Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals

Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals

1.12 Atomic-Level Dislocation Dynamics in Irradiated Metals Y N Osetsky Oak Ridge National Laboratory, Oak Ridge, TN, USA D J Bacon The University of Liverpool, Liverpool, UK Published by Elsevier Ltd 1.12.1 1.12.2 1.12.2.1 1.12.2.2 1.12.3 1.12.3.1 1.12.3.2 1.12.3.3 1.12.3.4 1.12.4 1.12.4.1 1.12.4.1.1 1.12.4.1.2 1.12.4.2 1.12.4.2.1 1.12.4.2.2 1.12.4.3 1.12.5 References Introduction Radiation Effects on Mechanical Properties Radiation-Induced Obstacles to Dislocation Glide Effects on Mechanical Properties Method Why Atomic-Scale Modeling? Atomic-Level Models for Dislocations Input Parameters Output Information Results on Dislocation–Obstacles Interaction Inclusion-Like Obstacles Temperature T ¼ K Temperature T > K Dislocation-Type Obstacles Stacking fault tetrahedra Dislocation loops Microstructure Modifications due to Plastic Deformation Concluding Remarks Abbreviations Symbols bcc DD DL EAM ESM fcc hcp IAP MBP MC MD MMM MS PAD PBC SFT SIA TEM b bL D G L t T vD « «˙ g w n rD t tc tP Body-centered cubic Dislocation dynamics Dislocation loop Embedded atom model Equivalent sphere method Face-centered cubic Hexagonal close-packed Interatomic potential Many-body potential Monte Carlo Molecular dynamics Multiscale materials modeling Molecular statics Periodic array of dislocations Periodic boundary condition Stacking fault tetrahedron Self-interstitial atom Transmission electron microscope 334 334 334 335 336 336 336 338 338 339 339 339 342 345 345 348 352 353 355 Dislocation Burgers vector Dislocation loop Burgers vector Obstacle diameter Shear modulus Dislocation length Simulation time Ambient temperature Dislocation velocity Shear strain Shear strain rate Stacking fault energy Angle between dislocation segments Poisson’s ratio Dislocation density Shear stress Critical resolved shear stress Peierls stress 333 334 Atomic-Level Dislocation Dynamics in Irradiated Metals 1.12.1 Introduction Structural materials in nuclear power plants suffer a significant degradation of their properties under the intensive flux of energetic atomic particles (see Chapter 1.03, Radiation-Induced Effects on Microstructure) This is due to the evolution of microstructures associated with the extremely high concentration of radiation-induced defects The high supersaturation of lattice defects leads to microstructures that are unique to irradiation conditions Irradiation with high energy neutrons or ions creates initial damage in the form of displacement cascades that produce high local supersaturations of point defects and their clusters (see Chapter 1.11, Primary Radiation Damage Formation) Evolution of the primary damage under the high operating temperature ($600 K to >1000 K) leads to a microstructure containing a high concentration of defect clusters, such as voids, dislocation loops (DLs), stacking fault tetrahedra (SFTs), gas-filled bubbles, and precipitates, and an increase in the total dislocation network density (see Chapter 1.13, Radiation Damage Theory; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects and Chapter 1.15, Phase Field Methods) These changes affect material properties, including mechanical ones, which are the subject of this chapter A general theory of radiation effects has not yet been developed, and currently the most promising way to predict materials behavior is based on multiscale materials modeling (MMM) In this framework, phenomena are considered at the appropriate length and times scales using specific theoretical and/or modeling approaches, and the different scales are linked by parameters/mechanisms/rules to provide integrated information from a lower to a higher level Research on the mechanical properties of irradiated materials, a topic of crucial importance for engineering solutions, provides a good example of this The lowest level treats individual atoms by first principles, ab initio methods, by solving Schroădingers equation for moving electrons and ions Calculations based on electron density functional theory (DFT) (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials) and its approximations, such as bond order potentials (BOPs), can consider a few hundred atoms over a very short time of femtoseconds to picoseconds Delivery of the resulting information to higher level models can be achieved through effective interatomic potentials (IAPs) (see Chapter 1.10, Interatomic Potential Development), in which the adjustable parameters are fitted to the basic chemical and structural properties obtained ab initio IAPs are required for atomic-scale modeling methods such as molecular statics (MS) and molecular dynamics (MD), which are used to simulate millions of atoms Time spanning nanoseconds to microseconds can be simulated by MD if the number of atoms is not large (see Section 1.12.3.3) This level can provide properties of point and extended defects and interactions between them (see Chapter 1.09, Molecular Dynamics) For mechanical properties, important interactions are between moving dislocations, which are responsible for plasticity, and defects created by irradiation Mechanisms and parameters determined at this level can then inform dislocation dynamics (DD) models based on elasticity theory of the continuum (see Chapter 1.16, Dislocation Dynamics) DD models can simulate processes at the micrometer scale and mesh with the mechanical properties of larger volumes of material used in finite elements (FEs) methods, that is, realistic models for the design of core components In this chapter, we consider direct interactions at the atomic scale between moving dislocations and obstacles to their motion The structure of the chapter is as follows First, we summarize the main features of the irradiation microstructure of concern Then we provide a short description of atomic-scale methods applied to dislocation modeling, bearing in mind the details presented in Chapter 1.09, Molecular Dynamics This is followed by a review of important results from simulations of the interaction between dislocations and obstacles We then describe how dislocations modify microstructure in irradiated metals Finally, we indicate some issues that will hopefully be resolved by atomic-scale modeling in the near future Our main aim is to give the reader a general picture of the phenomena involved and encourage further research in this area The following sources1–4 provide a more general and deeper understanding of dislocations