A PHASE FIELD MODEL FOR THE SOLIDIFICATION OF
MULTICOMPONENT ALLOYS
Pil-Ryung Cha, Dong-Hee Yeon and Jong-Kyu Yoon
School of Materials Science and Engineering, Seoul National University,
Seoul 151-742, Korea
ABSTRACT
A recently developed phase field model for the isothermal solidification of
multicomponent alloys is presented. Using this model, the 2-D dendritic growth of
binary and ternary alloys is simulated. The existing phase field model for the
solidification of alloys is constrained to binary alloy systems. Particularly, when the
solute atom is interstitial, the existing phase field model is not applicable. In this study,
it will be shown that the existing phase field model for the isothermal solidification of
binary alloys is not applicable to alloy systems with interstitial alloy elements and that
our model for the solidification of multicomponent alloys could be applied to any alloy
system with both interstitial and substitutional elements.
1.INTRODUCTION
Understanding dendritic solidification is of great importance as the microstructural
scales of dendrites control the segregation pattern and consequently determine the
properties of the materials. Although there have been significant developments in
understanding dendritic structures in past decades, our knowledge on the dendritic
growth phenomena is largely based on experiments and idealized theorectical models.
Therefore, the comprehensive description of dendritic solidification still remains a
formidable task.
Recently, the phase field model has been proposed to simulate solidification of pure
materials[1-3] and binary alloys[4-8]. The phase field model is a very useful method for
realistically simulating microstructure evolution with which diffusion in the solid and
liquid phases, coarsening of dendrites and the curvature and kinetics effect on the
moving S/L interface can be described. This model is very efficient, especially in
numerical treatment, because all governing equations are written in unified forms
without distinguishing the interface from solid and liquid phases. In the model, the
phase field  ( x, t ) characterizes the physical state(   1 in the solid,   0 in the
liquid and 0    1 at the interface) of the system at each position and time. The phase
field variable  changes steeply but smoothly at the S/L interface region, which avoids
direct tracking of the interface position.
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The proposed models for alloy solidification[4-8] are restricted to binary alloys.
Particularly, when the solute atom is interstitial, these models are not applicable[9,10].
The application of the phase field model for solidification to practical technology is
closely linked to our ability to model microstructural development in multicomponent
alloys (three or more components). Therefore, it is very important to develop a phase
field model for the solidification of multicomponent alloy systems with both
substitutional and interstitial elements.
In this study, a phase field model is proposed for the solidification of multicomponent
alloys applicable to both substitutional and interstitial alloys with thin interface limit.
Our model was developed based on the WBM model[4] for the solidification of binary
alloys and has the limitation for the interface thickness such as the WBM model for
binary alloys. The detailed derivation of this model can be found in Ref.[10]. The
previous work[9] of authors is the development of the phase field model for the
solidification of multicomponent alloys based on the model proposed by Kim, et al.[7]
and this model[9] has no limitation on the interface thickness.
2.DEVELOPMENT OF MULTICOMPONENT ALLOY PHASE FIELD MODEL
We consider an isothermal solution with n alloy elements which may exist as either a
solid or a liquid in a domain,  . We choose the mole number per unit volume as the
concentration variable. It is assumed that (i) the difference in the molar volume between
the solid and liquid phase is negligible, (ii) substitutional elements and the solvent
have the same molar volume( V S ), (iii) interstitial elements do not contribute to the
entire volume, that is, the molar volume of interstitial elements is negligible, and (iv)
the effect of substitutional vacancys is also negligible.
Because the mole number per unit volume is used as the concentration variable, the
two following constraints are imposed.
1
ci 
,
(1)

VS
iS
and
c
iI
i
 cVa  const. ,
(2)
where the symbols S and I indicate the substitutional and interstitial element,
respectively and , c i and cVa are the mole number of the i-th component and the mole
number of vacancies per unit volume in the interstitial sites, respectively. From Eqs. (1)
and (2), the concentrations of the solvent and interstitial vacancies are removed from
concentration variables, where the solvent element is the n-th element. Consequently,
the number of concentration variables is n-1.
The analysis begins based on the formulation through the free energy functional of
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Wheeler,Boettinger and McFadden[4]. The total free energy of the system is defined as
F  , c1 ,, cn1    f ( , c1 ,, cn1 ) 

