Cyclic Partial Phase Transformations In Low Alloyed Steels: Modeling and Experiments

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Cyclic Partial Phase

Transformations 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

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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. Gamsj¨ager Leoben University, Austria

Prof. dr. Z. G. Yang Tsinghua University, China Prof. dr. ir. E. Br ¨uck 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

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

isbn978-94-6191-771-3

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Chapter

1

Introduction

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 A3is 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

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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 transformnucle-ation starts by the nuclenucle-ation of the new phase,

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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.

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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 Chapter2, 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 Chapter4, 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 Chapter5to 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. Chapter6presents the in-situ observations of interface migration during the cyclic phase transformations in a Fe-C-Mn alloy.

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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.

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Chapter

2

The 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].

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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,

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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.2.1is 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 T1and T2, thus the

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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.2.2will 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

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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 phase

transformation 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

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Figure 2.3: The schematic alloying element concentration profiles of (a) the ferrite to austenite and (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 transfortransfor-mation during the diffusion-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 transfortransfor-mation 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

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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→γand Mγ→αare the interface mobility of the ferrite to austenite

transfor-mation and the austenite to ferrite transfortransfor-mation. ∆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

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∆G(cα₎_{= χ} α→γ cα−_cα eq(T2) (2.4) ∆G(cγ₎_{= χ} γ→α cγeq(T1) − cγ (2.5)

Whereχα→γandχγ→αare 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

Ji_α→γ = vα→γ cγeq(T2) − cα (2.6) Ji_γ→α = vγ→α cγ−_cα eq(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

J_α→γi = Dα∂cα ∂z −Dγ ∂cγ ∂z (2.8) J_γ→αi = Dα∂cα ∂z −Dγ ∂cγ ∂z (2.9)

In this work, an approximation is made for the concentration gradient ∂cα

∂z and ∂c∂zγ

at the interface, yielding

∂cα ∂z = cα₁ −_cα 3 2∆xα (2.10) ∂cγ ∂z = cγ₁ −_cγ 3 2∆xγ (2.11)

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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.2.1and2.2.

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 50µm, 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

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Table 2.1: Physical parameters used in the calculation. Parameter 1050K 1100K χγ→α, J/(at.%) 110.0 NA χα→γ, J/(at.%) NA 765 D_γ, 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 0.5 QG, KJ/mol 140.0 140.0 (a) (b)

Figure 2.4: The austenite fraction during (a) the ferrite to austenite transformation at 1100K and (b) the austenite to ferrite transformation at 1050K calculated by the mixed-mode model and 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,

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(a)

(b)

(c)

(d)

Figure 2.5: The evolution of carbon profile in (a) the austenite and (b) ferrite phase during the cyclic austenite to ferrite transformation at 1050K and in (c) the austenite and (d) ferrite phase during the cyclic ferrite to austenite transformation at 1100K according to the mixed mode model

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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 difdif-fusion-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

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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 Cγis 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. Cγis strongly dependent on the ratio between interface mobility and diffusion coefficient M/D .

Fixing the diffusion coefficient and varying the pre-exponential factor M0of

inter-face mobility, the carbon concentration at the interinter-face 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 Cγbecome 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

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(a)

(b)

Figure 2.6: (a) Carbon concentration at the interface in the austenite phase Cγduring the cyclic austenite to ferrite transformation and (b) carbon concentration at the interface in the ferrite phase Cαduring 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 Cαas 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

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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 = Cγeq−Cγ Cγeq−Cαeq (2.12) Hα→γ = Cα−_Cα eq Cγeq−Cαeq (2.13)

When Cγ= Cαeqfor the cyclic austenite to ferrite transformation and Cα = Cγeqfor 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

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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 to austenite 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 definitransforma-tion decrease as the phase transformatransforma-tion 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.

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Figure 2.8: The ratio of the ferrite to austenite transformation rate and the austenite to ferrite transformation rate (d f_{(d f}/dt)_/dt)α→γ

γ→α in the initial stage as a function of M

∗ 0, assuming M γ→α 0 = M α→γ 0 = M∗₀ In Fig.2.8, assuming Mγ→α₀ = Mα→γ₀ = M∗

0, the ratio of the ferrite to austenite

trans-formation rate and the austenite to ferrite transtrans-formation rate(d f_{(d f}/dt)_/dt)α→γ_γ→αin 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 transfortransfor-mation. 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∗

0increases 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_{(d f}/dt)_/dt)α→γ_γ→αin the initial stage is calculated as a function of M

α→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 ratio M

α→γ 0

Mγ→α₀ increases until

the ferrite to austenite transformation is completely diffusion-controlled. There is a certain value (less than 1.0) of M

α→γ 0

Mγ→α₀ at which the ferrite to austenite transformation is as

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Figure 2.9: The ratio of the ferrite to austenite transformation rate and the austenite to ferrite transformation rate (d f_{(d f}/dt)_/dt)α→γ

γ→α in the initial stage as a function of

Mα→γ₀ 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→γ0

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

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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 impingeimpinge-ment 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.

