Phase field modelling of the austenite to ferrite transformation in steels

Phase Field Modelling of the

Austenite to Ferrite

The research described in this thesis was performed in the department of Material Science and Technology, the Delft University of Technology.

Phase Field Modelling of the

Austenite to Ferrite

Transformation in Steels

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 8 januari 2007 om 15.00 uur

door

Maria Giuseppina MECOZZI

Dottore in Fisica

Prof. dr. Ir. S. Van der Zwaag

Toegevoegd promotor: Dr. ir. J. Sietsma Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof dr. ir. S. van der Zwaag, Technische Universiteit Delft, promotor Dr. ir. J. Sietsma, Technische Universiteit Delft, toegevoegd promotor Prof. dr. A.A. Howe, University of Sheffield, Sheffield, UK

Prof. dr. I. M. Richardson, Technische Universiteit Delft

Prof. dr. M. Militzer, University of British Columbia, Vancouver, Canada Prof. dr. ir. L.J. Sluijs, Technische Universiteit Delft

Prof. ir. L. Katgerman , Technische Universiteit Delft

ISBN-10: 90-77172-26-2 ISBN-13: 978-90-77172-26-1

Keywords: low-carbon steel, ferrite growth kinetics, phase field model, 2D and 3D microstructure simulation

Copyright  2006 by M.G. Mecozzi

All right reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanics, including photocopying, recording or by any information storage and retrieval system, without permission from the author

Contents

1. General introduction 1

1.1 The austenite to ferrite transformation kinetics 1

1.2 This thesis 4

References 8

2 Phase field theory 11

2.1 Introduction 12

2.2 Phase field equations 14

2.2.1 Determination of the phase field parameters in term of

physical parameters 19

2.3 Diffusion equations 23

2.4 Driving force calculation 26

2.4.1 Ortho-equilibrium 26

2.4.2 Para-equilibrium 28

2.5 Summary 30

References 30

3 Analysis of the austenite to ferrite transformation in a

C-Mn steel by phase field modelling 33

3.1 Introduction 34 3.2 Experimental procedure 35 3.3 Simulation conditions 36 3.4 Results 39 3.5 Discussion 46 3.6 Conclusions 49 References 50

4 Analysis of austenite to ferrite transformation in a

Nb micro-alloyed C-Mn steel by phase field modelling 53

4.1 Introduction 54

4.2 Materials 55

4.3 Simulation conditions 56

4.4 Experimental procedure 59

4.5 Results and discussion 59

4.6 Conclusions 71

transformation 73

5.1 Introduction 74

5.2 Simulation conditions 76

5.3 Results 79

5.3.1 Nucleation on triple lines 79

5.3.2 Nucleation on triple lines and grain surfaces 84

5.4 Comparison with 2D simulations 88

5.4.1 Transformation kinetics 88

5.4.2 Predicted microstructures 91

5.5 Conclusions 95

References 96

6 The effect of nucleation behaviour in phase field simulations

of the austenite to ferrite transformation 99

6.1 Introduction 100 6.2 Simulation conditions 102 6.3 Results 106 6.4 Discussion 111 6.5 Conclusions 118 References 118

7 The mixed mode character of the austenite to ferrite

transformation kinetics in phase field simulations 121

7.1 Introduction 122

7.2 simulation conditions 124

7.3 Evolution of the character of the transformation 127

7.4 Results 129

7.4.1 Transformation kinetics 129

7.4.2 Carbon distribution and soft impingement 132

7.4.3 Mixed mode character of the transformation kinetics 136

Chapter 1

General introduction

1.1 The austenite to ferrite transformation kinetics

The properties of steel strongly depend on its composition and microstructure. For a given steel chemistry, different steel microstructures may be produced in relation to specific thermal or thermo-mechanical treatment imposed during rolling, subsequent controlled cooling and coiling [1-2].

In C-Mn steel the face-centered cubic (fcc) austenite (γ) is the stable phase during annealing at high temperature; the temperature above which this phase is stable depends on the steel chemistry and varies for common steel grades between 1000 K and 1185 K. Upon cooling, γ phase transforms in different stable or metastable phases, again depending on the steel chemistry and on the cooling conditions. This explains why the quantitative understanding of the kinetics of the γ decomposition has been an important goal of many industrial and academic steel investigations for many years [3-4].

The body centered cubic (bcc) ferrite (α) phase is the first reaction product formed upon cooling from γ and therefore the γ-to-α transformation process has been extensively investigated; the specific attention towards this transformation has been also favored by the relative simplicity of the α product in comparison with the other stable and metastable phases or phase mixtures like pearlite, bainite and martensite. Nevertheless, a coherent, physically based description of the α growth from γ remains elusive.

modelled assuming that the carbon diffusion in austenite is the rate controlling process [5-7]. The interfacial reaction, which transforms the fcc lattice into the bcc lattice, is supposed to be fast enough not to affect the transformation kinetics; in this condition the C concentrations in α and γ at the interface at any time during transformation are equal to the equilibrium concentrations (local equilibrium) and the transformation is then said to be

diffusion-controlled. The other extreme is to assume that the transformation kinetics is

controlled by the rate of lattice transformation, and the carbon diffusivity in γ and α is fast enough to maintain a homogeneous carbon concentration in each phase: Christian [8] first introduced the concept of interface-controlled kinetics by expressing the interface velocity,

v, as the product of the driving pressure for the transformation, ∆Gγ/α, and the interface mobility, µ., which gives a measure for the mobility of lattice atoms at the interface. More recently the interface-controlled model was used to describe the γ to α transformation kinetics in different cooling conditions, also including ultra fast-cooling [9-11].

In reality both the long-range carbon diffusion in γ and the lattice transformation at the interface influence the transformation kinetics, which therefore has a mixed-mode character [12-14]. The interfacial conditions, i.e. the carbon content at the interface and the interface velocity, strongly depend on the nature of the phase transformation with respect to diffusion-controlled or the interface-controlled mode.

