Computer Simulation of Microstructure Evolution in Low Carbon

Prévia do material em texto

1. Introduction
The development of process models has received tremen-
dous attention in the steel industry over the past three
decades. Today a wealth of sophisticated models is avail-
able for most steel processing routes that is unparalleled
compared to other materials producing industries. In part
this is due to the large production volume of steel which
outnumbers that of any other metal and alloy by a wide
margin. Associated with this is a high degree of automati-
zation in the steel industry and process models play a cru-
cial part to produce steel with consistent quality and high
productivity. This model development has been made possi-
ble by the enormous progress in computer technology over
the last decades. Increasingly, many of these models are
based on sound physical principles and include sub-models
that track the microstructure evolution along a given
process path. In particular, this approach is of paramount
importance for the production of low carbon sheet steels
that are required for some of the most demanding applica-
tions of steel, e.g. in the automotive industry.1) Thus, the
present review paper emphasizes the microstructure models
for this family of steels.
In particular, hot rolling of steels has been an emphasis
of computer modelling. Starting with the pioneering work
of Sellars et al.2,3) in the late 1970’s mathematical models
were developed in the 1980’s primarily for plate rolling.4–7)
The processing parameters (e.g. strain rates in the roll-bite
and interstand times) of plate rolling can be comparatively
easily replicated with laboratory simulations to characterize
the individual microstructure phenomena of grain growth,
recrystallization, precipitation and phase transformation. In
parallel, models were developed for hot strip mills, i.e. the
most sophisticated processing facilities found in steel mills.
Senuma et al.8) reviewed the status of these models in 1992.
Since this review, their conclusion that computer modelling
would become a key technology for hot strip production has
become a reality. Virtually all major steel producers now
employ microstructure hot strip models routinely, at least
on an off-line basis.9–19) Moreover, novel strip processing
routes, i.e. compact strip processing (CSP)/thin slab casting
and direct rolling (TSDR), have been introduced since the
1990’s. This necessitated the extension of models, which
had originally been developed for integrated mills, to these
new processing conditions.
Another recent aspect of sheet steel production is the
move into advanced high strength steels (AHSS). This is
primarily driven by the increased demands of automobile
manufacturers to employ steels with improved properties in
terms of strength–ductility balance and crashworthi-
ness.1,20,21) The development of AHSS is an excellent exam-
ple of the versatility of steels to be tailored to a multiplicity
of properties by manipulating steel chemistry and process-
ing routes. An important advantage of steels as compared to
light weight metals and alloys (e.g. Al and Mg) is that the
austenite–ferrite transformation can be exploited to produce
microstructures that are associated with significantly im-
proved property profiles. This is indeed the key metallurgi-
cal tool for AHSS.
Processing windows for this new class of steels are gen-
erally tighter than those for conventional steels. Therefore,
the design and control of robust processing routes moti-
ISIJ International, Vol. 47 (2007), No. 1, pp. 1–15
Computer Simulation of Microstructure Evolution in Low Carbon
Sheet Steels
Matthias MILITZER
The Centre for Metallurgical Process Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4.
(Received on July 24, 2006; accepted on September 14, 2006 )
Models of microstructure evolution in steels are reviewed. The emphasis of the review is on low carbon
sheet steels both hot-rolled and cold-rolled and annealed. First the state-of-the-art on industrial microstruc-
ture process models is presented. The individual model concepts for grain growth, recrystallization, precipi-
tation and phase transformations are briefly discussed. The development from empirically-based models to
physically-based models is identified as a key issue to have increased predictive capabilities for these mod-
els over a wider range of steel grades and operational conditions. The challenges in the development of the
next generation of models are delineated. In particular, new aspects of microstructure evolution associated
with novel processing routes and advanced high strength steels are evaluated. Further, the majority of the
currently employed models are on the macro-scale but future microstructure models will increasingly be
meso-scale models that predict actual microstructures rather than a number of average parameters (e.g.
grain size, fraction transformed) to describe microstructure evolution.
KEY WORDS: mathematical model; microstructure; steel; hot rolling; annealing; grain growth; recrystalliza-
tion; precipitation; phase transformation.
1 © 2007 ISIJ
Review
vates the development of advanced microstructure models
that provide an increased insight into the underlying physi-
cal metallurgy principles. Further, AHSS are frequently
produced as cold-rolled annealed and/or coated sheets.
Here, the intercritical annealing step assumes a crucial role
to obtain the desired complex, multi-phase microstructures.
Intercritical annealing constitutes a paradigm shift from an-
nealing of conventional steels where no cycling through the
transformation region occurs. The added complexity of the
annealing routes for AHSS requires the development of
novel annealing models that also address austenite forma-
tion which has received comparatively little attention in
process models.
Currently available microstructure models for the pro-
duction of sheet and other steels are usually formulated on
the macro-scale, i.e. the microstructure is described with a
number of so-called internal state variables such as grain
size, fraction recrystallized and fraction transformed. How-
ever, the advances in computer technology make it now fea-
sible to routinely conduct modelling of the microstructure
on the meso-scale if not on the atomistic level. In particular
the meso-scale approach appears to open a new era of mod-
elling the microstructure evolution in sheet steels. In this
methodology actual microstructures can be predicted rather
than mean values which have traditionally been used to
characterize microstructures. The significance of these new
modelling strategies is also fueled by the realization that
material properties may markedly depend on morphology
and spatial distributions of microstructure constituents.22–24)
In this paper, first the models proposed for hot strip
rolling are reviewed. Special attention is given to the under-
lying metallurgical phenomena of grain growth, recrystal-
lization, precipitation and austenite decomposition. In a
second part, the status of annealing models is analyzed with
an emphasis on the austenite formation. Finally, novel
meso-scale model concepts and their application to the mi-
crostructure evolution in low carbon steels are delineated.
2. Microstructure Models for Hot Rolling
2.1. Hot Strip Mill Model Overview
In the past twenty years, mathematical modelling of hot
strip rolling has reached a relatively mature state with two
modelling philosophies competing,25,26) i.e. (i) microstruc-
ture engineering where the operational parameters of a mill
are related to the properties of hot band by accurately mod-
elling the microstructure evolution, (ii) artificial intelli-
gence methods such as expert systems, neural networks and
fuzzy logic. The former approach is designed to provide an
increasing insight into the underlying physical metallurgical
principles whereas the latter is a “black-box” approach that
makes use of the vast collection of mill data. Both ap-
proaches have their merits. Beynon,26) while advocating
physically-based models,yielded promising but
not yet satisfactory results.164) However, overall, PFMs ap-
pear to be a very promising tool for the modelling of the
microstructure evolution in steels. A particular advantage
of the PFM approach is that one modelling tool can be used
to track the microstructure evolution from casting to the
final processing step of sheet production, i.e. either coiling
or annealing.
5. Conclusions
(1) Sophisticated microstructure evolution models are
available for hot rolling of low carbon steels. Future devel-
opment of these models will have to emphasize the bainite
transformation stage that becomes increasingly important
for novel advanced high strength steels.
(2) Microstructure models for continuous annealing
have by far received less attention than those for hot rolling.
A new generation of microstructure annealing models will
have to be developed since intercritical annealing to pro-
duce multi-phase steels is now increasingly been conducted
in industry. In particular, modelling the austenite formation
kinetics appears to be a more challenging subject than pre-
viously envisioned.
(3) The development of improved microstructure mod-
els for low carbon steel sheets will significantly benefit
from the exciting opportunities that novel meso-scale mod-
elling approaches offer. In particular, phase field models
seem to be on the verge to provide a predictive tool that can
be employed from casting to coiling and annealing, respec-
tively.
ISIJ International, Vol. 47 (2007), No. 1
13 © 2007 ISIJ
Fig. 10. Comparison of PFM simulated 2D (left) vs. 3D (right) microstructures for continuous cooling transformation in a 0.1wt%C–0.49wt%Mn
steel at 0.4 K/s (top; total presented area is 45 mm�45 mm) and 10 K/s (bottom; total presented area is 33 mm�33 mm); ferrite (light gray),
austenite (dark gray)after 114).
