Phase Field Modelling of the Austenite to Ferrite

Transformation in Steels

The research described in this thesis was performed in the department of Material Science and Technology, the Delft University of Technology. The research described in this thesis was carried out in the first two years under the VESPISM project (contract number G5RD-CT-2000-00315), a Fifth Framework project funded by the European Commission, and in the second two years under the project number MC5.03172 in the framework of the Strategic Research Program of the Netherlands Institute for Metal Research (NIMR) in the Netherlands (www.nimr.nl).

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 Universita’degli studi di Roma (Italië)

geboren te Viterbo (Italië)

Dit proefschrift is goedgekeurd door de promotor: 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 Printed in the Netherlands

V

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 References 71

VI

5 3D phase field modelling of the austenite to ferrite 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

7.5 Discussion 138 7.6 Conclusions 143 References 144

Summary 145 Samenvatting 151

Acknowledgements 157 List of publications 159 Curriculum vitae 161

Chapter 1

1

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. In Fe-C steels the γ decomposition involves two distinctly different phenomena occurring simultaneously: a) the carbon partition between α and γ, influenced by the carbon diffusion in γ since the carbon diffusion in α is much faster, b) the construction of the bcc (α) lattice from the fcc (γ) lattice. In the literature, the kinetics of the γ to α transformation is often

Chapter 1

2

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 CG x xαγγ α∆ , calculated from the transient local carbon composition at the interface in

the α and γ side, Cxα and Cxγ . This is expressed by

( ),C Cv G x xαγγ αµ= ∆ (1.1)

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

interface, Cxα and Cxγ , change with time during transformation [12], due to the finite

interface mobility and the non-zero net carbon flux at the interface. For example Cxγ

change continuously from the initial carbon concentration, 0Cx and the equilibrium carbon

concentration in γ, Ceqxγ .

Much of the difficulty in the understanding the γ to α transformation kinetics in C-Mn steels arises from the still not clear effect on this transformation of the addition of a

Chapter 1

3

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 ( 6/ 10Mn CD Dγ 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 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 tφ , also called phase field or order parameters, are introduced to describe at any time,

Chapter 1

4

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 multiphase field model developed by Steinbach et al. and implemented in the MICRESS (MICrostructure Evolution Simulation Software) code was used in this thesis to analyse the austenite to ferrite transformation kinetics in a low carbon steels. Part of the work here reported was carried out under a fifth framework project funded by the European Commission: “Development of a physically-based research tool to conduct virtual experiments to solve problems in steel metallurgy (VESPISM)” (contract number G5RD-CT-2000-00315). The primary objective of the project was to allow an improvement in the development and production of advanced steel grades by the development, validation and preliminary application of the multiphase field model developed by Steinbach et al. The chosen phase field approach was used to calculate kinetic and morphological evolution in solidification, grain growth, recrystallisation and solid-state phase transformation in C-Mn steels. This thesis deals with the solid-state phase transformation work package of the project.

Chapter 1

5

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, ( ),i r tφ , is

described as the sum of the pairwise interaction with the order parameters ( ),j r tφ ,

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 kx of element k, with 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, kxγ and kxγ by

( )1k k ki ix x xα γφ φ= + − (1.2)

A distinctive feature of the model used here is that the /k kx xα 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. The phase field formulation used in this thesis does not incorporate in the driving pressure calculation the solute drag effect due to the Mn segregation at the interface; therefore the

Chapter 1

6

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. The phase field analysis reported in the chapters 3 and 4 of this thesis are limited to 2D space. The restriction on 2D was to a large degree due to the computational demands required for performing meaningful 3D simulations. Nevertheless, the incorporation of 3D aspects in the modelling of diffusional solid-state phase transformations seemed to be

Chapter 1

7

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.

In the final chapter, Chapter 7, a detailed analysis of the mixed-mode character of the γ to α transformation with respect to the diffusion and interface-controlled mode is performed. 2D and 3D phase field simulations are performed under isothermal conditions. Different interface mobilities and nucleus densities are employed since the former has a direct effect on the interface velocity and the latter has an effect on the time development of the carbon distribution in austenite. Since different nucleus densities are used in different simulation, the interface position is used, alternatively to the ferrite fraction, to describe the transformation kinetics. Soft impingement, i.e. of the overlap of diffusion fields arisen from two neighbouring growing α grains, is analysed in relation to different interface

Chapter 1

8

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) 7424-

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

Chapter 1

9

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 34. S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L.Margulies,

H.F. Poulsen, M. Th. Rekveldt, S. van der Zwaag, Science, 298 (2002) 1003-05

Chapter 1

10

Chapter 2

11

Chapter 2

Phase Field Theory

Abstract This chapter deals with the multi-phase field model used in this thesis to describe the austenite to ferrite transformation kinetics in a polycrystalline C-Mn steel. It is based on the early formulation by Steinbach and co-workers. Each grain of the system, either austenitic or ferritic, is described with an individual phase-field parameter. Each phase-field parameter has a set of attributes for describing the specific grain, here only its phase, i.e γ or α. The governing phase field equations, for the microstructural evolution of the multi-domain system, are derived by minimising the total free energy functional of the system. The coupling with the solute (here carbon and manganese) diffusion equation is also reported. The technique of the linearisation of the phase diagram for the driving pressure calculation is presented with a particular emphasis on the possible constrained equilibria that can be considered for manganese in relation to the very slow manganese diffusion.

Chapter 2

12

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. The phase field model was originally proposed to simulate the dendritic growth in pure undercooled melts [2-6] and later it was extended to the solidification of alloys with a single solid phase [7-8] or with two solid phases [9-10]. Many phase-field models are based on the classical phase-field approach proposed by Wheeler, Boettinger and McFadden (WBF model) for binary alloys [6], which starts from a double well description of the free energy density for pure material. For the extension to binary alloys, a continuous composition field is defined through the interface and the free energy density is extrapolated as a mixture in composition of the free energy densities of the pure materials. Within the diffuse interface two or more phases locally coexist, and since only a single continuous composition field is defined for the whole

Chapter 2

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. Depending on the formulation of the total free energy of the system, in principle the phase field technique can handle several phenomena affecting the austenite to ferrite transformation kinetics, like solute drag or transformation stress. These phenomena are considered explicitly in the phase field approach used in this thesis and thus they will affect the effective value of the interface mobility used as an adjustable parameter of the model.

Chapter 2

14

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 1 2( , ... )Nφ φ φ φ= . 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 ( ),i r tφ and

( ),j r tφ we have ( ), 0i r tφ ≠ and ( ), 0j r tφ ≠ , with ( ) ( ), 1 ,i jr t r tφ φ= − , while ( ), 0k r tφ = for

all k i≠ and k j≠ . Analogously in a triple point between the regions i, j, k, we have ( ),i r tφ ,

( ),j r tφ and ( ), 0k r tφ ≠ , while ( ), 0l r tφ = for all l k i j≠ ≠ ≠ .

Chapter 2

15

Generally for N grains we get the constraint

( )1

, 1N

ii

r tφ=

=∑ (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.

1iij

j i i

ddtφ δτ

δφ−

≠

= −∑ F (2.2)

The factor i

δδφ

−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 ( )1 2, ... Nφ φ φ φ= and its gradient

( )1 2, ... Nφ φ φ φ∇ = ∇ ∇ ∇ 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 potij ij

i jV

f f drφ φ φ φ φ ∇ = ∇ + ∑∫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

( ) ( )21,2

gradij ij i j j if φ φ ε φ φ φ φ∇ = ∇ − ∇ (2.4)

Chapter 2

16

where ijε 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 3 2 3 21 1 13 3 3

potij ij i j ij i i j j j if mφ β φ φ φ φ φ φ φ φ = − + − − +

(2.5)

The parameters ijβ 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

ijm represents the difference of the bulk free energy between two grains and thus 0ijm ≠ only

when the grain i and j have different phases. For a system of two grains of different phases, γ and α ( iφ φ= and 1jφ φ= − ) the eq. (2.5)

becomes

( ) ( ) ( )22 221 3 23

potf mφ βφ φ φ φ= − − − (2.6)

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

at 12

mφβ

= + 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, ... Mij ijm m x x x= . If m > 0 the phase ( )0γ φ = is stable; if m < 0 the phase ( )1α φ = is

stable.

Chapter 2

17

0 10

m > 0 m = 0 m < 0

β /16

f pot (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 14 2

ijpotij ij i j i j i j i j

mf φ β φ φ φ φ φ φ φ φ φ = − − + − < <

(2.7)

( ) for 0 and 1potijf φ φ φ= ∞ = =

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

becomes

( ) ( ) ( ) ( ) ( )11 2 1 1 arcsin 2 14 2

pot mf 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. The choice of the eq. 2.5 or eq. 2.7 for the potential term affects the solution of the phase field equation but does not change the physical basis of the model.

Chapter 2

18

0 10

m > 0 m = 0 m < 0

f pot (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 potij ij

i i i

f fδ φ φ φδφ φ φ

∂ ∂ − = −∇ − ∇ + ∂∇ ∂

F (2.9)

and then eq. (2.2) becomes for ( )potijf φ a double well potential

{ }1 2 2 2 2 4N

iij ij i j j i ij i j j i ij i j

j im

tφ τ ε φ φ φ φ β φ φ φ φ φ φ−

≠

∂ = ∇ − ∇ − − + ∂ ∑ (2.10)

and for ( )potijf φ a double obstacle potential

{ }1 2 2 2N

iij ij i j j i ij i j ij i j

j i

mtφ τ ε φ φ φ φ β φ φ φ φ−

≠

∂ = ∇ − ∇ − − + ∂ ∑ (2.11)

Chapter 2

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) ( , , ,ij ij ij ijmε β τ ) 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, ( ),r tφ , 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 iφ φ= and 1jφ φ= − becomes

( ) ( )1 2 2 14 1 12

mtφ 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

22

14 1 4 12

p mt r r rφτ 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 t t t′→ = + ∆ (2.14) 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

( )r X r s t→ = − (2.16)

Chapter 2

20

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 tφ 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 Xr X s s s

= = − + + (2.18)

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

p pr s

= (2.19)

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 dr dX r dXφ φ φ φ∂ ∂

= =∂ ∂

(2.20)

and

d dX d ds d vt dX dt dX dt dXφ φ φ φ∂

= = − = −∂

(2.21)

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

( ) ( )2 2

22

11 12

ds p d d mdt s dX dX

ε φ φτ ε β φ φ φ φ φ − + − = − − − + −

(2.22)

for − η/2< X < η/2 and η << s

Chapter 2

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

2ds pw

dt sετ

= + (2.23)

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

( ) ( )2

22

11 12

d dw mdX dX

φ φτ 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 1tanh2 2

Xφη

= − +

(2.25)

representing a diffuse interface, of constant thickness η, moving with a constant velocity v. We have from eq. (2.25)

( ) ( )2

2 2

6 18 11 , 12

d ddX dX

φ φφ φ φ φ φη η

= − − = − −

(2.26)

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

2

2

6 18 14 42

w mετ β φη 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

92

εβη

= (2.28)

The equation (2.27) thus becomes

Chapter 2

22

23mw η

τ= (2.29)

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

2 2

3ds mdt

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 k Gµ σ= − − ∆ (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 3

G mη εµ σ βτ

= = ∆ = (2.32)

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

( ) ( )22

18 1 61 12

Gtφ 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 22

618Niji

ij ij i j j i i j j i i jj i

Gtφ µ σ φ φ φ φ φ φ φ φ φ φ

η η≠

∆ ∂ = ∇ − ∇ − − − ∂

∑ (2.34)

In a similar way it is possible to derive the multiphase field equations for a double obstacle potential, which results in

Chapter 2

23

( )2

2 222

iij i j j i i j i j ijj ij

Gtφ π πµ σ φ φ φ φ φ φ φ φ

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 Axα and Axγ , obtained by solving the

diffusion equation for the ferrite and austenite phase, respectively

( ) ( );AA

A A A Axx D x D xt t

γαα α γ γ

∂∂= ∇ ∇ = ∇ ∇

∂ ∂ (2.36)

where ADα and ADγ 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 coexist with a relative amount given by the phase field parameters φi and φj. The local

Chapter 2

24

composition ( ),Ax r t becomes a continuous variable in r through the interface and it splits into

the austenite and ferrite composition, ( ),Ax r tγ and ( ),Ax r tα , as

( ) ( ) ( ) ( ) ( ), , , , ,A A Ai jx r t r t x r t r t x r tα γφ φ= + (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

( )1A A Ai ix x xα γφ φ= + − (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

( )1A

A A A ki i

x D x D xt α α γ γφ φ∂ = ∇ ∇ + − ∇ ∂

(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 AeqA Aeq

x xk kx 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 AeqAA A

i iAeqi

x kx D xt k

φ φφ

−∂ = ∇ ∇ − ∇ ∂ + − (2.41)

with

( ) ( )( )

*

1 1

A Aeq A Ai aA

i Aeqi

D k D DD

kγ γφ

φφ

+ −=

+ − (2.42)

For an Fe-C-Mn system the local composition x1…xM is represented by the carbon and manganese concentration, xC and xMn. Since the diffusivity of Mn atoms is much lower than

Chapter 2

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 eqCC C

i iC eqi

x kx D xt k

φ φφ

−∂ = ∇ ∇ − ∇ ∂ + − (2.43)

with

( ) ( )( )

*

1 1

C C eq C Ci aC

i C eqi

D k D DD

kγ γφ

φφ

+ −=

+ − (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 Cxα and Cxγ 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 60.0

0.2

0.4

0.6

0.8

1.0η φ

xC

xCγ

xCα

distance along the normal to the interface

φ

0.00

0.05

0.10

0.15

0.20

0.25

xC, x

Cγ , x

Cα (w

t %)

Figure 2.4 Carbon distribution in austenite, Cxγ , in ferrite, Cxα , and overall carbon concentration, xC, calculated by eq. (2.43)

Chapter 2

26

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, Cxγ , Cxα and Mnxγ , Mnxα can

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

( ) ( )

;1 1

C C C eqC C

C eq C eq

x x kx xk kγ αφ φ φ φ

= =+ − + −

(2.45)

( ) ( )

;1 1

Mn Mn MneqMn C

Mneq Mneq

x x kx xk kγ αφ φ 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, C eqk and Mn eqk , 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 , C Rxα , MnRxα and C Rxγ , MnRxγ is

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

pseudo-phase diagram for Mn MnRx xα= and the Fe-Mn pseudo phase diagram for C C Rx xα= . In

the same way the boundary line for the γ phase is linearised in the Fe-C pseudo-phase diagram

Chapter 2

27

for Mn MnRx xγ= and the Fe-Mn pseudo phase diagram for C C Rx xγ= . The derived slopes are Fe Cmα

− , Fe Mnmα− , Fe Cmγ

− and Fe Mnmγ− .

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, Ceqix and Mneq

ix with i = α, γ, is then given by

( )R

Ceq C Ri i Fe C

i

T Tx x

m −

−= + (2.47)

( )R

Mneq MnRi i Fe Mn

i

T Tx 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 ( ), ,C MnG x x T∆ is calculated using the

undercooling

Chapter 2

28

eqT T T∆ = − (2.49) where the local equilibrium temperature, Teq is given by

( ) ( ){0.5eq R Fe C C C R Fe Mn Mn MnRT T m x x m x xα α α α α α− − = + − + − +

( ) ( ) }}Fe C C C R Fe Mn Mn MnRm x x m x xγ γ γ γ γ γ− − + − + − (2.50)

The driving pressure is then expressed by

( ), ,C MnG 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 Ceqk . 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

( ) ( ) ( ){ }, 0.5C PE R Fe C C C R Fe C C C RaG x T S T m x x m x x Tγ γ γ α α

− − ∆ = ∆ + − + − − (2.52)

where C Rxγ and C Rxα is the equilibrium carbon content of austenite and ferrite at TR, and Fe Cmγ

− and Fe Cmα− are the slopes of the boundary lines of the γ and α phase linearised at TR in

Chapter 2

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 2.0900

950

1000

1050

1100

0.00 0.02 0.04900

950

1000

1050

1100

γ

α + γ

linearised γ lineγ line

Tem

pera

ture

(K)

wt% C

α

α + γ

linearised α lineα 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.

In the early study on phase field modelling in this thesis the austenite to ferrite transformation kinetics, reported in Chapter 3 and Chapter 4, we used the approached described in section 2.4.1 to derive the driving pressure for the transformation. Further studies showed that if the driving pressure is derived under para-equilibrium conditions, as described in section 2.4.2, a better agreement with the experimental data is obtained in the latest stage of the transformation when equilibrium is approached. Therefore in more recent studies, described in Chapter 5, Chapter 6 and Chapter 7, the transformation kinetics was modeled under para-equilibrium conditions.

Chapter 2

30

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 11. I. Steinbach, F. Pezzolla, B. Nestler, M. Seeβelberg, R. Prieler, G.J. Schmitz and J.L.L.

Rezende, Physica D, 94 (1996) 135-47

Chapter 2

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 14. S.G. Kim, W.T. Kim, T. Suzuki, Physical Review E, 58 (1998) 3316-23 15. B. Sundman, B. Jansson, J-O. Andersson, CALPHAD, 9 (1985) 153-90

Chapter 2

32

Chapter 3

33

Chapter 3

Analysis of the austenite to ferrite transformation in a C-Mn steel by phase field modelling

Abstract This chapter deals with the austenite (γ) decomposition to ferrite (α) during cooling of a Fe-0.10 C, 0.49 Mn (wt%) steel. A phase field model is used to simulate this transformation. The model provides qualitative information on the microstructure that develops on cooling, and quantitative data on both the ferrite fraction formed and the carbon concentration profile in the remaining austenite. The initial austenitic microstructure and the ferrite nucleation data, derived by metallographic examination and dilatometry, are set as input data of the model. The interface mobility is used as a fitting parameter to optimise the agreement between the simulated and experimental ferrite fraction curve derived by dilatometry. A good agreement between the simulated α-γ microstructure and the actual α-pearlite microstructure observed after cooling is obtained. The derived carbon distribution in austenite during transformation provides comprehension of the nature of the transformation with respect to the interface-controlled or diffusion-controlled mode. It is found that at the initial stage the transformation is predominantly interface-controlled but gradually a shift towards diffusion control takes place to a degree that depends on cooling rate.

Chapter 3

34

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]. The effect of both the interface mobility and the carbon diffusion is considered in the phase field modelling of solid-state transformations, and therefore the phase field approach can be considered as a mixed-mode model. The phase field model was originally proposed to simulate the dendritic growth in pure undercooled melts [8-11] and later it was extended to the solidification of alloys with a single solid phase [12-14] or with two solid phases [15-16]. The models developed for describing the alloy solidification may be divided into two groups depending on the definition of the free energy density of the interfacial region, which is of finite thickness, η. The first is the model proposed by Wheeler, Boettinger and McFadden (WBF model) [12], in which any point within the interfacial region is assumed to be a mixture of the liquid and solid phase with the same composition. The problem in this model is that it produces a chemical interface energy that depends on the interface thickness. Steinbach and co-workers on the other hand propose a model with a different definition of the free energy density in the interface [18-19] (multi domain model). In their model the interfacial region is assumed to be a mixture of liquid and solid phase with different compositions, defined by a constant ratio for each element. This model was augmented by Kim et al. [17] by a thermodynamic consistent derivation and it can be used for general multiphase and multicomponent problems.

Chapter 3

35

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 content in this phase. Since these two assumptions are rather approximative, in this work a

Chapter 3

36

more accurate method was used [22], which allows obtaining the ferrite and pearlite fractions, f

α and f p, 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

fV V

γα

α γ

−=

− (3.1)

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

00

3 1LV VL

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 ( eqf fα α= , eqp pf f= ) 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

( )2

2 222

iij i j j i i j i j ijj ij

Gtφ π πµ σ φ φ φ φ φ φ φ φ

η η ∂ = ∇ − ∇ + − + ∆ ∂

∑ (3.3)

Chapter 3

37

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 ( ),Cx r t

is determined by solving the equations

( ) ( );CC

C C C Cxx D x D xt t

γαα 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 CeqCC C

i iC eqi

x kx D xt k

φ φφ

−∂ = ∇ ∇ − ∇ ∂ + − (3.5)

with

( ) ( )( )

*

1 1

C Ceq C Ci aC

i C eqi

D k D DD

kγ γφ

φφ

+ −=

+ − (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 MneqMn Mneq

x xk kx 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 ( C Rxγ = 0.113 wt%, C Rax = 0.005

wt%, MnRxγ = 0.513 wt%, MnRax = 0.261 wt%, mγ

Fe-C = –323.73 K/wt %, mαFe-C = –-

10307.27 K/wt %, mγFe-Mn = –34.19 K/wt %, mα

Fe-Mn = –89.93 K/wt %).

Chapter 3

38

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 µm2 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)= µ0

αγ 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

DL

vγ= (see

eq.(2.45)) (Figure 2.2), to avoid numerical instability problems. Since Ld decreases as the cooling rate increases, also η has to be decreased.

Chapter 3

39

The carbon diffusivity values in γ and α, CDγ and CDα are taken from the literature [27].

3.4 Results The phase field simulations provide qualitative information on the microstructure that develops and quantitative information on both the fraction transformed (taken to be equal to the area fraction transformed in this 2D simulation) and the carbon concentration profile in the remaining austenite. These three aspects will be presented in turn and compared to the experimental results. Figure 3.1 shows the measured ferrite volume fraction as a function of temperature for the three cooling rates (solid lines). The experimental reproducibility is shown to be quite good. At low and medium cooling rates the experimental transformation curves start to overlap at a fraction of about 0.5 indicating that a situation close to equilibrium is reached at the later stages of the transformation. The experimental curves at 10 K/s are never close to the equilibrium state, as expected for such a cooling rate. The best-fit phase field simulations (lines) are also added in Figure 3.1, together with the equilibrium ferrite fraction as derived by the linearised phase diagram. The parameter values used for these fits are listed in Table 3.1. The equilibrium ferrite fraction as predicted by the linearised phase diagram is lower than the experimental ferrite fraction at 0.05 and 0.4 K/s in the latest stage of transformation when equilibrium is approached. A possible reason for this temperature shift is that in reality the transformation occurs without bulk partitioning of Mn. Therefore, the equilibrium ferrite fraction is also calculated assuming that C and Mn are in equilibrium only at the interface (local equilibrium), while far away from the interface the Mn content is the same in both phases, equal to the alloy composition (local equilibrium with negligible partitioning, LENP). The ferrite fraction under this condition is displayed in Figure 3.1 (dotted line). The fact that this curve is at lower temperature with respect to the full equilibrium curve, as is the experimental equilibrium curve, supports the assumption that Mn does not partition during transformation. At low and medium cooling rates the simulated transformation curves coincide at a fraction of about 0.75, which is larger than the intersection value of the experimental curves. For all cooling rates it was possible to get an acceptably good fit between the

Chapter 3

40

calculated and measured curves, except for the low temperature range for the highest cooling rate.

