Nuclear Science
NEA/NSC/R(2019)2
June 2020www.oecd-nea.org

State-of-the-Art Reporton Multi-Scale Modelling
Methods

Nuclear Energy Agency

NEA/NSC/R(2019)2
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11 June 2020

NUCLEAR ENERGY AGENCY
NUCLEAR SCIENCE COMMITTEEState-of-the-Art Report on Multi-Scale Modelling Methods

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2 │ NEA/NSC/R(2019)2
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STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Foreword
The Working Party on Multi-scale Modelling of Fuels and Structural Materials for Nuclear
Systems (WPMM) was established under the auspices of the Nuclear Energy Agency(NEA) Nuclear Science Committee (NSC) to review multi-scale models and simulations
as validated tools to predict the behaviour and performances of fuels and structural
materials, in support of nuclear systems design and fuel fabrication. The WPMM’s
objective is to promote the exchange of information on theoretical and computational
methods, experimental validation, and other topics related to modelling and simulation of
nuclear materials. The WPMM established the Expert Group on Structural Materials
Modelling in 2009 to provide targeted critical reference reviews of the state of the art in
relation to the use of multi-scale modelling so as to describe the changes induced byirradiation in structural nuclear materials. The aim of this expert group is to reliably
reproduce experimental data, while providing the keys to understand and interpret existing
experimental results, with a view to predicting the behaviour of structural nuclear materials
under unexplored conditions and supporting the choice and the development of newmaterials. The present volume presents a state-of-the-art review of physical multi-scalemodels rooted in computational physics, to describe the properties and behaviour ofmaterials of interest for the nuclear community.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│34 │ NEA/NSC/R(2019)2

Acknowledgements
The Nuclear Energy Agency (NEA) wishes to express its sincere gratitude to
Dr Veena Tikare (Sandia National Laboratory, United States), Chair of the Expert Groupon Multi-scale Modelling Methods, and to Dr Ram Devanathan (Pacific Northwest
National Laboratory, United States), both of whom co-ordinated this report. The NEA
would also like to thank the members of the Expert Group on Structural Materials
Modelling and the authors who contributed to this report. Special thanks are also due toProfessor Theodore M. Besmann (United States), Chair of the Working Party on Multiscale Modelling of Fuels and Structural Materials for Nuclear Systems.

List of authors
M. J. Caturla

Universidad de Alicante, Spain

L. J. Criscenti

Sandia National Laboratories, United States

R. Devanthan

Pacific Northwest National Laboratory, United States

D. M. Duffy

University College London, United Kingdom

N. M. Ghoniem

University of California, United States

M. Krack

Paul Scherrer Institute, Switzerland

T. D. Kühne

University of Paderborn, Germany

N. V. Luzginova

Nuclear Research and Consultancy Group, Netherlands

D. Perez

Los Alamos National Laboratory, United States

J. Y. R. Rashid

ANATECH Corp., United States

D. Seif

University of California, United States

R. E. Stoller

Oak Ridge National Laboratory, United States

V. Tikare

Sandia National Laboratories, United States

A. I. Topuz

Materials Innovation Institute (M2i) and University of Groningen,
Netherlands

B. P. Uberuaga

Los Alamos National Laboratory, United States

A. UldryPaul Scherrer Institute, Switzerland

E. van der Giessen

University of Groningen, Netherlands

A.F. Voter

Los Alamos National Laboratory, United States

M. J. Welland

Canadian Nuclear Laboratory, Canada

T. R. Zeitler

Sandia National Laboratories, United States
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2│5

Table of contents
List of abbreviations and acronyms ..................................................................................................... 9
Executive summary ............................................................................................................................. 13
1. Density functional theory ................................................................................................................ 16
1.1. Introduction................................................................................................................................. 16
1.2. The first-principles approach and the density functional theorems ............................................ 16
1.3. Choice of functionals .................................................................................................................. 19
1.4. Density functional theory implementations ................................................................................ 21
1.5. Scope of density functional theory ............................................................................................. 23
1.6. The use of density functional theory in nuclear materials research ............................................ 24
1.7. Conclusions and outlook ............................................................................................................. 25
References.......................................................................................................................................... 25
2. Ab-Initio molecular dynamics......................................................................................................... 29
2.1. Introduction................................................................................................................................. 29
2.2. Molecular dynamics .................................................................................................................... 30
2.3. An ab-initio potential .................................................................................................................. 31
2.4. Ab-initio molecular dynamics ..................................................................................................... 36
2.5. An efficient and accurate Car-Parrinello-like approach to Born-Oppenheimer molecular
dynamics ..................................................................................................................................... 39
2.6. Conclusion .................................................................................................................................. 45
Acknowledgments ............................................................................................................................. 46
References.......................................................................................................................................... 46
3. Classical interatomic potentials...................................................................................................... 51
3.1. Introduction................................................................................................................................. 51
3.2. Key issues in developing classical potentials ............................................................................. 52
3.3. Classification of classical potentials ........................................................................................... 53
3.4. State of the art in classical potentials for nuclear fuel cycle materials ....................................... 55
3.5. Challenges and future direction for development of classical potentials for nuclear
materials ...................................................................................................................................... 64
3.6. Conclusions................................................................................................................................. 66
References.......................................................................................................................................... 66
4. Molecular dynamics ........................................................................................................................ 68
4.1. Introduction to molecular dynamics ........................................................................................... 68
4.2. Modelling radiation damage using molecular dynamics ............................................................ 70
4.3. Limitations/challenges ................................................................................................................ 74
4.4. Future developments ................................................................................................................... 77
References.......................................................................................................................................... 78
5. Atomistic simulation methods for long-time dynamics in materials for nuclear energy
systems .................................................................................................................................................. 81
5.1. Introduction................................................................................................................................. 81
5.2. Parallel-replica dynamics ............................................................................................................ 83
5.3. Hyperdynamics ........................................................................................................................... 85
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5.4. Temperature accelerated dynamics ............................................................................................. 88
5.5. Adaptive kinetic Monte Carlo ..................................................................................................... 91
5.6. κ-Dynamics ................................................................................................................................. 92
5.7. Example applications .................................................................................................................. 93
5.8. Limitations and ongoing challenges ........................................................................................... 97
5.9. Advances from 2013 to 2018 ...................................................................................................... 98
5.10. Conclusion ................................................................................................................................ 99
Acknowledgments ........................................................................................................................... 100
References........................................................................................................................................ 100
6. Kinetic Monte-Carlo methods ...................................................................................................... 106
6.1. Introduction............................................................................................................................... 106
6.2. Background ............................................................................................................................... 106
6.3. Atomistic kinetic Monte Carlo.................................................................................................. 108
6.4. Object kinetic Monte Carlo....................................................................................................... 108
6.5. Event and first-passage kinetic Monte Carlo ............................................................................ 110
6.6. Other advanced models ............................................................................................................. 110
6.7. Conclusions............................................................................................................................... 111
Acknowledgements .......................................................................................................................... 112
References........................................................................................................................................ 113
7. Dislocation dynamics ..................................................................................................................... 116
7.1. Introduction............................................................................................................................... 116
7.2. Dislocation dynamics: Methodology ........................................................................................ 117
7.3. Interaction between dislocations and defects ............................................................................ 118
7.4. Collective behaviour: What can dislocation dynamics predict? ............................................... 122
7.5. Limitations, needs and challenges ............................................................................................ 124
References........................................................................................................................................ 126
8. Introduction to the phase-field modelling technique: A primer on the Allen-Cahn andCahn-Hilliard models ........................................................................................................................ 128
8.1. Introduction............................................................................................................................... 128
8.2. The Allen-Cahn model.............................................................................................................. 130
8.3. The Cahn-Hilliard model .......................................................................................................... 134
8.4. Phase stability, decomposition and nucleation ......................................................................... 136
8.5. Connection to equilibrium thermodynamics ............................................................................. 137
8.6. Summary and future directions ................................................................................................. 140
Acknowledgements .......................................................................................................................... 140
References........................................................................................................................................ 140
9. A review of the rate theory of defect clustering in irradiated materials .................................. 143
9.1. Introduction and background .................................................................................................... 143
9.2. Physics of radiation-induced point defects ............................................................................... 144
9.3. Classical nucleation theory ....................................................................................................... 146
9.4. Rate theory models of nucleation and growth .......................................................................... 151
9.5. Evolution of the size distribution and the Fokker-Planck approximation................................. 154
9.6. Stress effects in rate theory models and the dislocation bias .................................................... 155
9.7. Space-time rate theory .............................................................................................................. 156
9.8. Conclusions and research needs................................................................................................ 157
References........................................................................................................................................ 158

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10. Finite element based material modelling of nuclear fuel ......................................................... 161
10.1. Introduction ............................................................................................................................. 161
10.2. Modelling of UO2 fuel viscoplasticity ................................................................................... 163
10.3. Constitutive modelling of fuel pellet fracture ......................................................................... 170
10.4. Finite element implementation ............................................................................................... 175
10.5. Model limitations .................................................................................................................... 176
References........................................................................................................................................ 177
Conclusions ........................................................................................................................................ 178

TablesTable 3.1. Force field models for glass ................................................................................................. 61
Table 3.2. Force field model for glass-water interaction ....................................................................... 61
Table 9.1. Relevant energy barriers (in eV) for point defect processes in four materials. .................. 146

FiguresFigure 2.1. Deviations from the BO surface of liquid SiO2 with respect to total energies (upper
panel) and mean force deviations (lower panel).................................................................. 43
Figure 2.2. Partial pair-correlation functions g(r) of liquid Si (upper left panel) and liquid SiO2
at 3 000 K and 3 500 K respectively, using a DZVP Gaussian basis set............................. 44
Figure 2.3. (a) The kinetic energy distribution of metallic liquid Si64. (b) Velocity autocorrelation
function. (c) Fourier transform of the velocity autocorrelation function of 32 waterat 325 K. .............................................................................................................................. 45
Figure 4.1. A schematic representation of a cascade simulation. .......................................................... 71
Figure 4.2. Top view (a) and side view (b) of a crowdion cluster oriented in the <011> direction
in a body centred cubic metal. ............................................................................................. 72
Figure 4.3. An amorphised region of zircon, formed by a 30 keV U PKA cascade simulation ............ 74
Figure 4.4. Plots of the variation of electronic and nuclear stopping powers for an Fe ion
moving in Fe ........................................................................................................................ 76
Figure 4.5. The relaxed defect configuration for oxygen interstitials in MgO. ..................................... 77
Figure 5.1. Schematic illustration of the parallel replica dynamics method ......................................... 84
Figure 5.2. Schematic illustration of the hyperdynamics method ......................................................... 87
Figure 5.3. Schematic illustration of the temperature accelerated dynamics method ........................... 90
Figure 5.4. Transformation pathway of a 45-vacancy void to a stacking fault tetrahedron
at T=475 K ........................................................................................................................... 94
Figure 5.5. Schematic of the energy landscape for five-interstitial clusters in MgO ............................ 95
Figure 5.6. An example of the type of complex processes that can occur near damaged grainboundaries. .......................................................................................................................... 96
Figure 6.1. Schematic diagram of the activation barrier for an atom in position 1 to move to avacant site (position 2) ...................................................................................................... 107
Figure 6.2. Total rate as the sum of individual rates of event up to n events ...................................... 108
Figure 7.1. Illustration of the position of DD in multi-scale modelling .............................................. 116
Figure 7.2. Geometry of a single dislocation (a) and top-view on a slip plane (b) showing two
discretisation schemes of a dislocation: edge-screw (top right) and nodal (bottom left) .. 117
Figure 7.3. Dissociated dislocation with stacking fault (grey area) under an applied shear stress inthe vertical direction interacting with small precipipates in Cu, according to projected
2D DD simulations ............................................................................................................ 119
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Figure 7.4. Pathway of formation of an SFT from a triangular Frank loop (solid line) throughdissociation into a Shockley partial (dashed) and a stair-rod dislocation (dotted) ............ 119
Figure 7.5. Interaction between a mobile edge dislocation and an inclined interstitial
prismatic loop .................................................................................................................... 120
Figure 7.6. Simple geometrical interaction between a dislocation and a particle ............................... 121
Figure 7.7. Stress field around a void induced by an edge dislocation according to DD .................... 121
Figure 7.8. Snapshot of a screw dislocation moving through a 3D distribution of dispersoids,
occasionaly leaving behind small Orowan loops .............................................................. 122
Figure 7.9. (a) Simulation model for the effect of particles and sessile dislocation loops on plastic
deformation in ferritic grains, by means of (b) a unified facet representation of hardparticles or sessile loops .................................................................................................... 123
Figure 7.10. Shear-induced distribution of edge dislocations in a 2D polycrystalline material
containing randomly distributed obstacles with a normal distribution of strengths
to represent irradiation-induced defects ............................................................................ 124
Figure 8.1. Example applications of phase-field modelling ................................................................ 129
Figure 8.2. Schematic representation of a one-dimensional (1D) average atomic density field ......... 130
Figure 8.3. Free energy curves for coexisting phases showing the tie line between equilibrium
concentrations.................................................................................................................... 132
Figure 8.4. Demonstrative phase-field profiles and excess energy ..................................................... 133
Figure 8.5. Demonstrative free energy profile for a Cahn-Hilliard model. ......................................... 135
Figure 8.6. Demonstrative plots of a 2 phase binary system ............................................................... 139
Figure 9.1. Constrained distribution function, h(m), versus cluster size, m. ....................................... 149
Figure 9.2. Helium distribution at various traps versus time............................................................... 154
Figure 9.3. (a) Dislocation capture efficiencies versus Temperature for copper and nickel, (b)Distribution of helium atoms in polycrystalline tungsten under high heat flux ................ 157
Figure 10.1. Interfaces and behavioural regimes in LWR fuel ........................................................... 162
Figure 10.2. Plane cross section of the idealised model of porous material ........................................ 165
Figure 10.3. Normalised tension response curve for ceramic materials .............................................. 171
Figure 10.4. 2D failure/yield surface for UO2 fuel pellet represented in principal stress space.......... 173
Figure 10.5. Fuel cracking pattern in axisymmetric fuel rod representation ....................................... 175
Figure 10.6. 3D simulation of fuel pellets deformation in CEA’s Toutatis Code ............................... 176

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

List of abbreviations and acronyms
1D

One-dimensional

2D

Two-dimensional

3D

Three-dimensional

AIMD

Ab-initio molecular dynamics

aKMC

Adaptive kinetic Monte Carlo

AKMC

Atomistic kinetic Monte Carlo

AMD

Accelerated molecular dynamics

ASPC

Always stable predictor-corrector

BCA

Binary collision approximation

bcc

Body centred cubic

BCMD

Basin constrained molecular dynamics

BKL

Borts-Kalos-Liebowitz

BMH

Born-Mayer-Huggins

BO

Born-Oppenheimer

BOMD

Born-Oppenheimer molecular dynamics

CALPHAD

Computer Coupling of Phase Diagraphs and Thermochemistry

CASL

Consortium for Advanced Simulation of Light Water Reactors

CEA

Commissariat à l'énergie atomique et aux énergies alternatives (Alternative
Energies and Atomic Energy Commission, France)CP

Car-Parrrinello

CPMD

Car-Parrinello molecular dynamics

CRSS

Critical resolved shear stress

DCs

Displacement cascades

DD

Dislocation dynamics

DFT

Density functional theory

DMFT

Dynamical mean field theory

DZVP

Double-zeta valence polarisation

EAM

Embedded-atom method

EFDA

European Fusion Development Agreement

EGM3

Expert Group on Multi-scale Modelling Methods (NEA)

EKMC

Event kinetic Monte Carlo

EOM

Equations of motion

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EPRI

Electric Power Research Institute (United States)

fcc

Face centred cubic

FEM

Finite element method

F-P

Fokker-Planck

FPKMC

First passage kinetic Monte CarloFS

Finnis-Sinclair

GAWP

Gaussian Augmented Plane Waves

GFE

Gibb’s free energy

GGA

Generalised gradient approximation

GPW

Gaussian Plane Waves

HE

Hartree equations

HK

Hohenberg-Kohn

HLW

High-level waste

HV

Helium-vacancy

ICT

Information and communications technology

INL

Idaho National Laboratory

KIM

Knowledgebase of Interatomic Models 1

KMC

Kinetic Monte Carlo

KP

Kinchin PeaseKS

Kohn-ShamLANL

Los Alamos National Laboratory

LDA

Local density approximation

LKMC

Lattice kinetic Monte Carlo

LOCA

Loss-of-coolant accident

LWR

Light water reactor

MC

Monte Carlo

MD

Molecular dynamics

MEP

Minimum energy pathway

MPS

Missing-pellet-surface

MOX

Mixed oxides

MQ-NMR

Multiple-quantum magic angle spinning nuclear magnetic resonance

M2i

Materials Innovation Institute (Netherlands)

1

https://openkim.org

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

NEA

Nuclear Energy Agency

NEB

Nudged elastic band

ODS

Oxide dispersion strengethened

OKMC

Object kinetic Monte Carlo

OMC

Occupation matrix control

ORNL

Oak Ridge National Laboratory (United States)

OT

Orbital transformation

ParRepParallel replica dynamics

ParSplice

Parallel trajectory splicing

PAW

Projector augmented wave

PCI

Pellet cladding interaction

PCMI

Pellet-clad mechanical interaction

PDE

Partial differential equations

PIHD

Path integral hyperdynamics

PKA

Primary knock-on atoms

PNNL

Pacific Northwest National Laboratory (United States)

RCS

Replacement collision sequences

RHS

Right hand side

RIA

Reactivity insertion accident

RT

Rate theory

SEAKMC

Self-evolving atomistic kinetic Monte Carlo

SVV

Stress corrosion cracking

SCF

Self-consistent field

SCK

Smoluchowski-Chapman-Kolmogorov

SFE

Stacking fault energy

SFTs

Stacking fault tetrahedra

SIAs

Self-interstitial atoms

SIC

Self-interaction correction

SpecTAD

Speculative parallelisation temperature accelerated dynamics

SNL

Sandia National Laboratory (United States)

TAD

Temperature accelerated dynamics

TAMMBER

Temperature-accelerated Markov models with Bayesian estimation of rates

TEM

Transmission electron microscopy

TSTTransition state theory

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TF

Thomas-Fermi

UFF

Universal force fieldVR

Vashista-Rahman

WPMM

Working Party on Multi-scale Modelling of Fuels and Structural Materials
for Nuclear Systems (NEA)

XC

Exchange and correlation

ZBL

Ziegler-Biersack-Littmark

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Executive summary
1

2

Ram Devanathan and Veena Tikare
1
Pacific Northwest National Laboratory, United States
2
Sandia National Laboratories, United States
This report was commissioned by the Nuclear Energy Agency Expert Group on Multi-scale
Modelling Methods (EGM3) under the Working Party on Multi-scale Modelling of Fuels
and Structural Materials for Nuclear Systems (WPMM) of the Nuclear Science Committee(NSC). The goal is to provide to the nuclear energy community an overview of models andcomputer simulation methods of interest at different length and time scales for the materials
process. Given the large scope of such an endeavour, it has taken the dedicated efforts ofan international team of experts to do justice to this subject. The integration of these models
provides a logical path to develop predictive understandings of the performance ofmaterials used in the current fleet of nuclear reactors and to design new materials foradvanced fission and fusion reactors.
The report starts at the electronic structure level with a chapter on density functional theory(DFT) written by Uldry and Krack. The authors introduce the framework of DFT, theindependent electrons approximation, Hohenberg and Kohn theorems and the Kohn-Shamequations. The discussion also delves into the choice of exchange correlation functionals
and the differences between various implementations of DFT. In the area of nuclear
materials, DFT can be used to interpret experimental observations and to provide
parameters for the empirical potentials used in larger scale molecular dynamics (MD)simulations. DFT calculations can be used to fill knowledge gaps, such as thermodynamic
data for multicomponent systems where reliable experimental data is missing for materials
systems of interest.
In the second chapter, Kühne introduces ab-initio molecular dynamics (AIMD), evaluating
the interaction potential between atoms using parameter-free electronic structure
calculations. By treating the ions classically, this method makes it possible to accurately
determine both the static and dynamic properties of the system. The chapter presents BornOppenheimer MD and Car-Parrinello MD. By way of illustration, the author discusses
results from liquid silicon and silica. Given that one can simulate systems with hundredsof atoms over a time scale of the order of nanoseconds, it is possible now to tackle nuclear
materials problems using ab-initio methods that were previously considered impossible due
to computational resource limitations.
It is possible to increase the length and time scales of the simulation, although with somecompromise in terms of accuracy by performing classical MD simulations. Interatomic
potentials, presented in Chapter 3, written by Zeitler and Criscenti, are at the heart of MD
simulations. These potentials are analytical expressions of potential energy as functions of
the interatomic distance, bond angles and related measures. These expressions use
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 13

14 │ NEA/NSC/R(2019)2
parameters that are optimised by fitting them to known experimental properties, such as
lattice constants and elastic constants, as well as data from DFT, such as defect formationenergies. Such potentials can also be used in kinetic Monte Carlo (KMC) simulations. The
authors begin with a discussion of the different functional forms of potentials in use andthen focus on the state of the art in potentials for nuclear materials. Potentials for nuclear
fuel, cladding, glassy materials and waste forms are analysed in detail. The chapter also
explores the challenges in parameterising potentials with limited experimental data.
The fourth chapter on classical MD simulation, written by Duffy, continues the discussionof dynamical simulation of interacting atoms or ions by numerically integrating Newton’s
equations of motion. As an example of the close ties between chapters and the logicalprogression of scales, this chapter briefly touches upon the interatomic potentials coveredin the previous chapter. The early stages of radiation damage can be modelled using MD
simulations of displacement cascades. These simulations only account for nuclear stopping.
For materials that are sensitive to the effects of electronic excitations, the authors point out
the need to go beyond cascade simulations. With recent developments in reactive potentials
and the incorporation of electronic effects in MD simulations, there is reason to be
optimistic about the predictive modelling of the performance of irradiated materials.
Uberuaga, Perez and Voter take atomistic simulations a step further in the fifth chapter bypresenting accelerated dynamics methods that enable the study of the long-time evolutionof systems driven far from equilibrium. These methods fall under the categories ofaccelerated MD or adaptive KMC, and include parallel replica dynamics, hyperdynamics,
temperature accelerated dynamics and κ-dynamics. The authors provide illustrative
examples, such as non-equilibrium transport of Xe in irradiated UO2 nuclear fuel,aggregation of defects in MgO and enhanced vacancy-interstitial recombination near grainboundaries. The chapter concludes with a discussion of the limitations of accelerated
dynamics methods, and the potential to reach millisecond time scales and beyond withemerging computer architectures.
Chapter 6 also discusses the potential of KMC methods to reach much longer time scales
than classical MD. In Chapter 6, Caturla and Stoller describe four KMC methods relevantto this report, namely atomistic KMC, object KMC, event KMC, and first passage KMC.
The goal is to reach time scales of the order of seconds or even hours to enable directcomparison with experimental observations. The authors also present advanced methods,
such as on-the-fly KMC that is especially relevant when the number of events possible is
too large to be tabulated a priori or when the reactions strongly depend on the changing
local environment. It is also possible to use machine learning to predict energy barriers on
the fly in complex systems, where it is challenging to catalogue transition rates. There isconsiderable potential to enhance understanding of microstructural evolution underirradiation by combining KMC with other methods, including cluster dynamics anddislocation dynamics (DD).
In Chapter 7, Topuz, Luzginova and van der Giessen describe the DD method that lies
between atomistic dynamics and crystal plasticity. DD uses atomistic input from DFT
calculations, MD and KMC simulations and microscopy experiments, such as
microstructural information, defect properties, defect density and defect distribution. In
turn, DD provides the critical resolved shear stress and hardening rate as parameters to be
used in crystal plasticity calculations. The focus of this chapter is on dislocation interactions
with irradiation-induced defects. The authors provide keen insights into the limitations ofDD and the need for further improvements, especially in describing fracture of nuclear
materials.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

The discussion moves further along the scale towards the continuum level in the next
chapter. The phase-field model, presented in Chapter 8 by Welland, uses order parameters
to model the evolution of the microstructure. Here the atomic scale is bypassed, and thefocus is on microstructure and interface evolution at the mesoscale. This chapter presents
the Allen-Cahn and Cahn-Hilliard models. Data from thermodynamic databases andparameters from DFT and MD in the form of interfacial energies and diffusion coefficients
serve as much-needed input for the phase-field model. There is growing potential for
expanding the reach of this method using novel techniques, for example adaptive meshrefinement that reduce the computational cost.
Chapter 9 reviews the rate theory of defect clustering in irradiated materials, with Seif andGhoniem discussing the clustering of defects when materials are driven far fromequilibrium due to bombardment by energetic particles. This theory is valuable forunderstanding microstructure evolution under irradiation at experimentally-relevant lengthand time scales. The authors review the physics of defect production under irradiation toprovide an understanding of the processes that contribute to the generation and annihilationof defects. As in the case of the methods discussed earlier, DFT and MD play a valuablerole in providing key parameters. There is also potential to couple rate theory to DD so as
to understand embrittlement and failure of irradiated materials.
Finally, in Chapter 10, Rashid discusses the modelling of nuclear fuel using the finite
element method. This macroscale model is used to understand the thermos-mechanicalbehaviour of irradiated nuclear fuel by solving coupled displacement and temperatureequations. The chapter introduces the pellet-clad mechanical interaction problem and theconstitutive formulation used to model fuel pellet fracture. Common codes that are used to
implement the model and limitations of the method, such as the lack of microstructuralrepresentation, are also presented.
Overall, the modelling methods presented in the present report represent a progression ofscales from the atomic to the macro scale. There is considerable potential to link scales bypassing relevant parameters from the lower scale or by providing microstructural
information from a higher scale to lower scale models. Such multi-scale modelling canovercome the limitations imposed by the current use of empirical models in computercodes, and enable predictive understanding of materials used in nuclear reactors.

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1. Density functional theory
A. Uldry and M. Krack
Paul Scherrer Institute, Switzerland

1.1. Introduction
Density functional theory (DFT) forms the basis of the multi-scale ladder. It deals with thefundamental interactions in materials at the level of the electrons, and provides the meansof solving the resulting quantum many-body problem. DFT is a first-principles method in
the sense that, in its raw formulation, no empirical laws or adjustable parameters areneeded. Going from the DFT formulation to an actual ground state requires, however, aseries of approximations. The use of pseudopotentials may introduce certain types of errors,
and the accuracy and applicability of the method rest with the choice of exchangecorrelation functionals. The decision on the choice of functionals is usually made bycomparison with experimental results or based on past experience with similar materials.
Most functionals, including those popular within the nuclear materials community, benefit
to some degree from cancellations of errors. Despite such shortcomings, DFT has been an
extremely successful approach to understanding properties of matter and also as apredictive tool. Both the theoretical aspects of functionals and the computational
implementations of DFT are active fields of research and subject to future developments.

1.2. The first-principles approach and the density functional theorems
This section sketches the essential facts behind DFT. Proofs and in-depth analysis can befound in the two text books upon which this introduction is loosely drawn, Richard Martin's(2004), and Parr and Young's book (1989). We first introduce the fundamental interactions
that are taken into account. The central theorem of DFT, that all properties of an interacting
system are unique functionals of the ground state density, is briefly stated. Finally, theformulation of the DFT as an independent particles problem in an effective potential ispresented.
The starting point for a closed system of electrons and nuclei is to consider the kinetic
energy and the electrostatic Coulomb interactions of all particles present. The Hamiltonian�e and of the nuclei T�n ,would therefore include the kinetic energy operator of the electrons T��
the interactions between electrons Vee , and between nuclei Vnn , and the interactions
�ne. The wave functions of such a system depend
between the electrons and the nuclei V
simultaneously on the co-ordinates of all the electrons as well as that of the nuclei.

The very first step taken in the framework of DFT is to apply the Born-Oppenheimer
approximation (1927), also called the adiabatic approximation. It consists in neglecting the
kinetic energy of the nuclei𝑁𝑁�n = − �
T
𝐼𝐼

∇2𝐼𝐼
2𝑀𝑀𝐼𝐼

(1)

This is reasonable in view of the large mass of the nuclei and their relative inertia comparedto the electrons. As a consequence, the movements of the electrons are decoupled from that

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2of the nuclei. In classical molecular dynamics (MD) the nuclei are treated as point particles
within Newton's formalism. In DFT, the positions of the nuclei are viewed as parameters,with the electrons moving instantaneously with the nuclei. One thereby reduces the
problem to one of electrons interacting with one another via the Coulomb interaction, andwith an external potential generated by the nuclei. The DFT Hamiltonian for a system
1
consisting of 𝑁𝑁 nuclei and 𝑛𝑛 electrons in atomic units ([a.u.]: ℏ = 𝑚𝑚e = 𝑒𝑒 == 1) is4𝜋𝜋𝜖𝜖0

therefore given by�e + V�ee + V�ne�= V
�nn + T
H
𝑁𝑁

𝑁𝑁

𝐼𝐼

𝐽𝐽>𝐼𝐼

𝑛𝑛

𝑛𝑛

𝑛𝑛

𝑁𝑁

𝑛𝑛

𝑖𝑖

𝑖𝑖

𝑗𝑗>𝑖𝑖

𝐼𝐼

𝑖𝑖

∇2𝑖𝑖
1
𝑍𝑍𝐼𝐼
= ��
− � + ��− ��
|𝒓𝒓𝑖𝑖 − 𝑹𝑹𝐼𝐼 |2
�𝒓𝒓𝑗𝑗 − 𝒓𝒓𝑖𝑖 ��𝑹𝑹𝐽𝐽 − 𝑹𝑹𝐼𝐼 �𝑍𝑍𝐼𝐼 𝑍𝑍𝐽𝐽(2)The labels 𝑖𝑖, 𝑗𝑗 and 𝐼𝐼, 𝐽𝐽 run over all electrons, respectively all nuclei; 𝑟𝑟𝑖𝑖 and 𝑅𝑅𝐼𝐼 denote theposition of an electron, respectively of a nucleus of charge 𝑍𝑍𝐼𝐼 and mass 𝑀𝑀𝐼𝐼 . The
�nn cause merely an energy shift, which can beelectrostatic interactions between nuclei V
ignored for the present discussion. At this stage the problem we are set to solve� Ψ({𝒓𝒓𝑖𝑖 , 𝜎𝜎𝑖𝑖 }) = 𝐸𝐸 Ψ({𝒓𝒓𝑖𝑖 , 𝜎𝜎𝑖𝑖 })H(3)

is still beyond reach. Although the wave function Ψ({𝒓𝒓𝑖𝑖 , 𝜎𝜎𝑖𝑖 }) now only depends on the coordinates 𝒓𝒓𝑖𝑖 and spins 𝜎𝜎𝑖𝑖 of the electrons, this dependence is non-trivial, as all the coordinates of all the electrons are still simultaneously involved.

A standard way to address a many-body problem such as the one above can be invoked at
this stage. The independent electrons approximation is obtained by considering theelectrons as moving independently in an effective potential, as yet unspecified. The wave
function Ψ({𝒓𝒓𝑖𝑖 , 𝜎𝜎𝑖𝑖 }) for 𝑁𝑁 particles decouples into 𝑁𝑁 one-particle wave functions, andconsequently, equation (3) with the Hamiltonian (2) is replaced by 𝑁𝑁 equations of the type� eff Ψ𝑖𝑖𝜎𝜎 (𝒓𝒓) = 𝜖𝜖𝑖𝑖𝜎𝜎 Ψ𝑖𝑖𝜎𝜎 (𝒓𝒓) with H� eff = −H

∇2𝜎𝜎+ 𝑉𝑉eff
(𝒓𝒓)
2

(4)

Finding an effective potential that maps as accurately as possible the 𝑁𝑁 independent
particles problem (4) into the 𝑁𝑁 many-body problem (2) is a real challenge. A way to tackle
the many-body problem via the electron density 𝑛𝑛(𝒓𝒓) = ∑𝑖𝑖 𝛿𝛿(𝒓𝒓 − 𝒓𝒓𝑖𝑖 ), and its exact
reformulation as an independent particles problem, were provided respectively by
Hohenberg and Kohnin 1964, and Kohn and Sham in 1965.
The Hohenberg and Kohn theorems (1964) apply to any interacting particles in an external�ee in thepotential. This includes the system of electrons with Coulomb interactions V�electrostatic field of the nuclei Vne of the full many-body problem (2). The first theorem
states that all properties of the interacting system are completely determined by theelectronic ground state density 𝑛𝑛(𝒓𝒓), and that the total energy of the system 𝐸𝐸 is a uniquefunctional of the density: 𝐸𝐸[𝑛𝑛(𝒓𝒓)]. The second theorem concludes that the density which
minimises the total energy 𝐸𝐸[𝑛𝑛(𝒓𝒓)] is the exact ground state density, and similarly, that theexact ground state is the minimum of the functional.

The Hohenberg-Kohn theorems are wonderfully general but limited to establishing theexistence of a universal functional 𝐸𝐸[𝑛𝑛(𝒓𝒓)], which, minimised, gives the exact ground state.
The theorems provide no pointers as to the form of 𝐸𝐸[𝑛𝑛(𝒓𝒓)]. The only straightforward case

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is that of the external potential, the electrons-nuclei interaction. The functional 𝐸𝐸ext [𝑛𝑛(𝒓𝒓)]is directly given by𝑁𝑁�ext = − �𝐸𝐸ext [𝑛𝑛(𝒓𝒓)] = � 𝑉𝑉�ext 𝑛𝑛(𝒓𝒓) 𝑑𝑑𝒓𝒓 with V
𝐼𝐼

𝑍𝑍𝐼𝐼|𝒓𝒓 − 𝑹𝑹𝐼𝐼 |(5)

�e . No explicit
A particular difficulty is raised on the other hand by the kinetic energy T
relationship between the kinetic energy and the density is known. The next progress will
be made by using an independent particles formulation, which allows for a formulation of
the kinetic energy.
The essential step towards a practical DFT formalism was realised by Kohn and Sham
(1965). They write the density in terms of non-interacting particles 𝑛𝑛(𝒓𝒓) = ∑𝑖𝑖,𝜎𝜎|Ψ𝑖𝑖𝜎𝜎 |2.
Although this density does represent the electron density, the individual single-particle
wave functions Ψ𝑖𝑖𝜎𝜎 are not the wave functions of individual electrons, but that ofmathematical objects. A quantity
1
𝑇𝑇𝑠𝑠 = − �⟨Ψ𝑖𝑖σ |∇2 |Ψ𝑖𝑖σ ⟩
2

(6)

𝑖𝑖,𝜎𝜎

can be defined as the independent particles kinetic energy, which in general differs from�e . The Coulomb interaction can be turned into an
the many-body energy of the operator T
expression of the density by defining
1 𝑛𝑛(𝒓𝒓1 ) 𝑛𝑛(𝒓𝒓2 )𝐸𝐸H [𝑛𝑛] = �
𝑑𝑑𝒓𝒓1 𝑑𝑑𝒓𝒓2|𝒓𝒓1 − 𝒓𝒓2 |2(7)

the so-called Hartree energy. It too differs in principle from the full many-body version of
the Coulomb interaction 𝐸𝐸ee [𝑛𝑛]. An exact formulation of the functional 𝐸𝐸[𝑛𝑛] for theelectrons-nuclei problem can however be given as
𝐸𝐸[𝑛𝑛] = 𝑇𝑇𝑠𝑠 [𝑛𝑛] + 𝐸𝐸H [𝑛𝑛] + 𝐸𝐸ext [𝑛𝑛] + 𝐸𝐸xc [𝑛𝑛](8)𝐸𝐸xc [𝑛𝑛] = (𝑇𝑇[𝑛𝑛] − 𝑇𝑇𝑠𝑠 [𝑛𝑛]) + (𝐸𝐸ee [𝑛𝑛] − 𝐸𝐸H [𝑛𝑛])(9)with 𝐸𝐸xc [𝑛𝑛], called the exchange-correlation functional, defined as

So the problem of mapping the non-interacting problem to that of a full many-body problem
is delegated to the exchange-correlation functional. Assuming 𝐸𝐸xc [𝑛𝑛] is known, thetransformation (2) to (4) is now exact, with the effective Hamiltonian replaced by the KohnSham Hamiltonian
� KS = −H∇2
𝜕𝜕𝐸𝐸 [𝑛𝑛(𝒓𝒓)]𝑛𝑛(𝒓𝒓′ )�ext (𝒓𝒓) + �
�xc (𝒓𝒓) with V
�xc (𝒓𝒓) = xc+V
𝑑𝑑𝒓𝒓′ + V′
|𝒓𝒓 − 𝒓𝒓 |2𝜕𝜕𝜕𝜕(𝒓𝒓)(10)

These are the Kohn-Sham equations, which can be solved by iteration. A starting density
is calculated from a set of given wave functions. The density is then used to solve the KohnSham equations, which in turn deliver a new set of wave functions. The procedure isrepeated until self-consistency is achieved.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

1.3. Choice of functionals
The Kohn-Sham equations (10) allowed DFT to become a practical approach for finding
the ground state of interacting particles. Provided the functional 𝐸𝐸xc [𝑛𝑛] is known, the KohnSham equations are an exact formulation of the many-body problem (2). In practice, theexchange-correlation functional is not known and must be approximated. Over the years,
substantial developments have added different levels of sophistication to the exchangecorrelation energy functional, usually at an added computational cost; the different levels
are often quoted as the rungs of the Jacob's ladder (Perdew, Schmidt, 2001).
The notion of exchange and correlation is born from applying the condition of
�ee . The
antisymmetry of the many-body wave function, in particular on the Coulomb term V
alternative to DFT for electronic structure calculations, the Hartree-Fock method, builds inthe antisymmetry of the wave function by constructing the many-body wave function with
Slater determinants of single-particle wave functions obeying a Hamiltonian of the type(4). The exchange contribution appears in that case as a term lowering the energy,
11
𝐸𝐸x = − � � Ψ𝑖𝑖𝜎𝜎∗ (𝒓𝒓1 )Ψ𝑗𝑗𝜎𝜎∗ (𝒓𝒓2 )Ψ 𝜎𝜎 (𝒓𝒓 )Ψ 𝜎𝜎 (𝒓𝒓 ) 𝑑𝑑𝒓𝒓1 𝑑𝑑𝒓𝒓2|𝒓𝒓1 − 𝒓𝒓2 | 𝑗𝑗 1 𝑖𝑖 22

(11)

𝑖𝑖,𝑗𝑗,𝜎𝜎

the rest of the Coulomb term being the Hartree energy (7). The exchange term involves
electrons of same spins. The concept of “exchange hole” is often introduced at this stage
as describing the region around an electron that other electrons of the same spin avoid(Jones, Gunnarsson, 1989). Therefore, in Hartree-Fock the exchange is exact, but the manybody correlations are neglected. In DFT both correlations and exchange have to be taken
into account in the exchange-correlation functional.

Local density approximation
Because the Hartree term already explicitly included in the Kohn-Sham equations (10) is along-range interaction, it seems reasonable to assume that the missing contributions thatmust come into 𝐸𝐸xc are of local nature. The local-density approximation (LDA) (Perdew,
LDA
Zunger, 1981; Vosko, Wilk, Nusair, 1980) constructs 𝐸𝐸xc
[𝑛𝑛(𝒓𝒓)] from that of thehomogeneous electron gas having the same density. In the latter case the correlation andexchange are local and calculable typically by quantum Monte Carlo simulations(Ceperley, Alder, 1980). This seemingly too simple approach has proven surprisingly
successful for a wide range of systems. The success of the LDA has been explained (Jones,
Gunnarsson, 1989) by three factors: the sum rules related to the “exchange hole” are�ee depends only on the spherical average of theconserved in the LDA formulation; V“exchange hole”; and the errors introduced in the separate exchange and correlation energy
tend to cancel out. LDA is still widely used, although it is known to overbind andunderestimate lattice parameters; cohesive energies and bulk modulus are found too large.
Applied to pure iron, LDA's extension to spin polarised systems predicts a non-magnetic
or anti-ferromagnetic ground state, instead of the ferromagnetic body centred cubic (bcc)
structure (Wang, Klein, Krakauer, 1985).

Generalised gradient approximation
A step further is realised by the inclusion of the gradient of the density (or spin density) in
GGA [𝑛𝑛(𝒓𝒓)],∇𝑛𝑛(𝒓𝒓)]. The
the expression for the exchange-correlation functional 𝐸𝐸xc
generalised gradient approximation (GGA), unlike LDA, must be parametrised and
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20 │ NEA/NSC/R(2019)2
sometimes deviates from strict first principles; a popular parametrisation used for
d-electron materials like Fe and Fe-alloys as well as nuclear fuel materials like uranium
dioxide (UO2) is that of Perdew, Burke and Ernzerhof (PBE) (1996). An alternative is thatof Perdew and Wang (PW91) (1992). Newer parametrisations (Hammer, Hansen, Norskov,
1999; Wu, Cohen, 2006) have been proposed recently but, to our knowledge, have not been
widely used for nuclear materials. GGA is usually more accurate than LDA, although it has
been found to soften bonds too much in some highly co-ordinate atoms. Both LDA andGGA are failing, however, to make correct predictions for strongly-correlated materials,
like UO2. In particular, for the latter compound, a Mott insulator, LDA and GGA wrongly
predict a metallic ground state.

DFT+U
Strongly-correlated systems, like Mott insulators, are manifestations of a many-body
physics that is not captured by local or semi-local exchange-correlation functionals. The 5f
actinide oxides used as nuclear fuels can be tackled within DFT by the inclusion of an onsite Coulomb repulsion in the formalism. Different formulations exist (Dudarev et al.,
1998; Lichtenstein, Anisimov, Zaanen, 1995). In essence, the DFT+U functionals(Lichtenstein, Anisimov, Zaanen, 1995) belong to the class of orbital-dependentfunctionals, whereby the localised orbitals (typically d or f) are shifted relative to other
orbitals. The selective corrective term, called effective Hubbard U parameter, i.e. 𝑈𝑈eff =𝑈𝑈 − 𝐽𝐽, is usually entirely empirical and is often adjusted in order to reproduce anexperimentally known property, for instance the band gap of the studied material, althoughmethods exist to calculate U as well (Anisimov, Aryasetiaman, Lichtenstein, 1997;
Coconnioni, de Gironcoli, 2005). The DFT+U term introduces basically a penalty functionthat “discourages” a delocalisation of the d or f electrons, since it favours energeticallyeither empty or fully occupied orbitals. Unfortunately, this creates a manifold of possible
localisation patterns for the f electrons, which causes the occurrence of metastable states(Jollet et al., 2009; Dorado et al., 2009, 2013). This introduces technical problems with theconvergence to the ground state, since the self-consistently converged solution for actinide
materials using DFT+U depends on the initial 5f orbital occupations. Several recipes havebeen devised in the literature to tackle this problem, e.g. the occupation matrix control(OMC) (Dorado et al., 2009), the U ramping method (Meredig, 2010), the quasi-annealing
method (Geng et al., 2010), the controlled symmetry reduction method (Gryaznov, Heifets,
Kotomin, 2012) and combined methods using 5f occupation smearing with U ramping(FOUR) (Rabone, Krack, 2013).

Self-interaction correction
Another type of orbital-dependent functionals is covered by the label Self-Interaction
Corrected (SIC). In Hartree-Fock the spurious 𝑖𝑖 = 𝑗𝑗 term entering into the Hartree
expression (7) is automatically exactly compensated by the exchange term (11). In DFT
such a cancellation will have to occur via the exchange-correlation functionals. Formalisms
whereby the self-interaction is subtracted orbital by orbital for all occupied orbitals havebeen developed (Svane, Gunnarsson, 1990; Svane et al., 1998). In effect, the selfinteraction correction allows the localisation of states, particularly in rare-earth materials,
and therefore it leads to an improved treatment of the magnetic ground state.

Dynamical mean field theory
Dynamical mean field theory (DMFT) (Georges, 1996) is often cited as the most successfultreatment to date of strongly-correlated systems. Although it does include elements from
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

band structure, DMFT is not as such a DFT approach. The self-energy is determined selfconsistently within the Green's function formalism via the so-called Anderson singleimpurity problem, the latter requiring a sophisticated computational apparatus in order tobe solved. DMFT is therefore a powerful, but computationally very expensive approach.
First applications of DMFT variants like DFT+DMFT to nuclear fuel materials likeuranium and plutonium oxides as well as their mixed oxides (MOX) are being performed,
but the model system sizes are still limited to a few tenth atoms (Amadon, Condens, 2012).

Hybrid functionals
Hybrid functionals have been proposed as a mean to combine the exact exchange propertyof Hartree-Fock and the correlations from GGA. This type of approach is useful when theinterest lies in defect energies located in the band gap of a semi-conductor. Hybrid
functionals are not entirely first-principles, as a mixing parameter has to be defined, andare not easily evaluated for solids, since an explicit (orbital based) calculation of the
Hartree-Fock exchange is required which is computationally expensive. The electronic
properties of nuclear fuel materials like UO2 (Kudin, Sciseria, Martin, 2002) as well as ofother actinide oxides (Prodan, Scuseria, Matin, 2007) are better described with respect toplain DFT methods, but hybrid functionals share the convergence problems with DFT+U
methods due to metastable states while being computational much more demanding,making the exploration of many occupation pattern computationally very expensive.

Meta-generalised gradient approximation functionals
Various other exchange-correlation functionals are available, although not popular within
the nuclear materials community. Meta-GGA (Tao et al., 2003) comes in the third position
after LDA and GGA in the Jacob's ladder as it includes a non-empirical dependence on theorbital kinetic energy density, i.e. terms of order ∇2 𝑛𝑛(𝒓𝒓). It has been successfully tested onmolecules, hydrogen-bonded complexes, and ionic solids (Tao et al., 2003). Despite thefurther order and the extra cost, this approximation remains semi-local in nature.
Van der Waals interactions are not included in any of the local or semi-local treatments,
but can be considered in an empirical manner using the DFT-D method (Grimme et al.,2010), which provides a dispersion correction for the elements H to Pu for popularexchange and correlations functionals.
Incorporation of a random phase approximation (RPA) approach (Amadon, Applencourt,
Bruneval, 2014) could account for such long-range interactions straightforwardly, eventhough at very high computational cost. Van der Waals forces are usually not relevant in
the nuclear context.
As a conclusion, we would like to emphasise that physical properties may depend strongly
on the choice of functionals: comparison with experiments for the property underconsideration should dictate the choice of functionals.

1.4. Density functional theory implementations
Many different implementations of the resolution of the Kohn-Sham equations (10) havebeen made available to users. They differ most predominantly in the choice of basis sets
for the single-particle wave functions, their different use or not of pseudopotentials, andwhether the system is periodic or a cluster of atom.