Acknowledgements
The author thanks F. Fazeli, W. J. Poole, C. W. Sinclair
and Y. Bréchet for their helpful comments on this review.
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ISIJ International, Vol. 47 (2007), No. 1
15 © 2007 ISIJsuggests a “grey-box” methodol-
ogy that combines both approaches as the most probable
way to develop modelling tools with predictive capabilities
for industrial processes. As a matter of fact, microstructure
engineering models indeed require tuning to specific mill
operating conditions.
For the present review only the microstructure based
models are considered. Starting from the concepts proposed
by Sellars et al.2,3) to predict the microstructure evolution
during multipass hot rolling and using the similarities with
plate rolling,27) several groups9–18,28–32) have developed mi-
crostructure evolution models for hot strip rolling. Some of
these models are available as commercial software pack-
ages whereas others are used in-house. The concepts of
these integrated models are similar with phenomenological
approaches being used to describe austenite conditioning
during rough and finish rolling and austenite decomposition
on the run-out table. The predicted microstructure is then
empirically related to the mechanical properties of hot
band. However, in detail the formulation of the proposed
microstructure models can be vastly different for the indi-
vidual microstructure phenomena of grain growth, recrys-
tallization, precipitation and phase transformation.
The early work emphasized modelling the austenite mi-
crostructure during multi-pass hot deformation. Particularly
in the last decade, the emphasis has shifted to modelling the
austenite decomposition on the run-out table.31,33–37) The
complexity of the phase transformations in steels as well as
of heat transfer of jet impingement boiling38,39) may have
led to this delayed emphasis in developing microstructure
models for run-out table cooling. It is the microstructure re-
sulting from the austenite decomposition and, thus, run-out
table processing that primarily determine hot band proper-
ties.
Plain low carbon steels were the first class of steels in-
cluded in hot strip microstructure models. Based on the
large amount of experimental data available from labora-
tory simulations and the comparative simplicity of the mi-
crostructure evolution in these steels, models have been
proposed that account for the chemistry dependency in
terms of the C and Mn contents.11,26) Further, these models
have been extended to higher carbon grades.40) Microal-
loyed high strength low alloy (HSLA) steels are another
family of steels that is now firmly established in hot strip
mill models. In particular, medium strength HSLA steels
microalloyed with Nb and/or Ti have been extensively 
investigated.12,29–32,41–43) State-of-the-art models include
HSLA grades with minimum yield strength levels of
550 MPa and higher.31,35) Nakata and Militzer44) have re-
cently described such a model for a Ti–Nb steel with
780 MPa tensile strength. These high strength levels are
achieved with a combination of fine, at least in part, acicu-
lar ferrite structures (ferrite grain sizes of 3 mm and below)
and nano-scaled precipitates.45) More recently, progress has
been made to include advanced high strength steels, i.e.
dual-phase (DP) and transformation-induced plasticity
(TRIP) steels, into hot mill modelling framework.37,46) For
these steels sophisticated run-out table cooling paths have
to be employed thereby facilitating the formation of com-
plex multi-phase microstructures that contain combinations
of ferrite, bainite, martensite and retained austenite. Conse-
quently, modelling the austenite decomposition during run-
out table cooling of these steels is currently an area of sig-
nificant research activities.37,47–50)
Another direction for current hot strip mill model devel-
opments is the extension of these models that had been for-
mulated for operational conditions of integrated steel mak-
ers to novel strip processing routes such as asymmetric
rolling and compact strip processing (CSP)/thin slab cast-
ISIJ International, Vol. 47 (2007), No. 1
2© 2007 ISIJ
ing and direct rolling (TSDR). For plain carbon steels, this
extension is comparatively straightforward where the avail-
able microstructure models can simply be applied to the
different processing conditions of these new rolling prac-
tices.51–53) But modified microstructure sub-models have to
be proposed for microalloyed steels.54,55)
After this general review of microstructural hot strip mill
models, the individual models to describe austenite grain
growth, recrystallization, precipitation and austenite de-
composition during multi-pass hot rolling and run-out table
cooling are briefly discussed in the following.
2.2. Austenite Grain Growth
Austenite grain growth takes place during reheating and
after recrystallization in the interstand time between roll
passes. Most hot rolling models use an empirical power law
dn
g�dn
g ,0�Kt.................................(1)
where dg is the austenite grain size, dg ,0 the initial austenite
grain size (e.g. recrystallized grain size), t is time; the grain
growth exponent, n, and K are fit parameters where K de-
scribes the temperature dependency of grain growth fre-
quently with an Arrhenius expression. Values for n and K
have usually been concluded from laboratory studies of
grain growth that consist of heating the steel sample to a se-
lected reheat temperature and holding. Only very few stud-
ies are available that experimentally quantify grain growth
after completion of recrystallization.56) Further, grain
growth in non-isothermal systems can be a very complex
process, as recently summarized by Mishra and DebRoy.57)
For example, even in the classical grain growth experiments
austenite grain growth kinetics at the holding temperature
may depend on the heating rate employed to reach this tem-
perature. Thus, an appropriate selection of the grain growth
parameters that apply to hot rolling is a challenging task.
Manohar et al.58) provide a summary of these parameters
that have been proposed for plain carbon and microalloyed
steels. For example, even for simple CMn steels the em-
ployed grain growth exponents vary between 2 and 10.
From a fundamental perspective the effects of solute drag
and particle pinning on grain growth have to be considered
in detail along a given processing path. Analyzing the role
of AlN precipitates in plain carbon steels Militzer et al.59)
suggest that ideal, unpinned grain growth occurs for these
steels during hot rolling such that n�2 and K�3gM are
proposed where g and M are the grain boundary energy and
grain boundary mobility, respectively. There is a great deal
of uncertainty regarding the selection of the mobility term
which to a first approximation can be formulated with data
for grain boundary diffusion in iron. In reality, an effective
value for M should be employed that is smaller due to
solute drag. Further, solute drag can also lead to deviations
from parabolic grain growth behaviour. For example, grain
growth studies in zone-refined iron suggest n values that
range from 2 to 4.60) The situation is even more complex in
microalloyed steels where particle pinning may occur, e.g.
when TiN precipitates are present, and solute drag is greatly
magnified, e.g. due to Nb in solution. Model approaches to
address these situations are in principle available57,58,61) but
are difficult to translate into predictive tools for industrial
hot rolling processes. Fortunately, austenite grain growth is
probably the least significant metallurgical phenomenon for
strip rolling. Taking the aforementioned ideal grain growth
parameters as an upper limit, austenite grain growth is neg-
ligible during finish mill interstand times such that the
austenite grain size at the exit of the finish mill is given by
the recrystallized grain size.59) Substantial grain growth
may, however, occur on the delay table between roughing
and finishing mill. An exact quantification of this grain
growth is then particularly of interest for Nb-microalloyed
grades that show no or very little recrystallization in the fin-
ishing mill. However, currently available informationis in-
conclusive and additional investigations are required to
quantify the Nb solute drag effect on austenite grain
growth.56)
2.3. Recrystallization
Modelling of the softening behaviour during hot rolling
has been a centre piece of the overall microstructure model-
ling since the extent of softening that occurs in the roll-bite
and the interpass time determines the flow stress of the ma-
terial and, thus, the roll forces required to deform the work-
piece. Jonas62) has recently provided an overview on the ef-
fect of strain rate and interpass time on the amounts and
types of softening. Broadly speaking there are three cate-
gories of mills, i.e. reversing mills, tandem strip mills and
rod mills with distinctly different interpass times that are
inversely related to the speed, and thus strain rate, of the
workpiece, as summarized in Table 1; the softening mecha-
nisms strongly depend on these deformation conditions.
Even though softening is associated with recovery and re-
crystallization it is frequently discussed in terms of recrys-
tallization. There are three distinctly different recrystalliza-
tion processes, all of which can potentially be operative in
strip rolling, i.e. static recrystallization (SRX) during the
interpass time, dynamic recrystallization (DRX) in the roll-
bite and post-dynamic or metadynamic recrystallization
(MDRX) in the interpass time following DRX.
