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Comprehensive nuclear materials 1 14 kinetic monte carlo simulations of irradiation effects Comprehensive nuclear materials 1 14 kinetic monte carlo simulations of irradiation effects Comprehensive nuclear materials 1 14 kinetic monte carlo simulations of irradiation effects Comprehensive nuclear materials 1 14 kinetic monte carlo simulations of irradiation effects Comprehensive nuclear materials 1 14 kinetic monte carlo simulations of irradiation effects
1.14 Kinetic Monte Carlo Simulations of Irradiation Effects C S Becquart Ecole Nationale Supe´rieure de Chimie de Lille, Villeneuve d’Ascq, France; Laboratoire commun EDF-CNRS Etude et Mode´lisation des Microstructures pour le Vieillissement des Mate´riaux (EM2VM), France B D Wirth University of Tennessee, Knoxville, TN, USA ß 2012 Elsevier Ltd All rights reserved 1.14.1 1.14.2 1.14.3 1.14.4 1.14.4.1 1.14.4.2 1.14.4.3 1.14.4.4 1.14.5 1.14.6 1.14.7 1.14.8 1.14.9 References Introduction Modeling Challenges to Predict Irradiation Effects on Materials KMC Modeling KMC Modeling of Microstructure Evolution Under Radiation Conditions Irradiation Rate Transmutation Rate Diffusion Rate Emission/Dissociation Rate Atomistic KMC Simulations of Microstructure Evolution in Irradiated Fe–Cu Alloys OKMC Example: Ag Fission Product Diffusion and Release in TRISO Nuclear Fuel Some Limits of KMC Approaches Advanced KMC Methods Summary and a Look at the Future of Nuclear Materials Modeling Abbreviations AKMC BKL EKMC FP HTR KMC MC MD NEB NRT OKMC PKA RPV RTA SIA TRISO Atomistic kinetic Monte Carlo Boris, Kalos, and Lebowitz Event kinetic Monte Carlo Frenkel pairs High-Temperature Reactor Kinetic Monte Carlo Monte Carlo Molecular dynamics Nudged elastic band Norgett, Robinson, and Torrens Object kinetic Monte Carlo Primary knockon atom Reactor pressure vessel Residence time algorithm Self-interstitial atoms Tristructural isotropic fuel particle 1.14.1 Introduction Many technologically important materials share a common characteristic, namely that their dynamic behavior is controlled by multiscale processes For 393 394 395 396 397 397 397 398 399 404 406 407 408 408 example, crystal growth, plasma processing of materials, ion-beam assisted growth and doping of electronic materials, precipitation in structural materials, grain boundary and dislocation evolution during mechanical deformation, and alloys driven by high-energy particle irradiation all experience cluster nucleation, growth, and coarsening that impact the evolution of the overall microstructure and, correspondingly, property changes These phenomena involve a wide range of length and time scales While the specific details vary with each material and application, kinetic processes at the atomic to nanometer scale (especially related to nucleation phenomena) are largely responsible for materials evolution, and typically involve a wide range of characteristic times The large temporal diversity of controlling processes at the atomic to nanoscale level makes experimental identification of the governing mechanisms all but impossible and clearly defines the need for computational modeling In such systems, the potential benefits of modeling are at a maximum and are related to reduction in time and expense of research and development and introduction of novel materials into the marketplace Systems in which the materials microstructure can be represented by multiple particles experiencing 393 394 Kinetic Monte Carlo Simulations of Irradiation Effects Timescale µs–s ns–µs ps–ns 10À12 and $10À3 s The goal of this chapter is to describe the state of the art in kinetic Monte Carlo (KMC) simulation, as well as to identify a number of priority research areas, moving toward the goal of accelerating the development of advanced computational approaches to simulate nucleation, growth, and coarsening of radiation-induced precipitates and defect clusters (cavities and/or dislocation loops) It is anticipated that the approaches will span from atomistic molecular dynamics (MD) simulations to provide key kinetic input on governing mechanisms to fully three-dimensional (3D) phase field and KMC models to larger scale, but spatially homogeneous cluster dynamics models 1.14.2 Modeling Challenges to Predict Irradiation Effects on Materials The effect of irradiation on materials is a classic example of an inherently multiscale phenomenon, as schematically illustrated in Figure Pertinent processes span over more than 10 orders of magnitude in length scale from the subatomic nuclear to Nano/microstructure and local chemistry changes; nucleation and growth of extended defects and precipitates Irradiation temperature, n/g energy spectrum, flux, fluence, thermal cycling, and initial material microstructure inputs: s–year Decades Brownian motion and occasional collisions against one another and systems with other defects (dislocations, grain boundaries, surfaces, etc.) are in particular amenable to multiscale modeling Within a multiscale approach, atomistic simulations (utilizing either electronic structure calculations or semiempirical potentials) investigate controlling mechanisms and occurrence rates of diffusional and reactive interactions between the various particles and defects of interest, and inform larger length scale kinetic (Monte Carlo, phase field, or chemical reaction rate theory) models, which subsequently lead to the development of constitutive models for predictive continuum scale models Simulating long-time materials dynamics with reliable physical fidelity, thereby providing a predictive capability applicable outside limited experimental parameter regimes is the promise of such a computational multiscale approach A critical need is the development of advanced and highly efficient algorithms to accurately model nucleation, growth, and coarsening in irradiated alloys that are kinetically