Full text of DOI:10.1557/JMR.2002.0331

Cellular automaton model to simulate nucleation and growth
of ferrite grains for low-carbon steels
L. Zhang
College of Science, Northeastern University, Shenyang 110006, People’s Republic of China, and
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy
of Sciences, Shenyang 110016, People’s Republic of China
C.B. Zhang
The State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110006,
People’s Republic of China, and College of Science, Northeastern University, Shenyang
1
10006, People’s Republic of China
Y.M. Wang
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy
of Sciences, Shenyang 110016, People’s Republic of China
X.H. Liu and G.D. Wang
The State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang
1
10006, People’s Republic of China
(Received 29 January 2002, accepted 3 June 2002)
A two-dimensional cellular automaton model was developed for the simulation of
nucleation and growth of ferrite grains at various cooling rates in low-carbon steels.
The model calculates the diffusion of the solute and temperature fields in an explicit
finite method and incorporates local temperature and concentration changes into a
nucleation or growth function, which is utilized by the automaton in a probabilistic
fashion. The modeling provides an efficient way to understand how those physical
processes dynamically progress and affect nucleation and growth of ferrite grains.
I. INTRODUCTION
microstructure formation and in turn, the mechanical and
physical properties of products by modeling. In these
modeling works, Umemoto et al. studied the effect of
cooling rate and austenite grain size on the resultant fer-
Because microstructures of low-carbon steels, espe-
cially their grain sizes of ferrite, have strong influence on
their mechanical properties, considerable industrial im-
portance is placed on predicting a ␥–␣ phase transfor-
mation of steels during heat treatment. In controlled
rolling, which is an optimized reheating, hot rolling, and
cooling process, the chief aim is to obtain the microstruc-
tures consisting of small and uniform ferrite grains,
which have favorable mechanical properties in general,
and in particular yield preferable strength and toughness.
In recent years, many efforts have been undertaken to
understand how to obtain the refinement microstructure
of ferrite grains. However, the microstructure forma-
tion in steels is not yet fully understood because of the
complexity of the solid-state transformation involving
the combined complex behavior of heat–mass transfer,
nucleation and growth etc., in the liquid and/or solid
states and their interfaces during the ␥–␣ phase transfor-
mation. Although experimental observations could quali-
tatively explain the microstructure formation without
using modeling, quantitative understanding of this com-
bined behavior and its dynamic dependence on tempera-
ture and composition of steels is required to control the
1
,2
rite grain size. Jacot and Rappaz provided a two-phase
model for the prediction of microstructural evolution in
an Fe–C alloy, which undergoes a diffusive phase trans-
3
formation. However, their works could not provide an
complete understanding of nucleation and growth of fer-
rite grains.
In fact, nucleation and growth of grains has been simu-
lated in solidification by using a method called cellular
7
–10
11
automation (CA).
According to Wolfram, CA
could be an alternative model for solving the problems
for systems with small degrees of freedom, typically
handled by differential equations, or complex systems
with large degrees of freedom. There are five distinct
characteristics of CA: (i) There exists a grid of uniform
cells, (ii) which are updated every time step (iii) with a
finite value or state (iv) using deterministic rules, and (v)
every cell is connected to a neighborhood of surrounding
cells that control its evolution in time.
1
–6
Thus, if nucleation and growth of grains could be
simulated by the CA technique in solidification, why
could it not be used to predict the phase transformation,
J. Mater. Res., Vol. 17, No. 9, Sep 2002
© 2002 Materials Research Society
2251
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
from austenite to ferrite, involved with nucleation and
growth of ferrite grains in low-carbon steels? The aim of
the present work is to establish a two-dimensional CA
model that predicts nucleation and growth of ferrite
grains for low carbon steels Fe–⌺X –C (here X ס Si,
in Fig. 2. It could be assumed, as observed in experi-
ments, that ferrite grains nucleate preferentially on grain
boundaries of austenite. In such cases, ferrite grains will
initially nucleate on the grain boundaries of austenite,
then the nucleated ferrite grains grow into the inner part
of the austenite grains (as shown in Fig. 3). During con-
i
i
Mn, Ni, Cu, Cr). The model uses an explicit finite vol-
ume technique to calculate the solutal and thermal fields.
The local temperature and concentration changes are in-
corporated into a nucleation or growth function, which is
used by the automaton in a probabilistic fashion within a
small representative specimen. In addition, the simulated
tinuous cooling from T to T, the number of newly
e
1
formed ferrite grains n is
T
e
I͑TЈ͓͒1 − f ͔
n͑T͒ = ͐T
dTЈ .
