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Atomic-scale modelling of primary damage and properties of radiation defects in metals
Nuclear Instruments and Methods in Physics Research B 202 (2003) 31–43 www.elsevier.com/locate/nimb
Atomic-scale modelling of primary damage and properties of radiation defects in metals Yu.N. Osetsky *, D.J. Bacon Materials Science and Engineering, Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
Abstract Considerable success has been achieved in recent years in the understanding of radiation damage production in highenergy displacement cascades, the properties of the defects and evolution of radiation damage in metals. Two main reasons form the basis of this success. First, the significant increase in computing power has allowed simulation of realistic cascade energies with good statistics and relatively long-time evolution of defects to be carried out. Second, new experimental findings and corresponding theoretical calculations have allowed interpretation of a number of mechanisms and phenomena crucial for understanding and prediction of practically important radiation effects, such as void swelling, radiation growth, matrix hardening and plastic flow localisation. In this paper we review the most significant results in atomic-scale computer modelling related to these issues, mainly focusing on new achievements such as the formation of extended defect clusters, the dynamic properties of defect clusters, interaction between radiation defects and strengthening of material due to radiation defects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ah; 78.20.Bh; 78.20.Ap Keywords: Computer modelling; Multiscale modelling; Atomic-scale; Defect clusters; Displacement cascades; Radiation damage
1. Introduction Computer modelling of mechanisms and processes of radiation damage in metals has been one of the fastest-growing areas of materials research over the past decade and, due to growing needs of electricity production, either by extended life of existing nuclear plants or the possible generation of new installations, continues to expand. Several reasons for this are not hard to recognise. First,
*
Corresponding author. Tel.: +44-151-794-5773/4662; fax: +44-151-794-4675. E-mail address: [email protected] (Yu.N. Osetsky).
the new paradigm in study of materials, so-called multiscale materials modelling (MMM), has been developed and used in many applications of materials science over the past few years. The main idea of MMM is rather simple and involves decomposition of the whole physical phenomenon into stages linked in time and length scales, each to be studied by a particular technique. The time and length scales of radiation effects ranges from picoseconds to years and from nanometres to metres respectively, covering thereby 22 and 9 orders in magnitude of time and length respectively. The techniques employed, ranging from electronic and atomic interactions via first-principles ab initio calculations to modelling of behaviour via
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)01827-X
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finite elements methods, have been intensively developed, and their sophistication is such that many of the properties of materials can now be simulated with a high level of realism. Second, computer modelling enables some phenomena that are not amenable to experimental study to be investigated. Third, success in the first two aspects has been realised because computer power has grown enormously, thereby allowing many techniques, which are examples of purely theoretical exercises from the past, to be used now quite routinely. In the MMM approach to radiation effects the place of atomic-scale modelling (ASM) is to link atomic and continuum scales, and includes studies of primary defect production, defect properties, reactions between defects and mechanisms of interactions affecting the mechanical properties, e.g. interactions with moving dislocations. ASM also provides mechanisms and parameters necessary for higher scale levels; e.g. Monte Carlo and rate theory studies of microstructure evolution, and continuum modelling of dislocation dynamics for estimating the change in mechanical response due to microstructural evolution. In this paper we present an overview of recent results of atomicscale modelling in these areas. An important issue in any atomic-scale model is the validity of the interatomic potential (IAP), for the IAP determines all the properties of the simulated system. We refer to reviews of the subject [1–3], and only mention that the most commonly used IAPs are based on either the long-ranged pair potentials [4] created within the generalized pseudopotential model by Moriarty [5] or the many-body functions of the embedded atom model [1,3] and the Finnis–Sinclair approach [6]. Other more rigorous types, such as tight-binding models, bond-order potentials and general manybody potentials, are not being used extensively in the large-scale atomic modelling, for this would require too high computational resource.
2. Primary damage The study of primary defect production in displacement cascades due to high-energy particle irradiation is one of the most developed applica-
tion of ASM, particularly via molecular dynamics (MD) technique. Typical MD simulation of displacement cascades involves crystallites containing up to few million atoms, with periodic boundary conditions and, sometimes, with the application of electron–phonon coupling [7] or/and with a thermal bath on the crystal periphery [8,9]. Cascades of primary-knock-on-atom (PKA) energy, EPKA , up to 50 keV have been simulated in bcc, fcc and hcp metals over a wide temperature range [7–22]. The cascade region is usually simulated until its temperature decreases to the ambient, though in some cases short-term post-cascade annealing has been modelled to investigate the behaviour of clusters over a few hundred picoseconds [13,22]. The standard output of cascade simulations includes the total number of the surviving vacancies and self-interstitial atoms (SIAs) at the end of cascade cooling phase, the fraction of vacancies and SIAs formed in clusters (ev , ei ), the size distribution and structure of these clusters (sessile, glissile, Burgers vector) and atomic displacements (for characterisation of mixing). A few tens of events at each condition (PKA energy and direction, temperature) should be simulated to achieve statistically representative results [13,22]. Extensive information has been obtained during the last decade on the nature of primary damage in high-energy displacement cascades (HEDCs) in the bulk of pure bcc [9,10,21], fcc [7,8,11–13,22,23] and hcp [14–16] metals, some binary (Fe–Cu [17]) and ordered (Ni3 Al [18,19]) alloys, and near metal surfaces [15,16,20]. The basic processes in metals have been clarified and the results are in overall agreement with experimental observations. 2.1. Defect production One of the most significant findings of atomistic modelling is that an extensive amount of recombination of SIAs and vacancies occurs during the cooling down stage of HEDCs. As a consequence, in all metals and alloys modelled so far, the total number, NF , of point defects (vacancy or SIA) surviving at the end of the cascade process is significantly lower than the number of atomic displacements, NNRT , predicted by the NRT model [24]. The example for Cu simulated at 100 K with
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two different IAPs [22] and high statistics of treated cascades (from 25 to 40 cascades for each energy) is presented in Fig. 1. The defect production efficiency, NF =NNRT , depends strongly on PKA energy, EP , less on material and weakly on ambient temperature [21] as shown in Fig. 2 where a
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treatment within the commonly accepted formum lation NF ¼ AðEP Þ is presented for a number of metals. In general the parameter A is lower than the corresponding coefficient in the NRT formula [24] and the exponent m is lower than unity. In general the MD data demonstrate cascade efficiency 0.2 of NRT estimations. 2.2. Intracascade clustering
Fig. 1. Number, NF , of vacancy–SIA pairs surviving at the end of the cooling stage of displacement cascades of energy EP in Cu, normalised per the NRT-predicted number, NNRT , simulated with many-body (MBP) and long-range pair (LRPP) potentials in [22]. Symbols indicate the mean and bars the standard error.
Fig. 2. Total number of surviving defects (vacancies or interstitials) versus PKA energy. Data obtained by different authors with different IAPs. The solid line presents the NRT estimate for an average displacement threshold energy of 40 eV.
The occurrence of intracascade clustering is another important finding of atomistic modelling. The collapse in cascades of vacancy clusters into dislocation loops was demonstrated in some metals on the basis of low temperature experiments [25] and used in theoretical treatment of void swelling [26]. The direct formation of SIA clusters in cascades was predicted by both MD modelling of HEDCs [11] and diffusion-based theoretical calculations [27] and has been confirmed recently by TEM [28]. Now, the intracascade clustering of vacancies and SIAs has been substantiated in a variety of bcc, fcc and hcp metals using MD simulations. For example, in Cu, each cascade of EP ¼ 10–20 keV yielded one to two SIA clusters of mean size 5–6 defects [22]. The fraction of SIAs in clusters, ei , was found to be in the range of 50– 60%. The majority of SIA clusters have the form of glissile perfect dislocation loops. Clusters in sessile configuration have also been observed in MD simulations, to an extent which is metal-dependent. The fraction is lowest (a few per cent) in bcc metals, where it takes the form of metastable three-dimensional (3-D) clusters [29], and highest (up to 30–40%) in low stacking fault energy fcc metals, in the form of faulted Frank dislocation loops [13,22]. The sessile fraction in hcp metals lies between these extremes. Production of vacancy clusters is strongly dependent on the crystal structure. Thus large compact clusters are observed only in simulations of fcc and hcp metals. In fcc Cu they are perfect or truncated stacking fault tetrahedra (SFTs) [13,22,30–32] while in hcp Zr they are planar defects similar to prism or basal dislocation loops [14,15,33]. No large compact planar or 3-D vacancy clusters have been observed in bcc metals. Instead, very loose complexes of vacancies in
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Fig. 3. The fraction of vacancies and interstitials surviving in clusters containing P 3 defects versus PKA energy in Cu [22].
second and higher order neighbour shells have been reported [10,34]. The best statistics of cascade defects has been obtained for Cu [13,22] where, on average, each cascade of 10–20 keV PKA energy produces one cluster of about five vacancies, and in total 25% of surviving vacancies are in clusters. Data on the dependence of intracascade clustering on energy and temperature in Cu is presented in Fig. 3 [22]. 2.3. Temperature effects No significant effect of temperature on the total number of defects produced in cascades has been reported, but there is a statistically significant effect on defect clustering [9,16,18,21,22]. For example, in Cu the number of vacancy and interstitial clusters per cascade decreases at high temperature while the number of defects per cluster increases (see Fig. 3). The difference between vacancy and interstitial clustering was found to be in the temperature dependence of the defect fraction contained in clusters: ei increases with temperature while ev is rather insensitive.