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The one-particle wave functions must be expanded in a basis set in order to proceed withthe solution of the Kohn-Sham equation. The prototype choice is between localised sets oforbitals, or a plane waves basis set. In theory, the dimension of the function space is infinite.
In practice, however, the number of basis functions must be limited, of course. A localised
basis set will have basis functions that already look like the objects they will eventuallyconstruct, i.e. have large weights around the atomic positions. These basis sets are biased
but computationally cheap, since very few basis functions will be needed to represent theelectronic density, if the functions are well chosen. Plane waves, on the other hand, are freeof any bias and fully flexible. The convergence with an increasing number of plane waves
is also straightforward. However, many plane waves are needed to reconstruct the density
around the atom. The situation is worse near to the nuclei, where strong variations occur.
It is also not unusual to combine atomic-like functions in some regions of space with aplane waves basis set.
It is a general practice to treat core electrons, semi-core electrons and valence electrons
separately. When a plane waves basis is chosen, the potentials of the core electrons aresmoothed out, thereby reducing the oscillations and allowing the number of plane waves toremain manageable. These so-called pseudopotentials therefore introduce another level ofapproximation and inaccuracy. They come in different types (norm-conserving, ultra-soft,etc.) and must be tested for transferability in all possible situations.
A powerful approach combining the use of pseudopotentials and all-electron accuracy is
the Projector augmented-wave (PAW) technique (Blöchl, 1944). By means of projectors,
the all-electron (i.e. not pseudised) wave functions of the valence electrons arereconstructed. This is the method of choice nowadays in many cases. It has been applied
for Fe and alloys as well as actinide materials, where the computationally cheaper ultrasoft pseudopotentials are not accurate enough (Kresse, Furthmüller, 1996).
DFT codes also differ in their use of periodic boundary conditions or cluster approach. In
the latter case, a cluster of atoms, possibly surrounded by compensating charges, is usuallytreated in a localised basis set. In the former case a unit cell is given, which makes theapproach naturally suited for the study of crystal structures. The periodic wave functions
in a crystal can be expanded in plane waves whose wave vectors are the reciprocal lattice
vectors of the crystal. Plane waves are therefore a natural choice of basis for periodic
structures, although many codes very successfully use also local orbital approaches in thiscase.
Periodic systems have a real space (infinite number of electrons) and a reciprocal space
representation (infinite number of k-points), and codes rely on both representations.
Numerical scheme exists that limit the number of k-points that are necessary; metals inparticularly require a dense k-points coverage.
Some of the most popular codes used by the nuclear materials community are briefly
described in the following without any claim for completeness:
WIEN2k (Blaha et al., 1990; Schwarz, Blaha, 2003) uses a combination of atomic-likefunctions in the regions around the atoms and plane waves for the interstitial regions. The
full description of the basis set is “(L)APW+lo”, for (Linearised) Augmented Plane Wave
plus Localised Orbitals (Cottenier, 2013). This approach is a “full potential”,
computationally demanding all-electron approach that does not call on pseudopotentials,
and is thereby one of the most accurate ones available. This can be crucial, if properties
sensitive to the wave functions near the core are required, like spin-orbit coupling

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2(Laskowski et al., 2004). It is often used for benchmarking other DFT codes using
pseudopotentials or PAWs (Lejaeghere et al, 2016).
VASP (Kresse, Furthmüller, 1996) is a plane waves code with a choice of ultra-soft
pseudopotentials, and also a PAW implementation (Kresse, Joubert, 1999). The latter has
been a popular choice for Fe-alloys. VASP is probably the most popular DFT code for themodelling of nuclear materials.
ABINIT (Gonze et al., 2005) is, by contrast to VASP, a freely available code which offers
comparable capabilities.
Castep (Clark et al., 2005) is a pseudopotential plane waves code which includes a module
for the ab-initio determination of Nuclear Magnetic Resonance parameters (Pickard, Mauri,2001).
SIESTA (Soler et al., 2005) is a pseudopotential method using atom-centred basis sets inperiodic systems. Some caution is needed when using the code's ultra-soft pseudopotentials
in iron magnetic systems. The efficient basis sets, however, allow for calculations ofrelatively large systems compared to other approaches.
Dmol3 (Delley, 1990) uses numerical atomic orbitals as basis sets. The code has both allelectron and pseudopotential capabilities, and an implementation of the scalar relativistic
all-electron approach.
CP2K is a freely available code employing a hybrid basis set of Gaussian functions andplane waves basis (GPW) (Lippert, Hutter, Parrinello, 1997). It provides also a Gaussian
Augmented Plane Waves (GAPW) implementation (Lippert, Hutter, Parrinello, 1999)
which allows for all-electron calculations (Krack, Parrinello, 2000). The hybrid basis set
approach enables a Kohn-Sham calculation that scales linearly with the system size (Vande Vondele et al., 2005), like for the SIESTA code. CP2K has been applied for nuclear fuelmaterials like UO2 (Krack, 2015), but it provides also, besides the DFT module
CP2K/QUICKSTEP, (Van de Vondele et al., 2005) the force field implementation
CP2K/FIST (Bertolus et al., 2015).
The accuracy of the results in general will depend on the choice of functionals, and thesuitability of the PAW or pseudopotentials used. Moreover, the number of plane waves, orthe choice of localised basis functions, must be tested thoroughly, as well as the density of
k-points, and supercell size when relevant. Unconverged results are totally meaningless,irrespective of the accuracy of the method. One should carefully differentiate between
deficiencies of the selected DFT approach and shortcomings of the employed code and thecorresponding input parameter settings.

1.5. Scope of density functional theory
DFT determines the ground state of a collection of atoms. As such, it is therefore a zero
Kelvin formalism. To this day, the number of atoms considered is of the order of 1 000,with 200-300 typical for supercell alloys system. We describe in the following section only
the properties most routinely determined in the contest of multi-scale modelling in nuclear
materials.
The self-consistent result of the Kohn-Sham equations gives the density as a function of
position and the total energy of the system. So the most direct application of DFT is thetotal energy comparison between different atomic arrangements. Energy differences

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 23

24 │ NEA/NSC/R(2019)2
converge faster with respect to the basis set quality than absolute energies and in somecases benefit from cancellations of systematic errors.
The determination of forces on the atoms, formally the variation of the energy with respect
to the nuclei positions, is implemented in most codes. Most will also have an algorithm
whereby positions of the nuclei are modified at constant volume until the forces on theatoms are as small as desired. This ionic relaxation will usually drive the system into a
local minimum, with no guarantee of having reached a global minimum. When startingfrom a reasonable geometry and using an appropriate functional, bond lengths comparisonswith experiments are nonetheless accurate to the order of the percent.
Full structure optimisations, including volume relaxation, are also routinely implemented.
These algorithms are based on the expression of the macroscopic stress tensor in the DFT
formalism. Volume relaxation, however, has to be tackled carefully. The k-points
integration accuracy varies with volume changes, so that the density of k-points has to be
sufficient across the volume range. Incomplete basis sets introduce an error in the stresstensor, called “Pulay stress”. In the case of plane waves, varying the volume will effectively
change discontinuously the quality of the basis set. These problems can be reduced by using
a sufficiently large basis set. It is often recommended to perform first a constant volume
cell shape relaxation, then to fit an equation of state through the total energies at different
volumes. That fit also gives the bulk modulus. Comparisons with experiments show thatthe lattice parameters can agree with the experimental values to the order of the percent.
Elastic constants can also be deduced from DFT. They are calculated from the stress tensorsof several structure optimisations performed under well-chosen strains, which will dependon the symmetry of the crystal. Linear elastic constants can be determined within 10% of
the experimental values.
The interatomic force constants, formally the derivatives of the force with respect to nuclei
positions, are not directly measured quantities. They are often determined in the so-called“frozen phonon” method, whereby the total energy versus displacements is calculated.
Relativistic effects are not explicitly taken into account in Hamiltonian (2), nor are the spinorbit interactions related to such effects. The relativistic effects are limited to regions near
the core, and can be taken into account at the level of the pseudopotentials. In all-electron
codes the core electrons are usually treated fully relativistic using scalar-relativistic wave
functions, and the spin-orbit coupling is calculated for valence electrons in a secondvariational procedure.

1.6. The use of density functional theory in nuclear materials research
The role of DFT in the multi-scale approach is two-fold: to generate data for MD empiricalpotential parametrisation, and to investigate structures and provide measurable quantities
for model validation.
MD empirical potentials are parametrised and adjusted on equilibrium properties obtained
by DFT. These are typically equilibrium bond lengths, elastic constants and cohesiveenergy per atom. For the particular purpose of nuclear materials, point defect formationenergies should also be calculated. This is done usually by the {\it supercell} technique.
The defects are placed at the centre of a large supercell built from repeated blocks of theoriginal unit cell. The size of the supercell must be large enough that the defects are
effectively isolated from their images across the periodic boundaries.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2│ 25

DFT is also the indicated method for the investigations of electronic, structural andmagnetic properties. Partial density of states is used to analyse the effect of impurities(Klaver, Drautz, Finnis, 2006). Spin-dependent calculations are also used to identify phasestability in magnetic materials and structural aspects like clustering of impurities (Olsson,
Klaver, Domain, 2010).
As hinted in the previous paragraph, some of the information delivered by DFT can bedirectly compared with experiments, like the lattice parameters. Others, like the interactionenergies, can only, if at all, be compared to experiments via related measurable quantities.
Structural information, for instance, can be obtained through spectroscopic measurement(Idhil et al., 2012). The migration energy along particular paths can be calculated by DFT(Domain, Becquart, Foct, 2004) and these values can be used as parameters for methods atlarger length scales. Resistivity recovery measurements can give access to that information,
but there are difficulties in assigning defects to particular binding energies. In many cases
DFT and experiments are therefore part of the same model validation loop.

1.7. Conclusions and outlook
DFT allows the calculations of ground state properties from first principles. The
applicability of this powerful approach is, however, limited to a relatively small number ofatoms. The accuracy of DFT, within its own scope, is determined by that of the exchangecorrelation functional. No functional captures correctly all the properties usually required.
In particular, the strongly-correlated effects have to be included as an extension to the DFT.
Convergence of the results with respect to the controllable parameters of the calculations(size or type of the basis set, k-points sampling, and supercell size) should be done
carefully. It is also essential to remember that ionic and volume relaxation may not express
a global minimum. Moreover, the convergence to the ground state of magnetic systems isoften extremely slow depending on the algorithm implemented. Nevertheless, it may stilllead to the wrong magnetic state.
Therefore, the accuracy of the DFT-determined values should always be evaluatedcritically, and should always be related to experimental measurements if possible. It may
be worthwhile investigating the benefit versus computational cost of the newer exchangecorrelation functionals for some compounds, although the problem of strongly-correlated
systems will likely remain restricted to empirical treatments in the near future.
It is likely that the accuracy of molecular dynamics simulations depends more on the formof the potentials than on the absolute convergence and accuracy of the DFT parameters.
However, in evaluating the quality of empirical potentials one should remember thelimitations of the DFT on which they are based.

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2. Ab-Initio molecular dynamics
Thomas D. Kühne
Department of Chemistry and Paderborn Center for Parallel Computing, University of Paderborn,
Germany

2.1. Introduction
Computer simulations and molecular dynamics in particular, is a very powerful method toprovide detailed and essentially exact information of classical many-body problems. With
the advent of ab-initio molecular dynamics (AIMD), where the forces are computed onthe-fly by accurate electronic structure calculations, the scope of either method has been
greatly extended. This new approach, which unifies Newton’s and Schrödinger’s equations,allows for complex simulations without relying on any adjustable parameter. This review
is intended to outline the basic principles as well as a survey of the field. Beginning withthe derivation of Born-Oppenheimer molecular dynamics, the Car-Parrinello method aswell as novel hybrid scheme that unifies best of either approach are discussed. The
predictive power is demonstrated by a series of applications ranging from insulators tosemiconductors and even metals in condensed phases.
The geometric increase in performance of computers over last few decades, together withadvances in applied physics and mathematics, has led to the birth of a new way of doing
science that is in the intersection of theory and experiment. As a result, they are referred toas computational sciences and allow for computer experiments under perfectly controllable
and reproducible conditions. In this way, computer simulations have been very successful
in explaining a large variety of physical phenomena and guiding experimental work. In
addition, it is even possible to predict new phenomena by conducting experiments in silicothat would otherwise be too difficult, too expensive, or simply impossible to perform.
However, by far the most rewarding outcome of computer simulations is the invaluable
insight they provide into the behaviour and the dynamics of a system. The two mostcommon algorithms for such studies are the Monte Carlo (MC) (Metropolis et al., 1953)
and molecular dynamics (MD) (Alder, Wainwright, 1957; Rahman, 1964; Verlet, 1967]
algorithm. The latter is simply the numerical solution of Newton’s equation of motion,
which allows both equilibrium thermodynamic and dynamical properties of a system atfinite temperature to be computed. Since it also provides a ’window’ onto the atomic realtime evolution of the atoms, another role of MD is that of a computational microscope.
One of the most challenging, but very important aspect of MD simulations is the calculationof the interatomic forces. In classical simulations, they are computed from empiricalpotential functions, which have been parametrised to reproduce experimental or accurate
ab-initio data on small model systems. Even though great strides in elaborating theseempirical potentials have been made, often the transferability to systems or regions of thephase diagram different from the ones to which they have been fitted is restricted.
Furthermore, they are not able to simulate with sufficient predictive power chemicalbonding processes that take place in many relevant systems. Eventually, some of the mostimportant and interesting phenomena of modern physics and chemistry are intrinsically
non-classical. Therefore, a first principle based approach, such as AIMD (Marx, Hutter,2009), where the forces are calculated on-the-fly from accurate electronic structure
calculations, is very attractive since many of these limitations can in principle be removed.
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However, the increased accuracy and predictive power of AIMD simulations comes atsignificant computational cost. For this reason, density functional theory (DFT)(Hohenberg, Kohn, 1964) is to date by far the most commonly employed electronic
structure theory, but it is important to note that AIMD is a general concept that in principle
can be used in conjunction with any electronic structure method. Nevertheless, the ab-initio
approach is not without problems – the relevant energy scale is tiny, well below 𝑘𝑘B 𝑇𝑇, andin particular the attainable length and time scales are still one of its major limitations.

2.2. Molecular dynamics

The mathematical task of MD is to evaluate the expectation value 〈𝒪𝒪〉 of an arbitrary
operator 𝒪𝒪(𝑹𝑹, 𝑷𝑷) with respect to the configurational Boltzmann distribution〈𝒪𝒪〉 =∫ 𝑑𝑑𝑹𝑹 𝑑𝑑𝑷𝑷 𝒪𝒪(𝑹𝑹, 𝑷𝑷) 𝑒𝑒 −𝛽𝛽𝛽𝛽(𝑹𝑹,𝑷𝑷)∫ 𝑑𝑑𝑹𝑹 𝑑𝑑𝑷𝑷 𝑒𝑒 −𝛽𝛽𝛽𝛽(𝑹𝑹,𝑷𝑷)

where 𝛽𝛽 = 1⁄𝑘𝑘B 𝑇𝑇 is the inverse temperature. The total energy function
𝑁𝑁ion

𝐸𝐸(𝑹𝑹, 𝑷𝑷) = �
𝐼𝐼=1

𝑃𝑃𝐼𝐼2
+ Φ(𝑹𝑹𝐼𝐼 )
2𝑀𝑀𝐼𝐼

(1)

(2)

where the first term denotes the nuclear kinetic energy, Φ(𝑹𝑹𝐼𝐼 ) the potential energyfunction, 𝑁𝑁ion the number of ions and 𝑀𝑀𝐼𝐼 the corresponding masses, depends itself on
nuclear positions 𝑹𝑹 and momenta 𝑷𝑷.

One way to evaluate equation (1), at least in principle, is to directly solve such a highdimensional integral, whose integrand is very sharply peaked in many dimensions, by auniform sampling using the MC algorithm. However, such an MC algorithm is very
inefficient, if it would not be for importance sampling (Metropolis et al., 1953), which
satisfies the sufficient detailed balance condition by rejections.
On the other hand, assuming the ergodicity hypothesis, the thermal average 〈𝒪𝒪〉 can not
only be determined as the ensemble average of a MC simulation, but using MD equally asa temporal average1
� 𝑑𝑑𝑑𝑑 𝒪𝒪(𝑹𝑹(𝑡𝑡), 𝑷𝑷(𝑡𝑡))𝜏𝜏→∞ 𝜏𝜏

〈𝒪𝒪〉 = lim(3)

However, by propagating the classical many-body system in time according to Newton’s
equation of motion, the ions are treated only classically; this approximation is usually
negligible, except for very light atoms or low temperature, where nuclear quantum effects
may be important and adopting a quantum formalism such as imaginary-time path integrals(Feynman, Hibbs, 1965; Ceperley, 1995) is required.
Similar to MC, within MD some kind of importance sampling is naturally performed by
preferentially visiting phase space of low potential energy. Furthermore, as denoted by theadditional time dependence in equation (3), MD allows for additional insights from theionic real-time evolution, at least in a statistical average sense. It is neither the intention,
nor possible, to obtain ”exact” trajectories by MD due to the infamous Lyapunov
instability, which states that slightly perturbed trajectories are intrinsically exponentiallydiverging with time.

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The equipartition theorem
13� 𝑀𝑀𝐼𝐼 𝑹𝑹̇2𝐼𝐼 � = 𝑘𝑘B 𝑇𝑇
2
2

(4)

where 𝑀𝑀𝐼𝐼 is the atomic mass, 𝑘𝑘B the Boltzmann constant and 𝑇𝑇 the instantaneoustemperature, offers an elegant way to bridge the gap between molecular mechanics andthermodynamics. This opens the door to extract a vast variety of relevant static anddynamic, as well as transport properties from a MD simulation.
Nevertheless, any computational resource is finite, which limits the time and length scales
accessible by computer simulations. One way to partially bridge the gap between the
microscopic size of the simulated system and the macroscopic reality is to introduce
periodic boundary conditions. In this way surface effects are eliminated by effectively
simulating an infinite system, albeit with a finite periodicity that is identical with the length
𝐿𝐿 of the simulation cell. As a consequence, only phenomena whose characteristic
correlation length is much smaller than 𝐿𝐿 can be simulated. By similar means only
processes whose typical relaxation time is significantly smaller than the simulation time 𝜏𝜏can be studied. Even though great strides have been made to extend the time and lengthscales, it is apparent that techniques such as those reviewed here are clearly needed.

2.3. An ab-initio potential
In AIMD the forces 𝑭𝑭𝐼𝐼 = −∇𝑹𝑹𝐼𝐼 Φ(𝑹𝑹𝐼𝐼 ) are determined on-the-fly using first-principles
electronic structure methods. That means that AIMD is not relying on any adjustableparameter, but only on 𝑹𝑹𝐼𝐼 , which constitutes its predictive power. However, finding theantisymmetric ground state eigenfunctions 𝜓𝜓0 of the corresponding many-body
Hamiltonian at each MD step comes at a significant computational cost, which has to becarefully balanced against the size and sampling requirements of MD.

2.3.1. The many-body Schrödinger equation
Applying the so-called Born-Oppenheimer (BO) approximation (1927), which we have
implicitly assumed in the preceding subsection, Φ(𝑹𝑹𝐼𝐼 ) can written as
Φ(𝑹𝑹𝐼𝐼 ) = ⟨𝜓𝜓0 |ℋ𝑒𝑒 ({𝒓𝒓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 )|𝜓𝜓0 ⟩ + 𝐸𝐸𝐼𝐼𝐼𝐼 (𝑹𝑹𝐼𝐼 )(5)ℋ𝑒𝑒 ({𝒓𝒓𝑖𝑖 }; 𝑹𝑹𝑖𝑖 ) 𝜓𝜓0 ({𝒓𝒓𝑖𝑖 }) = 𝜀𝜀0 (𝑹𝑹𝐼𝐼 ) 𝜓𝜓0 ({𝒓𝒓𝑖𝑖 })(6)

where ℋ𝑒𝑒 ({𝒓𝒓𝑖𝑖 }; 𝑹𝑹𝑖𝑖 ) is the electronic many-body Hamiltonian, that depend on the electronic
co-ordinates {𝒓𝒓𝑖𝑖 } and parametrically on 𝑹𝑹𝐼𝐼 . Essentially, the BO approximation allows fora product ansatz of the total wavefunction consisting of the nuclear and electronic
wavefunctions. Due to the large separation of the nuclear and electronic masses, theelectrons can be expected to be in its instantaneous equilibrium with the much heavier
nuclei, so that the electronic subsystem can be treated independently at constant 𝑹𝑹𝐼𝐼 .
Nevertheless, we are left with a formidable task to solve the electronic, non-relativistic,
time-independent, many-body Schrödinger equation

which is a high-dimensional, non-linear eigenvalue problem, with eigenfunctions 𝜓𝜓0 ({𝒓𝒓𝑖𝑖 })and eigenvalues 𝜀𝜀0 (𝑹𝑹𝐼𝐼 ), respectively. To visualise the complexity of equation (6) let us
consider the following Gedankenmodell to represent the solution 𝜓𝜓0 ({𝒓𝒓𝑖𝑖 }) on a real-space
grid, where each co-ordinate is discretised by 100 mesh points. Ignoring spin and taking
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 31

32 │ NEA/NSC/R(2019)2𝜓𝜓0 ({𝒓𝒓𝑖𝑖 }) to be real, for 𝑁𝑁𝑒𝑒 electrons 106𝑁𝑁𝑒𝑒 grid points are required, so that the solution ofa single Si atom would require more grid points than the number of electrons in the whole
universe, not to mention solving such a large non-linear eigenvalue problem.

2.3.2. Density functional theory
Fortunately, this curse of dimensionality can be ingeniously bypassed by the use of DFT,
which is based on two celebrated papers of Hohenberg, Kohn and Sham (1964; 1965). The
former, the so-called Hohenberg-Kohn (HK) theorem, proves the existence of a one-to-one
mapping between the ground state density 𝜌𝜌0 (𝒓𝒓) and an external potential 𝑣𝑣(𝒓𝒓). In thisvain, 𝜌𝜌0 (𝒓𝒓), which depends on just three electronic degrees of freedom, is designated as
the principal quantity rather than the 3𝑁𝑁𝑒𝑒 -dimensional many-body wavefunction. As aconsequence, the nondegenerate ground state wavefunction 𝜓𝜓0 ({𝒓𝒓𝑖𝑖 }) = 𝜓𝜓[𝜌𝜌0 (𝒓𝒓)] andlikewise ℋ𝑒𝑒 [𝜌𝜌0 (𝒓𝒓)] are both unique functionals of 𝜌𝜌0 (𝒓𝒓), just as the ground state energy
𝐸𝐸0 = 𝐸𝐸 DFT [𝜌𝜌0 (𝒓𝒓)] = ⟨𝜓𝜓[𝜌𝜌0 (𝒓𝒓)|ℋ𝑒𝑒 [𝜌𝜌0 (𝒓𝒓)]|𝜓𝜓[𝜌𝜌0 (𝒓𝒓)⟩(7)

𝐸𝐸 DFT [𝜌𝜌0 ] = ⟨𝜓𝜓0 |ℋ𝑒𝑒 |𝜓𝜓0⟩ ≤ ⟨𝜓𝜓′|ℋ𝑒𝑒 |𝜓𝜓′⟩ = 𝐸𝐸 DFT [𝜌𝜌′](8)

𝐸𝐸 DFT [𝜌𝜌0 ] = min ⟨𝜓𝜓|ℋ𝑒𝑒 |𝜓𝜓⟩ = min ⟨𝜓𝜓[𝜌𝜌]|ℋ𝑒𝑒 [𝜌𝜌]|𝜓𝜓[𝜌𝜌]⟩ = min 𝐸𝐸 DFT [𝜌𝜌]

(9)The latter obeys the variational property

for which equality holds if and only if 𝜌𝜌0 = 𝜌𝜌′. As a consequence equation (6) can be solved
not only by iteratively diagonalising ℋ𝑒𝑒 [𝜌𝜌] within a self-consistent field (SCF) procedure,
but equally by minimising the quantum expectation value of ℋ𝑒𝑒 [𝜌𝜌]:
𝜓𝜓

𝜌𝜌

𝜌𝜌

In principle the minimisation has to be performed under the constraint that 𝜌𝜌(𝒓𝒓) is 𝑁𝑁representable, i.e. that it is arising from an antisymmetric 𝑁𝑁-body wavefunction 𝜓𝜓({𝒓𝒓𝑖𝑖 }).
Luckily, this had been solved, and it can be demonstrated that any single-particle density
can be written in terms of an antisymmetric many-body wavefunction (Gilbert, 1975;
Harriman, 1981). On the contrary, for the 𝑣𝑣-representability problem, which states that𝜌𝜌(𝒓𝒓) is the ground state density of a local potential 𝑣𝑣(𝒓𝒓), no such general solution is known.
The HK theorem just guarantees that there cannot be more than one potential for eachdensity, but does not exclude the possibility that there is no potential realising that density.
It is only known for discretised systems that every density in interacting ensemble is vrepresentable. Interestingly, the constructive proof of Levy and Lieb (Levy, 1982; Leib,
1983) shows that for an interacting system v-representability is not required for the proofof the HK theorem.
For the sake of simplicity in the following I will throughout assume atomic units andconfine myself to the physical relevant Coulomb system, for which
𝑁𝑁𝑒𝑒

1
1
𝑍𝑍𝐼𝐼
� + 𝑉𝑉�ℋ𝑒𝑒 = � ∇2𝑖𝑖 + �+�= 𝑇𝑇� + 𝑈𝑈|𝑹𝑹𝐼𝐼 − 𝒓𝒓𝑖𝑖 |2
�𝒓𝒓𝑖𝑖 − 𝒓𝒓𝑗𝑗 �
𝑖𝑖=1

𝑖𝑖<𝑗𝑗

(10)

𝐼𝐼,𝑖𝑖

� is the electron-electron
Where 𝑇𝑇� is the kinetic energy operator of the electrons, while 𝑈𝑈
interaction and 𝑉𝑉� = ∑𝑖𝑖 𝑣𝑣(𝒓𝒓𝑖𝑖 ) the electron-ion operator. The former two operators areuniversal and independent of the system, while the latter is system dependent, or nonuniversal. DFT explicitly recognises that it is indeed the potential 𝑣𝑣(𝒓𝒓), which distinguishes� oncenon-relativistic Coulomb systems and offers a prescription how to deal with 𝑇𝑇� and 𝑈𝑈STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

and for all (Capelle, 2006). Hence, even at this stage based on nothing but the HK theorem,
DFT is already of some practical use without having to solve the many-body Schrödinger
equation and without having to make a single-particle approximation. In principle it shouldbe even possible to calculate all observables, since the HK theorem guarantees that they
are all functionals of 𝜌𝜌(𝒓𝒓). Presuming the availability of physical sound and accurate
approximations one can write
𝐸𝐸 DFT [𝜌𝜌(𝒓𝒓)] = 𝑇𝑇[𝜌𝜌(𝒓𝒓)] + 𝑈𝑈[𝜌𝜌(𝒓𝒓)] + 𝑉𝑉[𝜌𝜌(𝒓𝒓)](11)

In the so-called Thomas-Fermi (TF) approximation (Thomas, 1927; Fermi, 1927) the fullelectron-electron interaction energy is approximated by the Hartree energy1𝜌𝜌(𝒓𝒓)𝜌𝜌(𝒓𝒓′ )(12)𝑈𝑈[𝜌𝜌(𝒓𝒓)] ≈ 𝑈𝑈𝐻𝐻 [𝜌𝜌(𝒓𝒓)] = � 𝑑𝑑𝒓𝒓 � 𝑑𝑑𝒓𝒓′|𝒓𝒓 − 𝒓𝒓′ |2that is the electrostatic interaction energy of 𝜌𝜌(𝒓𝒓). In addition, the kinetic energy isapproximated as𝑇𝑇[𝜌𝜌(𝒓𝒓)] ≈ � 𝑑𝑑𝒓𝒓 𝑡𝑡 hom [𝜌𝜌(𝒓𝒓)] = 𝑇𝑇 LDA [𝜌𝜌(𝒓𝒓)](13)

𝑇𝑇 LDA [𝜌𝜌(𝒓𝒓)] ≈ � 𝑑𝑑𝑑𝑑 𝑡𝑡𝑠𝑠hom [𝜌𝜌(𝒓𝒓)] = 𝑇𝑇𝑠𝑠LDA [𝜌𝜌(𝒓𝒓)](14)

where 𝑡𝑡 hom [𝜌𝜌(𝒓𝒓)] is the kinetic energy density of a homogeneous interacting system,
which is also known as the local-density approximation (LDA). Due to the fact that the
explicit form of 𝑡𝑡 hom [𝜌𝜌(𝒓𝒓)] is only known for a non-interacting system, 𝑡𝑡 hom [𝜌𝜌(𝒓𝒓)] is
further estimated by the single-particle approximation 𝑡𝑡 hom [𝜌𝜌(𝒓𝒓)], i.e.
where

𝑡𝑡𝑠𝑠hom [𝜌𝜌(𝒓𝒓)] =3(3𝜋𝜋 2 )2/3 𝜌𝜌(𝒓𝒓)5/3
10

In the end the TF energy functional
𝐸𝐸 TF [𝜌𝜌(𝒓𝒓)] = 𝑇𝑇𝑠𝑠LDA [𝜌𝜌(𝒓𝒓)] + 𝑈𝑈𝐻𝐻 [𝜌𝜌(𝒓𝒓)] + 𝑉𝑉[𝜌𝜌(𝒓𝒓)]

(15)

(16)

implies not only the single-particle approximation to the full electron-electron interaction,
but also the single-particle mean-field approximation 𝑇𝑇𝑠𝑠LDA [𝜌𝜌(𝒓𝒓)] to the exact kinetic
energy of the inhomogeneous interacting system. As a consequence, all many-bodycorrelation effects are neglected.
However, the HK theorem predicates that they are again a functional of 𝜌𝜌(𝒓𝒓). The additionof an approximation to the exact exchange and correlation (XC) energy results in a formallyexact theory, which is referred to as orbital-free DFT (Smargiassi, Madden, 1994). It istherefore important to recognise, that the HK theorem is nothing but the formal
exactification of the TF approximation. Similarly, the Kohn-Sham (KS) (1965) scheme canbe considered as the exactification of the self-consistent Hartree equations (HE) (1928),
which differs only in the kinetic energy from the TF approximation. In fact, for the fictitious
non-interacting system the kinetic energy is known exactly, even though only in terms ofan explicit single-particle orbital functional, i.e. as an implicit density functional
𝑁𝑁𝑒𝑒1
𝑇𝑇𝑠𝑠 [𝜌𝜌(𝒓𝒓)] = − � � 𝑑𝑑𝒓𝒓 𝜓𝜓𝑖𝑖∗ (𝒓𝒓)∇2 𝜓𝜓𝑖𝑖 (𝒓𝒓) = 𝑇𝑇𝑠𝑠 [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}]2
𝑖𝑖=1

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Here the fictitious single-particle orbitals, or simply KS orbitals, are denoted as 𝜓𝜓𝑖𝑖 (𝒓𝒓). As
we will see immediately they are eigenfunctions of a fictitious system, known as the KS
system. It therefore should be noted that they differ from the single-particle orbitals usedin wavefunction based methods and have no strict physical meaning, with two notable
exceptions: at the presence of the exact XC functional for the special case of an isolated
system with 𝑣𝑣(∞) = 0, (i) the highest occupied eigenvalue 𝜀𝜀𝑁𝑁 can be shown to be thenegative of the exact, many-body, first ionisation potential including relaxation effects, and(ii) that the lowest unoccupied eigenvalue 𝜀𝜀𝑁𝑁+1 is the negative of the electron affinity.
Beside these two exceptions, only the density has a real physical meaning and can be
written in terms of 𝜓𝜓𝑖𝑖 (𝒓𝒓) as𝑁𝑁occ

𝜌𝜌(𝒓𝒓) = � 𝑓𝑓𝑖𝑖 𝜓𝜓𝑖𝑖 (𝒓𝒓)𝜓𝜓𝑖𝑖∗ (𝒓𝒓)

(18)

𝑖𝑖=1

where 𝑁𝑁occ is the number of occupied orbitals and 𝑓𝑓𝑖𝑖 the occupation number of state 𝑖𝑖, so
that
𝑁𝑁occ

� 𝑓𝑓𝑖𝑖 = 𝑁𝑁𝑒𝑒

(19)

𝑖𝑖=1

Therewith, the KS energy functional is simply given by
𝐸𝐸 KS [𝜌𝜌(𝒓𝒓)] = 𝐸𝐸 KS [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}]= 𝑇𝑇𝑠𝑠 [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}] + 𝑈𝑈𝐻𝐻 [𝜌𝜌(𝒓𝒓)] + 𝑉𝑉[𝜌𝜌(𝒓𝒓)] + 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]𝑁𝑁1= − � 𝑓𝑓𝑖𝑖 � 𝑑𝑑𝒓𝒓 𝜓𝜓𝑖𝑖∗ (𝒓𝒓)∇2 𝜓𝜓𝑖𝑖 (𝒓𝒓)
2
𝑖𝑖=1

′)

𝜌𝜌(𝒓𝒓)𝜌𝜌(𝒓𝒓1+ � 𝑑𝑑𝒓𝒓𝑑𝑑𝒓𝒓′|𝒓𝒓 − 𝒓𝒓′ |2(20)+ � 𝑑𝑑𝒓𝒓 𝑣𝑣ext (𝒓𝒓)𝜌𝜌(𝒓𝒓) + 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]where 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)] = (𝑇𝑇[𝜌𝜌(𝒓𝒓)] − 𝑇𝑇𝑠𝑠 [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}]) − (𝑈𝑈[𝜌𝜌(𝒓𝒓)] − 𝑈𝑈𝐻𝐻 [𝜌𝜌(𝒓𝒓)] is the alreadymentioned and apparently unknown XC energy functional, whereas 𝑣𝑣XC (𝒓𝒓) is thecorresponding XC potential. This definition also shows that a significant of part 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]is due to correlation effects of the kinetic energy that is known explicitly only in terms of
the reduced two-particle density matrix (Dreizler, Gross, 1990).
Since up to the exactifying term 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)] equation (20) is identical to the HE, it is not
surprising that the corresponding Euler-Lagrangian equation1
�− ∇2 + 𝑣𝑣𝑠𝑠KS (𝒓𝒓)� 𝜓𝜓𝑖𝑖 (𝒓𝒓) = 𝜀𝜀𝑖𝑖 𝜓𝜓𝑖𝑖 (𝒓𝒓)2(21)also results in a similar fictitious single-particle equation. Since 𝑣𝑣𝑠𝑠KS (𝒓𝒓) = 𝑣𝑣H (𝒓𝒓) +𝑣𝑣XC (𝒓𝒓) + 𝑣𝑣(𝒓𝒓) is the effective potential of an artificial system, such that the ground state
density and therewith the energy equals those of the true interacting many-body system.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

This particular system is therefore called KS system, its effective potential KS potential
and the resulting set of self-consistent equations are referred to as KS equations:1
�− ∇2 + 𝑣𝑣H (𝒓𝒓) + 𝑣𝑣XC (𝒓𝒓) + 𝑣𝑣(𝒓𝒓)� 𝜓𝜓𝑖𝑖 (𝒓𝒓) = 𝜀𝜀𝑖𝑖 𝜓𝜓𝑖𝑖 (𝒓𝒓)2𝑁𝑁occ

� 𝑓𝑓𝑖𝑖 𝜓𝜓𝑖𝑖 (𝒓𝒓)𝜓𝜓𝑖𝑖∗ (𝒓𝒓) = 𝜌𝜌(𝒓𝒓)
𝑖𝑖=1

(22)

𝛿𝛿𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]= 𝑣𝑣XC (𝒓𝒓)𝛿𝛿𝛿𝛿(𝒓𝒓)

At self-consistency it is possible to express 𝐸𝐸0KS [𝜌𝜌(𝒓𝒓)] in terms of the single-particle KS
eigenvalues 𝜀𝜀𝑖𝑖 . Due to the fact that they are not the eigenvalues of the interacting manybody system, but of fictitious non-interacting KS system, 𝐸𝐸0KS [𝜌𝜌(𝒓𝒓)] is merely the sumof 𝜀𝜀𝑖𝑖 , but
𝑁𝑁occ

1

𝐸𝐸0KS [𝜌𝜌(𝒓𝒓)] = � 𝑓𝑓𝑖𝑖 𝜀𝜀𝑖𝑖 − � 𝑑𝑑𝒓𝒓𝑑𝑑𝒓𝒓′
2𝑖𝑖𝜌𝜌(𝒓𝒓)𝜌𝜌(𝒓𝒓′ )− � 𝑑𝑑𝒓𝒓 𝑣𝑣XC (𝒓𝒓)𝜌𝜌(𝒓𝒓) + 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]|𝒓𝒓 − 𝒓𝒓′ |(23)That is to say that in order to make genuine calculations the KS scheme systematically�, onto an equivalent fictitious singlemaps the full interacting many-body problem, with 𝑈𝑈
�𝑠𝑠 + 𝑉𝑉�H + 𝑉𝑉�XC , but without 𝑈𝑈�body problem, with an effective potential operator 𝑉𝑉�KS = 𝑈𝑈(Capelle, 2006):
𝐸𝐸XC [𝜌𝜌(𝒓𝒓)]TF �⎯⎯⎯⎯⎯⎯⎯� HK↓ 𝑇𝑇𝑠𝑠 [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}]𝑇𝑇𝑠𝑠 [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}] ↓HE �⎯⎯⎯⎯⎯⎯⎯� KS
𝐸𝐸XC [𝜌𝜌(𝒓𝒓)](24)

2.3.3. The exchange and correlation functional

In the previous subsection DFT has been outlined as an exact theory, presuming that theexact XC functional is known. Unfortunately, except for the uniform electron gas(Ceperley, Alder, 1980), this is not the case and one has to resort to more or less accurate
approximations. For the sake of brevity, only the main physical principles rather than thevarious rungs of “Jacob’s ladder to heaven” (Tao et al., 2003) will be discussed here.
On this account the following break-up is particularly convenient:1𝜌𝜌(𝒓𝒓)𝜌𝜌XC (𝒓𝒓, 𝒓𝒓′ )𝐸𝐸XC [𝜌𝜌(𝒓𝒓)] = � 𝑑𝑑𝒓𝒓 � 𝑑𝑑𝒓𝒓′= 𝐸𝐸X [𝜌𝜌(𝒓𝒓)] + 𝐸𝐸C [𝜌𝜌(𝒓𝒓)]|𝒓𝒓 − 𝒓𝒓′ |2(25)

where 𝐸𝐸X [𝜌𝜌(𝒓𝒓)] is the exchange energy due to Pauli repulsion, 𝐸𝐸C [𝜌𝜌(𝒓𝒓)] the electron
correlation energy and 𝜌𝜌XC (𝒓𝒓, 𝒓𝒓′ ) = 𝜌𝜌X (𝒓𝒓, 𝒓𝒓′ ) + 𝜌𝜌C (𝒓𝒓, 𝒓𝒓′ ) the XC hole. The former istherefore the energy lowering due to the antisymmetry requirement on the wavefunction ofa fermionic system and can be exactly calculated in terms of an explicit orbital or implicit
density functional𝜓𝜓𝑖𝑖∗ (𝒓𝒓)𝜓𝜓𝑗𝑗∗ (𝒓𝒓′ )𝜓𝜓𝑖𝑖 (𝒓𝒓)𝜓𝜓𝑗𝑗 (𝒓𝒓′ )1′[𝜌𝜌(𝒓𝒓)]= − � 𝑑𝑑𝒓𝒓 � 𝑑𝑑𝒓𝒓
𝐸𝐸X
= 𝐸𝐸X [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}]|𝒓𝒓 − 𝒓𝒓′ |2

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Equation (26) is the so-called Hartree-Fock exchange energy, but with KS orbitals.
However, the nonlocal form of the exact exchange energy comes at a considerable
computational burden to solve four-centre integrals, which is about two orders ofmagnitude more expensive than is the case for local or semi-local approximations to theexact XC functional. The correlation energy accounts for the additional energy lowering,
since electrons with opposite spins also avoid each other. However, contrary to theexchange part, no exact expression for 𝐸𝐸C [𝜌𝜌(𝒓𝒓)] is known, neither in terms of orbitals nor
densities.

Obviously, DFT would be of little use if one had to know 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)] exactly, but luckily it
is usually energetically substantially smaller than each of the remaining terms, which are
known. One can thus hope that reasonable simple approximations to 𝐸𝐸XC [𝜌𝜌(𝒓𝒓)], will stillallows for qualitatively correct estimates of 𝐸𝐸0 [𝜌𝜌(𝒓𝒓)], without relying on additional
adjustable parameters.

2.4. Ab-initio molecular dynamics

In the following let us assume that the potential energy function is calculated on-the-flyusing DFT, so that Φ(𝑹𝑹𝐼𝐼 ) = 𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ] = 𝐸𝐸 KS [{𝜓𝜓𝑖𝑖 [𝜌𝜌(𝒓𝒓)]}] + 𝐸𝐸𝐼𝐼𝐼𝐼 (𝑹𝑹𝐼𝐼 ). In any case,
AIMD (Marx, Hutter, 2009; Car, Parrinello, 1985; Payne et al., 1992; Parinello, 1997; Car,
2002; Tuckerman, 2002) comes in two fundamental flavours, which are outlined in thissection.

2.4.1. Born-Oppenheimer molecular dynamics
In Born-Oppenheimer MD (BOMD) the potential energy 𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ] is minimised at every
MD step with respect to {𝜓𝜓𝑖𝑖 (𝒓𝒓)} under the holonomic orthonormality
constraint �𝜓𝜓𝑖𝑖 (𝒓𝒓)�𝜓𝜓𝑗𝑗 (𝒓𝒓)� = 𝛿𝛿𝑖𝑖𝑖𝑖 . This leads to the following Lagrangian𝑁𝑁

1ℒBO �{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 , 𝑹𝑹̇𝐼𝐼 � = � 𝑀𝑀𝐼𝐼 𝑹𝑹̇2𝐼𝐼 − min 𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ] + � Λ 𝑖𝑖𝑖𝑖 ��𝜓𝜓𝑖𝑖 �𝜓𝜓𝑗𝑗 � − 𝛿𝛿𝑖𝑖𝑖𝑖 �{𝜓𝜓𝑖𝑖 }
2
𝑖𝑖=1

(27)

𝑖𝑖,𝑗𝑗

where Λ 𝑖𝑖𝑖𝑖 is a Hermitian Lagrangian multiplier matrix. By solving the according EulerLagrangian equations
𝑑𝑑

𝜕𝜕ℒ
𝜕𝜕ℒ
=
𝑑𝑑𝑑𝑑𝑹𝑹̇𝐼𝐼 𝜕𝜕𝑹𝑹̇𝐼𝐼 𝜕𝜕𝑹𝑹𝐼𝐼

𝜕𝜕ℒ=
𝑑𝑑𝑑𝑑𝑹𝑹̇ 𝜕𝜕⟨𝜓𝜓𝚤𝚤̇ | 𝜕𝜕⟨𝜓𝜓𝑖𝑖 |
𝑑𝑑

𝜕𝜕ℒ

(28)

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

one obtains the associated equations of motion (EOM)
𝑀𝑀𝐼𝐼 𝑹𝑹̈2𝐼𝐼 = −∇𝑹𝑹𝐼𝐼 �min 𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ] �{𝜓𝜓𝑖𝑖 }=−

��𝜓𝜓𝑖𝑖 �𝜓𝜓𝑗𝑗 �−𝛿𝛿𝑖𝑖𝑖𝑖 ��𝜕𝜕𝜕𝜕𝜕𝜕⟨𝜓𝜓𝑖𝑖 | 𝛿𝛿𝛿𝛿𝜕𝜕+ � Λ 𝑖𝑖𝑖𝑖�𝜓𝜓 �𝜓𝜓 � − 2 �− � Λ 𝑖𝑖𝑖𝑖 |𝜓𝜓𝑗𝑗 ���𝜕𝜕𝑹𝑹𝐼𝐼𝜕𝜕𝑹𝑹𝐼𝐼 𝑖𝑖 𝑗𝑗
𝜕𝜕𝑹𝑹𝐼𝐼 𝛿𝛿⟨𝜓𝜓𝑖𝑖 |
𝑖𝑖,𝑗𝑗

𝑖𝑖

𝛿𝛿𝛿𝛿
0≲−
+ � Λ 𝑖𝑖𝑖𝑖 |𝜓𝜓𝑗𝑗 �𝛿𝛿⟨𝜓𝜓𝑖𝑖 |

𝑗𝑗

(29)

𝑗𝑗

�𝑒𝑒 ⟨𝜓𝜓𝑖𝑖 | + � Λ 𝑖𝑖𝑖𝑖 |𝜓𝜓𝑗𝑗 �
= −𝐻𝐻
𝑗𝑗

The first term on the right hand side (RHS) of equation (29) is the so-called HellmannFeynman force. The second term, which is denoted as Pulay (1969), or wavefunction
force 𝐹𝐹WF , is a constraint force due to the holonomic orthonormality constraint, and is nonvanishing if and only if the basis functions 𝜙𝜙𝑗𝑗 explicitly depend on 𝑹𝑹𝐼𝐼 . The final term stems
from the fact that, independent of the particular basis set, there is always an implicit
dependence on the atomic positions through the expansion coefficient 𝑐𝑐𝑖𝑖𝑖𝑖 (𝑹𝑹𝐼𝐼 ) within thecommon linear combination of atomic orbitals 𝜙𝜙𝑖𝑖 :𝜓𝜓𝑖𝑖 (𝑹𝑹𝐼𝐼 ) = � 𝑐𝑐𝑖𝑖𝑖𝑖 (𝑹𝑹𝐼𝐼 )𝜙𝜙𝑗𝑗(30)

𝐹𝐹 = 𝐹𝐹HF + 𝐹𝐹WF + 𝐹𝐹NSC

(31)

𝑗𝑗

The second factor stems from the assumption that the KS orbitals are real, which is aninessential simplification. Nevertheless, the whole term vanishes whenever 𝜓𝜓𝑖𝑖 (𝑹𝑹𝐼𝐼 ) is aneigenfunction of the Hamiltonian within the subspace spanned by the not necessarily
complete basis set (Almlöf, Helgaker, 1981; Scheffler, Vigneron, Bachelet, 1985). Note,that this is a much weaker condition than the original Hellmann-Feynman theorem(Hellman, 1937; Feynman, 1939), which we hence have not availed throughout thederivation, except as an eponym for the first RHS term of equation (29). However, as the�𝑒𝑒 are only obtained at exact
KS functional is non-linear, eigenfunctions of its Hamiltonian 𝐻𝐻
self-consistency, which is why the last term of equation (29) is also referred to as non-selfconsistent force 𝐹𝐹NSC . Unfortunately, in any numerical calculation this cannot be assumed,which results in immanent inconsistent forces and to the inequality of equation (29).
Neglecting either 𝐹𝐹WF or 𝐹𝐹NSC , i.e. applying the Hellmann-Feynman theorem to a noneigenfunction leads merely to a perturbative estimate of the generalised forces (Bendt,
Zunger, 1983)

which, contrary to the energies, depends just linearly on the error in the electronic charge
density. As a consequence, it is much more exacting to calculate accurate forces than totalenergies. However, as a corollary of the BO approximation, the electronic, as well as theionic subsystems are adiabatically strictly separated from each other, and therefore does
not entail any restrictions on the maximum possible integration time step, so that time steps
up to the ionic resonance limit are feasible. This actually holds irrespective of the band gap,so, at least in principle, even metals can be straightforwardly treated.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 37

38 │ NEA/NSC/R(2019)2

2.4.2. Car-Parrinello molecular dynamics
In Car-Parrinello MD (CPMD) (1985), a coupled electron-ion dynamics is performed, inwhich the electronic degrees of freedom are added to the Lagrangian as classical ones:
𝑀𝑀

𝑁𝑁

𝑖𝑖=1

𝑖𝑖=1

1
1
ℒCP �{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 , 𝑹𝑹̇𝐼𝐼 � = 𝜇𝜇 ��𝜓𝜓̇ 𝚤𝚤 �𝜓𝜓̇ 𝚤𝚤 � + � 𝑀𝑀𝐼𝐼 𝑹𝑹̇2𝐼𝐼 − 𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ]
2
2

(32)

+ � Λ 𝑖𝑖𝑖𝑖 ��𝜓𝜓𝑖𝑖 �𝜓𝜓𝑗𝑗 � − 𝛿𝛿𝑖𝑖𝑖𝑖 �𝑖𝑖,𝑗𝑗Once again, applying the Euler-Lagrangian equations equation (28) entails an EOM, where
the electronic degrees of freedom inhere an artificial inertia 𝜇𝜇 and are propagated within afictitious Newtonian dynamics, such that the electrons follows the ions adiabatically:
𝑀𝑀𝐼𝐼 𝑹𝑹̈2𝐼𝐼 = −∇𝑹𝑹𝐼𝐼 �𝐸𝐸[{𝜓𝜓𝑖𝑖 }; 𝑹𝑹𝐼𝐼 ]�
=−𝜇𝜇𝜓𝜓̈ 𝚤𝚤 (𝒓𝒓, 𝑡𝑡) = −��𝜓𝜓𝑖𝑖 �𝜓𝜓𝑗𝑗 �−𝛿𝛿𝑖𝑖𝑖𝑖 �𝜕𝜕𝜕𝜕𝜕𝜕+ � Λ 𝑖𝑖𝑖𝑖�𝜓𝜓 �𝜓𝜓 �
𝜕𝜕𝑹𝑹𝐼𝐼𝜕𝜕𝑹𝑹𝐼𝐼 𝑖𝑖 𝑗𝑗
𝑖𝑖,𝑗𝑗

𝛿𝛿𝛿𝛿
+ � Λ 𝑖𝑖𝑖𝑖 |𝜓𝜓𝑗𝑗 �𝛿𝛿⟨𝜓𝜓𝑖𝑖 |

�
(33)

𝑗𝑗

�𝑒𝑒 ⟨𝜓𝜓𝑖𝑖 | + � Λ 𝑖𝑖𝑖𝑖 |𝜓𝜓𝑗𝑗 �
= −𝐻𝐻
𝑗𝑗

As a consequence of the BO approximation, the high frequency oscillations of equation(33) vanishes on ionic time scales, so that 𝜓𝜓̈ 𝚤𝚤 ≃ 0. Hence, similar to Ehrenfest dynamics(1927), the total derivative of the instantaneous, rather than the fully minimised,�𝑒𝑒 �Ψ0� of the Hamiltonian yields the forces that are consistent withexpectation value �Ψ0 �𝐻𝐻
the corresponding energies. This means, that owing to the absence of necessity to fully
minimise the energy functional but rather to simply evaluate it at the instantaneous time
step, 𝐹𝐹NSC is identical to zero by its very definition. Given a sufficiently small fictitious
mass, the constant of motion is strictly conserved and errors in the forces are negligible, inparticular if the ionic masses are renormalised by a constant mass tensor (Blöchl, Parrinello,
1992; Blöchl, 1994; Tangney, Scandolo, 2002). In this respect, CPMD combines most of
the advantages of BO and Ehrenfest MD in the sense that the KS functional are only
evaluated. There is no need to repeatedly solve it either by diagonalisation or, equivalently,
iterative minimisation. However, due to the finite accuracy of any integrator, the holonomic
orthonormality constraint of the orbitals has to be explicitly enforced. In order to ensure an
adiabatic energy-scale separation of the nuclear and the electronic degrees of freedom andto prevent energy transfer between them, the ionic phonon frequency 𝜔𝜔𝐼𝐼 has to be muchsmaller than the electronic analogue 𝜔𝜔𝑒𝑒 . As had been suggested in an eminent
phenomenological study of Pastore and Buda (1991)
Δ𝐸𝐸gap𝜔𝜔𝑒𝑒 ∝ �𝜇𝜇(34)

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

As a consequence, the maximum integration time step Δ𝑡𝑡max depends on the inertia
like √𝜇𝜇. The same also holds for the deviation from the BO surface (Borbemann, Schütte,1998)
�𝜓𝜓𝜇𝜇 (𝑟𝑟, 𝑡𝑡) − 𝜓𝜓0 (𝑟𝑟, 𝑡𝑡)� ≤ 𝐶𝐶 �𝜇𝜇(35)

Therefore, the fictitious mass, although physically completely meaningless, acts as acontinuous slider which allows to adjust any desired degree of accuracy, in terms ofdeviation from the BO surface, reciprocal to the computational efficiency in a wellcontrolled manner. But if a metallic system is treated, due to the fact that Car-Parrinello(CP) states are strictly not KS eigenstates, equation (34) is identical to zero and either athermostat for the electronic degrees of freedom (Blöchl, Parrinello, 1992; Sprik, 1991;
Blöchl, 2002) to counterbalance the exchange of energy, or an extended functional withfractional occupation numbers (Mermin, 1965; Gillan, 1989; Alavi et al., 1994; Marzari,
Vanderbilt, Payne, 1997) is necessary. In the end drawing a proper conclusion, if either
BOMD or CPMD is to be favoured, turns out to be very subtle (Tangney, 2006) anddepends largely on the definition of accuracy and on the particular application.