(2)
Q
2
results are verified by the experimental results.
Here I(TЈ) is ferrite nucleation rate per unit area of aus-
tenite grain boundary at a temperature TЈ, f is a grain
boundary area fraction occupied by the ferrite grains, and
Q is a cooling rate.
At the temperature TЈ, the ferrite nucleation rate I is
expressed as
II. MATHEMATICAL DESCRIPTION
With respect to an Fe–⌺X –C multicomponent alloy
i
with a given low carbon concentration c, Fe-⌺X could
i
1
2
be regarded as a super-element S; thus the multicom-
ponent S–C alloy is similar to a low-carbon Fe–C alloy.
As a temperature in a local zone of the S–C alloy speci-
men is decreased at a given cooling rate, the phase trans-
formations would take place:
K2
−
1/2
I = K (kTЈ͒ D exp −
,
1
␥
ͩ
␥
→␣+␥1 2
ͪ
RTЈ͑⌬GN
͒
(
3)
where D is the carbon diffusivity in austenite, K is
(
→
␥–Fe) → (␥–Fe) + (␣ + Fe)
␥
1
a constant related to the nucleation site density, K2
(␣ + Fe) + cementite
.
(1)
is a constant related to the austenite/ferrite interfacial
For the Fe–C alloy with the concentration c, it is as-
sumed that the equilibrium temperature beginning at a
␥→␣+␥1
energy, k is Boltzmann’s constant, and ⌬GN
is a
driving force of ferrite nucleation. It will be described in
Sec. II.B.
␥
→␣ phase transformation A is a temperature on
e3
the ␥-phase line of an Fe–C phase diagram correspond-
ing to the concentration c as shown in Fig. 1.
B. Driving force of ferrite nucleation
␥
→␣+␥1
A. The ␥→␣ phase transformation under
The driving force of ferrite nucleation ⌬GN(Fe) of
␥
→␣+␥1
␥→␣
continuous cooling
the Fe–C alloy is calculated by ⌬GN(Fe) ס ⌬GFe
−
␥
Fe
␥→␣
Fe
RT ln ␣ , where ⌬G
is a free energy change of Fe in
The ␥→␣ phase transformation under continuous
cooling could be considered as the sum of short-time
isothermal holdings at successive temperatures as shown
␥
the ␥→␣ phase transformation, and a is an activity
Fe
coefficient of Fe in austernite. For the Fe–⌺X –C alloy
i
presented in the simulation, because Fe–⌺X has been
i
regarded as the super-element S, which is similar to a
FIG. 2. Diagram showing the approximation of continuous cooling as
the sum of short isothermal holding.
FIG. 1. Phase diagram of a Fe–C alloy.
2
252
J. Mater. Res., Vol. 17, No. 9, Sep 2002
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
calculation of the driving force of ferrite nucleation for
C. Diffusion of the solute and temperature fields
the Fe–C alloy, the driving force of ferrite nucleation
The diffusion of the solute and temperature fields has
a large influence on the ␥→␣ phase transformation,
or the phase transformation is controlled by the diffu-
sion. Then diffusion kinetics is described as follows
␥→␣+␥1
⌬G
to the multicomponent alloy could be written as
N(S)
␥
͑
N S
→␣+␥1
␥→␣
␥
⌬
G
= ⌬GS
− RT ln aS
,
(4)
͒
␥→␣
where ⌬GS
is the free energy change of the super-
Ѩu
2
element S from the ␥ → ␣ phase transformation, and it is
= D
u
ٌ u
,
(9)
(10)
(11)
Ѩt
given by
Ѩc␷
2
␥
→␣
i
M
i
␥→␣
Fe͑S͒
=
D ٌ c
,
⌬
GS
= 141͚xi͑⌬T
− ⌬T ͒ + ⌬G
.
