Download - 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

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


4.3.1 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


5.3.1 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.


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.

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


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


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


∆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


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,


(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


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


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


(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


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


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.


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


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.


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


• 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.


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)


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


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


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


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


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.


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)


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


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)


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


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


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.

Chapter 4Analysis of the stagnant stage during

cyclic phase transformations


• 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].


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


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


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.


4.3.2 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


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


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


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


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


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


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.


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


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


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


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


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


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


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.


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


• 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


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.


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


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.


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


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


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.


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


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


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 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


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-


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


(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


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


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.


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


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


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



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


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


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.


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.


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.

Chapter 6In-situ observation of the cyclic phase

transformation


• 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


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


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].


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


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


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.


(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-


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


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


• 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


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


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


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.


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


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


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


(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.


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


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


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


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


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


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


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


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


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


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-


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


10−11

10−10

10−9

10−8

10−7

10−6

10−50

100

200

300

400

500

600

700

800

900


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


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.

Chapter 8Transformation stasis during the

isothermal bainitic ferrite formation in

Fe-C-X alloys


• 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


in Fe-C-X alloys

10−12

10−10

10−8

10−6

10−40

200

400

600

800


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


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


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-


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


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


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-


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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.


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 opwarmen 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


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


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


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


(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


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


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


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.

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


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


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


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


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


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


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


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,


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


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


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


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


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


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.

Bibliography

[1] C. Zener, “Theory of Growth of Spherical Precipitates from Solid Solution,”

Journal of Applied Physics, vol. 20, pp. 950–953, 1949.

[2] J. Sietsma and S. van der Zwaag, “A concise model for mixed-mode phase

transformations in the solid state,” Acta Materialia, vol. 52, pp. 4143–4152, 2004.

[3] E. Gamsjager, J. Svoboda, and F. Fischer, “Austenite-to-ferrite phase transforma-

tion in low-alloyed steels,” Computational Materials Science, vol. 32, pp. 360–369,

2005.

[4] E. Gamsjager, M. Militzer, F. Fazeli, J. Svoboda, and F. Fischer, “Interface mobility

in case of the austenite-to-ferrite phase transformation,” Computational Materials

Science, vol. 37, pp. 94–100, 2006.

[5] G. Krielaart, J. Sietsma, and S. van der Zwaag, “Ferrite formation in Fe-C alloys

during austenite decomposition under non-equilibrium interface conditions,”

Materials Science and Engineering A, vol. 237, pp. 216–223, 1997.

[6] H. Chen and S. van der Zwaag, “Modeling of soft impingement effect during

solid-state partitioning phase transformations in binary alloys,” Journal of Mate-

rials Science, vol. 46, pp. 1328–1336, 2011.

[7] H. Zurob, C. Hutchinson, A. Beche, G. Purdy, and Y. Brechet, “A transition from

local equilibrium to paraequilibrium kinetics for ferrite growth in Fe-C-Mn: A

195

196 Bibliography

possible role of interfacial segregation,” Acta Materialia, vol. 56, pp. 2203–2211,

2008.

[8] H. Zurob, C. Hutchinson, Y. Brechet, H. Seyedrezai, and G. Purdy, “Kinetic

transitions during non-partitioned ferrite growth in FeCX alloys,” Acta Materialia,

vol. 57, pp. 2781–2792, 2009.

[9] M. Enomoto and H. Aaronson, “Partition of Mn during the growth of proeutec-

toid ferrite allotriomorphs in an Fe-1.6 at. pct C-2.8 at. pct Mn alloy,” Metallurgical

and Materials Transactions A, vol. 18, pp. 1547–1557, 1987.

[10] O. Thuillier, F. Danoix, M. Goune, and D. Blavette, “Atom probe tomography of

the austeniteferrite interphase boundary composition in a model alloy,” Scripta

Materialia, vol. 55, pp. 1071–1074, 2006.

[11] T. Jia and M. Militzer, “Modelling Phase Transformation Kinetics in FeMn Al-

loys,” ISIJ International, vol. 52, pp. 644–649, 2012.

[12] J. Odqvist, M. Hillert, and J. Ågren, “Effect of alloying elements on the γ to α

transformation in steel. I,” Acta Materialia, vol. 50, pp. 3211–3225, 2002.