and modeling of plasticity issues 1.12.2 Radiation Effects on Mechanical Properties 1.12.2.1 Radiation-Induced Obstacles to Dislocation Glide Primary damage of structural materials is initiated by the interaction of high-energy atomic particles with material atoms to cause the energetic recoil and displacement of primary knock-on atoms (PKAs) PKA energy can vary from a few tens to tens of Atomic-Level Dislocation Dynamics in Irradiated Metals thousands of electron volt and the PKA spectrum can be calculated for a particular position in a particular installation.5 A PKA with energy >$1 keV gives rise to a displacement cascade that produces a localized distribution of point defects (vacancies and self-interstitial atoms, SIAs) and their clusters (see Chapter 1.11, Primary Radiation Damage Formation) Further evolution of these defects produces specific microstructures that depend on the irradiation type, ambient temperature, and the material and its initial structure (see Chapter 1.13, Radiation Damage Theory) This radiation-induced microstructure consists typically of voids, gas-filled bubbles, DLs (that can evolve into a dislocation network), secondary-phase precipitates, and other extended defects specific to the material, for example, SFTs in face-centered cubic (fcc) metals These features are generally obstacles to the dislocation motion Their size is typically6 in the range of nanometers to tens of nanometers and their number density may reach $1024 mÀ3 At this density, the mean distance between obstacles can be as short as $10 nm, and such a high density of small defects, particularly those with a dislocation character, makes the mechanisms of radiation effects on mechanical properties very different from those due to other treatments 1.12.2.2 Effects on Mechanical Properties Radiation-induced defects, being obstacles to dislocation glide, increase yield and flow stress and reduce ductility (see Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys for experimental results) Furthermore, if the obstacle density is sufficiently high to block dislocation motion, preexisting Frank-Reed dislocation sources are unable to operate and plastic deformation requires operation of sources that are not active in the unirradiated state These new sources operate at much higher stress and give rise to new mechanisms such as yield drop, plastic instability, and formation of localized channels with high dislocation activity and high local plastic deformation Understanding these phenomena is necessary for predicting material behavior under irradiation and the design and selection of materials for new generations of nuclear devices Obstacles induced by irradiation affect moving dislocations in a variety of ways, but can be best categorized as one of two types, namely inclusionlike obstacles and those with dislocation properties The first type includes voids, bubbles, and precipitates, for example They usually have relatively 335 short-range strain fields and their properties may not be changed significantly by interaction with dislocations (Copper precipitates in iron are an exception to this – see Sections 1.12.4.1.1–1.12.4.1.2.) Those that are not impenetrable are usually sheared by the dislocation and steps defined by the Burgers vector b of the interacting dislocation are created on the obstacle– matrix interface Unstable precipitates, such as Cu in Fe, may also suffer structural transformation during the interaction, which can change their properties These obstacles not usually modify dislocations significantly, although they may cause climb of edge dislocations (see Sections 1.12.4.1.1–1.12.4.1.2) Their main effect is to create resistance to dislocation glide Obstacles such as voids and bubbles are among the strongest, and as a result of their high density, they contribute significantly to radiation-induced hardening.7 Materials designed to exploit oxide dispersion strengthening (ODS) are produced with a high concentration of rigid, impenetrable oxide particles, which introduce extremely high resistance to dislocation motion.8 These obstacles are also considered here as their scale, typically a few nanometers, is similar to that of obstacles formed under irradiation The second obstacle type consists of those with a dislocation character, for example, DLs and SFTs, and so dislocation reactions occur when they are encountered by moving dislocations Loops have relatively long-range strain fields and hence interact with dislocations over distances much greater than their size SFTs are three-dimensional (3D) structures and have short-range strain fields Loops with perfect Burgers vectors are glissile, in principle, whereas SFTs and faulted loops, for example, Frank loops in fcc metals, are sessile In addition to causing hardening, the reaction of these defects with a gliding dislocation can modify both their own structure and that of the interacting dislocation As will be demonstrated in Section 1.12.4.2, their effect depends very much on the geometry of the interaction, that is, their position and orientation relative to the moving dislocation, and the nature of the mutual dislocation segment that may form in the first stage of interaction The contribution of these obstacles to strengthening can be significant, for their density can be high Modification of irradiation-induced microstructure due to plastic deformation is an additional possibly important effect If mechanical loading occurs during irradiation, it can contribute significantly to the overall microstructure evolution and therefore to change in material properties Accumulation of internal stress during irradiation is unavoidable in real structural 336 Atomic-Level Dislocation Dynamics in Irradiated Metals materials and so this effect should not be ignored The effects of concurrent deformation and irradiation on microstructure are far from clear, for only a few experimental studies of in-reactor deformation have been performed.9 This area, therefore, provides a good example of how atomic-scale modeling can help in understanding a little-studied phenomenon dislocation is effectively infinite in length If the model contains one obstacle, the length, L, of the model in the periodic direction represents the center-to-center obstacle spacing along an infinite row of obstacles It is the treatment of the boundaries in the other two directions that distinguishes one method from another A versatile atomic-scale model should allow for the following.10 1.12.3 Method Reproduction of