Thermodynamics of bcc metals in phase-field-crystal models.

Abstract

We examine the influence of different forms of the free-energy functionals used in the phase-field-crystal (PFC) model, and compare them with the second-order density-functional theory (DFT) of freezing, by using bcc iron as an example case. We show that there are large differences between the PFC and the DFT and it is difficult to obtain reasonable parameters for existing PFC models directly from the DFT. Therefore, we propose a way of expanding the correlation function in terms of gradients that allows us to incorporate the bulk modulus of the liquid as an additional parameter in the theory. We show that this functional reproduces reasonable values for both bulk and surface properties of bcc iron, and therefore it should be useful in modeling bcc materials. As a further demonstration, we also calculate the grain boundary energy as a function of misorientation for a symmetric tilt boundary close to the melting transition.

Full text

Page 1

Thermodynamics of bcc metals in phase-field-crystal models
A. Jaatinen,1,2C. V. Achim,1K. R. Elder,3and T. Ala-Nissila1,4
1Department of Applied Physics, COMP Center of Excellence, Helsinki University of Technology, P.O. Box 1100,
Helsinki FIN-02015 TKK, Finland
2Department of Materials Science and Engineering, Helsinki University of Technology, P.O. Box 6200, Helsinki FIN-02015 TKK, Finland
3Department of Physics, Oakland University, Rochester, Michigan 48309-4487, USA
4Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
?Received 30 April 2009; revised manuscript received 16 July 2009; published 9 September 2009?
We examine the influence of different forms of the free-energy functionals used in the phase-field-crystal
?PFC? model, and compare them with the second-order density-functional theory ?DFT? of freezing, by using
bcc iron as an example case. We show that there are large differences between the PFC and the DFT and it is
difficult to obtain reasonable parameters for existing PFC models directly from the DFT. Therefore, we propose
a way of expanding the correlation function in terms of gradients that allows us to incorporate the bulk
modulus of the liquid as an additional parameter in the theory. We show that this functional reproduces
reasonable values for both bulk and surface properties of bcc iron, and therefore it should be useful in modeling
bcc materials. As a further demonstration, we also calculate the grain boundary energy as a function of
misorientation for a symmetric tilt boundary close to the melting transition.
DOI: 10.1103/PhysRevE.80.031602PACS number?s?: 81.10.?h, 61.50.Ah, 61.72.Bb, 61.72.Mm
I. INTRODUCTION
Many macroscopic material properties are strongly influ-
enced by the complex spatial patterns that form on micro-
scopic length scales during nonequilibrium processing. Such
microstructures can include, for example, grain boundaries,
antiphase domain walls, magnetic domains, and composition
gradients. While these structures are not equilibrium states,
their lifetimes can often be months, years, decades, or even
longer and consequently they play a major role in determin-
ing real material properties. The precise nature of such mi-
crostructures can be controlled by the processing ?solidifica-
tion, ball milling, annealing, etc.? used to produce the
material.
Despite the need for understanding the nonequilibrium
processing routes that lead to microstructures that optimize
material behavior, it has proved extremely difficult to accu-
rately model microstructure formation. What makes model-
ing microstructure formation difficult is the emergence of
different time and length scales in the problem, e.g., grain
boundary formation takes place on the atomic length scale,
while solute transport occurs on diffusive time and length
scales. In most cases, molecular dynamics ?MD? simulations
with present-day computers are limited to time and length
scales that are orders of magnitude smaller than needed to
match experimental conditions. This limitation necessitates
the development of coarse-grained methods for modeling mi-
crostructure formation.
One such method is the so-called phase-field-crystal
?PFC? method ?1?, which models a field in the solid phase
that exhibits the periodic nature of the underlying crystal
lattice. The advantage of using this approach instead of spa-
tially uniform or slow-varying phase fields is that many
crystal-structure-related properties, e.g., multiple grain orien-
tations, dislocations, and anisotropy, naturally arise in this
model. On the downside, spatial grid resolution in PFC needs
to be subatomic, thus making it challenging to reach the
length scales accessible by conventional phase-field ap-
proaches.
Oneobviousinterpretation
parameter field used in the PFC model is that of a nonequi-
librium ensemble average of the atomic number density. This
interpretation makes it tempting to derive this phenomeno-
logical model from a microscopic theory. To this end, a con-
nection between the PFC model and the classical density-
functional theory ?DFT? of freezing ?2? was recently
established by Elder et al. in Ref. ?3?. In essence, it was
shown that the free-energy functional used in earlier PFC
studies can be obtained from the DFT of freezing by making
certain approximations to the original functional. Unfortu-
nately, the approximations are quite drastic and the PFC free-
energy functional is not an accurate description of the origi-
nal DFT. While it is possible in principle to directly use DFT
to describe microstructure formation, in practice this turns
out to be a very difficult task. The reason for this is that DFT
predicts very sharp density peaks that require an extremely
small spatial grid spacing as compared to the PFC density
maxima, which are much smoother and easier to handle nu-
merically.
In this paper we investigate several recently proposed ap-
proaches for selecting the PFC parameters to fit several mac-
roscopic or thermodynamic features. More specifically, we
examine the predictions for the fitting schemes proposed by
Elder et al. ?3? and Wu and Karma ?WK? ?4? and compare
them with DFT results. As our main result we present a
fitting procedure, containing one more parameter, which cap-
tures additional macroscopic features not included in the
prior fits. We demonstrate that the model gives a relatively
good description of many of the bulk and surface properties
of bcc iron. The model is further used to estimate the grain
boundary energy near the melting point of bcc iron at a spe-
cific density and temperature.
This paper is organized as follows. In Sec. II we briefly
review the oldest and simplest version of the DFT of freezing
and the methodology of PFC modeling, with the emphasis on
oftheperiodic order-
PHYSICAL REVIEW E 80, 031602 ?2009?
1539-3755/2009/80?3?/031602?10?
©2009 The American Physical Society031602-1