2
2
 d ,
2
(3)
where f is the free energy density of the system and  is a phase field taking a value
of zero in the liquid and smoothly changing to one in the solid. We require that
f (1, c1 ,, cn1 ) and f (0, c1 ,, cn1 ) are the the thermodynamic free energy densities
for homogeneous solid and liquid.
In order to derive a kinetic model, it is assumed that the system evolves in time so that
its total free-energy decreases montonically. Since concentration is the conserved field
variable and phase field variable is not conserved, the composition field is determined
by solving the Cahn-Hilliard diffusion equation[11], whereas the phase field variable is
determined by solving the Ginzburg-Landau equation[12]. This case corresponds to
Model C of Hohenberg and Halperin[13].
The evolution equation for phase field is

 M  2  2  f  ,
t


(4)
and the evolution equation for the k-th concentration variable is
n 1
ck
    M ki f ci
t
i 1
n 1 n 1
n 1
j 1 i 1
i 1
    M ki f cic j c j     M ki f ci 
,
(5)
n 1
where M is the phase-field mobility and M ki is determined using Dkj   M ki f ci c j .
i 1
The phase field equation (4) and the diffusion equation (5) can be derived in a
thermodynamically consistent way from the entropy functional. In this study, the free
energy density of the system is defined as follows:
f (c1 ,, cn1 )  h( ) f S (c1 ,, cn1 )  (1  h( )) f L (c1 ,, cn1 )  wg ( ),
(6)
where g ( ) is a double-well potential and w is the height of the double-well
potential.
The parameters  and w in the phase field equation are to be matched with the
interface energy  and interface thickness 2 . The relationships among  , w , 
and 2 are as follows[10].
1
  2  (0 )  (0)d0 ,
(7)
0
2 


2
d 0
0.9
0.1
( 0 )  (0)
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.
(8)
n 1
where (c1 ,, cn1 , )  f (c1 ,, cn1 , )   f cei ci , 0 is phase field profile at the
i 1
equilibrium state, f cei is the equilibrium chemical potential of i-th component, and
Z (0 ) is defined by
n 1
Z ( 0 )  h( 0 ) f S (c~1 ( 0 ),, c~n1 ( 0 ))  1  h( 0 ) f L (c~1 ( 0 ),, c~n1 ( 0 ))   f cei c~i .
(9)
i 1
where c~i is the concentration profile of i-th component in an equilibrium state. The
detailed derivation of the above relationships can be found in Ref.[10]
In order to determine the phase field mobility at the thin interface thickness condition
and low velocity limit, the growth velocity that chemical rate theory yields will be
used[14].
The chemical rate theory for one componental systems describes interface movement
by assuming thermally activating individual hopping processes of atoms across the
interface. The theory yields for the growth velocity,

 G 
(10)
V  V0 (T )  1  exp 
 ,
 RT 

is the maximal growth velocity when the driving force is infinite (no
where V0
backward hopping) and G is the driving force per mole(  F / ctot , where ctot is the
total moles of elements per unit volume) for the solidification.
In the low velocity limit, i.e., V  V0 , by a Taylor series expansion, Eq.(10)
becomes approximately
V  V0
FS / ctot
G
 V0
,
RT
RT
(11)
where ctot is the total moles of elements per unit volume and 1/ VS  iI ci . The
driving force for the solidification without the solute-drag effect in a multicomponent
system is given by[9]
FS  f (c ,, c
S
S
1
S
n 1
)  f ( c , , c
L
L
1
n 1
L
n 1
)
j 1
f L
c c
.
c Lj

L
j
L
j

(12)
Using Eq. (11), the phase field mobility will be determined.
Under the assumption of negligible diffusivity in solid, we will derive the
corresponding relationship for the present multicomponent alloy phase field model at a
thin interface limit condition. At a 1-D instantaneous steady state with an interfacial
velocity, V, the governing equations for concentrations become
dck
d  n1
d

   M ki
f ci  .
(13)
dx dx  i 1
dx 
The following equation can be derived from Eq. (13) to the first order in the Peclet
V
285
~ ~
number, P  2V / D ( D : average interface diffusivity)[10],
n 1


f L (c1 ,, cn1 )  f S (c1 ,, cn1 )   ciL  ciS f cLi  V ,
(14)
i 1
where  is a constant for a given temperature and is defined as follows[10]:
1

M
n 1 c L ,e  n 1
j
 d0 
dx

B ejk ck0  ckS ,e dxdc j ,



S ,e  
  dx 

cj
x
j 1
k 1

2


(15)
where ck0 and  0 are the k-th composition and phase field at an equilibrium state,
respectively.
It should be noted that the left hand side of Eq.(14) is the same as the driving force
for the solidification per unit volume. So, using Eq.(11) and Eq.(12), the phase field
mobility in the finite interface limit can be determined from the following equation
RTc tot