In the mixed-mode approach the interface velocity, v, can be formulated as the product of the intrinsic interface mobility, µ, and the driving pressure for the interface migration, calculated from the Gibbs free energy difference between γ and α across the interface,

(

C, C

)

Gαγ x x_{γ α}

∆ , calculated from the transient local carbon composition at the interface in the α and γ side, _xC

α andxγC. This is expressed by

(

C, C

)

v= ∆µ Gαγ x x_{γ α} (1.1)

Unlike the diffusion-controlled model, the carbon concentrations in γ and α at the interface, _xC

α and xγC, change with time during transformation [12], due to the finite interface mobility and the non-zero net carbon flux at the interface. For example _xC

γ change continuously from the initial carbon concentration, _{x and the equilibrium carbon 0}C

concentration in γ, _xC eq

γ .

substitutional element, like Mn. The addition of Mn to the binary Fe-C system makes the identification of the actual interface conditions non-trivial, even if the transformation kinetics is assumed to be diffusion controlled with local equilibrium for carbon at the interface. As results of the large difference in the diffusivities of C and Mn in γ at the temperature of interest (_DMn_/DC ₁₀ 6

γ γ ≈ − ), it is usually not possible to simultaneously satisfy the mass balance for C and Mn at the interface with a tie-line passing through the alloy composition. Even if a large free energy decrease would result from the complete equilibrium partitioning of Mn, due to the larger Mn solubility in γ than in α, the possibility exists that the γ-to-α transformation proceeds without Mn partitioning in the bulk of the parent and newly formed phase. Therefore, as an alternative to the ortho-equilibrium condition, where all the solute atoms redistribute according to ortho-equilibrium between the parent and new formed phase, different constrained equilibria have been defined for Mn, i.e. local equilibrium with negligible partitioning (LENP) [15-16] or para-equilibrium (PE) [17-18].

If a finite interface mobility is assumed, the effect of Mn segregation at the moving interface on reducing the effective driving pressure of the transformation (solute-drag) has to be also considered [19-21]. If the transformation model does not taken into account quantitatively the effect of solute drag, a different temperature dependence of the interface mobility than that expected for the intrinsic interface mobility will result.

In order to be able to translate the interface velocity evaluated in kinetics models into a measurable quantity, such as the ferrite fraction evaluated by dilatometry, the geometry of the parent and the newly formed phase has to be incorporated in the model. The simplest approach was that of Vandermeer [7] who considered the austenite grain as a sphere and the ferrite to nucleate uniformly along the outer surface. In this model the final ferrite grain size is intrinsically identical to the prior austenite grain size. A more refined model is the tetrakaidecahedron model in which the austenite grain is assumed to be a tetrakaidecahedron. This approach allows incorporation of the ferrite nucleation site density per austenite grain as a model parameter to reproduce the grain size depending on the cooling conditions [9].

The most elaborated model for modelling the austenite decomposition available at the moment is the phase field approach. Based on the construction of a Ginzburg-Landau free energy functional, the phase field model treats a multi-phase system, containing both bulk and interface regions, in an integral manner. One or more continuous field variables,

( )

,

i r t

t, and at each point, r, the different domains present in the system. Typically these field

variables have a constant value in the bulk regions and change continuously over a diffuse-interface of thickness η. Phase field models were originally proposed to simulate dendritic growth in pure undercooled melts and have been meanwhile successfully applied to describe solidification in alloys [22-26]. Initial applications of phase field modelling to the solid-state austenite-to-ferrite transformation have been reported only more recently [27-33]. Phase field models provide a powerful methodology to describe phase transformations. This technique can easily handle time-dependent growth geometries, and thus enables the prediction of complex microstructure morphologies. Since both the interface mobility and the carbon diffusion are incorporated in the phase field modelling of solid-state transformations, the phase field approach has to be considered as an example of mixed-mode model. It can incorporate strain effects for solid-state transformations and can account for solute drag and trapping by means of the interface mobility acting as a model parameter. A critical issue in phase field model is the treatment of the interface region. Many phase-field models are based on the classical phase-field approach proposed by Wheeler, Boettinger and McFadden [23]. Alternatively, Steinbach et al. propose their multi-domain model with a different definition of the free energy density in the interface [24-25] and different assumptions about alloy composition within the diffuse interface.

1.2 This

thesis

The details of the multiphase field model, specifically formulated for a dual phase (γ and α) polycrystalline system are described in Chapter 2. The time evolution of a set of continuous field variables, each representing a particular domain (grain) of the system, determines the microstructural evolution of the multi-domain (polycrystalline) system. A set of phase-field equations is solved coupled to a number of diffusion equations equal to the number of diffusive species present in the system. In C-Mn steel, since the diffusivity of Mn atoms is much lower than that of C atoms, it is reasonable to assume that Mn does not diffuse in the bulk of α and γ phase during transformation and then the phase field equations are coupled only with the carbon diffusion equation. When a new ferrite grain is formed, the time evolution of the order parameter representing the new grain, φ_i

( )

r t, , is described as the sum of the pairwise interaction with the order parameters φ_j

( )

r t, , representing all the neighbouring grains. The interface mobilities and interfacial energies and the driving pressure for the transformation are parameters of the phase field equations and they determine the kinetics of the microstructural evolution. The driving pressure of the transformation depends on the local carbon and manganese content within the diffuse interface. In the phase field model used the local concentration _{x of element k, with k} k

equal to C or Mn, becomes a continuous variable in r through the interface and it is built

up from the austenite and ferrite composition, _xk

γ and xγk by

(

1

)

k k k

i i

x =φxα + −φ xγ (1.2)

A distinctive feature of the model used here is that the_xk /_xk

α γ ratio is constant within the interface, given by the local equilibrium ratio. In this thesis “local equilibrium” simply indicates that equilibrium is imposed at the interface under either ortho-equilibrium (equilibrium for all components) or para-equilibrium (equilibrium only for the fast diffusing species). Although the code used for the solution of the phase field and diffusion equations allows the derivation of the driving pressure for a specific interface composition through the coupling with the Thermo-calc software, in this thesis a linearisation of the phase diagram is used to derive the driving pressure of the transformation under the assumption that it is proportional to the local undercooling at the interface. Using this approach, different constrained equilibria for Mn can be set for the driving pressure calculation.

interface mobility, which is used as fitting parameter to optimise the agreement between the experimental and simulated γ to α transformation kinetics, is not the intrinsic mobility of the interface but it is an effective parameter that also incorporated the solute drag effect. In the model nucleation is treated by imposing a new grain at selected places and at pre-imposed undercooling conditions. The initial dimension of the new grain is a single grid element. The nucleation mechanism is not predicted within the phase field approach as such, but follows from prescribed nucleation criteria derived from separated theory or experimental data.

Chapter 3 deals with the γ decomposition to α of a Fe-0.10 C, 0.49 Mn (wt%) steel during cooling at different cooling rates, ranging between 0.05 K/s and 10 K/s. The initial austenitic microstructure and the ferrite nucleation data are derived by metallographic examination and dilatometry, and they are set as input data of the model. All nuclei in the calculation domain, the number of which is derived from the experimental ferrite grain size, are set to form at a single temperature, estimated from the ferrite start temperature evaluated from the dilatometric curves. The interface mobility is used as a fitting parameter to optimise the agreement between the simulated and experimental ferrite fraction curve derived by dilatometry. The derived carbon distribution in austenite during transformation is studied in order to provide insight into the nature of the transformation with respect to the interface-controlled or diffusion-controlled mode.

