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Available online at www.sciencedirect.com
Acta Materialia 61 (2013) 2884–2894
www.elsevier.com/locate/actamat
Development of a kinetic model for bainitic isothermal
transformation in transformation-induced plasticity steels
S. Li a, R. Zhu a, I. Karaman a,b, R. Arro´yave a,b,⇑
b
a
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
Materials Science and Engineering Program, Texas A&M University, College Station, TX 77843, USA
Received 25 October 2012; received in revised form 17 January 2013; accepted 21 January 2013
Available online 27 February 2013
Abstract
In this work, we modify existing models to simulate the kinetics of bainitic transformation during the bainitic isothermal transformation (BIT) stage of a typical two-stage heat treatment – BIT is preceded by an intercritical annealing treatment – for TRIP steels. This
effort is motivated by experiments performed in a conventional TRIP steel alloy (Fe–0.32C–1.42Mn–1.56Si) that suggest that thermodynamics alone are not sufficient to predict the amount of retained austenite after BIT. The model implemented in this work considers
the non-homogeneous distribution of carbon – resulting from finite carbon diffusion rates – within the retained austenite during bainitic
transformation. This non-homogeneous distribution is responsible for average austenite carbon enrichments beyond the so-called T0 line,
the temperature at which the chemical driving force for the bainitic transformation is exhausted. In order to attain good agreement with
experiments, the existence of carbon-rich austenite films adjacent to bainitic ferrite plates is posited. The presence of this austenite film is
motivated by earlier experimental work published by other groups in the past decade. The model is compared with experimental results
and good qualitative agreement is found.
Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: TRIP steels; Thermodynamics; Kinetics; Bainitic transformation; Retained austenite
1. Introduction
1.1. TRIP-assisted steels
Low-alloy, high-ductility transformation-induced plasticity (TRIP)-assisted steels constitute a possible costeffective strategy for vehicle weight reduction, which in turn
can significantly improve the fuel economy of the vehicle
fleet. Many works have shown TRIP-assisted steels to have
outstanding mechanical performance compared to other
low-cost steel alloys [1–3]. TRIP-assisted steels (with Fe–
C–Mn–Si as their main constituents) have a multi-phase
microstructure consisting of ferrite, bainite and martensite
as well as finite amounts of retained austenite (RA). In fact,
one of the key characteristics of these low-alloy steels is the
presence of the stabilized (retained) austenite phase, which
⇑ Corresponding author. Tel.: +1 979 845 5416.
E-mail address: [email protected] (R. Arro´yave).
contributes to the enhancement of the overall ductility
through its martensitic transformation during deformation.
Since carbon is a very efficient austenite stabilizer, carbon
enrichment of RA ðwcC Þ is a effective way to control the
stability of austenite while keeping the amounts of other
alloying elements at a minimum. With respect to the other
phases, the proper control of wcC can improve the mechanical properties – especially elongation [3–5] – of these steels.
In addition to the control of the stability of RA, optimal
design of multi-phase TRIP steels requires the accurate
control of the volume fraction (and microstructural features) of the different constituent phases (ferrite, bainite
and martensite). This can in turn be achieved through the
careful design of appropriate heat treatments.
1.2. Two-stage heat treatment
Although there are many possible treatment strategies to
maximize the stability of the RA, as well as to control the
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.actamat.2013.01.032
S. Li et al. / Acta Materialia 61 (2013) 2884–2894
amount of other phases necessary to optimize the mechanical response of TRIP-assisted steels, the most common
approach is a heat treatment that consists of two stages
[6,7,4]: first, an isothermal hold within the ferrite + austenite two-phase field (intercritical annealing, IA), followed by
a rapid quench to a lower temperature at which the IA-austenite (partially) transforms isothermally into bainite (bainitic isothermal treatment, BIT), as shown in Fig. 1.
During IA, the initial pearlitic microstructure is dissolved, which results in a two-phase microstructure consisting of austenite and ferrite. The microstructural change
during IA has two main functions: first, the formation of
ferrite contributes to the overall ductility of the alloy; second, the partitioning of carbon from IA-ferrite to IA-austenite (ferrite has almost no C solubility) stabilizes the
latter against martensitic transformation upon quenching
to lower temperatures. After IA, the alloy is quenched
and held at a lower temperature (TBIT) to induce the (partial) isothermal transformation of IA-austenite into bainite.
The formation of almost C-free bainitic ferrite results in
carbon rejection into the remaining austenite – provided
carbide formation is suppressed or retarded – which contributes to the further stabilization of austenite upon
quenching to room temperature. This is a necessary condition in order to take advantage of the TRIP effect and the
enhanced ductility it provides.