controlled by elementary (diffusive) processes involving characteristic time scales between 50nm Long-range defect transport and annihilation at sinks Gas diffusion and trapping Radiation enhanced diffusion and induced segregation of solutes Cascade aging and local solute redistribution Defect recombination, He and H clustering, and generation migration Primary defect production and short-term annealing Atomic–nm Underlying microstructure (preexisting and evolving) impacts defect and solute fate nm–µm µm–mm mm–m Lengthscale Figure Illustration of the length and time scales (and inherent feedback) involved in the multiscale processes responsible for microstructural changes in irradiated materials Reproduced from Wirth, B D.; Odette, G R.; Marian, J.; Ventelon, L.; Young, J A.; Zepeda-Ruiz, L A J Nucl Mater 2004, 329–333, 103 Kinetic Monte Carlo Simulations of Irradiation Effects structural component level, and span 22 orders of magnitude in time from the subpicosecond of nuclear collisions to the decade-long component service lifetimes.1,2 Many different variables control the mix of nano/microstructural features formed and the corresponding degradation of physical and mechanical properties of nuclear fuels, cladding, and structural materials The most important variables include the initial material composition and microstructure, the thermomechanical loads, and the irradiation history While the initial material state and thermomechanical loading are of concern in all materials performancelimited engineering applications, the added complexity introduced by the effects of radiation is clearly the distinguishing and overarching concern for materials in advanced nuclear energy systems At the smallest scales, radiation damage is continually initiated by the formation of energetic primary knock-on atoms (PKAs) primarily through elastic collisions with high-energy neutrons Concurrently, high concentrations of fission products (in fuels) and transmutants (in cladding and structural materials) are generated and can cause pronounced effects in the overall chemistry of the material, especially at high burnup The PKAs, as well as recoiling fission products and transmutant nuclei quickly lose kinetic energy through electronic excitations (that are not generally believed to produce atomic defects) and a chain of atomic collision displacements, generating a cascade of vacancy and self-interstitial defects High-energy displacement cascades evolve over very short times, 100 ps or less, and small volumes, with characteristic length scales of 50 nm or less, and are directly amenable to MD simulations if accurate potentials are available The physics of primary damage production in high-energy displacement cascades has been extensively studied with MD simulations.3–8 (see Chapter 1.11, Primary Radiation Damage Formation) The key conclusions from the MD studies of cascade evolution have been that (1) intracascade recombination of vacancies and self-interstitial atoms (SIAs) results in $30% of the defect production expected from displacement theory, (2) many-body collision effects produce a spatial correlation (separation) of the vacancy and SIA defects, (3) substantial clustering of the SIAs and to a lesser extent the vacancies occur within the cascade volume, and (4) high-energy displacement cascades tend to break up into lobes or subcascades, which may also enhance recombination.4–7 Nevertheless, it is the subsequent diffusional transport and evolution of the defects produced during displacement cascades, in addition to solutes and 395 transmutant impurities, that ultimately dictate radiation effects in materials and changes in material microstructure.1,2 Spatial correlations associated with the displacement cascades continue to play an important role in much larger scales as processes including defect recombination, clustering, migration, and gas and solute diffusion and trapping Evolution of the underlying materials structure is thus governed by the time and temperature kinetics of diffusive and reactive processes, albeit strongly influenced by spatial correlations associated with the microstructure and the continuous production of new radiation damage The inherently wide range of time scales and the ‘rare-event’ nature of many of the controlling mechanisms make modeling radiation effects in materials extremely challenging and experimental characterization often unattainable Indeed, accurate models of microstructure (point defects, dislocations, and grain boundaries) evolution during service are still lacking To understand the irradiation effects and microstructure evolution to the extent required for a high fidelity nuclear materials performance model will require a combination of experimental, theoretical, and computational tools Furthermore, the kinetic processes controlling defect cluster and microstructure evolution, as well as the materials degradation and failure modes may not entirely be known Thus, a substantial challenge is to discover the controlling processes so that they can be included within the models to avoid the detrimental consequences of in-service surprises High performance computing can enable such discovery of class simulations, but care must also be taken to assess the accuracy of the models in capturing critical physical phenomena The remainder of this chapter will thus focus on a description of KMC modeling, along with a few select examples of the application of KMC models to predict irradiation effects on materials and to identify opportunities for additional research to achieve the goal of accelerating the development of advanced computational approaches to simulate nucleation, growth, and coarsening of microstructure in complex engineering materials 1.14.3 KMC Modeling The Monte Carlo method was originally developed by von Neumann, Ulam, and Metropolis to study the diffusion of neutrons in fissionable material