The cyclic phase transformation
Transcript of The cyclic phase transformation
Cyclic Partial PhaseTransformations In Low Alloyed
Steels:
Modeling and Experiments
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 27 juni 2013 om 10:00 uur
door
Hao Chen
Master of Engineering In Materials Science
Tianjin University, Tianjin, China
geboren te Anqing, China.
ii
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. S. van der Zwaag
Samenstelling promotiecommissie:
Rector Magnificus voorzitter
Prof. dr. ir. S. van der Zwaag Technische Universiteit Delft, promotor
Prof. dr. G. Purdy McMaster University, Canada
Prof. dr. M. Militzer University of British Columbia, Canada
Prof. dr. J. Ågren KTH - Royal Institute of Technology, Sweden
Prof. dr. E. Gamsjager Leoben University, Austria
Prof. dr. Z. G. Yang Tsinghua University, China
Prof. dr. ir. E. Bruck Technische Universiteit Delft
Prof. dr. ir. R. Benedictus Technische Universiteit Delft, Reservelid
The research carried out in this thesis is financially funded by ArcelorMittal.
Copyright c© 2013 by Hao Chen
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and retrieval
system, without the prior permission of the author.
Printed in The Netherlands by PrintPartners Ipskamp
isbn 978-94-6191-771-3
Author email: [email protected]; [email protected]
To my grandparents and Kun
iv
Contents
1 Introduction 1
1.1 Phase transformations in steels . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Content of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The cyclic phase transformation concept 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Local equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Paraequilibrium model . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1 Measured kinetics of the cyclic phase transformations . . . . . . . 32
3.4.2 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
v
vi Contents
3.5.1 Local equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5.2 Paraequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6.1 Stagnant stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6.2 Inverse transformation stages . . . . . . . . . . . . . . . . . . . . . 42
3.6.3 Non-equilibrium interface conditions . . . . . . . . . . . . . . . . 43
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Analysis of the stagnant stage during cyclic phase transformations 47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Fe-C alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Fe-C-Mn alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 Fe-C-M (M= Ni, Si, Cu, Co) alloys . . . . . . . . . . . . . . . . . . 56
4.3.4 Fe-C-Mn-M (M= Ni, Si, Co) alloys . . . . . . . . . . . . . . . . . . 58
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Indirect evidence for the existence of an interfacial Mn Spike 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.1 Effect of Mn concentration . . . . . . . . . . . . . . . . . . . . . . 70
5.3.2 Effect of the number of cycles prior to final cooling . . . . . . . . 73
5.3.3 Creating 2 Mn spikes to create 2 growth retardation stages . . . . 79
5.3.4 Linking growth retardation to a physical location of Mn spikes . 82
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 In-situ observation of the cyclic phase transformation 87
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Contents vii
6.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.1 Dilatometer results . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.2 In-situ HT LSCM observations . . . . . . . . . . . . . . . . . . . . 92
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Bainitic transformation during the interrupted cooling experiments 103
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3.1 Dissipation due to diffusion inside interface . . . . . . . . . . . . 106
7.3.2 Interface friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3.3 Chemical driving force . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3.4 Gibbs energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8 Transformation stasis during the isothermal bainitic ferrite formation in
Fe-C-X alloys 129
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2.1 Fe-Mn-C alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2.2 Fe-Mo-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2.3 Fe-Si-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
viii Contents
Summary 153
Samenvatting 157
A The effect of transformation path on stagnant stage 163
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.3 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.4 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B A mixed mode model with covering soft impingement effect 175
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2.1 Diffusion controlled growth model . . . . . . . . . . . . . . . . . . 177
B.2.2 The mixed-mode model . . . . . . . . . . . . . . . . . . . . . . . . 180
B.3 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Bibliography 194
Acknowledgments 211
Curriculum Vitae 215
List of Publications 217
Chapter 1Introduction
1.1 Phase transformations in steels
While steel has a history covering serval thousands of years, it is still one of the
most important structural materials in practical applications nowadays. Like many
other materials, the mechanical properties of steel are determined by its microstructure
and composition. However, due to the versatility in its microstructure the mechanical
properties of steel are much more adjustable than those of other materials. The versatile
microstructures in steel are obtained via the transformation of the iron lattice from face
centered cubic (FCC) to body centered cubic (BCC). During the lattice transformation,
there is also redistribution of carbon or other alloying elements between these two iron
lattices, which also influences the mechanical properties. In order to precisely tune the
mechanical properties of steel, it is required to deeply understand the mechanism of
the FCC to BCC transformation in steel.
In metallurgy, the FCC iron is termed “Austenite ”, which is thermodynamically
stable at elevated temperatures and enriched in carbon. The temperature A3 above
which only the austenite is stable is determined by the composition of the steel, and for
common steel grades A3 is between 727 C and 912 C. During a typical heat treatment,
the steel is first heated up to a temperature higher than A3 for austenization, and then
cooled down for the FCC to BCC transformation. Upon cooling the morphology
2 Chapter 1. Introduction
and carbon content of the BCC iron formed can vary significantly. Two BCC iron
microstructures are of interest here: (i) Allotriomorphic ferrite. Allotriomorphic ferrite
grains are equiaxed, and mainly grow from the austenite grain boundaries at relatively
high temperatures. It is also called “grain boundary ferrite ”. In this thesis, the
allotriomorphic ferrite will be called “ferrite ”for simplicity. The transformation from
austenite to ferrite is a time-dependent reconstructive reaction which requires large
scale displacement of the iron and carbon atoms, and the carbon will be rejected by
ferrite and diffuse into austenite due to the low carbon solubility in ferrite. From
a thermodynamical point of view the substitutional alloying elements should also
diffuse between austenite and ferrite to minimize the Gibbs energy. However, from a
kinetical point of view the substitutional alloying elements can not take part in long
range diffusion during the transformation due to their low diffusivities. It is generally
accepted that the rate of austenite to ferrite transformation is controlled by carbon
diffusion, and the chemical driving force is only dissipated by the diffusion process [1].
However, some recent studies qualitatively indicate that the transformation rate is also
influenced by the interface mobility [2–6] and partitioning of substitutional alloying
elements [7–13]; (ii) Bainite (bainitic ferrite). Bainitic ferrite is BCC iron with an non-
polygonal microstructure that forms in steels upon cooling to medium temperatures.
The mechanism of bainitic transformation is still heavily disputed [14–33], and two
competitive views: (a) the mechanism of bainitic ferrite formation is the same as that of
ferrite although their morphologies are totally different [34–39]. During the formation
of a bainitic ferrite plate, it is perceived that the carbon has to diffuse away from bainitic
ferrite to austenite, while the substitutional alloying elements do not partition. The
growth rate of bainitic ferrite is only determined by carbon diffusion ; (b) the bainitic
transformation is considered to be a diffusionless process [17, 20, 40, 41]. During the
growth of a bainitic plate there is no need for carbon diffusion, but carbon diffusion
may take place after the growth.
Generally speaking, during phase transformations there are two processes: nucle-
ation and growth. The phase transformation starts by the nucleation of the new phase,
1.1. Phase transformations in steels 3
and then the newly developed phase interfaces migrate into the parent phases. Even
with the most modern techniques, the nucleation process can not be measured directly
and precisely, thus its mechanism is still not very clear. The growth stage of phase
transformations in steels has been studied widely both experimentally and theoreti-
cally [13, 20, 30, 34–36, 38, 39, 41–52]. Despite abundant effort that has been paid in the
past, some key questions, in particular the role of alloying elements on the kinetics
of the moving interfaces and on the transformation kinetics, are not yet fully solved.
In this thesis, the kinetics of austenite decomposition into ferrite at high temperatures
and that into bainitic ferrite at medium temperatures in low alloyed steels are of in-
terest, and effort will be paid to improve the understanding of growth mechanism of
these two transformations. It has been found in the literature that the austenite to
ferrite transformation in low alloyed steels can be roughly described by the classical
diffusional models [1, 13], however, the fine details, like the degree of partitioning of
substitutional alloying elements [7, 8, 10, 13] and the exact value of interface mobil-
ity [53], are still disputed and needs clarification for improving the diffusional models.
Much more effort is also required to discriminate the existing views on the mechanism
of bainitic transformation.
Up to now, the kinetics of ferrite and banite formation are studied in conventional
isothermal or continual cooling experiments. In such experiments nucleation of new
grains and their growth occur simultaneously, and the unknown parameters such as
the spatial density and distribution of the nucleation sites and the variation in the
rate of nucleation during the total transformation, have a very large effect on the de-
tails of the growth reconstructed from the overall transformation curve. Therefore,
new experimental approaches are indeed required to clarify the fine details of growth
mechanism.
4 Chapter 1. Introduction
1.2 Content of this thesis
In this thesis, two new experimental approaches, the cyclic partial phase transformation
experiments at high temperatures and interrupted cooling experiments at medium
temperatures, are designed to study the growth kinetics of the austenite to ferrite and
bainitic transformation more accurately. The new experimental results are used to
discriminate between existing phase transformation models.
In Chapter 2, the cyclic partial phase transformation concept to study the austenite
to ferrite and the ferrite to austenite growth, is described in detail. The mixed mode
model and classical diffusion controlled growth model have been reformulated to
the conditions of the cyclic phase transformations, and then the models are applied to
simulate the cyclic phase transformations in a Fe-C alloy. Finally, a comparison between
the mixed-mode model and diffusional model is made, and the effect of interface
mobility on the transformation kinetics is discussed. In order to discriminate between
Paraequilibrium [54,55] and Local Equilibrium [56,57] models, a series of cyclic partial
phase transformation experiments in Fe-C-Mn alloys have been designed in Chapter
3. Interesting new transformation stages are observed and reported. The modeling
results are compared with the experimental results in details, and the effect of Mn on
the transformation kinetics is discussed. In Chapter 4, the cyclic phase transformations
in a series of Fe-C, Fe-C-M(M=Mn, Ni, Cu, Si,Co) and Fe-C-Mn-M(M= Ni, Si,Co) alloys
are simulated by Local Equilibrium model to illuminate the effect of alloying element
on the length of the stagnant stage newly discovered in Chapter 3. The effect of
heating/cooling rate on the length of stagnant stage is also investigated for an Fe-Mn-C
alloy. A series of experiments are designed in Chapter 5 to prove the existence of the
residual Mn spike after the cyclic partial phase transformations in Fe-Mn-C alloys,
which is theoretically predicted in Chapter 3. The effect of residual Mn spikes on
the austenite to ferrite transformation kinetics during the final cooling of the cyclic
experiments is systematically investigated. Chapter 6 presents the in-situ observations
of interface migration during the cyclic phase transformations in a Fe-C-Mn alloy.
1.2. Content of this thesis 5
The directly measured interface velocities for the austenite to ferrite and vice versa
are compared with predictions by Paraequilibrium and Local Equilibrium models.
In Chapter 7 a series of interrupted cooling experiments at medium transformation
temperatures are designed to study the nature of the bainitic transformation in low
alloyed steels. A so called Gibbs energy balance approach is proposed to theoretically
explain the newly observed features in the interrupted cooling experiments. In Chapter
8, the Gibbs energy balance approach is applied to model the transformation stasis
phenomena in a series of Fe-C-X alloys. The Gibbs energy balance model predictions
are compared with those of the T0 concept, and the physical origin of the occurrence
of transformation stasis is discussed. The main finding as reported in this thesis are
reported in the Summary. In addition to the findings as reported in the thesis chapters,
some of the additional transformation research results are reported in two appendices.
6 Chapter 1. Introduction
Chapter 2The cyclic phase transformation
concept
This chapter is based on
• H Chen, S van der Zwaag, Application of the cyclic phase transformation
concept for investigating growth kinetics of solid-state partitioning phase
transformations, Comp Mater Sci, 2010; 49:801-813.
2.1 Introduction
Both the austenite (γ) to ferrite (α) and the ferrite to austenite phase transforma-
tions in iron-based alloys are of great interest in the steel production, as the final
microstructure of products and their properties are determined by these solid-state
phase transformations. They have been widely investigated from both an experi-
mental and a theoretical perspectives. For more details about the austenite to fer-
rite [3–5, 8, 11, 42, 47, 50, 51, 58–69] and the ferrite to austenite phase transformations,
see Ref. [70–77]. Despite abundant efforts that has been paid in the past, the kinetics
of these two phase transformations is still not understood well [13].
Generally, integral models for both the austenite to ferrite and the ferrite to austenite
8 Chapter 2. The cyclic phase transformation concept
phase transformations involve two parts: nucleation and growth [78]. As for nucle-
ation, the classic nucleation theory (CNT) [78] is the most widely used approach to
estimate the nucleate rates in terms of parameters like the activation energy for nu-
cleation, the Zeldovich non-equilibrium factor, a frequency factor (the rate at which
atoms are added to the sub-critical nucleus) and the density of available nucleation
sites. However, all parameters used to calculate the nucleation rate are difficult or
even impossible to be measured experimentally by modern techniques [60]. In the
past, site-saturation nucleation model [79, 80] has been widely applied to modeling
the austenite to ferrite and the ferrite to austenite for the sake of simplicity, which
unavoidably affects the accuracy of these models and the kinetic parameter obtained
from models and experiments.
After nucleation, the interfaces of new particles migrate into the parent phase dur-
ing the growth process. There are two classic models for growth kinetics of phase
transformation: (i) diffusion-controlled growth model, in which the kinetics of trans-
formation is governed by diffusion processes only. The classical model for the diffusion-
controlled kinetics has been developed by Zener [1]; (ii) interface-controlled growth
model. Interface-controlled growth model states that the interface migration veloc-
ity itself is the rate-controlling factor. For more detail about the interface-controlled
growth model, see Ref [78]. However, a number of recent publications [2, 4, 5] have
shown that both the diffusion-controlled growth model and the interface-controlled
growth model can not fully describe the growth kinetics of solid-state partitioning
phase transformations in metals. Consequently, a mixed-mode model [2–4], which
takes both diffusion of alloying elements and interface mobility effect into account and
evaluates the relative importance of each in a more physically reasonable way, should
bring significant improvement in the modeling of the growth kinetics.
When modeling both the austenite to ferrite and the ferrite to austenite phase
transformation with the mixed mode approach, the interface mobility is an important
physical parameter. Although much work [4, 81–87] has been already done to obtain
interface mobility of the austenite to ferrite phase transformation in the past, signif-
2.2. Simulation conditions 9
icant discrepancies between the values for the interface mobility remain [53]. The
discrepancy can be attributed to several reasons: (i) the experimental transformation
curves from which the interface mobility is derived are affected by nucleation, which
can not be accounted for in a proper manner in the data analysis; (ii) soft impinge-
ment effects [6, 88, 89] at the later stage of phase transformation are also difficult to
be corrected. While the determination of the interface mobility for the austenite to
ferrite transformation is already difficult, determination of the interface mobility for
the reverse phase transformation, the ferrite to austenite transformation, is even more
complex. The reason for this is the complicating effect of pearlite dissolution and
resulting compositionally inhomogeneous starting stage during most of the austenite
formation which affects both later stage nucleation and growth kinetics. The aim of
this chapter is to propose an alternative cyclic transformation procedure in the austen-
ite/ferrite region from which both the mobility for the austenite to ferrite and that for
the ferrite to austenite transformation can be determined. The model to be presented
addresses both the diffusional transformation model approach and the mixed-mode
model approach.
2.2 Simulation conditions
Fig. 2.1 is the schematics of partial phase diagram of binary Fe-C alloys. In the cyclic
phase transformation simulation, the temperature will cycle between T1 (the austenite
to ferrite transformation) and T2 (the ferrite to austenite transformation), both of which
are located in the austenite + ferrite region in the phase diagram. Therefore, unlike
the standard phase transformation state analysis, the current analysis deals with two
incomplete reactions not involving nucleation. Two points in the computer simulation
should be mentioned here: 1. the cooling rate and heating rate is assumed to be infinite,
so there is no phase transformation during heating and cooling; 2. the holding time
should be enough to make the equilibrium situation established at T1 and T2, thus the
phase transformation at T1 and T2 will start from the equilibrium situation at T2 and
10 Chapter 2. The cyclic phase transformation concept
Figure 2.1: Schematic of the partial Fe-C phase diagram
T1. Fig. 2.2a is the schematic of the initial condition of the ferrite to austenite phase
transformation at T2 or the final condition of the austenite to ferrite transformation at
T1, Fig. 2.2b is the schematic of the initial condition of the austenite to ferrite phase
transformation at T1 or the final condition of the ferrite to austenite transformation at
T1.
2.3 Models
After Temperature jumps from T1 to T2 or from T2 to T1, the profile of alloying
elements in Fig. 2.2 will switch and the austenite/ferrite interface would move into the
ferrite phase or the austenite phase. In the mixed-mode model, both the interface mo-
bility and the finite diffusivities of the alloying elements are considered to have effect
on the kinetics of phase transformation, and the concentration of alloying elements at
the interface does not evolve according to local equilibrium assumption but depends
2.3. Models 11
Figure 2.2: (a) The schematic of the initial condition of the ferrite to austenite phase trans-formation at T2; (b) The schematic of the initial condition of the austenite to ferrite phasetransformation at T1
on the diffusion coefficient of alloying elements and interface mobility during phase
transformation. The schematic alloying element concentration profiles of the austenite
to ferrite and the ferrite to austenite transformation during the mixed-mode cyclic
phase transformation are shown in Fig. 2.3a and Fig. 2.3b. To solve diffusion equations
12 Chapter 2. The cyclic phase transformation concept
Figure 2.3: The schematic alloying element concentration profiles of (a) the ferrite to austeniteand (b) the austenite to ferrite transformation during the mixed-mode cyclic phase transfor-mation and (c) the ferrite to austenite and (d) the austenite to ferrite transformation during thediffusion-controlled cyclic phase transformation
in both the austenite and ferrite phase, two grids are constructed in the austenite and
ferrite phase containing N and M points with equidistant spacing and , then the dif-
fusion profile in the austenite and ferrite phase can be numerically calculated by the
Murray-Landis method described above. In classical diffusion-controlled model for
solid-solid partitioning phase transformations [1], local equilibrium is assumed to be
maintained at the interface during the entire phase transformation, which means that
chemical potential of all alloying elements is equal at the interface during the phase
transformation. Local equilibrium can be maintained only when the lattice transfor-
mation reaction during the phase transformation is infinitely fast. In other words,
the interface mobility M, the proportionality factor between the interface velocity and
the driving force, is assumed to be infinite. According to local equilibrium assump-
tion, the concentration of alloying elements at the interface, the key parameter in the
diffusion-controlled model, can be easily calculated from thermodynamic databases.
The schematic alloying element concentration profiles of the austenite to ferrite and the
2.3. Models 13
ferrite to austenite transformation during the diffusion-controlled cyclic phase trans-
formation are shown in Fig. 2.3c and Fig. 2.3d. Except for the difference in imposed
interface condition, the diffusion controlled growth model would be calculated as same
as the mixed mode model, thus only the details about the mixed-mode growth model
will be given here.
The mixed-mode model for phase transformation with only one diffusion flux in
the parent phase has been developed analytically in reference [2]. However, during the
cyclic phase transformation, there is diffusion flux in both the austenite phase and the
ferrite phase during the austenite to ferrite and the ferrite to austenite phase transfor-
mation. Therefore, a mixed-mode mode will be redefined here for both the austenite to
ferrite and the ferrite to austenite transformation during the cyclic phase transforma-
tion. Generally, the interface velocity of the ferrite to austenite phase transformation
and the austenite to ferrite can be written as
vα→γ = Mα→γ∆G(cα) (2.1)
vγ→α = Mγ→α∆G(cγ) (2.2)
Where Mα→γ and Mγ→α are the interface mobility of the ferrite to austenite transfor-
mation and the austenite to ferrite transformation. ∆G(cα) and ∆G(cγ) are the driving
force as a function of the alloying element concentration in the ferrite phase and the
austenite phase at the interface. The interface mobility, M, which is temperature de-
pendent, can be expressed as
M = M0exp(−QG/RT) (2.3)
Where M0 is a pre-exponential factor, QG is the activation energy for the atomic
motion. The driving force, ∆G(cα) and ∆G(cγ), are proportional to the deviation of the
mobile alloying element concentration in the ferrite phase and the austenite phase at
the interface from the equilibrium concentration, and can be expressed as
14 Chapter 2. The cyclic phase transformation concept
∆G(cα) = χα→γ(cα − cαeq(T2)
)(2.4)
∆G(cγ) = χγ→α(cγeq(T1) − cγ
)(2.5)
Whereχα→γ andχγ→α are proportionality factors which can be calculated by Thermo
calc [90]. The migration of the interface would cause a flux of alloying elements, and
this flux, proportional to interface velocity and the alloying element concentration
difference at the interface, can be expressed as
Jiα→γ = vα→γ
(cγeq(T2) − cα
)(2.6)
Jiγ→α = vγ→α
(cγ − cαeq(T1)
)(2.7)
Since there should be no accumulation of alloying elements at the interface, the flux of
alloying element should be balanced by the diffusion flux in both austenite and ferrite
phase, which can expressed as the following equations
Jiα→γ = Dα∂cα
∂z−Dγ
∂cγ∂z
(2.8)
Jiγ→α = Dα∂cα
∂z−Dγ
∂cγ∂z
(2.9)
In this work, an approximation is made for the concentration gradient ∂cα∂z and ∂cγ
∂z
at the interface, yielding
∂cα∂z
=cα1 − cα32∆xα
(2.10)
∂cγ∂z
=cγ1 − cγ32∆xγ
(2.11)
Combination of above equations yields expressions for the concentration at the
2.4. Results and Discussion 15
interface cα and cγ. The diffusion fields of neighboring grains will overlap at the later
stage of the partitioning phase transformations, which would slow down the kinetics
of phase transformation and thus is called soft impingement in the classic diffusion-
controlled growth model. To simulate soft impingement effect, symmetric growth of
the austenite (ferrite) phase at either side of the ferrite (austenite) phase is assumed.
Hence, the carbon mass-flux in the middle of the austenite (ferrite) grain must be zero.
The Fortran coded program operates as follows:
1. Calculate the interface velocity v and alloying element concentration at the inter-
face according to the mixed-mode model with Eq.2.1 and 2.2.
2. Insert the interface velocity and alloying element concentration at the interface
into Murray-Landis equation and calculate the new concentration profile at each
grid point in both the austenite and the ferrite phases.
3. Update the interface position and calculate the new grid spacing.
4. Save data and go back to 1.
2.4 Results and Discussion
In this work, the cyclic phase transformation (the austenite to ferrite transformation
at T1=1050 K and the ferrite to austenite transformation at T2=1100K) are theoretically
calculated for a binary Fe-0.3at.%C alloy. The physical parameter values are listed in
Tab. 2.1. In order to consider the phase transformation in terms of fraction, the size of
the total austenite and ferrite region is assumed to be 50 µm, and the specific volumes
of both phases are taken equal.
In Fig. 2.4, the austenite fractions during the ferrite to austenite transformation
at 1100K and the austenite to ferrite transformation at 1050K are simulated by the
16 Chapter 2. The cyclic phase transformation concept
Table 2.1: Physical parameters used in the calculation.
Parameter 1050K 1100Kχγ→α, J/(at.%) 110.0 NAχα→γ, J/(at.%) NA 765
Dγ, m2/s 1.0 × 10−12 4.0 × 10−12
Dα, m2/s 1.0 × 10−10 4.0 × 10−10
M0, mol m/s 0.5 0.5QG, KJ/mol 140.0 140.0
(a)
(b)
Figure 2.4: The austenite fraction during (a) the ferrite to austenite transformation at 1100Kand (b) the austenite to ferrite transformation at 1050K calculated by the mixed-mode modeland diffusion-controlled growth model
mixed-mode model varying the interface mobility and the diffusion-controlled growth
model. Increasing the interface mobility, the kinetics predicted by the mixed-mode
model would be closer to the prediction by the diffusion-controlled model. When the
assumed interface mobility value is 100 times the interface mobility given in the Tab. 2.1,
2.4. Results and Discussion 17
(a)
(b)
(c)
(d)Figure 2.5: The evolution of carbon profile in (a) the austenite and (b) ferrite phase during thecyclic austenite to ferrite transformation at 1050K and in (c) the austenite and (d) ferrite phaseduring the cyclic ferrite to austenite transformation at 1100K according to the mixed modemodel
18 Chapter 2. The cyclic phase transformation concept
the simulated result of the mixed-mode model is the same as that of diffusion-controlled
growth model. Therefore, it shows that the transformation is purely controlled by dif-
fusion when the interface mobility is large enough, which means diffusion-controlled
growth model is just one extreme case of the mixed-mode growth model.
Fig. 2.5a and Fig. 2.5b indicate the carbon profile in the austenite and ferrite phase
as a function of time during the cyclic austenite to ferrite transformation at 1050K
according to the mixed mode model calculation. The carbon concentration at the
interface in the austenite Cγ increases as the austenite/ferrite interface moves into the
austenite phase, which is the same as the normal austenite to ferrite transformation
(100% austenite to 100%ferrite). The slowly increasing Cγ at the interface would lead
to a decrease in the interface velocity during the phase transformation. Due to the
buildup of carbon concentration profile in both austenite phase and ferrite phase, the
grain size would play an important role in the growth kinetics at the later stage of phase
transformation. When the length of the diffusion field in the austenite or ferrite phase
is equal to the length of remaining austenite or growing ferrite, the diffusion fields
will overlap with the diffusion fields in their neighboring grains, which is beginning
to be visible in Fig. 2.5a at t=10s in the austenite phase and Fig. 2.5b at t=1s in the
ferrite phase. This is the soft impingement effect. Soft impingement starts earlier in
the ferrite phase as the diffusion of carbon in the ferrite phase is much faster than in the
austenite phase. The soft impingement in the austenite phase and ferrite phase would
affect the carbon concentration profile in the austenite and ferrite phase, and decrease
the interface migration velocity as the carbon profile in both the ferrite and austenite
phase should become less and less lean upon further growth of the ferrite phase into
the austenite phase. The interface migration velocity would approaches 0 when the
carbon profiles in both the austenite and ferrite phase become completely flat and
the carbon concentrations in them are equal to the equilibrium carbon concentrations
according to the phase diagram. Fig. 2.5c and Fig. 2.5d indicate the carbon profile in
the austenite and ferrite phase as a function of time during the cyclic ferrite to austenite
transformation at 1100K according to the mixed mode model calculation. The carbon
2.4. Results and Discussion 19
concentration at the interface in the ferrite Cα decreases as the interface migrates into
the ferrite phase, which would decrease the driving force for interface migration and
thus slow down the transformation. Due to the low interface mobility for the ferrite
to austenite transformation, the carbon in the austenite phase should diffuse across
the interface into the ferrite to generate enough driving force for interface migration
in the initial stage, and then the carbon in the ferrite phase would diffuse back to
austenite phase in the later stage. The ferrite to austenite transformation ceases as the
carbon concentrations in the ferrite and austenite phase are equal to the equilibrium
concentration at 1100K.
Unlike the normal austenite to ferrite transformation, the value of Cγ is lower than
the initial carbon concentration in the austenite Cγeq(1100K) at the beginning of the
cyclic austenite to ferrite transformation. This phenomenal can be explained in this
way: as the temperature decreases from 1100K to 1050K, a very sharp carbon gradient
will be generated immediately in the ferrite phase, and this carbon gradient would
make the interface move rapidly in order to keep mass balance in the interface. In
addition, according to the mixed-mode model, the driving force for interface migration
is inversely proportional to Cγ , thus Cγ would be lower than Cγeq(1100K) in order to
get enough driving force for interface migration if the interface mobility is not large
enough. Cγ is strongly dependent on the ratio between interface mobility and diffusion
coefficient M/D .
Fixing the diffusion coefficient and varying the pre-exponential factor M0 of inter-
face mobility, the carbon concentration at the interface in the austenite phase Cγ as a
function of time during the cyclic austenite to ferrite transformation has been indicated
in Fig. 2.6a. It shows that Cγ become closer and closer to the equilibrium carbon con-
centration in the austenite at 1050K as the interface mobility increases, in other words,
the growth mode become more and more diffusion-controlled. When M∗
0 = 100M0,
Cγ is almost equal to Cγeq(1050K) during the entire austenite to ferrite transformation,
which means the phase transformation is almost purely diffusion-controlled. This is
why the kinetics predicted by the mixed-mode model is more or less the same as that
20 Chapter 2. The cyclic phase transformation concept
(a)
(b)
Figure 2.6: (a) Carbon concentration at the interface in the austenite phase Cγ during the cyclicaustenite to ferrite transformation and (b) carbon concentration at the interface in the ferritephase Cα during the cyclic ferrite to austenite transformation as a function of time.
by the diffusion-controlled growth model as shown in Fig. 2.4. Fig. 2.6b indicates the
carbon concentration at the interface in the ferrite Cα as a function of time during the
cyclic ferrite to austenite transformation. It shows that Cα decrease as the interface
mobility increases, in other words, the growth mode would be also closer and closer to
pure diffusion-controlled. In [2], the mixed-mode character has been quantified by a
mode parameter S defined for the case of the normal austenite to ferrite transformation
2.4. Results and Discussion 21
starting from a compositionally homogenous austenite. As there is no carbon gradient
in the ferrite phase during the normal austenite to ferrite transformation, the carbon
concentration at the interface in the austenite phase will never be lower than the initial
carbon concentration in the austenite phase no matter how small is the interface mo-
bility. Therefore, the defined growth mode parameter S will change from 1 to 0 during
the normal austenite to ferrite transformation. S=0 means a pure diffusion-controlled
transformation, while S=1 means pure interface-controlled with the assumption that
the bulk carbon concentration in austenite is equal to the carbon concentration in the
growing ferrite. However, in the cyclic transformation in the intercritical region, the
homogenous starting condition dose not apply as the formation of pre-existing ferrite
must have led to solute partitioning and a composition difference at the interface. Ac-
cording to the definition of S in Ref [2], S would be larger than 1 in the initial stage of
the cyclic austenite to ferrite transformation when M∗
0 = M0 , which shows that S is
not effective in quantifying the growth mode any more in the cyclic phase transforma-
tion. Hence a new definition of the character of the growth mode is required. Here,
a new growth mode parameter H will be defined which applies to the cyclic phase
transformations in the intercritical region
Hγ→α =Cγ
eq − Cγ
Cγeq − Cα
eq(2.12)
Hα→γ =Cα− Cα
eq
Cγeq − Cα
eq(2.13)
When Cγ = Cαeq for the cyclic austenite to ferrite transformation and Cα = Cγ
eq for the
cyclic ferrite to austenite transformation, both Hγ→α and Hα→γ are equal to 1.0, which
means pure interface-controlled growth here. The magnitude of the diffusion coeffi-
cient in the austenite and ferrite phase does not have an effect on the transformation
rate in this condition, which is only controlled by the interface mobility. Also, there
is no composition variance at the interface during the growth. Therefore, H is more
physically reasonable and general to define the growth mode, but it has to be stressed
22 Chapter 2. The cyclic phase transformation concept
that the underlying concept of H is identical to that of S .
(a)
(b)
Figure 2.7: The growth mode parameter H as a function of time for (a) the cyclic ferrite toaustenite and (b) the cyclic austenite to ferrite transformation.
The growth mode parameter H as a function of time for the cyclic ferrite to austenite
and the cyclic austenite to ferrite transformation are calculated in Fig. 2.7. The growth
mode parameter H for both the ferrite to austenite and austenite to ferrite transforma-
tion calculated by the new definition decrease as the phase transformation proceeds,
which implies that the growth mode deviate more and more from pure interface-
controlled growth. As the interface mobility used here is not low enough, the growth
mode parameter H is not equal to 1.0 when the interface starts to migrate. Considering
the soft impingement effect, the growth mode H for both the ferrite to austenite and
the austenite to ferrite transformation would decrease to 0 when the thermodynamic
equilibrium is established.
2.4. Results and Discussion 23
Figure 2.8: The ratio of the ferrite to austenite transformation rate and the austenite to ferritetransformation rate
(d f/dt)α→γ(d f/dt)γ→α
in the initial stage as a function of M∗0, assuming Mγ→α0 = Mα→γ
0 =
M∗0
In Fig. 2.8, assuming Mγ→α0 = Mα→γ
0 = M∗
0 , the ratio of the ferrite to austenite trans-
formation rate and the austenite to ferrite transformation rate (d f/dt)α→γ(d f/dt)γ→α
in the initial stage
is calculated as a function of M∗
0, M∗
0 is the temperature independent pre-exponential
factor used in the calculation. Similar with the prediction of diffusion-controlled
growth model, the mixed model also predicts that the ferrite to austenite transfor-
mation is faster than the austenite to ferrite transformation. Unlike the prediction of
diffusion-controlled growth model, the mixed-mode model predicts that the (d f/dt)α→γ(d f/dt)γ→α
increases as the M∗
0 increases until both the ferrite to austenite and austenite to ferrite
transformation are purely controlled by diffusion.
In Fig. 2.9, the ratio of the ferrite to austenite transformation rate and the austenite to
ferrite transformation rate (d f/dt)α→γ(d f/dt)γ→α
in the initial stage is calculated as a function ofMα→γ
0
Mγ→α0
.
For a certain value of Mγ→α0 , the ratio of the ferrite to austenite transformation rate and
the austenite to ferrite transformation rate increases as the ratioMα→γ
0
Mγ→α0
increases until
the ferrite to austenite transformation is completely diffusion-controlled. There is a
certain value (less than 1.0) ofMα→γ
0
Mγ→α0
at which the ferrite to austenite transformation is as
fast as the austenite to ferrite transformation. Below this value, the ferrite to austenite
24 Chapter 2. The cyclic phase transformation concept
Figure 2.9: The ratio of the ferrite to austenite transformation rate and the austenite to ferrite
transformation rate(d f/dt)α→γ(d f/dt)γ→α
in the initial stage as a function ofMα→γ
0
Mγ→α0
.
transformation would be slower than the austenite to ferrite transformation. As the
Mγ→α0 increases, the transformation rate ratio would be closer and closer to the ratio
predicted by diffusion-controlled growth model, and the transformation rate ratio at
differentMα→γ
0
Mγ→α0
predicted by mixed mode model would be the same as those by diffusion
controlled model when the Mγ→α0 is infinite. Here, the comparison between the two
transformations is made by varying the temperature independent pre-exponential
factor since this allows a straightforward comparison of result independent of the two
transformation temperature chosen in the calculations. The comparison can be easily
transformed into a comparison by varying the interface mobility, since the ratio of the
two interface mobilities is proportional to the ratio of pre-exponential factors, where
the proportionality depends on the type of relationship between the interface mobility
and the pre-exponential factor.
In order to retrieve quantitative values for the interface mobility, the kinetic mod-
els are usually fitted to experimental transformation curves by varying the interface
mobility, and the value of interface mobility which can make the model fit experimen-
tal transformation curves with the minimum error is considered to be the interface
mobility of the phase transformation. Normally, the model is fitted to the entire exper-
2.5. Summary 25
imental transformation curves, but the initial stage of the cyclic phase transformation
is considered to be more appropriate for retrieving interface mobility for the follow-
ing reasons: (i) as discussed above, the transformation kinetics is more sensitive to
interface mobility in the initial stage. At the later stage of phase transformation, the
transformation rate predicted by the mixed-mode model should be almost the same as
the transformation rate predicted by the diffusion-controlled growth model, and the
effect of the interface mobility on the transformation kinetics is close to zero, which
means the mixed-mode model can fit the later stage of phase transformation by any
interface mobility. (ii) There is no soft impingement and hard impingement in the
initial stage of phase transformation, thus assumptions for correcting soft impinge-
ment and hard impingement are avoided for building models, which make the fitting
more accurate. Based on the discussion above, the initial stage of the phase trans-
formation should be used to retrieve the value of interface mobility, however, fitting
a kinetic model assuming site-saturation nucleation with the initial stage of normal
phase transformation would underestimate the value of interface mobility if the nu-
cleation process is not finished instantaneous before growth process. Therefore, the
initial stage of cyclic phase transformation without nucleation would be the proper
stage for retrieving interface mobility.
2.5 Summary
The cyclic phase transformation in the intercritical region is very promising for inves-
tigating the interface mobility of partitioning phase transformation more accurately as
assumption of nucleation is avoided during modeling growth kinetics. By analyzing
the cyclic phase transformation, the following conclusions can be reached:
1. Pure interface-controlled growth model and pure diffusion-controlled growth
model are just two extreme cases of the mixed-mode model, and the partitioning
phase transformation kinetics predicted by the mixed-mode model is always
equal or slower than that by pure diffusion-controlled model.
