond (Fig. 7). Over this time interval, the excess energy of the primary knock-on atom is transferred to the simulated crystallite. Note that the energy of the crystallite exhibits the highest fluctuations. Thereafter, the fluctuations and the number of defects produced decrease. Within 3 4 ps of
b
Fig. 7. Temporal variations of the number of point defects. The crystallite contains the grain boundaries 613 (à). The upper curve corresponds to the primary knock-on atom at a distance of 1.04 nm from the grain boundary and the lower curve is for a distance of 3.16 nm. The sample contains the grain boundaries 617 (b). The upper curve corresponds to the primary knock-on atom at a distance of 1.84 nm from the grain boundary and the lower curve is for a distance of 3.87 nm. The energy of the primary knock-on atom is 1 keV. The direction of the atomic velocity is normal to the grain boundary plane
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
b
[230]
[230] [320]
[320]
GB
GB
c
d
[230]
[230] [320]
25
[320]
GB
GB
Fig. 8. Projection of defect structure onto the plane (001) at different instants of time: 0.06 (à), 0.2 (b), 1 (c) and 10 ps (d ). Gray and black circles correspond to vacancies and interstitials, respectively. The distance from the primary knock-on atom to the grain boundary 613 is 1.81 nm. GB is the grain boundary
generation of a displacement cascade, the number of radiation-induced defects is stabilized. Temporal variations of the number of defects in displacement cascades for a crystallite with grain boundaries are shown in Fig. 7. The variation pattern for the number of defects (total number of defects at the peak time, curve falloff rates, total number of stable defects late in the simulations, and corresponding characteristic times) in crystallites with different types of grain boundaries are much the same. Figures 8 and 9 show different evolution stages of the simulated displacement cascades in vanadium crystallites with grain boundaries. The figures correspond to crystallites with grain boundaries 613 for the primary knock-on atom lying at a distance of 1.81 and 3.87 nm from the grainboundary plane, respectively. The direction of the initial velocity of the primary knock-on atom coincides with the
crystallographic direction [320]. For the primary knock-on atom lying at a distance of 1.81 nm from the grain-boundary plane, most of atomic displacements responsible for the formation of point defects are localized in the grain-boundary region. Note that a considerably smaller fraction of point defects, including the relaxation stage, passes through the grain boundary. This behavior of the atomic displacement cascade is due to the fact that in the grain-boundary region, defects are generated both by rearrangement of grainboundary atoms and by penetration of interstitial atoms into this region from the outside. For a distance of ~1.83 nm from the primary knock-on atom to the grain boundary, some 90 % of the defects produced lie in the grain-boundary region, whereas for a distance of ~3.8 nm, the number of defects reduces down to 55 % for both types of the grain interface (Table 4). Table 4
Influence of the distance of the primary knock-on atom from the grain boundary on the number of defects in the grain-boundary region 613
Type of grain-boundary
617
Distance of the primary knock-on atom from the grain boundary, nm
1.8
3.9
1.8
3.9
Relative number of defects in the grain-boundary region in an as-relaxed displacement cascade, %
89
54
86
55
26
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
[230]
[230] [320]
b
[320]
GB
GB
c
[230]
[230] [320]
d
GB
[320]
GB
Fig. 9. Projection of defect structure onto the plane (001) at different instants of time: 0.06 (a), 0.2 (b), 1 (c) and 10 ps (d ). Gray and black circles show vacancies and interstitials, respectively. The primary knock-on atom is at a distance of 3.87 nm from the grain boundary 613
Starting with a distance of ~3.8 nm from the primary knock-on atom to the grain boundary, scarcely any atomic displacement cascades intersect the grain-boundary region
Fig. 10. Number of clusters produced by displacement cascades along the crystallographic direction [100] versus energy of the primary knockon atom Åpka and size of the cluster
(Fig. 9). A similar situation is observed for a crystallite with the grain boundary 617. The calculations show that the number of stable defects (point defects and clusters thereof) formed late in the relaxation stage depends on the distance of the primary knock-on atom from the grain boundary. The greater the distance, the smaller the number of defects generated in the crystallite. As the distance from the grain boundary increases, the number of defects tends to a value characteristic of the material with ideal structure. An important feature of the radiation-affected zone is the number of point-defect clusters (vacancies and intrinsic interstitial atoms) formed there. A cluster is considered to be a group of point defects where the separation between the nearest neighbors is smaller than the radius of the second coordination sphere in the perfect vanadium lattice. The calculations show that a relatively small number of clusters is formed in the crystallite in the greater part of the relaxation stage. Note that clusters made up of three defects account for a high proportion of the total number of clusters. Analysis of these clusters shows them to be dumbbells oriented along the crystallographic direction [111]. This is inherent both to crystallites with ideal structure (Fig. 10) and to those with grain boundaries (Fig. 11).
