Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals

Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals Comprehensive nuclear materials 1 12 atomic level dislocation dynamics in irradiated metals

EAM Embedded atom model

ESM Equivalent sphere method

PAD Periodic array of dislocations

PBC Periodic boundary condition

SFT Stacking fault tetrahedron

SIA Self-interstitial atom

TEM Transmission electron microscope

Symbols

b Dislocation Burgers vector

b L Dislocation loop Burgers vector

«˙ Shear strain rate

g Stacking fault energy

w Angle between dislocation segments

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

microstruc-tures that are unique to irradiation conditions

Irra-diation 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,

Pri-mary Radiation Damage Formation) Evolution of

the primary damage under the high operating

tem-perature (600 K to >1000 K) leads to a

micro-structure containing a high concentration of defect

clusters, such as voids, dislocation loops (DLs),

stack-ing fault tetrahedra (SFTs), gas-filled bubbles, and

precipitates, and an increase in the total dislocation

network density (seeChapter1.13, Radiation

Dam-age 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

multi-scale 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

den-sity functional theory (DFT) (seeChapter 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 tural properties obtainedab initio IAPs are required foratomic-scale modeling methods such as molecular stat-ics (MS) and molecular dynamics (MD), which are used

struc-to simulate millions of astruc-toms Time spanning conds to microseconds can be simulated by MD if thenumber of atoms is not large (see Section 1.12.3.3).This level can provide properties of point andextended defects and interactions between them (seeChapter1.09, Molecular Dynamics) For mechanicalproperties, important interactions are between movingdislocations, which are responsible for plasticity, anddefects created by irradiation Mechanisms and para-meters determined at this level can then inform dislo-cation dynamics (DD) models based on elasticitytheory of the continuum (seeChapter1.16, Disloca-tion Dynamics) DD models can simulate processes atthe micrometer scale and mesh with the mechanicalproperties of larger volumes of material used in finiteelements (FEs) methods, that is, realistic models for thedesign of core components In this chapter, we considerdirect interactions at the atomic scale between movingdislocations and obstacles to their motion

nanose-The structure of the chapter is as follows First,

we summarize the main features of the irradiationmicrostructure of concern Then we provide a shortdescription of atomic-scale methods applied to dislo-cation modeling, bearing in mind the details pre-sented inChapter1.09, Molecular Dynamics This

is followed by a review of important results fromsimulations of the interaction between disloca-tions and obstacles We then describe how dislocationsmodify microstructure in irradiated metals Finally,

we indicate some issues that will hopefully beresolved by atomic-scale modeling in the near future.Our main aim is to give the reader a general picture ofthe phenomena involved and encourage furtherresearch in this area The following sources1–4provide

a more general and deeper understanding of tions and modeling of plasticity issues

disloca-1.12.2 Radiation Effects on Mechanical Properties1.12.2.1 Radiation-Induced Obstacles toDislocation Glide

Primary damage of structural materials is initiated bythe interaction of high-energy atomic particles withmaterial atoms to cause the energetic recoil anddisplacement of primary knock-on atoms (PKAs).PKA energy can vary from a few tens to tens of

thousands of electron volt and the PKA spectrum can

be calculated for a particular position in a particular

installation.5A 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

Chapter1.11, Primary Radiation Damage

Forma-tion) Further evolution of these defects produces

specific microstructures that depend on the

irradia-tion type, ambient temperature, and the material

and its initial structure (seeChapter1.13, Radiation

Damage Theory) This radiation-induced

micro-structure 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

reach1024

m3 At this density, the mean distance

between obstacles can be as short as10 nm, and such a

high density of small defects, particularly those with

a dislocation character, makes the mechanisms of

radi-ation 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

disloca-tion glide, increase yield and flow stress and reduce

ductility (see Chapter1.04, Effect of Radiation on

Strength and Ductility of Metals and Alloys for

experimental results) Furthermore, if the obstacle

den-sity is sufficiently high to block dislocation motion,

pre-existing 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

deforma-tion 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

inclusion-like obstacles and those with dislocation properties

The first type includes voids, bubbles, and

preci-pitates, for example They usually have relatively

short-range strain fields and their properties may not

be changed significantly by interaction with tions (Copper precipitates in iron are an exception tothis – seeSections 1.12.4.1.1–1.12.4.1.2.) Those thatare not impenetrable are usually sheared by the dislo-cation and steps defined by the Burgers vector b ofthe interacting dislocation are created on the obstacle–matrix interface Unstable precipitates, such as Cu in

