Modeling of Ferrite-Austenite Phase Transformation Using a ...

ISIJ International, Vol. 54 (2014), No. 2, pp. 422–429

Modeling of Ferrite-Austenite Phase Transformation Using a
Cellular Automaton Model

Dong AN,1) Shiyan PAN,1) Li HUANG,1) Ting DAI,1) Bruce KRAKAUER2) and Mingfang ZHU1)*

1) Jiangsu Key Laboratory for Advanced Metallic Materials, School of Materials Science and Engineering, Southeast

University, Nanjing, Jiangsu, 211189 China. 2) AO Smith Corporate Technology Center, Milwaukee, WI, 53224 USA.

(Received on August 29, 2013; accepted on November 5, 2013)

A two-dimensional (2D) cellular automaton (CA) model is proposed to simulate the ferrite-austenite
transformation in binary low-carbon steels. In the model, the preferential nucleation sites of austenite, the
driving force of phase transformation coupled with thermodynamic parameters, solute partition at the fer-
rite/austenite interface, and carbon diffusion in both the ferrite and austenite phases are taken into con-
sideration. The proposed model is applied to simulate the ferrite-to-austenite transformation during
isothermal heating at 760°C that is in the ferrite and austenite two-phase range, the austenite-to-ferrite
transformation during continuous cooling, and carbon diffusion during tempering at different temperatures
for an Fe-0.2969 mol.% C alloy. The results show that during the isothermal heating, austenite nucleates
and grows. The austenite grains are mostly located at the boundaries of ferrite grains. The carbon concen-
tration in austenite is higher than that in ferrite. The simulated microstructure agrees reasonably well with
the experimental observation. During the continuous cooling process, the austenite-to-ferrite transforma-
tion occurs accompanied with carbon diffusion. After cooling from the heating temperature of 760°C to
room temperature with a cooling rate of 2°C/s, the carbon concentration field is nearly uniform, while a
higher cooling rate of 5°C/s results in a non-uniform carbon concentration field. After tempering at differ-
ent temperatures for 20 min, the uniformity of carbon distribution increases with increasing tempering
temperature. The simulation results are used to understand the mechanisms of the observed experimental
phenomena that a cold-rolled low-carbon enameling steel presents different yield strengths after different
heat treatment processes.

KEY WORDS: modeling; cellular automaton; ferrite-austenite transformation; low-carbon steel.

1. Introduction ticity finite element method to describe the austenite-to-
ferrite transformation in the deformed austenite phase.8) Tong
Ferrite-austenite transformations in low-carbon steels et al.10) applied a q-state Potts model-based Monte Carlo
have attracted wide attention due to their fundamental role
in phase transformation and industrial importance. Various (MC) method to simulate the austenite-ferrite transforma-
experimental work1–6) has been carried out to study the
mechanisms of ferrite-austenite transformation and the rela- tion during the isothermal austenite decomposition, in which
tionship between microstructure and mechanical properties.
However, the microstructure evolution could scarcely be the non-equilibrium austenite-ferrite interface is mixed dif-
observed in real-time, and some microstructural features
cannot be examined with even the most sophisticated ana- fusion/interface controlled. In addition, they combined the
lytical tools.
MC model with a crystal plasticity finite element method to
During recent several decades, various numerical models
have been developed to simulate the microstructure evolu- study the influence of austenite deformation on the subse-
tion and solute diffusion during solid-solid phase transfor- quent isothermal austenite-ferrite transformation.11) Because
mation in steels. The phase field (PF) method7–9) has been
successfully applied to simulate the austenite-ferrite trans- of its simple structure and good computational efficiency,
formation. Yamanaka and co-workers7) constructed a PF
model to simulate the formation of Widmanstatten ferrite the cellular automaton (CA) method has also been adopted
plates during the isothermal austenite-to-ferrite transforma- by many researchers12–15) to study the mechanisms of solid-
tion. The PF model was also combined with the crystal plas- solid phase transformation. Kumar and co-workers12) devel-

