Comprehensive nuclear materials 1 15 phase field methods

1.15

Phase Field Methods

P. Bellon
University of Illinois at Urbana-Champaign, Urbana, IL, USA

ß 2012 Elsevier Ltd. All rights reserved.

1.15.1

Introduction

411

1.15.2
1.15.3
1.15.4
1.15.4.1
1.15.4.2
1.15.4.2.1
1.15.4.2.2
1.15.4.2.3
1.15.4.2.4
1.15.5
References

General Principles and Applications of PF Modeling
Quantitative PF Modeling
PF Modeling Applied to Materials Under Irradiation
Challenges Specific to Alloys Under Irradiation

Examples of PF Modeling Applied to Alloys Under Irradiation
Effects of ballistic mixing on phase-separating alloy systems
Coupled evolution of composition and chemical order under irradiation
Irradiation-induced formation of void lattices
Irradiation-induced segregation on defect clusters
Conclusions and Perspectives

412
418
420
420
421
421
423
426
427
428
430

Abbreviations
1D
CVM
KMC
ME
PF
PFM
SIA

One-dimensional
Cluster variation method

Kinetic Monte Carlo
Master equation
Phase field
Phase field model
Self-interstitial atom

1.15.1 Introduction
Electronic and atomistic processes often dictate the
pathways of phase transformations and microstructural 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 dynamics simulations (see Chapter 1.09, Molecular
Dynamics) and atomistic Monte Carlo simulations
(see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). While these atomistic
methods can now simulate quite accurately the evolution of specific alloy systems, these simulations are
nevertheless limited to small length scales, from a few
to 100 nm. Molecular dynamics is furthermore limited 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, Kinetic
Monte Carlo Simulations of Irradiation Effects).
An alternative modeling approach is to replace the
many microscopic degrees of freedom of the system
of interest by the few mesoscopic variables that are
sufficient to provide a realistic description. This
approach has been widely used in many disciplines,
and well-known examples are the Fourier and Fick
equations, which describe the diffusive transport

of heat and chemical species, respectively. This
approach is also commonly used in modeling the
evolution of point defects, in particular, during irradiation (see Chapter 1.13, Radiation Damage Theory and Sizmann1). The work of Cahn and Hilliard2–
5
and Landau and Lifshitz (see for instance Tole´dano and Toledano6) provided a way to include the
contributions of interfaces to chemical evolution,
thus making it possible to model heterogeneous
and multiphase materials. Kinetic models based on
these descriptions are broadly referred to as phase
field (PF) methods, since the microstructure of a
material is fully characterized by a few mesoscopic
field variables such as concentration, magnetization,
chemical order, or temperature. One key assumption
of this approach is that the variables chosen to
describe the state of the system vary smoothly across
any interface or, in other words, that interfaces are
diffuse. This assumption finds a natural justification
in the theory of critical phenomena, since the

411


412

Phase Field Methods

interface thickness diverges at the critical temperature.7 Diffuse interface models offer some advantages over sharp interface models,8 in particular,
for the modeling of complex microstructures. Furthermore, the PF approach can be extended to
include macroscopic variables other than the local
composition, making it possible to describe chemical order–disorder transitions, solid–liquid reactions, displacive transformations, and more

recently dislocation glide. PF methods and applications have been recently reviewed by Chen,9
Emmerich,10 and Singer-Loginova and Singer.11
This chapter focuses on solid–solid phase transformations, with a particular emphasis on transformations and microstructural evolution relevant to
irradiated materials. While conventional PF modeling 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 possess atomic resolution,12–26 the 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 pointdefect 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 nucleation 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 materials 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 irradiated materials are presented in Section 1.15.4.2.
Finally, conclusions and perspectives are given in
Section 1.15.5.

1.15.2 General Principles and
Applications of PF Modeling
The first step in PF modeling lies in choosing
and defining the fields of interest. These continuous
variables are functions of space and time, and they
are in most cases scalar fields, such as temperature,
or the concentration of some chemical species of
interest. In systems with solid–liquid interfaces, a
phenomenological field variable is introduced in
such a way that it varies continuously from 0 to 1 as
one goes from a fully solid to a fully liquid phase.
Multidimensional fields can be used as well, for
instance, to describe the local composition of a
multicomponent alloy, the local degree of chemical
order, or the local crystallographic orientation of
grains. These multidimensional fields may transform
like vectors under symmetry operations, thus leading
to a vectorial representation of the system and tensorial expressions for mobilities (as will be discussed
later), but there are cases for which the multidimensional fields cannot be reduced to vectors.27 In all
cases, an averaging procedure is necessary to define
continuous field variables for systems that are intrinsically discrete at the atomic scale. Various averages
can be used, including (1) a spatial average over
representative volume elements, which will correspond to the cells used for evolving the PF variables;
(2) a spatial and temporal average; or (3) a spatial
and ensemble average. The spatial averaging method

is used most often, although in many cases the
exact conditions of the averaging procedure are not
defined. Section 1.15.3 will cover a model where this
coarse-graining is performed explicitly and rigorously. The last two averaging procedures are rarely
explicitly invoked, although one of their advantages is
that a smaller volume can be used for the spatial
average, thanks to the additional averaging performed either in time or in the configuration space
of a system ensemble.
Turning now to the kinetic equations used to
describe the evolution of these field variables, an
important distinction is whether the field variable is
conserved or nonconserved. For the sake of simplicity, the following discussion focuses on alloy systems.
Let us consider two simple examples, one where the
field variable is the local composition, C(r,t), in a
binary A–B alloy system, and a second example,
also for a binary alloy system, but this time with a
fixed composition and where chemical ordering takes
place. The degree of chemical order is described
by the field S(r,t). For the sake of convenience, one


Phase Field Methods

may normalize that field such that S(r,t) ¼ 0 corresponds to a fully disordered state and S(r,t) ¼ Æ1 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 environment. This imposes the constraint that the time evolution 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;t Þ
¼ ÀrJ ðr;t Þ
½1Š
@t
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 ; t Þ ¼ dF =dCðr ; t Þ, 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;t Þ
dF
¼ Àr ÀMr
½2Š
@t
dCðr;t Þ
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;t Þ
dF
¼ ÀL
@t
dSðr;t Þ

½3Š

where L is the mobility coefficient for the nonconserved
field S(r,t). Two important consequences of eqns [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 in Section 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 inhomogeneity contribution as a perturbation of a homogeneous state, one finds that, in the limit of small
amplitude and long wavelength for this perturbation,
the lowest order correction to the homogeneous free

413

energy is proportional to the square of the gradient of
the field variable. For instance, returning to the simple example of an alloy described by the concentration field C(r,t), the total free energy can be written as
ð
½4Š
F fCðr ; t Þg ¼ dV ½f ðCÞ þ kðrCÞ2 Š
V

where f (C) is the free-energy density of a homogeneous alloy for the composition C, and k the gradient
energy coefficient, which is positive for an alloy system
with a positive heat of mixing. A similar expression can
be used in the case of a nonconserved order parameter,
for example, S(r,t), or more generally, in the case of an
alloy described by nC conserved order parameters and
nS nonconserved order parameters
ð 
F ¼ dV f ðC1 ; C2 . . . CnC ; S1 ; S2 . . . SnS Þ
V