the correct atomic configuration of the dislocation core and its movement under the action of stress Application of external effects such as applied stress or strain, and calculation of the resultant response such as strain (elastic and plastic) or stress and crystal energy Possibility of moving the dislocation over a long distance under applied stress or strain without hindrance from the model boundaries Simulation of either zero or non-zero temperatures Possibility of simulating a realistic dislocation density and spacing between obstacles Sufficiently fast computing speed to allow simulation of crystallites in the sizes range where size effects are insignificant 1.12.3.1 Why Atomic-Scale Modeling? First principle ab initio methods for self-consistent calculation of electron-density distribution around moving ions provide the most accurate modeling techniques to date They take into account local chemical and magnetic effects and provide significant potential for predicting material properties They are used with success in applications where the properties are limited to the nanoscale, for example, microelectronics, catalysis, nanoclusters, and so on (Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials) A typical scale for this is of the order of a few nm However, this leaves a significant gap between ab initio methods and those required to model properties of bulk materials arising from radiation damage These involve phenomena acting over much longer scales, such as interactions between mobile and sessile defects, their thermally activated transport, their response to internal and external stress fields and gradients of chemical potential Models for bulk properties are based on continuum treatments by elasticity, thermal conductivity, and rate theories where global defect properties such as formation, annihilation, transport, and interactions are already parameterized at continuum level The only technique that currently bridges the gap in the scales between ab initio and the continuum is computer simulation of a large system of atoms, up to 106–108 Atoms move as in classical Newtonian dynamics due to effective forces between them calculated from empirical interatomic potentials and respond to internal and external fields due to temperature, stress, and local imperfections Atomic-scale modeling has provided the results presented in this chapter In the following section, we present a short description of typical models for simulation of dislocations and their interactions with defects formed by radiation 1.12.3.2 Atomic-Level Models for Dislocations All models use periodic boundary conditions in the direction of the dislocation line so that the A comprehensive review of models developed so far is to be found in Bacon et al.,4 and so here we merely present a short summary of the pros and cons of some models used most commonly Historically, the earliest models consisted of a small region of mobile atoms surrounded in the directions perpendicular to the dislocation direction by a shell of atoms fixed in the positions obtained by either isotropic or anisotropic elasticity for displacements around the dislocation of interest.11 This model was used successfully to investigate dislocation core structure and, being simple and computationally efficient, can use a mobile region large enough to simulate interaction between static dislocations and defects and small defect clusters Its main deficiencies are its inability to model dislocation motion beyond a few atomic spacings because of the rigid boundaries (condition 3) and its restriction to temperature T ¼ K (condition 4) The desirability of allowing for elastic response of the boundary atoms due to atomic relaxation in the inner region, for example, when a dislocation moves, has led to the development of several quasicontinuum models The elastic response can be accounted for by using either a surrounding FE mesh or an elastic Green’s function to calculate the response of boundary atoms to forces generated by the inner region Such models are accurate but computationally Atomic-Level Dislocation Dynamics in Irradiated Metals inefficient and have not found wide application so far.4 Furthermore, their use for simulation of T > K (condition 4) is still under development.12 Nevertheless, quasicontinuum models, especially those based on Green’s function solutions, can be employed in applications where calculation of forces on atoms is computationally expensive and a significant reduction in the number of mobile atoms is desirable.13 The models now most widely applied to simulate dislocation behavior in metals are based on the periodic array of dislocations (PAD) scheme first introduced for simulating edge dislocations.14,15 In this, periodic boundary conditions are applied in the direction of dislocation glide as well as along the dislocation line, that is, the glide plane is periodic This means that the dislocation is one of a periodic, 2D array of identical dislocations The success of PAD models is because of their simplicity and good computational efficiency when applied with modern empirical IAPs, for example, embedded atom model (EAM) type They can be used to simulate screw, edge, and mixed dislocations.4,10,16 With a PAD model containing $106–107 mobile atoms, essentially all conditions 1–6 can be satisfied Their ability to simulate interactions with strong obstacles of size up to at least $10 nm makes PAD models efficient for investigating dislocation–obstacle interactions relevant to a radiation damage environment Practically all important radiation-induced obstacles can be simulated on modern computers using parallelized codes and most can even be treated by sequential codes Details of model construction for different dislocations can be found elsewhere.4,10,16 Here we just present an example of system setup for screw or edge dislocations in bcc and fcc metals interacting with dislocations loops and SFTs, as presented in Figure 337 There are two types of DL in an fcc metal: glissile perfect loops with bL ¼ 1/2h110i and sessile Frank loops with bL ¼ 1/3h111i There are two types of glissile loop with Burgers vectors 1/2h111i and h100i in a body-centered cubic (bcc) metal Visualizing interaction mechanisms is a strong feature of atomic scale modeling The main idea is to extract atoms involved in an interaction and visualize them to understand the mechanism Usually these atoms are characterized by high energy, local stresses, and lattice deformation The techniques used are