disloca-Fe, may also suffer structural transformation during theinteraction, which can change their properties Theseobstacles do not usually modify dislocations signifi-cantly, although they may cause climb of edge dis-locations (see Sections 1.12.4.1.1–1.12.4.1.2) Theirmain effect is to create resistance to dislocation glide.Obstacles such as voids and bubbles are among thestrongest, and as a result of their high density, theycontribute significantly to radiation-induced harden-ing.7 Materials designed to exploit oxide dispersionstrengthening (ODS) are produced with a high con-centration of rigid, impenetrable oxide particles, whichintroduce extremely high resistance to dislocationmotion.8These obstacles are also considered here astheir scale, typically a few nanometers, is similar to that

of obstacles formed under irradiation

The second obstacle type consists of those with adislocation character, for example, DLs and SFTs,and so dislocation reactions occur when they areencountered by moving dislocations Loops have rel-atively long-range strain fields and hence interactwith dislocations over distances much greater thantheir size SFTs are three-dimensional (3D) struc-tures and have short-range strain fields Loops withperfect Burgers vectors are glissile, in principle,whereas SFTs and faulted loops, for example, Frankloops in fcc metals, are sessile In addition to causinghardening, the reaction of these defects with a glidingdislocation can modify both their own structure andthat of the interacting dislocation As will be demon-strated inSection 1.12.4.2, their effect depends verymuch on the geometry of the interaction, that is, theirposition and orientation relative to the moving dislo-cation, and the nature of the mutual dislocation seg-ment that may form in the first stage of interaction.The contribution of these obstacles to strengtheningcan be significant, for their density can be high.Modification of irradiation-induced microstructuredue to plastic deformation is an additional possiblyimportant effect If mechanical loading occurs duringirradiation, it can contribute significantly to the overallmicrostructure evolution and therefore to change inmaterial properties Accumulation of internal stressduring irradiation is unavoidable in real structural

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

exper-imental 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

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

tech-niques 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,

cataly-sis, 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

proper-ties 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

gra-dients of chemical potential Models for bulk properties

are based on continuum treatments by elasticity,

ther-mal conductivity, and rate theories where global defect

properties such as formation, annihilation, transport,

and interactions are already parameterized at

contin-uum level The only technique that currently bridges

the gap in the scales betweenab initio and the

contin-uum 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

calcu-lated from empirical interatomic potentials and

respond to internal and external fields due to

tempera-ture, stress, and local imperfections Atomic-scale

mod-eling 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

dislocation is effectively infinite in length Ifthe model contains one obstacle, the length, L, ofthe model in the periodic direction represents thecenter-to-center obstacle spacing along an infiniterow of obstacles It is the treatment of the boundaries

in the other two directions that distinguishes onemethod from another A versatile atomic-scalemodel should allow for the following.10

1 Reproduction of the correct atomic configuration

of the dislocation core and its movement under theaction of stress

2 Application of external effects such as appliedstress or strain, and calculation of the resultantresponse such as strain (elastic and plastic) orstress and crystal energy

3 Possibility of moving the dislocation over a longdistance under applied stress or strain withouthindrance from the model boundaries

4 Simulation of either zero or non-zero temperatures

5 Possibility of simulating a realistic dislocationdensity and spacing between obstacles

6 Sufficiently fast computing speed to allow tion of crystallites in the sizes range where sizeeffects are insignificant

simula-A comprehensive review of models developed so far

is to be found in Baconet al.,4

and so here we merelypresent a short summary of the pros and cons of somemodels used most commonly Historically, the earliestmodels consisted of a small region of mobile atomssurrounded in the directions perpendicular to thedislocation direction by a shell of atoms fixed in thepositions obtained by either isotropic or anisotropicelasticity for displacements around the dislocation ofinterest.11This model was used successfully to inves-tigate dislocation core structure and, being simple andcomputationally efficient, can use a mobile regionlarge enough to simulate interaction between staticdislocations and defects and small defect clusters Itsmain deficiencies are its inability to model dislocationmotion beyond a few atomic spacings because of therigid boundaries (condition 3) and its restriction totemperatureT ¼ 0 K (condition 4)

The desirability of allowing for elastic response ofthe boundary atoms due to atomic relaxation in theinner region, for example, when a dislocation moves,has led to the development of several quasicontinuummodels The elastic response can be accounted for byusing either a surrounding FE mesh or an elasticGreen’s function to calculate the response of bound-ary atoms to forces generated by the inner region.Such models are accurate but computationally