Chapter 4 presents a phase field analysis of the effect of Nb addition on the γ → α transformation kinetics of a Nb micro-alloyed C-Mn steel during cooling. A single cooling rate (0.3 K/s) but different austenitisation temperatures are considered to investigate the effect of both Nb in solution and precipitated as NbC on the transformation kinetics. As in the previous chapter, the initial austenitic microstructure and the nucleus density are derived from experimental data. Unlike the previous chapter we assume that the ferrite nuclei form continuously over a temperature range of about 50 K below the nucleation start temperature, following the experimental nucleation work of Offerman et al. [34]. Again the interface mobility is used as a fitting parameter to optimise the agreement between the experimental and simulated ferrite fraction curve and it is taken as representative for the effect of NbC on the phase transformation.

essential mainly to achieve realistic diffusion profiles but also to take into account a more realistic nuclei distribution in quadruple points, triple lines and grain surfaces.

Chapter 5 is devoted to a first series of 3D PFM simulations of the austenite-to-ferrite transformation using the MICRESS code. The investigation employs the same alloy as previously evaluated in Chapter 3. Two of the three cooling rates already investigated in chapter 3 are considered to analyse the significance of 3D simulations as compared to 2D PFM calculations. The first case of cooling at 0.4 K/s leads to the activation of just one nucleation mode at triple lines. For the second case with cooling at 10 K/s also nucleation at grain surfaces is also considered to get a substantial α grain refinement as observed in the final microstructure. As already applied in Chapter 4, not all nuclei are set to form at a single temperature but over a nucleation temperature range set equal to 18 K for each nucleation mode. 3D simulations results are presented in detail for both cooling scenarios. Subsequently, the 3D results are compared to those obtained using the 2D PFM thereby allowing a critical analysis of 2D vs. 3D simulations. Based on this analysis the challenges and directions for future model developments are delineated.

In Chapter 6 we combine the 3D phase field model as presented in Chapter 5 with the best experimental indication of the actual ferrite nucleation behaviour to describe a representative transformation kinetics and resulting ferrite microstructure. Thus the nucleation temperature interval is employed as an adjustable parameter in addition to the effective interface mobility. A number of combinations of these two parameters is found to equally well represent the experimental curve. The comparison between the simulated and the experimental ferrite grain size distribution is used as additional experiment data to establish the most realistic combination nucleation temperature range and interface mobility.

mobility and nucleus density used in the simulation. The principal observations and conclusions of this work are summarised in the final chapter.

References

1. D.A. Porter, K.E. Easterling, Phase Transformation in Metals and Alloys, Chapman &

Hall, London, 1992

2. R. W. K. Honeycombe, H.K.D.H. Bhadeshia, Steel Microstructure and Properties, Edward Arnold, London 1995

3. M. Enomoto, ISIJ Intern, 32 (1992), 297-306

4. R.C. Reed, H.K.D.H. Bhadeshia, Mat. Sci. Technol. 8 (1992) 421-35

5. C. Zener, J. Appl. Phys., 20 (1949) 950-53

6. H.K.D.H. Bhadeshia, L.E. Svensson, B. Gretoft, Acta Metall., 33 (1985) 1271-83

7. R.A. Vandermeer, Acta Metall. Mater. 38 (1990) 2461-70

8. J.W. Christian, The Theory of Transformation in Metals and Alloys, Pergamon Press,

Oxorf, 1981, 476-479

9. Y. van Leeuwen, T.A. Kop, J. Sietsma, S. van der Zwaag, J. Phys IV France, 9 (1999) 401-409

10. T.A. Kop, Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 40 (2000) 713-18 11. Y. van Leeuwen, M. Onink, J. Sietsma, S. van der Zwaag, ISIJ Int 41 (2001) 1037-46

12. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Engin. A, 1997, vol. 237, 216-223

13. Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int.,2003, vol. 43, 767-73

14. J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143-52.

15. J.S. Kirkaldy, Can. J.Phys, 36 (1958) 907-916

16. G.R. Purdy, D.H. Weichert, J.S. Kirkaldy, AIME Met. Soc. Trans, 230 (1964) 1025-34

17. M. Hillert, Paraequilibrium, Int. Report, Swed. Inst. Metals Res., 1953

18. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Cambridge

University Press, Cambridge, UK, 1998, 349-67

19. G.R. Purdy, Y.J. M. Brechet Acta Metall. Mater. 43 (1995) 3763-74 20. M. Hillert, Acta Mater. 47 (1999) 4481-4505

21. F. Fazeli, M. Militzer, Metall. Mater. Trans. A, 36 (2005) 1395-1405

22. G. Caginalp, Physical Review A, 39 (1989) 5887-96

23. A.A. Wheeler, W.J. Boettinger, G.B. McFadden, Physical Review A, 45 (1992)

25. S.G. Kim, W.T. Kim, T. Suzuki, Physical Review E, 58 (1998) 3316-23

26. A. Karma, Physical Review E, 49 (1994) 2245-50

27. G. Pariser, P. Shaffnit, I. Steinbach, W. Bleck, Steel Research 72 (2001) 354-60.

28. D.H. Yeon, P.R. Cha, J.K. Yoon. Scripta Mater., 45 (2001) 661-68

29. I. Loginova, J. Odqvist, G. Amberg, J. Ågren. Acta Mater., 51 (2003) 1327-39

30. I. Loginova, J. Ågren, G. Amberg, Acta Mater. 52 (2004) 4055-63

31. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , M. Apel, P. Schaffnit, I. Steinbach,

Metall. Mater. Trans. 36A (2005) 2327-40

32. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , Acta Mater. 53 (2005) 1431-40

33. C.J. Huang , D.J. Browne , S. McFadden, Acta Mater. 54 (2006) 11-21

11

Chapter 2

Phase Field Theory

Abstract

2.1 Introduction

Many inhomogeneous systems, like multiphase materials, contain domains of well-defined phases of different composition and crystal structure or, in the case of single-phase systems, domains representing different grains of specific orientations. In non-equilibrium conditions a microstructural evolution of the system takes place, driven by the tendency of the system to reduce its total free energy.

The classical approach used for modelling the microstructural evolution in materials treats the region separating the different domains as a region of zero thickness; the motion of this sharp interface describes the kinetics of the relaxation towards the equilibrium condition. Unfortunately sharp-interface models are difficult to implement since they require the solution of diffusion equations subject to the moving boundary conditions at the interface. This approach, in which a boundary, the interface, has to be determined as part of the solution, is usually called free-boundary problem. The free-boundary approach can be successful for phase change with simple geometry but becomes impractical for more complicated two or three-dimensional systems.

A more convenient approach for describing many types of microstructural evolution processes is represented by the phase field models [1], based on the construction of a Ginzburg-Landau free-energy functional; unlike the classical model, the phase field approach treats the system, containing both bulk and interface regions in an integral manner. One or more continuous field variables, also called phase field or order parameters, are introduced to describe at any time, t, in each point, r, of the system the different domains present. Typically these field variables are

constant in the bulk regions and change continuously over a diffuse interface of thickness η.