based on analysis of nearest neighbors,17 central symmetry parameter,18 energy,19 stress,20 displacements,10 and Voronoi polyhedra.21 A relatively simple and fast technique, for example, was suggested for an fcc lattice.16 It is based on comparison of position of atoms in the first coordination of an atom with that of a perfect fcc lattice If all 12 neighbors of the analyzed atom are close to that position, it is assigned to be fcc If only nine neighbors correspond to perfect fcc coordination, the atom is taken to be on a stacking fault Other numbers of neighbors can be attributed to different dislocations Modifications of this method have been successfully applied in hexagonal close-packed (hcp)22 and bcc23 crystals Another improvement of this method for MD simulation at T > K was introduced24 in which the above analysis was applied periodically (every 10–50 time_ and over a certain steps depending on strain rate e) time period (100–1000 steps) A probability of an atom to be in different environment was estimated and the final state was assigned to the maximum over the analyzed period Such a probability analysis can be applied to other characteristics such as energy or stress excess over the perfect state and it provides a clear picture when the majority of thermal fluctuations are omitted L L t t Screw L S S t Ͻ111Ͼ Obstacle: loop or SFT Ad b L b = 1/3 dC Edge b = Ͻ100Ͼ dC t b = 1/2 Ͻ111Ͼ D b = 1/2 Ͻ110Ͼ A C B fcc bcc Figure Examples of periodic array of dislocation model setup for screw and edge dislocations in body-centered cubic and face-centered cubic crystals Examples of dislocation loops, a stacking fault tetrahedron, and sense of applied resolved shear stress, t, are indicated 338 Atomic-Level Dislocation Dynamics in Irradiated Metals 1.12.3.3 Input Parameters The IAP is a crucial property of a model for it determines all the physical properties of the simulated system Discussion on modern IAPs is presented in Chapter 1.10, Interatomic Potential Development and so we not elaborate on this subject here Another important property is the spatial scale of the simulated system The periodic spacing, Lg, in the direction of the dislocation glide has to be large enough to avoid unwanted effects due to interaction between the dislocation and its periodic neighbors in the PAD; 100–200b is usually sufficient.10 Furthermore, the model should be large enough to include all direct interactions between the dislocation and obstacle and the major part of elastic energy that may affect the mechanism under study MD simulations have demonstrated that a system with a few million atoms is usually sufficient to satisfy conditions for simulating interaction between a dislocation and an obstacle of a few nanometers in size The biggest obstacles considered to date are nm voids,23 10 nm DLs,25 and 12 nm SFTs26 in crystals containing $6–8 million mobile atoms It should be noted that static simulation (T ¼ K) usually requires the largest system because most obstacles are stronger at low T and the dislocation may have to bend strongly and elongate before breaking free.23 Simulation of a dynamic system, that is, T > K, introduces another important and limiting factor for atomic-scale study of dislocation behavior, namely the simulation time, t, which can be achieved with the computing resource available Under the action of increasing strain applied to the model, the time to reach a given total strain determines the minimum applied strain rate, e,_ that can be considered This parameter defines in turn the dislocation velocity Consider a typical simulation of dislocation–obstacle interaction in an Fe crystal, for which b ¼ 0.248 nm For L ¼ 41 nm, a model containing  106 mobile atoms would have a cross-section area of 5.73  10À16 m2; that is, a dislocation density rD ¼ 1.75  1015 mÀ2 For Lg ¼ 120b ¼ 29.8 nm, the model height perpendicular to the glide plane would be 19 nm At e_ ¼  106 sÀ1 , the steady state velocity, vD, of a single dislocation estimated from the Orowan relation _ D b is 11.6 m sÀ1 The time for the dislocation vD ¼ e=r to travel a distance Lg at this velocity would be 2.6 ns Thus, even if the dislocation breaks away from the void without traversing the whole of the glide plane, the total simulated time would be $1 ns The lowest strain rate for dislocation-obstacle interaction reported so far27 is 105 sÀ1 and it resulted in vD ¼ 48 cm sÀ1 This strain rate is about six to ten orders of magnitude higher than that usually applied in laboratory tensile experiments and more than ten orders higher than that for the creep regime This presents an unresolvable problem for atomic-scale modeling and even massive parallelization gains only three or four orders in e_ or vD We conclude that the possibilities of modern atomic-scale modeling are limited to dislocation velocity of at least $0.1 cm sÀ1 Nevertheless, atomic-scale modeling, particularly using MD (T > K), is a powerful, and sometimes the only, tool for investigating processes associated with lattice defect interactions and dynamics The main advantage of MD is that, if applied properly to a large enough system, it includes all classical phenomena such as evolution of the phonon system and therefore free energy, rates of thermally activated defect motion, and elastic interactions It is, therefore, one of the most accurate techniques for investigating the behavior of large atomic ensembles under different conditions We reemphasize that the realism of atomic-scale modeling is limited mainly by the validity of the IAP and restricted simulation time 1.12.3.4 Output Information Atomic-scale methods and particularly MD can provide a wide range of valuable information on the processes simulated The most important are Information on the physical state of the system This includes temperature and stress and their distribution; displacement of atoms and their transport; interaction energy and therefore force between defects; and evolution of internal, elastic, and free energies Extraction of this information is well understood and procedures can be found in Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys; Chapter 2.13, Properties and Characteristics of ZrC; Chapter 5.01, Corrosion and Compatibility and Chapter 1.09, Molecular Dynamics Detail of atomic mechanisms This includes analysis of the position and environment of individual atoms based on calculation of their energy, site stress, or local atomic configuration Atoms can then be identified with particular features such as constituents