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
27
à
b
c
d
Fig. 11. Distribution of clusters versus size of clusters in the cooling stage of the evolution of a displacement cascade for a primary knock-on atom at a distance of 1.81 (à) and 3.87 (b) from the grain boundary 613 and at a distance of 1.84 (c), 3.87 nm (d ) from 617
It follows from an analysis of cluster configurations shown in Fig. 11(a) that the cluster of dimensionality 2 is a bivacancy. Among the clusters of dimensionality 3, there is one consisting of vacancies alone. This is a vacancy complex. The other clusters of dimensionality 3 are dumbbell configurations made up of two interstitial atoms and a vacancy, with most of the clusters lying in the grain-boundary region and having the same spatial orientation. All clusters of higher dimensionalities are mixed configurations (vacancies plus interstitial atoms). Figure 11(b) shows two clusters of dimensionality 2 (bivacancies). In this case, dumbbell configurations made up of two interstitials and a vacancy lie predominantly in the bulk of the crystallite. Two clusters of dimensionality 4 are vacancy complexes and the third type of clusters is mixed configurations. In the grain-boundary region, there is a large cluster consisting of 40 point defects. It should be noted that the relaxation stage of the evolution of displacement cascades in crystallites with grain boundaries is characterized by formation of one or two clusters of fairly large size in the grain-boundary region. The closer the primary knock-on atom to the grain boundary, the larger the size of the cluster formed there. This is readily seen from a comparison of Fig. 11(a and b) and Fig. 11(c and d ).
5. Conclusions Extended grain boundaries substantially affect the evolution pattern of atomic displacement cascades produced under irradiation of materials. Grain interfaces accumulate most of radiation-induced defects and defect clusters of fairly large size that can appreciably interfere with propagation of atomic displacement cascades through grain boundaries. There is a threshold energy-dependent distance of the primary knock-on atom from the interface. Starting with this threshold distance, this type of grain boundary becomes an insurmountable obstacle for atomic displacement cascades generated by a primary knock-on atom of lower energy. The work was supported by Federal Target Program (State contracts Nos. 02.513.11.3129 and 02.513.11.3200). References [1] E. Àlonso, M.-J. Caturla, T. Diaz de la Rubia and J.M. Perlado, Simulation of damage production and accumulation in vanadium, J. Nuclear Materials, 276, No. 13 (2000) 221. [2] A. Souidi, C.S. Becquart, C. Domain, D. Terentyev, L. Malerba, A.F. Calder, D.J. Bacon, R.E. Stoller, Yu.N. Osetsky and M. Hou, Dependence of radiation damage accumulation in iron on underlying models of displacement cascades and subsequent defect migration, J. Nuclear Materials, 355, No. 13 (2006) 89.
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[3] D.J. Bacon, F. Gao and Yu.N. Osetsky, Computer simulation of displacement cascades and the defects they generate in metals, Nucl. Instrum. Meth. in Phys. Res. B, 153, No. 1 4 (1999) 87. [4] Yu.N. Osetsky and D.J. Bacon, Defect cluster formation in displacement cascades in copper, Nucl. Instrum. Meth. in Phys. Res. B, 180, No. 1 4 (2001) 85. [5] F.J. Pérez-Pérez and R. Smith, Modelling radiation effects at grain boundaries in bcc iron, Nucl. Instrum. Meth. in Phys. Res. B, 153, No. 1 4 (1999) 136. [6] K. Sugio, Y. Shimomura and T.D. de la Rubia, Computer simulation of displacement damage ñascade formation near sigma 5 twist boundary in silver, J. Phys. Soc. Jpn., 67, No. 3 (1998) 882. [7] Yu.R. Kolobov, G.P. Grabovetskaya, K.V. Ivanov and N.V. Girsova, Effect of grain boundary state and grain size on creep mechanisms in
ultrafine-grained nickel, Fiz. Met. Metalloved., 91, No. 5 (2001) 107 (in Russian). [8] Yu.R. Kolobov, R.Z. Valiev, G.P. Grabovetskaya et al. Grain Boundary Diffusion and Properties of Nanostructured Materials, Nauka, Novosibirsk, 2001 (in Russian). [9] M. Satou, S. Yip and K. Abe, Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties, J. Nucl. Mater., 307311, P. 2 (2002) 1007. [10] Handbook on Physical Quantities, Ed. by I.S. Grigoriev and E.Z. Meilikhov, CRC Press, Boca Raton, 1997. [11] A. Suzuki and Y. Mishin, Atomistic modeling of point defects and diffusion in copper grain boundaries, Interface Science, 11, No. 1 (2003) 131.
