Comprehensive nuclear materials 1 15 phase field methods

Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods

P Bellon

University of Illinois at Urbana-Champaign, Urbana, IL, USA

ß 2012 Elsevier Ltd All rights reserved.

1.15.4.2 Examples of PF Modeling Applied to Alloys Under Irradiation 4211.15.4.2.1 Effects of ballistic mixing on phase-separating alloy systems 4211.15.4.2.2 Coupled evolution of composition and chemical order under irradiation 423

1.15.4.2.4 Irradiation-induced segregation on defect clusters 427

Abbreviations

CVM Cluster variation method

KMC Kinetic Monte Carlo

ME Master equation

PF Phase field

PFM Phase field model

SIA Self-interstitial atom

1.15.1 Introduction

Electronic and atomistic processes often dictate the

pathways of phase transformations and

microstruc-tural evolution in solid materials For quantitative

modeling of these transformations and evolution, it

is thus effective, and sometime necessary, to rely on

methods using some representation of atoms and of

their dynamics, as for instance in molecular

dynam-ics simulations (see Chapter 1.09, Molecular

Dynamics) and atomistic Monte Carlo simulations

(see Chapter 1.14, Kinetic Monte Carlo

Simula-tions of Irradiation Effects) While these atomistic

methods can now simulate quite accurately the

evo-lution of specific alloy systems, these simulations are

nevertheless limited to small length scales, from a few

to 100 nm Molecular dynamics is furthermore

lim-ited to small time scales, typically in the nanosecond

range, although in some cases, new developments

have made it possible to obtain atomistic simulations

at much longer times (see Chapter 1.14, KineticMonte Carlo Simulations of Irradiation Effects)

An alternative modeling approach is to replace themany microscopic degrees of freedom of the system

of interest by the few mesoscopic variables that aresufficient to provide a realistic description Thisapproach has been widely used in many disciplines,and well-known examples are the Fourier and Fickequations, which describe the diffusive transport

of heat and chemical species, respectively Thisapproach is also commonly used in modeling theevolution of point defects, in particular, during irra-diation (seeChapter1.13, Radiation Damage The-oryand Sizmann1) The work of Cahn and Hilliard2–

5

and Landau and Lifshitz (see for instance dano and Toledano6) provided a way to include thecontributions of interfaces to chemical evolution,thus making it possible to model heterogeneousand multiphase materials Kinetic models based onthese descriptions are broadly referred to as phasefield (PF) methods, since the microstructure of amaterial is fully characterized by a few mesoscopicfield variables such as concentration, magnetization,chemical order, or temperature One key assumption

Tole´-of this approach is that the variables chosen todescribe the state of the system vary smoothly acrossany interface or, in other words, that interfaces arediffuse This assumption finds a natural justification

