1.12 Atomic-Level Dislocation Dynamics in
Irradiated Metals
Y. N. Osetsky
Oak Ridge National Laboratory, Oak Ridge, TN, USA

D. J. Bacon
The University of Liverpool, Liverpool, UK

Published by Elsevier Ltd.

1.12.1
1.12.2
1.12.2.1
1.12.2.2
1.12.3
1.12.3.1
1.12.3.2
1.12.3.3
1.12.3.4
1.12.4
1.12.4.1
1.12.4.1.1
1.12.4.1.2
1.12.4.2
1.12.4.2.1
1.12.4.2.2
1.12.4.3
1.12.5
References

Introduction

Radiation Effects on Mechanical Properties
Radiation-Induced Obstacles to Dislocation Glide
Effects on Mechanical Properties
Method
Why Atomic-Scale Modeling?
Atomic-Level Models for Dislocations
Input Parameters
Output Information
Results on Dislocation–Obstacles Interaction
Inclusion-Like Obstacles
Temperature T ¼ 0 K
Temperature T > 0 K
Dislocation-Type Obstacles
Stacking fault tetrahedra
Dislocation loops
Microstructure Modifications due to Plastic Deformation
Concluding Remarks

Abbreviations

Symbols

bcc
DD
DL
EAM
ESM
fcc
hcp
IAP

MBP
MC
MD
MMM
MS
PAD
PBC
SFT
SIA
TEM

b
bL
D
G
L
t
T
vD
«
«˙
g
w
n
rD
t
tc
tP

Body-centered cubic

Dislocation dynamics
Dislocation loop
Embedded atom model
Equivalent sphere method
Face-centered cubic
Hexagonal close-packed
Interatomic potential
Many-body potential
Monte Carlo
Molecular dynamics
Multiscale materials modeling
Molecular statics
Periodic array of dislocations
Periodic boundary condition
Stacking fault tetrahedron
Self-interstitial atom
Transmission electron microscope

334
334
334
335
336
336
336
338
338
339
339
339

342
345
345
348
352
353
355

Dislocation Burgers vector
Dislocation loop Burgers vector
Obstacle diameter
Shear modulus
Dislocation length
Simulation time
Ambient temperature
Dislocation velocity
Shear strain
Shear strain rate
Stacking fault energy
Angle between dislocation segments
Poisson’s ratio
Dislocation density
Shear stress
Critical resolved shear stress
Peierls stress

333


334


Atomic-Level Dislocation Dynamics in Irradiated Metals

1.12.1 Introduction
Structural materials in nuclear power plants suffer
a significant degradation of their properties under
the intensive flux of energetic atomic particles
(see Chapter 1.03, Radiation-Induced Effects on
Microstructure). This is due to the evolution of
microstructures associated with the extremely high
concentration of radiation-induced defects. The high
supersaturation of lattice defects leads to microstructures that are unique to irradiation conditions. Irradiation with high energy neutrons or ions creates
initial damage in the form of displacement cascades
that produce high local supersaturations of point
defects and their clusters (see Chapter 1.11, Primary Radiation Damage Formation). Evolution of
the primary damage under the high operating temperature ($600 K to >1000 K) leads to a microstructure containing a high concentration of defect
clusters, such as voids, dislocation loops (DLs), stacking fault tetrahedra (SFTs), gas-filled bubbles, and
precipitates, and an increase in the total dislocation
network density (see Chapter 1.13, Radiation Damage Theory; Chapter 1.14, Kinetic Monte Carlo
Simulations of Irradiation Effects and Chapter
1.15, Phase Field Methods). These changes affect
material properties, including mechanical ones,
which are the subject of this chapter.
A general theory of radiation effects has not yet
been developed, and currently the most promising
way to predict materials behavior is based on multiscale materials modeling (MMM). In this framework,
phenomena are considered at the appropriate length
and times scales using specific theoretical and/or
modeling approaches, and the different scales are
linked by parameters/mechanisms/rules to provide

integrated information from a lower to a higher level.
Research on the mechanical properties of irradiated
materials, a topic of crucial importance for engineering
solutions, provides a good example of this. The lowest
level treats individual atoms by first principles, ab initio
methods, by solving Schro¨dinger’s equation for moving
electrons and ions. Calculations based on electron density functional theory (DFT) (see Chapter 1.08, Ab
Initio Electronic Structure Calculations for Nuclear
Materials) and its approximations, such as bond order
potentials (BOPs), can consider a few hundred atoms
over a very short time of femtoseconds to picoseconds.
Delivery of the resulting information to higher level
models can be achieved through effective interatomic
potentials (IAPs) (see Chapter 1.10, Interatomic
Potential Development), in which the adjustable

parameters are fitted to the basic chemical and structural properties obtained ab initio. IAPs are required for
atomic-scale modeling methods such as molecular statics (MS) and molecular dynamics (MD), which are used
to simulate millions of atoms. Time spanning nanoseconds to microseconds can be simulated by MD if the
number of atoms is not large (see Section 1.12.3.3).
This level can provide properties of point and
extended defects and interactions between them (see
Chapter 1.09, Molecular Dynamics). For mechanical
properties, important interactions are between moving
dislocations, which are responsible for plasticity, and
defects created by irradiation. Mechanisms and parameters determined at this level can then inform dislocation dynamics (DD) models based on elasticity
theory of the continuum (see Chapter 1.16, Dislocation Dynamics). DD models can simulate processes at
the micrometer scale and mesh with the mechanical
properties of larger volumes of material used in finite
elements (FEs) methods, that is, realistic models for the

design of core components. In this chapter, we consider
direct interactions at the atomic scale between moving
dislocations and obstacles to their motion.
The structure of the chapter is as follows. First,
we summarize the main features of the irradiation
microstructure of concern. Then we provide a short
description of atomic-scale methods applied to dislocation modeling, bearing in mind the details presented in Chapter 1.09, Molecular Dynamics. This
is followed by a review of important results from
simulations of the interaction between dislocations and obstacles. We then describe how dislocations
modify microstructure in irradiated metals. Finally,
we indicate some issues that will hopefully be
resolved by atomic-scale modeling in the near future.
Our main aim is to give the reader a general picture of
the phenomena involved and encourage further
research in this area. The following sources1–4 provide
a more general and deeper understanding of dislocations and modeling of plasticity issues.

1.12.2 Radiation Effects on
Mechanical Properties
1.12.2.1 Radiation-Induced Obstacles to
Dislocation Glide
Primary damage of structural materials is initiated by
the interaction of high-energy atomic particles with
material atoms to cause the energetic recoil and
displacement of primary knock-on atoms (PKAs).
PKA energy can vary from a few tens to tens of


Atomic-Level Dislocation Dynamics in Irradiated Metals


thousands of electron volt and the PKA spectrum can
be calculated for a particular position in a particular
installation.5 A PKA with energy >$1 keV gives rise
to a displacement cascade that produces a localized
distribution of point defects (vacancies and
self-interstitial atoms, SIAs) and their clusters (see
Chapter 1.11, Primary Radiation Damage Formation). Further evolution of these defects produces
specific microstructures that depend on the irradiation type, ambient temperature, and the material
and its initial structure (see Chapter 1.13, Radiation
Damage Theory). This radiation-induced microstructure consists typically of voids, gas-filled bubbles,
DLs (that can evolve into a dislocation network),
secondary-phase precipitates, and other extended
defects specific to the material, for example, SFTs
in face-centered cubic (fcc) metals. These features
are generally obstacles to the dislocation motion.
Their size is typically6 in the range of nanometers
to tens of nanometers and their number density may
reach $1024 mÀ3. At this density, the mean distance
between obstacles can be as short as $10 nm, and such a
high density of small defects, particularly those with
a dislocation character, makes the mechanisms of radiation effects on mechanical properties very different
from those due to other treatments.
1.12.2.2

Effects on Mechanical Properties

Radiation-induced defects, being obstacles to dislocation glide, increase yield and flow stress and reduce
ductility (see Chapter 1.04, Effect of Radiation on
Strength and Ductility of Metals and Alloys for
experimental results). Furthermore, if the obstacle density is sufficiently high to block dislocation motion, preexisting Frank-Reed dislocation sources are unable to

operate and plastic deformation requires operation of
sources that are not active in the unirradiated state.
These new sources operate at much higher stress and
give rise to new mechanisms such as yield drop, plastic
instability, and formation of localized channels with
high dislocation activity and high local plastic deformation. Understanding these phenomena is necessary for
predicting material behavior under irradiation and the
design and selection of materials for new generations of
nuclear devices.
Obstacles induced by irradiation affect moving
dislocations in a variety of ways, but can be best
categorized as one of two types, namely inclusionlike obstacles and those with dislocation properties.
The first type includes voids, bubbles, and precipitates, for example. They usually have relatively

335

short-range strain fields and their properties may not
be changed significantly by interaction with dislocations. (Copper precipitates in iron are an exception to
this – see Sections 1.12.4.1.1–1.12.4.1.2.) Those that
are not impenetrable are usually sheared by the dislocation and steps defined by the Burgers vector b of
the interacting dislocation are created on the obstacle–
matrix interface. Unstable precipitates, such as Cu in
Fe, may also suffer structural transformation during the
interaction, which can change their properties. These
obstacles do not usually modify dislocations significantly, although they may cause climb of edge dislocations (see Sections 1.12.4.1.1–1.12.4.1.2). Their
main effect is to create resistance to dislocation glide.
Obstacles such as voids and bubbles are among the
strongest, and as a result of their high density, they
contribute significantly to radiation-induced hardening.7 Materials designed to exploit oxide dispersion
strengthening (ODS) are produced with a high concentration of rigid, impenetrable oxide particles, which

introduce extremely high resistance to dislocation
motion.8 These obstacles are also considered here as
their scale, typically a few nanometers, is similar to that
of obstacles formed under irradiation.
The second obstacle type consists of those with a
dislocation character, for example, DLs and SFTs,
and so dislocation reactions occur when they are
encountered by moving dislocations. Loops have relatively long-range strain fields and hence interact
with dislocations over distances much greater than
their size. SFTs are three-dimensional (3D) structures and have short-range strain fields. Loops with
perfect Burgers vectors are glissile, in principle,
whereas SFTs and faulted loops, for example, Frank
loops in fcc metals, are sessile. In addition to causing
hardening, the reaction of these defects with a gliding
dislocation can modify both their own structure and
that of the interacting dislocation. As will be demonstrated in Section 1.12.4.2, their effect depends very
much on the geometry of the interaction, that is, their
position and orientation relative to the moving dislocation, and the nature of the mutual dislocation segment that may form in the first stage of interaction.
The contribution of these obstacles to strengthening
can be significant, for their density can be high.
Modification of irradiation-induced microstructure
due to plastic deformation is an additional possibly
important effect. If mechanical loading occurs during
irradiation, it can contribute significantly to the overall
microstructure evolution and therefore to change in
material properties. Accumulation of internal stress
during irradiation is unavoidable in real structural


336


Atomic-Level Dislocation Dynamics in Irradiated Metals

materials and so this effect should not be ignored. The
effects of concurrent deformation and irradiation on
microstructure are far from clear, for only a few experimental studies of in-reactor deformation have been
performed.9 This area, therefore, provides a good
example of how atomic-scale modeling can help in
understanding a little-studied phenomenon.

dislocation is effectively infinite in length. If
the model contains one obstacle, the length, L, of
the model in the periodic direction represents the
center-to-center obstacle spacing along an infinite
row of obstacles. It is the treatment of the boundaries
in the other two directions that distinguishes one
method from another. A versatile atomic-scale
model should allow for the following.10

1.12.3 Method

1. Reproduction of the correct atomic configuration
of the dislocation core and its movement under the
action of stress.
2. Application of external effects such as applied
stress or strain, and calculation of the resultant
response such as strain (elastic and plastic) or
stress and crystal energy.
3. Possibility of moving the dislocation over a long
distance under applied stress or strain without

hindrance from the model boundaries.
4. Simulation of either zero or non-zero temperatures.
5. Possibility of simulating a realistic dislocation
density and spacing between obstacles.
6. Sufficiently fast computing speed to allow simulation of crystallites in the sizes range where size
effects are insignificant.