* Corresponding author: E-mail: [email protected] oped a CA model to simulate the competition between the
DOI: http://dx.doi.org/10.2355/isijinternational.54.422
nucleation and the early growth of ferrite from austenite. It

is found that the competition determines the variation of fer-

rite grain size with the cooling rate and austenite grain size.
Zhang et al.13) employed a CA model to investigate the

transformation of austenite to ferrite during continuous

cooling. An hexagonal lattice was used in the model to

reduce the anisotropy caused by the CA algorithm. The

model incorporates the local concentration changes with a

nucleation or growth function to obtain the final nucleation

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ISIJ International, Vol. 54 (2014), No. 2

number, ferrite grain size and the kinetics of ferrite forma- 3. Model Description, Governing Equations, and
tion based on the cooling rate or the undercooled tempera- Numerical Algorithms
ture. Taking both carbon diffusion and ferrite/austenite
interface dynamics into consideration, Lan et al.14) devel- 3.1. Model Description
oped a two-dimensional CA model to predict the growth The steel used in the experiment is a multi-component Fe
kinetics of ferrite grains. Moreover, Lan and Xiao et al.15)
combined the CA method with a crystal plasticity finite ele- alloy. For the sake of simplicity, a pseudo-binary Fe–C alloy
ment model to simulate the austenite-ferrite transformation is adopted in the present simulations. Figure 3 shows the ver-
in a C–Mn steel with an heterogeneously deformed micro- tical section of the phase diagram calculated using Pandat,16)
structure. The results show the inhomogeneous microstruc-
ture evolution occurring in the deformed austenite Fig. 1. The stress-strain curves of the samples cooled with differ-
decomposing process. However, limited work has so far ent cooling rates after heated at 760°C for 300 s. (Online
been reported regarding applying simulations to analyze the version in color.)
processing-microstructure-property relationships of steels.
Fig. 2. The stress-strain curve of the samples before and after tem-
In this paper, a two-dimensional CA model is proposed to pering at different temperatures. (Online version in color.)
simulate the ferrite (α) – austenite (γ) transformation during
isothermal heating and continuous cooling, as well as the
carbon diffusion in the subsequent tempering process. The
simulation results are applied to understand the mechanisms
of the experimentally observed phenomena that a cold-
rolled low-carbon enameling steel exhibits different yield
strengths after heat treatment with different cooling rates
and then tempering at different temperatures.

2. Experimental Phenomena

A cold-rolled enameling steel, containing 0.07 wt.% C,
0.43 wt.% Si, 1.22 wt.% Mn, 0.04 wt.% P and 0.001 wt.%
S, was heated at 760°C for 5 min, and then cooled by air
cooling (~5°C/s) and sand cooling (~2°C/s). After heated at
760°C for 5 min and then air cooled, the samples were then
tempered at 200°C–500°C for 20 min. Table 1 shows the
yield strengths of the cold rolled enameling steel before and
after the different heat treatment processes. It is found that
the yield strength of the sample by sand cooling is higher
than that of the one by air cooling. After tempering, the
yield strength increases with the tempering temperature
increasing.

Figures 1 and 2 present the stress-strain curves of the
samples with different heat treatment processes. As shown
in Fig. 1, the stress-strain curve of the sample by sand cool-
ing has an evident yield platform, while the one obtained by
air cooling does not have obvious yield platform. After the
samples by air cooling are tempered, the yield platform
becomes gradually obvious as tempering temperature
increases, as shown in Fig. 2.

It is known that yield platform is closely related to the
Cottrell atmosphere which is associated with carbon diffu-
sion and distribution. In the following sections, a CA model
is proposed and applied to simulate ferrite (α) – austenite (γ)
transformation during different heat treatment processes to
understand the mechanisms of the experimental phenomena
exhibited in Figs. 1 and 2.

Table 1. Yield strengths of the samples before and after different
heat treatment processes (MPa).

760°C for 5 min Air cooling + temping Fig. 3. The vertical section of the phase diagram calculated from
200°C 300°C 500°C the multi-component Fe alloy with 0.07 wt.% C, 0.43 wt.% Si,
As rolled Sand cooling Air cooling 310.3 337.3 387.6 1.22 wt.% Mn, 0.04 wt.% P and 0.001 wt.% S. (Online ver-
335.9 sion in color.)
356.0 291.2

