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 Comprehensive nuclear materials 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.

1.14.2 Modeling Challenges to Predict Irradiation Effects on Materials 394

1.14.4 KMC Modeling of Microstructure Evolution Under Radiation Conditions 396

1.14.5 Atomistic KMC Simulations of Microstructure Evolution in

1.14.6 OKMC Example: Ag Fission Product Diffusion and Release in

1.14.9 Summary and a Look at the Future of Nuclear Materials Modeling 408

Abbreviations

AKMC Atomistic kinetic Monte Carlo

BKL Boris, Kalos, and Lebowitz

EKMC Event kinetic Monte Carlo

FP Frenkel pairs

HTR High-Temperature Reactor

KMC Kinetic Monte Carlo

MD Molecular dynamics

NEB Nudged elastic band

NRT Norgett, Robinson, and Torrens

OKMC Object kinetic Monte Carlo

PKA Primary knockon atom

RPV Reactor pressure vessel

RTA Residence time algorithm

SIA Self-interstitial atoms

TRISO 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

example, crystal growth, plasma processing of materi-als, 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 over-all 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

Brownian motion and occasional collisions against one

another and systems with other defects (dislocations,

grain boundaries, surfaces, etc.) are in particular

ame-nable to multiscale modeling Within a multiscale

approach, atomistic simulations (utilizing either

elec-tronic structure calculations or semiempirical

poten-tials) investigate controlling mechanisms and

occurrence rates of diffusional and reactive

interac-tions 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

devel-opment of constitutive models for predictive

contin-uum scale models Simulating long-time materials

dynamics with reliable physical fidelity, thereby

providing a predictive capability applicable outside

limited experimental parameter regimes is the

prom-ise 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

10
12and10
3s 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 radia-tion-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 1 Pertinent processes span over more than 10 orders of magni-tude in length scale from the subatomic nuclear to

Irradiation temperature, n/g energy spectrum, flux, fluence, thermal cycling, and initial material microstructure inputs:

Long-range defect transport and annihilation

at sinks

Radiation enhanced diffusion and induced segregation of solutes

Nano/microstructure and local chemistry changes;

nucleation and growth of extended defects and precipitates

50nm

Lengthscale

Cascade aging and local solute redistribution

Gas diffusion and trapping

He and H generation

Defect recombination, clustering, and migration

Underlying microstructure (preexisting and evolving) impacts defect and solute

fate

Primary defect production and short-term annealing

Figure 1 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.

structural component level, and span 22 orders of

magnitude in time from the subpicosecond of nuclear

collisions to the decade-long component service

life-times.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

performance-limited 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

continu-ally initiated by the formation of energetic primary

knock-on atoms (PKAs) primarily through elastic

col-lisions with high-energy neutrons Concurrently, high

concentrations of fission products (in fuels) and

trans-mutants (in cladding and structural materials) are

gen-erated 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

col-lision 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

simula-tions if accurate potentials are available

The physics of primary damage production in

high-energy displacement cascades has been

exten-sively studied with MD simulations.3–8(seeChapter

1.11, Primary Radiation Damage Formation) The

key conclusions from the MD studies of cascade

evo-lution 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

va-cancy 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

displace-ment cascades tend to break up into lobes or

subcas-cades, which may also enhance recombination.4–7

Nevertheless, it is the subsequent diffusional

trans-port and evolution of the defects produced during

displacement cascades, in addition to solutes and

transmutant impurities, that ultimately dictate radia-tion effects in materials and changes in material micro-structure.1,2 Spatial correlations associated with the displacement cascades continue to play an important role in much larger scales as do 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 mech-anisms make modeling radiation effects in materials extremely challenging and experimental characteri-zation 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, theoreti-cal, 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 detri-mental 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 devel-opment of advanced computational approaches to simulate nucleation, growth, and coarsening of mi-crostructure in complex engineering materials

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

40 years ago by Besco,11Doran,12and 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

com-putational materials science, we shall focus our

atten-tion on KMC simulaatten-tion 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.9,10They provide a

solu-tion to the Master Equasolu-tion which describes a

physi-cal 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 Elcock17to 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

transi-tion 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

ther-mally activated eventx, such as a vacancy jump in

an alloy or the jump of a void can be expressed as:

GX¼ nX exp
Ea

kBT

½1

wherenXis the attempt frequency,kBis Boltzmann’s constant,T is the absolute temperature, and Eais 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 extract-ing a random number and comparextract-ing it to the relative probability The associated time-step length dt and average time-step lengthDt are given by:

dt ¼P
lnr

n GX andDt ¼P1

where r is a random number between 0 and 1 The RTA is also known as the BKL (Bortz, Kalos, Lebowitz) algorithm.19Other techniques are possible,

as described by Chatterjee and Vlachos.20The basic steps in a KMC simulation can be summarized thus:

1 Calculate the probability (rate) for a given event

to occur

2 Sum the probabilities of all events to obtain a cumulative distribution function

3 Generate a random number to select an event from all possible events

4 Increase the simulation time on the basis of the inverse sum of the rates of all possible events

Dt ¼ Pw

i N i R i

0

@

1

A, where w is a random deviate that assures a Poisson distribution in time-steps andN andR are the number and rate of each event i

