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Free Energy Functionals for Efficient Phase Field Crystal
Modelingof Structural Phase Transformations
Michael Greenwood,1,2 Nikolas Provatas,2 and Jörg Rottler1
1Department of Physics and Astronomy, University of British
Columbia,6224 Agricultural Road Vancouver, British Columbia, V6T
1Z1, Canada
2Department of Materials Science and Engineering, McMaster
University,1280 Main Street West, Hamilton, Ontario, L8S 4L7,
Canada
(Received 30 December 2009; revised manuscript received 20 June
2010; published 23 July 2010)
The phase field crystal (PFC) method is a promising technique
for modeling materials with atomic
resolution on mesoscopic time scales. While numerically more
efficient than classical density functional
theory (CDFT), its single mode free energy limits the complexity
of structural transformations that can be
simulated. We introduce a new PFC model inspired by CDFT, which
uses a systematic construction of
two-particle correlation functions that allows for a broad class
of structural transformations. Our approach
considers planar spacings, lattice symmetries, planar atomic
densities, and atomic vibrational amplitudes
in the unit cell, and parameterizes temperature and anisotropic
surface energies. The power of our
approach is demonstrated by two examples of structural phase
transformations.
DOI: 10.1103/PhysRevLett.105.045702 PACS numbers: 64.70.K�,
46.15.�x, 61.50.Ah, 81.10.Aj
Solid-state transformations involve structural changesthat
couple atomic-scale elastic and plastic effects withdiffusional
processes [1–3]. These phenomena are pres-ently impossible to
compute at experimentally relevanttime scales using molecular
dynamics simulations. Onthe other hand, mesoscale continuum models
wash outmost of the relevant atomic-scale physics that leads
toelasticity, plasticity, defect interactions, and grain bound-ary
nucleation and migration. Traditional phase field stud-ies of
precipitate and ledge growth [3,4] thus introducethese effects
phenomenologically. To our knowledge, thereare no phase field
models presently available that self-consistently model
polycrystalline interactions and elasticand plastic effects at the
atomic scale.
Classical density function theory (CDFT) provides aformalism
that describes the emergence of crystalline orderfrom a liquid or
solid phase through a coarse-graineddensity field [5].
Unfortunately, it requires high spatialresolution and is
inefficient for dynamical calculations[6] due to sharp density
spikes in the solid phases. Therecently introduced phase field
crystal (PFC) model hasbeen gaining widespread recognition as a
hybrid methodbetween CDFT and traditional phase field methods.
Thisnew formalism captures the essential physics of CDFTwithout
having to resolve the sharp atomic density peaks[7–11]. Despite
their success, existing PFC free energiesallow for only a limited
range of structural transformationsbetween different crystalline
states. Moreover, extensionsof the original PFC concept to crystal
symmetries such assquare [12,13] and fcc [6,14,15] have been
somewhat adhoc and not self-consistently connected to
materialproperties.
This Letter proposes an extension of the PFC model
bysystematically constructing phenomenological kernelswith energy
minima for targeted crystalline states. Our
approach preserves the PFC model’s numerical efficiency,and its
utility is demonstrated by dynamically simulatingthe growth of
solid phases into a liquid and the nucleationof precipitate phases
within a parent phase of differentcrystallographic symmetry. We
begin with a free energyfunctional,
�F½nð~rÞ� ¼ �Fid½nð~rÞ� þ�Fex½nð~rÞ�; (1)where �F is the free
energy difference with respect to auniform reference density �0
[16], and has contributionsfrom a noninteracting term �Fid and an
excess energy�Fex responsible for the formation of structured
phases.Equation (1) is expanded around the uniform reference
density using the dimensionless number density nð ~rÞ ¼�ð~rÞ=�o
� 1, where �ð ~rÞ is the coarse-grained local den-sity. The ideal
contribution to the energy is approximatedby expanding the
Helmholtz free energy of an ideal gas,
�Fid ¼ �okBTZ
d~r½1þ nð ~rÞ� ln½1þ nð ~rÞ� � nð ~rÞ
� �okBTZ
d~r
�nð ~rÞ22
� nð ~rÞ3
6þ nð ~rÞ
4
12
�: (2)
The excess energy is expanded to include only secondorder
correlations [9,16–18],
�Fex ¼ ��okBT2Z
d~rnð~rÞZ
d~r0½C2ðjr� r0jÞnð~r0Þ�; (3)
where C2ðj ~r� ~r0jÞ plays the role of a two-particle
directcorrelation function. The PFC model evolves the
di-mensionless number density field nð~rÞ in time using thetypical
conserved dissipative dynamics [7], @nð ~rÞ@t ¼Mr2½��F=�nð ~rÞ� þ
Anr2� where M is a kinetic mo-bility parameter and conserved noise
� with an amplitudeof An is added to facilitate nucleation from
metastable
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states [7,19]. Equation (3) is integrated in reciprocal
spaceusing a semi-implicit technique [8,11].