þ

nC
X
p¼1

kp ðrCp Þ2 þ

nS
X


q

ij ri Sq rj Sq

½5Š

q¼1

An implicit summation over the indices i and j is
assumed in the last term of eqn [5]. The number of
nonzero and independent gradient energy coefficients
q
ij for the nonconserved order parameters is dictated
by the symmetry of the ordered phase. Specific examples, for instance for the L12 ordered structure, can be
found in Braun et al.27 and Wang et al.29 The free energy
can also be augmented to include other contributions,
in particular those coming from elastic fields using the
elasticity theory of multiphase coherent solids pioneered by Khachaturyan,30 in the homogeneous modulus case approximation. This makes it possible to take
into account the effect of coherent strains imposed by
phase transformations or by a second phase, for example, a substrate onto which a thin film is deposited.31
Two important interfacial quantities, the excess
interface free energy and the interface width, can be
derived from eqn [5] for a system at equilibrium. We
follow here the derivation given by Cahn and
Hilliard.2 Considering the case of a binary alloy
where two phases may form, referred to as a and b,
and with respective B atom concentrations Ca and Cb,
the existence of an interface between these two
phases results in an excess free energy s
ð
s ¼ dV ½f ðCÞ þ kðrCÞ2 À CmeB À ð1 À CÞmeA Š ½6Š
V

where meA and meB are the chemical potential of A and
B species when the two phases a and b coexist at

414

Phase Field Methods

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
introduced as
Df ðCÞ ¼ f ðCÞ À ½CmeB þ ð1 À CÞmeA Š
¼ C½mB ðCÞ À meB Š þ ð1 À CÞ½mA ðCÞ À meA Š ½7Š
(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
field is given by
ds @Df
@k
½8Š
¼
À 2kDC À
ðrCÞ2
dC
@C
@C
At equilibrium, the excess free energy s is minimum,
and the concentration field must be such that
ds=dC ¼ 0. Thus,
@Df
@k
@

¼ 2kDC þ
ðrCÞ2 ¼
ðkðrCÞ2 Þ
½9Š
@C
@C
@C
Equation [9] must hold locally for any value of the
concentration field along the equilibrium profile joining the a- and b-phases, and this can only be satisfied if
kðrCÞ2 ¼ Df

½10Š

for all values of C(r). It is interesting to note that eqn
[10] means that the equilibrium concentration profile
is such that, at any point on this profile, the homogeneous and inhomogeneous contributions to the total
free energy are equal. The interfacial excess free
energy is thus given by
ð
ð
½11Š
s ¼ 2 dV Df ¼ 2 dV kðrCÞ2
V

V

This last integral over the spatial coordinates can
be rewritten as an integral over the concentration
field. Assuming a one-dimensional (1D) system for
simplicity,

Cðb

Cðb

dC krC ¼ 2

s¼2
Ca

dC

pffiffiffiffiffiffiffiffiffi
kDf

½12Š

Ca

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 Tc yields a symmetric
double-well potential for the homogeneous free
energy,2 which we write here as



2C À 1 2
2C À 1 2
Df ¼ Dfmax 1 À

1þ
½13Š
Cab
Cab

with Cab ¼ Cb À Ca ¼ 1 À 2Ca ¼ 2Cb À 1. Using
eqn [10], the equilibrium concentration profile can
now be obtained:
rffiffiffiffiffiffiffiffiffiffiffi !
Cab
2 Dfmax
1
tanh
x þ
½14Š
CðxÞ ¼
2
k
Cab
2
Integration along this equilibrium profile from eqn [11]
yields the interfacial energy
4 pffiffiffiffiffiffiffiffiffiffiffiffiffi
s ¼ Cab kDfmax
½15Š
3
Furthermore, the width of the equilibrium profile we,
which is defined as the length scale entering the argument of the hyperbolic tangent function in eqn [14], is
given by
rffiffiffiffiffiffiffiffiffiffiffi

Cab
k
½16Š
we ¼
2 Dfmax
In this conventional approach to PFMs, Dfmax and k
are phenomenological coefficients. Equations [15] and
[16] play an important role in assigning values to
these coefficients for a specific alloy system. The
excess interfacial energy may be known experimentally or it may be calculated separately, for instance
by ab initio calculations.32 If the interfacial width we
is also known, one can obtain Dfmax and k from an
inverse solution of eqns [15] and [16] (note that Cab
is given by the equilibrium phase diagram). Even if
we is not known, values for Dfmax and k can be chosen
to yield a prescribed value for s. In all cases, it is
important to recognize that any microstructural feature that develops during the simulations is expressed
in units of we. At elevated temperatures, as T ! Tc , s
vanishes2 while we goes to infinity, and therefore,
at high enough temperatures, interfaces are diffuse,
thus meeting this essential requirement underlying
the PF method.
The PF eqns [2] and [3] are usually solved numerically on a uniform mesh with an explicit time integration, using periodic boundary conditions, when
surface effects are not of interest. When the free
energy contains an elastic energy contribution, it is
quite advantageous to use semi-implicit Fourierspectral algorithms (see Chen9 and Feng et al.33 for
details). Variable meshing can also be employed, in
particular to better resolve interfaces when they tend
to be sharp, for instance at low temperatures.
A few examples selected from the literature serve

to illustrate the capacity of PFMs to successfully
reproduce a wide range of phenomena. In particular,
Khachaturyan30 and his collaborators34–38 proposed a
microelasticity theory of multiphase coherent solids,


Phase Field Methods

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 mixedspace 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 configurations where the Al-rich solid solution and the tetragonal y0 -Al2Cu coexist. For the elastic strain energy
calculations, the elastic constants of y0 -Al2Cu were
calculated ab initio. An important feature of this system 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 1 illustrates
that these three anisotropies, interfacial, elastic, and

kinetic, are required to reproduce the morphology
of y0 precipitates.

Isotropic

Figure 2 illustrates another effect of coherency
stress on microstructural evolution, this time for an
A1–L10 order–disorder transition in a Co–Pt alloy.40
The tetragonal distortion accompanying the ordering
reaction leads to the formation of self-organized
tweed patterns of coexisting (cubic) A1- and (tetragonal) L10-phases. As seen from Figure 2, the agreement
between experimental and simulated microstructures
is remarkable. Ni and Khachaturyan proposed recently
that, in order to minimize elastic energy during transformations involving symmetry changes and lattice
strain, a pseudospinodal decomposition is likely to
take place, leading to 3D chessboard patterns.41
PF modeling has also been used extensively to
study martensitic transformations,34–38,42–44 phase
transformations in ferroelectrics45–57 (see also the
recent review by Chen58 on that topic), transformations in thin films,47,59–65 grain growth and recrystallization,66–81 and microstructural evolution in the
presence of cracks or voids.82–84 A recent extension
of PFMs has been the inclusion of dislocations in the
models,85–87 by taking advantage of the equivalence
between dislocation loops and coherent misfitting
platelet inclusions.88 This approach has been applied,
for instance, to study the interaction between moving
dislocations and solute atoms,89 or to study the influence of dislocation arrays on spinodal decomposition
in thin films.61 Rodney et al.87 have pointed out,

Interface only

(b)

(a)
Int + strain

Strain only

(c)

Int + strain + kinetics

Experiment

50 nm

50 nm

(d)

415

(e)

(f)

Figure 1 Phase field simulations of y0 -phase precipitation in Al–Si–Cu alloys at 450  C, illustrating that strain, interfacial, and
kinetic anisotropies are required to reproduce experimental morphologies. Reprinted with permission from Vaithyanathan, V.;
Wolverton, C.; Chen, L. Q. Phys. Rev. Lett. 2002, 88(12), 1255031–1255034. Copyright by the American Physical Society.