26 Chapter 2. The cyclic phase transformation concept
2. The initial stage of the cyclic partitioning phase transformation is the appropriate
stage for retrieving the value of interface mobility.
3. According to the diffusion controlled phase transformation model, the ferrite
to austenite transformation should always be faster than the austenite to ferrite
transformation during the initial stages of the transformation. While the ratio of
the ferrite to austenite transformation rate and the austenite to ferrite transfor-
mation rate is a function of the interface mobility ratio according to the mixed
mode model prediction.
4. The new model allows the determination of the ratio of the two interface mobili-
ties from experimental transformation curves.
Chapter 3The kinetics of cyclic phase
transformations in a lean Fe-C-Mn
alloy
This chapter is based on
• H Chen, B Appolaire, S van der Zwaag, Application of cyclic partial phase
transformations for identifying kinetic transitions during solid-state phase
transformations: Experiments and Modeling , Acta Mater, 2011; 59:6751-6760.
3.1 Introduction
As discussed in Chapter 2, the kinetics of the austenite to ferrite transformation in the
binary Fe-C alloy is only determined by carbon diffusion and the value of interface
mobility. However, for ternary alloys Fe-C-M (M=Mn, Ni, Cr, Mo . . . ), the formation
of full local equilibrium at the moving interface is much more complicated than for
simple binary alloys due to the addition of the substitutional element M [58]. Based on
different assumptions for the partitioning mode of substitutional elements, two mod-
els have been proposed for describing the phase transformation kinetics in ternary
Fe-C-M alloys: (i) the local equilibrium (LE) model [56, 57], in which the interface is
28 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
assumed to migrate under full local equilibrium with the partitioning of both C and
M. Depending on the alloy composition and temperature, the transformation rate is
determined either by carbon diffusion or M diffusion. (ii) The paraequilibrium model
(PE) [54,55] which relies on constrained equilibrium: it indeed assumes that the phase
transformation in Fe-C-M alloys can proceed without any redistribution of M and that
the chemical potential of carbon across the interface should be constant. Hence, the
transformation rate is only determined by carbon diffusion. In the last decades, these
two models have been widely applied for describing the growth kinetics of partition-
ing phase transformations in Fe-C-M alloys, and their respective relevance has been
discussed at length [7, 9, 91, 92]. Although much effort has been paid to address this
issue, there are still many uncertainties about the growth mode of partitioning phase
transformations in Fe-C-M alloys [58].
As proved in Chapter 2, the cyclic phase transformation approach is quite promising
for investigating the growth kinetics as the nucleation effect is avoided. In this chapter, a
series of cyclic phase transformation experiments in the γ+α two-phase region of a lean
Fe-Mn-C alloy are done using dilatometry to study the effect of alloying elements on the
migrating austenite/ferrite interfaces. The corresponding cyclic phase transformations
experiments are also simulated by local equilibrium and paraequilibrium models. A
detailed comparison between the experiments and simulation is made. The growth
mode transitions during the γ → α transformation and vice versa are discussed, and
some suggestions for some improvements of the growth models are made.
3.2 Experimental
The material investigated here is a high purity Fe-0.17Mn-0.023C (wt. %) alloy with
impurities 0.009 wt. % Si , 0.006 wt. % Ni and 0.008wt. % Cu. A Bahr 805A dilatometer
is used to measure the dilation of the specimen (10 mm in length and 5 mm in diameter)
during the cyclic experiments. Two type S thermocouples, spaced 4 mm apart, were
3.2. Experimental 29
spot welded to the sample to have an accurate temperature measurement and to check
for the absence of a significant temperature gradient along the sample. The measured
temperature gradient along the sample was always smaller than 5 K. The heat treat-
ment procedures for the cyclic experiments in the Fe-0.17Mn-0.023C (wt. %) alloy can
be divided into type I (immediate) and type H (holding) experiments, as shown in
Figs. 3.1a-b. In both experiments, the as received material is first full austenization at
1000C and then cooled down to T1 for 20 min isothermal holding to create a mixed
ferrite-austenite microstructure with minimal compositional gradients. In type I exper-
iments, the temperature is cycled between T1 and T2 without any isothermal holding
at the two heating-cooling inversion temperatures. In type H experiments, the tem-
perature is also cycled between T1 and T2 but with isothermal holding (t = 20 min) at
both temperatures. Both T1 and T2 are located in the α+γ two-phase field in the phase
diagram. The cooling rate and heating rate during cycling in both I and H experiments
were 10C/min. Typically, 3-5 temperature cycles were imposed per experiment. The
experiment conditions are summarized in Tab. 3.1. The reported f γeq(T1) and f γeq(T2)
values in Tab. 3.1 are the Thermocalc calculated equilibrium fractions of austenite at
T1 and T2, respectively.
time
austenitization
A3
A1
temperature
T2
T1
time
austenitization
A3
A1
temperature
T2
T1
(a) (b)
Figure 3.1: The heat treatment procedures for (a) type I and (b) type H cyclic experiments.
30 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
Table 3.1: The cyclic experiments conditions (A3 = 896C and A1 = 729C).
Experiments T1 (C) f γeq(T1) T2 (C) f γeq(T2) Mode Figure1 860 20% 885 57.5% I 3.2a2 860 20% 895 97% I 3.2b3 870 28.3% 895 97% I 3.2c4 885 57.5% 895 97% I 3.2d5 860 20% 885 57.5% H 3.3
3.3 Models
The local equilibrium model and the paraequilibrium model for the cyclic partial phase
transformations in Fe-Mn-C alloys are summarized below.
3.3.1 Local equilibrium model
As any diffusion-controlled model, the LE model requires solving Fick’s second law
for all the alloying elements involved in the process, in both austenite and ferrite:
∂xφi∂t
= ∇ ·(Dφ
i ∇xφi)
(3.1)
where φ stands for α or γ, and i for C or M. xφi is the mole fraction of species i in phase
φ, and Dφi are the diffusion coefficients of C and M in α and γ, which are concentration
dependent in a way related to the thermodynamic description of the phases.
The partial differential equations must be provided with suitable initial and bound-
ary conditions. Zero-flux conditions are set at the “outer” boundaries for symmetry
and to account for interactions between neighboring grains. At the moving interface,
the local equilibrium assumptions provide the constitutive laws which determine the
interfacial concentrations. Indeed, in the LE model [56, 57], both carbon and substitu-
tional element M partition according to local equilibrium assumptions, which means
that the chemical potentials of carbon and M should be constant across the interface.
Hence:
µγi = µαi (3.2)
3.3. Models 31
where µφi is the chemical potential of element i at the interface in phase φ. Moreover,
mass balances must be satisfied for both C and M at the interface:
Jγi − Jαi = v (xγ∗i − xα∗i ) (3.3)
where xφ∗i is the interface concentration of i in phase φ, Jφi the diffusion flux of i in phase
φ, and v the interface migration velocity.
Based on Eqs (3.1-3.3), the equilibrium concentrations at the interface in both α
and γ as well as the migration velocity can be determined at every time step, and
consequently the position of interface as a function of time can be calculated. Due to
the large difference in the diffusivities of C and M, there are two different partitioning
modes of M during the phase transformations. In the first mode, the transformation
kinetics is fast and controlled by carbon diffusion. The concentration of M in the
growing phase is the same as that in the parent phase, but due to LE conditions a
“spike” of M is moving ahead of the interface. Thus, this mode has been termed
“Local equilibrium with negligible partitioning” (LE-NP) [56,57]. In the second mode,
the carbon concentration gradient in the parent phase is almost negligible while that
of M is large. Hence the transformation kinetics is slow and controlled by diffusion of
M. This mode has been termed “Local equilibrium with partitioning” (LE-P) [56, 57].
3.3.2 Paraequilibrium model
In the PE model [54, 55], the substitutional element M is supposed not to redistribute
among α and γ at the interface, whereas the chemical potential of C remains constant
across the interface. This constrained equilibrium is expressed as:
µγC = µαC (3.4)
(µγM − µαM) = −
xFe
xM(µγFe − µ
αFe) (3.5)
32 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
where the last equation proceeds from the constant ratio xM/xFe across the interface.
Solving the diffusion equations for C in both α and γ, together with the mass balance
for C at the interface and the previous PE conditions Eqs. (3.4-3.5) gives the interface
concentrations as well as the kinetics of the PE transformation.
3.4 Experimental results
3.4.1 Measured kinetics of the cyclic phase transformations
855 860 865 870 875 880 885 890 895 900
116
117
118
119
120
121
122
123
124
125
Temperature / °c
Leng
th c
hang
e /
µm
Type I−cycling between 885 °c and 860 °cA
1
A2
A7
A6
A5
A4
A3
A8
855 860 865 870 875 880 885 890 895 900
105
110
115
120
125
Temperature / °c
Leng
th c
hang
e /
µm
Type I−cycling between 895 °c and 860 °c
A8
A1
A2
A7
A6
A3
A4
A5
855 860 865 870 875 880 885 890 895 900
110
112
114
116
118
120
122
Temperature / °c
Leng
th c
hang
e /
µm
Type I−cycling between 895 °c and 870 °c
A3
A5
A4
A7
A8
A6
A1
A2
885 890 895
114
115
116
117
118
119
120
Temperature /°c
Len
gth
ch
ang
e /µ
m
Cycling between 895 °c and 885°c
A10
A9
A6
A2
A1
A3
A8
A5
A4
A7
(a) (b)
(c) (d)Figure 3.2: The dilation as a function of temperature during type I cyclic experiments between(a) 885C and 860C, (b) 895C and 860C, (c) 895C and 870C, (d) 895C and 885C.
Fig. 3.2a shows the dilation as a function of temperature during the type I cyclic
experiment between 860C and 885C (experiment 1, Tab. 3.1). Two distinctively dif-
ferent stages can be distinguished during the first heating cycle: (i) a linear thermal
expansion stage (A1-A2 in Fig. 3.2a) during which no phase transformation or interface
migration takes place and which is called the “stagnant stage” in the present work. (ii)
A contraction stage (A2-A3 in Fig. 3.2a) due to the α → γ transformation on heating,
during which the interface migrates into the existing ferrite phase. Once the maximum
temperature of 885C is reached, the specimen is cooled down immediately. Three
3.4. Experimental results 33
stages can now be observed during the cooling down process: (i) a nonlinear contrac-
tion (A3-A4 stage) due to a continuation of the α→ γ transformation, notwithstanding
cooling of the sample. To distinguish the α → γ transformations in the A3-A4 stage
from that in the A2-A3 stage, the A3-A4 stage is called the “inverse transformation
stage” in the present work; (ii) a linear thermal contraction (A4-A5 stage) without
obvious phase transformation taking place, again called a “stagnant stage”. (iii) A
nonlinear expansion (A5-A6 stage), which is attributed to the γ → α transformation.
Unlike the first cycle, a nonlinear expansion due to the “inverse” γ→ α transformation
is also found at the onset of the heating stage in the second cycle. The remainder of
the second cycle is very similar to that of the first cycle. The third and fourth cycles
exhibit the same features as that of the second cycle.
Fig. 3.2b indicates the dilatation as a function of temperature during a type I cyclic
experiment between 860C and 895C (experiment 2, Tab. 3.1). In this case the upper
temperature of 895C is close to the A3 temperature. While the total dilation is larger, all
stages of the transformation, including the stagnant and inverse transformation stages
are present. The magnitude of the inverse phase transformation stage in Fig. 3.2b
is however significantly smaller than that in Fig. 3.2a. Finally, in a new experiment
(experiment 3, Tab. 3.1) the lower cyclic transformation is increased to 870C. The
character of the transformation loop as shown in Fig. 3.2c is preserved, but the slowing
down of the transformation upon cooling to the lower transformation temperature is
less.
To prove that there is no phase transformation in the stagnant stage, the length
change of a sample, which was cyclically annealed between 895C and 885C (experi-
ment 4, Tab. 3.1) (later to be followed by full cyclic annealing between 895C and 860C)
, is shown in Fig. 3.2d. It shows that the length changes of A2-A3, A5-A6 and A8-A9
stages during cooling almost overlap with those during heating, which means the
length changes in these stages are only attributed to thermal contraction or expansion
and further support the conclusion that there is no or only a marginal phase transfor-
mation during the stagnant stages in Figs. 3.2a-c. The slope of the length change curve
34 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
in the inverse α → γ transformation stage is decreasing slightly with the number of
annealing cycles. The kinetics of the subsequent cyclic annealing between 895C and
860C (not shown here) is highly similar to that in Fig. 3.2b.
855 860 865 870 875 880 885 890 895 900
110
115
120
125
130
Temperature / °c
Leng
th c
hang
e /
µm
Type H−cycling between 885 °c and 860 °c
B3
B4
B5
B6
B1
B2
Figure 3.3: The dilation as a function of temperature during type H cyclic experiments between885C and 860C.
Fig. 3.3 shows the length change as a function of temperature during a type H
cyclic experiment between 885C and 860C (experiment 5, Tab. 3.1). The cyclic phase
transformation curve for H cyclic condition resemble the curves for type I experiments
with some interesting differences at and just after isothermal holding. The curve for
the H experiment exhibits stagnant stages and the normal phase transformation stage,
while the inverse phase transformation stage is absent. Instead, an abrupt length
change is observed after the isothermal holding at 885C and 860C and switching
from heating to cooling and vice versa.
Finally, Figs. 3.2-3.3 show that the length changes during cyclic transformations
are not fully reversible as minor length changes occur between subsequent cycles.
These minor changes are attributed to transformation plasticity [93–95]. Given the fact
that these changes are very minor and can not be attributed to specific stages of the
3.5. Simulation results 35
transformation cycle, they are ignored in the subsequent analysis.
3.4.2 Microstructure
In order to check that nucleation of new ferrite grains is unlikely to occur during
the cyclic experiments, the average grain size has been measured in samples having
undergone several cyclic partial phase transformations. For that purpose, the ferrite
grain boundaries were revealed by etching with a 2.5 vol.% Nital solution, and the mi-
crostructures after the cyclic phase transformations were analyzed by light microscopy.
The line intercept method was employed in three different directions in order to de-
termine the mean grain size. The average diameters of the ferrite grains after the type
I and H cyclic experiments are 54.8 and 51.4 µm, respectively. The average grain size
after different cyclic experiments was found to be effectively constant.
3.5 Simulation results
In this section, the cyclic phase transformation experiments are simulated using the
well known Dictra software [96] and imposing either local equilibrium (LE) or parae-
quilibrium (PE) conditions. A planar geometry was used here, and the half thickness of
the system was assumed to be 25 µm, which is close to the measured ferrite grain size
after the cyclic phase transformations. The cooling and heating rates in the simulation
were set at 10 K/min, which is the same as these in the experiments.
3.5.1 Local equilibrium
First, an isothermal phase transformation at 860C was simulated to obtain realis-
tic initial conditions in terms of C and Mn profiles for the subsequent cyclic phase
transformations. The C and Mn profiles during the isothermal phase transformation
indicate that the growth mode of isothermal phase transformation at 860C occurs
via local equilibrium with negligible partitioning (LE-NP), in agreement with the fact
36 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
that the nominal composition of the Fe-Mn-C alloys studied here lies in the LE-NP
region according to Thermo-Calc calculations. The interface stops migrating after 43 s.
However, Mn continues to diffuse in both α and γ phase while the interface is almost
immobile. At the end of the isothermal holding, no C gradient exists in both phases,
while there is a narrow Mn profile in front of the interface in both α and γ phase.
855 860 865 870 875 880 885 890 895 9008
10
12
14
16
18
20
22
Temperature/ °c
inte
rfac
e po
sitio
n/ µ
m
c3c
5
PELE
c2
c4
c1
c6
855 860 865 870 875 880 885 890 895 9008
10
12
14
16
18
20
22
Temperature /°c
inte
rfac
e po
sitio
n /µ
m
LE
PE
D2
D6
D1
D5 D
4
D3
(a)
(b)Figure 3.4: The α/γ interface position as a function of temperature during (a) the type I and (b)the type H cyclic phase transformations between 885C and 860C simulated under both localequilibrium and paraequilibrium conditions.
In Fig. 3.4, the α/γ interface position predicted by the LE model is plotted in blue
as a function of temperature during the type I and H cyclic experiments between
885C and 860C. The calculations related to type I cycles in Fig. 3.4a predict features
very similar to those observed in experiments. The sluggish stages C1-C2 and C4-C5
in Fig. 3.4a are comparable to the stagnant stages A1-A2 and A4-A5 in experiments
(Fig. 3.2), respectively, and the inverse phase transformation stage A3-A4 is reflected
3.5. Simulation results 37
accordingly by the C3-C4 stage. In Fig. 3.4b, the calculations related to the type H cycles
between 885C and 860C predict evolutions also comparable with those observed in
experiments. Indeed, the sluggish stages at the onset of cooling or heating predicted by
the LE model correspond to the stagnant stages B1-B2 and B4-B5 (Fig. 3.3). Moreover, no
inverse α→ γ transformation shows up in the simulation, which is also in agreement
with the experiments.
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
Distance/ µm
Car
bon
conc
entr
atio
n /w
t.%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e po
sitio
n /µ
m
c3c
5
c6
c1
c2
c4
0 5 10 15 20 250.08
0.1
0.15
0.2
0.25
0.3
0.32
Distance/ µm
Mn
conc
entr
atio
n /w
t.%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e po
sitio
n /µ
m c2
c6
c1
c3c
4c
5
(a)
(b)Figure 3.5: The evolution of C and Mn profiles in the C1-C3 stage during type I cyclic phasetransformations between 885C and 860C predicted by the local equilibrium model. Thepoints for which the C and Mn profiles are shown are indicated with the same type of symboland color in the interface position curve correspondingly.
In order to have a better understanding of the cyclic phase transformation kinetics
predicted by the LE model, the evolution of the C and Mn profiles during the successive
38 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
stages C1-C3, C3-C4 and C4-C6 are presented in Figs. 3.5, 3.6 and 3.7 respectively. As
shown in Fig. 3.5a, the C profiles in both α and γ differ only marginally from the initial
C profiles in the C1-C2 stage during heating, with almost no C gradient in austenite.
The Mn concentrations at the interface in both α and γ decrease during this stage,
which causes a significant depletion of Mn next to the interface in α (Fig. 3.5b): the Mn
concentrations are different between α at the interface and γ in the bulk, which means
that the system is shifting towards a slow LE-P mode. In the C2-C3 stage, the Mn
concentrations at the interface in α and γ are decreasing continuously upon heating,
making appear a depleted spike in α. At a certain temperature, the Mn concentration
at the interface in the growing γ becomes equal to the Mn concentration in bulk α. At
the same time, a positive C gradient is building up in γ. Both features mean that the
system evolves from slow LE-P to fast LE-NP in the C2-C3 stage. It is interesting to
note that the initial Mn spike in γ (C1) is left behind the interface in the C2-C3 stage,
since the interface migrates much faster than Mn diffuses in γ.
As shown in Figs. 3.6a and 3.6b, at the beginning of cooling in the C3-C4 stage,
the interface continues to migrate into α: the inverse α → γ transformation is thus
predicted by the LE model. In this stage, the Mn concentrations at the interface in both
α and γ increase as temperature decreases. When the depleted spike in α which has
developed during the previous stage shrinks slightly, a new Mn spike is building up in
γ: the Mn profile exhibits a zigzag shape at the interface position (green and red curves
in Fig. 3.6b) with two gradients on both interface sides which counterbalance each
other. Concomitantly, the C gradient in γ formed during heating diminishes during
the C3-C4 stage. These evolutions indicate that the system switches from fast LE-NP
to slow LE-P in the C3-C4 stage. It is worth noting that, at the same temperature, the
partitioning mode of Mn during the inverse α → γ transformation (transition from
LE-NP to LE-P) is different from that during the α→ γ transformation on heating (LE-
NP). This difference clearly illustrates in a single experiment that the transformation
kinetics is totally controlled by the interface conditions.
The evolution of C and Mn profiles in the C4-C6 stage is presented in Figs. 3.7a-b.
3.5. Simulation results 39
0 5 10 15 20 250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Distance/ µm
Car
bo
n c
on
cen
trat
ion
/wt.
%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e p
osi
tio
n /µ
m
c6
c3c
4c
5
c1
c2
0 5 10 15 20 250.08
0.1
0.15
0.2
0.25
0.3
0.32
Distance/ µm
Mn
co
nce
ntr
atio
n /w
t.%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e p
osi
tio
n /µ
m
c6
c3c
4c
5
c2
c1
(a)
(b)Figure 3.6: The evolution of C and Mn profiles in the C3-C4 stage during type I cyclic phasetransformations between 885C and 860C predicted by the local equilibrium model. Thepoints for which the C and Mn profiles are shown are indicated with the same type of symboland color in the interface position curve correspondingly.
After the inverse α → γ stage, the interface remains almost immobile: this is quite
comparable with the stagnant stage in the experiments. During the C4-C5 stage, there
is no carbon gradient in both α and γ, but there are still Mn gradients in both α and
γ with the zigzag shape already observed in the C3-C4 stage: the system is pinned in
the LE-P mode. Progressively, the interface Mn concentrations increase, destroying
the zigzag shape: when the Mn spike in γ is growing, the depleted Mn spike in α
40 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
diminishes, with an inversion of the Mn gradient in α at the interface (green and red
curves in Fig. 3.7b).
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
Distance/ µm
Car
bo
n c
on
cen
trat
ion
/wt.
%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e p
osi
tio
n /µ
m
c1
c6
c2
c3
c4
c5
0 5 10 15 20 250.08
0.1
0.15
0.2
0.25
0.3
0.32
Distance/ µm
Mn
co
nce
ntr
atio
n /w
t.%
850 860 870 880 890
12
14
16
18
20
Temperature / °c
inte
rfac
e p
osi
tio
n /µ
m
c6
c3c
5 c4
c2
c1
(a)
(b)Figure 3.7: The evolution of C and Mn profiles in the C4-C6 stage during type I cyclic phasetransformations between 885C and 860C predicted by the local equilibrium model. Thepoints for which the C and Mn profiles are shown are indicated with the same type of symboland color in the interface position curve correspondingly.
At the very beginning of the C5-C6 stage, the Mn concentration at the interface in
the growing α becomes the same as the Mn concentration of bulk γ, with a single Mn
spike moving in γ, ahead of the migrating interface. This is the signature that the
system has switched to fast LE-NP mode. Finally, during the γ → α transformation,
3.6. Discussion 41
the zigzag formed in the C4-C5 stage is left behind the interface in α, and is rapidly
smoothed out by the significant Mn diffusion in α.
3.5.2 Paraequilibrium
In Fig. 3.4, the α/γ interface position computed with the PE model is plotted with red
lines as a function of temperature during the type I and H cycles between 885C and
860C. As shown in Fig. 3.4a, the PE model predicts a very short inverse transformation
stage and no sluggish stage at all for the type I cycles, in total disagreement with the
experimental measurements (Fig. 3.2). Concerning the type H cycles in Fig. 3.4b,
the predicted kinetics in which no stagnant stage is detected, also deviates from the
experimental results in Fig. 3.3.
3.6 Discussion
3.6.1 Stagnant stages
These stagnant stages are easily mistaken to be incubation times for nucleation if they
are observed in usual experiments (100% parent phase→100% new phase). Unlike the
situation in these usual experiments, the cyclic partial phase approach reported here
always involves α+γmixtures at all stages of the cycle, which means that nucleation of
either α or γ is not compulsory during the cyclic phase transformations. Furthermore,
after several partial transformation cycles, the ferrite grain size has barely changed even
significantly, which can indirectly prove that there is no substantial nucleation during
the cyclic phase transformations. Even if there were new nucleation events during the
cyclic annealing, the existing α/γ interface could in principle migrate in the initial stage
of cooling or heating right after changing the direction of healing to cooling and vice-
versa . Therefore, the stagnant stage can be attributed to a real growth kinetic issue.
As shown above, the LE model considering Mn partitioning predicts sluggish phase
transformation stages during both type I and H cycles, which are very comparable to
42 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
the stagnant stages observed in the cyclic experiments. According to the LE prediction,
the kinetics of the sluggish stage is controlled by the Mn partitioning. Fig. 3.4 shows
that both γ→ α transformation at 860C and α→ γ transformation at 885C according
to the PE model are predicted to be incomplete, and to remain in the α+γ region. From
a thermodynamic point of view, it seems reasonable to assume that the kinetics of the
cyclic phase transformations could be modeled by a model based on paraequilibrium.
However, the paraequilibrium model does not predict any sluggish or stagnant stage
upon reversal of cooling-heating. Therefore, it can be concluded that the LE model
considering local Mn partitioning is more appropriate as it explains the stagnant stages
observed in the cyclic phase transformations.
3.6.2 Inverse transformation stages
A comparison between the A3-A4 stage in Fig. 3.2a and the B4-B5 stage in Fig. 3.3
makes it clear that the non-linear contraction stage A3-A4, which follows the reversal
from heating to cooling, is due to the α → γ transformation and not to experimental
errors. The lever rule can be used to calculate the fraction of α or γ during the cyclic
experiments. The γ fraction at A3 in type I experiments (Fig. 3.2a) is 49.8%, while the
equilibrium fraction of γ is 57.5%. Hence the transformation proceeds in the original
direction although the heating is changed to cooling. In contrast, in the H experiment
(experiment 5, Tab. 3.1 and Fig. 3.3), the equilibrium fraction of γ at 885C is reached
after the isothermal holding, and no inverse transformation stage is observed during
cooling and heating. It can be concluded that the occurrence of the inverse phase
transformations in type I experiments is caused by non-equilibrium conditions at the
heating-cooling reversal temperatures T1 and T2.
Both the LE model and the PE model can predict the inverse phase transformation
stage. However, compared with the PE prediction, the simulation by the LE model
shows a substantially longer inverse transformation stage, which is more comparable
with the experimental results. In the LE model, as there is Mn partitioning during the
3.6. Discussion 43
interface migration, it takes time to adjust the local conditions (including Mn and C
gradients) at the interface for the γ→ α transformation upon cooling. On the contrary,
in the PE simulation, as it is not necessary to alter the Mn profile at the interface,
the local conditions at the interface can easily be adjusted when the temperature is
changed. Hence, it is clear that the time for adjusting the local conditions at the interface
determines the duration of the inverse transformation stages in the simulations. If it
has not been possible in the present work to predict quantitatively the kinetics of the
inverse transformation stages, we have clearly elucidated the role played by the Mn
partitioning in the duration of these stages.
3.6.3 Non-equilibrium interface conditions
Although the LE model here can predict the features of the cyclic phase transforma-
tion kinetics qualitatively, there are still many discrepancies between experiments and
simulations. (i) The simulation predicts a much faster kinetics for both the γ→ α and
α→ γ transformations than the experiments indicate. (ii) There is difference between
the simulated sluggish stages (C1-C2, C4-C5, D1-D2 and D4-D5) and the corresponding
stagnant stages in the experiments. The interface in the sluggish stages is still migrating
slowly, while in the experiments the interface seems completely immobile or at least
slower than in the computed sluggish stages. This difference is very obvious between
B4-B5 stage in Fig. 3.3 and D4-D5 stage in Fig. 3.4b. (iii) The sluggish stages in the cal-
culations are shorter than the stagnant stages in the experiments. These discrepancies
could be attributed to the fact that the dissipation of Gibbs energy by interface friction is
not considered in the classical diffusion controlled growth model. It was indeed stated
in [2,66] for binary systems that the classical model always predicts kinetics faster than
kinetics calculated with the mixed-mode model considering a finite interface mobility.
It is very likely that the first and second discrepancies could be eliminated if a finite
interface mobility is included in modeling.
As the interface velocity is almost zero in the stagnant stages in Figs. 3.2, the dissipation
44 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
of Gibbs energy by interface migration is close to zero, and the interface concentrations
should be almost equilibrium concentrations according to the mixed-mode model. At
first sight, it seems that the mixed-mode model would predict the same interface con-
ditions in the stagnant stages as the LE model does, and the consideration of a finite
interface mobility would not affect the duration of the stagnant stages. However, it has
to be pointed out that, according to the mixed-mode model, the interface conditions
at A3 point in Figs. 3.2 should be off-equilibrium due to the fast interface migration
rate. It can be expected that the C concentration in austenite at A3 point predicted
by the mixed-mode model should be higher than those predicted by LE model, and
thus the starting temperature for the Mn non-partitioning growth predicted by the
mixed-mode model should be lower than that by LE model. Therefore, the mixed-
mode model would predict a longer sluggish phase transformation stages than the LE
model does, which would be more comparable with the experiments.
3.7 Conclusion
Cyclic partial phase transformation experiments in the intercritical region are very
promising for investigating the growth kinetics of partitioning phase transformations
more accurately as complicating concurrent nucleation processes do obscure the actual
growth kinetics. Unlike in the case of the usual full and single phase transformations,
two special stages have been observed in the cyclic phase transformations: the stagnant
stage and the inverse phase transformation stage. The stagnant stages (stages at which
the transformation does not proceed even for a substantional change in temperature) in
both the austenite to ferrite and ferrite to austenite transformations are mainly caused
by the Mn partitioning and are well described by the Local Equilibirum model. The
inverse phase transformation stages (stages at which the transformation proceeds in a
direction not in line with the change in heating to cooling or vice versa) were found to
be due to the equilibrium condition not being reached at the transition temperatures.
Moreover, mode switching from Mn partitioning to non-partitioning could be iden-
3.7. Conclusion 45
tified in the cyclic phase transformation experiments, both for the austenite to ferrite
and the ferrite to austenite transformations. In general, the LE model was found to
predict the observed experimental data for cyclic partial phase transformations rather
well. The PE model was found to be less capable of describing the observations, even
for conditions where PE was thermodynamically expected.
46 Chapter 3. The kinetics of cyclic phase transformations in a lean Fe-C-Mn alloy
Chapter 4Analysis of the stagnant stage during
cyclic phase transformations
This chapter is based on
• H Chen, M Goune, S van der Zwaag, Analysis of the stagnant stage in
diffusional phase transformations starting from austenite-ferrite mixtures ,Comp
Mater Sci, 2012; 55:34-43.
4.1 Introduction
In Chapter 3, a new stagnant stage phenomenon is observed during the cyclic
phase transformations in a Fe-0.17Mn-0.023C(all in wt.%), which was attributed to Mn
partitioning. In this chapter, the cyclic phase transformations in Fe-C, Fe-M-C (M=Mn,
Ni, Cu, Si, Co) and Fe-Mn-C-M (M=Ni, Si, Co) alloys have been simulated to investigate
the stagnant stage extensively. The effect of thermal history, alloy composition and
partitioning coefficient of the alloying elements on the stagnant stage is analysed in
detail.
48 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
4.2 Simulation conditions
In this work, the type I cyclic phase transformation approach is applied to investigating
the phase transformations starting from a mixture of austenite and ferrite. In type I
cyclic phase transformation, the temperature is cycled between T1 and T2 without any
isothermal holding at the two heating-cooling inversion temperatures. Both T1 and T2
are located in the α+γ two-phase field in the phase diagram. The temperature program
for type I simulation can be found in Fig. 3.1a [97]. The cyclic phase transformations are
simulated using the well known Dictra software [90,96] and imposing local equilibrium
(LE) conditions. A planar geometry was used, and the half thickness of the slab was
all assumed to be 25 µm. The configuration of LE model can be found in Ref [97].
4.3 Results and Discussion
In Fig. 4.1, a sketch of a typical cyclic phase transformation in a lean Fe-C-Mn al-
loy is given, indicating the inverse transformation, stagnant stage and normal phase
transformation stages. Point A is the starting point of the cyclic phase transformations.
AB and DE stages are inverse transformation stages during heating and cooling, re-
spectively. The inverse transformation stage is defined as part of the transformation
cycle during which the transformation proceeds in a direction inverse to a normal
transformation. i.e. austenite formation during cooling and ferrite formation during
heating. BC and EF stages are stagnant stages during heating and cooling, respectively.
The stagnant stage is defined as the part of the transformation cycle during which the
degree of transformation does not change even in the presence of a substantial tem-
perature change. CD stage is the normal ferrite to austenite transformation during
heating, and FA is the normal austenite to ferrite transformation during cooling.
In this work, only the stagnant stage is of interest, and the inverse transformation
stage will not be discussed. In order to describe the stagnant stage in a quantative
manner, the length of the stagnant stage is defined as
4.3. Results and Discussion 49
Figure 4.1: The sketch of a typical type I cyclic phase transformations. I=Inverse transformationstage(blue), S=Stagnant stage (red), N=Normal transformation stage (black).
S = Tstart − Tend (4.1)
where Tstart and Tend are the start and end temperatures of the stagnant stage, respec-
tively.
4.3.1 Fe-C alloy
In Fig. 4.2, a type I cyclic phase transformation in the austenite + ferrite region has
been simulated for Fe-0.02C (in wt.%) alloy. No signs of stagnant stage is found during
both heating or cooling.
In this simulation, three temperature cycles have been applied, and the interface po-
sitions of different cycles repeat each other very well. The average interface migration
rate during the austenite to ferrite transformation is the same as that during the ferrite
to austenite transformation. In the local equilibrium model, the interface mobility is
assumed to be infinite, which means the interface migration rate is only controlled by
50 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
875 880 885 890 895 900 905
6
8
10
12
14
16
18
20
22
Temperature / °c
Inte
rfac
e po
sitio
n /
µm
Fe−0.02C
Ae3
Equilibrium interface position at T2
Figure 4.2: The α/γ interface position as a function of temperature during the type I cyclicphase transformations in Fe-C alloy under local equilibrium conditions.The concentration of Cis in wt.%
.
C diffusion in austenite and ferrite during the cyclic phase transformations. However,
In ref [2,6,98], it was stated that the interface mobility in Fe-C alloys is probably finite,
which means the transformation kinetics is controlled by both diffusion of C and in-
terface mobility. Furthermore, in ref [3], a finite effective interface mobility has been
explicitly applied to simulating the growth kinetics in Fe-Mn-C alloys, and the finite
mobility approach was confirmed by comparing experimental results with modelling
results. If a finite interface mobility is applied for both α → γ and γ → α transfor-
mation in Fe-C alloys , the interface position of different temperature cycles may not
reproduce to the same degree. Until now, there are still many discrepancies regarding
the exact value of interface mobility [53], so the quantitative simulations, considering
finite interface mobilities, are not possible here. Although the classical local equilib-
rium model can not predict the transformation rate very precisely, the basic features
of the cyclic phase transformation can be well modeled by it [97]. Therefore, the local
equilibrium model is qualified for the investigation of the stagnant stage phenomenon.
4.3. Results and Discussion 51
4.3.2 Fe-C-Mn alloys
In this section, 115 cyclic phase transformations between 885C and 860C for a fixed
alloy composition of Fe-0.02C-0.2Mn (in wt.%) with different combinations of cooling
and heating rate are simulated. The range of cooling and heating rates is from 5K/min
to 1500K/min.
(a)
(b)Figure 4.3: Theα/γ interface position as a function of temperature during the type I cyclic phasetransformations between 860C and 885C simulated under local equilibrium conditions byvarying (a) cooling rate and (b) heating rate.
As an example, the α/γ interface position as a function of temperature during the
type I cyclic phase transformations between 860C and 885C simulated under local
equilibrium conditions by varying (a) cooling rate and (b) heating rate in Fig. 4.3. The
52 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
stagnant stage during the cooling from 885 C to 860 C is of interest here. The full
description of all features is presented in Chapter 3. In Fig. 4.3a, the heating rate
is fixed at 150K/min, while the cooling rate varies from 5K/min to 300K/min. The
starting interface positions for different cooling stages are the same due to a fixed
heating rate. It shows that the length of stagnant stage extends with increasing the
cooling rate. This can be explained as: as the cooling rate increases, the time for the
inverse transformation stage decreases, which leads to less inverse transformation. The
interface approaches equilibrium position (11 µm) during the inverse transformation,
so the magnitude of inverse transformation finally determines the interface position at
the end of the inverse transformation stage or at the beginning of stagnant stage. As
the diffusion of Mn is very slow, there is just a very sharp Mn spike at the interface and
the Mn concentrations in austenite and ferrite are both equal to nominal concentration
during the cyclic phase transformations. During the stagnant stage, there is no carbon
gradients in both austenite and ferrite, which is also predicted in [97]. The mean carbon
concentrations in austenite and ferrite are solely determined by the fraction of austenite
and ferrite. The further the interface position at the end of inverse transformation
stage is away from equilibrium position, the higher the carbon concentration in the
austenite at the end of inverse transformation. An increase in carbon concentration
in austenite will decrease growth mode transition temperature, at which the growth
mode of austenite to ferrite transformation switches from ”Local equilibrium with
partitioning” (LE-P) into ”local equilibrium with negligible partitioning” (LE-NP). The
growth mode transition temperature is the same as the end temperature of stagnant
stage Tend here. Therefore, increase in cooling rate would lead to an extension in the
length of the stagnant stage if the heating rate is fixed.