(5)
␷
␷
NM
Ѩt
i
M
Here x is a mole fraction of a alloy element X , ⌬T
Ѩc␥
Ѩn
Ѩc␣
i
i
␣
ր
␥
␣ ␥
ր
i
NM
D␥
− D␣
͒
= ␯ ͑c␣ − c␥ .
and ⌬T
Table I. ⌬G
are Zener parameters, which are listed in
ͯ ͯ
n
Ѩn
1
3
␥→␣
Fe(S)
␣ ␥
ր
␣ ␥
ր
is the free energy change of Fe in the
␥
→␣ phase transformation, and it is given by
Here, t is time, c is a carbon concentration in phase v (␣
␷
or ␥), and D is the associated diffusion coefficient (here
␥
Fe
→␣
2
␷
⌬
G
= 20853.06 − 466.35T − 0.046304T
␥|␣
␣
␥|␣
͑
S
͒
D and D are assumed to be two constants), c and c␥
are two concentrations of the ␣ and ␥ domains at the ␣/␥
␥
␣
+
71.147T lnT
.
(6)
interface, ␷ is the normal velocity of the ␣/␥ interface
movement resulting from a concentration difference be-
tween ␣ and ␥ sides at the ␣/␥ interface, and n means the
n
The activity coefficient of the super-element S in aus-
␥
tenite a can be expressed as
S
␥
S
␥
C
␥
C
normal to this interface pointing to the ␥ side. D is a
u
ln a =
͕
ln͓͑1 − Z x ͒
ր
͑1 − x ͔͒
͖
ր
͑Z − 1͒
.
(7)
␥
␥
diffusion constant of the temperature field in ␣ and ␥
phases; u characterizes a dimensionless temperature field
␥
Here x is a initial mole fraction of carbon in austenite,
C
1
4
of the specimen, which can be expressed as a dimen-
sionless variable by the relation
and Z is given by
␥
Z ס 14 − 12exp(W /RT )
.
(8)
␥
␥
u ס (T − T )/(L /C )
,
(12)
ϱ
at
p
Here R is a gas constant, and W (ס 1250 J/mol) is the
exchange energy among carbon atoms in austenite.
␥
where L is the latent heat of the phase transformation,
C is the heat capacity of the multicomponent alloy, and
p
at
Tϱ is the liquid temperature on the boundary B far from
the studied regions as shown in Fig. 3.
III. CELLULAR AUTOMATON MODELING
The cells of the cellular automaton are characterized
by certain attributes that determine the states of one cell,
and the CA evolution is defined by rules for changing
the attributes. Four key features must be specified in the
model: the neighborhood of a cell, boundary conditions,
nucleation rules, and transition rules. The neighborhood
determines which cells will be “seen” by any cell in the
CA lattice. In this work, the CA is a two-dimensional
hexagonal lattice with periodic boundary condition at
each boundary face, and the neighborhood is defined as
the nearest neighbors as seen in Fig. 4. The present simu-
lation is two-dimensional, and the simulated results could
be interpreted as what happens in a cross section of the
simulated austenite or ferrite grains. Each cell in the CA
lattice takes a state: ␣, ␥, or “␣/␥ interface” as shown in
Fig. 5, and an integer in a class of integer numbers from
FIG. 3. Schematic diagram of the phase transformation in which fer-
rite nucleated on the grain boundaries of austenite.
i
M
i
TABLE I. Values of ⌬T and ⌬T
for different alloy elements.
NM
i
M
i
Element
⌬T (per at.% xi)
⌬T (per at.% xi)
N
Si
0
−3
−37.5
−6
4.5
−19
Mn
Ni
Cu
Cr
−39.5
−18
−11.5
−18
1
to 100 is used to represent one crystallized orientation
of each cell. Then the description of the modeling pro-
cedure will be given as follows.
J. Mater. Res., Vol. 17, No. 9, Sep 2002
2253
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
In the beginning of the simulation, all cells in the CA
lattice are set as the ␥ phase with the same initial con-
centration, i.e., the composition of the alloy, and the same
initial temperature T , which is higher than A , corre-
nucleation probability pnuc of that cell becoming ferrite is
calculated as detailed in Sec. III.A. To that cell, a random
number r (0 ഛ r < 1) is generated in advance. If r ഛ
s
s
pnuc is satisfied for it, the cell will nucleate. In the mean-
ini
e3
sponding to the alloy composition. As the temperature is
decreased at a given cooling rate Q, the temperature
change of every cell is written as T → T − Q и ⌬t, where
time, it will release an amount of latent heat from the
phase transformation and precipitate an amount of sol-
precipitate
utes ⌬c(
into its neighboring austenite cells,
i,j)
precipitate
␥
␣*
␣*
⌬
t is a time step.
⌬c(
ס c − c , where c is the concentration
i,j)
(i,j) (i,j) (i,j)
Step 1: To a cell (i, j) on the grain boundaries of
on ferrite phase line of the phase diagram at the tempera-
ture T(i,j). Each cell on the grain boundaries of austenite
will also be scanned following the same steps as above.