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

signifi-cant 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

peri-odic array of dislocations (PAD) scheme first

intro-duced 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.4,10,16With a PAD model

containing106

–107mobile atoms, essentially all

con-ditions 1–6 can be satisfied Their ability to simulate

interactions with strong obstacles of size up to at

least10 nm makes PAD models efficient for

investi-gating dislocation–obstacle interactions relevant to a

radiation damage environment Practically all

impor-tant 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

dislo-cations 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 inFigure 1

There are two types of DL in an fcc metal: glissileperfect loops with bL¼ 1/2h110i and sessile Frankloops with bL¼ 1/3h111i There are two types ofglissile loop with Burgers vectors 1/2h111i andh100i in a body-centered cubic (bcc) metal

Visualizing interaction mechanisms is a strongfeature of atomic scale modeling The main idea is

to extract atoms involved in an interaction and lize them to understand the mechanism Usuallythese atoms are characterized by high energy, localstresses, and lattice deformation The techniquesused are based on analysis of nearest neighbors,17central symmetry parameter,18energy,19stress,20dis-placements,10 and Voronoi polyhedra.21 A relativelysimple and fast technique, for example, was suggestedfor an fcc lattice.16 It is based on comparison ofposition of atoms in the first coordination of anatom with that of a perfect fcc lattice If all 12 neigh-bors of the analyzed atom are close to that position, it

visua-is assigned to be fcc If only nine neighbors spond to perfect fcc coordination, the atom is taken to

corre-be on a stacking fault Other numcorre-bers of neighborscan be attributed to different dislocations Modifica-tions of this method have been successfully applied inhexagonal close-packed (hcp)22 and bcc23 crystals.Another improvement of this method for MD simu-lation atT > 0 K was introduced24

in which the aboveanalysis was applied periodically (every 10–50 time-steps depending on strain rate _e) and over a certaintime period (100–1000 steps) A probability of anatom to be in different environment was estimatedand the final state was assigned to the maximum overthe analyzed period Such a probability analysis can

be applied to other characteristics such as energy orstress excess over the perfect state and it provides aclear picture when the majority of thermal fluctua-tions are omitted

Obstacle:

loop or SFT

C Edge

L

Screw

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.

1.12.3.3 Input Parameters

The IAP is a crucial property of a model for it

determines all the physical properties of the

sim-ulated system Discussion on modern IAPs is

pre-sented 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

more, the model should be large enough to include

Further-all direct interactions between the dislocation and

obstacle and the major part of elastic energy that

may affect the mechanism under study MD

simula-tions have demonstrated that a system with a few

million atoms is usually sufficient to satisfy

condi-tions 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,25and 12 nm SFTs26in 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 lowT

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 whichb ¼ 0.248 nm

ForL ¼ 41 nm, a model containing 2  106

mobile atomswould have a cross-section area of 5.73 1016m2;

that is, a dislocation density rD¼ 1.75  1015

m2.For Lg¼ 120b ¼ 29.8 nm, the model height per-

pendicular to the glide plane would be 19 nm At

_e ¼ 5  106s1, the steady state velocity, vD, of a

single dislocation estimated from the Orowan relation

vD¼ _e=rDb is 11.6 m s1 The time for the dislocation

to travel a distanceLgat 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 be1 ns

The lowest strain rate for dislocation-obstacleinteraction reported so far27is 105s1and it resulted

invD¼ 48 cm s1 This strain rate is about six to tenorders of magnitude higher than that usually applied

in laboratory tensile experiments and more than tenorders higher than that for the creep regime Thispresents an unresolvable problem for atomic-scalemodeling and even massive parallelization gains onlythree or four orders in_e or vD We conclude that thepossibilities of modern atomic-scale modeling are lim-ited to dislocation velocity of at least0.1 cm s1.Nevertheless, atomic-scale modeling, particularlyusing MD (T > 0 K), is a powerful, and sometimesthe only, tool for investigating processes associatedwith lattice defect interactions and dynamics Themain advantage of MD is that, if applied properly to

a large enough system, it includes all classical nomena such as evolution of the phonon system andtherefore free energy, rates of thermally activateddefect motion, and elastic interactions It is, therefore,one of the most accurate techniques for investigatingthe behavior of large atomic ensembles under differ-ent conditions We reemphasize that the realism ofatomic-scale modeling is limited mainly by the valid-ity of the IAP and restricted simulation time.1.12.3.4 Output Information