2.5. An efficient and accurate Car-Parrinello-like approach to Born-Oppenheimer
molecular dynamics
Even though DFT-based AIMD has been very successful in describing a large variety ofphysical phenomena, its computational cost has limited the attainable length and timescales in spite of substantial progress. For a while it was believed that linear scaling
methods (Yang, 1991; Galli, Parrinello; 1992; Goedecker, 1999) could have offered asolution. Unfortunately, the crossover point at which linear scaling methods become
advantageous has remained fairly large, especially if high accuracy is needed (Ceriotti,
Kühne, Parrinello, 2008, 2009). Therefore, it would be very desirable to accelerate ab-initio
simulations with up to thousands of atoms, such that simulations as long as a few
nanoseconds can be routinely performed, thus making completely new phenomena
accessible to AIMD simulations. BOMD, in which the DFT functional is fully minimisedat each MD time step, does not seem to offer much room for further improvement. For thisreason, recently another direction has been followed to improve the efficiency at current
system sizes. In the spirit of CPMD (Car, Parrinello, 1985), some form of dynamics for theelectronic degrees of freedom is implemented, which automatically keeps the system close
to the instantaneous BO surface, but at variance to the original proposal in a localised
orbital representation (Iyengar et al., 2001; Sharma, Wu, Car, 2003; Herbert, Head-Gordon,
2004). The acceleration stems on the one hand from this more compact description of theelectronic wavefunctions, but is mainly due to the ability to reduce or even fully bypass theaforementioned SCF cycle. Nevertheless, just like in CPMD, these methods suffer fromrather short integration time steps. However, rather recently a novel Car-Parrinello-likeapproach to BOMD has been proposed, which overcomes this limitation and combines theaccuracy and long-time steps of BOMD with the efficiency of CPMD (Kühne et al., 2007).
From now on the general case will be considered, where the DFT KS orbitals are expanded
in a non-orthogonal basis set. Let 𝑀𝑀 be the dimension of the Hilbert space, i.e. the numberof basis functions, and 𝑺𝑺 the 𝑀𝑀 × 𝑀𝑀 overlap matrix. As usual the expansion coefficients of
the 𝑁𝑁 lowest occupied orbitals are arranged in a rectangular 𝑀𝑀 × 𝑁𝑁 matrix 𝑪𝑪. The density
matrix can then be written as 𝑷𝑷 = 𝑪𝑪𝑪𝑪𝑇𝑇 and must obey the idempotency condition 𝑷𝑷 = 𝑷𝑷𝑷𝑷𝑷𝑷
that is due to the fermionic nature of electrons, which compels the wavefunction to beantisymmetric in order to meet the Pauli Exclusion Principle. The potential energy surface
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 39

40 │ NEA/NSC/R(2019)2
on which the ions move is defined by the minimum of an appropriately chosen energyfunctional 𝐸𝐸DFT [𝑪𝑪, 𝑹𝑹𝐼𝐼 ], which is expressed as a functional of 𝑪𝑪 and a function of the ionic
co-ordinates 𝑹𝑹𝐼𝐼 . In this notation the BO EOM reads as follows:
𝑀𝑀𝐼𝐼 𝑹𝑹̈2𝐼𝐼 = −∇𝐼𝐼 min 𝐸𝐸DFT [𝑪𝑪, 𝑹𝑹𝐼𝐼 ]
𝐂𝐂

(36)

where the search for the minimum is restricted to the 𝑪𝑪’s that satisfy the orthonormality
condition 𝑪𝑪𝑇𝑇 𝑺𝑺𝑺𝑺 = 𝑰𝑰, which is equivalent to imposing the idempotency condition on 𝑷𝑷. As
before, the forces of equation (36) can be divided into three contributions, (i) the HellmannFeynman forces (Hellmann, 1937; Feynman, 1939), (ii) the Pulay forces (1969), which arepresent whenever the basis set depends on the ionic positions, and (iii) a residual term
(Bendt, Zunger, 1983), which is non-zero except when full self-consistency is reached. The
last term leads inevitably to poor energy conservation in BOMD unless a very tightconvergence criterion is imposed. In Car-Parrinello-like approaches this is circumvented
by the design of a coupled electron-ion dynamics, which maintains the system very close
to the BO surface, but at the cost of small integration time steps.

2.5.1. Density matrix propagation
Based on ideas of the original CP approach it is possible to design an improved dynamics
for the coupled system of electrons and ions (Kühne et al., 2007). However, contrary to theoriginal scheme, this novel method is not expressed as an explicit EOM for the 𝑪𝑪’s, butrather as an integration scheme for the electronic degrees of freedom. The knowledge ofthe previous 𝐾𝐾 values of 𝑪𝑪(𝑡𝑡𝑛𝑛−𝑙𝑙 ), where 𝑙𝑙 ∈ [1, 𝐾𝐾], determines the value of 𝑪𝑪(𝑡𝑡𝑛𝑛 ), suchthat at any instant of time the 𝑪𝑪’s are as close as possible to the instantaneous ground state.
As for the short-term integration of the electronic degrees of freedom, accuracy is crucial
so a highly accurate and efficient algorithm is required. Therefore, here the always stablepredictor-corrector (ASPC) method of Kolafa (2004, 2005) has been selected. This scheme
was originally devised to deal with classical polarisation, so that care must be taken thatduring the evolution the idempotency condition is always satisfied. The modified predictor
𝐾𝐾
2𝐾𝐾��𝐾𝐾−𝑚𝑚𝑝𝑝 (𝑡𝑡 )𝑚𝑚+1)𝑪𝑪𝑇𝑇 �(𝑡𝑡�𝑛𝑛−𝑚𝑚
𝑪𝑪 𝑛𝑛 ≅ � (−1)
𝑚𝑚 2𝐾𝐾−2 𝑪𝑪(𝑡𝑡
���
�����
���) 𝑺𝑺(𝑡𝑡𝑛𝑛−𝑚𝑚 )𝑪𝑪(𝑡𝑡𝑛𝑛−1 )
𝑛𝑛−𝑚𝑚
�
�
𝐾𝐾−1
𝑷𝑷(𝑡𝑡𝑛𝑛−𝑚𝑚 )
𝑚𝑚=1

(37)

uses the extrapolated contra-covariant density matrix 𝑷𝑷𝑷𝑷 as an approximate projector on tothe occupied subspace 𝑪𝑪(𝑡𝑡𝑛𝑛−1 ). In this way, the fact that the physically relevant contracovariant density matrix 𝑷𝑷𝑷𝑷 evolves much more smoothly and is therefore substantiallyeasier to predict than 𝑪𝑪 is ideally utilised. The modified predictor is followed by a corrector
step to minimise the error and to further reduce the deviation from the instantaneous groundstate. The corrector
𝐾𝐾(38)
2𝐾𝐾 − 1consists only of a single preconditioned minimisation step min[𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 )] of a properly
selected minimisation procedure. Apparently, the predictor can also be repeatedly applied,
in which case the ground state is even more closely approached, but at the cost of additional
electronic gradient calculations. However, as will be shown immediately in general this is
not necessary. The numerical coefficients of equation (37) were selected in order to ensure
time-reversibility up to 𝑂𝑂(ℎ𝐾𝐾+2 ) while 𝜔𝜔 was chosen to guarantee a stable relaxation
towards the minimum. Due to the fact that the energy is invariant under unitarytransformations within the subspace of occupied orbitals 𝑪𝑪, it must be ensured that this𝑪𝑪(𝑡𝑡𝑛𝑛 ) = 𝜔𝜔 min[𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 )] + (1 − 𝜔𝜔)𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 )with 𝜔𝜔 =STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

time-dependent gauge transformation is not strongly changed by min[𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 )], as in thiscase continuity between the 𝑪𝑪’s may be lost.

2.5.2. Electronic forces by orbital transformations

Moreover, the minimisation scheme must be very efficient in bringing the system as close
as possible to the instantaneous ground state and at the same time preserves theidempotency condition of the density matrix. For these reasons, the orbital transformation(OT) method of VandeVondele and Hutter (2003) has been chosen. Inspired by the formof the exponential transformation (Hutter, Parrinello, Vogel, 1994) an auxiliary variable X
is introduced, to parameterise the occupied orbitals
𝑪𝑪(𝑿𝑿) = 𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 ) cos(𝑼𝑼) + 𝑿𝑿𝑼𝑼−1 sin(𝑼𝑼)(39)

where 𝑼𝑼 = (𝑿𝑿𝑇𝑇 𝑺𝑺𝑺𝑺)1/2 and the variable 𝑿𝑿 has to obey the linear constraint 𝑿𝑿𝑇𝑇 𝑺𝑺𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 ) =0. Under this condition 𝑪𝑪(𝑿𝑿) leads to an idempotent density matrix for any choice of 𝑿𝑿,
provided that the reference orbitals 𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 ) are orthonormal. Thus, any finite step along thepreconditioned gradient direction will exactly fulfil the idempotency constraint byconstruction. Due to the linear constraint the minimisation with respect to 𝑿𝑿 is performed
in an auxiliary tangent space. Since this space is linear, no curved geodesics must be
followed, as is the case for variables such as 𝑪𝑪 that are nonlinearly constrained. In this way,large minimisation steps can be taken, especially if a good preconditioner is used (Gan,
Haynes, Payne, 2000). In fact, using an efficient, idempotency conserving direct minimiser
such as OT is decisive for the success of this approach. Since the ASPC integrator only
approximately preserves the idempotency constraint, it sporadically has to be explicitly
enforced, either by Cholesky decomposition or by single purification iterations (McWeeny,1960).

2.5.3. Total energies and forces
Having obtained the new wavefunction it is now possible to evaluate the energy and thenuclear forces, which are derived from the following approximate energy functional:1𝜌𝜌𝑝𝑝 (𝒓𝒓)𝜌𝜌𝑝𝑝 (𝒓𝒓′ )𝐸𝐸PC [𝜌𝜌𝑝𝑝 ] = Tr [𝑪𝑪𝑇𝑇 𝐻𝐻[𝜌𝜌𝑝𝑝 ]𝑪𝑪] − � 𝑑𝑑𝒓𝒓 � 𝑑𝑑𝒓𝒓′|𝒓𝒓 − 𝒓𝒓′ |2− � 𝑑𝑑𝒓𝒓 𝑉𝑉XC

[𝜌𝜌𝑝𝑝 ]𝜌𝜌𝑝𝑝

+ 𝐸𝐸XC[𝜌𝜌𝑝𝑝 ]

(40)

+ 𝐸𝐸𝐼𝐼𝐼𝐼

where 𝜌𝜌𝑝𝑝 is the density associated with 𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 ). 𝐸𝐸PC [𝜌𝜌] can be thought of as an
approximation to the Harris-Foulkes functional (Harris, 1985; Foulkes, Haydock, 1989)
and maintains the predictor-corrector flavour of this method. The validity of 𝐸𝐸PC [𝜌𝜌]
depends only on the efficiency of the minimiser and on the quality of the propagation
scheme. The ionic forces are calculated by evaluating the analytic gradient of 𝐸𝐸PC [𝜌𝜌] withrespect to the nuclear co-ordinates. However, as Δ𝜌𝜌 = 𝜌𝜌 − 𝜌𝜌𝑝𝑝 ≠ 0, besides the usual
Hellmann-Feynman and Pulay forces an extra term appears:− � 𝑑𝑑𝒓𝒓 ���

𝜕𝜕𝑉𝑉XC [𝜌𝜌𝑝𝑝 ]� Δ𝜌𝜌 + 𝑉𝑉𝐻𝐻 [Δ𝜌𝜌]� (∇𝐼𝐼 𝜌𝜌𝑝𝑝 )�
𝜕𝜕𝜌𝜌𝑝𝑝(41)

where 𝜌𝜌 is the corrected density evaluated using 𝑪𝑪(𝑡𝑡𝑛𝑛 ) and 𝜌𝜌𝑝𝑝 is the predicted density
calculated from 𝑪𝑪𝑝𝑝 (𝑡𝑡𝑛𝑛 ). Using variational density functional perturbation theory (Putrino,
Sebastiani, Parrinello, 2000; Benoit, Sebastiani, Parrinello, 2001), equation (41) can be
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 41

42 │ NEA/NSC/R(2019)2
efficiently computed very similar to employing the coupled-perturbed KS scheme.
However, due to the fact that usually only a single preconditioned minimisation step isperformed, 𝑪𝑪(𝑡𝑡𝑛𝑛 ) is just an approximate eigenfunction of 𝐻𝐻[𝜌𝜌𝑝𝑝 ] within the subspacespanned by the finite basis set used. This leads to an insignificant error in the forces,
provided that 𝑪𝑪(𝑡𝑡𝑛𝑛 ) is very close to the ground state.

2.5.4. Modified Langevin equation

The ability of this dynamics to maintain the system on the BO surface may varyconsiderably. It is essentially ideal in systems like water, but potentially somewhat lessperfect in liquid Si at high temperature, where swift bonding and rebonding processes
continuously take place. However, in all cases the dynamics is dissipative, most likely
because the employed propagation scheme is not symplectic. Nevertheless, it is possible torigorously remedy this downward drift if we assume that the forces arising from ourdynamics 𝑭𝑭PC can be modelled as 𝑭𝑭PC = 𝑭𝑭BO − 𝛾𝛾𝐷𝐷 𝑹𝑹̇𝐼𝐼 , which, as we shall see immediately,is an excellent assumption. The value of the intrinsic friction coefficient 𝛾𝛾𝐷𝐷 does not need
to be known but it can be bootstrapped by taking a cue from the work of Krajewski andParrinello (2001). The canonical distribution is sampled by using the following Langevintype equation
𝑀𝑀𝐼𝐼 𝑹𝑹̈𝐼𝐼 = 𝑭𝑭PC − 𝛾𝛾𝐿𝐿 𝑹𝑹̇𝐼𝐼 + 𝚵𝚵𝐼𝐼

(42)

𝑀𝑀𝐼𝐼 𝑹𝑹̈𝐼𝐼 = 𝑭𝑭BO − (𝛾𝛾𝐷𝐷 + 𝛾𝛾𝐿𝐿 )𝑹𝑹̇𝐼𝐼 + 𝚵𝚵𝐼𝐼

(43)

〈𝚵𝚵𝐼𝐼 (0)𝚵𝚵𝐼𝐼 (𝑡𝑡)〉 = 6(𝛾𝛾𝐷𝐷 + 𝛾𝛾𝐿𝐿 )𝑀𝑀𝐼𝐼 𝑘𝑘B 𝑇𝑇𝑇𝑇(𝑡𝑡)(44)where 𝑀𝑀𝐼𝐼 is the ionic mass, 𝛾𝛾𝐿𝐿 is a Langevin friction coefficient and 𝚵𝚵𝐼𝐼 = 𝚵𝚵𝐼𝐼𝐷𝐷 + 𝚵𝚵𝐼𝐼𝐿𝐿 an
additive white noise. Using the above assumption equation (42) is identically to:

In order to guarantee an accurate sampling of the Boltzmann distribution, the noise has toobey the fluctuation dissipation theorem:

The choice of 𝛾𝛾𝐿𝐿 is arbitrary, while the unknown 𝛾𝛾𝐷𝐷 has to be determined by requiring thatthe aggregate noise term generate the correct average temperature, i.e. fulfils the13equipartition theorem 〈 𝑀𝑀𝐼𝐼 𝑹𝑹̇2𝐼𝐼 〉 = 𝑘𝑘B 𝑇𝑇. As we see in a moment, this leads to correct a2
2
sampling of the Boltzmann distribution. In addition, since the initial dynamics is quite
accurate, 𝛾𝛾𝐷𝐷 is rather small and even dynamical properties can be very well reproduced.

2.5.5. Illustrative examples: liquid silicon, silica and water

For the purpose of demonstrating this new approach, it has been implemented in the mixed
Gaussian Plane Wave (GPW) (Lippert, Hutter, Parrinello, 1997) code QUICKSTEP (Krack,
Parrinello, 2004; VandeVondele et al., 2005), which is part of the publicly available suiteof programmes CP2K. 2 In order to illustrate that this method works well irrespective ofband gap, system size and type, calculations on metallic liquid silicon and liquid silica arepresented. Both systems are known to be very difficult, and are examples of liquid metals(Si) as well as of complex, highly polarisable, ionic liquids (SiO2). Furthermore, thesimulations have been performed at 3 000 K and 3 500 K respectively, which leads torapidly varying density matrix elements, thus making the propagation of the electronic
2. www.cp2k.org
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

degrees of freedom particularly challenging. Hence, the selected test cases can beconsidered as worst-case scenarios for any method.
All simulations have been performed at their experimental liquid densities using doublezeta valence polarisation (DZVP) basis sets, adequate density cutoffs, Goedecker-TeterHutter pseudopotentials (Goedecker, Teter, Hutter, 1996; Hartwigsen, Goedecker, Hutter,
1998; Krack, 2003) and the local density approximation to the exact exchange andcorrelation functional. For simplicity the Brillouin zone is sampled at the Γ-point only,
while equation (43) is integrated using the algorithm of Ricci and Ciccotti (2003), with atime step of ℎ = 1 fs. The friction coefficient 𝛾𝛾𝐿𝐿 was set equal to zero, while the values for𝛾𝛾𝐷𝐷 turned out to be in the range of 10−4 fs−1 . The new 𝑪𝑪’s are predicted using 𝐾𝐾 = 4 inequation (37), which ensures time-reversibility up to 𝑂𝑂(ℎ6 ).
First, the accuracy in terms of the energetic deviation from the BO surface is considered.
As can be seen in Figure 2.1. the energies are an upper bound to the ground state and are
displaced by a very small and approximately constant amount. It is also shown that, as
already mentioned, the deviation from the BO surface can be even further reduced byincreasing the number of corrector steps. In fact, it is actually possible to control thedeviation from the BO surface by varying the number of corrector steps in order to achievea preassigned accuracy level. However, in the following only simulations based on a single
corrector step will be reported.
Figure 2.1. Deviations from the BO surface of liquid SiO2 with respect to total energies(upper panel) and mean force deviations (lower panel)

Note: The deviation in the energies corresponds to a constant shift of 4.16 ∙ 10−4 Hartree per atom for one
corrector step and 3.5 ∙ 10−5 Hartree per atom for two corrector steps. The average mean force deviation isunbiased.
Source: Kühne, 2019.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 43

44 │ NEA/NSC/R(2019)2
Figure 2.2. Partial pair-correlation functions g(r) of liquid Si (upper left panel) and liquid
SiO2 at 3 000 K and 3 500 K respectively, using a DZVP Gaussian basis set

Source: Kühne, 2019.

Nevertheless, let us now to turn to more realistic problems such as those shown in Figure
2.2. Although these simulations have been performed with only a single corrector step, they
are still amazingly close to the BOMD reference results. It should be emphasised that evenin liquid Si, which is metallic and poses problems when using an ordinary CP scheme, asingle corrector step is sufficient. This establishes the efficiency of this method, since onlya single preconditioned gradient calculation with no additional minimisation step has to be
performed. The possible acceleration, in comparison with regular BOMD calculations,depends crucially on the system studied. In the undoubtedly difficult cases just presented a
speedup of two orders of magnitude compared to using a pure extrapolation scheme has
been observed. For simpler problems still an increase in efficiency of at least one order ofmagnitude can be expected.
In Figure 2.3. displays results, which prove that also dynamical properties can be evaluatedwith accuracy. To that extent the velocity autocorrelation function and its Fourier transform
at 325 K is presented. The results are in good agreement with accurate reference
calculations and are consistent with experiment, as well as ab-initio all-electron
calculations (Krack, Parrinello, 2000), showing that in spite of the stochastic nature ofequation (43) dynamical properties can also be simulated. This implies that also chemical
reactions and even non-equilibrium processes can be treated. In the same picture it isexplicitly verified that the previous assumptions are justified, and indeed a canonicalsampling is performed, by showing that the kinetic energy distribution is Maxwellian
distributed. To this end, a 64 atom liquid Si simulation is carried out for as long as 1 ns, toreduce the noise and to ensure a proper sampling of the relevant kinetic energy distribution
tails.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 2.3. (a) The kinetic energy distribution of metallic liquid Si64 (b) Velocity
autocorrelation function. (c) Fourier transform of the velocity autocorrelation function of
32 water at 325 K.

(a)

(b)

(c)

Note: Calculation details: (a) 1 ns trajectory, time step: 3.25 fs, basis set: DZVP Gaussian basis set, density
cutoff: 100 Ry (c) basis set: TZV2P Gaussian basis set, density cutoff: 280 Ry, exchange-correlation functional:
BLYP,. Langevin friction coefficients: 𝜸𝜸𝑳𝑳 = 𝟎𝟎 and 𝜸𝜸𝑫𝑫 ~𝟏𝟏𝟏𝟏−𝟖𝟖 fs−𝟏𝟏 .
Source: Kühne, 2019.

Due to space considerations only a fraction of the systems studied is reported here.
Nevertheless, in all cases this method has proven to be accurate and the gain in speed has
always been remarkable (Kühne, Krack, Parrinello, 2009; Camellone, Kühne, Passerone,
2009; Cuinotta et al., 2009; Kühne et al., 2011; Luduena, Kühne, Sebastiani, 2011a, 2011b).
Structure relaxations via dynamic annealing and geometry optimisation have also beensuccessfully performed (Carvati et al, 2007, 2009a, 2009b, 2009c, 2010, 2011). Contrary
to CPMD and related methods integration time steps up to the ionic resonance can be used.
Thanks to this development it is now possible to perform AIMD simulations on mediumsized systems up to a few nanoseconds, thus making a new class of problems accessible toab-initio simulations.

2.6. Conclusion
To conclude it should be noted, that with increasing length and time scales CPMD-based
approaches are expected to become more advantageous than BOMD, since otherwise
meeting the more and more stringent accuracy requirements of longer simulations andlarger system sizes would entail an ever tighter wavefunction convergence. The CarParrinello-like approach to BOMD (Kühne et al, 2007) just described extends the scope of
ab-initio simulations by combining the best of either method and allows for AIMD
simulations previously thought not feasible.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS│ 45

46 │ NEA/NSC/R(2019)2

Acknowledgements
The author would like to thank Matthias Krack and Michele Parrinello for various fruitful
discussions, as well as the whole CP2K-Team. In addition, he is indebted to the Graduate
School of Excellence MAINZ and the IDEE project of the Carl-Zeiss Foundation for
financial support. The generous allocation of computer time and support from CSCS, theICT Services of ETH Zurich and the DEISA consortium is kindly acknowledged.

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3. Classical interatomic potentials
T. R. Zeitler and L. J. Criscenti
Sandia National Laboratories, United States

3.1. Introduction
The behaviour of nuclear materials under operating conditions is difficult, expensive,
unsafe and in some cases impossible to observe experimentally. Computer simulation
provides an attractive alternative to direct experimental observation and has the distinct
advantage of allowing for systematic study of individual effects on materials properties.
Classical simulations have a rich history of providing atomic-scale information based on
simple energy minimisation (e.g. predicting stable defect configurations), molecular
dynamics (MD) (i.e. the time evolution of a system of atoms), and Monte Carlo (MC) (i.e.
stochastic) simulations that can then be used to drive the design of future experiments.
These techniques require a description of the potential energy of a system, which is usuallydescribed via a set of classical potentials (or force field).
A set of interatomic potentials consists of parameterised equations and parameter values
developed for specific interactions that allow for calculation of potential energy as afunction of a geometric variable (usually interatomic separation, but angles are sometimes
used) (Cygan, 2001). For example, a simple harmonic potential has the form E(rij)=k(rijro)2, where the parameters k and ro are defined for the interaction of two types of atoms iand j — e.g. O and H in an O-H bond—and the interatomic separation is given by rij. A
harmonic potential for the interaction of Si and O in a Si-O bond will have the same form,
but different parameter values. The potential energy of a system is therefore defined by thepotentials and the geometry of the system. In this way, the potential energy of a complex
system can be calculated by summing the contributions of each interaction according to theprescribed analytical expressions defined by the potentials (Cygan, 2001).
Interatomic potentials are parameterised and verified using experimental results andelectronic-scale ab-initio calculations. These potentials are then used as input for atomicscale simulations. The materials properties calculated from atomic-scale simulations (e.g.
thermal conductivity or defect populations) can then be used for higher-level (e.g. kinetic
Monte Carlo (KMC)) calculations. Of course, the reliability of the properties calculated byhigher-level methods depend heavily on the quality of the underlying potentials used forthe atomic-scale simulations. Clearly, developing reliable potentials is of great necessity tomulti-scale modelling endeavours and the ability of a higher-level method to accurately
describe physical processes. For example, a classical potential should be able to properly
reproduce point defect populations before a higher-level method could be expected toaccurately calculate the effects of defects on microstructure for a material undergoingneutron irradiation (Becquart, Domain, 2011).
If the ultimate goals of simulation are to predict material lifetimes or the long-term
microstructure evolution due to exposure to severe conditions experienced in nuclear
reactors (e.g. elevated temperatures, corrosive environments, cyclic loading [Devananthan
et al., 2001]), then an accurate description of atomic-scale interactions for nuclear materials
is necessary. Increased reliability and confidence in potential models will lead to improved
multi-scale modelling results. When an accurate multi-scale modelling approach is
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implemented, a feedback loop will then be available which will allow for increasing
efficiency in exploiting the properties of current materials (e.g. enhancing enrichment andfuel burnup in nuclear fuels) and a true engineering of novel nuclear materials. One of theindispensable keys to a proper multi-scale model is the development of quality interatomic
potentials.

3.2. Key issues in developing classical potentials
3.2.1. Fitting/verification
After assuming a form of the classical potential (e.g. harmonic, Buckingham), parameters
are fit in order to replicate experimental data (typically static elastic and dielectric constants
and lattice parameters, but also defect properties) or theoretical calculations. Parameters
are changed systematically, manually or using a computer program such as GULP (Gale,
Gulp, 1997), to minimise the difference between given data and those calculated with thenew potential. The number of parameters needed for a complete potential depends on theform of the potential. Potential parameters can be improved by including a greater numberof experimental comparisons.
A key test to ensure an accurate potential is model verification. This can be done bycomparison with experimental data that has not been used to initially fit the parameters,
though calculated properties (e.g. from ab-initio MD simulations) also provide additional
points of comparison. In most cases, some experimental data is used as fitting parameters
for development of the model, so it is not surprising when there is strong agreement withthese same calculated properties. However, more complicated properties (i.e. second-orderproperties) are increasingly used as both fitting parameters and points of verification. One
example is parameterising to vibrational power spectra.

3.2.2. Applicability/transferability
One of the key issues in potential development has been determining the extent to which a
set of interatomic potentials can reliably be applied to situations far removed from thefitting domain. This is true in terms of interatomic separations, temperature/energy range,
and composition (e.g. the transferability from pure to mixed systems). For example,
potential parameters that are fit to zero K lattice properties may not be able to fully
reproduce dynamical properties at high temperatures (e.g. predicting melting point). And
fitting parameters to a distinct range of temperature data does not guarantee a proper fit for
temperatures outside of that range. In this way, a lack of quality experimental data can limit
the development of a quality potential.
A lack of initial fitting parameters can lead to limitations in its range of applicability. A
single model that is applicable over a large range of temperature and composition is rare
(Tiwary van de Walle, Gronbech-Jensen., 2009). By fitting to a large number of initialstructures, transferability of a potential set to a new system (which can be fundamentallydifferent, but consist of the same components) is helped (Cygan, Liang, Kalinichev, 2004).
General force fields (such as the universal force field (UFF) (Rappe et al., 1992) or Dreiding(Mayo et al., 1990) force field) have been used for materials (Duren, Bae, Snurr, 2009;
Nalaparaju et al., 2009) that did not exist when the force fields were created.

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3.3. Classification of classical potentials
A set of classical potentials can be classified in a number of ways, including its descriptionof individual ions, what types of interactions are included, and the specific forms of thoseinteractions.

3.3.1. Description of ion
Two basic models of interatomic potentials have been used with nuclear materials, the coreshell model and the rigid ion model. The core-shell model accounts for polarisation effects
by describing ions as massive cores connected to massless charged shells by a spring (Dick,
Overhauser, 1958). In this case, the shells interact with one another via short-range
potentials, while long-range Coulomb interactions take place among all cores and shells.
Some examples of core-shell models are those of Jackson (1986) and Gotte (2007). In therigid ion model, all ions are described as massive point charges. MD simulations with rigid
ion models are faster because there are fewer interactions to calculate. Rigid ion models
include those of Morelon (2003), Sindzingre and Gillan (1988), and Karakasidis (1994).
Defect energy calculation benefits from a description of electronic polarisation of the latticeions. This is usually accounted for by use of core-shell models in which an ionic dipole canbe represented by the core-shell separation. However, due to the extremely high energy
considered in displacement cascade simulations (discussed below), core-shell models are
not suitable for this type of simulation (Morelon et al., 2003).

3.3.2. Interatomic potential types
Besides a model type, each set of potentials has general forms of energy description. The
electrostatic contribution to energy is long-range in nature and given by the well-known
form of the Coulomb energy:−𝐸𝐸𝑖𝑖𝑖𝑖 =1 𝑞𝑞𝑖𝑖 𝑞𝑞𝑗𝑗
4𝜋𝜋𝜖𝜖0 𝑟𝑟𝑖𝑖𝑖𝑖(1)

where qi is the charge assigned to atom i, and rij is the interatomic distance between atoms
i and j and ε0 is the permittivity of free space. The total contribution of electrostatics canbe obtained by summing over all atoms and their periodic images. The Ewald summation(1921) or Wolf’s direct method (1999) are typically used to efficiently calculate this sum.
In some cases, partial charges are used instead of formal charges.
Other potentials are restricted to apply only for a smaller range of interatomic separations(i.e. a cutoff separation is enforced) and are thus termed “short-range” contributions. The
simplest short-range description of forces is that of a two-body (“pair”) potential that isdefined only between two atoms. A number of pair potential forms exist, including
Buckingham (1938), Born-Mayer-Huggins (BMH) (Fumi, Tosi, 1964a, 1964b), LennardJones (L-J) (Frenekl, Smit, 2002), and Morse (1929).
Many-body potentials are used in cases where a simple two-body treatment is insufficient
to properly replicate fitting data. In a simple case, the potential energy of a three-body termis based on the positions of two neighbours. This formulation is able to represent somedirectionality in bonding. Some examples of three-body potential forms are StillingerWeber (1985) and the embedded-atom method (EAM) (Daw, Baskes, 1983).
Finally, complex reactive force fields, where explicit charge transfer and even the breakingof bonds are permitted, are becoming more popular as computational power continues to
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explode. Some examples of reactive force fields include REAXFF (van Duin et al. 2008)
and that of Garofalini (2001).

3.3.3. Specific forms of classical potentials
Buckingham
The Buckingham form of pair potentials is common for oxide materials and has the form:
𝐸𝐸𝑖𝑖𝑖𝑖 = 𝐴𝐴𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒 �−𝑟𝑟𝑖𝑖𝑖𝑖
𝐶𝐶𝑖𝑖𝑖𝑖
�− 6𝜌𝜌𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑗𝑗(2)where Aij, ρij, and Cij are parameters for the interactions of atoms of type i and j. The firstterm represents Pauli electron repulsion due to overlapping electron densities while thesecond describes van der Waals attraction. At the very short interatomic separations
experienced in high-energy displacement cascade simulations, the net Buckingham
potential is unphysically attractive, which has led to the development of splined potentials
that result in repulsion even at very small interatomic distances (Ziegler, Biersack, Littmark
et al., 1985; Govers et al., 2007).
The BMH potential energy description is similar to the Buckingham potential (Lewis,
Catlow, 1985). The Bushing-Ida (Yoshiaki, 1976) form of potential energy has been used
in a number of models (Basak, Sengupta, Kamath, 2003; Yamada et al., 2000a) for UO2
and can be written as the summation of Buckingham and Morse potentials (Watanabe etal., 2008).

Morse
Covalent bonding between anions and cations is sometimes accounted for via use of a
Morse potential of the generic form:

Lennard-Jones

𝐸𝐸𝑖𝑖𝑖𝑖 = 𝐷𝐷0 �1 − 𝑒𝑒 −𝛼𝛼(𝑟𝑟−𝑟𝑟0 ) �(3)

2

A simple and commonly used potential form is that of L-J:
𝐸𝐸𝑖𝑖𝑖𝑖 = 𝐷𝐷0 ��

12

𝑅𝑅0
�
𝑟𝑟𝑖𝑖𝑖𝑖

− 2�

6

𝑅𝑅0
� �
𝑟𝑟𝑖𝑖𝑖𝑖(4)where D0 is the depth of the potential energy well and R0 is the equilibrium atomic
separation. The first term describes electron repulsion and the second reflects London
dispersion (Cygan, 2001).

Vashishta-Rahman
A Vashishta-Rahman (VR) potential has the form:
𝐶𝐶𝑖𝑖𝑖𝑖𝑞𝑞𝑖𝑖 𝑞𝑞𝑗𝑗
𝐴𝐴𝑖𝑖𝑖𝑖 (𝜎𝜎𝑖𝑖 + 𝜎𝜎𝑗𝑗 )𝜂𝜂𝑖𝑖𝑖𝑖
𝑃𝑃
𝐸𝐸𝑖𝑖𝑖𝑖 =+�� − � 4� − 6
𝜂𝜂𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖
𝑟𝑟

(5)

𝑖𝑖𝑖𝑖

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where the first term is the Coulomb term, the second term represents electron repulsion, thethird term represents a monopole-induced dipole interaction, and the final term describes
van der Waals attraction (Gunay et al., 2011).

Ziegler-Biersack-Littmark
The Ziegler-Biersack-Littmark (ZBL) universal pair potential (1985) is widely used forsmall interatomic separations where a traditional Buckingham or BMH potential isunphysically attractive (Tiwary et al. 2009, 2001; Ackland, 1997). The ZBL potential isthen smoothly connected to the other form using some sort of (arbitrary) spline, such thatthe overall potential, as well as its derivative is always continuous. The morphology of adisplacement cascade is heavily influenced by this repulsion at small separations (Becquart
et al., 2000). An alternative to the ZBL potential is a Buckingham (4-range) potential,
sometimes used to avoid the attraction at unphysically small O-O separations (Jackson etal., 1986; Read, Jackson, 2010; Björkas, Nordlund, 2007).

Embedded-atom method
The EAM (Daw, Baskes, 1983, 1984) and Finnis-Sinclair (FS) (1984) are potential forms
for describing metallic bonding and represent significant advances in metallic potentials.
EAM consists of pair potentials plus an embedding energy to represent many-bodyinteractions due to the local environment. This embedding energy is analogous to thebonding term in FS. In addition to being a function of interatomic separation, the interaction
between two atoms is also a function of local environment. These potentials are densitydependent, which makes validation more challenging, especially when the potentials areextended to alloys.
The form of the EAM potential is:
1
𝐸𝐸𝑖𝑖 = 𝐹𝐹𝛼𝛼 �� 𝜌𝜌𝛽𝛽 (𝑟𝑟𝑖𝑖𝑖𝑖 )� + � 𝜙𝜙𝛽𝛽 (𝑟𝑟𝑖𝑖𝑖𝑖 )
2
𝑗𝑗≠𝑖𝑖

(6)

𝑗𝑗≠𝑖𝑖

where F is the embedding energy and is a function of atom i’s environment via the density
ρ of surrounding atoms. The potential is defined for atom types i and j and applies to allneighbours j of atom i within a prescribed cutoff distance. A pair potential 𝜙𝜙 is also definedfor the i-j interaction. The EAM formulism has been shown to be versatile for a number ofdifferent Fe-alloys (Jang, Lee, Hong, 2008).

3.4. State of the art in classical potentials for nuclear fuel cycle materials
3.4.1. Nuclear fuels
UO2
The most common nuclear fuel in use is UO2; likewise, the greatest number ofcomputational studies of nuclear fuels is for UO2. A number of different potentials havebeen used over the past 50 years to model UO2 structure, defect behaviour, and diffusion,as well as the effect of neutron irradiation on those properties. In many cases, incremental
advances in the UO2 potentials have led to a greater agreement between simulation results
and experimental data (or theoretical calculations).

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The first UO2 potential was introduced in 1962 (Benson, Freeman, Dempsey, 1963).
Catlow et al. later developed a series of improved UO2 potentials (1978, 1977) in the 1970s– these were core-shell models fit to elastic and dielectric constants and lattice parameter
and were designed to calculate defect formation energies. A set of rigid ion potentials(Walker, Catlow, 1981) of the BMH form was also developed by Catlow et al to be able torun MD at high temperatures in order to investigate superionic properties. In addition,
Grimes and Catlow also developed an extensive set of self-consistent core-shell potentials
to examine migration and solution energies of Xe, I, Ba, Kr, Rb, Br, He, Sr, Y, Te, Cs, Ba,
La, Ce, Zr, Ru in UO2. (See references 26-34 in Govers et al., 2007.)Jackson later modified (Jackson et al., 1986, 1985) Catlow’s potentials to eliminate theunphysical attraction of Buckingham energy at small separations by adding a new
functional form of the potential and combining it with the Buckingham form using
polynomial interpolations (splines). Sindzingre and Gillan (1988) later modified O-U
interactions for the same potential and added a splined (4-range) Buckingham for O-O
interactions. Karakasidis and Lindan (1994) revised these parameters after discovering that
the O-O spline potential was not stable.
Another core-shell model was developed for MD simulations of UO2 by Lindan and Gillan(1994) to look at oxygen diffusion at high temperature and the superionic phase. Yamada
et al. (2000a) developed an independent potential with non-integer charges (qU=+2.4) anda covalent U-O (Morse) term.
Morelon et al. (2003) used point defect energies (static and migration) for fitting to a BMHtype potential, eliminating U-U short-range interactions, zeroing the attractive term for UO interactions, and revising the partial charges of U and O species (Morelon et al., 2003).
These defect properties are important in displacement cascade simulations, though the
migration energies are more difficult to calculate (than defect energy formation) because
they require a determination of the transition structure of the migration process (Morelon
et al., 2003). The Morelon potential showed improved agreement with thermal expansion
data, especially above 2000 K (whereas the Yamada potential (2000a) is only good up to
about 1 500 K) where anharmonicity effects have been taken into account by the use of
defects and transition states were included in the parameterisation of potential parameters.
It also shows improved defect energies, but mediocre agreement with elastic properties.
An extensive two-part investigation of interatomic potentials by Govers et al. (2007, 2008)
for UO2 includes a thorough review of the history of UO2 potentials from 1962 to 2007. In
addition to detailing significant modifications of a number of independently-developed
potentials, the first part of the investigation compared 19 separate versions of potential
models on their ability to reproduce experimental and ab-initio properties including lattice
parameters; elastic constants; phonon dispersion modes; and the formation, migration andbinding energies for various defects using static calculations. The potentials reproduced thelattice parameters and elastic constants, though this is not surprising because mostparameters were originally fit to these properties. A wide range of values was calculated
for the other properties, leading the authors to conclude that no best potential existed in thegroup. However, they could conclude that neither the class (i.e. core-shell or rigid ion) nor
type (e.g. Buckingham, Morse) of model had a great impact on the accuracy of thepotentials; rather, it was the parameterisation of the model that was critical.
In the second part of the investigation, Govers et al. (2008) used MD simulations to test thesame potentials for their abilities to reproduce the temperature dependence of various
properties including lattice parameters, specific heat and bulk modulus, as well as melting
temperature. This was a test of the anharmonicity of the potentials, i.e. their abilities to
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provide information on non-equilibrium properties even though they were fit to static latticeproperties. While Govers et al. again conclude that no single best potential was found, they
make the generalisation that rigid ion models with partial-charge assignments and somecore-shell potentials were best able to reproduce experimental data (likely because thermal
expansion curves were part of the fitting processes in many of these cases). Core-shell
potentials are generally developed to be able to reproduce static polarisation effects andthus do not accurately reproduce anharmonicity effects (Govers et al., 2008). However,they did note that they expect challenges with partial-charge rigid ion models when dealing
with charged defects (e.g. how to properly determine the ionicity of the defect)—andsuggest that a potential developed with formal charges would be more appropriate in thatcase.
Since Govers’ extensive reviews, additional UO2 models have been published, as well as acomparison of various potentials for their ability to produce defects via energetic recoils(Devananthan, Yu, Weber, 2009). This study showed a dramatic effect of a chosen potential
on the calculated defect population. Because the size of defect clusters and distribution of
defects play a significant role in the later evolution of the defect site, a proper descriptionof the potential energy is crucial to be able to have confidence in higher-level calculations.
Also, for an electronic insulator such as UO2, defects play a major role in determining
thermal transport properties. This study also found that the potential of Yakub (2007) best
replicated equilibrium properties.
Tiwary et al. (2009) developed a UO2 potential that is applicable across all interatomic
distances by fitting to the results of ab-initio calculations of defect energies. They also
underlined the importance of a proper splining procedure to join multiple potentials thatare individually valid over separate spatial regions.
Using a novel methodology for potential development, Read and Jackson (2010) introduced
a new potential to more accurately represent the UO2 lattice following Frenkel defectcreation due to neutron irradiation and to correct the overestimated contribution of theattractive term in the cation-anion Buckingham potentials previously published. This model
was developed to enhance applicability over a wide range of structural data and a wide
range of other physical properties. In this core-shell model, the anion-anion Buckingham
potential was derived and the cation-anion short-range interactions were given a BMH
form. Though not included in the original parameterisation, calculated phonon dispersion
curves and defect energies compare well with literature values. This model has not yet been
applied to MD simulations.
A new potential by Gunay (2001) has been shown to be excellent at describing the transition
to the superionic state and melting in UO2. The potential has a VR form and is a rigid ion
model with partial-charge assignments.
Other UO2 models of note include those of Arima (rigid ion) (2005), Meis (core-shell)(1998), and Busker (core-shell) (Abramowski, Grimes, Bradford, 2000; Busker, Grimes,
Bradford, 2000, 2003). The Busker model is notable because it has good transferability
among a number of elements and has ability to study nonstoichiometry since it isparameterised for U3+ and U5+.
Because it is radioactive, UO2 is experimentally difficult to use for studying the effects ofradiation damage. However, CeO2 is similar to UO2 in terms of fundamental properties(e.g. lattice parameter, density, thermal diffusivity), so it has been proposed as a surrogatematerial for investigations of radiation damage in UO2 (Aidhy, Wolf, El-Azab, 2001). A
number of pair potentials have been developed for CeO2, including rigid ion BMH and
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core-shell Buckingham forms (Xu et al., 2010). Particularly of interest is that some havebeen parameterised for nonstoichiometry (Gotte et al., 2007).

PuO2/MOX fuelYamada et al. (2000b) have developed rigid ion potentials for mixed oxide (MOX) andnitride nuclear fuels, adding a Morse potential to the Buckingham potential. The potentials
were developed to be transferable among solid solutions of UO2 and PuO2 for MOX fuels.
The form of the rigid ion potential is Buckingham with an additional Morse term for cationanion pairs. Thermal and mechanical properties of MOX fuel were calculated via MD
simulations. This potential was later modified by Basak (2003) to better reproduce thermal
expansion data. The partial-charge potentials developed by Arima (2005) are also of the
BMH-type and have been used to examine thermal properties of (U, Pu)O2 MOX fuels via
MD simulation.
Tiwary et al. (2011) developed an approach for creating a consistent form of the potential
over all distances and applied it to UO2 (Tiwary et al., 2009) and then extended its use to
PuO2 and NpO2 (Tiwary et al., 2011) in a self-consistent manner such that MOX fuels (U,Pu, Np)O2 can be simulated over a wide range of experimental conditions. They fit thepotentials to an extensive database of experimental and ab-initio results and thoroughly
showed the potentials reliably reproduce properties outside of the fitting parameters. This
represents a major step forward in the production of transferable potentials.

Nitride fuel
A mixed nitride (U, Pu)N fuel potential was developed by fitting to thermal expansion andpressure expansion data to a BMH + Morse form (Kurosaki et al., 2000); and, calculating
thermal and mechanical properties with MD simulations. Radiation defect generation in
UN was studied via ab - initio calculations, the results of which can be used to develop
better quality classical potentials (Kotomin et al., 2008).

3.4.2. Cladding/structural elements
Fe and Fe-alloys
Ferritic and austentitic steels in nuclear reactors (e.g. pressure vessel ferritic steel andinternal structure austentitic steel) are subject to neutron irradiation that can lead to stresscorrosion cracking and eventually failure. Metal cladding encapsulates nuclear fuel and is
exposed to neutron irradiation, which can lead to swelling (due to void formation) andaffect diffusion properties in the material. The effects of irradiation on a number ofsteel/iron alloys are now routinely studied using MD displacement cascade simulations
where primary knock-on atoms (PKA) are given a high initial kinetic energy and based on
the time evolution of the system of interest, defects are introduced in a perfect crystal as
momentum is transferred to a PKA’s neighbour atoms.
The quality of the potentials describing interatomic interactions influences the ability of thesimulation to predict neighbour atom trajectories, diminished mechanical properties, anddefect generation and clustering (Devananthan, Yu, Weber, 2009; Malerba et al., 2010;
Malerba, 2006). Primary damage defects in iron include self-interstitial atoms (SIAs),
vacancies, and Frenkel pairs, and their clustering can give an indication of diffusion
mechanisms and nucleation sites for larger defects (Björkas, Nordlund, 2007). The resultsfrom MD cascade simulations can be used as input into higher-level (e.g. KMC orcontinuum) models, so accuracy in calculated properties is of great importance (Becquart,
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Domain, 2011). With this consideration, many recent studies rely on electronic-scale
ab - initio DFT calculations for development and verification of the interatomic potentials
used in MD simulations. This is particularly important because experimental observationof irradiation effects is nearly impossible due to the time and length scales at work (Bequart,
Domain, 2011).
Primarily, Fe has been studied as a model system for steel, though development of new
potentials now allows for study of a number of Fe-alloys. Bequat and Domain (2011) givean excellent review of the over 45-year history of damage formation investigations for Fe
and Fe-alloys (e.g. Fe-Cr and Fe-Cu). The effects of P (Ackland et al., 2004), Cu (Pasianot,
Malerba, 2007), and He (Seletskaia et al., 2007; Gao et al., 2011), which are known toembrittle Fe, have also been considered. Little research has been done on the Fe-C system(Becquart et al, 2007) due to difficulty of simulating ab - initio supercell sizes that are large
enough to depict the low C concentration present in ferritic steels. The effects of interatomic
potentials on irradiation damage in Fe are reviewed by Becquart et al. (2000).
Despite the seemingly simplistic structure of Fe, a number of potentials have beendeveloped, none of which are able to reproduce experimental data entirely. Of particularconcern are the well-known deficiencies in many Fe models for predicting defect energies(Malerba, 2006). Understanding the accumulation of damage due to neutron irradiation is
crucial to the understanding of the effects on Fe, but there are still discrepancies in predicted
defect cluster populations at the atomic scale (Björkas, Nordlund, 2007; Malerba, 2006)
that make predicting longer time scale damage problematic. Although ab-initio calculationsof defect structures and energies continue to be increasingly important for interatomic
potential development and verification (Becquart, Domain, 2009), there is still a lack offundamental understanding of how and why defect clustering occurs and how thatclustering affects defect mobility (Samaras, Victoria, Hoffelner, 2009).
A number of studies exist which compare radiation damage effects among multiple Fepotentials (Becquart et al., 2000; Malerba et al., 2010). One of these studies (Becquart etal, 2000) found significant differences in primary damage creation due to subtle differencesin the potentials. While many Fe potentials have been derived for equilibrium-type
calculations, they are later “hardened” (to eliminate the possibility of close contacts ofatoms) to be suitable for cascade simulations. Although all potentials in the study hadreasonable threshold displacement energies, they each have had different hardening
procedures, which leads to differences in repulsive potential for small separations. These
differences result in significant differences in cascade shape, as well as defect energies andpopulations. Even thermal properties are affected. It was concluded that the different
hardening methods play a role in these properties, underlining the importance of proper
potential development and the applicability of different potentials over different
interatomic separation ranges (Becquart et al, 2000).
The Mendelev set of interatomic potentials for body centred cubic-Fe (bcc-Fe) (Mendelev
et al, 2003) is of the EAM type and was derived based on a careful fitting procedure thattook into account a large set of experimental data and DFT-derived point defect energies.
A more recent analysis of the state of the art in Fe potentials indicates that Mendelev-type
potentials are the best choice for developing Fe-alloy potentials, as well as applying the
calculated properties to larger scale models (Malerba et al., 2020). This study also
concludes that the ability of classical interatomic potentials to capture SIA clusters anddislocations is still elusive. The study provides an extensive database of static and dynamic
properties (including defect formation and migration) for Fe based on the results ofsimulations for five published potential models.
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Some more recent models of note include those of Muller (2007) and Dudarev and Derlet(2005, 2011). Muller’s potential for Fe is able to reproduce the temperature-dependent body
centred cubic to face centred cubic (bcc-to-fcc) transition of Fe. The “magnetic” potentialof Dudarev and Derlet contains an explicit magnetic contribution to the interaction energy.