1.12.3.1

Why Atomic-Scale Modeling?

First principle ab initio methods for self-consistent
calculation of electron-density distribution around
moving ions provide the most accurate modeling techniques to date. They take into account local chemical
and magnetic effects and provide significant potential
for predicting material properties. They are used with
success in applications where the properties are limited
to the nanoscale, for example, microelectronics, catalysis, nanoclusters, and so on (Chapter 1.08, Ab Initio
Electronic Structure Calculations for Nuclear
Materials). A typical scale for this is of the order of a
few nm. However, this leaves a significant gap between
ab initio methods and those required to model properties of bulk materials arising from radiation damage.
These involve phenomena acting over much longer
scales, such as interactions between mobile and sessile
defects, their thermally activated transport, their
response to internal and external stress fields and gradients of chemical potential. Models for bulk properties
are based on continuum treatments by elasticity, thermal conductivity, and rate theories where global defect
properties such as formation, annihilation, transport,
and interactions are already parameterized at continuum level. The only technique that currently bridges
the gap in the scales between ab initio and the continuum is computer simulation of a large system of atoms,

up to 106–108. Atoms move as in classical Newtonian
dynamics due to effective forces between them calculated from empirical interatomic potentials and
respond to internal and external fields due to temperature, stress, and local imperfections. Atomic-scale modeling has provided the results presented in this chapter.
In the following section, we present a short description
of typical models for simulation of dislocations and
their interactions with defects formed by radiation.
1.12.3.2 Atomic-Level Models for
Dislocations
All models use periodic boundary conditions in
the direction of the dislocation line so that the

A comprehensive review of models developed so far
is to be found in Bacon et al.,4 and so here we merely
present a short summary of the pros and cons of some
models used most commonly. Historically, the earliest
models consisted of a small region of mobile atoms
surrounded in the directions perpendicular to the
dislocation direction by a shell of atoms fixed in the
positions obtained by either isotropic or anisotropic
elasticity for displacements around the dislocation of
interest.11 This model was used successfully to investigate dislocation core structure and, being simple and
computationally efficient, can use a mobile region
large enough to simulate interaction between static
dislocations and defects and small defect clusters. Its
main deficiencies are its inability to model dislocation
motion beyond a few atomic spacings because of the
rigid boundaries (condition 3) and its restriction to
temperature T ¼ 0 K (condition 4).
The desirability of allowing for elastic response of
the boundary atoms due to atomic relaxation in the

inner region, for example, when a dislocation moves,
has led to the development of several quasicontinuum
models. The elastic response can be accounted for by
using either a surrounding FE mesh or an elastic
Green’s function to calculate the response of boundary atoms to forces generated by the inner region.
Such models are accurate but computationally


Atomic-Level Dislocation Dynamics in Irradiated Metals

inefficient and have not found wide application
so far.4 Furthermore, their use for simulation of
T > 0 K (condition 4) is still under development.12
Nevertheless, quasicontinuum models, especially
those based on Green’s function solutions, can be
employed in applications where calculation of forces
on atoms is computationally expensive and a significant reduction in the number of mobile atoms is
desirable.13
The models now most widely applied to simulate
dislocation behavior in metals are based on the periodic array of dislocations (PAD) scheme first introduced for simulating edge dislocations.14,15 In this,
periodic boundary conditions are applied in the
direction of dislocation glide as well as along the
dislocation line, that is, the glide plane is periodic.
This means that the dislocation is one of a periodic,
2D array of identical dislocations. The success of
PAD models is because of their simplicity and good
computational efficiency when applied with modern
empirical IAPs, for example, embedded atom model
(EAM) type. They can be used to simulate screw,
edge, and mixed dislocations.4,10,16 With a PAD model

containing $106–107 mobile atoms, essentially all conditions 1–6 can be satisfied. Their ability to simulate
interactions with strong obstacles of size up to at
least $10 nm makes PAD models efficient for investigating dislocation–obstacle interactions relevant to a
radiation damage environment. Practically all important radiation-induced obstacles can be simulated on
modern computers using parallelized codes and most
can even be treated by sequential codes.
Details of model construction for different dislocations can be found elsewhere.4,10,16 Here we just
present an example of system setup for screw or edge
dislocations in bcc and fcc metals interacting with
dislocations loops and SFTs, as presented in Figure 1.

337

There are two types of DL in an fcc metal: glissile
perfect loops with bL ¼ 1/2h110i and sessile Frank
loops with bL ¼ 1/3h111i. There are two types of
glissile loop with Burgers vectors 1/2h111i and
h100i in a body-centered cubic (bcc) metal.
Visualizing interaction mechanisms is a strong
feature of atomic scale modeling. The main idea is
to extract atoms involved in an interaction and visualize them to understand the mechanism. Usually
these atoms are characterized by high energy, local
stresses, and lattice deformation. The techniques
used are based on analysis of nearest neighbors,17
central symmetry parameter,18 energy,19 stress,20 displacements,10 and Voronoi polyhedra.21 A relatively
simple and fast technique, for example, was suggested
for an fcc lattice.16 It is based on comparison of
position of atoms in the first coordination of an
atom with that of a perfect fcc lattice. If all 12 neighbors of the analyzed atom are close to that position, it
is assigned to be fcc. If only nine neighbors correspond to perfect fcc coordination, the atom is taken to

be on a stacking fault. Other numbers of neighbors
can be attributed to different dislocations. Modifications of this method have been successfully applied in
hexagonal close-packed (hcp)22 and bcc23 crystals.
Another improvement of this method for MD simulation at T > 0 K was introduced24 in which the above
analysis was applied periodically (every 10–50 time_ and over a certain
steps depending on strain rate e)
time period (100–1000 steps). A probability of an
atom to be in different environment was estimated
and the final state was assigned to the maximum over
the analyzed period. Such a probability analysis can
be applied to other characteristics such as energy or
stress excess over the perfect state and it provides a
clear picture when the majority of thermal fluctuations are omitted.

L

L

t

t
Screw

L
S

S
t

Ͻ111Ͼ


Obstacle:
loop or SFT

Ad

b

L

b = 1/3

dC

Edge

b = Ͻ100Ͼ

dC

t

b = 1/2 Ͻ111Ͼ

D
b = 1/2 Ͻ110Ͼ
A
C
B


fcc
bcc
Figure 1 Examples of periodic array of dislocation model setup for screw and edge dislocations in body-centered cubic
and face-centered cubic crystals. Examples of dislocation loops, a stacking fault tetrahedron, and sense of applied
resolved shear stress, t, are indicated.


338

Atomic-Level Dislocation Dynamics in Irradiated Metals

1.12.3.3

Input Parameters

The IAP is a crucial property of a model for it
determines all the physical properties of the simulated system. Discussion on modern IAPs is presented in Chapter 1.10, Interatomic Potential
Development and so we do not elaborate on this
subject here.
Another important property is the spatial scale of
the simulated system. The periodic spacing, Lg, in the
direction of the dislocation glide has to be large
enough to avoid unwanted effects due to interaction
between the dislocation and its periodic neighbors in
the PAD; 100–200b is usually sufficient.10 Furthermore, the model should be large enough to include
all direct interactions between the dislocation and
obstacle and the major part of elastic energy that
may affect the mechanism under study. MD simulations have demonstrated that a system with a few
million atoms is usually sufficient to satisfy conditions for simulating interaction between a dislocation
and an obstacle of a few nanometers in size. The

biggest obstacles considered to date are 8 nm voids,23
10 nm DLs,25 and 12 nm SFTs26 in crystals containing
$6–8 million mobile atoms. It should be noted that
static simulation (T ¼ 0 K) usually requires the largest
system because most obstacles are stronger at low T
and the dislocation may have to bend strongly and
elongate before breaking free.23
Simulation of a dynamic system, that is, T > 0 K,
introduces another important and limiting factor for
atomic-scale study of dislocation behavior, namely the
simulation time, t, which can be achieved with the
computing resource available. Under the action of
increasing strain applied to the model, the time to
reach a given total strain determines the minimum
applied strain rate, e,_ that can be considered. This
parameter defines in turn the dislocation velocity.
Consider a typical simulation of dislocation–obstacle
interaction in an Fe crystal, for which b ¼ 0.248 nm.
For L ¼ 41 nm, a model containing 2 Â 106 mobile atoms
would have a cross-section area of 5.73 Â 10À16 m2;
that is, a dislocation density rD ¼ 1.75 Â 1015 mÀ2.
For Lg ¼ 120b ¼ 29.8 nm, the model height perpendicular to the glide plane would be 19 nm. At
e_ ¼ 5 Â 106 sÀ1 , the steady state velocity, vD, of a
single dislocation estimated from the Orowan relation
_ D b is 11.6 m sÀ1. The time for the dislocation
vD ¼ e=r
to travel a distance Lg at this velocity would be 2.6 ns.
Thus, even if the dislocation breaks away from the
void without traversing the whole of the glide plane,
the total simulated time would be $1 ns.


The lowest strain rate for dislocation-obstacle
interaction reported so far27 is 105 sÀ1 and it resulted
in vD ¼ 48 cm sÀ1. This strain rate is about six to ten
orders of magnitude higher than that usually applied
in laboratory tensile experiments and more than ten
orders higher than that for the creep regime. This
presents an unresolvable problem for atomic-scale
modeling and even massive parallelization gains only
three or four orders in e_ or vD. We conclude that the
possibilities of modern atomic-scale modeling are limited to dislocation velocity of at least $0.1 cm sÀ1.
Nevertheless, atomic-scale modeling, particularly
using MD (T > 0 K), is a powerful, and sometimes
the only, tool for investigating processes associated
with lattice defect interactions and dynamics. The
main advantage of MD is that, if applied properly to
a large enough system, it includes all classical phenomena such as evolution of the phonon system and
therefore free energy, rates of thermally activated
defect motion, and elastic interactions. It is, therefore,
one of the most accurate techniques for investigating
the behavior of large atomic ensembles under different conditions. We reemphasize that the realism of
atomic-scale modeling is limited mainly by the validity of the IAP and restricted simulation time.
1.12.3.4

Output Information

Atomic-scale methods and particularly MD can provide a wide range of valuable information on the
processes simulated. The most important are
1. Information on the physical state of the system.
This includes temperature and stress and their

distribution; displacement of atoms and their
transport; interaction energy and therefore force
between defects; and evolution of internal, elastic,
and free energies. Extraction of this information is
well understood and procedures can be found in
Chapter 1.04, Effect of Radiation on Strength
and Ductility of Metals and Alloys;
Chapter 2.13, Properties and Characteristics of
ZrC; Chapter 5.01, Corrosion and Compatibility
and Chapter 1.09, Molecular Dynamics.
2. Detail of atomic mechanisms. This includes analysis of the position and environment of individual
atoms based on calculation of their energy, site
stress, or local atomic configuration. Atoms can
then be identified with particular features such as
constituents of defect clusters, stacking faults, dislocation cores, and so on. Having this information
at particular times provides unique knowledge of


Atomic-Level Dislocation Dynamics in Irradiated Metals

defect structure,
transformation.

motion,

interactions,

and

The information summarized in 1 and 2 can be

used to determine how the mechanisms involved
depend on parameters such as obstacle type and
size and dislocation type, material temperature, and
applied stress or strain.