A number of models9,10,31,63) have been developed for the
recrystallization kinetics that use the Johnson–Mehl-
Avrami–Kolmogorov (JMAK) approach, i.e.
..................(2)
where X is the fraction recrystallized, k the Avrami expo-
nent and t0.5 the time for 50% recrystallization. The above
general equation can be applied to all types of recrystalliza-
tion processes, i.e. SRX, DRX and MDRX. The detailed
X
t
t
k
� � �1 0 693
0 5
exp .
.
















ISIJ International, Vol. 47 (2007), No. 1
3 © 2007 ISIJ
Table 1. Typical ranges for strain rates and interstand times in
hot rolling operations62)
model formulations in terms of the parameters k and t0.5 can
vary significantly between different approaches proposed
thereby leading to potentially very different predictions for
a given steel and deformation schedule, as recently summa-
rized by Liu et al.53) However, a number of general observa-
tions can be made by comparing these different formula-
tions. The Avrami exponent is in general in the range be-
tween 1 and 2, i.e. it does not fall between 3 and 4 as would
be expected from the three dimensional growth geometry of
recrystallizing grains. These lower k values can, however,
be rationalized in terms of the heterogeneous nature of the
recrystallization process, as shown in detail by Chen and
Van der Zwaag64) who incorporate both microstructure and
texture components in the deformed state into a compre-
hensive model for recrystallization kinetics.
For SRX, the time for 50% recrystallization can in gen-
eral be expressed as a function of strain, e , strain rate, ė,
austenite grain size, dg, and temperature, T. Frequently, the
formulation
t0.5�Ada
gebė c exp(Qrex/RT) ......................(3)
is employed where R is the ideal gas constant and the pa-
rameters A, a, b, c and Qrex have to be obtained from labo-
ratory simulations. In particular the rate constant A and the
effective activation energy Qrex for recrystallization depend
on steel chemistry; the values for Qrex fall typically in the
range 200–400 kJ/mol. The parameters a, b and c that char-
acterize the role of austenite grain size, strain and strain
rate, respectively, vary from study to study and sometimes
they are also considered to depend on steel grade. The ef-
fect of steel chemistry on the recrystallization parameters
for t0.5 is in particular well studied for CMn and Nb
steels.43,65)
Overall, the above approach is empirical in nature and more
rigorous models are available66) to provide insight into the
mechanisms of static recrystallization. For example, the ad-
dition of alloying elements, i.e. C, Mn, Nb, Cr, Mo etc.
leads to a reduction of the recrystallization rates and this
can be attributed to solute drag which reduces effectively
the mobility of grain boundaries.66) As a result, at lower
rolling temperatures and decreasing interpass times, the
material may not completely recrystallize. An example for
the associated accumulation of strain in the workpiece is
shown for a number of Nb steel strips in Fig. 1.
DRX requires the application of a critical strain that is
larger than that for SRX. The critical strain, e c, for DRX is
usually described by the following equation
e c�Xdz
gZx ..................................(4)
Here, X , z and x are empirical parameters and Z is a tem-
perature corrected strain rate, i.e. the so-called Zener–Hol-
lomon parameter,
Z�ė exp(�Q/RT).............................(5)
where Q is an effective deformation activation energy. In
general DRX, is promoted during deformation at lower
strain rates and higher temperatures such that the regimes
of SRX vs. DRX have been delineated using a limiting
Zener–Hollomon parameter above which DRX cannot
occur31)
Zlim�h exp(�udg)�Z0 ........................(6)
where the parameters h , u and Z0 are steel grade specific.
An important postulation of this approach has been that
there is essentially no DRX taking place in the finishing
mill stands. In contrast, Jonas et al.62,63) propose that the ac-
cumulation of strain when rolling under conditions of par-
tial or no recrystallization may exceed the critical strain for
DRX. Using this philosophy they are able to describe the
mean flow stress (MFS) evolution deduced from mill logs
with the help of a DRX/MDRX softening mechanism, in
particular for those situations where the MFS is decreasing
in later finishing stands. Alternatively, extended recovery
and/or SRX may be proposed to explain the observed drop
in MFS since the rates of recovery and SRX increase with
strain accumulation as well. An argument in favour of a
mechanism which does not involve DRX could be made by
considering that the interpass times are approximately two
orders of magnitude larger than the residence times in the
roll bite. However, a resolution of these opposing views re-
garding the potential of DRX during strip finish rolling re-
quires further detailed investigations. Some of the chal-
lenges associated with this task include the inaccessibility
of the workpiece during hot rolling, the extrapolation of
empirical relationships obtained at lower strain rates in the
laboratory to the high strain rates of finish rolling and the
determination of the accumulated strain (see Fig. 1). For the
latter, frequently a simple linear relationship in terms of the
recrystallized fraction, X, is proposed such that the accumu-
lated strain, e a, in pass j is given by63)
e a�e j�(1�X)e j�1 ............................(7)
where e j is the applied strain in pass j and e j�1 is the accu-
mulated strain for the previous pass, j�1. More complex
ISIJ International, Vol. 47 (2007), No. 1
4© 2007 ISIJ
Fig. 1. Strain accumulation in austenite during hot rolling of Nb
steel strips.32)*
* Reprinted from Journal of Materials Processing Technology, 125–156, M. Pietrzyk, Through-process modelling of microstructure evolution in hot forming
of steels, 53–62, Copyright (2002), with permission from Elsevier.
schemes and relationships have been proposed to determine
the accumulated strain in the partial recrystallization
regime.67,68) In addition, considering the structural hetero-
geneity during partial recrystallization leads to significantly
different predictions for the overall recrystallization kinet-
ics.68) Clearly, from an academic point of view, there is a
need to modernize the approaches for recrystallization ki-
netics but this may be less significant from a more prag-
matic perspective of having a sufficiently accurate predic-
tion of recrystallization under industrial processing condi-
tions.
For microalloyed steels, in particularthose containing
Nb, the effect of potential strain-induced precipitation on
recrystallization may have to be considered. Recrystalliza-
tion stops if precipitation occurs due to complete pinning of
the grain boundaries and this situation has led to the intro-
duction of the so-called recrystallization stop temperature,
Tnr, below which static recrystallization does not
occur.67,69–71) The concept of Tnr is widely used even for
cases where just partial recrystallization is observed due to
solute drag.31,72) The definition of Tnr is fairly ambiguous
since the regime of partial or no recrystallization depends
on processing conditions such that the term Tnr remains
only credible for a well defined range of processing param-
eters. While in strip mills strain accumulation is commonly
observed because of incomplete recrystallization (see Fig.
1) many researchers argue that strain-induced precipitation
can be neglected in conventional strip rolling prac-
tices.31,41,72) Typically, laboratory investigations suggest
minimum precipitation start times of approximately 10 s
that have been observed at around 900°C. However, it has
been suggested that the higher strain rates in the finishing
mills as compared to those employed in the laboratory may
lead to an acceleration of the precipitation process because
of deformation-induced vacancies.73,74) Thus, it is still a
matter of debate whether strain-induced precipitation or
solute drag are the dominant mechanisms of strain accumu-
lation during strip rolling of Nb steels. However, for typical
finish rolling practices it can be expected that precipitation
of Nb(CN) in austenite is marginal as recently shown for a
hot-rolled coil where fine precipitates form in ferrite as ver-
ified by their orientation relationship with the ferrite
matrix.45)
In general, the interaction between recrystallization and
precipitation can be even more complex than that delin-
eated above. In addition, acceleration of recrystallization
due to particle stimulated nucleation and/or reduced solute
drag are possible scenarios.75) In any event, the JMAK ap-
proach cannot be used to describe recrystallization kinetics
that is affected by precipitation. Alternative models are
available that account for the effects of solute drag and pre-
cipitation on recovery and recrystallization. For example,
the model of Zurob et al.66) has been applied with some
success to describe the combined precipitation and soften-
ing behaviour in hot deformed Nb steels. In this approach,
suitable pinning and solute drag terms are included in the
softening kinetics that is described based on a dislocation
density formulation. Using the Taylor relationship, s f�r1/2,
the dislocation density, r , is translated into a flow stress, s f,
that is used to determine the recovery kinetics. The model
of Verdier et al.76) is employed to describe softening due to
recovery. The recovery rate is explicitly linked to precipita-
tion by a factor that accounts for the density of precipitates
that form at dislocation nodes. The dislocation density de-
creases continuously due to recovery and provides the input
for the driving pressure of recrystallization. The model also
includes the pinning pressure of precipitates with a mean
particle size rp and volume fraction fp based on a Zener drag
term, i.e. k fp/rp where k is a pinning constant. The analysis
of Zener drag for grain growth by Manohar et al.77) would
suggest k�5.9 but in practice k is frequently used as a fit
parameter; Zurob et al. postulate k�3. Further, an effective
grain boundary mobility, M, is introduced to account for
solute drag, i.e.