in the theory of critical phenomena, since the

411

interface thickness diverges at the critical

tempera-ture.7 Diffuse interface models offer some

advan-tages over sharp interface models,8 in particular,

for the modeling of complex microstructures

Fur-thermore, the PF approach can be extended to

include macroscopic variables other than the local

composition, making it possible to describe

chemi-cal order–disorder transitions, solid–liquid

reac-tions, displacive transformations, and more

recently dislocation glide PF methods and

applica-tions have been recently reviewed by Chen,9

Emmerich,10 and Singer-Loginova and Singer.11

This chapter focuses on solid–solid phase

transfor-mations, with a particular emphasis on

transforma-tions and microstructural evolution relevant to

irradiated materials While conventional PF

model-ing lacks atomic resolution, the main interest in this

technique comes from the fact that it can provide

the evolution of large systems, exceeding the

micrometer scale, over very long time scales, from

seconds to centuries Recent developments have led

to the introduction of PF models (PFMs) that

pos-sess atomic resolution,12–26the so-called PF crystal

models This model, which can be seen as a density

functional theory for atoms, appears very promising,

although at this time it is not clear whether it can

reproduce correctly the discrete nature of

point-defect jumps from one lattice site to a neighboring

lattice site The PF crystal model is not covered in

this chapter, so the interested reader should consult

the above references

This chapter is organized as follows.Section 1.15.2

introduces the key concepts and steps employed in

conventional, that is, phenomenological PF modeling,

and provides some illustrative examples Section

1.15.3 focuses on important recent developments

toward quantitative PF modeling, whereby evolution

equations are rigorously derived by coarse-graining a

microscopic model This approach provides a full

treatment of fluctuations and thus makes it possible to

study fluctuation-controlled reactions, such as

nucle-ation of a second phase The capability of PFMs to

reach large time and length scales makes them an

attractive tool for simulating the evolution of

materi-als relevant to nuclear applications, in particular, for

alloys subjected to irradiation Applying PF modeling

to these nonequilibrium materials, however, raises

new challenges, as is discussed in Section 1.15.4.1

Some selected results of PF modeling applied to

irra-diated materials are presented in Section 1.15.4.2

Finally, conclusions and perspectives are given in

or the concentration of some chemical species ofinterest In systems with solid–liquid interfaces, aphenomenological field variable is introduced insuch a way that it varies continuously from 0 to 1 asone goes from a fully solid to a fully liquid phase.Multidimensional fields can be used as well, forinstance, to describe the local composition of amulticomponent alloy, the local degree of chemicalorder, or the local crystallographic orientation ofgrains These multidimensional fields may transformlike vectors under symmetry operations, thus leading

to a vectorial representation of the system and sorial expressions for mobilities (as will be discussedlater), but there are cases for which the multidimen-sional fields cannot be reduced to vectors.27 In allcases, an averaging procedure is necessary to definecontinuous field variables for systems that are intrin-sically discrete at the atomic scale Various averagescan be used, including (1) a spatial average overrepresentative volume elements, which will corre-spond to the cells used for evolving the PF variables;(2) a spatial and temporal average; or (3) a spatialand ensemble average The spatial averaging method

ten-is used most often, although in many cases theexact conditions of the averaging procedure are notdefined.Section 1.15.3will cover a model where thiscoarse-graining is performed explicitly and rigor-ously The last two averaging procedures are rarelyexplicitly invoked, although one of their advantages isthat a smaller volume can be used for the spatialaverage, thanks to the additional averaging per-formed either in time or in the configuration space

of a system ensemble

Turning now to the kinetic equations used todescribe the evolution of these field variables, animportant distinction is whether the field variable isconserved or nonconserved For the sake of simplic-ity, the following discussion focuses on alloy systems.Let us consider two simple examples, one where thefield variable is the local composition, C(r,t), in abinary A–B alloy system, and a second example,also for a binary alloy system, but this time with afixed composition and where chemical ordering takesplace The degree of chemical order is described

by the field S(r,t) For the sake of convenience, one

may normalize that field such that S(r,t) ¼ 0

corre-sponds to a fully disordered state and S(r,t) ¼ 1 to

a fully ordered state The first field variable C(r,t) is

globally conserved – assuming here that the system

of interest is not exchanging matter with its

environ-ment This imposes the constraint that the time

evolu-tion of the field variable at r is balanced by the

divergence of the flux of species exchanged between

the representative volume centered on r and the

remainder of the system:

@Cðr;tÞ

One then makes use of linear response theory in the

context of thermodynamics of irreversible processes28

to linearly relate the flux J(r,t) to the driving

force responsible for this flux Here, this driving

force is the gradient of the chemical potential

mðr; tÞ ¼ dF=dCðr; tÞ, where F is the free energy of

the system for a compositional field given by C(r,t)

The resulting evolution equation is thus

@Cðr;tÞ

dFdCðr;tÞ

½2

where M is a mobility coefficient In contrast, for the

nonconserved order parameter S(r,t), its evolution is

directly related to the free energy change as S(r,t) varies,

so that by making use of linear response theory again

@Sðr;tÞ

dF

where L is the mobility coefficient for the nonconserved

field S(r,t) Two important consequences ofeqns [2] and

[3]are worth noticing First, although all extrema of the

system free energy (i.e., minima, maxima, saddle points)