In Fig. 4.3b, the cooling rate is fixed at 150K/min, while the heating rate varies
from 5K/min to 300K/min. The interface positions at 885 C are different due to
the different heating rates. It is indicated that both the magnitude of the inverse
transformation and the length of stagnant stage decrease with decreasing heating rate.
The interface position is approaching to the equilibrium position (11 µm) as the heating
4.3. Results and Discussion 53
rate decreases, which leads to less inverse transformation. It is interesting to note that
the inverse transformation stage almost disappears when the heating rate is lower
than 50K/min, although the interface positions do not reach equilibrium position at
T2. Similarly, the interface positions at the beginning of the stagnant stage determins
the length of stagnant stage in Fig. 4.3b. As the heating rate increases, the interface
position at the beginning of the stagnant stage deviates more from the equilibrium
position, the length of stagnant stage extends.
Figure 4.4: The contour for the length of stagnant stage
In Fig. 4.4, all simulation results are summerized qualitatively in a contour plot of
the length of the stagnant stage as a function of heating and cooling rate. It shows
the length of stagnant stage increases with increasing heating rate and cooling rate.
The effect of cooling rate on the length of stagnant stage decreases as the heating rate
decreases. The cooling rate has almost no effect on the length of stagnant stage if the
heating rate is very slow. This can be explained as follows: When the heating rate is very
slow, there is no inverse transformation and thus the interface position at the begining
of stagnant stage is the same for different cooling rates. It is the interface position
at the beginning of stagnant stage which finally determines the austenite conditions
and the length of stagnant stage. Therefore, the cooling rate would not influence the
54 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
length of stagnant stage if the heating rate is very slow. The effect of heating rate
on the length of stagnant stage diminishes as the cooling rate decreases. At a slow
cooling rate, the interface position would approach to equilibrium position during the
inverse transformation, thus the interface position at the beginning of stagnant stage
should be almost the same and close to equilibrium position although the heating rate
is different. Therefore, at a very slow cooling rate, the heating rate has no effect on the
interface position at the beginning of stagnant stage and thus length of the stagnant
stage.
Based on the discussion above, both the heating rate and cooling rate during the
cyclic phase transformation, which determines the interface positions at the beginning
of inverse transformation and stagnant stage, are considered to affect the length of stag-
nant stage. However, fundamentally speaking, it is the austenite and ferrite conditions
at the end of inverse transformation which determines the length of stagnant stage .
Any factor, which can change the austenite and ferrite conditions, would have effects
on the length of the stagnant stage. Hence, the cycling temperature and isothermal
holding time at T1 or T2 are likely to influence the austenite and ferrite conditions and
thus to affect the stagnant stage.
In addition to heat treatment parameters, the concentration of Mn is another inter-
esting factor which affects the length of stagnant stage. In Fig. 4.5a, the cyclic phase
transformations in Fe-C-Mn alloys with the same C concentration but different Mn
concentrations are simulated between 860C and 885C. The concentrations of Mn and
C are all in wt.%. The interface migrating distance during the cyclic phase transforma-
tion increases as the Mn concentration increases, which is because Mn is a γ stabilizer
and the Ae3 temperature decreases as the concentration of Mn increases. It shows that
the length of stagnant stage increses significantly due to the increase in Mn concentra-
tion. However, it is not proper to claim that the Mn concentration affects the length
of stagnant stage by Fig. 4.5a, as interval between the interface position and equilib-
rium interface position at the beginning of stagnant stage increases with increasing Mn
concentration, which means the carbon concentration in austenite at the beginning of
4.3. Results and Discussion 55
stagnant stage increases with increasing Mn concentration. The difference of carbon
concentration in austenite could be a factor affecting the length of stagnant stage.
860 865 870 875 880 885 890 895 900
6
8
10
12
14
16
18
20
22
Temperature / °c
inte
rfac
e po
sitio
n /
µm
Fe−0.1Mn−0.02CFe−0.2Mn−0.02CFe−0.3Mn−0.02C
A3
A3
A3
Equilibrium Interface position at 885 °c
860 865 870 875 880 885 890 895 9006
8
10
12
14
16
18
20
22
Temperature / °c
inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.1MnFe−0.02C−0.2MnFe−0.02C−0.3Mn
Equilibrium interface position at T2 Ae3Ae3Ae3
(a)
(b)
Figure 4.5: Theα/γ interface position as a function of temperature during the type I cyclic phasetransformations in Fe-C-Mn with the same C concentration but different Mn concentrationbetween (a) 860C and 885C and (b)different temperature combinations:T2 − T1 = 25C, andAe3 − T2 = 5C.The concentrations of Mn and C are all in wt.%
.
In order to keep the gap between the interface position and the equilibrium in-
terface position almost the same for different Mn concentrations, the cyclic phase
56 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
transformations are simulated for Fe-C-Mn alloys with the temperature combinations:
T2 − T1 = 25C, and Ae3 − T2 = 5C, as shown in Fig. 4.5b. In this simulation, the gap
between the equilibrium position and the interface position at T2 is almost the same
for different Mn concentrations, which means the carbon concentration in austenite is
almost the same for different Fe-C-Mn alloys. The carbon concentration in austenite is
not an affecting factor for the length of stagnant stage anymore. However, the length
of the stagnant stage still strongly depends on the Mn concentration, which means the
Mn concentration is really one of the key factors for determining the length of stagnant
stage.
4.3.3 Fe-C-M (M= Ni, Si, Cu, Co) alloys
In order to investigate the effect of ferrite stablizers and austenite stablizers on the
length of the stagnant stage, the cyclic phase transformations in Fe-C-M alloys (M=
Ni, Si, Cu, Co ) are simulated. Ni and Cu are austenite stablizers, while Si is a ferrite
stablizer. The heating and cooling rates are both fixed at 10K/min here. In Fig. 4.6, the
cyclic phase transformations are simulated for Fe-C-M (M= Ni, Cu, Si, Co) alloys with
the temperature combinations: T2−T1 = 25C, and Ae3−T2 = 5C. The concentrations
of M and C are all in wt.%.
It shows that the length of stagnant stage increases significantly as the Ni concen-
tration increases, while the concentration of Co almost has no effect on the length of
stagnant stage. The length of the stagnant stage during the cyclic phase transformation
in Fe-Cu-C and Fe-Si-C increases marginally with increasing Cu and Si concentrations.
In Fig. 4.7, the length of stagnant stage during the cyclic phase transformations in
Fe-M-C alloys as a function of the M concentrations is summarized. In Fig. 4.8, the
average ratio of the length of the stagnant stage and concentration of substitutional
elements S/C is ploted as a function of the partitioning coefficient of M. It shows that
the length of stagnant stage is also determined by partitioning coefficients of alloying
elements between austenite and ferrite. For austenite stablizers (Mn, Ni and Cu), the
4.3. Results and Discussion 57
860 865 870 875 880 885 890 895 900
8
10
12
14
16
18
20
22
Temperature / °c
Inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.1NiFe−0.02C−0.2NiFe−0.02C−0.3Ni
Ae3Ae3Ae3Equilibrium interface position at T2
865 870 875 880 885 890 895 9008
10
12
14
16
18
20
22
Temperature / °c
Inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.1CuFe−0.02C−0.2CuFe−0.02C−0.3Cu
Equilibrium interface position at T2 Ae3Ae3Ae3
880 885 890 895 900 905 910 915 920 9256
8
10
12
14
16
18
20
22
Temperature / °c
Inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.1SiFe−0.02C−0.2SiFe−0.02C−0.3Si
Ae3Equilibrium interface position at T2
Ae3 Ae3
875 880 885 890 895 900 9058
10
12
14
16
18
20
22
Temperature / °c
inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.1CoFe−0.02C−0.2CoFe−0.02C−0.3Co
Equilibrium interface position at T2 Ae3
(a) (b)
(c) (d)Figure 4.6: The α/γ interface position as a function of temperature during the type I cyclicphase transformations in Fe-M-C with the same C concentration but different M concentrationbetween temperature combinations: T2−T1 = 25C, and Ae3−T2 = 5C (a) Fe-C-Ni, (b)Fe-C-Cu,(c) Fe-C-Si and (d) Fe-C-Co. The concentrations of M and C are all in wt.%.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−2
0
2
4
6
8
10
12
14
Concentration of substitutional element /wt.%
Leng
th o
f sta
gnan
t sta
ge /
°c
Fe−C
Mn
Ni
Cu
Si
Co
Figure 4.7: The length of stagnant stage during the cyclic phase transformations in Fe-C-Malloys as a function of the M concentrations.
length of stagnant stage is proportional to their partitioning coefficients , while the
length of the stagnant stage increases as the partitioning coefficents of ferrite stablizers
decrease.
58 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
0.6 0.8 1 1.2 1.4 1.6 1.8 2−5
0
5
10
15
20
25
30
35
40
Partitioning coefficient
S/C
(°c
/wt.%
)
Ni
Mn
Si
Co
Cu
Figure 4.8: The average ratio of the length of stagnant stage and concentration of substitutionalelements S/C as a function of the partitioning coefficient of M.
4.3.4 Fe-C-Mn-M (M= Ni, Si, Co) alloys
In this section, the cyclic phase transformations in Fe-0.02C-0.2Mn-xM (M= Ni, Si, Co,
x=0.1, 0.2 and 0.3 ) are simulated to investigate the effect of a fourth alloying element
on the stagnant stage. The concentrations of Mn, M and C are all in wt.%. As the the
partitioning coefficient of Mn is larger than those of Ni, Si and Co, it is expected that the
Mn partitioning would still control the kinetics of Fe-C-Mn-M alloys and the length of
stagnant stage would not change significantly with the addition of the fourth alloying
element. In Fig. 4.9, the interface position as a function of temperature is simulated
during the cyclic phase transformations in Fe-0.02C-0.2Mn-xM (M= Ni, Si, Co) with
the temperature combination: T2 − T1 = 25C, and Ae3 − T2 = 5C. The concentration
of M varies from 0.1 wt.% to 0.3 wt.%. The cooling and heating rates are both fixed at
10K/min.
It shows that the length of stagnant stage increases significantly by adding Ni and
Si, and it is proportional to Ni and Si concentrations. The addition of Co has no effect at
all on the kinetics of the cyclic phase transformations. During the stagnant stage, there
is no carbon gradients in both austenite and ferrite, and the carbon concentrations in
austenite and ferrite are almost the same for Fe-0.02C-0.2Mn and Fe-0.02C-0.2Mn-xM
4.3. Results and Discussion 59
855 860 865 870 875 880 885 890 895 9006
8
10
12
14
16
18
20
22
Temperature / °c
Inte
rfa
ce
po
sitio
n /
mu
m
Fe−0.02C−0.2MnFe−0.02C−0.2Mn−0.1NiFe−0.02C−0.2Mn−0.2NiFe−0.02C−0.2Mn−0.3Ni
Ae3 Ae3 Ae3 Ae3
Equilibrium interface position at T2
865 870 875 880 885 890 895 900 905 910 9156
8
10
12
14
16
18
20
22
Temperature / °c
Inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.2MnFe−0.02C−0.2Mn−0.1SiFe−0.02C−0.2Mn−0.2SiFe−0.02C−0.2Mn−0.3Si
Ae3 Ae3 Ae3Ae3
Equilibrium interface position at T2
865 870 875 880 885 890 8956
8
10
12
14
16
18
20
Temperature / °c
inte
rfa
ce
po
sitio
n /
µm
Fe−0.02C−0.2MnFe−0.02C−0.2Mn−0.1CoFe−0.02C−0.2Mn−0.2CoFe−0.02C−0.2Mn−0.3Co
Equilibrium interface position at T2
Ae3
(a) (b)
(c)
Figure 4.9: The interface position as a function of temperature during the type I cyclic phasetransformations in quaternary Fe-0.02C-0.2Mn-xM (M= Ni, Si, Co, x=0.1, 0.2 and 0.3 ) alloys(a) Fe-C-Mn-Ni, (b)Fe-C-Mn-Si, (c) Fe-C-Mn-Co.
alloys. The carbon concentration is not an affecting factor for the length of stagnant
stage here, and the Mn and M concentration in austenite and ferrite would determine
the length of stagnant stage.
In Fig. 4.10a, the Mn profiles at the end of inverse transformation in Fe-0.02C-0.2Mn
and Fe-0.02C-0.2Mn-xNi(x=0.1, 0.2 and 0.3, in wt.%) alloys are presented. The Mn
profiles at the interface of both Fe-0.02C-0.2Mn and Fe-0.02C-0.2Mn-xNi alloys exhibit
a zigzag shape with two gradients on both interface sides which counterbalance each
other. It is this zigzag Mn profiles which pin the interface and is a proof that the
growth kinetics is controlled by Mn partitioning at the end of inverse transformation.
The Mn profile of Fe-0.02C-0.2Mn-xNi with different Ni concentration are quite similar
with that of Fe-0.02C-0.2Mn alloy. It is interesting to note that there is a Mn spike
left behind the interface. An indirect evidence for the existence of this Mn spike has
also been found in experimental dilatation curves, and it caused a specific ”growth
retardation” during the final cooling down following a cyclic phase transformations
60 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
0 5 10 15 20 250.1
0.15
0.2
0.25
0.3
0.35
Distance / µm
Mn
co
nce
ntr
atio
n /
wt.
%
Fe−0.02C−0.2MnFe−0.02C−0.2Mn−0.1NiFe−0.02C−0.2Mn−0.2NiFe−0.02C−0.2Mn−0.3Ni
0 5 10 15 20 250.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Distance / µm
Ni co
nce
ntr
atio
n /
wt.
%
Fe−0.02C−0.1NiFe−0.02C−0.2Mn−0.1NiFe−0.02C−0.2NiFe−0.02C−0.2Mn−0.2NiFe−0.02C−0.3NiFe−0.02C−0.2Mn−0.3Ni
0 5 10 15 20 250.1
0.15
0.2
0.25
0.3
0.35
Distance / µm
Mn
co
nce
ntr
atio
n /
wt.
%
Fe−0.02C−0.2MnF−0.02C−0.2Mn−0.1SiFe−0.02C−0.2Mn−0.2SiFe−0.02C−0.2Mn−0.3Si
0 5 10 15 20 250.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Distance / µm
Si co
nce
ntr
atio
n /
wt.
%
Fe−0.02C−0.1SiFe−0.02C−0.2Mn−0.1SiFe−0.02C−0.2SiFe−0.02C−0.2Mn−0.2SiFe−0.02C−0.3SiFe−0.02C−0.2Mn−0.3Si
0 5 10 15 20 250.1
0.15
0.2
0.25
0.3
0.35
Distance / µm
Mn
co
nce
ntr
atio
n /
wt.
%
Fe−0.02C−0.2MnFe−0.02C−0.2Mn−0.1CoFe−0.02C−0.2Mn−0.2CoFe−0.02C−0.2Mn−0.3Co
(a) (b)
(c) (d)
(e)
Figure 4.10: The profiles of alloying elements at the end of inverse transformation in Fe-0.2Mn-0.02C-xNi: (a) Mn profiles and (b) Ni profiles; Fe-0.2Mn-0.02C-xSi: (c) Mn profiles and (d) Siprofiles; Fe-0.2Mn-0.02C-xCo: (e) Mn profiles. The concentrations of Mn, M and C are all inwt.%.
in the intercritical austenite + ferrite region [99]. As this Mn spike does not affect the
growth kinetics during the cyclic phase transformations, it will not be investigated in
detail in this work. In Fig. 4.10b, the Ni profiles at the end of inverse transformation
in Fe-xNi(x=0.1, 0.2 and 0.3, in wt.%)-0.02C and Fe-0.02C-0.2Mn-xNi(x=0.1, 0.2 and
0.3, in wt.%) alloys are indicated. In all Fe-xNi-0.02C and Fe-0.2Mn-0.02C-xNi alloys,
the Ni profiles exhibit a zigzag shape at the interface, and a Ni spike is left behind
the interface. The zigzag Ni profile would pin the interface as the zigzag Mn profile
4.3. Results and Discussion 61
does, and the growth kinetics is also controlled by Ni partitioning. The Ni profile of
Fe-0.02C-0.2Mn-xNi alloys are quite similar with that of Fe-0.02C-xNi alloys if the Ni
concentration is the same, which implies that the addition of Mn almost has no effect
on the Ni partitioning.
In Fig. 4.10c and Fig. 4.10d , the Mn and Si profiles at the end of inverse transfor-
mation in Fe-0.02C-0.2Mn and Fe-0.02C-0.2Mn-xSi(x=0.1, 0.2 and 0.3, in wt.%) alloys
are indicated. Both zigzag Mn profile and a Mn spike in austenite left behind the
interface have been identified in Fig. 4.10c, and the addition of Si has no effect on the
Mn partitioning. An zigzag Si profile at the interface and a Si spike in austenite also
appear in Fig. 4.10d. The growth kinetics is also affected by Si partitioning. Different
from Mn profiles, the zigzag Si profile at the interface is composed of a depleted Si
spike in austenite and an enriched Si spike in ferrite, and the Si spike left behind in
austenite is depleted. This is because that the Si is ferrite stabilizer. The Si profile
of Fe-0.02C-0.2Mn-xSi is quite comparable with that of Fe-xSi-0.02C if the Si concen-
tration is the same, which also means that the addition of Mn does not affect the Si
partitioning. In Fig. 4.10e, the Mn profiles at the end of inverse transformation in Fe-
0.02C-0.2Mn and Fe-0.02C-0.2Mn-xCo(x=0.1, 0.2 and 0.3, in wt.%) alloys are indicated.
The Mn profiles are quite similar with that of Fe-0.02C-0.2Mn. The Co distributes ho-
mogeneously in both austenite and ferrite without partitioning. The growth kinetics in
Fe-0.02C-0.2Mn-xCo alloys is only controlled by Mn partitioning during the stagnant
stage.
Base on the analysis of alloying element profiles , the growth kinetics in Fe-0.02C-
0.2Mn-xM during the stagnant stage is controlled by both Mn and M partitioning. In
Fig. 4.11, the length of stagnant stage is summarized as a function of the concentration
of the fourth alloying element. It shows that the length of stagnant stage of Fe-0.2Mn-
0.02C-xM is more or less an addition of those of Fe-0.02C-0.2Mn and Fe-0.02C-xM.
Therefore, it can be concluded that the effect of substitutional alloying elements on the
length of stagnant stage is additive.
62 Chapter 4. Analysis of the stagnant stage during cyclic phase transformations
0 0.05 0.1 0.15 0.2 0.25 0.3 0.356
8
10
12
14
16
18
20
22
Addition of substitutional elements /wt.%
Len
gth
of
stag
nan
t st
age
/°c
Co
Ni
Si
Mn
Figure 4.11: The length of stagnant stage during the cyclic phase transformations in Fe-0.02C-0.2Mn-xM alloys as a function of the concentration of the fourth alloying element M
4.4 Conclusion
In this chapter, the stagnant stage during the cyclic phase transformations in Fe-C-
M alloys (M=Mn, Ni, Cu, Si, Co) and Fe-Mn-C-M (M=Ni,Si,Co) alloys have been
systematically studied. The stagnant stage is caused by partitioning of substitutional
alloying elements during the phase transformations. The length of the stagnant stage
is determined by austenite and ferrite conditions, partitioning coefficients of M and
concentrations of M. In quaternary alloys, the effect of substitutional alloying elements
on the length of stagnant stage seems to be additive.
Chapter 5Indirect evidence for the existence of
an interfacial Mn Spike
This chapter is based on
• H Chen, S van der Zwaag, Analysis of ferrite growth retardation induced by
local Mn enrichment in austenite: a cyclic phase transformation approach,Acta
Mater, 2013; 61:1338-1349
• H Chen, S van der Zwaag, Indirect evidence for the existence of the Mn
partitioning spike during the austenite to ferrite transformation,Phil Mag Lett,
2012; 92:86-92
5.1 Introduction
As shown in Fig. 3.6 of Chapter 3, the Local equilibrium model predict that after the
cyclic phase transformations the Mn spike is separated from the migrating interface
and left behind the interface in austenite. This Mn spike is called ”residual Mn spike”
here. The term ”spike” actually refers to the pointed concentration profile calculated
for a one dimensional transformation model. In reality, the ”spike” is a curved plane
of negligible thickness located at the austenite side of the moving austenite-ferrite
interface. To directly determine the predicted residual Mn spike experimentally, there
64 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
are two difficulties:(i) The Mn spike exists in front of the moving austenite/ferrite
interface at high temperature, while the modern chemical analysis technique like Atom
Probe Tomography [10, 100, 101] can only work at room temperature. It means that
the sample has to be quenched to room temperature, and only the chemical analysis
at the stationary martensite/ferrite interface can be measured. Furthermore, it willl be
very difficult to ascertain that the enriched interfacial region is exactly in the very small
sample volume probed by the atom probe [10]. (ii) The predicted Mn spike is extremely
thin, which is very difficult to be correctly detected by Atom Probe Tomography [10].
In this chapter, new experiments will be designed and performed in a series of Fe-Mn-C
alloys to indirectly prove the existence of residual Mn spike.
5.2 Experimental
The materials investigated here are a set of pure Fe-Mn-C model alloys and a reference
pure Fe-C alloy. The compositions of the alloys are presented in Tab. 5.1.
Table 5.1: Chemical compositions and cycling temperatures of the investigated alloys (Com-positions in wt. % and Temperature in C)
Alloy C Mn Fe T1 T2 T3
A 0.1 0 balance 815 855 NoneB 0.1 0.49 balance 785 842 810C 0.1 1.0 balance 765 822 NoneD 0.023 0.17 balance 860 885 None
A carefully tuned Bahr 805A dilatometer is used to measure the dilation of the
specimen (10 mm in length and 5 mm in diameter) during the cyclic experiments.
Two thermocouples, spaced 4 mm apart, were spot welded to the sample to have
an accurate temperature measurement and to check for the absence of a significant
temperature gradient along the sample. The measured temperature gradient along
the sample was typically of the order of 2-3 C. In this work, the standard type I
cyclic experiments [97] are performed, and the heat treatment procedure was indicated
in Fig. 3.1. After the cycling, the sample is finally cooled down from T2 to room
5.3. Results and Discussion 65
temperature. It is the final austenite to ferrite transformation during the final cooling
which is of interest here. The cooling rate and heating rate during the experiments
were always 10C/min. In some experiments, the standard type I cyclic experiment was
modified to prove assumed features of the relevant processes taking place at the moving
austenite-ferrite interface. In this work, based on the Thermo Calc calculation [90], the
cycling temperatures T1 and T2 were selected appropriately to make sure that both
austenite to ferrite and ferrite to austenite transformations can take place in the fast
LENP mode. Unlike the case of normal isothermal experiments, we found that in
the cyclic phase transformation experiments the LENP-LEP transition temperatures
for both austenite to ferrite transformation and ferrite to austenite transformations
depend on the local interface conditions instead of the nominal composition of the
bulk material. It means that the LENP-LEP transition temperatures change during
the cyclic phase transformations. In the cyclic experiments, we have to make sure
that the T1 temperature (for austenite to ferrite transformation) has to be at least 10
degrees lower than the minimum LENP-LEP transition temperature for austenite to
ferrite transformations during the cyclic phase transformations. In the same way, the
T2 temperature (for ferrite to austenite transformation) has to be at least 10 degrees
higher than the maximum LENP-LEP transition temperature calculated for ferrite to
austenite transformations during the cyclic phase transformations. The gap between
T1 or T2 and minimum or maximum LENP-LEP transition temperatures increase with
increasing nominal Mn concentration of the alloy.
5.3 Results and Discussion
Fig. 5.1a shows the dilation as a function of temperature during the final austenite to
ferrite transformation(in solid and red line) after type I cyclic phase transformations
between 885C and 860C. During the final γ → α transformation after the cyclic
phase transformations, a special growth retardation around 860C is identified. The
retarded growth stage occurs from 861C to 856C. The observation is not specific to
66 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
820 830 840 850 860 870 880 890 900
118
120
122
124
126
128
130
Temperature /°c
Len
gth
ch
ang
e /µ
m
Growthretardation
Cycling between 860°c and 885°c
820 830 840 850 860 870 880 890 900100
105
110
115
120
125
130
Temperature /°c
Len
gth
ch
ang
e /µ
m
Usual phase transformation
(a)
(b)
Figure 5.1: The dilatation as a function of temperature during (a) the final γ → α transforma-tion(in solid and red line) after the cyclic transformations between 860C and 885C and (b)the usual γ→ α transformation (in solid and red line).
the two selected temperatures T1 and T2, and is also observed for other temperature
combinations provided T1 and T2 are in the two phase region and sufficiently far apart
(typically >15C). In Fig. 5.1b, the dilation as a function of temperature during a usual
γ → α transformation experiment (in solid and red line) is shown. The dilatation of
heating stage is in dotted line. In this experiment the sample is heated up to 1000 C
held for 5 minutes for full austenization, and then cooled down to room temperature.
Both the heating and cooling rate are 10K/min. No growth retardation is observed in
the latter experiment, which indicates that the growth retardations in Fig. 5.1a should
be due to the prior partial phase transformations.
5.3. Results and Discussion 67
820 830 840 850 860 870 880 89010
15
20
25
Temperature /°c
Inte
rfac
e p
osi
tio
n /µ
m
a
b
c
d
Local equilibrium simulation
Growth retardation
820 830 840 850 860 870 880 89010
15
20
25
Temperature /°c
Inte
rfac
e p
osi
tio
n /µ
m
Paraequilibrium simulation
(a)
(b)
Figure 5.2: The α/γ interface position as a function of temperature during the final coolingdown(in solid and red line) after type I cyclic phase transformations between 860C and 885Csimulated under both (a) local equilibrium conditions and (b) paraequilibrium.
In Fig. 5.2, the interface position as a function of temperature predicted by (a) local
equilibrium model and (b) paraequilibrium model for the final cooling down after
type I cyclic phase transformations between 860C and 885C is indicated. It shows
that there is no growth retardation during the final cooling down according to parae-
quilibrium simulation, while an obvious growth retardation stage has been identified
around 860 C in the local equilibrium simulation, which is qualitatively comparable
to the experimental observation. The paraequilibrium model did not predict the novel
68 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
observation of growth retardation. Only the local equilibrium model considering Mn
partitioning predicts the new feature of growth retardation. A growth mode transition
from LE-P mode to LE-NP mode has been identified in LE model, and the transition
temperature is 877 C.
10 15 20 250
0.1
0.2
0.3
0.4
Mn
co
nce
ntr
atio
n (
wt.
%)
10 15 20 250
0.1
0.2
0.3
0.4
10 15 20 250
0.1
0.2
0.3
0.4
10 15 20 250
0.1
0.2
0.3
0.4
Distance /µm
a b
c d
Interface
Interface
Interface
Residual Mn spike
Interface
Residual Mn spike
(a) (b)
(c) (d)
Figure 5.3: The evolution of Mn profiles during the cyclic phase transformations and the finalcooling down predicted by local equilibrium model. The plots (a) to (d) correspond to thepoints marked a, b, c, d in Fig. 5.2a
In order to show the underlying reason for the growth retardation stage, the Mn
profiles during the cyclic phase transformations and final cooling down predicted
by the local equilibrium model are plotted in Fig. 5.3. In Fig. 5.3a, the Mn profile
(linked to the point a in Fig. 5.2a) during the isothermal transformation at 860 C
before the cyclic part of the experiment is shown. There is a very sharp Mn spike at
the migrating interface, which indicates the growth mode of the isothermal γ → α
transformation at 860C is local equilibrium with negligible partitioning. In Fig. 5.3b,
the Mn profile at the end of isothermal phase transformation stage (linked to the point
5.3. Results and Discussion 69
b in Fig. 5.2a)is shown. During the isothermal holding the Mn spike was diffusing out,
and the thickness of the Mn spike increased at the end of the isothermal holding.
After the isothermal transformation at 860 C, the cyclic phase transformation starts.
The Mn profile at 885 C during the first temperature cycle (linked to the point c in
Fig. 5.2a) is shown in Fig. 5.3c, in which two Mn spikes are observed: the first one
is in ferrite at the new interface position (about 12.5 µm), and the second one is in
the austenite at the old interface position (about 20 µm). The first Mn spike at the
α/γ interface corresponds to the local equilibrium with negligible partitioning growth
mode. The second Mn spike in the austenite is for the first time identified in this work,
and is the result of the imposed thermal cycle. It is called ”residual Mn spike” here. The
existence of the residual Mn spike can be explained as follows: Once the temperature
is increased from 860 C in the first temperature cycle, the interface starts to migrate
backward into ferrite, and the tie-line which determines interfacial Mn concentrations
in both austenite and ferrite will switch from that for γ → α transformation into that
for α→ γ transformation. Since the Mn diffusion in austenite is too slow to accompany
the change of tie-line and the interface migration, a Mn spike is left behind the interface
in austenite.
Once the highest transformation temperature of 885C is reached, in the simulation
the temperature is decreased immediately, which results in interface migration back
into austenite. During the cyclic phase transformation between 885C and 860C, the
interface migrates forward and backforward between about 12.5 µm and 20 µm. In
Fig. 5.3d, the Mn profile at 860C during the final cooling cycle (linked to the point d in
Fig. 5.2a)is indicated. It shows that the Mn spike at the moving interface has to migrate
through the residual Mn spike in austenite in order to complete the transformation as
a result of the further cooling down. When the interface migrates through the residual
Mn spike, the Mn concentration of the untransformed austenite in front of the moving
interface is higher than the norminal value (0.17wt. %), which slows down the γ → α
transformation rate. The passage of the moving interface through the residual Mn
spike in Fig. 5.3d leads to the growth retardation stage in Fig. 5.2a.
70 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
5.3.1 Effect of Mn concentration
If the growth retardation is indeed caused by the residual Mn spike in austenite, it can
easily be expected that the magnitude of growth retardation would also be affected by
Mn concentration in the alloys. In order to investigate the effect of Mn concentration
on the magnitude of growth retardation, a series of standard type I experiments are
performed in Fe-0.1C-xMn (x=0, 0.49, and 1.0, all in wt. %) alloys.
700 750 800 850100
105
110
115
Temperature/ °c
Leng
th c
hang
e /
µm
Cycling between 842 °c and 785 °c
Growthretardation
Fe−0.1C−0.49Mn
700 750 800 85085
90
95
100
105
Temperature / °c
Leng
th c
hang
e /
µm
Cycling between 822 °c and 765 °c
Growthretardation
Fe−0.1C−1.0Mn
(a)
(b)
Figure 5.4: The dilatations as a function of temperature during the finalγ→ α transformation(insolid line) after the type I cyclic phase transformations in a (a) Fe-0.1C-0.49Mn alloy and (b)Fe-0.1C-1.0Mn alloy (all in wt. %).
In Fig. 5.4a and b , the dilations as a function of temperature during the final
γ → α transformation(in solid line) after the type I cyclic phase transformations in
Fe-0.1C-0.49Mn and Fe-0.1C-1.0Mn alloys are indicated. It shows that the γ → α
transformation under the fast LENP mode can finish in less than 10 minutes. As
experimentally indicated in Ref [102], it took 37 days to form less than 5 % ferrite
5.3. Results and Discussion 71
during the γ → α transformation under the slow LEP mode, which is much slower
than the transformation rate of LE-NP mode observed here in the cyclic experiments.
The length of the stagnant stage increases as a function of Mn concentration, which
is in good agreement with computational simulations in [103]. It also shows that
the magnitude of growth retardation increases significantly with the increase in Mn
concentration. In the Fe-0.023C-0.17Mn alloy, the length of the growth retardation
stage is about 3 C, while it increases to 15 C in the Fe-0.1C-1.0Mn alloy.
740 760 780 800 820 840 860100
102
104
106
108
110
112
Temperature / °c
Le
ng
th c
ha
ng
e /
°c
No growth retardation
Fe−0.1C
Figure 5.5: The dilatation as a function of temperature during the final γ→ α transformation(insolid line) after type I cyclic phase transformations between 855C and 815C in a Fe-0.1C alloy.
In order to further prove that the growth retardation is due to the residual Mn
spike in austenite, a standard type I cyclic experiment is performed in a binary Fe-
0.1C alloy. In Fig. 5.5, the dilation as a function of temperature during the final
γ → α transformation(in solid line) after type I cyclic phase transformations between
855C and 815C in a Fe-0.1C alloy is indicated. It shows that there is no growth
retardation during the austenite to ferrite transformation upon final cooling in Fe-0.1C
alloy. This experiment further indicates that the growth retardation is linked to the Mn
partitioning.
In Fig. 5.6a, the simulated Mn profiles at the end of the type I cyclic phase trans-
formations for the three Fe-C-Mn alloys are shown. It shows that there are two Mn
72 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
spikes in each profile: (i) an extremely sharp Mn spike at the interface, which further
indicates that the transformation mode is LENP; (ii) a residual Mn spike left behind the
interface. The simulated residual Mn spikes in different alloys are quite different, and
the amplitude of the spike increases with increasing Mn concentration. It is interesting
to note that there are small Mn depletions to the left hand side of the residual Mn
spikes in the Fe-0.1C-0.49Mn and Fe-0.1C-1.0Mn alloys, while no depletion has been
found in Fe-0.023C-0.17Mn alloy. The occurrence of these small depletions is because
during the cyclic phase transformations, a negative Mn spike for the ferrite to austenite
transformation would appear when the interface moves backward into the ferrite. This
negative Mn spike leads to the small depletion in the left side of the simulated resid-
ual Mn spikes. Theoretically speaking, these small Mn depletions would accelerate
the austenite to ferrite transformation. However, the depletion was too small to be
experimentally observable. The average enrichment of Mn at the location of residual
Mn spike is calculated for the three alloys by integrating over the spike. The average
enrichment of Mn increases with increasing the Mn concentration. In Fig. 5.6b, the
length of growth retardation stage is plotted as a function of the average enrichment
of Mn. It is found that the experimentally determined length of growth retardation is
more or less proportional to the calculated enrichment of Mn, which is in good agree-
ment with expectation. It is interesting to note that the amplitudes of the simulated
residual Mn spikes here are all within the statistical error of atom probe tomography
(about 0.8 at.%) [10]. Even if the residual Mn spike in austenite at high temperature
can be preserved to room temperature and correctly located in martensite or ferrite, it
is still difficult to measure the simulated residual Mn spike accurately by atom probe
tomography. However, the simulated residual Mn spike is strong enough to retard
the moving austenite-ferrite interface, leading to macroscopically observable growth
retardation. In this work, only one dimensional residual Mn spikes are simulated for
the purpose of explaining the new experimental features qualitatively, however, please
note that in experiments the residual Mn spike is most likely to be a curved plane.
5.3. Results and Discussion 73
5 10 15 20 250
0.5
1
1.5
2
Distance / µm
Mn
co
nce
ntr
atio
n w
t.%
Fe−0.023C−0.17MnFe−0.1C−0.5MnFe−0.1C−1.0MnInterface
Residual Mn spike
-0,1 0,0 0,1 0,2 0,3 0,4-4
0
4
8
12
16
20
Leng
th o
f ret
arda
tion
/ O C
Average enrichment of Mn /wt.%
(a)
(b)
Figure 5.6: (a)The simulated residual Mn profiles in Fe-Mn-C alloys with different Mn concen-trations after the type I cyclic phase transformations; (b) The length of growth retardation inFe-Mn-C alloys as a function of the average enrichment of Mn in the residual Mn spikes
5.3.2 Effect of the number of cycles prior to final cooling
There are two prerequisites for the presence of growth retardation: (i) before the final
cooling residual Mn spikes must have been created in the austenite by the cyclic partial
phase transformations; (ii) The moving interface should impinge the residual Mn spike
face on. If the interface would not have retraced its path during the cyclic partial phase
74 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
transformations, the moving interface during the final cooling will not be parallel to the
plane of the residual Mn spike. It means that the interface will impinge the residual
Mn spike at a certain angle, which would decrease the macroscopically observable
magnitude of growth retardation to a lower and even indiscernible level. In order to
prove the importance of the second prerequisite not yet identified in the literature, a
new set of cyclic experiments is presented here.