416

Phase Field Methods

(a)

(b)

(c)
Time

(d)

(e)

(f)

Figure 2 Comparison between transmission electron microscopy experimental observations (a–c) and phase field modeling
(d–f) of formation of chessboard pattern in Co39.5Pt60.5 cooled from 1023 K to (a) 963 K, (b) 923 K, and (c) 873 K. The scale
bar corresponds to 30 nm. Reproduced from Le Bouar, Y.; Loiseau, A.; Khachaturyan, A. G. Acta Mater. 1998, 46(8), 2777–2788.

however, that the artificially wide dislocation cores
required by the above approach lead to weak shortrange interactions. These authors have introduced a
different PFM for dislocations, which allows for narrow dislocation cores. As an illustration of that model,
Figure 3 shows the development of dislocation loops
and their interaction with hard precipitates in a 3D
g/g0 single crystal. It is interesting to note that dislocation 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 homogeneous free energy difference Dfmax involves the free
energy of the unstable state separating the two minima
at Ca and Cb. It is thus been questioned90 whether this
quantity can be rigorously defined from thermodynamic principles. If one employs mean field techniques such as the cluster variation method (CVM)91–95
to derive the homogeneous free energy of an
alloy, Dfmax is 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 gradient
energy coefficients k and  in eqn [5]. In many
applications of PF modeling, these coefficients are
taken as phenomenological constants that can be
adjusted at will, as long as the microstructures are
scaled 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, for
instance C(r,t), a regular solution model,2,100 or equivalently a Bragg–Williams approximation,101 establishes that the gradient energy coefficient is not
arbitrary but that it is directly proportional to the
interaction energy between atoms, that is, to the heat
of mixing of the alloy. Furthermore, in the most
general case, k should in fact be composition and
temperature dependent. Starting from an atomistic
model, rigorous calculations of k are possible by

monitoring the intensity of composition fluctuations
as a function of their wave vector, and using the
fluctuation–dissipation theorem.100 In the case of a
simple Ising-like binary alloy, it is observed that k
varies as Cð1 À CÞ, where C is the local composition
of the alloy.100 Furthermore, when more than one field
variable is employed, care should be taken to consider
all possible contributions of field heterogeneities to
the free energy of the system, as the different fields
may be coupled. Symmetry considerations are
important to identify the nonvanishing terms, but it


Phase Field Methods

(a)

(b)

the interfacial anisotropy can be fitted to experiments
or to atomistic simulations.
Let us now return to the mobility coefficients M
and L introduced in eqns [2] and [3]. For the sake of
simplicity, many PF calculations are performed while
assigning an arbitrary constant value to these coefficients. An improvement can be made by relating
the mobility to a diffusion coefficient. In the case of
M, for instance, in order to make eqn [1] consistent
with Fick’s second law for an ideal binary alloy system, one should choose
M¼

(c)

(d)

(e)

(f)

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.

may remain challenging to assign values to these
nonvanishing terms that are consistent with the thermodynamics of the alloy considered.
Another important point is that interfacial energies are in general anisotropic. In order to obtain
realistic morphological evolution, it is often important, 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 constraints27,30 to determine the
number of independent coefficients in a general
expression of the inhomogeneity term, see eqn [5].
In both approaches, the different coefficients entering

417

Cð1 À CÞ ~
D
kB T

½17Š

where C is the average solute concentration and D~
the interdiffusion coefficient. In both cases, the
simulated times are expressed in arbitrary units
of MÀ1 or LÀ1, thus precluding a direct connection
with experimental kinetics. This problem is also
directly related to the lack of absolute physical length
scales in these simulations. Moreover, using a 1D
Bragg–Williams model composed of atomic planes,
Martin101 showed that M is not a constant but is
in fact a function of the local composition along
the equilibrium profile. A complete connection
between atomistic dynamics and M will be made
in Section 1.15.3. Similar to the discussion on
coupling between various fields for the gradient
energy terms, kinetic coupling is also expected in
general. The kinetic couplings between composition (a conserved order parameter) and chemical
ordering (a nonconserved order parameter) are
revealed by including sublattices into Martin’s 1D
model and deriving the macroscopic evolution of
the fields from the microscopic dynamics. In that
case, atoms jump between adjacent planes.102,103 As
a result, instead of the mere superposition of eqns

[2] and [3], the kinetic evolution of coupled concentration and chemical order in a binary alloy is
given by



@Cðr;t Þ
dF
¼r M1 r
@t
dCðr;t Þ



dF
þ r M2 r
dSðr;t Þ



@Sðr;t Þ
@F
dF
¼ ÀL1
þ r L2 r
@t
@Sðr;t Þ
dSðr;t Þ




dF
þ r L3 r
dCðr;t Þ

½18Š

where L1 is a mobility coefficient, and L2, L3, M1,
and M2 are second-rank mobility tensors, since they


418

Phase Field Methods

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 coworkers104 have 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 phenomenological approach.

1.15.3 Quantitative PF Modeling
The PF equations introduced in Section 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 we of
the appropriate field variable. As discussed in the
previous section, for the case of one scalar conserved
order parameter, this width we and the excess interfacial free energy s are directly related to the gradient energy coefficient k and the energy barrier
between the two stable compositions Dfmax (see
eqns [15] and [16]).
Beyond the difficulty of parameterizing k and
Dfmax to 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 temperatures. 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 variables, and practical questions about the merits of
PF modeling over atomistic simulations.

Another important problem related to the lack of
absolute length scale in conventional PF modeling
concerns the treatment of fluctuations. Fluctuations
arise owing to the discrete nature of the microscopic
(atomistic) models underlying PFMs. Furthermore,
fluctuations are necessary for a microstructure to
escape a metastable state and evolve toward its global
equilibrium state, such as during nucleation. Fluctuations, or numerical noise, will also determine the
initial kinetic path of a system prepared in an unstable
state. The standard approach for adding fluctuations
to the PF kinetic equations is to transform them into
Langevin equations, and then to use the fluctuation–
dissipation theorem to determine the structure and
amplitude of these fluctuations. For instance, in
the case of one conserved order parameter, the
Cahn–Hilliard diffusion equation, that is, eqn [2], is
transformed into the Langevin equation:


@Cðr;t Þ
dF
¼ Àr ÀMr
þ xðr;t Þ
½19Š
@t
dCðr;t Þ
where xðr;t Þ is a thermal noise term. The structure of
the noise term can be derived using fluctuation–
dissipation105,106:
hxðr;t Þi ¼ 0

hxðr;t Þxðr0 ;t 0 Þi ¼ À2kB TMr2 dðr À r0 Þdðt À t 0 Þ

½20Š

where the brackets h i indicate statistical averaging
over an ensemble of equivalent systems. However,
eqn [20] does not include a dependence of the
noise amplitude with the cell size, which is not
physical. Even if this dependence is added a posteriori,
it is observed practically that this noise amplitude
gives rise to unphysical evolution, as reported by
Dobretsov et al.107 While these authors have proposed
an empirical solution to this problem by filtering out
the short-length-scale noise in the calculation of the
chemical potentials, a physically sound treatment of
fluctuations requires a derivation of the PF equations
starting from a discrete description.
Recently, Bronchart et al.100 have clearly demonstrated how to rigorously derive the PF equations
from a microscopic model through a series of controlled approximations. We outline here the main
steps of this derivation. The interested reader is
referred to Bronchart et al.100 for the full derivation.
These authors consider the case of a binary alloy
system in which atoms migrate by exchanging their
position with atoms that are first nearest neighbors on
a simple cubic lattice. A microscopic configuration is
defined by the ensemble of occupation variables, or


Phase Field Methods

spin values, for all lattice sites, C ¼ fsi g, where
si ¼ Æ1 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):
Ã
X
@PðCÞ
¼À
W ðC ! Cij ÞPðCÞ
@t
i; j

þ

Ã
X
i; j

W ðCij ! CÞPðCij Þ

½21Š

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 Nd lattice 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 cn ¼ 0; 1=Nd ; . . . ; Nd =Nd . A mesoscopic configuration is fully defined on this coarse-grained system
~
by C ¼ fcn g. 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:


 2 Ã
~
X
@PðCÞ
a
ba
~
~
~
~
y
lmn ðCÞexp
¼À
ðmm ðCÞ À mn ðCÞÞ PðCÞ
@t
d
2d
n;m
þ gain term

½22Š

where a is the lattice parameter and d the number of
lattice planes per cell (i.e., Nd ¼ (d/a)3), y is the
~
attempt frequency of atom exchanges, lmn ðCÞ is a
mobility function that is directly related to the
microscopic jump frequency, b ¼ ðkB T ÞÀ1 , and
~
mn ðCÞ 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 righthand side of eqn [22] represents a loss term, and there
is a similar gain term, which is not detailed.
The mesoscopic ME eqn [22] can be expanded to
the second order using 1/Nd as the small parameter
for the expansion. The resulting Fokker–Planck equation is then transformed into a Langevin equation for
the evolution of the composition in each cell n:
ðnÞ

@cn a 2 y X
~
~
~
lnm ðC Þ½mm ðC Þ À mm ðC ފ þ zn ðt Þ ½23Š
¼ 2
@t
d kB T m

419

where the noise term zn ðt Þ is a Gaussian noise with
first and second moments given by
hzn ðt Þi ¼ 0
hzn ðt Þzn ðt 0 Þi ¼

2 a2 X
~
lnp ðC Þdðt À t 0 Þ
Nd d 2 p

hzn ðt Þzm ðt 0 Þi ¼

À2 a2
~
lnm ðC Þdðt À t 0 Þ
Nd d 2

ðnÞ

½24Š

While the structure of eqns [23] and [24] is quite
similar to that of the phenomenological eqns [19]
and [20], there are several key differences in these
two descriptions. First, thermodynamic quantities
such as the homogeneous free-energy density and
the gradient energy coefficient are now cell-size
dependent. These quantities can be evaluated separately using standard Monte Carlo techniques.100

Second, the mobility coefficients, and thus
the correlations in the Langevin noise, are functions of the local concentration, as well as of the
cell size.
Bronchart et al.100 applied their model to the study
of nucleation and growth in a cubic A1ÀcBc system
for various cell sizes, d ¼ 6a, d ¼ 8a, and d ¼ 10a. The
supersaturation is chosen to be small so that the
critical nucleus size is large enough to be resolved
by these cell sizes. As seen in Figure 4, for a given
supersaturation, the evolution of the volume fraction
of precipitates is independent of the cell size and
in very good agreement with fully atomistic kinetic
Monte Carlo (KMC) simulations (not shown in
Figure 4).
The above results are important because they
show that it is possible to derive and use PF
equations that retain an absolute length scale
defined at the atomistic level. The point will be
shown to be very important for alloys under irradiation. On the other hand, the work by Bronchart
et al.100 clearly highlights the difficulty in using
quantitative PF modeling when the physical length
scales of the alloy under study are small, as for
instance in the case of precipitation with large
supersaturation, which results in a small critical
nucleus size, or in the case of precipitation growth
and coarsening at relatively moderate temperature,
which results in a small interfacial width. In these
cases, one would have to reduce the cell size down
to a few atoms, thus degrading the validity of the
microscopically based PF equations since they are

derived by relying on an expansion with respect to
the parameter 1/Nd.


Phase Field Methods

Volume fraction

420

0.04

C = 0.160, d = 8a
C = 0.165, d = 8a
C = 0.170, d = 8a

0.02

C = 0.170, d = 6a
C = 0.170, d = 10a
0.00

0

2 ´ 106

4 ´ 106
−1

Time (unit: q )

Figure 4 Evolution of the volume fraction of precipitates with time for a three-dimensional binary alloy A1ÀcBc using the
microscopically derived phase field eqns [23] and [24]. Parameters a and d are the lattice parameter and the number of lattice
planes in h100i directions. For a given concentration, C ¼ 0.17, the precipitation kinetics is equally well resolved with three
different cell sizes. Reprinted with permission from Bronchart, Q.; Le Bouar, Y.; Finel, A. Phys. Rev. Lett. 2008, 100(1).
Copyright by the American Physical Society.

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 instance eqns [2] and [3]. In the context
of the thermodynamics of irreversible processes,28 the
mobility matrix is the matrix of Onsager coefficients.
Irradiation can, however, drive and stabilize a material
system into a nonequilibrium state,108 owing to ballistic mixing and permanent defect fluxes, and so it may
appear questionable at first whether linearized relaxation 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 activated diffusion of atoms and point defects. A superposition 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 not
include explicitly point defects. Vacancies and interstitials are, however, essential to the evolution of
irradiated materials, and it is thus necessary to
include them as additional field variables. The situation is more problematic with point-defect clusters,
which often play a key role in the annihilation of free
point defects. Since the size of these clusters cover a
wide range of values, it would be quite difficult to add
a new field variable for each size, for example, for
vacancy clusters of size 2 (divacancies), size 3 (trivacancies), size 4, etc. Moreover, under irradiation conditions leading to the direct production of defect
clusters by displacement cascades, additional length
scales are required to describe the distribution of
defect cluster sizes and of atomic relocation distances. These new length scales are not physically
related to the width of a chemical interface at equilibrium, we, and therefore, they cannot be safely
rescaled by we. This analysis clearly suggests that
one needs to rely on a PFM where the atomic scale
has been retained. This is, for instance, the case in the
quantitative PFM reviewed in Section 1.15.3.
Another possible approach is to use a mixed continuous–discrete description, as illustrated below in Section 1.15.4.2.4. We note that information on defect
cluster 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 field
variables. The difficulty is thus to develop a model
that can correctly integrate this external noise. It is