Due to the importance of the BIT treatment in dictating
the carbon enrichment in austenite, considerable effort has
been dedicated to the understanding of the thermodynamics and kinetics of this important phase transformation
[8–11]. Although there are still many aspects of the
transformation subject to debate, the incomplete nature
of bainitic transformation in many alloys over broad temperature ranges is a phenomenon that has been widely
observed by many groups for a long time, so this phenomenon can be considered as given. On the other hand, the
physical reason for the interruption of this phase transformation is still a controversial matter.
Recently, Aaronson and collaborators [11] examined
different theories as well as experimental data, and concluded that a possible explanation for the incomplete nature of the BIT reaction involves the cessation of growth
2885
due to a coupled-solute drag effect accentuated by overlapping carbon diffusion fields associated with nearby ferrite
crystals [11]. On the other hand, Bhadeshia and collaborators [12,13] explained the incomplete nature of the BIT
reaction as a consequence of the partitionless nature of
the austenite–ferrite transformation, followed by partitioning of carbon into the residual austenite and the subsequent
exhaustion of the chemical driving force for the (partitionless) face-centered cubic (fcc) ! body-centered cubic (bcc)
phase transformation.
A prediction that follows from Bhadeshia’s argument is
a definite limit in the carbon enrichment of the remaining
austenite during bainitic transformation. In fact, using
thermodynamic arguments, it is suggested that one must
take the elastic energy barrier for the growth of bainite
plates into account. Bhadeshia estimated the barrier to be
about 400 J mol1 between austenite and ferrite [13]. This
extra energy barrier leads to a further constraint in the
maximum carbon enrichment in austenite, and the locus
of this composition at different BIT temperatures is usually
denoted as T 00 . Beyond this critical enrichment, the transformation cannot continue. From comparisons between
experiments and thermodynamic models, because of the
non-homogeneous C-distribution, Chang and Bhadeshia
[14] also proposed that the maximum carbon enrichment
of austenite during BIT occurs when the carbon composition of austenite reaches the so-called T0 point, at which
the Gibbs energies of ferrite and austenite are equal. Over
the past few decades, many experiments have actually
shown that the level of enrichment of carbon in austenite
at the end of the (incomplete) BIT tends to be close to
the T0 or T 00 curves (see e.g. Ref. [11]).
Moreover, very recent atom probe tomography experiments by Caballero et al. [15–18] on so-called “slow bainite” show that bainitic ferrite is indeed supersaturated
with carbon – or at least has a much higher C concentration than would be expect if the bainite transformation
occurred through diffusional processes – at the early stages
of the transformation, providing further support for the
displacive (partitionless) nature of bainitic transformation,
at least in ultraslow bainite. It is reported that after
1273 K/15 min and 473 K/240 h the carbon content is as
high as 8–12 at.% (when wt.% is about 1.8–2.8), which is
a composition that lies between T 00 and paraequilibrium
Ae3 as the austenite film thickness (tA) is about 10–
50 nm; when tA is larger than 50 nm, the carbon content
is about 5–8 at.% (about 1.1–1.8 wt.%) [19]. These works
indicate that, within the
thicker austenite film, the carbon
T0
content is closer to wC0 . In the present work, the analysis
of the BIT process assumes that the transformation is partionless and that the thermodynamic arguments for the
interruption of BIT (i. e. T 00 curve) are valid.
1.3. Motivation for this work
Fig. 1. Schematic of the two-step heat treatment in TRIP-assisted steels.
IA: inter-critical annealing; BIT: bainite isothermal transformation; a:
ferrite; c: austenite; M: martensite; B: bainitic ferrite.
As mentioned above, the fact that the driving force for
bainitic transformation can be exhausted at some point
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S. Li et al. / Acta Materialia 61 (2013) 2884–2894
during the BIT process [13] has been validated through the
observation of incomplete bainitic transformations in socalled slow bainite steels [15,16,19]. However, investigations by the present authors and elsewhere [20–22] suggest
that thermodynamics alone cannot be used to understand
(and design) BIT treatments for TRIP and other classes
of steels. Our group has studied the low-alloy Fe–0.32C–
1.42Mn–1.56Si in some detail, and we have used thermodynamic arguments to interpret the observed austenite
carbon enrichment and resulting phase constitution [23].
Fig. 2 shows a thermodynamic analysis of a two-stage
heat treatment for Fe–0.32C–1.42Mn–1.56Si subjected to
a fixed TIA and different TBITs. The figure shows a vertical
section of the Fe–C–Mn–Si phase diagram with fixed Mn
and Si contents and varying amounts of C. The figure also
shows the evolution of the carbon enrichment in austenite
during the different stages of the two-stage heat treatment.