are stationary states, often in practice only the minima

can be obtained at steady state due to numerical errors

Second, the stationary state reached from some initial

state may not correspond to the absolute minimum of

the free energy In order to overcome this problem,

noise can be added to transform these deterministic

equations into stochastic (Langevin) equations, as will

be discussed inSection 1.15.3

Following the work of Cahn and coworkers2–4

and Landau and Ginzburg,6 the free energy F is

decomposed into a homogeneous contribution and

an heterogeneous contribution Treating the

inhomo-geneity contribution as a perturbation of a

homoge-neous state, one finds that, in the limit of small

amplitude and long wavelength for this perturbation,

the lowest order correction to the homogeneous free

energy is proportional to the square of the gradient ofthe field variable For instance, returning to the sim-ple example of an alloy described by the concentra-tion field C(r,t), the total free energy can be written asFfCðr ; t Þg ¼

homoge-be used in the case of a nonconserved order parameter,for example, S(r,t), or more generally, in the case of analloy described by nCconserved order parameters and

nSnonconserved order parameters

F ¼ð

q

ij for the nonconserved order parameters is dictated

by the symmetry of the ordered phase Specific ples, for instance for the L12ordered structure, can befound in Braun et al.27and Wang et al.29The free energycan also be augmented to include other contributions,

exam-in particular those comexam-ing from elastic fields usexam-ing theelasticity theory of multiphase coherent solids pio-neered by Khachaturyan,30in the homogeneous mod-ulus case approximation This makes it possible to takeinto account the effect of coherent strains imposed byphase transformations or by a second phase, for exam-ple, a substrate onto which a thin film is deposited.31Two important interfacial quantities, the excessinterface free energy and the interface width, can bederived fromeqn [5]for a system at equilibrium Wefollow here the derivation given by Cahn andHilliard.2 Considering the case of a binary alloywhere two phases may form, referred to as a and b,and with respective B atom concentrations Caand Cb,the existence of an interface between these twophases results in an excess free energy s

s ¼ð

V

dV ½f ðCÞ þ kðrCÞ2 Cme

B ð1  CÞme

A ½6

where meAand meBare the chemical potential of A and

B species when the two phases a and b coexist at

equilibrium At equilibrium, this excess free energy is

minimum A homogeneous free energy Df referenced

to the equilibrium mixture of a- and b-phases is

(Note that the ‘D’ symbol in Df in eqn [7]does not

refer to a Laplacian.) The variational derivative of

this excess energy with respect to the concentration

At equilibrium, the excess free energy s is minimum,

and the concentration field must be such that

Equation [9] must hold locally for any value of the

concentration field along the equilibrium profile

join-ing the a- and b-phases, and this can only be satisfied if

for all values of C(r) It is interesting to note thateqn

[10]means that the equilibrium concentration profile

is such that, at any point on this profile, the

homoge-neous and inhomogehomoge-neous contributions to the total

free energy are equal The interfacial excess free

energy is thus given by

This last integral over the spatial coordinates can

be rewritten as an integral over the concentration

field Assuming a one-dimensional (1D) system for

p

½12

In order to proceed further, it is necessary to assume a

functional shape for the concentration profile or for the

homogeneous free energy Df Expanding the free

energy near the critical point Tcyields a symmetric

double-well potential for the homogeneous free

energy,2which we write here as

!

þ1

Integration along this equilibrium profile fromeqn [11]

yields the interfacial energy

s ¼4

3Cab

ffiffiffiffiffiffiffiffiffiffiffiffiffikDfmax

p

½15Furthermore, the width of the equilibrium profile we,which is defined as the length scale entering the argu-ment of the hyperbolic tangent function ineqn [14], isgiven by

by ab initio calculations.32 If the interfacial width we

is also known, one can obtain Dfmax and k from aninverse solution ofeqns [15] and [16] (note that Cab

is given by the equilibrium phase diagram) Even if

weis not known, values for Dfmaxand k can be chosen

to yield a prescribed value for s In all cases, it isimportant to recognize that any microstructural fea-ture that develops during the simulations is expressed

in units of we At elevated temperatures, as T ! Tc, svanishes2 while we goes to infinity, and therefore,

at high enough temperatures, interfaces are diffuse,thus meeting this essential requirement underlyingthe PF method