Periodic structures emerge in the PFC model by theintroduction
of the kernel C2ðkÞ that results in free energyminima corresponding
to particular modes in the densityfield. Here we do not aim to
construct such kernels fromfirst principles, but rather seek the
simplest possible formsthat result in the desired crystal
structure. To this end, weexpand the free energy functional about a
uniform refer-ence density [18] and choose modes in the
correlationfunctions that will create an energy minimum for
thedesired structure. The original PFC model by contrastuses only a
single mode [7]. Figure 1 shows two examplesof direct pair
correlation functions that produce stablesquare (a) and simple
cubic (sc) (b) lattices for minimizedfree energies. The functions
C2ðkÞ are constructed in re-ciprocal space by combining multiple
peaks whose posi-tion, amplitude, and width are determined as
follows:
The number and position of peaks in the reciprocal
spacecorrelation function is determined by the desired unit cell.In
diffraction theory a reciprocal lattice has families ofpeaks
derived from the interplanar spacings. For a 2Dsquare lattice, for
instance, the unit cell contains twofamilies of planes, f10g and
f11g, with spacings �1 and�2 as shown in the inset of Fig. 1(a).
For a perfect crystal,there are an infinite number of peaks located
at integer
multiples of the wave vector ki ¼ 2�=�i, where i denotesthe
plane. For square and cubic lattices, it is sufficient, inour PFC
formalism, to keep only the lowest frequencymode for each family of
these peaks in the correlationfunction.Temperature enters our
correlation function via modu-
lation of the peak heights by a factor e��2k2i =2�i�i , where
theeffective temperature � acts as a control parameter. Theform of
this term is motivated by the effect of thermalmotion of atoms
about their equilibrium positions withamplitude �v, which modulates
the scattering intensity
by a Debye-Waller-Factor e��2vk2i =2. The peak heights inthe
correlation function are additionally influenced by
the(dimensionless) atomic density �i within a plane and thenumber
of planes �i in each family. We therefore enterthese effects into
the excess energy phenomenologically by
modulating the peaks in C2ðkÞ with a factor exp½� �2k2i
2�i�i�.
For example, in a square lattice the families of f11g andf10g
planes each consist of 4 sets of planes [i.e., the f11gfamily
contains (11), (�11), (1�1), and (�1 �1 )] and therefore�11 ¼ �10 ¼
4. The (11) plane has an atomic density�11 ¼ 1=
ffiffiffi2
pand the (10) plane has a density of �10 ¼ 1.