[13] G. Purdy, J. Ågren, A. Borgenstam, Y. Brechet, M. Enomoto, T. Furuhara, E. Gam-

sjager, M. Goune, M. Hillert, C. Hutchinson, M. Militzer, and H. Zurob, “ALEMI:

A Ten-Year History of Discussions of Alloying-Element Interactions with Migrat-

ing Interfaces,” Metallurgical and Materials Transactions A, vol. 42A, pp. 3703–3718,

2011.

[14] W. T. Reynolds, S. K. Liu, F. Z. Li, S. Hartfield, and H. I. Aaronson, “An Investiga-

tion of the Generality of Incomplete Transformation to Bainite in Fe-C-X Alloys,”

Metallurgical and Materials Transactions A, vol. 21A, pp. 1479–1491, 1990.

[15] H. I. Aaronson, G. Spanos, and W. T. Reynolds, “A progress report on the defi-

nitions of bainite,” Scripta Materialia, vol. 47, pp. 139–144, 2002.

Bibliography 197

[16] T. Furuhara, T. Yamaguchi, G. Miyamoto, and T. Maki, “Incomplete transfor-

mation of upper bainite in Nb bearing low carbon steels,” Materials Science and

Technology, vol. 26, pp. 392–397, 2010.

[17] H. Bhadeshia, Bainite in Steels. Institute of Materials, 2001.

[18] G. Purdy and M. Hillert, “Overview no. 38: On the nature of the bainite trans-

formation in steels,” Acta Metallurgica, vol. 32, pp. 823–828, 1984.

[19] Z. Yang and H. Fang, “An overview on bainite formation in steels,” Current

Opinion in Solid State and Materials Science, vol. 9, pp. 277–286, 2005.

[20] H. Bhadeshia and D. Edmonds, “The mechanism of bainite formation in steels,”

Acta Metallurgica, vol. 28, pp. 1265–1273, 1980.

[21] H. Bhadeshia and D. Edmonds, “The bainite transformation in a silicon steel,”

Metallurgical and Materials Transactions A, vol. 10, pp. 895–907, 1979.

[22] H. Bhadeshia, “The bainite transformation: unresolved issues,” Materials Science

and Engineering: A, vol. 273-275, pp. 58–66, 1999.

[23] M. Goune, O. Bouaziz, S. Allain, K. Zhu, and M. Takahashi, “Kinetics of bai-

nite transformation in heterogeneous microstructures,” Materials Letters, vol. 67,

pp. 187–189, 2012.

[24] L. Y. Lan, C. L. Qiu, D. W. Zhao, X. H. Gao, and L. X. Du, “Effect of austenite grain

size on isothermal bainite transformation in low carbon microalloyed steel,”

Materials Science and Technology, vol. 27, pp. 1657–1663, 2011.

[25] G. Sidhu, S. Bhole, D. Chen, and E. Essadiqi, “An improved model for bainite

formation at isothermal temperatures,” Scripta Materialia, vol. 64, pp. 73–76, 2011.

[26] R. Hackenberg and G. Shiflet, “A microanalysis study of the bainite reaction at

the bay in Fe-C-Mo,” Acta Materialia, vol. 51, pp. 2131–2147, 2003.

198 Bibliography

[27] M. Kang, M. Zhang, and M. Zhu, “In situ observation of bainite growth during

isothermal holding,” Acta Materialia, vol. 54, pp. 2121–2129, 2006.

[28] M. Peet, S. Babu, M. Miller, and H. Bhadeshia, “Three-dimensional atom probe

analysis of carbon distribution in low-temperature bainite,” Scripta Materialia,

vol. 50, pp. 1277–1281, 2004.

[29] D. Gaude-Fugarolas and P. Jacques, “A New Physical Model for the Kinetics of

the Bainite Transformation,” ISIJ International, vol. 46, pp. 712–717, 2006.

[30] M. Hillert, “Nature of local equilibrium at the interface in the growth of ferrite

from alloyed austenite,” Scripta Materialia, vol. 46, pp. 447–453, 2002.

[31] J. Wang, v. d. W. P, and v. d. Z. S, “On the influence of alloying elements on

the bainite reaction in low alloy steels during continuous cooling,” Journal of

Materials Science, vol. 5, pp. 4393–4404, 2000.