The PFeqns [2] and [3]are usually solved ically on a uniform mesh with an explicit time inte-gration, using periodic boundary conditions, whensurface effects are not of interest When the freeenergy contains an elastic energy contribution, it isquite advantageous to use semi-implicit Fourier-spectral algorithms (see Chen9 and Feng et al.33 fordetails) Variable meshing can also be employed, inparticular to better resolve interfaces when they tend

numer-to be sharp, for instance at low temperatures

A few examples selected from the literature serve

to illustrate the capacity of PFMs to successfullyreproduce a wide range of phenomena In particular,Khachaturyan30and his collaborators34–38proposed amicroelasticity theory of multiphase coherent solids,

which has been widely used to include a strain energy

in the overall free energy A method for systems with

strong elastic heterogeneity has been proposed by Hu

and Chen,39 which includes higher order terms that

are usually neglected in Khachaturyan’s approach

Figure 1 illustrates the anisotropic morphology of

Al2Cu precipitates growing in an Al-rich matrix.32

Bulk-free energies were calculated using a

mixed-space cluster expansion technique, with input from

first-principle calculations for about 40 different

ordered structures with full atomic relaxations

Interfacial energies were calculated at T ¼ 0 K from

first-principle calculations as well, using

configura-tions where the Al-rich solid solution and the

tetrag-onal y0-Al2Cu coexist For the elastic strain energy

calculations, the elastic constants of y0-Al2Cu were

calculated ab initio An important feature of this

sys-tem is that both elastic and interfacial energies are

strongly anisotropic, and the PF approach makes it

possible to include these anisotropies Furthermore,

when the high-aspect-ratio y0-phase forms, its growth

kinetics will be anisotropic as well, which can be

included in a phenomenological way by introducing

a dependence of the mobility on the orientation of

the precipitate–matrix interface.Figure 1illustrates

that these three anisotropies, interfacial, elastic, and

kinetic, are required to reproduce the morphology

of y0precipitates

Figure 2 illustrates another effect of coherencystress on microstructural evolution, this time for anA1–L10order–disorder transition in a Co–Pt alloy.40The tetragonal distortion accompanying the orderingreaction leads to the formation of self-organizedtweed patterns of coexisting (cubic) A1- and (tetrago-nal) L10-phases As seen fromFigure 2, the agreementbetween experimental and simulated microstructures

is remarkable Ni and Khachaturyan proposed recentlythat, in order to minimize elastic energy during trans-formations involving symmetry changes and latticestrain, a pseudospinodal decomposition is likely totake place, leading to 3D chessboard patterns.41

PF modeling has also been used extensively tostudy martensitic transformations,34–38,42–44 phasetransformations in ferroelectrics45–57 (see also therecent review by Chen58on that topic), transforma-tions in thin films,47,59–65grain growth and recrystal-lization,66–81 and microstructural evolution in thepresence of cracks or voids.82–84A recent extension

of PFMs has been the inclusion of dislocations in themodels,85–87by taking advantage of the equivalencebetween dislocation loops and coherent misfittingplatelet inclusions.88This approach has been applied,for instance, to study the interaction between movingdislocations and solute atoms,89or to study the influ-ence of dislocation arrays on spinodal decomposition

in thin films.61 Rodney et al.87 have pointed out,

Isotropic

however, that the artificially wide dislocation cores

required by the above approach lead to weak

short-range interactions These authors have introduced a

different PFM for dislocations, which allows for

nar-row dislocation cores As an illustration of that model,

Figure 3shows the development of dislocation loops

and their interaction with hard precipitates in a 3D

g/g0single crystal It is interesting to note that

dislo-cation loop initially expands by gliding in the soft g

channels, until the local stresses are large enough for

the dislocation to shear the hard g0-phase

The above presentation of the PF equations leaves

certain questions open First, the maximum

homoge-neous free energy difference Dfmax involves the free

energy of the unstable state separating the two minima

at Caand Cb It is thus been questioned90whether this

quantity can be rigorously defined from

thermody-namic principles If one employs mean field

techni-ques such as the cluster variation method (CVM)91–95

to derive the homogeneous free energy of an

alloy, Dfmaxis in fact very sensitive to the approximation

used, and generally decreases as the size of the largest

cluster used in the CVM increases.96 Kikuchi97–99

has argued that, in order to resolve this paradox, Df

should not be considered as the free energy of any

homogeneous state, but that it should be understood

as the local contribution to the free energy of the system

along the equilibrium composition profile

The second set of questions relates to the gradientenergy coefficients k and  in eqn [5] In manyapplications of PF modeling, these coefficients aretaken as phenomenological constants that can beadjusted at will, as long as the microstructures arescaled in units of k1/2 or 1/2

