NUMERICAL SIMULATION FOR  TRANSFORMAION IN
Fe-Mn-C SYSTEM BY PHASE-FIELD MODEL
Dong-Hee Yeon, Pil-Ryung Cha, Jong-Kyu Yoon
School of Materials Science and Engineering, Seoul National University,
Seoul 151-742, Korea
ABSTRACT
In Fe-Mn-C system, there is a large difference in mobility of solute Mn and C.
Substitutional element Mn diffuses much slower than interstitial element C in Fe. By
definition, paraequilibrium means that only the mobile elements are equilibrated while
the sluggish ones behave as a single element. The  transformation may be under
local equilibrium, paraequilibrium, or lying between these two limits. The evolution
mode could be determined by the diffusion velocity of solute elements in the matrix in
front of the moving interface and the interface migration. Phase-field model could be
applicable to simulate these phenomena without any constraints or boundary
conditions at the  interface. The objective of this study is to find out evolution
mode for paraequilibrium or full thermodynamic equilibrium by the phase-field model
of ternary system.
1. INTRODUCTION
Many researches have been carried out for the  transformation of Fe-Mn-C
system both theoretically and experimentally [1-5]. In  transformation of Fe-Mn-C
system, there is a regime that can be characterized by the absence of partitioning of
the substitutional alloying element Mn. This type of transformation is called
paraequilibrium. Under paraequilibrium, carbon diffuses at an appreciable rate but the
alloying element Mn is almost immobile relative to iron, the growing phase inherits
the alloy content form the parent phase. While the mobile element carbon is in the
local equilibrium at the phase interface, the chemical potentials of alloying
substitutional element have no physical meaning and thus these substituional elements
behave thermodynamically as if they were only a single element [4]. Many works
have been done mainly about the limit in which the transformation under
paraequilibrium can be possible thermodynamically and about the transition from
fully local equilibrium to paraequilibrium. But, both fully local equilibrium and
375
paraequilibrium states are thermodynamically unique states and the reaction of course
may choose any one of infinite numbers of undefined nonequilibrium states lying
between the equilibrium and paraequilibrium limits. Whether nature in fact attains
such states between two limits is a very subtle problem in kinetics. So the model is
needed that can select the state with considering the thermodynamics and kinetics of
the system correctly.
The Phase field model can be recommended as the unified model that is applied to
any systems under fully local equilibrium, paraequilibrium and nonequilibrium states
lying between two limits. It has been reported that phase field model is able to
describe successfully the interface kinetics such as solute trapping effect when it is
applied to solidification in binary system [6,7]. In order to describe the
paraequilbrium state, the phase field model for multicomponent alloy is needed and it
can distinguish the interstitial alloy elements from the substitutional alloy elements
because the large difference in atomic mobility is induced from the different diffusion
mechanism. However, the existing phase field models for alloys are restricted to
binary alloys. And, although these models are applicable to only binary system that
contains only substitutional alloy element, they are often applied to the system that
has interstitial alloy element such as carbon. [8,9] In this study, the 
transformation in Fe-Mn-C system is simulated by the phase field model for
multicomponent alloys. The result is compared with the simulation by the model that
does not distinguish between intestitial and substituiuional alloy elements. The phase
field model is certified by the comparison with the results of the system that is
composed of only substituional alloy elements, so the difference of atomic mobility is
small.
2. THE PHASE FIELD MODEL FOR MULTICOMPONENT ALLOYS
When the phase field model is applied to the multicomponent alloy system, the
phase field evolution and diffusion equation could be established from a LandauGinzberg free energy functional F(c1,c2,). For this purpose, (x,t) is postulated to
characterize the phase of the system. The phase field  is defined as a continuous
variable between =1 at one phase and =0 at the other phase. In the interfacial
region that has the finite width, (x,t) has the value between 0 and 1 and varies
steeply but smoothly. In multicompnent systems, the free energy functional
F(c1,c2,) is the functional of the phase field variable with solute concentration. [10]

2
2
F    f  , c1 , c2 , , , cn  
 dV
V
2


376
(1)
Here, f(c1,c2,) is the free energy density (free energy per unit volume) and it
may be written in the form,
f  , c1 , c2 ,  , cn   h  f  c1 , c2 ,  , cn   1  h  f  c1 , c2 ,  , cn   wg   ,
(2)
where f  and f  are the free energy densities of each phase, respectively and
functions of solute concentration ci. It is assumed that the phases in the interfacial
region are the mixture of phases that are composed of same concentration (WBM
Model) [6]. In equation (2), we choose h(), g() in the following forms.
h    3 6 2  15  10


(3)
g     1   
2
2
(4)
where g() is the double-well potential associated with phase change.
The evolution equation of phase field and diffusion equations are driven from
gradients of the above functional.