3.4.3. Waste forms
The containment of high-level waste (HLW) has been an active area of research anddevelopment for over 50 years (Weber et al., 2009). Borosilicate glasses and complex
ceramic composites have been developed to meet HLW containment issues. Research todate has focused on immobilising existing HLW that consists of all the liquid effluentsfrom the reprocessing of commercial and defence spent nuclear fuel. New research is
focused on creating new separation waste streams, recycling spent nuclear fuel, andminimising the volume and heat load of waste forms. New waste forms tailored to specificwaste streams and high waste loadings may be needed. Two useful review papers on
nuclear waste glasses and ceramic waste forms for actinides are from Grambow (2006) andLumpkin (2006), respectively.
Waste streams include: (1) the long-lived fission product 99Tc, (2) the principal heatgenerating isotopes 137Cs and 90Sr, (3) lanthanides, (4) minor actinides, (5) remaining
fission products, (6) volatile radionuclides, (7) the undissolved solids from fuel dissolution(Weber et al., 2009). Waste-form types include (1) glass, (2) ceramic, and (3) glassceramic. Classical MD has been used to look at the chemical durability of glass and various
single-phase ceramic waste forms. The main vulnerabilities of polyphase ceramic or glassceramic waste forms for radionuclide immobilisation are along grain boundaries and canbe better investigated using mesoscale models.

Glass waste forms
Glass waste forms are the most commonly used HLW form today. The amorphous andrelatively disordered structures of glasses can incorporate a wide range of chemicalelements. Borosilicate glass compositions may include up to 30-40 different elements.
Although borosilicate glass is used by most countries, Russia has chosen to use
aluminophosphate glass for HLW. One of the advantages of glass waste forms is that theparticles emitted in radioactive decay can be readily accommodated in their amorphous
structure, and the effects of α-decay are small at the ambient temperatures expected overthe decay times for actinides. Volume changes and changes in chemical durability due toα-decay are minimal. Potential problems with glass waste forms relative to crystalline
waste forms include (1) more rapid dissolution in groundwater, and (2) radioisotope
diffusion rates in the waste form could increase the local surface concentration ofradioisotopes exposed to water.
Classical MD simulations have been used to study glass structure and the interaction ofglass surfaces with water for many years. The choice of interatomic potentials to use inclassical MD simulations is of fundamental importance for the accuracy of simulated
physical and chemical properties such as structure, elastic constants, heat capacities andother thermodynamic properties (Pedone et al., 2006). Numerous interatomic potentials areavailable in the literature for silica glass and silica polymorphs (van Beest, Kramer, vanSanten, 1990; Duffrene, Kieffer, 1998; Demiralp, Cagin, Goddard, 1999; Vashista et al.,
1990; Takada et al., 2004). These force fields incorporate a variety of functions and terms
to describe different silica glass properties.

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However, fewer potentials are available for multicomponent glasses like those proposed
for HLW immobilisation, and there have not been many efforts to provide generalised
self-consistent force fields that are capable of modelling the structures,
mechanical, and chemical properties of silica-based glasses with different
compositions (Pedone et al., 2006). Table 3.1. lists classical potentials found in theliterature that have been developed and used to simulate multicomponent silica-based
glasses. Table 3.2. lists classical and reactive potentials that have been used to study
the interaction of multicomponent glass with water.
Table 3.1. Force field models for glass
Composition
Na, K, Ca, Si, B, O

Form of potential
Modified BMH used by Busing w/o dispersion
terms
Buckingham form known as BKS

SiO2, AlPO4
Al, O
Li, Cs, B, O
Mg, Ca, Al, Si, O
Na, Ca, Al, Si, B, Zr, O
Na, Ca, Al, Si, B, O

Modified BMH
BMH
Modified BMH used by Busing w/o dispersion
terms
BMH
BMH – improvement on D&G (1996)

Na, Si, O
Al, O
Na, Ca, Si, Al, O, Zr

Buckingham
Buckingham
Morse function with partial ionic charge model

Soules (1979)

Authors

Van Beest, Kramer, and van Santen(1990)
Blonski and Garofalini (1993)
Verhoef and den Hartog (1995)
Okuno and Kawamura (1995)
Delaye and Ghaleb (1996)
Abbas et al. (2003)
Cormier et al. (2003)
Du and Cormack (2004)
Adiga et al. (2006)
Pedone et al. (2006)

Table 3.2. Force field model for glass-water interaction
Composition
Si, Na, H2O
Si, O,H2O
Si, O,H2O
Si, O,H2O
Si, O,
H2O

BMH

FF for glass

Si-OH,

Buckingham (Teter’s)

Si-OH,

BMH + 3 bond angle
terms
Modified BMH + 3-bodypotential
Modified BMH + 3-bodypotential

Si-OH,
Si-OH,

FF for –OH and H2O
Shell model for H2O-OH: Morse potential
3-body term for Si-O-H; Coulomb subtracted
Morse potential for –OH group
Rahman, Stillinger & Lemberg form for Si-H, OH, and H-H
Modified BMH + 3-body potential
Modified BMH + 3-body potential

Authors
Leed and Pantano(2003)Du and Cormack(2005)
FeustonandGarofalini (1990)
Litton and Garofalini(2001)
Garofalini (2001)

From our review of available force fields, three stand out as potential candidates for future
research on the design of new HLW glass waste forms and their dissolution properties. The
first force field was specifically developed by Delaye and Ghaleb (1996) to simulate anuclear waste glass matrix. As in previous simulation studies of the structure of simple
glasses (e.g. Soules, 1979; Soules, Varshneya, 1981), potentials of the BMH form wereused to represent a glass comprising the major components of a nuclear waste glass (SiO2+ B2O3 + ZrO2 + Al2O3 + Na2O (+ CaO)). Three-body potentials were applied to O-Si-O,
O-B-O and Si-O-Si triplets. The local environmental structures showed overall agreementwith experimental results and the simulated densities, thermal expansion coefficients andviscosities of the simulated glasses were on the same order of magnitude as experimental
data. This force field has been used to examine a suite of nuclear waste glass compositions(Delaye, Louis-Achille, Ghaleb, 1997). Experimentally determined changes in glass
structure with increasing boron concentration were reproduced by the calculations.
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Additional work by this French group includes combined multiple-quantum magic angle
spinning nuclear magnetic resonance (MQ-NMR) spectroscopic and MD studies to
examine the Na environment in several two- and three-component glasses (Angeli, Delaye,Charpentier, Petit, Ghaleb and Faucon, 2000a) and changes in Al-O-Si angle withincreasing Ca content (Angeli, Delaye, Charpentier, Petit, Ghaleb and Faucon, 2000b).
Abbas et al. (2003) used this same force field model to compare the structural properties ofbulk glass with a glass surface. The simulations predict the migration of alkali cations
towards the surface, lower coordination numbers for trivalent elements in the subsurface
layer, oxygen enrichment in the outer layer, and larger tetrahedral ring structures on thesurface.
The second force field model is that of Pedone et al. (2006) who developed a generalised
self-consistent force field that is able to model both the structures and mechanicalproperties of silica-based glasses with different compositions. To produce a coherent set of
potential functions, these researchers performed empirical fitting to structural andmechanical properties of a large set of crystalline oxides. The potentials developed in thiswork consist of three terms: a long-range Coulomb potential, a short-range Morse function,
and a repulsive contribution (C/r12). The resulting potential is given by:
𝑈𝑈(𝑟𝑟) =𝑧𝑧𝑖𝑖 𝑧𝑧𝑗𝑗 𝑒𝑒 2
𝐶𝐶𝑖𝑖𝑖𝑖2+ 𝐷𝐷𝑖𝑖𝑖𝑖 ��1 − 𝑒𝑒 −𝑎𝑎𝑖𝑖𝑖𝑖(𝑟𝑟−𝑟𝑟0 ) � − 1� + 12
𝑟𝑟
𝑟𝑟

(7)

A rigid ionic model with partial charges is used to handle the partial covalency of silicate
systems in order to model the quenching of melts and structures and mechanical propertiesof glasses. A core-shell model might provide more accurate results for surfaces, but would
require a shorter-time step and lead to computationally more expensive quench simulations.
This force field has been parameterised to include numerous glass components including
many of importance in nuclear waste glasses (Na2O, CaO, Al2O3, SiO2 and ZrO2).
However, to our knowledge, this force field has neither been used to study glass surface
structures nor combined with a water model to examine glass surface hydroxylation orother interactions between glass surfaces and water. In earlier papers by the same group(Du, Cormack, 2005; Zeitler, Cormack, 2006), surface hydroxyl-silicon and hydroxylphosphorous potentials were developed to study the hydroxylation of silica andphosphosilicate glasses. These potentials include a short-range interaction described by a
Buckingham equation, a potential function for the hydroxyl group that has the Coulombsubtracted Morse form, and a three-body term introduced to reproduce the Si(P)-O-H bondangles on glass surfaces. The surface structures were used successfully to investigate theenergy of chemisorption of water on different surface sites and to determine the OH
coverage of glasses.
The third force field that shows promise for studying multicomponent nuclear waste glassdissolution is the reactive force field of Garofalini (2001). From the beginning, thisforce field was developed to investigate glass surface structures and the interaction of waterwith glass surfaces. It now includes parameters for Si, Al, and B, and allows for breaking
glass network bonds between Si-O, Al-O and B-O by reaction with water molecules(Garofalini, 2011). This force field has been demonstrated to successfully reproduce bulk
glass and hydroxylated surface structures, but is not the best for calculating the mechanics
or energetics of these systems. Mahadevan and Garofalini (2007, 2008) have developed a
new dissociative water potential to improve upon the previous force field model. However,this new potential is still being developed for multicomponent systems.

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Ceramic waste forms
In crystalline ceramic phases, radionuclides occupy specific atomic positions in the
periodic structures of constituent crystalline phases allowing high loadings of specificradionuclides (Weber et al., 2009). Waste-form phases tend to have complex structures,with a number of different coordination polyhedra of various sizes and shapes and withmultiple schemes to allow for charge balance with radionuclide substitutions. Ceramic
waste forms are designed to incorporate either actinide or Cs/Sr waste streams. These waste
forms are usually polyphase, however, in some cases single-phase ceramics such as
zirconolite, monazite, apatite, or NaZr-phosphate can incorporate nearly all theradionuclides (i.e. actinide or Cs/Sr) into a single structure. Unlike glass, crystalline phases
can be shown to survive several hundred million years or more in wet, thermal geologic
environments.
A major concern with the use of ceramic waste forms is the modification of crystalline
structures by irradiation (Weber et al., 2009; Grambow, 2006; Lumpkin, 2006). At highionisation doses and T, many materials undergo decomposition, phase separation andbubble formation under electron-beam irradiation on laboratory time scales. Single-phaseceramics generally exhibit a crystalline-to-amorphous transformation with volume changes
ranging from 5-18%. Because this is a critical concern for the use of ceramic waste forms,
classical MD studies have focused on the impact of irradiation on candidate crystalline
materials.
Classical potentials used to study the effects of the α-decay process, the collision of α-recoil
nuclei with surrounding atoms, and localised displacement cascades (DCs) typically are
BMH potentials or potentials that include Buckingham parameters to describe short-range
interactions. An example of the latter type of potential was developed for zirconolite(CaZrTi2O7), a matrix being considered for the long-term confinement of actinides, by
Veiller et al. (2001):
𝑈𝑈𝑖𝑖𝑖𝑖 �𝑟𝑟𝑖𝑖𝑖𝑖 � = 𝐴𝐴𝑖𝑖𝑖𝑖 exp �−𝑟𝑟𝑖𝑖𝑖𝑖
𝐶𝐶𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖
1 𝑍𝑍𝑖𝑖 𝑍𝑍𝑗𝑗 𝑒𝑒 2
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � ��− 6 +
𝑝𝑝𝑖𝑖𝑖𝑖𝜂𝜂
𝑟𝑟𝑖𝑖𝑖𝑖 4𝜋𝜋𝜀𝜀0 𝑟𝑟𝑖𝑖𝑖𝑖(8)

In this equation, rij is the interionic distance, Zie and Zje are the formal charges of the ions,
η=6.7 Å and Aij, pij, and Cij are adjustable parameters to be determined. The first two terms
correspond to the short-range Buckingham potentials and represent both the repulsive
interactions due to ion overlap and the attractive van der Waals interaction between them.
The last term is a screened Coulombic term where the complementary error function
reduces the ionic charges Zie and Zje as a function of rij.
The best mineral structure potentials include ion polarisation effects. For example, a shellmodel includes the electronic cloud linked to the core ion by a spring of constant stiffness.
However, DCs can only be modelled using rigid ion model potentials (Veiller,
Crocombette, Ghaleb, 2002). An important parameter that affects radiation damage is thethreshold displacement energy (Ed) which is the minimum kinetic energy necessary todisplace an atom from its equilibrium crystallographic site. It is possible to calculate Ed
through classical simulations; however, there are not very many experimental values for Ed
except for zircon (ZrSiO4). Different atoms in each crystal phase will have different Edvalues based on ionic mass, charge and nearest neighbours. To initiate the DC, one cationin the crystal structure is replaced by U4+ or U3+.
Veiller et al. (2002) compared simulation results for zirconolite with those for zircon(Crocombette, Ghaleb, 2001) and UO2 (Jackson, Huntington, Ball, 1995). A trend that is
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consistent with experimental observation is observed by this suite of studies. For zircon, an
amorphous core is created around the DC track with no surrounding defects. For UO2,
vacancies and interstitials are created throughout the crystal with no specific defects along
the DC trajectory. The behaviour of zirconolite lies between that of zircon and UO2. The
simulated irradiation of zirconolite does not result in as much amorphisation as observedin experiment; nor is the phase transition from monoclinic to fluorite structure reproducedby the simulations. These differences could be due to weaknesses in the empiricalpotentials, but there are other possible explanations for them as well (Veiller, Crocombette,
Ghaleb, 2002).
One disadvantage of using a rigid ion potential, is that the energy or heat generated by the(PKA, cannot be transferred to the electrons in the crystal structure. To compensate for thisPhillips and Crozier (2009) developed a two-temperature model to represent the interaction
between atoms and free electrons during thermal transients such as radiation damage andcascade simulations. This model communicates energy between electronic and atomic
subsystems using a Langevin thermostat, and does not impact the form of the classicalpotentials. This approach was used to study the effects of cascade damage in a gadolinium
pyrochlore (Gd2Zr2O7). The simulations showed that the electronic subsystem can act as aheat sink for the PKA and the inclusion of this system reduces the predicted cascade
damage in the crystal structure (Ismail et al., 2010). More research is required to moreaccurately depict the electron-ion interactions during a cascade event.

3.5. Challenges and future direction for development of classical potentials fornuclear materials
Range of application
A well-known limitation of classical potentials is that they are generally only applicable
over a small range of temperatures and/or interatomic separations. Even when potentials
that are good over different ranges are combined, the way that they are combined also
affects the resulting properties (Tiwary et al., 2009). Recent work by Tiwary et al. (2009)
has taken advantage of DFT calculations in order to develop a systematic method for
producing UO2 potentials that are good over a large range of interatomic separations. By
applying this procedure to MOX fuels and Fe-alloy claddings will lead to better potentials
for radiation damage studies.
While ever-increasing computer power has led to routine electronic-scale DFT
calculations, these ab-initio calculations are usually limited to predicting structures at 0 K.
For the case of defects, these configurations may be expected to be different at the highlocal temperatures due to irradiation. The use of ab-initio MD (AIMD) to address thisproblem, while computationally costly, may uncover some new information about defectstructure and therefore allow for better fitting of classical potential parameters.
Great strides have been made to make potentials that are applicable from room temperature
to above the melting point of a material, but based on recent reviews, clearly there aredifferent potentials that are best over different temperature ranges (Govers et al., 2007,2008).

Charge transfer/reactive models
While reactive models are a relatively new invention (even in the relatively short historyof classical simulation), their development for nuclear materials will represent a major step
forward in the ability to consider defect migration and charge transfer between ions. This
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may be particularly useful for the case of UO2, where U exists in different oxidation states.
It will also allow for more accurate surface reaction simulations. Some initial work has
been done for a charge transfer model for Fe (Samaras, Victoria, Hoffelner, 2009), which
could be used to predict oxygen mobility and oxidation mechanisms. However, additional
work is needed to develop new functional forms for interatomic potentials in mixed ioniccovalent systems of interest.

Density functional theory calculations
To an increasing extent, the results of DFT calculations are being used for developing andverifying interatomic potentials. This is especially true for defect configurations andmigration energies, where experimental data is lacking or impossible to obtain. However,there are still some limitations to DFT calculations. Because DFT-calculated properties
represent the lowest-level in multi-scale simulation schemes, it is extremely important thatthese limitations be addressed.
The choice of functional has been shown to influence the resulting structure – e.g. local
density approximation (LDA) predicts incorrect structure of low temperature Fe and does
not predict its magnetism; the use of generalised gradient approximation (GGA) resolves
this problem, though it is not immediately clear why this should be so (Becquart, Domain,2001). The choice of basis set or pseudo potential can also have an effect on some energetic
properties of Fe-alloys since core atoms may overlap in cases where SIAs are predicted to
be close to each other – in these cases, a more expensive “all electron” method may berequired (Becquart, Domain, 2009). Actinides are especially difficult in this case because
they contain f electrons (Devanathan et al., 2010).
DFT calculations also may suffer from the problem of applicable range. The stability ofdefect configurations is still difficult to predict—some configurations not included in theoriginal parameterisation cannot be reproduced by a set of potentials (Geysermans, 2008).
The applicability of these potentials for cases where DFT results do not exist is still not
clear (Malerba et al., 2010), though a recent study of polymer structures far fromequilibrium is promising (Mattsson et al., 2010).
Also, inclusion of the magnetic contribution to structure (e.g. in Fe-alloys) could play arole in calculated defect energies since spin-orbital coupling plays a major role in the
structure of some nuclear materials. This needs to be taken into account at the quantum
mechanical level (Samaras, Victoria, Hoffelner, 2009). Some initial work has already been
done (Dudarev, Derlet, 2005; Chiesa et al., 2001).

Lack of appropriate experimental data
Because of the lack of experimental results for primary damage due to displacement
cascades, it is difficult to test the reliability of new potentials for Fe and Fe-alloys. A review
of some popular Fe potentials shows that the defect population and distribution can besensitive to the potential (Becquart et al., 2000). Additional ab-initio calculations can help
to resolve these issues, which have initially been addressed by Tiwary et al. (2009)
Indeed, standard interatomic potentials have been called into question since increases incomputing power have allowed for routine ab-initio calculations that can better predict
defect configurations. Increased use of ab-initio calculation results as fitting parameters
has led to greater versatility and applicability in EAM potentials (Mendelev, Ackland,2007).

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In many cases, experimental data does not exist and atomic-scale results must be fed intohigher-level models in order to be able to make a reasonable comparison. However, thisintroduction of an additional model makes any discrepancy difficult to reconcile (Becquart,
Domain, 2009).
An additional consideration is that experimental data, even when available, may not bedirectly comparable to simulated results. Simulations are almost always done for systems
without impurities, whereas real materials invariably have impurities that can lead todiscrepancies in properties even among experimental data (Becquart, Domain, 2009).

3.6. Conclusions
Computer simulation provides an attractive alternative to direct experimental observation
for nuclear materials that are difficult to observe experimentally under operating
conditions. Classical simulations are used to predict material lifetimes or the long-term
microstructure evolution due to exposure to severe conditions experienced in nuclear
reactors. In addition, they are used to shed light on the use of glass and ceramic waste forms
to contain radioactive waste over extended periods of time. Classical potentials are
expressions that describe interatomic interactions. These potentials are fit to experimental
data and quantum-based DFT calculations. The capability of each potential to predictmaterial behaviour is often limited by the parameterisation of the potential; i.e. the range
of interatomic distances and conditions (T, P) used to fit the potential. Although there ismore work to be done to improve these potentials, their use has already served to illuminate
the properties and behaviours of nuclear materials, and will continue to contribute to ourdevelopment of new approaches to enhance enrichment and fuel burnup in nuclear fuels,
predict waste-form lifetimes and engineer novel nuclear materials.

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4. Molecular dynamics
Dorothy M. Duffy
Department of Physics and Astronomy, University College London, United Kingdom

4.1. Introduction to molecular dynamics
Molecular dynamics (MD) employs the numerical integration of Newton’s equations of
motion to calculate the time evolution of a system of interacting atoms. The time and lengthscales that can be studied using molecular dynamics make it particularly suitable for the
calculation of the primary damage produced by an irradiation event. In this paper we
describe the MD simulations, known as cascade simulations, used to model radiation
damage and the extent to which these simulations have contributed to our understanding ofradiation processes and the resulting defect structures. We discuss the limitations of thecurrent models, in particular the accuracy of the interatomic potentials, the limits to theradiation energy association with simulation cell size and the inadequate description of therole of electrons. We discuss how further development in these areas will improve thepredictive nature of the models.

4.1.1. Methodology
MD is an atomistic simulation methodology that is based on the numerical integration of
Newton’s equations of motion. A simulation is initiated by assigning co-ordinates andvelocities to a set of interacting atoms. The force on each atom is calculated from the known
positions of all other atoms therefore, in principle at least, the positions and velocities of
the atoms at a future time can be determined by numerical integration of the equations of
motion. A balance between accuracy and simulation time is achieved by choosing anappropriate time step for integration, which is generally a few (1-3) femtoseconds. Efficient
integration algorithms, such as the Verlet velocity algorithm, ensure that numerical errors
are minimised. The Verlet algoritm for the calculation of the co-ordinate r at time t isderived by summing the Taylor expansions of the cordinate r at times t + ∆t and t-∆t, thatis:f (t ) 2 ∆t 3 ∂ 3 r
r (t + ∆t ) = r (t ) + v (t )∆t +∆t ++ ........
2m
3! ∂t 3(1)

f (t ) 2 ∆t 3 ∂ 3 r∆t −+ .......
2m
3! ∂t 3(2)

f (t ) 2∆tm(3)

r (t − ∆t ) = r (t ) − v (t )∆t +r (t + ∆t ) ≈ 2r (t ) − r (t − ∆t ) +STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Therefore, the co-ordinates of all the atoms at a time t + ∆t can be calculated from the coordinates at times t and t - ∆t and the forces at time t (f(t)). The force on each atom (labelledi) is calculated from the derivatives of the potential energy (U) of the system of atoms withrespect to the co-ordinate of atom i:

fi = −

∂U
∂ri

(4)

Therefore, if the potential energy of the system of atoms, as a function of the co-ordinates,
and the initial co-ordinates are known, then the co-ordinates at a future time can beevaluated using equation (3).

4.1.2. Interatomic potentials
The most important parameters that limit the accuracy of MD simulations are theinteratomic potentials that describe the interactions between the atoms. In general, thepotential energy of a set of interacting atoms can be written:

U = ∑ Vi + ∑ Vij +ii , j ≠i

∑V + .........

ijk
i , j ≠i ,k ≠i , j(5)Therefore, in theory at least, the forces on all atoms can be determined if the positions are
known. In practice, however, determining the various interactions in equation (5) is
challenging. An alternative approach is to calculate U directly using ab-intio techniques,
such as density functional theory, and to employ the Born-Oppenheimer approximation to
calculate forces. Such ab-intio MD methods are under continuous development and it is
now possible to model a few thousand atoms for a few picoseconds. These system sizes arewell below those required for radiation damage therefore we focus here on classical MD inwhich an approximate potential energy function must be determined a priori.
In classical MD the challenge is to find approximation for the potential energy (U inequation (5)) that will retain the important physics but will not be prohibitively time
consuming to calculate. Two body interactions, such as the Lennard Jones potential,
truncate equation (5) after the first term. This allows for very large systems to be modelled
but it only gives a reasonable description for very simple systems, such as the noble gases.
The Coulomb interactions between charged atoms are calculated using efficientsummations techniques, such as the Ewald summation, to take account of the long-range
nature of this interaction. Covalent bonds in polymers and biomolecules are described wellby the first three terms in the equation (bond stretching, bond bending and torsion).
Interatomic potentials for materials that have wide ranging technological applications haveattracted particular attention, with over 30 different potentials being published for silicon(Bazant et al., 1997). Carbon has presented particular challenges and a number of potentials
have been developed that attempt to describe the change in the bond hybridisation withenvironment (e.g. REBO (Brenner, 1990) and the REAXFF (van Duin et al., 2001)
potentials). The effective charge on atoms in ionic crystals may also depend on theenvironment and variable charge potentials have been developed in order account for thiseffect to some extent (Rappel, Goddard, 1991). The many-body interactions necessary tocapture the elastic properties of metallic materials are included by embedded-atom typepotentials (Daw, Baskes, 1992) such as the successful Finnis-Sinclair model (Finnis,
Sinclair, 1984) which includes a term related to the local density of the atoms.

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In spite of dramatic improvements in potentials over the years, it is not generally possible
to ensure that any potential will describe all properties of a material equally well. Potentials
are derived by fitting to experimental data or ab-initio results and the ability of a potential
to predict a property that was not used in the fitting procedure (known as the transferability)
is the signal of a good potential. The potential chosen for a particular simulation will depend
on which properties make the dominant contribution to the features to be studied.

4.1.3. Time/length scales
The time and length scales that can be modelled using MD are strongly dependent on thecomplexity of the interatomic potentials. A record breaking one trillion atom simulation
has been performed on the Livermore’s Bluegene/L computer but this was to demonstrate
a capability rather than to extract results from the simulation (Germann, Kadau, 2008).
Typical large-scale simulations range from a few hundred thousand atoms to a few million
and typical time scales range from a few tens of picoseconds to a few nanoseconds. Manybody potentials for metals are efficient to calculate but the long-range Coulomb interactions
in ionic materials are more time consuming. Potentials that attempt to model the effects ofchemical environment (such as REAXFF and REBO potentials) result in relatively slow
simulation times therefore, as always, a balance needs to be found between accuracy andefficiency.

4.2. Modelling radiation damage using molecular dynamics
4.2.1. Advantages
Early models of radiation damage used the binary collision approximation (BCA), in
various forms to estimate the number of Frenkel pairs (NKP) created by a radiation eventwith a particular energy (E). The first of these, the Kinchin Pease (KP) model, was able toderive a simple formula (equation (2)) by making a number of assumptions (Kinchin,Pease, 1955).

N KP =

E
2Ed(6)

Here Ed is the displacement energy, which is the minimum energy required to displace anatom from its lattice site to a neighbouring interstitial site to produce a stable Frenkel pair.
It is clear that the KP model overestimates the number of defects because it neglects therecombination events that inevitably occur. An improved form (theNorgett−Robinson−Torrens model) was suggested by Norgett (1975) which considers only
the fraction of the energy not lost to inelastic collisions with electrons and takes someaccount of recombination. More sophisticated BCA models, based on numericalsimulations, have been developed that include accurate scattering potentials and electronicenergy loss (Robinson, Torrens 1974; Ziegler 2004). However, binary collision models are
only accurate for situations in which the collisions between the moving atoms and the atomsof the host crystal are spatially separated, which is only the case for light energetic atoms.
In the general case, the moving atoms interact with many host atoms simultaneously and it
is difficult to predict the number of defect pairs created by the resulting correlated atomic
motion.
The correlated atomic motion resulting from radiation damage events can be successfullymodelled using MD. MD is significantly more demanding, from a computational point of

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view, than BCA models, nevertheless, the length and time scales of the primary damage
events of radiation damage can readily be modelled. Primary damage refers to the defects
remaining after the excess energy imparted by the radiation event has dissipated and the
local temperature has returned to the ambient value, which typically occurs around a few
tens of picoseconds after the radiation event. The spatial extent of the primary damage isstrongly dependent on the mass and energy of the moving atom. Heavy atoms rapidly lose
energy to the host crystal and produce localised damage whereas light atoms, such as
helium, have a low cross section for nuclear collisions and travel long distances before
stopping.

4.2.2. Methodology
A cascade simulation is a particular type of MD simulation that is used to model radiation
damage in materials. The initial configuration is an arrangement of atoms with theequilibrium crystal structure and random velocities, corresponding to the simulation
temperature, assigned to the atoms. One atom (the primary knock-on atom (PKA)) in thesimulation cell is assigned a velocity corresponding to the kinetic energy (the PKA energy)of radiation event to be modelled (Figure 4.1.). The MD simulation is then evolved using astandard integration algorithm until the PKA energy has dissipated and the simulation
temperature converges to a constant value.
Figure 4.1. A schematic representation of a cascade simulation.

Note: The PKA (green) is assigned a velocity (white arrow) corresponding to the required PKA energy at thestart of the simulation.
Source: Duffy, 2019.

Atoms are displaced from their lattice positions during the simulation but many will returnto an alternative lattice site as the energy dissipates. Not all atoms return to lattice sites
however. Some remain in interstitial positions, leaving a corresponding number of vacantlattice sites. The identification of defects can be carried out by counting the number ofatoms in each Wigner Seitz cell of the lattice. An empty cell signals a vacancy and a cellwith two atoms signals an interstitial. Alternatively, a lattice site can be labelled as vacant
if there are no atoms within a predefined radius (ro) and an atom is labelled as an interstitial
if it is further than ro from any lattice site. The former (Wigner Seitz) method provides aunique method for defect counting whereas the second method is sensitive to the value ofro chosen as the cutoff. The second method does, however, capture details about the defectstructure that are missed by the Wigner Seitz method. Split interstitials (dumb-bells),
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crowdion clusters and stacking fault tetrahedra are examples of common defect clusters
that are created by cascade simulations of radiation damage in metals. A crowdion is anextended interstitial, in which an additional atom is inserted in a close packed direction ina metal. This has the effect of displacing several atoms, located along the close packed
direction, from their lattice sites. An example of a 30 crowdion cluster is shown in Figure
4.2.
Cascade simulations require a number of features which are not always available instandard MD programmes. One useful feature is a variable time step. The highly nonequilibrium nature of a cascade means that in a typical MD time step of 1 fs a 10 keV Fe
atom would move a distance of 0.35 Å, which will result in a large change in the force andpossibly a failed simulation. To avoid these effects it is necessary to use a very shortsimulation time step (possibly as low as 0.001 fs) in the initial stages of a high-energy
cascade simulation. As the simulation progresses the velocity of the PKA atoms falls
rapidly therefore progressively longer time steps can be employed. A variable time stepalgorithm imposes to limit the distance moved during one-time step by identifying the
maximum velocity of the ensemble of atoms and choosing an appropriate time step.
Figure 4.2. Top view (a) and side view (b) of a crowdion cluster oriented in the <011>
direction in a body centred cubic metal.

(a)

(b)

Note: The cluster has 170 interstitials (large orange spheres) and 140 vacancies (small blue spheres). Eachcrowdion has one excess interstitial.
Source: Duffy, 2019.

A further consideration for cascade simulations is how the excess energy of the PKA atom
is dissipated during the simulation. Standard thermostats are inappropriate because theycalculate the temperature from the average velocity of the atoms. The average velocity canbe extremely high in the early stages of a simulation therefore the excess energy will be
rapidly removed by the thermostat and the cascade will be quenched. There are twoalternatives: either the total energy is conserved (using a constant energy NVE simulation),
which will result in a temperature rise as the PKA energy dissipates through the cell, or theexcess energy is removed via a boundary thermostat, which rescales the velocities of theatoms that reside within a few Å of the cell boundary. Both methods are commonly used.
As with all MD simulations, the ability of the interatomic potentials to reproduce the effectof atomic interactions in real materials is crucial to the accuracy of the results. The
challenges associated with interatomic potentials will be discussed in the next section but
here we discuss an issue that concerns cascade simulations in particular – that of close
encounters between atoms due to the high velocity of some atoms. Conventional

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interatomic potentials are highly inaccurate at close interatomic separations as they use nearequilibrium conditions for fitting procedures. For cascade simulations it is necessary tomodify the short-range interaction potential to give a more realistic description of forces atclose separations and this is achieved by replacing the chosen potential with a screened
Coulomb potential, known as the Ziegler-Biersack-Littmark (ZBL) potential (V(r);equations (7)-(9), at close separations (Biersack and Ziegler 1982).

V (r ) =𝜙𝜙(𝜐𝜐) = 0.1818𝑒𝑒−3.2 𝑦𝑦+ 0.5099𝑒𝑒

Z1 Z 2 e 2 rφ( )a4πε 0 r−0.9423 𝑦𝑦a=+ 0.2802𝑒𝑒

0.8853a0
Z10.23 + Z 20.23(7)−0.4029 𝑦𝑦+ 0.2817𝑒𝑒−0.2016 𝑦𝑦

(8)
(9)

Here r is the interatomic separation, a0 is the Bohr radius and Z1 and Z2 are the atomic
numbers of the interacting atoms. The ZBL potential is joined to the standard potential bya spline function to ensure that the potential, and its derivatives, are continuous.

4.2.3. Results
MetalsMD has been an invaluable tool for studying radiation damage in metals since thepublication of the classic paper by Gibson et al. (1960). Early simulations were restricted
to small simulation cells and, consequently, low PKA energies. Nevertheless, interestingand important processes such as channelling, where atoms travel long distances between
lattice planes, and replacement collision sequences (RCS) were identified. In an RCS, amoving atom makes a direct collision with a neighbouring atom along a close packed
crystallographic direction. The initial atom occupies the neighbouring site and sets theneighbour in motion along the same direction so the sequence is repeated until the energy
is dissipated. Such sequences are effective mechanisms for generating well separated
vacancy-interstitial pairs with relatively low energy radiation events.
As computer power increased and the accuracy of interatomic potentials improved, thepower of cascade simulations increased dramatically. The relationship between the number
of Frenkel pairs and the PKA energy was established for several metals and compared to
the NRT model (see, for example, de la Rubia, 1996; Stoller et al., 1996, 1997; Stoller,2012). The structures of defect clusters created by high-energy radiation damage wereidentified. Dislocation loops, formed by the collapse of interstitial clusters, were observed
and stacking fault tetrahedra were observed in face centred cubic (fcc) metals. The
formation of defect clusters by radiation damage makes a significant contribution to thechange in mechanical properties as these clusters may be more or less mobile than isolated
defects. Vacancy clusters may also be nucleation sites for bubbles, as they act as trappingsites for gases created by transmutation reactions. Defect clusters may contribute toradiation embrittlement because they act as obstacles to dislocation motion. Indeed, thebarrier strength of different types of defect cluster, a parameter that is required for
dislocation dynamics simulations, can be calculated using molecular dynamics. Swelling,which occurs due to the differential volume relaxation between vacancies, can also be
calculated using MD simulations.
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Ceramics
Ceramic materials are employed for a wide range of nuclear applications, including nuclear
fuel and encapsulation materials for nuclear waste disposal. MD has made significant
contributions to the understanding of the fundamental processes in these materials and theradiation effects and issues are generally quite different than those in metals. Embrittlement
is not a concern, as ceramics are intrinsically brittle, but swelling can be problem as theresulting internal stresses may cause cracking. Amorphisation occurs readily in someceramic materials and this has been investigated using cascade simulations. The degree of
covalency has been identified as having a strong influence on the susceptibility of ceramicmaterials to amorphisation (Trachenko, 2006). An example of an amorphised region of
zircon, formed by a 30 keV U PKA cascade simulation is shown in Figure 4.3.
(Devanathan, 2009). The complex crystal structures of many nuclear ceramics means thata rich variety of possible defects structures are possible and MD simulations can help tocharacterise these defects (Zinkle et al., 2002).
Figure 4.3. An amorphised region of zircon, formed by a 30 keV U PKA cascade simulation

Source: Devanthan, 2009.

4.3. Limitations/challenges
4.3.1. Simulation size
The PKA energy that can be modelled by MD depends on the number of atoms in thesimulation cell as a sufficiently large cell must be employed to avoid self-interaction
effects. The PKA, and any subcascades it creates, must be confined within the simulation
cell. The energy deposited in the cell by the cascade event thermalises by atomic collisions
and diffuses to the boundaries of the simulation cell, where it can re-enter the cell throughthe periodic boundary and affect the results. A boundary thermostat, which dissipatesenergy in the outer layers of the simulation cell, inhibits energy diffusion across theperiodic boundary and permits smaller simulation cells to be used. As a rough guide, around
5 -10 ×104 atoms are required per keV of PKA energy but this number depends, to someextent, on the material being modelled. Recent simulations of 0.5 MeV cascades in Fe used
5×108 atoms in the simulation cell and were performed on 6×104 parallel processors(Zarkadoula et al., 2013). These simulations are particularly relevant to fusion technology
as 0.5 MeV is close to the average recoil energy from a 14.1 MeV neutron, produced bydeuterium-tritium fusion, in Fe.

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4.3.2. Interatomic potentials
As discussed in the introduction, the results obtained from any MD simulation are only as
reliable as the interatomic potential employed. For cascade simulations it is important thatthe potentials reproduce the melting temperature and thermal expansion coefficient in orderto obtain a realistic response of the material to the large energy density associated with theradiation event. The displacement energies and defect properties (energies and mobilities)
are also very important as these will influence the number of defects produced and thedegree of defect clustering and annihilation. The importance of steel, both ferritic andaustenitic, as a structural material for both fission and fusion reactors has led to thedevelopment of many interatomic potentials for iron and detailed comparisons between the
results obtained from cascades using different potentials (Becquart et al., 2000; Malerba
2006; Terentev et al., 2006; Malerba 2010). The conclusions from these studies were thatthe number of residual defects produced was largely potential independent, at least for themost recent potentials that focused on fitting parameters that are relevant to radiation
damage. However, details of the damage formation and, in particular, the fraction of defects
that formed part of clusters did depend on the choice of potentials. Even the method bywhich the equilibrium potential was joined to the high-energy ZBL potential seemed toaffect the details of the cascade evolution.

4.3.3. Electronic effects
Another concern about cascade simulations is that the electrons, and the effects they haveon damage production, are largely neglected. In effect, the electronic degrees of freedom
are integrated out of the problem and included only at the level of the interatomic potentials.
However, an atom moving through a material loses energy via both nuclear collisions,which are described well by MD, and inelastic collisions with electrons. A particular
example of the relative stopping power (the energy loss per unit distance) for an Fe atom
moving in an Fe lattice, calculated using the SRIM code, is shown in Figure 4.4. It is clear
from the figure that nuclear loss dominates at low energies but inelastic (electronic) losses
increase as the energy of the moving atom increases.

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Figure 4.4. Plots of the variation of electronic and nuclear stopping powers for an Fe ion
moving in Fe

Note: At low energies the nuclear losses dominate and at high energies the electronic losses dominate.
Source: Duffy, 2019.

The energy loss resulting from inelastic collisions in metals is often included as a friction
term that operates above some cutoff velocity (Nordlund et al., 1998). This, however, does
not take account of what happens to the energy that is deposited in the electrons. For weakcoupling between the lattice and the electrons, and a high electronic thermal conductivity,
the energy is transported rapidly from the cascade region and it should not affect theresidual damage (Flynn, Averbach, 1988, Finnis et al., 1991). However, if the coupling
between the lattice and the electrons is strong and the electronic thermal conductivity islow, the electronic energy can be confined close to the cascade for sufficient time to beredistributed back to the lattice. In this case, the residual defect number can either beenhanced or reduced depending on whether the electronic effects quench the cascade orcontribute to defect annealing. This effect has been demonstrated (Rutherford, Duffy, 2007)
using a methodology that couples the effective electronic temperature to an MD simulationof the lattice (Duffy, Rutherford, 2007). A comprehensive review of the electronic effects
in radiation damage has been produced by Race et al., (2010) and the introduction of theseeffects in radiation damage simulations is reviewed by Darkins and Duffy (2018).
The effects of excited electrons are even more significant in band gap materials than they
are in metals. Cations in some metal oxides, including UO2, exist in a number of different
charge states. Electrons excited to the conduction band by radiation, and the corresponding
holes, become mobile and thus they may get trapped at lattice defects, possibly created byother damage events. Trapped electrons will substantially modify the defect’s conformationand mobility. Figure 4.5. shows the calculated relaxed conformation of an O interstitial in
MgO with a net charge of -2e, -1e and 0. The O- interstitial was found to have a very low(~0.06 eV) migration barrier, which suggests that the interstitial with one trapped hole will
be extremely mobile (Mulroue, Duffy 2011). As the defect mobility dominates the
microstructure evolution, the electronic defects will have a strong influence on the evolving
material properties. Electrons, holes and excitons can also distort the lattice to such an
extent that they become trapped by their own distortion field (self-trapped) and the decay
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2of these self-trapped carriers can result in energy redeposition to the lattice, or even defectformation (Shluger, Stoneham 1993). Thus, it is clear that modelling radiation damage inband gap materials using cascade simulations and classical potentials neglects several
effects that could strongly influence the results (Klaumünzer, 2006; Duffy et al., 2012).
Figure 4.5. The relaxed defect configuration for oxygen interstitials in MgO.

Note: The interstitials have a net charge of a) -2e (cell centre configuration) b) -1e (<011> oriented dumb-bell)and c) 0 (<111> oriented dumb-bell).
Source: Duffy, 2019.

4.4. Future developments
The future requirements for MD simulations of radiation damage follow from thelimitations discussed in the previous section. Advances in computer power and thedevelopment of more efficient algorithms, particularly for reading/writing large datasets,
will extend the range of PKA energies that can be modelled, although simulations with
PKA energies higher than 500 keV are unlikely to become routine in the near future.
Interatomic potential development and evaluation is an area where much effort is required.
The KIM (Knowledge base of Interatomic Models) project (Tadmor et al., 2011) shouldmake an important contribution to interatomic potential development and evaluation in thenext few years. Much more needs to be done on developing potentials for real engineering
alloys although past experience in the development of potentials for Fe/Cr alloys (Olsson
et al., 2005; Caro et al., 2005), that aim to describe the observed phase behaviour, reveals
just how challenging this can be. It is important that the phase behaviour is correctly
described as radiation enhanced diffusion can accelerate the precipitation of stableintermetallic particles. Radiation enhanced segregation and depletion at grain boundaries
are important effects that rely on accurate mixed potentials. Oxide dispersion strengthened(ODS) steel offer a potential solution to the low high temperature strength of conventional
ferritic steel but modelling radiation effects in ODS steel will require potentials for theoxide (YTiO3) nanoparticles and, even more challenging, the interaction between the oxide
and the host metal. It is clear there is much yet to be done in the field of potential
development before the full potential of modelling radiation effects using MD can berealised.
A further area where MD could make significant contributions is in the examination of theeffect of microstructure on radiation damage. There is evidence from cascade simulationsin nanocrystalline metals (Samaras et al., 2002) and from temperature accelerated dynamics(Bai et al., 2010) that grain boundaries enhance defect recombination and there is
speculation that the trapping of defects at oxide metal interfaces in ODS steel will also
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enhance defect recombination and reduce radiation damage. However, there has been only
a limited number of simulations to date and a more comprehensive study of the effect ofgrain boundaries and other interfaces would help towards the extrapolation to engineering
materials.
A related, but just as significant, issue is the effect of radiation damage on the
microstructure of materials. This goes beyond the production of defect clusters and bubbles
that contribute to embrittlement by interacting with dislocations. It includes effects such as
precipitation of stable phase particles from supersaturated solution and the dissolution ofmetastable phase particles. Alloy steels contain a wide range of elements to enhance
properties and aid processing but these additives can form a number of intermetallic phasesthat may precipitate with the help of radiation enhanced diffusion. The microstructure
evolution in nuclear fuel is an extreme example of how radiation, fission gas production
and high thermal gradients can transform the microstructure of a ceramic material.
As discussed above, the electronic effects associated with radiation damage are notincluded in classical MD simulations. The topic attracted attention around 20 years ago but
it has largely been neglected since, although there has recently been revived interest in the
topic. The development of ab-initio techniques, such as time-dependent density functional
theory and time-dependent tight binding (Mason et al., 2007) , has helped in theunderstanding of excited states, but how such excitations affect the radiation damge on thetimescales associated with MD has largely been unexplored. This field is now ripe fordevelopment.
Validation of a computational model by comparison with experimental measurements isessential for maximum confidence in the results but this is generally challenging due to thewidely different length and time scales involved. The situation is improving due to thedevelopment of in situ transmission electron microscopy where samples can be observedby TEM during ion bombardment and this enables the development of the microstructure
to be directly observed. The JANNUS multibeam experimental facility (Serruys et al.,2009) is one example of a facility that is devoted to promoting interaction between
modelling and experiment.
In summary, the time and length scales that can be studied using classical MD simulations
make it suitable for the calculation of the primary damage produced by an irradiation event.
However, more research and development are required, particularly in areas of interatomic
potentials and the description of electronic effects, to improve the predictive nature of themodels.