Such an approach,however, is problematic for many reasons First,when one scalar field variable is employed, forinstance C(r,t), a regular solution model,2,100or equiv-alently a Bragg–Williams approximation,101 estab-lishes that the gradient energy coefficient is notarbitrary but that it is directly proportional to theinteraction energy between atoms, that is, to the heat

of mixing of the alloy Furthermore, in the mostgeneral case, k should in fact be composition andtemperature dependent Starting from an atomisticmodel, rigorous calculations of k are possible bymonitoring the intensity of composition fluctuations

as a function of their wave vector, and using thefluctuation–dissipation theorem.100 In the case of asimple Ising-like binary alloy, it is observed that kvaries as Cð1  CÞ, where C is the local composition

of the alloy.100Furthermore, when more than one fieldvariable is employed, care should be taken to considerall possible contributions of field heterogeneities tothe free energy of the system, as the different fieldsmay be coupled Symmetry considerations areimportant to identify the nonvanishing terms, but it

may remain challenging to assign values to these

nonvanishing terms that are consistent with the

ther-modynamics of the alloy considered

Another important point is that interfacial

ener-gies are in general anisotropic In order to obtain

realistic morphological evolution, it is often

impor-tant, and sometimes even absolutely necessary, to

include this anisotropy, for example, in the modeling

of dendritic solidification The symmetry of the mesh

chosen for numerically solving the PF equations

introduces interfacial anisotropy but in an unphysical

and uncontrolled way One possible approach to

introduce interfacial anisotropy is to let k vary with

the local orientation of the interface with respect to

crystallographic directions.11,32 Another approach is

to rely on symmetry constraints27,30to determine the

number of independent coefficients in a general

expression of the inhomogeneity term, see eqn [5]

In both approaches, the different coefficients entering

the interfacial anisotropy can be fitted to experiments

or to atomistic simulations

Let us now return to the mobility coefficients Mand L introduced ineqns [2] and [3] For the sake ofsimplicity, many PF calculations are performed whileassigning an arbitrary constant value to these coeffi-cients An improvement can be made by relatingthe mobility to a diffusion coefficient In the case of

M, for instance, in order to make eqn [1]consistentwith Fick’s second law for an ideal binary alloy sys-tem, one should choose

M ¼Cð1  CÞ

where C is the average solute concentration and D~the interdiffusion coefficient In both cases, thesimulated times are expressed in arbitrary units

of M1or L1, thus precluding a direct connectionwith experimental kinetics This problem is alsodirectly related to the lack of absolute physical lengthscales in these simulations Moreover, using a 1DBragg–Williams model composed of atomic planes,Martin101 showed that M is not a constant but is

in fact a function of the local composition alongthe equilibrium profile A complete connectionbetween atomistic dynamics and M will be made

in Section 1.15.3 Similar to the discussion oncoupling between various fields for the gradientenergy terms, kinetic coupling is also expected ingeneral The kinetic couplings between composi-tion (a conserved order parameter) and chemicalordering (a nonconserved order parameter) arerevealed by including sublattices into Martin’s 1Dmodel and deriving the macroscopic evolution ofthe fields from the microscopic dynamics In thatcase, atoms jump between adjacent planes.102,103As

a result, instead of the mere superposition of eqns[2] and [3], the kinetic evolution of coupled con-centration and chemical order in a binary alloy isgiven by

Figure 3 Phase field modeling of the evolution of a

dislocation loop (red line) in a g (dark phase)/g0(white phase)

under applied stress Reproduced from Rodney, D.; Le

Bouar, Y.; Finel, A Acta Mater 2003, 51(1), 17–30.

relate diffusional fluxes (vectors) to chemical

potential gradients (vectors) In the case of cubic

crystalline phases, second-rank tensors reduce to

scalars, but in many ordering reactions, noncubic

phases form, thus leading to anisotropic mobility

Vaks and coworkers104have also derived PFMs for

simultaneous ordering and decomposition starting

from microscopic models These works, however,

illustrate the fact that it would be quite difficult,

especially for multidimensional field variables, to

assign correct values to the kinetic coefficients for a

given alloy system by relying solely on a

phenomeno-logical approach

1.15.3 Quantitative PF Modeling

The PF equations introduced inSection 1.15.2, that

is,eqns [2] and [3], are phenomenological, and one

particular consequence is that they lack an absolute

length scale All scales observed in PF simulations

are expressed in units of the interfacial width weof

the appropriate field variable As discussed in the

previous section, for the case of one scalar conserved

order parameter, this width weand the excess

inter-facial free energy s are directly related to the

gradi-ent energy coefficigradi-ent k and the energy barrier

between the two stable compositions Dfmax (see

eqns [15] and [16])