of defect clusters, stacking faults, dislocation cores, and so on Having this information at particular times provides unique knowledge of Atomic-Level Dislocation Dynamics in Irradiated Metals defect structure, transformation motion, interactions, and The information summarized in and can be used to determine how the mechanisms involved depend on parameters such as obstacle type and size and dislocation type, material temperature, and applied stress or strain 1.12.4 Results on Dislocation– Obstacles Interaction 1.12.4.1 Inclusion-Like Obstacles 1.12.4.1.1 Temperature T ¼ K Voids in bcc and fcc metals at T ¼ K and >0 K are probably the most widely simulated obstacles of this type Most simulations were made with edge dislocations.10,25–34 A recent and detailed comparison of strengthening by voids in Fe and Cu is to be found in Osetsky and Bacon.34 Examples of stress–strain curves (t vs e) when an edge dislocation encounters and overcomes voids in Fe and Cu at K are presented in Figures and 3, respectively The four distinct stages in t versus e for the process are described in Osetsky and Bacon10 and Bacon and Osetsky.23 The difference in behavior between the two metals is due to the difference in their dislocation core structure, that is, dissociation into Shockley partials in Cu but no splitting in Fe (for details see Osetsky and Bacon34) 339 Under static conditions, T ¼ K, voids are strong obstacles and at maximum stress, an edge dislocation in Fe bows out strongly between the obstacles, creating parallel screw segments in the form of a dipole pinned at the void surface A consequence of this is that the screw arms cross-slip in the final stage when the dislocation is released from the void surface and this results in dislocation climb (see Figure 4), thereby reducing the number of vacancies in the void and therefore its size In contrast to this, a Shockley partial cannot cross-slip Partials of the dissociated dislocation in Cu interact individually with small voids whose diameter, D, is less than the partial spacing ($2 nm), thereby reducing the obstacle strength Stress drops are seen in the stress–strain curve in Figure The first occurs when the leading partial breaks from the void; the step formed by this on the exit surface is a partial step 1/6h112i and the stress required is small Breakaway of the trailing partial controls the critical stress tc For voids with D larger than the partial spacing, the two partials leave the void together at the same stress However, extended screw segments not form and the dislocation does not climb in this process Consequently, large voids in Cu are stronger obstacles than those of the same size in Fe, as can be seen in Figure and the number of vacancies in the sheared void in Cu is unchanged Cu-precipitates in Fe have been studied extensively23,27–29 due to their importance in raising the yield stress of irradiated pressure vessels steels35 and 200 Shear stress (MPa) 150 100 50 -50 D (nm): 0.0 0.9 1.0 1.5 2.0 3.0 4.0 5.0 0.5 1.0 6.0 1.5 Strain (%) Figure Stress–strain dependence for dislocation–void interaction in Fe at K with L ¼ 41.4 nm Values of D are indicated below the individual plots From Osetsky, Yu N.; Bacon, D J Philos Mag 2010, 90, 945 With permission from Taylor and Francis Ltd (http://www.informaworld.com) 340 Atomic-Level Dislocation Dynamics in Irradiated Metals 350 300 Shear stress (MPa) 250 200 150 100 50 -50 D (nm): 1.5 -100 0.0 0.2 0.4 0.6 0.8 Strain (%) 1.0 1.2 Figure Stress–strain dependence for dislocation–void interaction in Cu at K with L ¼ 35.5 nm Values of D are indicated below the individual plots From Osetsky, Yu N.; Bacon, D J Philos Mag 2010, 90, 945 With permission from Taylor and Francis Ltd (http://www.informaworld.com) 20 5.0 nm 16 4.0 nm [110], a 12 3.0 nm 2.0 nm 1.0 nm 0.9 nm -40 -30 -20 -10 [112], a 10 20 30 40 Figure [111] projection of atoms in the dislocation core showing climb of a1/2h111i{110} edge dislocation after breakaway from voids of different diameter in Fe at K Climb-up indicates absorption of vacancies The dislocation slip plane intersects the voids along their equator From Osetsky, Yu N.; Bacon, D J J Nucl Mater 2003, 323, 268 Copyright (2003) with permission from Elsevier the availability of suitable IAPs for the Fe–Cu system.36 These precipitates are coherent with the surrounding Fe when small, that is, they have the bcc structure rather than the equilibrium fcc structure of Cu Thus, the mechanism of edge dislocation interaction with small Cu precipitates is similar to that of voids in Fe The elastic shear modulus, G, of bcc Cu is lower than that of the Fe matrix and the dislocation is attracted into the precipitate by a reduction in its strain energy Stress is required to overcome the attraction and to form a 1/2h111i step on the Fe–Cu interface This is lower than tc for a void, however, for which G is zero and the void surface energy relatively high Thus, small precipitates ( nm) are relatively weak obstacles and, though sheared, remain coherent with the bcc Fe matrix after dislocation breakaway tc is insufficient to draw out screw segments and the dislocation is released without climb The Cu in larger precipitates is unstable, however, and their structure is partially transformed toward 8 6 4 2 0 -2 -2 -4 -4 -6 -6 -8 -8 -50 -45 -40 , ao -35 341 _ [112], ao _ , ao Atomic-Level Dislocation Dynamics in Irradiated Metals _ -2 -4 , ao Figure Position of Cu atoms in four consecutive ð110Þ planes through the center of a nm precipitate in Fe after dislocation breakaway at K The figure on the right shows the dislocation line in [111] projection after breakaway; climb to the left/right indicates absorption of vacancies/atoms by the dislocation From Bacon, D J.; Osetsky, Yu N Philos Mag 2009, 89, 3333 With permission from Taylor and Francis Ltd (http://www.informaworld.com) 1.2 - Voids in Fe 1.0 - Cu-precipitates in Fe - Voids in Cu tC Gb/L 0.8 0.6 Dvoid = 1.52 0.4 DOrowan = 0.77 0.2 0.0 10 15 20 25 30 (D-1 + L-1)-1, b Figure Critical stress tc (in units Gb/L) versus the harmonic mean of D and L (unit b) for voids and Cu-precipitates in Fe and voids in Cu at K the more stable fcc structure when penetrated by a dislocation at T ¼ K This is demonstrated in Figure by the projection of atom positions in four {110} atomic planes parallel to the slip plane near the equator of a nm precipitate after dislocation breakaway In the bcc structure, the {110} planes have a twofold stacking sequence, as can be seen by the upright and inverted triangle