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5. Atomistic simulation methods for long-time dynamics in materials fornuclear energy systems
1

2

2

B. P. Uberuaga , D. Perez and A. F. Voter
1
Materials Science and Technology Division, Los Alamos National Laboratory, United States
2
Theoretical Division, Los Alamos National Laboratory, United States

5.1. Introduction
Many important processes in materials systems are intrinsically atomistic in nature but
involve time scales that span many orders of magnitude, thus exceeding what can bedirectly simulated using molecular dynamics (MD). This is especially true for materials in
nuclear energy applications, in which defects created by collision cascades on the
femtosecond-picosecond time scale cause microstructural changes that continue to evolvefor years, in many cases leading to failure of the material. In this chapter, we review
atomistic methods for reaching long-time scales in systems like these. These accelerated
MD and adaptive kinetic Monte Carlo (aKMC) methods exploit the infrequent-event natureof the diffusive events that comprise this long-time evolution. In favourable cases, thesemethods can predict state-to-state evolution that approximates what would result from an
extremely long molecular dynamics simulation, and the most accurate of the methods cando this to arbitrary accuracy. We present some examples of applications of these methods
to problems relevant to nuclear energy materials, the subject of this volume. We thendiscuss situations that are difficult for the methods, causing them to be less efficient, andwe conclude with a short list of the most pressing issues in the further development of theseapproaches to make them as powerful and predictive as possible for realistic problems.
The evolution of radiation damage in materials spans many time and length scales. Whilethe initial damage production occurs on the atomic scale over picoseconds via collisioncascades, the damage ultimately manifests itself macroscopically, often in the form ofswelling or cracking, which can take years to develop. There is a wide range of phenomenathat bridge these two extremes, including defect diffusion, annihilation and aggregation,
the formation of interstitial loops and voids, and the development of a more complex
microstructure. As a result, no one simulation method can be employed to study the
problem of radiation damage on all relevant time and length scales. Rather, a combinationof many techniques must be used to address this problem. MD simulation, in which atom
positions are evolved by integrating the classical equations of motion in time, can probe
timescales of ps to ns, making it ideal for studying collision cascades. The initial, postthermal-spike damage produced in the collision cascade can be directly simulated using
MD. However, once that damage has been formed, diffusion and subsequent annihilationor aggregation of those defects can occur on much longer time scales, perhaps even seconds
or beyond, depending on the conditions (temperature, pressure, etc.) and the material. Suchphenomena must be accounted for to accurately predict the formation and evolution oflarger scale features such as interstitial dislocation loops and vacancy voids. These areimportant in the evolving microstructure, leading to the macroscopic response to radiation
damage such as swelling and cracking. While at some larger time and length scale, thematerial evolution can be described by higher-level models, for much longer times than the
sub-microsecond time scale that direct MD can reach, full atomistic simulation is necessary
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to understand the complex and often competing phenomena that must be accounted for andquantified to parameterise these higher-level models.
In this chapter we discuss some methods developed over the past 20 years for reachingthese longer time scales without sacrificing atomistic detail or dynamical accuracy. Theselong-time methods work for infrequent-event systems, in which the system spends a long
period of time in a given state (typically corresponding to a particular basin of the potential
energy surface), occasionally making a transition to a neighbouring state (e.g. through the
motion of a defect). These transitions are usually activated processes, whose rate is limited
by a potential energy barrier. The characteristic that the state residence time is much longer
than the time duration of the transition event (typically 1 ps) is what defines these as
infrequent-event systems.
These long-time methods are most valuable for treating what we often term “complex
infrequent event systems," and many if not most radiation damage systems fall into thisclass. These are characterised by an interplay of reaction pathways with different activationenergies (and hence different temperature dependences), so that one cannot simply raise
the temperature of the system to accelerate the dynamics in a meaningful way, andtransition pathways that are complicated enough that many important pathways might beomitted if one resorts to a simple model (e.g. lattice-based kinetic Monte Carlo (KMC) witha pre-calculated rate catalogue). The point of these long-time-dynamics methods is thatthey do not make assumptions about transition pathways or the nature of the states,
assumptions that would limit their accuracy. Consequently, these methods give dynamical
evolution from state to state that is roughly the same as that of a very long direct-MD
simulation. For the most accurate of these methods, parallel replica dynamics, the state-tostate evolution can be made arbitrarily accurate.
These methods generally fall into two classes: accelerated molecular dynamics (AMD)methods and aKMC approaches. In the AMD approach, the key concept is to let the
trajectory find its own way out of the current state of the system – as a trajectory in astandard MD simulation would – but to coax it into to finding this escape pathway more
quickly. The methods in this class are parallel replica dynamics (Section 5.3)
hyperdynamics (Section 5.4), and temperature accelerated dynamics (Section 5.5). The
aKMC approach, described in Section 5.6, is a generalisation of standard KMC. The goal
in aKMC is to find all possible, or all relevant, escape pathways from the current state of
the system, typically by performing a set of saddle-point searches. One of these pathways
is then selected to move the system to the next state. In Section 5.7. we briefly describe avery recently developed method in this long-time simulation class, κ-dynamics.
After presentation of the methods and a review of some recent advances in Sections 5.3.
through 5.7., in Section 5.8. we give some examples of applications of these methods toproblems relevant for nuclear materials. We note that some of the method presentation
draws from another review we wrote recently (Perez et al., 2009). We then conclude in
Sections 5.9. and 5.10. with a discussion of the current limitations of these methods and theareas in which we believe further development is needed to make the methods maximallyuseful for simulating atomistic evolution in nuclear energy materials.
Due to space constraints, the descriptions of the methods that follow omit manyfoundational details. The interested reader is encouraged to consult the original references(e.g. Voter, 1998, 1997a, 1997b; Sørensen, Voter, 2000; Henkelman, Jónsson, 2001, Lu,
Makarov, Henkelman, 2010) or more in-depth review articles (e.g. Uberuaga et al., 2005a;
Voter, 2006) for a more thorough treatment. The reader can also find some more discussion

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2of the relative merits of the methods and how to choose the best method for a particular
problem in Uberuaga and Voter (2006).
Finally, we note that the manuscript for this chapter was prepared in 2013, but not publisheduntil 2020. We have left the main body of the chapter largely unchanged, as we believe it
still represents a valuable introduction to the methods and review of how they can beapplied to nuclear-energy materials. We have added a short section near the end of thechapter that describes the important developments in the past five years.

5.2. Parallel replica dynamics
Parallel replica dynamics (ParRep) is the most general and accurate of the AMD methods.
It assumes only that the system will escape from the current state in a way that obeys firstorder kinetics; i.e. for any trajectory that has been in a state long enough to have lost its
memory of how it entered the state (longer than the correlation time τcorr), the probability
distribution function for the time of the next escape from that state is given by𝑝𝑝(𝑡𝑡) = 𝑘𝑘𝑒𝑒 −𝑘𝑘𝑘𝑘 ,(1)

where k is the rate constant for escape from the state. ParRep allows for the temporalparallelisation of the state-to-state dynamics of such a system on M processors. This is tobe contrasted with standard parallelisations of MD simulations in which spatial
decomposition schemes are used.
We briefly sketch the derivation here. For a state in which the total rate of escape is k,simultaneously explored on M processors, the effective escape rate for the first escape of
any replica is Mk. If the simulation time accumulated on one processor is t, the total timeon the M processors will then be tsum = Mt. Thus, using a simple change of variable, p(t)can be written as
𝑝𝑝(𝑡𝑡)𝑑𝑑𝑑𝑑 = 𝑀𝑀𝑀𝑀𝑒𝑒 −𝑀𝑀𝑀𝑀𝑀𝑀 𝑑𝑑𝑑𝑑
= 𝑘𝑘𝑒𝑒 −𝑘𝑘𝑘𝑘𝑠𝑠𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠= 𝑝𝑝(𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠 )𝑑𝑑𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠 ,

(2a)
(2b)
(2c)

implying that the probability to leave the state per unit MD time is the same whether thattime is accumulated on one or M processors. While this sketched derivation applies forprocessors of equal speed, we emphasise that the same conclusion can be shown to be valid
even if the processors run at different speeds and/or with time-varying speeds (Voter,1998).
Figure 5.1. shows a schematic of the algorithm. Starting with a system in a particular state,
it is replicated on each of the M processors. Each replica, after an independent random
momentum assignment, is evolved forward for a time Δtdeph ≥ τcorr to eliminate correlations
between replicas, a stage referred to as dephasing. After dephasing, each processor carries
out an independent MD trajectory, typically performed at a constant temperature. Together,
the trajectories explore phase space within the particular basin M times faster than a single
trajectory would. Once a transition is detected on any processor, all processors are stopped.
The simulation clock is then advanced by tsum, the accumulated trajectory time summed
over all replicas until the first transition occurred.

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Figure 5.1. Schematic illustration of the parallel replica dynamics method

Note: The four steps, described in the text, are (A) replication of the system into M copies, (B) dephasing of
the replicas, (C) propagation of independent trajectories until a transition is detected in any of the replicas, and(D) brief continuation of the transitioning trajectory to allow for correlated events such as recrossings or followon transitions to other states. The resulting configuration is then replicated, beginning the process again.
Source: Motalenti and German, 2002

ParRep also correctly accounts for correlated dynamical events (Chandler, 1978; Voter
Doll, 1985), i.e. there is no requirement that the system obeys transition state theory(Marcelin, 1915; Wigner, 1932; Eyring, 1935; Pechukas, 1981), unlike the other AMD
methods (or aKMC). This is accomplished by allowing the trajectory that made thetransition to continue for a further amount of time Δtdeph ≥ τcorr, during which recrossingsor follow-on events may occur. The simulation clock is then advanced by Δtcorr, the newstate is replicated on all processors, and the whole process is repeated.
The computational efficiency of the method is limited by both the dephasing stage, which
does not advance the system clock, and the correlated event stage, during which only one
processor accumulates time. (This is illustrated schematically in Figure 5.1., where dashedline trajectories advance the simulation clock but dotted line trajectories do not.) Thus, theoverall efficiency will be high when𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟(5)≫ Δ𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑ℎ + Δ𝑡𝑡𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ,𝑀𝑀where τrxn = 1/k.

An extension to ParRep allows the method to be applied to driven systems (Uberuaga,
Stuart, Voter, 2007). To result in valid dynamics, the processors must run at the same speed,
and the drive rate must be slow enough that at any given time the rates for the different
escape pathways in the system depend only on the instantaneous configuration of thesystem.
While the derivation of the ParRep method does not impose a particular definition of a“state” of the system, the operational definition often used in practice corresponds to asingle basin of the potential energy surface, i.e. a state is taken to be the ensemble of pointsof configuration space that converge to the same fixed point under a local minimisation of
the energy of the system (e.g. using a steepest-descent algorithm). An exponential
distribution of escape times is then obtained if the typical timescale for a transition out of
the state is long compared to the characteristic vibrational period of the system around that
fixed point. While this definition has the virtue of being conceptually and computationallysimple, it limits the range of possible applications to systems where the basins are deep

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

enough (relative to kBT) and well separated from each other and leaves many other, more
complex, systems out of reach. This limitation can sometimes be overcome by a moregeneral definition of a state. For example, in the case of pyrolysis of hexadecane, it was
shown that states could be defined as the ensemble of all configuration space points thatshare the same network of covalent bonds (Kum et al., 2004). In that case, these“superstates” contain a large number of simple energy basins of the potential energy
surface, each corresponding to a different conformation of the molecular backbone. There,
the method exploited the separation of timescale between the rapid changes of dihedral
angles of the backbone (intrasuperstate transitions) and the slow covalent bond breaking
process (intersuperstate transitions) rather than between the vibrational timescale and thatof sampling of the different dihedral angles. This enables one to ignore the “irrelevant” fasttransitions that would demand incessant dephasing and decorrelation and concentrate
directly on the real kinetic bottlenecks. This approach has been applied to other cases as
well, such as surface diffusion kinetics while the surface is in contact with a liquid phase(with its huge number of shallow states) (Perez et al., 2009) and diffusion of heteroepitaxial
surface clusters in metallic systems (Uche et al., 2009). This approach was recently
mathematically formalised (Le Bris et al., 2012), showing that the ParRep procedure gives
correct results for arbitrary state definitions, provided the dephasing and decorrelation
times are made long enough.
We will give specific examples below of the application of ParRep in situations relevant tonuclear energy materials, but we mention here that ParRep dynamics has been successfully
applied to a number of varied problems, including the diffusion of H2 in crystalline
C60 (Uberuanga et al., 2003a), the pyrolysis of hexadecane (Kum et al., 2004), the diffusionof defects in plutonium (Uberuanga et al., 2003b), the transformation of voids into stacking
fault tetrahedra in face centred cubic (fcc) metals (Uberuaga et al., 2007a), the stretchingof carbon nanotubes (Uberuaga, Stuart, Voter, 2007), grain boundary sliding in Cu (Mishin
et al., 2007), friction-force microscopy (Martini et al., 2009; Dong et al., 2011), thediffusion of Li through a polymer matrix (Duan et al., 2005), the fracture process in metals(Warner, Curtin, Qu, 2007), and the folding dynamics of small proteins (Zagrovic, Sorin,
Pande, 2001). As parallel-computing environments become even more common, ParRep
will become an increasingly important tool for reaching long-time scales in complex
infrequent-event systems.

5.3. Hyperdynamics
Another way to accelerate the state-to-state evolution of an infrequent-event system is toconstruct an auxiliary system in such a way that the escape dynamics of the latter are fasterthan those of the former, while enforcing that one maps onto the other by a suitable
renormalisation of time. Hyperdynamics (Voter, 1997a) realises this objective by buildingon the concept of importance sampling (Valleau, Whittington, 1977; Berne, Ciccotti,
Coker, 1998) and extending it into the time domain. In this approach, the auxiliary system
is obtained by adding a non-negative bias potential ΔVb(r) to the potential of the original
system V(r) so that, effectively, the height of the barriers between different states isreduced, as schematically shown in Figure 5.2. The relationship between the dynamical
evolution of the original and biased systems is recovered using transition state theory(TST). According to TST, the rate of escape of the original system out of a given state A isgiven by
𝑇𝑇𝑇𝑇𝑇𝑇
𝑘𝑘𝐴𝐴→=< |𝑣𝑣𝐴𝐴 |𝛿𝛿𝐴𝐴 (𝒓𝒓) >𝐴𝐴
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS

(6)

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86 │ NEA/NSC/R(2019)2
where δA(r) is a Dirac delta function centred on the dividing surface between state A andthe neighbouring state(s) (here the hypersurface is at r = 0), vA is the velocity normal to itand ⟨P⟩A is the canonical ensemble average of a quantity P for a system confined to state
A. By standard importance sampling manipulations, the last equation can be recast in aform where the averages are obtained on the biased potential instead. We find:
𝑇𝑇𝑇𝑇𝑇𝑇
𝑘𝑘𝐴𝐴→=< |𝑣𝑣𝐴𝐴 |𝛿𝛿𝐴𝐴 (𝒓𝒓)𝑒𝑒 𝛽𝛽∆𝑉𝑉𝑏𝑏 (𝑟𝑟) >𝐴𝐴𝑏𝑏< 𝑒𝑒𝛽𝛽∆𝑉𝑉𝑏𝑏 (𝑟𝑟) >𝐴𝐴𝑏𝑏,(7)

where β = 1/kBT and kB is the Boltzmann constant. If we impose the condition that the bias
potential must vanish at the dividing surface, the last equation can be rewritten as
𝑇𝑇𝑇𝑇𝑇𝑇
𝑘𝑘𝐴𝐴→=< |𝑣𝑣𝐴𝐴 |𝛿𝛿𝐴𝐴 (𝒓𝒓) >𝐴𝐴𝑏𝑏.
< 𝑒𝑒𝛽𝛽∆𝑉𝑉𝑏𝑏 (𝑟𝑟) >𝐴𝐴𝑏𝑏(8)This result is very appealing since the relative rates of escapes from A to other states isinvariant under the addition of the bias potential, i.e.
𝑘𝑘𝐴𝐴𝑇𝑇𝑇𝑇𝑇𝑇
𝑏𝑏 →𝐵𝐵
𝑘𝑘𝐴𝐴𝑇𝑇𝑇𝑇𝑇𝑇
𝑏𝑏 →𝐶𝐶=

𝑇𝑇𝑇𝑇𝑇𝑇
𝑘𝑘𝐴𝐴→𝐵𝐵
𝑇𝑇𝑇𝑇𝑇𝑇 .
𝑘𝑘𝐴𝐴→𝐶𝐶(9)

Thus, the state-to-state dynamics on the biased potential is equivalent to that on the original
potential as long as the time in each state is renormalised to account for the uniform relative
increase of all the rates introduced by the biased potential. This rescaling of time is obtainedby multiplying the MD timestep ΔtMD by the inverse Boltzmann factor for the bias potential,
so that n MD timesteps on the biased potential are equivalent to an elapsed time of𝑛𝑛

𝑡𝑡hyper = � ∆𝑡𝑡𝑀𝑀𝑀𝑀 𝑒𝑒 𝛽𝛽∆𝑉𝑉𝑏𝑏 �𝒓𝒓�𝑡𝑡𝑗𝑗��

(10)

𝑗𝑗=1

on the original potential. Even though the speedup is typically different for each state, thisdefinition of the hypertime gives an ongoing estimate of the accelerated time with statisticalerror bars that are unbiased, and in the long-time limit (e.g. after many transitions) it
converges on the exact value with vanishing relative error. The speedup for escape fromstate A is given by the average boost factor B,
𝐵𝐵hyper =𝑡𝑡hyper= < 𝑒𝑒 𝛽𝛽∆𝑉𝑉(𝒓𝒓) >𝐴𝐴𝑏𝑏 ,𝑡𝑡𝑀𝑀𝑀𝑀(11)

and the overall computational boost is this boost divided by the relative extra cost of
calculating the bias potential and associated forces.
If both the original and biased systems obey TST and the bias potential is zero on all theridgetops, so that the above derivation holds, hyperdynamics can provide considerable,
accurate time acceleration compared to direct-MD simulations. It has been successfully
applied to a variety of systems, including desorption of organic molecules from graphitic
substrates (Beker, Mignogna, Fichthorn, 2009), surface diffusion of metallic clusters(Voter, 1997b), heteroepitaxial growth (Miron, Fichthorn, 2005), microscopic studies of
sliding friction (Kim, Falk, 2010, 2011), plastic response of compressed nanopillars (Hara,

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Li, 2010) and the dynamics of biomolecules (Hamelberg, Mongag, McCammon, 2004;
Xin, Doshi, Hamelberg, 2010).
Figure 5.2. Schematic illustration of the hyperdynamics method
A bias potential (∆V(r)), is added to the original potential (V(r), solid line). Provided that ∆V(r) meets certain
conditions, primarily that it be zero at the dividing surfaces between states, a trajectory on the biased potential
surface (V(r)+∆V(r), dashed line) escapes more rapidly from each state without corrupting the relative escapeprobabilities. The accelerated time is estimated as the simulation proceeds.

Source: Uberuaga et al., 2019

In practice, the applicability of hyperdynamics is limited by the availability of bias
potentials that meet the necessary requirements and have reasonable computational
overhead. The construction of bias potentials that are valid, efficient and transferable
remains a challenge and a subject of active research. A review of bias potential forms as of2002, such as the Hessian-based bias (Vote, 1997b) and the flat bias (Steiner, Genilloud,Wilkins, 1998), can be found in Voter, Montalenti and Germann (2002).
An important advance on this front was taken by Miron and Fichthorn, with theintroduction of their “bond-boost” bias potential (2003). As the name suggests, the bondboost bias potential is composed of pairwise terms that tend to soften the bonds between
atoms. The key assumption here is that transitions between states will involve the formationor breaking of some bond so that the proximity to a transition state will be signalled by anunusually large distortion of a bond. If the overall bias potential is then designed to vanish
when any bond in the system distorts by more than some critical amount (say by more than
20% of its equilibrium length), then it should be possible to safely turn off the bias before
a dividing surface is reached. While this approach relies on the assumption that bondlengths alone are reliable indicators of the distance to a potential energy ridge, it provides
a very flexible and efficient way to carry out hyperdynamics.
The flexibility of this bond-boost approach can be further exploited to make hyperdynamics
more tolerant to low barriers. Indeed, the presence of low barriers is detrimental to theperformance of hyperdynamics because it will frequently cause the bias potential to vanish.
In cases where appropriate “superstates” can be defined, it is possible to leave the bias
potential turned on when the system crosses intrasuperstate barriers without affecting theaccuracy of the simulation. This power of this promising type of “bridging” approach,
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proposed by Miron and Fichthorn (2004), has been demonstrated for deposition of Co on
Cu(100) (Miron, Fitchorn, 2005).
Hamelberg et al (2004) have developed a variation on Steiner’s flat bias (Steiner,
Genilloud, Wilkins, 1998) that is effective for accelerating the dynamics in biomolecules.
The rugged landscape makes these very challenging systems for accelerated dynamics, andthis bias is typically applied aggressively enough that the bias potential is not zero at all the
key barriers. Hamelberg and co-workers (Xin, Doshi, Hamelberg, 2010) have recently
developed a way to extract the correct rate constant for a known pathway by appealing to
Kramers rate theory after performing multiple hyperdynamics simulations with bias
potentials that are non-zero at the dividing surface.
Hara and Li (2010) developed a bias potential to accelerate the nucleation of dislocations
in a system under stress, by constructing a form that is sensitive to the local shear distortion,
turning off when it reaches a critical value. Although this mechanism-specific approachruns the risk of slowing the rates for processes that the tailored bias is insensitive to(because the bias may be non-zero at the dividing surfaces for these other processes), thiscan be a powerful approach. They achieved boost factors greater than 1010 in simulationsof compressed nanopillars. Kim and Falk (2010, 2011) have developed bias potentials
suitable for the study of microscopic sliding friction, and have shown how in a driven
system, hyperdynamics can be effectively combined with parallel computation by
speculatively simulating future system configurations. Fichthorn and co-workers (2009)
have demonstrated that hyperdynamics can be carried out with electronic-structure-based
forces, although the much greater expense of electronic structure calculations compared toempirical potentials means that with present-day computers and electronic structure
methods, this is barely viable for most systems.
Chen and Horing (2007) have proposed a variation on the hyperdynamics approach, path
integral hyperdynamics (PIHD), which releases the requirements on the bias potential. The
bias is allowed to be non-zero at the dividing surfaces, and TST-violating correlated events
are allowed as well. These effects are properly accounted by appealing to a Langevin path
integral formalism to define a weight for each trajectory. As such, the calculation of therate constant for a given rare event requires multiple simulations, in contrast to the standard
hyperdynamics where a single trajectory is representative of the long-time evolution.
Recently, this PIHD approach has been shown to provide an effective strategy in a studyof diffusion in a periodic potential, including the case of a time-dependent forcing (Ikonen
et al., 2011), and in a study of barrier crossing by self-avoiding polymer chains (Shin et al.,2010).

5.4. Temperature accelerated dynamics
One natural way of speeding up the dynamics of a system is to simply raise the temperature.
However, while the rates of processes will increase with higher temperatures, the relative
probabilities of different events occurring will be different than at the original temperatureof interest. Correcting for this reordering is the basic idea behind temperature accelerated
dynamics (TAD) (Sørensen, Voter, 2000). In TAD, transitions are sped up by increasing
the temperature to some Thigh, but transitions that should not have occurred at the original
temperature Tlow are filtered out. The TAD method assumes that the system obeys harmonic
TST (Vineyard, 1957), in which the transition pathway is associated with a saddle point on
the energy landscape, and the rate constant has a simple Arrhenius temperature dependencewith a fixed prefactor. As a result, TAD is more approximate than the other AMD methods.

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However, for many applications, especially in solids, this additional approximation is
acceptable.
In each basin, the system is evolved at Thigh. When a transition is detected, the saddle point
for that transition is found. The trajectory is then reflected back into the basin andcontinued. This procedure is referred to as “basin constrained molecular dynamics”(BCMD). During the BCMD, a list of escape paths and escape times at Thigh for eachpathway is generated. Assuming that harmonic TST holds, and knowing the saddle point
energy for the transition, we can then extrapolate each escape time observed at Thigh toobtain a corresponding escape time at Tlow. This extrapolation, which does not requireknowing the preexponential factor, can be illustrated graphically on an Arrhenius-style plot(ln(1/t) vs. 1/T), as shown in Figure 5.3. The time for each event seen at Thigh extrapolated
to Tlow is
𝑡𝑡𝑙𝑙𝑜𝑜𝑤𝑤 = 𝑡𝑡ℎ𝑖𝑖𝑖𝑖ℎ 𝑒𝑒 𝐸𝐸𝑎𝑎(𝛽𝛽𝑙𝑙𝑙𝑙𝑙𝑙−𝛽𝛽ℎ𝑖𝑖𝑖𝑖ℎ ) ,(12)where β = 1/kBT and Ea is the activation energy.

As the BCMD is continued, a new shorter-time event may be discovered. With theadditional assumption that there is a minimum preexponential factor, νmin, which bounds
from below all the preexponential factors in the system, we can define a time at which the
BCMD trajectory can be stopped. This time has the property that the probability any
transition observed later would replace the first transition at Tlow is less than δ, an
uncertainty set by the user. This “stop” time is given by
𝑡𝑡ℎ𝑖𝑖𝑖𝑖ℎ,𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≡ln(1⁄𝛿𝛿 ) 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙,𝑠𝑠ℎ𝑜𝑜𝑜𝑜𝑜𝑜��,𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚ln(1⁄𝛿𝛿 )(13)

where tlow, short is the shortest transition time at Tlow. When this stop time is reached, thesystem clock is advanced by tlow, short, and the corresponding transition is accepted. The TAD
procedure is then started again in the new basin. Thus, in TAD, two parameters govern theaccuracy of the simulation: δ and νmin.
The average boost in TAD can be dramatic when barriers are high and Thigh/Tlow is large.
However, as TAD relies upon harmonic TST for validity, any anharmonicity error at Thighwill lead to inaccuracy in the predicted dynamics. This anharmonicity error can becontrolled by choosing a Thigh that is not too high.
A number of advances have led to increased efficiency in particular systems. “Synthetic”
mode (Sørensen, Voter, 2000), a KMC treatment of low-barrier transitions, cansignificantly improve the efficiency in cases where low-barrier events are repeated often.
Furthermore, if we know something about the minimum barrier to leave a given state, either
because we have visited the state before and have a lower bound on this minimum barrieror because the minimum barrier is supplied a priori, we can accept a transition and leave
the state earlier than the time given by equation (13). See Montalenti and Voter (2002) for
details.
The AMD methods in general do not scale particularly well with system size, but Shim etal. (2007, 2008) have recently shown how the TAD method can be spatially parallelised,
to make it more efficient for larger systems.
Temperature accelerated dynamics has been demonstrated to be very effective for studying
the long-time behaviour of defects produced by collision cascades in someceramics (Uberuaga et al., 2004, 2007b), and more nuclear material applications will be
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discussed below. Coupled with MD simulation of surface deposition events, TAD has been
applied to simulate the long-time-scale growth of Ag (Mishin, Sørensen, Voter, 2001),
Cu (Shim et al., 2008), Cu on Ag(100) (Sprague et al., 2002), and magnetron sputterdeposition growth of Mg-Al-O thin films (Georgieva, Voter, Bogaerts, 2011). Other
applications for which TAD has proven effective include defect diffusion on oxidesurfaces (Harris et al., 2004), the dissolution of boron-interstitial clusters in silicon (Cogoni
et al., 2005), and topological changes in carbon nanotube fragments at hightemperature (Uberuaga et al, 2012). Tsalikis et al., (2010a) have recently combined TAD
with advanced sampling techniques to study long-time dynamical properties of glassforming systems.
Figure 5.3. Schematic illustration of the temperature accelerated dynamics method

Note: Progress of the high-temperature trajectory can be thought of as moving down the vertical time line at
1/Thigh. For each transition detected during the run, the trajectory is reflected back into the basin, the saddle
point is found, and the time of the transition (solid dot on left time line) is transformed (arrow) into a time on
the low-temperature time line. Plotted in this Arrhenius-like form, this transformation is a simple extrapolation
along a line whose slope is the negative of the barrier height for the event. The dashed termination line connects
the shortest-time transition recorded so far on the low temperature time line with the confidence-modified
minimum preexponential (ν⋆ =ν / ln(1/δ)) on the y axis. The intersection of this line with the high-T time
min

min

line gives the time (tstop, open circle) at which the trajectory can be terminated. With confidence 1-δ, we cansay that any transition observed after tstop could only extrapolate to a shorter time on the low-T time line if it
had a preexponential factor lower than νmin.
Source: Uberuaga et al., 2019

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5.5. Adaptive kinetic Monte CarloIn the aKMC method, the ultimate goal of accelerating the state-to-state dynamics is thesame as in the AMD methods, but the conceptual approach is different. aKMC is a
generalisation of the KMCo method (Voter, 2006) which we briefly describe first.
In KMC, there is no classical trajectory at all. Instead, from a list of possible pathways andrates for escape from the current state, one escape path is chosen randomly with aprobability proportional to its rate constant, and the system is advanced along that path toa new state. The clock is then incremented in an appropriate way, by drawing a time fromthe exponential distribution (equation (1)) with exponent equal to the sum of the escaperates (Young, Elcock, 1996; Bortz, Kalos, Lebowitz, 1975; Gillespie, 1976). In a typical
implementation of KMC, sometimes called lattice KMC, it is assumed each atom can beassociated with a fixed lattice position, and the state-to-state transitions all occur alongwell-understood unit-process pathways. In the bulk, the key unit process is the jump of avacancy, and on a surface, it is the jump of an adatom to an adjacent site. In principle,
somewhat more complicated unit processes (e.g. surface exchange events (Feilbelman,
1990)) can be included, but typically they are not. With the additional assumption that therate constant for a unit-process pathway is influenced only by the nearby atomic
environment, one can compile a list of all possible pathways and their associated rate
constants, forming a rate catalogue (Voter, 1986) that is used to drive the lattice KMC
simulation. The rates are usually based on harmonic transition state theory (Vineyard,
1957), so that a required step is finding the saddle point connecting the initial and final statefor every entry in the rate catalogue.
An attractive formal feature of KMC is that if the rate catalogue is complete (all possible
escape paths from each state are included) and the rate constant for each path is exact(which might require going beyond the TST approximation), then the state-to-state
dynamical evolution for this system of states is exact (Gillespqie, 1976, 1977). In standard
KMC, however, this exact limit is essentially never achieved, and in most cases probably
not even well approximated, due to the simplifying on-lattice and unit-process assumptions.
In the aKMC approach (Henkelman, Jónsson, 2001; Middleton, Wales, 2004; El-Mellouhi,
Mousseua, Lewis, 2008) the aim is to generate a much more complete rate catalogue.
Rather than assuming a set of unit-process escape mechanisms, for each state that is visited
during the simulation, one carries out a systematic search for saddle points corresponding
to escape paths out of the basin. This can be accomplished efficiently using eigenvectorfollowing approaches such as the dimer method (Henkelman, Jónsson, 1999; Munro,
Wales, 1999) or the activation-relaxation technique (Barkeman, Mousseau, 1996; Beland
et al., 2011). A key advance, relative to the state of the art of the mid-1990s, is that thesemethods only require first derivatives of the potential to find the saddle point. When thistype of saddle search is carried out, it is not unusual to find a complex pathway involving
multiple atoms (e.g. see Henekelman and Jónsson (1999)), and the process may lead to an
off-lattice geometry as well. This behaviour is consistent with the complex pathways foundduring AMD simulations.
This aKMC approach comes much closer to the ideal of exact KMC than the lattice KMC
described above. The more searches that are performed, the more complete the aKMC
catalogue for the current state is, and the more likely that a kinetically appropriate pathway
is selected for the next event. Although it is difficult, if not impossible, to prove that all
saddle points surrounding a basin can be found in a finite number of searches, Henkelman
has recently discussed how to approximate the confidence that the relevant saddles have
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been found after a certain number of randomly initialised saddle searches, by monitoring
the rate, per search, at which redundant saddles are found (Xu, Henkelman, 2008).
The computational efficiency of aKMC can be increased in several ways. It can be readily
parallelised by performing saddle searches independently on different processors. Also, forlarge systems, each search can be spatially localised to a subset of the system, although thisshould be done with care, to avoid missing pathways that are more delocalised than
expected. Storing information about each state allows substantial savings when a state isrevisited (as is true for all the methods discussed in this chapter), and combining this withthe localisation concept and pattern recognition techniques can lead to significant
computational gain when locally equivalent, or locally similar, defects are identified (ElMellouhi, Mousseau, 2008; Xu, Henkelman, 2008; Konwar, Bhute, Chatterjee, 2011).
The aKMC approach has been applied to a variety of processes, including post-cascade
defect evolution (Xy, Stoller, Osetsky, 2013), surface diffusion and growth of metal
surfaces (Xu, Henkelman, 2008; Konwar, Bhute, Chatterjee, 2011; Henkelman, Jónsson,
2003; Scott et al., 2011), surface erosion by sputtering (Henkelman, Jónsson, 2003), selfdiffusion in copper grain boundaries (Pedersen et al., 2009), decomposition of methanol on
Cu(100) (Xu, Mei, Henkelman, 2009), elastic effects on the diffusion of vacancies insilicon (El-Mellouhi, Mousseau, 2008), and kinetics in a model glass-forming
system (Middleton, Wales, 2004). Using a related approach, an off-lattice self-learning
KMC developed specifically for surfaces, Rahman and co-workers have studied surface
diffusion of clusters on fcc(111) (Kara et al., 2009). Henkelman’s group has discussed theuse of electronic structure forces with aKMC (Xu, Henkelman, 2008), and shown that they
can be implemented effectively, although this remains very computationally expensive.
Jónsson and co-workers (Henkelman, Jónsson, 2003; Pedersen, Jónsson, 2010) havediscussed the effective use of aKMC in a distributed computing environment.

5.6. κ-Dynamics
Here we briefly describe a promising approach to long-time dynamics developed by Lu,
Makarov and Henkelman (2001), which they term κ-dynamics. Building on the conceptthat trajectories initiated at an arbitrary TST dividing surface offer a way to recover theexact rate constants for an infrequent-event system (Voter, Doll, 1985), they have designed
a method that combines importance sampling with parallelised dividing-surface trajectories
to give accelerated state-to-state transitions. They start by specifying a dividing surface that
totally encloses the initial state of the system, and which would be infrequently reached bya standard trajectory. Although the efficiency would be higher if this dividing surface wasclose to the ridgetop surrounding the state, it is not a formal requirement of the method.
Hence, simple and tractable definitions can be used to approximate this exceedingly
complex ideal surface. Using importance sampling, the TST rate constant for escapethrough this dividing surface is determined, and a set of representative points on this surface(i.e. Boltzmann distributed within this dividing surface subspace) are generated. These
representative points are used as the starting points for trajectories that are run forward andbackward in time as far as necessary to determine whether an escape trajectory has beenfound. An escape trajectory is one that in the outgoing direction proceeds directly to somenew state of the system without recrossing the surface, and in the reverse direction settles
into the initial state before deeply entering any other states, perhaps after some number ofrecrossings. Although many trajectories may be required before an escape trajectory isfound, this stage of the simulation is parallelisable, as is the importance sampling. In thelimit where the importance sampling produces a perfectly unbiased set of starting points on

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the dividing surface, κ-dynamics gives correct state-to-state dynamics. Using a bond-boost
bias potential form for their importance sampling, Lu et al (2010) applied this approach tosurface diffusion events on an Al(100) surface, demonstrating its accuracy, as well as its
ability to properly describe important events in which the final state is not connected to theinitial state, which requires that correlated dynamical events are treated correctly. Of the
AMD and aKMC methods discussed in this chapter, only ParRep and κ-dynamics can treat
these correlated events correctly (although in hyperdynamics these correlated events canoccur, and may be approximately correct).

5.7. Example applications
Long-time dynamics methods have been used to examine a number of phenomena in
nuclear materials or model materials meant to give basic insight that is applicable to nuclear
materials. Here we describe three examples of such simulations and how they provided
new insight into the properties and behaviour of these materials. In each of these example
cases, an AMD method was employed, but except for the first case (void transformation),in which correlated dynamical events were important, similar results could have beenobtained using an aKMC approach.

5.7.1. Void evolution in fcc metals
At high temperatures, the defects produced under irradiation can aggregate, leading to
detrimental effects such as swelling. This occurs when a vacancy bias is produced as aconsequence of interstitials effectively being removed from the system via the formationof interstitial loops. While it is in fact the loops themselves that cause the swelling, thecharacteristic signature of this effect is the formation of voids from the aggregation of theexcess vacancies. On very long-time scales, these voids can diffuse and aggregate to form
even larger clusters. Using AMD methods, primarily parallel replica, we examined thelong-time behaviour of voids in both Cu, as a simple model fcc material, and Pu.
These simulations revealed surprising behaviour. It had been assumed in the literature thatonce voids form, they could not transform to other types of vacancy clusters, such as
stacking fault tetrahedra (SFTs), as the barrier for that transformation was considered too
high (Zinkel, Seitzman, Wolfer, 1987). However, as described in Uberuaga, et al. (2007a)and shown in Figure 5.4., we found that there is such a high entropic preference for the SFT
that the free energy barrier at temperatures of 475 K was very small, meaning that thetransformation can occur on the timescale of microseconds at that temperature. As shownin Figure 5.4., from an energy landscape perspective, the transformation is very
complicated, with many local minima connected by a variety of barrier heights. However,once the process begins, the entropic driving force is so strong that it happens very quickly
on an MD time scale. That is, the system does not thermalise within each local minimum,
but rather progresses very quickly along the collection of states connecting the initial voidand the final SFT. This illustrates the complexity of the types of processes that can be found
via the AMD methods, but also the potential dangers of simulating these types of systems:
a method that effectively thermalised the system within each and every state (such as a
KMC simulation based upon this landscape) would most likely result in the wrong
dynamics. This transformation happens in both Cu and Pu (Uberuaga, Valone, 2008). It isinteresting to note that repeating this simulation with even a single He atom placed in thecentre of the void significantly slows down the transformation.

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Figure 5.4. Transformation pathway of a 45-vacancy void to a stacking fault tetrahedron at
T=475 K

Note: The figures show the structure of the vacancy cluster as the transformation proceeds. Red spheres arevacancies and blue spheres are interstitials, as determined by comparison to the perfect fcc lattice. The blue
points and solid line represent the energy profile for the process. The red points are an estimate of the freeenergy profile along the path. The large difference between the two is a consequence of the large change inentropy that results from the collapse of the void.
Source: Uberuaga et al. 2007a

5.7.2. Non-equilibrium transport processes
Under irradiation, extremely large amounts of energy are deposited into the material,
leading to mass transport mechanisms that would essentially be irrelevant underequilibrium conditions. This effect can take two forms. First, the defect configuration
produced directly by the cascade might be metastable, having kinetic properties
significantly different than the equilibrium structure. Second, as the defects produced
during collision cascades aggregate, they may form structures that are high in energy, again
metastable, but that also have unique kinetics.
Two examples illustrate these effects. In the case of the fissioning of U in UO2, fission
products are generated with large amounts of energy. When the fission product is Xe, theequilibrium configuration is a substitution on the U sublattice. However, after a fission
event, the final state of the Xe atom depends on the available defects and MD simulations
have shown that the final position of the Xe atom can be either interstitial or
substitutional (Parfitt, Grimes, 2009). Most studies of Xe diffusion in bulk UO2 havefocused on the substitutional state as it is the most thermodynamically favoured. However,based on these MD simulations, we examined the mobility of interstitial Xe using basin
constrained MD, the foundation of TAD (Liu et al., 2011). We found that Xe can diffusemuch faster interstitially than it can substitutionally via a complex concerted mechanismon the oxygen sublattice. In this mechanism, Xe pushes out one oxygen, replacing it, andSTATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

then is pushed back out into a new interstitial site. This process has a barrier significantly
lower than vacancy-assisted diffusion and may be very important for Xe transport inreactor, where Xe diffusion is known to be much faster than expected based on equilibriumtransport measurements.
A second example, illustrated in Figure 5.5., involves the aggregation of defects in MgO,
chosen here as a model oxide to understand radiation effects, though MgO also has been
proposed as a material in some advanced fuel concepts. Using a combination of MD and
TAD, we examined the clustering of defects produced during collision cascades (Uberuaga
et al., 2004; 2005b; 2006). We found that large interstitial clusters could adopt a number ofstructural variants and that higher energy structures, structures that would have essentiallyzero concentration at equilibrium, can form as a consequence of aggregation reactions andthat these high-energy clusters are very mobile. In fact, the fastest diffusing species wehave seen in MgO is a metastable structure of the six-interstitial cluster, which diffuses on
the ns time scale at room temperature. In contrast, the ground state structure of this same
six-interstitial cluster is extremely immobile, with an average hop time measured in years.
Figure 5.5. Schematic of the energy landscape for five-interstitial clusters in MgO

Note: Red indicates oxy- gen, blue magnesium, dark colours interstitials and light colours vacancies. There areat least three structural variants for each of the Mg2O3 and Mg3O2 clusters. Each variant exhibits unique
diffusive characteristics.
Source: Adapted from Uberuaga et al, 2006

The case shown in Figure 5.5. demonstrates the complexity associated with even thissimple oxide. Here, the landscape of two versions of a five-interstitial cluster, Mg2O3 andMg3O2, are shown. Two striking aspects are illustrated. First, for each of these clusters,
there are at least three structural variants with different relative stabilities. Second, each ofthese variants exhibits its own diffusive behaviour. Given that, during irradiation, theseclusters form from even higher energy states (dispersed interstitials), they can easily form
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from the defects produced during irradiation. A simple reaction-diffusion model allowed
us to determine that these types of clusters would have measurable effects on the nucleationand growth rate of interstitial loops within the material. Thus, it would be critical to include
these in mesoscale models of radiation damage evolution in materials such as MgO.

5.7.3. Defect interactions with grain boundaries
The key challenge in advanced nuclear energy concepts – both fission and fusion – isdeveloping materials that can withstand the harsh environments, particularly irradiation.
Proposed design concepts require materials that can sustain damage doses of 200 dpa or
higher (dpa, or displacements per atom, is a typical measure of damage, and indicates theaverage number of times an atom is displaced from its lattice site during irradiation). Thus,
it is imperative to design materials that can withstand these doses. One promising avenue
is to use internal interfaces – phase boundaries and grain boundaries – to promote defectrecombination and, effectively, self-healing.
We used MD and TAD to examine the interaction between irradiation-induced defects andgrain boundaries in Cu (Bai et al., 2010), again chosen as a model system that can beeffectively modelled on the atomistic scale. We simulated collision cascades near theboundaries with MD and the long-time evolution of those defects with TAD. The
simulations revealed three important phenomena. First, as seen before (Sugio, Shimomura,de la Rubia, 1998) during the cascades, the boundaries tend to absorb many moreinterstitials than vacancies, effectively reducing bulk defect recombination. This implies
that if no subsequent thermally activated processes were active, boundaries would worsenradiation tolerance by increasing the defect content compared to a single crystal. Second,at moderate temperatures, the interstitials at the boundary could interact with the residualvacancies in the bulk over relatively large distances (∼1 nm) with barriers that are verysmall compared to bulk vacancy diffusion. This so-called “interstitial emission”
mechanism, illustrated in Figure 5.6., leads to enhanced interstitial-vacancy recombinationcompared to what would be expected if the vacancies directly diffused to the boundary.
Finally, these results indicate that the sink strength of a boundary is not a fixed quantity,
but depends on the absorbed defect concentration and thus the irradiation conditions andthe microstructural state of the material (e.g. the density of boundaries). This has clear
consequences for mesoscale models of nanomaterials under irradiation.
Figure 5.6. An example of the type of complex processes that can occur near damaged grainboundaries.

Note: In this case, the grain boundary had absorbed interstitials during a collision cascade. Those absorbed
interstitials interact much more strongly with nearby vacancies than did the pristine grain boundary. Theinterstitials effectively “re-emit” from the boundary, annihilating those vacancies, with barriers much smaller
than the competing process of conventional vacancy diffusion.
Source: Adapted from Bai et al., 2010

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5.8. Limitations and ongoing challenges
While the methods described above have enjoyed considerable success so far, significant
challenges remain. The foremost issue is that in many systems, these methods fail toprovide significant acceleration due to the presence of large numbers of states connectedby low barriers. In essence, the methods described in this chapter achieve time accelerationby collapsing the time to the first event down to a time (or a time-equivalent incomputational work) that is on the order of the vibrational time scale. If low barriers cause
the time to the first event to be only marginally longer than this vibrational time scale, theboost factor will not be much greater than unity, and can even be much less than unity when
the overhead is taken into account (e.g. in TAD). Some strategies have been put forward to
mitigate this issue, such as super-state parallel replica dynamics (Perez et al., 2009), statebridging hyperdynamics (Miron, Fichthorn, 2004), TAD with synthetic mode (Søren,Voter, 2000), TAD with superbasin sampling (Tsalikis et al., 2010a), on-the-fly superbasin
detection (Tsalikis et al., 2010b), and incorporation of Tabu sampling (El-Mellouhi,
Mousseau, Lewis, 2008). For the aKMC methods, absorbing Markov-chain (Novotny,
1995), and related kinetic system acceleration methods can be employed (e.g.
see Chatterjee and Voter (2010) and references therein). However, we believe more work
is required before victory can be claimed, as many systems of interest remain out of reachwith the current methods. For example, an on-the-fly state definition algorithm that
automatically identifies an exploitably large separation of timescales would tremendously
extend the reach of parallel replica dynamics. Statistical analysis tools could also be used
to identify dynamically “irrelevant” states that could be ignored or lumped with others
without affecting the long-time dynamics. Many of these ideas are now being explored andwill hopefully lead to more general AMD methods in the next few years.
The second issue is the overall accuracy of the predictions that come out of these long-time
methods. We have limited our discussion to methods that give a high-quality descriptionof the long-time dynamics of a system, given particular interatomic potential that provides
the forces guiding the dynamics. The burden for providing results accurate enough thatthese methods can be reliably useful in guiding and interpreting experiment thus falls on
the quality of this interatomic potential, which unfortunately is often not up to the task. As
discussed above, we are beginning to see implementations of AMD and aKMC methods
with electronic structure methods, which can provide much more accurate forces. Further
development in this area will be valuable. However, this is still extremely expensive withpresent-day computers and methods, and a substantial number of systems we wish to study
will probably remain out of reach of this kind of direct electronic-structure treatment for
the foreseeable future. Thus, it remains important to develop advanced interatomic
potential forms, perhaps coupled tightly with electronic structure methodology to gain
greater accuracy at reasonable cost. Further, by extending this concept, it should be possible
to develop methods that also take specific advantage of the characteristics of infrequentevent systems and the AMD or aKMC method advancing the dynamics. An example of an
approach in this general class, designed for regular molecular dynamics, is the “learn on
the fly" method of Csanyi et al. (2004), in which spatially local potentials are repeatedly
refit, after each set of a few MD steps, to density functional theory results. In the case of
infrequent-event systems, it should be possible, for example, to take advantage of the longresidence time in a particular energy basin to fit a highly accurate “basin-specific"
interatomic potential for each state the system visits.
The final issue we will note is the trend towards increasingly parallel, and increasingly
specialised, computer architectures, which offer both new opportunities and new
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challenges. Machines at the exascale, anticipated to be less than five years away, willprobably have ~108 processing cores, each running slower than present-day processors tokeep the total power requirements under control. In addition to the challenge of harnessing
this many processors effectively without becoming hopelessly communication bound, thenonstandard architectures expected for these systems will make programming a non-trivial
task. We have begun to see this added difficulty in programming cell processors (Germann,
Kadau, Swaminarayan, 2009) and GPUs (Anderson, Lorenz, Travesset, 2008), forexample. On the other hand, the payoff can be substantial. Moreover, the increasing easewith which specialised processors can be designed and manufactured for specific tasks
offers the opportunity for co-designed computers and algorithms that achieve an efficiency
and speed that is not possible on general purpose machines. The Anton (Shaw et al., 2008)
computer, designed specifically to perform ultra-fast MD simulations for biologicalsystems, is an excellent example of this. It is capable of reaching more than a millisecondon realistic biomolecules. One could envision a similar project with the aim of ultra-fastsimulation of materials systems, and additional speed might be gained if the characteristics
of infrequent-event methods are directly incorporated.

5.9. Advances from 2013 to 2018
Since the time this manuscript was written in 2013, there have been a few noteworthy
advances in the methods, which we very briefly describe in this section. For a review ofsome recent applications, see references: (Zamora et al., 2016a; Perez, Uberuaga, 2015;
Ervin, Xu, 2018).
An approximate version of the hyperdynamics method has been developed, designed tomaintain constant boost for large systems. In standard hyperdynamics, because making thesystem larger means there are more escape pathways, the system is more frequently broughtnear a dividing surface, a condition requiring that the bias potential go to zero, as discussed
above. Consequently, for any valid form of bias potential, the boost in hyperdynamics
decays towards unity as the system is made very large. In local hyperdynamics (Kim, Perez,
Voter, 2015), a bias force is defined locally, rather than globally, to circumvent thisproblem. Although this approach is approximate, it gives strikingly accurate results, andopens the door to massively parallel hyperdynamics simulations on systems with millions
or billions of atoms.
A version of the TAD method exploiting speculative parallelisation (SpecTAD) has beendeveloped (Zamora et al., 2016a). The key concept here is that any time an attemptedtransition to a state j is observed during the high-temperature basin constrained MD in statei, a TAD simulation on a separate processor can be initiated in state j, on the chance that
this transition to state j will be the one that is ultimately accepted. This spawned process is
continued until/unless it becomes clear that state j is no longer a candidate for the acceptedtransition. SpecTAD can give substantially higher boost factors than TAD alone under theright conditions, as can be seen in a study of the radiation-damage-relevant defects in
MgAl2O4 (Zamora et al., 2016b).
A spatially parallelised version of ParRep has been developed (Martinez, Uberuaga, Voter,2014), using the same sort of sublattice approach mentioned above for spatially parallelised
TAD (Shim et al., 2007). For a certain range of system sizes and processor counts (e.g. 106
atoms, 107 processors, total defect event rate 105 s-1), this approach gives a substantiallygreater parallel speedup than is available from either direct ParRep or spatially parallel MD.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Additionally, a generalisation of parallel replica dynamics with greater parallel efficiency,
parallel trajectory splicing (ParSplice), has been developed (Perez, Huang, Voter, 2018).
In ParSplice, trajectory segments are generated in many states at the same time – any known
state becomes a candidate for execution of one or more segment-generating processes.
Building on the quasi-stationary distribution theory discussed in Le Bris et al., (2012), a
trajectory segment is “spliceable” if it begins by spending at least a correlation time (ordephasing time) in one state, and similarly ends with a full correlation time in one (same ordifferent) state. Segments are then spliced together end to end – i.e. a segment that ends ina particular state can be spliced to another segment that begins in that same state – to obtainan MD-accurate overall trajectory. Using information from the observed behaviour of thesystem, one can build a Markovian statistical model for predicting where new trajectory
segments will be most likely needed as the system evolves. This model improves theefficiency of ParSplice, but has no impact on the accuracy. For systems whose kinetics arecharacterised by superbasin trapping, ParSplice can give a substantial speedup comparedto ParRep (Perez, Huang, Voter, 2018). ParSplice has been effectively applied to systems
with radiation-induced disorder (Perriot et al., 2017) to fusion materials (Perez et al., 2017),
and to metallic nanoparticles (Huang et al., 2017).
Finally, AMD techniques have started to evolve beyond the original goal of generating
individual state-to-state trajectories towards the efficient generation of higher-scale models
of long-time evolution. A recent example of such an approach is TAMMBER (temperatureaccelerated Markov models with Bayesian estimation of rates) (Swinburne, Perez, 2018).
In TAMMBER, the objective is to use parallel-computing resources to most efficiently
create a Markovian (KMC-like) model of the state-to-state dynamics while precisely
accounting for the effect of the incompleteness of the model due to the finite amount of(A)MD that was used to parameterise it. The computational engine that underpins
TAMMBER is a combination of ParSplice and TAD: short burst of high-temperature MD
are carried out in a distribution of states chosen so as to optimally increase the validity timeof the model at some lower temperature (the amount of time a trajectory is expected to
evolve through known transitions before, statistically, a transition that is not yet part of themodel would occur). While the validity time is a global quantity, it can be inferred using
estimates of the contribution of the yet-unobserved pathways to total state-wise transition
rates; in TAMMBER, this is carried out using a Bayesian formalism that exploits the
Poisson statistics of first-escape times. This formalism also allows for the propagation of
the uncertainties to observables calculated from the model.