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
Atomic collision cascades in vanadium crystallites with grain boundaries S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich, A.V. Zheleznyakov and V.M. Chernov1 1
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia Bochvar All-Russia Research Institute for Inorganic Materials, Moscow, 123060, Russia
The effect of grain boundaries on the special features inherent to the evolution of atomic collision cascades and formation of radiationaffected regions in vanadium crystallites was investigated. The presence of grain boundaries in the material was found to have a considerable impact on the radiation damage pattern, specifically on the radiation-induced defect distribution and size. Grain boundaries were shown to serve as a barrier to propagation of atomic displacement cascades and to accumulate most of radiation-induced defects. Relaxed vanadium crystallites contained a relatively small number of clusters made up of point defects, i.e., vacancies and intrinsic interstitial atoms. Keywords: radiation-induced defects, molecular dynamics, grain boundaries
1. Introduction Investigations into structural damage to materials under irradiation are of particular interest for long-term predictions of changes in their mechanical properties. Primary evidence for the radiation effect is generation of atomic displacement cascades responsible for radiation-induced structural defects, formation of microstructures, and changes in the physical-mechanical properties of materials. Simulation of atomic displacement cascades, analysis of the resulting defects, investigation into their evolution and formation of relatively stable radiation-induced defects, as a rule, are performed for materials with ideal structure [1 4]. It is known, however, that extended interfaces may have a considerable impact on many of the properties of materials [58]. Experimental investigations into generation of atomic displacement cascades are extremely difficult to perform. Because of this, computer simulations appear to be the most suitable method for deriving information of primary radiation processes developing in materials under irradiation. We have performed a molecular-dynamics simulation of the effect of grain boundaries on the evolution pattern of atomic displacement cascades and formation of radiation
* Corresponding author Prof. Konstantin P. Zolnikov, e-mail: [email protected] Copyright © 2009 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2009.03.003
induced defects in vanadium crystallites. To this end, symmetrical tilt grain boundaries with different energies were chosen [5]. Vanadium crystallites were picked up because vanadium-base alloys exhibit the most promise for nuclear power engineering. For the purpose in hand, the influence of the distance of a primary knock-on atom from the grain boundary on the evolution pattern of displacement cascades in a vanadium crystallite was investigated. 2. Computational formalism Molecular-dynamics calculations were performed, using the Lammps code (http://lammps.sandia.gov) developed in National Sandia Laboratories, USA. Interatomic interactions in vanadium crystallites were described by means of an interatomic potential of the FinnisSinclair type placed at our disposal by M.I. Mendeleev (Ames Laboratory, USA). The potential used is of importance for a simulation of atomic collision cascades and enables such characteristics as the lattice parameter, elastic moduli, the energy of point defect formation and migration and so forth to be described with fairly high accuracy (Table 1). The simulated crystallites exhibited symmetrical tilt boundaries of two types: 613 (320)[001] and 6 17 (410)[001], allowing the use of periodic boundary conditions [11] employed for the simulation of atomic displacement cascades in this work. The choice of the simulated crystallite size
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
depended on the primary knock-on atom energy Åpka. In the calculations, the number of atoms in the computational grid was varied between 65 000 (for Epka < 500 eV) and 450 000 (for Epka > 500 eV). The minimum time it takes to initiate the evolution of a displacement cascade (for Epka < 50 keV) and formation of stable defects is usually 1015 ps. The minimum time increases moderately as Epka is further increased. By and large the computation time is determined by three parameters: primary knock-on atom energy, crystallite size and interatomic interaction potential. In the cascade evolution simulation, the integration time was changed three times. In the cascade nucleation stage, the integration time was 1017 s, whereas in the relaxation stage it was 1016 s. In the stable defect evolution stage, the integration time was 1015 s. Using special computer software for visualization of the structure and fields of the physical quantities involved, the evolution of damage zones was examined to reveal possible