From experiments, it has been observed that wc;IA
is
C
between 0.53 and 0.63 wt.% after 1045 K, 2 h of IA treatment [23]. The figure shows the T 00 curve used to predict
the maximum carbon enrichment of the RA after BIT at
three different TBITs (613, 643 and 693 K). Assuming reasonably accurate prediction of T 00 and T0 using the TCFE6
V6.2 database for the present alloy, the maximum carbon
enrichment in RA after BIT would correspond to the T 00
curve. However, experimental measurements of the carbon
content in RA suggest a greater enrichment than expected.
This observation is not really new; in fact, many other
works have consistently
found austenite carbon enrichT0
ments higher than wC0 after BIT [24–26]. Although the
issue has not been completely resolved, this deviation from
the thermodynamic limit for BIT has been explained in
Fig. 2. The phase diagram for designing Fe–0.32C–1.42Mn–1.56Si TRIPassisted steel microstructures. The blue curves are for the IA treatment; the
red curve is for T 00 , which is the thermodynamic limitation of the BIT
treatment. The experimental results are the carbon contents in RA after
1045 K, 2 h IA and 613, 643, and 693 K BIT heat treatments. (For
interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
terms of the competition between the austenite–ferrite
phase transformation and the carbon diffusion into the
remaining austenite [27,8].
The biggest discrepancy with expectations, however, is
shown in Fig. 3 [23]. In this figure, we show the calculated
volume fraction of RA after quenching from BIT, taking
into account the stability of RA after BIT. The calculations
considered both the T0 and T 00 limits for carbon enrichment. According to the thermodynamic analysis, increasing
TBIT would lead to a monotonous decrease in the fraction
of RA after BIT + quenching. The reason for this is clear
(see Fig. 2): as TBIT increases, the maximum carbon enrichment in austenite after BIT decreases as T0 has a negative
slope. Based on this simple analysis, and assuming athermal martensitic transformation, one would expect higher
BIT temperatures to result in lower volume fractions of
RA. On the other hand, experiments suggest that the fraction of RA actually increases with TBIT. This clear qualitative disagreement between the thermodynamic analysis and
observations suggests that a deeper analysis of not only the
thermodynamics but also the kinetics of BIT is necessary.
1.4. Description of the present paper
In this work, we carry out a detailed analysis of the thermodynamics and kinetics of the BIT stage and apply it to a
conventional TRIP steel alloy with the composition of Fe–
0.32C–1.42Mn–1.56Si. The present model is essentially a
modification of the model described in Refs. [28,22]. The
model’s objective is not to achieve a quantitatively accurate
prediction of the transformation times for BIT; rather, the
model is focused on investigating possible relaxations of
the thermodynamic limits for the BIT by taking into
account the competition between the austenite–ferrite
phase transformation during BIT and the rate of carbon
rejection from the newly formed bainite into the remaining
Fig. 3. The volume fraction of RA in an (Fe–0.32C–1.42Mn–1.56Si) alloy
after constant TIA of 1045 K and different BIT temperatures (the solid
curves are the theoretical predictions [23]).
S. Li et al. / Acta Materialia 61 (2013) 2884–2894
austenite. One of the significant assumptions in this work is
that no carbide (including cementite) is formed during the
entire BIT process due to the relatively high concentration
of Si in the alloy [6]. We describe the model in detail and
then propose a new upper bound for the carbon saturation
in RA as well as the maximum amount of RA after
quenching to room temperature for a given set of TIA
and TBIT.
2. Modeling the bainite transformation
Over the years, several models have been developed to
describe the kinetics of the BIT transformation. As
expected, the main characteristics of the models are defined
by the assumed nature of bainitic transformation. In models that accept the displacive nature of bainitic transformation, it is assumed that bainitic ferrite grows with a very
high supersaturation of carbon and that this carbon is
eventually rejected into the residual austenite. Since bainitic transformation usually occurs at rather low temperatures, it is often assumed that the substitutional elements
are not mobile enough [29,18,19]. This means that, to a first
approximation, the thermodynamic analysis of the transformation can be done assuming that paraequilibrium
conditions prevail [29]. As mentioned above, the transformation stops once the carbon concentration in the residual
austenite reaches the critical value (corresponding to the
T0 or T 00 curve) at which the driving force for the transformation is exhausted.
Although in many steel alloys it has been observed that
the austenite carbon enrichment is much closer to T0 or T 00
than to Ae3, there are still discrepancies, particularly with
regard to the observed carbon enrichment in austenite
[24,25]. This is atributed to the competition between phase
transformation and carbon diffusion [27].
2.1. Description of the model used in this work
Despite the complications in understanding bainitic
transformation, many phenomenological models with
empirical parameters have been proposed to simulate its
kinetics [9,30–32]. These models have been able to successfully reproduce the incomplete nature of the transformation.