5.10. Conclusion
Since their introduction 20 years ago, the AMD and aKMC methods have proven useful ina variety of situations, including some relevant to nuclear energy materials, where thetimescales of interest are out of reach of direct molecular dynamics and where the kinetics
are too rich to be adequately described with a limited list of pre-determined pathways.
When the activation barriers between the different states are high relative to the thermalenergy, any of the AMD methods can yield colossal accelerations, providing a view of
atomistic dynamics over unprecedented timescales. Further, by leveraging the particularstrength of each of the methods, or by generalising and combining them with othertechniques, an even wider variety of situations can be efficiently simulated. Nonetheless,there also remain a large number of systems where persistent low barriers severely limit
the available computational boost. In the previous section, we identified this as the greatest
limitation of the methods and the most important frontier for future development. Two
additional important areas for development are designing methods to provide fast and
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100 │ NEA/NSC/R(2019)2
accurate forces to guide the dynamics, perhaps by coupling with electronic structure
methods and exploiting the properties of the AMD and aKMC methods; and harnessing thefull power of specialised computer architectures and the upcoming exascale parallelism.

Acknowledgements
Work at Los Alamos National Laboratory (LANL) was supported by the US Departmentof Energy (DOE), Office of Basic Energy Sciences, Division of Materials Science, by the
LANL Laboratory Directed Research and Development programme, and by the Center for
Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center
funded by the US Department of Energy, Office of Basic Energy Sciences. LANL is
operated by Los Alamos National Security, LLC, for the National Nuclear Security
Administration of the US DOE under Contract No. DE-AC52-06NA25396.

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6. Kinetic Monte-Carlo methods
1

2

María J. Caturla and Roger E. Stoller
1
Dept. Física Aplicada, Facultad de Ciencias, Fase II, Universidad de Alicante, Spain
2
Materials Science and Technology Division, Oak Ridge National Laboratory, United States

6.1. Introduction
The different kinetic Monte-Carlo methods that are most widely used in the field of nuclear
materials are described in this chapter. These methods allow for calculations of defectevolution in time for time and length scales that can not be reached by atomistic models
such as molecular dynamics simulations. Therefore, the results of these calculations takeus closer to the experimental time and length scales. However, they require informationfrom other more fundamental models or from experiments. The KMC methods included inthis paper are classified as: atomistic kinetic Monte Carlo, object kinetic Monte Carlo andevent kinetic Monte Carlo. In the last section improved models such as parallel algorithms
are described.
One of the challenges of multi-scale modelling in nuclear reactor materials is to be able topredict microstructure evolution of irradiated materials for times that expand the lifetimeof the reactor. The intrinsic difficulty in developing these models lies in the fact that timescales from picoseconds to years must be studied. The initial damage produced by theenergetic particles, the so-called cascade damage, lasts only a few tens of picoseconds andextends only a few nanometers. However, it must be accurately described since the initial
distribution of damage is crucial for the later damage evolution. Molecular dynamics (MD)simulations with empirical potentials, have been very successful in modelling this initialstage of damage production. However, these models can only reach time scales ofnanoseconds and length scales of nanometers. For a description of these models see
Chapters 2 to 4 in this issue.
Kinetic Monte Carlo (KMC) models on the other hand, are able to simulate the evolution
in time of a set of objects that have associated certain events with known probabilities ofoccurrence. Unlike in MD simulations, the vibrations of each atom around its equilibrium
position is not modelled, and much longer time scales can be achieved. However, KMC
requires information of the types of events that can occur as well as the probability for eachone of these events. This information is obtained in different ways depending on thespecific KMC approach used. We will describe three different KMC methods: atomistic
kinetic Monte Carlo (AKMC), object kinetic Monte Carlo (OKMC) and event kinetic
Monte Carlo (EKMC). Firstly, a general background of these models will be given.

6.2. Background
KMC methods follow the kinetics of slow processes or rare events. Unlike MD simulations
where time scales of only a few nanoseconds can be achieved since the minimum time stepis on the order of femtoseconds, in KMC methods time scales of seconds or hours can bereached. This allows for direct comparison with experimental measurements, since it is at
the right time and also length scale. One of the first KMC simulations were performed byBeeler (1966), followed by those of Besco (1967) and Doran (1970, 1972) all in the field
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of radiation effects, since they studied short-term annealing of defects in face centred cubic(fcc) and body centred cubic (bcc) materials. Since then, different algorithms based on the
KMC approach have been developed and applied to many different fields.
The KMC method is based on knowing the transition rates between events. And thesetransition rates are obtained from energy barriers between two different states of the system.
For example, the migration of an atom from one lattice position, 1, to a neighbouring vacantsite, 2, will be driven by an activation energy as shown schematically in Figure 6.1. Thisimplies that all the possible barriers for the different transitions of the system to be studied
must be known. Generally, transition state theory (TST) is used to obtain the rates between
two different states (Marcelin, 1915). For a detailed and in-depth description of TST and
KMC see the review by A. Voter (2007).
Figure 6.1. Schematic diagram of the activation barrier for an atom in position 1 to move toa vacant site (position 2)

Source: Caturla et al., 2019

Once the activation energy is known, the rate for the event, Γe , is then obtained considering:Γ e = Γ e0 exp(− Ea /KT )(1)

where Γe0 is the jump or attempt frequency, Ea is the activation energy for that particular
event, (in the example above, a migration energy), K is Boltzmann's constant and T is thetemperature.
Once all the different type of events and event rates are known, the KMC evolves thesystem in time. The residence-time algorithm or Bortz-Kalos-Liebowitz (BKL) algorithm(Kalos, 1986) is used both in OKMC and in atomistic or lattice KMC. The algorithm
proceeds by selecting one event from all possible, with the appropriate weight for each
event. First, the total rate for all events is calculated as:

R= ∑ Γ e Ne

(2)that is the sum over all events of the probability of each event, Γe, times the number ofobjects that can undergo that event, Ne. An event is chosen randomly between 0 and R,
therefore ensuring that each event is weighted by the appropriate probability of occurrence.
Figure 6.2. shows a diagram of the total rate and the event picked for a particular simulation
step.

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108 │ NEA/NSC/R(2019)2
Figure 6.2. Total rate as the sum of individual rates of event up to n events

Source: Caturla et al., 2019

The time of the simulation is then increased by ∆t:Δt=− log ξR

(3)

where R is the total rate given by equation (2) and ξ is a random number between 0 and 1,
that is used to give a Poisson distribution of the time. Once the event has been selected, arandom particle is chosen from all those that can undergo that event. The particle is thenmoved and the total rate has to be computed again for the next simulation step.
A different KMC algorithm for advancing the time is used in the so-called EKMC and firstpassage kinetic Monte Carlo (FPKMC) methods, as explained later in this chapter.
However, both methods use, as in the case of AKMC and OKMC, a set of tabulated rates
for all possible events in the simulation.

6.3. Atomistic kinetic Monte Carlo
In AKMC atoms are fixed in a rigid lattice (Young, Elcock, 1966). Events are usually the
migration of an atom from a lattice position to a neighbouring position, considering theappropriate activation energies for migration. For further details on this method see
references (Bellon, 2003; Soisson 2006, 2007, 2010).
This method is used very successfully to study processes such as precipitation orsegregation in alloys driven by point defects (Soisson, 2006). It is also widely used to study
thin film deposition and growth (Gilmer et al., 2000; Huang, Gilmer, Diaz de la Rubia,
1998). AKMC is often also known as lattice kinetic Monte Carlo (LKMC) as in reference(Khrushcheva et al., 2003).
AKMC has a high computational cost, since the jump of every single atom is followed in
the simulation. That limits the total time that can be reached as well as the system size thatcan be modelled. However, it has other advantages with respect to the OKMC that will be
described below.

6.4. Object kinetic Monte Carlo
OKMC started to be widely used in the field of radiation damage after the work of H.
Heinisch in the early 90s (1990). Unlike in the case of AKMC, not all atoms are described,
but only the objects of interest. In the case of radiation, these objects are the defects
produced during the irradiation, that is vacancies, self-interstitials, impurities and theirclusters. The events these objects can perform are diffusion events, dissolution from acluster, interaction between different defects or defects with other objects such as grain
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boundaries or dislocations. The probabilities of these events are given by the migrationenergies and binding energies of the defects. For example, the probability of a defect oftype i undergoing a migration event is given by:Γ im = Γ i0 exp (− Eim/ KT )(4)

where, as mentioned above, Γi0 is the jump frequency, Eim is the migration energy for thatparticular defect, K is Boltzmann's constant and T is the temperature. When a migration
event is selected the object is moved a distance δ , the jump distance, which is often selected
between first and second nearest neighbours. When the object can migrate in any direction(three-dimensional migration) the jump is performed by randomly placing the object withina sphere of radius δ. When the migration of the object is restricted to one particular direction(one-dimensional migration), such as the case of self-interstitial clusters in metals, aparticular direction of motion (<111> for Fe and <110> for Cu, for example) with respect
to the simulation box is given to the object when it is created, and the jumps are performed
only along that direction and with a distance δ.
The probability of a defect of type i undergoing a dissolution event from a cluster is givenby:Γ id = Γ i0 exp(− (Eim+Eib )/ KT )(5)

where Eib is the binding energy of the defect to the cluster. This energy depends on thenumber of defects in the cluster.
The initial conditions of the simulation are the (x,y,z) co-ordinates of those defects
produced by the irradiation as well as their type. In case of a continuous irradiation, newdefects are introduced in the simulation box with a rate according to the dose rate of theexperiment that is being simulated. The positions and types of defects are obtained frommolecular dynamics, from binary collision approximation (BCA) calculations, such as
those obtained from SRIM (SRIM-ref) or Marlow (MARLOW-ref), or as a random
distribution of Frenkel-pairs, depending on the type of calculation. For example, whendamage is produced by electrons the last approximation can be used (Fu et al., 2005). In
the case of damage produced by light ions such as He, calculations using the binary
collision approximation are appropriate. However, for self-irradiation and heavy-ions it isnecessary to use those results obtained from MD simulations. Often times a combinationof BCA and MD calculations is used to obtain the distribution of defects during irradiationfor energies that can not be reached by MD alone. In this case the BCA is used to obtain
the energies of those recoils produced by the energetic particle along its path, but the final
defect distribution produced by those recoils is the one obtained from MD simulations. Thisapproximation is based on the existence of a threshold for sub-cascade formation.
Most of the OKMC calculations for radiation effects in metals describe the objects as points
in the simulation box with a capture radius that depends on the number of defects of that
object. This capture radius is normally defined spherical as:(6)
where n is the number of defects in the cluster and Ω is the atomic volume. This captureradius is used to define when two defects interact. Also when a defect dissolves from a

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cluster it is positioned outside this capture radius. When using this approach information
regarding the lattice structure is lost. It is however possible to keep the location of every
defect during the calculation with the consequent increase in memory for the calculation.
This is in fact done in models of dopant diffusion in silicon (Johnson, Carturla, Diaz de la
Rubia, 1998). Strain effects such as the bias interaction between interstitials anddislocations can be included in this capture radius, increasing the capture radius forinterstitials. However, it is also possible to include strain effects in OKMC using elasticity
theory (Sun et al., 2000; Hudson et al., 2005). Note that in the AKMC method describedabove there is no need to define a capture radius since all atoms, although in a rigid lattice,
are included in the calculation.
As mentioned above, the system evolves in this KMC following the residence-timealgorithm. Examples of applications of this method can be found in (Domain, Becquart,
Malerba, 2004; Becquart et al., 2010; or Caturla et al., 2000 among many others).

6.5. Event and first-passage kinetic Monte Carlo
Another type of algorithm to evolve the time in a kinetic Monte Carlo model is the oneused by the so-called EKMC and the FPKMC methods. Like in AKMC or OKMC thesemethods require a list of event rates. However, the evolution in time of those events is
performed differently. In these KMC models the time delay for all the events in thesimulation is calculated first. And the event with the shortest-time delay is picked first.
That event is then performed, and all the new time delays have to be calculated again for
the next simulation step. The time advances by the delay time of the event picked. Thisalgorithm is used in codes such as JERK (Lanore, 1974; DallaTorre et al., 2005) and has
been used very successfully to compute the electrical resistivity of irradiated Fe (Fu 2005)
and Fe in the presence of impurities such as carbon (Jourdan, Crocombette, 2012). In the
EKMC method some approximations are made in order to calculate the delay time for
events such as the interaction between to neighbouring objects. For more details on EKMC
see references (DallaTorre et al., 2005; Jourdan, Crocombette, 2012; Becquart et al., 2010).
A general method, without any approximations, was recently developed by Opplestrup et
al (2006, 2009) named FPKMC. In this method, as in the EKMC algorithm, the event
selected is the one that would occur in the shortest time from all possible events. However,FPKMC lacks the approximations included in EKMC. In FPKMC, each one of the walkers
is surrounded by a “protective domain” and when a walker is selected it is moved to theedge of that domain. That results in a tremendous computational gain with respect to
OKMC since many small jumps, that would have to be done in OKMC where the jump
distance is fixed, are automatically avoided in FPKMC. However, the efficiency of
FPKMC decreases significantly with respect to OKMC when the number of particles ishigh, since for each step, all the times associated to the jump of each walker to the edge of
their “protective domains” have to be calculated. Some applications and comparisons
between FPKMC and OKMC can be found in Donev et al. (2010).

6.6. Other advanced models
In recent years there has been considerable efforts to improve the KMC. In particular, there
is an increased interest and need to produce more accurate values for the energy barriers ofthe different states, especially in the case of OKMC. One of the great advancements in thisarea is what is known as on-the-fly KMC. The basic idea of this method is to compute therates of the specific processes at the same time as the kinetic algorithm is evolving. That is,

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the event rates are not tabulated before the KMC calculation starts. This is particularly
important for those systems where the type of events possible are very large and it is notpossible, a priori, to know or define every possible scenario. This is, for example, the caseof defect diffusion in alloys, in particular, in concentrated alloys, where the rate of aparticular reaction (a vacancy jump) will depend on the local environment.
The methodologies used to implement an on-the-fly KMC algorithm differs between
different groups and is adapted to the type of problem that needs to be solved. Probably the
first on-the-fly KMC model is the one by Henkelman and Jónsson (2001). The authors use
the dimer method (Helkelman, Jónsson, 1999) to obtain the saddle points between different
states and construct an event catalogue. Stress-assisted diffusion of hydrogen in iron has
been studied (Ramasubramanian et al., 2008) combining on-the-fly calculations of barriers
using empirical potentials with pre-calculated barriers with more accurate density
functional theory. For other examples of on-the-fly KMC see reference (Blackwell et al.,2013; Robinson et al., 2012; Vernon et al., 2011; El-Mellouhi, Mousseau, 2008).
A new method named self-evolving atomistic kinetic Monte Carlo (SEAKMC) also uses
an on-the-fly determination of reaction saddle points and energy barriers in a KMC model(Xu, Osetsky, Stoller, 2011). Longer simulation times can be reached in SEAKMC because
it focuses the defect analysis on defined “active volumes” which are regions that encompass
the defects of interest. The atomistic fidelity of SEAKMC to the underlying physics has
been demonstrated in direct comparisons with MD simulations and the ability of SEAKMC
to reach long times has identified new phenomena (Xu, Osetsky, Stoller, 2012; Xu,
Osetsky, Stoller, Terentyev, 2013). In addition, when the systems to be studied are very
complex, such as in defect evolution in concentrated alloys, special methods have beendeveloped to produce the catalogue of transition rates in a more efficient way. One suchmethod consists of using artificial neural networks to predict the values of the energy
barriers (Castin, Malerba, 2010).
The application of KMC simulations in more and more complex systems also requires
efficient methods of solving the KMC algorithm. Nowadays, with the existence ofsupercomputers, parallelisation would seem an obvious way to boost the time in thesecalculations. Parallelisation is relatively simple in an EKMC or FPKMC algorithm.
However, this is not the case in OKMC. In the OKMC algorithm described above the total
rate for all events, R, must be computed at each time step. And in order to compute this rate
all the events must be known. This would mean that all nodes should know of all events atevery time step, therefore making the parallelisation very inefficient. Recently, Martinez etal. (Martinez, 2011) have developed a synchronous parallel algorithm that, unlike previous
attempts, solves the same master equation as the serial algorithm. In this case, the total rateof all events on each processor is kept fixed for all processors by including null events.
SEAKMC has been parallelised using a hybrid OpenMP and MPI scheme.

6.7. Conclusions
KMC algorithms are useful simulation tools to follow the evolution in time of a set of
events of known probability of occurrence. They can reach time and length scales that allow
for direct comparison to experiments, and therefore, for validation of the models. In fact,
KMC can be used for validation of other, more fundamental results, such as DFT
calculations of migration energies and binding energies of defects (see for example Fu etal., 2005).

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112 │ NEA/NSC/R(2019)2
One of the major disadvantages of KMC is the need to know all possible events for aparticular system, as well as all rates associated to those events. This can only be achieved
in simple systems. However, the systems of interest for nuclear applications are never
simple, they are often multicomponent alloys with almost unavoidable impurities.
Fortunately, an increasing number of groups are developing new methodologies to tackle
these problems such as on-the-fly KMC algorithms, where the rates are not catalogued but
computed on-the-fly by different methods, including predictive models such as those basedon artificial neural networks.
Besides the effort to increase the accuracy of the KMC method for a particular problem,
there are also significant advances in terms of speeding up the KMC algorithm itself
without using approximations. Such is the case of FPKMC, which can be extremely
efficient for low radiation doses (low defect concentrations) or parallel OKMC.
In the view of this author, the development of KMC methods for radiation damage
applications will come from different fronts. On-the-fly methods are very recent andpromising developments, provided they can be implemented in an efficient manner to reachreasonable time and length scales. On the other hand, one important effect that is lacking
in KMC calculations for radiation damage applications is an efficient way of computing
defect diffusion including elastic interactions. These interactions are extremely importantto understand, for example, loop interactions in metals or dislocation-defect interactions,particularly under strain. This requires a combination of KMC and dislocation dynamics
calculations. Some examples already exist in the literature including one single dislocation(Wen, Takahashi, Ghoniem, 2009) but further developments are needed.
Besides the development of methods and algorithms, having more groups in the radiation
field using KMC models for real applications would also help improve the field. Even
though the KMC algorithm is fairly simple to implement, developing an efficient code,
capable of modelling many different conditions and many different species, is not straightforward. However, unlike for other simulation methods such as MD, there are few KMC
programmes available to researchers, at least in the case of radiation damage. An effort inthis respect is being done at the IMDEA Materials Institute in Madrid (Martín-Bragado etal., 2013) for AKMC and OKMC. Further developments along this line for other KMC
algorithms such as FPKMC or parallel OKMC would also be desirable. On the other hand,as mentioned above, standard KMC simulations rely on a set of parameters as input data.
Researchers could also benefit from a well-structured data base of parameters properly
documented. Such a project is also currently taking place at CETA-Ciemat 3. Finally, one
would expect that, in the spirit of the multi-scale modelling methodology often used insimulations of radiation damage, the combination of all these algorithms (AKMC, on-thefly KMC, FPKMC and OKMC), together with cluster dynamics, could reach the time andlength scales needed to accurately predict the effects of radiation.

Acknowledgements
We thank Dr Ignacio Martín-Bragado for reviewing and helpful comments on this chapter.

3. www.ceta-ciemat.es
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7. Dislocation dynamics
1,3

2

3

A. I. Topuz , N. V. Luzginova and E. van der Giessen
1
Materials Innovation Institute (M2i), Netherlands;
2
Nuclear Research and consultancy Group (NRG), Netherlands;
3
University of Groningen, Netherlands

7.1. Introduction
Dislocation dynamics (DD) is a methodology that, in terms of length and time scales, sits
in between atomistics and continuum plasticity. It is powerful for problems in which there
are too many dislocations to be treated by molecular dynamics (MD), but too few
dislocations for continuum plasticity to be applicable. DD applies to both face centred cubic(fcc) and body centred cubic (bcc) materials, by the application of appropriate slip systems
and constitutive rules.
As illustrated in Figure 7.1., DD requires input from MD or kinetic Monte Carlo (KMC)
etc. in terms of material parameters to describe dislocation evolution and microstructural
information, such as grainsize and defect density. Using this, it predicts quantities such as
critical resolved shear stress or hardening rate, which may serve as input for a continuum
plasticity description, or entire stress-strain curves. Also, it potentially may serve as aroutine to describe plastic deformation during other phenomena, such as crack growth or
fatigue.
Figure 7.1. Illustration of the position of DD in multi-scale modelling

Source: Topuz et al., 2019

This review will start with a brief review of the underlying assumptions of DD, thetheoretical models used in setting up the framework and its numerical most popular
implementation. Subsequently, we will describe how the method has been applied in thelast two decades to study the influence of irradiation-induced defects on plasticity. For this,we will first consider unit events, with individual dislocations interacting with individual
defects, and then present the predictions of plastic deformation as the result of the collective
behaviour of multiple dislocations.
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7.2. Dislocation dynamics: Methodology
7.2.1. Continuum description of dislocations
While dislocations are line defects in the atomic lattice, at the lengthscale of DD the atoms
are averaged out to an elastic continuum and the dislocations are retained as discrete
entities. A dislocation loop on a crystallographic plane is characterised by the tangent
vectoras a function of the local co-ordinate s along the loop, and the Burgers vectorof the displacement discontinuity across the plane, , inside the loop, see Figure 7.2.a.
Figure 7.2. Geometry of a single dislocation (a) and top view on a slip plane (b) showing two
discretisation schemes of a dislocation: edge-screw (top right) and nodal (bottom left)

Source: Topuz et al., 2019

In DD, the state of stress and strain associated with each dislocation is a continuum
elasticity problem. Most DD models adopt linear elasticity, in which case the dislocationline itself becomes a singularity. Closed-form analytical solutions currently exist only forspecial cases. For straight dislocations (or dislocation segments) the ones in an infinitely
large, isotropic elastic medium are best known (see, e.g. Hirth, and Lothe (1973)).
Comparison with atomistic simulations has shown that the continuum solutions areaccurate at distances of 5~6 times the magnitude of the Burgers vector, b, away from thecore of the dislocation. These elastic fields incorporate the long-range interaction between
dislocations but generally do not account for finite dimensions of the crystal. The
incorporation of traction-free boundary conditions is classically done by means of image
dislocations, but since 1995 a versatile method for general boundary-value problems isavailable that exploits superposition in linear elasticity (Van der Giessen, Needleman,
1995; Weygand et al., 2002).

7.2.2. Constitutive rules
Effects that occur on atomic length scales are not incorporated in the elastic stress fields
and need to be formulated in the form of constitutive rules. Such short-range effects include
the friction stress; dislocation mobility; annihilation of dislocation segments of opposite
character; the interaction of a dislocation with an obstacle or with a grain boundary; thenucleation of a dislocation (or its multiplication in two dimensions (2D)), etc. All of theseare essentially governed by the Peack-Koehler force. Line tension can be incorporated
either directly through the lengthening of a dislocation or through its local curvature. Crossslip of screw dislocations is generally treated within the framework of the Friedel-Escaig
model. Constitutive rules tend to be relatively simple and involve material parameters
whose value is obtained from smaller-scale computations, typically MD simulations, or
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from experiments. The difference between implementations for fcc versus bcc crystals is
the number of slip systems and the appropriate constitutive rules.

7.2.3. Numerical issues
A DD simulation involves an incremental time-marching algorithm comprising twoelementary steps: (1) update the dislocation structure, (2) determine the interactive forcesin the updated situation. Changes in the dislocation structure are governed by the PeachKoehler force along the dislocation line, which is a configurational force. It is computedfrom the stress field of all dislocations, the image (or applied) stress and line tension.
Neglecting the mass attributed to a dislocation line, the dislocation structure is generally
evolved in a quasi-static way.
Three classes of analyses can be identified: (a) real three-dimensional (3D); (b) projected
2D; (c) cross-sectional 2D. In (c), dislocations are assumed to be straight, infinitely longedge dislocations (with) or screw dislocations (with), so that the problem is truly
2D. In both (a) and (b), dislocations are considered to be loops on a plane, but in (b) only asingle slip plane is being considered. In either case, dislocations are discretised (see Figure
7.2.b.): either as a string of curved (parametric representation) or straight segments (nodal
representation), or in terms of segments that are either edge or screw. In practice, adaptive
re-meshing of the loop is used in order to capture large changes in curvature.

7.2.4. Input for dislocation dynamics for irradiation-induced defects
Theoretical or computational approach begins with ab-initio electronic structure
calculations which determine basic defect properties such as formation, binding andmigration energies of point defects or small defect clusters. In the next stage, MD can
describe especially primary damage production. Also, the properties of larger defects likevoids and nanoscopic particles are acquired from MD simulations. However, defect andsolute diffusion as well as clustering require larger periods (seconds) and distances(micrometrers) with respect to MD. Hence, the information obtained from ab-initio and
MD is introduced to KMC as input data. KMC plays an important role in informing DD
about the 3D spatial distribution, the number and the size of dislocation loops and defects(Wirth et al., 2004). Besides these computational strategies, experimental approaches liketransmission electron microscopy (TEM) can be used in order to get spatial distributions,volume fraction, average diameter, and density of defects (Bako et al., 2007).

7.3. Interaction between dislocations and defects
Metals inevitably contain defects as a consequence of production. The defects induced byirradiation as they emerge at the sizescale of individual dislocations, depend on many
processing conditions but also on crystallography. In fcc materials, stacking fault tetrahedra(SFT) are a predominant form of damage in low stacking fault energy (SFE) materials likeCu, while in high SFE fcc materials interstitials give rise to Frank sessile loops (Osetsky etal., 2006). Frank loops serve as hardening particles in bcc crystals as well, but in thesematerials irradiation also gives rise to voids. In this section, we summarise the state of theart in the DD modelling of the interaction between a single dislocation and these defects.

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Figure 7.3. Dissociated dislocation with stacking fault (grey area) under an applied shear
stress in the vertical direction interacting with small precipipates in Cu, according toprojected 2D DD simulations

Source: Adapted from Hardikar et al., 2001

The modelling of SFT’s requires a representation of partial dislocations, characterised by
the stacking fault energy γ. This energy level scales inversely with the separation distance
between adjoining Shockley partials. Hardikar et al. (2001) proposed a simple scheme byrepresenting two partial dislocations as individual dislocations that apart from long-range
elastic energy interact with each other via the energy cost corresponding to the stacking
fault they bound. They determined the value of γ by fitting the separation distance to thevalue predicted by molecular statics (through the quasicontinuum method). An example ofthe profile of the dislocation, showing leading and trailing edges, is shown in Figure 7.3.
Figure 7.4. Pathway of formation of an SFT from a triangular Frank loop (solid line)
through dissociation into a Shockley partial (dashed) and a stair-rod dislocation (dotted)

Source: Martinez et al., 2008

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A more generic approach was introduced recently by Martinez et al. (2008) in which
segments of a dissociated dislocation are characteried explicitly by the quantity , inaddition to and . Here, is the unit normal to the slip plane with opposite directions
for the two partials. This allows for a seamless unified treatment of partials and full
dislocations (for which) as well as Frank loops (a partial edge dislocation withnot contained in the slip plane). Traditionally, Frank’s energy criterion is used to govern
the dissociation of dislocations, but within the framework of DD these authors have shownthat it is more convenient to adopt a power dissipation criterion. Using their methodology,
Martinez et al. (2008) elucidated how the stacking fault contained in the Frank loop is
unfaulted in a way that naturally leads to a SFT, as shown Figure 7.4. The method is likely
to be amenable to study the interaction between a dislocation and an SFT, as done by meansof MD by Wirth et al. (2001), but has not been performed to the best of our knowledge.
Interstitials or vacancies are the results of fast neutron irradiation; as individual atomic
defects they are in the realm of MD or KMC, but once they cluster and collapse they canbe regarded as sessile prismatic dislocation loops. The stress field of a planar prismatic loop
is given in closed-form in (Khraishi et al., 2001) or in more general 3D configurations canbe computed in much the same way as for glide dislocations, as demonstrated, for example,
by Novokshanov and Roberts (2009). Figure 7.5. shows snapshots of a DD simulation of
the passage of a mobile dislocation over an 80nm prismatic loop, which can cross-slip as
well as glide. Depending on the height at which the mobile dislocation enters, twointeraction mechanisms were obtained leading to different strengths and a different
character of the retained prismatic loop. These simulations have also shown that randomly
distributed loops are weaker ‘obstacles’ than loops that are decorating dislocations andthereby locking them.
Figure 7.5. Interaction between a mobile edge dislocation and an inclined interstitial
prismatic loop

Source: Novokshanov and Roberts, 2009

The interaction between a moving dislocation and a dispersoid or another reinforcing
particle can be described by DD provided it is sufficiently large. Various ways to describe
the direct interaction between the dislocation and the particle can be found in the literature.
A rather sophisticated method is to add an explicit repulsive force field between particle
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and dislocation, as proposed by Bako et al. (2007), which will pin the dislocation against
the particle. This applies to incoherent and unshearable particles. For such particles, a
simpler approach, which is particularly popular in 2D projected simulations, is to represent
the particle geometrically as an impenetrable boundary. In this approach, as a dislocation
folds around the particle, it is easily seen that the line tension is the most dominant
contribution to the force on the particle, see Figure 7.6. Introduction of a maximum valueof this force is, in effect, an extra rule which allows to model shearable particles.
Figure 7.6. Simple geometrical interaction between a dislocation and a particle

Source: Adjanor et al., 2010

Voids are shearable particles in ultimate form. While it can be justified to neglect themismatch in elastic properties between the crystal and reinforcing particles, the treatmentof voids requires proper account of the traction-free conditions on the void surface. Hafez
Haghigat et al. (2009) have done so by making use of superposition in conjunction with afinite element method. Confrontation of the stress field around a 2nm void as induced byan edge dislocation at some distance away from it with MD simulations confirmed that thiseffect can indeed be captured by elasticity, see Figure 7.7. The internal free surface of thevoid attracts the dislocation, both as it moves towards the void and when it needs to breakaway from it. It is this interaction that gives rise to the void effectively acting as a particle,
with a strength that depends on the lattitude of the slip plane (Hafez Haghighat et al., 2009).
Figure 7.7. Stress field around a void induced by an edge dislocation according to DD

Source: Hafez Haghiaghat et al., 2009
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7.4. Collective behaviour: What can dislocation dynamics predict?
The emergent phenomenon caused by the collective behaviour of moving dislocations isplasticity. DD is geared to predicting plasticity on the basis of knowledge regarding theinteraction with individual defects and other microstructural features such as precipitates,
grain boundaries, twins and laths. There is a large body of literature on DD plasticity,
mostly dedicated to stress-strain behaviour, including characteristics like yield strength andhardening rate, but with limited yet increasing attention for the role of plasticity inembrittlement. For space reasons, in the following we concentrate entirely on ways inwhich irradiation-induced defects have been represented in such studies and some salient
findings.

7.4.1. 3D simulations
Fully 3D computations require a large amount of input data, including the position of initialdislocations and defects as well as their size and shape. For this reason (probably), moststudies so far have focused on the effect of a single type of defect at a time.
A very detailed study has been performed by Bako et al. (2007) in which the distributionof dispersoids in an Fe alloy were obtained from TEM images. The critical resolved shear
stress (CRSS) was computed as the applied stress at which a screw dislocation propagatedthrough the obstacle field (see Figure 7.8. just before the critical stage). The dislocation
was discretised using the nodal method, the dispersoids were assumed to be spherical andthe interaction with the dislocation treated as mentioned in the previous section. Hundred
different realisations of dispersoid distributions were analysed in order to get statistics on
the predicted CRSS, which translates into the initial yield strength via the Schmid factor.
Figure 7.8. Snapshot of a screw dislocation moving through a 3D distribution of dispersoids,
occasionaly leaving behind small Orowan loops

Source: Bako et al., 2007

In a similar spirit, Shin et al. (2009) performed simulations to study the increase of the yieldstrength of an fcc crystal due to spatial distribution of Frank loops. However, a simplified
interaction rule was used: each loop was treated as an obstacle with a barrier strength
proportional to the loop radius and independent of the impact location (that is, in contrast
with Figure 7.6.). As the final step of a multi-scale analysis, Diaz de la Rubia et al. (2000)
reported a similar DD simulation but now including both Frank loops and SFT’s. These
computations were the first to explain the development of dislocation-free channels. In the

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studies mentioned above the various defects were incorporated each by their own
representation. Robertson and Gururaj (2011) recently proposed a method in which both
hard precipitates and sessile loops were treated by approximating their shapes with facets,
see Figure 7.9.
In so far as DD studies go beyond the initial yield stress and predict stress-strain curves it
is important to note that in most cases (and certainly the ones cited here) the computations
are load controlled. This means that a uniform macroscopic stress is increased
incrementally and that strain at each instant is computed. Sometimes only plastic strain iscomputed from the area swept by the mobile dislocations (e.g. Robertson and Gururaj(2011)), sometimes the elastic part of the strain is added by application of Hooke’s law(e.g. Kraishi et al. (2001)). This kind of computation is notably different, though, from adisplacement or strain driven tensile test which gives rise to stress drops rather than strain
jumps. Simulation of a displacement driven test requires the solution of a boundary-value
problem (e.g. Weygand et al. (2001)).
Figure 7.9. (a) Simulation model for the effect of particles and sessile dislocation loops on
plastic deformation in ferritic grains, by means of (b) a unified facet representation of hardparticles or sessile loops

Source: Robertson and Gururaj, 2011

7.4.2. 2D simulations
The 3D computations mentioned above are able to capture many of the details of theinteractions between dislocations and (irradiation-induced) defects, but they are limited by
the number of dislocations that can be handled. Many more dislocations and larger strains
can be achieved with 2D simulations where all dislocations are either pure edge or purescrew, since dislocation lines reduce to points, but, obviously, the detailed interactions are
lost. In fact, 2D obstacles are imaginary entities that, somehow, are to represent the (spatial)
distribution of real defects and their variation in strength. Likewise, a 2D nucleation site is
the projection of a pinned segment that serves as a Frank-Read source. Nevertheless, withappropriately tuned rules, it has been possible, for example, to accurately capture the size
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dependence of the stress-strain response of thin polycrystalline films (Shishvan and Van
der Giessen (2010)). In this cited paper, the source parameters were fit from experimentsbut, in principle, it should be possible to extract them from 3D (or projected 3D)computations (in the spirit of the suggestion made by Hardikar et al. (2001)). To the bestof the authors’ knowledge this latter approach has not yet been applied to study the
influence of irradiation-induced defects.
Durgaprasad et al. (2011) have presented an experimentally-based implementation of 2D
DD simulations for irradiated materials (see Figure 7.10.). All defects were represented bypoint obstacles whose (“locking”) strength is assumed to obey a Gaussian distribution. The
value of the average strength was related by these authors to the radiation fluence via theconfrontation of single-crystal simulations with experimental yield strength values. For
simplicity, only single slip was considered and shear loading was strain controlled. The
grain boundaries in this work are assumed to be impenetrable and because of thisinteraction give rise to a significant Hall-Petch effect.
Figure 7.10. Shear-induced distribution of edge dislocations in a 2D polycrystalline material
containing randomly distributed obstacles with a normal distribution of strengths torepresent irradiation-induced defects

Source: Durgaprasad et al., 2011

7.5. Limitations, needs and challenges
Just like any computational model, DD has restrictions; some of these are inherent, anddefine the range of applicability of the model, others are limitations that are of morepractical nature and may be overcome in future implementations. We close the exposition
with improvements and extensions that are needed, especially for nuclear materials.

7.5.1. Inherent limitations•

DD only applies to phenomena occurring on length scales. Below this, thedislocation core becomes important, and this cannot be captured by linear elasticity.

•

The quality of the predictions is controlled by the extent to which physicalmechanisms are represtented by constitutive rules and by their accuracy. In
particular, the modelling of cross-slip and climb is not as robust as the desciptionof dislocation glide.

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The glide velocity of a dislocation needs to be smaller than the shear wave speed,since the stress fields adopted for individual dislocations are for quasi-static
dislocations.

7.5.2. Practical limitations (at this moment)•

The fields for individual dislocations and dislocation segments are well known interms of isotropic elasticity. In truly 2D computations the fields based on cubic
isotropy are also known, yet somewhat more cumbersome. Elastically anisotropic
fields in 3D, however, are only known formally in integral form, which makes themuseless.

•

The computing time increases rapidly with number of dislocations because theirlong-range interactions involves anproblem. Fast multipole methods havebeen developed to remedy this, but the break-even point lies at rather large numbersof dislocation segments.

•

The necessary information content about the initial microstructure, including
defects and the distribution of their properties, is very large and, in most cases,
unknown. It is common to generate random microstructures based on known global
measures such as densities, but these microstructures may not be realistic or evenattainable in practice. An alternative approach therefore is to perform DD
simulations of a preceding deformation process in order to obtain, for example,
more realistic dislocation structures that serves as initial microstructure for thedesired computation. At any rate, multiple realisations need to be analysed in orderto obtain statistically meaningful results.

•

The accuracy with which boundary conditions can be incorporated in 3D is quite
limited. Superposition methods with a finite element solution are versatile but rely
quite sensitively on the mesh size, especially where dislocations intersect
boundaries (cf. Weygand et al. (2001)). The treatment of free surfaces recently
proposed by Tang et al. (2006) is accurate, but restricted to traction-free boundary
conditions.

7.5.3. Needs•

Interfaces are known to have a significant effect on dislocation plasticity. Grainboundaries, for instance, can contribute significantly to hardening (leading to thewell-known Hall-Petch grain size effect). Till this date, DD studies of the effect ofradiation on yield and hardening have been carried out (almost) exclusively on
single crystals, which, at least, raises questions about the relative contribution ofirradiation-induced damage. It has been demonstrated in purely 2D studies thatgrain boundary effects can be incorporated to a first approximation in DD byassuming the boundaries to be impenetrable (e.g. Balint et al. (2008), Shishvan,
Van der Giessen (2010), Durgaprasad et al. (2011)) and refined grain boundarymodels are under construction. This work should be extended to 3D.

•

The interaction between dislocations and defects has been modelled at varying
levels of sophistication, but have all ignored the elastic mismatch between the metal
crystal and, for instance, particles. The influence of this mismatch may be small but
this can only be established once a method is available to incorporate it.

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As a further improvement of the previous item, there is a need for methods to study
the propagation of a dislocation through a shearable particle, as a function of its
strength, size and shape.

•

All DD studies referenced in the above have assumed that new dislocations are
generated by multiplication through the Frank-Read process. Although relevant,especially for single crystals, this ignores the possibility of nucleation of
dislocations from (free) surfaces and interfaces. The need for models of dislocation
nucleation is quite generic, and not less so in order to capture the possible
enhancement of nucleation from defects such as voids, grain boundaries and otherinterfaces as well as He bubbles.

•

By the very nature of DD, thermal effects cannot be represented directly but haveto be captured in constitutive rules. For some processes, like cross-slip, this has
been done by means of a probabilistic approach. However, while it is known that
thermal activation is key in the evolution of microstructures, to the best of theauthors’ knowledge this has not been incorporated yet in any DD model. As
dislocation crossing of obstacles is a dominant phenomenon in irradiated materials,
the incorporation of thermal acitivation seems to be of high priority.

•

To this date, DD studies have focused almost exclusively on the effect of irradiation
damage on plastic deformation, viz. yield and hardening, while the effects on
fracture properties are likely to be of equal importance from a nuclear engineering
point of view. Hence, irradiation embrittlement is still a relatively unexplored area.
Plastic properties, of course, play an important role in fracture, but there is no
simple one-to-one mapping. Especially in many popular nuclear materials which
possess rather limited plasticity, fracture is phenomenon that necessitates a multiscale approach (cf., e.g. Odette et al. (2003)). While the interaction between adislocation and a stationary crack has been investigated to quite some detail, there
have been only a handful of DD-based simulation studies of crack growth. The
recent work by Noronha and Ghoniem (2007) is the first attempt to study the effectof irradiation on the ductile-to-brittle transition temperature, but by using a very
simplified model. There is a lot of room for improvement here.

References
Adjanor, G. et al. (2010), “Overview of the RPV-2 and INTERN-1 packages: From primary damage tomicroplasticity”, J. Nucl. Mater. Vol. 406, pp. 175–186.
Bako, B. et al. (2007), “Discrete dislocation dynamics simulations of dislocation interactions with Y2O3
particles in PM2000 single crystals”, Phil. Mag, Vol. 81, pp. 3 645–3 656.
Balint, D. et al. (2008), “Discrete dislocation plasticity analysis of the Hall-Petch effect”, Int. J. Plasticity,Vol. 24, pp. 2 149–2 172.
Diaz de la Rubia, T. et al. (2000), “Multiscale modelling of plastic flow localization in irradiatedmaterials”, Nature, Vol. 406, pp. 871–874.
Durgaprasad, P.V. et al. (2011), “Application of dislocation dynamics to assess irradiation effect onmaterials with different grain sizes”, Nucl. Engrg. Design, Vol. 241, pp. 603–607.
Hafez Haghighat, S.M. et al. (2009), “Dislocation–void interaction in Fe: A comparison between molecular
dynamics and dislocation dynamics. J. Nucl. Mater., Vol. 386–388, pp. 102–105.

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Hardikar, K., V. B. Shenoy and R. Phillips (2001), “Reconciliation of atomic-level and continuum notions
concerning the interaction of dislocations and obstacles”, J. Mech. Phys. Solids, Vol. 49, pp. 1951–
1967.
Hirth, J.P. and J. Lothe (1968) Theory of Dislocations, McGraw-Hill, New York.
Khraishi, T.A. et al. (2001), “Modelling of irradiation-induced hardening in metals using dislocation
dynamics”, Phil. Mag. Lett., Vol. 81, pp. 583–593.
Martinez, E. et al. (2008), “Atomistically informed dislocation dynamics in fcc crystals”, J. Mech. Phys.
Solids, Vol. 56, pp. 869–895.
Noronha, S.J. and N. M. Ghoniem (2007), “Brittle–ductile transition in F82H and effects of irradiation”,
J. Nucl. Mater., Vol. 367–370, pp. 610–615.
Novokshanov, R. and S. Roberts (2009), “3D Dislocation dynamics modelling of interactions between
prismatic loops and mobile dislocations in pure iron”, J. Nucl. Mater. Vol. 386–388, pp. 64–66.
Odette, G.R. et al. (2003) “Cleavage fracture and irradiation embrittlement of fusion reactor alloys:
mechanisms, multiscale models, toughness measurements and implications to structural integrity
assessment” J. Nucl. Mater., Vol. 323, pp. 313–340.
Osetsky, Y.N., D. Rodney, and D.J. Bacon (2006), “Atomic-scale study of dislocation–stacking fault
tetrahedron interactions. Part I: mechanisms”, Phil. Mag., Vol. 86, pp. 2 295–2 313.
Robertson, C. and K. Gururaj (2011), “Plastic deformation of ferritic grains in presence of ODS particles
and irradiation-induced defect clusters: A 3D dislocation dynamics simulation study”, J. Nucl. Mater.,Vol. 415, pp. 167–178.
Shin, C., J. Kwon, and W. Kim (2009), “Prediction of radiation-induced yield stress increment in austenitic
stainless steels by using a computational approach”, J. Nucl. Mater., Vol. 386–388, pp. 610–612.
Shishvan, S.S. and E. Van der Giessen (2010), “Distribution of dislocation source length and the size
dependent yield strength in freestanding thin films”, J. Mech. Phys. Solids, Vol. 58 pp. 678–695.
Tang, M. et al. (2006), “A hybrid method for computing forces on curved dislocations intersecting freesurfaces in three-dimensional dislocation dynamics”, Model. Simul. Mat. Sci. Engrg., Vol. 14, pp.
1 139–1 151.
Van der Giessen, E. and A. Needleman (1995), “Discrete Dislocation Plasticity: A Simple Planar Model”,
Model. Simul. Mat. Sci. Engrg., Vol. 3, pp. 689–735.
Weygand, D. et al. (2002), “A. Aspects of boundary-value problem solutions with three-dimensional
dislocation dynamics”, Model. Simul. Mat. Sci. Engrg., Vol. 10, pp. 437–468.
Wirth, B.D. et al. (2001), “Mechanical property degredation in irradiated materials: A multiscale modelling
approach”, Nuclear Instruments and Methods in Physics Research B, Vol. 180, pp. 23–31.
Wirth, B.D. et al. (2004), “Multiscale modeling of radiation damage in Fe-based alloys in the fusion
environment”, J. Nucl. Mater., Vol. 329-333, pp. 103–111.
Zinkle, S.J. (2005), “Fusion Material Science: Overview of challenges and recent progress”, Physics ofPlasmas, Vol. 12, 058101.

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8. Introduction to the phase-field modelling technique: A primer on the
Allen-Cahn and Cahn-Hilliard models
M. J. Welland
Computational Techniques, Canadian Nuclear Laboratory, Canada

8.1. Introduction
Phase-field models have emerged as a robust modelling approach to understand and predictmicrostructure evolution with complex interface morphology including topology changes(Allen, Cahn, 1972; Cahn, Hilliard, 1958; Provatas, Elder, 2010; Steinbach, 2013; Bellon,
2012; Chen, 2002, Boettinger et al., 2002). Models of this type can quantitatively simulate
phenomena that involve phase changes and/or interfacial energy such as fracture (Borden
et al., 2012; Bourdin et al., 2008; O’Connor et al., 2016, Chakraborty et al., 2014), grainboundary movement (Kennouche et al., 2016; Zhang et al., 2016; Tonks et al., 2014 ;Ahmed et al., 2017), bubble evolution (Chakraborty et al., 2014; Millet et al., 2012; Li etal, 2010), melting/solidification (Welland et al., 2009; Böhler et al., 2014), and nano-scale
phenomena (Welland et al., 2015; Ulvestad et al., 2015), some of which are shown in Figure8.1. Models may be derived in a thermodynamically self-consistent fashion, integrated withthermodynamic databases, and used to demonstrate kinetic phenomena such as dendritic
solidification and solute trapping. They are readily coupled with other continuum-scaletransport equations such as heat and mass transport, elasto-mechanics, surface effects, etc.
This document is an introduction to the phase-field modelling concept and discussion of
the canonical Allen-Cahn and Cahn-Hilliard models. The theory refers to both equilibrium
thermodynamics and transport processes and a basic familiarity of both is assumed. Several
comprehensive reviews have been written on the current state of phase-field models(Provatas, Elder, 2010; Steinbach, 2013; Bellon, 2012; Chen, 2002, Boettinger et al., 2002)
to which the reader is referred for a more in-depth discussion. Similarly, there are several
review articles on phase-field modelling of irradiation damage (Devanthan et al., 2010; Liet al., 2017; Tonks et al., 2017).
Phase-field models represent the local phase at a point in space by a field variable which
varies smoothly and continuously in space between phases, representing a diffuse interface.
Contrasted to sharp interface models, this approach captures complex interface morphology
in a robust fashion while naturally including interfacial energy which dominates many
nano- and microscopic processes (Welland, Lewis, Thompson, 2008). However, thisrobustness has substantial computational expense, requiring small mesh sizes in typicalnumerical solver techniques, as the interface, which is typically very thin, must be
resolvable in the system.
Phase-field models are typically derived at the mesoscale which is characterised as theintermediate scale between the description of material as discrete components and acontinuum. This scale is large compared to atomic distances and includes enough discrete
components to be treated statistically such that thermodynamic properties such astemperature, concentration, pressure, and phase are robustly defined (Steinbach, 2013).
However, the scale is small compared to other characteristic, macroscale properties suchas the size of the system of interest, and the gradients of such thermodynamic properties.