 0.1 n1 n1
(16)


   ji f ci0  j (0 )d0 ,
V0
M 2
2 0.9 j 1 i 1
where ctot  1/ VS   ciS ,e , the matrix,  , is the inverse matrix of  ,  ij is
iI
defined as f cic j and  j (0 )  
0.1 n 1
0

k 1

B ejk ck0  ckS ,e
 d .
( )  (0)
3.NUMERICAL SIMULATION
In order to verify multicomponent alloy phase field model, numerical simulations for
the isothermal solidification of ternary alloy systems are presented. Based on the
verified phase field model for multicomponent alloys, 2D dendritic growth of binary
and ternary alloys with interstitial alloy elements will be presented.
3.1 Validation of the model
In order to confirm the validity of our model, the solidification of ternary alloy systems
is considered. We selected an alloy system with only substitutional alloy elements, FeCr-Ni and an alloy system with both substitutional and interstitial alloy elements, FeMn-C for the numerical simulation. The verification of our model will be conducted
from the two viewpoints. One is whether our model evolves from the non-equilibrium
state to the equilibrium state and produces the equilibrium volume fraction and
concentrations of solid and liquid phases. The other is whether our model realizes the
kinetics of solidification.
First, it will be shown that our model can produce exactly the equilibrium state of
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solidification. The Fe-9.0at%Cr-10.0at%Ni alloy system solidifies from liquid to
austenite(FCC) at 1757K and the Fe-1.0at%Mn-0.5at%C alloy system solidifies from
liquid to -ferrite(BCC) at 1780K. The surface excess energy,  , was set at 0.4J/m2,
and then, the limitation of the interface thickness[10] gives the condition of
2  0.973m in the Fe-9.0at%Cr-10.0at%Ni alloy system at 1757 K and
2  0.085m in the Fe-1.0at%Mn-0.5at%C alloy system at 1780 K when
thermochemical data[15] were used.
The interface thickness 2 was taken as 6nm in the Fe-Mn-C alloy system at 1780 K
and 60nm in the Fe-Cr-Ni alloy system, respectively. grid sizes of 1nm and 10nm for
the above alloy systems were used so that the interfacial region ranged over six grid
spacings. The total grid number was fixed at 200. The parameters of the phase field
equation,  and w were obtained from Eqs. (7) and (8). VD was assmumed to be 1
m/s. The diagonal terms of the solute diffusivity matrix were assumed to be the same in
each phase and each component, which was assumed to be 3 10 9 . The off-diagonal
terms were assumed to be zero for convenience, although the diffusivity matrix of each
component in each phase could be obtained in a general way. The phase field mobility
was obtained from Eq. (16).
Figure 1 shows the evolution of the Cr and Ni concentration profiles at different times
in the Fe-Cr-Ni alloy system, respectively. At 1 10 2 sec, the system reached
equilibrium and at this time, the concentrations of Cr and Ni in solid phase were
8.829at% and 9.936at%, respectively and those in liquid phase were 10.031at% and
10.38at%,rescpectively. The equilibrium concentrations in our model showed good
agreements with the equilibrium concentrations from thermodynamic data[15]. The
equilibrium volume fraction of austenite and liquid phases are 0.8564 and 0.1436,
respectively, which are almost identical to the values from thermodynamic data[15].
Fig. 1 The evolution of the concentration profile of Cr and Ni. The concentration
profiles are shown at t= 0, 3.75 10 4 , 7.50  10 4 , 1.25 10 3 , 1.625 10 3 ,
2.375  10 4 , 1.0 10 2 sec..
Figure 2 shows the evolution of the concentration profile of Mn and C at different
287
times for L/-ferrite solidification in the Fe-Mn-C alloy system at 1780K, respectively.
At 5.88 10 5 sec, the system reached equilibrium and the Mn and C concentrations in
the solid phase were 0.933at% and 0.263at%, respectively and in the liquid phase
1.263at% and 1.444at%, respectively. The equilibrium volume fractions of -ferrite and
liquid were 0.797 and 0.203, respectively. The concentrations of Mn and C and the
volume fraction of solid and liquid phase were in good agreements with the values from
the thermodynamic data[15].
Fig. 2 The evolution of the concentration profile of Mn and C. The concentration
profiles are shown at t= 0, 2.96  10 6 , 5.92 10 6 , 1.18 10 5 , 1.77 10 5 ,
2.35  10 5 , 5.88 10 5 sec..
Fig. 3 Variations of interface concentrations, calculated as a function of distance  *
between the interface and a solute sink in a liquid, which engulfs all the solute influx
from its neighbor. The solid line represents the calculated results from the present model
and the filled circles are the results from the analytic solution (16) for the classical sharp
interface model.
Second, it will be shown that our model can realize the kinetics of solidification
exactly. In order to study the kinetics of solidification, an 1D isothermal system reported
by Kim, et al.[6,7] is adopted for steady state simulations. In addition, the alloy system
for steady state solidification is adopted as a dilute solution so that the off-diagonal
diffusivities of the solutes could be neglected in the dilute solutions. In the classical
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sharp interface model with diffusion in liquid only, the exact solution[10] is as follows.
(1  k ie )(e Vx / Dii  e V
L

