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 1. 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 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.


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 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 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
½1Š
GX ¼ nX exp À
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
1
and Dt ¼ P
dt ¼ P
GX

GX
n

½2Š

n

where r is a random number between 0 and 1. 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:
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
sum1 of the rates of all possible events
0
@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.
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 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 1 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

½3Š

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

¼ Ea0 þ
½4Š
2
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 ¼ Ea0 þ

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 Â 10À5 cascades per second, with a
cascade vacancy escape probability of 0.60 and a

vacancy introduction rate of 1 Â 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 Â 104 s ($1 day), while an individual vacancy diffuses into the simulation volume every
1 Â 104 s ($3 h). AKMC simulations have also been
performed to study the effect of varying the cascade
introduction rate from 1.13 Â 10À3 to 1.13 Â 10À7
cascades per second (dpa rates from 1 Â 10À9 to
1 Â 10À13 dpa sÀ1). The simulated conditions should
be compared to those experienced by RPVs in light
water reactors, namely from 8 Â 10À12 to 8 Â 10À11
dpa sÀ1, and to model alloys irradiated in test reactors, which are in the range of 10À9–10À10 dpa sÀ1.
Figures 2 and 3 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 2 Â 106 atoms
(100a0 Â 100a0 Â 100a0) of which 6000 atoms are Cu
(0.3 at.%). Figure 2 demonstrates 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 considerably from cascade to cascade. Further, Figure 2 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 3 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 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.

cluster with one Cu atom and a tetravacancy cluster
with two Cu atoms. From 200 ms 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 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 2 ms of cascade aging.

401

(b)

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).

The evolution from 2 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,
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 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 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 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 Â 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 Â 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 Â 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 Â 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 2 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 4 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
0
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
Cu cluster size

50

0
2

4

6

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

Figure 4 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 4 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 Â 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 þ all first nearest neighbors), while 9 clusters
contain 23 or more atoms (number density of
3.8 Â 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 5 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 5 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 ðSj À Sk Þ þ Edumb
½5Š
E¼
i¼1;2

j
where i equals 1 or 2 and corresponds to first or
second nearest neighbor interactions, respectively,

and where j and k refer to the lattice sites and Sj
(respectively Sk) is the species occupying site j
(respectively k): Sj in {X, V} where X ¼ Fe or Cu.
A more detailed description of the model can be
found in Ngayam Happy et al.63
In this study, various Cu contents were simulated
(0, 0.18, 0.8, and 1.34 at.%) at three different temperatures (300, 400, and 500  C). Without going into
too much detail here, one can state that these AKMC
simulation results are qualitatively similar to those
presented in Figures 2–5, which showed the formation
of small, vacancy-solute clusters and copper enriched
cluster/precipitate formation at 300  C. Similarly, the
effect of decreasing dose rate in high Cu content alloys
was also found to accelerate Cu precipitation.
This model does show that the formation of the Cu
clusters/precipitates during neutron irradiation takes
place via two different mechanisms depending upon
the Cu concentration. In a highly Cu supersaturated
matrix, precipitation is accelerated by irradiation,
whereas in the case of low Cu contents, Cu precipitates form by induced segregation on vacancy clusters.
The influence of temperature was investigated for
an Fe–0.18 wt% Cu alloy irradiated at a flux of
2.3 Â 10À5 dpa sÀ1. At 400 and 500  C, neither Cu
precipitates nor Cu-vacancy clusters were formed,
in agreement with the results of Xu et al.87 At these
temperatures, the model indicates that the vacancy
clusters are not stable and induced segregation is thus
hindered. Another interesting result obtained with
this model is that the presence of Cu atoms in the
matrix was found to decrease the point defect cluster

sizes because of the strong interactions of Cu with
both vacancies and SIAs.