3. Defect cluster properties Recognition that intracascade clustering of both SIAs and vacancies is substantial has initiated ex-
tensive investigation into the atomic-level properties of such clusters. Two different approaches are employed. Static modelling – in which the atoms have no kinetic energy and equilibrium is computed by minimisation of the crystal potential energy – is used to study the energy and structure of clusters, their interaction with other defects and stress and displacement fields. SIA and vacancy clusters and dislocation loops containing up to several hundred point defects have been simulated in crystallites containing up to a few million atoms using periodic or fixed boundaries [35–47]. Dynamic modelling by MD – in which the crystal has non-zero temperature – is used to study transport properties and mechanisms of cluster motion in periodic crystallites. These simulations require smaller crystallites (up to a hundred thousand atoms) but last up to several tens nanoseconds. Motion of clusters of up to a few hundred SIAs and vacancies [39,41–43,47–51], and statics and dynamics of cluster–cluster [44,52–54] and cluster-point defect [55–57] interactions have been considered in bcc, fcc and hcp metals. 3.1. Vacancy clusters Stability of vacancy clusters is strongly dependent on crystal structure. The most stable vacancy clusters in fcc Cu are in the form of stacking-fault tetrahedra [37,41], which in MD modelling are stable up to 1000 K for at least a few ns. Faulted Frank vacancy dislocation loops (Burgers vector b ¼ 13h1 1 1i) in Cu are also very stable and tend to dissociate into SFTs. An example of stability of vacancy clusters in Cu from [42] is presented in Fig. 4(a). Studies of hcp Zr [36,45] show that planar vacancy clusters in the form of perfect dislocation loops in prism planes (b ¼ 13h1 1 2 0i) and faulted loops in the basal plane (b ¼ 16h2 0 2 3i) are stable. The situation in bcc metals, particularly in Fe, is more complex. Small clusters (<30 vacancies) can be metastable in the form of planar or 3-D, void-like configurations, and during high temperature annealing transform into cloud-like complexes which are mobile [35,59]. Larger clusters can be stable as compact 3-D voids or perfect edge dislocation loops (b ¼ 12h1 1 1i or h1 0 0i) [34,35,40].
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surface. The growth of large SFTs is suppressed due to difficulties in propagation of long jogs, and this limits the range of size distribution to a few nanometres. This mechanism explains experimental observation that SFT size distribution is independent of the irradiation conditions [60]. 3.2. Interstitial clusters The structure and energy of perfect and faulted interstitial dislocation loops in Cu and Zr, and perfect loops in Fe have been investigated by static modelling [38–40,43–47]. Furthermore, the thermally activated 1-D glide of perfect dislocation loops containing up to a few hundred SIAs, i.e. up to a few nanometres in diameter, has been studied by MD [39–41,44,48–52]. The following conclusions can be drawn.
Fig. 4. Binding energy of SIA and vacancy clusters in Cu [4,55]. Circles – faulted 13h1 1 1i Frank loops, diamonds – rhombusshaped 12h1 1 0i perfect loops, triangles – SFTs.
Attempts have been made to study nucleation of vacancy clusters in different metals under conditions of high vacancy supersaturation. It was demonstrated that vacancy-rich regions collapse into the stable configurations of SFTs and faulted Frank loops in Cu [30,31,38], basal and prism loops in Zr [34] and perfect loops in {1 1 1} and {1 0 0} planes in Fe [35], depending on the level of supersaturation. There was no evidence for the formation of compact 3-D void-like configurations in any of the crystal structures, which may be related to the limited time-scale of atomistic studies or the impossibility of void nucleation without the help of gaseous impurities. A recent significant achievement related to vacancy clusters provides an explanation of the growth mechanism of SFTs in Cu [42]. It was found that growth of an SFT occurs by stair-rod dislocation jog propagation over the SFTÕs facetted
(a) All types of SIA clusters in all metal structures have very high binding energy, of the order 2– 3 eV/SIA (Fig. 4(b)). They cannot dissociate thermally, but can transform from metastable into stable configurations. (b) The most stable clusters are nuclei of perfect dislocation loops, i.e. Burgers vector 12h1 1 0i in fcc (Cu), 12h1 1 1i and h1 0 0i in bcc (Fe) and 1 h1 1 2 0i in hcp (Zr). These clusters can be de3 scribed as compact planar sets of crowdions with their axis along the Burgers vector direction [38,43,44] and are intrinsically glissile. (c) Sessile, stable clusters are formed in fcc (13h1 1 1i faulted Frank loops) and hcp (12h0 0 0 1i faulted basal loops) metals but do not form in bcc crystals where all stable clusters are intrinsically glissile. Large rhombus-shaped clusters (with sides along h1 1 2i) in Cu and rectangular clusters in Zr (with sides along [0 0 0 1] and h1 0 1 0i directions) can dissociate along their glide prism. Such clusters become sessile due to the formation of stair rod dislocations at their corners [39,43,47,52]. (d) Small perfect dislocation loops perform fast thermally activated 1-D motion. Mobility decreases with increasing size, but the activation energy does not depend on size and is close to that of an individual crowdion of the corresponding type, e.g. 0.02 eV [41,43,48–52].
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The thermally activated motion of clusters is suppressed at large size, e.g. above 300 SIAs for 12h1 1 1i clusters in Fe and 120 SIAs for 1 h1 1 0i clusters in Cu [41,43,52]. 2 (e) Although the thermally activated motion of clusters is unlikely to be described by the conventional diffusion process, attempts have been made to parameterise such motion as correlated 1-D diffusion: the jump frequency of the cluster centre of mass (CCM), correlation factor and effective migration energy have been estimated for clusters containing up to 91 SIAs in Fe, Cu and Zr [41,43,44,50–52]. Qualitatively similar results were obtained in all cases and the example for Fe is presented in Fig. 5. It has been shown that the mechanism of cluster motion can be described as nearly stochastic jumps of crowdions with the probability to move the CCM being inversely proportional to the number of SIAs in the cluster [50]. The effective correlation factor is significantly larger than unity, and clusters preferentially move in one direction over large distance, e.g. up to
Fig. 5. Jump frequency of SIA clusters in Fe at function of temperature T . Tm is the melting temperature and the number of SIAs is indicated on the left of each Arrhenius-like dependence [42,51].
tens of nm over tens of ps. The physical basis for the fast thermally activated glide of SIA clusters is not yet clarified, but the parameters obtained can be used in higher level modelling of microstructure evolution and damage accumulation [60]. 3.3. Interactions where SIA clusters are involved The importance of glissile SIA clusters arises from their high stability and fast 1-D mobility, and also the long-range interactions that can occur along their glide path. The latter include interactions between themselves and with sessile clusters and dislocations, and are therefore of importance for models to predict the microstructural evolution during irradiation under cascade damage conditions. A great variety of reactions has been studied already by atomic modelling and can be separated into two groups. 3.3.1. Reactions To this group we attribute interaction between clusters and point defects, which may lead to growth and shrinkage; interaction between clusters, which may result in coalescence or annihilation, and interactions that can induce changes in clusters properties (e.g. sessile () glissile transformations). Interactions of this type are widely observed in cascade simulations where the probability of such events is increased by the high local concentration of defects. For example, growth, shrinkage and coalescence of glissile perfect SIA clusters due to interactions with point defects (vacancies and SIAs) and other clusters, and the formation of sessile 3-D clusters were observed in simulations of high-energy cascades in Fe [29] and Cu [13,22,23,61]. Due to existence of a stable stacking fault in Cu, the probability of observing sessile SIA clusters is much higher than in Fe. An example where a group of small glissile SIA clusters in a 20 keV cascade in Cu interact to create a faulted defect of SIAs lying in a {1 1 1} plane is presented in Fig. 6 [13,63]. MD study of glissile clusters with intersecting glide prisms in Fe and Cu has shown that the final result depends on cluster size, temperature and
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Fig. 6. Three 12h1 1 0i glissile SIA clusters coalesce to form the nucleus of a sessile 13h1 1 1i Frank loop during simulation of a 20 keV cascade in Cu [13]. The time after the PKA creation is indicated for each configuration. Dark spheres are interstitial atoms, light spheres are vacant sites. The projection direction is close to h1 1 1i.