900 950 1000 1050 1100 11500.0

0.2

0.4

0.6

0.8

1.0

LENP

Equilibrium

10 Ks-1

0.4 Ks-10.05 Ks-1

Ferri

te fr

actio

n

T (K)

Figure 3.1 Experimental transformation curves, (markers) and best fitting phase field predictions (lines) at different cooling rates; the dashed and dot curves represent the ferrite fractions determined in the ortho equilibrium and LENP condition respectively. The parameters of the phase field fits are given in Table 3.I.

Figure 3.2a shows the microstructure development at low cooling rate using the parameter values given in Table 3.1. Cooling rate Nucleation undercooling (K) Nucleus density Interface mobility pre- (K/s) triple points grain boundaries n (m-2) factor, µ0

αγ (m4 s-1 J-1) 0.05 15 - 9.1 x 107 2 x 10-7 0.4 35 - 2.8 x 109 2 x 10-7 10 45 70 1.8 x 1010 6 x 10-7

Table 3.1 Parameter values used in the phase field simulations that best fit the experimental ferrite fraction curves at each cooling rate (Figure 3.1)

Chapter 3

41

At this condition, nucleation is very rare and ferrite formation only occurs incidentally in some of the austenite grains. In the simulation the grains grow more or less spherically but the detailed shape reflects the austenitic microstructure present. Recent in-situ Laser Scanning Confocal Microscopy measurements of the austenite to ferrite transformation [28] have also indicated that prior austenite grain boundaries can guide the transformation front in the manner reflected in the figure. The final ferrite grain size (≈ 90 µm) is much larger than the austenite grain size (20 µm), as expected for such a low cooling rate.

Figure 3.2 Evolution of the microstructure (a) and carbon distribution (b), at four temperatures on cooling at 0.05 K/s; n = 9.07 x 107 m-2 and µ0

αγ = 2 x 10-7 m4 s-1 J-1. For the colour version refer to the appendix.

Figure 3.2b shows the corresponding carbon concentration during the transformation. The figure correctly shows the increase in austenitic carbon concentration with increasing fraction transformed. As can be derived from the uniform colour in the profiles, the carbon concentration gradient in the austenite is small, in accordance with the low cooling rate applied, indicating that the transient state of transformation is not far from the equilibrium state. Figure 3.3a presents the microstructure development during cooling at the medium rate using the simulation parameters values indicated in Table 3.1. Note that a smaller area is simulated in this case. The nucleus density is about 30 times higher than for figure 3.2a. The microstructure shows the expected behaviour. The ferrite nucleates at several triple

a b

Chapter 3

42

points simultaneously and each grain grows approximately isotropically. The growth rate along the grain boundary is higher than in the bulk, leading to a slight distortion of the round shape. The final ferrite grain size (≈ 20 µm) is approximately equal to the initial austenite grain size (20 µm). Upon hard impact a straight interface between impacting grains develops. At later stages of the transformation small austenite islands are trapped between the ferritic grains. These austenite islands will transform to pearlite islands upon further cooling but this effect is not included in the simulation.

Figure 3.3 Evolution of the microstructure (a) and carbon distribution (b), at four temperatures on cooling at 0.4 K/s; n = 2.84 x 109 m-2 and µ0

αγ = 2 x 10-7 m4 s-1 J-1. For the colour version refer to the appendix

Figure 3.3b shows the corresponding carbon concentration profiles. The figure correctly reflects the increase in austenitic carbon concentration with increasing fraction transformed. The carbon concentration gradients in the austenite grains are still small, in accordance with the medium cooling rate applied, indicating that the transient state of transformation is still not deviating strongly from the equilibrium state. Figure 3.4a displays the microstructure development at the highest cooling rate of 10 K/s. For this cooling rate (and higher nucleation rate) the transformation proceeds via decoration of the grain boundary by multiple ferrite grains of similar size. As nucleation at the triple points is easier than in the grain boundaries, the ferrite grains at the former triple

a b

Chapter 3

43

points are somewhat larger than those formed at the grain boundaries. The undercooling for the nucleation at the grain boundaries was set at a higher value than that at the triple points (Table 3.1). The grain boundaries between ferrite grains are perpendicular to the former austenite grain boundaries.

Figure 3.4 Evolution of the microstructure (a) and carbon distribution (b), at four temperatures on cooling at 10 K/s; n = 1.78 x 1010 m-2 and µ0

αγ = 6 x 10-7 m4 s-1 J-1. For the colour version refer to the appendix

The microstructure reflects the microstructure developing during cooling at this rate as derived from microstructures observed after interrupt quenching. The final ferrite grain size is considerably smaller than the austenite grain size. The carbon concentration maps are plotted in figure 3.4b. Now the maps clearly show carbon concentration gradients in trapped austenite islands, in accordance with expectations. In Figure 3.5 we show the actual microstructures observed after the three cooling rates imposed. The micrographs show similar microstructures as calculated in the simulations, including similar differences in the final ferrite grain size.

a b

Chapter 3

44

a b

c

Figure 3.5 Microstructures after cooling at 0.05 K/s (a), 0.4 K/s (b) and 10 K/s (c)

The simulated carbon concentration profiles along the AA lines in Figure 3.2c, 3.3c and 3.4c are shown respectively in Figures 3.6a, 3.6b and 3.6c. In all data sets we see that the maximum carbon concentration (at the interface) increases with transformation. This increase is the result of both the carbon enrichment and the decreasing temperature leading to higher local equilibrium values at the interface (given by the open circles).

Chapter 3

45

Figure 3.6 Carbon distribution along the line AA in Figure 3.2b, 3.3b, 3.4b at different temperatures on cooling at 0.05 K/s (a), 0.4K/s (b), 10 K/s (c) (solid lines). Comparison with the carbon content in austenite at the interface as predicted by the diffusion controlled and interface mobility-controlled model (markers)

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

wt %

Cdistance along AA (µm)

0 2 4 6 8 10 120.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

wt %

C

distance along AA (µm)

Diffusion controlled model Interface mobility controlled

mode

0 2 4 6 8 10 120.00

0.05

0.10

0.15

0.20

0.25

0.30

wt %

C

distance along AA (µm)

a b

c

Chapter 3

46

Figures 3.6a and 3.6b show that the carbon concentration profiles in the austenite are relatively flat, as is to be expected for conditions close to equilibrium. Figure 2.8c indicates well developed carbon concentration gradients as well as the first signs of overall enrichment at the later stages. The carbon concentrations calculated at the interface in the case of either diffusion control or interface control are also added in Figure 3.6 to provide insight in the nature of phase transformation with respect to the interface-controlled or diffusion-controlled mode. This aspect will be analysed in more detail in the discussion.

3.5 Discussions A major advantage of the phase field model over other physical transformation models, prescribing the morphology of the forming phase, is its ability to allow very different morphologies to form, depending on the transformation conditions. Hence it is of interest to compare the predicted microstructures with the experimental microstructures. For the lowest cooling rate of 0.05 K/s the transformed microstructure is shown in Figure 3.2. The figure shows that for this low cooling rate, and the corresponding low nucleation site density, the transformation does not start in each of the parent austenite grains. Once the transformation starts at a few isolated locations, the carbon level in the remaining austenite is increased (as diffusion is faster than the rate of interface migration) preventing the activation of other dormant nuclei. For these conditions most of the austenite grains transform fully to ferrite and the carbon is transported from grain to grain until it accumulates in the last remaining austenite regions. Here the austenite to ferrite transformation stops and the enriched regions will then transform into pearlite. It is clear that for these conditions alignment of initial nucleation sites in bands will lead to the formation of pearlite bands. Such an alignment of nucleation sites in bands is to be expected for hot-rolled carbon manganese steels when slab reheating conditions did not lead to the total disappearance of the Mn segregation, as in the steel of figure 3.5a. Hence the experimental microstructure for this low cooling rate (fig 3.5a) is in very good agreement with the phase field simulation, the only difference being the size of the ferrite grains. For the medium cooling rate of 0.4 K/s the model yields a microstructure in which the ferrite grain size is approximately equal to the parent austenite grain size, leading to the formation of relatively large and angular pearlite islands at the edges of the former

Chapter 3

47

austenite grains (figure 3.3a). The sizes of the ferrite grains and the austenite islands are more or less equal when the original austenite grain size does not vary too much. These results are in good agreement with the experimental observations (fig 3.5b). Finally, for the highest cooling rate of 10 K/s the model microstructure consists of many fine ferrite grains surrounding pearlite colonies of a more or less constant size, at the centre of the original parent austenite grain size. The ferrite grain size is significantly smaller than the austenite grain size, but only one pearlite colony is formed per austenite grain. These simulation results are again in good agreement with the observations (fig 3.5c), except that the pearlite colony is less rounded than predicted by the model. Hence, it may be concluded that the phase field model consistently reproduces the variety in microstructure observed experimentally. We now turn to the comparison of the simulated transformation kinetics and the experimental transformation kinetics as determined by dilatometry (Figure 3.1). As indicated in section 3.4 the agreement is qualitatively rather good. It is interesting to note that the best agreement is obtained for the 0.4 K/s cooling rate. For this cooling rate a uniform microstructure, with a random distribution of the pearlite colonies seems to form. Such a microstructure is the implicitly assumed and necessary microstructure for the derivation of the fraction curve from the dilation curve. A recent work on the analysis of γ → α transformation using dilatometry [29] has shown that if a banded ferrite/pearlite microstructure is formed after cooling, the method used for deriving the ferrite fraction curve from the dilatometric signal induces systematic errors. The modest fit at the cooling rate of 0.05 K/s for ferrite fractions of 0.3-0.5 might be due to the effect of banding. At this cooling rate a banded ferrite-pearlite microstructure is indeed formed after cooling (Figure 3.5a). For 10 K/s cooling rate the fit has been optimised at the early stages of the transformation where the fit is very good. However, at higher fractions transformed the simulation underestimates the actual transformation rate. Such a misfit could be solved by introducing intragranular nuclei in the austenite in the model. However, the observed uniformity in ferrite grain size suggests that all nucleation events occurred more or less simultaneously. Hence it seems that the temperature dependence of the interfacial mobility is not properly treated in the model yet. The phase field model also enables the carbon profile perpendicular to the moving interface to be monitored as shown in figure 3.6 for the three cooling conditions imposed. The curves show a steady increase in maximum interfacial carbon concentration with

Chapter 3

48

progression of the transformation. The increase is related to both the decrease in temperature and the decrease in austenite fraction. In case the transformation rate is purely controlled by the rate of carbon diffusion (diffusion controlled model), local equilibrium at the interface should be present at all times and the corresponding carbon concentration can be derived from the phase diagram. The predicted local equilibrium values are plotted in figure 3.6 (open symbols). The phase field predictions are always below the predictions of the diffusion-controlled model, except at low and medium cooling rates in the latest stage of transformation, when the microstructure approaches the equilibrium state. An alternative approach is to assume the interface mobility to be rate limiting and the carbon diffusion to be relatively rapid (interface controlled model). The carbon

concentration in austenite as predicted by the interface-controlled model, Cxγ is calculated

from the equation

0

1

C eqC x f xx

fα α

γα

−=

− (3.8)

assuming that the carbon transferred from ferrite to austenite is redistributed instantaneously, and that the carbon concentration in ferrite is equal to the equilibrium

value Ceqxα [4]. In eq.(3.8) x0 is the steel carbon content and f α is the ferrite fraction

obtained in the present simulation. These concentrations, calculated by eq. (3.8), are also indicated in figure 3.6 (closed symbols). The actual carbon concentration is quite close to the concentration for the interface-controlled mode at the initial stages. Indeed, just after nucleation, the small amount of carbon atoms rejected from the ferrite phase can easily diffuse into the bulk of austenite, and the lattice transformation is the rate-controlling process. As the ferrite grains grow, the amount of carbon expelled from ferrite increases and carbon diffusivity in the bulk becomes also important: the transformation kinetics evolves towards a mixed-mode character. Mixed mode models [6-7] assume that the interfacial concentration is determined by both the interfacial mobility and the carbon diffusivity. In a very recent paper, Sietsma and Van der Zwaag [7] have analysed the development of the interfacial concentration under mixed-mode conditions. The authors have shown that the phase transformation is interface mobility controlled at the early stages of transformation and shifts towards diffusion controlled in the later stages. This observation holds for all three cooling rates. Figure 3.6b, for medium cooling rate, clearly shows that the carbon

Chapter 3

49

concentrations for diffusion control and interface-mobility control approach each other when the transformation proceeds, which is a consequence of the phase distribution approaching equilibrium. This is also the case for low cooling rate, but is not shown in Figure 3.6a. In these conditions the distinction between interface-mobility control, diffusion control and mixed mode can no longer be made. Finally we address the parameter values used in the model calculations. The parameter values are listed in Table 3.1. As the understanding of nucleation is still insufficient to predict the nucleation undercooling as a function of steel composition and cooling rate, we used values based on the experimental dilatometry curves. The trend of increasing undercooling with increasing cooling rate is as expected and typical for the steel and the conditions. The nucleus density values increase significantly, from a relatively low value of 9.1 x 107 m-2 at low cooling rate to a value of 1.8 x 1010 m-2 at the highest cooling rate. These data should be compared to changes in nucleus densities observed experimentally at different transformation temperature using in situ neutron depolarisation measurements [30]. The measured nucleus densities varied from 7.6 x 1013 m-3 at 1000 K to 1.8 x 1014 m-3 at 940 K. The interfacial mobility pre-exponential factor equals to 2 x 10-7 m4s-1J-1 for low and medium cooling rate (Table 3.1). For high cooling rate a higher value (6 x 10-7 m4s-1J-1) has been used. Although only a constant value would have been physically correct, the required variation in the pre-factor value is considered to be small, and well within the uncertainty of the model. An increased interface mobility was necessary to balance the impossibility to set in 2 D simulation the nucleus density value expected at this cooling rate.

3.6 Conclusions The phase field model is able to describe the γ → α transformation kinetics in a C-Mn steel during cooling. It provides qualitative information on the microstructure that develops on cooling and quantitative data on both the ferrite fraction formed and the carbon concentration profile in the remaining austenite. An acceptable agreement is obtained between the experimental and simulated ferrite fraction curves at low and medium cooling rate, setting a cooling rate independent interface mobility and increasing the nucleus density as the cooling rate increases. At high cooling rate the fit has been optimised at the early stages of the transformation where the

Chapter 3

50

fit is very good. However, at higher fractions transformed the prediction underestimates the actual transformation. The introduction in the model of intragranular nuclei in the austenite could solve such a misfit. The simulated microstructures at different cooling rates also reproduce quite well the variety in microstructures observed experimentally. The ferrite grain size and morphology as well as the distribution of pearlite colonies (the remaining austenite regions in the model) observed experimentally are well simulated by the model. The derived carbon profiles in austenite show an increase in the austenite carbon content at the interface with progression of transformation, most pronounced at the highest cooling rate. It also provides insight in the nature of phase transformation with respect to the interface-controlled or diffusion-controlled mode. It is found that at the initial stages of the transformation is always nearly interface-controlled; then it evolves in different way depending on the morphologies formed on cooling and on the way the equilibrium is approached. References 1. C. Zener, J. Appl. Phys., 20 (1949) 950-53 2. H.K.D.H. Bhadeshia, L.E. Svensson, B. Gretoft, Acta Metall., 33 (1985) 1271-83 3. R.A. Vandermeer, Acta Metall. Mater. 38 (1990) 2461-70 4. Y. van Leeuwen, T.A. Kop, J. Sietsma, S. van der Zwaag, J. Phys IV France, 9 (1999)

401-409 5. T.A. Kop, Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 40 (2000) 713-18 6. Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int., 43 (2003) 767-73 7. J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143-4152 8. G. Caginalp, Physical Review A, 39 (1989) 5887-96 9. R. Kobayashy, Physica D, 66 (1993) 410-23 10. J. B. McFadden, A.A. Wheeler, R.J. Braun., S.R. Coriell, R.F. Sekerka, Physical

Review E, 48 (1993) 2016-24 11. J.B. Collins, H.Levine, Physical Review B, 31 (1985) 6119-22 12. A.A. Wheeler, W.J. Boettinger, G.B. McFadden, Physical Review A, 45 (1992) 7424-

39 13. G. Caginalp, W. Xie, Physical Review E, 48 (1993) 1897-1909 14. J.A. Warren, W.J. Boettinger, Acta Metallurgica Materialia, 43 (1996) 689-703

Chapter 3

51

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

16. A. Karma, Physical Review E, 49 (1994) 2245-50 17. S.G. Kim, W.T. Kim, T. Suzuki, Physical Review E, 58 (1998) 3316-23 18. I. Steinbach, F. Pezzolla, B. Nestler, M. Seeβelberg, R. Prieler, G.J. Schmitz and J.L.L.

Rezende, Physica D, 94 (1996) 135-47 19. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998) 73-86 20. D.H. Yeon, P.R. Cha and J. K. Yoon, Scripta Materialia, 45 (2001) 661-68 21. G. Pariser, P. Shaffnit, I. Steinbach, W. Bleck, Steel Research, 72 (2001) 354-60 22. T.A. Kop, J. Sietsma, S. van der Zwaag, J. Mat. Science, 36 (2001) 519-26 23. B. Sundman, B. Jansson, J-O. Andersson, CALPHAD, 9 (1985) 153-90 24. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn., 14 (1998) 10-18 25. M. Militzer, Material Science & Technology 2003, Austenite Formation and

Decomposition, Chicago, (2003) 195-211 26. W.F. Lange, M. Enomoto, H.I. Aaronson, Metall. Trans A, 19 (1988) 427-40 27. Handbook of Chemistry and Physics, CRC Press Inc. Boca Raton, 1989 28. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, M. Apel, P.Schaffnit, I. Steinbach,

Material Science & Technology 2003, Austenite Formation and Decomposition, Chicago, (2003) 353-66

29. T.A. Kop, J. Sietsma, S. van der Zwaag, Mater. Sci. Techn., 17 (2001) 1569-74 30. S.G.E. te Velthuis, M.T. Rekveldt, J. Sietsma, S. van der Zwaag, Physica B, 241-243

(1998) 1234-1236

Chapter 3

52

Chapter 4

53

Chapter 4

Analysis of austenite to ferrite transformation in a Nb micro-alloyed C-Mn steel

by phase field modelling

Abstract The phase field model is used to simulate the γ → α transformation in a Nb microalloyed C-Mn steel during cooling from different austenitisation temperatures. The initial austenitic microstructure and the nucleation conditions, derived by metallographic tests, are set as input data of the model. The interface mobility is taken as representative for the effect of NbC on the phase transformation. It is used as a fitting parameter to optimise the agreement between the experimental ferrite fraction curve, obtained by dilatometry, and the simulated ferrite fraction. A decrease of the γ/α interface mobility is found when the austenitisation temperature decreases from 1373 to 1173 K as a consequence of the presence of NbC precipitates during austenitisation. The presence of NbC has a distinct effect on the nature of the phase transformation with respect to the interface-controlled or diffusion-controlled mode.

Chapter 4

54

4.1 Introduction The properties of C-Mn steels strongly depend on the microstructure, which forms as a result of austenite (γ) decomposition during cooling from the hot rolling temperature. Upon cooling from the austenitic temperature range the proeutectoid ferrite (α) formation is usually the first to occur and has a strong influence on the subsequent diffusional or partly diffusional phase transformation products (pearlite, bainite). For this reason the γ to α transformation is of great interest in the steel production. Generally speaking, the γ to α transformation occurs in two steps: nucleation and growth. Upon nucleation a new interface is generated that separates the product α phase from the parent γ phase. This interface migrates into the surrounding parent phase during subsequent growth. In a recent work on ferrite nucleation in a C-Mn steel, Offerman et al. [1] showed that the number of ferrite nuclei increases rapidly during cooling just below the nucleation start temperature and thus new ferrite grains continuously form over a temperature range of about 50 K. The migration rate of the interface is determined by the diffusion of the alloying elements (mainly C and Mn) ahead of the interface and the intrinsic interface mobility, which is connected with the structural rearrangement (γ(FCC)→α(BCC)). The γ to α transformation is often classified as either diffusion controlled [2-4] or interface mobility controlled [5-6], depending on whether the majority of the free energy accompanying the transformation is dissipated by the solute diffusion in the bulk phases or by the structural rearrangement. In reality both processes will influence the transformation kinetics and the transformation can be expected to be of a mixed mode character [7-8]. In Nb micro-alloyed C-Mn steel the presence of Nb strongly affects the γ to α transformation. Already a very low Nb addition retards the ferrite formation considerably. It seems reasonable to believe that solute Nb segregates to the austenite/ferrite interface and reduces the ferrite growth kinetics due to solute drag effects [9]. It has been established that the precipitation of niobium carbide (NbC) during the austenitisation treatment refines the γ structure and promotes the occurrence of austenite to ferrite transformation at higher temperatures [10]. The precipitation of NbC during austenitisation could also have an effect on the γ decomposition kinetics [11]. Few studies have been done in this field. However there is some evidence, derived from recrystallisation studies, [12] suggesting that NbC formed during austenitisation can retard the progress of transformation by pinning the γ/α interface.