Beyond the difficulty of parameterizing k and

Dfmaxto accurately reflect the properties of a given

alloy system, the phenomenological nature of these

coefficients creates additional problems In particular,

as the number of mesh points used in a simulation

increases, the interfacial width, expressed in units

of mesh point spacing, remains constant if no other

parameter is changed Increasing the number of mesh

points thus increases the physical volume that is

simulated but does not increase the spatial resolution

of the simulations If the intent is to increase the

spatial resolution, one would have to increase k so

that the equilibrium interface is spread over more

mesh points Equilibrium interfacial widths in alloy

systems typically range from a few nanometers at

high temperatures to a few angstroms at low

tem-peratures In the latter case, if the interface is spread

over several mesh points, it implies that the volume

assigned to each mesh point may not even contain

one atom This raises fundamental questions about

the physical meaning of the continuous field

vari-ables, and practical questions about the merits of

PF modeling over atomistic simulations

Another important problem related to the lack ofabsolute length scale in conventional PF modelingconcerns the treatment of fluctuations Fluctuationsarise owing to the discrete nature of the microscopic(atomistic) models underlying PFMs Furthermore,fluctuations are necessary for a microstructure toescape a metastable state and evolve toward its globalequilibrium state, such as during nucleation Fluctua-tions, or numerical noise, will also determine theinitial kinetic path of a system prepared in an unstablestate The standard approach for adding fluctuations

to the PF kinetic equations is to transform them intoLangevin equations, and then to use the fluctuation–dissipation theorem to determine the structure andamplitude of these fluctuations For instance, inthe case of one conserved order parameter, theCahn–Hilliard diffusion equation, that is, eqn [2], istransformed into the Langevin equation:

@Cðr;tÞ

dFdCðr;tÞ

þ xðr;tÞ ½19where xðr;tÞ is a thermal noise term The structure ofthe noise term can be derived using fluctuation–dissipation105,106:

hxðr;tÞi ¼ 0hxðr;tÞxðr0;t0Þi ¼ 2kBTMr2dðr  r0Þdðt  t0Þ ½20where the brackets h i indicate statistical averagingover an ensemble of equivalent systems However,

eqn [20] does not include a dependence of thenoise amplitude with the cell size, which is notphysical Even if this dependence is added a posteriori,

it is observed practically that this noise amplitudegives rise to unphysical evolution, as reported byDobretsov et al.107While these authors have proposed

an empirical solution to this problem by filtering outthe short-length-scale noise in the calculation of thechemical potentials, a physically sound treatment offluctuations requires a derivation of the PF equationsstarting from a discrete description

Recently, Bronchart et al.100have clearly strated how to rigorously derive the PF equationsfrom a microscopic model through a series of con-trolled approximations We outline here the mainsteps of this derivation The interested reader isreferred to Bronchart et al.100for the full derivation.These authors consider the case of a binary alloysystem in which atoms migrate by exchanging theirposition with atoms that are first nearest neighbors on

demon-a simple cubic ldemon-attice A microscopic configurdemon-ation isdefined by the ensemble of occupation variables, or

spin values, for all lattice sites, C ¼ fsig, where

si ¼ 1 when the site i is occupied by an A or a

B atom, respectively The evolution of the probability

distribution of the microscopic states is given by the

following microscopic Master Equation (ME):