phe-Atomic-scale methods and particularly MD can vide a wide range of valuable information on theprocesses simulated The most important are

pro-1 Information on the physical state of the system.This includes temperature and stress and theirdistribution; displacement of atoms and theirtransport; interaction energy and therefore forcebetween defects; and evolution of internal, elastic,and free energies Extraction of this information iswell understood and procedures can be found inChapter 1.04, Effect of Radiation on Strengthand Ductility of Metals and Alloys;Chapter2.13, Properties and Characteristics ofZrC;Chapter5.01, Corrosion and Compatibility

andChapter1.09, Molecular Dynamics

2 Detail of atomic mechanisms This includes ysis of the position and environment of individualatoms based on calculation of their energy, sitestress, or local atomic configuration Atoms canthen be identified with particular features such asconstituents of defect clusters, stacking faults, dis-location cores, and so on Having this information

anal-at particular times provides unique knowledge of

defect structure, motion, interactions, and

transformation

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 T¼ 0 K

Voids in bcc and fcc metals atT ¼ 0 K and >0 K are

probably the most widely simulated obstacles of this

type Most simulations were made with edge

disloca-tions.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

pre-sented 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)

Under static conditions,T ¼ 0 K, voids are strongobstacles and at maximum stress, an edge dislocation

in Fe bows out strongly between the obstacles, ing parallel screw segments in the form of a dipolepinned at the void surface A consequence of this isthat the screw arms cross-slip in the final stagewhen the dislocation is released from the void surfaceand this results in dislocation climb (see Figure 4),thereby reducing the number of vacancies in the voidand therefore its size In contrast to this, a Shockleypartial cannot cross-slip Partials of the dissociateddislocation in Cu interact individually with smallvoids whose diameter, D, is less than the partialspacing (2 nm), thereby reducing the obstaclestrength Stress drops are seen in the stress–straincurve inFigure 3 The first occurs when the leadingpartial breaks from the void; the step formed by this

creat-on the exit surface is a partial step 1/6h112i and thestress required is small Breakaway of the trailingpartial controls the critical stress tc For voids with

D larger than the partial spacing, the two partialsleave the void together at the same stress However,extended screw segments do not form and the dislo-cation does not climb in this process Consequently,large voids in Cu are stronger obstacles than those ofthe same size in Fe, as can be seen inFigure 6and thenumber of vacancies in the sheared void in Cu isunchanged

Cu-precipitates in Fe have been studied sively23,27–29 due to their importance in raising theyield stress of irradiated pressure vessels steels35and

1.5 -50

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 (http://www.informaworld.com).

the availability of suitable IAPs for the Fe–Cu

sys-tem.36 These precipitates are coherent with the

sur-rounding Fe when small, that is, they have the bcc

structure rather than the equilibrium fcc structure of

Cu Thus, the mechanism of edge dislocation

inter-action 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 theFe–Cu interface This is lower thantcfor a void, how-ever, for which G is zero and the void surface energyrelatively high Thus, small precipitates (3 nm) arerelatively weak obstacles and, though sheared, remaincoherent with the bcc Fe matrix after dislocationbreakaway tc is insufficient to draw out screw seg-ments and the dislocation is released without climb.The Cu in larger precipitates is unstable, however,and their structure is partially transformed toward

350 300 250 200 150 100 50 0

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 (http://www.informaworld.com).

0

4 8 12 16

20

5.0 nm

4.0 nm 3.0 nm 2.0 nm 1.0 nm 0.9 nm

-Figure 4 [111] projection of atoms in the dislocation core showing climb of a1/2 h111i{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 more stable fcc structure when penetrated by

a dislocation at T ¼ 0 K This is demonstrated in

Figure 5by 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

break-away 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

out-side of the precipitate, but atoms represented by

circles are in a different sequence Atoms away fromthe Fe–Cu interface are seen to have adopted athreefold sequence characteristic of the {111} planes

in the fcc structure This transformation of Cu ture, first found in MS simulation of a screw disloca-tion penetrating a precipitate,37,38 increases theobstacle strength and results in a critical line shapethat is close to those for voids of the same size.34Under these conditions, a screw dipole is created

8 6 4 2 0 -2 -4 -6 -8

2009, 89, 3333 With permission from Taylor and Francis Ltd (http://www.informaworld.com).