13

interfacial phase mixture, all the coexisting phases have the same composition. In spite of its simplicity, a problem in this approach, especially in numerical simulation where a finite interface thickness is assumed, is that the parameters depend on the interface thickness.

Steinbach et al. propose in their multi-domain model a different definition of the free energy density in the interface [11-13]. In their model the interfacial region is assumed to be a mixture of phases with different compositions, but constant in their ratio. This model was augmented by Kim et al. [14] by a thermodynamic consistent derivation and it can be used for general multi-phase and multi-component problems.

In this chapter the multi-phase field model derived by Steinbach et al is presented and applied for describing the austenite to ferrite transformation kinetics in polycrystalline low carbon steels. The time evolution of a set of continuous field variables, each of them representing a particular domain (grain) of the system, is obtained by solving a set of phase field equations, derived by minimising the total free energy of the system, as reported in section 2.2.

The coupling with the solute (here carbon and manganese) diffusion equations is reported in section 2.3. The last section is dedicated to the way in which the chemical driving pressure of the transformation is calculated using the linearisation of the phase diagram with a particular emphasis to the possible constrained equilibria that can be considered because of the low diffusion of manganese.

A major advantage of the phase field model over other physical transformation models is its ability to allow very different morphologies to form, depending on the nucleation and growth conditions. The nucleation process is not predicted within the phase-field approach but follows from prescribed nucleation criteria derived form other theories or experimental data. Based on experimental data nuclei of a new phase are set to form in selected sites and at pre-imposed undercooling conditions.

2.2 Phase

field

equations

The multiphase field model derived by Steinbach et al [11-13] is used to study the γ to α transformation. A polycrystalline system of N grains is described by a set of N order parameters φi(r,t) , also represented by the vector φ =( , ... )φ φ φ1 2 N

G

. φi(r,t) is defined as follows:

φ i(r,t) = 1, if the grain i is present at the location r and time t;

φ i(r,t) = 0 if the grain i is not present at r and t;

φ i(r,t) changes continuously from 0 to 1 within a transition region or interface of width η ij

(see Figure 2.1). The interfacial thickness is taken to be the same for each pair of grains in contact, i.e. ηij = η.

Figure 2.1 Definition of the phase field parameter φi(r,t)

(a) representation of the microstructure (b) φ1(r) along the section AA

In the boundary between the regions defined by the phase field parameters φ_i

( )

r t, and

( )

,

j r t

φ we have φ_i

( )

r t, ≠ and 0 φ_j

( )

r t, ≠ , with 0 φ_i

( )

r t, = −1 φ_j

( )

r t, , while φ_k

( )

r t, = for 0 all k i≠ and k≠ . Analogously in a triple point between the regions i, j, k, we have j φ_i

( )

r t, ,

( )

,

j r t

15 Generally for N grains we get the constraint

( )

1 , 1 N i i r t φ = =

∑

(2.1)

Each order parameter has a set of attributes relevant to describe the system of interest. These attributes can be the lattice structure and orientation, the lattice strain, the electric and magnetic polarization etc. In the present work only the lattice structure, i.e. bcc (α) or fcc (γ), is considered since isotropic structural and physical properties are assumed.

In this approach the microstructural change of a polycrystalline system is described by the

time evolution of N order parameters φi(r,t), which is obtained minimising the total free energy

of the system F , i.e.

1 i ij j i i d dt φ _τ δ δφ − ≠ = −

∑

F (2.2) The factor i δ δφ

− F is the thermodynamic driving force, which drives the system towards the

equilibrium, and τij is a frictional coefficient associated with dissipation effects during the i→j

transformation.

F is assumed to be a functional of the phase field vector φG=

(

φ φ φ₁, ..._{2 N}

)

and its gradient

(

1, 2... N

)

φ φ φ φ

∇ = ∇ ∇JJJG ∇ and it is expressed as the volume integral over the volume V of the

system of a free energy density, which is given by the sum of a gradient and potential energy term, i.e.

(

)

(

)

( )

, , N grad , pot ij ij i j V f f dr φ φG ∇ =JJJG

∫

∑

_ φ φG ∇ +JJJG φG _ F (2.3)

The gradient energy term is not null only in the interface regions and then provides the contribution of the interface to the free energy; it is given by

where ε_ij are the gradient energy coefficients.

Many phase field models, particularly in solidification modelling, use a double well form for the potential energy term, i.e.

( )

2 2 1 3 2 1 3 2 1 3 3 3 pot ij ij i j ij i i j j j i f φ =β φ φ −m _ φ φ φ+ − φ φ φ− + _   G (2.5)

The parameters β_ij determine the energy barrier for the transition between the two grains and

it will be related later in this section to the surface energy and the interface width. The terms

ij

m represents the difference of the bulk free energy between two grains and thus m_ij ≠ only 0 when the grain i and j have different phases.

For a system of two grains of different phases, γ and α (φ φ_i = and φ_j = − ) the eq. (2.5) 1 φ

becomes

( )

₂

(

)

2 2

(

)

₂ 1 3 2 3 pot f φ =βφ −φ − m − φ φ (2.6)

This potential energy term function of φ has two minima at φ= and at 0 φ = and a maximum 1

at 1

2

m

φ

β

= + and it is shown in Figure 2.2 from φ changing between 0 and 1.

β determines the energy barrier for the transition between the phases α and γ and, as will be

shown in section 2.2.1, it is related to the surface energy, σ, and the interface width, η; m is

related to the driving pressure for the phase transformation, ∆G(x1_…xM_{, T), and then depends}

on the local composition x1…xM (for a system with M components) and temperature T,

(

1_{, ...}2 M

)

ij ij

17 0 1 0 m > 0 m = 0 m < 0 β /16 f po t (φ )

φ

Figure 2.2 Double-well potential for a system of two grains of different phases

Another potential expression that has been employed in phase field models is the so-called double obstacle potential, given by

( )

(

)

1

(

)

arcsin for 0 1 4 2 ij pot ij ij i j i j i j i j m f φ =β φφ − _φ φ− φ φ + φ φ− _ < <φ   G (2.7)

( )

for 0 and 1 pot ij f φG = ∞ φ = φ=

For a system of two grains of different phases, γ and α (φ φ_i = and φ_j = − ), eq. (2.7) 1 φ

becomes

( )

(

1

)

(

2 1

) (

1

)

1arcsin 2

(

1

)

4 2 pot m f φ =βφ −φ − _ φ− φ −φ + φ− _   (2.8)

where m is the thermodynamic driving pressure for the transformation and β is the potential

energy coefficient. The double obstacle potential is represented in Figure 2.3.