on the Manhattan Project9,10 and was first applied to simulate radiation damage of metals more than 396 Kinetic Monte Carlo Simulations of Irradiation Effects 40 years ago by Besco,11 Doran,12 and later Heinisch and coworkers.13,14 Monte Carlo utilizes random numbers to select from probability distributions and generate atomic configurations in a stochastic process,15 rather than the deterministic manner of MD simulations While different Monte Carlo applications are used in computational materials science, we shall focus our attention on KMC simulation as applied to the study of radiation damage The KMC methods used in radiation damage studies represent a subset of Monte Carlo (MC) methods that can be classified as rejection-free, in contrast with the more classical MC methods based on the Metropolis algorithm.9,10 They provide a solution to the Master Equation which describes a physical system whose evolution is governed by a known set of transition rates between possible states.16 The solution proceeds by choosing randomly among the various possible transitions and accepting them on the basis of probabilities determined from the corresponding transition rates These probabilities are calculated for physical transition mechanisms as Boltzmann factor frequencies, and the events take place according to their probabilities leading to an evolution of the microstructure The main ingredients of such models are thus a set of objects (which can resolve to the atomic scale as atoms or point defects) and a set of reactions or (rules) that describe the manner in which these objects undergo diffusion, emission, and reaction, and their rates of occurrence Many of the KMC techniques are based on the residence time algorithm (RTA) derived 50 years ago by Young and Elcock17 to model vacancy diffusion in ordered alloys Its basic recipe involves the following: for a system in a given state, instead of making a number of unsuccessful attempts to perform a transition to reach another state, as in the case of the Metropolis algorithm,9,10 the average time during which the system remains in its state is calculated A transition to a different state is then performed on the basis of the relative weights determined among all possible transitions, which also determine the time increment associated with the selected transition According to standard transition state theory (see for instance Eyring18) the frequency Gx of a thermally activated event x, such as a vacancy jump in an alloy or the jump of a void can be expressed as: Ea ẵ1 GX ẳ nX exp kB T where nX is the attempt frequency, kB is Boltzmann’s constant, T is the absolute temperature, and Ea is the activation energy of the jump During the course of a KMC simulation, the probabilities of all possible transitions are calculated and one event is chosen at each time-step by extracting a random number and comparing it to the relative probability The associated time-step length dt and average time-step length Dt are given by: Àlnr and Dt ¼ P dt ¼ P GX GX n ½2 n where r is a random number between and The RTA is also known as the BKL (Bortz, Kalos, Lebowitz) algorithm.19 Other techniques are possible, as described by Chatterjee and Vlachos.20 The basic steps in a KMC simulation can be summarized thus: Calculate the probability (rate) for a given event to occur Sum the probabilities of all events to obtain a cumulative distribution function Generate a random number to select an event from all possible events Increase the simulation time on the basis of the inverse sum1 of the rates of all possible events @Dt ¼ P w A, Ni Ri where w is a random deviate that i assures a Poisson distribution in time-steps and N and R are the number and rate of each event i Perform the selected event and all spontaneous events as a result of the event performed Repeat Steps 1–4 until the desired simulation condition is reached 1.14.4 KMC Modeling of Microstructure Evolution Under Radiation Conditions KMC models are now widely used for simulating radiation effects on materials.21–50 Advantages of KMC models include the ability to capture spatial correlations in a full 3D simulation with atomic resolution, while ignoring the atomic vibration time scales captured by MD models In KMC, individual point defects, point defect clusters, solutes, and impurities are treated as objects, either on or off an underlying crystallographic lattice, and the evolution of these objects is modeled over time Two general approaches have been used in KMC simulations, object KMC (OKMC) and event KMC (EKMC),35,36 which differ in the Kinetic Monte Carlo Simulations of Irradiation Effects treatment of time scales or step between individual events Within the OKMC designation, it is also possible to further subdivide the techniques into those that explicitly treat atoms and atomic interactions, which are often denoted as atomic KMC (AKMC), or lattice KMC (LKMC), and which were recently reviewed by Becquart and Domain,45 and those that track the defects on a lattice, but without complete resolution of the atomic arrangement This later technique is predominately referred to as object Monte Carlo and used in such codes as BIGMAC27 or LAKIMOCA.28 More recently, several algorithmic ideas have been identified that, in combination, promise to deliver breakthrough KMC simulations for materials computations by making their performance essentially independent of the particle density and the diffusion rate disparity, and these will be further discussed as outstanding areas for future research at the end of the chapter KMC modeling of radiation damage involves tracking the location and fate of all defects, impurities, and solutes as a function of time to predict microstructural evolution The starting point in these simulations is often the primary damage state, that is, the spatially correlated locations of vacancy, self-interstitials, and transmutants