symbols near the outside of the precipitate, but atoms represented by circles are in a different sequence Atoms away from the Fe–Cu interface are seen to have adopted a threefold sequence characteristic of the {111} planes in the fcc structure This transformation of Cu structure, first found in MS simulation of a screw dislocation penetrating a precipitate,37,38 increases the obstacle strength and results in a critical line shape that is close to those for voids of the same size.34 Under these conditions, a screw dipole is created 342 Atomic-Level Dislocation Dynamics in Irradiated Metals and effects associated with this, such as climb of the edge dislocation on breakaway described above for voids in Fe, are observed.23,27 The results above were obtained at T ¼ K by MS, in which the potential energy of the system is minimized to find the equilibrium arrangement of the atoms The advantage of this modeling is that the results can be compared directly with continuum modeling of dislocations in which the minimum elastic energy gives the equilibrium dislocation arrangement An early and relevant example of this is provided by the linear elastic continuum modeling of edge and screw dislocations interacting with impenetrable Orowan particles39 and voids.40 By computing the equilibrium shape of a dislocation moving under increasing stress through the periodic row of obstacles, as in the equivalent MS atomistic modeling, it was shown that the maximum stress fits the relationship tc ẳ Gb ẵlnD1 ỵ L1 ị ỵ D 2pAL ½1Š where G is the elastic shear modulus and D is an empirical constant; A equals if the initial dislocation is pure edge and (1 À n) if pure screw, where n is Poisson’s ratio Equation [1] holds for anisotropic elasticity if G and n are chosen appropriately for the slip system in question, that is, if Gb2/4p and Gb2/4pA are set equal to the prelogarithmic energy factor of screw and edge dislocations, respectively.39,40 The value of G obtained in this way is 64 GPa for h111i{110} slip in Fe and 43 GPa for h110i{111} slip in Cu.41 The explanation for the D- and L-dependence of tc is that voids and impenetrable particles are ‘strong’ obstacles in that the dislocation segments at the obstacle surface are pulled into parallel, dipole alignment at tc by self-interaction.39,40 (Note that this shape would not be achieved at this stress in the line-tension approximation where self-stress effects are ignored.) For every obstacle, the forward force, tcbL, on the dislocation has to match the dipole tension, that is, energy per unit length, which is proportional to ln(D) when D (L and ln(L) when L ( D.39 Thus, tcbL correlates with Gb2ln(D1 ỵ L1)1 The correlation between tc obtained by the atomic-scale simulations above and the harmonic mean of D and L, as in eqn [1], is presented in Figure A fairly good agreement can be seen across the size range down to about D < nm for voids in Fe and 3–4 nm for the other obstacles The explanation for this lies in the fact that in the atomic simulation, as in the earlier continuum modeling, obstacles with D > 2–3 nm are strong at T ¼ K and result in a dipole alignment at tc Smaller obstacles in Fe, for example, voids with D < nm and Cu precipitates with D < nm, are too weak to be treated by eqn [1] Thus, the descriptions above and the data in Figure demonstrate that the atomicscale mechanisms that operate for small and large obstacles depend on their nature and are not predicted by simple continuum treatments, such as the line-tension and modulus-difference approximations that form the basis of the Russell–Brown model of Cuprecipitate strengthening of Fe,42 often used in predictions and treatment of experimental observations The importance of atomic-scale effects in interactions between an edge dislocation and voids and Cu-precipitates in Fe was recently stressed in a series of simulations with a variable geometry.43 In this study, obstacles were placed with their center at different distances from the dislocation slip plane An example of the results for the case of nm void at T ¼ K is presented in Figure The surprising result is that a void with its center below the dislocation slip plane is still a strong obstacle and may increase its size after the dislocation breaks away This can be seen in Figure 7, where a dislocation line climbs down absorbing atoms from the void surface More details on larger voids, precipitates, and finite temperature effects can be found in Grammatikopoulos et al.43 1.12.4.1.2 Temperature T > K In contrast to the T ¼ K simulations above, modeling by MD provides the ability to investigate temperature effects in dislocation–obstacle interaction (The limit on simulation time discussed in Glide plane Dz = R R/2 -R/2 -R Climb-up - atoms left inside void Climb-down - atoms taken off void Figure Schematic representation of the configurations studied for voids of radius R in Grammatikopoulos et al.43 and the corresponding shape of the edge dislocation line seen in [111] projection after breakaway Dz is the distance of the center of the void from the dislocation glide plane From Grammatikopoulos, P.; Bacon, D J.; Osetsky, Yu N Model Simulat Mater Sci Eng 2011, 19, 015004 With permission from IOP Publishing Ltd Atomic-Level Dislocation Dynamics in Irradiated Metals Section 1.12.3.3 prevents study of the creep regime controlled by dislocation climb.) Results on the temperature dependence of tc from simulation of interaction between an edge dislocation and and nm voids in Fe,29,30,34 Cu-precipitates in Fe,27,29 and voids in Cu30,34 are presented in Figure In general, the strength of all the obstacles becomes weaker with increasing temperature, although the mechanisms involved are not the same for the different obstacles The temperature-dependence of void strengthening in Fe has been analyzed by Monnet et al.44 using a mesoscale thermodynamic treatment of MD data in the point obstacle approximation to estimate activation energy and its temperature dependence In this way, the obstacle strength found by atomic-scale modeling can be converted into a mesoscale parameter to be used in higher level modeling in the multiscale framework More investigations are required to define mesoscale parameters for more complicated cases such as voids in Cu and Cu-precipitates in Fe Void strengthening in Cu exhibits specific behavior in which the temperature-dependence is strong at low T < 100 K but rather weak at higher T (for more details see Figure in Osetsky and Bacon34) The reason for this is as yet unclear