5 Perform the selected event and all spontaneous events as a result of the event performed

6 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 radi-ation effects on materials.21–50 Advantages of KMC models include the ability to capture spatial correla-tions in a full 3D simulation with atomic resolution, while ignoring the atomic vibration time scales cap-tured by MD models In KMC, individual point defects, point defect clusters, solutes, and impurities are treated

as objects, either on or off an underlying crystallo-graphic 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

treatment of time scales or step between individual

events Within the OKMC designation, it is also

possi-ble 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,45and those that track the defects

on a lattice, but without complete resolution of the

atomic arrangement This later technique is

predomi-nately 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

microstruc-tural evolution The starting point in these simulations

is often the primary damage state, that is, the

spa-tially 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

gov-erned 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

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 energiesT, or introduced on the basis of the number of FP expected from displace-ment 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 1 MeV (much lower energy electrons may not produce any FP, whereas higher energy ones may produce small displacement cascades with the forma-tion 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 subcascade¼0:8T

2ED

½3 whereT is the damage energy, that is, the fraction of the energy of the particle transmitted to the PKA as kinetic energy andEDis the displacement threshold energy (e.g., 40 eV for Fe and reactor pressure vessel (RPV) steels51)

The rate of producing transmutations can also be included in KMC models, as deduced from the reac-tion rate density determined from the product of the neutron cross-section and neutron flux Like the irra-diation 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

defects, the migration barriers can be from

experi-mental data, that is, from diffusion coefficients, or

theoretically, using either ab initio calculations as

described in Caturla et al.49

and Becquart and Domain50or 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 Soissonet al.,52

Le Bouar and Soisson,53and 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

finalEf and the initialEias follows:

Ea¼ Ea0þEf
Ei

2 ¼ Ea0þDE

whereEa0is 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

frequen-cies Indeed, detailed molecular statics calculations

suggest that this represents an upper-bound influence

of the effect,55and 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 energiesEiandEf, as

well as the energy of the moving species at the saddle

pointEa0can be determined using interatomic

poten-tials as described in Becquartet al.,26

Bonnyet 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

relax-ation 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

environ-ment Without considering symmetries, this number

issN

, wheres is the number of species in the system

In spite of using the fast techniques that were

devel-oped to find saddle pointson the fly such as the dimer

method,57the nudged elastic band (NEB) method,58

or eigen-vector following methods,59 this number

quickly becomes unmanageable Ideally, the

alterna-tive should be to find patterns in the dependence of

the energy barriers on the configuration This is the

approach chosen by Djurabekova and coworkers,56 using artificial intelligence systems For more complex alloys, for which no interatomic potentials exist,Eiand

Efcan be estimated using neighbor pair interactions.60– 63

A recent example of the fitting procedure of a neigh-bor 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 mechanisma priori (e.g., activation–relaxation technique (ART) or dimer methods)65–68have been devised The accuracy of the simulations is thus improved as fewer assumptions are made within the model However, interatomic poten-tials 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

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 techni-ques (AKMC, OKMC, and EKMC) have been used so far to model microstructural evolutions during radia-tion damage In atomistic KMC, the evoluradia-tion of

a complex microstructure is modeled at the atomic scale, taking into account elementary atomic anisms In the case of diffusion, the elementary mech-anisms leading to possible state changes are the diffusive jumps of mobile point defect species, includ-ing 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 Hinet 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, dis-locations, grain boundaries) and their clusters (‘pure’ clusters, such as voids, SIA clusters, He or C clus-ters), as well as mixed clusters such as clusters con-taining both He atoms, solute/impurity atoms, and

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 intervalDt 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

radi-ation effects in structural materials and nuclear fuels,

this chapter next considers two examples, namely the

use of AKMC simulations to predict the coupled

evo-lution of vacancy clusters and copper precipitates

dur-ing low dose rate neutron irradiation of Fe–Cu alloys

and the use of an OKMC model to predict the

trans-port 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,74–76Cu

precipitation in a-Fe has been studied extensively

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 pri-mary 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 Monasterioet al.,34

and Wirthet al.77

Separation of the vacancy and SIA cluster diffusional time scales natu-rally leads to the nearly independent evolution of these two populations, at least for the relatively low dose rates that characterize RPV embrittlement.1,78–81 The relatively short time (100 ns at 290C) evolu-tion 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 migra-tion and hence the SIA/SIA clusters are the only diffus-ing defects These OKMC simulations, which are described in detail elsewhere,77thus provide a database

of initially ‘aged’ cascades for longer time AKMC cas-cade 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 popula-tion produced in ‘aged’ high-energy displacement cascades and obtained from the OKMC to the ulti-mate annihilation of vacancies at the simulation cell boundary, and including the introduction of new cas-cade damage and fluxes of mobile point defects The potential energy of the local vacancy, Cu–Fe, envi-ronment determines the relative vacancy jump prob-ability 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 ener-getics are described using Finnis–Sinclair N-body type potentials The iron and copper potentials are from Finnis and Sinclair84and Acklandet al.,85

