PHYSICAL
REVIEW
E
VOLUME
48,
NUMBER 3 SEPTEMBER
1993
Phase-field
models for anisotropic
interfaces
G. B.
McFadden,
A.
A.
Wheeler,
*
R. 3.
Braun,
and S. R. Coriell
National Institute
of
Standards and
Technology,
Gaithersburg, Maryland
90899
R. F.
Sekerkat
Department
of
Physics,
Carnegie
Mellon
University,
Pittsburgh,
Pennsylvania
1M18
(Received
23 February
1993)
The
inclusion
of
anisotropic
surface free
energy
and anisotropic
linear interface
kinetics
in phase-
field models
is studied
for the
solidification of
a
pure
material. The formulation is
described for
a
two-dimensional
system
with a
smooth crystal-melt interface and for
a
surface free
energy
that
varies
smoothly
with
orientation, in
which case a
quite
general dependence
of the surface
free
energy
and
kinetic coefBcient
on
orientation can be
treated;
it is
assumed that
the
anisotropy
is
mild
enough
that
missing
orientations
do
not occur.
The method of matched
asymptotic
expansions
is used
to
recover the
appropriate
anisotropic
form
of
the
Gibbs-Thomson
equation
in
the
sharp-interface
limit
in
which the
width
of
the diffuse interface is thin
compared
to
its local radius of curvature.
It is found
that
the
surface free
energy
and the thickness of the diffuse interface
have
the same
anisotropy,
whereas
the kinetic coefFicient has
an
anisotropy
characterized
by
the
product
of the
interface thickness with the intrinsic
mobility
of the
phase
field.
PACS
number(s):
64.
70.
Dv,
68.
10.
Gw,
81.
30.
Bx,
82.
65.
Dp
I. INTRODUCTION
Phase-field
models
[1
—
6]
provide
a convenient basis for
the numerical
solution of
complicated
solidification prob-
lems. In a
phase-field
model,
in
addition to the
custom-
ary
energy
and/or
concentration
variables,
an additional
variable,
the
phase
field,
p,
is
introduced to label
explic-
itly
the
liquid
and solid
phases.
The
phase
field
takes
on a
constant
value
in
each bulk
phase,
e.
g.
,
y
=
0
in
the solid
phase
and
p
=
1 in
the
liquid phase.
The
transformation from
solid to
liquid
occurs
over a thin
transition
region
where
y
varies
smoothly
from zero to
one. The usual thermodynamic
functions
describing
the
system
can then be
modified to
incorporate gradient
en-
ergy
terms;
in
particular,
terms
proportional
to
~Vp~
can
contribute
to
the
surface
excess
quantities
that
play
a
fundamental role in
Gibbs's
formulation of surface
ther-
modynamics
[7].
In
this
sense,
phase-Beld
models
are
nat-
ural
outgrowths
of
disuse-interface models
dating
back
to work
by
van der Waals
[8],
by
Cahn
and Allen
[9,
10],
and
by
Cahn and Hilliard
[11,
12].
From
a
computational
viewpoint,
phase-field
models
are
similar
in some
ways
to
the
enthalpy
method
[13],
in
that
explicit tracking
of
the
solid-liquid
interface
is avoided. .
Phase-field
models,
however,
are more versatile
than
enthalpy
methods
since
such eBects as undercooling
of
the melt and
departures
&om
thermodynamic
equilibrium
at
the
interface are
in-
cluded automatically
(see,
e.
g.
,
[14]).
An
important example
of
the
utility
of phase-field
models is
given
by
the
numerical
studies of
dendritic
growth
by
Kobayashi
[15
—
18]
and
by
Wheeler,
Murray,
and
Schaefer
[19].
The
phase-field
treatments seem to
capture
successfully
a
broad
variety
of
dendritic
growth
phenomena,
including
the
correct
relation
between
Peclet
number and
undercooling,
the
emission
of
sidearms,
and
the
coarsening
behavior
of
sidearms
that
are further
re-
moved
from
the
tip.
The
anisotropy
of surface free
en-
ergy
or of
interface
kinetics is
generally thought
to
play
a
fundamental role in the
dynamics
of dendritic
growth
[20,
21].
Since
in
a phase-field formulation the interface is
dif-
fuse,
the
proper
incorporation of
surface
free-energy
anisotropy
requires
careful consideration. Phase-field
models with
anisotropy
have been considered
previously
for
specific
choices of
gradient
energy. Caginalp
and
Fife
[22,
23]
have
considered
models in which the
isotropic
"square
gradient"
expression
is
replaced
by
a
more
gen-
eral
quadratic
form
with
diferent coeKcients in each
co-
ordinate direction. For
an
isothermal
system
this leads
to
an
elliptical equilibrium
shape.
In order to obtain more
complicated anisotropies, I
anger
[2]
proposed
the
addi-
tion to the
gradient
energy
of
terms
involving
the
squares
of
higher
derivatives of the
phase field,
and
gave
an
exam-
ple
leading
to cubic
anisotropy.
Cahn
and Kikuchi
[24]
have considered
discrete
forms of
difFuse-interface
mod-
els,
and
have
also
included
anisotropic
eKects
through
the
choice of nearest-neighbor interactions
(see
also
[25,26]).
Both
Kobayashi
[17]
and
Wheeler,
Murray,
and Schaefer
[19]
include
anisotropy
by
allowing
the coefficient
of
the
gradient
energy
to
depend
on the local orientation of the
gradient
of the
phase
field.
Early
numerical calculations
were
performed
by
Smith
[27]
and
by
Umantsev,
Vino-
grad,
and
Borisov
[28]
in
which
no
explicit
anisotropy
was
included in the
models; rather,
anisotropy
was
provided
implicitly
by
the
underlying
grid
used in the numerical
calculations.
In this
paper
we
present
an
asymptotic
analysis
in the
sharp-interface limit
of
the model
studied
by
Kobayashi
[17]
and
Wheeler,
Murray,
and
Schaefer
[19],
including
an
anisotropic
mobility.
We consider a
two-dimensional
phase-field
description for
the solidification of a
single-
component
material of uniform
density,
and
assume that
1063-651X/93/484,
'3)/2016{9)/$06.
00 48 2016
1993
The American
Physical Society