1.14.6 OKMC Example: Ag Fission
Product Diffusion and Release in
TRISO Nuclear Fuel
The second example demonstrates mesoscale KMC
model simulations of the diffusion of silver (Ag)
through the pyrolytic carbon and silicon carbide
containment layers of a TRISO nuclear fuel particle.
The model atomically resolves Ag, but provides a
nonatomic, mesoscale medium of carbon and silicon
carbide that includes a variety of defect features
including grain boundaries, the carbon–silicon carbide interfaces, cracks, precipitates, and nanocavities.
These defect features can serve as either fast diffusional pathways or traps for the migrating silver. The
model consists of a 2D slab geometry incorporating
the pyrolytic carbon and silicon carbide layers, with
incident silver atoms placed at the innermost pyrolytic carbon layer, as described in more detail in
Meric de Bellefon and Wirth.88
The key input parameters to the model (diffusion
coefficients, trap binding energies, interface characteristics) are determined from available experimental
data, or parametrically varied, until more precise
values become available from lower length scale
modeling. The predicted results, in terms of the
time/temperature dependence of silver release during postirradiation annealing and the variability of
silver release from particle to particle have been
compared to available experimental data from the
German High-Temperature Reactor (HTR) Fuel
Program,89 as shown below in Figure 7, and studies
performed by the Japan Atomic Energy Research

Institute ( JAERI).90
Figure 6 presents KMC simulation results, which
shows the effect of different grain geometries in SiC
on silver release during postirradiation annealing. In
this model, the grains are considered to have a rectangular geometry. The smaller dimension is parallel
to the interfaces and has a fixed length of 1 mm. The
longer dimension, parallel to the radial direction in
a TRISO fuel particle, has a variable length that is
uniformly distributed among grains over a range from
1 to 40 mm, as shown in the upper plot of Figure 6.
Such a grain distribution mimics a highly columnar
structure, as is often observed experimentally.91
The diffusion coefficient for silver transport
within the grain boundaries has been assumed to be
three orders of magnitude higher than in bulk SiC.
As expected, within this model, the presence of
grains provides fast diffusion paths for silver transport and accelerates the released fraction. Adding a


Kinetic Monte Carlo Simulations of Irradiation Effects

405

1

Ag release fraction

Decreasing grain
width range

0.1
IPyC

OPyC

SiC

Fixed grain
height (1 µm)

0.01

Grain width range

0.001

0

50

1–40 µm
1–1 µm
0.1–0.5 µm
5–10 µm
No grains

100
150
200
Time (h) during 1700 ЊC PIA

250

Figure 6 Kinetic Monte Carlo simulations of the fractional release of Ag through the pyrolytic carbon and silicon carbide
layers at 1700  C, which demonstrate the influence of the grain geometry in SiC on silver release. The rectangular to
columnar grains have a height of 1 mm and width in the range of 1–40 mm (along the entire SiC layer).

1

Released fraction of silver

AVR experiments:
AVR 82/9 (1600 ЊC)
AVR 74/11 (1700 ЊC)
AVR 76/18 (1800 ЊC)
Simulation:
Particle 23 (1600 ЊC)
Particle 24 (1700 ЊC)
Particle 25 (1800 ЊC)

0.1

0.01

0.001
0

50

100

150

200

250

Heating time (h)
Figure 7 Kinetic Monte Carlo simulations of the fractional release of silver, as compared with the measured fractional
release of silver at 1600, 1700, and 1800  C. The microstructure of the 1600 and 1700  C simulations is identical and
contains an isotropic grain geometry in silicon carbide that consists of 1-mm long square grains with a grain boundary
diffusivity 100 higher than in bulk. The microstructure of the 1800  C simulation is similar except that the SiC grains
characteristics have been modified to match the measured release at 1800  C. Faster transport through grain
boundaries is needed, which is obtained by implementing a highly columnar structure with grains that are
0.5 mm wide and with (radial) lengths between 10 and 40 mm, and a high grain boundary diffusivity that is 2000 times
higher than in bulk.