cluster orientation [45,53]. In general, for small clusters and high temperature, the probability that one of them changes its Burgers vector to coalesce with the other increases. Large clusters may form immobile complexes and the probability for this is higher in Cu than in Fe. Important interactions may occur when the mobility of glissile SIA clusters is reduced or when they become immobile without change of structure. Such effects were observed for 12h1 1 1i SIA cluster in Fe interacting with a vacancy [56] and an impurity atom [62]. In particular, a significant decrease in the cluster mobility occurs when an impurity atom interacts with the cluster and becomes trapped at its edge. This effect has been observed in the simulation of a carbon solute atom and a 12h1 1 1i glissile cluster in Fe [62]. Whilst the cluster became essentially immobile (at 300 K), the carbon atom maintained its mobility around the cluster edge by core diffusion. The mobility of a cluster–solute complex is controlled by the solute, which, at high enough temperature, could be
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dragged by the cluster one-dimensionally towards the latterÕs own sink in the microstructure. The process of SIA cluster growth and shrinkage is material-dependent. In bcc Fe a vacancy can be absorbed by the cluster edge and thus reduce the cluster size [56]. Vacancy–SIA cluster interaction in Cu is more complicated and strongly depends on cluster size [57,58]. A vacancy can recombine with interstitial at the edge of small clusters (<49 SIAs) or simply reduce cluster mobility if it lies inside the glide prism, as in Fe. However, larger clusters do not absorb a vacancy because of dissociation of the bounding dislocations on its glide prism. Although interaction energy between a vacancy and the stacking-fault ribbon formed by dissociation is rather high (up to 1 eV), the vacancy is not absorbed but simply prevents mobility of such clusters [57,58]. The situation in hcp metals, particularly in Zr, falls between of these two extremes [58]. A vacancy can annihilate on a f1 0 1 0g face of an SIA cluster, where dissociation is weak, but behaves as in Cu on a (0 0 0 1) face, which is strongly dissociated. Such peculiarities in SIA cluster–vacancy interaction should lead to a difference in evolution of SIA clusters, i.e. cluster growth is suppressed in bcc metals and is easier in fcc and hcp ones. This may help in understanding the difference in damage accumulation in bcc, fcc and hcp metals (see e.g. [64]). 3.3.2. Creation of specific microstructures Interaction between glissile clusters/loops with parallel Burgers vector, b, leads to formation of spatially extended complexes. Modelling by statics and MD in [53,54] has revealed that 12h1 1 1i SIA clusters in Fe, with no overlapping glide prisms, attract each other and form a stable complex. The nature of the clusters does not change, i.e. they are still glissile, but the mobility of the complex depends on the number and size of the clusters: the complex retains mobility if the number of clusters is small (<5) and the total number of SIAs is <300, but becomes immobile for larger numbers. A complex can grow further, attracting more glissile clusters with the same b, thereby creating a raft of clusters/dislocation loops. The interaction energy obtained in the atomistic modelling was
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compared with that estimated from elasticity theory, and whereas the long-range interaction can be approximated well by the theory, the short-range interaction is stronger in the atomistic model [54]. This type of interaction depends on the crystal lattice structure, for in Cu glissile SIA clusters with b ¼ 12h1 1 0i repel each other at large separation but attract at short distances, creating an immobile complex, even for two small clusters [53]. The reasons for such behaviour are not yet understood and cannot be clarified within the isotropic elasticity theory. Nevertheless, the effects found in modelling are qualitatively consistent with experiments, for rafts of dislocation loops are observed in bcc [64–66] but not fcc metals. Another pertinent interaction is that between a glissile SIA cluster and an edge dislocation when both have the same b. This interaction was studied by atomistic modelling and isotropic elasticity in
[54,55] for b ¼ 12h1 1 1i in Fe and b ¼ 12h1 1 0i in Cu. The situation is more complicated in Cu due to the enhanced dissociation of perfect loop in the stress field of a dislocation. As can be seen in Fig. 7 the interaction energy, EINT , is significantly higher a predicted by elasticity theory at short distances where such dissociation may occur in clusters of size too small to dissociate in isolation (e.g. [43]). In general, these results confirm that, in both Fe and Cu, short-range interactions are too complicated to consider in elasticity theory. However, for long-range interactions, when cluster–dislocation separation is much larger than the cluster size and the defects have little effect on each others properties, elasticity calculations give good results when dislocation parameters such as core energy and radius are estimated from atomic-level modelling. This interaction is very long-range, e.g. in Fe EINT > 0:1 eV for a cluster of 37 SIAs (diameter 1 nm) at a distance of about 170 nm, implying a very effective mechanism for the edge dislocation decoration by glissile SIA loops, as observed in bcc and fee metals [64,67–69].
4. Impact on mechanical properties
Fig. 7. Interaction energy versus distance between an hexagonal 12h1 1 0i 49-SIA loop and a 12h1 1 0i dissociated edge dislocation in Cu. Circles indicate modelling results, solid line full integration of dislocation stress field over the loop and dashed line the infinitesimal loop approximation [54].
This is a relatively new area of atomistic modelling for radiation damage research and involves both static and dynamic approaches. Models employ periodicity along the dislocation line and with the boundaries in the other two directions either fixed, with atoms displaced according to elasticity theory [54,55,70,71], or flexible, with atoms able to respond to unbalanced forces using either lattice or elastic Green functions [72,73]. The static modelling has been used to study interactions between edge dislocations and defect clusters (voids, SFTs, SIA clusters) in bcc and fcc metals. The results of static modelling can be compared with models based on the elasticity theory of dislocations, e.g. stress–strain response, strain energy and plastic strain during the dislocation–obstacle interaction [70,71], and used to obtain parameters necessary for higher level methods, e.g. continuum dislocation dynamics modelling. Current progress in computer power allows static and dynamic modelling of systems of realistic size, e.g. spacing
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between dislocations and between obstacles about 100 nm in crystals of about ten million atoms over a few hundred ps [70,71]. So far, such studies have been limited to dynamics of edge dislocations in cubic metals. Interaction between a moving dislocation and small glissile clusters (4–37 SIAs) in Ni was studied in [74] for different dislocation–cluster geometries. Mechanisms such as cluster drag, cluster absorption on the dislocation line or at previously created jogs, and dislocation pinning and unpinning were demonstrated. The drag of an SIA cluster by a moving edge dislocation, both having the same b, was also observed in Fe [23,70]. The glissile character of clusters allows the decorated dislocation to maintain high velocity under the applied stress. Thus, decoration by SIA clusters/loops with the same b does not pin the dislocation directly, but by being dragged does increase the interaction crosssection of the dislocation with other microstructural features. A few simulations have been made for the dynamic interaction of a dislocation with vacancy clusters. Thus, interactions between a dissociated edge dislocation and an SFT in Cu have been modelled at zero and non-zero temperatures [70, 75,76]. In general, the SFT acts as a rather strong obstacle to dislocation glide, and may be destroyed by being absorbed by the dislocation, thereby forming a superjog, which reduces the mobility of the dislocation. As an example, we present in Fig. 8 the configuration of an SFT in Cu after a 12h1 1 0i edge dislocation has passed through under strain applied at zero temperature [75]. Very recently, the dynamics of dislocation glide and the interaction of an edge dislocation with b ¼ 12h1 1 1i in Fe with a row of voids have been investigated [23,70,71]. An example of information obtained from the static simulation of the interaction between a dislocation gliding in a {1 1 0} plane and spherical voids each of 339 vacancies (diameter 2 nm) and spacing 40 nm is presented in Fig. 9. The dislocation glided under gradually increasing shear strain and the variation of shear stress, crystal energy and plastic displacement was determined. This information can be used directly in continuum models of dislocation dynamics.