Chapter 4

55

This work deals with the kinetics of the austenite to ferrite transformation during controlled cooling of a niobium-micro-alloyed C-Mn steel. The transformation kinetics is investigated experimentally using dilatometry on both C-Mn-Nb steel and an analogous C-Mn steel. The phase field model derived by Steinbach and co-workers [13-14] is used to simulate the γ to α transformation during controlled cooling. Phase field simulations are performed at the same cooling conditions as the experimental tests. The initial austenitic microstructure and the ferrite nucleation data, derived by metallographic examination, are set as input data of the model. The effective interface mobility is taken as representative for the effect of NbC on the phase transformation. It is used as a fitting parameter to optimise the agreement between the simulated and experimental ferrite fraction curve derived by dilatometry. The effect of the solute and precipitated Nb on the apparent α/γ interface mobility is investigated.

4.2 Materials Two steels are investigated in this work: the first is a Nb-containing steel, referred to as C-Mn-Nb steel, the second is a Nb-free steel, referred to as C-Mn steel. Their chemical compositions are given in Table 4.1. The Thermo-Calc software [15] was used to calculate the equilibrium phase fractions as a function of temperature. The results obtained are reported in Figure 4.1. In the calculations we assume that the niobium nitride formation does not play a role in the transformation kinetics of this steel, due to the presence of Al, which forms nitrides and removes all solute nitrogen in the steel. C Mn Nb Si Ni Al N C-Mn-Nb 0.136 1.200 0.020 0.017 0.027 0.053 0.004 C-Mn 0.170 0.926 0.000 0.003 0.023 0.041 0.003

Table 4.1 Chemical composition of the analysed steels (wt %)

Chapter 4

56

900 1000 1100 1200 1300 14000.0

0.2

0.4

0.6

0.8

1.0

NbC

C-Mn-Nb steel

C-Mn steel

ferrite NbC

Temperature (K)

Ferri

te fr

actio

n

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

NbC fraction

Figure 4.1 Mole fraction of the phases present in equilibrium in the investigated steels as a function of temperature

Niobium carbide precipitates in the C-Mn-Nb steel below 1342 K. The C-Mn steel has been chosen because the equilibrium fraction of ferrite as a function of temperature is identical to that of the C-Mn-Nb steel. The transition temperature, Ae3, has the values 1092 K for the C-Mn-Nb steel and 1091 K for the C-Mn steel. The cementite precipitation kinetics was not analysed in the present work and for this reason the equilibrium cementite fraction is not reported in the figure.

4.3 Simulations conditions 2D phase field simulations of the γ to α transformation during continuous cooling were performed for a Fe-C-Mn ternary alloy of the composition given in Table 4.1. Due to the low Nb content in the C-Mn-Nb steel, the niobium carbide precipitation causes only a negligible change in the carbon content; hence niobium is not considered as element of the system, and NbC precipitates are not included in the simulation as a distinct phase. The effect of niobium on the transformation kinetics during cooling is taken into account by adjusting the mobility values to fit the experimental ferrite fraction curves.

Chapter 4

57

The initial austenitic microstructure was generated by means of a Voronoi tessellation, the mathematical algorithm able to best reproduce the microstructure observed in polycrystalline materials. Periodic boundary conditions were set. The calculation domain size, A, and the number of austenite grains in the domain were set depending on the initial austenite grain size, which was experimentally determined. Since the grain-boundary curvature and its effect on grain-boundary motion is incorporated in phase-field modelling, grain growth of ferrite [17] is in principle accounted for. However, in the processing conditions that are regarded in the present study, no significant grain-growth effects have been observed. Possible clustering of nuclei or very small grains in the early stages of the transformation cannot be studied with the present modelling technique. Consequently, the ferrite nucleus density in the simulations has been chosen in agreement with the experimentally observed grain size, using the relation

2

4

f

dα

α

ρ π= (4.1)

where dα is the measured ferrite grain size and fα is the final ferrite fraction (0.80 and 0.76 respectively for C-Mn-Nb steel and C-Mn steel). The total number of nuclei in the calculation domains was equal to the integer value of NSA, which, divided by A, gives the nucleus density values used in the simulation, ρS (see Table 4.2).

TA (K) 1173 1273 1373 C-Mn-Nb steel 16.3 x109 m-2 10.0 x 109 m-2 1.1 x 109 m-2

C-Mn steel 6.7 x 109 m-2 2.7 x 109 m-2 1.5 x 109 m-2

Table 4.2 Nucleus densities, ρS, used in the simulations at different austenitisation temperatures

We assume that nucleation takes place preferentially at the triple points and subsequently at the grain boundary. However, the choice of the particular nucleation sites does not have a significant effect on the generated final microstructure. Following the work of Offerman et al. [1], we assumed that the ferrite nuclei form continuously over a temperature range of about 50 K below the nucleation start temperature. Figure 4.2 shows the nucleation distribution used in the simulation for the austenitisation temperature of 1273 K, compared

Chapter 4

58

to Offerman’s experimental data. The temperature of the start of nucleation was set equal to the ferrite start temperature estimated from the experimental ferrite fraction curve.

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

num

ber o

f nuc

lei n

orm

alise

dto

the

max

imum

val

ue

Undercooling respect to the start of nucleation (K)

experimental data [1] simulation data for TA = 1273 K

Figure 4.2 Nucleation distribution used in the simulation for the austenitisation temperature of 1273 K compared to Offerman’s experimental data [1].

We assume that the α/γ interface mobility, µαγ, is temperature dependent, according to the relation µαγ(T)= µ0

αγ exp(−Qµ/RT). In the present calculations the activation energy Qµ was set equal to 140 kJ/mol, the value used by Krielaart and Van der Zwaag in a study on the transformation behavior of binary Fe–Mn alloys [18]. The temperature-independent pre-exponential factor of the interface mobility was used as a fitting parameter. In order to resolve the carbon concentration gradient within the interface, the interface thickness η has to be at least 6 times the grid spacing, but at the same time, in order to avoid numerical instability problems, η has to be lower than the diffusion length of carbon in austenite. A mesh size of 0.14 µm was used in all simulations and η was set equal to 6 times the mesh size. The interface energy values [19] and the carbon diffusivity values in γ and α, DC

γ and DCα, [20] are taken from the literature.

Chapter 4

59

4.4 Experimental procedures The γ to α transformation kinetics in the analysed steels was determined experimentally by dilatometry using a Bähr 805A/D dilatometer. The samples, 4 mm in diameter and 10 mm in length, were 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 inhomogeneity in the sample, respectively. In all tests the temperature differences along the sample length were smaller than 10 K. The samples were heated at a rate of 0.8 K/s to three different austenitisation temperatures, TA, 1173, 1273 and 1373 K, maintained at this temperature for 1200 s and then cooled to room temperature at a rate of 0.3 K/s. The ferrite fraction curves are derived from the dilatometric signal using the method described in [21], taking into account the carbon enrichment of austenite during transformation and the density difference between pearlite and ferrite. The prior austenite grain size was evaluated in samples quenched from temperatures at which 5 % of α is formed at the γ grain boundary. The ferrite grain size was determined from the microstructural analysis of the samples after continuous cooling to room temperature. All samples were etched with 2 % Nital. For the quantitative analysis of the fractions of solute and precipitated niobium in C-Mn-Nb steel as a function of the austenitisation temperature, samples were electrochemically brought into solution using an ethylene diamine tetra-acetic acid (EDTA) solution. In this process the precipitates do not dissolve. The electrolyte is filtered using a filter with a 100 nm pore size. The residue is mixed with Na2SO4. This mixture is melted and dissolved in a solution of sulphuric and oxalic acid. The niobium concentration of this solution as well as the niobium concentration in the electrolyte are determined using a Perkin Elmer Inductive Coupled Plasma – Optical Emission Spectrometer (ICP-OES).

4.5 Results and discussion Figure 4.3 shows the amount of niobium precipitated as NbC in the C-Mn-Nb steel measured as a function of the austenitisation temperature (markers). The calculated niobium content present in NbC at the equilibrium is also reported in Figure 3 (line). Although 1373 K is above the temperature at which all niobium is in solution in austenite according to equilibrium, the kinetics of dissolution of niobium carbide at this temperature

Chapter 4

60

is not fast enough to allow the complete dissolution of all niobium carbides during holding for 1200 s at 1373 K. The difference between the measured value of niobium in NbC and the theoretical value of niobium expected to precipitate in NbC at the equilibrium decreases as the austenitisation temperature decreases (the amount of niobium carbides that has to be solubilised decreases). In the remainder the experimental value will be used.

900 1000 1100 1200 1300 14000.000

0.005

0.010

0.015

0.020

wt %

Nb

in N

bC

Temperature (K)

Equilibrium (Thermo-Calc) Measured after austenitisation

and quenching

Figure 4.3 C-Mn-Nb steel: equilibrium concentration of niobium precipitated as niobium carbide (line) and measured values after austenitisation at 1173, 1273 and 1373 K and quenching to room temperature (markers)

The dependence of the austenite grain size on the austenitisation temperature, TA, is reported in Figure 4.4. In C-Mn steel the austenite grain size systematically increases with increasing austenitisation temperature. On the other hand in C-Mn-Nb steel the austenite grain size remains constant when the temperature increases from 1173 K to 1273 K, and starts to increase only above the latter temperature. This is due to the presence at low and intermediate austenitisation temperatures of niobium carbide precipitates, which pin the austenite grain boundaries. The pinning effect of the precipitates is barely present at 1373 K and the austenite grains grow therefore more rapidly at this temperature.

Chapter 4

61

1150 1200 1250 1300 1350 14000

20

40

60

80

100

120

140

160 C-Mn-Nb steel C-Mn steel

auste

nite

gra

in si

ze, d

γ (µm

)

austenitisation temperature (K)

Figure 4.4 Austenite grain size as a function of the austenitisation temperature Figure 4.5 shows the measured ferrite volume fraction as a function of temperature derived by dilatometry for different austenitisation temperatures.

850 900 950 1000 10500.0

0.2

0.4

0.6

0.8

1.0

1273 K

1173 K1373 K

C-Mn steel

C-Mn-Nb steel

ferri

te fr

actio

n

Temperature (K)

Figure 4.5 Experimental transformation curves during cooling from different austenitisation temperatures in the C-Mn-Nb steel (black markers) and in the C-Mn steel (gray markers).

Chapter 4

62

The ferrite start temperature for both steels is reported as a function of the austenite grain size in Figure 4.6. In both steels the ferrite start temperature decreases with increasing prior austenite grain size (fewer ferrite nucleation sites are present). In the Nb-containing steel the ferrite start temperature as well as the austenite grain size remain unchanged when the austenitisation temperature increases from 1173 K to 1273 K. However as soon as the austenite grain size starts to increase (dγ= 97 µm for TA = 1373 K), the decrease of the ferrite start temperature is more pronounced than in the Nb-free steel. The niobium in solution is held responsible for the additional delay of the austenite decomposition. The retarding effect of dissolved Nb on the γ to α transformation is also reported in a work of Lee et al. [22].

0 20 40 60 80 100 120 140 160

1020

1025

1030

1035

1040

1045

1050

ferri

te st

art t

empe

ratu

re, T

S (K)

austenite grain size, dγ (µm)

C-Mn-Nb steel C-Mn steel

Figure 4.6 Ferrite start temperatures of the investigated steels as a function of the austenite grain size

Figure 4.7 shows the micrographs of the samples of the C-Mn-Nb steel and C-Mn steel respectively after dilatometric tests at the three different austenitisation temperatures. The microstructure of the C-Mn-Nb steel observed after cooling from 1173 K and from 1273 K consists of ferrite plus pearlite (Figure 4.7 a-b); when the austenitisation temperature increases to 1373 K some Widmanstätten ferrite is also detected in samples after cooling (Figure 4.7c). In the C-Mn steel some Widmanstätten ferrite is present in the samples quenched from 1273 K and from 1373 K (Figure 4.7e-f).

Chapter 4

63

Figure 4.7 Microstructures of the C-Mn-Nb steel after cooling at 0.3 K/s from 1173 K (a), 1273 K (b) and 1373 K(c) and the C-Mn steel after cooling at 0.3 K/s from 1173 K (d), 1273 K (e) and 1373 K (f)

In both steels a strong tendency to band formation is observed. Due to the segregation of alloying elements, such as Mn, during casting, a banded ferrite plus pearlite microstructure

a

b

c

e

d

f

Chapter 4

64

is formed after hot rolling and controlled cooling. This anisotropy is not removed during the austenitisation treatment, in particular for the lower austenitisation temperatures (Figure 4.7a, b, d, e). As shown previously by Kop et al. [23], bands parallel to the dilatometric sample axis tend to enhance the pearlite signal in the dilatometric curve. The tendency to band formation is not taken into account in the analysis of dilatometry curves nor has it been incorporated in the phase field model. The ferrite grain sizes of both steels are reported in Figure 4.8 as a function of the austenitisation temperature. Although the C-Mn-Nb samples austenitised at 1173 K and 1273 K have the same prior austenite grain size, a finer ferrite grain size was measured in the former than in the latter. The niobium carbide precipitates at the austenite grain boundaries in the sample austenitised at 1173 K possibly act as nucleation sites for the ferrite, leading to the production of a finer ferrite grain size. The larger austenite grain sizes produced at the highest austenitisation temperature (all Nb is in solution and austenite grain growth takes place) decrease the surface area for nucleation of ferrite grains and then a coarser ferrite grain is produced. The obtained values for ρ depend on both the ferrite start temperature and the initial austenite grain size.

1150 1200 1250 1300 1350 14000

5

10

15

20

25

30

35

C-Mn steel

C-Mn-Nb steel

ferri

te g

rain

size

, dα

austenitisation temperature, K

Figure 4.8 Ferrite grain size as a function of the austenitisation temperature

Chapter 4

65

Using the experimentally derived input parameters, the austenite grain size, the undercooling and the number of nuclei, the model simulations could now be adjusted to the experimental data using µ0

αγ as the only adjustable fitting parameter.

850 900 950 1000 10500.0

0.2

0.4

0.6

0.8

1.0

TA = 1373 K

TA = 1273 K TA = 1173 K

ferri

te fr

actio

n

Temperature (K)

850 900 950 1000 10500.0

0.2

0.4

0.6

0.8

1.0

TA = 1273 K

TA = 1373 KTA = 1173 K

ferri

te fr

actio

n

Temperature (K)

Figure 4.9 Experimental transformation curves (markers) and best fitting phase field simulations (lines) at different austenitisation temperatures for the C-Mn-Nb steel (a) and for C-Mn steel (b).

b

a

Chapter 4

66

Figure 4.9 shows the measured ferrite volume fraction curves in the C-Mn-Nb steel (solid line) and the best fitting phase field simulations (markers) for different TA. The phase field model is used for ferrite formation only; the calculations are stopped at the experimental onset of pearlite formation. For all simulations the agreement with the experimental data is very good for ferrite fractions lower than 0.4. The deviation of the simulated curves from the experimental ones at higher ferrite fraction might be related to the presence of a banded microstructure at low austenitisation temperature. Geometry effects could also explain this deviation; 2D simulations predicted faster transformation kinetics than that expected in 3D because the interfacial area associated with the growth in 2D (cylindrical when translated in 3D-space) is larger than that associated with the spherical growth in 3D. The Widmanstätten ferrite formation, which is known to have faster kinetics than the allotriomorphic ferrite formation, at about 950 K in the C-Mn-Nb steel austenitised at 1373 K [24] explains why the simulated fraction underestimates the experimental ferrite in the latest stage of transformation. The Widmanstätten ferrite formation is not included in the model.

0.000 0.005 0.010 0.015 0.020

1.0x10-7

2.0x10-7

3.0x10-7

4.0x10-7

5.0x10-7

6.0x10-7 C-Mn-Nb steel C-Mn steel

µ 0 (m4 J-1

s-1)

measured wt % Nb in NbC

Figure 4.10 Best fit µ0 value as a function of the measured amount of niobium precipitated as NbC during austenitisation

Figure 4.10 shows µ0

αγ values used in the best fitting simulations as a function of the measured amount of Nb that is precipitated in NbC. The increase of precipitated niobium,

Chapter 4

67

as the austenitisation temperature decreases, causes a distinct decrease of the pre-exponential value of the interface mobility. This result is explained assuming that the fine precipitates stable in austenite during austenitisation at 1173 and 1273 K, which were not dissolved during the austenitisation treatment and were effective for the austenite grain refinement at these temperatures, also pin the α/γ interface during ferrite formation on cooling. It shows that the retarding effect of NbC is stronger than the solute-drag effect caused by Nb in solution.

1040 K

1010 K

1025 K

950 K

Figure 4.11 Evolution of the microstructure in the C-Mn-Nb steel at four temperatures on cooling from TA = 1273 K. For the colour version refer to the appendix.

Figure 4.11 represents a typical simulated microstructure of the C-Mn-Nb steel developed at different temperatures during cooling (the austenitisation temperature is 1273 K). Not all nuclei form at the same temperature but nucleation takes place over a nucleation

7 µm 7 µm

7 µm 7 µm

Chapter 4

68

temperature range of 45 K. This value affects the ferrite grain size distribution at the end of transformation; this can however not be quantified for comparison with the experimental one due to the low number of ferrite grains. The average ferrite grain size at the end of transformation (9.8 µm) is approximately equal to the experimental ferrite grain size (10 µm) and the simulated final ferrite plus austenite microstructure reproduces quite well the experimental final ferrite plus pearlite microstructure (the austenite to pearlite transformation is not included in the model).

1040 K

1010 K

1025 K

950 K

Figure 4.12 Carbon distribution in the C-Mn-Nb steel at four temperatures on cooling from TA = 1273 K. For the colour version refer to the appendix.

Figure 4.12 shows the corresponding carbon concentration during the transformation. The figure correctly shows the increase in austenitic carbon concentration with increasing fraction transformed. As can be derived from the uniform colour in the profiles, the carbon

7 µm

7 µm 7 µm

7 µm

A

A

Chapter 4

69

concentration gradient in the austenite is small, in accordance with the low cooling rate applied. The phase field model also allows monitoring the carbon concentration profile perpendicular to the moving interface and thus provides comprehension of the nature of the transformation with respect to the diffusion-controlled mode [2-4] and interface-controlled mode [5-6]. Figure 4.13a shows the carbon profile along the line AA in Figure 4.12 at subsequent stages of the transformation during cooling from 1273 K. Each profile corresponds to a specific time and temperature during transformation. The investigated temperature range corresponds to that in which a good agreement between the experimental and simulated ferrite fraction is obtained (see Figure 4.9). Figure 4.13b represents the analogous carbon profile during cooling from 1373 K. The curves show a steady increase in maximum interfacial carbon concentration with progression of the transformation. The increase is related to both the decrease in temperature and the decrease in austenite fraction. The equilibrium value of carbon in austenite for each temperature is also reported in Figure 4.13 (open symbols). Those values represent the carbon concentration at the γ/α interface assuming an infinite interface mobility, implying that the transformation is purely controlled by rate of carbon diffusion and local equilibrium at the interface is present at all times (diffusion controlled model). The carbon concentration values at the interface as predicted by the interface controlled model are also reported in Figure 4.13 (closed symbols). In this case 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. For TA equal to 1273 K the carbon concentration predicted by the phase field model is quite close to the concentration for the interface-controlled mode. In contrast for TA = 1373 K the predicted transformation kinetics has a mixed mode character. Hence the presence of Nb precipitates in austenite has an effect on the nature of the austenite to ferrite transformation during cooling; as a consequence of the decrease of the interface mobility (see Figure 4.10), the transformation kinetics changes from having mixed mode character (Figure 4.13b) to be interface mobility controlled (Figure 4.13a) upon decreasing the austenitisation temperature from 1373 K to 1273 K or 1173 K.

Chapter 4

70

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

Diffusion controlled model Interface controlled model

wt %

C

distance from the interface (µm)

a

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 Diffusion controlled model Interface controlled model

wt %

C

distance from the interface (µm)

b

Figure 4.13 Carbon distribution in austenite at different temperatures on cooling from 1273 K (a) and 1373 K (b); phase field simulations (solid lines) and carbon content at the interface as calculated by the diffusion controlled and interface controlled model (markers)

Chapter 4

71

4.6 Conclusions The phase field model is able to accurately describe a large part of the γ → α transformation kinetics in a Nb micro-alloyed C-Mn steel during cooling. An acceptable agreement is obtained between the experimental and simulated ferrite fraction curves setting the pre-exponential value of the interface mobility constant during cooling. A marked increase in the pre-exponential value of the interface mobility results when the austenitisation temperature is increased from 1173 K to 1373 K. This is due to the pinning effect that the precipitates, formed in austenite during austenitisation, have on the α/γ interface during transformation, which is stronger than solute-drag retardation by Nb in solution. The simulated microstructures from different austenitisation temperatures also reproduce the experimentally observed microstructures quite well. The Nb being either in solution or precipitated as NbC in austenite has a distinct effect on the nature of the phase transformation with respect to the interface-controlled or diffusion-controlled mode. When NbC precipitates in austenite during holding at low or medium austenitisation temperature, the austenite to ferrite transformation is always nearly interface-controlled as a consequence of the low interface mobility; on the other hand, when Nb is dissolved in austenite at high austenitisation temperature, the transformation kinetics has a mixed mode character.

References 1. S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L.Margulies,

H.F. Poulsen, M. Th. Rekveldt, S. van der Zwaag, Science, 298 (2002) 1003-05 2. C. Zener, J. Appl. Phys., 20 (1949) 950-53 3. H.K.D.H. Bhadeshia, L.E. Svensson, B. Gretoft, Acta Metall., 33 (1985) 1271-83 4. R.A. Vandermeer, Acta Metall. Mater. 38 (1990) 2461-70 5. Y. van Leeuwen, T.A. Kop, J. Sietsma, S. van der Zwaag, J. Phys IV France, 9 (1999)

401-409 6. T.A. Kop, Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 40 (2000) 713-18 7. Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int., 43 (2003) 767-73 8. J. Sietsma, S. van der Zwaag, Acta Materialia 52 (2004) 4143-4152

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9. K.J. Lee, K.B. Kang, J.K. Lee, O. Kwon, R.W. Chang, Proc. of Int. conf. On Mathematical Modelling of Hot Rolling of Steel, (1990) 435-445

10. T. Gladman, Recrystallization and Grain Growth of Multi-Phase and Particle containing Material, ed. N. Hansen, A.R. Jones, T. Leffers, (1980) 183

11. R.W.K. Honeycombe, Proc. of Int. Conf. of HSLA Steel, (1985) 243-250 12. C.M. Sellars, HSLA Steels Metallurgy and Application, (1986) 73-81 13. I. Steinbach, F. Pezzolla, B. Nestler, M. Seeβelberg, R. Prieler, G.J. Schmitz and J.L.L.