0 1 0 m > 0 m = 0 m < 0 f po t (φ ) φ

Figure 2.3 Double-obstacle potential for a system of two grains of different phases

In order to derive the phase field equation, the functional derivative in eq. (2.2) is calculated. We have from eq. (2.3)

(

,

)

( )

grad pot ij ij i i i f f δ _{φ φ φ} δφ φ φ  ∂ _{∂ } _ − = −∇_ − _{ } ∇ + _ ∂∇ ∂   F _(2.9)

and then eq. (2.2) becomes for pot

( )

ij

f φ a double well potential

{

}

1 2 2 2 _{2 4} N i ij ij i j j i ij i j j i ij i j j i m t φ _τ − _{ε φ φ φ φ β φ φ φ φ φφ} ≠ ∂ _{=  ∇ − ∇ −  − +}     ∂

∑

(2.10)

and for pot

( )

ij

f φ a double obstacle potential

19

2.2.1 Determination of the phase field parameters in term of physical parameters

In this section we derive the relation between the parameters in the phase field equation (2.7) ( ,ε β τ_{ij ij}, ,_ij m_ij) and physical material parameters: the grain boundary or interface mobility, µij,

the grain boundary or interface energy σij, the change of Gibbs energy for the transformation

∆Gij. In order to do so, we consider the simplified system of two grains of phase α, the newly

formed phase, and phase γ, the parent phase. The system is described by a single order parameter, φ

( )

r t, , which is equal to 1 if at the location r at time t the phase α is present and 0 if at r at t the phase γ is present. The double well potential (eq.2.6) is used for the potential

energy term. The phase field equation (2.10) with φ φ_i = and φ_j = − becomes 1 φ

(

)

(

)

1 2 2 _{4 1} 1 ₁ 2 m t φ τ ε φ−  β φ φ φ φ φ  ∂ _{= ∇ −}  ₋  ₋ _{− −}       ∂      (2.12)

Assuming that there is no orientation dependence of the order parameter, eq.(2.12) becomes

(

)

(

)

2 2 2 1 4 1 4 1 2 p m t r r r φ τ ∂ −ε _ ∂ + ∂ _φ = − β −φ φ_ −φ_+ −φ φ ∂ ∂ ∂    (2.13)

where p = 0, 1 or 2 when the dimension of the system is 1, 2, or 3, respectively.

In one dimension (p=0) the differential operator on the left hand side of eq. (2.13) is invariant under general translation in time and space, which is represented by

t→ = + ∆ (2.14) t′ t t

r→ = + ∆ r′ r r (2.15)

Assuming that the interface profile maintains the same shape during propagation, in the steady state condition the phase field parameter profile is equal at all times in a reference system that moves with the interface. This means that, if the translation

( )

is applied, where s(t) is the interface position at the time t, φ is invariant under general translation in time (eq. (2.14)). In one dimension eq. (2.13) is therefore invariant under the translations (2.14) and (2.16). Then in one dimension there is a particular solution of the form

( )

r t, X r t

( )

, r s t

( )

φ =φ_ _=φ_ − _{ (2.17)}

In a higher dimension, due to the extra term (p/r)(∂/∂r), eq. (2.13) is not invariant under the

translation given by eq. (2.16). However, since φ varies only within the transition region of

thickness η with η<< s, we can apply a Taylor expansion according to

1 ... p p p X r X s s s   = = _ − + _ +   (2.18)

If the terms of magnitude (2η/s) or less are neglected, this becomes

p p

r = (2.19) s

Therefore, eq. (2.13) is approximately invariant under the translations (2.14) and (2.16) also in higher dimensions.

The partial derivatives in eq. (2.13) may be transformed in ordinary derivatives by

2 2 2 2 , d d r dX r dX φ φ φ φ ∂ ₌ ∂ ₌ ∂ ∂ (2.20) and d dX d ds d v t dX dt dX dt dX φ φ φ φ ∂ _{= = − = −} ∂ (2.21)

where v is the interface velocity. Then eq. (2.13) becomes

21

In the steady-state condition there is no time dependence of φ on both sides of the transition

region because on one side of the interface the transformation is complete and on the other it

has not started yet. The time dependence of φ comes only implicitly through the movement of

the shape-preserving interface. Therefore the parameter w, defined by

2 ds p w dt s ε τ = + (2.23)

is time independent. Combining the eqs. (2.22) and (2.23) yields

(

)

(

)

2 2 2 1 1 1 2 d d w m dX dX φ φ τ ε β φ φ φ φ φ − − = − − _ − _+ −   (2.24)

The solution of eq. (2.24) that satisfies the boundary conditions φ → 1 for X → ∞ and φ → 0

for X → − ∞ is 1 3 1 tanh 2 2 X φ η   = _− _+   (2.25)

representing a diffuse interface, of constant thickness η, moving with a constant velocity v.

We have from eq. (2.25)

(

)

22 2

(

)

6 18 1 1 , 1 2 d d dX dX φ _{φ φ} φ _{φ φ φ} η η   = − − = − _ − _   (2.26)

Substituting the derivatives in eqs. (2.26) into the equation (2.24), we obtain

2 2 6 18 1 4 4 2 w ε m τ β φ η η  _  =_ − _ − _+     (2.27)

Since w is time independent, the first term at the right hand side of eq. (2.27) has to be zero, or

2 2 9 2 ε β η = (2.28)

2 3 m w η τ = (2.29)

Substituting the expression for w, eq. (2.23), into eq. (2.29) we have

2 ₂ 3 ds m dt η ε κτ η   = − _ − _   (2.30)

where κ =p/s is the interface curvature.

The comparison of eq. (2.30) with the Gibbs-Thomson equation,

[

]

v= −µ σk− ∆G (2.31)

allows the derivation of relationships between the phase field parameters (ε, β, τ, η, m) and the

physical parameters, the interface mobility µ, the interface energy σ and the change of Gibbs

energy for the transformation ∆G, which are given by

2 , 2 , 3 G 3m η ε µ σ β τ = = ∆ = (2.32)

Substituting the eqs (2.32) into eq. (2.13), the phase field equation in terms of physical parameters is given by

(

)

(

)

2 2 18 1 6 1 1 2 G t φ µ σ φ φ φ φ φ φ η η    ∂ _{= ∇ − −}  ₋ ₊ ∆ ₋      ∂      (2.33)

which is the simplified case of two grains of γ and α phase. The general multiphase field equations for a double well potential are given by

(

2 2

)

(

)

2 6 18 N ij i ij ij i j j i i j j i i j j i G t φ _{µ σ φ φ φ φ φφ φ φ φφ} η η ≠ ∆   ∂ ₌  _{∇ − ∇ − −} ₋     ∂

∑

_{   } (2.34)

23

(

)

2 2 2 2 2 i ij i j j i i j i j ij j _ij G t φ _{µ σ φ φ φ φ} π _{φ φ} π _φφ η η     ∂ ₌  _{∇ − ∇ + − + ∆}      ∂

∑

_{   } (2.35)

In the present work a system of N grains of phase γ or α is considered. When neighboring

grains have different phases the interfacial mobilities and interfacial energies, µij and σij in

equations (2.34) or (2.35) are given by the γ/α interface mobility µ and interface energy σ;

∆Gij is the driving pressure for the transformation, i.e. ∆Gγα (x1(r,t)… xM(r,t),T). The driving

pressure is a function of the local chemical composition x1(r,t)… xM(r,t), for a system of M

components, and the temperature T. When neighboring grains have the same phase (austenite or ferrite), µij and σij are the grain boundaries mobilities and energies, µγγ or µαα and σγγ or σαα

respectively. In that case ∆Gij is zero and the driving pressure for grain growth is given by the

respective grain boundary energy times the curvature term (the term within the square brackets in equations (2.34) or (2.35)). Grain growth is assumed to be of secondary importance in this work and it is minimized by choosing an artificially small value for the γ/γ and α/α grain boundary mobilities.