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Fig. 8. Configuration of a 21-vacancy SFT (a) before and (b) after interaction with a 12h1 1 0i dislocation in Cu. (c)–(e) are views after 45° rotations around the vertical axis. Light spheres indicate the position of atoms inside the SFT and dark spheres are atoms in fcc crystal lattice sites [70].
In addition to the static simulations typified by Fig. 9, simulations of a dislocation gliding under stress and meeting a row of voids have also been made over wide temperature, stress and void size ranges and the main observations are as follows. 1. At non-zero temperature a straight edge dislocation can move at shear stresses significantly below the Peierls stress, rP (for the iron model used rP 35 MPa [71]). The mechanism of motion can be described as thermal activation of
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lead to the complete absorption of a void after several dislocations have passed through it.
Fig. 9. Shear stress, excess of potential energy in the model crystallite and plastic displacement versus strain during propagation of a 12h1 1 1i edge dislocation through a row of 2 nm voids with spacing 40 nm in Fe [71]. Vertical arrows indicate different stages: I – the dislocation enters the voids, II – dislocation is pinned by the voids and bows between them and III – the dislocation is released from the voids.
kink nucleation and propagation with a low activation energy 0.01 eV: the stress mainly affects the correlation in the motion of the kinks. At low stress (<0:1rP ) this motion is similar to the random jumps of kinks, while at high stress ( P rP ) the motion of the whole dislocation line becomes uniform. 2. The dislocation–void interaction depends on void size, dislocation velocity and density, applied stress and temperature. 3. Interaction between an edge dislocation and voids leads to climb due to absorption of vacancies from a void. The height of climb depends on the obstacle size and strain rate and could
Similar modelling has also been performed for the Fe–Cu system, which is relevant for pressure vessel steels where small Cu-precipitates coherent with the bcc Fe matrix form under irradiation. For an 12h1 1 1i edge dislocation intersecting coherent Cu precipitates of diameter up to 4 nm, the Cu precipitates are weaker obstacles than voids [71]. This can be seen in Fig. 10, where the configuration of the dislocation line just before it is released from either a 2 nm void or precipitate of the same size is presented. However, the rate of increase of critical stress with precipitate size is stronger than that for voids, implying that large precipitates can induce stronger hardening. In fact, the increase in yield stress may be even more pronounced due to a dislocation-induced bcc ) fcc transformation of the precipitate. This was first observed in static modelling of screw dislocations [77] and has now been found for edge dislocations [71]. The climb due to dislocation–precipitate interaction was also observed and it was found to be strain rate dependent – the higher the rate the more significant is the edge dislocation climb. Clearly, there are still many questions to be answered with regard to the large variety of interactions that can occur between dislocations and radiation defects. At least one important conclusion can be drawn now: a moving edge dislocation can sweep out both vacancy and interstitial clusters by absorbing individual point defects. This process may allow understanding of clear channel formation in irradiated metals after a yield drop [78,79].
5. Conclusions Atomic-scale computer modelling can be successfully applied to study the main radiation effects in metals: primary damage production, microstructure evolution and changes in deformation behaviour. It can be used to investigate small-scale (1–100 nm) fast (<100 ns) processes far beyond the resolution of experimental techniques. The greatest advantage of MD-type computer modelling is
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Fig. 10. Shape of the dislocation line corresponding to the maximum (critical) stress when a 12h1 1 1i edge dislocation cuts through a row of either 2 nm voids (339 vacancies) or coherent Cu-precipitates (339 Cu-atoms). The maximum stress is 207 MPa for the voids and 126 MPa for the precipitates.
that it offers the possibility of identifying and determining the fundamental processes involved in the production, accumulation and interaction of defect clusters with other defect clusters and dislocations. It has been demonstrated unambiguously that defect clusters can be formed directly in displacement cascades. The specific properties of SIA clusters such as high stability, fast thermally activated one-dimensional glide and long-range interactions along their glide prism play an extremely important role in the whole microstructure evolution. The use of atomistic simulation to determine mechanisms of formation of specific microstructures such as rafts of dislocation loops and decoration of edge dislocations by dislocation loops, and mechanisms of interaction between of clusters between themselves and with other mi-
crostructural features, such as other radiation-induced defects, impurities, dislocations, etc., can provide necessary and unique information for explanation and prediction of radiation effects in materials. Acknowledgements The authors acknowledge financial support from the UK EPSRC. References [1] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 289. [2] H. Rafii-Tabar, Phys. Rep. 325 (2000) 239.
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[3] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443. [4] Yu.N. Osetsky, A.G. Mikhin, A. Serra, Philos. Mag. A 72 (1995) 361. [5] J.A. Moriarty, Phys. Rev. B 38 (1988) 3199; Phys. Rev. B 42 (1990) 361. [6] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45. [7] K. Nordlund, L. Wei, Y. Zhong, R.S. Averback, Phys. Rev. B 57 (1998) R13965. [8] T. Diaz de la Rubia, M.W. Guinan, J. Nucl. Mater. 174 (1990) 151. [9] F. Gao, D.J. Bacon, P.E.J. Flewitt, T.A. Lewis, J. Nucl. Mater. 249 (1997) 77. [10] A.F. Calder, D.J. Bacon, J. Nucl. Mater. 207 (1993) 25; W.J. Phythian, A.J.E. Foreman, R.E. Stoller, D.J. Bacon, A.F. Calder, J. Nucl. Mater. 223 (1995) 245; R.E. Stoller, J. Nucl. Mater. 276 (2000) 22; N. Soneda, S. Ishino, T. Diaz de la Rubia, Philos. Mag. Lett. 81 (2001) 649. [11] A.J.E. Foreman, C.A. English, W.J. Phythian, Philos. Mag. A 66 (1992) 655. [12] K. Nordlund, R.S. Averback, J. Nucl. Mater. 276 (2000) 194. [13] Yu.N. Osetsky, D.J. Bacon, Nucl. Instr. and Meth. B 180 (2001) 85. [14] S.J. Wooding, L.M. Howe, F. Gao, A.F. Calder, D.J. Bacon, J. Nucl. Mater. 254 (1998) 191; S.J. Wooding, D.J. Bacon, W.J. Phythian, Philos. Mag. A 72 (1995) 1261; S.J. Wooding, D.J. Bacon, Philos. Mag. A 76 (1997) 1033. [15] N. de Diego, D.J. Bacon, Philos. Mag. A 80 (2000) 1393. [16] F. Gao, D.J. Bacon, L.M. Howe, C.B. So, J. Nucl. Mater. 294 (2001) 288. [17] C.S. Becquart, C. Domain, J.C. van Duysen, J.M. Raulot, J. Nucl. Mater. 294 (2001) 274. [18] F. Gao, D.J. Bacon, Philos. Mag. A 75 (1997) 1603. [19] M. Spaczer, A. Caro, M. Victoria, T. Diaz de la Rubia, Phys. Rev. B 50 (1994) 13204. [20] M. Ghaly, K. Nordlund, R.S. Averback, Philos. Mag. A 79 (1999) 795. [21] D.J. Bacon, F. Gao, Yu.N. Osetsky, J. Nucl. Mater. 276 (2000) 1. [22] Yu.N. Osetsky, D.J. Bacon, B.N. Singh, J. Nucl. Mater., in press. [23] Yu.N. Osetsky, D.J. Bacon, B.N. Singh, B. Wirth, J. Nucl. Mater., in press. [24] M.J. Norgett, M.T. Robinson, I.M. Torrens, Nucl. Eng. Des. 33 (1975) 50. [25] M.A. Kirk, I.M. Robertson, M.L. Jenkins, C.A. English, T.J. Black, J.S. Vetrano, J. Nucl. Mater. 149 (1987) 21. [26] R. Bullough, B.L. Eyre, K. Krishan, Proc. R. Soc. Ser. A 346 (1975) 81. [27] C.H. Woo, B.N. Singh, H. Heinisch, J. Nucl. Mater. 174 (1990) 190. [28] M.A. Kirk, M.L. Jenkins, H. Fukushima, J. Nucl. Mater. 276 (2000) 50. [29] F. Gao, D.J. Bacon, Yu.N. Osetsky, P.E.J. Flewitt, T.A. Lewis, J. Nucl. Mater. 276 (2000) 213.