Rezende, Physica D, 94 (1996) 135-47 14. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998 73-86 15. B. Sundman, B. Jansson, J-O. Andersson, CALPHAD, 9 (1985) 153-90 16. MICRESS®, Software developed in ACCESS is an independent research center

associated with the Technical University of Aachen 17. E. Cotrina, A. Iza-Mendia, B. Lopez, I. Gutierrez, Metall. Mat. Trans. A, 35A (2004)

93-101 18. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn., 14 (1998) 10-18 19. W.F. Lange, M. Enomoto, H.I. Aaronson, Metall. Trans., 19A (1988) 427-40 20. Handbook of Chemistry and Physics, CRC Press Inc. Bona Raton, 1989 21. T.A. Kop, J. Sietsma, S. van der Zwaag, J. Mat. Science 36 (2001) 519-26 22. K.J. Lee, J.K. Lee, Scripta Materialia, 40 (1999) 831-836 23. T.A. Kop, J. Sietsma, S. van der Zwaag, Mater. Sci. Techn., 17 (2001) 1569-74 24. T.A. Kop, P.G. Remijn, V. Svetchnikov, J. Sietsma, S. van der Zwaag, J. Mat. Science

36 (2001) 1863-71

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73

Chapter 5

3D phase field modelling of the austenite to ferrite transformation

Abstract The phase field method has been used to simulate the austenite-to-ferrite phase transformation in steel, for the first time in three-dimensional (3D) space. The effect of the exact morphology of the initial 3D austenitic microstructure is shown to be small, but the nucleus density and the temperature range in which nucleation takes place, both derived from experimental observations, are of primary importance for the kinetics. When 3D simulations are compared to 2D simulations, it becomes evident that 2D simulations predict faster transformation rates. In the often-used practice of considering the interface mobility as an adjustable model parameter, systematically too low values will be obtained if 2D simulations are compared with experiments. Moreover, unrealistic features can be observed in the simulated 2D microstructures, whereas the 3D microstructures show a realistic representation of the actual microstructure. The overall conclusion is therefore that phase field modelling is distinctly more powerful when applied in 3D space.

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74

5.1 Introduction The austenite-to-ferrite transformation in steels is one of the most studied phase transformations because of its outstanding technological relevance in manipulating and tailoring steel properties. Status and efforts in modelling the austenite decomposition kinetics have recently been reviewed [1] with an emphasis on approaches that can also be used to predict the transformation kinetics under industrial processing conditions. These model approaches can be grouped into semi-empirical formulations [2-4] using the well established Johnson-Mehl-Avrami-Kolmogorov theory [5-7] in conjunction with the additivity rule [8], and more fundamental formulations based on either diffusion-controlled, interface-controlled or mixed-mode models [9-15]. In the mixed-mode models both long range diffusion and interface reaction rate are taken explicitly into account [16]. An important emphasis of these modelling efforts has recently been to account quantitatively for the role of substitutional alloying elements in delaying the austenite decomposition by considering solute drag [17-19]. All these models are phenomenological in nature and, while actually designed at the mesoscopic scale, describe the fraction transformed such that they fall effectively into the macroscopic length scale. These phenomenological approaches can be translated into truly mesoscale models by employing front tracking methods, as e.g. demonstrated for austenite formation [20]. In addition to these sharp interface models the phase field model approach, which employs a diffuse interface, can be used for mesoscale modelling. 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 [21-25]. Initial applications of phase field modelling to the solid-state austenite-to-ferrite transformation have been reported [26-32]. 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. It essentially replicates the mixed-mode philosophy, 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. In the approach proposed by Wheeler et al. [22] it is assumed that the interface region is a mixture of the two phases having the same composition. The problem with this approach is that the parameters depend on the interface thickness. Alternatively, it has been proposed that the free energy density in the interface is based on a mixture of the two phases with different composition that are defined by a constant ratio

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for each element [23]. This approach is now commercially available in the MICRESS (MICrostructure Evolution Simulation Software) code. Even though a number of details have yet to be clarified, phase field models appear to be a very promising tool for the austenite-ferrite and other solid-state transformations with the potential to be developed into predictive models for industrial processing conditions. At present, a number of important advances has been made in describing the austenite-to-ferrite transformation kinetics with phase field models. In particular, satisfactory agreement of calculated and measured transformation kinetics has been obtained when suitable interfacial mobilities had been assumed [30-32]. The recent work by Huang et al. [32] provides a detailed analysis that includes isothermal and continuous cooling transformation as well as massive transformation. In addition, ferrite grain coarsening behind the transformation front is addressed. However, all these models appear to be descriptive and have yet to be brought to a stage that reliable quantitative predictions can be made for commercial steels. A particular drawback of the previously reported austenite-ferrite phase field models is that they are exclusively 2D simulations. The restriction on 2D is to a large degree due to the computational demands to conduct meaningful 3D simulations. Such simulations have so far primarily been carried out for solidification where a much more advanced state in accounting quantitatively for the role of alloying elements has been attained [33-35]. In the case of solid-solid transformations, 3D phase field simulations have been reported for martensitic transformations in FeNi and AuCd alloys [36-39]. Although it seems essential to incorporate 3D aspects also in the modelling of diffusional solid-state phase transformations, for instance to achieve realistic diffusion profiles, or to take into account 3D structural features like quadruple points where four grains meet, due to computational limitations no such studies have been published in the past. However, the progress in computer technology makes it feasible to conduct 3D phase field calculations on a routine basis. The present chapter is devoted to a first series of 3D phase field simulations of the austenite-to-ferrite transformation using the MICRESS code. Two cases for continuous cooling transformation kinetics in a low-carbon steel with distinctly different ferrite nucleation conditions are considered to analyse the significance of 3D simulations as compared to 2D phase field calculations. The first case of cooling at 0.4 K/s leads to the activation of just one nucleation mode, i.e. at triple lines, whereas for the second case with cooling at 10 K/s also nucleation at grain surfaces has to be taken into account. After a brief introduction of the phase field formulation employed and the required translation between 2D and 3D microstructure features, the 3D simulations results are presented in

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detail for both cooling scenarios. Subsequently, the 3D results are compared to those obtained using the 2D phase field model thereby allowing a critical analysis of 2D vs. 3D simulations. Based on this analysis the challenges and directions for future model developments are delineated.

5.2 Simulation conditions 3D phase field calculations are performed to describe the austenite-to-ferrite transformation using the MICRESS code. The investigations employ the same process conditions as previously evaluated with a 2D PFM approach as described in chapter 3. In particular, the austenite decomposition in a Fe-0.1wt%C-0.49wt%Mn alloy with an initial austenite grain size of 20 µm (EQAD=equivalent area diameter) is investigated for cooling rates of 0.4 K/s and 10 K/s. The system is considered in the para-equilibrium limit [40], where the substitutional sublattice remains configurationally frozen consistent with the assumption that only carbon diffusion is relevant. The driving pressure for the reaction is determined from a linearised quasi-binary Fe-C phase diagram that has been established based on data obtained from Thermo-Calc. In detail, the model employs the following thermodynamic approximations where the phase boundaries are given by

1073 10350( 0.00886)CT xα α= − − (5.1)

1073 106.2( 0.279)CT xγ γ= − − (5.2)

Here the temperatures Tα and Tγ (in K) describe the phase boundaries for phases α and γ

with carbon concentrations respectively Cxα and Cxγ (in wt%). In this approximation, the

Ae3-temperature, i.e. Tγ for the nominal carbon concentration 0Cx , is 1106 K for the

investigated system ( 0Cx = 0.1wt%). According to this linearised phase diagram, at each

temperature the partition factor kC can be determined. The driving pressure is calculated using the undercooling, ∆T, which is expressed as

0.5 ( ) ( )C C CT T x T k x Tγ γ α γ ∆ = + − (5.3)

where T is the actual temperature and Cγ is the concentration in the austenite portion of the calculation cell. The driving pressure is then predicted from

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G S T∆ = ∆ ∆ (5.4)

where for the proportionality factor ∆S the value of 0.346 × 106 J/Km3 derived by Thermo-Calc is employed. The interfacial energy of the α-γ interface is taken to be 0.5 J/m2. The values for carbon diffusion coefficients assuming Arrhenius relations are summarized in Table 5.1.

0CDγ (m2/s) CQγ (kJ/mol) 0

CDα (m2/s) CQα (kJ/mol)

2.2×10-4 122.5 1.5×10-5 142.1 Table 5.1 Diffusion parameters used for the present simulations [41] Adjustable parameters in the model are the mobility of the interface and a number of nucleation parameters. The mobility is assumed to obey an Arrhenius relationship 0 exp( / )Q RTµ µ= − (5.5)

with an activation energy Q of 140 kJ/mol [13] and the pre-exponential µ0 is employed as a fitting parameter and further on called mobility factor. In terms of nucleation conditions the following can be adjusted: nucleation modes, minimum undercooling, shield distance (the minimum inter-nucleation site distance in the model) and shield time, which define an area or volume around a given nucleus for which for the specified time no further nucleation occurs. Further, for each nucleation mode a maximum number of nuclei can be specified in total as well as per individual time step. The nucleation parameters are designed to adjust nuclei density, distribution of nucleation sites, maximum nucleation temperature (or minimum undercooling ∆TN for a nucleation mode) as well as the spread of nucleation temperatures δT for a given mode. The nuclei density can be determined from the desired (or targeted) ferrite grain size in the final microstructure assuming that each nucleus will result in a ferrite grain in the final microstructure (which can be achieved in the model when eliminating or minimizing ferrite grain coarsening by making suitable assumptions for energies and mobilities of ferrite-ferrite grain boundaries). For the austenite-to-ferrite transformation nucleation can be assumed to occur on austenite grain boundaries and two nucleation modes, namely on triple lines and on grain surfaces, are considered in the phase field simulations as described in detail in the results sections.

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In addition to these physical parameters, there are a number of numerical parameters that have to be carefully chosen to ensure stability and convergence of the calculations. In particular, node size, ∆x, time step, ∆t, and interface thickness, η, are critical here. These values have been evaluated using a sensitivity analysis. The interface thickness is taken consistently to be 4 nodes, i.e. η = 4∆x, which is the de-facto minimum for a phase field simulation. This minimization of the interface thickness is desirable for reducing the required computational effort and can be justified here since curvature effects are of minor importance in the present study. The phase field calculations are carried out under a regime of driving pressure averaging across the interface region to increase stability of the calculations and periodic boundary conditions are implemented for the calculation domain. Node size selection and an automatic time stepping mechanism that uses a suitable numerical factor, τ, are employed such that convergence is reached within the limits of the error bars of dilatometry experiments. The 3D calculations have been made for simplified austenite grain geometries as illustrated in Figure 5.1, i.e. (a) a packing with a central cubic grain (to be denoted as “cube”), (b) a packing with a central tetrakaidecahedron-shaped grain (to be denoted as “tetrakaidecahedron”), (c) a film structure consisting of two layers of grains that adapt a 2D arrangement used in previous 2D calculations [30] (to be denoted as “film”), and (d) a random structure obtained by a Voronoi construction (to be denoted as “random”). To translate 2D microstructure information into 3D the following considerations can be made. The average 2D grain size, d, quantified as an equivalent area diameter (EQAD) is smaller than the average equivalent volume diameter by a factor 1.2 [42]. Then the relationship between number of nuclei, N0, in 2D with a domain size (area) of A to the number of nuclei, N, in 3D with a domain size (volume) V can be obtained from

20 4

N d fAπ= (5.6)

and

3(1.2 )6

N d fVπ= (5.7)

where f is the final ferrite fraction (here f ≈ 0.9). The ratio of nuclei is then

30 2 (1.2)3

N AdN V

= (5.8)

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and specifically for square and cubic domains with the same domain length L one can write

30 2 (1.2) 1.153

N d dN L L

= = (5.9)

a b

c d Figure 5.1 Initial austenite microstructures; a) cube, b) tetrakaidecahedron, c) film, d)

random

5.3 Results

5.3.1 Nucleation on triple lines

Experimental measurements have shown that for the steel simulated the continuous cooling transformation at 0.4 K/s provides a condition where the initial austenite grain size and the resulting ferrite grain size are practically identical [30]. Thus, on average about one ferrite nucleus forms per austenite grain, suggesting that only corner nucleation takes place. However, the MICRESS code does not permit to restrict nucleation sites to grain corners, i.e. quadruple points. As a result, this situation is simulated by assuming that nucleation at

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triple lines is the only operational nucleation mode; this approach is consistent with the assumptions for the associated 2D calculations where nucleation is restricted to triple points. Table 5.2 summarizes calculation details for the four different initial austenite microstructures including domain size, nuclei densities and the resulting ferrite grain sizes. The node size employed for these calculations is ∆x = 0.3 µm. In all cases, the first nucleus forms at 1092 K and the nucleation temperature spread, δT, is 18 K. The selection of this value for δT has been motivated by the nucleation investigations of Offerman et al. [43] who obtained a typical temperature spread of corner nucleation in the order of 10 – 30 K. During the time of cooling through the nucleation temperature range new nuclei are introduced continuously one by one into the calculation domain such that the last nucleus forms at 1074 K. Within the limits set by the domain sizes and the requirements for periodic boundary conditions, nuclei densities are obtained that are very similar in all four cases. The resulting ferrite grain sizes are 17 – 18 µm, which can be considered as the same when experimentally measured grain sizes are taken as comparison criterion; the accuracy of grain size measurements is frequently limited to ±10%. γ microstructure Domain size,

µm3 Number of γ grains

dγ, µm

Number of nuclei

Nuclei density, µm-3

dα,, µm

Cube 30×30×30 4 20 5 1.85×10-4 18 Tetrakaidecahedron 45×45×45 12 20 15 1.65×10-4 18 Film 37.5×37.5×30 7 19 8 1.90×10-4 17 Random 33×33×33 6 19 7 1.95×10-4 17

Table 5.2 Microstructural data for the simulations and resulting ferrite grain sizes for 0.4 K/s

Taking a mobility factor of 3.5×10-7m4/Js the predicted transformation kinetics are shown in Figure 5.2. Further, Figure 5.2 shows for comparison also the experimental data obtained by dilatometry [30]. As a matter of fact, the mobility factor has been selected to provide a reasonable fit to these experimental data. Excellent agreement of model and experiment can be achieved up to a fraction transformed of 0.4. For the later transformation stages, the current model predicts higher transformation rates than experimentally observed. Figure 5.2 emphasizes the temperature range where the simulated transformation kinetics is primarily taking place such that the final ferrite fraction obtained

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at lower temperatures is not shown. But it should be noted that the final ferrite fraction of approximately 0.9 is the same in simulation and experiment. As indicated in Figure 5.2, the simulated transformation rate is independent of the assumed initial austenite microstructure in terms of grain morphology provided the ferrite nuclei density and spatial distribution are approximately the same. The latter is achieved by taking similar initial austenite grain sizes and aiming for similar resulting ferrite grain sizes.

1020 1040 1060 1080 11000.0

0.2

0.4

0.6

0.8

1.0

ferri

te fr

actio

n

Temperature (K)

Experimental Cube Tetrakaidecahedron Random film

Figure 5.2 Effect of initial austenite microstructure on transformation behaviour at

0.4K/s. The details of the microstructure evolution with ferrite shown in red are summarized in Figure 5.3 as a function of temperature for the tetrakaidecahedron initial austenite microstructure where the largest number of ferrite triple line nuclei, i.e. 15, is realized with the present phase field simulation. At 1085 K, four newly formed ferrite grains are visible and their nucleation positions are confirmed to be located at grain boundary triple lines. The size of these four ferrite grains scales with their nucleation temperature. Even though only four grains are visible, already six nuclei have formed when 1085 K is reached but the last two nuclei are still too small to be visible in the representation scale of Figure 5.3. At 1080 K, the number of ferrite grains has increased to 10 and some of the new nuclei have formed at positions that are located close to the domain boundaries. Most notably there is one ferrite grain located near the corner point of the domain such that this grain is cut into

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8 pieces as a result of the periodic boundary conditions. Further, substantial growth of the four initial nuclei located more in central positions of the domain is clearly identified and a more or less spherical growth geometry applies to each individual ferrite grain under these nucleation conditions. At 1075 K all but one of the 15 nuclei are present and the beginning of impingement of the growing ferrite can be seen. The final figure in this series, i.e. at 1060 K when a ferrite fraction of approximately 0.75 has been formed, displays the situation when the transformation approaches the saturation regime with substantial impingement of ferrite grains.

1085 K 1080 K

1075 K 1060 K Figure 5.3 Predicted 3D microstructure evolution for 0.4K/s using the initial

tetrakaidecahedron configuration, ferrite shown in red and interfaces in green. For the colour version refer to the appendix.

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In addition to fraction transformed and the spatial distribution and morphology of the ferrite grains, the evolution of the carbon concentration within the remaining austenite is obtained from the phase field simulations. The character of the transformation can be evaluated with the local interfacial carbon concentration. Comparing the maximum carbon concentration that is recorded in the interface region with those which would be applicable to an either solely interface or solely diffusion controlled growth mode the mixed-mode character of the austenite-to-ferrite transformation can be quantified. In case of an interface-controlled growth there will be no gradients in the carbon distribution such that the carbon concentration in austenite, and thus the interfacial concentration, are just obtained from the fraction transformed and mass balance. In the diffusion-controlled reaction local equilibrium is established at the interface and the interfacial concentration is

then given by Ceqxγ . An example for this analysis is shown in Figure 5.4 where results of

three different interfacial positions are compared to the limits of interface and diffusion controlled growth, respectively.

1065 1070 1075 1080 1085 10900.05

0.10

0.15

0.20

0.25

0.30

0.35

interface controlled

diffusion controlled

Carb

on c

once

ntra

tion

(wt%

)

Temperature (K)

Figure 5.4 Predicted interfacial carbon concentrations (symbols) for cooling at 0.4K/s;

solid lines indicate values for diffusion and interface-controlled reactions, respectively

Initially the reaction is interface controlled and moves gradually closer to the diffusion case without, however, reaching it. Further, the apparent degree of this mixed-mode character depends on the interfacial position selected because each grain experiences in

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detail different growth geometries, i.e. different levels of soft impingement, that lead to an individual and locally varying degree of the mixed-mode character.

5.3.2 Nucleation on triple lines and grain surfaces

In the case of transformation at the larger cooling rate of 10 K/s experiments have shown that the transformation involves lower transformation temperatures and additional nucleation modes, i.e. at grain surfaces, leading to a substantial ferrite grain refinement and a final grain size of approximately 9 µm [30]. In the present case detailed calculations have been limited for the two cases with the smaller domain sizes, i.e. cube and random initial austenite structures. This limitation is largely due to additional computational demands since a step size of 0.2 µm has to be employed to reach convergence for this large cooling rate scenario. To reflect the desired ferrite grain refinement the number of nuclei has been taken as 35 for the cube case and 47 for the random case, both leading to a ferrite grain size of approximately 9 µm. The nucleation temperature spread is increased compared to the case of nucleation at triple lines only, since additional nucleation modes will be activated at lower temperatures. The nucleation temperature for a given mode is rather insensitive to cooling rate [44]. Therefore, the first nuclei form in the simulations again at 1092 K but the overall nucleation temperature spread is increased to 35 K, i.e. the last nucleus is introduced at 1057 K. This nucleation temperature spread has then to be divided between the two operational nucleation modes. Most notably, the nucleation at triple lines should be similar to the low cooling rate case in terms of number of nuclei and spread of nucleation temperatures. Thus, the number of nuclei forming at triple lines is taken to be the same as for the low cooling rate case, e.g. for the random configuration 7 ferrite grains nucleate at triple lines and the remaining 40 form at grain surfaces for a total of 47 ferrite grains. Further, in the present calculations the following nucleation temperature ranges are employed: 1092 - 1076 K at triple lines and 1072 - 1057 K at grain surfaces. This nucleation spread of δT = 16K/15K (over a total temperature range of 35 K) provides an excellent fit to the experimental data of Mecozzi et al. [30] when a mobility factor of 24×10–7m4/Js is taken, as shown in Figure 5.5. Two sets of experimental data are shown to illustrate the accuracy of the measurements thereby indicating the precision required for the simulations in terms of selecting the adjustable physical parameters, i.e. nucleation conditions and mobility factor. The selection of the mobility factor requires some discussion since it is for 10 K/s approximately one order of magnitude larger than that employed for 0.4 K/s. Clearly, this required apparent increase in the mobility factor as

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transformation temperature is lowered indicates that the interfacial mobility is used here as an effective value, i.e. in a similar fashion as proposed previously [14, 32]. The observed trends of mobility as a function of temperature replicate those reported for other simulations. For example, in the phase field simulations of Huang et al. [32] it had been assumed that the pre-exponential mobility factor increases linearly by about two orders of magnitude when the temperature is lowered from 1130 to 960 K. In essence, physical models have yet to be developed to account in detail for this trend and may be related to a potential solute drag effect of substitutional alloying elements (here Mn), as proposed by Fazeli and Militzer [19].

900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0

ferri

te fr

actio

n

Temperature (K)

Experimental data Model (cube) Model (random)

Figure 5.5 Comparison of predicted transformation kinetics with experimental data for

cooling at 10 K/s. The ferrite evolution is shown in Figure 5.6. The nucleation at triple lines leads to a fairly random distribution of ferrite grains originating from this nucleation mode, as seen at 1067 K when nucleation at triple lines is completed. Subsequent nucleation at grain surfaces occurs with some degree of inhomogeneity. The clustering of nucleation sites at grain surfaces, however, appears to be restricted to areas at the grain surfaces that are least affected by the triple-line nucleated ferrite (cf. microstructures at 1060 and 1017 K, respectively). This is a very reasonable result since the carbon concentration around the growing ferrite that had nucleated earlier is increased, thereby reducing the local undercooling. The degree of clustering for the grain surface nuclei can be modified by

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selecting in detail different nucleation parameters, but this does not alter the transformation kinetics markedly since the overall number of nuclei and, thus, the nuclei density is kept constant in accordance with the final ferrite grain size.