overlaps of different parts of the displacement cascade. The analysis was based on general assumptions about overlaps of parts of the cascade, because no unambiguous criterion for identifying a cascade overlap has hitherto been established. The crystallite damage analysis was performed with allowance made for the number of Frenkel pairs and population of clusters formed by a displacement cascade. Frenkel pair is taken to mean an interstitial atom and a closest stable vacancy. There are two ways of identifying positions of vacancies and interstitial atoms in the simulated crystallite [2]: (1) to look at the occupancy of spheres of a certain radius by displaced atoms and (2) to use WignerSeitz cells. The radius of the sphere (center of the sphere coincides with the lattice site) was chosen to be 0.3 of the vanadium crystal lattice parameter. If the atom leaves the sphere or WignerSeitz cell, it is considered to be a displaced atom. If there are no atoms inside the sphere or cell, a vacancy is believed to form there. If there are two or more atoms inside the sphere or cell, the situation corresponds to an interstitial atomic configuration. Note that both of the ways provide fairly close results for considering the number of structural defects at a certain instant of time [2]. In the simulation of atomic displacement cascades, point defects (vacancies and intrinsic interstitial atoms) may occur in the immediate vicinity of one another. If the distance between any closest point defects is smaller than a certain threshold value, they are said to belong to the same cluster. However, there is no clear criterion for determining the threshold distance. It is suggested here that the threshold distance be the radius of the second coordination sphere in the perfect vanadium lattice [2]. Then the sum of stable vacancies and interstitial atoms (intrinsic interstitials) in a cluster is calculated. Notably, the number of vacancies coincides with that of interstitial atoms.
21 Table 1
Properties of vanadium calculated with the use of MIM* potential Properties
Experiment
Lattice parameter, nm
Calculation (ÌIÌ)
0.3039 [9]
0.3030
Binding energy, eV/atom
5.31*
5.016
Elastic modulus C11 , GPa
229 [9]
228
Elastic modulus C12 , GPa
119 [9]
119
Elastic modulus C44 , GPa
43 [9]
42
Vacancy formation energy
Efv , eV
2.48*
2.49
Vacancy migration energy + vacancy formation energy Emv Efv , eV
3.19*
3.27
Interstitial formation energy Efi ( 100 ), eV
3.22*
3.21
Interstitial formation energy Efi ( 110 ), eV
3.09*
3.09
2.81*
2.81
2.81*
2.83
2190 [10]
3119
Interstitial formation energy Efi ( 111 ), eV Crowdion formation energy eV/atom
E fc,
Melting temperature Tm, K * M.I. Mendeleev, private communication
3. Atomic displacement cascades in the perfect vanadium lattice Before generation of atomic displacement cascades, the simulated vanadium crystallites were relaxed at 10 K. As soon as the primary knock-on atom was initiated, a chain of atomic displacements began to develop in the crystallite. The number of displaced atoms and the size of the radiationaffected zone increased simultaneously until the energy transferred by the primary knock-on atom to its nearest neighbors was distributed over the bulk of the simulated crystallite, so that no atoms with energy above that of the threshold displacement was left. This stage in the development of the cascade is called ballistic. Late in this stage, the
Fig. 1. Peak time versus primary knock-on atom energy Åpka
22
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
b
Fig. 2. Number n of point defects versus Å pka at the peak time (a) and in the cooling stage (b)
number of defects produced by atomic displacements is at its maximum. Then a recombination (relaxation) stage begins, where the number of defects reduces down to a certain relatively stable value after which only diffusion processes will change the number and distribution pattern of the defects in the crystallite. It is to be noted that the kinetic temperature in the recombination stage reaches high values (thermal peak) in the cascade region. Late in the ballistic stage, the essential characteristics of the displacement cascade are as follows: (1) the maximum number of the defects produced, (2) the peak time (time it
takes to generate a maximum number of defects), (3) the volume of the region where a cascade develops and (4) the cascade density (number of defects per unit volume of the cascade region at the peak time). These characteristics depend on the primary knock-on atom energy and, generally speaking, on the crystallite temperature. The latter is ignored here. The peak time as a function of cascade energy is presented in Fig. 1. As can be seen from the figure, the curve slope decreases as the energy increases. This behavior is likely to be due to splitting of the main cascade into subcas-
à
b
c
d