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ISIJ International, Vol. 54 (2014), No. 2

a thermodynamic phase diagram calculation software, based equation can be derived:
on the composition of the experimental steel described in
Section 2. The relevant thermodynamic parameters used in Δgm / Vm = RTM (ke −1) ⎣⎡uα (T ) − ueα (T )⎦⎤ / Vm ..... (3)
the simulations are taken from Fig. 3. The initial concentra-
tion of the pseudo-binary Fe–C alloy used for simulations is where TM is the allotropic transformation temperature, uα(T)
taken as u0=0.2969 mol.% C (indicated by the dotted line in is the actual concentration in ferrite at temperature T, ke is
Fig. 3). This composition will produce the equilibrium γ the equilibrium partition coefficient, defined as ke =
phase fraction of 0.12 at the temperature of 760°C, which is
identical with that of the experimental steel. ueγ (T ) / ueα (T ), where ueα (T ) and ueγ (T ) are the equilibrium

In the present work, the phase transformations of both concentrations in ferrite and austenite at temperature T,
α→γ and γ →α during heat treatment are simulated, while respectively.
the coarsening of ferrite grains is ignored. The heat treat-
ment processes used in the simulations are divided into three The equilibrium concentrations of ferrite and austenite,
parts:
(1) Isothermal heating process: ueα (T ) and ueγ (T ), are determined by fitting the thermody-

The heating temperature is 760°C and holding time is namic data in the phase diagram of the pseudo binary Fe–C
300 s. During isothermal heating, ferrite-to-austenite phase alloys shown in Fig. 3. The equilibrium concentrations of
transformation, solute carbon partition at the α/γ interface, ferrite and austenite varying with temperature can be written
and carbon diffusion in both ferrite and austenite phases take in the following polynomial forms:
place. Based on experimental observations, the nucleation
sites of austenite are preferentially distributed at the ferrite ∑ueα (T ) = 9 Bk T k ......................... (4)
grain boundaries. k
(2) Continuous cooling process: =1

The continuous cooling processes are simulated from ∑ueγ (T ) = 9 Fk T k ......................... (5)
760°C to room temperature with the cooling rates of 2°C/s k
and 5°C/s, corresponding to the cases of sand cooling and air =1
cooling, respectively. During the cooling process, austenite-
to-ferrite phase transformation, carbon partition at the α/γ where the coefficients B1=1.9369×10–24, B2=1.0019×10–21,
interface, and carbon diffusion in both ferrite and austenite B3=4.4444×10–19, B4=1.5343×10–16, B5=3.1797×10–14,
phases take place. For the sake of simplicity, the phase B6=–1.2132×10–16, B7=1.7459×10–19, B8=–1.1201×10–22,
transformation of austenite to pearlite is not considered. As B9=2.6974×10–26; F1=1.4970×10–22, F2=8.1092×10–20,
shown in Fig. 3, the eutectoid temperature is about 686°C. F3=3.7673×10–17, F4=1.3621×10–14, F5=2.9566×10–12,
Considering the kinetic effect on phase transformation, the F6=–1.0041×10–14, F7=1.2849×10–17, F8=–7.3405×10–21,
end temperatures of γ→α phase transformation are taken as F9=1.5789×10–24. When the temperature is above the eutec-
650°C and 600°C for the cooling rates of 2°C/s and 5°C/s, toid temperature of 686°C, the equilibrium concentrations of
respectively. When the temperature is cooled down below ferrite and austenite are temperature dependent calculated
the end temperature of γ→α phase transition, the retained using Eqs. (4) and (5). While when the temperature is below
austenite is assumed to be transformed to supersaturated fer- the eutectoid temperature, the equilibrium concentrations of
rite, while carbon diffusion continues to the room tempera- ferrite and austenite are taken as the constant values identi-
ture with temperature dependent diffusion coefficients. cal with those calculated at the eutectoid temperature.
(3) Tempering process:
According to Eqs. (1) and (3), and ignoring the effect of
The case cooled with 5°C/s is then isothermally heated at interface curvature, the velocity of interface migration, vn, is
200°C–500°C, respectively, and held for 20 min. During the determined by
tempering process, carbon diffusion takes place, but no
phase transformation is considered. vn = Meff ⋅ (−Δgm / Vm )