Phase Field Methods

well documented that, for nonlinear dissipative systems, 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 development of PFMs for alloys under irradiation is the
fact that in nearly all traditional models the mobility matrix is oversimplified, for instance Mirr ¼
Cð1 À CÞD~irr =kB T , 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 coupling between these different field variables. As
discussed at the end of Section 1.15.2, this approximation raises concerns because it misses the fact that
these kinetic coefficients are related since they originate from the same microscopic mechanisms. This is,
in particular, the case for the coupled evolution of
point defects and chemical species in multicomponent alloys. This coupling is of particular relevance
to the case of irradiated alloys since irradiation can
dramatically alter segregation and precipitation reactions owing to the influence of local chemical environments 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 irradiation.118 This evolution is the sum of a thermally
activated term, for which the classical Cahn diffusion
model can be used, and a ballistic term:


ð
dCðxÞ
dF
¼ Mirr r2
À Gb CðxÞ À wR ðx À x 0 ÞCðx 0 Þdx 0 ½25Š
dt
dC

421

where Mirr is the thermal atomic mobility, here accelerated by the irradiation, F the free energy of the
system, Gb the jump frequency of the atomic relocations forced by the nuclear collisions, and wR is the
normalized distribution of relocation distances, characterized by a decay length R. Since most of these
atomic relocations take place between nearest neighbor atoms, in a first approximation one may assume
that R is small compared to the cell size. In this case,
the second ballistic term in eqn [25] reduces to a
diffusive term:
dCðxÞ
dF
¼ Mirr r2

À Gb a 2 r2 C
dt
dC

½26Š

In this case, the model thus reduces to the one initially introduced by Martin,119 and the steady state
reached under irradiation is the equilibrium state
that the same alloy would have reached at an effective
irr
temperature Teff ¼ T ð1 þ Gb =Girr
th Þ, where Gth is an
average atomic jump frequency, enhanced by the
point-defect supersaturation created by irradiation.
In particular, in the case of an alloy with preexisting
precipitates, depending upon the irradiation flux and
the irradiation temperature, this criterion predicts
that the precipitates should either dissolve or continuously coarsen with time.
Some relocation distances, however, extend beyond
the first nearest neighbor distances,120,121 and it is
interesting to consider the case where the characteristic distance R exceeds the cell size. An analytical model
by Enrique and Bellon118 revealed that, when R
exceeds a critical value Rc, irradiation can lead to
the dynamic stabilization of patterns. To illustrate
this point, one performs a linear stability analysis
of this model in Fourier space, assuming here that
the ballistic jump distances are distributed exponentially. The amplification factor w(q) of the Fourier
coefficient for the wave vector q is given by
oðqÞ=M ¼ À ð@ 2 f =@C 2 Þq 2 À 2kq4
À gR2 q 2 =ð1 þ R2 q 2 Þ

½27Š

where f (C) is the free-energy density of a homogeneous alloy of composition C, k the gradient energy
coefficient, and g ¼ Gb =M is a reduced ballistic jump
frequency. The analysis is here restricted to compositions and temperatures such that, in the absence of
irradiation, spinodal decomposition takes place, that is,
@ 2f/@C 2 < 0.
The various possible dispersion curves are plotted
in Figure 5. Unlike in the case of short R, it is now
possible to find irradiation intensities g such that the


422

Phase Field Methods

Thermal
w(q)

Total

qmin

128
q

Irradiation
Figure 5 Sketch of the dispersion curve given by the
linear stability analysis eqn [27], in the case when the

ballistic relocation distance R is large. The total dispersion
curve is decomposed into its thermal and irradiation
components. Wave vectors below qmin are stable against
decomposition.

ballistic term in eqn [27] is greater than j@ 2 f =@C 2 j at
small q, but smaller than that at large q. In such cases,
the amplification factor is first negative for small q
values, but it becomes positive when q exceeds some
critical value qmin, while for larger q, the amplification factor is negative again. Therefore, decomposition is still expected to take place, but only for wave
vectors larger than qmin, that is, for wavelengths smaller than 2p/qmin. It can thus be anticipated that coarsening will saturate, since at large length scales, the
alloy remains stable with respect to decomposition.
Enomoto and Sawa122 have investigated this model
using a 2D PFM based on eqn [25]. The interest here
is that the PFM, unlike the above linear stability
analysis, includes both linear and nonlinear contributions to the evolution of composition inhomogeneity
and also permits following the morphology of the
decomposition. Using this model, Enomoto and Sawa
have confirmed the existence of the patterning regime,
see Figure 6, and showed that this patterning can take
place in the whole composition range. The PFM
approach allows for a direct determination of the patterning length scale as a function of the irradiation
conditions, as illustrated in Figure 7.
Similar results have also been obtained using a
variational analysis of eqn [25], leading to the dynamical phase diagram displayed in Figure 8. As seen in
this diagram, when the characteristic length for the
forced relocation is smaller than the critical value Rc,
the system never develops patterns at steady state.
Above Rc, patterning takes place when the irradiation

64

0

Figure 6 Irradiation-induced compositional patterning
in a binary alloy with (a) CB ¼ 50% and (b) CB ¼ 35%,
using a two-dimensional phase field model based on
eqn [25] with 1282 cells. Reproduced from Enomoto, Y.;
Sawa, M. Surf. Sci. 2002, 514(1–3), 68–73.

conditions are chosen so as to result in an appropriate
g value. Another result obtained from the KMC
simulations is that the steady state reached by an
alloy is independent of its initial state.
Experimental tests performed on a series of dilute
Cu–M alloys, with M ¼ Ag, Co, Fe, have confirmed
some of the key predictions of the above simulations
and analytical modeling. In particular, irradiation conditions that result in atomic relocation distances
exceeding a few angstroms do lead to the dynamical
stabilization of precipitates at intermediate irradiation
temperature.123 These results provide also a compelling rationalization of the puzzling results reported by
Nelson et al.124 on the refinement of g0 precipitates in
Ni–Al alloys under 100-keV Ni irradiation at 550  C.
The origin of the above irradiation-induced compositional patterning lies in the finite range of
the atomic mixing forced by nuclear collisions.125
Enrique and Bellon126,127 have shown that the effect
of this finite-range dynamics can be formally recast as
effective finite-range repulsive interaction between
like particles. It is interesting to note that PF

Phase Field Methods

5

Patterning

423

g2

g1

R/Ö(C/A)

I(k,t)/(S02L04)

t/t0 = 500

2.5

Solid solution
(Rc,gc)

1
Macroscopic phase separation
0.1

0

0

(a)

5
kL0

10

101

<k>L0

-1/3

1
g /(A2/C)

Figure 8 Analytical dynamical phase diagram yielding the
most stable steady state in a phase-separating A50B50 alloy
as a function of the forced relocation distance R and the
relative ballistic jump frequency g. Insets are (111) sections
of three-dimensional kinetic Monte Carlo (KMC) results; the
lateral size of the KMC inset is 17 nm. Reprinted with
permission from Enrique, R. A.; Bellon, P. Phys. Rev. Lett.
2000, 84(13). Copyright by the American Physical Society.