Fig. 3. Point defects in a simulated crystallite containing 450 000 atoms at different instants of time: 0.1 (à), 0.2 (b), 0.4 (c), 2 ps (d ). Gray and black circles correspond to vacancies and interstitials, respectively. The initial direction of the primary knock-on atom is along [320] and the primary knockon atom energy is 1 keV
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
23 b
)
)
[410]
[320]
[001]
[001]
[230]
[140]
Fig. 4. Projections of symmetrical tilt grain boundaries onto the plane (001) for the grain boundaries 613(à) and 617(b). Shaded are periodicity regions along a grain boundary. Black and gray circles show atoms lying in adjacent atomic planes
cades initiated by secondary knock-on atoms with energy below that of the primary knock-on atom. The secondary knock-on atoms generate cascades with shorter peak time compared to the main cascade and, on the whole, decrease the peak time. The maximum number of defects in the vicinity of an atomic displacement cascade as a function of energy of the primary knock-on atom is shown in Fig. 2(a). The relation is near-linear, with the number of defects in the cascade increasing with increase in the energy of the cascade. However, the number of defects produced in a relaxed crystallite is a nonlinear function of energy of the primary knock-on atom (Fig. 2(b)). The decrease in the rate of growth of the curve with increase in the primary knock-on atom energy (Fig. 2(b)) is attributable to the splitting of the main cascade into subcascades. The relative position of point defects in the simulated vanadium crystallite in different evolution stages of an atomic displacement cascade is shown in Fig. 3. The initial velocity of the primary knock-on atom was oriented along the crystallographic direction [320]. The primary knockon atom energy was 1 keV. Figure 3(a and b) corresponds to the ballistic stage of the evolution of the cascade. Figure 3 (c and d) illustrates the recombination stage and the cooling stage of the evolution of radiation damage, respectively.
4. The influence of grain boundaries on the formation of atomic displacements in a cascade Symmetrical tilt or twin grain boundaries formed from two crystals of bilateral symmetrical structure are the simplest configuration of all grain boundaries. Table 2 gives parameters of 6 types of symmetrical tilt grain boundaries formed in bcc crystals: directions of rotation axes, angles )
Fig. 5. Energy of symmetrical tilt grain boundaries in a vanadium crystallite versus the tilt angle ) before (open circles) and after relaxation (full circles)
Table 2
Symmetrical tilt grain boundary parameters in vanadium crystallite Normals to planes (320)(230)
Rotation axis
(530)(350) (210)(120) (310)(130)
[001]
)
6
22.62q
13
28.07q
17
36.87q
5
53.13q
5
(410)(140)
61.93q
17
(510)(150)
67.38q
13
Fig. 6. Mean potential energy per atom Eð along the Y-direction normal to the 613 grain boundary plane
24
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 Table 3
The distance of the primary knock-on atom from the symmetrical tilt grain boundary Type of grain boundary Distance of the primary knock-on atom from the grain boundary, nm Direction of primary knock-on atom velocity
(320)[001] 613
(410)[001] 617
[320]
[410]
1.04, 1.81, 2.65, 3.16, 3.87, 4.42
of rotation of one grain relative to another, and the number 6 of coincidence lattice sites. Figure 4 shows projections of unrelaxed crystallite structures with the grain boundaries 613 and 617 onto the plane (001). During relaxation of the initial bicrystal, the atoms near the grain boundary are found to experience the largest displacements. It is from the atomic displacement that the grain-boundary width can be estimated. Atoms away from this region retain bilateral symmetrical order relative to the boundary. Bicrystals with different types of grain boundaries were relaxed at room temperature to produce stable structure. The grain-boundary energy ÅGB before and after relaxation is presented in Fig. 5. These calculations show no correlation between the grain-boundary energy and tilt angle. The grain boundary (410) [001] 617 exhibits the highest energy, whereas the lowest energy is obtained for the grain boundary (320) [001] 613. That is why these grain boundaries were chosen to investigate the effect of extended interfaces on the evolution pattern of atomic displacement cascades and subsequent formation of radiation-affected regions. The grain-boundary width is generally several atomic layers. Due to disordering of these layers, many of the physical properties (thermal expansion, electrical resistivity, elastic moduli and other characteristics) near the grain boundary may differ essentially from the bulk properties upon relaxation and exhibit a high degree of anisotropy. The grainboundary width can be determined from calculations of changes in the potential atomic binding energy Åð in the à
1.02, 1.84, 2.57, 3.24, 3.87, 4.53