3.2. Interface Migration { } ..... (6)
According to the interface dynamic theory,17) the velocity
= Meff ⋅ −RTM (ke −1) ⎣⎡uα (T ) − ueα (T )⎤⎦ / Vm
of interface migration, vn, is determined by
Therefore, the increment of α phase fraction can be eval-
−Δgm / Vm = vn / Meff + σ K ................... (1) uated by Δϕ = vnΔx/Δt, where ϕ is α phase fraction, Δx and
Δt are the grid spacing and time step, respectively. Equation
where Δgm is the driving force of phase transformation, Vm (6) can be used for the simulation of both α→γ and γ→α
is the mole volume, Meff is the interface mobility, σ is the phase transformations. For the phase transformation of
interface energy, and K is the interface curvature. In the
present model, the effect of interface curvature is neglected. α→γ, the value of vn is negative, and thus the fraction of fer-
The interface mobility, Meff, is calculated by18) rite is decreased, i.e., Δϕ <0. According to the lever law, the
mean concentration u is calculated by
Meff = M0exp (−Q / RT )Vm .................... (2)
u = ϕuα + (1 −ϕ )uγ .......................... (7)
where M0 is the pre-exponential factor, Q is the activation
energy, and R is the gas constant. Defining p(ϕ) = ϕ + ke(1 – ϕ), Eq. (7) can be simplified
as u = uαp(ϕ). Thus uα in Eq. (6) is calculated by
With the assumption of dilute solution, the following
uα = u / p (ϕ ) .............................. (8)

3.3. Carbon Diffusion

During α – γ phase transformation, solute partition between
ferrite and austenite at the α/γ interface is considered
according to uγ = keuα. According to the Fick’s second law,
the governing equation of carbon diffusion in 2D can be

written as

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ISIJ International, Vol. 54 (2014), No. 2

∂u = ∂ ⋅ ⎡⎣⎢D(ϕ) ⋅ ∂(u / p(ϕ)) ⎤ + ∂ ⋅ ⎡⎢D(ϕ) ⋅ ∂(u / p(ϕ)) ⎤ solution sequences of the different processes are described
∂t ∂x ∂x ⎦⎥ ∂y ⎣ ⎥ below.
∂y ⎦
I. The solution sequence for the simulation of isothermal
.......................................... (9) heating process is as follows:

where D(ϕ) is the diffusion coefficient associated with the (1) Initialize the simulation system with domain length,
fraction of ferrite. Similar to the calculation of mean con- grid spacing, initial uniform carbon composition field, and
ferrite grains with different orientations.
centration u, D(ϕ) can be calculated by
(2) Set austenite nucleus at ferrite grain boundaries. The
D(ϕ) = ϕ Dα + ke (1 −ϕ ) Dγ .................. (10) grain index, phase fraction ϕ, and mean concentration u of
austenite are also initialized with the corresponding values.
where Dα and Dγ are carbon diffusivities in the α and γ phases,
respectively. Equation (9) is derived based on the mass con- (3) Calculate the increment of γ phase fraction by solving
servation and solute equilibrium at the α/γ interface. On the Eq. (6) and Δϕ = GγvnΔt/Δx. When ϕ =0, the cell is trans-
right-hand side of Eq. (9), an equivalent concentration u/p(ϕ) formed to austenite.
is used, which ensures the solute equilibrium at the α/γ
interface. On the other hand, the derivative of mean concen- (4) Calculate carbon diffusion in both α and γ phases by
tration with respect to time, ∂u/t, on the left-hand side of Eq. solving Eqs. (9) and (10).
(9) includes the effect of solute partition due to α – γ tran-
sition. Thus, Eq. (9) facilitates the problem of discontinuous (5) Time step from Step (3) until the end of the simula-
carbon concentrations and solute partition at the α/γ inter- tion.
face in a straightforward manner, and the entire domain can
be treated as a single phase for solute transport calculation. II. The solution sequence for the simulation of continuous
Equation (9) is solved using an explicit finite difference cooling process is as follows:
scheme with a time step determined by the carbon diffusivity
in ferrite, and the zero-flux boundary condition is adopted. (1) Initialize the simulation system with the microstruc-
ture and carbon concentration field calculated at the end of
3.4. Numerical Solution Sequence the isothermal heating process.