Step 2: The cells remaining at the ␣/␥ interface are
identified. Each ␣/␥ interface cell will be scanned again,
based on its carbon concentration, temperature, and the
states of its neighboring cells, to determine whether or
not the phase transformation (nucleation or growth of its
neighbor ferrite cells) can take place. In this model, the
phase transformation from ferrite to austenite is not
considered.
␥
austenite with a carbon concentration c(i,j), its Ae3,(i,j)
␥
corresponding to its c( is capable of being obtained
i,j)
from the Fe–C phase diagram. If the cell’s tempera-
ture T(i,j) is decreased below Ae3,(i,j), T(i,j) ഛ Ae3,(i,j), a
Step 3: Each cell transformed to ferrite from austenite,
such as one cell (i, j), will precipitate an amount of sol-
utes into its neighboring austenite cells, and the cell (i, j)
has a concentration determined by the ferrite line of the
phase diagram corresponding to its temperature. Mean-
while, the cell will also release an amount of latent heat
of phase transformation, and its temperature field is in-
creased according to u(i,j) → u(i,j) + latent, which “latent”
is dimensionless latent heat.
Step 4: The diffusion of the solute and temperature
␥(␣ or ␥/␣)
(i,j)
fields takes place. The concentration c
and the
FIG. 4. One “cell” with its six neighbors 1–6. LCA is the distance
between two adjacent cells.
temperature field u(i,j) of each cell in the CA lattice, such
as one cell (i, j), are calculated as detailed in Sec. III.B.
The temperature T(i,j) of the cell is given by Tϱ
u(i,j) C /L → T(i,j).
+
p
at
Step 5: The temperature of every cell in the CA lattice,
such as one cell (i, j), is decreased continuously, T(i,j)
−
⌬
t и Q → T(i,j). If the temperature of one cell remaining
austenite is a little lower than the temperature of cement-
ite formation, cementite will form.
A. Nucleation
In the present work, it is assumed a number of poten-
tial nucleating sites exists on the austenite grain bound-
aries, each of which can become instantaneously active
when some rules are satisfied. According to Eq. (2), as
there is an undercooling temperature ␦T ס A − T to one
e3
cell on the grain boundaries of austenite, a new nucleus
density ␦n is given by
␦
T
␦
n =
I
ր
Q и ͑1 − f ͒d␦T
.
(13)
(14)
͐
0
Here f takes the following value
1
␣
FIG. 5. Schematic illustration of ␣/␥ interface cells, which separate
the ␣ and ␥ domains corresponding to three states (␣/␥ interface, ␣ and
␥
*
␥*
␣*
f = ͑c − c͒
ր
͑c − c ͒
␣
ր
␥
,
ͭ
␥
) of cells in the CA lattice.
0
␥
2
254
J. Mater. Res., Vol. 17, No. 9, Sep 2002
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
* ␣*
␥
where c and c are the concentrations on the austenite
Case III: If both the cell (i, j) and its one neighboring cell
and ferrite phase lines of the phase diagram, respectively,
corresponding to the temperature of the studied cell as
shown in Fig. 1. The nucleation probability of each cell
on the grain boundaries of austenite, pnuc corresponding
to a possible nucleation event, is given by
nei belong to the interface,
␣
␣*
nei,
␣
␥*
nei,
␹
= ͓ f D
c
+ ͑1 − f ͒D
c
͑i, j͔͒
ր
c
nei,
͑
i, j
͒
␣
␥
nei,
͑
i, j
͒
͑
i, j
͒
and
␣
␣*
␣
␥*
͑i, j͒
pnuc ס ␦n и S␥
.
(15)
␹͑i, j͒ = ͓ f D c + ͑1 − f ͒D c
͔
ր
c͑i, j͒
,
(20)
␣
␥
͑
i, j͒
Here S is a surface area of one cell on the grain bound-
aries of austenite of the specimen. To a cell on the grain
␥
with
␣
i, j
␣
boundaries of austenite, a random number r (0 ഛ r < 1)
s
s
f ͑ + f
͒
͑
nei, i, j
͒
is generated. If r ഛ p is satisfied for that cell, it will
f ␣
=
,
s
nuc
2
become ferrite or it will nucleate.