(a)

(b)

(c)

(d)

(e)

(f)

100

10-1 1
10
(b)

102
t/t0

103

Figure 7 (a) Spherically averaged and rescaled structure
factor for a low ballistic mixing frequency g ¼ 0.0005 (○) and
a higher ballistic mixing frequency g ¼ 0.05 (●). (b) Evolution
of the first moment of this structure factor as a function of
rescaled time, demonstrating that the alloy undergoes
continuous coarsening with the low mixing rate, but
stabilizes at the finite length scale (irradiation-induced
patterning) for the higher mixing rate. Reproduced from
Enomoto, Y.; Sawa, M. Surf. Sci. 2002, 514(1–3), 68–73.

simulations of alloys with Coulomb interactions also
predict a patterning of the microstructure.128 The
parallel with the treatment of finite-range mixing is
in fact quite strong, since a screened Coulomb repulsion is described by a decaying exponential, as also
assumed for the probability of finite-range ballistic
exchanges in deriving eqn [27]. The contribution of
the Coulomb repulsion to the linear stability analysis
is thus proportional to 1/(q2 þ qD2), where qD is the

Figure 9 Phase field modeling of the evolution of ordered
precipitates in the presence of electrostatic (repulsive)
interactions, with increasing time from A to F. The average
precipitate size reaches a finite value at equilibrium.
Reprinted with permission from Chen, L. Q.; Khachaturyan,
A. G. Phys. Rev. Lett. 1993, 70(10), 1477–1480. Copyright
by the American Physical Society.

screening wavelength. For reasons similar to the ones
discussed in the case of finite-range mixing, it is then
anticipated that these interactions will suppress
coarsening. This is confirmed by PF simulations, as
illustrated in Figure 9.
1.15.4.2.2 Coupled evolution of composition
and chemical order under irradiation

Many engineering alloys contain ordered phases or
precipitates to optimize their properties, in particular mechanical properties. It is thus important to


424

Phase Field Methods

investigate how these optimized microstructures
evolve under irradiation. It is anticipated that, under
appropriate conditions, ballistic mixing can lead to
the dissolution of precipitates, and to the disordering
of chemically ordered phases.129 Matsumura et al.130
used a 1D PF approach on a model binary alloy
system to specifically investigate what evolution irradiation may produce. In that model, the composition
field is represented by the globally conserved order
parameter X(r), while the degree of order is represented by the nonconserved order parameter S(r).
X is chosen to vary from À1 to þ1 for pure A and
pure B composition, respectively, and S takes a value
ranging from 0 to 1 for fully disordered and fully
ordered phases, respectively. The free energy functional of the system is written as
F ½fX ðrÞ; SðrÞ; T gŠ
9
8
H ðt Þ
>
2>
>
>
>
ðrX
Þ
f
ðX
;

S;
T
Þ
þ
ð>
=
<
2
dr
¼
>
>
>
>
K ðt Þ
2
>
>
:
ðrSÞ ;
þ
½28Š
2
where f (X, S, T ) is the mean field free energy of
a homogeneous alloy, and H and K are positive
constants of the interfacial energy coefficients in
the presence of varying field X and S, respectively.
The homogeneous free-energy density is given by a
Landau expansion

f ðX ; S; T Þ

2
3
2
2
2
2
Þ
À
bðT
Þfx
ðT
Þ
À
X
gS
ðX
À
x
m
0
aðT Þ4
5
¼ f0 þ
2
þ bðT Þ2 x ðT Þ2 S 4
1

½29Š

where f0 is the mean field free energy of the disordered phase with composition xm, and a, b, x12 are
positive constants depending on temperature. The
equilibrium phase diagram for this model system is
given in Figure 10. Notice, in particular, that at low
temperature and for compositions sufficiently far
from the equiatomic composition, an ordered phase
coexists with a disordered phase.
The kinetic evolution of these two fields is governed by
@X
¼D0mix fr2 X
@t



þ LðT ; fÞr

2

dF ðfX ; S; T gÞ
Àm
dX


½30Š

and
@S
dF ðfX ; S; T gÞ
¼ ÀefS À MðT ; fÞ

@t
dS

½31Š

where f is the atomic displacement rate, m is the
chemical potential, D0mix and e are positive coefficients
characterizing the efficiency of mixing and disordering

1.2
Disorder

Temperature, T/Tc

1.0

0.8

x0(T)
Order

0.6

x2(T)

x2(T)
0.4
Order
+
disorder

0.2

x1(T)

Order
+
disorder

x1(T)

x0(T)
0.0
-1.0

x0(T)
-0.5

0.0
Composition, X

0.5

1.0

Figure 10 Equilibrium phase diagram for model alloy system given by eqns [28] and [29]. The dotted lines correspond
to the metastable extrapolation of the order–disorder transition into the miscibility gap. Reprinted with permission from
Matsumura, S.; Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184–6193. Copyright by the American Physical Society.

Phase Field Methods

by irradiation, and L and M are the mobility coefficients for the conserved and nonconserved fields.
Note that here there is no kinetic coupling between
these two fields. There is, however, thermodynamic
coupling through the expression chosen for the
homogeneous free-energy density, eqn [29].
Although point defects are not explicitly used
as PF variables, the dependency of the mobility coefficients M and L with temperature, irradiation
flux, and sink density (cS) is obtained from a rate
theory model for the vacancy concentration under

425

irradiation in a homogeneous alloy.1 The steady-state
phase diagrams for two irradiation flux values are
given in Figure 11. At the higher flux, the phase
diagram is composed of homogeneous disordered
and ordered phases only. At low enough temperature,
the ballistic mixing and disordering dominate the
evolution of the alloy, leading to the destabilization
of the ordered phase at and near the stoichiometric
composition X ¼ 0, and to the disappearance of the
two-phase coexistence domains for off-stoichiometric
compositions.

1.2
CS = 10−5

Disorder

f = 10−5f c

Temperature, T/Tc

1.0

0.8

x1irr

x1irr

Order

0.6

0.4
x2

x0irr (T,f )

O+D

0.2
x1
0.0
-1.0

-0.5

Disorder

O+D
x1

0.0
Composition, X

(a)

0.5

1.0

1.2
CS = 10−5

Disorder

f = 10−4f c

Temperature, T/Tc

1.0

0.8

Order

x1irr

x1irr

0.6

0.4

x0irr (T,f )
x2

x2

0.2
x1
0.0
-1.0
(b)

-0.5

Disorder

0.0
Composition, X

x1
0.5

1.0

Figure 11 Steady-state phase diagrams under irradiation for (a) a low irradiation flux and (b) an irradiation flux 10 times
larger. The two-phase field is barely present in (a), and is no longer stable in (b). Reprinted with permission from Matsumura, S.;
Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184–6193. Copyright by the American Physical Society.