The CA algorithm used in the present work is described (2) Calculate the increment of the α phase fraction by
solving Eq. (6) and Δϕ = GαvnΔt/Δx. When ϕ =1, the cell is
as follows. The simulation system is divided into a uniform transformed to the ferrite phase.

orthogonal arrangement of cells. Each cell has the following (3) Calculate carbon diffusion in both the α and γ phases
variables: (1) grain orientation, I; (2) α phase fraction, ϕ by solving Eqs. (9) and (10) using the temperature depen-
(the cell represents α phase or γ phase, when ϕ =1 or ϕ =0, dent carbon diffusivities in α and γ phases.
respectively); (3) the symbol ϕint=1 indicates the cells at the
ferrite/austenite (α/γ) interface; and (4) mean carbon con- (4) Time step from Step (2) until the end of the simula-
centration, u. The velocity of interface migration is calculat- tion.

ed using Eq. (6). It is assumed that the kinetics of interface III. The solution sequence for the simulation of tempering
process is as follows:
migration along grain boundaries is 2.5 times faster. Then,
(1) Initialize the simulation system with the microstruc-
the increment of new phase fraction, Δfnew, is evaluated by ture and carbon concentration field calculated at the end of
the continuous cooling process.
Δfnew = GnewvnΔt / Δx ...................... (11)
(2) Repeat calculating carbon diffusion in the domain by
solving Eqs. (9) and (10) until the end of the simulation.

The physical property parameters used in the simulations
are listed in Table 2.

where Gnew is a geometrical factor that is introduced to elim- 4. Results and Discussion
inate the artificial anisotropy caused by the CA square cell.
4.1. Isothermal Heating Process
Gnew is related to the states of neighboring cells and defined In the present work, the calculation domain consists of a
by
400×316 square grid with Δx=0.3 μm. The domain size
∑ ∑Gnew= ⎡ 1 ⎛ 4 SmI + 1 S4 II ⎞⎤ equals the size of SEM micrographs. At the beginning of the
min ⎢1, 3 ⎜⎝ m =1 2 ⎠⎟⎥⎦ , simulation, the computational domain is initialized with a
⎣ m=1 m uniform carbon concentration u=u0 and ferrite phase (ϕ =1,
and ϕint=0) with various grain orientations, as shown in Fig. 4.
⎧0( fnew < 1) .... (12)
⎨⎩1 fnew
S I , S II (= = 1)

where S I and SII indicate the state of the nearest neighbor Table 2. Physical property parameters.
cells and the second-nearest neighbor cells, respectively,
and fnew is the new phase fraction of neighbor cells. Symbol Definition and Unit Value

At the end of each time step, the fraction of the new phase R Gas constant (J mol–1 K–1) 8.314

of each interface cell is updated. When the fraction of new

phase equals one, the interface cell transforms its state from Dα Carbon diffusivity in ferrite (m2 s) 2.2×10–4 exp(–122 500/RT)9)
interface to the new phase. This transformed new phase cell
Dγ Carbon diffusivity in austenite (m2 s) 1.5×10–5 exp(–142 100/RT)9)
in turn captures a set of its neighbors of the parent phase to
Vm Mole volume (m3 mol–1) 7.2×10–6
be the new α/γ interface cells. The phase transformation will
thus continue in the next time step. Then, carbon redistribu- Meff Interface mobility (m4 J–1 s–1) 0.035exp(–14 700/RT)Vm19)
tion and diffusion is calculated by solving Eq. (9). Since the
simulations involve different heat treatment processes, the TM Allotropic transformation 1 168.64
temperature, A3(K)

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ISIJ International, Vol. 54 (2014), No. 2