In this work, we use the model developed by Bhadeshia
[13] and further modified by Gaude-Fugarolas and Jacques
[28,22] to describe the kinetics of bainitic transformation.
This model assumes that the nature of the transformation
from austenite to bainitic ferrite is displacive (partionless)
in nature and thus the bainitic ferrite is supersaturated with
carbon during the early stages of the transformation
[28,22,33,34,10].
The model further assumes that, at the start of the BIT
reaction, bainite nucleates at the post-IA-austenite grain
boundary. During this so-called primary nucleation, the
bainite subunit grows under partitionless conditions and
the carbon content of the bainitic ferrite is much higher
than what could be expected under paraequilibrium
2887
conditions [19]. On the other hand, the model assumes that
the incipient formation of a bainitic ferrite nucleus occurs
via partitioning. As soon as the primary bainite subunit
forms, the excess carbon in bainitic ferrite starts being
rejected, whilst further nucleation of bainite subunits
occurs simultaneously in an autocatalytic fashion (see
Fig. 4). The model assumes that the primary–secondary
nucleation of bainite subunits occurs during most of bainitic transformation period [22].
The present model assumes that two distinct conditions
must be satisfied in order for bainitic transformation to
occur. The formation of an incipient ferrite nucleus
requires a so-called nucleation barrier [13] to be overcome.
For this to happen, the maximum driving force for ferrite
nucleation, DGM, must exceed this nucleation barrier.
According to Bhadeshia and co-workers [35,36], the maximum driving force (J mol1) can be estimated using:
DGM ¼ RTln a waFe =aðwcFe Þ
ð1Þ
a
where a wFe corresponds to the activity of Fe in ferrite under paraequilibrium conditions and aðwcFe Þ corresponds to
the chemical activity of Fe in austenite obtained through
a parallel tangent construction. R is the gas constant and
T is the heat treatment temperature.
Although the calculation of the driving force for the
nucleation of bainitic ferrite is apparently at odds with
the assumption of a partitionless transformation, it should
be noted that the model assumes that the nucleation of the
bainitic ferrite occurs in a partitional manner, although the
subsequent growth of the bainite subunit is partitionless. It
should also be noted that this assumption can be relaxed
and instead it can be assumed that the nucleus actually
forms in a partitionless manner. The end result is then only
reflected in the time required for the completion of the BIT
transformation, as has been verified by unpublished work
by the present authors.
One of the key quantities that defines the beginning of
the nucleation process is the actual barrier for bainite
Fig. 4. Schematic illustration of bainitic transformation; the primary
nucleation occurs at the austenite grain boundary and autocatalytic
(secondary) nucleation can be observed on top of the primary bainite
subunit [22].
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S. Li et al. / Acta Materialia 61 (2013) 2884–2894
nucleation. Bhadeshia [13] estimated this quantity based on
a vast database of experimental data for the bainite start
temperature as well as on existing thermodynamic models.
This quantity thus depends on the thermodynamic database used to analyze the experiments. Using the published
experimental results [37,38] and the Fe-based thermodynamic database (TCFE6 version 6.2), the nucleation energy
(J mol1) is given by the following expression:
DGN ¼ 4:736T 4769
ð2Þ
where T corresponds to the BIT temperature in kelvin.
One of the key characteristics of this model is that it is
assumed that the primary and secondary nucleation of
bainitic ferrite continues as long as the chemical driving
force for the partitionless transformation of austenite into
ferrite overcomes the so-called barrier for nucleation of
bainite [13]. Following Jacques’s model [22], the transformation criterion (F) is presented as:
(
tanh DGMRTGN ; if ðDGM GN Þ < 0
F¼
ð3Þ
0;
if ðDGM GN Þ P 0
DGM is the maximum driving force calculated using Eq. (1)
and GN is given by Eq. (2) [13].
As mentioned above, the primary and secondary (autocatalytic) nucleation of bainite subunits dominates most of
the transformation process. The primary nucleation starts
at the austenite grain boundary and the number of nucleation events (dIp) during the time interval ds is [22]:
dI p ¼ N 0 sc m F ds
ð4Þ
where N0 is the surface density of the potential nucleation
sites, which is estimated as 2 104 nuclei m2; m is the attempt frequency (=kB T/⁄, kB being the Boltzmann constant); and sc is the remaining austenite grain boundary
area (m2), which is a function of the mean lineal intercept
measure of the austenite grain size ðL; mÞ and the volume
fraction of the residual austenite (vc) [22,39]:
sc ¼
2 23
vc
L
ð5Þ
In this model, it is assumed that the bainite subunit is
lenticular in shape and grows as soon as it forms. Experimental observations report that the aspect ratio of the bainite subunit is about 0.025 [40] and that the thickness can
be described by the empirical formula tB = 2.0 107
(T 528)/150 (m) [39]. The volume of the bainite subunit (m3) is then:
3
T 528
2
18
ð6Þ
V UB ¼ ð20tB Þ ptB ¼ 3:2 10 p 150
More subunits may form on top of these primary bainite
plates through autocatalytic nucleation. The autocatalytic
nucleation rate (dIa/d s) is related to the number of the
primary nucleation (Ip), the diameter of the grain (Dc),
the length of a single bainite subunit (lB) and the effective
driving force. The process is formulated as:
dI a ¼ ba I p 2Dc
F ds
p lB
ð7Þ
where ba is the fitting parameter and is reported as 1.5 [22].