750 760 770 780 790 800 810 820 830 840 850102
104
106
108
110
112
114
116
118
Temperature / °c
Leng
th c
hang
e /
µm
1 cycle
No growth retardation
Time
T
A3
T2
T1
A1
Fe−0.49Mn−0.1C
0 5 10 15 20 250.2
0.3
0.4
0.5
0.6
0.7
0.8
Distance /µm
Mn
Co
nce
ntr
atio
n /w
t.%
1 cycle
Residual Mn spike
Ferrite
Interface
Austenite
(a)
(b)
Figure 5.7: (a)The dilatation as a function of temperature during the final γ → α transfor-mation(in solid line) after the type I cyclic phase transformations with one temperature cyclebetween 842C and 785C in a Fe-0.49Mn-0.1C alloy; (b) The simulated Mn profile at the startingpoint of final cooling after the cyclic phase transformation with one temperature cycle between842C and 785C in a Fe-0.49Mn-0.1C alloy
First, a series of cyclic phase transformation experiments with different number of
temperature cycle are performed. In Fig. 5.7a, the dilation as a function of tempera-
5.3. Results and Discussion 75
ture during the final γ → α transformation(in solid line) after the type I cyclic phase
transformations with only one temperature cycle between 842C and 785C in a Fe-
0.49Mn-0.1C alloy is presented. Interestingly, different from the experiment with three
temperature cycles as shown in Fig. 5.4b , no growth retardation stage is observed in
Fig. 5.7a. In order to analyze this phenomena, an one dimension LE simulation has
been made, and all settings in the simulations are the same as those in the experiment.
In Fig. 5.7b, the simulated Mn profiles at the starting point of final cooling after the
cyclic phase transformation with one temperature cycle is shown. It shows that there is
already a residual Mn spike left in austenite after one temperature cycle, which means
that the first prerequisite for presence of growth retardation can be fulfilled by one tem-
perature cycle. Therefore, it can be deduced that the absence of growth retardation in
Fig. 5.7a could be attributed to not meeting the second prerequisite. The change in path
for the growing ferrite (i.e. its non-retraceability) can be explained as follows: in the
first temperature cycle, the first isothermal austenite to ferrite transformation at 785C
starts from nucleation of ferrite, while nucleation is not necessary at the onset of the
second austenite to ferrite transformation during final cooling. The interface migration
direction in the second austenite to ferrite transformation could be different from that
in the first isothermal austenite to ferrite transformation, which create residual Mn
spike in the austenite and determine its location. In other words, the interface planes
of the second ferrite formation stage will impinge the residual Mn spike at a certain
angle. This non-parallel impingement would decrease the effectiveness of retarding
interface by residual Mn spike, and not lead to observable growth retardation. After
the first temperature cycle, nucleation is not necessary anymore for both the austenite
to ferrite and ferrite to austenite transformation, and then the interfaces would become
more and more capable of retracing their paths. Therefore, it can be expected that the
interfaces could impinge the residual Mn spikes properly only when the number of
temperature cycle is more than 1.
In order to further prove the point that ferrite re-nucleation affects the interface mi-
gration direction, one more experiment is designed. In this experiment, the specimen
76 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
(Fe-0.49Mn-0.1C alloy) is firstly cycled three times between 842C and 785C, and then
reheated to 900 C for fully austenization. After austenization for only 10 seconds, the
specimen is cooled down to room temperature for the austenite to ferrite transforma-
tion. In this experiment, after three temperature cycles, the austenite/ferrite interface
migration direction become stable, and a residual Mn spike has been created at a cer-
tain position in the austenite. Reaustenization at the later stage would not destroy the
residual Mn spike, but new nucleations of ferrite are needed for the austenite to ferrite
transformation during final cooling. If the newly formed austenite/ferrite interfaces
during the final cooling would still migrate in the same paths as those during the cyclic
phase transformations, it can be expected that the growth retardation will appear at
785C. In Fig. 5.8, the dilations as a function of temperature during the cyclic phase
transformations between 842C and 785C (black and dash line), reaustenization (blue
and dotted line) and final cooling (red and solid line) in a Fe-0.49Mn-0.1C alloy are indi-
cated for this experiment. It shows that the growth retardation after full austenization
disappears, which indicates that the newly formed austenite/ferrite interfaces have a
different migration direction from that perpendicular to the residual Mn spikes.
680 700 720 740 760 780 800 820 840 860 880 90085
90
95
100
105
110
Temperature /°c
Len
gth
ch
ang
e /µ
m
The cyclic phase transformationsReaustenizationFinal cooling
Time
T
T2
A1
T1
A3
Figure 5.8: The dilatations as a function of temperature during the cyclic phase transformationsbetween 842C and 785C (black and dash line), reaustenization (blue and dotted line) and finalcooling (red and solid line) in a Fe-0.49Mn-0.1C alloy.
In order to study the effect of the number of cycles within the two phase region
on the growth retardation, a series of standard type I cyclic experiments with various
5.3. Results and Discussion 77
number of temperature cycle are performed. The number of temperature cycle ranged
from 1 to 6. In Fig. 5.9a, the dilations as a function of temperature during the final
γ → α transformation(in solid line) after type I cyclic phase transformations with 6
temperature cycles between 842C and 785C in a Fe-0.49Mn-0.1C alloy are shown. It
is clear that the magnitude of growth retardation in this experiment is much larger
than that in the experiment with three temperature cycles in Fig. 5.4b.
Applying the level rule, the fraction of ferrite as a function of temperature during
the final cooling stage are calculated for the cyclic experiments with various number
of temperature cycle. In Fig. 5.9b, the fractions of ferrite as a function of temperature
during the final γ → α transformation after type I cyclic phase transformations with
1-6 temperature cycles between 842C and 785C in a Fe-0.49Mn-0.1C alloy are indi-
cated. The starting fraction of ferrite at the onset of the cooling is about 0.2 for all the
experiments. The figure shows that the magnitude of growth retardation increases as
the number of cycling increases, and no growth retardation appears in the experiment
with only one temperature cycle. Fig. 5.9c shows the transformation rate as a function
of ferrite fraction during the final γ→ α transformation after type I cyclic phase trans-
formations with 1-6 temperature cycles between 842C and 785C in a Fe-0.49Mn-0.1C
alloy. One negative peak and two positive peaks appear in experiments with more
than one temperature cycle. The negative peak is related to the growth retardation.
Please note that the ferrite fractions at which the negative peaks appear decrease as
the number of temperature cycling increases, a feature which also follows from the
simulation will be explained now.
It is expected that after every temperature cycle a residual Mn spike will be created
in austenite, and thus the magnitude of Mn accumulation increases with increasing
the number of temperature cycles. In Fig. 5.9d, the residual Mn spikes after 1, 2,
and 6 temperature cycles are simulated by the LE model. After the first temperature
cycle, there is one residual Mn spike left behind in austenite. During the second
temperature cycle, the second residual Mn spike is created, and the first Mn spike will
spread out. The first Mn spike also overlaps with the second Mn spike, which leads
78 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
750 760 770 780 790 800 810 820 830 840 850102
104
106
108
110
112
114
116
118
Temperature / °c
Leng
th c
hang
e /
µm
Growthretardation
T1
T2
Time
T 6 cyclesA
3
A1
Fe−0.49Mn−0.1C
700 720 740 760 780 800 820
0.2
0.4
0.6
0.8
1
Temperature /°c
f α
1cycle2cycles3cycles4cycles6cycles
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.005
0.01
0.015
fα
df α/d
T
1cycle2cycles3cycles4cycles6cycles
6 8 10 12 14 16 18 200.2
0.3
0.4
0.5
0.6
0.7
0.8
Distance /µm
Mn
Co
nce
ntr
atio
n /w
t.%
1 cycle2 cycles6 cycles
Interface
Residual Mn spikes
(a)
(b)
(c)
(d)Figure 5.9: (a)The dilatations as a function of temperature during the final γ→ α transforma-tion(in solid line) after type I cyclic phase transformations with 6 temperature cycles between842C and 785C in a Fe-0.49Mn-0.1C alloy;(b) the fraction of ferrite as a function of temper-ature and (c) the transformation rate as a function of ferrite fraction during the final γ → αtransformation(in solid line) after type I cyclic phase transformations with 1- 6 temperaturecycles between 842C and 785C in a Fe-0.49Mn-0.1C alloy; (d)the simulated residual Mn spikesin austenite after the cyclic phase transformations with 1, 2 and 6 temperature cycles.
5.3. Results and Discussion 79
to one thicker Mn spike. Increasing the number of temperature cycles, the width of
the simulated residual Mn spike extends. In Fig. 5.9d, the broadened residual Mn
spike after 6 temperature cycle is shown. In the experiments, the residual Mn spikes
of cycle 2 and higher are co-located and become effective for decelerating the interface
migration during the final cooling. Increasing number of temperature cycles, the width
of simulated residual Mn spike extends, and its position shifts towards the moving
interface. Hence in simulations with increasing number of cycles slightly less ferrite
has formed before the moving interface impinges with the residual Mn spike, which is
in good agreement with the experimental results in Fig. 5.9c.
5.3.3 Creating 2 Mn spikes to create 2 growth retardation stages
5 10 15 200.2
0.3
0.4
0.5
0.6
0.7
0.8
Distance /µm
Mn
co
nce
ntr
atio
n /w
t.%
2I2I2I1I
Spike 2
Interface
Spike 1
Figure 5.10: The simulated Mn profiles after the type ”2I2I” and ”2I1I” cyclic experiments in aFe-0.49Mn-0.1C alloy.
In this section, a special cyclic phase transformations heat treatment is presented
to create two residual Mn spikes in austenite in order to induce 2 growth retardation
stages in the final cooling curve. In the first experiment, the sample is firstly cycled
two times between T1 and T2 to create the first residual Mn spike in austenite, and then
cycled between T2 and T3 two times to create the second residual Mn spike in austenite.
This is called a Type ”2I2I” experiment. In Fig. 5.10, the residual Mn spikes created by
the type ”2I2I” experiment (solid line) in a Fe-0.49Mn-0.1C alloy have been simulated
80 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
by LE model, T1=785C, T2=842C and T3=810C. It shows that indeed two residual
Mn spikes have been created according to the simulation, which is in agreement with
the expectation. The residual Mn spike 1 in Fig. 5.10 is created by the cyclic phase
transformations between 785C and 842C, while the cyclic phase transformations
between 842C and 810C builds up the residual Mn spike 2. During the final cooling,
the austenite/ferrite interface has to firstly migrate through the residual Mn spike 2,
which would lead to a growth retardation stage around 810C. After the interface
passes through the Mn spike 2, the interface will impinge the residual Mn spike 1, and
this will lead to the growth retardation at 785C.
700 750 800 850100
105
110
115
Temperature / °c
Leng
th c
hang
e /
µm
A1
Growth retardation 2
Growth retardation 1
A3
T2
T3
T1
T
Time
Fe−0.1C−0.49Mn
Type 2I2I
Type 2I2I
700 750 800 850100
105
110
115
Temperature / °c
Leng
th c
hang
e /
µm
Growth retardation 2
Growth retardation 1
A3
T2
T3
T1
A1
T
Time
Fe−0.1C−0.49Mn
Type 2I1I
Type 2I1I
(a)
(b)
Figure 5.11: The dilatation as a function of temperature during (a) type ”2I2I” and (b) type”2I1I” cyclic phase transformation experiments in a Fe-0.49Mn-0.1C alloy, T1=785C, T2=842Cand T3=810C
In Fig. 5.11a, the dilation as a function of temperature during the type ”2I2I” cyclic
phase transformations in a Fe-0.49Mn-0.1C alloy is shown. A schematic of the heat
treatment is indicated in the insert. The cycling temperatures are the same as those
5.3. Results and Discussion 81
in simulations. It shows that during the final cooling there are indeed two growth
retardations at 785C and 810C, which is in perfect agreement with our simulations.
Interestingly, the growth retardation at 785C in Fig. 5.11a is much weaker than the
one as shown in Fig. 5.4b. By comparing the heat treatments of these two experiments,
it is found that the second cyclic phase transformations between 842C and 810C
decreases the magnitude of growth retardation at 785 C in Fig. 5.11a. There are
two possible explanations for the reduced magnitude of growth retardation caused
by the first set of cyclic phase transformations: (i) The first residual Mn spike 1 as
shown in Fig. 5.10 in austenite created by the first set of cyclic phase transformations is
diffusing out during the second set of cyclic phase transformations, which decreases its
effectiveness of retarding the interface during the final cooling; (ii) it is to be expected
that the interface migration direction would be affected to some degree by passing
through the residual Mn spike 2 created by the second cyclic phase transformations.
Hence, the interface will not impinge the residual Mn spike 1 created by the first cyclic
phase transformations properly, which also reduces its effectiveness of retarding the
interface.
Based on the above analysis, it can be expected that the strength of the residual Mn
spike 2 created by the second cyclic phase transformations affects the magnitude of
the growth retardation caused by the first set of cyclic phase transformations. In order
to illustrate this point, a second experiment is designed here. In this experiments, the
specimen is firstly cycled two times between 842C and 785C, and then cycled one time
between between 842C and 810 C before final cooling down to room temperature.
This experiment is called a type ”2I1I” experiment, and its calculated Mn profile is
shown in Fig. 5.10 (dashed line).
In Fig. 5.11b, the dilation as a function of temperature during the type ”2I1I”
cyclic phase transformations in a Fe-0.49Mn-0.1C alloy, T1=785C and T2=842C and
T3=810C. A schematic of the heat treatment is indicated in the insert. It shows that
during the final cooling there is significant growth retardation at 810C, which means
that the residual Mn spike 2 caused by the single temperature cycle of the second
82 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
cyclic phase transformation in type ”2I1I” experiment is still effective for retarding the
interface migration. This new observation can be explained as: after the first cyclic
phase transformations between 842C and 785C, the interfaces are able to retrace their
paths. During the second cyclic phase transformations between 842C and 810 C,
although only one temperature cycling is applied, the interfaces would still migrate in
the same paths as nucleation is not necessary in the second cyclic phase transforma-
tion. Therefore, the residual Mn spike created by the single temperature cycle is still
effective for growth retardation. This new observation further clearly indicates that the
disappearance of growth retardation is not simply due to insufficient temperature cy-
cles, instead, it is fundamentally caused by the change in interface migration direction
by nucleation.
It is also interesting to note that the magnitude of growth retardation at 785C caused
by the first cyclic phase transformations in Fig. 5.11b is larger than that in Fig. 5.11a.
This can be simply explained as follows: compared with type ”2I1I” experiment, in
type ”2I2I” experiment the interface has to pass through a stronger residual Mn spike
2 before it impinge with the residual Mn spike 1, and it would possibly change the
interface migration direction more. Hence it leads to less growth retardation at 785C
in Fig. 5.11a.
5.3.4 Linking growth retardation to a physical location of Mn spikes
Until now, all the experimental results show that the growth retardations appear at the
lower cycling temperature T1 or T3 , and it seems that the presence of growth retar-
dation is determined by cycling temperature. However, according to the theoretical
simulations, the growth retardation is caused by the residual Mn spikes, which means
that the growth retardation does not occur at a particular temperature but at a partic-
ular position of the interface (i.e. when it meets the residual Mn spike). In order to
prove that growth retardation is not temperature determined but position determined,
two final cyclic experiments are presented here.
5.3. Results and Discussion 83
680 700 720 740 760 780 800 820 840102
104
106
108
110
112
114
Temperature/°c
Le
ng
th c
ha
ng
/µ
m
Cyling between 842°c and 785°cFinal cooling at 10°c/minCyling between 842°c and 785°cFinal cooling at 300°c/min
Growthretardation
Figure 5.12: The dilatation as a function of temperature during the final cooling at a rate of10C/min and 300C/min after the type I cyclic phase transformations in a Fe-0.1C-0.49Mn alloy.
In the new experiments, two identical samples (Fe-0.1C-0.49Mn alloy) are firstly
cycled three times between 842C and 785C with a cooling and heating rate of 10K/min
to yield similar sets of Mn spikes in the austenite grains. After these three temperature
cycles, the first sample is cooled down to room temperature at a rate of 10K/min, while
the second sample is cooled down to room temperature at a rate of 300K/min. In
Fig. 5.12, the dilation as a function of temperature during the final cooling is indicated.
As predicted, the experimental data shows that the growth retardation in the second
experiment appears at a lower temperature than the first experiment, which proves that
the appearance of growth retardation is not determined by the cycling temperature.
In order to better analyze the results, the transformation rates as a function of
temperature and fraction of ferrite during the final cooling are plotted in Fig. 5.13. The
figures clearly show that the negative peaks (i.e. the growth retardation) in these two
experiments appear at different temperatures but at a similar fraction of ferrite. Hence
the appearance of growth retardation is determined by the physical location of the
residual Mn spike instead of the temperature.
84 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
660 680 700 720 740 760 780 800 8200
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Temperature /°c
df α
/dT
dT/dt=10°c/mindT/dt=300°c/min
PeakPeak
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.002
0.004
0.006
0.008
0.01
0.012
0.014
fα
df α
/dT
dT/dt=10°c/mindT/dt=300°c/min
Peak
(a)
(b)Figure 5.13: The transformation rates as a function of (a) temperature and (b) fraction of ferriteduring the final cooling at a cooling rate of 10C/min and 300C/min after the type I cyclicphase transformations between 842C and 785C in a Fe-0.1C-0.49Mn alloy
5.4 Conclusion
In this chapter, the newly discovered ”growth retardation” stage in Fe-Mn-C alloys
has been systematically investigated using dilatometry and the cyclic phase transfor-
mation concept. The growth retardation stage in the final cooling curve is shown to be
due to the residual Mn spikes (i.e. curved plane of local Mn enrichment) left behind
in austenite by the previous cyclic phase transformations. The magnitude of growth
retardation increases with increasing the Mn concentration and the number of temper-
ature cycles during the cyclic phase transformations. For the growth retardation to take
5.4. Conclusion 85
place in a macroscopically observable manner, it was deduced that the austenite-ferrite
interfaces have to retrace their path during the cyclic phase transformations within the
two phase region. Direct evidence of this path reversibility will be presented in the
next chapter. All features of the observed growth retardation stages are qualitatively
described by the conventional LE model, while PE model does not work at all.
86 Chapter 5. Indirect evidence for the existence of an interfacial Mn Spike
Chapter 6In-situ observation of the cyclic phase
transformation
This chapter is based on
• H Chen, E Gamsjager, S Schider, S van der Zwaag, In-situ observation of
austeniteferrite interface migration in a lean Mn steel during cyclic partial phase
transformations , Acta Mater, 2013; 61:2414-2424.
6.1 Introduction
As shown in Chapter 3 and 5, when recording such partitial cyclic transforma-
tion using dilatometry in Mn steels, two special features are observed [97, 99]: (i) a
stagnant stage in which the degree of transformation does not vary while the temper-
ature changes; Given the excellent qualitative agreement with prediction of a Local
Equilibrium (LE) model [1, 56, 57], it was deduced that the stagnant stage is due to a
growth mode transition from Local Equilibrium-Negligible Partitioning (LE-NP) to Lo-
cal Equilibrium-Partitioning (LE-P). While less likely, the stagnant stage could also be
caused by renewed ferrite or austenite nucleation upon reversal of the heating/cooling
cycle. (ii) the growth retardation stage , which was found to be due to the interaction
88 Chapter 6. In-situ observation of the cyclic phase transformation
of the moving interface with the residual Mn spike created by prior interface pas-
sage. From an analysis of the phenomenon of growth retardation, it was deduced that
the path of the moving interface during the cyclic phase transformations should be
retraceable. In-situ observation of interface migration during the cyclic phase transfor-
mations is required to directly prove this deduction. In this chapter, High Temperature
Laser Scanning Confocal Microscopy (HT LSCM) is applied to directly observe the
movement of austenite-ferrite interface during the cyclic partial phase transformations
in a lean Mn containing steel, and to demonstrate the interfacial mobility during the
stagnant stage as well as the retraceability of the interface during the cycling. Further-
more, the kinetics obtained from the in-situ observation of migrating interfaces will be
compared with dilatometric study as well as the prediction for interfacial movement
from a LE transformation model.
6.2 Experimental
The material investigated is the Fe-0.107C-0.173Mn (in at.%)alloy used earlier. The
average ferrite grain size at room temperature was 50 µm. To measure the phase
transformation kinetics during the cyclic partial phase transformations, two different
techniques are used: Dilatometry and High Temperature Scanning Laser Confocal
Microscopy. A Bahr 805A dilatometer is used to measure the dilatation of the specimen
(10 mm in length and 5 mm in diameter) during the cyclic experiments, and the
dilatation signal can be translated into the average phase transformation kinetics within
the bulk of the sample using the rule of mixtures (Level rule). In addition to the
temperature-control thermocouple, a second thermocouple, placed at 4 mm from the
first, was used to check the temperature gradient along the length of the specimen.
The measured temperature gradient was always smaller than 5 C. The dilatometer
experiment is performed under high vacuum.
A HT LSCM Yonekura VL 2000 DX equipped with a mirror furnace SVF 17 SP
6.2. Experimental 89
Figure 6.1: Schematic sketch of HT LSCM Yonekura VL 2000DX and Mirror furnace SVF 17 SP
(Fig. 6.1) is applied to directly observe the local interface migration on the surface of the
sample (5 mm in diameter and 1 mm in height) during the cyclic phase transformation
experiment. The LSCM has been invented by Minski [104], while the infrared furnace
setup is based on the work by Emi et.al [105]. The HT LSCM has been frequently
applied to study phase transformations in metals (see e.g. [106,107]). To avoid oxidation
on the surface of the sample, the LSCM experiments are performed under a high purity
argon atmosphere. A type H cyclic heat treatment involving an extended hold time at
both the lower and the upper cycling temperature as shown in Fig. 6.2 is applied for
both set of experiments. Unlike conventional type H experiments [97], the sample has
not been preaustenized in the first annealing cycle in order to avoid oxidization of the
sample surface in the hot chamber of the LSCM. Instead, the sample in the current type
H cyclic experiment is first heated up to the highest holding temperature T2 (880 C )
for 120s isothermal ferrite to austenite transformation, and then the temperature was
cycled between T2 and the lower holding temperature T1 (840 C) with 60s isothermal
holding at T1 and 120 s at T2 . Both T2 and T1 are located in the austenite + ferrite
region in the phase diagram. The heating and cooling rates for all experiments were
90 Chapter 6. In-situ observation of the cyclic phase transformation
0.5 C/s.
Figure 6.2: Schematic sketch of heat treatment procedures for the type H experiments
6.3 Results
6.3.1 Dilatometer results
In the Fig. 6.3, the length change as a function of temperature during the type H cyclic
experiment between 840C and 880C is shown. During heating to and isothermal
holding at 880C, the initial microstructure (almost 100% ferrite due to the ultra low
carbon concentration) partially transforms into austenite, and a mixture of austenite +
ferrite is created. The specimen with a mixture of austenite and ferrite was then cooled
down at a rate of 0.5 C/s to 840C, during which two stages are observed: (i) A linear
thermal contraction stage (H1-H2 as shown in Fig. 6.3) without phase transformation or
interface migration, which has been named stagnant stage ; (ii) A nonlinear expansion
stage (H2-H3 stage as shown in the Fig. 6.3) due to the austenite to ferrite transformation
upon cooling, during which the austenite/ferrite interfaces moves into austenite. Once
the lower temperature of 840C was reached, the specimen was kept at this temperature
6.3. Results 91
Figure 6.3: The dilatation as a function of temperature during the type H cyclic experimentsbetween 880C and 840C in a Fe-Mn-C alloy
for 60s. During isothermal holding, the austenite/ferrite interfaces continue to migrate
into austenite, which is accompanied by a volume dilatation (H3-H4 as shown in
Fig. 6.3). After the isothermal holding at 840 C, the specimen was heated up to
880C. During the heating, there are also two stages: (i) a linear thermal expansion
due to increase in temperature, during which no phase transformation occurs (H4-H5
as shown in Fig. 6.3). It is also called stagnant stage. (ii) a nonlinear contraction, which
is caused by the ferrite to austenite transformation (H5-H6 as shown in Fig. 6.3). The
second and third transformation cycles show a similar character as that in the first
cycle. It is worth noting that the length of the specimen tends to increase marginally
with increasing number of cycles, which is attributed to the transformation plasticity
effects [93, 95].
92 Chapter 6. In-situ observation of the cyclic phase transformation
6.3.2 In-situ HT LSCM observations
During the first heating, it is observed that the austenite nucleates at ferrite grain
boundaries, and that the newly formed austenite/ferrite interfaces grow into ferrite.
The austenite/ferrite interfaces continue to migrate into the ferrite during the isother-
mal holding at 880 C, and a mixture of ferrite and austenite is created at the end of
holding. In the live recorded video (see the video in supplementary file) the move-
ment of austenite/ferrite interfaces is quite obvious, yet qualitative image analysis of
successive stills was required to accurately determine the position of the interface. The
interface position was recorded as a function of time and temperature during the entire
thermal cycling procedure. In the following we will describe in detail the characteris-
tic behaviour of two moving interfaces during the cyclic transformation and impose
interface positions on the micrographs. The behaviour of these two interfaces reflects
that of many more grains observed on the sample surface.
In Fig. 6.4a, the microstructure at the end of isothermal holding at 880C is shown.
There is a mixture of austenite and ferrite in the system, and the dotted lines are
the austenite/ferrite interfaces. It is worth noting that the austenite/ferrite interfaces
already stopped at the end of the isothermal holding, which means that the isother-
mal ferrite to austenite transformation at 880C has come to a completion. After the
isothermal holding at 880C, the specimen is cooled down to 840C at a rate of 0.5C
/s. Now two different processes for the austenite to ferrite transformation starting
from a mixture of austenite + ferrite during the cooling are possible: (i) There is no
new ferrite nucleation. Instead, the existing austenite/ferrite interfaces migrate into
the austenite; (ii) The transformation proceeds via the nucleation of new grains while
the old ferrite-austenite interfaces present remain at their position in the sample and
turn into ferrite-ferrite grain boundaries.
In Fig. 6.4b-f, the evolution of microstructure during the cooling stage in the first
cycle of type H experiment is shown. It is shown that during the cooling from 880C to
870C the austenite/ferrite interfaces does not move at all, nor is there nucleation of new
6.3. Results 93
Figure 6.4: The evolution of microstructure during the cooling in the first cycle of type Hexperiment between 880C and 840C in a Fe-Mn-C alloy
94 Chapter 6. In-situ observation of the cyclic phase transformation
Figure 6.5: The evolution of microstructure during the heating in the first cycle of type Hexperiment between 880C and 840C in a Fe-Mn-C alloy
6.3. Results 95
Figure 6.6: The evolution of microstructure during the cooling in the second cycle of type Hexperiment between 880C and 840C in a Fe-Mn-C alloy
96 Chapter 6. In-situ observation of the cyclic phase transformation
grains. This stage of non interfacial movement corresponds to so called stagnant stage
in the dilatometer results, and the duration of the stagnant stage (the temperature
range of stagnant stage) measured in dilatometer is exactly the same as that in the
in-situ observation. Once the temperature is lower than 870C, the austenite/ferrite
interfaces start to move into austenite, but there is still no new ferrite grain to appear.
As shown in Fig. 6.4c, at 860C the austenite/ferrite interfaces have moved to new
positions in the austenite. This in-situ observation directly proves that the ferrite
formation from a mixture of austenite and ferrite proceeds via migration of existing
austenite/ferrite interfaces instead of by new ferrite nucleation. In Fig. 6.4d-f, the
austenite/ferrite interfaces continue to move into the austenite. It is shown that during
the interface migration, the shape of the austenite/ferrite interface changes due to
hard impingement of the growing ferrite grains. In Fig. 6.4g and h, the intermediate
positions of the two austenite/ferrite interfaces (I and II) during this cooling stage are
indicated. The dashed lines signify the starting austenite/ferrite interfaces, and the
temperature interval between plotted interface positions is 5C. The figure shows that
during the phase transformation the interfaces move in a more or less planar manner
and that orientation of the austenite/ferrite interfaces, i.e. the non impinging interface,
varies only slightly.
At the later stage of isothermal holding at 840C, the austenite/ferrite interfaces
as shown in Fig. 6.4f are immobile again, which means the austenite to ferrite trans-
formation finished. The specimen is then heated up to 880C, and the evolution of
the microstructure during the heating is shown in Fig. 6.5a-e. It is very clear that the
austenite/ferrite interfaces are all immobile until the temperature reaches 850C, which
is in agreement with the stagnant stage upon heating observed in the dilatometer ex-
periments. The duration of the stagnant stage during heating in the in-situ experiments
(about 10 C, or 20 seconds) is almost the same as that observed in the dilatometer ex-
periment. The austenite/ferrite interfaces start to move backwards into ferrite once the
temperature is above 850C. In Fig. 6.5e, the position of austenite/ferrite interfaces at
the end of isothermal holding at 880C is shown. It shows that after the first cycle the
6.3. Results 97
interfaces did not move back exactly to the original position shown in Fig. 6.4a. The
passages of the two austenite/ferrite interfaces (I and II) during this heating stage are
indicated in Fig. 6.5f and g.
In the Fig. 6.6a-e, the evolution of microstructure upon cooling in the second cycling
is indicated. It shows again that the austenite/ferrite interfaces do not move until the
temperature is lower than 870C. It is interesting to note that at the end of isothermal
holding at 840C in the second cycle the austenite/ferrite interfaces now move to exactly
the same position as that in the first cycle as shown in Fig. 6.4f. The passages of the two
austenite/ferrite interfaces (I and II) during this cooling stage are indicated in Fig. 6.6f
and g.
In Fig. 6.7, the position of austenite/ferrite interfaces as a function temperature
during the type H cyclic experiment between 880C and 840C is summarized. The in-
terface positions as shown in Fig. 6.4a are the starting points A1 and B1 in Fig. 6.7. A1-A2
and B1-B2 are comparable with the stagnant stage H1-H2 upon cooling in Fig. 6.7, while
A4-A5 and B4-B5 stages are relevant to H4-H5 stage during heating. The transformation
curves in Fig. 6.7 have a comparable shape as the transformation curves derived from
dilatometry.
98 Chapter 6. In-situ observation of the cyclic phase transformation
(a)
(b)
Figure 6.7: The experimentally obtained interface position as a function of temperature duringthe cycle of type H experiment between 880 C and 840 C in a Fe-Mn-C alloy
6.4 Discussion
One of the important findings in the in-situ observation here is that during the cyclic
partial phase transformations the transformations proceed via existing austenite/ferrite
6.4. Discussion 99
interfaces migration instead of new nucleation. The finding directly proves that the
stagnant stage should not be interpreted as a result of incubation for nucleation, and
is indeed a growth kinetics issue.
Figure 6.8: The interface position as a function of temperature during the cycle of type Hexperiment between 880 C and 840 C in a Fe-Mn-C alloy simulated by LE and PE model
As for the growth kinetics, much effort has been made to validate PE and LE mod-
els by comparing experimentally observed transformation rate with their predictions,
but there are still many discrepancies. According to Thermo Calc calculation, both
LE and PE growth are thermodynamically possible at 880Cand 840C, and thus both
models can be applied to simulating the interface migration kinetics during the cyclic
phase transformations here. In Fig. 6.8, the position of the austenite/ferrite interface
as a function of temperature during the cyclic phase transformation between 880C
and 840C is plotted for both the PE and LE model. It is clearly shown that in the
PE simulation the austenite/ferrite interface immediately starts to move upon cooling
or heating from the transition temperatures, which is totally different from the exper-
imental results from dilatometer and HT LSCM. Instead, the LE model predicts two
sluggish transformation stages S1-S2 and S4-S5, which are comparable with the stag-
100 Chapter 6. In-situ observation of the cyclic phase transformation
nant stages observed in both experiments. According to the evolution of simulated
Mn profiles during cyclic phase transformations, it was found that the sluggish stages
are related to a Mn enrichment zone adjacent to the migrating interface. Additional
cyclic phase transformation experiments for Fe-C-xMn (x is mole fraction of Mn, and
the mole fraction of C is fixed) and Fe-C-Mn-xNi (x is mole fraction of Ni, and the mole
fractions of C and Mn are fixed) alloys [108] has shown that the duration of stagnant
stage increases with increasing mole fraction of the partitioning substitutional alloying
elements Mn or Ni, and no stagnant stage was observed in Fe-C alloy. The stagnant
stage is clearly related to partitioning of substitutional alloying elements.
In Fig. 6.9, the experimentally observed positions of austenite/ferrite interfaces as a
function of time during the austenite to ferrite transformation and ferrite to austenite
transformation in the first cycle of the cyclic phase transformation are compared with
those simulated by the LE and PE model. It shows that for both austenite to ferrite and
ferrite to austenite transformations the equilibrium migration distance of interface I is
different from that of interface II, while in the initial stage (before soft impingement
starts) the migration rates for both interfaces are quite close to each other. The difference
in the equilibrium migration distance is due to difference in the size of the grain in
which the interface moves back and forth. According to model calculations in Ref [6], at
a certain temperature the grain size should not have an effect on the interface migration
rate in the initial stage of the phase transformation without soft impingement effect,
while it is obvious that the equilibrium migration distance at a certain temperature
is determined by the grain size. This model prediction is in good agreement with
the in-situ observation here. The dilatometer result in Fig. 6.9 is obtained based on
the assumption that during the cyclic phase transformations the geometry is planar,
and it is considered to be an average kinetics. It shows that for the initial stages of
both austenite to ferrite transformation and ferrite to austenite transformations the
average kinetics measured by dilatometry is also quite comparable to those obtained
from in-situ observations. As accurate corrections of the transformation kinetics for
soft impingement effect in the absence of a full 3D description of the grains is difficult,
6.4. Discussion 101
(a)
(b)
Figure 6.9: The interface position as a function of time during (a) the austenite to ferritetransformation and (b) the ferrite to austenite transformation in the type H experiment between880 C and 840 C in a Fe-Mn-C alloy.
the initial stages of both the austenite to ferrite transformation and ferrite to austenite
transformation are taken as the proper stages for validating the different theoretical
models. It is worth pointing out that due to the limited resolution of HT LSCM the
initial stage of growth is very difficult or impossible to be observed in the normal
experiments in which the interface migration starts from a small nuclei [106], while in
the cyclic phase transformations the initial stage of growth can be easily measured as
102 Chapter 6. In-situ observation of the cyclic phase transformation
the interfaces already exist and can be detected all along the temperature cycling.
In [99], a growth retardation stage in dilatometric experiments has been discovered
during the final austenite to ferrite transformation after several cyclic partial phase
transformations in Fe-C-Mn alloys, and the growth retardation was analysed to be
due to the interaction between the moving interfaces and the residual Mn spikes left
behind by prior transformation cycles. For the growth retardation to take place in a
macroscopically observable manner, it was deduced that the austenite/ferrite interfaces
had to migrate in a retraceable way during the cyclic phase transformations. If the
interface migration would not be retraceable, the interfaces would not impinge the
residual Mn spike head-on, and no significant growth retardation would occur. The
in-situ observation in this work directly proves that the austenite/ferrite interfaces
indeed migrate in a retraceable manner during the cyclic phase transformations.
6.5 Conclusion
In this chapter, for the first time the austenite/ferrite interface migration during cyclic
partial phase transformations has been observed in-situ by LSCM. The interface migra-
tion kinetics directly observed from LSCM is in perfect qualitative and almost perfect
quantitative agreement with the mechanisms and kinetics as derived by dilatometry.
The in-situ observation directly proves that in the cyclic experiments the transforma-
tions proceed via migration of existing austenite/ferrite interface instead of nucleation
of new grains, and that the interface migration is retraceable. There is now direct
evidence of the interface being immobile during the stagnant stage. The LE model pre-
dicts the kinetics of both the austenite to ferrite and ferrite to austenite transformations
rather well.
Chapter 7Bainitic transformation during the
interrupted cooling experiments
This chapter is based on
• H Chen, A Borgenstam, J Odqvist, I Zuazo, Goune Mohamed, J Ågren,S van der
Zwaag, Application of interrupted cooling experiments to study the mechanism
of bainitic ferrite formation in steels, Acta Materialia, accepted for publication,
2013.
7.1 Introduction
The kinetics of the bainitic ferrite formation in lean and more alloyed steels is of great
interest for the production of advanced high strength steels, and has been widely stud-
ied both experimentally and theoretically [19,27,33–35,37,109–115]. Despite abundant
efforts, the mechanism of bainitic ferrite formation is still one of the most controver-
sial topics in the field of phase transformations [13, 14, 19, 37, 43]. Generally speaking,
there are two competing views on the mechanism of bainitic transformation:(i)In the
first one, the bainitic transformation is considered to be a diffusionless transforma-
tion [17, 20], which in principle is the same as the martensitic transformation. During
the growth of a bainitic plate there is no need for carbon diffusion, but the diffusion
103
104Chapter 7. Bainitic transformation during the interrupted cooling experiments
may take place after the growth; (ii)In the second view the bainitic transformation
is considered as a diffusional transformation [18, 36, 37, 110, 116]. During the bainitic
ferrite formation the carbon diffuses away from the growing ferrite into austenite, and
the transformation rate is controlled by the diffusion of carbon and possibly other
alloying elements. In the last decades, much effort has been paid to perform conven-
tional isothermal experiments using modern techniques to validate either two views.