Since the isothermal holding temperature of 760°C is transformation might take place mainly in this period. After
higher than the eutectoid temperature (686°C), during the
isothermal heating process, ferrite-to-austenite phase trans- holding about 10 s, the γ fraction approaches gradually the
formation happens. According to the experimental observa- equilibrium value of 0.12.
tions, the nucleation of austenite mostly appears at the
ferrite grain boundaries. The number of the austenite nuclei The evolution of the carbon concentration field during
is also determined based on the experimental metallographs.
With holding time increasing, the α phase transforms to γ isothermal heating at 760°C is shown in Fig. 7. Different
phase accompanied with carbon diffusion, and austenite colors represent different carbon concentration levels, and
grains grow until the fraction of γ phase reaches the equi-
librium value (about 0.12). Figure 5 presents a comparison the black lines represent the boundaries of different grains.
of the simulated and experimental microstructures after iso-
thermal holding at 760°C for 300 s. Figure 5(a) shows the Since the carbon concentration in the γ phase is about two
simulated morphology in which the deep and light colors orders of magnitude larger than that in the α phases, the col-
indicate ferrite and austenite phases, respectively, and the or levels in Figs. 7(a)–7(c) are adjusted to show clearly the
black lines represent the boundaries of different grains. Fig-
ure 5(b) shows the SEM image obtained by quenching the evolution of the carbon concentration field in the α phase,
sample after heating at 760°C for 300 s to retain the austen- and the maximum values of the color legends are lower than
ite morphology. It is found that the simulated microstructure
compared reasonably well with the experimental observation. the real concentration in the γ phase. However, the color leg-
end of Fig. 7(d) depicts the actual carbon concentration lev-
Figure 6 shows the simulated austenite phase fraction
varying with time during the isothermal heating at 760°C. els in both the γ and α phases. It can be seen that the carbon
As shown, at the early stage of heating, the increment of γ concentration in the α phase is much lower than that in the
fraction is quite limited. Then the γ fraction increases γ phase. In the early stage of isothermal heating, austenite
evidently within 0.1 s–10 s, which implies that the α→γ nucleates at the ferrite grain boundaries. The area near the

γ/α interface exhibits the lower carbon concentration,
because of the solute partition between α and γ phases. As
α→γ phase transformation proceeds, more carbon atoms are
absorbed by the growing austenite grains. Driven by the car-

bon concentration gradients, carbon diffusion happens.

After the fraction of γ phase reaches the equilibrium value,
the phase transformation is completed, while carbon diffu-

Fig. 4. The initialized ferrite grains with different grain orienta- Fig. 6. Simulated austenite phase fraction varying with time during
tions. (Online version in color.) the isothermal heating at 760°C. (Online version in color.)

Fig. 5. Comparison of microstructures after isothermal heating at 760°C for 300 s: (a) simulation and (b) SEM image
obtained by quenching the sample after heating at 760°C for 300 s to retain the austenite morphology. (Online ver-
sion in color.)

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ISIJ International, Vol. 54 (2014), No. 2

Fig. 7. Simulated evolution of carbon concentration field during isothermal heating at 760°C for: (a) 0.1 s; (b) 1 s; (c) 10 s;
and (d) 300 s. (Online version in color.)

Fig. 8. Simulated carbon distribution after isothermal heating at 760°C for 300 s and cooled down to the room temperature
with different cooling rates: (a) 2°C/s and (b) 5°C/s. (Online version in color.)

sion continues, resulting in gradually uniform carbon con- 4.2. Continuous Cooling Process
centration fields in both α and γ phases. The carbon concen- During the continuous cooling process, the temperature of
trations in both phases tend to be stabilized with the new
equilibrium values, as shown in Fig. 7(d). The minimum and the calculation domain decreases from 760°C to room tem-
maximum values in the legend of Fig. 7(d) correspond to the perature with different cooling rates of 2°C/s and 5°C/s.
equilibrium concentrations in the α phase and γ phases, During cooling, austenite-to-ferrite phase transformation,
respectively. accompanied with carbon diffusion, takes place. As
described in Section 3.1, the end temperatures ofγ →α phase

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ISIJ International, Vol. 54 (2014), No. 2

Fig. 9. Simulated carbon distribution after tempering at different temperatures for 20 min: (a) 200°C; (b) 300°C; and (c)
500°C. (Online version in color.)