[32] R. Larn and J. Yang, “The effect of compressive deformation of austenite on

the bainitic ferrite transformation in Fe-Mn-Si-C steels,” Materials Science and

Engineering: A, vol. 278, pp. 278–291, 2000.

[33] F. Fazeli and M. Militzer, “Modelling Simultaneous Formation of Bainitic Ferrite

and Carbide in TRIP Steels,” ISIJ International, vol. 52, pp. 650–658, 2012.

[34] D. Quidort and Y. Brechet, “Isothermal growth kinetics of bainite in 0.5% C

steels,” Acta Materialia, vol. 49, pp. 4161–4170, 2001.

[35] A. Borgenstam, M. Hillert, and J. Ågren, “Metallographic evidence of carbon

diffusion in the growth of bainite,” Acta Materialia, vol. 57, pp. 3242–3252, 2009.

[36] H. I. Aaronson, W. T. Reynolds, and G. R. Purdy, “The incomplete transforma-

tion phenomenon in steel,” Metallurgical and Materials Transactions A, vol. 37A,

pp. 1731–1745, 2006.

Bibliography 199

[37] H. I. Aaronson, G. R. Purdy, D. Malakhov, and W. T. Reynolds, “Tests of the zener

theory of the incomplete transformation phenomenon in Fe-C-Mo and related

alloys,” Scripta Materialia, vol. 44, pp. 2425–2430, 2001.

[38] M. Hillert, “Diffusion in growth of bainite,” Metallurgical and materials transactions

A, vol. 25A, pp. 1957–1966, 1994.

[39] M. Hillert, “Solute drag in grain boundary migration and phase transforma-

tions,” Acta Materialia, vol. 52, pp. 5289–5293, 2004.

[40] G. Rees and H. Bhadeshia, “Bainite transformation kinetics Part 1 Modified

model,” Materials Science and Technology, vol. 8, pp. 985–993, 1992.

[41] F. Caballero, C. Garcia-Mateo, M. Santofimia, M. Miller, and C. Garcıa de Andres,

“New experimental evidence on the incomplete transformation phenomenon in

steel,” Acta Materialia, vol. 57, pp. 8–17, 2009.

[42] I. Loginova, J. Odqvist, G. Amberg, and J. Ågren, “The phase-field approach and

solute drag modeling of the transition to massive austenite to ferrite transforma-

tion in binary Fe-C alloys,” Acta Materialia, vol. 51, pp. 1327–1339, 2003.

[43] M. Hillert, A. Borgenstam, and J. Ågren, “Do bainitic and Widmanstatten ferrite

grow with different mechanisms?,” Scripta Materialia, vol. 62, pp. 75–77, 2010.

[44] F. Caballero, M. Miller, and C. Garcia-Mateo, “Carbon supersaturation of ferrite

in a nanocrystalline bainitic steel,” Acta Materialia, vol. 58, no. 7, pp. 2338–2343,

2010.

[45] C. R. Hutchinson, A. Fuchsmann, and Y. Brechet, “The Diffusional Formation of

Ferrite from Austenite in Fe-C-Ni Alloys,” Metallurgical and Materials Transactions

A, vol. 35, pp. 1211–1221, 2004.

[46] I. Loginova, J. Ågren, and G. Amberg, “On the formation of Widmanstatten

ferrite in binary FeC phase-field approach,” Acta Materialia, vol. 52, pp. 4055–

4063, 2004.

200 Bibliography

[47] Y. Liu, F. Sommer, and E. Mittemeijer, “Abnormal austeniteferrite transformation

behaviour in substitutional Fe-based alloys,” Acta materialia, vol. 51, pp. 507–519,

2003.

[48] H. Guo and G. Purdy, “Scanning Transmission Electron Microscopy Study of In-

terfacial Segregation of Mn during the Formation of Partitioned Grain Boundary

Ferrite in a Fe-C-Mn-Si Alloy,” Metallurgical and Materials Transactions A, vol. 39,

pp. 950–953, 2008.

[49] H. Azizi-Alizamini and M. Militzer, “Phase field modelling of austenite forma-

tion from ultrafine ferrite-carbide aggregates in Fe-C,” International Journal of

Materials Research, vol. 101, pp. 534–541, 2010.