Therefore, the increase of the bainite volume (m3) is
4Gc!a
dV B ¼ V c ðI P þ I a Þ V UB exp ds
ð8Þ
RT
4Gc!a is the diffusionless driving force that enables austenite to transform into bainite; Vc is the volume fraction
of the unstable residual austenite against bainitic transformation. Before the completion of bainitic transformation,
the austenite adjacent to bainitic ferrite is enriched with
carbon – this is the so-called inter-bainitic austenite film
[14]. Because of the high carbon content, the austenite film
is stable against bainitic transformation and this austenite
volume fraction is excluded from Vc in Eq. (8) (see below
for more detailed discussion).
2.2. The formation of inter-bainite C-enriched austenite film
Following the nucleation and (almost instantaneous)
growth of bainite subunits, carbon is rejected from the
newly formed bainitic ferrite and diffuses into the surrounding residual austenite. As the transformation continues, the residual austenite becomes further enriched with
carbon, exhausting the driving force for the (partitionless)
growth and reducing the transformation rate.
If the carbon diffusion in austenite is extremely fast and
without any constrain, the carbon rejected from the forming bainite subunits would be distributed almost instantaneously throughout the remaining austenite. In this case,
it is to be expected that the transformation
stop
would
when the carbon content reaches the T 00
T0
wC0
line, as
shown in Fig. 5a). On the other hand, if the phase transformation is much faster than the diffusion of carbon out of
the bainitic ferrite and into the residual austenite, it is
expected that the carbon would accumulate close to the
bainite–austenite interface, as shown in Fig. 5b. In this
case, the residual austenite beyond the bainite subunit
would be under-enriched with carbon and the transformation could proceed to completion. In this case, with very
limited carbon solubility, the carbon content in the RA
would be expected to be as high as wCpc , that is, it would
approach the paraequilibrium composition on austenite
side.
In a real system, we would expect different mechanisms
to contribute to the non-homogeneous distribution of carbon outside the bainitic ferrite. First, after moving across
the ferrite/austenite interface, the carbon distribution in
the residual austenite can be characterized by Fick’s second
law. If the diffusion rate is not infinite, an austenite region
highly enriched with carbon would be expected to be found
adjacent to the bainite subunit [14,41]. This so-called
enriched austenite film prevents the further transformation
of austenite into bainite in the region immediately adjacent
to the already formed bainite subunit. On the other hand,
S. Li et al. / Acta Materialia 61 (2013) 2884–2894
2889
less) fcc ! bcc transformation has the “wrong” sign and
the transformation cannot proceed. Beyond this
excluded region, it is assumed that bainite nucleates
as
T0
soon as the local carbon content is 0 below wC0 .
T
5. The carbon content higher than wC0 will be constrained
between bainite plates, and the rest of the carbon diffusing out of bainitic ferrite will go to the non-transforming
austenite.
As shown in Fig. 6, the carbon content in austenite film
is assumed to be distributed according to the assumptions
above. The carbon distribution in the film can be
approached by the analytical solution [14,42]:
x
C Þerfc pffiffiffiffiffi
wC ðxÞ ¼ wC þ ðwCpc w
ð9Þ
2 Dt
C and wCpc are the carbon content (wt.%) in
where wC ; w
austenite film, the average carbon content in residual
austenite and the paraequilibrium carbon content in the
austenite side, respectively. D is the average carbon diffusion coefficient. According to the conservation of carbon,
the increased carbon in austenite when the bainite is being
decarburized (td) can be predicted as:
Z
Z
4wC dx ¼
4wC dx
ð10Þ
austenite
bainite
The carbon distribution in austenite at td can be predicted
using Eqs. (9) and (10). Following the postulates above, the
thickness of the austenite film (as Fig. 6) can be obtained.
Fig. 5. The schematic diagrams of two different conditions for the rate of
bainitic transformation relative to carbon diffusion into RA: the carbon
diffusion is (a) fast and (b) slow compared to bainitic transformation.
in regions away from this enriched austenite film, the driving force for the transformation remains high. The
enriched austenite films then act effectively as a carbon
sink, which allows the transformation to progress even
beyond the T 00 limit. The overall carbon content
in the
T0
RA would then be expected to lie between wC0 and wpc
C ,
as has been shown by many experiments [14,15,25].