transformation are chosen as 650°C and 600°C for the cool- The color legends in Fig. 9 are also set to be identical with
ing rates of 2°C/s and 5°C/s, respectively. When the temper- that of Fig. 7(d). It is noted that with tempering temperature
ature is cooled down below the end temperature of γ→α increasing, the uniformity of carbon distribution increases
phase transition, the retained austenite is assumed to be gradually. The maximum and minimum carbon concentra-
transformed to supersaturated ferrite through allotropic tions in Figs. 9(a)–9(c) are 3.2291 mol.% and 0.0966 mol.%,
transformation without considering the austenite to pearlite 2.9819 mol.% and 0.1794 mol.%, 0.2996 mol.% and
transformation. The diffusion of carbon continues to room 0.2960 mol.%, respectively. Comparing Figs. 9(a) with 8(b),
temperature with the temperature dependent carbon diffu- it can be found that the tempering at 200°C for 20 min has
sivity in the ferrite phase. less effect on the carbon concentration field. When the tem-
pering temperature is increased to 500°C, the carbon distri-
Figure 8 shows the simulated carbon distribution after bution becomes much more uniform as shown in Fig. 9(c)
isothermal heating at 760°C for 300 s and cooled down to that is very close to the one after isothermal heating at
room temperature with different cooling rates of 2°C/s and 760°C for 300 s and cooled with 2°C/s as shown in Fig. 8(a).
5°C/s. The legends in Fig. 8 are identical with that in Fig. This is due to the fact that the carbon diffusivity increases
7(d) for comparison. As shown, the carbon concentration with temperature. The relative high carbon diffusivity at
field for the case of 2°C/s is more uniform than that cooled 500°C results in a more sufficient carbon diffusion. As dis-
with 5°C/s. For the latter case, some regions at the bound- cussed in the previous section, sufficient carbon diffusion is
aries have relative higher carbon concentrations as shown in beneficial to forming high concentration Cottrell atmo-
Fig. 8(b). Apparently, these regions should be the original spheres, leading to a more obvious yield platform and
austenite grains before cooling. The maximum and increased yield strength. Based on this mechanism, it can be
minimum carbon concentrations are 0.3001 mol.% and reasonably explained that the yield platform becomes grad-
0.2879 mol.% for the 2°C/s cooling rate (Fig. 8(a)), while ually evident and yield strength increases with increasing
they are 3.2367 mol.% and 0.0946 mol.% for the 5°C/s cool- tempering temperature as shown in Fig. 2 and Table 1.
ing rate (Fig. 8(b)), respectively. It is evident that the cool-
ing process with lower cooling rate provides more sufficient 5. Conclusions
time for carbon diffusion, which is beneficial for carbon
diffusion to the locations of dislocations to form high con- (1) A 2D CA model is proposed to simulate the ferrite-
centration Cottrell atmospheres. It is well known that high to-austenite phase transformation. The model involves the
concentration Cottrell atmospheres have a crucial impor- preferential nucleation sites of austenite, the driving force of
tance on pinning dislocations, leading to a obvious yield phase transformation, carbon solute partition at the ferrite/
platform in the stress-strain curve, and an increased yield austenite interface, and carbon diffusion in both the ferrite
strength.20) Accordingly, it is understandable that the sample and austenite phases. The present model is able to simulate
cooled by sand cooling (2°C/s) has a more obvious yield ferrite-to-austenite phase transition during isothermal
platform and a higher yield strength compared with the one heating in the temperature range of ferrite-austenite phase
by air cooling (5°C/s), as shown in Fig. 1 and Table 1. coexistence, austenite-to-ferrite phase transition during con-
tinuous cooling, and carbon diffusion during tempering pro-
4.3. Tempering Process cesses of binary low-carbon steels.
The sample cooled by air cooling (5°C/s) undergoes an
(2) During isothermal heating at 760°C that is within
additional tempering process at different temperatures. the temperature range of ferrite-austenite phase coexistence,
Since the tempering temperatures are below the eutectoid ferrite-to-austenite phase transition takes place accompanied
temperature, there is no α-γ phase transformation. As shown with carbon redistribution. Austenite nucleates at the bound-
in Fig. 8(b), at the end of the continuous cooling process aries of ferrite grains. The growing austenite absorbs carbon
with the cooling rate of 5°C/s, the carbon concentration field atoms from ferrite at the α/γ interface, resulting in carbon
is still quite uneven. Therefore, carbon diffusion occurs dur- concentration gradients and then carbon diffusion in both
ing the tempering process. phases. After holding for 300 s, the carbon concentrations
in both phases are nearly uniform and stabilized with the
Figure 9 presents the simulated carbon concentration
fields after tempering at different temperatures for 20 min.

© 2014 ISIJ 428

ISIJ International, Vol. 54 (2014), No. 2

corresponding equilibrium values. The simulated micro- Corporate Technology Center, USA, and NSFC (Grant No.
structure compared reasonably well with that obtained 51371051).
experimentally.
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