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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.

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Chapter

3

The kinetics of cyclic phase

transformations in a lean Fe-C-Mn

alloy

This chapter is based on

• H Chen, B Appolaire, S van der Zwaag, Application of cyclic partial phase transformations for identifying kinetic transitions during solid-state phase transformations: Experiments and Modeling , Acta Mater, 2011; 59:6751-6760.

3.1 Introduction

As discussed in Chapter2, 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

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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 Chapter2, 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 theγ+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 theγ → 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 B¨ahr 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

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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 experitreat-ments 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 1000◦

C 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 T1and 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 T1and T2are located in theα + γ two-phase field in the phase

diagram. The cooling rate and heating rate during cycling in both I and H experiments were 10◦_{C/min. Typically, 3-5 temperature cycles were imposed per experiment. The}

experiment conditions are summarized in Tab. 3.1. The reported feqγ(T1) and feqγ(T2)

values in Tab.3.1 are the Thermocalc calculated equilibrium fractions of austenite at T1and T2, respectively. time austenitization A3 A1 temper a tur e T2 T1 time austenitization A3 A1 temper a tur e T2 T1 (a) (b)

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Table 3.1:The cyclic experiments conditions (A3= 896◦C and A1 = 729◦C).

Experiments T1(◦C) feqγ(T1) T2(◦C) feqγ(T2) Mode Figure

1 860 20% 885 57.5% I 3.2a 2 860 20% 895 97% I 3.2b 3 870 28.3% 895 97% I 3.2c 4 885 57.5% 895 97% I 3.2d 5 860 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:

∂xφ_i ∂t = ∇ · Dφ_i ∇_xφ i (3.1)

whereφ stands for α or γ, and i for C or M. xφ_i is the mole fraction of species i in phase φ, and Dφ_i 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:

µγ_i = µα

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3.3. Models 31

whereµφ_i is the chemical potential of element i at the interface in phaseφ. Moreover, mass balances must be satisfied for both C and M at the interface:

Jγ_i −_Jα

i = v (x γ∗

i −xα∗i ) (3.3)

where xφ∗_i is the interface concentration of i in phaseφ, Jφ_i 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 amongα and γ at the interface, whereas the chemical potential of C remains constant across the interface. This constrained equilibrium is expressed as:

µγ_C = µα C (3.4) (µγ_M−µα M)= − xFe xM(µ γ Fe−µαFe) (3.5)

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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 Length change / µ m

Type I−cycling between 885°c and 860°c A 1 A 2 A 7 A 6 A 5 A 4 A 3 A 8 855 860 865 870 875 880 885 890 895 900 105 110 115 120 125 Temperature /°c Length change / µ m

Type I−cycling between 895°c and 860°c A 8 A 1 A 2 A 7 A 6 A 3 A 4 A 5 855 860 865 870 875 880 885 890 895 900 110 112 114 116 118 120 122 Temperature /°c Length change / µ m

Type I−cycling between 895°c and 870°c

A 3 A 5 A 4 A 7 A 8 A 6 A 1 A 2 885 890 895 114 115 116 117 118 119 120 Temperature /°c Length change / µ m

Cycling between 895 °c and 885°c A 10 A 9 A 6 A 2 A 1 A 3 A₈ A 5 A 4 A 7 (a) (b) (c) (d)

Figure 3.2:The dilation as a function of temperature during type I cyclic experiments between (a) 885◦C and 860◦C, (b) 895◦C and 860◦C, (c) 895◦C and 870◦C, (d) 895◦C and 885◦C.