produced in displacement cascades resulting from irradiation and obtained from MD simulations, along with the displacement or damage rate which sets the time scale for defect introduction The rates of all reaction–diffusion events then control the subsequent evolution or progression in time and are determined from appropriate activation energies for diffusion and dissociation; moreover, the reactions and rates of these reactions that occur between species are key inputs, which are assumed to be known The defects execute random diffusion jumps (in one, two, or three dimensions depending on the nature of the defect) with a probability (rate) proportional to their diffusivity Similarly, cluster dissociation rates are governed by a dissociation probability that is proportional to the binding energy of a particle to the cluster The events to be performed and the associated time-step of each Monte Carlo sweep are chosen from the RTA.17,18 In these simulations, the events which are considered to take place are thus diffusion, emission, irradiation, and possibly transmutation, and their corresponding occurrence rates are described below 1.14.4.1 Irradiation Rate The ‘irradiation’ rate, that is, the rate of impinging particles in the case of neutron and ion irradiation, is usually transformed into a production rate (number 397 per unit time and volume) of randomly distributed displacement cascades of different energies (5, 10, 20, keV) as well as residual Frenkel pairs (FPs) New cascade debris are then injected randomly into the simulation box at the corresponding rate The cascade debris can be obtained by MD simulations for different recoil energies T, or introduced on the basis of the number of FP expected from displacement damage theory In the case of KMC simulation of electron irradiation, FPs are introduced randomly in the simulation box according to a certain dose rate, assuming most of the time that each electron is responsible for the formation of only one FP This assumption is valid for electrons with energies close to MeV (much lower energy electrons may not produce any FP, whereas higher energy ones may produce small displacement cascades with the formation of several vacancies and SIAs) The dose is updated by adding the incremental dose associated with the scattering event of recoil energy T, using the Norgett–Robinson–Torrens expression8 for the number of displaced atoms In this model, the accumulated displacement per atom (dpa) is given by: Displacement per subcascade ¼ 0:8T 2ED ½3 where T is the damage energy, that is, the fraction of the energy of the particle transmitted to the PKA as kinetic energy and ED is the displacement threshold energy (e.g., 40 eV for Fe and reactor pressure vessel (RPV) steels51) 1.14.4.2 Transmutation Rate The rate of producing transmutations can also be included in KMC models, as deduced from the reaction rate density determined from the product of the neutron cross-section and neutron flux Like the irradiation rate, the volumetric production rate is used to introduce an appropriate number of transmutants, such as helium that is produced by (n, a) reactions in the fusion neutron environment, where the species are introduced at random locations within the material 1.14.4.3 Diffusion Rate Usually the rates of diffusion can be obtained from the knowledge of the migration barriers which have to be known for all the diffusing ‘objects’; that is, for the point defects in AKMC, OKMC, and EKMC or the clusters in OKMC or EKMC For isolated point 398 Kinetic Monte Carlo Simulations of Irradiation Effects defects, the migration barriers can be from experimental data, that is, from diffusion coefficients, or theoretically, using either ab initio calculations as described in Caturla et al.49 and Becquart and Domain50 or MD simulations as described in Soneda and Diaz de la Rubia.22 Since the migration energy depends on the local environment of the jumping species, it is generally not possible to calculate all of the possible activation barriers using ab initio or even MD simulations Simpler schemes such as broken bond models, as described in Soisson et al.,52 Le Bouar and Soisson,53 and Schmauder and Binkele,54 are then used Another kind of simpler model is based on the calculation of the system configurational energies before and after the defect jump In this model, the activation energy is obtained from the final Ef and the initial Ei as follows: Ef À Ei DE ẳ Ea0 ỵ ẵ4 2 where Ea0 is the energy of the moving species at the saddle point The modification of the jump activation energy by DE represents an attempt to model the effect of the local environment on the jump frequencies Indeed, detailed molecular statics calculations suggest that this represents an upper-bound influence of the effect,55 and although this is a very simplified model, the advantage is that this assumption maintains the detailed balance of jumps to neighboring positions The system configurational energies Ei and Ef , as well as the energy of the moving species at the saddle point Ea0 can be determined using interatomic potentials as described in Becquart et al.,26 Bonny et al.,44 Wirth and Odette,55 and Djurabekova et al.56 when they exist However, at present, this situation is only available for simple binary or ternary alloys This approach allows one to implicitly take into account relaxation effects as the energy at the saddle point which is used in the KMC and is obtained after relaxation of all the atoms The challenge in that case is the total number of barriers to be calculated, which is determined by the number of nearest neighbor sites included in the definition of the local atomic environment Without considering symmetries, this number is sN, where s is the number of species in the system In spite of using the fast techniques that were developed to find saddle points on the fly such as the dimer method,57 the nudged