direction normal to the grain-boundary plane (Fig. 6). The plot was constructed by breaking the simulated crystallite into thin layers normal to the grain-boundary plane and the atomic binding energy per atom inside a layer was averaged. The binding energy per atom for the perfect vanadium lattice at 10 K was 5.014 eV. The grain-boundary width was found from the condition that the binding energy inside the boundary differs from its value for the perfect lattice and as a rule is higher by more than 0.01 eV. An atomic displacement cascade begins to develop at the instant the primary knock-on atom is generated. To gain insight into the influence of grain boundaries on the evolution of displacement cascades, we examined the distance of the primary knock-on atom from the interface. The energy of the primary knock-on atom was 1 keV in all cases studied and the direction of the velocity of the primary knock-on atom was chosen to be normal to the grain-boundary plane. The distances of the primary knock-on atom from the grainboundary plane are listed in Table 3. The special features of the evolution of atomic displacement cascades in materials with grain boundaries are in many ways similar to the processes involved in materials with ideal structure. In particular, the largest number of defects in both cases is generated within half a picosecond (Fig. 7). Over this time interval, the excess energy of the primary knock-on atom is transferred to the simulated crystallite. Note that the energy of the crystallite exhibits the highest fluctuations. Thereafter, the fluctuations and the number of defects produced decrease. Within 3 4 ps of
b
Fig. 7. Temporal variations of the number of point defects. The crystallite contains the grain boundaries 613 (à). The upper curve corresponds to the primary knock-on atom at a distance of 1.04 nm from the grain boundary and the lower curve is for a distance of 3.16 nm. The sample contains the grain boundaries 617 (b). The upper curve corresponds to the primary knock-on atom at a distance of 1.84 nm from the grain boundary and the lower curve is for a distance of 3.87 nm. The energy of the primary knock-on atom is 1 keV. The direction of the atomic velocity is normal to the grain boundary plane
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
b
[230]
[230] [320]
[320]
GB
GB
c
d
[230]
[230] [320]
25
[320]
GB
GB
Fig. 8. Projection of defect structure onto the plane (001) at different instants of time: 0.06 (à), 0.2 (b), 1 (c) and 10 ps (d ). Gray and black circles correspond to vacancies and interstitials, respectively. The distance from the primary knock-on atom to the grain boundary 613 is 1.81 nm. GB is the grain boundary
generation of a displacement cascade, the number of radiation-induced defects is stabilized. Temporal variations of the number of defects in displacement cascades for a crystallite with grain boundaries are shown in Fig. 7. The variation pattern for the number of defects (total number of defects at the peak time, curve falloff rates, total number of stable defects late in the simulations, and corresponding characteristic times) in crystallites with different types of grain boundaries are much the same. Figures 8 and 9 show different evolution stages of the simulated displacement cascades in vanadium crystallites with grain boundaries. The figures correspond to crystallites with grain boundaries 613 for the primary knock-on atom lying at a distance of 1.81 and 3.87 nm from the grainboundary plane, respectively. The direction of the initial velocity of the primary knock-on atom coincides with the
crystallographic direction [320]. For the primary knock-on atom lying at a distance of 1.81 nm from the grain-boundary plane, most of atomic displacements responsible for the formation of point defects are localized in the grain-boundary region. Note that a considerably smaller fraction of point defects, including the relaxation stage, passes through the grain boundary. This behavior of the atomic displacement cascade is due to the fact that in the grain-boundary region, defects are generated both by rearrangement of grainboundary atoms and by penetration of interstitial atoms into this region from the outside. For a distance of ~1.83 nm from the primary knock-on atom to the grain boundary, some 90 % of the defects produced lie in the grain-boundary region, whereas for a distance of ~3.8 nm, the number of defects reduces down to 55 % for both types of the grain interface (Table 4). Table 4
Influence of the distance of the primary knock-on atom from the grain boundary on the number of defects in the grain-boundary region 613
Type of grain-boundary
617
Distance of the primary knock-on atom from the grain boundary, nm
1.8
3.9
1.8
3.9
Relative number of defects in the grain-boundary region in an as-relaxed displacement cascade, %
89
54
86
55
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S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028 à
[230]
[230] [320]
b
[320]
GB
GB
c
[230]
[230] [320]
d
GB
[320]
GB