[50] Y. Liu, D. Wang, F. Sommer, and E. Mittemeijer, “Isothermal austeniteferrite

transformation of Fe0.04at.% C alloy: Dilatometric measurement and kinetic

analysis,” Acta Materialia, vol. 56, pp. 3833–3842, 2008.

[51] M. Pernach and M. Pietrzyk, “Numerical solution of the diffusion equation with

moving boundary applied to modelling of the austeniteferrite phase transfor-

mation,” Computational Materials Science, vol. 44, pp. 783–791, 2008.

[52] T. Kop, J. Sietsma, and S. van der Zwaag, “Dilatometric analysis of phase transfor-

mations in hypo-eutectoid steels,” Journal of materials science, vol. 6, pp. 519–526,

2001.

[53] M. Hillert and L. Hoglund, “Mobility of α/γ phase interfaces in Fe alloys,” Scripta


[54] A. Hultgren, “Isothermal transformation of austenite,” Trans ASM, vol. 39,

pp. 915–1005, 1947.

[55] M. Hillert, Introduction to Paraequilibrium. Stockholm: KTH, Sweden, 1953.

Bibliography 201

[56] J. Kirkaldy, “Diffusion in multicomponent metallic systems. II. Solution for two-

phase systems with applications to transformation in Steel,” Can J Phys, vol. 36,

pp. 907–916, 1958.

[57] D. E. Coates, “Diffusion-controlled precipitate growth in ternary systems I,”

Metallurgical Transactions, vol. 3, pp. 1203–1212, 1972.

[58] A. Van der Ven and L. Delaey, “Models for precipitates growth during the austen-

ite to ferrite transformation in Fe-C and Fe-C-M alloys,” Progress in Materials

Science, vol. 40, pp. 181–264, 1996.

[59] M. Mecozzi, J. Sietsma, and S. van der Zwaag, “Phase field modelling of the

interfacial condition at the moving interphase during the austenite to ferrite

transformation in CMn steels,” Computational Materials Science, vol. 34, pp. 290–

297, 2005.

[60] S. E. Offerman, N. H. van Dijk, J. Sietsma, S. Grigull, E. M. Lauridsen, L. Mar-

gulies, H. F. Poulsen, M. T. Rekveldt, and S. van der Zwaag, “Grain nucleation

and growth during phase transformations.,” Science, vol. 298, pp. 1003–1005,

2002.

[61] H. Guo, G. R. Purdy, M. Enomoto, and H. I. Aaronson, “Kinetic Transitions and

Substititional Solute ( Mn ) Fields Associated with Later Stages of Ferrite Growth

in Fe-C-Mn-Si,” Metallurgical and Materials Transactions A, vol. 37A, pp. 1721–

1729, 2006.

[62] J. Odqvist, B. Sundman, and J. Ågren, “A general method for calculating devia-

tion from local equilibrium at phase interfaces,” Acta Materialia, vol. 51, pp. 1035–

1043, 2003.

[63] J. Svoboda, E. Gamsjager, F. Fischer, Y. Liu, and E. Kozeschnik, “Diffusion

processes in a migrating interface: The thick-interface model,” Acta Materialia,

vol. 59, pp. 4775–4786, 2011.

202 Bibliography

[64] Y. Lan, D. Li, and Y. Li, “Modeling austenite decomposition into ferrite at different

cooling rate in low-carbon steel with cellular automaton method,” Acta Materialia,

vol. 52, pp. 1721–1729, 2004.

[65] Y. Liu, F. Sommer, and E. Mittemeijer, “Kinetics of the abnormal austenitefer-

rite transformation behaviour in substitutional Fe-based alloys,” Acta Materialia,

vol. 52, no. 9, pp. 2549–2560, 2004.

[66] H. Chen, Y. Liu, Z. Yan, Y. Li, and L. Zhang, “Consideration of the growth mode

in isochronal austenite-ferrite transformation of ultra-low-carbon Fe–C alloy,”

Applied Physics A, vol. 98, pp. 211–217, 2010.

[67] M. Militzer, M. Mecozzi, J. Sietsma, and S. Vanderzwaag, “Three-dimensional

phase field modelling of the austenite-to-ferrite transformation,” Acta Materialia,

vol. 54, pp. 3961–3972, 2006.