[30] C.C. Matthai, D.J. Bacon, J. Nucl. Mater. 135 (1985) 173. [31] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, J. Nucl. Mater. 165 (1988) 286. [32] K. Nordlund, F. Gao, Appl. Phys. Lett. 74 (1999) 2720. [33] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, J. Nucl. Mater. 195 (1992) 83. [34] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, J. Nucl. Mater. 184 (1991) 127; V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, Mater. Sci. Forum 97–99 (1992) 97. [35] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, J. Nucl. Mater. 170 (1990) 66; J. Nucl. Mater. 173 (1990) 229. [36] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, Phys. Stat. Sol. (b) 155 (1989) 373. [37] V.G. Kapinos, Yu.N. Osetsky, P.A. Platonov, J. Nucl. Mater. 195 (1992). [38] Yu.N. Osetsky, M. Victoria, A. Serra, S.I. Golubov, V. Priego, J. Nucl. Mater. 251 (1997) 34. [39] Yu.N. Osetsky, A. Serra, B.N. Singh, S.I. Golubov, Philos. Mag. A 80 (2000) 2131. [40] B.D. Wirth, G.R. Odette, D. Maroudas, G.E. Lucas, J. Nucl. Mater. 276 (2000) 33. [41] Yu.N. Osetsky, D.J. Bacon, A. Serra, Philos. Mag. Lett. 79 (1999) 273. [42] Yu.N. Osetsky, A. Serra, M. Victoria, V. Priego, S.I. Golubov, Philos. Mag. A 79 (1999) 2259; Philos. Mag. A 79 (1999) 2285. [43] Yu.N. Osetsky, D.J. Bacon, A. Serra, B.N. Singh, S.I. Golubov, J. Nucl. Mater. 276 (2000) 65. [44] N. de Diego, Yu.N. Osetsky, D.J. Bacon, Metall. Trans. 33A (2002) 783. [45] J. Marian, B. Wirth, J.M. Perlado, Phys. Rev. Lett. 88 (2002) 255507. [46] N. de Diego, Yu.N. Osetsky, D.J. Bacon, in press. [47] M.A. Puigvi, Yu.N. Osetsky, A. Serra, Philos. Mag. A, in press. [48] Yu.N. Osetsky, F. Gao, D.J. Bacon, MRS Symp. Proc. 540 (1999) 691. [49] Yu.N. Osetsky, A. Serra, V. Priego, MRS Symp. Proc. 527 (1998) 59. [50] N. Soneda, T. Diaz de la Rubia, Philos. Mag. A 81 (2001) 331. [51] A.V. Barashev, Yu.N. Osetsky, D.J. Bacon, Philos. Mag. A 80 (2000) 2709. [52] Yu.N. Osetsky, D.J. Bacon, A. Serra, B.N. Singh, S.I. Golubov, Philos. Mag. A 83 (2002) 61. [53] Yu.N. Osetsky, A. Serra, V. Priego, J. Nucl. Mater. 276 (2000) 202. [54] Yu.N. Osetsky, D.J. Bacon, F. Gao, A. Serra, B.N. Singh, J. Nucl. Mater. 283–287 (2000) 784. [55] Yu.N. Osetsky, D.J. Bacon, A. Serra, B.N. Singh, MRS Symp. Proc. 653 (2001) Z3.4. [56] M. Pelfort, Yu.N. Osetsky, A. Serra, Philos. Mag. Lett. 81 (2001) 803. [57] M.A. Puigvi, A. Serra, N. de Diego, Yu.N. Osetsky, D.J. Bacon, Philos. Mag. A, in preparation.
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[70] J.O. Stiegler, E.E. Bloom, Radiat. Eff. 8 (1971) 33; M. Kiritani, J. Nucl. Mater. 216 (1994) 220. [71] Yu.N. Osetsky, D.J. Bacon, in: 2nd REVE (REactor for Virtual Experiment) meeting, Segovia, Spain, October 2001, unpublished. [72] Yu.N. Osetsky, D.J. Bacon, Philos. Mag., in preparation. [73] S. Rao, C. Hernandez, J.P. Simmons, T. Parthasarathy, C. Woodward, Philos. Mag. A 77 (1998) 231. [74] S.I. Golubov, X. Liu, H. Huang, C.H. Woo, Comp. Phys. Comm. 137 (2001) 312. [75] D. Rodney, G. Martin, Phys. Rev. Lett. 82 (1999) 3272; Phys. Rev. B 61 (2000) 8714. [76] Yu.N. Osetsky, D.J. Bacon, in: International Workshop on Dislocation-Defect Interactions in Irradiated Metals, Toledo, Spain, April 2000, unpublished. [77] T. Diaz de la Rubia, H. Zbib, T.A. Kraishi, B.D. Wirth, M. Victoria, M.J. Caturla, Nature 406 (2000) 871. [78] T. Harry, D.J. Bacon, Acta Mater. 50 (2002) 195; Acta Mater. 50 (2002) 209. [79] M. Victoria, N. Baluc, C. Bailat, Y. Dai, M.I. Luppo, R. Shaublin, B.N. Singh, J. Nucl. Mater. 276 (2000) 114.
Atomic-scale modelling of primary damage and properties of radiation defects in metals Yu.N. Osetsky *, D.J. Bacon Materials Science and Engineering, Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
Abstract Considerable success has been achieved in recent years in the understanding of radiation damage production in highenergy displacement cascades, the properties of the defects and evolution of radiation damage in metals. Two main reasons form the basis of this success. First, the significant increase in computing power has allowed simulation of realistic cascade energies with good statistics and relatively long-time evolution of defects to be carried out. Second, new experimental findings and corresponding theoretical calculations have allowed interpretation of a number of mechanisms and phenomena crucial for understanding and prediction of practically important radiation effects, such as void swelling, radiation growth, matrix hardening and plastic flow localisation. In this paper we review the most significant results in atomic-scale computer modelling related to these issues, mainly focusing on new achievements such as the formation of extended defect clusters, the dynamic properties of defect clusters, interaction between radiation defects and strengthening of material due to radiation defects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ah; 78.20.Bh; 78.20.Ap Keywords: Computer modelling; Multiscale modelling; Atomic-scale; Defect clusters; Displacement cascades; Radiation damage
1. Introduction Computer modelling of mechanisms and processes of radiation damage in metals has been one of the fastest-growing areas of materials research over the past decade and, due to growing needs of electricity production, either by extended life of existing nuclear plants or the possible generation of new installations, continues to expand. Several reasons for this are not hard to recognise. First,
*
Corresponding author. Tel.: +44-151-794-5773/4662; fax: +44-151-794-4675. E-mail address: [email protected] (Yu.N. Osetsky).
the new paradigm in study of materials, so-called multiscale materials modelling (MMM), has been developed and used in many applications of materials science over the past few years. The main idea of MMM is rather simple and involves decomposition of the whole physical phenomenon into stages linked in time and length scales, each to be studied by a particular technique. The time and length scales of radiation effects ranges from picoseconds to years and from nanometres to metres respectively, covering thereby 22 and 9 orders in magnitude of time and length respectively. The techniques employed, ranging from electronic and atomic interactions via first-principles ab initio calculations to modelling of behaviour via
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)01827-X
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finite elements methods, have been intensively developed, and their sophistication is such that many of the properties of materials can now be simulated with a high level of realism. Second, computer modelling enables some phenomena that are not amenable to experimental study to be investigated. Third, success in the first two aspects has been realised because computer power has grown enormously, thereby allowing many techniques, which are examples of purely theoretical exercises from the past, to be used now quite routinely. In the MMM approach to radiation effects the place of atomic-scale modelling (ASM) is to link atomic and continuum scales, and includes studies of primary defect production, defect properties, reactions between defects and mechanisms of interactions affecting the mechanical properties, e.g. interactions with moving dislocations. ASM also provides mechanisms and parameters necessary for higher scale levels; e.g. Monte Carlo and rate theory studies of microstructure evolution, and continuum modelling of dislocation dynamics for estimating the change in mechanical response due to microstructural evolution. In this paper we present an overview of recent results of atomicscale modelling in these areas. An important issue in any atomic-scale model is the validity of the interatomic potential (IAP), for the IAP determines all the properties of the simulated system. We refer to reviews of the subject [1–3], and only mention that the most commonly used IAPs are based on either the long-ranged pair potentials [4] created within the generalized pseudopotential model by Moriarty [5] or the many-body functions of the embedded atom model [1,3] and the Finnis–Sinclair approach [6]. Other more rigorous types, such as tight-binding models, bond-order potentials and general manybody potentials, are not being used extensively in the large-scale atomic modelling, for this would require too high computational resource.