elastic band (NEB) method,58 or eigen-vector following methods,59 this number quickly becomes unmanageable Ideally, the alternative should be to find patterns in the dependence of the energy barriers on the configuration This is the Ea ẳ Ea0 ỵ approach chosen by Djurabekova and coworkers,56 using artificial intelligence systems For more complex alloys, for which no interatomic potentials exist, Ei and Ef can be estimated using neighbor pair interactions.60– 63 A recent example of the fitting procedure of a neighbor pair interactions model can be found in Ngayam Happy et al.63 A discussion of the two approaches applied to the Fe–Cu system has been published by Vincent et al.64 Also note that in the last 10 years, methods in which the possible transitions are found in some systematic way from the atomic forces rather than by simply assuming the transition mechanism a priori (e.g., activation–relaxation technique (ART) or dimer methods)65–68 have been devised The accuracy of the simulations is thus improved as fewer assumptions are made within the model However, interatomic potentials or a corresponding method to obtain the forces acting between atoms for all possible configurations is necessary and this limits the range of materials that can be modeled with these clever schemes The attempt frequency (nX in eqn [1]) can be calculated on the basis of the Vineyard theory69 or can be adjusted so as to reproduce model experiments 1.14.4.4 Emission/Dissociation Rate The emission or dissociation rate is usually the sum of the binding energy of the emitted particle and its migration energy As in the case of migration energy, the binding energies can be obtained using either experimental studies, ab initio calculations, or MD As stated previously, three kinds of KMC techniques (AKMC, OKMC, and EKMC) have been used so far to model microstructural evolutions during radiation damage In atomistic KMC, the evolution of a complex microstructure is modeled at the atomic scale, taking into account elementary atomic mechanisms In the case of diffusion, the elementary mechanisms leading to possible state changes are the diffusive jumps of mobile point defect species, including point defect clusters Typically, vacancies and SIAs can jump from one lattice site to another lattice site (in general first nearest neighbor sites) If foreign interstitial atoms such as C atoms or He atoms are included in the model as in Hin et al.,70,71 they lie on an interstitial sublattice and jump on this sublattice In OKMC, the microstructure consists of objects which are the intrinsic defects (vacancies, SIAs, dislocations, grain boundaries) and their clusters (‘pure’ clusters, such as voids, SIA clusters, He or C clusters), as well as mixed clusters such as clusters containing both He atoms, solute/impurity atoms, and Kinetic Monte Carlo Simulations of Irradiation Effects interstitials, or vacancies These objects are located at known (and traced) positions in a simulation volume on a lattice as in LAKIMOCA or a known spatial position as in BIGMAC and migrate according to their migration barriers In the EKMC approach,72,73 the microstructure also consists of objects The crystal lattice is ignored and objects’ coordinates can change continuously The only events considered are those which lead to a change in the defect population, namely clustering of objects, emission of mobile species, elimination of objects on fixed sinks (surface, dislocation), or the recombination between vacancy and interstitial defect species The migration of an object in its own right is considered an event only if it ends up with a reaction that changes the defect population In this case, the migration step and the reaction are processed as a single event; otherwise, the migration is performed only once at the end of the EKMC time interval Dt In contrast to the RTA, in which all rates are lumped into one total rate to obtain the time increment, in an EKMC scheme the time delays of all possible events are calculated separately and sorted by increasing order in a list The event corresponding to the shortest delay, ts, is processed first, and the remaining list of delay times for other events is modified accordingly by eliminating the delay time associated with the particle that just disappeared, adding delay times for a new mobile object, etc To illustrate the power of KMC for modeling radiation effects in structural materials and nuclear fuels, this chapter next considers two examples, namely the use of AKMC simulations to predict the coupled evolution of vacancy clusters and copper precipitates during low dose rate neutron irradiation of Fe–Cu alloys and the use of an OKMC model to predict the transport and diffusional release of fission product, silver, in tri-isotropic (TRISO) nuclear fuel These two examples will provide more details about the possible implementations of AKMC and OKMC models 1.14.5 Atomistic KMC Simulations of Microstructure Evolution in Irradiated Fe–Cu Alloys Cu is of primary importance in the embrittlement of the neutron-irradiated RPV steels It has been observed to separate into copper-rich precipitates within the ferrite matrix under irradiation As its role was discovered more than 40 years ago,74–76 Cu precipitation in a-Fe has been studied extensively 399 under irradiation as well as under thermal aging using atom probe tomography, small angle neutron scattering, and high resolution transmission electron microscopy Numerical simulation techniques such as rate theory or Monte Carlo methods have also been used to investigate this problem, and we next describe one possible approach to modeling microstructure evolution in these materials The approach combines an MD database of primary damage production with two separate KMC simulation techniques that follow the isolated and clustered SIA diffusion away from a cascade, and the subsequent vacancy and solute atom