1067 K 1060 K

1017 K 840 K

Figure 5.6 Predicted 3D microstructure evolution for 10K/s using the initial random

configuration nucleation on triple lines and grain surfaces, ferrite is shown in red and the interfaces in green. For the colour version refer to the appendix.

The kinetics itself is also for this case confirmed to be of mixed-mode character, as shown in Figure 5.7. The degree of this character varies in detail from grain to grain according to the variation in local nucleation and growth conditions.

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940 960 980 1000 1020 1040 1060 10800.0

0.2

0.4

0.6

0.8

1.0

interface controlled

diffusion controlled

Carb

on c

once

ntra

tion

(wt%

)

Temperature (K)

Figure 5.7 Predicted interfacial carbon concentrations (symbols) for cooling at 10K/s;

solid lines indicate values for diffusion and interface-controlled reactions, respectively.

Considering the final microstructure predicted as a result of the austenite decomposition, a quite reasonable ferrite microstructure appearance can be rationalized. This can also be verified in terms of the grain size distribution, as shown in Figure 5.8. Here, the results from three initial configurations (random, cube and tetrakaidecahedron) with a total of 199 grains have been combined for better statistical relevance. A log-normal distribution is approximately obtained where the largest (smallest) grains in the distribution are not more than a factor of 3 larger (smaller) than the mean grain size. This is in reasonable agreement with the experimentally observed ferrite grain size distribution, which is added in Figure 5.8 for comparison purpose, although the latter is slightly broader. This difference seems to indicate that the grain size distribution could be used for further optimisation of the temperature range for nucleation. In the distribution obtained by the simulations, the tail towards larger grain sizes is composed from the ferrite nucleated first at triple lines. The majority of grains have, however, formed at grain surfaces and because of the associated lower nucleation temperatures as compared to nucleation at triple lines, remain as the smaller grains in the distribution.

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0 10 20 30 40 500.00

0.05

0.10

0.15

0.20

0.25

0.30

frequ

ency

d (µm)

Figure 5.8 Predicted (grey bars) and measured (black bars) ferrite grain size

distribution for 10K/s.

5.4 Comparison with 2D simulations

5.4.1 Transformation kinetics

To further evaluate the relevance of the predicted 3D microstructures it is useful to compare the calculations with 2D simulations and to evaluate 2D cuts through the 3D predictions. The latter reflects what is commonly done in metallographic investigations of samples obtained in laboratory simulations or industrial processing. 2D calculations have been performed, taking selected 2D cuts from the 3D initial austenite microstructure as the starting structure. The results have then been compared to the selected 2D cuts from the 3D simulation structures. Care has been taken to translate the nucleation conditions of the 3D simulations into the 2D nuclei density according to equations (16) and (17), respectively. Further, to translate the 3D nucleation scenario for the more complex situation at 10 K/s with two nucleation modes into 2D it has been assumed that the ratio of nuclei at triple lines to nuclei at grain surfaces is the same in 2D as in 3D (or as close as possible considering that the numbers of nuclei have to be integers). All other parameters of the 2D calculations remain unchanged including mobility factor and nucleation temperature range.

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Figure 5.9 provides a comparison of the 2D and 3D predictions of the transformation kinetics for a cooling rate of 0.4 K/s where just nucleation at triple lines occurs using the tetrakaidecahedron configuration.

1050 1060 1070 1080 1090 11000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ferri

te fr

actio

n

Temperature (K)

3D 2D (13) 2D (76) 2D (87) 2D (127) 2D (146) 2D (base)

Figure 5.9 Comparison of 3D and 2D predictions for 0.4 K/s and tetrakaidecahedron

geometry with δT=18K; numbers in brackets indicate cut plane Figure 5.10 is a similar representation for 10 K/s where the random configuration has been employed. Clearly, the 2D kinetics is independent of the actual cut plane and this replicates the 3D finding that the predicted kinetics is rather independent of the detail of the initial austenite microstructure as long as the nuclei density is comparable. In addition, Figures 5.9 and 5.10 show the 2D predictions using a 2D initial austenite microstructure obtained by a Voronoi construction that had been employed in previous 2D PFM simulations [30]. This particular 2D scenario is referred to as the 2D base case and gives again a very similar prediction for the kinetics as obtained from the 2D cuts. Comparing the 2D and 3D kinetics it is evident that for the same set of parameters 2D calculations predict a faster transformation than the 3D calculations. For 10 K/s this leads also to an apparent ferrite saturation fraction that is markedly larger in 2D than in 3D, i.e. approximately 0.9 in 2D vs. 0.8 in 3D. This can be attributed to the fact that the transformation temperature range is lowered and extended for 10 K/s as compared to 0.4 K/s, leading to a too low mobility for the transformation to complete. For the given set of parameters, the transformation

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predicted with the 2D simulations is shifted to higher temperatures such that the para-equilibrium ferrite fraction can still be closely approached.

900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0

ferri

te fr

actio

n

Temperature (K)

3 D 2D (17) 2D (37) 2D (129) 2D (base)

Figure 5.10 Comparison of 2D and 3D transformation kinetics for 10K/s and random

case with δT=35K; numbers in brackets indicate cut plane The increased transformation rate in the 2D calculations forms a very reasonable result since 2D calculations can be seen as a cylindrical growth regime in 3D, whereas the actual 3D case resembles in the present case a more spherical growth regime, in particular in the initial stages of the transformation process. Comparing the interfacial area for cylindrical growth with that for spherical growth it is evident that this area is, in the initial transformation stages, significantly larger in case of cylinders. Thus, a cylindrical growth geometry leads initially to larger transformation rates as compared to a spherical growth geometry. Consequently, when comparing 2D simulations with experiments one will have to employ a lower apparent mobility than that used in 3D calculations to replicate the same kinetics. Figure 5.11 illustrates the corrected 2D kinetics with reference to the 3D kinetics.

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900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0

10 K/s

0.4 K/s

ferri

te fr

actio

n

Temperature (K)

3D 2D

Figure 5.11 Predicted 2D and 3D transformation kinetics using corrected mobility

terms for the 2D simulations The corrected 2D mobility factors have been decreased from 3.5×10−7m4/Js to 1.5×10−7m4/Js for 0.4 K/s, and from 24×10−7m4/Js to 6×10−7m4/Js for 10 K/s, respectively. The required amount of the correction scales with the differences in transformation temperatures of 2D vs. 3D calculations, as shown in Figures 5.9 and 5.10, i.e. less than 10 K for 0.4 K/s, but 15 to 35 K for 10 K/s, when considering fractions transformed between 0.2 and 0.5, where a rapid progress in the transformation occurs. Further, the overall transformation curves are very similar for 2D and 3D simulations. This suggests that a simple adjustment of the mobility factor as an effective value in 2D simulations is sufficient, at least in the present morphologically rather simple case. For more complex growth geometries it may be expected that an adjustment of the mobility would be a means with limits that, in general, can only provide an approximate description of the transformation kinetics.

5.4.2 Predicted microstructures

The predicted final microstructures for 0.4 K/s are compared in Figure 5.12 where selected 2D cuts through the 3D microstructure, as obtained for the tetrakaidecahedron case, are shown together with the microstructures obtained in equivalent 2D simulations for these

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92

particular cuts, i.e. without adjusting the mobilities in 2D. Having a casual look at these microstructures only slight differences seem to be present. The grains look fairly rounded and their number remains below 10 such that it is very difficult to make any statistically relevant comments. However, it is useful to appreciate a number of details in these rather simplified structures that show distinct features of the 2D vs. 3D approach. In 2D calculations all nuclei have formed in the plane of view whereas this is in general not the case for the 3D simulations. In the latter new ferrite grains may have nucleated in planes above or below the 2D cut shown and subsequently grown into the plane of view. As a result, the number of grains seen in a particular plane may be different for 2D and 3D simulations. The cut for node position x = 76 illustrates this situation; there are 7 ferrite grains in 2D whereas only 5 ferrite grains are visible in the 2D cut from the 3D simulation. The cut at x = 127 depicts a situation where in both cases 7 ferrite grains are in the plane.

76

127

Figure 5.12 Comparison 2D (left) and 3D (right) microstructures for 0.4 K/s; numbers indicate cut plane, total presented area is 45µm×45µm, white represents ferrite, orange austenite and blue the interfaces.

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This discussion makes clear that the distribution of grain sizes is better represented in 3D simulations than in 2D simulations. Another, distinct feature of the 3D calculations is that grain boundaries and α-γ interfaces, respectively, may be oriented nearly parallel to the 2D cut, such that an extended grain boundary/interface area is displayed in the 2D cut, as is evident for some of the boundaries/interfaces in the x = 76 plane. This behaviour may be viewed as an artefact but is just a straightforward consequence of the PFM approach as it employs a diffuse interface. In contrast, for the 2D calculations the displayed interface thickness (and resulting area) is always given by the nominal interface thickness of the calculation. To appreciate in more detail the comparison of 2D and 3D calculations the evaluation of the high cooling rate case of 10 K/s is instructive because additional nucleation modes are activated such that more ferrite grains have to be considered, leading to a statistically more relevant analysis. The final microstructures, as obtained for the random configuration, are shown for a number of selected 2D representations in Figure 5.13 similar to what is shown in Figure 5.12 for 0.4 K/s. For better comparison, the final microstructure predicted by 3D simulations is shown together with 2D predictions at the same ferrite fraction level of 0.85. Here, some important differences can be recognized when comparing the 2D cuts from the 3D simulations with the 2D microstructures. The most striking feature is that the 3D calculations result in clearly different and more realistic structures from a morphological point of view. Realistic grain shapes can be widely seen in the 2D cuts obtained from the 3D simulation. The remaining austenite, which defines the areas of the second transformation products, e.g. pearlite and martensite, are true islands as expected from ferrite nucleation at prior austenite grain boundaries, even though not all prior austenite grain boundaries have been covered by ferrite, see e.g. the cut at x = 129. This is a remaining evidence of the aforementioned inhomogeneous nucleation at grain surfaces but appears overall to be a rather minor point. In addition to similar nucleation issues as discussed above for 0.4 K/s, the 2D simulations lead to grain and second phase morphologies that seem rather unrealistic. The ferrite grains appear more often than not as squashed circles and only in a small minority of cases reasonable triple point configurations have formed. The distribution of the second phase is that of long elongated channels with a number of narrow inlet-type features developing between ferrite grains. This unrealistic morphology can be expected also to influence the diffusion behaviour, and consequently the transformation kinetics.

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17

37

129 Figure 5.13 Comparison of 2D microstructures (left) with the associated cuts through

the 3D microstructure (right) for 10K/s; numbers indicate cut plane, total presented area is 33µm×33µm, white represents ferrite, orange austenite and blue the interfaces.

In summary, 3D phase field simulations have been carried out for the austenite-to-ferrite transformation and compared to 2D phase field simulations. An evaluation of these simulations suggests that the additional computational effort of the 3D calculations is well justified. In 3D the predicted microstructure morphologies are in much better agreement with experimental findings than the morphologies obtained by 2D simulations. This

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morphological aspect will become even more critical when also non-equiaxed microstructures are considered, e.g. when simulating the transformation to Widmanstätten ferrite. Further, while in 2D and 3D simulations the transformation kinetics and ferrite grain size can approximately be matched with experimental data by suitable parameter selection in terms of interface mobilities and nucleation conditions, it appears to be difficult to obtain a complete matching of 3D results with 2D simulations. This indicates that the unrealistic growth geometries employed in 2D simulations cannot be completely mitigated by suitable effective values for other physical parameters. In the present case with primarily homogeneous equiaxed ferrite microstructures these geometrical aspects may appear to be of second order, i.e. adjusted apparent 2D mobility factors are smaller but of the same order of magnitude as those obtained in the 3D simulations. However, when microstructure inhomogeneities are of significance, e.g. banded microstructures, 3D growth geometries may be essential to capture these features at all. In any event, the interpretation of physical parameters concluded from 2D simulations will be limited. However, also in the present 3D simulations the interfacial mobility is used as an effective parameter that essentially increases as temperature decreases. Even though this is consistent with other simulation results reported in the literature [14, 32] physical based models of the solute-interface reaction have yet to be connected to PFM simulations that meaningful temperature trends for the mobility per se can be employed. 5.5 Conclusions Phase field simulations, aimed at predicting phase transformations in metallic microstructures, performed in 3D space, represent a distinctly more realistic picture of the phase transformation and the resulting microstructure than 2D simulations. First, the different types of nucleation sites are better represented. Secondly, the 2D morphology leads to distinctly faster transformation rates than in 3D, and the resulting mobility values in 2D are therefore too small, when extracted from comparisons with experiments. Thirdly, the morphology of the remaining austenite towards the end of the simulation is well described in 3D, but was found to be unrealistic in 2D. Simulations in 3D space therefore lead to significantly more accurate morphologies of the final, non-ferritic transformation products (pearlite, bainite, martensite) than 2D simulations.

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References 1. M. Militzer, Austenite Formation and Decomposition. E.B. Damm, M.J. Merwin,

editors. Warrendale (PA): TMS, 2003.195-211 2. I. Tamura I, C. Ouchi, T. Tanaka, H. Sekine. Thermomechanical Processing of High

Strength Low Alloy Steels. London: Butterworth and Co., 1988, 17-48 3. M. Militzer, E.B. Hawbolt, T.R. Meadowcroft, Metall. Mater. Trans. A 31 (2000)

1247-59 4. M. Umemoto, A. Hiramatsu, A. Moriya, T. Watanabe, S. Nanba, N. Nakajima, G.

Anan, Y. Higo, ISIJ International 32 (1992) 306-315 5. A. Kolmogorov, Izv. Akademii Nauk USSR Ser. Matemat. 3 (1937) 355-359 6. W. Johnson, R. Mehl, Transactions AIME 135 (1939); 135: 416-458 7. M. Avrami, J. Chem. Phys. 8 (1940) 212-224 8. J.W. Christian, The Theory of Transformation in Metals and Alloys, 2nd edition.

Oxford: Pergamon Press, (1982) 476-479 9. R.A. Vandermeer, Acta Metall. Mater. 38 (1990) 2461-70 10. R.G. Kamat, E.B. Hawbolt, L.C. Brown, J.K. Brimacombe. Metall. Trans A. 23 (1992)

2469-2480 11. V.M.M. Silalahi, M. Onink, S. van der Zwaag. Steel Research, 66 (1995) 482-489 12. J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsjäger, N.K. Simha. Acta Mater. 49 (2001)

1249-59 13. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn. 14 (1998) 10-18 14. T.A. Kop, Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 40 (2000) 713-18 15. Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int.,2003, vol. 43, pp. 767-73 16. J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143-52 17. G.R. Purdy, Y.J.M. Brechet, Acta Metall. Mater, 43 (1995) 3763-74 18. M. Hillert, Acta Mater. 47 (1999) 4481-4505 19. F. Fazeli, M. Militzer. Metall. Mater. Trans. A 36 (2005) 1395-.1405 20. A. Jacot, M. Rappaz, R.C. Reed, Acta Mater. 46 (1998) 3949-62 21. G. Gaginalp, Physical Review A, 39 (1989) 5887-96 22. A.A. Wheeler, W.J. Boettinger, G.B. McFadden, Physical Review A, 45 (1992) 7424-

39 23. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998) 73-86 24. S.G. Kim, W.T. Kim, T. Suzuki, Physical Review E, 58 (1998) 3316-23 25. Karma, Physical Review E, 49 (1994) 2245-50

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26. G. Pariser, P. Shaffnit, I. Steinbach, W. Bleck, Steel Research 72 (2001) 354-60 27. D.H. Yeon, P.R. Cha, J.K. Yoon. Scripta Mater., 45 (2001) 661-68 28. I. Loginova, J. Odqvist, G. Amberg, J. Ågren. Acta Mater., 51 (2003) 1327-39. I.

Loginova, J. Ågren, G. Amberg, Acta Mater. 52 (2004) 4055-63 29. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , M. Apel, P. Schaffnit, I. Steinbach,

Metall. Mater. Trans. 36A (2005) 2327-40 30. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , Acta Mater. 53 (2005) 1431-40 31. C.J. Huang , D.J. Browne , S. McFadden, Acta Mater. 54 (2006) 11-21 32. A. Karma, W.J. Rappel, Phys. Rev. Lett. 77 (1996) 4050-53 33. A. Karma, W.J. Rappel, Phys. Rev. E 157 (1998) 4323-49 34. H.J. Jeong , N. Goldenfeld, J.A. Dantzig. Phys. Rev. E 64 (2001) 041602 1-14 35. Y. Wang, A.G. Khachaturyan, Acta Mater. 45 (1997) 759-73 36. A. Artemev, Y. Jin, A.G. Khachaturyan, Acta Mater. 49 (2001) 1165-77 37. Y.M. Jin, A Artemev, A.G. Khachaturyan Acta Mater.49 (2001) 2309-20 38. Wang YU, Jin YM, Khachaturyan AG. Acta Mater. 52 (2004) 1039-50 39. M. Hillert, Paraequilibrium, Int. Report, Swed. Inst. Metals Res., 1953 40. Handbook of Chemistry and Physics. Bona Raton: CRC Press Inc., 1989 41. A. Giumelli, M. Militzer, E.B. Hawbolt, ISIJ International 39 (1999) 271-80 42. S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L.Margulies,

H.F. Poulsen, M. Th. Rekveldt, S. van der Zwaag, Science, 298 (2002) 1003-05 43. M. Militzer, R. Pandi, E.B. Hawbolt, Metall. Mater. Trans. A 27 (1996) 1547-56

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Chapter 6

99

Chapter 6

The effect of nucleation behaviour in phase field simulations of

the austenite to ferrite transformation

Abstract 3D phase field simulations of the austenite (γ) to ferrite (α) transformation during continuous cooling at different cooling rates were performed for a Fe-0.10 C- 0.49 Mn (wt%) steel. The initial austenitic microstructure and the nuclei density are input data based on experimental observations. Ferrite nuclei are assumed to form continuously over a temperature range of δT. The model employs an effective interfacial mobility with an activation energy of 140 kJ/mol and a pre-exponential factor, µ0, that, together with δT, is used as adjustable parameters to match a reference transformation curve for each employed cooling rate. A number of combinations of values (µ0, δT) is found to equally well represent the same kinetics as the reference 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 of nucleation temperature range and interface mobility.

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6.1 Introduction Notwithstanding the continuous development of novel construction materials, the classical hot rolled C-Mn steel retains its position as a very important construction material or as input material for cold rolled steel grades. Its properties depend on its microstructure which itself is a function of both the steel chemistry and the thermo-mechanical conditions imposed during rolling, subsequent controlled cooling and coiling. Of all the processes taking place during this final production stage, the decomposition of the parent austenite phase into ferrite and pearlite remains of prime importance. Hence it is not surprising that the decomposition of austenite has been modeled extensively, using a wide range of approaches. In the early but still popular transformation models based on the Johnson-Mehl-Avrami-Kolmogorov approach [1-3], all attention is focused on the kinetics and all microstructural aspects are essentially ignored. However, even in the simplest JMAK model nucleation and growth are recognized as being the two relevant and intrinsically different processes. When fitting the JMAK equations to the experimental data, information may be obtained about both the nucleation and the growth behaviour. In particular, the JMAK exponent is shown to be related to the nucleation conditions [4], even though only two modes of nucleation kinetics are considered: site saturation and continuous nucleation at a constant rate. In the case of site saturation all nuclei are present and active at the start of the transformation and their number density remains constant during the entire transformation. In the case of continuous nucleation the number of activated nucleation sites increases at a constant rate during the transformation, with a rate of nucleus formation depending on temperature and the fraction parent phase still present. More refined transformation models do however incorporate the relevant features of the parent microstructure. The simplest approach is that of Vandermeer [5] 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 and, assuming ferrite growth controlled by carbon diffusion in austenite, provides a satisfactory description of austenite decomposition in most Fe-C alloys. However more sophisticated growth parameters (i.e. interface mobility, solute-drag effect) have to be introduced as adjustable parameters to describe the growth rate of ferrite in more complex alloys, including Fe-C-Mn steels [6-8]. 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 modeling

Chapter 6

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parameter to reproduce the grain size depending on the cooling conditions [9]. The most elaborated model for modeling the austenite decomposition available at the moment is the phase field approach [10, 11]. Such a model allows a proper description of the initial microstructure not only in 2D but also in 3D [12-14]. Within this approach the location of nuclei can be specified and varied depending on the cooling conditions. The ferrite growth is described via a mixed mode approach, i.e. both the diffusion of partitioning elements and the apparent mobility of the austenite/ferrite interface are accounted for. Recently, Militzer et al. [14], showed that the 3D phase field approach not only describes the kinetics well but is also capable of yielding a realistic morphology for the resulting microstructure. The different types of nucleation sites are better represented in 3D than in 2D. While in 2D simulations all nuclei form in the plane of view, the ferrite grains present in 2D cuts of 3D microstructure may be nucleated in planes above or below the plane of view, exactly like for metallographic sections of experimental samples. However, even in these advanced modeling studies the nucleation behaviour has usually been implemented in a relatively simplistic manner, i.e. either site saturation or a period of continuous nucleation is assumed [13- 14]. A major reason for the simple approximation for the actual nucleation kinetics is the experimental complexity of detecting the nuclei, given their small dimension and the short period during which they exist as such. However, in a recent high intensity micro-beam XRD experiment Offerman et al [15] succeeded in directly recording the development of the number of ferrite grains formed in a C-Mn steel during transformation, which reflects the nucleation rate for a low cooling rate. Their experiment showed that the nucleation takes place over a well defined temperature interval during the early stage of the transformation. The observed behaviour did not resemble site saturation, nor continuous nucleation behaviour. In the present work 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. The advantage of the phase field approach is that it predicts the microstructure in combination with the transformation kinetics. Thus the nucleation temperature interval is employed as an adjustable parameter in addition to the effective interface mobility. The most realistic combination of these two is then derived by using both the experimentally determined transformation kinetics and the grain size distribution as comparison criteria.