F
(5)
 M 
t

n 1
J k   M ki 
i 1
F
ci
(6)
In the equations (5) and (6), M is the phase field mobility and Mki is the mobility
matrix. From equations (5) and (6), it is possible to get governing equations.

 M   2 2  f
t
(7)
2 2


 M      h  f  f   g  




ck
 n1

     M ki f ci 
t
 i 1

In the above equations, f and fCi mean
(8)
f
f
,
respectively. Since equation (8)
 c i
should be the same form with traditional diffusion equation in one phase region, we
n 1
can determine Mki using Dkj   M ki f ci c j . The parameter  in equation (7) and (8)
i 1
should be related to the interfacial properties, the interfacial energy  and interface
thickness 2 respectively. The parameters are determined at the sharp interface limit

when we define interfacial region as the region
3 2

where changes from 0.1 to 0.9 at - to . The phase field mobility can be
V 
determined at the sharp interface limit as
 F , where F is the driving
M  2
condition,  
 
, 2  2.2 2
force per unit volume and V is the interface velocity. The phase field mobility can be
377
calculated with the relationship between driving force and interface velocity V   M i F
where Mi is interface mobility. [6,10]
3. CONSIDERATION OF INTERSTITIAL ALLOY ELEMENT
The existing phase field models cannot be applied to the system that contains
interstitial alloy element because these models use the Cahn-Hilliard type diffusion
equation and they usually use atomic fraction of solute atoms as the concentration
variables. These models cannot afford to distinguish the different diffusion
mechanisms of substitutional alloy elements and interstitial alloy elements. The
diffusion of substitutional solute atoms always accompanies with the diffusion of
solvent atoms while the diffusion of interstitial solute atoms does not. If one uses the
atomic fraction of solute atoms as the concentration variables, the increase of
concentration of interstitial alloy elements means the decrease of the concentration of
solvent atoms. In order to consider the interstitial alloy element in the diffusional
transformation, the site fraction is used as the independent variables [3,11]. However,
the definition of the site fraction in the mixture phases is a delicate problem. So, we
use moles per unit volume as the concentration variables.
It is assumed that the solvent atoms and the substitutional elements contribute to the
total molar volume of the alloy and there is no contribution from the interstitial
elements. There is no constriction on the movement of the interstitial atoms and
total moles of substitutional atoms are conserved. The two constraints can be written
as follows.
1
ci 
 const .
(9)