......................(8)
where Mpure is the intrinsic grain boundary mobility that is
concluded from data for grain boundary diffusion in iron,
cm is the solute concentration and am a solute drag parame-
ter that is associated with the jump frequency of solutes
across the grain boundary and the binding energy of solutes
to the grain boundary. The evolution of the particle size dis-
tribution and the solute concentration can be rationalized
with a precipitation model. Zurob et al. utilize the simpli-
fied precipitation model proposed by Deschamps and
Brechet78) where the precipitation kinetics is separated 
into nucleation/growth and growth/dissolution/coarsening
regimes by introducing a suitable transition criterion. After
having concluded a number of material parameters (e.g.
grain boundary energy, solute-boundary binding energy
etc.) from the literature, the combined recovery-recrystal-
lization-precipitation model of Zurob et al.66) employs two
fit parameters that are related to the density of potential nu-
cleation sites for precipitates and newly recrystallized
grains, respectively. Alternative models, in particular, for
the interaction of precipitation and recrystallization can be
found in the literature.69–71) While in detail these models
differ from that proposed by Zurob et al. philosophically
they are very similar. All these models are phenomenologi-
cal in nature with the intent to incorporate an ever increas-
ing physical basis. The limitation of this approach is fre-
quently given by the lack of knowledge regarding the quan-
tification of solute drag, particle pinning and the inhomoge-
neous nature of the recrystallization process.
Aside from the recrystallization kinetics and thus the
amount of fraction recrystallized the resulting recrystallized
grain size is an important output of a recrystallization
model. The significance of the grain size prediction is mag-
nified by the lack of substantial austenite grain growth after
completion of recrystallization in the finishing mill. The
models for recrystallized grain size fall into the categories
of SRX vs. DRX. For SRX, the recrystallized grain size is
described by an empirical approach of the form
dSRX�Ldq
ge�p exp(�Qgx/RT)....................(9)
where L , q, p and Qgx are parameters. The statically recrys-
tallized grain size increases with the initial grain size and
decreases as the applied strain increases; the effect of strain
M
M
c� �
�
1
1
pure
m mα




ISIJ International, Vol. 47 (2007), No. 1
5 © 2007 ISIJ
rate is small79) and in most models not considered. From a
more fundamental perspective, dSRX is expected to be inde-
pendent of deformation temperature. Nevertheless, a num-
ber of authors29,31) propose values for Qgx that are not zero
but about one order of magnitude smaller than Qrex. Fre-
quently, it is assumed that new recrystallized grains form
predominantly at grain boundaries such that Eq. (9) can be
reformulated by using the effective grain boundary area in-
stead of dg.9) Further, Sun et al.80) propose q�1/3 based on
grain boundary nucleation, but significantly different values
(e.g. 0.67) have also been proposed for q.10) The effect of
strain on dSRX is captured with values of p that typically fall
into the range 1/3 to 1.31,53) In case of partial recrystalliza-
tion, a number of formulations for the austenite grain size
as a function of the initial grain size, dSRX and X can be
found in the literature.10,54) Regardless of differences in de-
tails all these descriptions replicate a successive grain re-
finement as shown in Fig. 2 for Nb steel strips.
For DRX, the recrystallized grains size can be expressed
as a function of the Zener–Hollomon parameter, i.e.10,29,54)
dDRX�LDRXZw .............................(10)
where LDRX and w are parameters. Interestingly, the de-
pendence of dDRX on Z is not well established since values
of w are reported that are both larger and smaller than zero.
2.4. Precipitation
In addition to the interaction of precipitation with recrys-
tallization precipitates are used in many ways to tailor the
properties of low carbon sheet steels, i.e. precipitation hard-
ening, grain size and texture control. The status of precipi-
tation research has recently been reviewed by Senuma.75)
Senuma’s paper includes also a brief overview on precipita-
tion models such that the reader is referred to this paperfor
more information. Most of these models are based on clas-
sical nucleation and growth theories.81) An emphasis of the
modelling effort had been on the precipitation behaviour in
austenite and this may have been fueled by the aforemen-
tioned interaction of strain-induced precipitation with re-
crystallization of austenite. Elaborate physical based mod-
els have been developed for the strain-induced precipitation
in Nb microalloyed steels.75,82) However, models are also
available for precipitation during transformation at the
moving austenite/ferrite interface,83) i.e. interphase precipi-
tates, and in ferrite.84)
An interesting development in describing precipitation in
ferrite during coiling of hot strip is to not track the evolu-
tion of the particle size distribution per se but instead to di-
rectly address the precipitation strengthening evolution for
the cooling path of a coil.85) For this purpose, the approach
taken by Shercliff and Ashby86) is utilized that assumes that
particle coarsening is the relevant rate-limiting step to de-
scribe precipitation kinetics during age hardening. Then a
temperature corrected time, P, can be introduced such
that85,86)
......................(11)
where Qp is the effective activation energy for diffusion of
the precipitating alloying elements (e.g. V, Nb, Ti, Cu). The
precipitation peak strength contribution, Dsp, occurs at a
particular value, Pp, that together with Qp can be deter-
mined from the peak times in the age hardening curves ob-
tained at different temperatures. For microalloyed HSLA
steels, the solubility limit of V, Nb and Ti, respectively, in
ferrite is negligible compared to the bulk concentration
such that Dsp is just a function of microalloying content
and independent of temperature. Integrating P along the
cooling path of a coil the precipitation strength contribution
can, to a first approximation, be obtained from31)
...................(12)
This approach had been seen to predict the coiling tempera-
ture windows for V and Nb microalloyed steels, as illus-
trated in Fig. 3.31,85) However, some limitations of this
rather simplified methodology have been noted for a
∆
∆
σ
σ
�
�
1 9 4
1 4
1 6
1 2
. ( / )
( / )
/
/
p p
p
P P
P P
P
t Q RT
T
�
�exp( / )p
ISIJ International, Vol. 47 (2007), No. 1
6© 2007 ISIJ
Fig. 2. Austenite grain size evolution during hot rolling of Nb
steel strips.32)*
Fig. 3. Prediction of normalized precipitation strength in V and
Nb microalloyed hot band as a function of coiling tem-
perature assuming a 30°C/h coil cooling.31)
* Reprinted from Journal of Materials Processing Technology, 125–156, M. Pietrzyk, Through-process modelling of microstructure evolution in hot forming
of steels, 53–62, Copyright (2002), with permission from Elsevier.
0.14wt%Ti–0.05wt%Nb steel44) and for precipitation hard-
ening by Cu.87) In these cases, the complexity of nucleation,
growth and coarsening of the precipitates has to be consid-
ered in more detail and a single parameter formulation, e.g.
with P, may then be of limited use. Recent modelling work
by Gouné et al.88) suggests that in these more complex sce-
narios the maximum precipitation strength may reach
higher values during non-isothermal treatments than can be
attained during isothermal aging. Thus, there is significant
scope for future precipitation modelling since this may
translate into novel processing routes that potentially opti-
mize precipitation hardening.