[68] Q. Liu, W. Liu, and X. Xiong, “Correlation of Cu precipitation with austenite-

ferrite transformation in a continuously cooled multicomponent steel: An atom

probe tomography study,” Journal of Materials Research, vol. 27, pp. 1060–1067,

2012.

[69] F. Fazeli and M. Militzer, “Application of solute drag theory to model ferrite

formation in multiphase steels,” Metallurgical and Materials Transactions A, vol. 36,

pp. 1395–1405, 2005.

[70] G. Speich, V. Demarest, and R. Miller, “Formation of austenite during intercritical

annealing of dual-phase steels,” Metallurgical Transactions A, vol. 12A, pp. 1419–

1428, 1981.

[71] V. Savran, Y. Leeuwen, D. Hanlon, C. Kwakernaak, W. Sloof, and J. Sietsma,

“Microstructural Features of Austenite Formation in C35 and C45 alloys,” Met-

allurgical and Materials Transactions A, vol. 38, pp. 946–955, 2007.

Bibliography 203

[72] E. D. Schmidt, E. Damm, and S. Sridhar, “A Study of Diffusion- and Interface-

Controlled Migration of the Austenite/Ferrite Front during Austenitization of a

Case-Hardenable Alloy Steel,” Metallurgical and Materials Transactions A, vol. 38A,

pp. 698–715, 2007.

[73] J. Huang, W. Poole, and M. Militzer, “Austenite formation during intercritical

annealing,” Metallurgical and Materials Transactions A, vol. 35, pp. 3363–3375,

2004.

[74] Z. Li, Z. Yang, C. Zhang, and Z. Liu, “Influence of austenite deformation on

ferrite growth in a FeCMn alloy,” Materials Science and Engineering A, vol. 527,

pp. 4406–4411, 2010.

[75] H. Azizi-Alizamini, M. Militzer, and W. Poole, “Austenite Formation in Plain

Low-Carbon Steels,” Metallurgical and Materials Transactions A, vol. 42A, pp. 1544–

1557, 2011.

[76] R. Wei, M. Enomoto, R. Hadian, H. Zurob, and G. Purdy, “Growth of

austenite from as-quenched martensite during intercritical annealing in an

Fe0.1C3Mn1.5Si alloy,” Acta Materialia, vol. 61, pp. 697–707, 2013.

[77] Y. Xia, M. Enomoto, Z. Yang, Z. Li, and C. Zhang, “Effects of alloying elements

on the kinetics of austenitization from pearlite in FeCM alloys,” Philosophical

Magazine, vol. 93, pp. 1095–1109, 2012.

[78] J. Christian, The theory of transformation in metals and alloys. Oxford-pergamon

press, 1981.

[79] F. Liu, F. Sommer, C. Bos, and E. J. Mittemeijer, “Analysis of solid state phase

transformation kinetics: models and recipes,” International Materials Reviews,

vol. 52, pp. 193–212, 2007.

204 Bibliography

[80] F. Liu and R. Kirchheim, “Nano-scale grain growth inhibited by reducing grain

boundary energy through solute segregation,” Journal of Crystal Growth, vol. 264,

no. 1-3, pp. 385–391, 2004.

[81] T. A. Kop, Y. V. Leeuwen, J. Sietsma, and S. van der Zwaag, “Modelling the

Austenite to Ferrite Phase Transformation in Low carbon steels in terms of the

interface mobility,” ISIJ International, vol. 40, pp. 713–718, 2000.

[82] M. Mecozzi, J. Sietsma, and S. van der Zwaag, “Analysis of austenite to ferrite

transformation in a Nb micro-alloyed CMn steel by phase field modelling,” Acta


[83] G. P. Krielaart and S. van der Zwaag, “Kinetics of austenite-ferrite transformation

in Fe-Mn alloys containing low manganese,” vol. 14, pp. 10–18, 1998.

[84] J. Wits, T. Kop, Y. van Leeuwen, J. Sietsma, and S. van der Zwaag, “A study on

the austenite-to-ferrite phase transformation in binary substitutional iron alloys,”

Materials Science and Engineering: A, vol. 283, pp. 234–241, 2000.