Page 2

its connection to the DFT. These models are then applied to
Fe and used to predict both bulk thermodynamics and sur-
face properties in Sec. III. These calculations highlight the
strengths and the weaknesses of the two different ap-
proaches. Finally, this information is exploited to develop a
free-energy functional that maintains the computational sim-
plicity of the PFC formulation yet provides better predictions
for bulk and surface properties of bcc iron.
II. THEORY
The PFC model is a method for modeling periodic sys-
tems, and it has already been applied to many different prob-
lems in materials science. Originally, the model was phe-
nomenologically postulated in the spirit of Ginzburg-Landau
?GL? theory by Elder et al. ?1? in which the dynamics of the
number density field ??? was assumed to be dissipative and
driven to minimize a free-energy functional. The rationale
for proposing the free-energy functional was that it should be
minimized when ? forms a periodic structure as occurs in
crystalline phases. Perhaps the simplest free-energy func-
tional that has this feature was derived by Swift and Hohen-
berg ?5? for Rayleigh-Bénard convection and it can be writ-
ten as
F???r?? =?dr?
1
2??r??? + ??q0
2+ ?2?2???r? +g
4??r?4?,
?1?
where ??r? is a conserved order-parameter field related to
the number density field and ?, ?, q0, and g are phenomeno-
logical constants ?1?. To model nonequilibrium phenomena,
one also needs an equation of motion for the order-parameter
field ??r?. As with most phase-field models for conserved
fields, the dynamics are assumed to be dissipative, con-
served, and driven to minimize the free energy in the usual
manner, i.e.,
??
?t= M?2?F
??,
?2?
where M is a mobility parameter. In the present work, how-
ever, no dynamical processes are considered.
The justification for functional ?1? is the fact that for a
certain parameter range, it is minimized by a periodic field
corresponding to the bcc solid phase in three dimensions,
while at some other parameter range the free energy is mini-
mized by a uniform field corresponding to the liquid phase.
In the model, the bcc structure always has a lower free en-
ergy than other structures such as the much more common
fcc structure ?see ?4? for the full phase diagram?. It is tempt-
ing to use Eq. ?1? for modeling bcc materials because it has
been shown to reproduce many crystal-structure-related
properties of real materials, and it is fairly simple to use. The
central issue here is how to choose the phenomenological
parameters to model a specific material. To address this ques-
tion, it is useful to consider the classical DFT of freezing.
The classical DFT of freezing is the statistical-mechanical
theory of classical inhomogeneous fluids applied to the
freezing transition. This approach to modeling the freezing
transition was pioneered by Ramakrishnan and Yussouff in
the late 1970s ?6?. A key component in this theory is the
intrinsic Helmholtz free energy F, which is a unique func-
tional of the one-particle number density field ??r?. The in-
terpretation of this functional is that, for any given density
field, the functional derivative ?F/???r? equals the intrinsic
chemical potential which gives rise to the given density field
?7?.
For an ideal gas, the intrinsic free energy is known exactly
as
Fid= kBT?dr??r??ln???r??T
3? − 1?,
?3?
where ?Tis thermal de Broglie wavelength. For an interact-
ing system, however, calculating F exactly requires calculat-
ing the partition function. To approximate the functional, one
first writes F as
F = Fid+ Fxs,
where Fidis given by Eq. ?3? and Fxsis the contribution to F
from the interactions between particles.
Many different approximations to Fxshave been proposed
in the literature. We present here the simplest way to come
up with a nonlocal expression for Fxs, which is often referred
to as the second-order theory. Knowing that Fxsacts as a
generating functional for the family of direct correlation
functions,
?4?
?
?nFxs???
???r1? ¯ ???rn?= − c?n??r1, ... ,rn;????,
we expand it to the second order around a uniform reference
density ?0, leading to
kBT=?dr???r?ln???r?/?0? − ???r?? + ??0?dr ???r?
2??dr1dr2???r1?c?2??r1,r2,?0????r2?,
?5?
?F
−1
?6?
where ?F=F???−F??0?, ???r?=??r?−?0, and ?0is the
chemical potential of the reference liquid. The function c?2?
entering Eq. ?6? is the two-body direct correlation function,
which is related to the total pair correlation function of the
liquid through the Ornstein-Zernike relation. At this level of
approximation, the only microscopic information entering
the functional is the pair correlation function of the liquid at
density ?0.
To study freezing, one writes the density in terms of plane
waves as
??r? = ?0?1 + ??????
?G?
? ˆGeiG·r+ c.c.?,
?7?
where ???is the fractional average density difference be-
tween the solid and the liquid phases, ?G? is a sufficient set
of nonzero reciprocal lattice vectors, and c.c. is the complex
conjugate. The set of Fourier amplitudes ? ˆG?or combinations
of them? acts as order parameters in this approach, such that
the liquid and the crystalline states are defined by ? ˆG=0 and
? ˆG?0, respectively. At each ???, F is then minimized with
JAATINEN et al.
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-2