406

Kinetic Monte Carlo Simulations of Irradiation Effects

columnar structure in those grains further increases
the release of silver; the released fraction increases
from 50 to 80% when the grain distribution shifts
from an isotropic structure (grains are 1 mm squares)
to a highly columnar structure (length of grains uniformly distributed from 1 to 40 mm) after 270 h of
heating at 1700  C.
Figure 7 presents KMC simulation results of the
transport of silver through a given PyC/SiC/PyC

microstructure during postirradiation thermal annealing at 1600, 1700, and 1800  C, as well as results
from three German experimental release measurements performed at annealing temperatures of 1600,
1700, and 1800  C. The simulated microstructures
include reflective interfaces, trapping cavities, and a
grain boundary structure in the SiC layer. The
microstructures for the 1600 and 1700  C simulations
(particle 23 and 24) are identical and contain an
isotropic grain geometry in SiC that consists of
1-mm long square grains with a grain boundary diffusivity 100Â higher than in bulk. In the 1800  C simulation (particle 25), the SiC grain characteristics are
varied to match the measured release at 1800  C.
Faster transport through grain boundaries is required
to match the experimental results, which is obtained
by implementing a highly columnar structure in
which the grains are 0.5 mm-wide and have a radial
length between 10 and 40 mm into SiC, as well as a
much higher grain boundary diffusivity that is 2000
times higher than in bulk.

1.14.7 Some Limits of KMC
Approaches
AKMC is a versatile method that can be used to
simulate the evolution of materials with complex
microstructure at the atomic scale by modeling the
elementary atomic mechanisms. It has been used
extensively to study phase transformations such as
precipitation, phase separation, and/or ordering in
many systems, as discussed in a recent review.92
Despite the fact that the algorithm is fairly simple,
the method is most of the time nontrivial to implement in the case of realistic materials (as opposed to
AB alloys for instance). Indeed, the determination of

the total potential energy of the system, that is, the
construction of the cohesive model when the chemistry of the system under study is complex and involves
many species or a complex crystallographic structure,
is difficult to obtain. Furthermore, the knowledge of
all the possible events and the rates at which they

occur, that is, the possible migration paths as well as
their energies is nontrivial. On rigid lattices, the
migration paths are more obvious to determine and
cluster expansion type methods may be extended to
determine the saddle point energies as a function of
the local chemical environment. This can, however,
take a very large amount of calculation time when
there is a drastic difference in the local environment.
Furthermore, complicated correlated motions such as
the adatom diffusion on the (100) surfaces of fcc
metals which occurs by a two-atom concerted displacement, in which the adatom replaces a surface
atom, which in turn becomes an adatom, cannot be
modeled within the simple scheme usually followed
in AKMC of jumps to 1nn neighbor sites.
Another drawback is that to be efficient, it is
tempting to use rigid lattices as a large number of
KMC steps have to be performed. This can lead to an
approximate (or even completely unrealistic) treatment of microstructure elements such as incoherent
carbide precipitates, SIA clusters, or interstitial dislocation loops. Note, however, that it is possible to
perform off-lattice AKMC, which will of course
require more time consuming simulations, as proposed recently by Mason et al.93 to investigate phase
transformation in Al–Cu–Mg alloys. The authors
noticed that the use of flexible lattices instead of
rigid ones affected the mobility of the vacancies as

well as the driving force of the reaction and therefore
the rate at which phase separation took place. Furthermore, note that off-lattice AKMC also requires
an equilibrium continuous cohesive model, which is
difficult to build for multicomponent alloys.
At the moment, OKMC methods have been mostly
used to investigate the annealing of the primary damage as in Heinisch and Singh14 or Domain et al.28 or the
effect of temperature change in the damage accumulation,94 but its strongest contribution in the field seems
to be the study of parameters or assumptions such as
the motion, 3D versus 1D motion, mobility of the SIA
clusters,95–98 or corroboration of theoretical assumptions such as the analytical description of the sink
strength.99 They have been used also to model as well
as reexamine simple experiments such as He desorption in W100 or in Fe101 as well as the influence of C in
isochronal annealing experiments. It can also be used to
determine the production rate or source term (i.e., the
‘irradiation flux’) in mean field rate theory (MFRT)
models, as discussed in the chapter on MFRT. As
no spatial correlation is explicitly considered in these
techniques, the source term has to take into account
intracascade agglomeration and recombination. The