VS
iS
c
iI
i
 cVa  const.
(10)
VS is the molar volume of substitutional alloy elements, cVa and ci is the mole
numbers of vacancies per unit volume in the interstitial sites and S, I denote
substitutional element and interstitial element respectively. It is assumed that all
substitutional elements have the same molar volume and interstitial elements have no
volume. We did not consider the vacancies in the substitutional sites.
From the above assumptions, we can evaluate correct driving force of solute
diffusion. The free energy per unit volume can be written as the following,
n
f   ci  i
(11)
i 1
From equation (8), if we do not consider the off diagonal term of atomic mobility
matrix, the driving force of solute diffusion per unit volume is f ci . For the case of
378
substitutional elements, the driving force per unit volume is  i   Fe  and for
interstitial elements, f ci   i .
4. NUMERICAL SIMULATION OF  TRANSFORMAION
The isothermal  transformation in Fe-Mn-C system is simulated by numerical
method. The overall compositions of each element are Mn 1.0% and C 1.0% and the
temperature of the system is 873K. In this system, the carbon diffuses 107~109 times
faster than Mn. So the ferrite grows under paraequilibrium. The simulation is
performed that the ferrite, which occupies initially 10% of whole system with the
same composition in austenite, grows into austenite phase. The result is shown in Fig.
1. Figure 2 is the same result in which the concentration is expressed as a function of
the atomic fraction.
1450
Mn
(Mole per unit Volume)
12000
1440
C
(Mole per unit Volume)
10000
Initial profile
0.5E-4sec
1.0E-4sec
1.5E-4sec
2.0E-4sec
1.2sec
8000
6000
4000
2000
1430
0.0
1.0x10
-2
2.0x10
-2
3.0x10
-2
4.0x10
0
-2
0.0
1.0x10
Position(  m)
-8
-8
2.0x10 3.0x10
Position(  m)
-8
4.0x10
-8
Fig. 1. Concentration Profile as a Function of Mole per Unit Volume
12
Initial profile
0.5E-4sec
1.0E-4sec
1.5E-4sec
2.0E-4sec
1.2sec
1.1
8
1.0
C (at %)
Mn(at %)
10
0.9
6
4
2
0
0.8
0.0
-2
-2
1.0x10 2.0x10 3.0x10
Position(  m)
-2
4.0x10
-2
-2
0.0
1.0x10
-2
-2
2.0x10 3.0x10
Position(  m)
-2
4.0x10
-2
Fig. 2. Concentration Profile as a Function of Atomic Fraction
From the above results, we can see that the phase-field model describes the phase
transformation under paraequilibrium. The result without considering interstitial
element is shown in Fig. 3. Comparing with Fig. 2, there is no change of the atomic
379
fraction of Mn and only concentration of carbon changes. This means that there is a
change of the atomic fraction of Fe and only Fe atoms move without movement of Mn
atoms. During the growth of ferrite, the difference of chemical potentials of carbon is
very small as compared with that of Mn across the interface. Kirkaldy et. al said that
under paraequilibrium as the mobile atom is fully local equilibrium state at the
interface and the chemical potential of immobile atom is discrete [2]. The small
difference in chemical potential of carbon might be kinetic effect.
1.02
8
7
Initial Profile
1.25E-5
2.50E-5
3.75E-5
Final Profile
(8.6sec)
5
C (at %)
Mn (at % )
6
1.00
4
3
2
1
0.98
0
0.0
1.0x10
-2
2.0x10
-2
3.0x10
-2
4.0x10
-2
0.0
Position (  m)
1.0x10
-2
-2
2.0x10 3.0x10
Position (  m)
-2
4.0x10
-2
Fig. 3. Concentration Profile as a Function of Atomic Fraction during 
Transformation of Fe-Mn-C System Without Considering the Interstial Atom
In order to compare with the above calculation, another  Transformation in the
Fe-Cr-Ni system, which contains only substitutional elements, is studied. The overall
compositions of each element are Cr 23.0% and Ni 9.04% and the temperature of the
system is 1373K. This system was verified thermodynamically and kinetically by the
concept of local equilibrium [12,13]. In this system, Cr diffuses only 10~102 times
faster than Ni. So Both Cr and Ni atoms are partitioned at the interface and the
transformation is controlled by diffusion of Ni atoms because Ni moves slower. The
simulation is performed that the austenite, which occupies initially 10% of whole
system with the same composition with ferrite, grows into ferrite phase. Figure. 4
show that the transformation is governed by Ni diffusion.
32
20
initial profile
0.02sec
0.08sec
final profile
(1.24sec)
28
26
Ni( at % )
Cr ( at % )
30
24
22
20
10
18
16
0.0
1.0x10
-2
2.0x10
-2
3.0x10
Position (  m)
-2
4.0x10
0
-2
0.0
1.0x10
-2
2.0x10
-2
3.0x10
Position (  m)
-2
4.0x10
-2
Fig. 4. Concentration Profile as a Function of Atomic in Fe-Mn-C System
380
When the system reaches equilibrium state, the equilibrium volume fractions of
ferrite and austenite are 20.11% and 79.89% respectively and equilibrium
concentrations of Cr and Ni in ferrite are 29.26at% and 5.69at% respectively and
those in austenite are 21.42at% and 9.88at% [13]. The result performed by the phase
field model is that the equilibrium volume fractions of ferrite and austenite are
20.44% and 79.56% respectively and equilibrium concentrations of Cr and Ni in
ferrite are 29.22at% and 5.70at% respectively and those in austenite are 21.43at% and
9.90at% respectively. From this result, the phase field model reflects thermodynamics
and solute conservation properly. For Fe-Mn-C system, it takes too long to reach the
equilibrium state. So, we calculate the equilibrium volume fraction of each phase and
their compositions by increasing the diffusivities of Mn hypothetically in both phases.
The result predicted by the thermodynamic calculation is that the ferrite (Fe-0.511at%
Mn-0.077at%C) occupies 86.6% of total volume and the austenite (Fe-4.172at%Mn6.984at%C) occupies 13.4% at equilibrium state. The result by phase field model is
that the compositions of Mn and C in ferrite is Fe-0.507at% Mn-0.0737 at %C and
that in austenite is Fe-4.169at%Mn-6.965at%C respectively and volume fractions of
ferrite and austenite are 86.5% and 13.5% respectively.
From the results presented, we can say that the transformation mode between fully
local equilibrium and paraequilibrium can be described properly using the phase field
model. The selection of transformation mode depends on the relationship between the
interface mobility and the diffusivities of the solute atoms.
5. CONCLUSION
Using the phase field model, the transformation of Fe-Mn-C system is
simulated. It is shown that the phase field model can describe not only
thermodynamics effects of the phase transformation but also the kinetics such as
partitioning of solute atoms, so it is possible to simulate various modes of partitioning
behavior at the interface without extra boundary condition for moving interface. By
consideration of interstitial elements, we can simulate the transformation under
paraequilibrium properly.
ACKNOWLEDGEMENTS
The authors appreciate Prof. Kyu-Hwan Oh in Seoul National University for his help
in thermodynamic evaluation Authors are grateful to the financial support of Brain
Korea 21 program supported by Ministry of Education, Korea.
381
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