Page 3

respect to all the amplitudes, ? ˆG, and phase stability and
coexistence regions are obtained using a common tangent
construction for the free-energy density as a function of the
average density.
Formally, the expansion in Eq. ?6? is only valid when the
parameter ???r? is small, which is certainly not true in the
solid phase. Nevertheless, it has been shown that even at this
level of approximation the classical DFT is capable of de-
scribing a liquid-solid phase transition. The agreement of the
theory with computer simulations or experiments varies from
case to case. For a system of hard spheres, the liquid-fcc
transition parameters obtained from the second-order theory
are only a few percent off from those obtained from Monte
Carlo simulations. In contrast for a classical one-component
plasma, the theory is not capable of describing the transition
from a liquid to a bcc structure ?2?.
A connection between the functionals given in Eqs. ?1?
and ?6? can be obtained ?3? by first defining the dimension-
less density deviation n as
n?r? =??r? − ?0
?0
.
?8?
Plugging this in Eq. ?6? and ignoring the trivial term related
to the chemical potential of the liquid, the free-energy
change becomes
=?dr??1 + n?r??ln?1 + n?r?? − n?r??
2?dr?dr? n?r??0c?2???r − r???n?r??. ?9?
?F?n?r??
kBT?0
−1
To obtain a functional similar to Eq. ?1?, two approximations
to Eq. ?9? are made. First, the local part is expanded as a
fourth-order power series as
?1 + n?ln?1 + n? − n ?1
2n2−a
6n3+
b
12n4.
?10?
This expansion differs from the one in Ref. ?3? in two re-
spects. First, here both the local and the nonlocal parts of the
free energy are expanded around the same reference density.
Second, we have included the constants a and b in the ex-
pansion. The Taylor series expansion made in Ref. ?3? is
obtained from Eq. ?10? by setting a=b=1. However, as n is
not necessarily a small variable, including the constants a
and b allows for more freedom in fitting to any desired prop-
erties.
Another approximation we make to Eq. ?6? is to expand
the nonlocal part in terms of gradients. This is most easily
done by expanding the direct correlation function in k space
as
C?k? ? C0+ C2k2+ C4k4,
?11?
where C?k???0c ˆ?k? and C0, C2, and C4are constants. We
note that the same expansion can be obtained also by ex-
panding n?r?? around n?r? up to fourth order, revealing the
fact that the assumption underlying this approximation is that
n varies slowly compared with the range of c?2?. The con-
stants C0, C2, and C4could, in principle, be obtained from an
expansion of c ˆ?k?, but in PFC studies so far they have been
treated as parameters that are fitted with some other proper-
ties of c ˆ?k?, e.g., height and position of the main peak ?3?.
With approximations ?10? and ?11?, the free-energy func-
tional becomes
=?dr?n?r?1 − C0+ C2?2− C4?4
12n?r?4?,
?F?n?r??
kBT?0
2
n?r? −a
6n?r?3
+
b
?12?
which is already mathematically quite similar to Eq. ?1?. In
theAppendix we show how Eq. ?1? is exactly recovered from
Eq. ?12? and how the constants entering these two function-
als are related. However, for the remainder of the present
paper, we will stick to the notation of Eq. ?12? in order to
emphasize the connection to the DFT rather than the Swift-
Hohenberg equation.
The calculation presented here thus shows that it is pos-
sible to derive Eq. ?1? from a microscopic DFT. The practical
importance of this derivation is that with Eq. ?12? it is pos-
sible to use the fast numerical methods, including semi-
implicit operator splitting ?8? and amplitude equations ?9?
that are not trivially generalized for use with the DFT. How-
ever, as the density of the solid phase is extremely inhomo-
geneous and rapidly varying, the accuracy of Eq. ?12? is
questionable. Nevertheless, in the following section we ad-
dress the question as to whether or not the parameters in the
PFC model can be chosen to match bulk and surface prop-
erties in Fe and compare the results with those of the classi-
cal second-order DFT and experiments.
III. CASE STUDY: IRON
In this section, we show some results of calculations of
material properties of iron using functionals given in Eqs. ?9?
and ?12?. Iron is a good candidate for this study, because it
freezes into a bcc structure, which is the structure that Eq.
?12? is known to give rise to. It is also a technically very
important material, and preceding attempts to model iron by
PFC exist in the literature ?4?.
A. Second-order density-functional theory of freezing
To begin with, we first show the results of calculations
performed using Eq. ?9?. As an input to this DFT calculation,
we use a direct correlation function c ˆ?k? shown in Fig. 1
derived from the embedded atom method-molecular dynam-
ics ?EAM-MD? simulated structure factor from Ref. ?4?. This
correlation function was simulated at a temperature of T
=1772 K and density of ?0=0.0801 Å−3. Since the accuracy
of the EAM-MD calculations is questionable in the small k
limit, the k=0 mode of c ˆ?k? was fit to the theoretical predic-
tion of Tsu and Takano ?10? for the isothermal compressibil-
ityof ironat
T=1833 K,
?10−11m2N−1. Using these values, together with Jimbo
and Cramb’s ?11? experimental expression for the density of
liquid iron at T=1833 K, we obtain the estimate ?0c ˆ?0?=1
−??T?0kBT?−1?−49, which will be used in this work.
which is
?T=1.04
THERMODYNAMICS OF bcc METALS IN PHASE-FIELD-…
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-3

Page 4

To calculate the bulk thermodynamic properties of solid
iron from Eq. ?9?, we have employed a simple Picard itera-
tion procedure. One unit cell with periodic boundary condi-
tions is used in these calculations. The initial configuration is
a sum of discrete delta functions at the lattice sites in the unit
cell. Based on the density profile at the ith iteration, we
generate the one-body direct correlation function c?1??r;?n??,
which we use to calculate the ?i+1?th density profile accord-
ing to ln?1+n??c?1?, such that the average density of the
system is conserved in each iteration step. For a given aver-
age density and lattice spacing, a few hundred iterations are
required for convergent results. At each average density, the
calculation is repeated with several sizes of the unit cell to
minimize the free-energy density with respect to the lattice
spacing. Thus, our calculation does not impose any restric-
tion on the vacancy concentration of the system.
The results of the free-energy density calculations for liq-
uid and bcc phases are shown in Fig. 2 and example plots of
?100? faces of density profiles of bcc and fcc phases are
shown in Fig. 3. Coexistence densities are obtained from the
free-energy densities by the standard common tangent con-
struction. Most importantly, we find a transition from liquid
to bcc at ?l/?0=1.097 and ?bcc/?0=1.126, giving a coexist-
ence gap ??bcc−?l?/?l=2.7% that is consistent with experi-
ment. More specifically this gap gives a volume change on
melting of ?V=0.29 Å3per atom in reasonable agreement
with the experimental value of ?V=0.38 Å3per atom ?12?.
The lattice spacing of the bcc coexisting with the liquid turns
out to be abcc=2.95 Å. However, this lattice spacing corre-
sponds to a negative vacancy concentration, which we esti-
mate to be cV?−16% ?i.e., ?2.32 atoms per unit cell?. This
results from the fact that as there is no constraint to the
number of atoms per lattice sites, the minimum free-energy
distance between lattice sites is defined by the positions of
the peaks in c ˆ?k?, and not by the average density. In DFT
studies, this problem is usually avoided by varying the ref-
erence density, and thus the positions of the peaks, along
with the average density of the solid ?2?. Of course, if the
DFT were an accurate approximation of the underlying sys-
tem, it would be able to predict the vacancy concentration
without additional adjustments, as in the case of hard spheres
in the fundamental measure DFT ?13?. However, that is not
the case for the current approximation.
The bulk modulus of the different phases can be calcu-
lated from the free-energy curves according to the thermody-
namic relation
BT= ?2??2?F/V?
??2 ?
T
.
?13?
The results are BT
BT
We note that these values were obtained by imposing no
l=98.0 GPa for the liquid at ?=?0, and
bcc=101.7 GPa for the bcc at coexistence with the liquid.
02468 1012
−50
−40
−30
−20
−10
0
10
k ( Å−1)
ρ0c(k)
246
−1
0
1
FIG. 1. ?Color online? Full c ˆ?k? used in our DFT calculations
?black solid line? and the expansions used in our different PFC
models: green solid line is the eighth-order fit ?EOF?, blue dashed
line is the GL fit, and red dotted line is the three parameter fit to C0
and the first maximum.
00.05 0.10.150.2
0
0.2
0.4
0.6
0.8
1
(ρ − ρ0) / ρ0
∆F / (kBTVρ0)
FIG. 2. Free energies of liquid ?solid line? and bcc ?dashed line?
phases in second-order DFT. Crosses show the coexistence points
obtained by double tangent construction.
0
0.5
1
0
0.5
1
0
10
20
30
40
x / aBCC
y / aBCC
1+n(x,y,0)
0
0.5
1
0
0.5
1
0
100
200
300
400
x / aFCC
y / aFCC
1+n(x,y,0)
(b)
(a)
FIG. 3. Local densities in ?100? crystal planes of bcc coexisting
with liquid ?above? and fcc coexisting with bcc ?below? from DFT.
JAATINEN et al.
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-4