␣
i, j)
␣
here f ( or f
is calculated by
nei,(i, j)
B. Growth
␥
*
i, j
␥*
nei,
c͑
͒
͒
− c͑
i, j
͒
c
c
͑
͑
i, j
͒
− cnei,͑i, j͒
Growth of ferrite is controlled by the diffusion of the
solute and temperature fields, and it continues until Ae3
of the ␣/␥ interface cell, such as a cell (i, j) becomes
lower than its temperature T(i,j) or when cementite phase
forms in front of the growing ferrite grains and prevents
their further growth.
␣
i, j
␣
nei,
f ͑
=
or f
=
,
͒
␥*
i, j
␣*
i, j
͑
i, j
͒
␥*
nei,
␣*
nei, i, j
͑
͒
c͑
− c͑
͒
i, j
͒
− c
(
21)
␥
*
i,j)
␣* ␥*
(i,j) nei,(i,j)
␣
where c( or c [c
or c * ] is the concentration
nei,(i,j)
of the cell (i, j) (its one neighbor cell) on the austenite
phase line or the ferrite phase line of the phase diagram
at the temperature T(i,j) [Tnei,(i,j)].
Step 2: At time t, a growth length l(
cell (i, j) toward its one neighboring ␣/␥ interface cell nei
Diffusion of the solute and temperature fields
(
i,j)
i,j)→nei
of the ferrite
Step 1: The diffusion of the solute and temperature
fields among those cells of the CA lattice is described by
an explicit finite approach. The change of the tempera-
ture field of the cell (i, j) during one time step ⌬t due to
the diffusion could be calculated in the following way:
is calculated by
t
͑
i, j
i, j
͒
͒
→nei
͑
i, j
͒
͒
→nei
l͑
͑t͒ =
v͑
dt
.
(22)
͐t0 i, j
6
2
⌬t
2
Here t is the time of the cell (i, j) becoming ferrite, and
0
u͑i, j͒
+
D ͓u
nei,͑i, j͒
− u͑i, j͔͒ → u͑i, j͒
,
(16)
(i,j)→nei
i,j)
͚
u
␷(
is the growing velocity of the cell (i, j) toward
3L
nei=1
CA
the neighbor cell nei and it can be calculated by Eq. (11).
Step 3: The time step ⌬t is an important measure in
modeling because the state of each cell is changed rhyth-
mically with its increase, and it could be defined as
where LCA is the distance between two adjacent cells as
shown in Fig. 4.
Calculating the diffusion of the solute field is rather
more complex than that of the temperature field. Equa-
tion (17) could be used to describe the concentration
variation of the ␣ cell, ␥ cell, or ␣/␥ interface cell (i, j)
due to the diffusion from its six neighboring cells
2
CA CA CA
2
2
1
LCA
L
L
L
⌬
t = min
ͩ
,
,
,
ͪ
,
(23)
5
vmax Du D␣ D␥
where ␷max is the maximum among the growth velocities
of all ferrite cells and “min” means the minimum value of
four ones in the round bracket.
Step 4: For a rapid cooling procedure, the relationship
between growth of ferrite and the solute diffusion must
be taken into account. The growth time step ⌬t
one ferrite cell (i, j) toward its one neighbor ␣/␥ interface
cell nei calculated by ⌬tgrow
6
2
⌬t
2
CA
c͑i, j͒
+
͓␹
nei=1
c
nei,(i, j) nei, i, j
− ␹͑i, j͒c(i, j)͔ → c͑i, j͒
.
Α
͑
͒
3L
(17)
(
i,j)→nei
of
grow
Case I: If the cell (i, j) with its one neighboring cell nei
is in the same phase ␷ (␣ or ␥),
(i,j)→nei
(i,j)→nei
ס LCA/␷(i,j)
. If it is
smaller than the time step of solute diffusion in austenite
␹nei,(i,j) ס ␹(i,j) ס D␷
.
(18)
2
⌬
tdiff (⌬tdiff ס L /D ), there the solute diffusion exists
CA
␥
in cell nei during growth of the ferrite cell. However, if
Case II: If the cell (i, j) belongs to the phase ␷, whereas
its one neighbor cell nei belongs to the interface,
(i,j)→nei
⌬tdiff ജ ⌬t
, growth could have already finished for
grow
the ferrite cell (i, j) toward the cell nei before the solute
diffusion in the cell nei would be able to take place.
␷*
␹nei,(i,j) ס D и c
/cnei,(i,j) and ␹(i,j) ס D␷
nei,(i,j)
.