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6.2 Simulation conditions Phase field simulations were performed to study the γ to α transformation kinetics during continuous cooling in a 3D simulation space. The MICRESS® code [16] was used to solve numerically the phase field equations 2.10 (or 2.11) and and the carbon diffusion equation 2.43. The simulations are performed for an Fe - 0.10 wt % C - 0.49 wt % Mn steel, for which the austenite to ferrite transformation kinetics was already investigated previously using the phase field approach, both in 2D and 3D [13,14]; experimental continuous cooling transformation data are already reported in Figure 3.1. Two different cooling rates are included in this study: 0.4 and 10 K/s. Table 6.1 summarises the data of the linearised quasi-binary Fe-C phase diagram in the para-equilibrium condition for the driving pressure calculation according eq. 2.52. The proportionality factor ∆S calculated using Thermo-calc [17] is equal to 3.5 x 105 J/Km3. The equilibrium transition temperature between austenite and ferrite (Ae3-temperature), calculated from the linearised γ line at the nominal carbon composition of 0.1 wt%, is equal to 1106 K. TR.(K) C Rxα (wt%) C Rxγ

(wt%) mαFe-C (K/wt%) mγ

Fe-C (K/wt%)

1073 0.009 0.279 -10350 -186.2 Table 6.1 Data of the linearised Fe-C pseudo binary diagram under para-equilibrium

condition [17] The calculations were made for a simplified 3D austenitic grain geometry, i.e. a packing with a central tetrakaidecahedron-shaped grain, shown in Figure 6.1. In Chapter 5 it was shown that the transformation kinetics is independent of the assumed initial γ microstructure if the nuclei density and distribution are the same. Table 6.2 summarises calculation details. The domain size was adjusted to have an average austenite grain size of 20 µm, equal to the value measured in metallographic cross sections of samples austenitised at 1273 K [13].

Chapter 6

103

Figure 6.1 Tetrakaidecahedron shape initial austenite microstructures used in 3D

simulations. For the colour version refer to the appendix.

γ microstructure Domain size µm3

Number of γ grains dγ (EQAD) µm

boundary condition

Tetrakaidecahedron 45×45×45 12 20 periodic

Table 6.2 Calculation parameters for the initial γ microstructures The mesh size, ∆z, was set to 0.3 µm and 0.2 µm for the simulations at 0.4 K/s and 10 K/s respectively. The reduction of the mesh size for higher cooling rate was necessary in order to avoid numerical instability problems due to the carbon gradient within the diffuse interface. In all simulations η was set equal to four times the grid size, i.e. η =4∆z, which is the de-facto minimum for the interface thickness in phase field simulations [14]. The nucleation parameters were defined depending on the cooling conditions. The nucleus density, ρ, is calculated from the measured ferrite grain size, dα, using the relation

( )31.2

6

f

dα

α

ρ π= (6.1)

where fα is the final ferrite fraction (fα = 0.9). The average ferrite grain size in 3D is a factor 1.2 larger than the average ferrite grain size as measured in a 2D metallographic section. [18].

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At 0.4 K/s cooling rate, when the measured α grain size is approximately equal to the initial γ grain size [13], the number of nuclei formed in the calculation domain is set equal to the number of austenite grains. For each γ grain one α nucleus is set to form at sites with low activation energy. In case of the MICRESS code this situation is simulated by assuming that nucleation at triple lines (TLs) is the only operational nucleation mode. At 10 K/s cooling rate a substantial α grain refinement is observed in the final microstructure; the measured ferrite grain size is equal to 9 µm [13]. This suggests that additional nucleation at less favorable sites, i.e. the grain surfaces (GSs), occurred at lower temperatures. Based on the nucleation study of Offerman et al [15], the nuclei are allowed to form during cooling over a nucleation temperature range, δT, which is controlled in the model by the value of the so-called “shield time”, i.e. the time interval after a nucleation event within which no further nucleation can take place. The number and the distribution of nuclei formed at a given temperature is controlled by the value of the “shield distance”, which is the radius of the zone centered in a nucleus in which nucleation is disabled. In the simulations at 0.4 K/s the nucleation temperature range for nucleation at triple lines, δTTL, is varied between 0 K (all nuclei form at a single temperature) and 24 K. In the simulation at 10 K/s, where nucleation at the grain surfaces also occurs at lower temperature, the total nucleation temperature range is larger compared to the case of nucleation at triple lines only. A first set of simulations at 10 K/s are run by setting all nuclei to form at grain surfaces at a constant nucleation rate within the temperature range δTGS, between 0 and 61. In a second set of simulations the total nucleation temperature range is divided between two operational nucleation modes at triple lines and at grain surfaces, δTTL and δTGS, respectively. The number of nuclei forming at the TLs was taken to be approximately the same as for the low cooling rate case, the remaining nuclei were set to form at the GSs. Table 6.3 summarises the nucleation conditions of simulations at both cooling rates. Figure 6.2 shows the number of nuclei formed during cooling at 10 K/s as a function of temperature for a total nucleation temperature range of 61 K. Due to the impossibility to evaluate the ferrite nucleation start temperature from experimental observations (ferrite is only recorded after a minimum fraction of about 1% is formed) the ferrite nucleation start temperature is set equal to the temperature at which ferrite is first detected in the dilatometric tests.

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105

# of nuclei in the domain cooling rate, K/s

at the TLs at the GSs nuclei density, ρ, m-3 dα , µm

0.4 15 - 1.7 x 1014 18 - 118

10 17 101

12.9 x 1014 9

Table 6.3 Nucleation conditions

1020 1040 1060 1080 11000

20

40

60

80

100

120

Num

ber o

f nuc

lei

Temperature (K)

nucleation at TL and GS nucleation at GS only

Figure 6.2 Number of nuclei as a function of temperature for simulations at 10 K/s for

nucleation at triple lines and grain surfaces and at grain surfaces only; the reported example is for a total nucleation temperature interval of 61 K.

The α/γ interface mobility, µ, is assumed to be temperature dependent, according to the relation µ (T)= µ0 exp(−Qµ/RT). In the present calculations the activation energy Qµ is taken to be 140 kJ/mol, the value used by Krielaart and van der Zwaag in a study on the transformation behaviour of binary Fe–Mn alloys [19]. For each employed cooling rate µ0 and δT are adjustable parameters to match a reference transformation curve, which is defined based on experimental ferrite fraction curves derived by dilatometry [13]. The simulated ferrite fraction curve that fits the experimental one by setting all nuclei to form at a single temperature and with an adjusted mobility value, is taken as the reference transformation curve.

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106

6.3 Results Figure 6.3 shows the transformation kinetics at 0.4 K/s for different values of the temperature range for nucleation at triple lines, δTTL.

980 1000 1020 1040 1060 1080 11000.0

0.2

0.4

0.6

0.8

1.0

µ0 = 2.4x10-7 m4 J-1 s-1

Ferri

te fr

actio

n

Temperature (K)

δT = 0 K δT = 12 K δT = 24 K

Figure 6.3 Triple line nucleation mode at 0.4 K/s: effect of temperature range, δTTL, on

the transformation kinetics for a given mobility. The condition µ0 = 2.4 × 10−7 m4J−1s−1 and δTTL = 0 K gives the reference kinetics at 0.4 K/s cooling rate, which fits the experimental behaviour in Figure 3.1 quite well. All 15 nuclei are set to form at the triple lines starting at 1092 K (minimum undercooling equal to 14 K). The maximum nucleation temperature is estimated from the experimental ferrite fraction curves [11]. As expected, increasing the nucleation temperature range delays the transformation for a given mobility. However, increasing the interface mobility can be used to compensate for the increase of the nucleation temperature range and to obtain the same transformation kinetics. Figure 6.4 shows the transformation kinetics for a number of combinations of values (µ0, δTTL) that result in the same kinetics as the reference curve. While, by a suitable adjustment of the apparent interfacial mobility, the net effect of the nucleation temperature range on the total transformation kinetics is small, the increase of the nucleation temperature range has a strong effect on the microstructure formed during cooling and on the final ferrite grain size distribution.

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107

980 1000 1020 1040 1060 1080 11000.0

0.2

0.4

0.6

0.8

1.0

Ferri

te fr

actio

n

Temperature (K)

δT (K) µ0 (m4 J-1 s-1)

0 2.4 x 10-7

18 3.5 x 10-7 24 3.8 x 10-7

Figure 6.4 Triple line nucleation mode at 0.4 K/s: change of the pre-exponential factor

of the interface mobility at different δTTL to have the same reference kinetics

Figures 6.5a-b present the microstructure evolution during cooling at 0.4 K/s, with δT = 0 K. For clarity the initial austenite microstructure is not shown in the figure. The ferrite nuclei form at the triple lines simultaneously and each grain grows approximately isotropically. The size of all 15 grains is approximately the same, leading to a narrow grain size distribution around a mean value of 20 µm. Figures 6.5c-d show the microstructure evolution during cooling at the 0.4 K/s, for δT = 24 K. In this case the size of each grain is related to its nucleation temperature. When the last nucleus forms at 1068 K, the first grain nucleated at 1092 K has a size of 27 µm. This phenomenon will be analysed more quantitatively for the simulations at 10 K/s in the discussion section. Figure 6.6 shows the transformation kinetics at 10 K/s setting all 118 nuclei to form at the grain surfaces. Different combinations of µ0 and the nucleation temperature range δTGS are selected to obtain the same kinetics as the reference curve for δTGS = 0 K. Unlike the situation for a cooling rate of 0.4 K/s, the necessary variation of µ0 at 10 K/s to compensate the increase of the nucleation temperature range affects the final ferrite fraction, since the equilibrium can not be reached in later transformation stages during relatively fast cooling when decreasing the interface mobility.

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108

a

c

b

d

Figure 6.5 Effect of δT on the simulated 3D microstructure during cooling at 0.4 K/s. δT = 0 K: a) 1082 K, b) 1072 K; δT = 24 K: c) 1082 K, d) 1072 K. For the colour version refer to the appendix

Figure 6.7 shows the transformation kinetics at 10 K/s obtained by setting both triple lines and grain surfaces as active nucleation sites during transformation. Again different combinations of µ0 and the nucleation temperature ranges δTTL and δTGS are chosen to obtain the same kinetics as the reference curve.

Chapter 6

109

900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0

Ferri

te fr

actio

n

Temperature (K)

µ0 (m4J-1s-1) δTGS (K)

13.5 x 10-7 0 K 21.1 x 10-7 34 K 24.0 x 10-7 61 K

Figure 6.6 Grain surface nucleation mode at 10 K/s: change of the pre-exponential

factor of the interface mobility for different values of δTGS to have the same transformation kinetics.

900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0

Ferri

te fr

actio

n

Temperature (K)

µ0 (m4J-1s-1) δTTL/δTGS (K)

16.3 x 10-7 0/0 24.2 x 10-7 16/15 29.0 x 10-7 27/30

Figure 6.7 Two nucleation modes at 10K/s: change of the pre-exponential factor of the

interface mobility for different values of δTTL and δTGS to have the same transformation kinetics.

Chapter 6

110

17 nuclei form at the triple lines, approximately the same number as for the low cooling rate case, and the remaining 101 nuclei form at the grain surfaces. Since the nucleation temperature for a given mode is assumed to be rather insensitive to the cooling rate, the first nuclei at the triple lines form again at 1092 K, like at the low cooling rate. The remaining nuclei at the grain surfaces form at lower temperature. The temperature at which the first nuclei at the grain surfaces form, is set in order to avoid an overlap of the temperature ranges for nucleation at triple lines and grain surfaces, as shown in Figure 6.2. When all nuclei at the triple lines are set to form at a single temperature (δTTL = 0 K) all nuclei at the grain surfaces are assumed to form at an additional undercooling of 10 K. The values of the interface mobility used to fit the reference kinetics are larger than those used in the simulations with a single nucleation mode at the grain surfaces; this is a consequence of the initially higher nucleation rate in simulations with only nucleation at grain surfaces (see Figure 6.2).

0 10 20 30 40 50 60 700.0

5.0x10-7

1.0x10-6

1.5x10-6

2.0x10-6

2.5x10-6

3.0x10-6

nucleation at TL (0.4 K/s) nucleation at TL & GS (10 K/s) nucleation at GS (10 K/s)

µ 0 (m4 J-1

s-1)

δT total (K)

Figure 6.8 Interface mobility as a function of the nucleation temperature spread to

reproduce the reference transformation kinetics at each cooling rate. These results are summarised in Figure 6.8 where the pre-exponential factor of the interface mobility, µ0, is plotted as a function of the total nucleation temperature range, δTtotal. In the figure the data obtained at 0.4 K/s, where only nucleation at triple lines

Chapter 6

111

occurs, are also added. µ0 increases with δTtotal for both cooling rates, but the µ0 values used to reproduce the kinetics at 10 K/s are higher than those used for the lower cooling rate. The dependence of µ0 on the cooling rate is physically unexpected but had been observed in earlier studies as well [20].

6.4 Discussions Early publications on phase field modelling of the austenite to ferrite transformation showed a highly satisfactory agreement between the experimental and simulated transformation kinetics when a suitable choice of the interface mobility was made [13, 21]. This ability was confirmed in the present work. Unlike most transformation kinetics models, phase field simulations are able to reproduce the morphology of the ferrite phase produced after cooling at different cooling rates [13]. The major limit of these early studies was to restrict the modelling to a 2D space. Recently, Militzer et al. [14] reported the results of a first series of 3D phase field simulations of the austenite to ferrite transformation during continuous cooling. They showed that 3D simulations not only describe the transformation kinetics better than 2D simulations but also lead to a more realistic picture of the final microstructure [14]. In the present work 3D simulations offer the further advantage of representing different types of nucleation modes better than in 2D simulations; in 2D simulations all nuclei form and grow in the plane of calculation while ferrite grains that appear in 2D cuts of 3D microstructure may be nucleated above or below the plane of view, exactly like in 2D sections of metallographic samples. Another important aspect of selecting the nucleation sites is related to whether they are randomly distributed or occur in clusters. A homogeneous nuclei distribution in the calculation domain makes the entire domain to be homogenously transformed, in agreement with the experimental microstructure [13], while the presence of nuclei clusters and the consequent untransformed austenite region in the final microstructure can artificially reduce the final ferrite fraction, as illustrated in Figure 6.9.

Chapter 6

112

900 950 1000 1050 11000.00.10.20.30.40.50.60.70.80.91.0

Ferri

te fr

actio

n

Temperature (K)

homogeneus nuclei distribution nuclei clusters

.

a

b

c

Figure 6.9 Effect of nuclei distribution on the transformation kinetics (a);microstructure developed with a homogeneous nuclei distribution (b) andwith the presence of nuclei clusters (c). For the colour version refer tothe appendix

Chapter 6

113

One of the main results of the present investigation is that the effective interfacial mobility required to fit a given transformation curve, depends strongly on the nucleation temperature range assumed: µ0 goes up as δT increases. The nucleation rate affects the transformation kinetics, which explains the different required mobilities at 10 K/s for nucleation at triple lines and grain surfaces as compared to nucleation at grain surfaces only (Figure 6.8). In order to combine both the nucleation temperature width and the nucleation rate in a single parameter, an average nucleation temperature was calculated, weighting each nucleation temperature Ti with the number of nuclei Ni formed at this temperature, according to

i iin

total

T NT

N⟨ ⟩ = ∑ (6.2)

Figure 6.10 shows µ0 as a function of the average undercooling with respect to the maximum nucleation temperature, Tn, n n nT T T⟨∆ ⟩ = − ⟨ ⟩ . At 10 K/s a single relationship

was found between µ0 and ⟨∆Tn⟩, independent of the operational nucleation modes. At 0.4 K/s much lower interface mobility is required to fit the transformation. This means that the pre-exponential factor for the interface mobility depends on the cooling rate. Clearly, this is in contradiction with the physical concept of the interface migration. The temperature dependence of µ0 may be related to the solute-drag effect of Mn atoms that segregate at the moving interface, as proposed by Fazeli and Militzer [22]. In the present phase field approach the solute drag term is not included in the driving pressure formulation. Hence the interface mobility, used explicitly as an adjustable parameter to fit a reference kinetics in agreement with the experimental data, does not represent the intrinsic mobility of the interface but it is an effective parameter, which includes the solute drag effect. If µ0 values are normalised to the value set when all nuclei form at the same temperature, i.e. nT⟨∆ ⟩ = 0

K, the dependence of this normalised mobility factor on nT⟨∆ ⟩ follows the same trend for

both cooling rates, as shown in Figure 6.11. The increase of the average nucleation undercooling, nT⟨∆ ⟩ , by increasing the nucleation

temperature range or decreasing the nucleation rate (in both cases nT⟨ ⟩ decreases) requires

a similar relative increase of the interface mobility to reproduce the target transformation kinetics. The increase of nT⟨∆ ⟩ from 0 K to 40 K requires an increase of µ0 by a factor 2.

Due to the difficulty in obtaining reliable nucleation data experimentally, this effect on the µ0 value can be regarded as relatively small.

Chapter 6

114

0 5 10 15 20 25 30 35 40 45 500.0

5.0x10-7

1.0x10-6

1.5x10-6

2.0x10-6

2.5x10-6

3.0x10-6

µ 0 (m4 J-1

s-1)

Tn - <Tn>

nucleation at TL (0.4 K/s) nucleation at TL & GS (10 K/s) nucleation at GS (10 K/s)

Figure 6.9 Interface mobility as a function of the average undercooling with respect to

the maximum nucleation temperature.

0 5 10 15 20 25 30 35 40 45 501.0

1.2

1.4

1.6

1.8

2.0

2.2

µ 0/µ0 (T

n = <

T n> )

Tn - <Tn>

nucleation at TL (0.4 K/s) nucleation at TL & GS (10 K/s) nucleation at GS (10 K/s)

Figure 6.10 Normalised interface mobility as a function of the average undercooling

with respect to the maximum nucleation temperature.

Chapter 6

115

From the previous figures and discussion it follows that the comparison with a given reference transformation curve does not allow establishing the correct combination of µ0 and δT unambiguously and additional data are required. The δT range can be found directly from dedicated experiments, such as the nucleation investigation of Offerman et al. [15], or indirectly from the final ferrite grain size distribution, as will be shown here. Figure 6.12 shows the ferrite grain size distribution obtained in simulations at 10 K/s for various δT ranges with nucleation at grain surfaces only and two nucleation modes, respectively. The total number of 118 ferrite grains in the final microstructure offers a good statistics for the evaluation of the ferrite grain size distribution. All the grain size distributions are fitted with a lognormal distribution defined by [22]

2

2

(ln )1( ) exp22

d Mf dSd S

αα

α π −

= −

(6.3)

where M is the natural logarithm of the median grain size and S the standard deviation of the equivalent Gaussian (normal) distribution. The mean value and the standard deviation of the variable dα, dα

µ and dασ respectively, are given by

( ) 2 / 2d M Sd d f d d e

α α α αµ∞ +

−∞= =∫ (6.4)

( ) ( ) ( )2 22 2d 1M S Sd dd f d d e e

α αα α ασ µ∞ +

−∞= − = −∫ (6.5)

(The mean value dαµ is different from the peak value of the distribution

2peak M Sd eα−= ).

δTTL, K δTGB, K δTtotal, K dα

µ , µm dασ , µm

- 0 0 10.1 0.8 - 35 35 10.2 2.3 Nucleation at grain surfaces - 61 61 11.1 6.1 0 0 11 10.6 0.9 16 15 35 9.9 2.8

Nucleation at triple lines and grain surfaces

26 29 61 9.7 5.6 Table 6.4 The mean value and the standard deviation of the ferrite grain size

calculated by fitting the simulated ferrite grain size distributions using eq.(6.3)

Chapter 6

116

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

µ0 (m4J-1s-1) δTGS (K)

13.5 x 10-7 0

Freq

uenc

y

dα (µm)

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

µ0 (m4J-1s-1) δTGS (K)

21.1 x 10-7 35

Freq

uenc

y

dα (µm)

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

µ0 (m4J-1s-1) δTGS (K)

24.2 x 10-7 61

Freq

uenc

y

dα (µm)

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0 µ0 (m

4J-1s-1) δTTL/δTGS (K)16.3 x 10-7 0/0

Freq

uenc

y

dα (µm)

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

µ0 (m4J-1s-1) δTTL/δTGS (K)

24.2 x 10-7 16/15Fr

eque

ncy

dα (µm)

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

µ0 (m4J-1s-1) δTTL/δTGS (K)

29.0 x 10-7 26/29

Freq

uenc

y

dα (µm)

Figure 6.11 Simulated ferrite grain size distributions (grey bars) obtained at 10 K/s with

different µ0 and nucleation temperature spread for grain surface nucleation only (left side) and triple line and grain surface nucleation (right side); lognormal fit according equation 136. (lines)

Chapter 6

117

The values dαµ and dα

σ are reported for the different simulated grain size distributions in

Table 6.4.The standard deviation of the distribution depends, in a first approximation, only on the total nucleation temperature range but not on the presence of a specific nucleation mode. In Figure 6.13 the experimental ferrite distribution for the sample yielding the experimental reference kinetics curve [11] is plotted. The 2D grain size values as derived from metallographic measurements are converted into 3D grain size values using the method proposed by Matsuura and Itoh [24]. The log-normal fit gives for the dα

µ and

dασ the values of 11.0 µm and 5.3 µm, respectively. The simulated grain size distribution

obtained with a total temperature range of about 61 K fits then quite well with the experimental one. This result suggests that if both nucleation at triple lines and grain surface occurs for a cooling rate of 10 K/s, a nucleation temperature range of about 30 K for each active nucleation mode may correctly reproduce the experimental grain size distribution.