In order to predict the maximum carbon content in RA,
taking into account finite carbon diffusion rates, several circumstances are postulated:
1. Carbon diffuses only in the x direction. Because the
aspect ratio of bainite plate is small (about 1/40), carbon
diffusion in the radial direction can be ignored.
2. Carbon diffusion across the bainitic ferrite–austenite
interface is estimated under paraequilibrium conditions.
3. Compared to austenite, the chemical potential of carbon
in bainitic ferrite is high. This means carbon tends to
diffuse from bainitic ferrite toward austenite. More
importantly, bainitic ferrite forms a structural barrier
that constrains carbon diffusion.
4. In order to have a sufficient driving force for phase
transition,
the carbon content in austenite must be lower
T0
than wC0 . Otherwise, the driving force for the (partition-
2.3. Model parameters
In this work, a number of assumptions are made regarding the initial conditions, such as the size of the austenite
grains (10 lm). Also, to estimate the initial state of bainitic
transformation, full equilibrium after IA treatment is
Fig. 6. Schematic diagram of the proposed carbon distribution in
austenite film.
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S. Li et al. / Acta Materialia 61 (2013) 2884–2894
assumed. Several parameters are observed or fitted from
experimental results, such as L, N0 and ba.
Fig. 7 shows continuous cooling transformation (CCT)
diagrams for bainitic transformation with different parameters for the present model. As shown in the figure, the
transformation is faster with a larger number of initial
nucleation sites N0 or a larger autocatalytic nucleation
parameter ba. When considering the numerical model, it
should be understood that the selection of N0 affects the
prediction of the incubation time and ba decides the slope
of the curve. One important observation is that these two
parameters affect the transformation time but not the
transformation mechanism, and both the final volume fraction of bainite and the carbon enrichment in RA are the
same in all cases. This is to be expected because the transformation is controlled by the carbon enrichment in the
RA, as described in Eq. (9). It should be emphasized that
the purpose of this work is to identify the mechanism by
Fig. 7. CCT diagrams for bainitic transformation at 613 K with different
fitting parameters, N0 (nuclei m2) and ba; the inputs are the equilibrium
state of austenite at 1045 K; the variation of (a) carbon content in
austenite ðwcC Þ and (b) volume fraction of bainite (VfBai) with transformation time.
which the carbon enrichment in the RA exceeds the thermodynamic limit, rather than to predict an accurate time
scale for the transformation. Therefore, N0 and ba are
taken as 2 104 and 1.5 in this work.
3. Experimental details
Billets of the model TRIP-assisted steel with a nominal
composition of Fe–1.5Mn–1.5Si–0.3C in wt.% are cast into
the dimensions of 25.4 25.4 177.8 mm3. Inductively
coupled plasma atomic emission spectroscopy is conducted
on the as-cast materials to confirm that the actual composition is Fe–0.32C–1.42Mn–1.56Si. The as-cast billets are
first heated up to 1223 K for 1 h to austenitize the microstructure, then quickly processed non-isothermally using
equal channel angular processing (ECAP) at a rate of
25.4 mm s1, while the billets are at 1223 K and the ECAP
die is at 573 K. After ECAP, the billets are air cooled to
room temperature. The fast extrusion rate is selected to
maintain the temperature of the billets as close to 1223 K
as possible such that no phase transformation occurs during extrusion. Two passes are conducted using route C
(180° rotation between the passes along the longitudinal
axis of the billet) with the same extrusion conditions. The
purpose of such high-temperature ECAP is, similar to
hot rolling or forging, to eliminate the dendritic banding
in the microstructure inherited from casting and to achieve
a homogeneous microstructure with a relatively small
phase size. Route C is selected to obtain nearly equiaxed
fine grains [43,44].
Dog-bone-shaped tension samples with gage dimensions
of 8 3 1 mm3 are cut from the ECAP billets using electrical discharge machining (EDM). Samples are first sealed
in quartz tubes with partial pressure of argon, then are
exposed to IA treatment in a furnace at 1045 K for 2 h, followed by quenching in a salt bath for bainitic isothermal
transformation (BIT) at 613, 643 and 693 K for specific
durations (varying from 4 min to 1 h) to form carbides,
and finally quenched in water.