2.3 Diffusion

equations

In the bulk of the α or γ phase, as well as within the diffuse α/α or γ/γ grain boundary regions,

the local concentration of the element A is given by _xA

α and xγA, obtained by solving the

diffusion equation for the ferrite and austenite phase, respectively

(

)

;

(

)

A A A A x A A x D x D x t t γ α α α γ γ ∂ ∂ _{= ∇ ∇ = ∇ ∇} ∂ ∂ (2.36) where _DA

α and DγA are the diffusivity of the element A in the phase α and γ respectively. In the

diffuse interface between the ferritic grain i and the austenitic grain j the bcc and fcc structures

composition _{x r t becomes a continuous variable in r through the interface and it splits into} A

( )

, the austenite and ferrite composition, _{x r t}A

( )

,

γ and x r tαA

( )

, , as

( )

,

( ) ( )

, ,

( ) ( )

, ,

A A A

i j

x r t =φ r t x r t_α +φ r t x r t_γ (2.37) In the diffuse interface only phase field parameters φi and φj are not zero and related by φj = 1

− φi. Then eq. (2.37) becomes

(

1

)

A A A

i i

x =φx_α + −φ x_γ (2.38)

The dependence of all variables on r and t is omitted in eq. (2.38).

The diffusion of the element A is expressed as the sum of fluxes in ferrite and austenite phase weighted by the phase field parameters φi and φj according to

(

1

)

A A A A k i i x D x D x t φ α α φ γ γ ∂ _{= ∇ ∇ + − ∇ }   ∂ (2.39)

Furthermore, it is assumed that carbon atoms redistribute between α and γ at the interface according to a partitioning ratio equal to the partitioning ratio at equilibrium kA eq_{, i.e.}

A Aeq A Aeq A Aeq x x k k x x α α γ γ = = = (2.40)

Using eq. (2.38) and eq. (2.40) the carbon diffusion eq. (2.39) becomes

( )

(

₍

)

₎

* 1 1 1 A Aeq A A A i Aeq i i x k x D x t φ φ k φ   −  ∂ _{= ∇} _{∇ − ∇ }   ∂ _ _ + − _ (2.41) with

( )

(

₍

₎

)

* 1 1 A Aeq A A i a A i _Aeq i D k D D D k γ φ γ φ φ + − = + − (2.42)

For an Fe-C-Mn system the local composition x1_…xM _{is represented by the carbon and}

25

that of C atoms, it is reasonable to assume that Mn does not diffuse in the bulk of α and γ phase during transformation and only long-range diffusion of carbon atoms occurs in the remaining austenite. The phase field equation (2.34) or (2.35) is coupled only with the carbon diffusion equation

( )

(

₍

)

₎

* 1 1 1 C C eq C C C i _{C eq} i i x k x D x t φ φ k φ   ₋  ∂ _{= ∇} _{∇ − ∇ }   ∂ _ _ + − _ (2.43) with

( )

(

₍

₎

)

* 1 1 C C eq C C i a C i C eq i D k D D D k γ φ γ φ φ + − = + − (2.44)

Figure 2.4 shows the carbon concentration profile xC across the interface region between a

ferritic and austenitic grain. In the bulk of α and γ grain xC _{reduces to x}C

α and xγC respectively

while in the interface region it is given by the solution of eq. (2.43). The phase field parameter is also reported in Figure 2.4.

0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 η φ xC xC_γ xC_α

distance along the normal to the interface

φ 0.00 0.05 0.10 0.15 0.20 0.25 x _C , x _C γ _, x _C α _(w t % )

Figure 2.4 Carbon distribution in austenite, _xC

γ , in ferrite, xαC, and overall carbon

2.4 Driving force calculation

While in the Fe-C system the thermodynamics is uniquely determined by the temperature and chemical composition of the system, for Fe-C-Mn the situation is not straightforward. Different possible constrained equilibria may be considered for manganese.

2.4.1 Ortho-equilibrium

At first we consider the situation where both C and Mn redistribute in the α and γ phase within

the interface region. Using eqs. (2.38) and (2.40), for A = C, Mn, _xC

γ , xαC and xγMn , xαMn can

be written in terms of the overall composition, xC and xMn, and φ , i.e.

(

1

)

;

(

1

)

C C C eq C C C eq C eq x x k x x k k γ = _{φ+ −φ} α = _{φ+ −φ} (2.45)

(

1

)

;

(

1

)

Mn Mn Mneq Mn C Mneq Mneq x x k x x k k γ = _{φ+ −φ} α = _{φ+ −φ} (2.46)

where xC is obtained by solving the carbon diffusion according to eq. (2.43) and xMn is the

alloy Mn content, since this element does not diffuse in bulk of γ and α phase.

In order to calculate the equilibrium partitioning ratios, _kC eq _{and k}Mn eq_{, the Fe-C-Mn ternary}

phase diagram is linearised at a reference temperature, TR. Figure 2.5 shows the isothermal

section of the ternary Fe-C-Mn diagram at TR _{= 1120 K as determined by the Thermo-calc}

software [15]. The alloy chemical composition (0.1 wt % C , 0.5 wt % Mn) is indicated by the point within the dual phase region. The investigated alloy is located in the γ + α region; the tie

line giving the chemical composition of the α and γ phase at TR _{, x}C R

α , xαMn R and xγC R, xγMn R is

indicated in the isothermal section. The boundary line for the α phase is linearised in the Fe-C

pseudo-phase diagram for _xMn _xMn R

α

= and the Fe-Mn pseudo phase diagram for _xC _xC R

α

= . In

27

for _xMn _xMn R

γ

= and the Fe-Mn pseudo phase diagram for _xC _xC R

γ

= . The derived slopes are

Fe C

m_α − _{, m}Fe Mn

α − , mγFe C− and mγFe Mn− .

Figure 2.5 Isothermal section of the ternary Fe-C-Mn diagram at TR = 1120 K as determined by the Thermo-calc software [15].