[85] S. Vooijs, Y. V. Leeuwen, J. Sietsma, and S. van der Zwaag, “On the mobility of

the austenite-ferrite interface in Fe-Co and Fe-Cu,” Metallurgical and Materials

Transactions A, vol. 31A, pp. 379–385, 2000.

[86] A. Kempen, F. Sommer, and E. Mittemeijer, “The kinetics of the austeniteferrite

phase transformation of Fe-Mn: differential thermal analysis during cooling,”

Acta Materialia, vol. 50, pp. 3545–3555, 2002.

[87] H. Song and J. Hoyt, “A molecular dynamics simulation study of the velocities,

mobility and activation energy of an austenite–ferrite interface in pure Fe,” Acta

Materialia, vol. 60, no. 10, pp. 4328–4335, 2012.

[88] H. Wang, F. Liu, T. Zhang, G. Yang, and Y. Zhou, “Kinetics of diffusion-controlled

transformations: Application of probability calculation,” Acta Materialia, vol. 57,

pp. 3072–3083, 2009.

Bibliography 205

[89] K. Fan, F. Liu, X. Liu, Y. Zhang, G. Yang, and Y. Zhou, “Modeling of isothermal

solid-state precipitation using an analytical treatment of soft impingement,” Acta


[90] J.-O. Andersson, T. Helander, L. Hoglund, P. F. Shi, and B. Sundman, “THERMO-

CALC and DICTRA, Computational Tools For Materials Science,” Calphad,

vol. 26, pp. 273–312, 2002.

[91] C. R. Hutchinson, H. S. Zurob, and Y. Brechet, “The Growth of Ferrite in Fe-C-X

Alloys : The Role of Thermodynamics , Diffusion , and Interfacial Conditions,”


[92] K. Oi, C. Lux, and G. Purdy, “A study of the influence of Mn and Ni on the

kinetics of the proeutectoid ferrite reaction in steels,” Acta Materialia, vol. 48,

pp. 2147–2155, 2000.

[93] L. Taleb, N. Cavallo, and F. Waeckel, “Experimental analysis of transformation

plasticity,” International Journal of Plasticity, vol. 17, pp. 1–20, 2001.

[94] L. Taleb and S. Petit, “New investigations on transformation induced plasticity

and its interaction with classical plasticity,” International Journal of Plasticity,

vol. 22, pp. 110–130, 2006.

[95] H. Han and D. Suh, “A model for transformation plasticity during bainite trans-

formation of steel under external stress,” Acta Materialia, vol. 51, pp. 4907–4917,

2003.

[96] A. Borgenstam, L. Hoglund, J. Ågren, and A. Engstrom, “DICTRA, a tool for

simulation of diffusional transformations in alloys,” Journal of Phase Equilibria,

vol. 21, pp. 269–280, 2000.

[97] H. Chen, B. Appolaire, and S. van der Zwaag, “Application of cyclic partial

phase transformations for identifying kinetic transitions during solid-state phase

206 Bibliography

transformations: Experiments and modeling,” Acta Materialia, vol. 59, pp. 6751–

6760, 2011.

[98] H. Chen and S. van der Zwaag, “Application of the cyclic phase transformation

concept for investigating growth kinetics of solid-state partitioning phase trans-

formations,” Computational Materials Science, vol. 49, no. 4, pp. 801–813, 2010.

[99] H. Chen and S. van der Zwaag, “Indirect evidence for the existence of the Mn

partitioning spike during the austenite to ferrite transformation,” Philosophical

Magazine Letters, vol. 2, pp. 86–92, 2012.

[100] O. Dmitrieva, D. Ponge, G. Inden, J. Millan, P. Choi, J. Sietsma, and D. Raabe,

“Chemical gradients across phase boundaries between martensite and austenite

in steel studied by atom probe tomography and simulation,” Acta Materialia,

vol. 59, pp. 364–374, 2011.

[101] L. Yuan, D. Ponge, J. Wittig, P. Choi, J. Jimenez, and D. Raabe, “Nanoscale austen-

ite reversion through partitioning, segregation and kinetic freezing: Example of

a ductile 2GPa FeCrC steel,” Acta Materialia, vol. 60, pp. 2790–2804, 2012.