1.12.4 Results on Dislocation–
Obstacles Interaction
1.12.4.1

Inclusion-Like Obstacles

1.12.4.1.1 Temperature T ¼ 0 K

Voids in bcc and fcc metals at T ¼ 0 K and >0 K are
probably the most widely simulated obstacles of this
type. Most simulations were made with edge dislocations.10,25–34 A recent and detailed comparison of
strengthening by voids in Fe and Cu is to be found
in Osetsky and Bacon.34 Examples of stress–strain
curves (t vs. e) when an edge dislocation encounters
and overcomes voids in Fe and Cu at 0 K are presented in Figures 2 and 3, respectively. The four
distinct stages in t versus e for the process are
described in Osetsky and Bacon10 and Bacon and
Osetsky.23 The difference in behavior between the
two metals is due to the difference in their dislocation
core structure, that is, dissociation into Shockley
partials in Cu but no splitting in Fe (for details see
Osetsky and Bacon34).

339


Under static conditions, T ¼ 0 K, voids are strong
obstacles and at maximum stress, an edge dislocation
in Fe bows out strongly between the obstacles, creating parallel screw segments in the form of a dipole
pinned at the void surface. A consequence of this is
that the screw arms cross-slip in the final stage
when the dislocation is released from the void surface
and this results in dislocation climb (see Figure 4),
thereby reducing the number of vacancies in the void
and therefore its size. In contrast to this, a Shockley
partial cannot cross-slip. Partials of the dissociated
dislocation in Cu interact individually with small
voids whose diameter, D, is less than the partial
spacing ($2 nm), thereby reducing the obstacle
strength. Stress drops are seen in the stress–strain
curve in Figure 3. The first occurs when the leading
partial breaks from the void; the step formed by this
on the exit surface is a partial step 1/6h112i and the
stress required is small. Breakaway of the trailing
partial controls the critical stress tc. For voids with
D larger than the partial spacing, the two partials
leave the void together at the same stress. However,
extended screw segments do not form and the dislocation does not climb in this process. Consequently,
large voids in Cu are stronger obstacles than those of
the same size in Fe, as can be seen in Figure 6 and the
number of vacancies in the sheared void in Cu is
unchanged.
Cu-precipitates in Fe have been studied extensively23,27–29 due to their importance in raising the
yield stress of irradiated pressure vessels steels35 and

200


Shear stress (MPa)

150

100

50

0

-50

D (nm):
0.0

0.9 1.0 1.5 2.0 3.0 4.0 5.0
0.5

1.0

6.0
1.5

Strain (%)
Figure 2 Stress–strain dependence for dislocation–void interaction in Fe at 0 K with L ¼ 41.4 nm. Values of D are
indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from
Taylor and Francis Ltd. ().



340

Atomic-Level Dislocation Dynamics in Irradiated Metals

350
300

Shear stress (MPa)

250
200
150
100
50
0
-50

D (nm):

1 1.5 2

3

4 5 6

7

8

-100

0.0

0.2

0.4

0.6
0.8
Strain (%)

1.0

1.2

Figure 3 Stress–strain dependence for dislocation–void interaction in Cu at 0 K with L ¼ 35.5 nm. Values of D are
indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from
Taylor and Francis Ltd. ().

20
5.0 nm

16

4.0 nm

[110], a

12

3.0 nm

8
2.0 nm
4

1.0 nm
0.9 nm

0
-40

-30

-20

-10

0
[112], a

10

20

30

40

Figure 4 [111] projection of atoms in the dislocation core showing climb of a1/2h111i{110} edge dislocation after
breakaway from voids of different diameter in Fe at 0 K. Climb-up indicates absorption of vacancies. The dislocation slip
plane intersects the voids along their equator. From Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2003, 323, 268.

Copyright (2003) with permission from Elsevier.

the availability of suitable IAPs for the Fe–Cu system.36 These precipitates are coherent with the surrounding Fe when small, that is, they have the bcc
structure rather than the equilibrium fcc structure of
Cu. Thus, the mechanism of edge dislocation interaction with small Cu precipitates is similar to that of
voids in Fe. The elastic shear modulus, G, of bcc Cu is
lower than that of the Fe matrix and the dislocation is
attracted into the precipitate by a reduction in its
strain energy. Stress is required to overcome the

attraction and to form a 1/2h111i step on the
Fe–Cu interface. This is lower than tc for a void, however, for which G is zero and the void surface energy
relatively high. Thus, small precipitates ( 3 nm) are
relatively weak obstacles and, though sheared, remain
coherent with the bcc Fe matrix after dislocation
breakaway. tc is insufficient to draw out screw segments and the dislocation is released without climb.
The Cu in larger precipitates is unstable, however,
and their structure is partially transformed toward


8

8

6

6

4


4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8
-50

-45

-40

<111>, ao

-35

4

341

_
[112], ao

_
<112>, ao

Atomic-Level Dislocation Dynamics in Irradiated Metals

2 _ 0 -2 -4
<110>, ao

Figure 5 Position of Cu atoms in four consecutive ð110Þ planes through the center of a 4 nm precipitate in Fe after
dislocation breakaway at 0 K. The figure on the right shows the dislocation line in [111] projection after breakaway; climb to
the left/right indicates absorption of vacancies/atoms by the dislocation. From Bacon, D. J.; Osetsky, Yu. N. Philos. Mag.
2009, 89, 3333. With permission from Taylor and Francis Ltd. ().

1.2
- Voids in Fe
1.0

- Cu-precipitates in Fe
- Voids in Cu


tC Gb/L

0.8

0.6

Dvoid = 1.52

0.4

DOrowan = 0.77

0.2

0.0
5

10

15

20

25

30

(D-1 + L-1)-1, b
Figure 6 Critical stress tc (in units Gb/L) versus the harmonic mean of D and L (unit b) for voids and Cu-precipitates in

Fe and voids in Cu at 0 K.

the more stable fcc structure when penetrated by
a dislocation at T ¼ 0 K. This is demonstrated in
Figure 5 by the projection of atom positions in four
{110} atomic planes parallel to the slip plane near the
equator of a 4 nm precipitate after dislocation breakaway. In the bcc structure, the {110} planes have a
twofold stacking sequence, as can be seen by the
upright and inverted triangle symbols near the outside of the precipitate, but atoms represented by

circles are in a different sequence. Atoms away from
the Fe–Cu interface are seen to have adopted a
threefold sequence characteristic of the {111} planes
in the fcc structure. This transformation of Cu structure, first found in MS simulation of a screw dislocation penetrating a precipitate,37,38 increases the
obstacle strength and results in a critical line shape
that is close to those for voids of the same size.34
Under these conditions, a screw dipole is created


342

Atomic-Level Dislocation Dynamics in Irradiated Metals

and effects associated with this, such as climb of the
edge dislocation on breakaway described above for
voids in Fe, are observed.23,27
The results above were obtained at T ¼ 0 K by MS,
in which the potential energy of the system
is minimized to find the equilibrium arrangement
of the atoms. The advantage of this modeling is that

the results can be compared directly with continuum
modeling of dislocations in which the minimum elastic
energy gives the equilibrium dislocation arrangement.
An early and relevant example of this is provided by
the linear elastic continuum modeling of edge and
screw dislocations interacting with impenetrable Orowan particles39 and voids.40 By computing the equilibrium shape of a dislocation moving under increasing
stress through the periodic row of obstacles, as in the
equivalent MS atomistic modeling, it was shown that
the maximum stress fits the relationship
tc ¼

Gb
½lnðDÀ1 þ LÀ1 Þ þ DŠ
2pAL

½1Š

where G is the elastic shear modulus and D is an
empirical constant; A equals 1 if the initial dislocation
is pure edge and (1 À n) if pure screw, where n is
Poisson’s ratio. Equation [1] holds for anisotropic elasticity if G and n are chosen appropriately for the slip
system in question, that is, if Gb2/4p and Gb2/4pA are
set equal to the prelogarithmic energy factor of screw
and edge dislocations, respectively.39,40 The value of G
obtained in this way is 64 GPa for h111i{110} slip in Fe
and 43 GPa for h110i{111} slip in Cu.41
The explanation for the D- and L-dependence of
tc is that voids and impenetrable particles are ‘strong’
obstacles in that the dislocation segments at the obstacle surface are pulled into parallel, dipole alignment
at tc by self-interaction.39,40 (Note that this shape

would not be achieved at this stress in the line-tension
approximation where self-stress effects are ignored.)
For every obstacle, the forward force, tcbL, on the
dislocation has to match the dipole tension, that is,
energy per unit length, which is proportional to ln(D)
when D (L and ln(L) when L ( D.39 Thus, tcbL
correlates with Gb2ln(DÀ1 þ LÀ1)À1. The correlation
between tc obtained by the atomic-scale simulations
above and the harmonic mean of D and L, as in
eqn [1], is presented in Figure 6. A fairly good
agreement can be seen across the size range down
to about D < 2 nm for voids in Fe and 3–4 nm for the
other obstacles. The explanation for this lies in the fact
that in the atomic simulation, as in the earlier continuum modeling, obstacles with D > 2–3 nm are strong
at T ¼ 0 K and result in a dipole alignment at tc.

Smaller obstacles in Fe, for example, voids with D < 2
nm and Cu precipitates with D < 3 nm, are too weak
to be treated by eqn [1]. Thus, the descriptions above
and the data in Figure 6 demonstrate that the atomicscale mechanisms that operate for small and large
obstacles depend on their nature and are not predicted by simple continuum treatments, such as the
line-tension and modulus-difference approximations
that form the basis of the Russell–Brown model of Cuprecipitate strengthening of Fe,42 often used in predictions and treatment of experimental observations.
The importance of atomic-scale effects in interactions between an edge dislocation and voids and
Cu-precipitates in Fe was recently stressed in a series
of simulations with a variable geometry.43 In this
study, obstacles were placed with their center
at different distances from the dislocation slip plane.
An example of the results for the case of 2 nm void at
T ¼ 0 K is presented in Figure 7. The surprising

result is that a void with its center below the dislocation slip plane is still a strong obstacle and may
increase its size after the dislocation breaks away.
This can be seen in Figure 7, where a dislocation
line climbs down absorbing atoms from the void
surface. More details on larger voids, precipitates,
and finite temperature effects can be found in
Grammatikopoulos et al.43
1.12.4.1.2 Temperature T > 0 K

In contrast to the T ¼ 0 K simulations above, modeling by MD provides the ability to investigate
temperature effects in dislocation–obstacle interaction. (The limit on simulation time discussed in

Glide plane
Dz =

R

R/2

0

-R/2

-R

Climb-up - atoms left inside void

Climb-down - atoms taken off void
Figure 7 Schematic representation of the configurations
studied for voids of radius R in Grammatikopoulos et al.43

and the corresponding shape of the edge dislocation line
seen in [111] projection after breakaway. Dz is the distance
of the center of the void from the dislocation glide plane.
From Grammatikopoulos, P.; Bacon, D. J.; Osetsky, Yu. N.
Model. Simulat. Mater. Sci. Eng. 2011, 19, 015004. With
permission from IOP Publishing Ltd.