Kinetic Monte Carlo Simulations of Irradiation Effects

amount of agglomeration can be obtained by annealing
the cascade debris using OKMC.102
In the OKMC, the evolution of individual objects
is simulated on the basis of time scales that encompass individual atomic diffusive jumps, dominated by
the very fast events. This method is not efficient at
high temperatures and/or high doses. The difficulty
is the inability to model sufficiently high doses necessary for macroscopic materials behavior due to the

focus on fast dynamics.
The time-step between events is much longer
in EKMC models, which require that a reaction
(e.g., clustering among like defects, annihilation
among opposite defects, cluster dissolution, or new
cascade introduction) occur within each Monte Carlo
sweep. EKMC can therefore simulate much longer
times and therefore simulate materials evolution
over higher doses. It is most efficient when few objects
are present in the simulation box. But questions relate
to whether the time-steps are too large to reliably
capture the underlying fast dynamics and whether
the assumed binary interactions are sufficient to reliably calculate interaction probabilities. Further,
EKMC models developed to date have not included
all of the relevant microstructural evolution mechanisms, but they do represent an interesting approach in
the limit of long time-step Monte Carlo simulations.

1.14.8 Advanced KMC Methods
To overcome some of the limitations described
above, two techniques have been recently derived:
the first-passage Green’s function and synchronous
parallel KMC. The first-passage Green’s function
approach has been successfully used in various subareas of computational science but so far has escaped
the widespread attention of materials scientists. Preliminary work indicates that constructive use of the
first-passage Green’s function approach for modeling
radiation microstructures is possible46,47 but will require considerable effort to develop a time-dependent
formulation of the method. However, the potential
payoff is well worth it: preliminary estimates show
that it should be possible to boost the effective performance of the (exact) Monte Carlo simulations by
several orders of magnitude. The speedup of the simulation with the first-passage Green’s function approach can be estimated from a rough argument

based on the number of events required to process
the diffusion of a vacancy from one void to another in
Oswald coarsening. For example, voids in irradiated

407

alloys are separated by $0.2 mm. But as the atomic
jump distance is typically on the order of 0.25 nm, the
ratio between the required diffusion length and the
atomic jump distance is around 103. The ripening
process consists mainly of vacancies detaching from
one void and diffusing to a neighbor. If this is done
with a direct hop simulation, then $106 random walk
diffusion hop events would be required, from vacancy
emission to absorption. Each such event requires the
generation of one or more random numbers and
changes in bookkeeping tables that store current positions for each defect. Using the first-passage Green’s
function algorithm, the vacancy in most cases will
reach the vicinity of a void in several (10–30) steps,
each of which will require <10 times the number of
calculations for a simple hop. Thus, it is possible to
conclude that the simulation of the ripening of voids
(which is similar to modeling radiation-induced precipitation processes) at this spacing would require
about three orders of magnitude less computer time
than the current KMC programs.
At the moment, the first-passage Green’s function
KMC appears to work very well for some specific
cases such as the one mentioned above, but when one
tries to model a more realistic case such as the continuous introduction of displacement cascades in
which all the defects are very close to each other

and diffuse with very different diffusivities, the possible ‘protective domains’ become very small and the
technique is not very efficient.
One additional difficulty with KMC simulations is
the fact that the current state-of-the-art simulation
codes utilize serial computing only. Thus, there exists
a critical need to accelerate the maturation of multiscale modeling of fusion reactor materials, namely
the development of advanced and highly efficient
Monte Carlo algorithms for the simulation of materials evolution when controlling processes occur with
characteristic time scales between 10À12 and $100 s.
There has recently been some activity associated with
synchronous parallel KMC48; however, the problem
of dealing with highly inhomogeneous regions and
species diffusing at rates that are disparate by many of
orders of magnitude tends to greatly reduce the parallel algorithm performance. Clearly, more effort is
needed on the development of advanced algorithms
for KMC simulations. Further, it is imperative that
the algorithms developed be highly efficient in
today’s massively parallel computing architectures.
At the moment, no single KMC method can
efficiently treat the complex microstructures and
kinetic evolution associated with radiation effects in


408

Kinetic Monte Carlo Simulations of Irradiation Effects

multi-component materials, nor efficiently balance
the computational requirements to treat inhomogeneous domains consisting of very different defect
densities. It is possible that a combination of different

techniques in the course of a single simulation will be
the most efficient pathway.