Fig. 9. Projection of defect structure onto the plane (001) at different instants of time: 0.06 (a), 0.2 (b), 1 (c) and 10 ps (d ). Gray and black circles show vacancies and interstitials, respectively. The primary knock-on atom is at a distance of 3.87 nm from the grain boundary 613
Starting with a distance of ~3.8 nm from the primary knock-on atom to the grain boundary, scarcely any atomic displacement cascades intersect the grain-boundary region
Fig. 10. Number of clusters produced by displacement cascades along the crystallographic direction [100] versus energy of the primary knockon atom Åpka and size of the cluster
(Fig. 9). A similar situation is observed for a crystallite with the grain boundary 617. The calculations show that the number of stable defects (point defects and clusters thereof) formed late in the relaxation stage depends on the distance of the primary knock-on atom from the grain boundary. The greater the distance, the smaller the number of defects generated in the crystallite. As the distance from the grain boundary increases, the number of defects tends to a value characteristic of the material with ideal structure. An important feature of the radiation-affected zone is the number of point-defect clusters (vacancies and intrinsic interstitial atoms) formed there. A cluster is considered to be a group of point defects where the separation between the nearest neighbors is smaller than the radius of the second coordination sphere in the perfect vanadium lattice. The calculations show that a relatively small number of clusters is formed in the crystallite in the greater part of the relaxation stage. Note that clusters made up of three defects account for a high proportion of the total number of clusters. Analysis of these clusters shows them to be dumbbells oriented along the crystallographic direction [111]. This is inherent both to crystallites with ideal structure (Fig. 10) and to those with grain boundaries (Fig. 11).
S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich et al. / Physical Mesomechanics 12 12 (2009) 2028
27
à
b
c
d
Fig. 11. Distribution of clusters versus size of clusters in the cooling stage of the evolution of a displacement cascade for a primary knock-on atom at a distance of 1.81 (à) and 3.87 (b) from the grain boundary 613 and at a distance of 1.84 (c), 3.87 nm (d ) from 617
It follows from an analysis of cluster configurations shown in Fig. 11(a) that the cluster of dimensionality 2 is a bivacancy. Among the clusters of dimensionality 3, there is one consisting of vacancies alone. This is a vacancy complex. The other clusters of dimensionality 3 are dumbbell configurations made up of two interstitial atoms and a vacancy, with most of the clusters lying in the grain-boundary region and having the same spatial orientation. All clusters of higher dimensionalities are mixed configurations (vacancies plus interstitial atoms). Figure 11(b) shows two clusters of dimensionality 2 (bivacancies). In this case, dumbbell configurations made up of two interstitials and a vacancy lie predominantly in the bulk of the crystallite. Two clusters of dimensionality 4 are vacancy complexes and the third type of clusters is mixed configurations. In the grain-boundary region, there is a large cluster consisting of 40 point defects. It should be noted that the relaxation stage of the evolution of displacement cascades in crystallites with grain boundaries is characterized by formation of one or two clusters of fairly large size in the grain-boundary region. The closer the primary knock-on atom to the grain boundary, the larger the size of the cluster formed there. This is readily seen from a comparison of Fig. 11(a and b) and Fig. 11(c and d ).
5. Conclusions Extended grain boundaries substantially affect the evolution pattern of atomic displacement cascades produced under irradiation of materials. Grain interfaces accumulate most of radiation-induced defects and defect clusters of fairly large size that can appreciably interfere with propagation of atomic displacement cascades through grain boundaries. There is a threshold energy-dependent distance of the primary knock-on atom from the interface. Starting with this threshold distance, this type of grain boundary becomes an insurmountable obstacle for atomic displacement cascades generated by a primary knock-on atom of lower energy. The work was supported by Federal Target Program (State contracts Nos. 02.513.11.3129 and 02.513.11.3200). References [1] E. Àlonso, M.-J. Caturla, T. Diaz de la Rubia and J.M. Perlado, Simulation of damage production and accumulation in vanadium, J. Nuclear Materials, 276, No. 13 (2000) 221. [2] A. Souidi, C.S. Becquart, C. Domain, D. Terentyev, L. Malerba, A.F. Calder, D.J. Bacon, R.E. Stoller, Yu.N. Osetsky and M. Hou, Dependence of radiation damage accumulation in iron on underlying models of displacement cascades and subsequent defect migration, J. Nuclear Materials, 355, No. 13 (2006) 89.
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