Page 5

constraint on the relation between lattice spacing and aver-
age density.
To our knowledge, the bulk modulus of bcc iron has been
measured experimentally only up to about T=820 °C, where
it reaches a value of BT
from the room temperature up to T=820 °C there is a mono-
tonically decreasing trend in BT
that this trend is linear up to the melting temperature, we get
an estimate BT
agreement with our DFT result.
When further increasing the density, this theory is also
able to describe a transition from bcc to fcc structure, taking
place at ?bcc/?0=1.772 and ?fcc/?0=1.789.At such high den-
sity, it comes as no surprise that the physical properties of the
fcc phase given by this density expansion do not agree very
well with experimental data.
In addition to bulk thermodynamic properties, we have
also calculated the bcc-liquid surface density profile and free
energies on ?100? and ?110? crystal planes. For these calcu-
lations, we use a quasi-one-dimensional slab, whose size is
one unit cell in the directions perpendicular to the surface
and 128 unit cells parallel to the surface. Half of the system
is then initialized as solid, and half of it as liquid. The sur-
face density profile is found using the same Picard iteration
procedure as for the bulk properties, with the exception that
mixing between successive iterations is now required to sta-
bilize the calculation. Surface free energies are then calcu-
lated by subtracting the bulk contribution from the total free
energy of the system containing the interface. Surface ener-
giesturnouttobe
=91.93 ergs/cm2, and ?111=86.9 ergs/cm2, giving an an-
isotropy coefficient ?4=??100−?110?/??100+?110?=1.5%. A
plot of the ?x,y? averaged density of ?110? surface is shown
in Fig. 4. The surface free energies disagree with the results
of MD simulations by about a factor of 2. To be more pre-
cise, the MD gives ?100=177.0 ergs/cm2, ?110=173.5, and
?4=1.0%, using the same potential that was used for obtain-
ing c ˆ?k? used in the present work ?4?.
bcc=135 GPa ?14?. On the other hand,
bccvs T ?14?. If we assume
bcc?105 GPa at T=1772 K in very good
?100=89.20 ergs/cm2,
?110
B. Phase-field-crystal model
After calculating the main physical properties of iron us-
ing the full second-order DFT, we now turn our attention to
the PFC model to see how well it is able to capture the
properties of DFT. Before presenting the results of our PFC
calculations, we point out that it should be obvious that the
density deviation field n is neither small nor slowly varying
in space. Therefore approximations ?10? and ?11? may not be
good approximations to the classical DFT results.
We used a Taylor series expansion, i.e., setting a=b=1 in
Eq. ?10? and fitting the three coefficients in Eq. ?11? to k
=0 limit, and height and position of the first peak in c ˆ?k?, in
the spirit suggested in ?3?. For our calculations, we have used
both the analytical one-mode approximation
n?r? = n0+ ?1 + n0?4u?cos?qx?cos?qy? + cos?qx?cos?qz?
+ cos?qy?cos?qz??,
?14?
and a numerical annealing based on Eq. ?2?. In Eq. ?14?, n0is
the average density and u is the density wave amplitude cor-
responding to the principal set of the reciprocal lattice vec-
tors. With this fitting procedure we found the solid phase to
be unstable at all average densities. The reason for this is the
fact that, as we already pointed out, approximations ?10? and
?11? are not consistent with the original DFT.
Calculations were performed to separately examine the
validity of the two approximations used, i.e., Eqs. ?10? and
?11?. We found that replacing the gradient expansion with the
full nonlocal term is not sufficient to stabilize the solid. On
the other hand, if we expand c ˆ?k?, but leaving the logarithmic
term unexpanded, we do find that there exists a stable bcc
solution. The problem is that the bcc only stabilizes at a
much higher density than in the DFT, and the free-energy
difference between solid and liquid is very much smaller
than in the DFT. The coexistence densities are ?l/?0=1.358
and ?bcc/?0=1.361, giving a coexistence gap of only ??bcc
−?l?/?l?0.2%. In this approximation the density profile of
the bcc phase is very much like the one-mode approximation
and significantly different from the DFT result shown in Fig.
3. Finally, we found that if n is expanded around the average
density ? ¯ instead of ?0, as in the original derivation ?3? of the
PFC from the classical DFT, then a stable one-mode-like bcc
structure appears at ?bcc/?0?1.409. The coexistence gap ob-
tained by the double tangent construction is even smaller
than 0.1% in this case, with the exact coexistence densities in
a given simulation slightly depending on ? ¯ chosen.
These results indicate that deriving PFC parameters from
the second-order DFT ?which itself predicts some unphysical
behavior? as presented in Ref. ?3? leads to very inaccurate
results. This naturally leads to the question, is it possible to
fit the parameters of the PFC free energy ?i.e., either Eq. ?1?
or Eq. ?12?? in such a manner that reasonable predictions can
be made? To examine this possibility we consider the ap-
proach taken by WK who proposed such a method for the
case of iron ?4?. In their method they used the height, the
position, and the second derivative of the first peak in C?k?
??0c ˆ?k? to fit the linear parts of the free energy. The nonlin-
ear coefficients are fitted such that for average density ?0the
free energy of the solid phase equals that of the liquid phase,
with a correct amplitude of density fluctuations obtained
from a molecular dynamics ?MD? simulation.
−10−50510
−1
0
1
2
3
z / aBCC
〈n〉
FIG. 4. Density profile of ?110? bcc-liquid interface in second-
order DFT. z here is the axis perpendicular to the surface, measured
in equilibrium lattice spacings of the bcc phase, and ?n?
=A−1??n?x,y,z?dxdy.
THERMODYNAMICS OF bcc METALS IN PHASE-FIELD-…
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-5