Incorporating both the decay of correlation peak heightsthrough
an effective temperature � and the inclusion ofonly the lowest
frequency peaks of each family of planes inthe excess energy leads
to the broad real-space densitypeaks that give this model its
numerical efficiency.The peaks in the reciprocal space correlation
function
are represented by Gaussians exp½� ðk�kiÞ22�2i
� of finite width�i rather than the �—peaks of a perfect
lattice. Theparameter �i accounts for changes in the free energy
dueto interfaces, defects, and strain. Varying �i changes thewidth
of a liquid-solid interface, which directly affects thesurface
energy [20]. The relationship between the interfacewidth and the
peak width in the correlation function is1=�i / Wi as illustrated
in Fig. 2. This result can be arrivedat by the inverse Fourier
transform ofC2ðkÞ and agrees wellwith the traditional phase field
models, which incorporatesurface energy through square gradient
terms in the orderparameter.In summary, each family of planes i in
the unit cell
contributes a peak to the kernel in the form of C2ðkÞi
¼exp½�ð�2k2i Þ=ð2�i�iÞ� expf�½ðk� kiÞ2�=ð2�2i Þg. To fur-ther
simplify the construction, the correlation functionsin this work
(Fig. 1) comprise the numerical envelope ofthe superposition of all
relevant peaks for the crystalstructure, to avoid shifts in the
peak positions and corre-spondingly changes to the stable structure
that would resultfrom a simple sum. This provides a general and
robustmethod by which to generate desired crystal
structures,incorporating temperature dependence through
modulationof the correlation peak heights. Through the use of �1
and�2 we can also change the anisotropy of a given
crystalstructure.The phase diagram is constructed from the free
energy
curves obtained for each bulk phase. Free energy curves
FIG. 1 (color online). (a) Direct pair correlation function for
asquare lattice at two effective temperatures (� ¼ 0:82, solid and�
¼ 0:5, dotted, see text). The two peaks in the correlation
func-tion each represent one of two planes in the unit cell with
inter-planar spacings �1 and �2 (see inset). (b) Correlation
functionfor a simple cubic (sc) lattice at two different
temperatures. In3D, a cubic lattice is represented by three planes
in the unit cell.
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are calculated by an iterative relaxation technique for
eachstructure using the kernel for the structure of interest.
Forthe liquid state, the energy is calculated by imposing aconstant
density field, and the energy per unit volume iscalculated by
numerically integrating Eq. (1). For the solidstate, the structures
corresponding to the correlation func-tion are fit using a Gaussian
density field and then relaxed.For example, for the correlation
function of Fig. 1(a),density fields for square and triangular
lattices with lattice
spacings of 1 and 2=ffiffiffi3
pare relaxed using the conserved
dynamics to allow density peaks to obtain an
amplitudecorresponding to the local minimum energy state.
Thisprocess is repeated for a series of mean densities to pro-duce
energy-density curves for each phase at a giveneffective
temperature � [Fig. 3(b), inset]. The doubletangent construction is
performed for many values of �,giving the phase diagrams in Fig.
3(a) for the squarecorrelation function and in Fig. 3(b) for the
fcc correlationfunction.
In both 2D and 3D, the phase that possesses the sym-metry of the
underlying correlation function emerges atlow temperatures and
intermediate densities. As the effec-tive temperature is increased
the lower frequency modes ofthe correlation function dominate over
the higher fre-quency modes, which changes the particular
structurethat minimizes the free energy. For example, the
square(fcc) crystal transforms into a triangular (bcc) crystal
sincethe triangular (bcc) structure minimizes the energy in
thesingle mode approximation. Note that our model alsopredicts
transformation of coexisting liquid and solidphases into a new
single solid phase at a peritectic pointin both two and three
dimensions. This remarkable featureopens an important window into
the study of structural
phase transformations. Coexistence between fcc crystalswith the
liquid phase is also obtained, as is coexistence ofthe liquid and
sc phases using the kernel shown in Fig. 1(b).Our model is applied
to two important examples of
solid-state processes using the phase diagram of Fig. 3.The
first is the growth of two structurally different latticesin
coexistence, initialized in rectangular domains, into aliquid phase
as illustrated in the inset of Fig. 3(a). Thesquare phase is
oriented such that the f10g planes are incontact with the liquid
phase. Since each peak in the two-particle correlation function
represents a single family ofplanes, the surface energy of the
interface of the squarephase {10}-facet is derived from the width
�2 of thesecond peak of the correlation function, while the
surfaceenergy of triangular phase facets are derived directly
fromthe width �1 of the first peak. Anisotropy of surface energycan
be controlled by increasing or decreasing �1 whileholding �2
constant. This effect also extends to the solid-solid surfaces
between boundaries of different structure(for example,
triangle-square boundaries). The trianglestructure is derived
directly from a single mode approxi-mation, therefore the magnitude
of the anisotropy is deter-
FIG. 2. Effect of peak width �i in the correlation functions
onplanar interface width Wi. Left: Three values of �i and
theirresulting interface widths. Top right: Peak shapes
correspondingthe interface widths to the left. Bottom right:
Dependence ofsimulated interface width on the correlation peak
width (seetext).