The microstructure of the heat-treated samples is
characterized using scanning electron microscopy (SEM),
X-ray diffraction (XRD) and wavelength-dispersive spectroscopy (WDS). The SEM samples are mechanically polished and etched using 2% Nital before examination. The
XRD samples are chemically polished in a solution of
14 ml of 40% HF, 100 ml of 30% H2O2 and 100 ml of
distilled water before the measurements. A Bruker-AXS
D8 X-ray diffractometer with Cu Ka radiation (wavelength = 0.15406 nm) is utilized to determine the volume
fraction and carbon content of RA following SAE SP453 [45]. A superconducting quantum interference device
(SQUID) magnetometer is also used to validate the volume
fraction of RA using the method reported in Ref. [46]. A
Cameca SX50 electron microprobe equipped with four
wavelength-dispersive X-ray spectrometers is utilized to
determine the Mn and Si concentrations in ferrite and
austenite (martensite at room temperature) on samples
S. Li et al. / Acta Materialia 61 (2013) 2884–2894
2891
Table 1
The chemical composition of the austenite after the same IA and different BIT heat treatments;
wcC1 and wcC2 are the carbon contents before and after BIT treatment, respectively.
Treatment
IA
BIT
wcMn
wcSi
wcC1
wcC2
A
613 K, 1 h
1.30
B
1045 K, 2 h
643 K, 15 min
1.48
1.49
0.53–0.63
1.22
C
693 K, 4 min
1.15
after IA treatments. Samples are mechanical polished without etching before the WDS examination.
4. Results
According to the previous analysis of the experimental
results,
the possible carbon content in the IA-austenite
c
wC;1 is between 0.53 and 0.63 wt.% [23]. Fig. 8 corresponds to the phase diagram for BIT treatment based on
equilibrium IA calculations and experimental
results for
c
the carbon content in IA-austenite wC;1 ¼ 0:53 wt:% .
With these two initial conditions (equilibrium vs. experimental measurements), the predicted TMs and T 00 are
slightly changed, but the kinetic limitations are different
by only about 0.07 wt.%. Here, TMs is calculated as the
temperature at which the free energy difference between
austenite and ferrite is sufficient to overcome the energy
barrier [47,23]. This means that a higher wcC;1 significantly
pushes bainitic transformation to a higher carbon content,
which increases the probability of bainite formation. This
also explains why, after IA, the non-equilibrium alloy can
result in a high volume fraction of RA even when the
Mn content is relatively low [23].
In the experiments, the composition of the RA is measured as shown in Table 1, which includes the inputs
Fig. 9. The volume fraction of RA after TIA = 1045 K and different TBIT;
(a) phase diagram for BIT; (b) the volume fraction of RA.
Fig. 8. Phase diagram for BIT treatment based on different conditions: the
equilibrium at 1045 K and the experimentally determined chemical
composition after IA at 1045 K for 2 h.
(wcMn and wcSi ) for calculating the phase diagram as
Fig. 9a. Two wcC;1 values, 0.53 (case 1) and 0.63 (case 2)
wt.%, are tested. After longer BIT treatments, the carbon
enrichment reaches beyond the thermodynamic limit T 00
towards, but not higher than, the prediction by the kinetic
model. Also, when TBIT is 693 K, the transformation is faster than the other two cases (613 and 643 K) and is actually
closer to the kinetic model at the end of BIT.
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S. Li et al. / Acta Materialia 61 (2013) 2884–2894
From Fig. 9a, in order to stabilize the austenite against
the martensitic transformation after BIT treatment, the
carbon content should be higher than 1.36 wt.%. If the carbon is insufficient, a certain amount of austenite will transform into martensite. The volume fraction of this
martensite is estimated by the Koistinen–Marburger relation [23]. The details of the three treatments are listed in
Table 1. The volume fraction of the RA is predicted and
compared to the experimental results as shown in
Fig. 9b. The experimental results are between the predictions of the thermodynamic and kinetic models. Furthermore, there are two or three turning points in each
kinetic result: 625 and 677 K in case 1 (the carbon composition of IA-austenite is assumed to be 0.53 wt.%) and 585,
693 and 709 K in case 2 (carbon composition of IA-austenite is assumed to be 0.63 wt.%). These are the critical temperatures for martensitic transformation before and after
BIT treatment. In case 1, austenite transforms into martensite below 625 K during quenching process after IA.
After BIT, the carbon enrichment is not high enough to
suppress martensitic transformation if TBIT is higher than
677 K (as Fig. 9a). This means austenite is not stable while
this alloy is cooling to room temperature. Within this temperature range, the increase in the RA with temperature is
due to the decrease in the maximum volume of bainite. The
same behavior can be seen in case 2, although the range
over which the volume fraction of RA increases with BIT
is wider. This makes sense as a greater carbon enrichment
makes austenite more stable against martensitic
transformation.
Here we should stress that the qualitative agreement
with experiments is non-trivial. This is so because of the
presence of the inter-bainitic carbon-rich austenite film.