Page 6

We will now adapt a scheme for obtaining the parameters
for the free energy used in the present work. This scheme is
a simplified version of the small-? analysis based scheme in
Ref. ?4? and since it is inspired by the Ginzburg-Landau
theory ?15?, we call this fitting method the GL PFC model.
First, for the linear part we fit the parameters in Eq. ?11? to
reproduce the numbers given in Ref. ?4?, 1/S?km?=1
−C?km?=0.332, C??km?=−10.40 Å2, and km=2.985 Å−1,
which is the position of the first maximum. The parameters
turnouttobe
C0=−10.9153,
=−0.1459 Å4. We then use the one-mode approximation
?Eq. ?14?? to find how to choose the nonlinear coefficients a
and b, such that the amplitude u in Eq. ?14? that minimizes
the free energy is us=0.72, as defined from a MD simulation
?4?, and that the free energies of liquid and bcc phases at that
minima are equal. The resulting relations are
C2=2.6 Å2, and
C4
a =
3
2S?km?us
,
b =
4
30S?km?us
2,
?15?
which with the present numbers give a=0.6917 and b
=0.085 40, which are notably far from unity, i.e., the values
obtained by expanding the second-order DFT ?3?.
The results of the numerical calculations for the free en-
ergy are shown in Fig. 5. There is a stable body-centered-
cubic solid phase, with coexistence densities ?l/?0=0.889
and ?bcc/?0=1.043. Again, the density profile of the bcc
phase looks similar to the one-mode approximation in con-
trast to the DFT result with unphysical negative densities in
between the maxima. The volume difference between the bcc
and the liquid phases is ?V=2.07 Å3per atom, which is
significantly larger than the experimental value of 0.38. This
is most likely due to the fact that in this fitting procedure the
absolute value of k=0 value of ?0c ˆ?k? ?i.e., ?C0?? is five times
smaller than the experimental value, implying a significantly
larger isothermal compressibility of the liquid state. The bulk
moduli of the phases at coexistence, as calculated from the
free-energy curves, are indeed quite small: BT
and BT
moduli impose a larger coexistence gap can be obtained by
expanding the free energies of liquid and bcc phases to the
second order in density around the point where liquid coex-
ists with bcc. By making certain approximations, we see that
the coexistence gap scales roughly as the inverse of BT, in an
agreement with a comparison of the current fitting method
with experiments ??Vexpt/?VGL?BGL
does, however, predict surface free energies that are in rea-
sonable agreement with MD, as shown earlier by Wu and
Karma ?4?. These are ?100=207.11 ergs/cm2and ?110
=201.67 ergs/cm2, giving
=1.3% compared with the MD simulation results in Table I.
We note that two approximations are employed when
choosing the nonlinear parameters by Eqs. ?15?. First, the
one-mode approximation was used instead of the full nu-
merical minimization, resulting in slightly inaccurate density
wave amplitudes and free energies. Second, the condition of
equal free energies at ?=?0only holds at the limit of weakly
first-order transition. More generally, the proper conditions
are ?i? the free energy of the liquid at ?=?0has a common
tangent with free energy of the solid at a higher density ?bcc
and ?ii? the amplitude of the first mode of density fluctua-
tions of the solid phase at ?=?bccis us. In an attempt to
improve this fitting method, we have also fitted the nonlinear
coefficients numerically, without making the previously de-
scribed approximations. The resulting coefficients are a
=0.5687 and b=0.0660. With this parameter set, the physical
properties we investigated are ?l/?0=1.000 ?as fitted?,
?bcc/?0=1.134, BT
?100=196.40 ergs/cm2, ?110=196.41, and ?111=184.85.
Thus, the conclusion about the fitting method remains unal-
tered: surface free energies are quantitatively well captured,
but bulk free energies as a function of the average density are
not.
l=18.6 GPa
bcc=22.2 GPa. An explanation to why the smaller bulk
l
/Bexpt
l
?. The procedure
?4=??100−?110?/??100+?110?
l=23.3 GPa, and BT
bcc=26.3 GPa while
−0.2 −0.10 0.10.2
−0.2
−0.1
0
0.1
0.2
0.3
(ρ − ρ0) / ρ0
∆F / (kBTVρ0)
FIG. 5. Free energies of liquid ?solid line? and bcc ?dashed line?
phases in the GL PFC model. Crosses show coexistence points ob-
tained by the double tangent construction.
TABLE I. Comparison of physical quantities of iron in different models with experiments and molecular
dynamics simulations.
QuantityExpt./MDDFT GL PFCEOF PFC
Expansion in melting ?Å3/atom?
Solid bulk modulus ?GPa?
Liquid bulk modulus ?GPa?
Surface energy ?100? ?ergs/cm2?
Surface energy ?110? ?ergs/cm2?
Surface energy ?111? ?ergs/cm2?
Anisotropy ?4?%?
0.38 ?12?
105 ?14?a
96.2 ?10?
177.0 ?4?
173.5 ?4?
173.4 ?4?
1.0 ?4?
0.292.07
22.2
18.6
207.1
201.7
194.8
1.3
0.43
94.5
93.2
165.7
161.5
157.2
1.3
101.7
98.0
91.9
89.2
86.9
1.5
aLinear extrapolation from lower temperatures.
JAATINEN et al.
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-6