evolution, as discussed in more detail in Odette and Wirth,21 Monasterio et al.,34 and Wirth et al.77 Separation of the vacancy and SIA cluster diffusional time scales naturally leads to the nearly independent evolution of these two populations, at least for the relatively low dose rates that characterize RPVembrittlement.1,78–81 The relatively short time ($100 ns at 290 C) evolution of the cascade is modeled using OKMC with the BIGMAC code.77 This model uses the positions of vacancy and SIA defects produced in cascades obtained from an MD database provided by Stoller and coworkers82,83 and allows for additional SIA/ vacancy recombination within the cascade volume and the migration of SIA and SIA clusters away from the cascade to annihilate at system sinks The duration of this OKMC is too short for significant vacancy migration and hence the SIA/SIA clusters are the only diffusing defects These OKMC simulations, which are described in detail elsewhere,77 thus provide a database of initially ‘aged’ cascades for longer time AKMC cascade aging and damage accumulation simulations The AKMC model simulates cascade aging and damage evolution in dilute Fe–Cu alloys by following vacancy – nearest neighbor atom exchanges on a bcc lattice, beginning from the spatial vacancy population produced in ‘aged’ high-energy displacement cascades and obtained from the OKMC to the ultimate annihilation of vacancies at the simulation cell boundary, and including the introduction of new cascade damage and fluxes of mobile point defects The potential energy of the local vacancy, Cu–Fe, environment determines the relative vacancy jump probability to each of the eight possible nearest neighbors in the bcc lattice, following the approach described in eqn [4] The unrelaxed Fe–Cu vacancy lattice energetics are described using Finnis–Sinclair N-body type potentials The iron and copper potentials are from Finnis and Sinclair84 and Ackland et al.,85 respectively; and the iron–copper potential was developed 400 Kinetic Monte Carlo Simulations of Irradiation Effects by fitting the dilute heat of the solution of copper in iron, the copper vacancy binding energy, and the iron–copper [110] interface energy, as described elsewhere.55 Within a vacancy cluster, each vacancy maintains its identity as mentioned above, and while vacancy–vacancy exchanges are not allowed, the cluster can migrate through the collective motion of its constituent vacancies The saddle point energy, which is Ea0 in eqn [4], is set to 0.9 eV, which is the activation energy for vacancy exchange in pure iron calculated with the Finnis–Sinclair Fe potential.84 The time (DtAKMC) of each AKMC sweep (or step) is determined by DtAKMC ¼ (nPmax)À1, where Pmax is the highest total probability of the vacancy population and n is an effective attempt frequency This is slightly different than the RTA, in which an event chosen at random sets the timescale as opposed to always using the largest probability as done here In this work, n ¼ 1014 sÀ1 to account for the intrinsic vibrational frequency and entropic effects associated with vacancy formation and migration, as used in the previous AKMC model by Odette and Wirth.21 As mentioned, the possible exchange of every vacancy (i) to a nearest neighbor is determined by a Metropolis random number test15 of the relative vacancy jump probability (Pi/Pmax) during each Monte Carlo sweep Thus at least one, and often multiple, vacancy jumps occur during each Monte Carlo sweep, which is different from the RTA Finally, as mentioned above, as the total probability associated with a vacancy jump depends on the local environment, the intrinsic timescale (DtAKMC) changes as a function of the number and spatial distribution of the vacancy population, as well as the spatial arrangement of the Cu atoms in relation to the vacancies The AKMC boundary conditions remove (annihilate) a vacancy upon contact, but incorporate the ability to introduce point defect fluxes through the simulation volume that result from displacement cascades in neighboring regions as well as additional displacement cascades within the simulated volume The algorithms employed in the AKMC model are described in detail in Monasterio et al.34 and the remainder of this section will provide highlights of select results The AKMC simulations are performed in a randomly distributed Fe–0.3% Cu alloy at an irradiation temperature of 290 C and are started from the spatial distribution of vacancies from an 20 keV displacement cascade The rate of introducing new cascade damage is 1.13  10À5 cascades per second, with a cascade vacancy escape probability of 0.60 and a vacancy introduction rate of  10À4 vacancies per second, which corresponds to a damage rate of this simulation at $10À11 dpa sÀ1 Thus a new cascade (with recoil energy from 100 eV to 40 keV) occurs within the simulated volume (a cube of $86 nm edge length) every 8.8  104 s ($1 day), while an individual vacancy diffuses into the simulation volume every  104 s ($3 h) AKMC simulations have also been performed to study the effect of varying the cascade introduction rate from 1.13  10À3 to 1.13  10À7 cascades per second (dpa rates from  10À9 to  10À13 dpa sÀ1) The simulated conditions should be compared to those experienced by RPVs in light water reactors, namely from  10À12 to  10À11 dpa sÀ1, and to model alloys irradiated in test reactors, which are in the range of 10À9–10À10 dpa sÀ1 Figures and show representative snapshots of the vacancy and Cu solute atom distributions as a function of time and dose at 290 C Note, only the Cu atoms that are part of vacancy or Cu atom clusters are presented in the figure The main simulation volume consists of  106 atoms (100a0  100a0  100a0) of which 6000 atoms are Cu (0.3 at.