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 atT 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

Oro-wan particles39and voids.40By computing the

equilib-rium 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 ¼2pALGb ½lnðD1þ L1Þ þ D ½1

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 n) if pure screw, where n is

Poisson’s ratio.Equation [1]holds for anisotropic

elas-ticity ifG and n are chosen appropriately for the slip

system in question, that is, ifGb2/4p and Gb2/4pA are

set equal to the prelogarithmic energy factor of screw

and edge dislocations, respectively.39,40The value ofG

obtained in this way is 64 GPa forh111i{110} slip in Fe

and 43 GPa forh110i{111} slip in Cu.41

The explanation for theD- and L-dependence of

tcis that voids and impenetrable particles are ‘strong’

obstacles in that the dislocation segments at the

obsta-cle 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, tcbLcorrelates withGb2ln(D1þ L1)1 The correlation

betweentcobtained 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 aboutD < 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

contin-uum modeling, obstacles withD > 2–3 nm are strong

at T ¼ 0 K and result in a dipole alignment at t

Smaller obstacles in Fe, for example, voids withD < 2

nm and Cu precipitates withD < 3 nm, are too weak

to be treated byeqn [1] Thus, the descriptions aboveand the data inFigure 6demonstrate that the atomic-scale mechanisms that operate for small and largeobstacles depend on their nature and are not pre-dicted by simple continuum treatments, such as theline-tension and modulus-difference approximationsthat form the basis of the Russell–Brown model of Cu-precipitate strengthening of Fe,42 often used in pre-dictions and treatment of experimental observations.The importance of atomic-scale effects in inter-actions between an edge dislocation and voids andCu-precipitates in Fe was recently stressed in a series

of simulations with a variable geometry.43 In thisstudy, 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 surprisingresult is that a void with its center below the disloca-tion slip plane is still a strong obstacle and mayincrease its size after the dislocation breaks away.This can be seen in Figure 7, where a dislocationline climbs down absorbing atoms from the voidsurface More details on larger voids, precipitates,and finite temperature effects can be found inGrammatikopouloset al.43

1.12.4.1.2 Temperature T> 0 K

In contrast to the T ¼ 0 K simulations above, eling by MD provides the ability to investigatetemperature effects in dislocation–obstacle interac-tion (The limit on simulation time discussed in

R

Glide plane

Climb-up - atoms left inside void

Climb-down - atoms taken off void

Section 1.12.3.3prevents study of the creep regime

controlled by dislocation climb.) Results on the

tem-perature dependence oftcfrom simulation of

inter-action between an edge dislocation and 2 and 6 nm

voids in Fe,29,30,34Cu-precipitates in Fe,27,29and voids

in Cu30,34are presented inFigure 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 Monnetet al.44

using amesoscale thermodynamic treatment of MD data in

the point obstacle approximation to estimate

activa-tion energy and its temperature dependence In this

way, the obstacle strength found by atomic-scale

mod-eling 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 seeFigure 5in Bacon and Osetsky23) Small

precipitates, D < 3 nm, are stabilized in the bcc

coherent state by the Fe matrix, as noted above

for T ¼ 0 K, and are weak, shearable obstacles Theresulting temperature-dependence oftcis small.Larger precipitates were seen to be unstable at

T ¼ 0 K with respect to a dislocation-induced formation toward the fcc structure This transforma-tion is driven by the difference in potential energy ofbcc and fcc Cu The free energy difference betweenthese two phases of Cu decreases with increasing Tuntil a temperature is reached at which the transfor-mation does not occur Thus, large precipitates arestrong obstacles at low T and weak ones at high T.This is reflected in the strong dependence oftconTshown in Figure 8 More explanation of this effectcan be found in Bacon and Osetsky.23These simula-tion results showing the different behavior of smalland large Cu-precipitates suggest that the yield stress

trans-of underaged or neutron-irradiated Fe–Cu alloys,which contain small, coherent Cu-precipitates, shouldhave a weakT-dependence, whereas that in an over-aged or electron-irradiated alloy, in which the popu-lation of coherent precipitates has a larger size, should

be stronger Some experimental observations supportthis.45 One is a weak change in the temperaturedependence of radiation-induced precipitate harden-ing in ferritic alloys observed after neutron irradiationwhen only small (<2 nm) precipitates are formed Theother is the experimentally-observed temperatureand size dependence of deformation-induced trans-formation of Cu-precipitates in Fe.46

Other obstacles with inclusion properties, such asgas-filled bubbles and other types of precipitates,have been studied less intensively, and we presentjust a few examples here