FIG. 3. (a) Phase diagram for a square lattice
correlationfunction showing coexistence between liquid, solid, and
trian-gular phases. The inset contains stripes of square and
triangularlattices in a liquid phase quenched to a temperature � ¼
0:79.The square phase density was initialized to nsq ¼ 0:067 and
thetriangular phase density to ntri ¼ 0:076. The mean density of
thesystem is set to �n ¼ 0:07 and the surface energy parameters
are�1 ¼
ffiffiffi2
pand �2 ¼ 1. (b) Phase diagram for a fcc correlation
function. The insets show the peritectic point and the
energycurves for the liquid, fcc and bcc states at � ¼ 0:06.
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mined strictly by the underlying structure. Small changesin the
strength of the anisotropy lead to small changes inthe angle of the
solid liquid interfaces at vertices betweenthe three phases.
The second application of interest is the nucleation of
astructural phase by a quench into the second phase regionof the
phase diagram. To illustrative this, the simple 2Dcase of
square-triangle structural transformations is shownhere. A liquid
with a homogeneous density field isquenched into the triangular
region of the phase diagram[point A on the quench path in Fig.
3(a)]. Small amplitudenoise facilitates the nucleation of randomly
oriented trian-gular phase seeds in the liquid phase. These seeds
grow,impinge and coarsen over time. A small grain is illustratedin
Fig. 4(a). After the triangular grains undergo somecoarsening, the
material is quenched into the square por-tion of the phase diagram
(point B). A thermodynamicdriving force leads to the nucleation,
growth, and coarsen-ing of the square phase. The nucleation events
occur atvertex positions in the triangular grain structures, as
ob-served in experiments [1]. The driving force for the
trans-formation is greatest at these positions and can overcomethe
nucleation barrier. Two such nucleation points areshown in Fig.
4(b). The orientation of a nucleus andcoherency strains can inhibit
the growth of the new phasethat precipitates, eventually to be
dominated by another
more favorable precipitate. This effect is illustrated inpanels
(c) and (d) of Fig. 4. These results demonstratethat our approach
is capable of modeling the distributionof square or fcc precipitate
orientations, an importantphenomenon that can be compared to
experiments.We introduced new PFC free energy functionals for
efficient numerical study of solid-state transformations.In
contrast to earlier work [13], we find that two-pointcorrelations
are sufficient to generate stable cubic lattices,and higher order
terms in the expansion of�Fex can still beneglected. The
correlation functions are systematicallybuilt up from fundamental
principles and desired crystallo-graphic properties of phases of
interest at a finite tempera-ture. Our approach can model
peritectic transitions as wellas the nucleation and growth of
second-phase precipitateswith different crystalline structures. Our
model can bevalidated against numerous experiments, as well as
MDsimulations of triple junctions.This work has been supported by
the Natural Science
and Engineering Research Council of Canada (NSERC).
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FIG. 4 (color online). 2D nucleation from triangles to
squareswith noise amplitude of An ¼ 0:01 (a) A liquid with density
�n ¼0:09 is quenched into the triangular region of the phase
diagram(� ¼ 0:82), marked (A) in Fig. 3(a), produces grains
withtriangular symmetry. (b)–(d) A further quench into the
squareportion of the phase diagram (� ¼ 0:6), marked (B) in Fig.
3(a),illustrating the nucleation and subsequent growth of
squaregrains at the vertices of the triangular grains.
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