Atomic-Level Dislocation Dynamics in Irradiated Metals

Section 1.12.3.3 prevents study of the creep regime
controlled by dislocation climb.) Results on the temperature dependence of tc from simulation of interaction between an edge dislocation and 2 and 6 nm
voids in Fe,29,30,34 Cu-precipitates in Fe,27,29 and voids
in Cu30,34 are presented in Figure 8. In general, the
strength of all the obstacles becomes weaker with
increasing temperature, although the mechanisms
involved are not the same for the different obstacles.
The temperature-dependence of void strengthening
in Fe has been analyzed by Monnet et al.44 using a
mesoscale thermodynamic treatment of MD data in
the point obstacle approximation to estimate activation energy and its temperature dependence. In this
way, the obstacle strength found by atomic-scale modeling can be converted into a mesoscale parameter to
be used in higher level modeling in the multiscale
framework. More investigations are required to define
mesoscale parameters for more complicated cases
such as voids in Cu and Cu-precipitates in Fe. Void
strengthening in Cu exhibits specific behavior in
which the temperature-dependence is strong at low
T < 100 K but rather weak at higher T (for more
details see Figure 8 in Osetsky and Bacon34). The

reason for this is as yet unclear.
Interestingly, MD simulation has been able to
shed light on thermal effects in strengthening due
to Cu-precipitates in Fe, as in Figure 8 (for more
details see Figure 5 in Bacon and Osetsky23). Small
precipitates, D < 3 nm, are stabilized in the bcc
coherent state by the Fe matrix, as noted above
1.0

2 and 6 nm obstacles:
.
ε = 5 × 106 s-1
Voids in Cu

0.8

Voids in Fe

tc Gb/L

Cu-prpt in Fe
0.6

6 nm

0.4

2 nm

0.2


0.0
0

100

200

300

400

500

600

700

T (K)

Figure 8 Plot of tc versus T for voids and Cu-precipitates
in Fe and voids in Cu. D is as indicated, L ¼ 41.4 nm, and
e_ ¼ 5 Â 106 sÀ1 .

343

for T ¼ 0 K, and are weak, shearable obstacles. The
resulting temperature-dependence of tc is small.
Larger precipitates were seen to be unstable at
T ¼ 0 K with respect to a dislocation-induced transformation toward the fcc structure. This transformation is driven by the difference in potential energy of

bcc and fcc Cu. The free energy difference between
these two phases of Cu decreases with increasing T
until a temperature is reached at which the transformation does not occur. Thus, large precipitates are
strong obstacles at low T and weak ones at high T.
This is reflected in the strong dependence of tc on T
shown in Figure 8. More explanation of this effect
can be found in Bacon and Osetsky.23 These simulation results showing the different behavior of small
and large Cu-precipitates suggest that the yield stress
of underaged or neutron-irradiated Fe–Cu alloys,
which contain small, coherent Cu-precipitates, should
have a weak T-dependence, whereas that in an overaged or electron-irradiated alloy, in which the population of coherent precipitates has a larger size, should
be stronger. Some experimental observations support
this.45 One is a weak change in the temperature
dependence of radiation-induced precipitate hardening in ferritic alloys observed after neutron irradiation
when only small (<2 nm) precipitates are formed. The
other is the experimentally-observed temperature
and size dependence of deformation-induced transformation of Cu-precipitates in Fe.46
Other obstacles with inclusion properties, such as
gas-filled bubbles and other types of precipitates,
have been studied less intensively, and we present
just a few examples here.
The effect of chromium precipitates on edge
dislocation motion in matrices of either pure Fe
or Fe–10 at.% Cr solid solution was studied by
Terentyev et al.47 Cr and Cr-rich precipitates have
the bcc structure and are coherent with the matrix.
Unlike Cu-precipitates in Fe, G of Cr is higher than
that of both matrices and so the dislocation is
repelled by Cr precipitates. Under increasing strain,
the dislocation moves until it reaches the precipitate–

matrix interface where it stops until the stress reaches
the maximum, tc, just before the dislocation enters the
precipitate (see Figure 2 in Terentyev et al.47). The t
versus e behavior is similar to that for voids in Cu, but
without stress drops associated with partial dislocations, and no softening effects similar to voids and Cuprecipitates in Fe were observed. At tc, the dislocation
shears the obstacle without acquiring a double jog.
Only 2.8 and 3.5 nm precipitates in the size range
D ¼ 0.6–3.5 nm had tc comparable with values given


344

Atomic-Level Dislocation Dynamics in Irradiated Metals

by eqn [2] (see Figure 4 in Terentyev et al.47); the
others were much weaker. Separate contributions
from the chemical strengthening (CS) and shear modulus difference (SMD) between Fe and Cr were estimated and their sum was found to be close to the tc
found in simulation. It was also found that tc for the
alloy with Cr precipitates in an Fe–Cr solid solution is
the sum of tc for the same precipitate in a pure Fe
matrix and the maximum stress for glide of the dislocation motion in the Fe–Cr solid solution alone.
Helium-filled bubbles created by vacancies and
helium formed in transmutation reactions are common features of the irradiated microstructure of
structural materials (see Chapter 1.13, Radiation
Damage Theory). However, there is a lack of information on the properties of He bubbles and their
contribution to changes in mechanical properties.
The main problem is the uncertainty regarding the
equilibrium state of bubbles of different sizes and at
different temperatures, that is, their He-to-vacancy
ratio (He/Vac). A small (0.5 M-atom) model was

used48–50 to simulate interaction between an edge
dislocation and a row of 2 nm cavities with He/Vac
ratios of up to 5 in Fe at T between 10 and 700 K. It
was found that tc has a nonmonotonic dependence on
the He/Vac ratio, dislocation climb increases with
this ratio, and interstitial defects are formed in the
vicinity of the bubble. Recent work to clarify the
equation of state of bubbles using a new Fe–He
three-body interaction potential51 has shown that
the equilibrium concentration of He is much lower
than expected; for example the He/Vac ratio is $0.5
for a 2 nm bubble at 300 K in Fe.52 Simulation of an
edge dislocation interacting with 2 nm bubbles using
the new potential for He/Vac ratio in the range 0.2–2
and T between 100 and 600 K has now been performed53 and preliminary conclusions drawn. The
dislocation interaction with underpressurized bubbles (He/Vac < 0.5) is similar to that with voids
described above, that is, the dislocation climbs up
by absorbing some vacancies on breakaway and tc
increases with increasing values of He/Vac ratio up
to 0.5. The interaction with overpressurized bubbles
(He/Vac > 0.5) is different. The dislocation climbs
down and tc decreases with increasing value of
He/Vac ratio. At the highest ratio, the dislocation
stress field induces the bubble to emit interstitial
Fe atoms from its surface into the matrix toward the
dislocation before it makes contact. The bubble pressure is reduced in this way and interstitials are
absorbed by the dislocation as a double superjog. Equilibrium bubbles are therefore the strongest. Some of

these conclusions, such as formation of interstitial
clusters around bubbles with high He/Vac ratios, are

similar to those observed earlier,48–50 others are not.
More modeling is necessary to clarify these issues.
As noted in Section 1.12.2.2, impenetrable obstacles such as oxide particles and incoherent precipitates represent another class of inclusion-like
features. Although these obstacles are usually preexisting and not produced by irradiation, they are considered to be of potential importance for the design
of nuclear energy structural materials and should
be considered here. Atomic-level information on
their effect on dislocations is still poor, however,
and we can only refer to some recent work on this.
The interaction between an edge dislocation and a
rigid, impenetrable particle in Cu was simulated by
Hatano54 using the Cu–Cu IAP as for the Fe–Cu
system36 and a constant strain rate of 7 Â 106 sÀ1 at
T ¼ 300 K. The particle was created by defining a
spherical region in which the atoms were held immobile relative to the surrounding crystal. The Hirsch
mechanism2 was found to operate. In the sequence
shown in Figures 1 and 2 of Hatano,54 several stages
can be observed such as (1) the dislocation under
stress approaching the obstacle from the left first
bows round the obstacle to form a screw dipole; (2)
the screw segments cross-slip on inclined {111}
planes at tc; (3) they annihilate by double cross-slip,
allowing the dislocation, now with a double superjog,
to bypass the obstacle; (4) a prismatic loop with the
same b is left behind and (5) the dragged superjogs
pinch-off as the dislocation glides away, creating a
loop of opposite sign to the first on the right of the
obstacle. tc varies with D and L as predicted by the
continuum modeling that led to eqn [1], but is over 3
times larger in magnitude. Hatano argues that this
could arise from either higher stiffness of a dissociated

dislocation or a dependence of tc on the initial position of the dislocation. It is also possible that the
requirement for the dislocation to constrict and the
absence of a component of applied stress on the crossslip plane results in a high value of tc.
Simulation of 2 nm impenetrable precipitates in
Fe has been carried out by Osetsky (2009, unpublished). The method is different from that used by
Hatano54 in that the precipitate, constructed from Fe
atoms held immobile relative to each other, was treated as a superparticle moving according to the total
force on precipitate atoms from matrix atoms. The
interaction mechanism observed is quite different
from those reported earlier for Cu,54 for the Hirsch
mechanism and formation of interstitial clusters does


Atomic-Level Dislocation Dynamics in Irradiated Metals

not occur. Instead, the mechanism observed was close
to the Orowan process, with formation of an Orowan
loop that either shrinks quickly if the obstacle is small
and spherical, or remains around it if D is large
(!4nm) or becomes elongated in the direction perpendicular to the slip plane. It is interesting to note
that if the model of a completely immobile precipitate in Hatano54 is applied to Fe, the same Hirsh
mechanism is observed as that in the earlier study.
Comparison of strengthening due to pinning of
a 1/2h111i{110} edge dislocation by 2 nm spherical
obstacles of different nature simulated at 300 K is
presented in Figure 9. One can see that the coherent
Cu precipitate is the weakest whereas a rigid impenetrable precipitate when Orowan loop is stabilized by
its shape is the strongest. A surprising result is that an
equilibrium He-bubble is a stronger obstacle than the
equivalent void. The reason for this is not clear yet.