If, on the other hand, carbon diffused much faster than
the rate of bainitic transformation, we could expect a
monotonic behavior in which an increase the BIT temperature would result in a decrease in the carbon enrichment
of austenite, with a corresponding decrease in the volume
fraction of RA after quenching from BIT. Fig. 9b shows
that the experimental volume fraction of the RA is closer
to the thermodynamic limit at low TBIT, while the experiments are closer to the “kinetic” limit at higher TBIT. These
results may not be conclusive, but we suggest that they can
be attributed to the size of the bainite plate being proportional to T3. At higher temperatures, the amount of carbon
rejected from bainite that needs to diffuse within the RA is
large, even if the carbon diffusion is rapid. On the other
hand, at low temperatures, the amount of carbon rejected
into austenite every time a bainite plate is formed is significantly less. In this case, the carbon distribution in the RA
is expected to be more homogeneous and therefore the
transformation is expected to stop
once the average carbon
T0
content in austenite reaches wC0 .
Fig. 10 contains the evolution of the phase constitution
after BIT and the complete heat treatment. In this work,
the fitting parameters (N0, ba) are not adjusted and the
model is not able to predict the transformation time
Fig. 10. The phase constitution evolution during BIT treatment; the
inputs are obtained from experiments and detailed thermodynamic
analysis [23].
precisely, so it is normalized by the time at which bainitic
transformation finishes. Even though the growing bainitic
ferrite continues to provide carbon, the austenite is not stable enough to suppress the martensitic transformation at
the end of the treatment until t/tfinish = 0.57. After this critical time point, only bainitic transformation is able to consume the austenite.
As shown in Fig. 11, thepredicted
average carbon con
tents in the austenite films wfilm
C
after 1045 K + 2 h and
613 K-TBIT treatment are less than the average of wCpc
T0
and wC0 . Before the completion of bainitic transformation,
the average carbon content in the residual austenite is
increasing with time. Therefore, the film becomes
Fig. 11. The average carbon content in the austenite film in different
thickness; the calculation is based on Fe–0.32C–1.42Mn–1.56Si after IA
treatment, 1045 K full equilibrium and BIT at 613 K.
S. Li et al. / Acta Materialia 61 (2013) 2884–2894
2893
5. Conclusion
In this work, the kinetic model for bainitic isothermal
transformation is implemented by considering the finite
diffusion of carbon within the RA. the resulting nonhomogeneous carbon distribution and the presence of a
carbon-rich austenite film adjacent to bainitic ferrite plates.
In agreement with experimental observations, the present
model predicts carbon enrichments in austenite beyond
the possible enrichment according to the thermodynamic
limit for the fcc ! bcc displacive bainitic transformation.
The model calculations also provide qualitative agreement
with experimental observations that suggest a temperature
range in which the volume fraction of RA actually
increases with TBIT, in contrast with what one would
expect using thermodynamic arguments alone. Moreover,
the volume fraction of RA predicted using the kinetic
model for bainitic transformation is in much better agreement with volume fractions measured/estimated by the
authors. Our thermodynamic ([23]) and kinetic (this work)
predictions seem to provide lower and upper bounds
(respectively) for the expected volume fraction of RA after
the two-stage heat treatment of conventional TRIP steel
alloys.
Acknowledgements
Fig. 12. The volume fraction of RA as a function of heat treatment
temperature based on: (a) T 00 and (b) the kinetic limit. For these
calculations, it is assumed that austenite reaches an equilibrium state
during IA.
increasingly thick and the average carbon content
decreases. The results quantitatively agree with experimental results: (i) wcC ¼ 1:8 2:8 wt.% while tA is 10–50 nm
[19]; and (ii) the carbon content in the austenite film is
higher than in bulky RA [48].
Fig. 12 shows contour plots of the volume fraction of
RA as a function of TBIT and TIA considering the so-called
thermodynamic and kinetic limits. The figures show very
different trends: when thermodynamics alone are considered, there is a monotonic decrease in the volume fraction
of RA as the treatment temperatures increase. On the other
hand, when the non-homogeneous distribution of carbon is
considered (austenite film), the region where the maximum
volume fraction of RA does not change monotonically
with temperature. Moreover, the kinetic limit predicts
much higher volume fractions of RA, which is in better
agreement with experimental observations.
This study is funded by The U.S. National Science
Foundation, Division of Civil, Mechanical, and
Manufacturing Innovation, Materials and Surface Engineering Program, Grant No. 0900187, and partially
through the NSF-International Materials Institute Program through Grant No. DMR-08-44082, Office of Special
Programs, Division of Materials Research. The authors
thank Dr. Pedro Rivera of University of Cambridge for
discussions during the preparation of this manuscript.
Dr. Paul Mason from Thermo-Calc is acknowledged for
his advice regarding the use of the computational thermodynamics software.
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