2.5. Austenite Decomposition
Austenite decomposition on the run-out table of a hot
mill plays a dominant role regarding the final microstruc-
ture of the produced hot band. After hot working, austenite
can decompose upon cooling, depending on alloying con-
tent and cooling rate, into a variety of transformation prod-
ucts, i.e. ferrite, pearlite, bainite and martensite. The de-
composition of austenite in steels has been extensively in-
vestigated for many decades and is one of the most widely
studied phase transformations. A review of the status of
modelling the austenite decomposition kinetics in low car-
bon steels has recently been provided.89) Thus, only the
main points will be summarized here.
In particular, model approaches are available which pro-
vide predictive tools for processing of conventional com-
mercial low carbon steels with ferrite-pearlite microstruc-
tures. The JMAK analysis (see Eq. (2)) has become a work-
horse in modelling the austenite decomposition. Referring
to Eq. (2), in this case X represents the fraction transformed
and the characteristic time t0.5 is replaced by a rate constant
b�0.693/(t0.5)
k. The JMAK equation has been used to suc-
cessfully describe ferrite, pearlite and bainite formation.
Usually, k can be taken as independent of temperature with
the following nominal values being proposed by Tamura et
al.90): k�1 for ferrite, k�4 for pearlite and k�4 for bainite.
For ferrite and pearlite, the proposed values for k are based
on an extensive number of experimental studies. For exam-
ple, for ferrite k-values have been found to be in the range
of 0.8 to 1.4.31,91,92) The situation is more complex for bai-
nite where different k values have been determined ranging
from 2 to 7.93,94) The parameter b can be written such that95)
...............................(13)
where b0 is a function of temperature, deff is the effective
austenite grain size and m the grain size exponent. The 
effective grain size is introduced to account for the effect 
of transformation from work-hardened austenite, i.e.
deff�dg exp(�e) where e is the strain accumulated due to
partial or no recrystallization in the finishing stands.96) The
effect of strain accumulation on austenite decomposition il-
lustrates the need to link microstructure models for rough
and finish rolling with that for run-out table cooling. Nomi-
nal values have been proposed for the grain size exponent,
i.e. m�1 for ferrite, m�2 for pearlite and m�0.65 for bai-
nite90) but a range of values can be found in the literature,
e.g. for ferrite m values of 1.3–2.2 have been proposed.31)
Thus, m together with the function b0 have to be considered
parameters that depend also on steel chemistry.
To address the non-isothermal character of run-out table
processing paths the JMAK theory is combined with the
additivity principle that holds as long as the Avrami expo-
nent, k, is a constant.97) As a result, the fraction transformed
during continuous cooling can be obtained from90)
......(14)
where j�dT/dt is the instantaneous cooling rate and Ts the
transformation start temperature. While the additivity prin-
ciple is in general reasonably well fulfilled for the forma-
tion of ferrite and pearlite, it is usually of very limited use
for the bainite reaction.98)
Further, when considering the overall austenite decompo-
sition kinetics the formation of different transformation
products have to be considered. Jones and Bhadeshia99) pro-
pose a simultaneous transformation model using an ex-
tended JMAK analysis. However, their analysis indicates
very little overlap in the formation of different transforma-
tion products. As a result, sequential transformation
models31,100) appear to be very useful and more straightfor-
ward in their formulation. The philosophy of this approach
is that at any given time just one reaction occurs with the
sequence being ferrite–pearlite–bainite–martensite. The oc-
currence of these reactions is separated by suitable transi-
tion conditions. A typical transformation model for run-out
table cooling of ferrite-pearlite steels consists of five sub-
models: (i) transformation start, (ii) ferrite growth, (iii) ini-
tiation of pearlite formation, (iv) pearlite growth and (v)
ferrite grain size. To evaluate the transformation start clas-
sical nucleation theory can be employed. However the lack
of detailed nucleation information and the realization that a
measurable transformation start criterion requires some ini-
tial growth of the ferrite nuclei has led to an alternative
transformation start modelthat considers carbon diffusion
controlled early growth of ferrite nucleated at temperature
TN at grain corners to describe the temperature, Ts, of meas-
urable transformation start.101) Measurable transformation
X
d
T
T
dT
m
T
T k
k
� � �1
1 0
1
exp
( )
( )
/
eff
s
∫














β
ϕ
β
β
� 0 ( )T
d m
eff
ISIJ International, Vol. 47 (2007), No. 1
7 © 2007 ISIJ
Fig. 4. Transformation start in DP (0.07 wt% C, 1.85 wt% Mn,
0.16 wt% Mo; TAe3�1 095 K) and TRIP (0.19 wt% C,
1.49 wt% Mn, 1.95 wt% Si; TAe3�1 145 K) steels. Solid
lines give model predictions and symbols indicate experi-
mental data.89)
start is assumed to coincide with nucleation site saturation
which is reached when carbon enrichment at the austenite
boundaries attains a critical level, c*, above which ferrite
nucleation is inhibited. This approach provides a satisfac-
tory description of the transformation start during continu-
ous cooling by using TN and c* as adjustable parameters
and remains applicable to advanced high strength steels, as
illustrated in Fig. 4. Subsequent ferrite growth is then de-
scribed using the JMAK theory. A number of different
functions have been provided to quantify the temperature
dependence of the rate parameter b0.
31,100) Rate constants
can be expressed by using classical nucleation theory and
diffusion controlled growth, as proposed by Umemoto et
al.100) Alternatively, empirical formulae have been intro-
duced with a minimum of two adjustable parameters.31) Fer-
rite growth stops when the formation of pearlite is initiated.
The transition from ferrite to pearlite formation can be ra-
tionalized by considering cementite nucleation at the mov-
ing austenite–ferrite interface. For nucleation to take place
the interface velocity must be below a critical value and dif-
ferent formulations have been proposed for this crite-
rion.100,102) The criterion adopted by Umemoto et al.,100) i.e.
the interface does not move in the incubation time for nu-
cleation a distance which is larger than the critical nucleus
size, is usually employed in overall austenite decomposition
models.100,103) Subsequent pearlite growth is described
using the JMAK equation and additivity similar to the fer-
rite growth modelling approach. However, the pearlite sub-
models are not significant for low carbon steels with carbon
contents of up to 0.1 wt% where pearlite volume fractions
are usually 5% and lower.
An important prediction of the transformation model is
the ferrite grain size which is determined in the early stages
of the transformation process. Thus, Suehiro et al.9) have
proposed to express the ferrite grain size, da, as a function
of the transformation start temperature, e.g.31)
da�(F exp(B�E/Ts))
1/3 ......................(15)
where F is the final ferrite fraction and Ts is in K. B and E
are parameters which depend on steel chemistry and B is in
general also a function of the initial austenite microstruc-
ture.31) Figure 5 summarizes the results for the ferrite grain
size predictions from work-hardened austenite in four
HSLA steels microalloyed with Nb and Ti, i.e. HSLA 50
(0.02 wt% Nb, 0.01 wt% Ti), HSLA 60 (0.035 wt% Nb),
HSLA 80 (0.08 wt% Nb, 0.05 wt% Ti) and HSLA 90
(0.05 wt% Nb, 0.14 wt% Ti). To a first approximation, the
general trends for the ferrite grain size predictions obtained
under run-out table conditions can be found by taking
B�55 and E�51 000 K for low carbon and HSLA steels of
up to 450 MPa yield strength, i.e. HSLA 50 and 60.104) For
HSLA steels with yield strengths of 550 MPa and higher,
i.e. HSLA 80 and 90, B�22.3 and E�18 100 K have been
determined.44) Good agreement is obtained with grain sizes
observed in industrially processed coils. Observed grain
sizes are approximately 5, 4, 3 and 2 mm for the HSLA 50,
60, 80 and 90 steels, respectively. These grain sizes are pre-
dicted for a cooling rate of approximately 100°C/s which is
typical for the impingement zone of run-out table water
jets. This austenite-to-ferrite/pearlite modelling framework
remains valid also for ferrite transformation in advanced
high strength steels.48,105)
In addition to these semi-empirical models, significant
efforts have been made to develop more fundamentally
based austenite-to-ferrite transformation models. Conven-
tionally, it is assumed that ferrite growth is controlled by
carbon diffusion in remaining austenite. Suitable carbon
diffusion models have been developed and validated for
Fe–C alloys.106–108) However, a more detailed analysis re-
veals that simple carbon diffusion models can only be ap-
plied for Fe–C alloys with a carbon content of 0.2 wt% and
higher.89) For low carbon steels the ferrite transformation
depends on both long range carbon diffusion and the inter-
face reaction such that mixed-mode models have been pro-
posed.109,110) The rate of interface migration is then ex-
pressed with the concept of mobility, i.e.
n�MDGint ................................(16)
where M is the intrinsic interface mobility and DGint is the
driving pressure at the interface. Equation (16) is coupled
to the diffusion equation for carbon to provide a framework
to describe the transformation kinetics. Critical to the im-
plementation of the mixed-mode model is the appropriate
selection of the mobility term, growth geometry and ther-
modynamic condition. The intrinsic mobility is frequently
expressed with an Arrhenius relationship, i.e.