@PðCÞ

X i; j

where the * symbol in the summation indicates that it

is restricted to microscopic states that are connected

to C through one exchange of the i and j nearest

neighbor atoms, resulting in the configuration Cij

The next step is to coarse-grain the atomic lattice

into cells, each cell containing Ndlattice sites It is

then assumed that local equilibrium within the cells

is achieved much faster than evolution across cells

The composition of the cell n, cn, is given by the

average occupation of its lattice site by B atoms, and

thus cn¼ 0; 1=Nd; ; Nd=Nd A mesoscopic

config-uration is fully defined on this coarse-grained system

by C~¼ fcng A chemical potential can be defined

within each cell and, if this chemical potential varies

smoothly from cell to cell, the microscopic ME,eqn

[21], can be coarse-grained into a mesoscopic ME:

where a is the lattice parameter and d the number of

lattice planes per cell (i.e., Nd¼ (d/a)3

), y is theattempt frequency of atom exchanges, lmnðC~

Þ is amobility function that is directly related to the

microscopic jump frequency, b¼ ðkBT Þ1, and

mnðC~

Þ is the chemical potential in cell n The *

symbol over the summation sign indicates that the

summation over m is only performed over cells that

are adjacent to the cell n; the first term on the

right-hand side ofeqn [22]represents a loss term, and there

is a similar gain term, which is not detailed

The mesoscopic MEeqn [22]can be expanded to

the second order using 1/Ndas the small parameter

for the expansion The resulting Fokker–Planck

equa-tion is then transformed into a Langevin equaequa-tion for

the evolution of the composition in each cell n:

lnpðC~Þdðt  t0Þ

Bronchart et al.100applied their model to the study

of nucleation and growth in a cubic A1cBc systemfor various cell sizes, d ¼ 6a, d ¼ 8a, and d ¼ 10a Thesupersaturation is chosen to be small so that thecritical nucleus size is large enough to be resolved

by these cell sizes As seen in Figure 4, for a givensupersaturation, the evolution of the volume fraction

of precipitates is independent of the cell size and

in very good agreement with fully atomistic kineticMonte Carlo (KMC) simulations (not shown in

Figure 4)

The above results are important because theyshow that it is possible to derive and use PFequations that retain an absolute length scaledefined at the atomistic level The point will beshown to be very important for alloys under irra-diation On the other hand, the work by Bronchart

et al.100 clearly highlights the difficulty in usingquantitative PF modeling when the physical lengthscales of the alloy under study are small, as forinstance in the case of precipitation with largesupersaturation, which results in a small criticalnucleus size, or in the case of precipitation growthand coarsening at relatively moderate temperature,which results in a small interfacial width In thesecases, one would have to reduce the cell size down

to a few atoms, thus degrading the validity of themicroscopically based PF equations since they arederived by relying on an expansion with respect tothe parameter 1/Nd

1.15.4 PF Modeling Applied to

Materials Under Irradiation

1.15.4.1 Challenges Specific to Alloys

Under Irradiation

The PFMs discussed so far are broadly applied to

materials as they relax toward some equilibrium

state In particular, the kinetics of evolution is given

by the product of a mobility by a linearized driving

force, see for instanceeqns [2] and [3] In the context

of the thermodynamics of irreversible processes,28the

mobility matrix is the matrix of Onsager coefficients

Irradiation can, however, drive and stabilize a material

system into a nonequilibrium state,108owing to

ballis-tic mixing and permanent defect fluxes, and so it may

appear questionable at first whether linearized

relax-ation kinetics is applicable A sufficient condition,

however, is that these different fields undergo linear

relaxation locally, and this condition is often met even

under irradiation A complicating factor arises from

the presence of ballistic mixing, which adds a second

dynamics to the system on top of the thermally

acti-vated diffusion of atoms and point defects A

superpo-sition of linearized relaxations for these two dynamics

is valid as long as they are sufficiently decoupled in

time and space, so that in any single location, the

system will evolve according to one dynamic at a

time KMC simulations indicate that, for dilute alloys,

this decoupling is valid except for a small range of

kinetic parameters where events from different

dynamics interfere with one another.109

A second issue is that PFMs, traditionally, do notinclude explicitly point defects Vacancies and inter-stitials are, however, essential to the evolution ofirradiated materials, and it is thus necessary toinclude them as additional field variables The situa-tion is more problematic with point-defect clusters,which often play a key role in the annihilation of freepoint defects Since the size of these clusters cover awide range of values, it would be quite difficult to add