Interestingly, MD simulation has been able to shed light on thermal effects in strengthening due to Cu-precipitates in Fe, as in Figure (for more details see Figure in Bacon and Osetsky23) Small precipitates, D < nm, are stabilized in the bcc coherent state by the Fe matrix, as noted above 1.0 and nm obstacles: ε = × 106 s-1 Voids in Cu 0.8 Voids in Fe tc Gb/L Cu-prpt in Fe 0.6 nm 0.4 nm 0.2 0.0 100 200 300 400 500 600 700 T (K) Figure Plot of tc versus T for voids and Cu-precipitates in Fe and voids in Cu D is as indicated, L ¼ 41.4 nm, and e_ ¼  106 sÀ1 343 for T ¼ K, and are weak, shearable obstacles The resulting temperature-dependence of tc is small Larger precipitates were seen to be unstable at T ¼ K with respect to a dislocation-induced transformation toward the fcc structure This transformation is driven by the difference in potential energy of bcc and fcc Cu The free energy difference between these two phases of Cu decreases with increasing T until a temperature is reached at which the transformation does not occur Thus, large precipitates are strong obstacles at low T and weak ones at high T This is reflected in the strong dependence of tc on T shown in Figure More explanation of this effect can be found in Bacon and Osetsky.23 These simulation results showing the different behavior of small and large Cu-precipitates suggest that the yield stress of underaged or neutron-irradiated Fe–Cu alloys, which contain small, coherent Cu-precipitates, should have a weak T-dependence, whereas that in an overaged or electron-irradiated alloy, in which the population of coherent precipitates has a larger size, should be stronger Some experimental observations support this.45 One is a weak change in the temperature dependence of radiation-induced precipitate hardening in ferritic alloys observed after neutron irradiation when only small ( 0.5) is different The dislocation climbs down and tc decreases with increasing value of He/Vac ratio At the highest ratio, the dislocation stress field induces the bubble to emit interstitial Fe atoms from its surface into the matrix toward the dislocation before it makes contact The bubble pressure is reduced in this way and interstitials are absorbed by the dislocation as a double superjog Equilibrium bubbles are therefore the strongest Some of these conclusions, such as formation of interstitial clusters around bubbles with high He/Vac ratios, are similar to those observed earlier,48–50 others are not More modeling is necessary to clarify these issues As noted in Section 1.12.2.2, impenetrable obstacles such as oxide particles and incoherent precipitates represent another class of inclusion-like features Although these obstacles are usually preexisting and not produced by irradiation, they are considered to be of potential importance for the design of nuclear energy structural materials and should be considered here Atomic-level information on their effect on dislocations is still poor, however, and we can only refer to some recent work on this The interaction between an edge dislocation and a rigid, impenetrable particle in Cu was simulated by Hatano54 using the Cu–Cu IAP as for the Fe–Cu system36 and a constant strain rate of  106 sÀ1 at T ¼ 300 K The particle was created by defining a spherical region in which the atoms were held immobile relative to the surrounding crystal The Hirsch mechanism2 was found to operate In the sequence shown in Figures and of Hatano,54 several stages can be observed such as (1) the dislocation under stress approaching the obstacle from the left first bows round the obstacle to form a screw dipole; (2) the screw segments cross-slip on inclined {111} planes at tc; (3) they annihilate by double cross-slip, allowing the dislocation, now with a double superjog, to bypass the obstacle; (4) a prismatic loop with the same b is left behind and (5) the dragged superjogs pinch-off as the dislocation glides away, creating a loop of opposite sign to the first on the right of the obstacle tc varies with D and L as predicted by the continuum modeling that led to eqn [1], but is over times larger in magnitude Hatano argues that this could arise from either higher stiffness of a dissociated dislocation or a dependence of tc on the initial position of the dislocation It is also possible that the requirement for the dislocation to constrict and the absence of a component of applied stress on the crossslip plane results in a high value of tc Simulation of nm impenetrable precipitates in Fe has been carried out by Osetsky (2009, unpublished) The method is different from that used by Hatano54 in that the precipitate, constructed from Fe atoms held immobile relative to each other, was treated as a superparticle moving according to the total force on precipitate atoms from matrix atoms The interaction mechanism observed is quite different from those reported earlier for Cu,54 for the Hirsch mechanism and formation of interstitial clusters does Atomic-Level Dislocation Dynamics in Irradiated Metals not occur Instead, the mechanism observed was close to the Orowan process, with formation of an Orowan loop that either shrinks quickly if the obstacle is small and spherical, or remains around it if D is large (!4nm) or becomes elongated in the direction perpendicular to the slip plane It is interesting to note that if the model of a completely immobile precipitate in Hatano54 is applied to Fe, the same Hirsh mechanism is observed as that in the earlier study Comparison of strengthening due to pinning of a 1/2h111i{110} edge dislocation by nm spherical obstacles of different nature simulated at 300 K is presented in Figure One can see that the coherent Cu precipitate is the weakest whereas a rigid impenetrable precipitate when Orowan loop is stabilized by its shape is the strongest A surprising result is that an equilibrium He-bubble is a stronger obstacle than the equivalent void The reason for this is not clear yet Little is known on the interactions between screw dislocations and inclusion-like obstacles In fact, we are aware of only one published study of screw dislocation–void interaction in fcc Cu.55 It was found that voids are quite strong obstacles and their strength and interaction mechanisms are strongly temperature dependent Thus at low temperature, the 1/2h110i {111} screw dislocation keeps its original slip plane 250 MD modeling in Fe: Edge dislocation 1/2Ͻ111Ͼ{110} Crystal 074 000 atoms Temperature 300 K Obstacle size nm Obstacle spacing 42 nm 225 tc (MPa) 200 175 