1.14.9 Summary and a Look at the
Future of Nuclear Materials Modeling
This chapter has attempted to illustrate the power
of KMC for modeling radiation effects in structural
materials and nuclear fuels, following an introduction
and review of the Monte Carlo technique. Monte
Carlo modeling, first developed by Metropolis and
coworkers9,10 during the Manhattan project does provide a physically satisfying technique to simulate the
stochastic evolution of defect evolution in materials
science and in fact has been used to simulate irradiation effects on materials for four decades. There are
three main types of KMC modeling used in irradiation effects, namely event Monte Carlo, object Monte
Carlo, and atomistic Monte Carlo.
This chapter has focused on describing the atomistic KMC and OKMC methods by providing two
examples of successful KMC simulations to predict
the coupled evolution of vacancy clusters and copper
precipitates during low dose rate neutron irradiation
of Fe–Cu alloys and the transport and diffusional
release of the fission product, silver, in TRISO nuclear fuel. These examples clearly demonstrate the
power and ability of KMC models to capture the
spatial correlations that can be an important component of microstructural evolution in nuclear materials. Yet, the further widespread application of KMC
models will require algorithmic developments that
can more readily treat the wide range of time scales
inherent in microstructural evolution and yet effectively incorporate the rare-event dynamics in integrating system performance to realistic time and
irradiation dose exposures.
Thus, the challenges that must be overcome in
future nuclear materials modeling include:
 bridging the inherently multiscale time and length

scales which control materials degradation in nuclear environments;
 dealing with the complexity of multicomponent
materials systems, including those in which the
chemical composition is continually evolving due
to nuclear fission and transmutation;
 discovering the unknown to prevent technical
surprises;

 transcending ideal materials systems to engineer
materials and components; and
 incorporating error assessments within each modeling scale and propagating the error through the
scales to determine the appropriate confidence
bounds on performance predictions.
Successful overcoming of these challenges will
result in nuclear materials performance models that
can predict the properties, performance, and lifetime
of nuclear fuels, cladding, and components in a variety of nuclear reactor types throughout the full life
cycle, and provide the scientific basis for the computational design of advanced new materials. While the
current chapter is focused on the KMC modeling
methodology, it is important to note the challenges
of predictive materials models of irradiation effects.
High performance computing at the petascale, exascale, and beyond is a necessary and indeed critical
tool in resolving these challenges, yet it is important
to realize that exascale computing on its own will not
be sufficient. This is best recognized from a simple
example considering the computational degrees of
freedom in a MD simulation. Assuming that reliable,
multicomponent interatomic potentials existed for
the nuclear fuel rod and cladding in a nuclear
power plant and that a constant time-step of

2 Â 10À15 s could sufficiently capture the physics of
high-energy atomic collisions to conserve energy;
then to simulate 1 day of evolution of 1 cm tall,
1 cm diameter fuel pellet clad and zirconium clad
would require $6 Â 1022 atoms for $4 Â 1019 timesteps. For comparison, the LAMPPS MD code using
classical force fields has been benchmarked with
40 billion atoms (4 Â 1010) and 100 time-steps on
10 000 processors of the RedStorm at Sandia
National Laboratory with a wall clock time of 980 s
and on 64 000 processors of the BlueGene Light at
Lawrence Livermore National Laboratory with a
wall clock time of 585 s.103 Thus, even assuming
optimistic scaling and parallelization, brute force
atomistic simulation of the first full power day of a
nuclear fuel pellet in a reactor by MD will remain
well beyond the reach of high performance computing capabilities for the next decade.

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