(19)
␷
J. Mater. Res., Vol. 17, No. 9, Sep 2002
2255
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
Step 5: For an ␣/␥ interface cell nei, if its temperature
T(i,j) is lower than its Ae3,(i,j) and its generated “trans-
formed” probability r (0 ഛ r < 1) is smaller than a
nucleated ferrite grains begin to grow, and more and
more ferrite grains nucleate [Fig. 6(b)]. Finally, Fig. 6(c)
displays the morphology with many large and small fer-
rite grains.
tra
i,j)→nei
tra
(
“capture” probability p
of its neighbor ferrite cell
cap
(
i, j) toward the cell nei, the cell nei will become ferrite.
As described in Sec. II.A., the model assumes that fer-
rite grains nucleate on the boundaries of austenite grains,
thus each cell on the boundaries of austenite grains is a
potential nucleation site. Under cooling, the nucleation
probability of ferrite becomes large for each potential
nucleation site, and it means that many cells may nucle-
ate. Nevertheless, the nucleated ferrite cells will precipi-
tate an amount of solutes into their neighboring austenite
grains and release certain latent heat. Therefore, the con-
centrations of the austenite cells neighboring upon the
ferrite grains become enriched. In addition, those of
the other austenite cells (inner austenite grains and on the
boundaries of austenite grains), which do not neighbor
upon the ferrite grains, may also be increased for the
solute diffusion. In the meantime, the temperature of
the austenite cells will become high, resulting from the
temperature field diffusion. For those austenite cells,
equilibrium temperatures beginning at the phase trans-
formation Aes could be decreased because their concen-
trations are increased from the solutal diffusion or
receiving precipitated solutes, but their actual tempera-
tures are increased from the thermal diffusion. At a given
time, the equilibrium temperatures beginning at the phase
transformation of some austenite cells may be lower than
their actual temperatures. Therefore, because undercool-
(i,j)→nei
The capture probability p
is given by
cap
͑
i, j
i, j
͒
͒
→nei
l͑
͑
͒
i, j →nei
cap
p
=
.
(24)
LCA
C. Cementite formation
As the temperature decreases, more and more ferrite
forms, and carbon solutes build up in front of the grow-
ing ferrite grains. Thus, many cells become undercooled
with respect to cementite formation, and the formed ce-
mentite will stop growth of the ferrite grains.
IV. RESULTS AND DISCUSSIONS
Figures 6(a)–6(b) show the microstructure evolution
and final morphology—from nucleation of the ferrite
cells on the austenite grain boundaries and growth of the
nucleated ferrite cells toward their neighbor austenite
cells—of a Fe–⌺X alloy [chemical composition (wt%):
C 0.15, Si 0.09, Mn 0.4, Cr 0.02, Ni 0.02, Cu 0.01] under
continuous cooling (Q ס 1.0 K/s) from temperature
1
ferent colors (replaced by the different grey scales in
black and white pictures) in the figures represent the
different austenite or ferrite grains. The modeled speci-
men size is 0.6 × 0.6 mm (corresponding to 200 ×
i
320 K to the final temperature, room temperature. Dif-
ing temperatures (⌬T ס A − T) of the cells remaining
es
austenite are decreased, the nucleation probabilities of
the potential nucleation cells on the grain boundaries
of austenite are also decreased. Simply stated, diffusion
affects nucleation of ferrite. Meanwhile, although the
austenite cells may not nucleate, they will be able to
become ferrite, resulting from growth of their neighbor-
ing ferrite cells. Growth of ferrite grains can proceed if
2
00 cells).
In the beginning, a few ferrite grains nucleate on the
grain boundaries of austenite [as shown in Fig. 6(a);
some nucleation sites of ferrite are marked by white
squares]. Then, as the temperature is decreased, the
FIG. 6. Ferrite grains nucleating on the surface of the austenite grains and their growth under the cooling rate Q ס 1 K/s. The final temperature
of the sample is room temperature. (a), (b), and (c) correspond to the three stages of nucleation beginning on the grain boundaries of austenite,
phase transformation processing and the final ferrite grain structure. In (a), some nucleation sites of ferrite are marked by white squares (ᮀ).