Little is known on the interactions between screw
dislocations and inclusion-like obstacles. In fact,
we are aware of only one published study of screw
dislocation–void interaction in fcc Cu.55 It was found
that voids are quite strong obstacles and their strength
and interaction mechanisms are strongly temperature
dependent. Thus at low temperature, the 1/2h110i
{111} screw dislocation keeps its original slip plane
250
MD modeling in Fe:
Edge dislocation 1/2Ͻ111Ͼ{110}
Crystal
2 074 000 atoms
Temperature
300 K
Obstacle size
2 nm
Obstacle spacing 42 nm

225

tc (MPa)

200
175
150
125
100
75
50


Cu-prpt

Void

He-bubble
He/Vac = 0.5

Rigid
particle

Orowan
mechanism

Figure 9 Critical stress for an edge dislocation
penetrating through different 2 nm obstacles in Fe at
T ¼ 300 K. L ¼ 41.4 nm and e_ ¼ 5 Â 106 sÀ1 .
Table 1

345

when it breaks away from the void. However, at high
T > 300 K, cross-slip is activated and plays an important role in dislocation–void interaction. Several
depinning mechanisms involving dislocation crossslip on the void surface were simulated and formation
of DLs was observed in some cases. Interestingly, the
void strength increases with increasing temperature
and the authors explain this by changing interaction
mechanisms. Intensive cross-slip was observed54 that
propagated through the periodic boundaries along
the dislocation line direction, with the result that

the void was interacting with its images. Similar
effects have been observed in the interaction of a
screw dislocation and SIA loops and SFTs, and the
possible significance of this for understanding the
simulation results has been discussed elsewhere.4
1.12.4.2

Dislocation-Type Obstacles

Extensive simulations of interactions between moving
dislocations and dislocation-like obstacles such as
DLs and SFTs has demonstrated that the reactions
involved follow the general rules of dislocation–
dislocation reaction, for example, Frank’s rule for
Burgers vectors,1,2 even though the reacting segments
are of the nanometer scale in length. Results of these
interactions are in the range from no effect on both
dislocation and obstacle to complete disappearance of
the obstacle and significant modification of the dislocation. A detailed analysis of reactions was made for
SFTs an fcc metal56 and later for SIA loops in Fe.57 In
general, five types of reaction were identified, as summarized in Table 1. The outcomes in Table 1 were
observed for different obstacles under different reaction conditions such as interaction geometry, strain
rate, ambient temperature, and so on. We give some
examples in the following section.
1.12.4.2.1 Stacking fault tetrahedra

Reactions of type R1 have been observed for both
screw and edge dislocations and all the defects with
dislocation character. Interestingly, the strength
effect of this reaction varies from minimum to


Description of main reactions between dislocations and obstacles with dislocation character4

Reaction

Dislocation type

Overall result

R1
R2
R3
R4
R5

Edge or screw
Edge or screw
Edge
Screw
Edge and screw

Dislocation and obstacle remain unchanged
Obstacle changed but dislocation unchanged
Partial or full absorption of obstacle by edge dislocation (superjog formation)
Temporary absorption of obstacle by screw dislocation (helix formation)
Dislocation drags glissile defects


346


Atomic-Level Dislocation Dynamics in Irradiated Metals

maximum. For example, it is insignificant in the case
of a 1/2h110i{111} edge dislocation interacting with
an SFT58 and maximum for a screw dislocation interacting with a DL when the loop is fully absorbed into
a helical turn on the dislocation.57 The mechanism
for the way both the obstacle and dislocation remain
unchanged is different for each case. An edge dislocation interacting with an SFT close to its tip creates
a pair of ledges on its surface that are not stable
and annihilate athermally.56,58 An example of this
reaction is presented in Figure 10. If the dislocation
slip plane is far enough from the SFT tip in the
compressive region of the dislocation (for details
of geometrical definitions see Bacon et al.4), the
ledges can be stabilized.56,58,59 This can be seen in
Figure 11 (1). If the dislocation passes through the
SFT several times in the same slip plane, it can
detach the portion of the SFT above the slip plane,
as shown in Figure 11 (2–4). Both the above mechanisms are common for small SFTs, low T, fast dislocations, and the position of the SFT tip above the
slip plane of an edge dislocation. If, however, the
SFT tip is below the dislocation slip plane, and T is
high enough and the dislocation speed low enough,
D

reaction R3 can be activated. The stages of this reaction are presented in Figure 12. An example of
effects of SFT orientation and temperature for the
interaction of an edge dislocation with an SFT is
presented in Figure 13. In this study, the dislocation
slip plane intersected a 4.2 nm SFT through its geometrical center at the applied e_ ¼ 5 Â 106 sÀ1 in a
wide temperature range from 0 to 450 K.59 Reaction

R1 was observed (see Figure 10) at all temperatures
when the SFT was oriented with its tip up relative to
the dislocation slip plane (orange triangles up in
Figure 13) and at the two lowest temperatures
when it was oriented in the opposite sense. At
T ¼ 300 K and orientation with tip down, a couple
of ledges were formed on the SFT surface (see
Figure 11). It may be noted that the R2 mechanism
requires higher applied stress even though the temperature is increased. At higher T ¼ 450 K, the interaction mechanism is changed and the whole portion
of the SFT above the slip plane is absorbed by the
dislocation (Figure 12), that is, reaction R3 occurs,
creating a pair of superjogs on the dislocation line.
Some vacancies were also found to form to accommodate the glissile configuration of the superjogs.

dC
Ad
C

A

(a)

(b)

(c)

(d)

Figure 10 An example of reaction R1 for an edge dislocation passing through 4.2 nm SFT (136 vacancies) oriented with
apex above the slip plane in Cu at 300 K. From Osetsky, Yu. N.; Rodney, D.; Bacon, D. J. Philos. Mag. 2006, 86, 2295.

With permission from Taylor and Francis Ltd. ().


Atomic-Level Dislocation Dynamics in Irradiated Metals

0

1

2

3

347

4

Figure 11 Shear of a 2.4 nm SFT (45 vacancies) by an edge dislocation in Cu at 300 K. 0: Initial SFT; (1): creation of
two ledges (reaction R1); (2–4): evolution of the configuration due to additional passes of the dislocation. From Osetsky,
Yu. N.; Stoller, R. E.; Matsukawa, Y. J. Nucl. Mater. 2004, 329–333, 1228. Copyright (2004) with permission from Elsevier.

B

A
C

ad

dA


ab

dA

Cd
Cd

(a)

(b)
Ca

Cd
Cb

dA

ad

bd

Cd

(c)

(d)

Figure 12 An example of reaction R3 for an edge dislocation and 4.2 nm SFT (136 vacancies) with its tip below the
slip plane in Cu at 450 K. From Osetsky, Yu. N.; Rodney, D.; Bacon, D. J. Philos. Mag. 2006, 86, 2295. With permission from
Taylor and Francis Ltd. ().


This is discussed later in Section 1.12.4.3. More
details on interactions between screw and edge dislocations and SFT can be found elsewhere.60–63
In general, it can be concluded that the SFTs
created under irradiation, that is, <4 nm in size,6
are very stable objects and unlikely to be eliminated
by a simple interaction with either edge or screw
dislocations. Numerous attempts have been made to
find a mechanism responsible for formation of clear,
defect-free channels in irradiated fcc materials.64
One of the most used models considers absorption
of an SFT by screw and mixed 60 dislocations.65
The absorption by conversion of an SFT into a helical turn on a screw dislocation has been observed by
in situ transmission electron microscope (TEM)
deformation experiments,64,66–68 but only partial
absorption has been found in MD modeling. As
observed and discussed elsewhere,4,59 SFT separation
into parts due to temporary absorption of part of an
SFT as a helical turn and its expansion along a screw
dislocation line can occur, but complete annihilation
of an SFT has not been reproduced by atomic-scale

modeling of bulk material. Possible reasons, including
inability of MD to reproduce the whole set of experimental conditions such as stress state, scale, strain
rate, and so on, were discussed in Matsukawa et al.66–68
An alternative interpretation was suggested as a result
of MD simulation of interaction between a screw
dislocation and an SFT in a thin film,26 that is, the
conditions realized experimentally for in situ TEM
deformation. A thin film of fcc Cu was simulated and

a 1/2h110i screw dislocation with an end on each
surface was moved toward the SFT (size 12 and
18 nm) placed in the film center. A number of steps
occurred that resulted in elimination of the vast
portion of the SFT26:
1. The dislocation glided toward the SFT and partially absorbed it as a helical turn.
2. Edge segments of the turn glided toward the free
surfaces and were annihilated there.
3. Glide of edge segments provided mass transport;
in this particular case, transport of vacancies to the
free surfaces.


348

Atomic-Level Dislocation Dynamics in Irradiated Metals

175
4.2 nm SFT-edge disl.:
- Tip down

150

- Tip up

CRSS (MPa)

125

R2 (ledges formation)


100

75

R3 (partial absorption)

50
R1 (no SFT damage)
25
0

100

200
300
Temperature (K)

400

500

Figure 13 Temperature dependence of the critical stress for an edge dislocation penetrating through a 4.2-nm SFT (136
vacancies) in one of two different orientations in Cu. The type of reaction is indicated for all cases.

4. The screw character of the dislocation was
restored but in another slip plane. The dislocation
then glided away leaving behind a small portion of
the original SFT.
Although this mechanism provides a mechanistic

understanding of in situ TEM observations, it can
operate only between surfaces or interfaces where
the ends of the screw dislocation can cross-slip and,
therefore, cannot be applied to bulk material and is
unlikely to be responsible for the clear channel formation observed in bulk samples. We will return to
this question later.
1.12.4.2.2 Dislocation loops

DLs are common objects formed in metals under irradiation. Depending on metal properties and irradiation
conditions, loops of different types can be formed. They
are mainly interstitial in nature although vacancy loops
can appear in some metals under specific conditions.
Here we consider only interstitial DLs. The shortest
Burgers vector in bcc metals is bL ¼ 1/2h111i and
loops with this are the most common. In neutron and
heavy-ion irradiated Fe, loops can have bL ¼ h100i,69
particularly at T above about 300  C. Note that both
have a perfect Burgers vector and are glissile. The most
common loops in fcc metals have bL ¼ 1/2h110i and
are also perfect and glissile. In metals and alloys
with a low stacking fault energy, Frank loops with
bL ¼ 1/3h111i can form. They are faulted and sessile.
In the following section, we consider examples of
reactions R1–R4 in fcc and bcc metals.

1.12.4.2.2.1

Face-centered cubic metals

Interaction between a 1/2h110i{111} edge dislocation and a 1/2h110i SIA loop in Ni was first studied

by Rodney and Martin,70 and then followed a series
of more detailed investigations involving glissile and
sessile SIA loops and screw and edge dislocations.16,70–74 Reaction R3 was observed with small
loops having bL ¼ 1/2h110i intersecting the dislocation slip plane.70,75 Glissile clusters were attracted
by the dislocation core and absorbed there athermally, creating a pair of superjogs. Superjog segments have different structure, as depicted in
Figure 14, and to accommodate this a few vacancies
are formed, as in the case of the R3 reaction with an
SFT (see Figure 12). Each superjog has different
mobility, for example, the Lomer-Cottrell segment
on the left of Figure 14(b) has high Peierls stress.
Therefore, although the jogged dislocation continues gliding under applied stress, it has a significantly lower velocity because it now experiences
higher effective phonon drag.71
An interesting example of an R2 reaction was
observed during interaction between a 1/2h110i
screw dislocation and a 1/3h111i Frank loop.16 If
the loop is not too large and the dislocation is not
too fast, the dislocation can absorb the whole loop
into a helical turn (see stages in Figure 15). The
turn can expand only along the dislocation line and,
therefore, if applied stress is maintained, the helix
constricts; finally a perfect DL, with the same b as
the dislocation, is released as the screw breaks away.


Atomic-Level Dislocation Dynamics in Irradiated Metals

349

z
y

x

AC

(a)

bC

bC

db
dC

Ab

bC
Ad

db

Ab

Ad

db/AC

AC

Ab


dC

(b)
Figure 14 (a) Structure of an edge dislocation after absorption of a 37-interstitial loop in Ni at 300 K, resulting
in reaction R2 and (b) the corresponding Burgers vector geometry of the superjogs labeled using the Thompson
tetrahedron notation. Reprinted with permission from Rodney, D.; Martin, G. Phys. Rev. B 2000, 61, 8714. Copyright (2000)
by the American Physical Society.