a new field variable for each size, for example, forvacancy clusters of size 2 (divacancies), size 3 (triva-cancies), size 4, etc Moreover, under irradiation con-ditions leading to the direct production of defectclusters by displacement cascades, additional lengthscales are required to describe the distribution ofdefect cluster sizes and of atomic relocation dis-tances These new length scales are not physicallyrelated to the width of a chemical interface at equi-librium, we, and therefore, they cannot be safelyrescaled by we This analysis clearly suggests thatone needs to rely on a PFM where the atomic scalehas been retained This is, for instance, the case in thequantitative PFM reviewed in Section 1.15.3.Another possible approach is to use a mixed continu-ous–discrete description, as illustrated below inSec-tion 1.15.4.2.4 We note that information on defectcluster sizes and relocation distances should be seen

as part of the noise imposed by the external forcing,here the irradiation, on the evolution of the fieldvariables The difficulty is thus to develop a modelthat can correctly integrate this external noise It is

well documented that, for nonlinear dissipative

sys-tems, the external noise can play a determinant role

and, for large enough noise amplitude, may trigger

nonequilibrium phase transformations.110–113

One last and important challenge in the

develop-ment of PFMs for alloys under irradiation is the

fact that in nearly all traditional models the

mobil-ity matrix is oversimplified, for instance Mirr¼

Cð1  CÞD~irr=kBT , which is a simple extension

to eqn [17] where D~ has been replaced by D~irr to

take into account radiation-enhanced diffusion In

the common case of multidimensional fields, for

instance for multicomponent alloys, or for alloys

with conserved and nonconserved field variables,

the mobility matrix is generally taken as a diagonal

matrix, thus eliminating any possible kinetic

cou-pling between these different field variables As

discussed at the end ofSection 1.15.2, this

approxi-mation raises concerns because it misses the fact that

these kinetic coefficients are related since they

origi-nate from the same microscopic mechanisms This is,

in particular, the case for the coupled evolution of

point defects and chemical species in

multicompo-nent alloys This coupling is of particular relevance

to the case of irradiated alloys since irradiation can

dramatically alter segregation and precipitation

reac-tions owing to the influence of local chemical

envir-onments on point-defect jump frequencies While

new analytical models have been developed recently

using mean field approximations to obtain expressions

for correlation factors in concentrated alloys,114–117

work remains to be done to integrate these results

into PFMs

1.15.4.2 Examples of PF Modeling Applied

to Alloys Under Irradiation

1.15.4.2.1 Effects of ballistic mixing on

phase-separating alloy systems

Consider the simple case where the external forcing

produces forced exchanges between atoms (such

relocations are found in displacement cascades), and

let us assume for now that these relocations are

ballistic (i.e., random) and take place one at a time

For this case, one can use a 1D PFM to follow the

evolution of the composition profile C(x) during

irra-diation.118This evolution is the sum of a thermally

activated term, for which the classical Cahn diffusion

model can be used, and a ballistic term:

initi-th Þ, where G irr

th is anaverage atomic jump frequency, enhanced by thepoint-defect supersaturation created by irradiation

In particular, in the case of an alloy with preexistingprecipitates, depending upon the irradiation flux andthe irradiation temperature, this criterion predictsthat the precipitates should either dissolve or contin-uously coarsen with time

Some relocation distances, however, extend beyondthe first nearest neighbor distances,120,121 and it isinteresting to consider the case where the characteris-tic distance R exceeds the cell size An analytical model

by Enrique and Bellon118 revealed that, when Rexceeds a critical value Rc, irradiation can lead tothe dynamic stabilization of patterns To illustratethis point, one performs a linear stability analysis

of this model in Fourier space, assuming here thatthe ballistic jump distances are distributed exponen-tially The amplification factor w(q) of the Fouriercoefficient for the wave vector q is given by

oðqÞ=M ¼  ð@2f =@C2Þq2 2kq4

 gR2q2=ð1 þ R2q2Þ ½27where f (C) is the free-energy density of a homoge-neous alloy of composition C, k the gradient energycoefficient, and g¼ Gb=M is a reduced ballistic jumpfrequency The analysis is here restricted to composi-tions and temperatures such that, in the absence ofirradiation, spinodal decomposition takes place, that is,

@2f/@C2< 0

The various possible dispersion curves are plotted

inFigure 5 Unlike in the case of short R, it is nowpossible to find irradiation intensities g such that the