M�M0 exp(�Q/RT).........................(17)
Krielaart and Van der Zwaag111) propose values for M0 and
Q based on experimental observations in binary Fe–Mn al-
loys, i.e. M0�5.8 cm mol/Js and Q�140 kJ/mol. While the
value for Q is generally accepted there is much debate re-
garding the pre-exponential factor.110,112–114) This term is
usually employed as an adjustable parameter leading to an
apparent mobility that decreases as the temperature in-
creases. To rationalize this effective mobility it has been
proposed that alloying elements (e.g. Mn, Mo, Si) exert a
solute drag effect on the moving austenite-ferrite inter-
face115) and Eq. (8) can be used to quantify this situation.
Alternatively, Fazeli and Militzer116) incorporate a solute
drag effect into a mixed-mode model by replacing the driv-
ing pressure in Eq. (16) with an effective driving pressure
where the thermodynamic driving pressure is reduced by a
ISIJ International, Vol. 47 (2007), No. 1
8© 2007 ISIJ
Fig. 5. Predicted ferrite grain sizes for run-out table cooling of
HSLA steels.89)
solute drag pressure. The latter depends on the interface ve-
locity and is a function of the solute-interface binding en-
ergy and the jump frequency of solutes across the interface.
In applying their model to the continuous cooling transfor-
mation of low carbon steels these two physical parameters
are employed together with the pre-exponential term M0 of
the intrinsic interface mobility as adjustable parameters.
Figure 6 illustrates the application of this model to the fer-
rite transformation in selected AHSS. Overall, the model
employs four fit parameters similar to the semi-empirical
JMAK approach. The advantage of the new approach is
then that the adjustable parameters are clearly defined in
their physics thereby facilitating a better quantification of
the role of alloying elements. This will gain increased sig-
nificance for AHSS due their higher alloying contents.
In addition, for these steels sub-models are required to
describe the bainite and martensite reactions. Sub-models
for Widmanstätten ferrite, bainite and martensite are in-
cluded in the sequential model of Umemoto et al.100) There
is, however, no explicit description of transformation start
and ferrite grain size. The initiation of Widmanstätten fer-
rite, bainite and martensite is associated with critical driv-
ing pressures which serve as transition conditions. Growth
rates of Widmanstätten ferrite and bainite are described by
using the concept developed by Trivedi.117) The fraction of
martensite is calculated as a function ofundercooling
below the martensite start temperature using the Koistinen-
Marburger equation.118) However, validation of this compre-
hensive model with experimental and/or industrial data has
focused on ferrite–pearlite steels and very limited compari-
son has been provided for steels where bainite is present.119)
More recently, Samoilov et al.47,120) have developed a simi-
lar approach using phenomenological nucleation and
growth approaches for the overall austenite decomposition
kinetics with the goal of applying the model to run-out
table cooling of DP steels. Some limited validation of the
model, which employs a number of fit parameters, is pre-
sented for plain carbon and Cr-alloyed steels using labora-
tory and industrial data. However, the overall status of mod-
elling the bainite transformation for complex industrial run-
out table cooling conditions is best characterized by a re-
cent statement of Siodlak et al.121) that hardly any useful
equation can be found in the literature. Very similarly, bai-
nite formation has also to be considered for the austenite
decomposition during annealing of AHSS, in particular
TRIP steels.122,123) For example, Iung et al.122) have devel-
oped an overall transformation model for annealing of cold
rolled TRIP steels that replicates the modelling philosophy
of Umemoto et al.100)
The basic challenge in modelling the kinetics of bainite
formation arises from understanding its underlying mecha-
nisms, which have been debated for a long time with dis-
placive93) and diffusional ledgewise growth124) being the
two proposed growth mechanisms. In addition, in multi-
phase steels bainite will form from an austenite–ferrite mi-
crostructure rather than a fully austenitic structure. Models
have been developed based on both proposed transforma-
tion mechanisms.125,126) The analysis by Minote et al.125) for
classical CMnSi TRIP steels suggests that the diffusional
model provides a better fit for bainite transformation at
higher temperature (�350°C) whereas for lower tempera-
tures (�350°C) the displacive mechanism seems to provide
a better description. Further, a more in-depth analysis re-
vealed that both model approaches employ a number of 
adjustable parameters that can be tuned such that both
transformation mechanisms may be used to equally well de-
scribe experimental data for the bainite transformation.127)
As a result, both model frameworks have been used to 
address the bainite transformation stage in TRIP
steels.50,123,125) One of the underlying assumptions of these
models is that bainitic ferrite forms in these steels without
cementite precipitation. In reality, however, some cementite
may also form and this will affect the bainite transforma-
tion kinetics. As a result, efforts are on their way to model
the bainite transformation by taking into account simultane-
ous cementite precipitation.128,129) Overall, much work re-
mains to be done to bring modelling the bainite transforma-
tion kinetics onto a truly satisfactory level.
3. Microstructure Models for Annealing
3.1. Annealing Model Overview
The microstructure evolution during annealing of con-
ventional steels is comparatively simple such that mi-
crostructure process modelling for continuous annealing
lines and/or hot dip galvanizing lines has received much
less attention than that for hot rolling. The major mi-
crostructure phenomenon during annealing of ferritic and
ferritic–pearlitic steels is that the cold-rolled material re-
crystallizes during heating to and holding at the peak tem-
perature. For deep drawable steel sheets, e.g. Al-killed
steels and interstitial-free (IF) steels, the control of precipi-
tates is critical during annealing. Models for recrystalliza-
tion and precipitation are available and they are in their na-
ture very similar to those proposed for hot rolling.75,130–134)
Consequently, just a very brief summary of these models
will be provided here. However, with the introduction of ad-
vanced high strength steels, e.g. DP and TRIP steels, a par-
adigm shift occurs as these materials require an intercritical
annealing step. During intercritical annealing, the initial
ferrite–pearlite microstructure transforms partially to
austenite. During subsequent cooling the austenite portion
ISIJ International, Vol. 47 (2007), No. 1
9 © 2007 ISIJ
Fig. 6. Ferrite growth kinetics during continuous cooling of DP
(0.07 wt% C, 1.85 wt% Mn, 0.16 wt% Mo) and TRIP
(0.19 wt% C, 1.49 wt% Mn, 1.95 wt% Si) steels with an
initial austenite grain size of 24 and 20 mm, respectively.
Solid lines indicate predictions using the mixed-mode
model incorporating solute drag and symbols experimen-
tal data.89)
of the resulting ferrite–austenite two phase structure de-
composes such that the desired microstructure is obtained,
e.g. ferrite–martensite in DP steels. This austenite decom-
position step is very similar to that discussed for run-out
table cooling apart from the possibility of the formation of
new ferrite in an epitaxial growth process since no ferrite
nucleation is required and potential differences in growth
morphologies. In summary, austenite formation is the new
metallurgical aspect in developing annealing models for
AHSS. Therefore, the status of austenite formation models
is emphasized in this review.