0 10 20 30 40 500.00.10.20.30.40.50.60.70.80.91.0

Freq

uenc

y

dα (µm)

Figure 6.12 Experimental ferrite grain size distribution (grey bars) and lognormal fit

according equation 6.3 (line) At lower cooling rate only nucleation at triple lines would occur but the temperature interval would be comparable to that for nucleation at triple lines as concluded for 10 K/s. The most realistic combination µ0 - δT is then determined from the trend of Figure 6.8, giving for µ0 the values of µ0 = 4.2 × 10−7 m4J−1s−1 and µ0 = 29 × 10−7 m4J−1s−1 at 0.4 and 10 K/s, respectively.

Chapter 6

118

6.5 Conclusions A critical parameter study for a 3D phase field model for the austenite-to-ferrite transformation has shown that the pre-exponential factor of the effective interface mobility, µ0, and the nucleation temperature range, δT, are strongly coupled. An increase of the δT range leads to an increase of µ0 to replicate a reference (experimental) kinetics. The interaction between µ0 and δT depends on the cooling rate and, for a given cooling rate, on the nucleation rates. However, the dependence of the relative mobility factor on the average undercooling for nucleation ⟨∆Tn⟩ follows the same trend for both cooling rates. The experimental ferrite grain size distribution is used to establish the most realistic combination of the values for µ0 and δT since the nucleation temperature interval markedly affects the standard deviation of the ferrite grain size distribution; in particular it was found that the width of the ferrite grain size distribution increases with the nucleation temperature spread. The values of δT for each nucleation mode are estimated from the grain size distribution while the fit of the transformation kinetics is used, together with the estimated δT to derive the effective interface mobility values for each cooling rate investigated.

References 1. A. Kolmogorov, Izv. Akademii Nauk USSR Ser. Matemat. 3 (1937) 355-359 2. W. Johnson, R. Mehl, Transactions AIME 135 (1939); 135: 416-458 3. M. Avrami, J. Chem. Phys. 8 (1940) 212-224. 4. I. Tamura I, C. Ouchi, T. Tanaka, H. Sekine. Thermomechanical Processing of High

Strength Low Alloy Steels. London: Butterworth and Co., 1988, 17-48. 5. R.A. Vandermeer, Acta Metall. Mater. 38 (1990) 2461-70 6. Y. van Leeuwen, S.I. Vooijs, J. Sietsma, S. van der Zwaag, Metall Mat Trans A 29

(1998) 2925-31 7. . G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn., 14 (1998) 10-18 8. T.A. Kop, Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 40 (2000) 713-18 9. Y. van Leeuwen, T.A. Kop, J. Sietsma, S. van der Zwaag, J. Phys IV France, 9 (1999)

401-409 10. Steinbach, F. Pezzolla, B. Nestler, M. Seeβelberg, R. Prieler, G.J. Schmitz and J.L.L.

Rezende, Physica D, 94 (1996) 135-47

Chapter 6

119

11. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998) 73-86 12. G. Pariser, P. Shaffnit, I. Steinbach, W. Bleck, Steel Research 72 (2001) 354-60 13. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , M. Apel, P. Schaffnit, I. Steinbach,

Metall. Mater. Trans. 36A (2005) 2327-40 14. M. Militzer, M.G. Mecozzi, J. Sietsma, S. van der Zwaag, Acta Mat 54 (2006) 3961-72 15. S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L.Margulies,

H.F. Poulsen, M. Th. Rekveldt, S. van der Zwaag, Science, 298 (2002) 1003-05 16. B. Sundman, B. Jansson, J-O. Andersson, CALPHAD, 9 (1985) 153-90 17. MICRESS®, Software developed in ACCESS , Aachen (Germany) 18. A. Giumelli, M. Militzer, E.B. Hambolt. ISIJ Int 29 (1999) 271-80 19. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn., 14 (1998) 10-18 20. Y. van Leeuwen, M. Onink, J. Sietsma, S. van der Zwaag, ISIJ Int 41 (2001) 1037-46 21. M.G. Mecozzi, J. Sietsma, S. van der Zwaag , Acta Mater. 53 (2005) 1431-40 22. F. Fazeli, M. Militzer. Metall. Mater. Trans. A 36 (2005) 1395-1405 23. S.K. Kurtz, F.M.A. Carpay, Journal Appl Phys, 51 (1980) 5725-44 24. K. Matsuura, Y. Itoh , Mater Trans JIM, 32 (1991) 1042

Chapter 6

120

Chapter 7

121

Chapter 7

The mixed mode character of the austenite to ferrite transformation kinetics

in phase field simulations

Abstract 2D and 3D phase field simulations are performed to study the mixed-mode character of the γ to α transformation with respect to the interface-controlled mode and diffusion-controlled mode. For all the conditions investigated, the transformation kinetics is interface controlled in the initial stage of transformation and shift toward a diffusion controlled as the ferrite grows. The rapidity of change from interface control to diffusion control depends on the interface mobility and also on soft impingement. Also the geometric conditions of the growing ferrite affects the transformation character, explaining the differences in transformation kinetics as derived by 2D and 3D simulations.

Chapter 7

122

7.1 Introduction The austenite (γ) to ferrite (α) transformation in Fe-C-Mn alloys involves the structural rearranging of γ (fcc) to α (bcc). This process, involving the individual jumping of Fe and Mn atoms across the γ / α interface, causes the movement of the γ/α interface and it is also referred to as interface migration. The γ to α transformation also involves the long-range redistribution of C atoms. The partitioning of C between γ and α causes a pile-up of C ahead of the moving interface at the γ side. The magnitude of this pile-up depends on the C diffusivity in the bulk of the γ phase and on the rapidity of the interface migration. If the lattice transformation is assumed to be a relatively fast process the C concentrations in α and γ at the interface at any time during transformation are equal to the equilibrium

concentrations Ceqxα and Ceqxγ . Since the C diffusion in the α phase is very high compared

to that in the γ phase, the C content in the α phase is assumed to be constant and equal to C

eqxα . On the contrary in the γ phase a C gradient is present, leading to carbon diffusion

from the α/γ interface towards the bulk of the γ phase. The growth of α phase is driven by the requirement of maintaining local equilibrium at the interface. The interface velocity v is calculated from the mass balance

int

C C

Ceq Ceq

D xv

x x zγ γ

γ α

∂=

− ∂ (7.1)

where CDγ is the C diffusivity in γ and z is the co-ordinate for the direction of growth.

Under such conditions the transformation kinetics is called diffusion controlled [1]. The other extreme is to assume the diffusion of C in γ to be relatively fast, thus causing the C concentration in austenite to be homogeneous. The C concentration in γ at the interface

is equal to the average C concentration in γ, Cxγ . In this condition the velocity of lattice

transformation controls the transformation kinetics. Christian [2] first introduced the concept of the 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, i.e.

v Gαγµ= ∆ (7.2)

Chapter 7

123

The interface mobility, defined by eq. 7.2, i.e. as the proportionality constant between the interface velocity and the driving pressure for the transformation, thus gives a measurement of the compliance of the lattice to the driving pressure. In reality the γ-to-α transformation has a mixed-mode character; neither the interface mobility nor the C diffusivity in γ has an infinite value, and both of them affect the transformation kinetics. In an early study of Van Leeuwen et al. [3], based on 2D simulation of γ to α transformation in Fe-C alloys, the mixed-mode character of the transformation was quantified by a mode parameter, which varies between 0 and 1

depending on the actual C concentration in γ at the interface, Cxγ . The authors analysed the

effect of the overall C content, 0Cx , on the character of the transformation; in particular

they found that the transformation changes from predominantly interface controlled at low

0Cx to predominantly diffusion controlled when 0

Cx is greater than a value, which depends

on the /CDγ µ ratio. The character of the γ to α transformation was recently analysed using

a simplified model by Sietsma and Van der Zwaag [4]. It was shown that a single parameter, Z, including the relevant thermodynamic and kinetics parameters, determines the character of the transformation, which was found to be interface controlled at the early stages of transformation and to shift towards diffusion controlled in the later stages. The phase field model, originally proposed to investigate the solidification of pure materials and alloys [5-6], was recently successfully used for describing solid-state transformation in steels [7-13]. In modelling the γ-to-α transformation kinetics, the phase field approach assumes finite values for both the C diffusion in γ and the interface mobility and thus it provides a suitable tool for analysing the mixed mode character of the transformation. This technique is able to predict complex ferrite morphologies and the carbon diffusion field in the parent austenite, and makes the analysis of the geometry effect on the transformation character possible; in particular the effect of the overlap of diffusion field arising from neighboring growing α grains (soft impingement) can be taken into account in describing the γ to α transformation kinetics; this modelling is then the most promising approach for analysing the mixed-mode character of the γ-to-α transformation, since experimental observation is not feasible. In this chapter 2D and 3D phase field simulations, based on the multiphase-field model, developed by Steinbach et al. [14-15], are used to investigate the mixed-mode character of the γ to α transformation. Different geometries for the growing ferrite are employed: at first we start by considering a “simplified geometry” consisting of a single α grain

Chapter 7

124

nucleating in the bulk of a γ grain; periodic boundary conditions are set. We also consider a polycrystalline γ microstructure with several nuclei forming at triple points or at the grain boundary. As already discussed in chapter 5 and chapter 6, the predicted kinetics is rather independent of the choice of particular nuclei sites as long as the nucleus density is maintained the same. The effect of the interface mobility and of the morphology of the growing grains on the transformation kinetics is analysed. The interaction between a particular diffusion field in the austenite phase and the transformation kinetics is also investigated.

7.2 Simulation conditions Isothermal γ-to-α transformation kinetics in a Fe-C-Mn steel is studied using the phase field model. 1000 K is the selected transformation temperature. At first a “simplified geometry”, consisting of a single α grain nucleating in the bulk of the γ matrix, is used to study the isotropic α growth in 2D and 3D space, as shown in Figure 7.1a and 7.1c. Further 2D and 3D simulations are run employing a polycrystalline γ microstructure as the initial microstructure and setting several α nuclei to form at the same time as shown in Figures 7.1b and 7.d. As already discussed in chapter 5 and chapter 6, the choice of particular nuclei sites does not affect the transformation kinetics as long as the nucleus density and distribution is maintained the same. The calculation parameters for 2D and 3D simulations are reported in Table.7.1. Periodic boundary conditions are set for all simulations. Initial

microstructure Grid size δz, µm

Domain size l p, µmp

Number of α nuclei

Nucleus density ρ, µm-p

2D Single γ grain 0.1 100 x 100 1 1.0 x 10-4 Polycrystalline γ 0.2 75 x 75 16 2.8 x 10-3 3D Single γ grain 0.2 30 x 30 x 30 1 3.7 x 10-5 Polycrystalline γ 0.3 45 x 45 x 45 118 1.3 x 10-3 Table 7.1 Calculation parameters for 2D and 3D simulations using different initial

microstructure and nucleation densities ( p= dimensionality)

Chapter 7

125

Figure 7.1 Different geometries used in 2D (a, b) and 3D (c, d) simulations: a simplified geometry (one α nucleus nucleated in the bulk of a single γ grain) (a, c) and a polycrystalline γ with several nuclei all nucleated at the same time (b, d). For the colour version refer to the appendix.

Since different ferrite nucleus densities are used in different calculations, the interface position, intr , is used to describe the transformation kinetics instead of the ferrite fraction

fα . The relation between intr and fα is derived assuming a spherical shape for the

growing α grains. If p is the space dimension, here 2 or 3,

( ) ( )1p

intfr p g p α

ρ

=

(7.3)

a

d

b

c

Chapter 7

126

with ρ the nucleus density and ( ) 12gπ

= and ( )1333

4g

π =

.

The parameters of the linearised phase diagram and the proportionality constant ∆S for the driving pressure calculation are evaluated using Thermo-calc® [16] under para-equilibrium conditions and their values are reported in Table 7.2. Refer to chapter 2 for the meaning of these symbols. TR.(K) C Rxα (wt%) C Rxγ

(wt%) mαFe-C (K/wt%) mγ

Fe-C (K/wt%) ∆S (J/Km3)

1073 0.009 0.279 -10350 -186.2 3.5 x 105 Table 7.2 Thermodynamic data under para-equilibrium condition

Using the data of Table 7.2 the equilibrium volume fraction of ferrite at 1000 K is derived to be equal to 0.867. The equilibrium ferrite fraction is translated into different interface

positions at the equilibrium, eqintr , depending on the nucleus density and on the space

dimension, as summarized in Table 7.3. 2D 3D Single γ grain Polycrystalline γ Single γ grain Polycrystalline γ

ρ (µm-p) 1.0 x 10-4 2.8 x 10-3 3.7 x 10-5 1.3 x 10-3 eq

intr (µm) 52.5 9.9 17.8 5.4

Table 7.3 Interface position at the equilibrium, for different simulations

The interface energy, σ, is taken to be 0.5 J/m2. The values for carbon diffusion coefficients assuming in D=D0 exp(-Q/RT) are reported in Table 7.4.

0CDα (m2/s) 0

CQα (kJ/mol) 0CDγ (m2/s) 0

CQγ (kJ/mol)

2.2×10-4 122.5 1.5×10-5 142.1

Table 7.4 Diffusion parameters used for the present simulations [17]

Chapter 7

127

The α/γ interface mobility, µ, is assumed to be temperature dependent, according to the relation µ (T) = µ0 exp(−Qµ/RT). In the present calculations different values for the pre-exponential µ0 are used, while the activation energy Qµ is taken to be 140 kJ/mol [18].

7.3 Evolution of the character of the transformation The mixed-mode character is quantified by the mode parameter S, as defined in ref. [4], i.e.

Ceq C

Ceq C

x xS

x xγ γ

γ γ

−=

− (7.4)

where Cxγ is the actual carbon concentration in γ at the interface; Ceqxγ is the carbon

concentration in γ at the interface in the condition of local equilibrium as assumed by the

diffusion controlled model; Cxγ is the carbon concentration in γ at the interface for an

homogeneous carbon distribution within this phase, the value that would result from the interface-controlled mode. In terms of the diffusion and kinetics parameters, i.e. the carbon

diffusivity in γ, CDγ and the interface mobility, µ, the condition C Ceqx xγ γ= is verified when

the µ is large enough to ensure local equilibrium at the interface at any time during

transformation; the other condition C Cx xγ γ= is verified when CDγ is large enough to

redistribute the carbon instantaneously in the bulk of the γ phase.

The parameter S has a value between 0 and 1: S = 0 that means C Ceqx xγ γ= as valid in the

diffusion-controlled mode; S = 1 that means C Cx xγ γ= , as valid in the interface-controlled

mode. In order to evaluate the S parameter at any time during transformation the carbon concentrations in eq. (7.4) have to be calculated as a function of time. The carbon concentration in austenite in equilibrium with ferrite at the transformation temperature is a constant value and it is calculated from the linearised phase diagram; using the data of

Table 7.2 we get 0.672 wt %Ceqxγ = at 1000 K. Cxγ changes during the transformation, depending on the ferrite fraction formed, f α, and is

calculated by the mass balance between α and γ, i.e.

Chapter 7

128

0

1

C eqC x f xx

fα α

γα

−=

− (7.5)

where x0 is the steel carbon content. In deriving eq (7.5) the carbon concentration in α,

Cxα , is assumed to be constant during transformation and equal to the equilibrium value Ceqxα . From the linearised phase diagram at 1000 K it follows that 0.016 wt %Ceqxα =

In order to calculate the S parameter care has been taken to evaluate the carbon

concentration at the interface, Cxγ . Unlike the classical kinetics models, describing the

interface as a region of zero thickness, in the phase field model the carbon concentration in γ at the interface does not have a single value within the diffuse interface. Combining eqs.

(2.38) and (2.40) for carbon, it is possible to obtain Cxγ in terms of the overall carbon

concentration and phase field parameter calculated for a specific grid position, i.e.

1 (1 )

CC

C eq

xxkγ φ

=− −

. (7.6)

According to eq. (7.6), different values for Cxγ may be derived depending on the value of φ,

which in the interface region varies between 0 and 1. Figure 7.2 shows the carbon content in austenite along the normal of the interface, including the values within the interface of thickness η. Figure 7.2 also reports the phase

field parameter at the same position. In the reported example three different values for Cxγ

are identified within the interface. Therefore a criterion for the calculation of the interface carbon composition at each time must be established. Assuming an isotropic carbon diffusion field around a ferrite grain and assuming also that the carbon diffusion field around a particular grain in a polycrystalline microstructure is the same for every nucleus in the calculation domain, since all nuclei form at the same time, the interface carbon

composition in γ at a given time is obtained from the average of Cxγ values, calculated

using eq. (7.6), in all cells with φ in the range [0.35-0.65].

Chapter 7

129

0 2 4 60.0

0.2

0.4

0.6

0.8

1.0η φ

xCγ

distance along the normal to the interface

φ

0.10

0.12

0.14

0.16

0.18

0.20

0.22

xC

γ (wt %

)

Figure 7.2 Phase field parameter, φ, and carbon content in γ phase and across the

interface region of thickness η.

7.4 Results 7.4.1 Transformation kinetics In Figure 7.3 the interface positions, rint, derived in 2D simulations employing two nucleus densities, ρ, 1.0 x 10-4 µm-2 and ρ = 2.8 x 10-3 µm-2 and for µ0 = 2.4x10-7 m4 J-1 s-1 and µ0 = 2.4x10-6 m4 J-1 s-1, are plotted as a function of tµ0, where t is the time. The two nucleus densities are applied using the simplified geometry for the lower value and the polycrystalline γ microstructure for the higher value. The plot rint vs tµ0, instead vs t, allows a first estimation of the character of the transformation. At the highest nucleus density the two curves representative of the low and high µ0 overlap, which means that the transformation is mainly controlled by the interface mobility and no significant carbon diffusion effect is present. At the lower nucleus density the two curves representative of the low and high µ0 deviate for tµ0 larger than 7.8 x 10-5 m4 J-1 (corresponding to t = 325 s at low µ0 and at t = 32.5 s at high µ0). The decrease of the interface velocity at µ0 = 2.4 x 10-6 m4 J-1 s-1, and the consequent deviation between the two curves in

Chapter 7

130

Figure 7.3 is due to the gradual shift of the character of the transformation towards diffusion control. This aspect will be more extensively discussed in the next section.

0.0 1.0x10-4 2.0x10-4 3.0x10-4 4.0x10-4 5.0x10-40

10

20

30

40

50

60

ρ =2.8 x 10-3 µm-2

(polycrystalline γ)

ρ = 1.0 x 10-4 µm-2 (simplified geometry)

Inte

rface

dist

ance

(µm

)

t µ0 (m4 J-1)

Figure 7.3 2D simulations: interface position as a function of tµ0 for a simplified

geometry and a polycrystalline γ microstructure employing different nucleus densities, ρ. µ0 = 2.4x10-7 m4 J-1 s-1 (closed markers) and µ0 = 2.4x10-6 m4 J-1 s-1 (open markers)

Figure 7.4 shows the interface position, rint, as a function of tµ0, for 3D simulations using, as in Figure 7.3, two nucleus densities and two interface mobilities. As in the 2D simulations, at the highest nucleus density the transformation kinetics is rather insensitive to the diffusion field in γ and depends only on the interface mobility as can be concluded from the overlap for rint as a function of tµ0. On the contrary at the lowest nucleus density the two kinetics deviate above a given time, which means that the different diffusion fields produced at low and high mobility affect the transformation kinetics. The effect is seen to be distinctly smaller than in 2D simulations. The interface velocities calculated from the slopes of the curves interface position vs time before reaching the plateau value are reported for 2D and 3D simulations in Table 7.5 and Table 7.6, respectively.

Chapter 7

131

0.0 5.0x10-5 1.0x10-4 1.5x10-402468

101214161820

ρ = 3.7 x 10-5 µm-3 (simplified geometry)

ρ =1.3 x 10-3 µm-3

(polycrystalline γ)

Inte

rface

dist

ance

(µm

)

t µ0 (m4 J-1)

Figure 7.4 3D simulations: interface position as a function of tµ0 for a simplified

geometry and a polycrystalline γ microstructure employing different nucleus densities, ρ. µ0 = 2.4x10-7 m4 J-1 s-1 (closed markers) and µ0 = 2.4x10-6 m4 J-1 s-1 (open markers)

ρ (µm-2)

µ0 (m4 J-1 s-1)

1 x 10-4

2.8 x 10-3 µm-2

2.4x10-7 0.050 0.042 2.4x10-6 0.449 0.399

Table 7.5 Interface velocity (µms-1) for 2D simulations employing different nucleus

densities and interface mobilities.

ρ (µm-3)

µ0 (m4 J-1 s-1)

3.7 x 10-5

1.3 x 10-3

2.4x10-7 0.044 0.041 2.4x10-6 0.411 0.406

Table 7.6 Interface velocity (µms-1) for 3D simulations employing different nucleus densities and interface mobilities.

Chapter 7

132

For a given interface mobility employed the interface velocity does not depend on the space dimension. 7.4.2 Carbon distribution and soft impingement Figure 7.5 shows the C diffusion field arisen from several α grains growing in a polycrystalline γ microstructure. The selected times, 100 s and 10 s for low and high interface mobility respectively, is such to have in both cases the average interface position at about 4 µm, from the nucleation site.

a

b

Figure 7.5 Carbon distribution map in 2D simulations employing a polycrystalline g geometry; µ0 = 2.4x10-7 m4 J-1 s-1 (a) and µ0 = 2.4x10-6 m4 J-1 s-1 (b). For the colour version refer to the appendix

All ferrite nuclei are set to form at the same time and then the carbon distribution around a growing grain is the same for all grains of the calculation domain. The carbon-concentration gradient in austenite is larger for the higher interface mobility value and that is true for all times during the transformation. This is shown in more detail Figure 7.6, where the carbon profiles along the line AB in Figure 7.5a-b are reported. As the transformation proceeds the carbon concentration in γ at the interface increases, approaching to the equilibrium carbon composition in γ. This reflects the previously mentioned shift in the character from interface controlled to diffusion controlled. The growth of ferrite grains is also accompanied by an increase of the carbon concentration at

A A

B B

Chapter 7

133

the mid-distance between the two growing interfaces, as a consequence of the overlap of the diffusion fields arisen from the neighboring growing grains, i.e. the soft impingement.