Page 7

C. Eighth-order fitting method for the phase-field-crystal
model
The results of the preceding section indicate that it is quite
difficult to simultaneously obtain accurate results for the sur-
face energies, miscibility gap, isothermal compressibility of
the liquid state, and the bulk modulus using the PFC-type
free-energy functional given in Eq. ?12?. The analysis of WK
?4,15? indicates that the factors defining the surface free en-
ergy in the PFC model are us, km, S?km?, and C??km?. On the
other hand, C0is an important quantity in defining the bulk
moduli of the phases and the size of the coexistence gap. To
improve the PFC model, we would like to include all the
four quantities C0, km, S?km?, and C??km? in the correlation
function of this model. Unfortunately, there are only three
parameters in Eq. ?11?, making it impossible to fit in all the
four quantities. For this reason, we propose a method of ex-
panding the correlation function
?0c ˆ?k? ? C?km? − ??km
km
We call this expansion the eighth-order fitting ?EOF?
method, because the expansion includes gradients up to the
eighth order. Using EOF, it is possible to fit all the important
properties mentioned earlier, by choosing the parameters as
2− k2
2 ?
2
− EB?km
2− k2
km
2 ?
4
.
?16?
? = −km
2C??km?
8
,
?17?
EB= C?km? − C0− ?.
?18?
By plugging in the parameters of WK and C0=−49, we get
?=11.583 and EB=38.085. As can be seen from Fig. 1, this
polynomial fits the c ˆ?k? curve up to the first peak much better
than the previously described fourth-order polynomials. The
nonlinear parameters a and b are obtained by the one-mode
approximation in order to avoid the unnecessary complica-
tion of the numerical fitting procedure. For n0=0 a one-mode
approximation to the expansion gives exactly the same result
as in the fourth-order expansion. Thus, the nonlinear coeffi-
cients are again given by Eqs. ?15?.
The free-energy curves of different phases in the EOF are
shown in Fig. 6. The coexistence between the liquid and the
bcc crystal structure is located at ?l/?0=0.975 and ?bcc/?0
=1.009, giving a very reasonable ??bcc−?l?/?l=3.4% for the
coexistence gap.Although the coexistence gap is still slightly
larger than in the DFT, we consider this result as a significant
improvement over the previous methods, which overestimate
the coexistence gap by almost an order of magnitude. In fact,
the volume change in melting is fortuitously even closer to
the experimental value than in the DFT, being ?V
=0.43 Å3per atom ?experiment: ?V=0.38 Å3per atom
?12?; DFT: ?V=0.29 Å3per atom?. The density field of the
bcc phase looks again very much like the one-mode approxi-
mation, as can be seen from Fig. 7. Examining the density
profile in more detail, we find that the amplitudes of the
Fourier modes go rapidly to zero as k increases. For ex-
ample, the amplitude corresponding to the second star of the
reciprocal lattice vectors is only about 1% of the amplitude
of the first set of modes.
The bulk moduli of liquid and bcc phases at coexistence,
as calculated from the free-energy curves, are also an im-
provement to the previous method, BT
BT
Thesurface
=165.66 ergs/cm2,
?110=161.50 ergs/cm2,
=157.16 ergs/cm2. These numbers are in better agreement
with simulations than the ones obtained from the previous
fitting scheme, and the anisotropy coefficient ?4turns out to
have the same value of ?4=1.3%. The fact that the surface
free energies are improved over the previous method is prob-
ably related to the fact that the difference between the aver-
l=93.2 GPa and
energies
and
bcc=94.5 GPa.freeare
?100
?111
−0.1−0.0500.050.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
(ρ − ρ0) / ρ0
∆ F / (kBTVρ0)
FIG. 6. Free energies of liquid ?solid line? and bcc ?dashed line?
phases in the EOF PFC model. Crosses show coexistence points
obtained by double tangent construction.
0
0.5
1
0
0.5
1
−5
0
5
10
x / aBCC
y / aBCC
1+n(x,y,0)
0
0.5
1
0
0.5
1
−5
0
5
10
x / (2π/q)
y / (2π/q)
1+n(x,y,0)
(b)
(a)
FIG. 7. Local density n in ?100? crystal planes of bcc coexisting
with liquid ?above? in the EOF PFC model. Below is the one-mode
approximation with n0=0.009 and u=0.734 for comparison.
THERMODYNAMICS OF bcc METALS IN PHASE-FIELD-…
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-7