Note that the same mechanism occurs when a screw
dislocation intersects a glissile 1/2h110i loop with
Burgers vector different from that of the dislocation,
that is, absorption into a helical turn and release of a
loop with the same Burgers vector as the dislocation.76 In the case when the Frank loop is large or
the dislocation (either screw or edge) is fast, the
dislocation simply shears the loop and creates a
step on its surface (see Figure 8 in Rodney16).
The probability of this reaction is higher for an
edge dislocation, whereas transformation of the
loop into a perfect loop is more probable for the
screw dislocation.73
The strengthening effect due to dislocation–SIA
loop interactions can be significant, especially when a
helical turn is formed on a screw dislocation. The
total contribution of dislocation–SIA loop interaction
to the flow stress under irradiation has not been
considered so far because of the large number of
possible reactions and the sensitivity of their outcomes in terms of mechanism and strengthening
effect to parameters such as interaction geometry,
loop size and Burgers vector, strain rate, and temperature. An estimate of the flow stress contribution for
the case of reaction R2 in Ni was made in Rodney and

Martin.75

1.12.4.2.2.2

Body-centered cubic metals

Many atomic-scale simulations of interaction
between a 1/2h111i{110} dislocation and an SIA
loop in bcc metals have been made, particularly
using EAM potentials for Fe. They have shown that
when the defects come into contact, a new dislocation
segment is formed with one of two possible Burgers
vectors 1/2h111i and h100i, and further evolution
depends on features such as loop size, b, position,
dislocation character, temperature, and strain rate.
Thus, many different outcomes have been observed
by atomic scale modeling, but here we can present
only a few common examples.
Reaction R3 is most common for small SIA
loops, that is, loop absorption on the dislocation
line resulting in a pair of superjogs. It occurs when
the loop is initially below the dislocation slip plane
(tension region of the dislocation strain field) and
bL ¼ 1/2h111i is inclined to the slip plane. An
important feature of the interaction is the ability
of such loops to glide quickly toward the core of
the approaching dislocation. SIA loops up to 5 nm
in diameter have been simulated.77–80 When small
loops (a few tens of SIAs) reach the core, they are
fully absorbed athermally creating a double superjog

of the size equivalent to the number of SIAs in the


350

Atomic-Level Dislocation Dynamics in Irradiated Metals

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15 Unfaulting of a 1/3h111i Frank loop by interaction with a screw dislocation in Ni at 300 K and transformation into
a helical turn on the dislocation line: (a) initial configuration, (b) first cross-slip, (c) and (d) successive cross-slip events,
(e) configuration at the end of unfaulting, (f) configuration after relaxation and elongation of the helical turn. From Rodney, D.
Nucl. Instrum. Meth. Phys. Res. B 2005, 228, 100. Copyright (2005) with permission from Elsevier.

loop. This process does not pin the dislocation and
these loops are very weak obstacles to dislocation
glide.77 Large loops (more than $100 SIAs) also
glide to make contact with the dislocation line but
are not absorbed athermally. Instead, a new dislocation segment is formed due to the following energetically favorable Burgers vector reaction between the
dislocation and loop:

1
1
½111Š À ½111Š ¼ ½010Š
2
2

½2Š

Five stages of the interaction are presented in
Figure 16 for a 5-nm loop containing 331 SIAs.
The Burgers vectors are indicated and Figure 16(b)
corresponds to the occurrence of the reaction of
eqn [2]. The new segment with b ¼ [010] cannot
glide in the dislocation slip plane ð110Þ and therefore
acts as a strong obstacle to further glide of the dislocation. Under increasing applied strain, the dislocation
bows out until two long segments of 1/2[111] screw
dislocations are formed as shown in Figure 16(c).
Figure 16(d) shows the same configuration in [111]


Atomic-Level Dislocation Dynamics in Irradiated Metals

(010)

(111)

t = 50 ps

(d)


[111]

_
(111)

_
(111)

(a)

(111)

351

(b)

t = 655 ps

t = 55 ps

(e)

(c)

t = 655 ps

t = 765 ps

Figure 16 Visualization of stages in the interaction between a 1=2½111Š edge dislocation and a 5 nm (331 SIAs) 1=2½111Š
loop in Fe at 300 K. From Bacon, D. J.; Osetsky, Yu. N.; Rong, Z. Philos. Mag. 2006, 86, 3921. With permission from

Taylor and Francis Ltd. ().

projection. High stress at junctions connecting the
dislocation line and loop, the remainder of the original loop, and the new segment induces the latter to
slip down on the ð101Þ plane, and glide of this [010]
segment over the loop surface results in the following
reaction with the remaining loop:
1
1
½111Š þ ½010Š ¼ ½111Š
½3Š
2
2
This concludes with the formation of a pair of superjogs on the original dislocation and results in complete absorption of the 5 nm loop. Large loops are
strong obstacles in this reaction, stronger than voids
with the same number of vacancies.4,80
It should be noted, however, that the interaction
just described depends rather sensitively on temperature because of the low mobility of screw segments
in the bcc metals. Cross-slip of the screw segments is
required to allow the [010] segment to glide down.
Simulations at T ¼ 100 K showed that although the
stages in Figure 16(a)–16(c) still occurred, the [010]
segment did not glide under the strain rate imposed
in MD and the screw dipole created by the bowing
dislocation was annihilated without the loop being
transformed according to eqn [3]. The resulting reaction was of type R1, for both dislocation and loop
were unchanged after dislocation breakaway.78 More
details and examples can be found.4,77–82
Competition between reactions R1, R2, and R3 for a
1/2h111i{110} edge dislocation and 1/2h111i and

h100i SIA loops has been considered in detail.83,84 We
cannot describe all the reactions here but some pertinent
features are underlined; note that the favorable Burgers
vector reaction between a 1/2h111i dislocation and a
h100i loop results in a 1/2h111i segment, for example,

1
1
½111Š þ ½100Š ¼ ½111Š
2
2

½4Š

Thus, a perfect loop with bL ¼ [100] can be converted
into a sessile complex of 1=2½111Š and 1=2½111Š loop
segments joined bya [100] dislocation segment.83 Similar
conjoined loop complexes were observed in simulations
of interactions between two glissile 1/2h111i loops85,86 in
bcc Fe. Competition between reaction R3 on one side
and R1 and R2 on the other was discussed in Bacon and
Osetsky.82 The earlier conclusion that small loops
( 1 nm) can be absorbed easily and not present strong
obstacles to dislocation glide was confirmed. The
strength and reaction mechanism for larger loops depends on their size and the loading conditions. At low
T 100 K, both 1/2h111i and h100i loops are strong
obstacles that are not absorbed by 1/2h111i dislocations.
As with obstacles that result in the true Orowan mechanism, the dislocation unpins by recombination of the
screw dipole, and the critical stress is determined by the
loop size, similar to eqn [1]. At higher Tand/or low strain

rate (<107 sÀ1), the mechanism changes from Orowanlike bypassing to complete loop absorption, irrespective
of bL. The absorption mechanism for R3 involves propagation of the reaction segment over the loop surface. This
requires cross-slip of the arms of the screw dipole drawn
out on the pinned dislocation and involves dislocation
reactions. Thermally activated glide and/or decomposition of the pinning segment, in turn, depends on loop
size, temperature, and bL. Therefore, the obstacle
strength of a 1/2h111i loop is, in general, higher than
that of a h100i loop because a h100i dislocation segment
associated with the former (see eqn [3], Figure 16 and
related text) is much less mobile than a 1/2h111i segment
involved in reactions with the latter (eqn [4]).


352

Atomic-Level Dislocation Dynamics in Irradiated Metals

Much less is known about interaction mechanisms
involving screw dislocations in bcc metals. We are
aware of only two studies reported to date that considered 1/2h111i and h100i SIA loops.87,88 The main
feature in these cases is the ability of a 1/2h111i
screw dislocation to absorb a complete or part loop
into a temporary helical turn before closing the turn
by bowing forward and breaking away. As in the case
of fcc metals described above, this provides a powerful route to reorienting the Burger vectors of different loops to that of the dislocation. An example of
such a reaction is visualized in Figure 17.88 Other
reactions observed in these studies87,88 include a
reforming of the original glissile loop into a sessile
complex of two segments having different b, as
described above (reaction R2), and complete restoration of the initial loop (reaction R1).

As is clear from the examples discussed above, the
obstacle strength associated with different mechanisms can depend strongly on parameters such as
loop size and Burgers vector, interaction geometry,
e_ and T. Unlike the situation revealed for inclusionlike obstacles discussed in Section 1.12.4.1, there
does not appear to be a simple correlation between
size and strength for loops. Some data related to particular sets of conditions can be found in Terentyev
et al.,83,84,88 as well as a comparison of the obstacle
strength of voids and DLs.84 There are cases when
loop strengthening is compatible to or even exceeds
that of voids containing the same number of point
defects, making DLs an important component of
radiation-induced hardening.
1.12.4.3 Microstructure Modifications due
to Plastic Deformation
In this section we consider some other cases, such as
those involved in reaction R5 ignored above, and
refocus some conclusions already made on other

reactions. We note here that so far the need to investigate DD in irradiated metals was led mainly by the
need to create multiscale modeling tools for predicting changes in mechanical properties due to irradiation. This is desirable for practical estimations in
engineering support of real nuclear devices. However, there is another consequence of dislocation
activity that is related to microstructure changes
that occur. While this may not be important in postmortem experiments on irradiated materials, it can be
important in real devices operating under irradiation.
It is obvious that internal and external stresses can
accumulate during irradiation of complicated devices
due to high temperature, radiation growth, swelling,
and transition periods of operation, for example, shut
down and restart. Creep is usually taken into account
but microstructure changes due to dislocation activity during irradiation are not. This activity affects the

whole process of microstructure evolution and
should therefore be taken into account in predicting
the effects of irradiation. The validity of this statement is demonstrated by recent in-reactor straining
experiments on some bcc and fcc metals and alloys.9
It is not possible at the moment to formulate unambiguous conclusions and more experimental work
will be necessary for this. Nevertheless, it is clear
that dislocation activity during irradiation directly
affects the radiation damage process. In the following
section, we describe some mechanisms that can contribute to this at the atomic-scale level.
First, consider reaction R5, omitted in Section
1.12.4.2, for dislocation obstacles. Drag of glissile
interstitial loops by moving edge dislocations was
first observed in the MD modeling of bcc and fcc
metals.89 It is well known that SIA clusters in the
form of small perfect DLs exhibit thermally activated
glide in the direction of their Burgers vector.90,91 It is
characterized by very low activation energy $0.01–
0.10 eV. An edge dislocation, having a long-range

C
B
C
D
z

x

A

B

A

y

(a)

(b)

(c)

(d)

Figure 17 Interaction between a 1/2[111] screw dislocation, gliding in the Ày direction shown by the double arrow
in (a), and a [010] SIA loop. The loop is absorbed (b,c) and then reformed in (d) with b ¼ 1/2[111]. From Terentyev, D. A.;
Bacon, D. J.; Osetsky, Yu. N. Philos. Mag. 2010, 90, 1019. With permission from Taylor and Francis Ltd.
().