2
256
J. Mater. Res., Vol. 17, No. 9, Sep 2002
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
FIG. 7. With the same initial austenite grain structure, the different cooling rates have effects on the final grain structure of ferrite. (a–d) Opti-
cal micrographs showing ferrite grains formed at various cooling rates: (a) Q ס 0.05 K/s, d␣ ס 106 ␮m; (b) Q ס 0.17 K/s, d␣ ס 79 ␮m;
(
c) Q ס 1.5 K/s, d ס 48 ␮m; and (d) Q ס 5.0 K/s, d ס 33 ␮m). (e–h) Simulated structures corresponding to the cooling rates Q ס 0.05 K/s,
␣
␣
Q ס 0.17 K/s, Q ס 1.5 K/s, and Q ס 5.0 K/s; (e) d␣ ס 108 ␮m, (f) d␣ ס 75 ␮m, (g) d␣ ס 43 ␮m, and (h) d␣ ס 38 ␮m.
J. Mater. Res., Vol. 17, No. 9, Sep 2002
2257
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
undercooling temperatures exist in their neighboring
austenite cells and cementite does not form in front of the
growing ferrite grains. Certainly, growth of the ferrite
grains is also affected by the diffusion of the solute and
temperature fields for which uniform distribution of
the solute and temperature fields is a great help to suffi-
cient growth of the ferrite grains. If the given cooling
rate is small enough, the uniformity structure of ferrite
grains could be obtained. However, because the cooling
rate could not be small enough in the model, distribution
of the solute or temperature fields is not uniform in the
modeled specimen. Then the number of the nucleated
ferrite grains may be large in some zones of the speci-
men, or the ferrite grains might grow more quickly in the
other zones. Thus, the final structure as shown in
Fig. 6(c) is obtained, where the size of some grains
is larger than that of others, and the resultant ferrite grain
size is not uniform. In general, the diffusion of the solute
and temperature fields plays an important role in nuclea-
tion and growth, or nucleation and growth is diffusion-
controlled. Besides these grains there are some small
deep black regions inside ferrite grains or on the grain
boundaries of ferrite, which are cementite. The formation
of cementite could be understood by the following
fact: The small regions are enriched in solutes while the
ferrite phase forms around them. Before the enriched
solutes can diffuse, the temperatures of the regions
remaining austenite decrease down the temperature of
cementite formation; then cementite phase forms in the
regions.
phase transformation corresponding to the alloy compo-
sition). At this time, many potential nucleation sites on
the grain boundaries of austenite could become ferrite
as there exist undercooling temperatures for them, and
the large undercooling temperatures also make nuclea-
tion probabilities of those potential sites high. Thus, the
final nucleation sum of ferrite increases because the sum
of the potential nucleation cells and the nucleation prob-
abilities of them are both improved. In addition to the
nucleation number increased, the diffusion of the solute
and temperature fields also has a great influence on
growth of the ferrite grains. Because the diffusion of the
temperature field is much quicker than that of the solute
field and the temperature distribution becomes quickly
uniform to the small modeled specimen, the diffusion of
the solute field has larger influence on nucleation and
growth of ferrite. At a small enough cooling rate, the
concentration distribution could be uniform due to
enough diffusion of the solute field. At a large cooling
rate, before diffusion of solute field could take place, the
growth of the ferrite grains toward their neighboring aus-
tenite cells could have finished. Because the temperature
decreases quickly, however, the cementite phase would
form in front of the growing ferrite grains and it
would stop their further growth if the temperature were
lower than the temperature of cementite formation to the
enriched solute cells neighboring them. Thus, the grain
size of ferrite is not able to become very large, and the
uniformity and fine grain size of ferrite is formed. Ac-
cording to this, the morphologies shown in Figs. 7(d)–
7(h) corresponding to the cooling rates Q ס 0.05 K/s,
Q ס 0.17 K/s, Q ס 1.25 K/s, and Q ס 5.0 K/s, respec-
tively, are calculated. Figure 8 illustrates the variation of
the average grain size of ferrite to the different cooling
Figure 7 displays the effect of the cooling rate on the
resultant structure of ferrite grains under the same initial
austenite grain size (126 ␮m) from the temperature
1
323 K, and the final temperature is room temper-
2
ature. Figures 7(a)–7(d) are the optical micrographs that
show ferrite grains formed by continuous cooling trans-
formation at various rates, and Figs. 7(e)–7(h) are the
simulated microstructure of ferrite grains at those rates.