However, both schools seem to be able to explain those experimental results to their
own satisfaction [27, 33–35, 113, 114]. Clearly the experimental conditions imposed in
the past have not allowed a clear discrimination between the correctness of both views
that could be accepted by both schools.
In [117] , it was found that the initial kinetics of austenite to ferrite transformation
starting from a mixture of austenite and ferrite is determined by the distribution of
alloying element at the austenite/ferrite interface created earlier. In other words, the
measured austenite to ferrite transformation kinetics starting from the austenite and
ferrite mixture can be used to demonstrate that substitutional element partitioning
does take place at the austenite-ferrite interface .
In this chapter, the concept behind the cyclic partial transformation will be extended
to study the bainitic ferrite formation. A series of type IC (Isothermal and Cooling)
dilatometric interrupted cooling experiments, in which the bainitic ferrite formation
starts from a mixture of austenite and bainitic ferrite, are performed for two Fe-C-Mn
and Fe-C-Mn-Si alloys. A so called Gibbs energy balance approach, in which the
dissipation of Gibbs energy due to diffusion inside the interface and interface friction
is assumed to be equal to the available chemical driving force, is proposed to explain
the special features observed in the interrupted cooling experiments.
7.2 Experimental
The materials investigated here are an Fe-3Mn-0.1C (wt. %) alloy and an Fe-3Mn-0.1C
-1.5Si(wt. %) alloy. In the interrupted cooling experiments, first an isothermal partial
7.2. Experimental 105
bainitic transformation is imposed to create a mixture of austenite and bainitic ferrite,
and then the mixture is slowly cooled down for further bainitic ferrite formation. This
kind of interrupted cooling experiment is called type IC (Isothermal and Cooling) ex-
periment. The temperature program for type IC experiment is sketched in Fig. 7.1. The
preceding austenization temperature and time for all type IC experiments are 1000C
and 5 minutes, respectively. The cooling and heating rate during the type IC exper-
iments are both 1C/s. Unlike conventional experiments the bainitic transformation
kinetics during the cooling stage, which is expected to reflect the austenite/bainitic
ferrite interface condition at the end point of isothermal holding stage, is the feature of
interest here. This kinetics can be used to deduce the evolution of the austenite/bainitic
ferrite interface condition during the isothermal bainitic transformation. In order to
study this evolution, different isothermal holding times are applied. It is expected
that different austenite/bainitic ferrite interface conditions are obtained at the end of
isothermal bainitic transformations.
Time
Austenization
Ф=1 /S
Bs
Type IC experiment
Ms
Tem
peratu
re
Figure 7.1: The temperature program for the type IC experiment
A Bahr 805A dilatometer is used to measure the dilatation of the specimen (10mm
106Chapter 7. Bainitic transformation during the interrupted cooling experiments
in length and 5mm in diameter) during the type IC experiment, and the dilatation
signal can be transferred into phase transformation kinetics using the rule of mixture.
In order to check the temperature difference along the length of the specimen, two
thermocouples, spaced 4 mm apart, were used in the experiments, and the measured
temperature gradient was typically smaller than 5C. The samples were mechanically
polished and etched in 2 % Nital solution for light optical microscopy (LOM). Trans-
mission electron microscopy (TEM) studies were conducted on a JEOL 2100F operating
at 200kV. The TEM samples were prepared by electropolishing at 60-80mV in a Struers
TenuPol-5 using a solution containing 950ml acetic acid and 50ml of perchloric acid.
7.3 Model
7.3.1 Dissipation due to diffusion inside interface
During the diffusional phase transformations, the interaction between solutes and mi-
grating interface gives rise to a retardation of the interface migration, and it changes the
local interface condition. There are two different approaches to modeling of the inter-
action between solutes and migrating interfaces:(i) Solute drag model. This model was
originally developed for grain growth by Cahn and Lucke [118, 119], and then Purdy
and Brechet extended it to phase transformations [120]; (ii) The Hillert-Sundman’s
model [121], which is specially for phase transformations. These two concepts have
been widely used in modeling of phase transformation kinetics [11,12,42,69,122–125].
According to Purdy and Brechet [120] a triangular potential well is assumed inside
the interface. In Fig. 7.2, the schematics of the potential well for austenite stabilizer
(i.e. Mn and Ni) and ferrite stabilizer (i.e. Si and Al) are shown. µ0α and µ0
γ are the
chemical potential of the solute in ferrite and austenite, respectively; E0 is the binding
energy; ∆E is the half of solute chemical potential difference between austenite and
ferrite, ∆E=(µ0γ-µ0
α)/2, and the magnitude of ∆E depends on the partition coefficient of
the solutes and temperature. −δ and δ are the boundary positions of the interface, and
7.3. Model 107
0
gm
d- d
2 ED
0
am
0E E- D
0 0
2E
g am m-D =
ag
0
gm
d- d
2 ED0
am
0E E- D
0 0
2E
g am m-D =
a
g
(a)
(b)
Figure 7.2: The schematics of potential well for (a) austenite stabilizer and (b) ferrite stabilizerinside interface
the thickness of the interface is 2δ.
Due to the potential well inside interface, the solutes favorably diffuse/segregate
into the interface during phase transformations. The governing equation for solute
108Chapter 7. Bainitic transformation during the interrupted cooling experiments
diffusion inside an interface moving with a quasi-steady velocity v is:
∂∂x
[D∂X∂x
+DXRT
∂E∂x
+ vX] = 0 (7.1)
Where X is the concentration of solutes, x is distance, D is the diffusion coefficient of
solute inside interface, R is gas constant, T is temperature, v is the interface velocity, E
is the free energy of interaction of the solute with the interface.
Based on the diffusion equation and the potential well, the solute profile inside
interface at a given velocity can be obtained. The dissipation of Gibbs energy due to
diffusion inside interface can be calculated from Cahn’s equation [119, 120]:
PVm = −
∫ +δ
−δ
(X − X0)(dE/dx)dx (7.2)
Where P is the solute drag force , which arises from the asymmetric solute distribution
inside the interface, Vm is the molar volume, X0 is the nominal concentration of solutes
in the alloys.
The dissipation of Gibbs energy can also be calculated from Hillert and Sundman’s
equation [12, 121]:
∆Gdi f fm = −
∫ +∞
−∞
(uM − u0M)[d(µM − µFe)/dx)]dx (7.3)
Where ∆Gdi f fm is the dissipation of Gibbs energy due to diffusion inside the interface,
uM is the u fraction of solute M defined as xM/(1 − xc), u0M is the initial u fraction of
solute M, µM and µFe are the chemical potential of M and Fe, respectively.
The equations by Cahn and Hillert-Sundman have been compared by Hillert [39],
and it was found that they give the same result if the same physical parameter values
are used in both calculations. In both equations, the dissipation inside the interface
and dissipation due to the spike of solutes in front of interface are both taken into
account.
7.3. Model 109
7.3.2 Interface friction
In the classical diffusional models, it is generally assumed that the interface friction is
zero, which means that the value of interface mobility is infinite and all the available
driving force is only dissipated by diffusion of solutes. However, in reality the value
of interface mobility is finite [2,12], and the dissipation caused by the interface friction
can be written as:
∆G f rictionm = vVm/M (7.4)
Where M is intrinsic interface mobility and Vm is the molar volume.
7.3.3 Chemical driving force
The chemical driving force per mole of atoms can generally be calculated from:
∆Gchemm =
n∑i
x0i [µγ/αi (xγ/αi ) − µα/γi (xα/γi )] (7.5)
Where x0i is the composition of material transfered over the interface, when there is no
diffusion in α, x0i =xα/γi , ∆Gchem
m is the chemical driving force, i is the element in the alloys,
n is the number of elements, xα/γi and xγ/αi are the mole fractions of i at the interface
on the ferrite and austenite side respectively(when i is subtitutional alloying elements,
xγ/αi is chosen to be the concentration at the root of the spike, thus the spike is included
in the calculations), µα/γi and µγ/αi are the chemical potential of i in ferrite and austenite.
If formation of bainitic ferrite is controlled by carbon diffusion, the growth rate of
a plate can be calculated from the Zener-Hillert equation [38]:
vDγ
C
=RT(xγ/αC − x0
C)2
8Vmσx0C
(7.6)
where v is the growth rate, DγC is the diffusion coefficient of C in austenite, σ is the
interfacial energy, xγ/αC is the mole fraction of C at the interface in austenite, x0C is the
mole fraction of C in the austenite.
110Chapter 7. Bainitic transformation during the interrupted cooling experiments
Coupling Eq 7.5 and Eq 7.6, the chemical driving force ∆Gchemm will be a function
of growth rate v and x0C. x0
C depends on the fraction of austenite transformed here,
and it increases with increasing fraction of austenite transformed. All calculations for
chemical driving force are performed in Thermo Calc TQ programming interface [90].
7.3.4 Gibbs energy balance
Based on the above equations, the dissipation of Gibbs energy due to diffusion inside
interface, the interface friction and the chemical driving force can be calculated. The
chemical driving force should be equal to the dissipation of Gibbs energy due to
diffusion inside the interface and interface friction:
∆Gchemm = ∆Gdi f f
m + ∆G f rictionm (7.7)
7.4 Experimental results
In order to obtain a mixture of bainitic ferrite and austenite, the isothermal tempera-
tures for the type IC experiments have to be lower than the bainite start temperature,Bs,
but higher than the martensite start temperature, Ms. Both experiments and modeling
work have been done to determine the proper isothermal temperatures for type IC
experiments in Fe-3Mn-0.1C -1.5Si and Fe-3Mn-0.1C alloys. The isothermal tempera-
tures for the Fe-3Mn-0.1C -1.5Si alloy were chosen to be 530 C and 500 C, while those
for the Fe-3Mn-0.1C alloy were 550 C and 570 C.
In Fig. 7.3a, the length change and fraction of austenite transformed as a function
of time during the isothermal transformation at 530C for Fe-3Mn-0.1C-1.5Si alloy
is shown. During the isothermal holding, the length of the sample increases as a
function of time, which is due to the austenite to bainitic ferrite phase transformation.
The fraction of bainitic ferrite is proportional to the relative increase in the length of
sample. Fig. 7.3 shows the kinetics in the initial stage of the isothermal bainitic ferrite
transformation is very fast, while in the later stage the transformation is extremely
7.4. Experimental results 111
300 350 400 450 500 55025
30
35
40
45
50300 350 400 450 500 550
25
30
35
40
45
50
20min5min3min2min
90s
1min
Leng
th ch
ange
/m
Temperature / oC
t=30s t=1min t=90s t=2min t=3min t=5min t=20min
(a)
(b)Figure 7.3: (a) The length change and fraction of austenite transformed as a function of timeduring the isothermal transformation at 530C for the Fe-3Mn-0.1C-1.5Si alloy; (b) The lengthchange as a function of temperature during the type IC experiment with different isothermalholding times at 530 C for Fe-3Mn-0.1C -1.5Si alloy
sluggish but not complete. It seems that there is a sharp transition from fast to extremely
sluggish transformation mode during the isothermal bainitic transformation, but the
precise transition point can not be established unambiguously.
In Fig. 7.3b, the length change as a function of temperature during the type IC
experiment with different isothermal holding times at 530 C for Fe-3Mn-0.1C -1.5Si
is indicated. It clearly shows that for the Fe-3Mn-0.1C -1.5Si alloy the slope of length
change during the second cooling stage following the isothermal bainitic ferrite for-
mation is significantly affected by the isothermal holding time. This means that the
112Chapter 7. Bainitic transformation during the interrupted cooling experiments
kinetics of bainitic ferrite formation during the cooling is affected by the prior isother-
mal bainitic transformations. The kinetics of bainitic ferrite formation upon cooling
can be divided into two categories: (i) When the isothermal holding is very short i.e.30
s or 1 minute, the length of the sample continues immediately to increase upon cool-
ing, which means the bainitic ferrite transformation can still proceed with a high rate
upon further cooling. The transformation rate at the beginning of cooling decreases
slightly with increasing isothermal holding time; (ii) When the isothermal holding time
exceeded 5 minutes, during the initial stage of cooling the length of sample decreases
linearly as a function of temperature, which is mainly due to thermal contraction. After
this holding time the bainitic ferrite formation is suppressed or very sluggish. Upon
further cooling, the bainitic ferrite formation starts with a high transformation rate
again. The initial stage of cooling, during which the transformation is suppressed or
very sluggish, is called ”stagnant stage” here. It is shown that the length of the stagnant
stage increases with increasing isothermal holding time. The same behavior is also
observed when the isothermal temperature is 500C for the same type of experiment.
The physical reason behind the stagnant stage during the bainitic ferrite formation will
be discussed in the following sections.
In Fig. 7.4, typical optical microstructures of the Fe-3Mn-0.1C -1.5Si alloy, isother-
mally heat treated for different times and then quenched down to room temperature,
are shown. In addition to bainitic ferrite plates there is also a small fraction of grain
boundary ferrite in both samples.
From Fig. 7.3b, it is not possible to conclude whether the transformation is sluggish
or completely stopped during the stagnant stage. In order to investigate this further,
a two step isothermal transformation experiment (called Type II) was performed.
In such an experiment, two isothermal transformations are imposed in succession
followed by linear cooling. The behavior during the second linear cooling is of interest
and taken to reflect the conditions at the austenite-bainite interfaces. In Fig. 7.5a,
the length change of a Fe-3Mn-0.1C-1.5Si sample during a Type II experiment, first
isothermally heat treated at 530 C for 5 minutes followed by 20 minutes at 520 C, as a
7.4. Experimental results 113
(a)
(b)Figure 7.4: The typical optical microstructure of the Fe-3Mn-0.1C-1.5Si samples (a) Isothermalannealed for 30s at 530C and quenched;(b) Isothermal annealed for 20minutes at 530C andquenched.
function of temperature is shown. The figure shows that after 5 minutes of isothermal
holding at 530 C the length of the sample increases marginally during the 20 minutes
isothermal holding at 520 C. The transformation has not completely stopped, but
less than 5 % austenite has been transformed during the 20 minutes of isothermal
holding. In Fig. 7.5b, the relative length changes as a function of time during the
isothermal transformation at 530 C and 520 C in the type II experiments are shown.
114Chapter 7. Bainitic transformation during the interrupted cooling experiments
250 300 350 400 450 500 55020
25
30
35
40
45
50250 300 350 400 450 500 550
20
25
30
35
40
45
50
Leng
th ch
ange
/m
Temperature / oC
0 200 400 600 800 1000 12000
5
10
15
200 200 400 600 800 1000 1200
0
5
10
15
20
Leng
th ch
ange
/m
Time /s
520 OC
530
(a)
(b)
Figure 7.5: (a)The length change of a Fe-3Mn-0.1C-1.5Si sample as a function of temperatureduring a type II experiments; (b)The length change of a Fe-3Mn-0.1C-1.5Si sample as a functionof time during the isothermal transformation at 530 C and 520 C
It shows that the transformation rate at 520 C is extremely slow, and there is no clue
to when the transformation would completely stop at this temperature. As shown
in Fig. 7.5a, compared with those in the type IC experiment, the kinetics of bainitic
ferrite formation upon cooling in the type II experiments is considerably slower, and
the length of stagnant stage is much longer. This indicates that the kinetics of bainitic
ferrite formation upon cooling is significantly affected by the 20 minutes isothermal
7.4. Experimental results 115
holding at 520 C although only a very small additional fraction of austenite has been
transformed.
Figure 7.6: The length changes as a function of temperature during the type IC experimentwith different isothermal holding times at 550 C for the Fe-3Mn-0.1C alloy.
In Fig. 7.6, the length changes as a function of temperature during the type IC
experiment with different isothermal holding times at 550 C for the Fe-3Mn-0.1C
alloy are shown. The figure shows that there is also a stagnant stage in Fe-3Mn-0.1C
alloy, and the basic features are similar as those of type IC experiments in Fe-3Mn-0.1C
-1.5Si alloy. Unlike the Fe-3Mn-0.1C -1.5Si alloy, there is the possibility of carbide
formation during the bainitic ferrite formation in the Fe-3Mn-0.1C alloy, which would
accelerate the transformation kinetics upon further cooling. In Fig. 7.7, typical TEM
of the Fe-3Mn-0.1C -1.5Si and Fe-3Mn-0.1C samples, which were isothermally heat
treated at 530 C for 20 minutes and 550 C for 60 minutes respectively and then
quenched down to room temperature, are shown. TEM observations show that there
116Chapter 7. Bainitic transformation during the interrupted cooling experiments
αααααααα
αααααααα
αααααααα
αααααααα αααααααα
(a)
(b)
Figure 7.7: The transmission electron micrographs of (a)the Fe-3Mn-0.1C -1.5Si alloy isother-mally heat treated at 530 C and then quenched ; (b) Fe-3Mn-0.1C alloy isothermally heattreated at 550 C and then quenched. αM is martensite, αB is bainitic ferrite
is no carbide formation in either alloy during the bainitic transformation. The TEM
observations, at even higher magnifications than those shown in Fig. 7.7, also show
that there is no carbide.
7.5 Theoretical analysis
In the type IC experiments, it is expected that once the isothermal transformation
a mixture of bainitic ferrite and austenite is created, the bainitic ferrite formation
will continue immediately according to both the diffusionless theory and diffusional
7.5. Theoretical analysis 117
transformation theory with the paraequilibrium assumption. However, as shown
in the type IC experiments for both Fe-3Mn-0.1C -1.5Si and Fe-3Mn-0.1C alloys, for
longer isothermal holding times a so called stagnant stage is observed. A similar
stagnant stage was also experimentally observed during the grain boundary ferrite
transformation starting from a mixture of austenite and ferrite [97, 117], and it was
found that the immobility of the interface was due to a thin layer of Mn enrichment
in front of the austenite/ferrite interface [117]. At first sight, it is expected that the
stagnant stage during the bainitic transformation here is also caused by the Mn spike
(enrichment) and Si spike (depletion) ahead of the interface. The diffusion coefficient
of Si and Mn in austenite(DFCCMn = 5.3× 10−22,DFCC
Si = 5.2× 10−21,m2/s) can be calculated
by DICTRA based on Mob2 database [126], and thus the thickness of the Mn and Si
spike can be roughly estimated by (2Dt)0.5. After 100 seconds, the thickness of the
Mn and Si spikes are estimated to be 0.5 and 1.7 nm, respectively. Therefore, the Mn
and Si spike can be considered as a reason for retarding the growth of bainitic ferrite
plates. However, during the bainitic ferrite formation the nucleation of bainitic plates
is continuous, which means that during cooling there are possibly newly nucleated
bainitic plates with no Mn or Si spikes. The presence of a spike is not sufficient in order
to explain why the newly nucleated bainitic plates cannot move during the initial stage
of cooling. The Gibbs energy balance approach will now be applied to model the phase
transformation kinetics during the type IC experiments in the Fe-Mn-C and Fe-Mn-Si-
C alloys studied here. The type IC experiments in the Fe-Mn-C alloy, in which only
the dissipation due to Mn diffusion inside the interface is considered, is taken as the
starting point.
In Fig. 7.8a, the dissipation of Gibbs energy due to Mn diffusion inside the interface
at 550 C for the Fe-Mn-C alloy as a function of interface velocity is plotted. The
thickness of interface 2δ is here assumed to be 0.5nm, and the diffusion coefficient of
Mn inside the interface DIntMn is assumed to be the geometric average of the diffusion
coefficient of Mn in austenite DFCCMn , in ferrite DBCC
Mn , and in the ferrite grain boundary
DGBMn. The value of DFCC
Mn and DBCCMn can be calculated using DICTRA, while the value of
118Chapter 7. Bainitic transformation during the interrupted cooling experiments
10−11
10−10
10−9
10−8
10−7
10−6
10−50
100
200
300
400
500
600
700
800
900
Interface velocity, m/s
Gib
bs
en
erg
y, J
/mo
l
E0Mn=9.9KJ/mol
E0Mn=8KJ/mol
E0Mn=11KJ/mol
∆Gmdiff +∆G
mfriction
−3 −2 −1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Normalized distance, x/ δ
X/X
0
Interface boundary Interface boundary
γ
αv=1.5x10−9m/s
v=1x10−7m/s
(a)
(b)
Figure 7.8: (a)The dissipation of Gibbs energy due to Mn diffusion inside the interface at 550 Cas a function of interface velocity; (b)The Mn profiles inside interface with the EMn
0 =9.9kJ/molat a high interface velocity (1 × 10−7 m/s) and a low interface velocity (1.5 × 10−9 m/s)
DGBMn is obtained from [127]. At 550 C, DInt
Mn is about 10−18m2/s, which is about 4 order of
magnitude higher than DFCCMn . To our knowledge, there is no experimentally determined
value of the intrinsic interface mobility for the austenite to ferrite transformation yet,
while the value of the effective interface mobility has been measured by different
research groups [3,4,53,84]. In the present calculation, the value of the effective interface
mobility will be used for calculations, and it is expected that it is an underestimation of
the intrinsic interface velocity. The value of the effective interface mobility used here
is derived from Wits et.al’s equation [84]. The binding energy of Mn, EMn0 , has been
7.5. Theoretical analysis 119
1E-8 1E-7 1E-6 1E-50
200
400
600
800
Chem
ical d
riving
force
, J/m
ol
0,00
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
Chemical driving force Mole fraction of C
Mole fraction of C
1E-9 1E-8 1E-7 1E-6 1E-50
200
400
600
800
Chem
ical d
riving
force
, J/m
ol
Interface velocity, m/s
f =0 f =0.4 f =0.7 f =0
(a)
(b)
Figure 7.9: (a)The chemical driving force and the mole fraction of C at the interface in austeniteas a function of interface velocity for Fe-3Mn-0.1C alloy at 550 C; (b)The chemical drivingforce as a function of interface velocity with different fractions of ferrite for Fe-3Mn-0.1C alloyat 550 C
experimentally measured by Enomoto et.al [128], and three different values of binding
energy are applied in the calculation to indicate the effect of binding energy on the
dissipation.
In Fig. 7.8b, the Mn profiles inside and interface with EMn0 =9.9kJ/mol at high and
low velocities are shown. It shows that at the high velocity (1×10−7 m/s) the Mn profile
is flat inside interface, and leads to very small dissipation as already shown in Fig. 7.8a .
At a low velocity (1.5×10−9 m/s), there is considerable amount of Mn segregation inside
120Chapter 7. Bainitic transformation during the interrupted cooling experiments
the interface, which leads to a large dissipation as shown in Fig. 7.8a. It is interesting to
note that at a low interface velocity the Mn concentration at the interface boundary on
the austenite side is higher than the nominal concentration, which contribute to part
of the total dissipation, known as dissipation due to a spike [12, 129].
Based on Zener-Hillert’s equation, the growth rate of a bainitic ferrite plate depends
on the mole fraction of C in austenite at the interface xγ/αC and the mole fraction of C
in the austenite, x0C. In Fig. 7.9a, the chemical driving force and the mole fraction of
C at the interface in austenite as a function of interface velocity for the Fe-Mn-C alloy
at 550 C is shown. The mole fraction of C in austenite x0C is fixed as the nominal
value, which means that the fraction of austenite transformed is zero. It shows that
the xγ/αC increases with increasing interface velocity, while the chemical driving force
decrease. As the mole fraction of C in austenite x0C depends on the fraction of austenite
transformed, it is expected that the chemical driving force is also affected by it. As there
is a small fraction of grain boundary ferrite formed in the experiments, the fraction
of transformed austenite includes both bainitic ferrite and grain boundary ferrite. In
Fig. 7.9b, the chemical driving force as a function of interface velocity with different
fractions of austenite transformed for Fe-Mn-C alloy at 550 C is shown. It indicates
that at a certain interface velocity the chemical driving force decreases with increasing
the fraction of austenite transformed.
In Fig. 7.10a, the dissipation of Gibbs energy with a binding energy EMn0 =9.9 kJ/mol
and chemical driving force for different fractions of ferrite at 550C are plotted together.
According to Equation 7.7, the dissipation of Gibbs energy has to be balanced by the
chemical driving force. In other words, the point of intersection between chemical
driving force and dissipation of Gibbs energy in Fig. 7.10a would be the solution of
Equation 7.7. When fα =0.0 (the carbon concentration in austenite is the nominal con-
centration), there is only one intersection point between dissipation of Gibbs energy
and chemical driving force. The intersection point gives a very high interface velocity
(about 1.5 µm/s), which means that the bainitic ferrite plates can grow at a very high
speed at the beginning of isothermal transformation at 550C. Based on the calculated
7.5. Theoretical analysis 121
10−11
10−10
10−9
10−8
10−7
10−6
10−50
100
200
300
400
500
600
700
800
Interface velocity /m/s
Gib
bs E
nerg
y /J
/mol
∆GMdiff
∆Gchem , fα=0
∆Gchem , fα=0.2
∆Gchem , fα=0.4
Fe−3Mn−0.1C
T=550°c
10−11
10−10
10−9
10−8
10−7
10−6
10−50
200
400
600
800
1000
Interface velocity, m/s
Gib
bs e
nerg
y, J
/mol
∆Gmdiff ,T=550°c
∆Gchem ,T=550°c
∆Gmdiff ,T=540°c
∆Gchem ,T=540°c
∆Gmdiff ,T=535°c
∆Gchem ,T=535°c
∆Gmdiff ,T=520°c
∆Gchem ,T=520°c
515 520 525 530 535 540 545 550 55510
−10
10−9
10−8
10−7
10−6
10−5
Temperature / °c
Inte
rfac
e ve
loci
ty /m
/s
Fe−3Mn−0.1C
(a)
(b)
(c)
Figure 7.10: (a) The dissipation of Gibbs energy with a binding energy EMn0 =9.9 kJ/mol and
chemical driving force with different fraction of ferrite at 550C for the Fe-3Mn-0.1C alloy;(b)The dissipation of Gibbs energy with a binding energy EMn
0 =9.9 kJ/mol and chemical drivingforce with fα =0.4 during cooling for the Fe-3Mn-0.1C alloy; (c) The predicted interface velocityas a function of temperature during the cooling stage
Mn profile at this high interface velocity, there is no Mn diffusion inside the interface,
which means the transformation is only controlled by carbon diffusion. This kind
of condition is similar to ”Paraequilibrium” [54, 55]. However, strictly speaking, the
122Chapter 7. Bainitic transformation during the interrupted cooling experiments
interface condition here deviates from paraequilibrium as a finite interface mobility
is assumed for the calculations. It is worth noting that the interface velocity would
increase to about 30µm/s if the interface mobility is assumed to be infinite. When fα
increases to 0.2, this results in an increase in carbon concentration in the austenite, the
chemical driving force still intersects with the dissipation at a very high interface ve-
locity although the magnitude of chemical driving force has deceased. Compared with
the interface velocity at fα =0, the interface velocity at fα =0.2 decreases marginally.
At fα =0 and 0.2, the bainitic ferrite plates migrate in a fast growth mode without
Mn diffusion inside the interfaces. When fα increases to 0.4, it is found that there are
three intersections between dissipation of Gibbs energy and chemical driving force.
The slowest interface velocity (around 0.8 nm/s) is the correct solution, as the interface
velocity can not increase to the other two high values. It means that the overall trans-
formation now become very sluggish according to model prediction. The calculated
Mn profile shows that there is significant Mn diffusion inside interface. Therefore,
based on the model prediction, during the isothermal bainitie transformation there is
growth mode transition from fast growth mode without diffusion inside interface to
sluggish growth mode with diffusion inside interface. The implications of this change
in growth rate regime for the actual lengthening and thickening rates of the bainite
plates remains to be studied. Additional magnetic measurements of in situ bainite
formation for the same steel (not shown here) show that the isothermal transformation
stops effectively at around fα =0.40. However, the classical paraequilibrium model
predicts that the transformation should stop at fα =0.94, which is much higher than
the experimentally determined value. This kind of phenomena is called ”incomplete
transformation”, and it has been experimentally investigated in different alloys sys-
tems [14]. The present Gibbs energy balance approach can well predict the incomplete
transformation phenomena in Fe-Mn-C studied here. It should be mentioned that the
value of binding energy EMn0 was adjusted a little bit to fit with experimental results,
but it is in the range of experimentally determined value by Enomoto et.al [128].
After the isothermal transformation, the samples are cooled down in the exper-
7.5. Theoretical analysis 123
iments. It is expected that the chemical driving force will increase with decreasing
temperature, and it will intersect with the dissipation of Gibbs energy at a high in-
terface velocity if the dissipation is temperature independent. However, as shown in
Fig. 7.10b, in reality the dissipation also increases during cooling. There is a competi-
tion between chemical driving force and dissipation during the cooling. In the initial
stage of cooling, the dissipation increase faster than the chemical driving force does,
which means that the interface is still frozen at a very slow velocity. This predicted
initial sluggish stage upon cooling is qualitatively comparable with the stagnant stage
observed in experiments. In Fig. 7.10c, the predicted interface velocity during the cool-
ing is plotted as a function of temperature. It shows that in the initial stage of cooling
the interface velocity is very slow, but at lower temperature it dramatically increases
to a value which is about 3 orders of magnitude higher. The modeling results are in
excellent qualitative agreement with the experimental results in the Fe-3Mn-0.1C alloy.
Using the same parameters as those used for the Fe-Mn-C alloy, the same model
calculations are also made for the Fe-Mn-Si-C alloy, in which dissipation due to both
Mn and Si diffusion inside the interface are considered. In Fig. 7.11a, the dissipation of
Gibbs energy due to Mn and Si diffusion inside the interface at 530 C for the Fe-Mn-
Si-C alloy as a function of interface velocity is plotted. It shows that there is almost no
dissipation due to Si diffusion inside interface when the interface velocity is very low,
which is because the chemical potential difference of Si between austenite and ferrite
is very small. The velocities at which the dissipation reaches the maximum value for
Mn and Si are different due to the difference in the values of their interfacial diffusion
coefficients. The total dissipation in the Fe-Mn-Si-C alloy is the sum of dissipation of
Gibbs energy due to Mn and Si. The addition of Si does not lead to significant increase
in the maximum total dissipation.
In Fig. 7.11b, the Si profiles inside interface at a high velocity and a low velocity are
shown. It shows that at a high velocity the Si profile is homogeneous inside interface,
which leads to a very small energy dissipation as shown in Fig. 7.11a. At a slow velocity,
there is considerable amount of Si segregation inside interface, which leads to large
124Chapter 7. Bainitic transformation during the interrupted cooling experiments
10−11
10−10
10−9
10−8
10−7
10−6
10−50
100
200
300
400
500
600
700
800
900
Interface velocity /m/s
Gib
bs
en
erg
y, J
/mo
l
Dissipation due to SiDissipation due to MnDissipation due to Mn and Si
−3 −2 −1 0 1 2 30
0.5
1
1.5
2
2.5
3
Normalized distance, x/ δ
X/X
0
Interface boundary
α γ
Interface boundary
v=1x10−8m/s
v=1x10−5m/s
(a)
(b)
Figure 7.11: (a)The dissipation of Gibbs energy due to Si diffusion inside interface at 530 Cwith a ESi
0 =12.3 kJ/mol as a function of interface velocity; (b)The Si profiles inside interface at ahigh interface velocity (1 × 10−5 m/s) and a low interface velocity (1 × 10−8 m/s)
dissipation as shown in Fig. 7.11a. At a slow interface velocity the Si concentration at
the interface boundary on austenite side is lower than the nominal concentration as Si
is ferrite stabilizer. This depletion of Si leads to a part of the total dissipation.
In Fig. 7.12, the predicted interface velocity during the cooling is plotted as a
function of temperature for the Fe-Mn-Si-C alloy. The fraction of bainite at the end of
isothermal holding is 0.57.There is a very sluggish growth stage at the beginning of
cooling, which is qualitatively in agreement with the experimental results. It is worth
pointing out that the lengthening rate of bainitic plates in the Fe-Mn-Si-C alloy has
7.6. Discussion 125
480 490 500 510 520 530 54010
−10
10−9
10−8
10−7
10−6
10−5
Temperature / °c
Inte
rfa
ce
ve
locity /m
/s
Fe−3Mn−1.5Si−0.1C
Figure 7.12: The predicted interface velocity during the cooling stage of type IC experimentsin the Fe-Mn-Si-C alloy as a function of temperature
been measured by HT LSCM [108], and was found to be lower than the prediction
by the paraequilibrium model with an infinite interface mobility but higher than the
prediction by the present model with an effective interface mobility measured by Wits
et.al [84]. As already mentioned, it is expected that the intrinsic interface mobility
should be higher than the effective interface mobility.
7.6 Discussion
During the growth of a bainitic ferrite plate there is some local plastic strain which
also causes dissipation, called mechanical dissipation here. In order to better describe
the experimental data, Bhadeshia [17] assumed that the mechanical dissipation during
the bainitic ferrite transformation is a temperature independent constant, 400 J/mol .
However, based on model predictions Bouaziz et.al [130] claimed that the mechani-
cal dissipation should be temperature and composition dependent as the mechanical
properties (such as yield point) of austenite is determined by them. Due to lack of
precise information about the magnitude of mechanical dissipation, in the current
126Chapter 7. Bainitic transformation during the interrupted cooling experiments
work only dissipation due to diffusion inside the interface and interface friction, called
”chemical dissipation” here, is considered. As mentioned above, in order to fit to
experimental results, the value of the binding energy, as a fitting parameter, is slightly
increased from an experimentally determined value to increase calculated dissipation.
It is expected that the fitted value of binding energy would be closer to the experimen-
tally determined value if the mechanical dissipation could be accurately included and
evaluated in the Gibbs energy balance model.
As shown in experimental results, the length of stagnant stage increases with in-
creasing isothermal holding time. This can be explained as follows: In the later stage of
isothermal bainitic transformation, the interfaces migrate at a very slow velocity, and
the Mn or Si spikes can form in front of the interfaces. The width of Mn or Si spikes
will extend with increasing isothermal holding time, which would prolong the length
of stagnant stage. As shown in Fig. 7.5, in type II experiment there is almost no ferrite
formation during the 20 minutes isothermal holding at 520C, while compared with
the type IC experiments the transformation kinetics during following cooling stage in
the type II experiment is significantly retarded. This retardation is also due to the effect
of solute spikes.
7.7 Conclusion
In this chapter, a series of interrupted cooling experiments were designed to study
the mechanism of bainitic ferrite transformation in the Fe-3Mn-0.1C and Fe-3Mn-1.5Si-
0.1C alloys. A special feature ”stagnant stage” is observed, and its length is not a
constant but increase with further holding time. The observation can be taken as a firm
evidence of diffusional rearrangement at the austenite-bainite interface. A so called
Gibbs energy balance approach is proposed to model the bainitic ferrite transformation
kinetics during the interrupted cooling experiments, and a kinetics transition from fast
growth mode without diffusion of Mn or Si inside the interfaces to slow growth mode
with diffusion inside the interfaces is predicted. The model calculations for interrupted
7.7. Conclusion 127
cooling experiments for both alloys predict with surprising accuracy the occurrence of
a stagnant stage after a well defined holding time. It is found that the stagnant stage is
caused by the slow growth mode. The existence of solute spikes in front of interfaces
is also effective for retarding the growth of bainitic ferrite plates.
128Chapter 7. Bainitic transformation during the interrupted cooling experiments
Chapter 8Transformation stasis during the
isothermal bainitic ferrite formation in
Fe-C-X alloys
This chapter is based on
• Hao Chen, Kangying Zhu, Lie Zhao, Sybrand van der Zwaag. Analysis of
transformation stasis during the isothermal bainitic ferrite formation in Fe-C-X
alloys , Acta Mater, accepted for publication, 2013.
8.1 Introduction
The effect of alloying element X (X=Mn,Ni,Co, Cr, et.al) on the kinetics of phase
transformations in steels has always been of great interest due to its practical im-
portance in the advanced steel design and production. Over the past decades, ex-
tensive modeling and experimental work has been done to elucidate the effect of
alloying elements on the kinetics of interface migration during the austenite to fer-
rite phase transformation in steels, of which the understanding is much more im-
proved [7–9, 11–13, 45, 69, 91, 92, 97, 108, 131, 132]. However, much less effort has been
paid to investigate the effect of alloying elements on the interface migration during
129
130Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
the bainitic ferrite formation, which is also a very important phase transformation in
steels.