The effect of chromium precipitates on edgedislocation motion in matrices of either pure Fe

or Fe–10 at.% Cr solid solution was studied byTerentyev et al.47

Cr and Cr-rich precipitates havethe bcc structure and are coherent with the matrix.Unlike Cu-precipitates in Fe,G of Cr is higher thanthat of both matrices and so the dislocation isrepelled by Cr precipitates Under increasing strain,the dislocation moves until it reaches the precipitate–matrix interface where it stops until the stress reachesthe maximum,tc, just before the dislocation enters theprecipitate (seeFigure 2in Terentyevet al.47

) Thetversuse behavior is similar to that for voids in Cu, butwithout stress drops associated with partial disloca-tions, and no softening effects similar to voids and Cu-precipitates in Fe were observed Attc, the dislocationshears 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 t comparable with values given

6 nm

= 5 × 10 6 s -1

ε.

t Gb/Lc

Figure 8 Plot of t c 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  10 6 s1.

by eqn [2] (see Figure 4 in Terentyevet al.47

); theothers were much weaker Separate contributions

from the chemical strengthening (CS) and shear

mod-ulus difference (SMD) between Fe and Cr were

esti-mated and their sum was found to be close to thetc

found in simulation It was also found thattcfor the

alloy with Cr precipitates in an Fe–Cr solid solution is

the sum of tcfor the same precipitate in a pure Fe

matrix and the maximum stress for glide of the

dislo-cation motion in the Fe–Cr solid solution alone

Helium-filled bubbles created by vacancies and

helium formed in transmutation reactions are

com-mon features of the irradiated microstructure of

structural materials (see Chapter 1.13, Radiation

Damage Theory) However, there is a lack of

infor-mation 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

dislocation and a row of 2 nm cavities with He/Vac

ratios of up to 5 in Fe atT between 10 and 700 K It

was found thattchas 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 is0.5

for a 2 nm bubble at 300 K in Fe.52Simulation 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

per-formed53 and preliminary conclusions drawn The

dislocation interaction with underpressurized

bub-bles (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

pres-sure is reduced in this way and interstitials are

absorbed by the dislocation as a double superjog

Equi-librium bubbles are therefore the strongest Some of

these conclusions, such as formation of interstitialclusters around bubbles with high He/Vac ratios, aresimilar to those observed earlier,48–50others are not.More modeling is necessary to clarify these issues

As noted inSection 1.12.2.2, impenetrable cles such as oxide particles and incoherent pre-cipitates represent another class of inclusion-likefeatures Although these obstacles are usually preex-isting and not produced by irradiation, they are con-sidered to be of potential importance for the design

obsta-of nuclear energy structural materials and should

be considered here Atomic-level information ontheir 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 arigid, impenetrable particle in Cu was simulated byHatano54 using the Cu–Cu IAP as for the Fe–Cusystem36 and a constant strain rate of 7 106

s1at

T ¼ 300 K The particle was created by defining aspherical region in which the atoms were held immo-bile relative to the surrounding crystal The Hirschmechanism2 was found to operate In the sequenceshown inFigures 1 and 2 of Hatano,54several stagescan be observed such as (1) the dislocation understress approaching the obstacle from the left firstbows round the obstacle to form a screw dipole; (2)the screw segments cross-slip on inclined {111}planes attc; (3) they annihilate by double cross-slip,allowing the dislocation, now with a double superjog,

to bypass the obstacle; (4) a prismatic loop with thesameb is left behind and (5) the dragged superjogspinch-off as the dislocation glides away, creating aloop of opposite sign to the first on the right of theobstacle tcvaries withD and L as predicted by thecontinuum modeling that led toeqn [1], but is over 3times larger in magnitude Hatano argues that thiscould arise from either higher stiffness of a dissociateddislocation or a dependence oftcon the initial posi-tion of the dislocation It is also possible that therequirement for the dislocation to constrict and theabsence of a component of applied stress on the cross-slip plane results in a high value oftc

Simulation of 2 nm impenetrable precipitates in

Fe has been carried out by Osetsky (2009, lished) The method is different from that used byHatano54in that the precipitate, constructed from Featoms held immobile relative to each other, was trea-ted as a superparticle moving according to the totalforce on precipitate atoms from matrix atoms Theinteraction mechanism observed is quite differentfrom those reported earlier for Cu,54for the Hirschmechanism and formation of interstitial clusters does