Before dealing with austenite formation, a brief outline is
provided regarding recrystallization and precipitation mod-
els for annealing of low carbon steels. Similar to the hot de-
formation models softening during annealing is frequently
described just with a recrystallization model using the
JMAK approach (cf. Eq. (2)).130–133) The Avrami exponent k
assumes values in the range of 0.5–2. The rate parameter
expressed by t0.5 can similarly to Eq. (3) be given by the fol-
lowing form
t0.5�ACR(e) exp(Qrex/RT) .....................(18)
where Qrex is depending on steel type in the range of 100 to
500 kJ/mol130,135) and the pre-exponential term ACR can be
expressed as a function of the true strain, e , that replicates
the amount of cold reduction. Both parameters reflect that
addition of alloying elements (C, Mn, Mo, Nb etc.) delays
recrystallization during annealing. Further, as an alternative
to the JMAK approach the Speich–Fischer equation has
been used as an empirical modelling tool for recrystalliza-
tion with similar fit qualities.130)
To describe recrystallization during continuous heating
of the cold-rolled steel the JMAK equation for isothermal
annealing can be integrated along the heating path using the
additivity principle. In general this has been seen to provide
satisfactory prediction for the fraction recrystallized during
these non-isothermal annealing conditions, as illustrated in
Fig. 7 for a DP steel. However, it has been shown, in partic-
ular for IF steels, that the Avrami exponent k can be a func-
tion of annealing temperature132) such that the additivity
rule is technically not applicable anymore. Thus, new re-
crystallization models have been proposed that are based on
several physical parameters, e.g. grain boundary mobility,
dislocation density, grain size etc.66,132,134) This facilitates a
clearer physical interpretation of the recrystallization
process and indicates that limitations of the JMAK ap-
proach are related to cases where solute drag and/or precip-
itation effects on recrystallization are changing with an-
nealing temperature. For example, Ye et al.132) found in
their analysis a grain boundary mobility with an apparent
dependence on cold reduction that can be rationalized in
terms of solute drag. An explicit inclusion of solute drag
into a recrystallization model can be facilitated by the
aforementioned model of Zurob et al.66) where the grain
boundary mobility depends on a solute drag parameter, am
(see Eq. (8)). These more physically based models permit
also to include the recovery stage into annealing models,
e.g. by using the approach proposed by Verdier et al.76) An
alternative formulation of a physical based recovery-recrys-
tallization-precipitationmodel has been proposed by Liu et
al.134) specifically for annealing of Al-killed steels. Figure
8 illustrates calculations obtained in this model framework
clearly indicating the three distinct recrystallization types
depending on the recrystallization-precipitation interaction.
At high temperature, recrystallization takes place before
precipitation (type I) and at low temperature after comple-
tion of precipitation (type III). At intermediate tempera-
tures (type II) there is an overlap of recrystallization and
precipitation leading to a characteristic halt in the recrystal-
lization process. This is very similar to the recrystallization
behaviour discussed for hot deformation of Nb-micro-
alloyed steels when strain-induced precipitation oc-
curs.66,69–71) Obviously accounting for the recrystallization-
precipitation interaction requires suitable sub-models for
precipitation kinetics. For a more comprehensive overview
of the precipitation models developed for annealing the
reader is referred to the recent review by Senuma.75)
3.2. Austenite Formation
An additional microstructure phenomenon that needs to
be considered for intercritical annealing is austenite forma-
tion. This is a key processing step for producing cold-rolled
ISIJ International, Vol. 47 (2007), No. 1
10© 2007 ISIJ
Fig. 7. Recrystallization behaviour for a DP (0.06wt%C–
0.155wt%Mn–1.86wt%Mo) steel under continuous heat-
ing conditions.133)
Fig. 8. Volume fraction recrystallized during isothermal anneal-
ing of an Al-killed steel (0.051wt%C–0.211wt%Mn–
0.048wt%Al–0.0038wt%N).134)*
* Reprinted from 37th Mechanical Working and Steel Processing Conference Proceedings, Vol. XXXIII, C. Liu, A. J. C. Burghardt, T. H. Jacobs and J. J. F.
Scheffer, A recrystallization model for Al-killed low carbon steels, 963–969, Copyright (1996), with permission from AIST.
annealed and/or coated advanced high strength steels.
While a significant body of work exists regarding intercriti-
cal annealing and austenite formation the latter transforma-
tion has overall received much less attention than austenite
decomposition. In particular during the 1970’s and 1980’s
extensive research on DP steels included a number of stud-
ies on the austenite formation.136–140) According to Speich
et al.136) austenite formation can be classified into three
stages: (1) very rapid growth of austenite into pearlite lead-
ing to complete dissolution of pearlite, (2) slower growth of
austenite into ferrite controlled primarily by carbon diffu-
sion in austenite, (3) very slow equilibration of the austen-
ite–ferrite mixture by redistribution of slow diffusing sub-
stitutional alloying elements. For industrial processing of
AHSS, stages 1 and 2 appear primarily of interest. In con-
trast, more recent studies by Minote et al.125) for a classical
TRIP steel (0.2wt%C–1.5wt%Mn–1.5wt%Si) suggest that
for specimens annealed for 5 min at 800°C full equilibrium
with redistribution of Si and Mn is attained, i.e. they found
excellent agreement with the experimentally observed
austenite fraction and that predicted by Thermo-Calc. Nev-
ertheless, it may be more appropriate to make predictions
based on a constrained equilibrium without redistribution
of substitutional alloying elements, i.e. the so-called parae-
quilibrium.141) Paraequilibrium predictions may be similar
to that obtained by assuming full equilibrium (i.e. so-called
orthoequilibrium) and, furthermore, one has to be very
careful with the interpretation of attaining an apparent equi-
librium.133) In any event, as a starting point for developing
microstructure models for intercritical annealing, it is use-
ful to determine equilibrium volume fractions for austenite
by employing available thermodynamic models. However,
more recent investigations on Al-TRIP steels suggest limi-
tations of the predictive capability of currently available
thermodynamic data when complex steel chemistries, e.g.
TRIP steels with Al alloying in excess of 1 wt%, are con-
sidered that fall outside the range of previously widely
studied chemistries.142)
Regardless of the predictive power of thermodynamic
models, kinetic models are required to model the austenite
formation during intercritical annealing in continuous an-
nealing/hot dip galvanizing lines since holding times may
be insufficient to reach equilibrium.143) Physical based mod-
els are available for the pearlite-to-austenite formation144)
but have not made an important impact on intercritical an-
nealing models since in practice the slower ferrite-to-
austenite transformation has been assumed to determine the
volume fraction of austenite. In particular diffusion models
have been developed for the ferrite-to-austenite formation
after completion of the pearlite-to-austenite forma-
tion.139,140,145,146) Katsamas et al.146) apply their model to in-
tercritical annealing of CMnSi TRIP steels. This approach
provides good agreement with experimental data of anneal-
ing cycles that consist of heating at 10°C/s to selected peak
temperatures and isothermal holding. Alternatively, the
JMAK approach has been applied to the isothermal austeni-
tization kinetics and used to predict with some success for
plain carbon steels intercritical annealing times for holding
at the peak temperature.147)
A recent study by Caballero et al.148) indicates limitations
of these approaches since the austenite formation from fer-
rite is in general non-additive. This observation is consis-
tent with the work of Huang et al.133) on a DP600 steel
(0.06wt%C–0.155wt%Mo–1.86wt%Mn) where it was not
only demonstrated that the austenite formation kinetics is
non-additive but that it can be further complicated by an
overlap with ferrite recrystallization. As shown in Fig. 7,
ferrite recrystallization in this steel is extended during con-
tinuous heating to temperatures in excess of 750°C, i.e.
deep into the intercritical region, for heating rates of 10°C/s
and higher which are of practical relevance for typical an-
nealing lines. In addition to kinetic effects, Huang et al.133)
have shown that the non-additive character of the reaction is
associated with significant modifications of the growth
morphologies when, for example, the heating rate to an in-
tercritical holding temperature is varied from 1 to 100°C/s.
At this point, no austenite formation models are available
that can take these heating rate effects into account and/or
address potentially simultaneous ferrite recrystallization.