150 125 100 75 50 Cu-prpt Void He-bubble He/Vac = 0.5 Rigid particle Orowan mechanism Figure Critical stress for an edge dislocation penetrating through different nm obstacles in Fe at T ¼ 300 K L ¼ 41.4 nm and e_ ¼  106 sÀ1 Table 345 when it breaks away from the void However, at high T > 300 K, cross-slip is activated and plays an important role in dislocation–void interaction Several depinning mechanisms involving dislocation crossslip on the void surface were simulated and formation of DLs was observed in some cases Interestingly, the void strength increases with increasing temperature and the authors explain this by changing interaction mechanisms Intensive cross-slip was observed54 that propagated through the periodic boundaries along the dislocation line direction, with the result that the void was interacting with its images Similar effects have been observed in the interaction of a screw dislocation and SIA loops and SFTs, and the possible significance of this for understanding the simulation results has been discussed elsewhere.4 1.12.4.2 Dislocation-Type Obstacles Extensive simulations of interactions between moving dislocations and dislocation-like obstacles such as DLs and SFTs has demonstrated that the reactions involved follow the general rules of dislocation– dislocation reaction, for example, Frank’s rule for Burgers vectors,1,2 even though the reacting segments are of the nanometer scale in length Results of these interactions are in the range from no effect on both dislocation and obstacle to complete disappearance of the obstacle and significant modification of the dislocation A detailed analysis of reactions was made for SFTs an fcc metal56 and later for SIA loops in Fe.57 In general, five types of reaction were identified, as summarized in Table The outcomes in Table were observed for different obstacles under different reaction conditions such as interaction geometry, strain rate, ambient temperature, and so on We give some examples in the following section 1.12.4.2.1 Stacking fault tetrahedra Reactions of type R1 have been observed for both screw and edge dislocations and all the defects with dislocation character Interestingly, the strength effect of this reaction varies from minimum to Description of main reactions between dislocations and obstacles with dislocation character4 Reaction Dislocation type Overall result R1 R2 R3 R4 R5 Edge or screw Edge or screw Edge Screw Edge and screw Dislocation and obstacle remain unchanged Obstacle changed but dislocation unchanged Partial or full absorption of obstacle by edge dislocation (superjog formation) Temporary absorption of obstacle by screw dislocation (helix formation) Dislocation drags glissile defects 346 Atomic-Level Dislocation Dynamics in Irradiated Metals maximum For example, it is insignificant in the case of a 1/2h110i{111} edge dislocation interacting with an SFT58 and maximum for a screw dislocation interacting with a DL when the loop is fully absorbed into a helical turn on the dislocation.57 The mechanism for the way both the obstacle and dislocation remain unchanged is different for each case An edge dislocation interacting with an SFT close to its tip creates a pair of ledges on its surface that are not stable and annihilate athermally.56,58 An example of this reaction is presented in Figure 10 If the dislocation slip plane is far enough from the SFT tip in the compressive region of the dislocation (for details of geometrical definitions see Bacon et al.4), the ledges can be stabilized.56,58,59 This can be seen in Figure 11 (1) If the dislocation passes through the SFT several times in the same slip plane, it can detach the portion of the SFT above the slip plane, as shown in Figure 11 (2–4) Both the above mechanisms are common for small SFTs, low T, fast dislocations, and the position of the SFT tip above the slip plane of an edge dislocation If, however, the SFT tip is below the dislocation slip plane, and T is high enough and the dislocation speed low enough, D reaction R3 can be activated The stages of this reaction are presented in Figure 12 An example of effects of SFT orientation and temperature for the interaction of an edge dislocation with an SFT is presented in Figure 13 In this study, the dislocation slip plane intersected a 4.2 nm SFT through its geometrical center at the applied e_ ¼  106 sÀ1 in a wide temperature range from to 450 K.59 Reaction R1 was observed (see Figure 10) at all temperatures when the SFT was oriented with its tip up relative to the dislocation slip plane (orange triangles up in Figure 13) and at the two lowest temperatures when it was oriented in the opposite sense At T ¼ 300 K and orientation with tip down, a couple of ledges were formed on the SFT surface (see Figure 11) It may be noted that the R2 mechanism requires higher applied stress even though the temperature is increased At higher T ¼ 450 K, the interaction mechanism is changed and the whole portion of the SFT above the slip plane is absorbed by the dislocation (Figure 12), that is, reaction R3 occurs, creating a pair of superjogs on the dislocation line Some vacancies were also found to form to accommodate the glissile configuration of the superjogs dC Ad C A (a) (b) (c) (d) Figure 10 An example of reaction R1 for an edge dislocation passing through 4.2 nm SFT (136 vacancies) oriented with apex above the slip plane in Cu at 300 K From Osetsky, Yu N.; Rodney, D.; Bacon, D J Philos Mag 2006, 86, 2295 With permission from Taylor and Francis Ltd (http://www.informaworld.com) Atomic-Level Dislocation Dynamics in Irradiated Metals 347 Figure 11 Shear of a 2.4 nm SFT (45 vacancies) by an edge dislocation in Cu at 300 K 0: Initial SFT; (1): creation of two ledges (reaction R1); (2–4): evolution of the configuration due to additional passes of the dislocation From Osetsky, Yu N.; Stoller, R E.; Matsukawa, Y J Nucl Mater 2004, 329–333, 1228 Copyright (2004) with permission from Elsevier B A C ad dA ab dA Cd Cd (a) (b) Ca Cd Cb dA ad bd Cd (c) (d) Figure 12 An example of reaction R3 for an edge dislocation and 4.2 nm SFT (136 vacancies) with its tip below the slip plane in Cu at 450 K From Osetsky, Yu N.; Rodney, D.; Bacon, D J Philos Mag 2006, 86, 2295 With permission from Taylor and Francis Ltd (http://www.informaworld.com) This is discussed later in Section 1.12.4.3 More details on interactions between screw and edge dislocations and SFT can be found elsewhere.60–63 In general, it can be concluded that the SFTs created under irradiation, that is,