426

Phase Field Methods

The model has also been used to study the dissolution of ordered precipitates under irradiation.
In agreement with prior lattice-based mean field
kinetic simulations,131 it is found that two different
dissolution paths are possible, depending upon the
composition and irradiation parameters. Ordered
precipitates may either disorder first and then slowly
dissolve or they may dissolve progressively while
retaining a finite degree of chemical order until their
complete dissolution. These two kinetic paths have
indeed been observed experimentally in Nimonic
PE16 alloys irradiated with 300-keV Ni ions.132,133
1.15.4.2.3 Irradiation-induced formation of
void lattices

The formation of voids in irradiated solids results from
the clustering of vacancies, which can be assisted by
vacancy clusters produced directly in displacement
cascades and by the presence of gas atoms. Vacancy
supersaturation under irradiation may locally reach a
level large enough to trigger clustering owing to the

biased elimination of interstitials on sinks, especially
since interstitial atoms and small interstitial clusters
usually migrate much faster than vacancies. Evans134
discovered in 1971 that under irradiation voids may
self-organize into a mesoscopic lattice. The symmetry
of the void lattice is identical to that of the underlying
crystal, but with a void lattice parameter about two
orders of magnitude larger than the crystalline lattice
parameter (see also the reviews by Jager and Trinkaus135 and Ghoniem et al.136 for irradiation-induced
patterning reactions). It has been suggested that the
formation of the void lattice results from the 1D migration of self-interstitial atoms (SIAs) and of clusters of
SIAs, although elastic interactions between voids could
also contribute to self-organization.137 This 1D migration of SIAs would stabilize the formation of voids
along directions of the SIAs migration by a shadowing

effect.138–140 The model proposed by Woo141 indicates,
in particular, that the mean free path of SIAs needs to
exceed a critical value for a void lattice to be stable.
Atomistic KMC simulations have been performed142
to evaluate the dynamics of void formation, shrinkage,
and organization during irradiation. Due to the large
difference in mobility of vacancies and interstitials,
the slow evolution of the microstructure, and the
large range of length scales, assumptions had to be
used, in particular, regarding the void position and
size. Recently, Hu and Henager143 have approached
the problem of void lattice formation in a pure metal
using a PFM. Their model relies on the traditional
approach presented in Section 1.15.2 for the evolution
of the vacancy field, but it makes use of continuumtime random-walk kinetics for modeling the fast

transport of interstitials. 2D simulations indicate that
irradiation can stabilize a void lattice if the ratio of
SIA to vacancy diffusion coefficients is large enough
(see Figure 12) and if the defect production rate is not
too large (see Figure 13). It would be interesting to
extend this first model to include interstitial clusters.
The model lacks an absolute length scale, for the reasons discussed in Section 1.15.2, and thus nucleation of
new voids is treated in a deterministic and phenomenological manner based on the local vacancy concentration. It would clearly be beneficial to use a quantitative
PFM of the type presented in Section 1.15.3 to treat
void nucleation. This would also then make it possible
to directly compare the void size stabilized by irradiation with experimental observations.
We note also that Rokkam et al.144 recently introduced a simple PFM for void nucleation and coarsening in a pure element subjected to irradiation-induced
vacancy production. In addition to the local vacancy
concentration, these authors introduced a nonconserved order parameter to model the matrix–void
interface, similar to the nonconserved order parameter

t * = 16 000

t * = 16 000

t * = 16 000

(a)

(b)

(c)

t * = 16 000

(d)

Figure 12 Phase field simulations of void distributions for a low generation rate of vacancies and self-interstitial atoms
(SIAs), g_ V ¼ g_ SIA ¼ 10À5 , for different diffusivity ratios between SIA and vacancies, DSIA =DV , (a) 10, (b) 102, (c) 103, and (d) 104.
Reproduced from Hu, S.; Henager, C. H., Jr. J. Nucl. Mater. 2009, 394, 155–159.


Phase Field Methods

t * = 16 000

t * = 16 000

t * = 16 000

t * = 16 000

(a)

(b)

(c)

(d)

427

Figure 13 Phase field simulations of void distributions for a high diffusivity ratio between self-interstitial atoms (SIAs)
and vacancies, DSIA =DV ¼ 104 , and various generation rates of vacancies and SIAs, g_ V ¼ g_ SIA , (a) 2 Â 10À3, (b) 5 Â 10À4,
(c) 10À4, and (d) 10À5. Reproduced from Hu, S.; Henager, C. H., Jr. J. Nucl. Mater. 2009, 394, 155–159.

used for solid–liquid interfaces. It is shown144,145 that
this model reproduces many known phenomena,
such as nucleation, growth, coarsening of voids, as
well as the formation of denuded zones near sinks
such as free surfaces and grain boundaries. This phenomenological model is currently limited by the
absence of interstitial atoms in the description. It
may also suffer from the fact that the void–matrix
interfaces are intrinsically treated as diffuse, whereas
real void–matrix interfaces are essentially atomically sharp. This problem is further discussed in the
following section.
1.15.4.2.4 Irradiation-induced segregation
on defect clusters

In order to circumvent the problems raised in the
previous paragraph and in Section 1.15.4.1 for the
inclusion of defect clusters in a PFM, Badillo et al.146
have recently proposed a mixed approach that combines discrete and continuum treatments of the
defect clusters, so that each cluster is treated as a
separate entity. Point-defect cluster size is treated as
a discrete quantity for cluster production, whereas
the long-term fate of clusters is controlled by a
continuum-based flux of free point defects. New
field variables are thus introduced to describe the
size of these clusters:
p

p

p

Nc;A p
Nc;V p
Nc;B
¼ d ; Cc;A ¼ d ; Cc;B ¼ d
½32Š
N
N
N
p
where Nc;V is the number of vacancies in the vacancy
cluster in the cell p, and Nd is the number of substitup
p
tional lattice sites per cell; Nc;A and Nc;B are the
numbers of A and B interstitial atoms, respectively,
forming the interstitial cluster in the cell p. Each cell
contains at most one cluster.
The production of point defects by irradiation
takes place at a rate dictated by the irradiation flux
p
Cc;V

f in dpa sÀ1 and by the simulated volume. In the case
of irradiation conditions leading to the intracascade
clustering of point defects, the total number of point
defects created in a displacement cascade, the fraction of those defects that are clustered, and the size
and spatial distribution of these clusters are used as
input data. The production of Frenkel pairs is treated
in the same way as defect clusters, except that the
variables affected are the free vacancy and interstitial

concentrations.
This treatment of defect and defect cluster production makes it possible to compare irradiation conditions with identical total defect production rates,
that is, identical dpa sÀ1 values, but with varied fractions of intracascade defect clustering and varied
spatial distribution of these clusters. Furthermore, it
is also very well suited for system-specific modeling,
since all the above information can be directly and
accurately obtained from molecular dynamic simulations sampling the primary recoil spectrum.121 In
particular, one can build a library of such displacement cascades, so that the PFM will inherit the
stochastic character in space and time of the production of defect clusters by displacement cascades.
The continuous flux of free point defects to the
clusters results in the growth or shrinkage of
these clusters, which translates into the continuous
p
p
evolution of the cluster field variables Cc;V , Cc;A , and
p
Cc;B . When any cluster field variable drops below
1=N d , this cluster is assumed to have dissolved, and
the remaining one point defect is transferred to the
corresponding free point-defect variable of that cell.
For the sake of simplicity, defect clusters are treated as
immobile, but the approach can be extended to
include mobility, in particular for small interstitial
clusters. Further details are available in Badillo et al.146
The potential of the above approach is illustrated
by considering a 2D A8B92 alloy with a zero heat of