From the figures, we can clearly see that a large cooling
rate makes the ferrite grain size finer than a small cool-
ing rate does. The phenomenon can be explained by the
following. To make the ferrite grain size uniform and fine,
increasing nucleation number and controlling growth of
ferrite are two efficient ways. If the cooling rate is small,
the diffusion of the solute field could go on a quite long
time. The average concentrations of the cells remaining
austenite become high because they receive the precipi-
tated solutes from the ferrite grains. Then their equilib-
rium temperatures beginning at the phase transformation
become lower. In some cases, because their temperatures
are higher than their equilibrium temperatures begin-
ning at the phase transformation, they cannot nucleate.
Nevertheless, under rapid cooling, the temperature is
decreased down to quite a low level (which is much
lower than the equilibrium temperature beginning at
rates. Here, the average grain size of ferrite d is calcu-
␣
num
i
lated by d ס ∑ d /num (num is the number of the
ferrite grains, and d is the grain size of the ith ferrite
␣
iס1 ␣
i
␣
FIG. 8. Effect of cooling rate on ferrite grain size.
2
258
J. Mater. Res., Vol. 17, No. 9, Sep 2002
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252

L. Zhang et al.: Cellular automaton model to simulate nucleation and growth of ferrite grains for low-carbon steels
grain). The figure indicates that the quicker the cooling
rate is, the finer the final grain size of ferrite is after the
phase transformation.
of low-carbon steels. In addition, the time step ⌬t
[Eq. (23)] selected is proven to be an important measure
for correctly simulating the phase transformation in the
CA model.
V. CONCLUSION
A two-dimensional cellular automaton model has been
developed to model the combined complex behavior of
heat–mass transfer, nucleation of ferrite, and growth
of nucleated grains during the ␥–␣ phase transfor-
mation of low-carbon steels. This modeling gives us
physical insight into how dynamically they interact and
constrain each other and what their combined effects on
the microstructural evolution are. The simulated results
are compared with the optical micrographs showing fer-
rite grains formed at various cooling rates, and they are in
good agreement with those micrographs. Moreover, CA
modeling can demonstrate the microstructural evolution
of multigrains involving nucleation and growth in nearly
reality, and it can, also give the microstructure at any
time. Nevertheless, it is investigated only through ex-
periments, The phase transformation of the low carbon
ACKNOWLEDGMENT
We acknowledge financial support of this work by the
Key Project of the National Fundamental Research of
China under the Grant No. G1998061509.
REFERENCES
1
. M. Umemoto, H. Ohtsuka, and I. Tamura, Acta Metall. 34, 1377
1986).
(
2. M. Umemoto, Z.H. Guo, and I. Tamura, Mater. Sci. Technol. 3,
249 (1987).
3
4
. A. Jacot and M. Rappaz, Acta Mater. 45, 575 (1997).
. M. Kumar, R. Sasikumar, and P.K. Nair, Acta Mater. 46, 6291
(
1998).
. W.F. Lange, III, M. Enomoto, and H.I. Aaronson, Metall. Trans.
9A, 429 (1988).
6. M. Militzer, R. Pandi, and E.B. Hawbolt, Metall. Mater. Trans. A,
7A, 1547 (1996).
5
1
2
steel Fe–⌺X –C is proven to be a mass–heat diffusion-
i
7
8
. M. Rappaz and Ch-A. Gandin, Acta Metall. 41, 345 (1993).
. Ch. Charbon, A. Jacot, and M. Rappaz, Acta Metall. 42, 3953
controlled process, and the diffusion processes have an
important influence on the nucleation and growth, and in
turn, the microstructure formation of ferrite grains. With
the same initial grain size, the final morphology of ferrite
depends mainly on the cooling rate during the phase
transformation. A larger cooling rate is a great help to
obtain finer ferrite grains in the CA model, which is in
agreement with experiment. Thus, it is concluded that the
present model can evaluate the ␥–␣ phase transformation
(
1994).
9. S.A. Spittle and S.G.R. Brown, J. Mater. Sci. 30, 3989 (1995).
10. M.F. Zhu and C.P. Hong, ISIJ 41, 436 (2001).
1
1
1
1. S. Wolfram, Phys. D 10D, 7 (1984).
2. H.K.D.H. Bhadeshia, Met. Sci. 15, 178 (1981).
3. H.I. Aaronson, H.A. Domian, and G.M. Pound, TMS-AIME 236,
753 (1966).
1
4. Y. Saito, G. Goldbeck-Wood, and H. Muller-Krumbhaar, Phys.
Rev. A. 38, 214 (1998).
J. Mater. Res., Vol. 17, No. 9, Sep 2002
2259
http://journals.cambridge.org
Downloaded: 22 Mar 2015
IP address: 169.230.243.252