During the isothermal bainitic transformations at temperatures between Bs (Bai-
nite start temperature) and B f (Bainite finish temperature), it has been observed quite
often that the bainitic ferrite formation temporarily ceased before the fraction of bai-
nite reached the thermodynamic fraction of ferrite predicted for the paraequilibrium
condition [54, 55]. This kind of phenomenon is called ”incomplete transformation
phenomenon” or ”transformation stasis” [14,20,36,133]. The transformation will con-
tinue with a very slow rate until the carbide starts to form and consume the carbon
in austenite. The incomplete transformation phenomenon has been considered as one
of the key indications for the diffusionless transformation mechanism [20]. Accord-
ing to the diffusionless theory, it is expected that the bainitic ferrite formation should
cease when the carbon concentration in austenite increases to a limit given by T0 in-
stead of paraequilibrium condition. T0 is the temperature at which the free energy of
austenite is equal to that of ferrite when the chemical concentration in austenite and
ferrite are the same. In order to better describe the incomplete transformation phe-
nomenon experimentally observed in Fe-Mn-Si-C alloys, a strain energy of 400 J/mol
due to the displacive nature of bainitic ferrite formation is assumed in the calculation
of the thermodynamical limit for diffusionless transformation, leading to the concept
of T′0 [17, 20] to predict the stasis. The underlying concept for the T′0 approach leads to
the conclusion that the incomplete transformation should be a general characteristic
of isothermal bainitic ferrite formation in steel irrespective of its composition. The
transformation stasis has also been experimentally investigated in a series of Fe-C-X
(X is the substitutional alloying elements, such as Mn, Si, Mo, et.al) alloys [14, 133].
It was not observed in Fe-C-Si, Fe-C-Cu and Fe-C-Ni alloys, while it was observed in
the Fe-C-Mn and Fe-C-Mo alloys. Based on the experimental results, it was concluded
that the transformation stasis is not a general characteristic of isothermal bainitic ferrite
formation in Fe-C-X alloys. It was deduced that the occurrence of transformation stasis
is attributed to segregation of alloying elements into the migrating austenite/bainitic
8.2. Results 131
ferrite interface, and the interaction of solutes and migrating interface gives rise to a
retardation of the interface migration. This kind of retardation is similar to the so-
lute drag effect during grain growth, and it was called coupled-solute drag effect by
Aaronson [36].
Both explanations for the occurrence of transformation stasis seem reasonable al-
though the basic assumptions in them are totally different. This chapter aims to shed
some new light on this long-standing controversial issue, and find out the physical
reason behind the transformation stasis phenomenon. In the analysis transformation
curves reported in the open literature are reevaluated using the model presented in
the previous chapter.
8.2 Results
8.2.1 Fe-Mn-C alloy
In the literature, the kinetics of isothermal bainitic ferrite formation has been exper-
imentally investigated in a series of Fe-C-Mn alloys with different Mn concentra-
tions [14, 16]. The transformation stasis was observed in a Fe-0.1C-3Mn (all in wt%)
alloy but not in the Fe-0.15C-1.5Mn-0.2Si and Fe-0.05C-1.5Mn-0.2Si alloys. In this sec-
tion, the Gibbs energy balance approach presented in Chapter 7 will be applied to
model the transformation stasis phenomenon in these Fe-C-Mn alloys. We focus first
on the Fe-0.1C-3Mn alloy as the transformation stasis for this alloy is well documented
in different sources.
In Fig. 8.1a, the total dissipation, dissipation due to solute drag effect and dissipation
due to Mn spike are calculated for Fe-0.1C-3Mn alloy at 550 C with a binding energy
EMn0 =9.9kJ/mol. The thickness of interface 2δ is here assumed to be 0.5nm, and the
diffusion coefficient of Mn inside the interface DIntMn is assumed to be the geometric
average of the diffusion coefficient of Mn in austenite DFCCMn , in ferrite DBCC
Mn , and in
the ferrite grain boundary DGBMn. The value of DFCC
Mn and DBCCMn can be calculated using
132Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
10−12
10−10
10−8
10−6
10−40
200
400
600
800
Interface velocity /m/s
Dis
sipa
tion
/J/m
ol Total dissipation
Dissipation due to solutedrag
Dissipation due tosolutedrag
−3 −2 −1 0 1 2 30
1
2
3
4
5
Norminized distance, x/ δ
C/C
0
Interface boundary Interface boundary
γα
v=1.5x10−9m/s
v=1x10−7m/s
(a)
(b)Figure 8.1: (a) The total dissipation, dissipation due to solute drag effect and dissipation due toMn spike as a function of interface velocity for Fe-0.1C-3Mn alloy at 550 C;(b) The Mn profilesinside interface with EMn
0 =9.9kJ/mol at two different interface velocities.
DICTRA [126], while the value of DGBMn is obtained from [127].
The dissipation due to solute drag is caused by the asymmetric diffusion profiles
inside the interface. It becomes 0 when the interface velocity becomes 0. This is the same
as the dissipation in the classic solute drag model for grain boundary migration.Unlike
the case of grain boundary migration, during phase transformations there is possibly
a spike of alloying elements in front of the migrating interfaces, which would also
lead to dissipation of Gibbs energy, called dissipation due to a spike [12, 129]. the
magnitude of the energy dissipation due to a spike is determined by the partitioning
coefficient of the alloying element kα/γi =exp(−∆E/RT). The total dissipation, ∆Gdi f fm , is
8.2. Results 133
10−12
10−10
10−8
10−6
10−40
200
400
600
800
1000
Interface velocity /m/s
Dis
sipa
tion
/J/m
ol
E0Mn=5kJ
E0Mn=9.9kJ
E0Mn=12kJ
10−12
10−10
10−8
10−6
10−40
200
400
600
800
Interface velocity /m/s
Dis
sipa
tion
/J/m
ol
Fe−1Mn−0.1CFe−2Mn−0.1CFe−3Mn−0.1C
(a)
(b)Figure 8.2: (a) The total energy dissipation assuming different values of binding energy asa function of interface velocity for the Fe-3Mn-0.1C alloy; (b)The total energy dissipationassuming EMn
0 =9.9kJ/mol as a function of interface velocity in the Fe-0.1C-xMn alloys withdifferent Mn concentrations.
the sum of these two dissipation terms. As shown in Fig. 8.1a, the dissipations due to
spike and solute drag depend on the interface velocity but in a different way. The total
dissipation reaches a maximum value at a specific velocity.
When the interface velocity is very fast (≥ 10−7m/s), the Mn profile inside the in-
terface is flat (Mn profile at 10−7m/s in Fig. 8.1b). The dissipations due to the spike
and due to solute drag are both small, leading to a very small total dissipation as
shown in Fig. 8.1a. When the interface velocity is slow (≤ 10−7m/s), there is a consid-
erable amount of Mn segregation inside the interface, as shown in Fig. 8.1b. The Mn
concentration at the interface boundary on the austenite side is higher than the nom-
134Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
inal concentration, which would lead to a Mn spike in front of the austenite/bainitie
interface.
In Fig. 8.2a, the total energy dissipation is calculated for Fe-3Mn-0.1C alloys at
550 C for different binding energy values. The maximum total energy dissipation
increases with increasing binding energy value. This is because the dissipation due
to solute drag is proportional to the value of binding energy. Increasing the Mn
concentration, it is expected that the dissipations due to spike and solute drag increase.
In Fig. 8.2b, the total dissipation at 550 C is calculated for different Mn concentrations.
The total dissipation indeed increases with increasing Mn concentration. Therefore,
the magnitude of total dissipation at a certain temperature is determined by the value
of binding energy and alloying element concentration.
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
200
400
600
800
Interface velocity /m/s
Gib
bs e
ne
rgy /J/m
ol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.2
∆Gmchem ,fα=0.4
T=550°cFe−0.1C−3Mn
Figure 8.3: The dissipation of Gibbs energy using a binding energy EMn0 =9.9 kJ/mol and
chemical driving force for three bainitic ferrite fraction ( fα =0, 0.2, 0.4) as a function of interfacevelocity at 550C for the Fe-3Mn-0.1C alloy
In Fig. 8.3, the total dissipation calculated using a binding energy EMn0 =9.9 kJ/mol
is plotted together with the available chemical driving force for three bainitic ferrite
fraction ( fα =0, 0.2, 0.4) for a Fe-0.1C-3Mn alloy. The available chemical driving force
decreases with increasing bainitic ferrite fraction. This is due to the fact that the carbon
concentration in austenite increases as the bainitic transformation proceeds. In an
8.2. Results 135
actual transformation, the available chemical driving force has to be balanced by the
total dissipation. In other words, the intersection between the value for the chemical
driving force and that for the total dissipation yields the predicted interfacial velocity
at that stage of the transformation. When fα = 0, there is only one intersection point
between total dissipation and chemical driving force curves, which gives a very high
interface velocity( about 30µm/s). Based on the calculated Mn profile inside interface at
this velocity, there is no Mn diffusion inside interface. It means that the transformation
at this stage is only controlled by carbon diffusion. When fα increases to 0.2 and
the carbon concentation in austenite increases, the magnitude of chemical driving
force deceases, yet the chemical driving force curve still intersects the total dissipation
curve at a very high interface velocity. Compared with the interface velocity at fα =0,
the interface velocity at fα =0.2 decreases marginally. At fα =0.2, the bainitic ferrite
plates also migrate in a fast mode without Mn diffusion inside interface. When the fα
increases to 0.4, there are three intersection points between dissipation and chemical
driving force curves. The slowest interface velocity (around 0.8 nm/s) is the correct
solution, which is about 3 orders of magnitude lower than those at fα =0 and 0.2.
Based on the Gibbs energy balance approach, it is predicted that there is a kinetic
transition from the fast growth mode without Mn diffusion inside interface to the
sluggish growth mode with Mn diffusion inside the interface during the isothermal
bainitic ferrite formation at 550 C for the Fe-0.1C-3Mn alloy. The fraction of bainitic
ferrite at which the kinetic transition occurs according to the Gibbs energy balance
(GEB) approach is called f GEBα . For this alloy and this transformation temperature
f GEBα =0.4. In contrast, the Paraequilibrium model predicts the isothermal bainitic ferrite
formation at 550 C in the Fe-0.1C-3Mn alloy to proceed with a high rate until a bainitic
ferrite fraction of 0.95. The fraction of bainitic ferrite predicted by the Paraequilibrium
model is called f PEα . The f GEB
α value at 550 C for the Fe-0.1C-3Mn alloy is much lower
than the f PEα value, which is in accordance with the transformation stasis (incomplete
transformation) phenomenon observed in experiments.
As shown in Fig. 8.2a, the value of the binding energy affects the magnitude of
136Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
8 9 10 11 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
fGEB
Degree of IC transformation
Binding energy, kJ/mol
f
PE
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Deg
ree
of IC
tran
sfor
mat
ion
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
fGEB
fPE
Degree of IC transformation
Mn concentration, wt.%
f
0.0
0.2
0.4
0.6
0.8
1.0
Deg
ree
of IC
tran
sfor
mat
ion
(a)
(b)Figure 8.4: (a) The f GEB
α , f PEα , the magnitude of incomplete transformation as a function of bind-
ing energy at 550 C for the Fe-3Mn-0.1C alloy; (b) The degree of incomplete transformation,f PEα , f GEB
α for a series of Fe-xMn-0.1C at 550C
maximum total dissipation, thus it is to be expected that f GEBα is affected by the value of
binding energy. In Fig. 8.4a, the f GEBα , f PE
α and the degree of incomplete transformation
are plotted as a function of binding energy at 550 C for the Fe-0.1C-3Mn alloy. The
degree of incomplete transformation is defined as ( f PEα − f GEB
α )/ f PEα . Increasing the value
of the binding energy results in a decrease of the f GEBα and an increase in the degree
8.2. Results 137
of incomplete transformation. It should be pointed out that the binding energy has
no effect on f PEα , as it is only determined by the alloy composition and the tempera-
ture. Enomoto et.al [128] have tried to experimentally measured the binding energy
of different alloying elements at high temperatures, and they found that the value of
binding energy depends on the type of alloying elements. For the bainitic transforma-
tion temperatures, no data for the binding energies is available. In the remainder of
this work, the binding energy is set as the only fitting parameter in the model.
As shown in Fig. 8.2b, increasing the Mn concentration will increase the total en-
ergy dissipation and decrease the chemical driving force. It can be easily deduced that
the degree of incomplete transformation should also be affected by the Mn concentra-
tion according to the Gibbs energy balance approach. Assuming EMn0 = 9.9kJ/mol, in
Fig. 8.4b, the degree of incomplete transformation, f PEα and f GEB
α at 550C are calculated
by the Gibbs balance approach for a series of Fe-0.1C-xMn alloys. The degree of incom-
plete transformation is indeed determined by the Mn concentration. The incomplete
transformation phenomenon almost disappear when the Mn concentration is less than
2 wt.%, and the bainitic transformation can not start when the Mn concentration is
higher than 3.5 wt.%. In contrast, the bainitic ferrite fraction according to the PE pre-
diction only depends marginally and smoothly over the entire 1-5% Mn composition
range.
101
102
103
104
105
1060
10
20
30
40
50
60
70
80
90
100
Time /s
f α
T=550°cT=500°cT=450°cT=400°c
Fe−3Mn−0.1C
Figure 8.5: The experimentally measured kinetics of isothermal bainitic transformation atdifferent temperatures in the Fe-0.1C-3Mn alloy.
Using a point counting method on selected micrograps, the isothermal bainitic fer-
138Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
10−11
10−10
10−9
10−8
10−7
10−6
10−50
200
400
600
800
1000
1200
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.75
∆Gmchem ,fα=0.8
Fe−0.1C−3Mn
T=500°c
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−50
200
400
600
800
1000
1200
1400
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,f
α=0
∆Gmchem ,f
α=0.8
∆Gmchem ,f
α=0.85
Fe−0.1C−3Mn
T=450°c
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−60
200
400
600
800
1000
1200
1400
1600
1800
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.8
∆Gmchem ,fα=0.85
Fe−0.1C−3Mn
T=400°c
(a)
(b)
(c)Figure 8.6: The dissipation of Gibbs energy with a binding energy EMn
0 =9.9 kJ/mol and chemicaldriving force with different fraction of ferrite for the Fe-0.1C-3Mn alloy at (a)500 C,(b) 450Cand (c) 400C
rite formation kinetics in the Fe-3Mn-0.1C alloy was reconstructed [14]. In Fig. 8.5, the
experimental data from Reynolds et.al is reproduced. At 550 C, the transformation
curve shows three stages:(i)an initial stage (t ≤ 103 seconds) during which the trans-
formation rate is very rapid. Carbides are not present in this stage; (ii) a second stage
8.2. Results 139
100
101
102
103
104
105
1060
10
20
30
40
50
60
70
80
90
100
Time /s
f α
T=600°cT=580°cT=500°c
Fe−1.5Mn−0.2Si−0.15C
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
100
200
300
400
500
600
700
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,f
α=0
∆Gmchem ,f
α=0.8
∆Gmchem ,f
α=0.84
∆Gmchem ,f
α=0.85
Fe−0.15C−1.5Mn−0.2Si
T=600°c
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
100
200
300
400
500
600
700
800
900
1000
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.85
∆Gmchem ,fα=0.9
Fe−0.15C−1.5Mn−0.2Si
T=580°c
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
200
400
600
800
1000
1200
1400
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.9
∆Gmchem ,fα=0.94
Fe−0.15C−1.5Mn−0.2Si
T=500°c
(a)
(b)
(c)
(d)Figure 8.7: (a) The experimentally measured kinetics of isothermal bainitic ferrite formation atdifferent temperatures in the Fe-0.15C-1.5Mn-0.2Si alloy; The dissipation of Gibbs energy witha binding energy EMn
0 =9.9 kJ/mol and chemical driving force with different fraction of ferritefor the Fe-0.15C-1.5Mn-0.2Si alloy at (b)600 C, (c) 580C and (d)500C
(103≤ t ≤ 104seconds ), the transformation stasis, during which the transformation is
extremely sluggish; (iii) a third stage ( t ≥ 104 seconds ) during which transformation
140Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
proceeds to completion at a very slow rate. During the third stage, there is carbide
formation, which consumes the carbon in austenite and thus increases the chemical
driving force for the bainitic transformation. The bainitic transformation in this stage
is extremely slow, and it takes about 12 days to form 60 % bainitic ferrite. It is worth
noting that if the isothermal transformation curve is plotted on a linear X axis, there are
only two stages: (i) the initial rapid transformation stage; (ii) the extremely sluggish
transformation stage covering both the second and third stage in the Fig. 8.5. The ex-
periments show that at 550 C transformation stasis occurs at fα =0.4, which is in good
agreement with the f GEBα predicted by GEB approach assuming a EMn
0 =9.9kJ/mol. The
binding energy of Mn used here is within the experimentally determined range [128].
Assuming EMn0 =9.9kJ/mol, the Gibbs energy balance approach is applied to predict
the kinetics of isothermal bainitic ferrite formation at 500 C, 450 C and 400C for
the Fe-0.1C-3Mn alloy. The results are shown in Fig. 8.6b, c and d. The total energy
dissipation and the available chemical driving force increase with decreasing temper-
ature. At 500 C, the GEB model predicts f GEBα =0.75, which is in good agreement with
experiments. At 450 C and 400 C, the GEB model predicts f GEBα =0.83, while the ex-
perimental results suggest an almost complete transformation. The difference may be
due to inaccuracy of the experimental determination of the bainitic ferrite fraction at
such high bainitic ferrite fractions, or may have a physical origin as is to be discussed
later.
In the literature, the kinetics of isothermal bainitic ferrite formation has also been
reported for Fe-0.15C-1.5Mn-0.2Si and Fe-0.05C-1.5Mn-0.2Si alloys at three different
temperatures [16]. In Fig. 8.7a, the experimentally determined kinetics of the isother-
mal bainitic ferrite formation is reproduced. It shows that at best there is a weak
stasis at a high bainitic ferrite fraction at 600 and 580 C, while it is absent at 500C.
In Fig. 8.7b,c,d, assuming the same EMn0 =9.9kJ/mol and ESi
0 =12.3kJ/mol, the Gibbs en-
ergy balance approach is applied to predict the kinetics of isothermal bainitic ferrite
formation at 600 C,580 C and 500C in the Fe-0.15C-1.5Mn-0.2Si alloy. The calcula-
tions show that f GEBα =0.9 at these three temperatures, which is in accordance with the
8.2. Results 141
experimental results 1.
Based on the above results, it can be concluded that the Gibbs energy balance
approach can well describe the transformation stasis phenomenon in the Fe-Mn-C
alloys, and the effect of Mn concentration on the occurrence of stasis is well qualitatively
predicted.
8.2.2 Fe-Mo-C
The kinetics of isothermal bainitic ferrite formation has also been systematically mea-
sured in a series of Fe-Mo-C alloys as a function of temperature [133]. Again the Gibbs
energy balance approach is applied to predict the transformation stasis phenomenon.
In Fig. 8.8a, the experimentally measured isothermal bainitic ferrite formation kinetics
at three different temperatures below the bay temperature for the Fe-0.19C-1.81Mo
(all in wt.%) alloy was reproduced. According to the Paraequilibrium model, the f PEα
values at these temperatures are around 0.9, which is higher than the experimentally
determined fractions at the transformation stasis. In Fig. 8.8b, the total dissipation is
plotted together with the available chemical driving force for the isothermal bainitic
ferrite formation at 585 C in the Fe-0.19C-1.81Mo alloy. In order to fit the experimental
results, the binding energy of Mo was adjusted to 30 kJ/mol.The figure shows that the
total dissipation almost approaches zero when the interface velocity is slower than
10−10 m/s, which means that there is almost no dissipation due to Mo spike. This is
because Mo almost does not partition between austenite and ferrite, and the chemical
potential of Mo in austenite is very close to that in ferrite (∆E is small). The modeling
results show that the growth mode of bainitic ferrite plates would shift from fast mode
into sluggish mode when the fraction of bainitic ferrite reaches 0.5, which is quite
comparable with the experimental value( f=0.45). In Fig. 8.8c and d, the total dissi-
pation is plotted together with the available chemical driving force for the isothermal
bainitic ferrite formation at 600 and 570C in the Fe-0.19C-1.81Mo alloy. The value of
1Work is ongoing to prove the validity of the model using more Mn levels but these results could notbe included in the thesis in time
142Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
Mo binding energy is chosen to be the same as that at 585 C. The transformation stasis
phenomenon at these two temperatures is still successfully predicted by the Gibbs
energy balance approach. Again the predicted f GEBα values are in good agreement with
experimental data.
In Fig. 8.9a, the experimentally measured isothermal bainitic ferrite formation kinet-
ics at three different temperatures below the bay temperature for the Fe-0.22C-0.23Mo
(all in wt.%) alloy are reproduced. It clearly shows that the transformation stasis oc-
curs only when the fraction of bainitic ferrite reaches around 0.8. The paraequilibrium
fraction of ferrite at these temperatures are also around 0.85 at these temperatures.
It means that the bainitic transformation is almost complete in the Fe-0.22C-0.23Mo
alloy. In Fig. 8.9b, c and d, the total dissipation is plotted together with the available
chemical driving force for the isothermal bainitic ferrite formation at 665, 655 and 635
C. It shows that the total dissipations in the Fe-0.22C-0.23Mo alloy are much lower
than those predicted in the Fe-0.19C-1.81Mo alloy. The f GEBα are all around 0.8 at these
temperatures,which is in good agreement with the experiments. The Gibbs energy
balance approach apparently can predict the incomplete transformation behavior of
Fe-0.13C-0.46Mo alloy successfully.
8.2.3 Fe-Si-C
In [14], the kinetics of isothermal bainitic ferrite formation has been systematically
measured in Fe-0.11C-1.83Si and Fe-0.38C-1.73Si (all in wt.%) alloys. In the Fe-0.38C-
1.73Si alloy there is pearlite formation during the isothermal transformation, while
there is only bainitic ferrite formation in the Fe-1.83Si-0.11C alloy. The isothermal
bainitic ferrite formation in the Fe-1.83Si-0.11C alloy will be of interest here, and it will
be simulated by the Gibbs energy balance model. According to the Paraequilibrium
model, the isothermal bainitic ferrite formation at the four (675, 625, 579 and 528 C)
temperatures proceeds at a high rate until the fraction of bainitic ferrite reaches about
0.95. In Fig. 8.10a, the experimentally measured isothermal bainitic ferrite formation
8.2. Results 143
101
102
103
104
105
1060
10
20
30
40
50
60
70
80
90
100
Time /s
f α
T=600°cT=585°cT=570°c
Fe−1.81Mo−0.19C
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
100
200
300
400
500
600
700
800
900
1000
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.4
∆Gmchem ,fα=0.5
∆Gmchem ,fα=0.6
Fe−0.19C−1.81Mo
T=585°c
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
0
100
200
300
400
500
600
700
800
900
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.2
∆Gmchem ,fα=0.4
Fe−0.19C−1.81Mo
T=600°c
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
0
100
200
300
400
500
600
700
800
900
1000
1100
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.6
∆Gmchem ,fα=0.7
Fe−0.19C−1.81Mo
T=570°c
(a)
(b)
(c)
(d)Figure 8.8: (a) The experimentally measured isothermal bainitic ferrite formation kinetics atthree different temperatures below the bay temperature for the Fe-0.19C-1.81Mo (all in wt.%)alloy; The total dissipation and the available chemical driving force for the isothermal bainiticferrite formation in the Fe-0.19C-1.81Mo alloy at (b)585 C;(c) 600 C and (d) 570 C
curves are reproduced. It shows that the isothermal bainitic ferrite formation at these
four temperatures is very fast, and the austenite almost fully transforms into bainitic
144Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
100
101
102
103
104
1050
10
20
30
40
50
60
70
80
90
100
Time/s
f α
T=665°cT=655°cT=645°cT=635°c
Fe−0.22C−0.23Mo
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
0
50
100
150
200
250
300
350
400
450
500
550
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.2
∆Gmchem ,fα=0.5
∆Gmchem ,fα=0.75
∆Gmchem ,fα=0.8
Fe−0.22C−0.23Mo
T=665°c
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
100
200
300
400
500
600
Interface velocity /m/s
Gib
bs e
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.75
∆Gmchem ,fα=0.8
Fe−0.22C−0.23Mo
T=655°c
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−40
100
200
300
400
500
600
700
Interface velocity /m/s
Gib
bs E
nerg
y /J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.8
∆Gmchem ,fα=0.85
Fe−0.22C−0.23Mo
T=635°c
(a)
(b)
(c)
(d)Figure 8.9: The experimentally measured isothermal bainitic ferrite formation kinetics at threedifferent temperatures below the bay temperature for the Fe-0.22C-0.23Mo (all in wt.%) alloy;The total dissipation and the available chemical driving force for the isothermal bainitic ferriteformation in the Fe-0.22C-0.23Mo alloy at (b)665 C;(c) 655 C and (d) 635 C
ferrite in 100 seconds. It means that there is no transformation stasis phenomenon in
the Fe-0.11C-1.83Si alloy. In Fig. 8.10b and c, the total dissipation is plotted together
8.2. Results 145
100
101
102
1030
10
20
30
40
50
60
70
80
90
100
Time /s
f α
T=675°cT=625°cT=579°cT=528°c
Fe−1.83Si−0.11C
10−10
10−9
10−8
10−7
10−6
10−5
10−40
200
400
600
800
1000
1200
Interface velocity, m/s
Gib
bs e
nerg
y, J
/mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.90
∆Gmchem ,fα=0.95
Fe−0.11C−1.83Si
T=579°c
10−10
10−9
10−8
10−7
10−6
10−5
10−40
200
400
600
800
1000
1200
1400
Interface velocity, m/s
Gib
bs e
nerg
y, j/
mol
∆Gmdiff
∆Gmchem ,fα=0
∆Gmchem ,fα=0.9
∆Gmchem ,fα=0.95
Fe−0.11C−1.83Si
T=528°c
(a)
(b)
(c)
Figure 8.10: (a) The experimentally measured kinetics of isothermal bainitic ferrite formationat different temperatures in the Fe-0.11C-1.83Si alloy; The dissipation of Gibbs energy with abinding energy ESi
0 =12.3 kJ/mol and chemical driving force with different fraction of ferrite forthe Fe-0.11C-1.83Si alloy at (b)579 C, (c) 528C
with the available chemical driving force for the isothermal bainitic ferrite formation
at 579 and 528 C in the Fe-0.11C-1.81Si alloy. In the model calculations here, the
binding energy of Si is chosen to be 12.3 kJ/mol. At 579 C, the GEB approach predicts
that at fα=0.9 the chemical driving force can still overcome the dissipation barrier,
and the growth rate of bainitic ferrite is very fast. The growth rate of bainitic plates
will decrease to about 0.012 µm/s at fα=0.95, which means that the transformation
almost stopped according to the GEB model. At 528 C, the chemical driving force
146Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
can overcome the dissipation barrier even at fα=0.95. The GEB appraoch prediction
slightly underestimates the bainitic ferrite fraction, but the results are in qualitative
agreement with experiments.
8.3 Discussion
In the results section, it was shown that the Gibbs energy balance approach based on a
diffusional theory can well predict the transformation stasis phenomenon in Fe-C-Mn,
Fe-C-Mo and Fe-C-Si alloys rather well. The transformation stasis phenomenon is due
to diffusion of alloying elements into the migrating austenite/bainitic ferrite interfaces,
which causes dissipation of Gibbs energy. The degree of incomplete transformation
( f PEα − f GEB
α )/ f PEα depends on the segregation tendency (i.e. the binding energy), the
partitioning coefficient, and concentration of alloying elements, which determine the
total dissipation. According to the Gibbs energy balance approach, the transforma-
tion stasis phenomenon can happen in any Fe-C-X alloy (X is substitutional alloying
element) provided the diffusion of X at the austenite/bainitic ferrite interface can lead
to a considerable amount of energy dissipation. The model automatically implies that
there should never be a transformation stasis phenomenon in binary Fe-C alloys.
In the literature the T′0 concept based on diffusionless theory has also been used
successfully to describe the incomplete transformation (transformation stasis) phe-
nomenon in Fe-C-Mn-Si alloys [17, 20]. The underlying physics of the T′0 concept
is totally different from that of the Gibbs energy balance model here. According to
T′0 concept, as bainitic transformation is considered as a diffusionless transformation,
the bainitic transformation should completely stop when the carbon concentration in
austenite reaches T′0 line which is the critical limit for diffusionless transformation. Ac-
cording to the GEB model, upon the transformation stasis the austenite/bainitic ferrite
interfaces does not stop but only migrate at a extremely low speed.
In Fig. 8.11a, the bainitic ferrite fraction at the stasis is plotted for the Fe-3Mn-0.1C
alloy as a function of the isothermal transformation temperature. The figure shows the
8.3. Discussion 147
350 400 450 500 550 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature / °c
f α
PET
0T
0’
GEBExperiments
Fe−3Mn−0.1C
0 0.5 1 1.5 2 2.5 3 3.5 4350
400
450
500
550
600
Carbon concentration /wt.%
Tem
pera
ture
/°c
PET
0
T0’
GEBExperiments
Fe−3Mn−0.1C
450 500 550 600 6500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature / °c
f α
PET
0
T0’
GEBExperiments
Fe−1.5Mn−0.2Si−0.15C
0 0.5 1 1.5 2 2.5 3450
500
550
600
650
Carbon concentration /wt.%
Tem
pera
ture
/°c
PET
0
T0’
GEBExperiments
Fe−1.5Mn−0.2Si−0.15C
(a)
(b)
(c)
(d)Figure 8.11: The bainitic ferrite fraction at the transformation stasis determined by experimentsand those predicted by the GEB approach, T0, T′0 and paraequilibrium boundaries for (a) the Fe-3Mn-0.1C alloy and (c) the Fe-1.5Mn-0.15C-0.2Si alloy; The concentration of carbon in austeniteat the transformation stasis predicted by the GEB approach and derived from experiments areplotted together with T0, T′0, paraequilibrium lines for (b) the Fe-3Mn-0.1C alloy and (d) theFe-1.5Mn-0.15C-0.2Si alloy
148Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
550 560 570 580 590 600 610 620 630 640 6500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature, °c
f α
PET
0
T0’
GEBExperiments
Fe−1.81Mo−0.19C
0 0.5 1 1.5 2 2.5550
560
570
580
590
600
610
620
630
640
650
Carbon concentration, wt.%
Tem
pera
ture
/°c
PET
0
T0’
GEBExperiments
Fe−1.81Mo−0.19C
600 610 620 630 640 650 660 670 680 690 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature / °c
f α
PET
0
T0’
GEBExperiments
Fe−0.22C−0.23Mo
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2600
610
620
630
640
650
660
670
680
690
700
Carbon concentration, wt%
Tem
per
atu
re /°
c
PET
0
T0’
GEBExperiment
(a)
(b)
(c)
(d)Figure 8.12: The bainitic ferrite fraction at the transformation stasis determined by experimentsand those predicted by the GEB approach, T0, T′0 and paraequilibrium boundaries for (a) the Fe-1.81Mo-0.19C alloy and (c) the Fe-0.23Mo-0.22C alloy; The concentration of carbon in austeniteat the transformation stasis predicted by the GEB approach and derived from experiments areplotted together with T0, T′0, paraequilibrium lines for (b) the Fe-1.81Mo-0.19C alloy and (d)the Fe-0.23Mo-0.22C alloy.
8.3. Discussion 149
experimental data, the f GEBα values calculated for 4 temperatures and three continuous
curves based on the PE model and the T0 and T′0 concept. At at 550 and 500 C both
the GEB and the T′0 models describe the data well. At 450 and 400 C , the fit is not
so good, but there is some doubt about the correctness of the experimental data as the
claimed bainitic ferrite fraction is even higher than the PE prediction 2. The data of
Fig. 8.11a are replotted in Fig. 8.11b in a Temperature-Carbon concentration plot. The
GEB and T′0 curves overlap and cover the high temperature data. Both the GEB, T0, T′0
and experimental data differ strongly from the PE line.
In Fig. 8.11c and Fig. 8.11d, the corresponding plots are shown for the alloy with
the lower Mn concentration: Fe-0.15C-1.5Mn-0.2Si. Now the experimental data, the
GEB prediction and the PE prediction nicely overlap over quite a temperature range
(Fig. 8.11c). Clearly the transformation proceeds almost to completion and the stasis is
rather small if present at all. This behavior is rather different from those predicted by the
T0 and T′0 concepts, which predict a significant stasis for transformation temperatures
of 500 C and higher. Fig. 8.11d is a replot of the same data in a Temperature-Carbon
concentration plot. There is an insignificant difference between the experimental data
and the GEB predictions. Again the T0, T′0 lines show a significant deviation both in
position and slope.
Fig. 8.12 is a similar plot as Fig. 8.11 but now for the two Mo levels analyzed.
Fig. 8.12a and b belong to the high Mo data (Fe-0.19C-1.81Mo) while Fig. 8.12c and d
belong to the low Mo data (Fe-0.22C-0.22Mo). The same conclusion as for the Fe-C-Mn
system can be drawn: at the higher Mo level the experimental data, the GEB, the T0
and T′0 predictions more or less agree and there is a substantial stasis. At the lower
Mo level the experimental data, the GEB and the PE predictions overlap, while the T0
and T′0 predictions really do not match the observations and predict the occurrence of
stasis which is not present at all.
In Fig. 8.13 the same analysis is shown for an Fe-0.11C-1.83Si alloy. Fig. 8.13a
2Magnetometer work is ongoing to precisely measure the fraction of bainitic ferrite at 450 and 400C in the Fe-3Mn-0.1C alloy
150Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
500 520 540 560 580 600 620 640 660 680 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature / °c
f α
PET
0
T0’
GEBExperiments
Fe−0.11C−1.83Si
0 0.5 1 1.5 2 2.5 3500
520
540
560
580
600
620
640
660
680
700
Carbon concentration /wt.%
Tem
pera
ture
/°c
PET
0
T0’
GEBExperiments
Fe−0.11C−1.83Si
(a)
(b)
Figure 8.13: (a) The bainitic ferrite fraction at the transformation stasis determined by experi-ments and those predicted by the GEB approach, T0, T′0 and paraequilibrium boundaries for theFe-1.83Si-0.11C alloy;(b) The concentration of carbon in austenite at the transformation stasispredicted by the GEB approach and derived from experiments are plotted together with T0, T′0,paraequilibrium lines for the Fe-1.83Si-0.11C alloy.
and b closely resemble Fig. 8.11c and d for the Fe-0.15C-1.5Mn-0.2Si alloy. Again the
experimental data, the GEB and the PE predictions fit rather well, yet the T0 and T′0
predictions deviate unacceptably.
Finally, it should be stated that although the T0 or T′0 concept can describe the
transformation stasis for the Fe-3Mn-0.1C rather well, there is a major conceptual flaw
in the T0 or T′0 concept in the sense that it predicts the occurrence of stasis also in
lean alloys and even in binary Fe-C alloys. There is a huge amount of experimental
data available to show that this is not the case. In contrast, the GEB approach can
well predict the transformation stasis phenomenon in all the alloys studied here, and
8.4. Conclusion 151
the effect of alloying element concentration on the transformation stasis is also well
evaluated.
8.4 Conclusion
In this work, a Gibbs energy balance approach is applied to model the transforma-
tion stasis phenomenon in steels, and its predictions are in good agreement with
experiments on Fe-C-Mn, Fe-C-Si, Fe-C-Mo alloys of different substitutional element
concentrations. The following conclusions can be reached:
• During the isothermal bainitic ferrite formation there is a sharp growth mode
transition from a fast mode without diffusion of alloying element inside the
interface to a sluggish mode with diffusion inside the interface, and the transfor-
mation stasis is caused by a transition to the sluggish transformation mode.
• The transformation stasis is not a general phenomenon of isothermal bainitic
transformation in steels, and it can occur only when the diffusion of alloying
elements leads to considerable dissipation of Gibbs energy.
• The T′0 concept failed in predicting the effect of alloying element concentration
on transformation stasis phenomenon.
152Chapter 8. Transformation stasis during the isothermal bainitic ferrite formation
in Fe-C-X alloys
Summary
This thesis aims to shed some new light onto the fine details of the kinetics of the
austenite to ferrite and bainitic transformation in steels, such as interface mobility and
partitioning of alloying element. For this purpose, novel experimental approaches are
designed to allow a clear discrimination between the correctness of the existing models
for both ferrite and bainitic ferrite formations, and new modeling approaches are
proposed to explain the special features observed in the newly designed experiments.
In Chapter 2, a new type of heat treatment involving cyclic partial phase transfor-
mations has been analyzed to determine the growth kinetics of the austenite to ferrite
phase transformation and vice versa more accurately. The mixed mode model and the
diffusion-controlled phase transformation model, including the soft impingement ef-
fects at the later stage of phase transformation, are reformulated for the cyclic austenite
to ferrite and ferrite to austenite transformation. A new growth mode parameter H is
defined for the mixed-mode model. The analysis shows that not all partitioning phase
transformations starts as pure interface-controlled growth but all of them shift gradu-
ally towards pure diffusion control when thermodynamic equilibrium is approached.