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Figure 8.1. Example applications of phase-field modelling(a): Evolution of intergranular fission gas bubbles in U-Mo for calculation of the effective thermal
conductivity. (b): Intergranular fracture with the presence of bubbles along the grain boundary. (c): Noncongruent melting of (U,Pu)O2 fuel in laser flash experiments. (d): Hydrogen accomodation in PdH
nanocubes compared to experimental results.

(a)

(b)

(c)

(d)

Source: (a) Hu et al., 2015, (b) Chakraborty, Tonks, Pastore, 2014, (c) Böhler et al., 2014, (d) Ulvestad et al.,2015

The governing equations for the phase-field models, along with any other coupled physics,
is derivable through the Theory of Irreversible Processes, also known as Non-equilibrium
Thermodynamics. Through the governing equations, the system is allowed to evolve so as
to generate entropy based on the energy of the system as a whole and is represented by asystem of coupled partial differential equations (PDE). This derivation provides a robust
mathematical link between equilibrium thermodynamics and the time-dependant evolutionof the system with multiple coupled phenomena such as heat and mass transport in thepresence of non-congruent phase change (Welland et al., 2009; de Groot, Mazur, 1984;
Welland, 2012).

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Both the Allen-Cahn and Cahn-Hilliard models have non-uniform, diffuse interface
solutions controlled by two factors which together determine the interfacial energy andwidth. The first factor deals with the local state of the material within the diffuse interface
which is not in either of the equilibrium phases and therefore has a non-optimal energy.
The second factor is related to the relative state of the surrounding material and ismathematically captured by the inclusion of a gradient energy term in the free energyfunctional of the system, which is sometimes referred to as the Ginzburg-Landau functional(Cahn, Hilliard, 1958; Provatas, Elder, 2010). Minimisation of the excess and gradientenergies leads to shrinking and widening of the diffuse interface respectively resulting in aminimal, spatially-dependant profile with a characteristic interface width, and a non-zero
interfacial energy.

8.2. The Allen-Cahn model
The Allen-Cahn equation, also referred to as the non-conserved phase-field model or ModelA, is useful for representing a first-order phase transformation such as
melting/solidification and evaporation/condensation (Allan, Cahn, 1972; Provatas, Elder,2010; Hohenberg, Halperin, 1977). These transformations feature a discontinuoustransition in the first derivative of the free energy, e.g. a latent heat of transformation.
A set of state variables are introduced akin to reaction co-ordinates for the phase transition,
and are not conserved quantities. Historically, these variables were envisioned as
representing the ordering / symmetry of the phase. This ordering is schematically depicted
in Figure 8.2. in which the average atomic density is periodic to the left and becomessmooth to the right corresponding to the transition of, e.g. a solid to a liquid. In modern
applications it is common to treat the phase-field variables as abstract ‘phase indicators’ tospatially represent the local phase and disregard potential physical meaning (Steinbach,2013).
Figure 8.2. Schematic representation of a one-dimensional (1D) average atomic density field

Note: The figure illustrates a crystal-like ordering on the left and fluid-like disordering on the right with adiffuse interface between them. The phase-field variable 𝜙𝜙 can be envisioned as the amplitude of the oscillation.
Source: Welland, 2019

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A fundamental example of the Allen-Cahn model is a two-phase material, which requires
only a single phase-field variable 𝜙𝜙. The free energy of the system is represented as theintegral over the system volume, 𝑉𝑉, of the local free energy density and additional terms
present in the interface,
𝐹𝐹 = �

𝑓𝑓(𝜙𝜙, … )�����𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝜖𝜖 2+ 𝑊𝑊𝜙𝜙 2 [1 − 𝜙𝜙]2 + |∇𝜙𝜙|2 𝑑𝑑𝑑𝑑 ,�����������������
2(1)

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

where 𝑓𝑓(𝜙𝜙, … ) is the phase-dependent free energy density, and the ellipses indicate otherstate variables such as concentration, temperature, etc. The first interfacial term is a doublewell potential which has minima at 𝜙𝜙 = 0,1. The form of this term is not unique and othermodels are in use such as the double-obstacle function (Steinbach, 2013). The interfacial
terms are controlled by parameters 𝑊𝑊 and 𝜖𝜖 whose product and ratio are related to theinterfacial energy and width respectively as described below.
An interpolation function, 𝑝𝑝(𝜙𝜙), which interpolates between single-phase properties, canbe envisioned as the local volume fraction of a phase. This function also ensures each phaseis stable with respect to infinitesimal variations in 𝜙𝜙. A simple and commonly used form
for 𝑝𝑝(𝜙𝜙) is,𝑝𝑝(𝜙𝜙) = 𝜙𝜙 3 [6𝜙𝜙 2 − 15𝜙𝜙 + 10],which satisfies 𝑝𝑝(𝜙𝜙 = 0) = 0, 𝑝𝑝(𝜙𝜙 = 1) = 1, and

(2)

𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
(𝜙𝜙 = 0) =(𝜙𝜙 = 1) = 0. The
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

free energy density of the multiphase system can then be interpolated between those of thetwo pure phases, 𝑓𝑓 𝛼𝛼 and 𝑓𝑓 𝛽𝛽 as,𝑓𝑓(𝜙𝜙, … ) = [1 − 𝑝𝑝]𝑓𝑓 𝛼𝛼 (… ) + 𝑝𝑝𝑓𝑓 𝛽𝛽 (… ).

(3)

The system may be visualised as in Figure 8.3. where the free energy curves for the 𝛼𝛼 and𝛽𝛽 phases are shown as functions of the concentration, 𝑐𝑐. The optimal, equilibrium state is
the lowest common tangent of the curves which imply the partitioning of species between
phases to 𝑐𝑐 𝛼𝛼 and 𝑐𝑐 𝛽𝛽 . The phase function 𝑝𝑝 is then the distance along the lowest common
tangent (tie line).

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132 │ NEA/NSC/R(2019)2
Figure 8.3. Free energy curves for coexisting phases showing the tie line between equilibrium
concentrations

Source: Welland, 2019

The change in the energy of the system with respect to a change in the phase-field profile
is given by the functional derivative,𝛿𝛿𝛿𝛿 𝜕𝜕𝜕𝜕
𝜕𝜕(𝜙𝜙 2 [1 − 𝜙𝜙]2 )=+ 𝑊𝑊− 𝜖𝜖 2 ∇2 𝜙𝜙.
𝛿𝛿𝛿𝛿 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕In equilibrium, the variation of 𝐹𝐹 with 𝜙𝜙 must be zero,

(4)

𝛿𝛿𝛿𝛿
= 0. Furthermore, for a planar𝛿𝛿𝛿𝛿

system in equilibrium with two coexisting phases, the driving force for local phase𝜕𝜕𝜕𝜕transformation is also zero,= 0. The shape of 𝜙𝜙 is then controlled by the interfacial𝜕𝜕𝜕𝜕

terms,𝑊𝑊

𝜕𝜕(𝜙𝜙 2 [1 − 𝜙𝜙]2 )− 𝜖𝜖 2 ∇2 𝜙𝜙 = 0,𝜕𝜕𝜕𝜕which has a solution with the normal co-ordinate 𝑥𝑥,
Where𝜙𝜙(𝑥𝑥) =1 1
𝑥𝑥+ tanh ,
2 2
2𝑑𝑑

𝜖𝜖 2
𝑑𝑑 = �
,
2𝑊𝑊

(5)

(6)

(7)

and is referred to as the interfacial width. Equation (6) is plotted in Figure 8.4. with the sizeof a factor of 𝑑𝑑 indicated.
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 8.4. Demonstrative phase-field profiles and excess energy
Above: equilibrium solution with the interface width shown. Below: Corresponding excess energy which
integrates to the interfacial energy.

Source: Welland, 2019

Inserting the equilibrium profile in equation (6) into the energy functional in equation (1),
we can determine the total free energy of the system including the interface. The difference
between this energy and that of a system without the interfacial terms is identified as theinterfacial energy, 𝜎𝜎,1 𝜖𝜖 2 𝑊𝑊𝜎𝜎 = �
,3
2(8)

and is shown as the shaded area in Figure 8.4.
The phase-field parameters in equation (1) are related to the material properties of interface
width and energy by inverting equations (7) and (8),𝜖𝜖 2 = 6𝜎𝜎𝜎𝜎,

𝑊𝑊 = 3𝜎𝜎⁄𝑑𝑑.

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(9)
(10)

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Finally, there is also a phase-field parameter related to the kinetics of phase change. Since
the phase-field variable is non-conserved, it is driven directly from the reduction of freeenergy. The governing equation for this parameter is simply,
𝜕𝜕𝜕𝜕
𝛿𝛿𝛿𝛿
= −𝑀𝑀𝜙𝜙
,
𝜕𝜕𝜕𝜕
𝛿𝛿𝛿𝛿

(11)

where 𝑀𝑀𝜙𝜙 is a mobility term which can be related to interface attachment kinetics (Karma,
1996), and the variational derivative is given in equation (4).

The Allen-Cahn formula can be generalised to multiphase systems through the introductionof additional phase-field variables, 𝜙𝜙1 , 𝜙𝜙2 … and the modification of the interfacial terms
in equation (1) to include multi-well potentials and other gradient energy terms ( Nestler,
Garcke, Stinner, 2005; Zhang, Steinbach, 2012; Welland, Wolf, Guyer, 2014; Choudhury,
Nestler, 2012). Other thermodynamic variables may be added to this model, such as
multicomponent diffusion, displacement variables to represented elasticity and plasticity,and electromagnetic properties (Zhang, Steinbach, 2012; Welland, Wolf, Guyer, 2014;
Choudhury, Nestler, 2012). In the case of coupling between phase transitions and anotherstate variable such as concentration, it is required that equation 11) be coupled to therelevant transport equation; i.e. for a binary system undergoing solidification / melting, theAllen-Cahn equation should be coupled to a mass balance equation.

8.3. The Cahn-Hilliard model
The Cahn-Hilliard model does not introduce an additional order parameter, but instead
relies on a free energy potential which is concave in a state variable, commonly
concentration, along with a gradient energy term for the same variable (Cahn, Hilliard,
1958). The state variable is usually conserved, and hence this model is also sometimes
referred to as the conserved phase-field model, or Model B.
The free energy functional of a canonical Cahn-Hilliard equation is,
𝐹𝐹 = � 𝑓𝑓(𝑐𝑐, … ) + 𝜅𝜅|∇𝑐𝑐|2 𝑑𝑑𝑑𝑑 ,(12)where 𝜅𝜅 is the gradient energy coefficient which acts on the concentration field, in contrast
to the phase-field variable as in the Allen-Cahn equation. The free energy density isrepresented by 𝑓𝑓(𝑐𝑐, … ), an example of which is shown in Figure 8.5. which corresponds toa binary, two-phase system with a miscibility gap in the phase diagram between the stablebulk phases at concentrations 𝑐𝑐 𝛼𝛼 and 𝑐𝑐 𝛽𝛽 . The equilibrium solution varies continuously
between 𝑐𝑐 𝛼𝛼 and 𝑐𝑐 𝛽𝛽 as indicated by the tie line, but in this case the material between theequilibrium phases still lies on the free energy curve. Therefore, there is an excess energy
enclosed between the tie line and the free energy curve which serves the same function as
the double-well potential in the Allen-Cahn equation.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 8.5. Demonstrative free energy profile for a Cahn-Hilliard model.

Note: The tie line shows the equilibrium miscibility gap between concentrations 𝑐𝑐 𝛼𝛼 and 𝑐𝑐 𝛽𝛽 . The difference
between the tie-line and the free energy curve is shown as Δ𝑓𝑓 and penalises material between phases. Alsolabelled are the metastable region in which classical nucleation needs to occur to trigger the phase change, andthe unstable region wherein infinitesimal fluctuation of composition can trigger spinodal decomposition.
Source: Welland, 2019

Since the excess energy described above depends on the free energy profile and is not of astandard form, the equilibrium profile across the interface does not in general have ananalytical solution. Practically however, the double-well potential is a reasonableapproximation and the concentration profile resembles the hyperbolic tangent solution inequation (6).
It is possible to obtain an estimate of the interfacial width, through consideration of theequilibrium solution to equation (12). Since this equation does not depend explicitly on thespatial co-ordinate, the equilibrium solution must satisfy the Bertrami Identity and it canbe shown that,Δ𝑓𝑓 = 𝜅𝜅|∇𝑐𝑐|2 ,(13)

for which the integration constant can be eliminated, given that far from the interface bothΔ𝑓𝑓 and ∇𝑐𝑐 are zero. One can then estimate the interface width as (Cahn, Hilliard, 1958),d=

𝑐𝑐 𝛽𝛽 − 𝑐𝑐 𝛼𝛼,max(∇𝑐𝑐)𝜅𝜅.
= �𝑐𝑐 𝛽𝛽 − 𝑐𝑐 𝛼𝛼 ��max(Δ𝑓𝑓)(14)

The governing equation for the evolution of the phase is simply that of the corresponding
conserved quantity, in this case the concentration. However, since the free energy of thesystem is now a functional of that quantity, the thermodynamic driving force for mass flux
must be the functional derivative,
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136 │ NEA/NSC/R(2019)2
𝛿𝛿𝛿𝛿
,
𝛿𝛿𝛿𝛿
𝜕𝜕𝜕𝜕
− 2𝜅𝜅∇2 𝑐𝑐.
=
𝜕𝜕𝜕𝜕

μ=
The mass flux is then,

𝐽𝐽⃗ = −𝑀𝑀∇𝜇𝜇,

(15)

(16)

where the mobility 𝑀𝑀 is related to the diffusivity of the species. The mass balance, which
includes the movement of the interface, is then,∂c= −∇ ⋅ 𝐽𝐽⃗.
∂t

(17)

The Cahn-Hilliard equation is a fourth order PDE, which can be problematic for the Finite
Element Method numerical solvers. Therefore, it is common to split this equation into two
second-order PDEs, equations (15) and (7), which is then solved as a coupled system.

8.4. Phase stability, decomposition and nucleation
A key difference between the Allen-Cahn and Cahn-Hilliard equations is in phasenucleation/decomposition. Mathematically, these processes are related to the stabilities of
their solutions to infinitesimal fluctuations. Physically, these processes can be triggered bythermal fluctuations, radiation or other, typically stochastic processes which can be
introduced into simulations to some degree.
Classical nucleation theory requires the embryo of a new phase to be of a sufficient size
such that the liberation of energy from the transformation exceeds the interfacial energy
incurred between the original and new phases. Since the embryo is larger than the scale ofthe interface it must be fully resolved within phase-field models. The random fluctuationmust therefore be of a size comparable to the critical nucleus size. This may be
implemented through Langevin noise terms in the governing equations, or ad hoc
placement of nuclei according to some physical observation (Welland et al., 2009;
Simmons et al., 2004; Jokisaari, Permann, Thornton, 2016).
This nucleation barrier allows both models to represent non-equilibrium scenarios such as
undercooling. Although the formation of an embryo poses difficulty in phase-field models,
the stability of an embryo is well captured and complicating phenomena such as latent heat
evolution and depletion zones can be simulated once an embryo is introduced (Jokisaari,
Permann, Thornton, 2016). While the standard Allen-Cahn model always requires anembryo of a critical size, this is only true in the Cahn-Hilliard model for the convex regionsof the free energy density as shown in Figure 8.5.
For a system with a concentration in the concave region of Figure 8.5., spinodal
decomposition can occur. The phases decompose spontaneously, splitting the average statevariables between them which, due to the concavity of the free energy profile, results in alower overall energy. Since the interfacial energy is related to the free energy profile, itdoes not prevent this decomposition. Mathematically, this result follows from Von
Neumann stability analysis applied to the stationary points of equations (11) and (17) whichshows that infinitesimal fluctuations, as may occur due to numerical error, are energeticallyfavourable and therefore grow without a barrier.
STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

8.4.1. Solution of the phase-field equations
Both the Allen-Cahn and Cahn-Hilliard equations are PDEs and, with the exception of afew simple cases, require numerical solutions. This may be accomplished through standard
real-space numerical methods such as the Finite Element, or Finite Difference methods.
Such real-space methods are a mature and common approach and offer adaptive meshrefinement techniques which are useful for resolving the interface over larger spatial lengthscales (Greenwood et al., 2018). Spectral methods, in which the solution is achieved in
Fourier space, offer excellent convergence, however require a regular mesh and impose
periodicity of the domain of interest although this may be overcome through the use of‘embedded domains’ (Chen, 2002).
In order to resolve the interface shape, and accurately capture variation of the system on
the scale of the interface, it is necessary to have a mesh be a fraction of the interface widthd. The actual interface width of a system may be measured experimentally through, e.g.
transmission electron microscopy, and is typically nanometers in scale. When structures ofinterest are many times larger than a realistic interfacial width, the simulations may become
prohibitively computationally expensive. Therefore, it is common to exaggerate d to belarger than the real interface width (capillary length), but still small compared to the featuresize of the problem. In some cases, this can result in exaggerated non-equilibrium effects
such as solute trapping, which may be compensated for through the introduction of an antitrapping term to the mass flux equation (Karma, 2001). While a larger interface width
results in a more computationally tractable problem, the sensitivity of the solution to the
choice of d then must be checked.

8.5. Connection to equilibrium thermodynamics
There is a close connection between phase-field models and CALPHAD (Computer
Coupling of Phase Diagrams and Thermochemistry). CALPHAD includes databases ofthermodynamic data for pure and solution phases from which phase diagrams may beobtained through free energy minimiser software. The results of CALPHAD calculations
are typically for bulk systems and insensitive to mesoscale thermodynamic effects.
Integration with phase-field models in a robust fashion rooted in the Theory of Irreversible
Processes, introduces interfacial energy and kinetic / non-equilibrium effects and therefore
provides an avenue for exploration of these phenomena.
Thermodynamic potentials are introduced directly into the free energy functional throughthe free energy density terms in equations (1) and (12). However, there remain two main
issues to be addressed: how the model discovers the optimal state of a new phase during
nucleation, and how the interfacial energy is implicitly related to the thermodynamics ofthe coexisting phases.
As discussed above, nucleation in either the Cahn-Hilliard or Allen-Cahn model involves
an energetic barrier associated with interface creation, which must be overcome by theliberation of energy by the phase transformation. In this case, information about the stateof the new phase must be known beforehand from the phase diagram or a coupled freeenergy minimiser. As an example, in a multicomponent system the species concentrations
within the new phase must be known in order to know when the phase change should
initiate.
For Cahn-Hilliard models within the spinodal decomposition region, the concavity of thethermodynamic potential allows the system to discover the optimal conditions by itself.
The interfacial energy is determined from the thermodynamic potential as the distance
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between the energy curve from the tie line. Through this model it is also straightforward toinclude other sources of interfacial energy, such as coherent elastic strain, wherein thestrain varies continuously through the interface (Welland et al., 2015; Ulvestad et al., 2015).
However, this definition of the interfacial energy is only strictly valid near the critical pointof the miscibility gap where the concavity disappears (Steinbach, 2013).
In the Allen-Cahn model, excess energy is added explicitly through the double-well term.
It may then be desirable to avoid implicit energy contributions so that they can be entirely
controlled explicitly. This becomes challenging within the interfacial region where, e.g. theconcentration is varying between equilibrium states. One approach to eliminating implicit
energy contributions is the interface dissipation method pioneered by Steinbach et al. where
the concentrations of each species in each phase is represented as unknown variables, anduses an ‘interface dissipation’ process to transfer mass between phases. This method has
the drawback of an increasing number of unknown variables, but is able to accommodatea wide variety of expressions for the pure phase potentials (Steinbach, 2013; Zhang,
Steinbach, 2012).
Alternately, a composite potential, 𝑓𝑓(𝜙𝜙, … ), may be defined which combines the purephase potentials weighted by their phase fractions. Early models considered theconcentrations to be equal in all constituent phases throughout the interface (Wheeler,
Boettinger, McFadden, 1992),
𝑓𝑓(𝜙𝜙, 𝑐𝑐) = [1 − 𝑝𝑝]𝑓𝑓 𝛼𝛼 (𝑐𝑐) + 𝑝𝑝𝑓𝑓 𝛽𝛽 (𝑐𝑐).

(18)

In this model, the pure phases are being evaluated at concentrations away from theirequilibria which implies variation in the chemical potential throughout the interface, andan excess energy associated with neither phase being in the equilibrium state. This is
illustrated in Figure 8.6. where the free energy curves along with the lowest common
tangent (tie line) are depicted, along with the composite potential showing a rise in energy
along the tie line.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 8.6. Demonstrative plots of a 2 phase binary system
Top: Free energy potentials with the global minimum indicated by the lowest common tangent construction.
Bottom left: The composite profile with equal concentrations, showing the excess energy along the green
lowest common tangent line. Bottom right: The Grand Potential formulation of the composite potential whichis flat along the lowest common tangent line, indicating the lack of implicit interfacial energy.

Source: Welland et al., 2017

More recently, a technique, generalised as the Grand Potential approach, was derived on
the premise that the species chemical potentials are equal through the interface, which
implies each constituent phase is evaluated at its equilibrium state,𝑓𝑓(𝜙𝜙, 𝑐𝑐) = [1 − 𝑝𝑝]𝑓𝑓 𝛼𝛼 (𝑐𝑐 𝛼𝛼 ) + 𝑝𝑝𝑓𝑓 𝛽𝛽 �𝑐𝑐 𝛽𝛽 �,𝜕𝜕𝜕𝜕𝜕𝜕𝑓𝑓𝛼𝛼(19)𝜕𝜕𝑓𝑓𝛽𝛽subject to the equality of the chemical potentials, = 𝛼𝛼 = 𝛽𝛽 . This approach maintains
𝜕𝜕𝑐𝑐
𝜕𝜕𝑐𝑐
𝜕𝜕𝜕𝜕
a constant chemical potential through the interface and eliminates the excess energy.
However, a problem with the approach is how to determine the phase-concentrations 𝑐𝑐 𝛼𝛼
and 𝑐𝑐 𝛽𝛽 that satisfy the chemical potential criterion. This may be done through a nested nonlinear solver or through coupling with a free energy minimisation routine (Nestler, Garcke,
Stinner, 2005; Choudhury, Nester, 2012). Alternately, if the equilibrium points are knowna priori, the composite function may be evaluated approximately though, e.g. a local
quadratic approximation of the free energy curve of the minimiser of the composite scheme(Choudhury, Nestler, 2012; Welland, Tenuta, Prudil, 2017).

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8.6. Summary and future directions
Phase-field models have shown versatility and robustness in their applications to a wide
variety of phenomena. Such models are effective in capturing interfacial energy andcomplex interface morphologies and can be coupled with other multiphysics simulations
such as heat and mass transport, elasticity, fluid flow, etc, in a thermodynamically selfconsistent fashion. Efficient integration with CALPHAD thermodynamic potential
databases is an ongoing area of development, especially with regards to understanding theimplicit effects on interfacial energy. Adaptive mesh refinement is also an active andimportant area of development which holds promise for scaling-up the reach of phase-fieldmodels by reducing computational expense. The diffuse interface concept has been applied
to other models such as the included-phase model which offer efficient implementations
for a subset of phenomena (Prudil, Welland, 2017). There exist a number of phase-field
implementations in a variety of platforms and there have been recent attempts to create
benchmark problems to compare these implementations (Steinbach, 2013; Jokisaari et al.,2017; Schwen et al., 2017).

Acknowledgements
The author would like to thank Dr Veena Tikare and Dr Nana Ofori-Opoku for reading thismanuscript.

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O’Connor, D.T et al. (2016), "Phase transformation and fracture in single Lix FePO4 cathode particles: aphase-field approach to Li-ion intercalation and fracture", Model. Simul. Mater. Sci. Eng., Vol. 24,
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Prudil, A.A. and M.J. Welland (2017), "A novel model of third phase inclusions on two phase boundaries",
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9. A review of the rate theory of defect clustering in irradiated materials
Dariush Seif and Nasr M. Ghoniem
University of California, United States

Abstract
9.1. Introduction and background
Clustering of atomic defects leads to changes in the microstructure of materials, and hence
induces drastic variations in their properties. We present here a review of the theory ofatomic defect clustering under non-equilibrium conditions, particularly encountered during
irradiation of materials with energetic particles. These conditions are met in a wide rangeof technical applications, ranging from nuclear and fusion energy to microelectronics andsurface engineering. Rate theory methods, which dwell in the intermediate region linking
the atomic and macro-scales, provide a useful tool in determining the evolution of irradiatedmicrostructures at experimentally observable time and length scales.
In this chapter, we begin with a brief description of the cascade-damage environment andthe physics of point defects. Classical nucleation theory is then presented in the context ofpoint defect clustering. This method is shown to be adequate for obtaining nucleation rates
for stable clusters with or without internal solute gas generation. To obtain average cluster
densities, rate theory methods are utilised. Starting from a basic understanding of stochastic
fluctuations in defect fields, it is shown that one can formulate master equations. When thetransition probabilities can be replaced by average macroscopic reaction rates, it is shownthat a deterministic set of rate equations for the concentrations of specific defect cluster
sizes is sufficient. Following this idea, a reduced-set rate theory model is presented andshown to describe materials under irradiation with high helium generation. Next, acontinuum Fokker-Planck approximation is shown for cases when atomic clusters grow or
shrink by single atomic defects. When internal sink densities are high, the interaction ofpoint defects with sink stress fields is known to cause a substantial imbalance in the loss of
vacancies and interstitials to sinks. This bias is typically implemented in rate theory models
to gaugege the effect of point defect segregation on void swelling and creep. We presentan ‘effective medium’ rate theory approach to calculate dislocation bias factors in metals
using point defect-dislocation interactions based on properties obtained from atomistic
simulations. By incorporating point defect fluxes, spatial resolution is attained in the model.
The chapter concludes with an assessment of the current state and future of rate-theorymodels.
Clustering of atomic species in materials is a process that embodies a rich variety ofphysical mechanisms, and which has significant implications to a wide range of material
technologies. Strong deviations from thermodynamic equilibrium conditions drive atomic
species into agglomeration, as an attempt to reduce the system’s free energy. Materials
inevitably contain atomic defects, in the form of vacancies, self-interstitials, impurity atoms
or insoluble gaseous species. In some situations, these defects are externally introduced
into the material. Examples of this can be found in structures under neutron irradiation, orin materials that are processed by ion beams, plasmas, high-speed deformation or otherhigh-energy sources. Apart from the valuable insight gained by understanding mechanisms
of atomic clustering, advancing a multitude of cutting-edge technologies is dependent on
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finding ways to modify the process of atom agglomeration. Energetic neutrons bombard a
number of critical materials in nuclear fission reactors. Materials used as nuclear fuels,
structural materials, the pressure vessel as well as instrumentation materials, are all
subjected to the generation of non-equilibrium concentrations of atomic defects. Energetic
neutron collisions with lattice atoms produce intrinsic lattice defects, as well as extrinsic
impurities or insoluble gas atoms. Likewise, in a fusion energy system, many materials will
be driven out of equilibrium as a result of intense neutron bombardment. Examples are thefirst wall and blanket structures, plasma interactive components, magnet materials,
instrumentation and other special-purpose materials.
In this chapter, we present a review of the theoretical developments in the rate theory
treatment of atomic clustering. The theory is applicable to atomic species, which broadly
include lattice or surface atoms, molecules, impurities or point defects. The focus here will
be to apply the theory to the study of nucleation and growth of precipitates by atomic
processes and develop methods to solve for the evolution of their size distribution. To
accomplish this task, we begin with a brief description of the cascade-damage environment
and the physics of point defects. Classical nucleation theory is then presented in the context
of point defect clustering to attain nucleation rates of stable clusters. From a basic
derivation of stochastic fluctuations in defect fields, the development of master equations
at the atomic level is achieved. A rate theory model is then presented for high helium
generation rates using a reduced set of deterministic rate equations for defect species. The
Fokker-Planck continuum approximation is then introduced for cases where theagglomeration of atomic clusters is primarily a result of simple atom absorption oremission. Lastly, the effects of stress are discussed in terms of internal point defect sinks
and a spatial rate theory solution is developed to calculate dislocation bias factors inirradiated materials.

9.2. Physics of radiation-induced point defects
The generation of point defects in materials stems from two primary types of neutroninteractions with bulk materials, known as elastic and inelastic interactions. These arecaused by fundamentally different processes and lead to the production of different typesof defects within materials. Elastic interactions occur when incoming neutrons transfer theirkinetic energy to a host lattice atom. If the energy transferred by the neutron in this collision
is greater than the displacement threshold energy, Eth, of the lattice atom, then the atom will
be ejected from its lattice site. This initial ejected atom is called the primary knock-on atom
(PKA). This process leads to the production of a vacancy and a self-interstitial atom (SIA)
within the host lattice. Upon ejection from its lattice site, if the PKA still carries greater
kinetic energy than Eth, it will lead to the ejection of a secondary knock-on atom, and soon. This successive chain of elastic collisions and scattering is referred to as cascade
damage and has been studied extensively both experimentally and computationally.
Inelastic interactions occur under high-energy irradiation conditions leading to (n, α)transmutation reactions within the host lattice. These interactions lead to the production ofsignificant quantities of solute gas atoms within the bulk, namely helium and hydrogen.
Helium is highly insoluble in metals and therefore highly mobile inside matrix materials.
The physics of point defects has been a topic of numerous studies within the field ofradiation damage for some time. It is well known that both vacancies and SIA's agglomerate
into clusters of different sizes. Vacancies combine to form three-dimensional (3D) voids inthe vicinity of the primary damage site while SIA's typically combine to form twodimensional (2D) glissile or sessile dislocation loops on the periphery. Studies have shown

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that glissile loops are highly mobile and perform one-dimensional (1D) random motion in
their slip direction. The production of Frenkel pairs and resulting clusters populate themicrostructural atmosphere are known to lead to hardening, swelling and creep phenomena
in irradiated metals.
In any rate theory study, it is of prime importance to properly identify all critical processes
that contribute to the production or loss. In radiation damage rate theory, production terms
may be due to collision cascades, emission from clusters, etc., while loss terms may include
recombination, absorption into clusters, etc. With the exception of the cascade production
term, each process is given a rate proportional to the Boltzmann factor for that process.
Thus, it is required that the energetics of all processes be known a priori.
The formation energy of a defect is typically defined as the difference in the total energy(at zero temperature) of the system containing the defect and that of a system of the same
number of atoms in their lowest energy configurations. These energies are commonly used
to determine equilibrium concentrations of defects. As an example, the formation energyof a carbon impurity atom in an iron crystal containing N atoms, can be calculated as(1)Here, E(N+1) is the energy of the system containing the defect and µFe and µC represent thechemical potentials of iron and carbon, respectively. The chemical potential represents theenergy per atom in a perfect, infinite lattice. Calculations of formation energies are best
performed using first-principles techniques.
The migration energy, Em, of a defect represents the activation energy of diffusion within
the crystal lattice. Lattice vibrations may result in an atomic jump to a neighbouring site
only if the diffusing defect is carrying kinetic energy equal to or greater than the migrationenergy. Accurate calculations of migration energies may be obtained using first-principles
calculations with minimum energy pathway (MEP) search methods such as the nudged
elastic band (NEB) method and the dimer method. These methods allow relaxations of thelattice around the defect as the defect treks the reaction co-ordinate to the final state. The
point of maximum energy along the path represents the saddle point and the migrationenergy can be calculated as the difference in energy between the saddle point configurationand the initial state.
The aggregation of point defects following collision cascades leads to the formation of
defect clusters in materials. The stability of radiation-induced defect clusters against
dissociation is governed by the binding energy, EB. The binding energy is defined as theenergy required to dissociate a bound system into its constituent parts. These energies canbe calculated in terms of formation energies under the assumption that the products ofdissociation are infinitely separated. For a dissociative reaction of type C→A+B, thebinding energy can be calculated as(2)
The activation energy required for the emission of a point defect from a cluster is taken as
the binding energy plus the migration energy of the point defect. A representative set ofvalues for the formation and migration energies of defects is shown in Table 9.1. Values inbold represent results from first-principles calculations.

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Table 9.1. Relevant energy barriers (in eV) for point defect processes in four materials.
Values in bold represent results of first-principles studies.
Process

Fe

W

Cu

Ni

Vacancy Formation

2.12(a), 2.15(b)3.11(c), 3.16(d)1.28(e), 1.3(f)1.62 (g), 1.60 (h,i)

Vacancy Migration

0.67(j), 0.67(b), 0.87(k)1.66(c), 1.78(b), 1.8(l)0.72(f)1.30(h), 1.29(i)Divacancy Formation

3.18(m)6.70(n)2.0(o)

3.02(g)Divacancy Migration

0.62(j)1.95(p)0.55(k)0.59(q)Interstitial Type

[110]

[111]

[100]

[100]

Interstitial Formation

3.86(r), 3.93(b)9.82(c), 9.548(b)2.73(s)4.91(i), 4.93(s)Interstitial Migration

0.34(j), 0.30(t), 0.167(k)0.013(b)0.078(u), 0.13(k)0.16(s)He Interstitial Form.

4.40(v), 3.37(w)6.19(d), 6.16(c)1.97(x)4.06(y)He Interstitial Mig.

0.06(v)

0.07(d)0.5(z), 0.63(α)

0.66(α), 0.81(β)

He Subst. Form.

4.23(v), 4.08(w)

4.77(d), 4.70(c)2.58(γ)3.13(g), 3.23(γ)

Source: Corresponding sources for the values represented in the table can be found in the References section
on page 158 (see the corresponding superscripts following each reference).

9.3. Classical nucleation theory
9.3.1. Becker-Döring theory
Classical nucleation theory, as developed by Becker and Döring in 1935, is a method todevelop a set of equations to model phenomena such as condensation, evaporation and ingeneral, any two-phase system (gas-liquid, crystalline-amorphous, etc.). It is especially
useful in computations of equilibrium size distributions of phases. In the case of vacancy
clustering, assumptions of the theory are as follows:
1. Atomic clusters originate when two atoms come together.
2. Clusters can grow or shrink by the absorption or emission of single vacancies,
respectively.
3. Small clusters may dissociate into monomers, however once a critical size isreached, the cluster will grow to a macroscopic size.
If we let C(m,t) represent the number of clusters containing m vacancies at time t pervolume, then we can express their rate of change as(3)
where R(m|m') is the transition probability rate that a cluster of size m' will grow to size m.
We can next define the impingement frequency βv(m) and emission frequency αv(m) of
single vacancies to clusters. These quantities are functions of the vacancy concentration
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and diffusion coefficient, as well as the cluster surface area, A(m). Under the second
assumption above, the transition probabilities can then be expressed in terms of Dirac delta
functions, δ, as(4)(5)
Substituting these into equation (3) and evaluating the integral, we obtain(6)
At equilibrium, ∂C/∂t=0, we can invoke the principle of thermodynamic reversibility, i.e.
that R(m±1|m)=R(m|m±1). The corresponding concentration is called the constrained
equilibrium function, C0. This allows αv(m) to be solved for as(7)which is assumed to be valid in non-equilibrium conditions as well. Plugging the expression
for αv(m) into equation (6) gives(8)This is known as the difference differential equation for nucleation. Expanding each of theterms in the brackets in Taylor series about n, it can be shown that equation (8) can be
written as(9)
We note this expression has the form of Fick's second law in a force field with a variablediffusion coefficient. This is consistent with the view that nucleation is random walkprocess through cluster size phase space. Viewing β v(m) as an effective diffusion
coefficient in size space and applying the continuity equation, the cluster nucleation
current, I(m) can be taken as(10)
and the continuity equation in an m-dimensional size space can be given as(11)
Thus we have developed a rate equation for the concentration of m-sized voids.

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9.3.2. Extension to radiation damage
Following the observations by Cawthorne and Fulton (1967) that the formation of voids
occurs in metals exposed to fast neutrons, interest in the application of classical nucleation
theory to radiation damage grew rapidly in the early 1970s. The first such undertakings
were done independently by Katz and Wiedersich (1971) and Russell (1971). These studies
extended the classical theory to irradiation cases, where in addition to vacancies,
interstitials can also contribute to growth and shrinkage processes. Also, radiation-induced
vacancies and interstitials are produced continuously, leading to the supersaturation of theirconcentrations beyond equilibrium, ultimately changing the size distribution evolution. The
Gibb's free energy (GFE) of a system containing a concentration distribution, Neq(m), ofclusters of sizes m, can be expressed as(12)
where G0 is the free energy of the perfect lattice, gm is the GFE required to form a void ofsize m, and the last term is the absolute temperature multiplied by the configurational
entropy. We note that gm is typically estimated as the void formation energy (Efm) under theassumption of negligible volume and entropy changes in the formation process. Also, Wm
represents the number of possible ways in which a void of size m can be distributed in thelattice, which can be shown to be(13)
The chemical potential of a void of size m can be defined as the partial derivative of G withrespect to the equilibrium concentration of size m voids, or(14)
The chemical potential of a single vacancy can therefore be taken as µv=Evf+kTln[ΩCv],
where the vacancy formation energy can be substituted from the equilibrium vacancy
concentration, given as Cveq =Ω-1exp(-Evf/kT). The vacancy chemical potential cantherefore be expressed as(15)
where Sv=Cv/Cveq, is the vacancy supersaturation in the lattice. For the vacancies and voids
to be in chemical equilibrium, the chemical potential of voids of size m must be equal to mtimes the chemical potential of an individual vacancy. Therefore, the equilibrium void
distribution function can be solved for from equations (14) and (15) as(16)

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Under non-equilibrium conditions, we can define the nucleation current (in the phase spaceof cluster size) as the rate at which clusters of size m grow to m+1. This can be expressedas(17)
Here, the emission of interstitials as a void growth process is neglected due to the large
formation energy of interstitials. By the condition of thermodynamic reversibility, the rateof capture of vacancies to clusters of size m should equal the rate of emission of vacanciesof clusters of size m+1, and therefore αv can be solved for from equation (7), with C0(m)replaced with Neq(m). Plugging this back into equation (17), we obtain(18)
where the function h(m) is defined in(19)
The ratio of impingement frequencies βi/βv is referred to as the arrival-rate ratio and playsa large role in determining the critical void size, mc, which occurs at the minimum of h(m).
Figure 9.1. shows a plot of the constrained distribution function, h(m), versus cluster size
for a range of arrival-rate ratios. This figure illustrates the fact that nucleation theory isheavily dependent on the parameters used. This is a major disadvantage since in most cases,
many parameters are not known to a high degree of accuracy.
Figure 9.1. Constrained distribution function, h(m), versus cluster size, m.

Source: Russell, 1971

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150 │ NEA/NSC/R(2019)2
Approximating the difference in the brackets of equation (18) as the differential d(N/h)/dm,
the resulting differential equation can be solved under the boundary conditions that N/hgoes to 1 as m goes to 1 and goes to 0 for infinitely large cluster sizes. The nucleation
current can then be calculated as(20)
where the first term on the right hand side is called the Zeldovich factor, and corrects forthe fact that some fraction of clusters, which have grown past the critical size, will still
shrink to smaller sizes.
The extension of the theory to include the effects of gas atoms resulting from transmutation
reactions was also accomplished by Katz and Wiedersich (1973) and Russell (1972). The
derivation of these heterogeneous nucleation models follows the same structure as thepreceding model, under the assumption that gas atoms are static nucleation sites for voids.
The number of clusters with j gas atoms per unit volume is taken as Mj, making the total
concentration of gas atoms, M, equal to Σj=1jMj. The GFE expression takes the modified
form:
(21)
where Njeq is the equilibrium distribution of gas-vacancy clusters per unit volumecontaining m vacancies and j gas atoms. The configurational term can be calculated as(22)
The presence of gas atoms in voids leads to an internal pressure that aids in stabilising voids
against dissociation. In a study of cavity growth in the presence of helium, Odette andLangley (1975) found that cavities would not grow above a critical size if they contain lessthan a certain number of helium atoms. Furthermore, a bimodal distribution of cluster sizes
was found under irradiation in the presence of gas atoms. The Calculations of Hayns andWood (1979) also found initial unimodal distributions evolve into bimodal for both
diffusion and surface reaction limited growth kinetics under irradiation. Typically observed
bimodal distributions consist of one class with cluster radii of a few nanometers and anotherclass with much larger size (Mansur and Coghlan, 1983). The presence of helium has theeffect of leading to smaller values of the critical size, as the pressure reduces the effect of
surface tension against collapse. Thus, the reversible work of formation gmj will contain apositive contribution due to surface tension and a negative contribution due to the gaspressure. Assuming the gas atoms in the matrix are at an effective pressure peq,
corresponding to the temperature and their concentration, the reversible work of adding gas
atoms to a cluster at pressure p is jkTln(peq/p). Assuming the ideal gas law holds, thepressure can be determined from pV=jkT, where V=mΩ. Estimates for peq can be obtained
from statistical mechanics models. The reversible work of cluster formation can then be
expressed as(23)

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where γ is the surface energy and the cluster radius R can be taken as (3mΩ /4π)1/3. For
chemical equilibrium to be satisfied, the condition mµv=µmj must be satisfied, where thechemical potential of gas-filled clusters can be taken as(24)
Using equation (15) and assuming chemical equilibrium, the equilibrium distribution ofgas atoms in voids, can be solved for as(25)
where ξ=(4π/kT)(3Ω/4π)2/3γ. The exponential term represents the free energy of cluster
formation. Whereas in the homogeneous void nucleation case this was a function of malone, this term represents an energy surface dependent on both m and j. Solution of thenucleation current Ij follows equation (20), with the modification that the critical cluster
size and h(m) are now a function of j as well. For more information, Mansur and Coghlan(1983) provide an extensive review of the effect of helium on the critical void size anddevelop analytical solutions for various conditions.

9.4. Rate theory models of nucleation and growth
9.4.1. General clustering rate theory
Under the non-equilibrium conditions that drive atomic defect clustering, the size of eachindividual cluster grows or shrinks by a process of accretion or emission. The “step” whichchanges the cluster size is controlled by fluctuations in the neighbourhood of the cluster.
Thus, an appropriate framework for these events is the theory of stochastic processes.
Defect clusters are either “born” or they “die” with each event. Many physical phenomena
have similar characteristics, and are generally described as birth-and-death processes. In
this description, the size of clusters is a randomly distributed variable with each size
containing any number of atomic species.
If the size of the transition step is by single atomic defects, then ∆x << x. Here, ∆x is thestep size and x is a vector of components representing the cluster constituents. Clustering
may occur via multiple atom aggregation, and ∆x is still smaller than x. Finally, the typicalcase of coalescence is associated with a transition, where ∆x is of the same order of x. In
the master equation description, one assigns to the system a set of transition probabilities
describing the process of cluster size variation in size space. Consider that the cluster size,x, is discrete, and is represented by a sequence of numbers, each describing the number ofa particular specie. Thus, instead of the vector notation, we use the index notation, Cijk…,
representing the fractional concentration of defect clusters containing i defects of type 1, j
defects of type 2, and so on. The types (i,j,k,…) may be lattice atoms, vacancies, gas atoms,
etc. The notation can also be extended to more complex defect phases in the solid. Since
Cijk… is fractional, it is precisely the probability of finding a defect cluster of these
constituents. The transition probabilities, ω, depend on a set of integers, ∆ijk…, which may
be positive, negative or zero, and they describe the step size, ∆x.

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The conditional probability of a transition describing the change of the number (i,j,k,…)due to a step, ∆ijk… , is given by(26)
The fundamental relationship represented by equation (26) is the Smoluchowski-ChapmanKolmogorov (SCK) equation, applied to defect clustering. Taking the limit as ∆t→0, themaster equation is obtained, which describes the birth-and-death process of the cluster.
Thus, the most general case of defect clustering can be described by the master equation.
It has been shown that if a Poisson distribution describes the transition probabilities ω, theirmicroscopic statistical averages become identical to the macroscopic system averages usedin reaction rate theories. Thus, the equilibrium laws of reaction rate theory become exact
consequences of the master equation. Averaging over transition probabilities leads to the
popular reaction rate theory, which now discards important aspects of defect fluctuations.
These are related to the system size, range over which fluctuations extend, and correlation
length over which two parts of the system can feel each other. It can be shown that in asystem that is macroscopically homogeneous, locally, fluctuations break this homogeneityand lead to the emergence of spatial defect patterns. As soon as the system deviates fromthe uniform state described by the spatially averaged rate equations, non-linear couplings
between neighbouring volume elements take place. As a result, a spatially uniform
description of defect concentrations becomes inadequate.

9.4.2. Reduced-set rate theory for high helium generation
The starting point of the theory of microstructure evolution is the SCK equation for a
Markovian process. In the continuum limit in size space, equation (26) can be expressed as(27)
where C(x,t) is the probability distribution for the stochastic variable x at time t andω(x,x’,t) is the transition probability per unit time from state x to state x’ at time t. If thetransition probabilities can be replaced by average macroscopic reaction rates, then (27)can be reduced to a deterministic set of rate equations for the concentrations of a specificdefect cluster sizes. This mean-field approximation does not take into account the statisticalnature of defect production, cascade effects, and the arrival and absorption of single andmultiple defects at defect clusters. Since defect clustering in irradiated solids is driven by
the concentration of three types of monomers (vacancies, interstitials, and helium), wewould have a coupled hierarchy of discrete equations for the probability distribution, C,
using rate or master equations. Each equation equates the concentration rates to the sum of
all production and loss rates of the species, stemming from the known physical phenomena(trapping/de-trapping, recombination, etc). For the case of radiation-induced heliumvacancy (HV) clustering, rate theory models have in the past been developed. Following a
study by Ghoniem et al. (1985), rate equations for the concentrations of vacancies (Cv),self-interstitials (Ci), helium interstitials (Cg), helium substitutionals (Cgv), di-heliuminterstitials (C2g), di-helium mono-vacancy (C2gv) and the critical nucleus (C*) can be
expressed as

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(28)
(29)
(30)
(31)
(32)
(33)
(34)
respectively. Here, the critical nucleus is taken as any cluster containing three helium
atoms. The production of Frenkel pairs is taken as the displacement damage rate, G, times
the average fraction of surviving pairs, f. Helium atoms are generated at a rate GH.
Annihilation at sinks occurs at rates proportional to the concentrations of sinks, which forgrain boundaries and precipitates, are CGB and Cppt, respectively. Absorption of interstitials
and vacancies at dislocations is accounted for in the equivalent sink concentrations Csi andCsv, respectively. Expressions for the concentration rates of helium atoms at precipitates(Mppt) and grain boundaries (MGB) are given as
(35)
(36)
The reaction frequencies of self-interstitials (α), helium (β) and vacancies (γ) and thethermal emission probabilities of vacancies (e1,e2) in the above equations are given as(37)
where the factor of 48 is the assumed combinatorial number, ν is the atomic jump frequency
and Em and Eb represent migration and binding energies, respectively. The factor δ is called
the resolution frequency and represents the probability per displacement for dissolving ahelium atom back into the matrix, b, times the displacement damage rate.
Solving the above coupled rate equations in time gives insight into the roles of various
mechanisms on cluster nucleation rates and their effect on helium transport to grainboundaries. During the initial stages of irradiation, helium atoms are generated asinterstitials, but soon are trapped when vacancies become available. These calculations
reveal the primary mechanism for reducing helium transport to grain boundaries is thegrowth of bubbles as available sinks. This is shown in Figure 9.2., where the rate of helium
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absorption at grain boundaries reaches saturation while the concentration of helium inmatrix bubbles continues to grow.
Figure 9.2. Helium distribution at various traps versus time

Source: Ghoniem et al., 1985

In a study by Stoller and co-workers (2008), a comparison between similar rate theory
studies and kinetic Monte Carlo methods was presented. Their findings expose the strong
parameter dependence of rate theory methods while arguing that it remains more
independent and resistant to errors due to simulation cell sizes.

9.5. Evolution of the size distribution and the Fokker-Planck approximation
In certain applications, the spacing between the states is small, and the hierarchy of discrete
master equations can be replaced by an equivalent continuum Fokker-Planck (F-P)
equation. It is interesting to note that except for small size interstitial loops, large defectclusters have low mobility inside irradiated solids. On the other hand, when we consider
the formation of surface atomic clusters during plasma or ion beam deposition, the mobilityof large clusters becomes significant. The transition probability from a defect cluster of sizex to x’ may be re-defined as(38)
We may distinguish between slow and fast variables in the transition probabilities, ω, in
the SCK equation, as (x, t) and (r=x-x’, τ), respectively. By expanding the function W(r=xx’, τ)C(r=x-x’, τ) in a Taylor series for the slow variable, truncating to second order, andintegrating over an appropriate correlation time, T, we obtain the F-P equation
(39)
where

and

represent thefirst and second moments of the transition probability, respectively. The first moment

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represents the elements of a drift vector, while the second is used to derive the elements ofa dispersion tensor. The indices i and j represent the dimension in cluster size space (1D
for single specie cluster, 2D for a two specie cluster, etc.). The correlation time, T, is chosen
to represent the appropriate physics of the relevant transition (for example, inverse ofarrival frequency for single-atom transitions or cascade production frequency for cascadeinduced transitions). A review of solution methods for the F-P equation can be found in
Ghoniem (1998).