respec-tively; and the iron–copper potential was developed

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

else-where.55Within a vacancy cluster, each vacancy

main-tains its identity as mentioned above, and while

vacancy–vacancy exchanges are not allowed, the

clus-ter can migrate through the collective motion of

its constituent vacancies The saddle point energy,

which isEa0ineqn [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 DtAKMC¼ (nPmax)
1, 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 1014s
1to 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

dif-ferent from the RTA Finally, as mentioned above, as

the total probability associated with a vacancy jump

depends on the local environment, the intrinsic

time-scale (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

(annihi-late) a vacancy upon contact, but incorporate the

ability to introduce point defect fluxes through the

simulation volume that result from displacement

cas-cades 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

ran-domly distributed Fe–0.3% Cu alloy at an irradiation

temperature of 290C and are started from the

spa-tial distribution of vacancies from an 20 keV

displace-ment cascade The rate of introducing new cascade

damage is 1.13 10
5 cascades per second, with a

cascade vacancy escape probability of 0.60 and a

vacancy introduction rate of 1 10
4vacancies per second, which corresponds to a damage rate of this simulation at 10
11dpa s
1 Thus a new cascade (with recoil energy from 100 eV to 40 keV) occurs within the simulated volume (a cube of86 nm edge length) every 8.8 104s (1 day), while an individ-ual vacancy diffuses into the simulation volume every

1 104s (3 h) AKMC simulations have also been performed to study the effect of varying the cascade introduction rate from 1.13 10
3 to 1.13 10
7 cascades per second (dpa rates from 1 10
9 to

1 10
13dpa s
1) The simulated conditions should

be compared to those experienced by RPVs in light water reactors, namely from 8 10
12 to 8 10
11 dpa s
1, and to model alloys irradiated in test reac-tors, which are in the range of 10
9–10
10dpa s
1 Figures 2 and 3 show representative snapshots

of the vacancy and Cu solute atom distributions

as a function of time and dose at 290C 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 2 106

atoms (100a0 100a0 100a0) of which 6000 atoms are Cu (0.3 at.%).Figure 2demonstrates the aging evolution

of a single cascade (increasing time at fixed dose prior

to introducing additional diffusing vacancies or new cascade), while Figure 3 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 do vary consider-ably from cascade to cascade Further, Figure 2 is representative of the results obtained with the previ-ous AKMC models of Odette and Wirth21 and Becquart and coworkers,24,26which demonstrated the formation and subsequent dissolution of vacancy-Cu clusters Figure 3 represents a significant extension

of that previous work21,24,26and demonstrates the for-mation 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 configura-tion from an aged 20-keV cascade Within 200ms at

290C, 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 clus-ters are associated with copper atoms, a divacancy

cluster with one Cu atom and a tetravacancy cluster

with two Cu atoms From 200ms to 2 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 2 ms after the cascade

By this time, 14 of the original vacancies have

dif-fused 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 2 ms of

cas-cade aging

The evolution from 2 to 48.8 ms involves the dif-fusion of isolated vacancies and di- and trivacancy clusters, along with the thermal emission of vacancies from the di- and trivacancy clusters Over this time,

7 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 clus-ter population at 48.8 ms, which now consists of three isolated vacancies and four vacancy clusters, includ-ing 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 num-ber of vacancies has been further reduced from 31 to

21 of the original 52 vacancies, the vacancy cluster

(a)

(c)

(b)

(d)

Figure 2 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) 2 ms, (c) 48 ms, (d) 55 ms,

(e) 83 ms, and (f) 24.5 h.

Figure 3 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).

population has been reduced to three vacancy

clus-ters (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 1 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

ther-modynamically unstable in dilute Fe alloys at 290C

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’

algo-rithm, and the rate of new displacement damage and

the diffusing supersaturated vacancy flux under

irra-diation Figure 2(f ) shows the configuration about

8.8 104s (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

super-saturated solution following the initial 24 h of

evolu-tion, 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 distri-bution 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 5 or more

Cu atoms.Figure 3(c)shows the evolution at 1 mdpa and 7.2 107s (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 2 and 4.4 mdpa, respectively The growth of the Cu clusters

is clearly evident whenFigure 3(d) and 3(e) is com-pared 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

inFigure 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 of0.5 nm

Figure 4shows the size distribution of Cu atom clusters at 5.34 mdpa, corresponding to the configu-ration shown inFigure 3(f ) The vast majority of the

2 0

350

40

30

20

10

0

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

300

250

200

150

100

50

4 6 8 10 12 14 16 18 20 22

Cu cluster size

Cu cluster size

24 26 28 30 32 34 36 38 40 42

Figure 4 Cu cluster size distribution obtained at 5.34 mdpa ( Figure 2(f) ) and 290C, at a nominal dose rate of

10
11dpa s
1and a vacancy introduction rate of 10
4s
1.