Further, present models focus on stage 2 of the austenite
formation. However, in DP steels the pearlite-to-austenite
transformation may already produce an austenite fraction
that replicates the desired martensite fraction of approxi-
mately 10%. Thus, even from this perspective there is a
need to revisit models for stage 1 of the austenite forma-
tion. It can be expected that such models will also provide
insight into the observed role of heating rate. The required
models will have to include sub-models for nucleation as
well as have to address morphological aspects. Thus, mod-
elling on the meso-scale may be appropriate.
4. Meso-scale Modelling Approaches
The models reviewed so far fall all into the macroscopic
length scale where some average microstructure property,
e.g. volume fraction of phases and grain size, for an entire
specimen, workpiece and/or a selected but macroscopic
part thereof is considered. However, with the advancement
in computer power over the last decades it is now possible
to conduct microstructure modelling on the meso-scale, i.e.
on the length scale of the microstructure features thereby
predicting actual microstructures. These meso-scale model-
ling tools can be broadly grouped into different categories,
namely Monte Carlo simulations, cellular automata, front
tracking models and phase field models. While in detail
these models differ significantly in philosophy, methodol-
ogy and numerical methods they all can be applied to de-
scribe the microstructure phenomena of interest for low
carbon steels. It is worthnoting that a Monte Carlo (MC)
method had been already used in the early 1990’s to simu-
late austenite grain size evolution in plate rolling.4) To eval-
uate the usefulness and status of these meso-scale models
for low carbon steels it is of primary importance to review
the application of these models to the austenite–ferrite
transformations as the crucial microstructure phenomenon.
Here, MC simulations, cellular automata and phase field
models are primarily of relevance.
In the MC method, the free energy model is the core part
that determines the transition of the state in a MC cell. MC
simulations are probabilistic in nature since they operate by
comparing an interaction energy between neighbouring
cells before and after a proposed change in state; the case
ISIJ International, Vol. 47 (2007), No. 1
11 © 2007 ISIJ
with the lowest interaction energy will be accepted with
some probability. The MC method is widely used for mod-
elling grain growth and recrystallization but can also be
used to model diffusional phase transformations, as re-
cently shown for the austenite-to-ferrite transformation in
binary Fe–C alloys with 2D MC simulations by Tong et
al.149–151) They have developed a MC method to describe
the mixed-mode isothermal austenite-to-ferrite transforma-
tion149) and subsequently extended their model to also in-
clude the ferrite formation from work-hardened austen-
ite.150) For this purpose, they coupled the MC method with
a crystal plasticity finite element method. Further, the MC
method has been used to simulate the deformation-induced
ferrite transformation.151) To date, MC modelling of austen-
ite decomposition is at a very early stage; the currently pre-
sented results may just be taken as proof that the MC
method can be employed to describe this type of diffusional
phase transformation and that reasonable physical trends,
e.g. regarding the role of work hardened austenite, can be
obtained. However, MC simulations of the austenite-to-fer-
rite transformation have yet to be conducted in 3D. Further,
they have so far not been compared with experimental data
and, consequently, the predictive capability of this method
has still to be evaluated.
Cellular automata (CA) combine the elegance of MC
simulations with ease of calibration to experimental data.
Thus, CA are an attractive tool for simulating microstruc-
ture evolution. A cellular automaton consists of an arrange-
ment of deterministic finite-state machines that work syn-
chronously according to a given set of rules depending only
on information provided locally by the states of neighbour-
ing cells. Similar to the MC simulations 2D CA schemes
have been proposed for the austenite-to ferrite transforma-
tion.152–155) Nucleation and growth models have been incor-
porated into the CA algorithm and the mixed-mode charac-
ter of the reaction can be replicated. A number of adjustable
parameters, e.g. in terms of nucleation, are employed to
generate a reasonable fit with experimental data. Figure 9
provides an example for the comparison of an experimen-
tally observed microstructure with a model generated mi-
crostructure for isothermal transformation in a plain carbon
steel.
MC and CA models can in principle employ any empiri-
cal and also physical-based relationships. However, the
phase field model (PFM) approach offers the advantage of
an inherently physical-based meso-scale modelling by em-
ploying a phase field equation (or multi-phase field equa-
tions as required), e.g.156)
.........................................(19)
where f is the phase field parameter that is 0 in the parent
phase (e.g. austenite) and 1 in the product phase (e.g. fer-
rite) and changes in the interface region of width h from 0
to 1, m is the interface mobility, s the interfacial energy
and DG the driving pressure. The proper solution of this
equation necessitates to employ a diffuse interface rather
than a sharp interface that is used for example in the CA
modelling. The thickness of the diffuse interface may be or-
ders of magnitude larger than that of the actual interface.
Therefore, the treatment of the interface region is a critical
issue in PFMs and different approaches are available that
either treat the interface region as a mixture of the two
phases having the same composition157) or as a mixture of
the two phases with different composition that are defined
by a constant ratio for each element.158) As a result, differ-
ent PFM formulations can be found in the literature and a
number of details have yet to be clarified to evaluate the
pros and cons of these formulations for modelling a partic-
ular microstructure phenomenon. Phase field models were
originally proposed to simulate dendritic growth in under-
cooled melts but their initial applications to the solid-state
austenite–ferrite transformation have been recently re-
ported.113,114,156,159–164) In general, phase field models pro-
vide a powerful methodology to describe phase transforma-
tions. This technique can easily handle time-dependent
growth geometries, and thus enables the prediction of com-
plex microstructure morphologies. Combining the phase
field equation with that of carbon diffusion the PFM ap-
proach essentially replicates the mixed-mode philosophy.
Further it can incorporate strain effects and account for
solute drag and trapping by means of the interface mobility
acting as a model parameter. At present, a number of im-
∂
∂
∇


















φ
µ σ φ
η
φ φ φ
η
φ φ
t
G
� � � � � �2
2
18
1
1
2
6
1( ) ( )
∆
ISIJ International, Vol. 47 (2007), No. 1
12© 2007 ISIJ
Fig. 9. Experimentally observed (a) and CA model generated (b) microstructure after isothermal transformation at 670°C for 300 s in a plain car-
bon steel (0.19wt%C–0.22wt%Si–0.69wt%Mn).153)*
* Reprinted from Scripta Materialia, 50, S. Kundu, M. Dutta, S. Ganguly and S. Chandra, Prediction of phase transformation and microstructure in steel
using cellular automaton technique, 891–895, Copyright (2004), with permission from Elsevier.
portant advances have been made in describing the austen-
ite-to-ferrite transformation kinetics with PFMs. In particu-
lar, satisfactory agreement of calculated and measured
transformation kinetics has been obtained when suitable in-
terfacial mobilities are assumed.113,114,156) The adopted in-
terface mobilities are effective values that essentially de-
crease with increasing temperature thereby indicating that
they are presumably affected by solute drag, as discussed in
Sec. 2.5. The recent work by Huang et al.113) provides a de-
tailed analysis that includes isothermal and continuous
cooling transformation as well as massive transformation.
In addition, ferrite grain coarsening behind the transforma-
tion front is addressed. However, all these models appear to
be descriptive and have yet to be brought to a stage that re-
liable quantitative predictions can be made for commercial
steels. A particular drawback of these austenite–ferrite
phase field models is that they are usually 2D simulations.
However, first 3D simulations of the austenite-to-ferrite
transformation have recently been conducted by Militzer et
al.114) As shown in Fig. 10, 3D simulations lead to much
more realistic microstructures as compared to 2D simula-
tions. From a morphological point of view realistic grain
shapes are apparent in the 2D cuts from 3D simulations
whereas in 2D simulations frequently unrealistic grain
shapes result, e.g. the remaining austenite develops long-
elongated channels with a number of narrow inlet-type fea-
tures between ferrite grains that often appear as squished
circles (see Fig. 10(c)). The 3D aspect of microstructure
modelling will become of even more significance when
complex growth geometries have to be considered. An im-
mediate example for the latter is given by the austenite for-
mation in the aforementioned Mo-bearing DP steel. An ini-
tial attempt to incorporate this morphological complexity
into a PFM has been made in 2D and