[102] J. B. Gilmour, G. R. Purdy, and J. S. Kirkaldy, “Partition of Manganese During the

Proeutectoid Ferrite Transformation in Steel,” Metallurgical Transactions, vol. 3,

pp. 3213–3222, 1972.

[103] H. Chen, M. Goune, and S. van der Zwaag, “Analysis of the stagnant stage

in diffusional phase transformations starting from austeniteferrite mixtures,”

Computational Materials Science, vol. 55, pp. 34–43, 2012.

[104] Minski M, “Laser Scanning Confocal Microscopy,” US Patent No. 3013467, 1961.

[105] J. Cho, H. Shibata, M. Suzuki, and T. Emi, “Bull Inst Adv Mater Process,” Tohoku

Univ (Japan), vol. 53, p. 47, 1997.

Bibliography 207

[106] E. Schmidt, D. Soltesz, S. Roberts, A. Bednar, and S. Sridhar, “The Austen-

ite/Ferrite Front Migration Rate during Heating of IF Steel,” ISIJ International,

vol. 46, pp. 1500–1509, 2006.

[107] M. Reid, D. Phelan, and R. Dippenaar, “Concentric Solidification for High Tem-

perature Laser Scanning Confocal Microscopy,” ISIJ International, vol. 44, pp. 565–

572, 2004.

[108] H. Chen and S. van der Zwaag, “The effect of alloying element on stagnant

stage,” To be submitted, 2013.

[109] I. a. Yakubtsov and G. R. Purdy, “Analyses of Transformation Kinetics of Carbide-

Free Bainite Above and Below the Athermal Martensite-Start Temperature,”


[110] G. Purdy, “Bainite: defect signatures,” Scripta Materialia, vol. 47, pp. 181–185,

2002.

[111] H. Guo, X. Gao, Y. Bai, M. Enomoto, S. Yang, and X. He, “Variant selection of

bainite on the surface of allotriomorphic ferrite in a low carbon steel,” Materials

Characterization, vol. 67, pp. 34–40, May 2012.

[112] M. Enomoto, “Partition of carbon and alloying elements during the growth of

ferrous bainite,” Scripta Materialia, vol. 47, pp. 145–149, 2002.

[113] D. Quidort and Y. Brechet, “The role of carbon on the kinetics of bainite trans-

formation in steels,” Scripta materialia, vol. 47, pp. 151–156, 2002.

[114] N. Luzginova, L. Zhao, and J. Sietsma, “Bainite formation kinetics in high carbon

alloyed steel,” Materials Science and Engineering: A, vol. 481-482, pp. 766–769,

2008.

[115] K. M. Wu, M. Kagayama, and M. Enomoto, “Kinetics of ferrite transformation in

an Fe-0 . 28 C-3 Mo alloy,” Materials Science and Engineering A, vol. 343, pp. 143–

150, 2003.

208 Bibliography

[116] M. Hillert, L. Hoglund, and J. Ågren, “Role of carbon and alloying elements

in the formation of bainitic ferrite,” Metallurgical and Materials Transactions A,

vol. 35A, pp. 3693–3700, 2004.

[117] H. Chen, W. Xu, M. Goune, and S. van der Zwaag, “Application of the stagnant

stage concept for monitoring Mn partitioning at the austenite-ferrite interface in

the intercritical region for FeMnC alloys,” Philosophical Magazine Letters, vol. 92,

no. 10, pp. 547–555, 2012.

[118] K. Lucke and K. Detert, “A quantitative theory of grain boundary motion and

recrystalization in metals in the presence of impurities,” Acta Metallurgica, vol. 5,

pp. 628–637, 1957.

[119] Cahn JW, “The impurity-drag effect in grain boundary motion,” Acta Metallur-

gica, vol. 10, pp. 789–798, 1962.

[120] G. Purdy and Y. Brechet, “A solute drag treatment of the effects of alloying

elements on the rate of the proeutectoid ferrite transformation in steels,” Acta

Metallurgica et Materialia, vol. 43, pp. 3763–3774, 1995.

[121] M. Hillert and B. Sundman, “A treatment of the solute drag on moving grain

boundaries and phase interfaces in binary alloys,” Acta Metallurgica, vol. 24,

pp. 731–743, 1976.