The equilibrium composition of each phase, C eq

i

x and Mneq

i

x with i = α, γ, is then given by

(

R

)

C eq C R i i Fe C i T T x x m − − = + (2.47)

(

R

)

Mneq Mn R i i Fe Mn i T T x x m − − = + (2.48)

For each temperature T the equilibrium partitioning ratio is determined using eqs. (2.47) and (2.48).

From each configuration φ(r,t), the driving pressure _{∆G x x}

(

C, Mn,_T

)

_{is calculated using the}

eq T T T

∆ = − (2.49)

where the local equilibrium temperature, Teq is given by

(

)

(

)

{

0.5 eq R Fe C C C R Fe Mn Mn Mn R T =T + _m_α − x_α −x_α +m_α − x_α −x_{α } +

(

)

(

)

}

}

Fe C C C R Fe Mn Mn Mn R m_γ − x_γ x_γ m_γ − x_γ x_γ   +_ − + − _{ (2.50)}

The driving pressure is then expressed by

(

C, Mn,

)

G x x T S T

∆ = ∆ ∆ (2.51)

∆S is the entropy difference between the phase α and the phase γ.

2.4.2 Para-equilibrium

As an alternative for the condition described above where both Mn and C redistribute within the interface and the thermodynamic data are derived by the ternary Fe-C-Mn system, we can consider the case where the γ → α transformation proceeds without Mn partitioning between ferrite and austenite at the interface; only C partitions at the interface according to an

equilibrium partitioning coefficient _kC eq_{. The para-equilibrium quasi-binary phase diagram,}

derived using Thermo-calc under the additional constrain of no Mn partitioning between phases, is reported in Figure 2.6.

In order to derive the driving pressure for the γ → α transformation, the quasi-binary phase

diagram of Figure 2.6 is linearised at the reference temperature TR_{. The driving pressure is}

given by

(

C,

)

PE

{

R 0.5 Fe C

(

C C R

)

Fe C

(

C C R

)

}

a G x T S T _m_γ − x_γ x_γ m_α − x_α x _ T ∆ = ∆ + _ − + − _− (2.52) where _xC R

γ and xαC R is the equilibrium carbon content of austenite and ferrite at TR, and

Fe C

m_γ − _{and m}Fe C

29

the para-equilibrium Fe-C phase diagram. ∆SPE _{is the entropy difference between the α and γ}

phase, as derived by Thermo-Calc under para-equilibrium condition.

0.0 0.5 1.0 1.5 2.0 900 950 1000 1050 1100 0.00 0.02 0.04 900 950 1000 1050 1100 γ α + γ linearised γ line γ line T em pe rature (K ) wt% C α α + γ linearised α line α line

Figure 2.6 Pseudo-binary Fe-C phase diagram derived from Thermo-calc software. The linearised α and γ lines at the reference temperature of 1173 K are also indicated. Expanded scale for low C content in the inset.

2.5 Summary

A phase field method for diffusional phase transformation in C-Mn was presented. The diffusion model of the components present in the system (here C and Mn) is based on the superposition of the fluxes of the individual component in different phases. The concentration of the elements in two or more coexisting phases in two or more boundary regions, are assumed to be connected by equilibrium partition. The driving pressure for the transformation is assumed to be proportional to the local undercooling at the interface, which is calculated from the local concentration of different components at the interface using a linearised phase diagram. Different constrained equilibria for Mn were considered for the driving pressure calculation.

Reference

1. L. D., Landau M.I.Khalatnikov, Dokl. Akad. Nauk SSSR 96, 469, 1954 (English translation in The Collected Works of L. D. Landau, edited by D. ter Haar (Pergamon, Oxford, 1965)

2. G. Caginalp, Physical Review A, 39, (1989) 5887-96 3. R. Kobayashy, Physica D, 66 (1993) 410-23

4. J. B. McFadden, A.A. Wheeler, R.J. Braun., S.R. Coriell, R.F. Sekerka, Physical

Review E, 48 (1993) 2016-24

5. J.B. Collins, H.Levine, Physical Review B, 31 (1985) 6119-22

6. A.A. Wheeler, W.J. Boettinger, G.B. McFadden, Physical Review A, 45 (1992) 7424-39

7. G. Gaginalp, W. Xie, Physical Review E, 48 (1993) 1897-1909

8. J.A. Warren, W.J. Boettinger, Acta Metallurgica Materialia, 43 (1996) 689-703

9. A.A.Wheeler, G.B. McFadden, W.J. Boettinger, Proc. Royal Society London Ser.A, 452 (1996) 495-525

10. A. Karma, Physical Review E, 49 (1994) 2245-50

31

12. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998) 73-86

13. I. Steinbach. Advances in Materials Theory and Modeling - Bridging Over Multiple

Length and Time Scales, San Francisco: MRS, 2001, AA7.14.1- AA7.14.6

Chapter 3

Analysis of the austenite to ferrite transformation

in a C-Mn steel by phase field modelling

Abstract

3.1 Introduction

In recent years there has been a strong development in modelling solid state transformations in steels and alloys, mainly because of the strong dependence of the final properties of the material on the microstructural evolution during processing. In C-Mn steels the austenite (γ) to ferrite (α) transformation during cooling is the most important transformation since it determines to a large extent the microstructure and therefore the properties of the final product. In the literature, the kinetics of the γ to α transformation is often modelled assuming that the carbon diffusion in austenite is the rate controlling process and the interface mobility is high enough not to affect the transformation kinetics (diffusion controlled transformation mode, DCM) [1-3]. The other extreme is to assume that the transformation kinetics is controlled only by the interface mobility and the carbon diffusivity in austenite is large enough to maintain a homogeneous carbon concentration in austenite (the interface controlled mode, ICM) [4-5]. In reality both processes will influence the transformation kinetics and the transformation is of a mixed mode character [6-7].

Few studies are present in literature on the application of the phase field model to simulate the γ to α transformation. Yeon et al. [20] simulated the isothermal austenite to ferrite transformation in a Fe-Mn-C system under para-equilibrium (only C redistributes during transformation while the Mn content remains equal in ferrite and austenite) using the WBF model. Pariser et al. [21] applied the model of Steinbach and co-workers for modelling the austenite to ferrite transformation in an ultra-low carbon steel and in an interstitial-free steel during cooling at a medium cooling rate.

In this chapter the phase field model derived by Steinbach and co-workers [18-19] is used to simulate the γ to α transformation during controlled cooling of a 0.10 wt %C, 0.49 wt % Mn steel. The transformation kinetics was experimentally investigated at three different cooling rates, low (0.05 K/s), medium (0.4 K/s) and high (10 K/s) cooling rate, using dilatometry. Phase field simulations were performed at the same cooling rates as the experimental tests. The input data of the simulations were consistent with the experimental conditions and adjusted to fit the experimental transformation kinetics. A good agreement between the simulated α-γ microstructure and the actual α-pearlite microstructure observed after cooling was obtained. the carbon distribution near the α/γ interface is analysed in terms of mixed-mode character of the transformation.