%) Figure demonstrates the aging evolution of a single cascade (increasing time at fixed dose prior to introducing additional diffusing vacancies or new cascade), while Figure demonstrates the overall evolution with increasing time and dose The aging of the single cascade is representative of the average behavior observed, although the number and size distribution of vacancy-Cu clusters vary considerably from cascade to cascade Further, Figure is representative of the results obtained with the previous AKMC models of Odette and Wirth21 and Becquart and coworkers,24,26 which demonstrated the formation and subsequent dissolution of vacancy-Cu clusters Figure represents a significant extension of that previous work21,24,26 and demonstrates the formation of much larger Cu atom precipitate clusters that results from the longer term evolution due to multiple cascade damage in addition to radiation enhanced diffusion Figure 2(a) shows the initial vacancy configuration from an aged 20-keV cascade Within 200 ms at 290 C, the vacancies begin to diffuse and cluster, although no vacancies have yet reached the cell boundary to annihilate Eleven of the initial vacancies remain isolated, while thirteen small vacancy clusters rapidly form within the initial cascade volume The vacancy clusters range in size from two to six vacancies At this stage, only two of the vacancy clusters are associated with copper atoms, a divacancy Kinetic Monte Carlo Simulations of Irradiation Effects (a) (b) (a) (c) (d) (c) (d) (e) (f) (e) (f) Figure Representative vacancy (red circles) and clustered Cu atom (blue circles) evolution in an Fe–0.3% Cu alloy during the aging of a single 20 keV displacement cascade, at (a) initial (200 ns), (b) ms, (c) 48 ms, (d) 55 ms, (e) 83 ms, and (f) 24.5 h cluster with one Cu atom and a tetravacancy cluster with two Cu atoms From 200 ms to ms, the vacancy cluster population evolves by the diffusion of isolated vacancies through and away from the cascade region, and the emission and absorption of isolated vacancies in vacancy clusters, in addition to the diffusion of the small di-, tri-, and tetravacancy clusters Figure 2(b) shows the configuration about ms after the cascade By this time, 14 of the original vacancies have diffused to the cell boundary and annihilated, while 38 vacancies remain The vacancy distribution includes six isolated vacancies and seven vacancy clusters, ranging in size from two divacancy clusters to a ten vacancy cluster The number of nonisolated copper atoms has increased from 223 in the initial random distribution to 286 following the initial ms of cascade aging 401 (b) Figure Representative vacancy (red) and clustered Cu atom (blue) evolution in an Fe–0.3% Cu alloy with increasing dose at (a) 0.2 years (97 udpa), (b) 0.6 years (0.32 mdpa), (c) 2.1 years (1.1 mdpa), (d) 4.0 years (2.0 mdpa), (e) 10.7 years (4.4 mdpa), and (f) 13.7 years (5.3 mdpa) The evolution from to 48.8 ms involves the diffusion of isolated vacancies and di- and trivacancy clusters, along with the thermal emission of vacancies from the di- and trivacancy clusters Over this time, additional vacancies have diffused to the cell boundary and annihilated, and 20 additional Cu atoms have been incorporated into Cu or vacancy clusters Figure 2(c) shows the vacancy and Cu cluster population at 48.8 ms, which now consists of three isolated vacancies and four vacancy clusters, including a 4V–1Cu cluster, a 6V–4Cu cluster, a 7V cluster, and an 11V–1Cu cluster Figure 2(d) and 2(e) shows the vacancy and Cu cluster population at 54.8 and 82.8 ms, respectively During this time, the total number of vacancies has been further reduced from 31 to 21 of the original 52 vacancies, the vacancy cluster 402 Kinetic Monte Carlo Simulations of Irradiation Effects population has been reduced to three vacancy clusters (a 4V–1Cu, 7V, and 9V–1Cu), and 30 additional Cu atoms have incorporated into clusters because of vacancy exchanges Over times longer than 100 ms, the 4V–1Cu atom cluster migrates a short distance on the order of nm before shrinking by emitting vacancies, while the 7V and 9V–1Cu cluster slowly evolve by local shape rearrangements which produces only limited local diffusion Both the 7V and 9V–1Cu cluster are thermodynamically unstable in dilute Fe alloys at 290 C and ultimately will shrink over longer times The vacancy and Cu atom evolution in the AKMC model is now governed by the relative rate of vacancy cluster dissolution, as determined from the ‘pulsing’ algorithm, and the rate of new displacement damage and the diffusing supersaturated vacancy flux under irradiation Figure 2(f ) shows the configuration about 8.8  104 s ($24 h) after the initial 20 keV cascade Only 17 vacancies now exist in the cell, an isolated vacancy which entered the cell following escape from a 500 eV recoil introduced into a neighboring cell plus two vacancy clusters, consisting of 7V–1Cu and 9V–1Cu Three hundred and forty-five Cu atoms (of the initial 6000) have been removed from the supersaturated solution following the initial 24 h of evolution, mostly in the form of di- and tri-Cu atom clusters Figure 3(a) shows the configuration at about 0.1 mdpa (0.097 mdpa) and a time of 7.1  106 s ($82 days) Ten vacancies exist in the simulation cell, consisting of eight isolated vacancies and one 2V cluster, while 807 Cu atoms have been removed from solution in clusters, although the Cu cluster size distribution is clearly very fine The majority of Cu clusters contain only two Cu atoms, while the largest cluster consists of only five Cu atoms Figure 3(b) shows the configuration at a dose of 0.33 mdpa and time of 2.1  107 s (245 days) Only one vacancy exists in the simulation volume, while 1210 Cu atoms are now part of clusters, including 12 clusters containing or more Cu atoms Figure 3(c) shows the evolution at mdpa and 7.2  107 s ($2.3 years) Again, only one vacancy exists in the simulation cell, while 1767 Cu atoms have been removed from supersaturated solution A handful of well-formed spherical Cu