Atomic-Level Dislocation Dynamics in Irradiated Metals

elastic field, can interact with such clusters and, if bL is
parallel to the dislocation glide plane, can drag or
push them as it moves under applied stress/strain.
The dynamics of this process have been investigated
in detail and correlations between cluster and dislocation mobility analysed.92–94 Additional friction due to
cluster drag reduces dislocation velocity, an effect that
is stronger in fcc than bcc metals because of features of
cluster structure.89 The maximum speed at which a
dislocation can drag an SIA cluster is achieved by a
compromise between dislocation-cluster interaction

force and cluster friction and varies at T ¼ 300 K from
$180 m sÀ1 in fcc Cu to >1000 m sÀ1 in bcc Fe for
loops containing a few tens of SIAs.89 An important
consequence of this drag process is that a moving
dislocation can sweep glissile clusters and transport
them through the material.
Other reactions that may affect microstructure
evolution involve both inclusion- and dislocationlike obstacles. The relevant reaction for the latter is
denoted R3 in Table 1. In this case, an edge dislocation climbs and the formation of superjogs by defect
absorption changes its structure and total line length.
This changes its mobility and its cross-section for
interaction with other defects, such as point defects,
their clusters and impurities, and this in turn affects
microstructure evolution. Reactions of type R2 can
also be important for they change properties of obstacles. In thermal aging without stress, obstacles such as
voids, SIA clusters and SFTs evolve towards their
equilibrium low-energy state, that is voids/precipitates into faceted near-spherical shapes, SFTs into
regular tetrahedron shape, and so on, whereas shearing creates interface steps on voids/precipitates and
creates ledges on SFT faces. These surface features
change the properties of the defects by putting them
into a higher energy state.
Reaction R4 introduces another mechanism of
mass transport, for the helical turn (representing the
absorbed defects) can only extend or translate in the
direction of its Burgers vector, that is, along the screw
dislocation line. The case of dislocation-SFT interaction in a thin film considered in Section 1.12.4.2.1
has demonstrated that this may introduce completely
new mechanisms. This effect can also play a role when
a dislocation ends on an internal interface where it can
cross-slip. Reaction R4 also orders the orientation of

DLs left behind by a gliding screw dislocation for it
changes bL of these loops to b of the dislocation,
irrespective of their initial orientation.
Finally, we describe a case when several of the
above mechanisms may have a significant effect on

353

microstructure changes if operating at the same time
on different defects. It is known from experimental
studies6 that under neutron irradiation, Cu accumulates a high density of SFTs and this density saturates
with dose at a high level ($1024 mÀ3), close to conditions under which displacement cascades overlap.
Taking into account that SIA clusters are necessarily
accumulated in the system, this high saturation density implies that annihilation reactions between the
vacancy population in SFTs and SIA loops is suppressed. MD modeling in which an SIA cluster was
placed between two SFTs about 10 nm apart and
intersecting the loop glide cylinder has confirmed
that annihilation reaction does not occur even after
50 ns at T < 900 K.95 The result is not surprising for
each vacancy of an SFT is distributed over the four
faces of the tetrahedron within the stacking faults.
However, simulations show that an annihilation
reaction can be promoted by the involvement of a
gliding dislocation. Two cases have been considered.
In one, an edge dislocation under applied stress
dragged a 1/2h110i SIA loop toward an SFT placed
7 nm below the dislocation slip plane and intersecting
the glide cylinder of the loop. The overlapping fractions of SFT and dragged SIA loop annihilated by
recombination. Different obstacle sizes (SFT from 45
to 61 vacancies and SIA loops from 37 to 91 SIAs) and

geometries with different levels of overlap were
simulated and recombination occurred in all cases.
In the other situation, the same SFT and SIA loop
10–20 nm apart were placed on the slip plane of a
screw dislocation. On approaching the obstacles, the
dislocation absorbed a portion of each to form two
helical turns (reaction R4). The turns of vacancy and
interstitial character had opposite sign and the smaller was annihilated by recombination with part of the
larger. On moving farther ahead, the dislocation
released the unrecombined portion of the remaining
helix to leave a small defect. This was usually part
of the original SIA loop because a loop can be
completely absorbed as a helical turn, whereas only
a part of an SFT can be absorbed in this way. Thus,
the overall result of interstitial loop drag under
applied stress was annihilation of a significant part
of both clusters by a reaction between helices of
opposite signs.

1.12.5 Concluding Remarks
Atomic-scale simulation by computer has become a
powerful tool for investigation of material properties


354

Atomic-Level Dislocation Dynamics in Irradiated Metals

and processes that cannot be achieved by experimental techniques or other theoretical methods. This is
due to the increasing power of computers, the development of new efficient modeling codes, and the

extensive usage of ab initio calculations for probing
atomic mechanisms and generating data for design of
new IAPs. Moreover, increasing length and time
scales attainable by atomistic modeling provides
overlap with experimental scales in some cases,
thereby allowing direct verification of modeling
results.26 In this chapter, we have described only a
selection of the results obtained within the last
decade by atomic-scale modeling of DD in irradiated
bcc and fcc metals. The examples presented and
references cited demonstrate how detailed insight
into mechanisms can be gained by such modeling.
For some obstacles to dislocation motion, for example, many inclusion-like obstacles, strengthening is
controlled by the dislocation line shape at breakaway
and can be parameterized using the existing elasticity theory models. In other cases, for example,
dislocation-like obstacles, reactions and their results,
including obstacle strength, depend very much on the
material, and dislocation character and core structure; dislocation behavior is also sensitive to conditions such as interaction geometry, temperature, and
strain rate. The variety of outcomes for dislocationlike obstacles is complicated and wide, and although
features of these reactions can be understood within
the framework of the continuum theory of dislocations, for example, Frank’s rule, a general formalization of the reactions in terms of obstacle strength and
reaction product does not exist. Nevertheless, it has
been seen that the insight gained by simulation
has allowed the outcome of reactions to be classified
in a meaningful way (Table 1). This will allow for
validation of higher-level DD modeling of these reactions using the continuum approximation. An excellent example of the way in which this can be done has
been provided by Martinez and coworkers,96,97 who
used MD and DD to simulate the same dislocation–
SFT interactions. The continuum modeling of these
nanometer-scale obstacles was verified by the atomic

simulation, and this enabled a large number of interaction geometries and conditions to be investigated
successfully by DD. Unfortunately, successful overlaps in scale of atomistic modeling and experiment
or/and continuum modeling are still rare. Efforts in
all techniques are necessary to progress understanding of mechanisms and their parameterization for
predictive modeling tools that can be applied to
irradiated materials.

Investigations such as those just discussed bring
assurance that atomic-scale modeling is correct at
least qualitatively and is invaluable in cases where
scale overlap of techniques is not yet achieved. Two
main problems exist with regard to the quality of its
quantitative outcomes. One is concerned with time
scale. As already mentioned, the limit on time scale is
the main disadvantage of current atomic-scale modeling. The maximum simulation time achieved so
far is of the order of a few hundred nanoseconds.
For dislocation studies, this allows dislocations with
velocity as low as 0.5 m sÀ1 to be modeled – and it
may be that some processes are insensitive to velocity at this level27 – but the overall strain rate (105 sÀ1)
is fast compared with experiment, and the interaction
time ($100 ns) with an obstacle is too short for
thermally-activated processes to be sampled. Development of new methods for keeping atomic-level
accuracy over at least microsecond to second time
scales is necessary to progress to the next step toward
predictive modeling for engineering applications.
An example of a new generation technique for
simulating realistic strain rates whilst retaining
atomic-scale detail was published recently.98 The
new technique combines atomic-scale modeling for
estimating vacancy migration barriers in the vicinity

of an edge dislocation and kinetic Monte Carlo
(MC) for simulating vacancy kinetics in a crystal
with a specified dislocation density. The technique
was successfully applied to simulate the process of
power-law creep over a macroscopic time scale with
microscopic fidelity. The other problem is concerned
with accuracy in describing interatomic interactions.
Much of the research described in this chapter, based
as it is on empirical EAM IAPs, is more related to the
behavior of model metals with bcc or fcc crystal
structure in general than to the elements Fe or Cu
in particular. This difficulty will become more acute
with the demand for more sophisticated, radiationresistant alloys, and future investigations of chemical
effects on plasticity will require IAPs that incorporate chemistry in a meaningful way. More on this is
presented in Chapter 1.10, Interatomic Potential
Development.
We would like to conclude on an optimistic note.
It is clear that significant progress has been achieved
over the last decade in understanding the details of
the atomic-scale mechanisms involved in dislocations
dynamics in structural metals in a reactor environment. The small, nanoscale nature of the obstacles
created by radiation damage is such that the techniques described here provide uniquely valuable


Atomic-Level Dislocation Dynamics in Irradiated Metals

information, despite the limitations they currently
experience.

Acknowledgments


22.
23.
24.
25.
26.

Much of the work described in this chapter was
carried out with support of the Division of Materials
Sciences and Engineering, U.S. Department of
Energy, under contract with UT-Battelle, LLC (YO,
modeling and results analysis).

27.
28.
29.
30.

References

31.
32.

1.
2.
3.
4.

5.
6.


7.

8.

9.

10.
11.
12.
13.
14.

15.
16.
17.
18.
19.
20.
21.

Hirth, J. P.; Lothe, J. Theory of Dislocations; Krieger:
Malabar, FL, 1982.
Hull, D.; Bacon, D. J. Introduction to Dislocations; 5th
edition, Butterworth-Heinemann: Oxford, 2011.
Bulatov, V. V.; Cai, W. Computer Simulations of
Dislocations; Oxford University Press: Oxford, 2006.
Bacon, D. J.; Osetsky, Yu. N.; Rodney, D. In Dislocations
in Solids; Hirth, J. P., Kubin, L., Eds.; Elsevier: Amsterdam,
2009; Vol. 15, pp 1–90.

Griffin, P. J.; Kelly, J. G.; Luera, T. F.; Barry, A. L.; Lazo, M. S.
IEEE Trans. Nucl. Sci. 1991, 38, 1216.
Singh, B. N.; Horsewell, A.; Toft, P. J. Nucl. Mater. 1999,
272, 97; Singh, B. N.; Horsewell, A.; Toft, P.; Edwards, D. J.
J. Nucl. Mater. 1995, 224, 131; Zinkle, S. J.; Singh, B. N.
J. Nucl. Mater. 2006, 351, 269.
Klueh, R.; Alexander, D. J. Nucl. Mater. 1995, 218,
151; Rieth, M.; Dafferner, B.; Rohrig, H.-D. J. Nucl. Mater.
1998, 258–263, 1147.
Benn, R. C. In Optimization of Processing, Properties, and
Service Performance Through Microstructural Control;
Bramfitt, B. L., Benn, R. C., Brinkman, C. R., Vander Voort,
G. F., Eds.; ASTM International: West Conshohocken, PA,
1988; STP 979, paper ID STP29131S.
Singh, B. N.; Edwards, D. J.; Ta¨htinen, S.; et al. Risø
Report No. Risø-R-1481 (EN); Oct 2004; p 47; Trinkaus,
H.; Singh, B. N. Risø-R-1610 (EN); Apr 2008; p 68.
Osetsky, Yu. N.; Bacon, D. J. Model. Simulat. Mater. Sci.
Eng. 2003, 11, 427.
Cotterill, R. M.; Doyama, M. Phys. Rev. 1966, 145, 465.
Golubov, S. I.; Osetsky, Yu. N. Unpublished, 2005.
Yang, L. H.; So¨derlind, P.; Moriarty, J. A. Mater. Sci. Eng.
A 2001, 309–310, 102.
Baskes, M. I.; Daw, M. S. In 4th International Conference
on the Effects of Hydrogen on the Behavior of Materials,
Jackson Lake Lodge, Moran, WY; Moody, N.,
Thompson, A., Eds.; The Minerals, Metals, Materials
Society: Warrendale, PA, 1989.
Daw, M. S.; Foiles, S. M.; Baskes, M. I. Mater. Sci. Rep.
1992, 9, 251.