Page 8

age densities of the two phases is smaller. The profile of the
?110? bcc-liquid surface in EOF is illustrated in Fig. 8 with
comparison to the GL fitting method. The two are in good
agreement.
Overall, the results obtained by the EOF are very encour-
aging, even though the density is still unphysically wide-
spread around the lattice sites and can even be negative in
between them. The field n should probably be interpreted as
a locally averaged, or weighted density difference, where the
higher-order Fourier modes are washed out by the weighting
procedure. If the model is interpreted in this way, it is a
promising choice for bcc materials, because the numerical
advantages to the DFT model are significant. We may also
argue that, in the uniform limit, both the PFC dynamical
equation ?2? with the present functional, and the dynamical
DFT equation of Marconi and Tarazona ?16? linearize to
?n ˆ?k?
?t
= − k2?1 − ?0c ˆ?k??n ˆ?k?.
?19?
Because the function c ˆ?k? is approximately the same in the
two models up to the first peak, the dynamics of the longer
wavelengths should be no different in the two models, al-
though the shorter wavelengths vanish faster in the PFC
model. On the other hand, in the uniform solid the two mod-
els can be made to reach equilibrium with the same value of
n ˆ?km?.
Finally, we note that although here we used MD simula-
tion data as an input for the EOF-PFC model, this might not
be necessary if one wishes to generalize this model to other
bcc materials. All that is needed from the correlation func-
tion is the k=0 limit, the maximum value C?km?, the position
of the maximum value km, and the second derivative at the
peak C??km?. The k=0 limit is related to the compressibility
of the liquid through the compressibility equation; for C?km?
we have the Hansen-Verlet freezing criterion, which states
that simple liquids freeze when S?km? reaches 2.9?0.1
?2,7,17?; and kmis simply related to the lattice spacing of the
solid. If these three were known, the only thing we would
need is C??km?, which could also be fitted with an experi-
mental value for the surface free energy. For the nonlinear
coefficients we also need the first mode amplitude us, for
which it is possible to obtain an estimate from the Linde-
mann melting criterion ?2?. Thus, if we consider the values of
usand S?km? to be universal, this would fix all the nonlinear
and one of the linear coefficients of the model. After that and
scaling out the length scale km, we would be left with a
model that should give a well description of many material
properties with only two parameters EBand ?.
D. Grain boundary energy in Fe
The free-energy functional and parameters presented in
the preceding section now provide a model of Fe that gives a
very good approximation to the surface tension, its aniso-
tropy, the liquid state compressibility, the bulk modulus, and
the miscibility gap. The model can now be exploited to make
predictions for other quantities such as the grain boundary
energy. Numerical simulations were conducted to measure
the energy ??? per unit area of a ?001? symmetric tilt grain
boundary as a function of grain misorientation angle ???. The
calculations were performed at a density of ??−?0?/?0
=0.030, which is just above the density of the solid at coex-
istence. Sample configurations are shown in Fig. 9 for both
low and high angle boundaries. The free energy per unit area
of the boundaries is shown as a function ? in Fig. 10.
As can be seen in Fig. 10 the grain boundary energy in-
creases with increasing mismatch angle, mainly due to a cor-
responding increase in the density of dislocations. For small
angles Read and Shockley ?18? derived the following expres-
sion for the grain boundary energy:
? =
Ga
4??1 − ???A − ln????,
?20?
where G is the rigidity modulus, a is the lattice constant, ? is
Poisson’s ratio, and A is a complex factor that Read and
Shockley estimated to be 0.45. In two dimensions the param-
eter A can be obtained by assuming that the dislocation
core size is chosen to minimize ?. This gives A=3/2
−ln?2???−0.34, which agrees very well with numerical cal-
culations ?1?. The inset in Fig. 10 shows a fit of the numeri-
cally obtained data with the Read-Shockley equation for
small angles. The best fit gives G=31.7 GPa and A=−0.41,
assuming ?=1/3. Independently from the grain boundary
−10−505 10
−2
−1
0
1
2
z / aBCC
〈n〉
FIG. 8. Profiles of ?110? bcc-liquid surface in PFC models. Solid
line is obtained from the present EOF PFC functional, and dashed
line is from the GL fitting method. z and ?n? are defined as in Fig.
4.
(b)(a)
FIG. 9. Sample grain boundaries for misorientation angles of
1.79° and 36.86° for the above and below configurations, respec-
tively. In these figures the grayscale corresponds to n?x,y,0?, with
the z axis parallel to the grain boundary.
JAATINEN et al.
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-8

Page 9

calculations, we have also performed numerical shearing ex-
periments with our PFC model, from which we estimate the
shear modulus to be G?53 GPa, in moderate agreement
with the fitted value.
Experimentally, the grain boundary energy of ? iron has
been determined to be 468 mJ/m2?19?. As real materials
contain more large-angle grain boundaries than small-angle
ones, it is most reasonable to compare the experimental
value with the large-angle grain boundary energies obtained
from our model. Therefore, the experimental value agrees
reasonably well with our result, where the maximum grain
boundary energy is approximately 380 mJ/m2. Interestingly,
according to ?19? the solid-liquid surface free energy of iron
is 204 mJ/m2, so the ratio between solid-liquid and grain
boundary energies ?GB/?s−l?2.29 agrees well with the
present result ?GB/?s−l?2.16.
The only previous study of the angular dependence of
grain boundary energy in iron, at least to our knowledge, has
been conducted using a modified analytical embedded atom
method by Zhang et al. ?20? at zero temperature. In this limit
the data are an order of magnitude greater than reported here
and inconsistent with the Read-Shockley ?RS? equation.
IV. CONCLUSIONS
We have studied free-energy functionals used in phase-
field-crystal ?PFC? models and compared them with the
second-order density-functional theory ?DFT? of freezing, by
using iron as an example case. The comparison of PFC and
DFT revealed quite significant differences between these two
theories. Most importantly, the derivation of PFC from the
DFT cannot be considered a reliable way of obtaining the
parameters for PFC. By fitting the parameters in a more ad
hoc way, our results confirm the argument of Wu and Karma
that by fitting in the correct height and second derivative of
the first peak in the direct correlation function, and the cor-
rect amplitude of density fluctuations in the solid corre-
sponding to the principal set of reciprocal lattice vectors, one
obtains a very reasonable estimate for the surface free energy
of the material. However, as our results show that bulk
moduli of the phases and coexistence gap between solid and
liquid phases turn out to be less accurate. To fix this problem,
we have presented a way of expanding the correlation func-
tion up to eighth order. Our results show that this EOF
method solves the problems with bulk moduli and coexist-
ence gap. We have also applied the EOF-PFC model to cal-
culating grain boundary energies of iron near its melting
point. We consider the EOF as a likely candidate for being
used in future PFC studies, at least if any quantitative accu-
racy of the results is desired.
ACKNOWLEDGMENTS
This work was supported in part by the Academy of Fin-
land through its Center of Excellence COMP grant, Tekes
through its MASIT33 project, and EU Grant No. STRP
016447 MagDot. A.J. acknowledges financial support from
the Finnish Academy of Science and Letters. K.R.E. ac-
knowledges support from NSF under Grant No. DMR-
0413062. We also wish to thank CSC-Scientific Computing
Ltd. for computational resources.
APPENDIX: DERIVATION OF THE SWIFT-HOHENBERG
FREE ENERGY FROM THE DFT
In Sec. II we show how a functional that resembles the
Swift-Hohenberg free energy ?Eq. ?1?? is obtained from the
classical density-functional theory. Here, we show how Eq.
?1? is exactly recovered from Eq. ?12? and how the constants
entering these functionals are related. We begin by rewriting
Eq. ?12? as
=?dr?
−a
6n?r?3+
?F?n?r??
kBT?0
n?r?
2?
1
S?km?+?
12n?r?4?,
km
4?km
2+ ?2?2?n?r?
b
?A1?
where kmis position of the first peak in C?k? ?or S?k?? and
S?km? is the peak value of the structure factor. ? is given by
Eq. ?17? for the GL-PFC model, or ?=C?km?−C?0? for the
Taylor series expansion. We then rewrite the field n as
n?r? = n ¯ +??r?
?0
,
?A2?
where n ¯ is a constant and ??r? is an order-parameter field
with the unit of the number density. By plugging this in Eq.
?A1?, defining F as the difference in free energy from F?n ¯?
-3.5 -3.5-3-3-2.5-2.5-2-2-1.5-1.5-1-1
11
1.51.5
22
2.52.5
33
3.5 3.5
FIG. 10. Grain boundary energy ? for Fe as a function of mis-
orientation angle. The solid points correspond to mismatch angles
of 0°–45°, while the open points correspond to 90°−? for angles
between 45° and 90°. In the inset, ?/? is plotted against ln???,
where in this instance ? is in radians. The solid line is a best fit to
a straight line given by ?/?=−?0.44089+1.087 ln???? J/m2
THERMODYNAMICS OF bcc METALS IN PHASE-FIELD-…
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-9