[122] A. Beche, H. Zurob, and C. Hutchinson, “Quantifying the Solute Drag Effect of Cr

on Ferrite Growth Using Controlled Decarburization Experiments,” Metallurgical

and Materials Transactions A, vol. 38, pp. 2950–2955, 2007.

[123] M. Enomoto, “Influence of solute drag on the growth of proeutectoid ferrite in

Fe–C–Mn alloy,” Acta Materialia, vol. 47, pp. 3533–3540, 1999.

[124] H. I. Aaronson, W. T. Reynolds, and G. R. Purdy, “Coupled-solute drag effects

on ferrite formation in Fe-C-X systems,” Metallurgical and Materials Transactions

A, vol. 35, pp. 1187–1210, 2004.

Bibliography 209

[125] M. Hillert and M. Schalin, “Modeling of solute drag in the massive phase trans-

formation,” Acta Materialia, vol. 48, no. 2, pp. 461–468, 2000.

[126] A. Borgenstam and M. Hillert, “Massive transformation in the Fe–Ni system,”


[127] J. Fridberg, L. Torndahl, and M. Hillert, “diffusion in iron,” jernkont Ann, vol. 153,

pp. 263–276, 1969.

[128] M. Enomoto, C. White, and H. Aaronson, “Evaluation of the effects of segregation

on austenite grain boundary energy in Fe–C–X alloys,” Metallurgical and Materials

Transactions A, vol. 19, pp. 1807–1817, 1988.

[129] M. Hillert, “Overview: Solute drag, solute trapping and diffusional dissipation

of Gibbs energy,” Acta Materialia, vol. 47, pp. 4481–4505, 1999.

[130] O. Bouaziz, P. Maugis, and J. Embury, “Bainite tip radius prediction by analogy

with indentation,” Scripta Materialia, vol. 54, no. 8, pp. 1527–1529, 2006.

[131] H. Luo, J. Shi, C. Wang, W. Cao, X. Sun, and H. Dong, “Experimental and numer-

ical analysis on formation of stable austenite during the intercritical annealing

of 5Mn steel,” Acta Materialia, vol. 59, pp. 4002–4014, 2011.

[132] G. Sheng and Z. Yang, “Transition between partitioned and unpartitioned growth

of proeutectoid ferrite in FeCXi systems,” Materials Science and Engineering: A,

vol. 465, no. 1-2, pp. 38–43, 2007.

[133] W. Reynolds, F. Li, C. Shui, and H. Aaronson, “The incomplete transforma-

tion phenomenon in Fe-C-Mo alloys,” Metallurgical and Materials Transactions A,

vol. 21, pp. 1433–1463, 1990.

[134] M. Santofimia, C. Kwakernaak, W. Sloof, L. Zhao, and J. Sietsma, “Experimental

study of the distribution of alloying elements after the formation of epitaxial

ferrite upon cooling in a low-carbon steel,” Materials Characterization, vol. 61,

pp. 937–942, 2010.

210 BIBLIOGRAPHY

[135] M. Gomez, C. I. Garcia, and A. J. Deardo, “The Role of New Ferrite on Retained

Austenite Stabilization in Al-TRIP Steels,” ISIJ International, vol. 50, no. 1, pp. 139–

146, 2010.

[136] E. Ahmad, M. Sarwar, T. Manzoor, and N. Hussain, “Effect of rolling and epitaxial

ferrite on the tensile properties of low alloy steel,” Journal of Materials Science,

vol. 41, no. 17, pp. 5417–5423, 2006.

[137] S. Offerman, N. van Dijk, J. Sietsma, E. Lauridsen, L. Margulies, S. Grigull,

H. Poulsen, and S. van der Zwaag, “Solid-state phase transformations involving

solute partitioning: modeling and measuring on the level of individual grains,”


[138] D. Crespo, T. Pradell, M. T. Clavaguera-Mora, and N. Clavaguera, “Mi-

crostructural evaluation of primary crystallization with diffusion-controlled

grain growth,” Physical Review B, vol. 55, pp. 3435–3444, 1997.

[139] C. Bos and J. Sietsma, “A mixed-mode model for partitioning phase transforma-

tions,” Scripta Materialia, vol. 57, pp. 1085–1088, 2007.

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.