9.6. Stress effects in rate theory models and the dislocation bias
The effect of stress on point defect diffusion has been a topic of great interest for sometime. It is well known that the presence of internal point defect sinks in irradiated materials
plays a significant role in the determining the size distribution. Point defect sinks include
dislocations, voids, grain boundaries, precipitates, and so on, each carrying their own
associated stress fields. Internal stresses in materials alter the energy landscape for point
defect diffusion, leading to various macroscopic changes in material properties over time.
A comprehensive theory of sink strengths was presented by Brailsford and Bullough (1981)
for various sinks in periodic and random arrays. In general, for a point defect of type η, theassociated sink strength for a sink of type α (dislocation, void, etc.) can be defined as(40)
where ρα represents the sink density in the material and Zηα is the capture efficiency of η
to α. The capture efficiency represents a ratio of the point defect flux into the sink with theassociated drift interaction, to the flux into the core due only to random walk diffusion.
Essentially, it is a measure of the effect of drift in the diffusion equation.
It is well known that swelling in irradiated metals is greatly enhanced by the preferential
absorption of interstitials rather than vacancies, to dislocations. This preference arises fromthe stronger interactions interstitials have with dislocation stress fields combined with theirhigh mobility in the matrix. A measure of this preference for interstitials is called thedislocation bias factor and is computed from the ratio of capture efficiencies of dislocations
to interstitials and vacancies. The bias can be defined as(41)
where Zi and Zv are the capture efficiencies of an interstitial and vacancy to a dislocation,
respectively. These parameters are of utmost importance specifically in radiation damage
rate theory (RT) codes simulating void swelling and radiation-creep. In the past, many
authors have developed analytical solutions for the dislocation bias. However, obtaining
solutions to diffusion equations with non-spherically symmetric sinks has proven a difficult
task. Wolfer and Ashkin (1975) developed a solution for spherical sinks under theassumption that the point defect-sink interaction is strictly radially dependent. This cameafter Ham’s solution (1959) to the steady-state diffusion equation near screw and edge
dislocations with point defects modelled as centres of dilatation. For interactions with bothradial and angular dependence, Wolfer and Ashkin (1976) developed a perturbation
method, extending Ham's work to also include the modulus interaction and externallyapplied stresses. In all of these studies solutions were reached under numerous assumptions,
including approximate treatments of both the point defect fields and interaction energy.

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9.7. Space-time rate theory
Recently, we have developed a finite element rate theory model to calculate capture
efficiencies and bias factors in irradiated metals. The model implements the diffusive flux
term allowing for two-dimensional spatial resolution which is not typically observed inmean-field rate theory calculations. Additionally, the use of atomistically obtained point
defect representations (dipole tensors) lead to a much more accurate interaction energy
expression than has been used in past studies. Departing from traditional mean-field rate
theory and the assumption of continuous sink densities, we consider the ‘effective medium’
approach. In this method, we may consider a single dislocation centred in a finite medium,with vacancy and interstitial concentrations maintained constant at an outer boundary.
These boundary concentrations can be obtained as steady-state solutions using a mean-field
rate theory approach. In the vicinity of a defect with an associated strain field, the flux of adiffusing species, η, is given by Fick's first law as(42)
where Dη and Cη are the diffusion coefficient and concentration of species η, respectively,β = (kT)-1, and Eint is the elastic interaction energy of the point defect with the strain field.

The capture radius of a species can then be defined such that the condition, ||β∇Eint||=b-1, issatisfied. The capture radius defines the approximate distance from the core at which thedrift force due to the interaction energy matches the thermal motion of the defect. It is at
this radius that the calculations of the flux are computed, and the diffusion coefficient canbe taken as a constant. It is important to note that the capture radius is a function oftemperature, material properties, and the defect's dipole tensor, which will be defined inwhat follows. By applying the continuity equation to equation (42), we obtain the rateequation for the concentration as(43)
where P represents production terms for defect η (Frenkel-pair production rate, mission
from clusters, etc) and L represents loss terms (recombination, absorption at sinks, etc.). In
the effective medium approach, the effect of production and loss terms are accounted forin the outer boundary concentrations, and thus equation (43) can be expressed as(44)Here, the interaction energy term represents the elastic interaction between the diffusing
species and a single edge dislocation. To calculate this interaction, we have recently
developed an atomistic-continuum coupling method in which an elastic representation ofpoint defects is made possible by the dipole tensor, which is calculated from the results ofmolecular statics calculations using interatomic potentials. To first order, the kth componentof the continuum displacement field can be calculated in terms of the Green's tensor
function G(x) and the defect's dipole tensor P(1) as(45)

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where repeated indices imply summation. Implementing the anisotropic elastic Green's
function and displacements from molecular statics calculations, a linear least squares
solution can be constructed to solve for the components of the dipole tensor from (45). In
the strain field of an edge dislocation, the elastic interaction energy can be expressed as
Pij(1)uj,i⊥, where uj,i⊥ represents the displacement gradient tensor due to the dislocation. Upon
solving for the concentration rates and fluxes, the dislocation bias factor defined byequation (41) can be calculated by first computing the capture efficiencies for the interstitialand vacancy. This is accomplished via line integrations of the dot product of the flux andunit normal along the capture radius. In Figure 9.3.a., the results of these calculations show
a decrease in capture efficiencies (and bias) as the temperature is increased. This is a resultof thermal fluctuations increasingly competing with drift forces at higher temperatures. In
Figure 9.3.b., the results of a separate spatial rate theory model simulating polycrystalline
tungsten under high heat flux is shown (Crosby and Ghoniem, 2011). Here, a heat transfer
analysis is coupled with the rate equations, and is solved within the framework of the finite
element method using a transient segregated solver. By implementing finite element
techniques, we have shown that spatial resolution can be obtained via the flux term ofinterstitials and vacancies.
Figure 9.3. (a) Dislocation capture efficiencies versus Temperature for copper and nickel, (b)Distribution of helium atoms in polycrystalline tungsten under high heat flux

(a)

(b)

Source: (a) Seif and Ghoniem, 2011, (b) Crosby and Ghoniem, 20119.8. Conclusions and research needs
The theory of defect clustering is deeply rooted in statistical physics, and shows fruitfulresults for the study of a variety of interesting physical phenomena. Under irradiation, thenucleation rate of critical-size voids is shown to be adequately described using classicalnucleation theory. To obtain cluster densities, however, rate theory methods need to be
utilised. Starting from a basic understanding of stochastic fluctuations in defect fields, it isshown that one can formulate master equations at the microscopic or atomic level. Theseequations can be treated in a more macroscopic sense by equivalent rate equations, in thelimit of Poisson probability distributions for transitions between states. When the transition
probabilities can be replaced by average macroscopic reaction rates, then the SCK equation
can be reduced to a deterministic set of rate equations for the concentrations of specificdefect cluster sizes. A reduced-set rate theory model was presented and shown to describe
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materials under irradiation with high helium generation. When the diffusion of large defectclusters in the bulk is slow, and the transitions between states are just between nearest
neighbours, a useful approximation is obtained, which is shown to be the Fokker-Planck
continuum theory. In materials with large dislocation densities, the interaction of point
defects with internal stress fields is known to cause an imbalance in the loss of vacancies
and interstitials to dislocation sinks. This dislocation bias is implemented in rate theorymodels to gauge the bias effect on void swelling and creep. We have shown a novel‘effective medium’ rate theory approach to calculate bias factors in metals using point
defect-dislocation interactions based on properties obtained from atomistic simulations. By
incorporating the flux term, a finite element method was developed which allowed for
spatial resolution in the model.
In the decades following early radiation damage rate theory studies, the availability of highperformance computational facilities has grown significantly, enabling large-scaleatomistic simulations. The rate theory approach, however, remains unique and useful inmultiple ways. With parameters obtained from atomistic methods such as density
functional theory and molecular dynamics, rate theory models allow for simulation at theexperimentally observable time and length scales unlike atomistic methods. In many cases
this allows for additional parameters to be taken directly from known experimental data.
Models using the mean field assumption allow for various system statistics to be obtained,
however, we have shown that spatial rate theory solutions are also attainable which may be
applied to simulations of heterogeneous microstructures. For the continued progress andrelevance of rate theory models there are several aspects that need to be addressed andimproved upon. With such high sensitivity to physical parameters, emphasis must be placedon developing a large database of defect energetics obtained using the most accurate firstprinciples methods. Activation barriers for important processes must be known to a highdegree of accuracy to ensure the reliability of these models. An important next step is tocouple spatial rate theory with other mesoscale methods such as dislocation dynamics,
where phenomena such as the embrittlement of nuclear materials can be investigated interms of dislocation motion processes. Additionally, efforts are needed to develop methods
that can incorporate the evolution of the cluster size distribution as a function of space andtime in an efficient computational manner that does not compromise the basic physics
underlying the rate theory.

References
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Becquart, C.S. and C. Domain (2007), “Ab initio calculations about intrinsic point defects and He in W”,
Nuc. Instr. and Meth. in Phys. Res. B, Vol. 255, pp. 23–26. (c)STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2│ 159

Brailsford, A.D. and R. Bullough (1981), “The theory of sink strengths”, Philos. Trans. R. Soc. London A,
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Caturla, M.J., C.J. Ortiz and C-C.Fu (2008), “Helium and point defect accumulation: (ii) kinetic
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Derlet, P.M., D. Nguyen-Manh, and S.L. Dudarev (2007), “Multiscale modelling on crowdion and vacancy
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Cu, Ag, Au, Ni, Pd, Pt, and their alloys”, Phys. Rev. B, Vol. 33(12). (e)Fu, C.C., and F. Willaime (2008), “First principles calucations in iron: structure and mobility of defectclusters and defects complexes for kinetic modelling”, C.R. Physique, Vol. 9, pp. 335–342. (j)Fu, C.C., and F. Willaime (2005), “Ab initio study of helium in α−Fe: Dissolution, migration, and clusteringwith vacancies”, F., Phys. Rev. B, Vol. 72.(a)Ghoniem, N.M., J.N. Alhajji, and D. Kaletta (1985), “The Effect of Helium Clustering on Its Transport toGrain Boundaries”, J. Nuc. Mat., Vol. 136, p. 192.
Ghoniem, N.M. (1999), “Clustering theory of atomic defects”, Rad. Eff. and Def. in Sol., Vol, 148:1-4, p.
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Ham, F.S. (1959), “Stress-Assisted Precipitation on Dislocations”, J. Appl. Phys., Vol. 30, p. 915.
Hayns, M.R. and M.H. Wood (1979), “A numerical study of evolving void-size distribution functions”, J.
Nuc. Mat., Vol. 87, p. 97.
Johnson, R.A. (1983), “Point-defect calculations for tungsten”, Phys. Rev. B, Vol. 27(4). (p)Katz, J.L. and H. Wiedersich (1971), “Nucleation of Voids in Materials Supersaturated with Vacancies
and Interstitials”, J. Chem. Phys.,Vol. 55, p. 1 414.
Katz, J.L. and H. Wiedersich (1973), “Effect of insoluble gas molecules on nucleation of voids in materials
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Kuhalainan, J., M. Manninen and E. Kautto (1996), “Helium, neon and argon in copper studied witheffective medium theory”, J. Phys.: Cond. Matt., Vol. 8:10317. (z)Mansur, L.K. and W.A. Coghlan, “Mechanism of helium interaction with radiation effects in metals andalloys: A review”, J. Nuc. Mat., Vol. 119, pp. 1-25.

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Megchiche, E.H., C. Mijoule and M. Amarouche (2010), “First principles of calculation of vacancyvacancy interactions in nicel: thermal expansion effects”, J. Phys.: Cond. Matt, Vol. 22:485502. (q)Mhaskar, S.P., and Naik, P.K., “Effect of surface relaxation on point defects in the vicinity of W (100)surfaces”, Phys. Stat. Sol. (b), Vol. 105, pp. 685–689, 1981. (n)Mishin, Y. et al (1999), “Interatomic Potentials for Al and Ni from Experimental Data and Ab initio
Calculations”, in Multiscale Modelling of Materials. Symposium, Cambridge University Press,
Cambridge, pp. 535–540. (i)Morishita, K. et al. (2000), “A molecular dynamics simulation study of small cluster formation andmigration in metals”, J. Nuc. Mat., Vol. 283-287, pp. 753–757. (u)Odette, G.R. and S.C. Langley (1975), “Modeling of synergistic effects of displaced atom of transmutant
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Rad. Eff. and Tritium Tech. for Fusion Reactors, Gatlinburg, Tennessee, CONF-750989.
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Russell, K.C. (1972), “Thermodynamics of gas-containing voids in metals”, Acta Met., Vol. 20, p. 899.
Schultz, H. (1991), “Atomic Defects in Metals”, in Atomic Defects in Metals - Fe. Landolt-Börnstein Group III Condensed Matter, Vol. 25, H. Ullmaier (ed.). (t)Seif, Dariush, and Nasr M. Ghoniem (2013), "Effect of anisotropy, SIA orientation, and one-dimensionalmigration mechanisms on dislocation bias calculations in metals", J. Nucl. Mat., Vol. 442.1-3, p. 633.
Seletskaia, T. et al. (2005), “Magnetic Interactions Influence the Properties of Helium Defects in Iron”,
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Stoller, R.E. et al. (2008), “Mean field rate theory and object kinetic Monte Carlo: A comparison of kinetic
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Wilson, W.D, M.I. Baskes and C.L. Bisson (1976), “Atomistics of helium buddle formation in a facecentered-cubis metal”, Phys. Rev. B, Vol. 13(6). (x)Wolfer, W.G. and M. Ashkin (1975), “Stress-induced diffusion of point defects to spherical sinks”, J. Appl.
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of Mod. Phys. B, Vol. 23(8). (d)Zhao, P. and Y. Shimomura (1999) “Molecular dynamics calculations of properties of the selfinterstitials
in copper and nickel”, Comp. Mat. Sci., Vol. 14, pp. 84–90. (s)STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

10. Finite element based material modelling of nuclear fuelJoseph Y. R. Rashid
ANATECH Corp., United States

10.1. Introduction
In light water reactors (LWRs) technology, the term “nuclear fuel” is understood to apply
to a system of fuel rods each of which is composed of two major sub-elements: the ceramic
UO2 fuel pellets and the zirconium-based alloy cladding. The former is the fission energy
source, and the latter is the heat transfer medium, which also serves as the primarycontainment of fission products. Modelling of the thermo-mechanical behaviourr of bothmaterials, individually and as interacting components of a fuel rod, is one of the mostimportant elements governing reactor performance. In a full-length LWR fuel rod, the fuelcolumn consists of several hundred cylindrical pellets, which, during normal operation,
experience fracturing and relocation, swelling, densification, and viscoplastic hot pressing.
As contiguous bodies, the fuel pellets are subject to rigid body motion and cannot be
isolated, geometrically and functionally, from the cladding. Consequently, they derive theircoherency from cladding confinement and, as such, their thermo-mechanical modelling
requires special numerical techniques to transform a geometrically discontinuous
behaviourr to a computationally stable continuum. The finite element modelling andsimulation of this thermo-mechanical behaviourr produces a non-linearlinear system of
equations, where the fundamental solution variables are the strongly coupled displacement
and temperature fields. For the coupled displacement and temperature solutions, thetemperature response will depend on the displacements through the pellet-cladding gap
thermal conductance, and the mechanical response variable – the stress field – will dependon the temperature through the material properties and behaviouralral models. Thisdependency is reflected in both, the constitutive relations and the nodal force-displacementtemperature system of governing equations.
A recent review of the current state of the art in LWR nuclear fuel modelling is presented
by Rashid, Yagnik, Montgomery (2011), placing special emphasis on pellet-cladmechanical interaction (PCMI) which governs fuel rods integrity during normal operations
and under accident conditions. The purpose of the present paper is to describe the materials
modelling of fuel viscoplasticity and fracture (Rashid, Tang, Johansson, 1974; Rashid,1974), which are the behavioural regimes that determine the intensity of PCMI and
cladding failure potential in LWRs. The computational implementation via the finite
element method (FEM) has achieved a mature level of practical application, but is confined
to two-dimensional simulation only (Rashid, Montgomery, Dunham, 2004), with threedimensional modelling limited to a few special effects studies (Brochard, Bentejac,
Hourdequin, 1997). However, full three-dimensional FEM simulations of generalutility have recently been undertaken (Larzelere, 2010, CASL), but still remain in theformative stage.
The nuclear fuel rod material and spatial domains are: the ceramic fuel pellets, individuallyas contiguous cylindrical bodies or as a continuous fuel column; the void volume and pelletcladding gap interface; the metal cladding; and the surrounding coolant channel. The
thermo-mechanical response of such fuel element is intimately connected with thesurrounding material; therefore, to convey to the reader a coherent picture of fuel behaviour
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modelling from which an assessment of adequacy or future improvements can be made, it
is instructive to describe the physical environment in which nuclear fuel operates. This is
depicted pictorially in Figure 10.1., which shows the various physical processes andbehavioural regimes that result from, and are driven by, the response of the fuel. These
regimes are depicted through inserts in Figure 10.1., and the interaction regimes
represented in this figure are greatly influenced by the fuel’s constitutive behaviour andthermo-mechanical response.
Figure 10.1. Interfaces and behavioural regimes in LWR fuelSource: Rashid, Y.R., S.K. Yagnik and R.O. Montgomery, 2011.

The mechanical response of the fuel pellets to the temperature spatial distribution causes
the pellet to hourglass and fracture radially and axially in a semi-random pattern, withradially oriented cracks as dominant feature. Circumferential cracks can occur undercooling during power reduction, and the resulting fracture pattern becomes a threedimensional (3D) fragmentation pattern. As the pellet fragments relocate radially outward
under time varying power history, the resulting change in the fuel-cladding gap can produce
a change in the gap conductance, with a corresponding change in the heat transport to the
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cladding and the surrounding coolant. Pellet-cladding contact eventually begins and theresulting mechanical interaction (PCMI) can exert localised hoop stress at the cladding
inner surface at radial pellet cracks, which, in the presence of fission products can lead tostress corrosion cracking (SCC) failure of the cladding. This failure mechanism came to be
known by the acronym PCI, which characterises the chemical effect of PCMI, and is theprimary failure mechanism under normal operations. The PCMI/PCI effects are further
aggravated by the combined damage of pellet radial cracks and manufacture-induced
chipped pellet corners, known as missing-pellet-surface (MPS). Under transient conditions,cladding heat-up can occur due to inefficient heat transfer to the coolant caused by one ofseveral transient regimes such as power-coolant mismatch, reactivity insertion accident(RIA) or loss of coolant accident (LOCA). The resulting cladding deformations during such
events lead to a loss of confinement of the fractured fuel pellets and induce rigid-body like
relocation motion. A high fidelity fuel model must have the capabilities to account for allof these behaviour regimes (Rashid, Montgomery, Dunham, 2004).
Under the thermal-hydraulic and the thermo-mechanical behaviour regimes represented in
Figure 10.1., both fuel and cladding material elements are subject to viscoplasticity andfracture, evolving differently in the two materials, with the ceramic fuel exhibiting the more
complex constitutive behaviour. In the present mathematical construct, emphasis is given
to the ceramic fuel pellet, which, with appropriate restrictions on certain model parameters,
can be generalised to the cladding material. In this respect, we identify two unique featuresthat distinguish the constitutive behaviour of fuel pellets from that of the cladding, as wellas metallic materials in general. The first unique feature is that the ceramic pellet, being a
porous compact of sintered uranium oxide powder, exhibits non-zero volumetricviscoplastic dilatation, as compared to zero plastic volume change in metals. The second
unique feature is that fracture in the fuel pellet is an inherent behaviour regime undernormal operating conditions, and consequently is modelled as a constitutive property in the
stress-strain relations. This results in non-linear time-dependent force-displacement
equations during all deformation stages, including small deformations where linear
behaviour is expected in other materials. In contrast, fracture in the cladding is calculated
as an end limit state, for comparison with a failure criterion, and, unlike the fuel pellet, isseldom treated during the course of the computations as a propagating fracture.
FEM is uniquely suited for the computational simulation of the above described thermomechanical behaviour, especially in the treatment of pellet fracture, which is modelled as
multi-axial direction-dependent distributed damage. This type of modelling fracture frees
the computational simulation of crack evolution and propagation from the rigid constraintsof the computationally averse classical fracture mechanics. The constitutive formulation
presented below reflects this unique adaptability of the FEM to the treatment of theinterdependency between the material behaviour and the numerical simulation of theoverall global response.

10.2. Modelling of UO2 fuel viscoplasticity
Fuel viscoplastic behaviour under plastic flow and creep is modelled by first prescribing ayield potential function and flow rule that admit plastic volume change, which wascharacteried experimentally as hot pressing (Kaufman, 1961). A yield potential function ischosen that is essentially an extension of the von Mises yield condition to include a
dependence on the first invariant of the stress tensor. Similarly, a creep flow rule ispostulated in which a creep volumetric compression is assumed to be related to a Poisson’s
ratio type parameter. Further more, the yield surface is non-symmetric in the stress space,
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as will be shown later, and the material behaves elastically in tension until it fractures. The
yield function has the form

F = J 2 + αJ12 − k 2 = 0,

(1)

The associated flow rule of this function isεijp = λ (S ij + 2αJ 1 δ ij )(2)

And the plastic volumetric incremental change isεiip = 6λαJ 1(3)Equation (3) states that the plastic volume change is proportional to and of the same signas J1, hence, equation (1) provides for plastic densification under isotropic compression.
The quantities J1 and J2 in Yield function (1) are defined as follows:

J1 = σ ii ,
J2 =

1
S S ,
2 ij ij(4)(5)

where σij are the co-ordinate stresses, Sij are the deviatoric stresses given by,
1
Sij = σ ij − σ kkδ ij .
3

(6)

10.2.1. Determination of αAn assessment of this parameter is made by considering an idealised model of porous
material. In doing so, the following assumptions are made:
1. Macroscopically, the porous material is viewed as a homogeneous continuum. This
continuum consists of numerous cells structured as spherical pores surrounded byspherical shells.
2. These shells are closely packed as shown in Figure 10.2. and the interstitial material
between the shells transmits forces but does not contribute to the strength of thematerial.
3. The cell is much larger than a microscopic element; therefore, its stress and strain
fields can be analysed within the framework of continuum mechanics.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 10.2. Plane cross section of the idealised model of porous material (not to scale)

Source: Rashid, Y.R., H.T. Tang and E.B. Johansson (1974).

For the array shown in Figure 10.2., 74% of the material volume is in the spherical shells
and 26% is in the interstitial material. Under hydrostatic compression, the spherical shell
will yield if

𝑃𝑃 = 2𝑌𝑌𝑠𝑠 ln(𝑟𝑟2 /𝑟𝑟1 ),(7)

where, Ys is the yield strength of the solid material surrounding the pores; r2 and r1, areouter and inner radii of the shell; and P is the pressure required to yield the shell. Equation(7) is obtained straightforwardly by solving the equilibrium equation subject to the vonMises yield criterion (Hill, 1950). The volume fraction of the pore referred to the sphericalshell is (r1/r2)3 and referred to the total volume is 0.74(r1/r2)3. If we let ρ be the theoreticaldensity fraction referred to the total volume, such that 1-ρ = 0.74(r1/r2)3, and substituting
for r1/r2 into equation (7) gives,P=

2
Ys ln{0.74 / (1 − ρ )}3

(8)This relation can be used together with the yield criterion to determine a value for α. In thecase of hydrostatic compression, J2 = 0, J1 = 3P, and equation (1) has the form
9αP 2 = K 2 .

(9)At initial yielding K = (1/√3) Y(ρ0) and eq. (9) gives

{

}

2

α = Y (ρ 0 ) / 3 (3) P ,(10)

where, ρ0 is the initial density fraction, and Y(ρ0) is the uniaxial yield stress of the porous
material at density fraction ρ0. Substituting equation (8) into equation (10) and setting ρ =ρ0 yields2Y (ρ 0 ) / Ys
α = .
 2 3 ln{0.74 / (1 − ρ 0 )}STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS

(11)

│ 165

166 │ NEA/NSC/R(2019)2
The numerator in equation (11) is the ratio of the flow stress at the initial density to theflow stress of the solid material surrounding the pores. It is seen that as ρ0 becomes larger,
Y(ρ0) approaches Ys, and if ρ0 = 1, α = 1, which corresponds to von Mises material. In theabove,

ρ0:

Initial (as-manufactured) fractional density of the fuelY(ρ0):

Initial yield strength of the fuel

YS:

Yield strength of solid (100% dense) UO2

Except for YS, the parameters in equation (11) are known for a typical fuel design. Because
it is virtually impossible to determine YS experimentally, and porosity dependent yieldstrength data is not available for extrapolation to zero porosity, it is necessary to estimate
YS in order to make use of the above expression for α. However, it is possible to avoid thisprocess by allowing Y(ρ0) to approach YS as ρ0 approaches unity and then calculate an
asymptotic value for α. Using this procedure α is calculated as follows:

α = limρ0 → 1[1 (2 3 ln(0.74 (1 − ρ )))] = 0.005
2

0

(12)

This value of α was found to give best results with respect to observed behaviour of PCMI.

10.2.2. Creep law
Utilising a creep flow rule, the creep strain rates are determined from the following pair offlow rules:eijc = JS ij ,(13)εiic = Lσ ii ,

(14)

and

where, J and L are general functions of stress, strain, temperature and εij are the deviatoriccstrain rate tensor defined by1eijc = εijc − εkkc δ ij .
3

(15)

Combining equations (13), (14), and (15) yields
13εijc = JS ij + Lσ kk δ ij .

(16)

Contracting both sides of equation (13) and substituting the following effective strain andeffective stress expressions:

3
2

1/ 2




εec =  eijc eijc (17)

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

3
2

1/ 2




σ e =  S ij S ij  ,

(18)

gives

J = εec / σ e .

(19)where J is seen to be the specific creep rate in shear, which is generally a function of time,
stress, temperature and neutron flux.
Returning to the second flow rule which defines the dilatational creep rate, equation (14),
we observe that the function L is the specific creep rate under hydrostatic stress, which, likeJ can be derived from creep tests generated under 3D stress conditions. Such tests, however,are limited and therefore assumptions have to be made to permit stress analysis. In order tomake use of a single creep rate function, namely J, the dilatational creep response isassumed to have the same time dependence as the creep in shear, implying that L would
differ from J by a constant. This leads to the assumption that Poisson’s ratio for creep
remains constant with time but may take on a different value for creep from the elastic
value. On this basis, the combined flow rule, equation (16), becomes:εijc = JBijklσ kl ,(20)

Bijkl = (0.5 +ν 0 )δ ik δ jl − (0.5 −ν 0 )δ ijδ kl(21)

where,and ν0 is a new parameter which governs the dilatational creep response. The value of thisparameter falls in the range

0 ≤ ν 0 ≤ 0.5 −ν(22)where, ν is the Poisson ratio for creep. From equations (21) and (22) it is clear that if thecreep volumetric compression parameter ν0 is zero then the material is incompressible
under creep, in which case the combined flow rule, equation (20), reduces to the usual flow
rule, equation (13).

10.2.3. Determination of ν0
The value of ν0 ranges from zero for incompressible material to (0.5 – ν), where ν is the
Poisson ratio for creep of the porous material. In order to establish a value for thisparameter, we employ the same idealised material model used in the determination of theplastic compression parameter α. Assume that the material surrounding the pores obeys acreep law of the formε = Aσ n ,(23)where, A and n are material constants. Under hydrostatic compression, the tangential creep
rate at the outer surface of the spherical shell is solved in Hill (1950) and has the form
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168 │ NEA/NSC/R(2019)2nn
 n
A 3  1εt =    P
3/ n
2  2n   (r2 / r1 ) − 1(24)

where, r2, r1 and P are defined in equation (7). Substituting the expression 1-ρ = 0.74(r1/r2)3
for r1/r2 into equation (24), we obtain
n

n

 n
A 3  1εt =   
 P
1/ n
2  2n   {0.74 / (1 − ρ )} − 1

(25)

Equation (25) is equal to r r2 at r = r2 with r being defined as the rate of the radialdisplacement of the spherical shell. Viewing the spherical shell as the basic element

composing the porous material, εt of equation (25) is then the uniaxial strain rate of theporous material. Consequently, the volumetric strain rate isε p = 3εt(26)From equation (23) and the modified creep law, the volumetric strain rate under hydrostatic
compression isεn = 6ν 0 AP n

(27)

Thus, treating the porous material as a homogeneous material, the creep parameter ν0 canbe determined by substituting equation (25) into equation (26) and setting ε p = εn , whichyieldsn13 
1
ν0 =   

1/ n
4  2   {0.74 / (1 − ρ )} − 1

n

(28)

As ρ approaches 1 ν0 approaches 0, and at 100% density the material becomes creep
incompressible.

10.2.4. Derivation of constitutive equations
Consider a material element where it is assumed that the strain rate tensor εij can be
expressed in incremental form as the sum of four components: elastic, time-independent
plastic, creep and free expansion; namely,d εij = d εije + d εijp + d εijc + d εijf

(29)

The elastic strains are related to the incremental stress by()

d ε ije = d C ijkl σ kl = C ijkl d σ kl + dC ijkl σ kl

(30)

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

or by the inverse relation()

d σ ij = d D ijkl ε ekl = D ijkl d ε ekl + dD ijkl ε ekl(31)

Equations 30 and 31 admit the variation of the material tensors Cijkl and Dijkl withtemperature and neutron flux. For isotropic material, which characterises UO2 pellets, they
reduce to

C ijkl =1− υδ ik δ jk − υ δ ij δ kl
EE

υE
D ijkl = E δ ik δ jl + ( )(δ δ1− υ
1 + υ 1 − 2υ) ij kl

(32)
(33)

where, E is Young's modulus and υ is Poisson's ratio.
The plastic strain increments must obey a flow rule of the form(dεijp = d λ ∂ F ∂ σij

)

(34)

where, F is defined in equation 1 and dλ is a positive scalar function which depends on thestate of stress, strain, temperature, hardening and neutron flux. Combining equations 20,
21, 29, 30, 31 and 34 the incremental stress-strain relations may be written as
d σ ij = Dijkl  d ε kl − d λ ∂∂σF − d t Bklmn J σ mn − d Cklmn σ mn − d ε klf 
kl



(35)

In the above equation, the scalar function dλ remains to be determined. This is done withthe aid of the consistency condition which ensures that the stresses remain on the yieldsurface during plastic flow, and is given by

d F = ∂∂σF d σ ij + ∂∂εFp dε ijp + ∂∂F
K dK = 0
ij

ij

(36)

This equation admits the dependence of the yield function on the temperature and neutron
flux through the parameter K. Making use of equations 34 through 36, and after somemanipulation, the final form of the incremental stress-strain law is found to bed σ ij = D ijkl  d ε kl − η ∂F Λ mn d ε*mn  − dψ ij∂ σ kl(37)

F dK  D∂F∂F − ∂F∂F dψ ij = η D ijkl ∂∂σF ∂∂K
mnpq ∂ σ
p


∂
σ
∂
σ
∂ ε mn
kl
mn
pq
mn 


(38)

d ε kl* = d ε kl − d t Bklmn J σ mn − d Cklmn σ pq − dε klf(39)

where,

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170 │ NEA/NSC/R(2019)2

∂F − ∂F∂F 
Λ kl = D ijkl ∂∂σF  D mnpq ∂ σ∂F ∂ σp∂σ∂ ε mn
ij 
mn
pq
mn 
η = 0 when
η =1

when

(40)

F < 0
∂F d σ + ∂F dK ≥ 0
ij
∂K
∂ σij

F = 0 and(41)

Rather than continue with the cumbersome indicial notation, we switch to the moreconvenient matrix notation, which is highly adaptable to implementation in the finite
element method, as well as for formulating pellet fracture as will be shown later. After
substituting for the various derivatives in equations 37 through 40, and casting the resultsin matrix notation, we obtain the following final form of the incremental stress-strain matrix
equations:

{

(

) }

∆ σ~ n = H≈ ∆ ε~ n − ∆ ε~ nf − J n B≈ + ∆ C≈ σ~ n−1(+ 2η KQ∆K I + J n M≈ C≈
in which[) (S + 2 β I δ )−1(

M= I≈ + D
Jn B
+ ∆C
≈
≈
≈
≈

(

(42)

1n ~

~ n

)] D
−1

(43)

≈

)(

)

(

)(

H
= M
− (η Q ) M
S + 2 β I1n δ~ S~ n + 2 β I1n δ~ T M
≈
≈
≈ ~ n
≈

(

) (

)

Q = S~ n + 2 β I1n δ~ T M≈ S~ n + 2 β I1n δ~ + 2 E ′ S~ n + 2 β I1n δ~ T S~ n + 2 β I1n δ~
3

)

(44)
(45)

In the above, the subscript n refers to time step tn, ∆ε n is the incremental strain vector,∆ σ~ n is the incremental stress vector, ∆ ε fn is the free expansion incremental strain~

vector, S is the deviatoric stress vector, δ is the Kronecker delta in vector form, D is the~≈elasticity matrix, I is the identity matrix, ∆ C is the change in C due to a change in≈≈≈temperature and/or fluence; the other symbols have been defined. Equation 42 is the matrix
equation that is modified for pellet fracture, as shown next, and is utilised in the derivationsof finite element stiffness and force matrices.

10.3. Constitutive modelling of fuel pellet fracture
Cracking under normal operations is an inherent behaviour of solid fuel pellets, and as suchit plays an important role in the fuel relocation and fuel-clad mechanical interaction
mechanisms. It is modelled as a multi-axial direction-dependent distributed damage,(distributed over the element integration-point volume), in the constitutive relations and in
the formulation of the equilibrium equations.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

10.3.1. Uniaxial response
A typical representation of the tensile response of the material is displayed in Figure 10.3.
which shows an initially linear behaviour up to a peak point followed by a strain softening
regime. The peak point on this figure is the crack initiation point; however, the crack is not
fully formed until the end of the strain softening regime, which can extend to a strain thatis five to ten times larger than the fracture initiation strain. The modelling of this post-peak
response of the material is very important to the behaviour of the fuel pellet, particularly
under cyclic power histories where pellet expansion and contraction, which influences thegap conductance and consequently the thermal response, are greatly affected by that
regime. Experimental stress-strain data in tension for fuel pellets is totally lacking.
However, Figure 10.3. was constructed using the observed response of ceramic materials
which generally exhibit the strain softening response shown in the figure. By knowing thefracture stress or strain and the elastic modulus and making an assumption on the extent of
the strain softening regime, a normalised form of the stress-strain curve can be constructed
as shown in Figure 10.3. It should be noted, however, that the model is not very sensitive
to the shape of the strain softening curve. Using the analytical fit in Figure 10.3., thenormalised stress is expressed as

σ = A + B e − C ε / εf
σf

(46)

where, εf and σf are the fracture strain and stress respectively and A, B and C are functionsof the ratio ε0/εf, where ε0 is the crack opening strain when the crack is assumed to be fully
formed.
Figure 10.3. Normalised tension response curve for ceramic materials

Source: Rashid, 2019

For strains greater than ε0, the material is not allowed to support any tensile stress normal
to the crack plane. In the strain softening range between the peak point ε = εf and ε = ε0,
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172 │ NEA/NSC/R(2019)2
the material loses its stiffness and sheds the tensile stress gradually. This behaviour,
however, is valid only for previously un-cracked material, but once the crack becomes fully
formed, the material loses its tensile stiffness capacity in the normal direction to the crack
regardless of the magnitude of the tensile strain. However, under strain reversals during
power cycling, the crack is allowed to close and develop compressive stiffness even underresidual tensile strain. This compressive stiffness recovery occurs gradually, from a verysmall residual value until the strain across the crack plane reaches ε0, then gradually
increasing to full value when the calculated crack strain becomes negative. This stiffness
history is derived using a non-dimensional crack history parameter e(t), as will be discussed
below. Equation (46) can be written as follows for ε > εf.
σ = σ ε εf = A + B e− C ε / εfσf
ε εf σf(47)

recognising that σ/ε = E and σf/εf = E0 , where E is the post-peak (cracking) modulus andE0 is the initial (elastic) modulus, and introducing the definition of e(t), equation (47)becomes(e (t ) = E E 0 = (ε f ε ) A + B e − C ε / ε f

)

(48)

The function e(t) takes on the values of unity and zero respectively for the conditions ε ≤ εfand ε ≥ ε0. In the range εf < ε < ε0, e(t) will have a value between unity and zero according
to equation (48). In this sense, e(t) can be regarded as the memory function of the crack.
This expression for e(t) will be used to modify the constitutive relations for the fuel pellet.

10.3.2. Multi-axial fracture
The preceding discussion dealt with the uniaxial response in tension. To generalise thisone-dimensional (1D) response to two- and 3D stress states, the smeared-cracking model(Rashid, 1974) is utilised in which cracking is modelled as a direction-dependent damage
distributed over the volume assigned to the element integration point. The crack is assumed
to initiate at a point when the calculated principal strain reaches the material fracture strain
as a temperature-dependent material property. The crack surface is the perpendicular planeto the principal direction. A failure/yield surface is illustrated in Figure 10.4., choosing, forconvenience, a two-dimensional (2D) stress state. Stress states in the compressioncompression quadrant of Figure 10.4. are characterised as elasto-plastic regimes, and forstress states in the positive-positive or positive-negative quadrants of the failure surface,
the material responds elastically until the stress state reaches the failure surface where thematerial will fail in the cracking mode; the tangent to the failure surface determines theorientation of the crack surface.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 10.4. 2D failure/yield surface for UO2 fuel pellet represented in principal stress space

Source: Rashid, 2019

Consider the strain state ε n , expressed in the principal co-ordinate system, then the~transformation of the co-ordinate strains ε n to ε n is
~
~

ε

~n

= Pεn
≈~

(49)

where P is function of the direction cosines of the principal direction unit vectors.

≈

Transforming the incremental quantities ∆ ε n and ∆ σ n to the principal axes gives:~~∆εn = P∆εn
~
≈ ~

(50)

T



∆ σ n =  P −1  ∆ σ n
~
~
≈ 

(51)

The transformed vectors ∆ ε n and ∆ σ n and the untransformed vectors ∆ ε n and~~~∆ σ n must, by conservation of energy, satisfy the inner product relation:~∆ ε Tn ∆ σ n = ∆ ε Tn ∆ σ n
~

~

~

~

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODS

(52)

│ 173

174 │ NEA/NSC/R(2019)2
Re-writing equation (42) in short form

∆ σn = H ∆ ε n − R n
~
~
≈ ~

(53)

where, R n is an internal stress vector which contains the rest of the terms in equation~(42). Transforming equation (53) to the principal co-ordinate system yieldsT∆ σ n = H ∆ ε n −  P −1  R n~≈ ~≈  ~(54)

where,T
H =  P −1  H P −1
≈ ≈  ≈ ≈(55)

Now the memory function e(t) introduced in equation (48) can be generalised to describe
the history of a crack in a principal plane, i.e. ei(t) describes the history of the crack whose
normal is in the ith principal direction, namely,(ei (t ) = (εf εi ) A + B e− C ε / εf

) , i = 1, 2, 3

(56)

Applying equation (56) to (53) leads to the following incremental stress-strain equation
expressed in the principal co-ordinate system:∆σn = E~≈T
H E ∆ ε n −  P −1 
~
≈ ≈
≈ 

T

T


  −1 
R n +  I − E   P  R n − σ n −1 ~~ ≈ ≈   ≈  ~

(57)

where, E is a diagonal matrix whose diagonal elements are functions of the three possible≈

crack histories ei(t), i = 1,2,3, defined by equation (56). Specifically, for the axisymmetric
2D case, e1 0
0 e
2
E = 
0 0
≈

0 0

0
0
e3
0

0 
0 
0 

e1e3 

(58)

In plane stress E defines two possible orthogonal cracks. In an axisymmetric stress state,≈

however, a third crack surface can develop along a radius, which implies the existence ofan infinite number of such cracks. Figure 10.5. shows the three possible crack types thatcan develop in the fuel pellet in axisymmetric representation.

STATE-OF-THE-ART REPORT ON MULTI-SCALE MODELLING METHODSNEA/NSC/R(2019)2

Figure 10.5. Fuel cracking pattern in axisymmetric fuel rod representation

Source: Rashid, 2019

In the present development the cracks can follow independent histories, including crackclosing and possible healing if the fuel temperature rises above the sintering temperature.
Finally, equation (57) is transformed back to the original co-ordinate system with the result∆ σ n = ZT H Z ∆ ε n − ZT R n −  I − ZT  σ n −1
~
~
≈ ≈ ≈
≈ ~
≈ ≈  ~(59)

where,Z = P −1 E P
≈
≈ ≈≈

(60)
−1





∆K 1 + J M C   S ′ + 2 β + J δ 
R n = H ∆ ε~ nf +  J n B + ∆ C  σ~ n−1  + 2η KQ
n
1 

~
~
≈
≈ ≈   n
≈
 ≈



(61)

10.4. Finite element implementation
Implementation of the above formulation in finite element analysis follows standard
procedures, which are familiar to most numerical analysts, and need not be repeated here.
However, the state of the art of finite element modelling of nuclear fuel is limited to 2D
simulation, and the best known finite element based fuel performance code, and mostfunctional as an industry tool, is EPRI’s Falcon (Rashid, Montgomery, Dunham, 2004).
Encouraging 3D finite element simulation of fuel response was developed by CEA using
the 3D Toutatis code, Figure 10.6., (Brochard, Bentejac, Hourdequin, 1997). It should be

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176 │ NEA/NSC/R(2019)2
noted, however, that such 3D simulations still remain as special-effects research studies
rather than a routine fuel performance evaluation tool.
Figure 10.6. 3D simulation of fuel pellets deformation in CEA’s Toutatis Code

Source: Brochard, Bentejac, Hourdequin, 1997

With the exception of the Falcon code cited above, the nuclear industry’s current practicein fuel performance modelling and analysis utilises finite-difference-based 1D axiallystacked radial representation, referred to in the literature as 1½-D simulation. The
movement towards higher dimensional simulation is discouraged by the large efforts
needed for re-licensing fuel performance codes. However, efforts at developing 3D fuelbehaviour codes have been initiated by the US Department of Energy at INL and ORNL
within the NEAMS and CASL programmes (Larzelere, 2010), which are aimed at enabling
science-based nuclear energy systems through advanced modelling and simulation, theoutcome of which would be a new generation of high fidelity fuel behaviour codes.

10.5. Model limitations
The fuel’s viscoplastic and fracture behaviour described above models the material as acontinuum. With respect to pellet cracking, the constitutive model formulates pellet
fracture in a framework that avoids computational instabilities which are inherent in acontiguous assembly of individual pellets due to rigid-body motion of the pellets and thecracking-induced pellet fragments. The above constitutive formulation of pellet fracture
produces regular fracture patterns, which are clearly at variance with the observed random
cracking. This limitation, however, is not particularly conducive to improvement throughresearch since there is no known mathematical construct by which random processes
become deterministic phenomena.
Another limitation of the model is that the evolution of the microstructure of the fuelmaterial during irradiation is not represented in the model. This includes the high burnup
structure and the evolution of the highly porous zone at the pellet periphery and the fission
gas pressurised pores within the grains and at the grain boundaries. These porosities, unlike
the uniformly distributed manufacturing porosities described above, are operationallyinduced randomly distributed pores that play a major role in fuel fragmentation during lossof coolant accidents. Characterising the associated fuel microstructure requires meso-scale
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and even atomistic-scale modelling approaches for which atomistic theory andcomputational simulations can be of great value (Stan, 2009).

References
Brochard, J., F. Bentejac and N. Hourdequin (1997), “Nonlinear finite element studies of the pelletcladding mechanical interaction in a PWR fuel,” in CW/4, Transactions of the 14th International
Conference on Structural Mechanics in Reactor Technology (SMiRT 14), Lyon, France, August 17-22,
1997.
Hill, R. (1950), The Mathematical Theory of Plasticity, Clarendon Press, Oxford.
Kaufman, S.K. (1961), The hot-pressing behavior of sintered Iow-density pellets of U02, ZrO2-U02, ThO2,
and ThO2-UO2, WAPD-TM-751, Westinghouse Electric Corporation, May, 1961.
Larzelere, A.R. (2010), Nuclear Energy Advanced Modeling and Simulation (NEAMS), DOE Presentation,
April 29, www.energy.gov/sites/prod/files/NEAC042910-NEAM.pdf.
Rashid, Y.R. (1974), “Mathematical Modeling and Analysis of Fuel Rods”, Nuclear Engineering and
Design, Vol. 29, pp. 22-32, November 1974.
Rashid, Y.R., H.T. Tang and E.B. Johansson (1974), "Mathematical Treatment of Hot Pressing of Reactor
Fuel," Nuclear Engineering and Design, Vol. 29, No. 1, November 1974.
Rashid, Y.R., R.O. Montgomery and R.S. Dunham (2004), Fuel Analysis and Licensing Code: FALCON
MOD01, Volume 1: Theoretical and Numerical Bases, EPRI Report, December 2004, -1011307.
Rashid, Y.R., S.K. Yagnik and R.O. Montgomery (2011), “LWR Fuel Performance Modeling and MultiDimensional Simulation – Current Status and Future Needs”, Journal of Materials (JOM), August,2011.
Stan, M. (2009), “Multi-Scale Models and Simulations of Nuclear Fuels”, Nuclear Engineering andTechnology, Vol. 41, No.1, February 2009.

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Conclusions
The most important deficiency in our understanding of irradiated material behaviour is
microstructure evolution. The initial microstructure of fuels and structural materials is
structurally and chemically complex. By necessity, approximations must be made torepresent this complexity in a computational model. As the material undergoes irradiation,
fission products are introduced and the chemistry changes. In addition, defects and fission
products migrate under a non-uniform temperature and stress field leading to anisotropic
swelling, gas release, segregation of species, nucleation and growth of new phases, andrecrystallisation. At the macroscale, chemical, mechanical and thermal effects can combine
to drive complex process, such as pellet-clad interaction, resulting in material failure.
Modelling these interrelated processes in a reliable fashion remains an ongoing challenge.
This report describes computer simulation methods and models for studying nuclear fuels
and structural materials across length and time scales. While some aspects of materials
degradation under the combined effects of stress, elevated temperature and radiation canbe isolated and studied at a given scale, it is essential to integrate models across scales toobtain a predictive understanding of the performance of materials.
At the most fundamental level, electronic structure calculations provide valuable
information about point defect properties and interfacial energies. Ab-initio molecular
dynamics (AIMD) enables the accurate simulation of a small collection of atoms over thenanosecond time scale. One can construct simple models of interfaces to understandsegregation, interfacial transport and the early stages of phase formation. This method is
especially relevant to the study of dissimilar material interfaces that are ubiquitous innuclear materials but are challenging to model with existing interatomic potentials.
Classical molecular dynamics (MD) and accelerated dynamics methods offer the promiseof extending the length and time scale of the simulation far beyond that possible with
AIMD. However, the lack of reliable interatomic potentials remains a constraint. Electronic
structure calculations can be used to determine properties of interest that are not available
experimentally. A database of such properties can be used with machine learning to develop
reliable potentials for complex materials systems. Most of the existing potentials are nonreactive and are unsuitable for modelling reactive chemical processes. At the same time,
the available reactive force fields have not been rigorously tested and validated for nuclear
materials. Existing potentials are parameterised for equilibrium properties but are often
used to simulate systems driven far from equilibrium and outside the range of applicabilityof some potentials. This is especially a concern when fixed charge models are used todescribe environments where charge transfer occurs. Even the widely studied Fe-Cr systemdoes not have a completely reliable potential. It is therefore quite challenging at present tostudy alloys with more than two components by MD simulations. There is clearly acontinuing need to develop, test and validate potentials for nuclear materials. It is alsoessential to continue the development of on-the-fly kinetic Monte Carlo methods that use
machine learning to predict barriers instead of tabulating all possible events a priori. Suchan approach is valuable when the local environment evolves rapidly and causes changes tothe reaction rates and pathways.
There is considerable potential to connect density functional theory, MD and experimental
data – e.g. transmission electron microscopy observations – with dislocation dynamics andcrystal plasticity, to understand the mechanical behaviour of materials in extreme
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environments. In addition, dislocation dynamics can be coupled with rate theory to provide
insights into the failure of nuclear reactor materials. Information from the atomic level canbe coarse-grained using the phase-field approach in order to extend the understanding to
the continuum scale and explore nuclear materials phenomena across scales. The
methodical integration of models will help us gradually replace empirical models in current
computer codes with physics-based predictive models.
Finally, there is a pressing need for uncertainty quantification, verification and model
validation in nuclear materials research. These must be performed for models at each scale.
Validation requires experimental data. If such data is not available, validation experiments
and lower length scale simulations must be designed to generate the needed data. In
addition, the links and parameters passing between models must also be rigorously
validated. Machine learning methods are being increasingly adopted in materials research.
The nuclear materials community must combine the power of data analytics with physicsbased models to gain new insights into materials degradation. The emergence of newcomputer architectures poses a challenge in relation to the reprogramming of existingcodes, but it also offers the promise of reaching length and time scales well beyond what
is possible at present.

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