2. Primary damage The study of primary defect production in displacement cascades due to high-energy particle irradiation is one of the most developed applica-
tion of ASM, particularly via molecular dynamics (MD) technique. Typical MD simulation of displacement cascades involves crystallites containing up to few million atoms, with periodic boundary conditions and, sometimes, with the application of electron–phonon coupling [7] or/and with a thermal bath on the crystal periphery [8,9]. Cascades of primary-knock-on-atom (PKA) energy, EPKA , up to 50 keV have been simulated in bcc, fcc and hcp metals over a wide temperature range [7–22]. The cascade region is usually simulated until its temperature decreases to the ambient, though in some cases short-term post-cascade annealing has been modelled to investigate the behaviour of clusters over a few hundred picoseconds [13,22]. The standard output of cascade simulations includes the total number of the surviving vacancies and self-interstitial atoms (SIAs) at the end of cascade cooling phase, the fraction of vacancies and SIAs formed in clusters (ev , ei ), the size distribution and structure of these clusters (sessile, glissile, Burgers vector) and atomic displacements (for characterisation of mixing). A few tens of events at each condition (PKA energy and direction, temperature) should be simulated to achieve statistically representative results [13,22]. Extensive information has been obtained during the last decade on the nature of primary damage in high-energy displacement cascades (HEDCs) in the bulk of pure bcc [9,10,21], fcc [7,8,11–13,22,23] and hcp [14–16] metals, some binary (Fe–Cu [17]) and ordered (Ni3 Al [18,19]) alloys, and near metal surfaces [15,16,20]. The basic processes in metals have been clarified and the results are in overall agreement with experimental observations. 2.1. Defect production One of the most significant findings of atomistic modelling is that an extensive amount of recombination of SIAs and vacancies occurs during the cooling down stage of HEDCs. As a consequence, in all metals and alloys modelled so far, the total number, NF , of point defects (vacancy or SIA) surviving at the end of the cascade process is significantly lower than the number of atomic displacements, NNRT , predicted by the NRT model [24]. The example for Cu simulated at 100 K with
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two different IAPs [22] and high statistics of treated cascades (from 25 to 40 cascades for each energy) is presented in Fig. 1. The defect production efficiency, NF =NNRT , depends strongly on PKA energy, EP , less on material and weakly on ambient temperature [21] as shown in Fig. 2 where a
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treatment within the commonly accepted formum lation NF ¼ AðEP Þ is presented for a number of metals. In general the parameter A is lower than the corresponding coefficient in the NRT formula [24] and the exponent m is lower than unity. In general the MD data demonstrate cascade efficiency 0.2 of NRT estimations. 2.2. Intracascade clustering
Fig. 1. Number, NF , of vacancy–SIA pairs surviving at the end of the cooling stage of displacement cascades of energy EP in Cu, normalised per the NRT-predicted number, NNRT , simulated with many-body (MBP) and long-range pair (LRPP) potentials in [22]. Symbols indicate the mean and bars the standard error.
Fig. 2. Total number of surviving defects (vacancies or interstitials) versus PKA energy. Data obtained by different authors with different IAPs. The solid line presents the NRT estimate for an average displacement threshold energy of 40 eV.
The occurrence of intracascade clustering is another important finding of atomistic modelling. The collapse in cascades of vacancy clusters into dislocation loops was demonstrated in some metals on the basis of low temperature experiments [25] and used in theoretical treatment of void swelling [26]. The direct formation of SIA clusters in cascades was predicted by both MD modelling of HEDCs [11] and diffusion-based theoretical calculations [27] and has been confirmed recently by TEM [28]. Now, the intracascade clustering of vacancies and SIAs has been substantiated in a variety of bcc, fcc and hcp metals using MD simulations. For example, in Cu, each cascade of EP ¼ 10–20 keV yielded one to two SIA clusters of mean size 5–6 defects [22]. The fraction of SIAs in clusters, ei , was found to be in the range of 50– 60%. The majority of SIA clusters have the form of glissile perfect dislocation loops. Clusters in sessile configuration have also been observed in MD simulations, to an extent which is metal-dependent. The fraction is lowest (a few per cent) in bcc metals, where it takes the form of metastable three-dimensional (3-D) clusters [29], and highest (up to 30–40%) in low stacking fault energy fcc metals, in the form of faulted Frank dislocation loops [13,22]. The sessile fraction in hcp metals lies between these extremes. Production of vacancy clusters is strongly dependent on the crystal structure. Thus large compact clusters are observed only in simulations of fcc and hcp metals. In fcc Cu they are perfect or truncated stacking fault tetrahedra (SFTs) [13,22,30–32] while in hcp Zr they are planar defects similar to prism or basal dislocation loops [14,15,33]. No large compact planar or 3-D vacancy clusters have been observed in bcc metals. Instead, very loose complexes of vacancies in
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Fig. 3. The fraction of vacancies and interstitials surviving in clusters containing P 3 defects versus PKA energy in Cu [22].
second and higher order neighbour shells have been reported [10,34]. The best statistics of cascade defects has been obtained for Cu [13,22] where, on average, each cascade of 10–20 keV PKA energy produces one cluster of about five vacancies, and in total 25% of surviving vacancies are in clusters. Data on the dependence of intracascade clustering on energy and temperature in Cu is presented in Fig. 3 [22]. 2.3. Temperature effects No significant effect of temperature on the total number of defects produced in cascades has been reported, but there is a statistically significant effect on defect clustering [9,16,18,21,22]. For example, in Cu the number of vacancy and interstitial clusters per cascade decreases at high temperature while the number of defects per cluster increases (see Fig. 3). The difference between vacancy and interstitial clustering was found to be in the temperature dependence of the defect fraction contained in clusters: ei increases with temperature while ev is rather insensitive.
3. Defect cluster properties Recognition that intracascade clustering of both SIAs and vacancies is substantial has initiated ex-
tensive investigation into the atomic-level properties of such clusters. Two different approaches are employed. Static modelling – in which the atoms have no kinetic energy and equilibrium is computed by minimisation of the crystal potential energy – is used to study the energy and structure of clusters, their interaction with other defects and stress and displacement fields. SIA and vacancy clusters and dislocation loops containing up to several hundred point defects have been simulated in crystallites containing up to a few million atoms using periodic or fixed boundaries [35–47]. Dynamic modelling by MD – in which the crystal has non-zero temperature – is used to study transport properties and mechanisms of cluster motion in periodic crystallites. These simulations require smaller crystallites (up to a hundred thousand atoms) but last up to several tens nanoseconds. Motion of clusters of up to a few hundred SIAs and vacancies [39,41–43,47–51], and statics and dynamics of cluster–cluster [44,52–54] and cluster-point defect [55–57] interactions have been considered in bcc, fcc and hcp metals. 3.1. Vacancy clusters Stability of vacancy clusters is strongly dependent on crystal structure. The most stable vacancy clusters in fcc Cu are in the form of stacking-fault tetrahedra [37,41], which in MD modelling are stable up to 1000 K for at least a few ns. Faulted Frank vacancy dislocation loops (Burgers vector b ¼ 13h1 1 1i) in Cu are also very stable and tend to dissociate into SFTs. An example of stability of vacancy clusters in Cu from [42] is presented in Fig. 4(a). Studies of hcp Zr [36,45] show that planar vacancy clusters in the form of perfect dislocation loops in prism planes (b ¼ 13h1 1 2 0i) and faulted loops in the basal plane (b ¼ 16h2 0 2 3i) are stable. The situation in bcc metals, particularly in Fe, is more complex. Small clusters (<30 vacancies) can be metastable in the form of planar or 3-D, void-like configurations, and during high temperature annealing transform into cloud-like complexes which are mobile [35,59]. Larger clusters can be stable as compact 3-D voids or perfect edge dislocation loops (b ¼ 12h1 1 1i or h1 0 0i) [34,35,40].
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surface. The growth of large SFTs is suppressed due to difficulties in propagation of long jogs, and this limits the range of size distribution to a few nanometres. This mechanism explains experimental observation that SFT size distribution is independent of the irradiation conditions [60]. 3.2. Interstitial clusters The structure and energy of perfect and faulted interstitial dislocation loops in Cu and Zr, and perfect loops in Fe have been investigated by static modelling [38–40,43–47]. Furthermore, the thermally activated 1-D glide of perfect dislocation loops containing up to a few hundred SIAs, i.e. up to a few nanometres in diameter, has been studied by MD [39–41,44,48–52]. The following conclusions can be drawn.
Fig. 4. Binding energy of SIA and vacancy clusters in Cu [4,55]. Circles – faulted 13h1 1 1i Frank loops, diamonds – rhombusshaped 12h1 1 0i perfect loops, triangles – SFTs.