The diffusion-controlled model predicts that the ferrite to austenite transformation
should be faster than the austenite to ferrite transformation, while the simulation of
the mixed-mode model shows that the transformation rate ratio is a function of inter-
face mobility value.
A series of cyclic partial phase transformation experiments have been performed in
154 Summary
Chapter 3 to investigate the growth kinetics of the austenite to ferrite phase transfor-
mation and vice versa in a lean Fe-C-Mn alloy. Unlike the usual phase transformation
experiments (100% parent phase → 100% new phase), in the case of cyclic partial
transformations two special stages are observed: a stagnant stage in which the de-
gree of transformation does not vary while the temperature changes and an inverse
phase transformation stage, during which the phase transformation proceeds in a di-
rection contradictory to the temperature change. The experimental results have been
analyzed using paraequilibrium (PE) and local equilibrium (LE) diffusional growth
models. Only the local equilibrium model was shown to be able to predict the new
features of the cyclic phase transformation kinetics. The stagnant stage was found
to be due to Mn partitioning, while the inverse phase transformation is caused by
non-equilibrium conditions when switching from cooling to heating and vice versa.
In Chapter 4, the stagnant stage during the austenite to ferrite transformation start-
ing from a mixture of austenite and ferrite has been systematically investigated using
local equilibrium model and the cyclic phase transformation concept. The stagnant
stage is identified in M containing alloys (M is substitutional alloying element, and the
partitioning coefficient of M between austenite and ferrite is not equal to 1, M=Mn, Ni,
Cu, Si), and the length of stagnant stage is found to be affected by the concentration of M
and the partitioning coefficient. There is no stagnant stage in Fe-C and Fe-C-Co alloys.
The length of the stagnant stages increases with increasing heating and cooling rates
during the cyclic phase transformations. The stagnant stage in quaternary alloys was
also investigated, and it was found that the effect of substitutional alloying elements
on the length of stagnant stage in multi-component alloys seems to be additive.
A series of new cyclic phase transformation dilatometric experiments has been
designed in Chapter 5 to investigate the newly discovered “growth retardation stage”
during the final austenite-ferrite transformation systematically and in detail, and to
provide further evidence for the existence of residual Mn spikes in austenite created
during the prior cyclic phase transformations. The magnitude of growth retardation
increases with increasing Mn concentration, and is absent in the binary Fe-C alloy.
155
New experiments also prove that growth retardation does not occur at a particular
temperature but only occur when the moving austenite-ferrite interface hits the residual
Mn spike. The magnitude of growth retardation is proportional to the number of
prior temperature cycles in the cyclic phase transformations, and there is no growth
retardation in experiments with only one temperature cycle. The fact that the growth
retardation can be observed by dilatometry implies that the the interface moved back
and forth in a retraceable manner. The classical Local Equilibrium model can still
qualitatively predict the newly observed features of growth retardation, while the
Paraequilibrium model does not work.
Chapter 6 presents In-situ High Temperature Laser Scanning Confocal Microscopy
(HT LSCM) observations of the austenite-ferrite interface migration during cyclic phase
transformations in a Fe-Mn-C alloy. It has been confirmed that during the cyclic phase
transformations the transformation proceeds via the migration of existing austenite-
ferrite interfaces. The interfaces migrate in a retraceable way. For the first time the
so-called stagnant stage has been observed directly. The new in-situ observations
show that the interface migration rates for interfaces in different grains are comparable
with each other prior to soft impingement, while the equilibrium migration distances
for different interfaces can be quite different depending on the local grain size. The
average interface velocities as measured by HTLSCM are in very good agreement
with the velocities derived from dilatometric data, and those are predicted by a Local
Equilibrium (LE) transformation model.
All new assumptions and conclusions drawn in Chapter 3 and 5 regarding the
details of phenomenon occurring at the moving austenite-ferrite interfaces have been
validated in the experiments reported in Chapter 6.
In Chapter 7, new interrupted cooling experiments have been designed to study the
kinetics of bainitic ferrite formation starting from a mixture of austenite and bainitic
ferrite. It is found that the kinetics of bainitic ferrite formation during the cooling
stage is determined by the isothermal holding time. The formation rate of bainitic
ferrite at the beginning of the cooling decreases with increasing the prior isothermal
156 Summary
holding time. An unexpected stagnant stage during the cooling stage appears when
the isothermal holding time increases to a critical point. There are two reasons for the
occurrence of the stagnant stage: (i) a solute spike in front of interface; (ii) a kinetic
transition. A so called Gibbs energy balance approach, in which the dissipation of
Gibbs energy due to diffusion inside the interface and interface friction is assumed to
be equal to the available chemical driving force, is applied to theoretically explain the
stagnant stage. A kinetics transition from a fast growth mode without diffusion of
Mn and Si inside the austenite-bainitic ferrite interfaces to a slow growth mode with
diffusion inside the interface is predicted. The stagnant stage is caused by a transition
to a slow growth mode. The Gibbs energy balance approach describes the experimental
observations very well.
The transformation stasis phenomenon during the isothermal banitic ferrite forma-
tion has been investigated in a series of Fe-C-X (X is substitutional alloying element)
alloys in Chapter 8. The Gibbs energy balance (GEB) approach is applied to model
the transformation stasis phenomenon in Fe-C-X alloys, and the theoretical predictions
are compared to experimental observations. The good agreement over several alloy
systems demonstrates that the transformation stasis is caused by diffusion of alloying
elements into the migrating austenite/bainitic ferrite interfaces. It is found that the oc-
currence of transformation stasis during isothermal bainitic ferrite formation depends
on the concentration and partitioning coefficient of the alloying element X as well as
its binding energy. The GEB model clearly outperforms the diffusionless T0 model.
Samenvatting
Dit proefschrift heeft als doel om nieuw inzicht te verschaffen in belangrijke details
van de kinetiek van de transformatie van austeniet naar ferriet en die van austeniet
naar bainiet, en in het bijzonder in de rol van de grensvlakmobiliteit en de locale
herverdeling van legeringselementen. Hiertoe zijn nieuwe experimentele procedures
bedacht die het mogelijk maken om een onderscheid te maken in de juistheid van
de bestaande conflicterende modellen. Ook zijn er zijn nieuwe theoretische modellen
ontwikkeld die de voor het eerst waargenomen nieuwe verschijnselen verklaren.
In Hoofdstuk 2 wordt een nieuw type warmtebehandeling, cyclisch partieel trans-
formeren, gentroduceerd voor de bestudering van de kinetiek van de transformatie
van austeniet naar ferriet en omgekeerd. Het mixed mode model en het diffusionele
fasetransformatie model zijn opnieuw geformuleerd voor de thermische randvoor-
waarden van cyclisch partieel transformeren. Een nieuwe parameter, H, waarmee
het instantane karakter van het groeiproces beschreven wordt, is gedefinieerd. De
analyse laat zien dat niet alle fasetransformaties beginnen als grensvlakmobiliteit-
gedomineerde transformaties. Onafhankelijk van hoe de transformatie begonnen is,
eindigen alle transformaties als diffusie bepaalde transformaties wanneer het thermo-
dynamisch evenwicht bijna bereikt is. Het diffusie-gebaseerde model voorspelt dat de
transformatie van ferriet naar austeniet sneller verloopt dan de austeniet naar ferriet
transformatie. Het mixed mode model laat zien dat de verhouding in de kinetiek van
beide processen een functie is van hun beider grensvlakmobiliteiten.
158 Samenvatting
In Hoofdstuk 3 worden een serie cyclisch partieel transformatie-experimenten
beschreven ter bepaling van de transformatiekinetiek van austeniet naar ferriet en vice
versa. De experimenten zijn uitgevoerd aan een laag-gelegeerde Fe-C-Mn legering.
Twee nieuwe verschijnselen die niet voorkomen in conventionele fasetransformaties
waarbij de uitgangstoestand uit slechts een enkele fase bestaat die vervolgens volledig
overgaat in het reactieproduct, zijn in deze cyclisch partile transformatieexperimenten
waargenomen: een stagnatie-fase, waarin de transformatie niet verloopt ondanks de
aanwezigheid van een behoorlijke drijvende kracht, en een inverse-fase, waarin de
transformatie verloopt in een richting die strijdig is met de richting van de opgelegde
temperatuursverandering. De experimentele waarnemingen zijn geanalyseerd met het
para-evenwichts (PE) model en met het lokaal-evenwicht (LE) diffusiemodel. Het LE
model was het enige van beide modellen waarmee de nieuwe verschijnselen goed en
semi-kwantitatief beschreven konden worden. De stagnatie-fase kon toegeschreven
worden aan de lokale Mn verrijking aan het grensvlak en de inverse-fase bleek het
gevolg van het bestaan van niet-evenwichtscondities aan het grensvlak op het mo-
ment van overgang van koeling naar opwarmen en omgekeerd.
In Hoofdstuk 4 is het verschijnsel van de stagnatie-fase dat op kan treden tijdens
cyclische partile austeniet-ferriet fasetransformaties als de metaalkundige uitgangstoe-
stand een combinatie van zowel austeniet als ferriet is, in meer detail geanalyseerd
aan de hand van het lokaal-evenwicht diffusiemodel. De lengte van de stagnatie-
fase in ternaire Fe-C-X (waarbij X een substitutioneel legeringselement is met een
verdelings-coefficient ongelijk aan 1, M=Mn, Ni, Cu, Si) is kwantificeerd. De lengte
van de stagnatie-fase bleek af te hangen van de concentratie van M en diens herverdel-
ingscofficint. Er treedt geen stagnatie-fase op in Fe-C en Fe-C-Co legeringen. De lengte
van de stagnatie-fase neemt toe met toenemende opwarm- en afkoelsnelheden. Ken-
merken van de stagnatie-fase zijn ook onderzocht voor quarternaire legeringen en er is
gevonden dat het effect van de legeringselementen op de lengte van de stagnatie-fase
additief lijkt te zijn.
Aanvullende dilatometrische experimenten op basis van het cyclisch partieel trans-
159
formeren concept, die als doel hadden om het verschijnsel van de stagnatie-fase nader
en in meer detail te onderzoeken, zijn beschreven in Hoofdstuk 5. De experimenten
hadden tot doel om indirect bewijs te leveren voor het bestaan van aaneengesloten Mn-
rijke zones als gevolg van eerder opgelegde partieel cyclische fasetransformaties. De
mate van groeivertraging (i.e. stagnatie) bleek af te hangen van het Mn gehalte in het
staal. Geen groeivertraging werd waargenomen in de binaire Fe-C legering. De nieuwe
experimenten lieten ook zien dat het moment van groeivertraging niet afhangt van de
temperatuur maar optreedt als het bewegende grensvlak weer in contact komt met de
Mn-rijke zone welke in eerder cycli gecre–eerd waren. De mate van groeivertraging
is evenredig met het aantal eerdere partile transformatiecycli. Geen groeivertraging
treedt op als maar een transformatiecyclus opgelegd wordt. Uit het feit dat de groeiver-
traging zelfs waargenomen kan worden in macroscopische dilatometer experimenten
kan geconcludeerd worden dat de grensvlakken een zelfde weg bewandelen tijdens
hun cyclische passages door het materiaal. Het klassieke lokaal-evenwicht (LE) dif-
fusiemodel kan de waarnemingen, zij het kwalitatief, zeer goed beschrijven, maar dat
lukt niet met het para-evenwichtsmodel.
In Hoofdstuk 6 worden in-situ Hoge Temperatuur Scanning Laser Confocale Mi-
croscopie (HT SLCM) waarnemingen aan bewegende austeniet-ferriet grensvlakken
tijdens cyclische fase transformaties in Fe-C-Mn legeringen gepresenteerd. De waarne-
mingen laten zien dat tijdens cyclische fasetransformaties de transformatie verloopt
via verplaatsing van bestaande grensvlakken. De grensvlakken verplaatsen zich als
voorspeld op een zich reproducerende manier. Deze eerste in-situ waarnemingen laten
zien dat de verplaatsingssnelheid in korrels van verschillende afmetingen identiek is
totdat soft impingement optreedt. De totale verplaatsingsafstand tot het bereiken van
de waarde behorende bij de evenwichtsfractie hangt echter duidelijk wel af van de kor–
relgrootte. De gemiddelde grensvlaksnelheden zoals gemeten met HT SLCM komen
goed overeen met de snelheden berekend uit de dilatometrische data. Ze komen ook
goed overeen met de via de het transformatiemodel berekende waardes.
Het is zeer tevredenstemmend te mogen constateren dat alle in hoofdstuk 3 en 5
160 Samenvatting
geformuleerde hypotheses over het gedrag van bewegende austeniet-ferriet grensvlakken
bevestigd werden door de experimenten van hoofdstukken 6.
In Hoofdstuk 7 worden nieuwe, onderbroken-afkoelingsexperimenten gerappor-
teerd die ontworpen zijn om de kinetiek van de van verdere bainietvorming vanuit
een mengsel van austeniet en bainiet duidelijker in kaart te brengen. De kinetiek van
de bainietvorming tijdens verdere afkoeling bleek af te hangen van de duur van de
voorafgaande isotherme gloeibehandeling. De bainietvormingssnelheid neemt af met
toenemende duur van de isotherme behandeling. Een onverwachte stagnatie-fase in
de bainietvorming treedt op als de isotherme behandeling langer duurt dan een kri-
tische tijd. Een dergelijke stagnatie-fase kan het gevolg zijn van een ophoping van
legeringselementen aan het grensvlak of van een verandering in het transformatiege-
drag. Een zogenaamde Gibbs vrije energie balans analyse, waarbij de dissipatie van
de Gibbs vrije energie als gevolg van diffusie in en naar het grensvlak en de intrin-
sieke weerstand tegen de grensvlakverplaatsing vergeleken wordt met de beschikbare
chemische drijvende kracht, bleek in staat het verschijnsel van de stagnatie-fase te
verklaren. Het model voorspelt een abrupte overgang van een snelle transformatie
waarbij de diffusie van Mn en Si geen tred houdt met de beweging van het grensvlak,
naar een veel tragere transformatie waarbij dit wel het geval is. De waargenomen
stagnatie-fase na een bepaalde gloeitijd wordt toegeschreven aan deze verandering in
transformatiesnelheid. Het gehanteerde model beschrijft de waarnemingen heel goed.
De abrupte verandering in de transformatiesnelheid tijdens isotherme bainietische
gloeibehandelingen ruim voordat de evenwichtsfractie ferriet bereikt is, is in Hoofd-
stuk 8 verder onderzocht voor een serie Fe-C-X legeringen waarbij X weer een sub-
sititutioneel legeringselement is. De Gibbs vrije energie balans (GEB) benadering
is gebruikt om het transformatie gedrag van Fe-C-X legeringen te voorspellen en
de voorspellingen zijn vergeleken met in de literatuur gerapporteerde experimentele
waarnemingen. De goede overeenkomst tussen het voorspelde gedrag en de exper-
imentele waarnemingen aan een reeks van Fe-C-X legeringen ondersteunt de con-
clusie dat de plotselinge en vroegtijdige afname van de transformatiesnelheid het
161
gevolg is van de diffusie van legeringselementen naar bewegende austeniet-bainiet
grensvlakken. Een eventueel optreden van een voortijdige stagnatie van de trans-
formatie tijdens isotherme gloeibehandelingen hangt af van de concentratie en de
herverdelingscofficint van het betreffende legeringselement en van de bindingsenergie
aan het grensvlak. Het GEB model is duidelijk veel beter in staat de waarnemingen
te beschrijven dan het bestaande T0 model waarbij de transformatie geacht wordt te
verlopen zonder diffusionele herverdeling van de legeringselementen.
162 Samenvatting
Appendix AThe effect of transformation path on
stagnant stage
This appendix is based on
• H Chen, W Xu, M Goune, S van der Zwaag, Application of the stagnant stage
concept for monitoring Mn partitioning at the austenite-ferrite interface in the
intercritical region for Fe-Mn-C alloys,Phil Mag Lett, 2012; 92:547-555.
A.1 Introduction
In Chapter 4, it was theoretically predicted that the length of stagnant stage was directly
determined by theγ/α interface conditions, inversely, the length of stagnant stage could
be used as a tool to deduce the γ/α interface conditions.
In this appendix, mixtures ofγ andαhave been obtained in the intercritical region of
a Fe-0.17Mn-0.023C (wt. %) alloy via two thermal routes: (i) direct isothermal holding
and (ii) isothermal holding after full austenization. The length of stagnant stage
during subsequent cooling is used to discriminate the γ/α interface conditions after the
isothermal α → γ and γ → α transformations. The phase transformation kinetics has
been measured by dilatometry, and then simulated using Local Equilibrium (LE) [56,57]
and Paraequilibrium (PE) models [54, 55]. The evolutions of the local γ/α interface
164 Appendix A. The effect of transformation path on stagnant stage
conditions during the isothermal α→ γ and γ→ α transformation are compared.
A.2 Experimental
Figure A.1: The optical micrograph of the as-received Fe-0.023C-0.17Mn alloy
The material investigated here is a Fe-0.17Mn-0.023C (wt. %) alloy, and the as-
received microstructure consists of equiaxed ferrite with a very small (less than 1
vol. %) fraction of fine pearlite, as shown in Fig. A.1 . The heat treatment procedures
in this work can be divided into type A and type B. In the type A experiments, the
sample was heated directly up to 875 C in the intercritical region with a heating
rate of 10 C/min for isothermal α → γ transformation. In the type B experiments,
the sample was firstly heated up with a heating rate of 10 C/min to 1000 C with
3 minutes isothermal holding for full austenization, and then the sample was cooled
down to the same temperature 875 C in the intercritical region for isothermal γ → α
transformations. After the isothermal α→ γ and γ→ α transformations, the samples
were cooled down to room temperature with a cooling rate of 10 C/min. A Bahr 805A
dilatometer is used to measure the dilation of the specimen (10 mm in length and 5 mm
A.3. Result and Discussion 165
in diameter) during the phase transformations.
A.3 Result and Discussion
Fig. A.2a shows the dilation as a function of temperature during the type A and type
B experiments in the Fe-0.17Mn-0.023C (wt. %) alloy. The temperature for isothermal
holding is 875 C, and the isothermal time is set as 1 hour to reach the equilibrium
fraction of austenite and ferrite during the isothermal phase transformations. In the
type A experiment, the pearlite would firstly transform into austenite, and then the
newly formed austenite grains grow into the ferrite grains during the isothermal hold-
ing, which is accompanied by a contraction at the holding temperature as shown in
Fig. A.2a. Due to the low carbon concentration in the alloy here, the volume fraction of
pearlite is less than 0.01, and the pearlite to austenite transformation as such can not be
distinguished from the ferrite to austenite transformation by dilatometry. In the type
B experiment, the sample was firstly fully austenized, and then the austenite would
transform into ferrite during the isothermal holding, which leads to an expansion as
shown in Fig. A.2a. The fraction of ferrite as a function of time during the α→ γ in the
type A experiment and γ → α transformation in the type B experiment is calculated
from the dilation according to the level rule, as shown in Fig. A.2b. The initial part
ferrite to austenite transformation is not isothermal, and the volume fraction of ferrite
transformed to austenite during heating is estimated to be about only 0.07. It is shown
that during the isothermal holding the fraction of austenite and ferrite is approaching
to the equilibrium fraction at 875C for both type A experiments and type B exper-
iments. According to a ThermalCalc calculation using the TCFE6 database [90], the
thermodynamic equilibrium fraction of ferrite at 875C for Fe-0.17Mn-0.023C (wt. %)
is 65 %, which is close to the observed quasi-equilibrium fraction of 65.5 % (type A)
and 62.5 %(type B).
After the isothermal transformations, the samples are cooled down to room tem-
perature at a slow cooling rate of 10C/min. In Ref [108,134,135], it was demonstrated
166 Appendix A. The effect of transformation path on stagnant stage
800 850 900 950100
105
110
115
120
125
130
135
Temperature/°c
Le
ng
th c
ha
ng
e /
µm
Type A experimentType B experiment
0 500 1000 1500 2000 2500 3000 35000
0.2
0.4
0.6
0.8
1
Time /s
f α
The α to γ transformation The γ to α transformation
820 830 840 850 860 870 8800
0.2
0.4
0.6
0.8
1
Temperature /°c
f α
Type A Type B
(a)
(b)
(c)
Figure A.2: (a) The dilation as a function of temperature during the type A and type Bexperiments; (b) The fraction of ferrite as a function of time during the α → γ and γ → αtransformations; (c) The fraction of ferrite as a function of temperature during the γ → αtransformation upon cooling.
experimentally that the retained ferrite after intercritical annealing grows epitaxially
given the presence of many α/γ interfaces in the sample without new nucleations dur-
ing further cooling. The newly formed ferrite is called “epitaxial ferrite”, and it plays a
A.3. Result and Discussion 167
big role in the mechanical properties of dual phase steels [136]. It is to be expected that
upon the imposed slow cooling the γ→ α transformation will immediately start from
a mixture of γ and α in the intercritical region as nucleation is not required. However,
as shown in Fig. A.2a, at the initial stage of cooling for both type A and B experiments
there is a linear contraction only, which is mainly caused by thermal expansion effect
only. In Fig. A.2c, at the beginning of cooling there is almost no ferrite formation , and
this is the so called stagnant stage [97]. Interestingly, the length of stagnant stage in
the type A experiment is much longer than that in the type B experiments.
The type A and type B experiments were simulated using the well known Dictra
software [126] and imposing either local equilibrium (LE) or paraequilibrium (PE) con-
ditions. In LE model, the interface is assumed to migrate under full local equilibrium
with the partitioning of both C and M (M is substitutional alloying element, M=Mn
in this work). The transformation rate in LE model is determined by either Mn or C
partitioning, which depends on the temperature and composition. In PE model, it is
assumed that there is no partitioning of M and that the chemical potential of C across
the interface is constant. Hence, the transformation is controlled by C diffusion . In
our simulations, a planar geometry was used, and the half thickness of the system was
assumed to be 25 µm, which is close to the measured ferrite grain size. The cooling and
heating rates in the simulations were set to the same values those in the experiments.
In the simulation of type A experiments, the rapid pearlite to austenite transforma-
tion is not simulated, and the starting condition is taken to be a mixture of ferrite and
austenite with a pearlitic composition. In Fig. A.3a, the fraction of ferrite as a function
of holding time predicted by LE model and PE model is indicated. The PE model
predicts that the isothermal α → γ and γ → α transformation should come to an end
within 50 seconds.The LE model shows that the transformation shifts into significantly
sluggish stage within 100 seconds, during which the interface is almost pinned and the
transformation kinetics is controlled by Mn diffusion. Both models overestimate the
observed transformation rates. This discrepancy between experiments and theories
could be attributed to the fact that the dissipation of Gibbs energy by interface friction
168 Appendix A. The effect of transformation path on stagnant stage
0 50 100 1500
0.2
0.4
0.6
0.8
1
Time /s
f α
Isothermal α to γ transformation, PE modelIsothermal γ to α transformation, PE modelIsothermal α to γ transformation, LE modelIsothermal γ to α transformation, LE model
820 830 840 850 860 870 8800
0.2
0.4
0.6
0.8
1
Temperature /°c
f α
Type A, LE modelType B, LE modelPE model
(a)
(b)
Figure A.3: (a) The fraction of ferrite as a function of time during the isothermal α → γ andγ → α transformation at 875C predicted by LE model and PE model and (b) The fractionof ferrite as a function of temperature upon cooling after the isothermal α → γ and γ → αtransformation at 875C predicted by LE and PE model
is not considered in the classical PE and LE models. In Refs [2,6], it was shown that the
diffusional models using an infinite interface mobility always predicted kinetics faster
than that predicted by a mixed-mode models considering a finite interface mobility.
The paraequilibrium fraction of ferrite predicted by the PE model for both α → γ
and γ → α transformation at 875C is 65 %, while the LE model predicts that the
fractions of ferrite after isothermal α→ γ and γ→ α transformation at 875C are 65 %
and 60 %, respectively. In PE model, it is assumed that there is no Mn partitioning, and
at a certain temperature there is only one paraequilibrium tie-line which determines
the interface concentrations of C during the phase transformations. This paraequilib-
rium condition can be reached quickly due to the high C diffusivity. Therefore, the
paraequilibrium fraction of ferrite at a certain temperature is not affected by the path
A.3. Result and Discussion 169
(α→ γ or γ→ α) through which the paraequilibrium condition is reached. However,
in local equilibrium, both Mn and C are considered to partition at the moving interface,
and thermodynamic equilibrium can not be reached in a limited time due to the low
Mn diffusivity. Instead of full thermodynamic equilibrium, only local equilibrium at
the interface is reached at the end of isothermal holding shown in Fig. A.3a, although
the fraction of α is close to that predicted by thermodynamic equilibrium. The tie-line
that determines the interface concentrations of M and C depends not only on the ther-
modynamic properties of γ and α but also on the kinetics of interface migration. In
general, at a certain temperature, there are a number of possible tie-lines, under which
the chemical potential of all elements are equal in austenite and ferrite. Given a certain
tie-line, the two interface migration velocities can be obtained by solving C and Mn
diffusion equations separately with mass balance at the interface, respectively. Only
the tie-line, for which these two interface migration velocities are the same, will be the
correct and operating tie-line. As the kinetics of the interface migration during the
α → γ transformation is different from that of the γ → α transformation ( Fig. A.3),
different tie-lines are in operation. Hence, the fraction of ferrite predicted by local
equilibrium model for a given temperature is affected by the path through which the
local equilibrium is reached.
In Fig. A.3b, the ferrite fraction during cooling after the prior isothermal α→ γ and
γ → α transformation at 875C predicted by LE and PE model is indicated. The PE
model predicts that there is no stagnant stage at the beginning of cooling in both type
A and B simulations, which is in conflict with the experimental results. The prediction
of LE model considering Mn partitioning shows that there are stagnant stages for both
type A and B experiments, but that the length of stagnant stage in type A experiment
is much longer than that in type B experiments. The stagnant stages predicted by the
LE model are very comparable with those observed in the experiments.
In order to illustrate the underlying physics, in Fig. A.4, the evolution of C and Mn
profiles during the isothermal α → γ and γ → α transformation at 875C predicted
by the LE model is presented. In the initial stage (1s and 5s) of isothermal α → γ
170 Appendix A. The effect of transformation path on stagnant stage
0 5 10 15 20 250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Distance /µm
Car
bo
n c
on
cen
trat
ion
/\wt%
t=1st=5st=50st=500st=3600s
Ferrite Austenite
0 5 10 15 20 250.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance /µm
Mn
co
nce
ntr
atio
n /w
t.%
t=1st=5st=50st=500st=3600s
AusteniteFerrite
0 5 10 15 20 250
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Distance /µm
Car
bo
n c
on
cen
trat
ion
/wt.
%
t=1st=5st=50st=500st=3600s
AusteniteFerrite
0 5 10 15 20 250.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance /µm
Mn
co
nce
ntr
atio
n /w
t.%
t=1st=5st=50st=500st=3600s
Ferrite Austenite
(a) (b)
(c) (d)Figure A.4: The evolution of C and Mn profiles during the isothermal α → γ transformationand isothermal γ → α transformation. (a) and (b) are the C and Mn profiles during α → γtransformation, (c) and (d) are the C and Mn profiles during γ→ α transformation.
transformation, there is a positive C gradient in the austenite, and a depleted Mn spike
appears at the interface.This kind of C and Mn profile indicate that the transformation
mode in the initial stage of isothermal α → γ transformation is local equilibrium-
negligible partitioning (LE-NP) mode. At t=50s, the positive C gradient in austenite
disappears, and instead of a depleted Mn spike there is a zigzag type Mn profiles at
the interface with different gradients on both interface sides. Such C and Mn profiles
indicate that the transformation mode is now in local equilibrium-partitioning (LE-P)
mode.Therefore, the transformation mode switches from fast LE-NP into LE-P mode
during the isothermal α → γ transformation. At the end of the isothermal α → γ
transformation(t=3600s), the carbon is distributed homogeneously in austenite (with a
concentration of 0.0566 wt.%) and ferrite (with a concentration of 0.0028 wt.%), and a
zigzag Mn profile exists at the interface. During the isothermal γ→ α transformation,
there is also a growth mode transition from LE-NP to LE-P, while the evolution of C and
A.3. Result and Discussion 171
Mn profiles at the interface is different from that of isothermal α → γ transformation.
At the end of the isothermal γ→ α transformation(t=3600s), the distribution of carbon
in austenite and ferrite is also homogeneous, the carbon concentration in austenite
and ferrite is 0.0543 wt.% and 0.00256wt.%, respectively, which are a little bit lower
than those of α → γ transformation. The Mn profile at the interface also evolves
into a zigzag shape at the end of γ → α transformation. The shape of the zigzag
Mn profile at the end of α → γ transformation is significantly different from that of
γ→ α transformation, since the tie-lines selected during these two transformations are
different. The interface conditions at t=3600s for both α→ γ and γ→ α transformation
are the starting conditions for the following γ → α transformation upon cooling, and
the difference in the length of stagnant stage is due to the different local interface
conditions at the end of isothermal α→ γ and γ→ α transformation.
As shown in Fig. A.4, the carbon concentration in austenite at the end of the
α→ γ transformation is quite close to that of the γ→ α transformation, while the Mn
distributions at the local interface differ significantly. In a separate simulation, it was
shown that the small difference in C concentration only contributes to 1C difference
in the length of stagnant stage. Therefore, it can be concluded that the duration of the
stagnant stage is mainly caused by the degree of partitioning of Mn at the interface.
In Fig. A.5, the evolutions of Mn profiles upon cooling after the isothermal α → γ
and γ → α transformations are indicated. The profiles at T=875C in Fig. A.5a and
Fig. A.5b are the Mn profile at the end of isothermal α→ γ and γ→ α transformations,
respectively. In both cases, the retained austenite consists of a layer of Mn enriched
austenite with a negative Mn diffusion gradient and a thick body of austenite with the
nominal (uniform) Mn concentration. The Mn enriched austenite layer for route A in
Fig. A.5a is much thicker than that for route B in Fig. A.5b. In a type A simulation, the
interface has to migrate through a much thicker Mn enriched layer in order to complete
the transformation as a result of the further cooling down. Based on the evolution of
Mn profiles, the transformation mode shifts from LE-NP into LE-P during the interface
migration through the Mn enriched austenite layer. The difference in thickness of Mn
172 Appendix A. The effect of transformation path on stagnant stage
enriched layer leads to stagnant stages of 10 and 5 C for route A and B, respectively,
which is in good agreement with experimental results.
14 14.5 15 15.5 16 16.5 17 17.5 180.1
0.15
0.2
0.25
0.3
0.35
0.4
Distance /µm
Mn
co
nce
ntr
atio
n /w
t.%
T=875°cT=872°cT=865°c
14 14.5 15 15.5 16 16.5 17 17.5 180.1
0.15
0.2
0.25
0.3
0.35
0.4
Distance /µm
Mn
co
nce
ntr
atio
n w
t.%
T=875°cT=872°cT=870°c
(a)
(b)
Figure A.5: The evolution of Mn profiles upon cooling after (a) the isothermal α→ γ transfor-mation and (b) the isothermal γ→ α transformation
A.4. conclusion 173
A.4 conclusion
Due to different tie-line selections, the α → γ and γ → α transformation in the inter-
critical region leads to different Mn partitioning profiles at the interface, which causes
the difference in the length of stagnant stage during the further transformation upon
cooling. The length of the stagnant stage is directly determined by the very local parti-
tioning effect at the interface, and it can be used as a tool to monitor the local interface
condition.
174 Appendix A. The effect of transformation path on stagnant stage
Appendix BA mixed mode model with covering
soft impingement effect
This appendix is based on
• H Chen, S van der Zwaag, Modeling of soft impingement effect during solid-
state partitioning phase transformations in binary alloys, J Mater Sci, 2011;
46:1328-1336.
B.1 Introduction
The diffusional phase transformation can be divided into two stages [89, 137]: (i) the
first stage of the phase transformation in which the diffusion fields in front of op-
posing interfaces in the parent phase do not overlap; (ii) the second stage in which
the diffusion fields start to overlap, and the phase transformation slows down, the
so-called soft impingement effect [89, 137]. As diffusion-controlled growth models
have been proposed for a very long time, the analytical diffusion controlled growth
model for the first, non-overlapping diffusion fields stage have been well developed
and are widely applied to describe the kinetics of phase transformation [1, 60, 66]. For
the second stage, initially a so called mean field approximation [138] was used to take
the soft impingement effect into account in the diffusion controlled growth models.
176 Appendix B. A mixed mode model with covering soft impingement effect
Later, in order to treat the overlap of diffusion filed in a more strict way, a number
of diffusion controlled growth models [88, 89, 137], assuming a linear diffusion field
in front of the interface, have been developed to describe the soft impingement effect
more accurately. In Ref [2], assuming a linear diffusion field in front of the interface, an
analytical mixed mode model has been developed to indicate the mixed mode charac-
ter of the diffusional phase transformation. However, a recent work by Bos et.al [139]
has shown that the original mixed-mode model underestimate the diffusional phase
transformation kinetics because of the linear diffusion field approximation. Also, the
soft impingement effect at the later stage of diffusional phase transformations is not
considered in the original mixed mode model. In this work, based on the polynomial
method, a precise diffusion profile expression is introduced to reformulate the ana-
lytical diffusion-controlled growth model and the analytical mixed mode model with
considering soft impingement effect, and the newly reformulated analytical models
are validated by a comparison with a fully numerical solution. Furthermore, the effect
of soft impingement on the overall phase transformation kinetics is investigated for
both the diffusion controlled growth model and mixed mode model, and results are
compared.
B.2 Models
Fig. B.1a and b are the schematics which illustrate the evolution of diffusion fields as
the interface migrates from the α phase into the β phase for the diffusion controlled
growth model and the mixed mode model, respectively. The main difference between
the diffusion controlled growth model and the mixed mode model is that the solute
concentration at the interface in the diffusion controlled growth model is assumed to
be fixed during the entire phase transformation, while it should change physically in
the mixed mode model. Referring to Fig. B.1, we define t2 as the time after which the
diffusion fields start to overlap, x0 as the interface position, L as the length of diffusion
field, Cβαeq and Cαβ
eq as the equilibrium concentration in the β and α phase, C0 as the bulk
B.2. Models 177
concentration, Cm as the carbon concentration at the center of the β phase, and 2X as
the thickness of the parent phase.
Figure B.1: The schematics of the diffusion fields evolution during the partitioning phasetransformation for (a) the diffusion controlled growth model and (b) the mixed mode model.
B.2.1 Diffusion controlled growth model
In the classical diffusion-controlled model for solid-solid partitioning phase transfor-
mations [1], local equilibrium is assumed to be maintained at the interface during the
entire phase transformation, which means that chemical potential of all alloying ele-
ments is equal and there is no chemical Gibbs energy difference at the interface itself
during the phase transformation. Local equilibrium can be maintained only when the
178 Appendix B. A mixed mode model with covering soft impingement effect
interface mobility value is infinite.
When x0 ≤ x ≤ x0 + L , the diffusion profile in front of the interface as a function of
position x for the non-overlapping diffusion stage in the diffusion controlled growth
model is described in a quadratic form here:
C(x) = A1 + A2(x − x0) + A3(x − x0)2 (B.1)
Where A1, A2 and A3 are the pre-factors, C(x) is the solute concentration as a function
of position.
The boundary conditions in the first stage can be described as:
C(x = x0) = Cβαeq (B.2)
C(x = x0 + L) = C0 (B.3)
∂C∂x
∣∣∣x=x0+L = 0 (B.4)
Based on the boundary conditions above, the pre-factors can be determined, and
the quadratic diffusion profile for the non-overlapping diffusion stage can be written
as:
C (x) = C0 +(Cβα
eq − C0
) (1 −
x − x0
L
)2
, x0 ≤ x ≤ x0 + L (B.5)
C (x) = C0, x0 + L ≤ x (B.6)
In order to obtain a general definition of the diffusion field ahead of the interface
which encompasses both the linear and quadratic type, the diffusion field is written
here as
C (x) = C0 +(Cβα
eq − C0
) (1 −
x − x0
L
)n−1
,n > 1 (B.7)
B.2. Models 179
when n = 2, it would become linear approximation as applied in Ref [88,89], when
n = 3, the diffusion field would be quadratic.