3.2 Experimental

procedures

The γ → α transformation in a 0.10 wt % C, 0.49 wt % Mn steel was experimentally analysed by dilatometry using a Bähr 805A/D dilatometer. The sample, 4 mm in diameter and 10 mm in length, was heated by a high-frequency induction coil. Two thermocouples, spot welded on the sample, were used to control the heating power during the test and to check the temperature homogeneity in the sample, respectively. In all tests the temperature differences along the sample length were smaller than 10 K.

The sample was heated at 1273 K at a heating rate of 0.8 K/s, maintained at this temperature for 120 s and then cooled to room temperature at a cooling rate of 0.05, 0.4 or 10 K/s.

The lever rule, which is generally used to obtain the fraction of transformed austenite from the dilatometric signal, assumes that the specific volume of ferrite, Vα, is equal to that of pearlite, Vp, and that the specific volume of austenite, Vγ, is independent of the carbon

more accurate method was used [22], which allows obtaining the ferrite and pearlite fractions, fα and fp, separately. In the analysis, the temperature range of transformation is

divided into two regions: a high-temperature region (T > Tt) in which only ferrite forms,

and a low-temperature region (T < Tt) in which only pearlite forms. The temperature Tt is

defined as the temperature at which the (second) point of inflection occurs in the length change curve with respect to the temperature. The point of inflection indicates an increased transformation rate, which is expected at the start of the pearlite formation. The ferrite and pearlite fractions are calculated using two separate equations, one valid for T > Tt and the

other for T < Tt. The experimental ferrite fraction, used in this study, is calculated for T >

Tt from the equation

(

V V

)

f V V γ α α γ − = − (3.1)

where the atomic volume V is calculated from the measured dilatation, ∆L, using the

equation 0 0 3 1 L V V L λ  ∆  = _ + _   (3.2)

where V0 is the initial average atomic volume and λ is a scaling factor, equal to 1 in an

ideal case. To determine λV0, the dilatometric signal before (f γ = 1) and after

transformation ( _{f f}eq

α = α , fp = fpeq) was used; a linear interpolation of the values for λV0

obtained before and after transformation was considered in the transformation temperature range. In the present study the pearlite fraction will not be considered.

3.3 Simulation

conditions

2D phase-field simulations of the austenite to ferrite transformation during continuous cooling were performed for a Fe - 0.1 wt % C - 0.49 wt % Mn ternary alloy. A code, developed by Access, was used to solve numerically the phase field equation

under the assumption that the potential term of the free energy density has a double obstacle structure, eq. (2.7). For each configuration φi(r,t), the carbon composition x r tC

( )

,

is determined by solving the equations

(

)

;

(

)

C C C C x C C x D x D x t t γ α α α γ γ ∂ ∂ = ∇ ∇ = ∇ ∇ ∂ ∂ (3.4)

respectively in the bulk of grain i for an ferritic and austenitic grain respectively and the equation

( )

(

₍

)

₎

* 1 1 1 C C eq C C C i _{C eq} i i x k x D x t φ φ k φ   −  ∂ _{= ∇} _{∇ − ∇ }   ∂ _ _ + − _ (3.5) with

( )

(

₍

₎

)

* 1 1 C C eq C C i a C i _{C eq} i D k D D D k γ φ γ φ φ + − = + − (3.6)

within the interface between a ferritic grain i and an austenitic grain j with φj = 1 − φi. The

equilibrium partition ratio for C is derived from the linearised phase diagram for ortho-equilibrium as described in 2.4.1.

Since the diffusivity of Mn atoms is much lower than that of C atoms, Mn-partitioning is limited to the interface thickness η, according to a partitioning ratio equal to the partitioning ratio at equilibrium kMn eq_{, i.e.}

Mn Mneq Mn Mneq Mn Mneq x x k k x x α α γ γ = = = (3.7)

In the bulk of the austenite and ferrite phases the Mn-content remains constant and equal to the overall Mn-content.

The thermodynamic data for the driving force calculation were taken from the linearised Fe-C and Fe-Mn pseudo-phase diagrams at 0.49 wt % Mn and 0.1 wt % C respectively as described in 2.4.1. The reference temperature, at which the linearisation was performed, was TR = 1120 K; the carbon and manganese contents in austenite and ferrite at TR and the corresponding slopes were calculated using Thermo-Calc (_xC R

The same three cooling rates as used in the experiments were simulated: a low (0.05 K/s), a medium (0.4 K/s) and a high (10 K/s) cooling rate. The initial austenitic microstructure was generated by means of a weighted Voronoi tessellation diagram calculation, which is implemented in the code. Periodic boundary conditions were set. The calculation domain size was 60 x 60 µm2 _{in the simulation at 0.05 K/s and 37.5 x 37.5 µm}2 _{in the simulations}

at 0.4 and 10 K/s. The number of grains was adjusted to have an austenite grain size of 20 µm in all simulations, typical for the austenitisation temperature of 1273 K.

We assume that at low and medium cooling rates nucleation occurs at the triple points only, while at the high cooling rate it occurs at the triple points and at the grain boundaries. Nucleation is treated in the model by imposing a new grain at a place, where critical supersaturation for the corresponding grain is exceeded. The initial dimension of the new grain is a single grid element. This nucleation mechanism is not predicted within the phase field formalism, but according to prescribed nucleation criteria for individual phases or grains.

The nucleation temperature is defined by the nucleation undercooling ∆Tn, estimated

comparing the experimental ferrite fraction curves with the equilibrium ferrite fraction curve as predicted by Thermo-Calc [23]. The nucleus density was changed depending on the cooling rate and adjusted in agreement with the experimentally observed final microstructures.

We assumed that the α/γ interface mobility, µαγ, is temperature dependent, according to the

relation µαγ(T)= µ0αγ exp(−Qµ/RT). In the present calculations the activation energy Qµ was

set equal to 140 kJ/mol, the value found by Krielaart and Van der Zwaag in a study on the transformation behaviour of binary Fe–Mn alloys [24]. This value has also been used by Militzer in his simulations of ferrite formation [25]. The pre-exponential factor of the interface mobility was considered in view of the agreement between simulations and experiment, and will be discussed later. The interface energy values were taken from literature [26].

In order to resolve the carbon concentration gradient within the interface, the interface thickness η was set equal to 7.5 times the mesh size. A finer mesh size was used at high cooling rate than at medium and low cooling rate, respectively 0.075 and 0.15 µm, because

η has to be smaller than the diffusion length of carbon in austenite,

C d D L v γ = (see

eq.(2.45)) (Figure 2.2), to avoid numerical instability problems. Since Ld decreases as the