Attempts have been made to study nucleation of vacancy clusters in different metals under conditions of high vacancy supersaturation. It was demonstrated that vacancy-rich regions collapse into the stable configurations of SFTs and faulted Frank loops in Cu [30,31,38], basal and prism loops in Zr [34] and perfect loops in {1 1 1} and {1 0 0} planes in Fe [35], depending on the level of supersaturation. There was no evidence for the formation of compact 3-D void-like configurations in any of the crystal structures, which may be related to the limited time-scale of atomistic studies or the impossibility of void nucleation without the help of gaseous impurities. A recent significant achievement related to vacancy clusters provides an explanation of the growth mechanism of SFTs in Cu [42]. It was found that growth of an SFT occurs by stair-rod dislocation jog propagation over the SFTÕs facetted
(a) All types of SIA clusters in all metal structures have very high binding energy, of the order 2– 3 eV/SIA (Fig. 4(b)). They cannot dissociate thermally, but can transform from metastable into stable configurations. (b) The most stable clusters are nuclei of perfect dislocation loops, i.e. Burgers vector 12h1 1 0i in fcc (Cu), 12h1 1 1i and h1 0 0i in bcc (Fe) and 1 h1 1 2 0i in hcp (Zr). These clusters can be de3 scribed as compact planar sets of crowdions with their axis along the Burgers vector direction [38,43,44] and are intrinsically glissile. (c) Sessile, stable clusters are formed in fcc (13h1 1 1i faulted Frank loops) and hcp (12h0 0 0 1i faulted basal loops) metals but do not form in bcc crystals where all stable clusters are intrinsically glissile. Large rhombus-shaped clusters (with sides along h1 1 2i) in Cu and rectangular clusters in Zr (with sides along [0 0 0 1] and h1 0 1 0i directions) can dissociate along their glide prism. Such clusters become sessile due to the formation of stair rod dislocations at their corners [39,43,47,52]. (d) Small perfect dislocation loops perform fast thermally activated 1-D motion. Mobility decreases with increasing size, but the activation energy does not depend on size and is close to that of an individual crowdion of the corresponding type, e.g. 0.02 eV [41,43,48–52].
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The thermally activated motion of clusters is suppressed at large size, e.g. above 300 SIAs for 12h1 1 1i clusters in Fe and 120 SIAs for 1 h1 1 0i clusters in Cu [41,43,52]. 2 (e) Although the thermally activated motion of clusters is unlikely to be described by the conventional diffusion process, attempts have been made to parameterise such motion as correlated 1-D diffusion: the jump frequency of the cluster centre of mass (CCM), correlation factor and effective migration energy have been estimated for clusters containing up to 91 SIAs in Fe, Cu and Zr [41,43,44,50–52]. Qualitatively similar results were obtained in all cases and the example for Fe is presented in Fig. 5. It has been shown that the mechanism of cluster motion can be described as nearly stochastic jumps of crowdions with the probability to move the CCM being inversely proportional to the number of SIAs in the cluster [50]. The effective correlation factor is significantly larger than unity, and clusters preferentially move in one direction over large distance, e.g. up to
Fig. 5. Jump frequency of SIA clusters in Fe at function of temperature T . Tm is the melting temperature and the number of SIAs is indicated on the left of each Arrhenius-like dependence [42,51].
tens of nm over tens of ps. The physical basis for the fast thermally activated glide of SIA clusters is not yet clarified, but the parameters obtained can be used in higher level modelling of microstructure evolution and damage accumulation [60]. 3.3. Interactions where SIA clusters are involved The importance of glissile SIA clusters arises from their high stability and fast 1-D mobility, and also the long-range interactions that can occur along their glide path. The latter include interactions between themselves and with sessile clusters and dislocations, and are therefore of importance for models to predict the microstructural evolution during irradiation under cascade damage conditions. A great variety of reactions has been studied already by atomic modelling and can be separated into two groups. 3.3.1. Reactions To this group we attribute interaction between clusters and point defects, which may lead to growth and shrinkage; interaction between clusters, which may result in coalescence or annihilation, and interactions that can induce changes in clusters properties (e.g. sessile () glissile transformations). Interactions of this type are widely observed in cascade simulations where the probability of such events is increased by the high local concentration of defects. For example, growth, shrinkage and coalescence of glissile perfect SIA clusters due to interactions with point defects (vacancies and SIAs) and other clusters, and the formation of sessile 3-D clusters were observed in simulations of high-energy cascades in Fe [29] and Cu [13,22,23,61]. Due to existence of a stable stacking fault in Cu, the probability of observing sessile SIA clusters is much higher than in Fe. An example where a group of small glissile SIA clusters in a 20 keV cascade in Cu interact to create a faulted defect of SIAs lying in a {1 1 1} plane is presented in Fig. 6 [13,63]. MD study of glissile clusters with intersecting glide prisms in Fe and Cu has shown that the final result depends on cluster size, temperature and
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Fig. 6. Three 12h1 1 0i glissile SIA clusters coalesce to form the nucleus of a sessile 13h1 1 1i Frank loop during simulation of a 20 keV cascade in Cu [13]. The time after the PKA creation is indicated for each configuration. Dark spheres are interstitial atoms, light spheres are vacant sites. The projection direction is close to h1 1 1i.
cluster orientation [45,53]. In general, for small clusters and high temperature, the probability that one of them changes its Burgers vector to coalesce with the other increases. Large clusters may form immobile complexes and the probability for this is higher in Cu than in Fe. Important interactions may occur when the mobility of glissile SIA clusters is reduced or when they become immobile without change of structure. Such effects were observed for 12h1 1 1i SIA cluster in Fe interacting with a vacancy [56] and an impurity atom [62]. In particular, a significant decrease in the cluster mobility occurs when an impurity atom interacts with the cluster and becomes trapped at its edge. This effect has been observed in the simulation of a carbon solute atom and a 12h1 1 1i glissile cluster in Fe [62]. Whilst the cluster became essentially immobile (at 300 K), the carbon atom maintained its mobility around the cluster edge by core diffusion. The mobility of a cluster–solute complex is controlled by the solute, which, at high enough temperature, could be
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dragged by the cluster one-dimensionally towards the latterÕs own sink in the microstructure. The process of SIA cluster growth and shrinkage is material-dependent. In bcc Fe a vacancy can be absorbed by the cluster edge and thus reduce the cluster size [56]. Vacancy–SIA cluster interaction in Cu is more complicated and strongly depends on cluster size [57,58]. A vacancy can recombine with interstitial at the edge of small clusters (<49 SIAs) or simply reduce cluster mobility if it lies inside the glide prism, as in Fe. However, larger clusters do not absorb a vacancy because of dissociation of the bounding dislocations on its glide prism. Although interaction energy between a vacancy and the stacking-fault ribbon formed by dissociation is rather high (up to 1 eV), the vacancy is not absorbed but simply prevents mobility of such clusters [57,58]. The situation in hcp metals, particularly in Zr, falls between of these two extremes [58]. A vacancy can annihilate on a f1 0 1 0g face of an SIA cluster, where dissociation is weak, but behaves as in Cu on a (0 0 0 1) face, which is strongly dissociated. Such peculiarities in SIA cluster–vacancy interaction should lead to a difference in evolution of SIA clusters, i.e. cluster growth is suppressed in bcc metals and is easier in fcc and hcp ones. This may help in understanding the difference in damage accumulation in bcc, fcc and hcp metals (see e.g. [64]). 3.3.2. Creation of specific microstructures Interaction between glissile clusters/loops with parallel Burgers vector, b, leads to formation of spatially extended complexes. Modelling by statics and MD in [53,54] has revealed that 12h1 1 1i SIA clusters in Fe, with no overlapping glide prisms, attract each other and form a stable complex. The nature of the clusters does not change, i.e. they are still glissile, but the mobility of the complex depends on the number and size of the clusters: the complex retains mobility if the number of clusters is small (<5) and the total number of SIAs is <300, but becomes immobile for larger numbers. A complex can grow further, attracting more glissile clusters with the same b, thereby creating a raft of clusters/dislocation loops. The interaction energy obtained in the atomistic modelling was
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compared with that estimated from elasticity theory, and whereas the long-range interaction can be approximated well by the theory, the short-range interaction is stronger in the atomistic model [54]. This type of interaction depends on the crystal lattice structure, for in Cu glissile SIA clusters with b ¼ 12h1 1 0i repel each other at large separation but attract at short distances, creating an immobile complex, even for two small clusters [53]. The reasons for such behaviour are not yet understood and cannot be clarified within the isotropic elasticity theory. Nevertheless, the effects found in modelling are qualitatively consistent with experiments, for rafts of dislocation loops are observed in bcc [64–66] but not fcc metals. Another pertinent interaction is that between a glissile SIA cluster and an edge dislocation when both have the same b. This interaction was studied by atomistic modelling and isotropic elasticity in
[54,55] for b ¼ 12h1 1 1i in Fe and b ¼ 12h1 1 0i in Cu. The situation is more complicated in Cu due to the enhanced dissociation of perfect loop in the stress field of a dislocation. As can be seen in Fig. 7 the interaction energy, EINT , is significantly higher a predicted by elasticity theory at short distances where such dissociation may occur in clusters of size too small to dissociate in isolation (e.g. [43]). In general, these results confirm that, in both Fe and Cu, short-range interactions are too complicated to consider in elasticity theory. However, for long-range interactions, when cluster–dislocation separation is much larger than the cluster size and the defects have little effect on each others properties, elasticity calculations give good results when dislocation parameters such as core energy and radius are estimated from atomic-level modelling. This interaction is very long-range, e.g. in Fe EINT > 0:1 eV for a cluster of 37 SIAs (diameter 1 nm) at a distance of about 170 nm, implying a very effective mechanism for the edge dislocation decoration by glissile SIA loops, as observed in bcc and fee metals [64,67–69].