Page 10

and ignoring terms that are constant or linearly proportional
to ?, we get
F???r?? =?dr?
?0
+kBT?
?0km
+1
4
3?0
??r?
2?kBT
2+ ?2?2???r? −kBT
?S?km?−1− an ¯ + bn ¯2?
4?km
3??r?4?.
?0
2?a
6−n ¯b
3???r?3
kBTb
?A3?
From here, we see that the cubic term can be made to vanish
by choosing
n ¯ =
a
2b.
?A4?
After making this substitution, Eq. ?A3? becomes
F???r?? =?dr?
+kBT?
?0km
??r?
2?
2+ ?2?2???r? +1
kBT
?0?
1
S?km?−a2
4b?
4?km
4
kBTb
3?0
3??r?4?.
?A5?
This equation is mathematically exactly of the same form as
Eq. ?1?, and the parameters can now be identified as
q0= km,
?A6?
? =kBT
?0?
1
S?km?−a2
4b?,
?A7?
? =kBT?
?0km
4,
?A8?
g =kBTb
3?0
3.
?A9?
In order to compare the present work with the work of Wu
and Karma ?4?, it is interesting to note that in our GL fit,
where a and b are given by Eqs. ?15? and ? is given by Eq.
?17?, we get for the linear parameters
? =− 103kBT
32?0S?km?,
?A10?
? = −kBTC??km?
8?0km
4
,
?A11?
which are exactly the same results as given by Eqs. ?41? and
?42? of Ref. ?4?. For the nonlinear coefficient g we get
2kBT
45?0
g =
3us
2S?km?,
?A12?
which is also the same as given by Wu and Karma, after a
minor error in Eq. ?45? of Ref. ?4? is corrected. Thus, our GL
fit results in the same Swift-Hohenberg equation as the
small-? analysis of Wu and Karma, even though the proce-
dures for obtaining the parameters are different.
?1? K. R. Elder, M. Katakowski, M. Haataja, and M. Grant, Phys.
Rev. Lett. 88, 245701 ?2002?; K. R. Elder and M. Grant, Phys.
Rev. E 70, 051605 ?2004?.
?2? Y. Singh, Phys. Rep. 207, 351 ?1991?.
?3? K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant,
Phys. Rev. B 75, 064107 ?2007?.
?4? ?a? K.-A. Wu and A. Karma, Phys. Rev. B 76, 184107 ?2007?;
?b? K.-A. Wu, Ph.D. thesis, Northeastern University, 2006.
?5? J. Swift and P. C. Hohenberg, Phys. Rev. A 15, 319 ?1977?.
?6? T. V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19, 2775
?1979?.
?7? J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids,
3rd ed. ?Academic, Amsterdam, 2006?.
?8? G. Tegze, G. Bansel, G. I. Tóthb, T. Pusztai, Z. Fan, and L.
Gránásy, J. Comput. Phys. 228, 1612 ?2009?; B. P. Vollmayr-
Lee and Andrew D. Rutenberg, Phys. Rev. E 68, 066703
?2003?; J. Zhu, L.-Q. Chen, J. Shen, and V. Tikare, ibid. 60,
3564 ?1999?.
?9? B. P. Athreya, N. Goldenfeld, and J. A. Dantzig, Phys. Rev. E
74, 011601 ?2006?.
?10? Y. Tsu and K. Takano, 88th Spring Conference ?Japan Institute
of Metals, Sendai, 1981?, Vol. 88, p. 86; T. Itami and M. Shi-
moji, J. Phys F: Met. Phys. 14, L15 ?1984?.
?11? I. Jimbo and A. W. Cramb, Metall. Mater. Trans. B 24, 5
?1993?.
?12? M. I. Mendelev, S. Han, D. J. Srolovitz, G. J. Ackland, D. Y.
Sun, and M. Asta, Philos. Mag. 83, 3977 ?2003?.
?13? Y. Rosenfeld, M. Schmidt, H. Löwen, and P. Tarazona, J.
Phys.: Condens. Matter 8, L577 ?1996?.
?14? D. J. Dever, J. Appl. Phys. 43, 3293 ?1972?; J. J. Adams, D. S.
Agosta, R. G. Leisure, and H. Ledbetter, ibid. 100, 113530
?2006?.
?15? K.-A. Wu, A. Karma, J. J. Hoyt, and M.Asta, Phys. Rev. B 73,
094101 ?2006?.
?16? U. M. B. Marconi and P. Tarazona, J. Phys.: Condens. Matter
12, A413 ?2000?.
?17? J.-P. Hansen and L. Verlet, Phys. Rev. 184, 151 ?1969?.
?18? W. T. Read and W. Shockley, Phys. Rev. 78, 275 ?1950?.
?19? L. E. Murr, Interfacial Phenomena in Metals and Alloys ?Ad-
dison Wesley, New York, 1975?.
?20? J.-M. Zhang, Y.-H. Huang, X.-J. Wu, and K.-W. Xu, Appl.
Surf. Sci. 252, 4936 ?2006?.
JAATINEN et al.
PHYSICAL REVIEW E 80, 031602 ?2009?
031602-10