clusters are visible, with the largest containing 13 Cu atoms With increasing dose, the free Cu concentration in solution continues to decrease as Cu atoms join clusters and the average Cu cluster size grows Figure 3(d) and 3(e) shows the clustered Cu atom population at about and 4.4 mdpa, respectively The growth of the Cu clusters is clearly evident when Figure 3(d) and 3(e) is compared At a dose of 4.4 mdpa, 48 clusters contain more than 10 Cu atoms, and the largest cluster has 28 Cu atoms The accumulated dose of 5.34 mdpa is shown in Figure 3(f ) At this dose, more than one-third of the available Cu atoms have precipitated into clusters, the largest of which contains 42 Cu atoms, corresponding to a precipitate radius of $0.5 nm Figure shows the size distribution of Cu atom clusters at 5.34 mdpa, corresponding to the configuration shown in Figure 3(f ) The vast majority of the 350 40 250 Number of clusters Number of clusters 300 200 150 30 20 10 100 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Cu cluster size 50 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Cu cluster size Figure Cu cluster size distribution obtained at 5.34 mdpa (Figure 2(f)) and 290 C, at a nominal dose rate of 10À11 dpa sÀ1 and a vacancy introduction rate of 10À4 sÀ1 Kinetic Monte Carlo Simulations of Irradiation Effects Cu clusters consist of di-, tri-, tetra-, and penta-Cu atom clusters However, as shown in the inset of Figure and as visible in Figure 3(f ), a significant number of the Cu atom clusters contain more than five Cu atoms Indeed, 29 clusters contain 15 or more Cu atoms (a number density of 1.2  1024 mÀ3), which corresponds to a cluster containing a single atom with all first and second nearest neighbor Cu atoms and a radius of 0.29 nm An additional 45 clusters contain at least nine Cu atoms (atom ỵ all first nearest neighbors), while clusters contain 23 or more atoms (number density of 3.8  1023 mÀ3) This AKMC simulation is currently continuing to reach higher doses However, the initial results are consistent with experimental observations and show the formation of a high number density of Cu atom clusters, along with the continual formation and dissolution of 3D vacancy-Cu clusters Figure shows a comparison of varying the dose rate from 10À9 to 10À13 dpa sÀ1 Each simulation was performed at a temperature of 290 C and introduced additional vacancies into the simulation volume at the rate of 10À4 sÀ1 The effect of increasing dose (a) (b) (c) Figure Comparison of the representative vacancy (red) and clustered Cu atom (blue) population at a dose of $1.9 mdpa and 290 C as a function of dose rate, at (a) 10À11 dpa sÀ1, (b) 10À9 dpa sÀ1, (c) 10À13 dpa sÀ1 403 rate at an accumulated dose of 1.9 mdpa is especially pronounced when comparing Figure 5(c) (10À13 dpa sÀ1) with Figure 5(a) (10À11 dpa sÀ1) and Figure 5(b) (10À9 dpa sÀ1) At the highest dose rate, a substantially higher number density of small 3D vacancy clusters is observed, which are often complexed with one or more Cu atoms Vacancy cluster nucleation occurs during cascade aging (as described in Figure 2) and is largely independent of dose rate, but cluster growth is dictated by the cluster(s) thermal lifetime at 290 C versus the arrival rate of additional vacancies, which is a strong function of the damage rate and vacancy supersaturation under irradiation Thus, the higher dose rates produce a larger number of vacancies arriving at the vacancy cluster sinks, resulting in the noticeably larger number of growing vacancy clusters Also, there is a corresponding decrease in the amount of Cu removed from the solution by vacancy diffusion In contrast, the effect of decreasing dose rate is greatly accelerated Cu precipitation Already at 1.9 mdpa, a number of large Cu atom clusters exist at a dose rate of 10À13 dpa sÀ1, with the largest containing 35 Cu atoms, as shown in Figure 5(c) The increased Cu clustering caused by a decrease in dose rate results from a reduction in the number of cascade vacancy clusters, which serve as vacancy sinks Thus, a higher number of free or isolated vacancies are available to enhance Cu diffusion required for the clustering and precipitation of copper While these flux effects are anticipated and have been predicted in rate theory calculations performed by Odette and coworkers,78,79 the spatial dependences of cascade production and microstructural evolution, in addition to correlated diffusion and clustering processes involving multiple vacancies and atoms are more naturally modeled and visualized using the AKMC approach While the results just presented in Figures 2–5 have shown the formation of subnanometer Cu-vacancy clusters and larger growing Cu precipitate clusters that result from AKMC simulations, which only consider vacancy-mediated diffusion, Becquart and coworkers have shown that Cu atoms in tensile positions can trap SIAs and therefore the Cu clustering behavior may also be influenced by interstitialmediated transport Ngayam Happy and coworkers63,86 have developed another AKMC model to model the behavior of FeCu under irradiation In this model, diffusion takes place via both vacancy and selfinterstitial atoms jumps on nearest neighbor sites The migration energy of the moving species is also determined using eqn [4], where the reference activation 404 Kinetic Monte Carlo Simulations of Irradiation Effects energy Ea0 depends only on the type of the migrating species Ea0 has been set equal to: the ab initio vacancy migration energy in pure Fe when a vacancy jumps towards an Fe atom (0.62 eV); the ab initio solute migration energy in pure Fe when a vacancy jumps towards a solute atom (0.54 eV for Cu); and the ab initio migration-60 rotation energy of the migrating atom in pure Fe when a dumbbell migrates (0.31 eV) Ei and Ef are determined using pair interactions, according to the following equation: X X eðiÞ Sj Sk ị ỵ Edumb ẵ5 Eẳ iẳ1;2 j