4. Impact on mechanical properties
Fig. 7. Interaction energy versus distance between an hexagonal 12h1 1 0i 49-SIA loop and a 12h1 1 0i dissociated edge dislocation in Cu. Circles indicate modelling results, solid line full integration of dislocation stress field over the loop and dashed line the infinitesimal loop approximation [54].
This is a relatively new area of atomistic modelling for radiation damage research and involves both static and dynamic approaches. Models employ periodicity along the dislocation line and with the boundaries in the other two directions either fixed, with atoms displaced according to elasticity theory [54,55,70,71], or flexible, with atoms able to respond to unbalanced forces using either lattice or elastic Green functions [72,73]. The static modelling has been used to study interactions between edge dislocations and defect clusters (voids, SFTs, SIA clusters) in bcc and fcc metals. The results of static modelling can be compared with models based on the elasticity theory of dislocations, e.g. stress–strain response, strain energy and plastic strain during the dislocation–obstacle interaction [70,71], and used to obtain parameters necessary for higher level methods, e.g. continuum dislocation dynamics modelling. Current progress in computer power allows static and dynamic modelling of systems of realistic size, e.g. spacing
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between dislocations and between obstacles about 100 nm in crystals of about ten million atoms over a few hundred ps [70,71]. So far, such studies have been limited to dynamics of edge dislocations in cubic metals. Interaction between a moving dislocation and small glissile clusters (4–37 SIAs) in Ni was studied in [74] for different dislocation–cluster geometries. Mechanisms such as cluster drag, cluster absorption on the dislocation line or at previously created jogs, and dislocation pinning and unpinning were demonstrated. The drag of an SIA cluster by a moving edge dislocation, both having the same b, was also observed in Fe [23,70]. The glissile character of clusters allows the decorated dislocation to maintain high velocity under the applied stress. Thus, decoration by SIA clusters/loops with the same b does not pin the dislocation directly, but by being dragged does increase the interaction crosssection of the dislocation with other microstructural features. A few simulations have been made for the dynamic interaction of a dislocation with vacancy clusters. Thus, interactions between a dissociated edge dislocation and an SFT in Cu have been modelled at zero and non-zero temperatures [70, 75,76]. In general, the SFT acts as a rather strong obstacle to dislocation glide, and may be destroyed by being absorbed by the dislocation, thereby forming a superjog, which reduces the mobility of the dislocation. As an example, we present in Fig. 8 the configuration of an SFT in Cu after a 12h1 1 0i edge dislocation has passed through under strain applied at zero temperature [75]. Very recently, the dynamics of dislocation glide and the interaction of an edge dislocation with b ¼ 12h1 1 1i in Fe with a row of voids have been investigated [23,70,71]. An example of information obtained from the static simulation of the interaction between a dislocation gliding in a {1 1 0} plane and spherical voids each of 339 vacancies (diameter 2 nm) and spacing 40 nm is presented in Fig. 9. The dislocation glided under gradually increasing shear strain and the variation of shear stress, crystal energy and plastic displacement was determined. This information can be used directly in continuum models of dislocation dynamics.
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Fig. 8. Configuration of a 21-vacancy SFT (a) before and (b) after interaction with a 12h1 1 0i dislocation in Cu. (c)–(e) are views after 45° rotations around the vertical axis. Light spheres indicate the position of atoms inside the SFT and dark spheres are atoms in fcc crystal lattice sites [70].
In addition to the static simulations typified by Fig. 9, simulations of a dislocation gliding under stress and meeting a row of voids have also been made over wide temperature, stress and void size ranges and the main observations are as follows. 1. At non-zero temperature a straight edge dislocation can move at shear stresses significantly below the Peierls stress, rP (for the iron model used rP 35 MPa [71]). The mechanism of motion can be described as thermal activation of
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lead to the complete absorption of a void after several dislocations have passed through it.
Fig. 9. Shear stress, excess of potential energy in the model crystallite and plastic displacement versus strain during propagation of a 12h1 1 1i edge dislocation through a row of 2 nm voids with spacing 40 nm in Fe [71]. Vertical arrows indicate different stages: I – the dislocation enters the voids, II – dislocation is pinned by the voids and bows between them and III – the dislocation is released from the voids.
kink nucleation and propagation with a low activation energy 0.01 eV: the stress mainly affects the correlation in the motion of the kinks. At low stress (<0:1rP ) this motion is similar to the random jumps of kinks, while at high stress ( P rP ) the motion of the whole dislocation line becomes uniform. 2. The dislocation–void interaction depends on void size, dislocation velocity and density, applied stress and temperature. 3. Interaction between an edge dislocation and voids leads to climb due to absorption of vacancies from a void. The height of climb depends on the obstacle size and strain rate and could
Similar modelling has also been performed for the Fe–Cu system, which is relevant for pressure vessel steels where small Cu-precipitates coherent with the bcc Fe matrix form under irradiation. For an 12h1 1 1i edge dislocation intersecting coherent Cu precipitates of diameter up to 4 nm, the Cu precipitates are weaker obstacles than voids [71]. This can be seen in Fig. 10, where the configuration of the dislocation line just before it is released from either a 2 nm void or precipitate of the same size is presented. However, the rate of increase of critical stress with precipitate size is stronger than that for voids, implying that large precipitates can induce stronger hardening. In fact, the increase in yield stress may be even more pronounced due to a dislocation-induced bcc ) fcc transformation of the precipitate. This was first observed in static modelling of screw dislocations [77] and has now been found for edge dislocations [71]. The climb due to dislocation–precipitate interaction was also observed and it was found to be strain rate dependent – the higher the rate the more significant is the edge dislocation climb. Clearly, there are still many questions to be answered with regard to the large variety of interactions that can occur between dislocations and radiation defects. At least one important conclusion can be drawn now: a moving edge dislocation can sweep out both vacancy and interstitial clusters by absorbing individual point defects. This process may allow understanding of clear channel formation in irradiated metals after a yield drop [78,79].
5. Conclusions Atomic-scale computer modelling can be successfully applied to study the main radiation effects in metals: primary damage production, microstructure evolution and changes in deformation behaviour. It can be used to investigate small-scale (1–100 nm) fast (<100 ns) processes far beyond the resolution of experimental techniques. The greatest advantage of MD-type computer modelling is
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Fig. 10. Shape of the dislocation line corresponding to the maximum (critical) stress when a 12h1 1 1i edge dislocation cuts through a row of either 2 nm voids (339 vacancies) or coherent Cu-precipitates (339 Cu-atoms). The maximum stress is 207 MPa for the voids and 126 MPa for the precipitates.
that it offers the possibility of identifying and determining the fundamental processes involved in the production, accumulation and interaction of defect clusters with other defect clusters and dislocations. It has been demonstrated unambiguously that defect clusters can be formed directly in displacement cascades. The specific properties of SIA clusters such as high stability, fast thermally activated one-dimensional glide and long-range interactions along their glide prism play an extremely important role in the whole microstructure evolution. The use of atomistic simulation to determine mechanisms of formation of specific microstructures such as rafts of dislocation loops and decoration of edge dislocations by dislocation loops, and mechanisms of interaction between of clusters between themselves and with other mi-
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