i
c ( x)  c  c
L
i

i
1  (1  k ie )(1  e V
*
*
/ DiiL
/ DiiL
)
.
(16)
)
If V and  * are given, the concentration profile of liquid phase can be obtained from
Eq. (16) and the concentration at the interface can be determined at x  0 . The alloy
system for the numerical calculations was adopted as Fe-1.0at%Mn-0.5at%C as used
above.
Figure 3 shows variations in the steady state Mn and C concentration at the interface,
calculated as a function of the distance  * between interface and solute sink. The
system temperature was 1780K which is between the solidus and liquidus temperatures.
The solid line is the calculated results from the present model and the solid circles are
the results from the analytic solution (16) for the classical sharp interface model. The
two results are in excellent agreement with each other. The error in the interface
concentrations was within 1%.
3.2 Dendritic growth of a ternary alloy
So far, it was shown that our model showed exactly the thermodynamic equilibrium
and realized the kinetics of solidification. We have developed a 2-D calculation program
based on the above verified 1-D model and applied it to the analysis of isothermal
dendritic growth. In order to include the effects of anisotropy in our model, we follow
the ideas reported by Kobayashi[1], Warren and Boettinger[5]. The detailed equations
for 2-D dendritic growth can be found in Ref.[10].
The dendrite shapes of Fe-0.5mol%C binary alloy and Fe-0.5mol%C-0.005mol%Mn
ternary alloy are shown in Fig. 4. The growth rate of the dendrite for the binary alloy is
larger than that for the ternary alloy, which comes from the decrease of equilibrium
liquidus and solidus lines due to the addition of the third element. In the ternary alloy,
the secondary arms are more well-developed and complex in comparison with those of
the binary alloy. The dendrite shape is significantly affected by a small amount of Mn.
Because the diffusivity of Mn is much smaller than carbon, it is easy to be enriched in
front of the interface and reduce the interface stability.
The calculated pattern for the dendritic growth of binary and ternary alloys shows
many features consistent with observations of dendritic growth of binary alloy presented
by Warren and Boettinger[5]. First the spine of the primary stalk has a low
concentration, with the regions of the dendrite between the secondary arms having the
highest. Second, we note the tendency of the sidebranches to begin their growth in a
direction not perpendicular to the primary stalk, and to later develop a growth axis
perpendicular to the primary stalk. Behind the dendrite tip, the root of the side arm is
289
significantly narrower than the further out secondary arm. This feature is commonly
observed in real dendrites. It is interesting to note the existence of spines of low
concentration also on the secondary branches, which are as low as the spines in the
primary stalks.
Fig. 4 Plots of the C concentration profiles for the Fe-C and Fe-Mn-C alloys. The
calculation was done on a 5001000 grid, and then reflected about y axis. The colors
range from blue (lowest concentration) to green (central concentration) to blue (highest
concentration). All the colors figures employ this color scheme.
4.CONCLUSION
A phase field model is presented for the solidification of a multicomponent alloy system
with a low interface velocity and a thin interface conditions. The model could be
applicable to any alloy systems with both substitutional and interstitial alloy elements
unlike the existing phase field model for the solidification of binary alloy systems.
Because in multicomponent alloy systems, the interface kinetics coefficient could not be
defined and the second response function does not have a simple form as in dilute
binary alloy systems[14], the growth velocity from the chemical rate theory was used in
order to determine the phase field mobility at the finite interface limit condition.
This model was applied to simulate isothermal growth of large scale dendritc
microstructures in a highly supersaturated liquid, and to predict the pattern of the solute
distribution within the solid and liquid. The research presented in this paper is the first
attempt to model the dendritic growth of multicomponent alloys.
ACKNOWLEDGEMENTS
The authors appreciate Prof. Kyu-Hwan Oh in Seoul National University for his help in
thermodynamic evaluation and also Prof. Seong-Gyoon Kim in Kunsan National
290
University for his helpful discussions. Authors are grateful to the financial support of
Brain Korea 21 program supported by Ministry of Education, Korea.
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