During the partitioning phase transformation, a mass balance law should always
hold:
∫ x0
0Cα
eqdx +
∫ x0+L
x0
C(x)dx +
∫ X
x0+LC0dx =
∫ X
0C0dx (B.8)
Combination of Eq B.7 and Eq B.8 yields the expression for the diffusion length in
front of the interface:
L =nx0(C0 − Cαβ
eq )
Cβαeq − C0
(B.9)
Since no accumulation of solutes will occur at the interface, the fluxes forwards and
away from the interface should be equal, which can be expressed as
v(Cβα
eq − Cαβeq
)= −D
∂C∂x
(B.10)
Finally, the interface position as a function of time and the diffusion growth coeffi-
cient can be expressed as
x0 = λ√
Dt (B.11)
λ =
√2(n − 1)Ω2
n(1 −Ω)(B.12)
Ω =Cβα
eq − C0
Cβαeq − Cαβ
eq
(B.13)
Where λ is the diffusion growth coefficient, Ω is the degree of super-saturation.
In the overlapping diffusion stage, the diffusion profile in front of the interface and
boundary conditions are written in the same form as Eq B.1, Eq B.2 and Eq B.4, but the
Eq B.3 is different and should be written as
180 Appendix B. A mixed mode model with covering soft impingement effect
C (x = x0 + L) = Cm (B.14)
The moment at which Cm starts to increase beyond C0 is the start of soft impinge-
ment, and the Cm increases until the equilibrium concentration is approached at the
final stage of partitioning phase transformation.
Similarly, the general polynomial diffusion profile for the overlapping diffusion
stage can be written as
C (x) = Cm +(Cβα
eq − Cm
) (1 −
x − x0
L
)n−1
,n > 1 (B.15)
The mass conservation law leads to
∫ x0
0Cα
eqdx +
∫ X
x0
C (x)dx =
∫ X
0C0dx (B.16)
Combination of Eq B.15 and Eq B.16 yields the expressions for the Cm and interface
migration velocity:
Cm =1
n − 1
[nx0
L
(C0 − Cαβ
eq
)+ nC0 − Cβα
eq
](B.17)
v = D
(Cβα
eq − Cm
)(n − 1)(
Cβαeq − Cαβ
eq
)L
(B.18)
Where the diffusion length L = X − x0.
B.2.2 The mixed-mode model
In mixed-mode mode [2, 5], both the interface mobility and the finite diffusivity are
considered to have effect on the kinetics of phase transformation, and the concentra-
tion of alloying elements at the interface does not evolve according to local equilibrium
assumption but depends on the diffusion coefficient of alloying elements and inter-
face mobility during the phase transformation. The mixed mode model will also be
B.2. Models 181
reformulated in two stages here.
Generally , the interface velocity in the mixed-mode model can be written as
v = M∆G (B.19)
Where M is the interface mobility, ∆G is the driving force for interface migration and
dependent on the solute concentration at the interface in the parent phase.
The interface mobility, M, which is temperature dependent, can be expressed as
M = M0 exp(−QG/RT) (B.20)
Where M0 is a pre-exponential factor, QG is the activation energy for the atomic
motion.
The driving force, ∆G, can be expressed as
∆G =
p∑i=1
Cαi
(µβi − µ
αi
)(B.21)
Where p is the number of alloying elements in the system, Cαi is the concentration
of the alloying element i in the α phase, µβi and µαi are the chemical potential of the
alloying element i in the β and α phase, respectively.
In this work, only one alloying element will be considered, thus the driving force,
∆G , can be approximated to be proportional to the deviation of the mobile alloy-
ing element concentration in the parent phase at the interface from the equilibrium
concentration, and can be expressed as
∆G = χ(Cβα
eq − Cβ)
(B.22)
Where χ is proportionality factors which can be calculated by Thermo-Calc and Cβ is
the solute concentration at the interface in the β phase.
In the mixed mode model, the diffusion profile in the first stage is still written in the
polynomial way, the boundary conditions in the non-overlapping stage are expressed
182 Appendix B. A mixed mode model with covering soft impingement effect
as
C (x = x0) = Cβ (B.23)
C (x = x0 + L) = C0 (B.24)
∂C∂x
∣∣∣x=x0 = 0 (B.25)
Based on the boundary conditions, the general polynomial diffusion profile for the
first non-overlapping stage in the mixed mode model can be derived as
C (x) = C0 +(Cβ− C0
) (1 −
x − x0
L
)n−1
, x0 ≤ x ≤ x0 + L (B.26)
C (x) = C0, x0 + L ≤ x (B.27)
Appling the mass balance law, the expression for the diffusion length can be ob-
tained as
L =nx0
(C0 − Cαβ
eq
)Cβ − C0
(B.28)
As there is no accumulation of solutes at the interface, the following equation can
be derived for the mixed mode model:
v(Cβ− Cαβ
eq
)= Mχ
(Cβα
eq − Cβ) (
Cβ− Cαβ
eq
)= −D
∂C∂x
(B.29)
The solute concentration at the interface can be obtained by solving Eq B.29:
Cβ =
(ZC0 + ∆C0
(Cαβ
eq + Cβαeq
))+
√(ZC0 + ∆C0
(Cαβ
eq + Cβαeq
))2− (Z + 2∆C0) ×
(ZC2
0 + 2∆C0Cαβeq Cβα
eq
)(Z + 2∆C0)
(B.30)
B.3. Numerical calculation 183
Where ∆C0 = C0 − Cαβeq and Z = D(n−1)
Mx0χn
The equation for the interface concentration is in the same form as that in the
original mixed-mode model in which a linear diffusion field is assumed, however, the
parameter Z in the original mixed mode model is just one case of that in the present
work. When n = 2, the mixed mode model presented here is the same as the original
mixed-mode model.
The original mixed mode model does not take the effect of soft impingement into
account, while this effect will be considered in this work. Appling the same method
described in the diffusion controlled growth model, the interface concentration can be
derived as
Cβ =
((Cαβ
eq + Cβαeq
)− Z
)+
√((Cαβ
eq + Cβαeq
)− Z
)2− 4
(Cαβ
eq Cβαeq − Z
XC0−x0Cαβeq
X−x0
)2
(B.31)
where Z = DnM(X−x0)χ
The solute concentration at the center of β phase Cm can be written as
Cm =1
n − 1
[nx0
L
(C0 − Cαβ
eq
)+ nC0 − Cβ
](B.32)
B.3 Numerical calculation
In this work, the Murray-Landis method is chosen for the numerical solution, and the
finite difference equation is written as
ci, j+1 − ci, j
∆t= D
(ci−1, j − 2ci, j − ci+1, j
)∆x2 + v
n − in − 1
(ci+1, j − ci−1, j
)2∆x
(B.33)
Where Cti is the concentration at grid point i at time t, Ct+∆t
i is the concentration at
grid point i at time t + ∆t. The second term on the right side of Eq.B.33 accounts for the
time dependence of the grid points. The diffusion controlled growth model and the
184 Appendix B. A mixed mode model with covering soft impingement effect
mixed mode model will be combined with the Murray-Landis method to simulate the
partitioning phase transformation here.
B.4 Results and Discussion
0 50 100 1500
1
2
3
4
5
6
7
8
9
10
t/s
Siz
e o
f fe
rrit
e (µ
m)
Mixed mode model
quadraticdiffusion field
lineardiffusion field
0 50 100 1500
1
2
3
4
5
6
7
8
9
10
t/s
Siz
e of
ferr
ite p
hase
(µm
)
lineardiffusion field
quadraticdiffusion field
Diffusion−controlled growth model
(a)
(b)
Figure B.2: The thickness of ferrite phase as a function of time during the austenite to ferritetransformation in a Fe-1.0 at.%C alloy at T =1050K predicted by (a) the mixed mode model and(b) the diffusion-controlled growth model with linear and quadratic diffusion field approxima-tions.
To illustrate the effect of soft impingement here, the austenite to ferrite transforma-
tion in a binary Fe-1.0 at.%C alloy at 1050K is investigated. At the given temperature,
Thermo-Calc gives χ=110 J/(at.%), and the equilibrium carbon concentration in the
austenite phase and ferrite phase are 2.05 at.% and 0.09 at.%, respectively. The diffu-
sion coefficient of carbon in austenite is 1.14× 10−12m2/s , and the interface mobility M
is taken to be 5.4 × 10−8 m mol/Js. In order to consider the phase transformation in a
B.4. Results and Discussion 185
finite medium, the finite thickness of the austenite phase 2X is assumed to be 20 µm ,
and the specific volumes of both phases are taken equal.
In Fig. B.2a, the thickness of ferrite phase is calculated as a function of time by the
mixed mode model with linear diffusion field and quadratic diffusion field. The solid
line in the figure is the modeling results with soft impingement correction, and the dot-
ted line is the results without soft impingement correction. It is shown that the mixed
mode model with soft impingement correction predicts the interface migration stops
when the fraction of ferrite reaches the equilibrium value, while the mixed mode model
without soft impingement correction shows that the interface migration will not stop
in finite time. Furthermore, the figure indicates that the mixed mode model with linear
diffusion field predicts a slower kinetics than the model with quadratic diffusion field
does. Fig. B.2b shows the results predicted by the diffusion controlled growth model,
which indicates that the diffusion controlled growth model with soft impingement
correction also predicts the transformation to finish as the thermodynamic equilibrium
is approached.
Fig. B.3 shows the fraction of ferrite as a function of time obtained by the analytical
models and the fully numerical solutions. It shows that both the analytical mixed mode
model and the analytical diffusion controlled mode with a quadratic diffusion field
predict the kinetics more precisely than those with linear diffusion field assumption.
Comparing the present mixed mode model assuming a quadratic diffusion profile
with the mixed mode model assuming an exponential diffusion profile in [139], the
derived equations for solute concentration at the interface during the non-overlapping
diffusion field stage are the same. However, as mentioned above, the exponential
diffusion profile is only valid in the infinite medium, while the quadratic diffusion
profile can also be applied in the overlapping diffusion field stage, as discussed in
model section. It has to be mentioned here that the quadratic diffusion profile is just
one case of polynomial diffusion profiles, in general, the exact diffusion profile can be
approximated as
186 Appendix B. A mixed mode model with covering soft impingement effect
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t/s
Fra
ctio
n o
f fe
rrit
e
Numerical calculationQuadratic diffusion fieldLinear diffusion field
Mixed mode model
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t/s
Fra
ctio
n o
f fe
rrit
e
Numerical calculation Quadratic diffusion fieldLinear diffusion field
Diffusion controlled growth model
(a)
(b)
Figure B.3: The fraction of ferrite as a function of time during the austenite to ferrite transfor-mation in a Fe-1.0 at.%C alloy at T =1050K predicted by (a) the mixed mode model and (b) theanalytical model
C (x) = A1 + A2 (x − x0) + A3(x − x0)2 + ..... + An(x − x0)n−1,n > 2 (B.34)
Eq B.34 is a (n−1) order polynomial diffusion profile, in which n parameters have to
be determined, which means n boundary conditions are needed to solve the problem.
Except Eq B.2 and Eq B.3, (n − 2) extra boundary conditions can be written as
∂mC∂xm
∣∣∣x=x0+L = 0,m = 1, 2.....n − 2 (B.35)
Based on the above boundary conditions, the parameters in Eq B.34 can be obtained.
Although the accuracy of the model could possibly be increased increasing the order
of the diffusion profile, it also make the case more complicated. As shown in Fig. B.3,
the accuracy of the quadratic diffusion profile is quite close to that of the numerical
B.4. Results and Discussion 187
solution, which means the accuracy will not be further improved significantly by
increasing the order of the diffusion profile. Therefore, the quadratic diffusion field
approximation is applied to the analytical models in all the following calculations.
0 10 20 30 40 50 60 70 80 90 1001
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Cm
(at
.%)
t/s0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
8
Dif
fusi
on
len
gth
Soft impingment starting point The mixed mode model
0 10 20 30 40 50 60 70 80 901
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Cm
(at
.%)
t/s0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
Dif
fusi
on
len
gth
(µm
)
Soft impingment starting point Diffusion controlled growth model
(a)
(b)
Figure B.4: The solute concentration at the center of the austenite phase and the diffusion lengthas a function of time predicted by (a) the mixed mode model and (b) the diffusion-controlledgrowth model
Considering the soft impingement effect at the later stage, the solute concentration at
the center of the austenite phase and diffusion length, which are the two key parameters
in the overlapping diffusion stage, are calculated as a function of time by the mixed
mode model and diffusion controlled growth model in Fig. B.4a and b, respectively.
The vertical dotted line in the figure indicates the critical point at which the diffusion
fields in the neighboring grains start to overlap. Before the critical point is reached,
the diffusion length extends as the interface migrates into the austenite phase, and the
solute concentration at the center of the austenite phase is not affected by diffusion
and is fixed at the bulk concentration of the Fe-C alloys. After the critical point, the
188 Appendix B. A mixed mode model with covering soft impingement effect
diffusion length would starts to shrink, and the solute concentration at the center of
the austenite phase would begin to increase towards the equilibrium concentration.
In the model considering the soft impingement, both the solute concentration at the
center of the austenite phase and the diffusion length can be used to estimate the soft
impingement starting point, while only the diffusion length can be the effective factor
for detecting soft impingement in the model without soft impingement correction.
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
Size of ferrite (µm)
Dif
fusi
on
len
gth
(µm
) Ω=0.3
Ω=0.5
Ω=0.8
Ω=0.1
Figure B.5: The diffusion length as a function of the thickness of ferrite phase for differentdegrees of super-saturation predicted by the diffusion controlled growth model, the dottedline indicates the start of soft impingement.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ω
δf/f
eq
Figure B.6: The ratio of ferrite transformed during the overlapping diffusion stage and theequilibrium fraction of ferrite as a function of the degree of super-saturation predicted by thediffusion controlled growth model
As discussed in the original mixed mode model [2], the bulk concentration C0 affects
the evolution of diffusion length during the austenite to ferrite phase transformation
at a certain temperature, and it is concluded that the smaller the bulk concentration C0,
B.4. Results and Discussion 189
the shorter the diffusion length. However, at different temperature, the equilibrium
concentrations in the austenite and ferrite phase are different, which also affects the
diffusion length evolution. Therefore, it is necessary and meaningful to summarize all
the effect factors into one factor to obtain a general law. In Fig. B.5, the diffusion lengths
as a function of the thickness of ferrite phase for different degrees of super-saturation
are calculated by the diffusion controlled growth model. The value of diffusion length
is only affected by the degree of super-saturation, in which the bulk concentration
and the equilibrium concentration in both the austenite and ferrite phase are included.
It shows that the magnitude of the diffusion length decreases with increasing the
degree of super-saturation. Actually, decreasing the bulk concentration at a certain
temperature discussed in the original mixed mode model [2] is just one specific case
of increasing the degree of super-saturation according to Eq B.13, and the Fig. B.5 can
be considered as a master curve for estimating the diffusion length.
In order to investigate whether the overlapping diffusion stage or the non-overlapping
diffusion stage dominates the transformation kinetics, the ratio of ferrite transformed
during the overlapping diffusion stage and the total equilibrium fraction of ferrite as
a function of the degree of super-saturation is calculated by the diffusion controlled
growth model in Fig. B.6. It is indicated that the ferrite transformed during the over-
lapping diffusion stage decreases with increasing the degree of super-saturation. This
can be easily understood in this way: as the super-saturation increases, the solute
concentration difference between the growing ferrite phase and the bulk concentration
would be smaller, which means less carbon has to be rejected from the ferrite phase
into the austenite phase as the interface migrates into the austenite phase and thus
less carbon would pile up in front of the interface, so the overlapping diffusion stage
would be shorter.
As shown in Fig. B.5 and Fig. B.6, the diffusion length in the diffusion controlled
growth model is as a simple function of the degree of super-saturation, while the
magnitude of diffusion length in the mixed mode model is not just determined by
the degree of super-saturation, the diffusion coefficient and the interface mobility
190 Appendix B. A mixed mode model with covering soft impingement effect
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
10
t/s
Len
gth
(µm
)
Diff
M*=M
Diff
M*=M
M*=0.1M
M*=0.1M
M*=0.05M
M*=0.05M
Figure B.7: The diffusion length (solid line) and size of ferrite (dotted line) as a function of timeduring the austenite to ferrite phase transformation at 1050K in Fe-1.0at.%C alloys predictedby the mixed mode model with different interface mobility and diffusion-controlled growthmodel, Diff signify diffusion controlled growth model.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
fraction of ferrite
dif
fusi
on
len
gth
(µm
)
M*=0.05M M*=0.1M
M*=M
Diff
Equilibrium fraction of ferrite
Figure B.8: The diffusion length as a function of the fraction of ferrite phase during the austeniteto ferrite phase transformation at 1050K in Fe-1.0at.%C alloys predicted by the mixed modemodel with different interface mobility and diffusion-controlled growth model, Diff signifydiffusion controlled growth model.
would also play a role. As discussed in recent work [98], the ratio of the interface
mobility and the diffusion coefficient has a strong effect on the kinetics of partitioning
phase transformation, thus there is no doubt that the soft impingement effect during
the partitioning phase transformation is also determined by this ratio. Compared
with the diffusion coefficient, the value of interface mobility during partitioning phase
transformation is still not known accurately [53]. Therefore, fixing the value of diffusion
coefficient and varying the value of interface mobility, in Fig. B.7, the diffusion length
and size of ferrite phase during the austenite to ferrite phase transformation at 1050K
B.4. Results and Discussion 191
0 1 2 3 4 5 6 7 8 9 101
1.2
1.4
1.6
1.8
2
2.2
2.4
position (µm)
Car
bo
n c
on
cen
trat
ion
(at
.%)
Diffusion controlled growth model
Interface moving direction
0 1 2 3 4 5 6 7 8 9 101
1.2
1.4
1.6
1.8
2
2.2
position (µm)
Car
bo
n c
on
cen
trat
ion
(at
.%)
Interface moving directionM*=M
0 1 2 3 4 5 6 7 8 9 101
1.2
1.4
1.6
1.8
2
position (µm)
Car
bo
n c
on
ccen
trat
ion
(at
.%)
M*=0.1MInterface moving direction
0 1 2 3 4 5 6 7 8 9 101
1.2
1.4
1.6
1.8
2
position (µm)
Car
bo
n c
on
cen
trat
ion
(at
.%)
M*=0.05M
Interface moving direction
(a)
(b)
(c)
(d)Figure B.9: The carbon profile evolution in the austenite phase predicted by the diffusioncontrolled growth model and the mixed mode model with different interface mobilities.
in Fe-1.0at.%C alloys as a function of time are predicted by the mixed mode model and
the diffusion controlled growth model. As the temperature and bulk concentration
of the Fe-C alloys are fixed, the degree of super-saturation is a constant, thus only
the effect of interface mobility and diffusion coefficient on the soft impingement is
192 Appendix B. A mixed mode model with covering soft impingement effect
investigated in Fig. B.7. It is indicated that the increase in the interface mobility has a
small effect on the diffusion length as a function of time but this leads to a significant
increase in the interface migration rate, which would make the soft impingement start
earlier. Also, increasing the interface mobility, the kinetics predicted by the mixed mode
model would become closer to that by the diffusion controlled growth model, since
the transformation kinetics is more and more controlled by the solutes diffusion, more
discussion about this point can be found in Ref [98]. In Fig. B.8, the diffusion length
as a function of the fraction of ferrite phase is calculated by the mixed mode model
with different interface mobility and diffusion controlled growth model. Although
the soft impingement starts earlier with increasing interface mobility as shown in
Fig. B.7, the ferrite transformed during the soft impingement stage decreases as shown
in Fig. B.8, which means the dominance of the soft impingement effect on the overall
transformation kinetics reduces. In order to indicate the effect of interface mobility on
the carbon profile evolution, in Fig. B.9, the carbon profile in the austenite phase is
calculated by the diffusion controlled growth model and the mixed mode model with
different interface mobilities.
The carbon concentration at the interface in the austenite phase, which determines
the magnitude of driving force for interface migration, is a key physical parameter in
the mixed mode, thus it is calculated by the mixed mode model in Fig. B.10a. It shows
that the interface concentration increases as the interface migrates, and the driving
force would decrease to 0 and interface stops migrating when the equilibrium con-
centration is approached. The interface concentration predicted by the mixed mode
model without soft impingement correction [139] would not be equal to equilibrium
concentration even when the equilibrium fraction is approached. It has to be partic-
ularly pointed out the interfacial carbon concentration is C0 when the ferrite fraction
is zero. However, in the initial stage, the interfacial carbon concentration increase ex-
tremely fast, which leads to a false appearance that the interfacial carbon concentration
is higher than C0 when ferrite fraction is zero as shown in Fig. B.10a. In Fig. B.10b,
the newly defined growth mode parameter H =Cβαeq −Cβ
Cβαeq −Cαβeq[98] as a function of fraction
B.4. Results and Discussion 193
0 0.1 0.2 0.3 0.4 0.5 0.61
1.2
1.4
1.6
1.8
2
Fraction of ferrite
Carb
on
co
ncen
trati
on
(at.
%)
M*=M
M*=0.1M
M*=0.05MEquilibrium concentration
0 0.1 0.2 0.3 0.4 0.5 0.60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Fraction of ferrie
H
M*=M
M*=0.1M
M*=0.05M
(a)
(b)
Figure B.10: (a) the carbon concentration at the interface and (b) the growth mode H as afunction of the fraction of ferrite predicted by the mixed mode model.
of ferrite phase for different interface mobility is presented. For a certain interface
mobility, the H decreases as the phase transformation proceeds, which indicates that
the growth mode become more diffusion controlled. As the soft impingement effect
is corrected in the present mixed mode model, the growth mode parameter would
decrease to 0 when the thermodynamic equilibrium is established. Also, the growth
mode is changed correspondingly with the variation of interface mobility, and the
194 Appendix B. A mixed mode model with covering soft impingement effect
growth mode is approaching to the pure diffusion controlled growth with increasing
the ratio of interface mobility and the diffusion coefficient. In other words, it can also
be stated that the dominance of soft impingement on the transformation kinetics is
determined by the growth mode if the degree of super-saturation is fixed.
B.5 Conclusion
Applying the polynomial diffusion field and considering the diffusion field overlap
quantitatively at the later stage of phase transformation, the mixed mode model and
diffusion controlled growth model are reformulated in an analytical form to be more
accurate and physically reasonable in the present work. The effect of soft impinge-
ment on the overall partitioning phase transformation kinetics is solely determined by
the super-saturation according to diffusion controlled growth, and it decreases with
increasing the super-saturation. However, in the mixed mode model, the effect of soft
impingement on the overall phase transformation kinetics is determined by both the
degree of the super-saturation and the growth mode.
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Acknowledgments
First of all, I would like to express my sincere gratitude to Prof Sybrand van der
Zwaag, for his continuous support and remarkable supervision throughout my PhD
study. Sybrand always guided my research in the right direction, and kept giving
me encouragement and advice about my PhD research and future career. I especially
thank him for his high efficiency of correcting my manuscripts. Sybrand’s education
philosophy were invaluable for me in the past four years, and will definitely be helpful
for my future career. Thank you Sybrand.
I greatly appreciate the financial support from Arcelor Mittal, and continuous project
management from Prof. Mohamed Goune. I am also very grateful to Dr.Kangying
Zhu for stimulating discussions and help in experiments, to Dr. Ian Zuazo for TEM
measurements and discussions, to Dr. Astrid Perlade, Dr. Thierry Lung and many
others for interesting discussions and hospitality during my visit at Metz.
During my PhD study, I was so fortunate to collaborate with several well respected
scientists in the field of phase transformation. Since the early stage of my PhD, I started
to collaborate with Dr.Benoıt Appolaire (Onera, France) and his student (Dr.Natalya
Perevishchikova) on phase field modeling of the cyclic phase transformations. I would
like to take this opportunity to thank Benoıt and Natalya for sharing their knowledge
of phase field theory with me, stimulating discussions, and hospitality during my visit
at Nancy. Benoıt is also acknowledged for his insightful comments and deep discus-
sions on my first Acta Paper. Since 2011, I began to work together with Prof. Ernst
212 ACKNOWLEDGMENTS
Gamsjager on interface mobility and HT LSCM study of interface migration, and really
learned a lot from Ernst about phase transformation. Ernst is always very friendly and
supportive to me! I have been kindly invited by Ernst to visit his group several times,
and my visit at Leoben is always pleasant (nice food and beer) and also scientifically
fruitful (stimulating discussions). Thank you Ernst! I am also very grateful to Mr.
Siegfried Schider for his great contribution to HT LSCM experiments and making the
lab so enjoyable to work in. In the last year of my PhD, I went to KTH for 3 months
internship, and worked together with Prof John Ågren, Prof. Annika Borgenstam,
Dr. Joakim Odqvist on modelling of bainitic transformation. I am very grateful to
John, Annika and Joakim for hospitality and interesting discussions. I would also
like to thank Prof. Mats Hillert for his private lessons and discussions about phase
transformation in steels. I really benefited a lot. Besides senior scientists, I also had
pleasant collaborations with young scientists Dr. Jaroslaw Opara (Institute for Ferrous
Metallurgy, Poland), Mr. Casper Versteylen (TU Delft) , Mr. Takayuki Otsuka (LSPM
University Paris 13), Dr. Xiangliang Wan (Wuhan University of Science and Technol-
ogy, China). I am so grateful to Casper, Jarek , Takayuki, Xiangliang for interesting
discussions and continuous help during my PhD study.
I would like to take this opportunity to thanks Dr. Lie Zhao (3me, TU Delft) and
Mr.Tjerk Koopmans (3me, TU Delft) for their help in magnetometer experiments and
interesting discussions. I am also very grateful to Prof. Masato Enomoto(Ibaraki Uni-
versity, Japan) for providing me the very interesting Fe-C-Mn and Fe-C-Mn-Si alloys
used in Chapter 7. Mr.Nico Geerlofs is acknowledged for his assistance in dilatome-
ter experiments. Thanks also goes to Prof.Zhigang Yang (Tsinghua University) for
calculating NPLE lines for me, Prof. Kaiming Wu (Wuhan University of Science and
Technology, China) for correcting my first Chinese manuscript and inviting me to visit
his steel research group. A special thanks goes to the supervisor of my master the-
sis, Prof. Yongchang Liu (Tianjin Unverisity, China) , who brought me into the field
of materials science. I also thank Prof. Liu for his hospitality during my visits at Tianjin.
ACKNOWLEDGMENTS 213
I would also like to express sincere gratitude to Prof.Gary Purdy (McMaster Uni-
versity, Canada), Prof. Matthias Militzer (University of British Columbia, Canada),
Prof. John Ågren, Prof. Ernst Gamsjager, Prof. Zhigang Yang, Prof. Ekkes Bruck (TU
Delft), Prof. Rinze Benedictus (TU Delft) for serving on the PhD committee.
My sincere acknowledgment also goes to friends and colleagues in the most inter-
national group NOVAM. I am very grateful to Shanta for her continuous support and
help in the past four years. For Dr.Stephane Forsik , I would like to thank him for help-
ing me improve my English, introducing me European culture and inviting me to visit
Paris. My officemates, Dr. Marek Prajer and Xiaojun Xu, are also acknowledged for
creating such a nice working atmosphere. Thanks goes to Dr. Mingxin Huang for his
help at the early stage of my PhD, stimulating scientific discussions, continuous sug-
gestions and encouragements during the course of my PhD research. Thank Jason, Xu,
Jie, Cong, Qi , Maruti, Jianwei, Theo, Ranjita, Mina, Qingbao, Martino, Michiel, Nijesh,
Zeljka, Hamideh, Jesus, Ugo, Santiago , Jimmy, Hongli , Jasper, Mladen, Richardo, Pim,
Antonio, Jeroen, Christian, Nora, Maarten, Fre, Kevin, Frederik, Renee and all other
former Novam members for interesting discussions, help, coffee and lunch. Thank
all my non-academic friends for help, all kinds of discussions, traveling together, fun,
beers, food in the past four years.
I would like to thank my family for their love, support, encouragement. Last but
certainly not least, a special thanks goes to kun, my wife, for her unwavering love,
understanding, quiet patience, support and encouragement, which make this journey
easy and comfortable.
214 ACKNOWLEDGMENTS
Curriculum Vitae
Hao ChenBorn on October 25, 1986
Anqing, Anhui province, China
Sept. 2003- Jul. 2007
Bachelor of Engineering in Materials Science, in School of Materials Science and
Engineering, Tianjin University, Tianjin, China.
Sept. 2007- Jun. 2009
Master of Engineering in Materials Science, in School of Materials Science and
Engineering, Tianjin University, Tianjin, China.
Jul. 2009- Jun. 2013
PhD candidate in the group Novel Aerospace Materials, Faculty of Aerospace
Engineering, Delft University of Technology, Delft, The Netherlands.
216 CURRICULUM VITAE
List of Publications
Journal
18. Ernst Gamsjager, Hao Chen, Sybrand van der Zwaag, Application of the cyclic
phase transformation concept for determining the effective austenite/ferrite in-
terface mobility, to be submitted, 2013.
17. Hao Chen, Kangying Zhu, Lie Zhao, Sybrand van der Zwaag, Analysis of trans-
formation stasis during the isothermal bainitic ferrite formation in Fe-C-Mn and
Fe-C-Mn-Si alloys, Acta Materialia, accepted for publication, 2013.
16. Hao Chen, Annika Borgenstam, Joakim Odqvist, Ian Zuazo, Goune Mohamed,
John Ågren, Sybrand van der Zwaag, Application of interrupted cooling ex-
periments to study the mechanism of bainitic ferrite formation in steels, Acta
Materialia, DOI: 10.1016/j.actamat.2013.04.020, 2013.
15. Hao Chen, Ernst Gamsjager, Siegfried Schider, Hamideh Khanbareh, Sybrand
van der Zwaag, In situ observation of austenite-ferrite interface migration in
a lean Mn steel during cyclic partial phase transformations, Acta Materialia,
61(2013)2414-2424.
14. Jaroslaw Opara, Roman Kuziak, Hao Chen, Sybrand van der Zwaag, A two-
dimensional Cellular Automata model to simulate microstructure development
and carbon redistribution during the phase transformation of austenite to ferrite
218 LIST OF PUBLICATIONS
using realistic angular starting microstructures, Computer Methods in Materials
Science, 12(2012)207- 217.
13. Hao Chen, Sybrand van der Zwaag, Analysis of ferrite growth retardation
induced by local Mn enrichment in austenite: a cyclic phase transformation
approach, Acta Materialia, 61(2013)1338-1349.
12. Hao Chen, Wei Xu, Goune Mohamed, Sybrand van der Zwaag, Application of
the stagnant stage concept for monitoring Mn partitioning at the austenite-ferrite
interface in the intercritical region for Fe-Mn-C alloys, Philosophical Magazine
letters 92(2012) :547-555.
11. Hao Chen, Goune Mohamed, Sybrand van der Zwaag, Analysis of the stag-
nant stage in diffusional phase transformations starting from austenite-ferrite
mixtures,Computational Materials Science 55(2012):34-43.
10. Hao Chen, Sybrand van der Zwaag, Indirect evidence for the existence of the Mn
partitioning spike during the austenite to ferrite transformation, Philosophical
Magazine letters 92(2012)86-92.
9. Hao Chen, Benoıt Appolaire, Sybrand van der Zwaag, Application of cyclic
partial phase transformations for identifying kinetic transitions during solid-state
phase transformations: Experiments and modeling, Acta Materialia 59(2011):
6751-6760.
8. Hao Chen, Benoıt Appolaire, Sybrand van der Zwaag, Interface motion and
interface mobility of the partitioning phase transformations in Fe-Mn-C and
Fe-C alloys: A cyclic phase transformation approach, Materials Science Forum
706(2012): 1367-1372.
7. Hao Chen, Sybrand van der Zwaag, Modeling of soft impingement effect dur-
ing solid- state partitioning phase transformations in binary alloys, Journal of
Materials Science 46(2011): 1328-1336.
LIST OF PUBLICATIONS 219
6. Hao Chen, Sybrand van der Zwaag, A mixed-mode model considering soft im-
pingement effects for solid-state partitioning phase transformations, Solid State
Phenomena 172(2011): 561-566.
5. Hao Chen, Sybrand van der Zwaag, Application of the cyclic phase transforma-
tion concept for investigating growth kinetics of solid-state partitioning phase
transformations, Computational Materials Science 49(2010): 801-813.
4. Hao Chen, Yong Chang Liu, Ze Sheng Yan, Yanli Li and Lifang Zhang, Con-
sideration of the growth mode in isochronal austenite-ferrite transformation of
ultra-low-carbon Fe-C alloy, Applied Physics A 98(2010):211-217.
3. Hao Chen, Yong Chang Liu, Dong Jiang Wang, Zesheng Yan, Jicheng Fu, Qingzhi
Shi, A JMAK-like approach for isochronal austenite–ferrite transformation kinet-
ics in Fe-0.055 wt%N alloy, Materials Science and Technology 26(2010): 572-578.
2. Chen Wei, Yong Chang Liu, Li Ming Yu, Hao Chen, Xun Wang, Effects of Al on the
failure mechanism of the Sn-Ag-Zn eutectic solder, Microelectronics Reliability
50(2010):1142-1145.
1. Yong Chang Liu, Hao Chen, Z M Gao, Y H Zhang, Q Z Shi, Evolution of cellular
spacing during directional solid-state ferrite–austenite transformation of Fe-Mn-
Al alloy, Journal of Crystal Growth 311(2009): 3761-3764.
Conference proceedings
3. Sybrand van der Zwaag, Hao Chen, Elucidating the role of partitioning of substi-
tutional alloying elements at the austenite-ferrite interface during phase transfor-
mations in lean steels, Proceedings of International Symposium on Automobile
Steel (ISAS2013) , Anshan, China.
2. Hao Chen, Sybrand van der Zwaag, An experimental study of the stagnant
stage in bainite phase transformations starting from austenite-bainite mixtures,
220 LIST OF PUBLICATIONS
Proceedings of TMP 2012 , Sheffield, UK.
1. Hao Chen, Sybrand van der Zwaag, Simulation of the stagnant stage during
the austenite to ferrite transformation in cyclic partial phase transformations,
Proceedings of TMP 2012 , Sheffield, UK.
Conference presentations
11. Accepted for oral presentation: Hao Chen, Sybrand van der Zwaag, The effect
of alloy composition on the duration of the stagnant stage during cyclic partial
austenite-ferrite transformations, Euromat 2013, Sevilla, Spain.
10. Accepted for highlight presentation: Hao Chen, Sybrand van der Zwaag, Appli-
cation of interrupted cooling experiments to unravel the mechanism of bainitic
ferrite formation in low alloy steels, Euromat 2013, Sevilla, Spain.
9. Accepted for oral presentation: Hao Chen, Sybrand van der Zwaag, New insights
into the correctness of diffusional and diffusionless concepts for the bainitic ferrite
formation in Fe-C-Mn and Fe-C-Mn-Si alloys, Thermec conference 2013, Las
Vegas, USA.
8. Oral presentation: Hao Chen, Sybrand van der Zwaag, Application of the
cyclic phase transformation concept for investigating the growth mechanism
of austenite-ferrite phase transformation: Experiments and Dictra simulations,
Computational Thermodynamics and Kinetics Seminar and Workshop 2013, KU
Leuven, Belgium.
7. Oral presentation: Hao Chen, Sybrand van der Zwaag, Analysis of the stag-
nant stage in diffusional phase transformations starting from austenite-ferrite
mixtures, TMP conference 2012, Sheffield, UK.
6. Oral presentation: Hao Chen, Sybrand van der Zwaag, An investigation into the
C and Mn partitioning at the austenite-ferrite interface in the intercritical region
LIST OF PUBLICATIONS 221
for Fe-Mn-C alloys, MSE conference 2012, Darmstat, Germany.
5. Oral presentation: Hao Chen, Ernst Gamsjager, Sybrand van der Zwaag, Appli-
cation of the cyclic phase transformation concept for determining the effective
austenite/ferrite interface mobility, MSE conference 2012, Darmstat, Germany.
4. Oral presentation: Hao Chen, Benoıt Appolaire, Sybrand van der Zwaag, The
cyclic phase transformations in lean carbon-manganese steels, Euromat 2011,
Montpellier, France.
3. Oral presentation: Hao Chen, Benoıt Appolaire, Sybrand van der Zwaag, Inter-
face motion and interface mobility of partitioning phase transformations: a cyclic
transformation approach, Thermec conference 2011, Quebec city, Canada.
2. Oral presentation: Hao Chen, Benoıt Appolaire, Sybrand van der Zwaag, The
cyclic phase transformations concept, ALEMI workshop 2011, Vancouver, Canada.
1. Poster: Hao Chen, Sybrand van der Zwaag, A mixed-mode model considering
soft impingement effect, PTM conference 2010, Avignon, France.