428

Phase Field Methods

mixing, so that at equilibrium, it always forms a
random solid solution. The production of interstitials
is, however, biased so that only A interstitial atoms
are created. This could, for instance, simulate an alloy
where there is a rapid conversion of B interstitial atoms
into A interstitial atoms via an interstitialcy mechanism. The preferential transport of A interstitial atoms
to defect clusters should lead to an enrichment of
A species around defect clusters, since these clusters
act as defect sinks. The effect of the primary recoil
spectrum on this irradiation-induced segregation is
studied by comparing two cases: the first one where a
small fraction of cascades, 1/Ncas ¼ 5 Â 10À4, produces
defect clusters, and the second one where that fraction
is 100 times higher, 1/Ncas ¼ 5 Â 10À2. In both cases,
however, the displacement rate per atom per second is
the same, here 10À7 dpa sÀ1. Figure 14 shows instantaneous concentration maps of the A solute atoms for
1/Ncas ¼ 5 Â 10À4. In this case, the PFM uses 64 Â 64
cells, each containing 7 Â 7 lattice sites. Segregation of
A species is clearly observed at a few locations, typically 2–5. This number is close to the average number
of defect clusters. The sharp peaks with high levels
of segregation correspond to segregation of existing
defect clusters, either interstitial or vacancy ones.
This is confirmed by visualizing the defect and defect
cluster fields, see Figure 15. The broader segregation
profiles in Figure 14 are the remnants of sharp segregation profiles after their corresponding clusters shrank

CA

1.0
400

0.5

300
0.0
0

200
100
200

100
300
400

0

Figure 14 The concentration field of A atom (CA ) for an
A8B92 alloy with zero heat of mixing where all interstitials are
created as A atoms. The two-dimensional model system
contains 448 Â 448 lattice sites, decomposed into 64 Â 64
cells for defining the phase field variables, each cell containing
7 Â 7 lattice sites. Irradiation displacement rate is 10À7
dpa sÀ1; the cascade frequency rate is 1/Ncas ¼ 5 Â 10À4, and
the irradiation dose is 6 dpa. Reproduced from Badillo, A.;
Bellon, P.; Averback, R. S. to be submitted.

and disappeared. As a result, the nonequilibrium segregation that build up on those clusters is being washed

out by vacancy diffusion, as expected for an A–B alloy
system with zero heat of mixing. In the case where
defect sinks have a finite lifetime, as in the present case,
one should thus expect a dynamical formation and
elimination of segregated regions.
In the case of much higher defect cluster production rate, 1/Ncas ¼ 5 Â 10À2, a very different microstructure is stabilized by irradiation, as illustrated in
Figure 16. Now a high density of clusters is present,
typically 40 interstitial clusters and 20 vacancy clusters, as seen in Figure 17(a) and 17(b), and the
segregation measured on these clusters is reduced
by about one order of magnitude compared to the
previous case. These results are reminiscent of the
experimental findings reported by Barbu and Ardell147
and Barbu and Martin,148 showing that, with irradiation conditions producing displacement cascades
(e.g., 500-keV Ni ion irradiation), the domain of
irradiation-induced segregation and precipitation in
undersaturated Ni–Si solid solutions is significantly
reduced compared to the case where irradiation produces only individual point defects (e.g., 1-MeV electron irradiation).

1.15.5 Conclusions and Perspectives
Thanks to fundamental advances, coupled with the
development of efficient algorithms and fast computers, the PF technique has become a very powerful
and versatile tool for simulating phase transformations and microstructural evolution in materials, as
illustrated in this chapter. This technique provides
simulation tools that are complementary to atomistic
models, such as molecular dynamics and lattice
Monte Carlo simulations, and to larger scale
approaches, such as finite element models. With
some modifications, it can also be employed for materials subjected to irradiation.
In the case of materials subjected to irradiation,
specific issues need to be addressed to fully realize

the potential of PF modeling. First, a proper description of point defects and atom transport requires
mobility matrices (or tensors) that capture the kinetic
coupling between these different species. In particular, the models reviewed in this chapter do not
account for the correlated motion of point defects
and atoms, thus leading to unphysical correlation
factors in the mobility coefficients. These correlation
effects, however, play an essential role in phenomena


0.0004
0.0003
0.0002
0.0001
0.0000
0

400

0.00020
0.00015
0.00010
0.00005

300
200

400
300

0

100

200

100

200

300

0.08
0.06
0.04
0.02
0.00
0

400
300
200
100
0

400

400
300
200
100

(d)

300

0

0.05
0.04
0.03
0.02
0.01
0.00
0

100
200

400

(b)

CClus Int

CClus Vac

0

400

(c)

100

200

100
300

(a)

429

Ci

CV

Phase Field Methods

200

100
300
400

0

~V, (b) free interstitials Cp þ Cp
Figure 15 Defect concentration fields corresponding to Figure 14: (a) free vacancies C
int A

int B
(A and B atoms), (c) clustered vacancies Cc;V , and (d) clustered interstitials Cc;A þ Cc;B (A and B atoms). Reproduced
from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.

CA

1.0
0.5

400
300

0.0
0

200

100

100

200
300
400

0

Figure 16 Concentration field of A atom CA for the same
A8B92 alloy as in Figure 14, except at a higher cascade
frequency, 1/Ncas ¼ 5 Â 10À2. Notice the significant

reduction in segregation on defect clusters compared to
Figure 14. Reproduced from Badillo, A.; Bellon, P.;
Averback, R. S. to be submitted.

such as irradiation-induced segregation and precipitation and are thus required in PFMs aiming for
system-specific predictive power. Second, it remains
challenging to include in a PFM all the elements of
the microstructure relevant to evolution under irradiation, namely point-defect clusters, dislocations,
grain boundaries, and surfaces, although it has been
shown here that models handling adequately a subset
of these microstructural elements are now becoming
available. Third, the numerical integration of the
evolution equations is more challenging than for
conventional PFMs in the sense that the continuous
defect production, as well as the large difference in
vacancy and interstitial mobility, usually prevents the
use of long integration time steps, even in coarse
microstructures. Finally, materials under irradiation
constitute nonequilibrium systems that are quite
sensitive to the amplitude and the structure of fluctuations, in particular the fluctuations resulting from


Phase Field Methods

0.08
0.06
0.04
0.02
0.00
0

400
300
200
100

CClus Int

CClus Vac

430

100

200

400
300
200
100
200

300
(a)

0.05
0.04
0.03
0.02
0.01

0.00
0

100
300

400

0

(b)

400

0

Figure 17 Defect cluster concentration fields corresponding to Figure 16: (a) clustered vacancies, Cc;V and (b) clustered
interstitials Cc;A þ Cc;B (A and B atoms). Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.

point defect and point-defect cluster production,
and from ballistic mixing of species. A self-consistent
and tractable PFM that would include both thermal and irradiation-induced fluctuations is still missing. Such a model would be very beneficial for the
study of microstructural evolution under irradiation,
especially that involving the nucleation of new phases,
defect clusters, or gas bubbles see Chapter 1.13,
Radiation Damage Theory; Chapter 1.14, Kinetic
Monte Carlo Simulations of Irradiation Effects; and
Chapter 1.09, Molecular Dynamics.

Acknowledgments

The author gratefully acknowledges stimulating
discussions with Robert Averback, Arnoldo Badillo,
Yan Le Bouar, Alphonse Finel, and Maylise Nastar.
The author also thanks Robert Averback for his critical reading of the manuscript. The support from the
US DoE-BES under Grant DEFG02-05ER46217 is
acknowledged.

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