Rodney, D. Acta Mater. 2004, 52, 607.
Honneycutt, D. J.; Andersen, H. C. J. Phys. Chem. 1987,
91, 4950.
Kelchner, C.; Plimpton, S. J.; Hamilton, J. C. Phys. Rev. B
1998, 58, 11085.
Wirth, B. D.; Bulatov, V.; Diaz de la Rubia, T. J. Nucl.
Mater. 2000, 283–287, 773.
Samaras, M.; Derlet, P. M.; Van Swygenhoven, H.;
Victoria, M. J. Nucl. Mater. 2006, 351, 47.
Gil Montoro, J. G.; Abascal, J. L. F. J. Phys. Chem. 1993,
97, 4211.

33.
34.
35.

36.
37.
38.
39.
40.
41.

42.
43.
44.
45.
46.
47.
48.

49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.

355

Voskoboinikov, R. E.; Osetsky, Yu. N.; Bacon, D. J. Mater.
Sci. Eng. A 2005, 400–401, 45.
Bacon, D. J.; Osetsky, Yu. N. Philos. Mag. 2009, 89, 3333.
Osetsky, Yu. N. Unpublished, 2003.
Terentyev, D. A.; Osetsky, Yu. N.; Bacon, D. J. Scripta
Mater. 2010, 62, 697.
Osetsky, Yu. N.; Matsukawa, Y.; Stoller, R. E.; Zinkle, S. J.
Philos. Mag. Lett. 2006, 86, 511.
Bacon, D. J.; Osetsky, Yu. N. J. Nucl. Mater. 2004,
329–333, 1233.
Osetsky, Yu. N.; Bacon, D. J.; Mohles, V. Philos. Mag.
2003, 83, 3623.
Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2003, 323,
268.
Osetsky, Yu. N.; Bacon, D. J. Mater. Sci. Eng. A 2005,
400–401, 374.

Terentyev, D.; Bacon, D. J.; Osetsky, Yu. N. J. Phys.
Condens. Matter 2008, 20, 445007.
Xiangli, L.; Golubov, S. I.; Woo, C. H.; Huang, H. CMES
2004, 5, 527.
Hatano, T.; Matsui, H. Phys. Rev. 2005, 72, 094105.
Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010,
90, 945.
Buswell, J. T.; English, C. A.; Hetherington, M. G.;
Phythian, W. J.; Smith, G. D. W.; Worral, G. W. In Effect of
Radiation on Materials; Packan, N. H., Stoller, R. E.,
Kumar, A. S., Eds.; ASTM: Philadelphia, PA, 1990; STP
1046, p 127.
Ackland, G. A.; Bacon, D. J.; Calder, A. F.; Harry, T. Philos.
Mag. A 1997, 75, 713.
Harry, T.; Bacon, D. J. Acta Mater. 2002, 50, 195.
Harry, T.; Bacon, D. J. Acta Mater. 2002, 50, 209.
Scattergood, R. O.; Bacon, D. J. Philos. Mag. 1975, 31, 179.
Scattergood, R. O.; Bacon, D. J. Acta Metall. 1982, 30,
1665.
Bacon, D. J. In Fundamentals of Deformation; Fracture;
Bilby, B. A., Miller, K. J., Willis, J. R., Eds.; Cambridge
University Press: Cambridge, 1985; p 401.
Russell, K. C.; Brown, L. M. Acta Metall. 1972, 20, 969.
Grammatikopoulos, P.; Bacon, D. J.; Osetsky, Yu. N.
Model. Simulat. Mater. Sci. Eng. 2011, 19, 015004.
Monnet, G.; Bacon, D. J.; Osetsky, Yu. N. Philos. Mag.
2010, 90, 1001.
Konstantinovic´, M. J.; Minov, B.; Kutnjak, Z.; Jagodic´, M.
Phys. Rev. B 2010, 81, 140203.
Lozano-Perez, S.; Jenkins, M. L.; Titchmarsh, J. M. Philos.

Mag. Lett. 2006, 86, 367.
Terentyev, D. A.; Bonny, G.; Malerba, L. Acta Mater. 2008,
56, 3229.
Schaublin, R.; Baluc, N. Nucl. Fusion 2007, 47, 1690.
Schaublin, R.; Chiu, Y. L. J. Nucl. Mater. 2007, 362, 152.
Hafez Haghighat, S. M.; Schaublin, R. J. Comput. Aided
Mater. Des. 2007, 14, 191.
Stoller, R. E.; Golubov, S. I.; Kamenski, P. J.;
Seletskaia, T.; Osetsky, Yu. N. Philos. Mag. 2010, 90, 935.
Stewart, D.; Osetsky, Yu. N.; Stoller, R. E. J. Nucl.
Mater. in press 2011.
Osetsky, Yu. N. Unpublished, 2009.
Hatano, T. Phys. Rev. 2006, 74, 020102.
Hatano, T.; Kaneko, T.; Abe, Y.; Matsui, H. Phys. Rev. B
2008, 77, 064108.
Osetsky, Yu. N.; Rodney, D.; Bacon, D. J. Philos. Mag.
2006, 86, 2295.
Terentyev, D.; Grammatikopoulos, P.; Bacon, D. J.;
Osetsky, Yu. N. Acta Mater. 2008, 56, 5034.
Osetsky, Yu. N.; Stoller, R. E.; Matsukawa, Y. J. Nucl.
Mater. 2004, 329–333, 1228.
Osetsky, Yu. N.; Stoller, R. E.; Rodney, D.; Bacon, D. J.
Mater. Sci. Eng. A 2005, 400–401, 370.


356

Atomic-Level Dislocation Dynamics in Irradiated Metals

60. Wirth, B. D.; Bulatov, V. V.; Diaz de la Rubia, T. J. Eng.

Mater. Sci. Technol. 2002, 124, 329.
61. Hirataniy, M.; Zbib, H. M.; Wirth, B. D. Philos. Mag. 2002,
82, 2709.
62. Szelestey, P.; Patriarca, M.; Kaski, K. Model. Simulat.
Mater. Sci. Eng. 2005, 13, 541.
63. Robach, J. S.; Robertson, I. M.; Lee, H.-J.; Wirth, B. D.
Acta Mater. 2006, 54, 1679.
64. Singh, B. N.; Ghoniem, N. M.; Trinkaus, H. J. Nucl. Mater.
2002, 307–311, 159.
65. Kimura, H.; Maddin, R. In Lattice Defects in Quenched
Metals; Cotterill, R. M. J., Ed.; Academic Press: New York,
1965; p 319.
66. Matsukawa, Y.; Osetsky, Yu. N.; Stoller, R. E.; Zinkle, S. J.
Mater. Sci. Eng. A 2005, 400–401, 366.
67. Matsukawa, Y.; Osetsky, Yu. N.; Stoller, R. E.; Zinkle, S. J.
J. Nucl. Mater. 2006, 351, 285–294.
68. Matsukawa, Y.; Osetsky, Yu. N.; Stoller, R. E.; Zinkle, S. J.
Philos. Mag. 2007, 88, 581.
69. Horton, L. L.; Bentley, J.; Farrel, K. J. Nucl. Mater. 1982,
108–109, 222.
70. Rodney, D.; Martin, G. Phys. Rev. B 2000, 61, 8714.
71. Rodney, D.; Martin, G.; Bre´chet, Y. Mater. Sci. Eng. A
2001, 309–310, 198.
72. Rodney, D. Nucl. Instrum. Meth. Phys. Res. B 2005, 228,
100.
73. Rodney, D. Compt. Rendus Phys. 2008, 9, 418.
74. Nogaret, T.; Robertson, C.; Rodney, D. Philos. Mag. 2007,
87, 945.
75. Rodney, D.; Martin, G. Phys. Rev. Lett. 1999, 82, 3272.
76. Osetsky, Yu. N. Unpublished, 2007.

77. Nomoto, A.; Soneda, N.; Takahashi, A.; Isino, S. Mater.
Trans. 2005, 46, 463.
78. Bacon, D. J.; Osetsky, Yu. N.; Rong, Z. Philos. Mag. 2006,
86, 3921.
79. Bacon, D. J.; Osetsky, Yu. N. JOM 2007, 59, 40.
80. Terentyev, D.; Malerba, L.; Bacon, D. J.; Osetsky, Yu. N.
J. Phys. Condens. Matter 2007, 19, 456211.
81. Bacon, D. J.; Osetsky, Yu. N. J. Math. Mech. Solids 2009,
14, 270.

82.

Bacon, D. J.; Osetsky, Yu. N. In Proceedings of the
International Symposium on Materials Science:
Energy Materials – Characterization, Modelling and
Application; Anderson, N. H., et al., Eds.; Risø
National Laboratory: Denmark, 2008; pp 1–12,
ISBN 978-87-550-3694-9.
83. Terentyev, D.; Grammatikopoulos, P.; Bacon, D. J.;
Osetsky, Yu. N. Acta Mater. 2008, 56, 5034.
84. Terentyev, D. A.; Osetsky, Yu. N.; Bacon, D. J. Scripta
Mater. 2010, 62, 697.
85. Osetsky, Yu. N.; Serra, A.; Priego, V. J. Nucl. Mater. 2000,
276, 202.
86. Marian, J.; Wirth, B. D.; Perlado, J. M. Phys. Rev. Lett.
2002, 88, 255507.
87. Liu, X.-Y.; Biner, S. B. Scripta Mater. 2008, 59, 51.
88. Terentyev, D. A.; Bacon, D. J.; Osetsky, Yu. N. Philos.
Mag. 2010, 90, 1019.
89. Osetsky, Yu. N.; Bacon, D. J.; Rong, Z.; Singh, B. N.

Philos. Mag. Lett. 2005, 84, 745.
90. Osetsky, Yu. N.; Bacon, D. J.; Serra, A.; Singh, B. N.;
Golubov, S. I. Philos. Mag. A 2003, 83, 61.
91. Anento, N.; Serra, A.; Osetsky, Yu. N. Model. Simulat.
Mater. Sci. Eng. 2010, 18, 025008.
92. Rong, Z.; Bacon, D. J.; Osetsky, Yu. N. Mater. Sci. Eng. A
2005, 400–401, 378.
93. Bacon, D. J.; Osetsky, Yu. N.; Rong, Z.; Tapasa, K. In
IUTAM Symposium on Mesoscopic Dynamics in
Fracture Process; Strength of Materials; Kitagawa, H.,
Shibutani, Y., Eds.; Kluwer: Dordrecht, 2004; p 173.
94. Rong, Z.; Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2005,
85, 1473.
95. Osetsky, Yu. N. Unpublished, 2008.
96. Martinez, E.; Marian, J.; Perlado, J. M. Philos. Mag. 2008,
88, 809; Philos. Mag. 2008, 88, 841.
97. Marian, J.; Martinez, E.; Lee, H.-J.; Wirth, B. D. J. Mater.
Res. 2009, 24, 3628.
98. Kabir, M.; Lau, T. L.; Rodney, D.; Yip, S.; Van Vliet, K. J.
Phys. Rev. Lett. 2010, 105, 09551.