Fig. 3.2a shows the dilation as a function of temperature during the type I cyclic experiment between 860◦

C and 885◦

C (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-A2in 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 885◦

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3.4. Experimental results 33

stages can now be observed during the cooling down process: (i) a nonlinear contrac-tion (A3-A4stage) due to a continuation of theα → 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 theγ → 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 860◦C and 895◦C (experiment 2, Tab. 3.1). In this case the upper temperature of 895◦C is close to the A3temperature. 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 870◦

C. 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 895◦

C and 885◦

C (experi-ment 4, Tab.3.1) (later to be followed by full cyclic annealing between 895◦C and 860◦C) , 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

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in the inverseα → γ transformation stage is decreasing slightly with the number of annealing cycles. The kinetics of the subsequent cyclic annealing between 895◦C and 860◦C (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 Length change / µ m

Type H−cycling between 885°c and 860°c

B 3 B 4 B 5 B 6 B 1 B 2

Figure 3.3:The dilation as a function of temperature during type H cyclic experiments between 885◦C and 860◦C.

Fig. 3.3 shows the length change as a function of temperature during a type H cyclic experiment between 885◦

C and 860◦

C (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 885◦

C and 860◦

C 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

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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 860◦C 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 860◦

C occurs via local equilibrium with negligible partitioning (LE-NP), in agreement with the fact

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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 bothα and γ phase.

8558 860 865 870 875 880 885 890 895 900 10 12 14 16 18 20 22 Temperature/ °c interface position/ µ m c₃ c 5 PE LE c 2 c 4 c 1 c 6 8558 860 865 870 875 880 885 890 895 900 10 12 14 16 18 20 22 Temperature /°c interface position / µ m LE PE D₂ D₆ D 1 D 5 _D 4 D 3 (a) (b)

Figure 3.4: Theα/γ interface position as a function of temperature during (a) the type I and (b) the type H cyclic phase transformations between 885◦C and 860◦C simulated under both local equilibrium and paraequilibrium conditions.

In Fig. 3.4, theα/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 885◦

C and 860◦

C. 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

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3.5. Simulation results 37

accordingly by the C3-C4stage. In Fig.3.4b, the calculations related to the type H cycles

between 885◦C and 860◦C 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-B2and B4-B5(Fig.3.3). Moreover, no

inverseα → γ transformation shows up in the simulation, which is also in agreement with the experiments.

0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 Distance/ µm Carbon concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c 3 c 5 c 6 c 1 c 2 c 4 0 5 10 15 20 25 0.08 0.1 0.15 0.2 0.25 0.3 0.32 Distance/ µm Mn concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c2 c 6 c 1 c 3 c 4 c 5 (a) (b)

Figure 3.5: The evolution of C and Mn profiles in the C1-C3stage during type I cyclic phase

transformations between 885◦C and 860◦C predicted by the local equilibrium model. The points for which the C and Mn profiles are shown are indicated with the same type of symbol and 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

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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 bothα 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 inα (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 inα. At a certain temperature, the Mn concentration at the interface in the growingγ becomes equal to the Mn concentration in bulk α. At the same time, a positive C gradient is building up inγ. 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 inγ 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.

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3.5. Simulation results 39 0 5 10 15 20 25 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Distance/ µm Carbon concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c6 c 3 c 4 c 5 c 1 c₂ 0 5 10 15 20 25 0.08 0.1 0.15 0.2 0.25 0.3 0.32 Distance/ µm Mn concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c 6 c 3 c 4 c₅ c 2 c₁ (a) (b)

transformations between 885◦C and 860◦C predicted by the local equilibrium model. The points for which the C and Mn profiles are shown are indicated with the same type of symbol and color in the interface position curve correspondingly.

After the inverse α → γ stage, the interface remains almost immobile: this is quite comparable with the stagnant stage in the experiments. During the C4-C5 stage, there

is no carbon gradient in bothα and γ, but there are still Mn gradients in both α and γ with the zigzag shape already observed in the C3-C4stage: 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

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diminishes, with an inversion of the Mn gradient inα at the interface (green and red curves in Fig.3.7b). 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 Distance/ µm Carbon concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c 1 c 6 c 2 c₃ c 4 c 5 0 5 10 15 20 25 0.08 0.1 0.15 0.2 0.25 0.3 0.32 Distance/ µm Mn concentration /wt.% 850 860 870 880 890 12 14 16 18 20 Temperature / °c interface position / µ m c₆ c₃ c 5 c₄ c 2 c₁ (a) (b)

transformations between 885◦

C and 860◦

C predicted by the local equilibrium model. The points for which the C and Mn profiles are shown are indicated with the same type of symbol and color in the interface position curve correspondingly.

At the very beginning of the C5-C6 stage, the Mn concentration at the interface in

the growingα 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 theγ → α transformation,

Cyclic Partial Phase Transformations In Low Alloyed Steels: Modeling and Experiments