0 5 10 15 200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Start ofhard impingment

wt %

C

distance along AB (µm)

a

0 5 10 15 200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

wt %

Cdistance along AB (µm)

b

Figure 7.6 Carbon distribution along the line AB in Figure 7.5a, 7.5b, at different time during transformation; µ0 = 2.4x10-7 m4 J-1 s-1 (a) and µ0 = 2.4x10-6 m4 J-1 s-1 (b)

A criterion to evaluate the start of soft impingent is by relating the penetration length of carbon in γ, λd, i.e. the effective length over which the diffusion gradient extends into the bulk of this phase, to the distance between two neighboring moving interfaces. Assuming that the diffusion distance of carbon in γ around a grain is the same for all the grains in the microstructure, and that is the case when all the grain nucleate at the same time, soft impingement starts to play a role in the transformation if the distance between the interfaces of two neighboring growing grains becomes less than 2λd. Figure 7.7 schematically shows the carbon distribution profile in the α and γ phase along a

line AB in Figure 7.5 connecting the center of two neighboring growing grains; Cboundxγ is

the carbon concentration in the point H, at the mid-distance between the two neighboring moving interfaces. The penetration length, λd, is defined in Figure 7.7 as the length

containing the same amount of carbon in the region MNH above Cboundxγ in the real carbon

profile, assuming constant the carbon concentration gradient across this length.

Chapter 7

134

8 10 12 14 16

0.0

0.1

0.2

0.3

0.4

0.5N

HM

α γ

xCγ bound

xCα

xCγ

λd

xC

z

Figure 7.7 Carbon distribution across a growing α/γ interface and definition of the carbon penetration length in austenite, λd.

The penetration lengths calculated from the carbon distributions of Figure 7.6a and 7.6b are reported as a function of the interface position in Figure 7.8. The penetration length is constant up to a value of the interface position, marked with circles in the Figure 7.8 that only slightly depends on the interface mobility employed. Above these values a rapid increase of the penetration length is detected as a consequence of approaching the equilibrium. Figure 7.9 shows the carbon concentration at the mid-distance of the two

moving interfaces, Cboundxγ , as a function of the interface position for both the interface

mobilities. The increase of Cboundxγ above the initial carbon concentration, marked with

squares in Figure 7.9, starts earlier than the rapid increases of the penetration length.

Chapter 7

135

0 2 4 6 8 1005

101520253035404550

λ d (µm

)

Interface position (µm)

µ0 = 2.4 x 10-7 m4J-1s-1

µ0 = 2.4 x 10-6 m4J-1s-1

Figure 7.8 High ρ: λd as a function of the interface position for µ0 = 2.4x10-7 m4 J-1 s-1

and µ0 = 2.4x10-6 m4 J-1 s-1; the circles mark the start of rapid increase of λd

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

xC γ boun

d (wt %

)

interface position (µm)

µ0 = 2.4 x 10-7 m4J-1s-1

µ0 = 2.4 x 10-6 m4J-1s-1

Figure 7.9 High ρ: C

boundxγ for µ0 = 2.4x10-7 m4 J-1 s-1 and µ0 = 2.4x10-6 m4 J-1 s-1) as a

function of the interface position; the circles mark the start of increase of C

boundxγ

Chapter 7

136

7.4.3 Mixed mode character of the transformation kinetics Figure 7.10 shows the S-parameter as a function of the interface position in 2D simulations for µ0 = 2.4x10-7 m4 J-1 s-1 for two different nucleus densities, ρ. For each curve a square

and a circle marks the start of the increase of Cboundxγ and of λd respectively. For the low

nucleus density the two interface positions corresponding to the increase of λd and Cboundxγ

are derived from the analysis of the carbon distribution analogous to that performed for the high nucleus density (Figures 7.8 and 7.9).

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

S

interface position (µm)

ρ = 1.0 x 10-4 µm-2, simplified geometry ρ = 2.8 x 10-3 µm-2, polycrystalline γ

Figure 7.10 2D simulations: mixed mode character of the γ to α transformation kinetics

employing different nucleus densities, ρ. µ0 = 2.4x10-7 m4 J-1 s-1; the

squares and the circles mark the start of the increase of Cboundxγ and λd,

respectively

For all the nucleation densities employed the transformation kinetics is nearly interface controlled with the exception of the latest stage of transformation when a sharp decrease of the S-parameter is detected in coincidence to the rapid increase of the penetration length. Figure 7.11 shows the S-parameter as a function of the interface position for µ0 = 2.4 x 10-6 m4 J-1 s-1 for two different values of the nucleus densities, ρ.

Chapter 7

137

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

S

interface position (µm)

ρ = 1.0 x 10-4 µm-2, simplified geometry ρ = 2.8 x 10-3 µm-2, polycrystalline γ

Figure 7.11 2D simulations: mixed mode character of the γ to α transformation kinetics

for a simplified geometry and a polycrystalline γ microstructure employing different nucleus densities, ρ. µ0 = 2.4x10-6 m4 J-1 s-1; the squares and the

circles mark the start of the increase of Cboundxγ and λd , respectively

The transformation is interface controlled (S = 1) only just after nucleation. As the ferrite grains grow the transformation kinetics evolves towards a diffusion-controlled mode. A more rapid shift of the transformation character towards diffusion controlled is detected in coincidence with the rapid increase of the penetration length. As for the low mobility case,

the start of the increase of Cboundxγ and λd is marked by a square and a circle respectively

only for a polycrystalline γ microstructure. The occurrence of the interface instability does not allow the analysis of the carbon distribution at the highest interface mobility. Figure 7.12 shows the S-parameter as a function of the interface position in 3D simulations for both µ0 = 2.4x10-7 m4 J-1 s-1 and µ0 = 2.4x10-6 m4 J-1 s-1. For each mobility value two different nucleus densities are considered, one for a simplified α plus γ geometry and the other for a polycrystalline γ microstructure.

Chapter 7

138

0 2 4 6 8 10 12 14 16 18 200.0

0.2

0.4

0.6

0.8

1.0

S

interface position (µm)

ρ = 3.7 x 10-5 µm-3

ρ = 1.3 x 10-3 µm-3

Figure 7.12 3D simulations: mixed mode character of the γ to α transformation kinetics

for a simplified geometry and a polycrystalline γ microstructure employing different nucleus densities, ρ and µ0 = 2.4x10-6 m4 J-1 s-1 (black markers) and µ0 = 2.4x10-6 m4 J-1 s-1 (gray markers)

As for 2D simulations the transformation kinetics shifts towards a mixed mode character as the interface mobility increases. Compared to the 2D results, for the same value of interface mobility the transformation kinetics in 3D simulations is more interface controlled, i.e. the S-parameter is higher in 3D simulations than in 2D simulations, especially if the higher interface mobility is used.

7.5 Discussions The γ-to-α transformation kinetics is controlled both by the carbon diffusion in the parent γ and by the rapidity of the lattice transformation at the interface, represented by a finite value of the interface mobility, µ. Phase field modelling provides a suitable tool to investigate the mixed mode character of the transformation since it employs a finite value for both the carbon diffusivity in γ and for the interface mobility. In this chapter 2D and 3D

Chapter 7

139

simulations, employing different ferrite nucleus densities and interface mobilities are used to analyse the character of the γ-to-α transformation. The transformation kinetics is here investigated through the time evolution of the interface position, rint, estimated from the ferrite fraction derived from the model simulations, assuming a spherical geometry for the growing ferrite grains. If the time is multiplied by the interface mobility µ0, and the interface position is plotted vs µ0t, it is found that the curves derived for low and high µ0 overlap up to a given time, or µ0t, during the transformation. This means that in this time interval eq. 7.2 is verified and the transformation is interface controlled. When a lower nucleus density is used and the equilibrium is reached at the higher interface position, above a given time a significant deviation between the curve at low and high mobility is found, which means that the transformation character shift towards diffusion controlled. Following the work of Sietsma and Van der Zwaag [4] the character of the transformation is also quantified by the parameter S, as defined in eq. (7.4). For all the conditions investigated in the initial stage of transformation the transformation kinetics is interface controlled; in fact, independently of the interface mobility and the carbon diffusivity in γ, just after nucleation the small amount of carbon atoms rejected from the ferrite phase can easily diffuse into the bulk of austenite and the transformation is interface controlled. For high interface mobility as the ferrite grains grow and the amount of carbon expelled from ferrite increases a rapid shift towards the diffusion-controlled character is found as shown in Figure 7.11. For low interface mobility the transformation maintains the interface-controlled character untill the start of soft impingement and the approach of the equilibrium condition (Figure 7.10).

The start of soft impingement is accompanied both by an increase of Cboundxγ , the carbon

concentration at the mid-distance between two neighboring growing grains, above the initial carbon composition in γ and by a sharp increase of λd, the penetration length of carbon in γ, which is also consequence of approaching the equilibrium. The increase of

Cboundxγ and λd does not occur simultaneously but the increase of C

boundxγ occurs earlier than

the increase of λd (see Figures 7.8-7.9). From the plots of S vs rint of Figure 7.11 and 7.12,

where both the increase of Cboundxγ and λd are marked by squares and circles, it is clear that

soft impingement starts to play a role in the transformation kinetics in coincidence of the sharp increases of λd, which causes a rapid decrease of S. In fact the increase of the penetration length implies an effectively smaller slope of the carbon concentration

Chapter 7

140

gradient, which makes the transport of C-atoms in γ slower, with a consequent shift of the transformation character towards diffusion control. In the work of Sietsma and Van der Zwaag [4] it was found that the transformation kinetics is governed by a single parameter Z given by

CD AZ

V

αγ

αµχ= (7.7)

assuming the driving pressure G∆ to be proportional to the deviation from the equilibrium concentration, i.e.

( )∆ C eq CG x xγ γχ= − (7.8)

with χ the proportionality factor. Vα and Aα are respectively the volume and the surface of the growing α. For a planar interface the value of the parameter Z determines the concentration at the

interface, Cxγ , which is given by (see ref. [4] for more detail)

( ) ( ) ( )( )2 2

0 0 0 0 0 0 0

0

2 2,

2

Ceq Ceq Ceq Ceq Ceq Ceq

CZx x x x Zx x x x Z x Zx x x x

xZ x

α γ α γ α γ

γ

+ ∆ + + + ∆ + − + ∆ + ∆ =+ ∆

(7.9)

with 0 0C Ceqx x xα∆ = − and 0

Cx is the carbon concentration in γ far away from the interface,

which is assumed to be equal to the alloy carbon composition. Z is therefore the parameter that is decisive for the mixed-mode character of the phase

transformation. If Z = 0, eq. (7.9) yields C Ceqx xγ γ= , resulting in S = 0 or diffusion control;

on the contrary if Z → ∞ and 0C Cx xγ = , S = 1 or interface control results.

The geometry conditions of the growing α grain, represented by the ratio /A Vα α , also govern the character of the transformation. The carbon diffusion in the bulk of the γ phase

is favoured by large /A Vα α and then for large /A Vα α the transformation becomes more

interface controlled. If a spherical shape is assumed for the α growing grain, /A Vα α is equal to / intp r , where p is the space dimension and rint is the interface position, which

equals the radius of the grain. For this reason the transformation kinetics in 3D simulations has a more interface-controlled character than in 2D simulations.

Chapter 7

141

In order to calculate the Z parameter from the phase field results the proportionality

constant χ and the ratio /A Vα α must be determined. In order to derive the constant χ, the driving pressure G∆ , calculated using eq. (2.52), was

plotted as a function of deviation from the equilibrium concentration, ( )∆ C Ceq Cx x xγ γ γ= − ;

the curves obtained for all simulations were linear fitted through zero to derive from the slope the χ parameter value, as reported in Figure 7.13.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-5x107

-4x107

-3x107

-2x107

-1x107

0

3D2D

∆G (J

/m3 )

∆xCγ = [ xCeq

γ-xC

γ ] (wt %)

ρ =1.0 x 10-4 µm-2 µ0 = 2.4 x 10-7 m4J-1s-1

ρ =2.8 x 10-3 µm-2 ρ =1.0 x 10-4 µm-2 µ0 = 2.4 x 10-6 m4J-1s-1

ρ =2.8 x 10-3 µm-2 7.4 x 10-7 J m-3 wt-1 SLOPE

Figure 7.13 Driving pressure of the transformation, ∆G , calculated using eq. (2.52)

plotted as a function of deviation from the equilibrium concentration, ∆ Cxγ

The ratio /A Vα α is derived from the ratio between the number of cells with 1φ = (α

phase) and the number of cells with 0 1φ< < (α/γ interface) multiplied by the grid

dimension and divided by the number of cells in the interface thickness. Figure 7.14 shows the parameter S as a function of Z for 2D and 3D simulations employing different interface mobilities and nucleus densities. The proportionality factor χ is set equal to 7.4 x 107 J m-3 wt%-1, as derived from the plot of Figure 7.13. For all the simulations the S-parameter follows the same trend when plotted vs Z up to the start of soft impingement.

Chapter 7

142

0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.60.70.80.91.0

3D2DS

Z (wt %)

ρ =1.0 x 10-4 µm-2 µ0 = 2.4 x 10-7 m4J-1s-1

ρ =2.8 x 10-3 µm-2 ρ =1.0 x 10-4 µm-2 µ0 = 2.4 x 10-6 m4J-1s-1

ρ =2.8 x 10-3 µm-2

Figure 7.14 S parameter as a function of Z parameters calculated for all simulations

with χ=7.4 x 107 J m-3 wt%-1.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

S

Z (wt %)

from eq 7.9 2D phase field simulation,

ρ =1.0 x 10-4 µm-2 µ0 = 2.4 x 10-7 m4J-1s-1

Figure 7.15 S vs Z plot derived from the interface carbon composition in γ , given by eq.

(7.9) (line) compared to the same plot derived from the interface carbon

composition in γ obtained by a 2D phase field simulation with µ0 = 2.4x10-7

m4 J-1 s-1 and ρ =1 x 10-4 µm-2 (markers). In both cases 0C Cx xγ ≈

Chapter 7

143

Figure 7.15 reports the S vs Z plot derived, as the ref. [4], using the eq. (7.9) to calculate

the carbon composition in γ at the interface and assuming 0C Cx xγ ≈ (line). The same plot,

derived using for the carbon composition in γ at the interface the value obtained in a 2D phase field simulation (µ0 = 2.4x10-7 m4 J-1 s-1 and ρ =1 x 10-4 µm-2), is also reported (markers). Figure 7.15 shows that the agreement with the analytical model is rather well but not perfect, due to some simplifying assumptions in the analytical model, like the

constant value during transformation of Cxα ( C Ceqax xα = ) and also the assumption 0

C Cx xγ ≈

for the S-parameter calculation [4]. The curvature effect is too small to impair the agreement.

7.6 Conclusions Phase field simulation provides a suitable tool to determine the character of the γ-to-α transformation with respect to the mixed-mode character of phase transformation. For all the conditions investigated, the transformation kinetics is interface controlled in the initial stage of transformation and shifts towards diffusion control as the ferrite grows. The rapidity of change from interface control to diffusion control depends on the interface mobility and also on soft impingement, which causes a sharp shift towards a diffusion-controlled character by affecting the carbon diffusion field in the parent γ. For high interface mobility the rapid shift towards the diffusion-controlled character is found already at the early stage of the transformation. For low interface mobility the transformation maintains the interface-controlled character until the start of soft impingement. The geometry of the growing ferrite also affects the transformation character. In particular a large ratio between the interfacial area and the volume of the growing ferrite makes the transformation more interface controlled. That explains why the transformation kinetics in 3D simulations is more interface controlled than in 2D simulations.

Chapter 7

144

References 1. C. Zener, J. Appl. Phys., 20 (1949) 950-53 2. J.W. Christian, The Theory of Transformation in Metals and Alloys, Pergamon Press,

Oxorf, (1981) 476-479 3. Y. van Leeuwen, T.A. Kop, J. Sietsma, S. van der Zwaag, J. Phys IV France, 9 (1999)

401-409 4. J. Sietsma, S. van der Zwaag, Acta Mater., 52 (2004), 4143-52 5. G. Caginalp. Phys. Rev. A 39 (1989) 5887-96 6. A.AWheeler , WJ Boettinger, GB McFadden. Phys. Rev. A 45 (1992) 7424-39 7. G. Pariser, P. Shaffnit, I. Steinbach, W. Bleck Steel Research 72 (2001) 354-60 8. DH. Yeon, PR. Cha, JK. Yoon. Scripta Mater., 45 (2001) 661-68 9. I. Loginova I, J. Odqvist, G. Amberg, J. Ågren , Acta Mater., 51 (2003) 1327-39 10. I. Loginova I, J. Ågren, G. Amberg. Acta Mater. 52 (2004) 4055-63 11. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, M.Apel, P. Schaffnit, I. Steinbach.

Metall. Mater. Trans. 36A (2005) 2327-40 12. M.G. Mecozzi, J. Sietsma, S. van der Zwaag. Acta Mater. 53 (2005) 1431-40 13. C.J. Huang, D.J. Browne, S. McFadden. Acta Mater. 54 (2006) 11-21 14. I. Steinbach, F. Pezzolla, B. Nestler, M. Seeβelberg, R. Prieler, G.J. Schmitz and J.L.L.

Rezende, Physica D, 94 (1996), 135-47 15. J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D, 115 (1998) 73-86 16. B. Sundman, B. Jansson, J-O. Andersson, CALPHAD, 9 (1985) 153-90 17. Handbook of Chemistry and Physics, CRC Press Inc. Boca Raton, 1989 18. G.P. Krielaart, S. van der Zwaag, Mater. Sci. Techn., 14 (1998) 10-18

Summary

145

Summary

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 treatments imposed during rolling, subsequent controlled cooling and coiling. 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: for the common steel grades the austenite single-phase is stable above a temperature ranging between 1000 K and 1185 K. During cooling, the γ phase transforms into different stable or metastable phases, depending on the steel composition and on the cooling conditions. For many years γ decomposition was analysed using semi-empirical models in which the experimental fractions of product phases were fitted for instance the Johnson-Mehl-Avrami-Kolmogorov equation with two adjusted parameters. The limit of this model is that the values of these fitting parameters, describing the grow kinetics of the specific product phase, are only valid within the condition in which the experimental data are derived. More physically based models are required for a better prediction of the steel microstructure not restricted to a specific combination of composition and process schedule. The most important transformation to be studied using a physically based model is the transformation of the γ phase to the body centered cubic (bcc) ferrite (α) phase. The α phase is the first reaction product formed upon cooling from γ and thus affects the subsequent formation of other phases in the steels. In Fe-C steels, with a carbon content higher than the maximum carbon solubility in α, the γ-to-α transformation involves two distinctly different phenomena occurring simultaneously: a) the carbon partitioning between α and γ, influenced by the carbon diffusion in γ (the carbon diffusion in α is much faster than that in γ), b) the construction of the bcc (α) lattice from the fcc (γ) lattice. 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, which means that the interfacial reaction, which transforms the fcc lattice into the bcc lattice, is a sufficiently fast process not to affect the transformation

Summary

146

kinetics; in this condition the carbon concentrations in α and γ at the interface at any time during transformation are equal to the equilibrium concentrations and the transformation is then said to be diffusion-controlled. The other extreme is to assume that the transformation kinetics is controlled only by the interface mobility and that the carbon diffusivity in austenite is large enough to maintain a homogeneous carbon concentration in austenite; the transformation is then said to be interface-controlled. The interface-controlled model introduces the concept of interface mobility as a proportionality factor between the interface velocity and the driving pressure of the transformation. This parameter, which gives a measure of the rapidity of lattice transformation, is assumed to obey an Arrhenius relationship with a pre-exponential temperature -ndependent factor and activation energy for atoms crossing the interface. Both the long-range carbon diffusion in γ and the lattice transformation at the interface has to be taken into account to describe the γ-to-α transformation, since in reality it has a mixed-mode character. The carbon concentration at the interface in the austenite has a value between the average carbon concentration in γ and the equilibrium carbon concentration in this phase and strongly depends on the nature of the phase transformation with respect to the diffusion-controlled or the interface-controlled mode. Besides the character of the transformation, another aspect that has to be considered is the role of the addition of a substitutional alloying elements, like Mn, in the transformation process. 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 a result of the large difference in the diffusivities of C and Mn in γ at the temperature of interest

( 6/ 10Mn CD Dγ γ−≈ ), it is usually not possible to simultaneously satisfy the local equilibrium

for C and Mn at the interface. 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 equilibrium between the parent and newly formed phase, different constrained equilibria have been defined for substitutional elements. For finite interface mobility, the effect of Mn segregation at the moving interface on the driving pressure of the transformation (solute-drag) has to be also considered.

Summary

147

The interface-controlled, diffusion-controlled and the mixed-mode models are phenomenological in nature and allow the prediction of the fraction transformed in relation to specific thermal treatments, therefore they effectively address the macroscopic length scale. These phenomenological approaches can be translated into a mesoscopic scale by employing front tracking methods. In addition to these sharp interface models, which involve several mathematical complications, the phase field model can be used for modelling the phase transformations at a mesoscopic scale. Based on the construction of a Landau-Ginzburg 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 tφ , also called phase field or order parameters, are

introduced to describe at any time, t, and at each point, r, the different domains present in the computation system, which represents the material on a microstructural scale. Typically these field variables have a constant value in the bulk regions and change continuously over a diffuse-interface of thickness η. In this thesis the multiphase field model developed by Steinbach et al. and implemented in the MICRESS (MICrostructure Evolution Simulation Software) code was used to analyse the austenite to ferrite transformation in a C-Mn steels. 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 the carbon diffusion equation assuming that Mn does not diffuse in the bulk of α and γ phase during transformation due to the much lower diffusivity of Mn atoms compared to that of C atoms. The interface mobilities, the 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 C and Mn content. In the early studies, reported in Chapter 3 - 4 we assumed that both C and Mn partition between α and γ in the diffuse interface and contribute to the driving pressure calculation. In Chapter 5 - 7 the system is considered in the para-equilibrium condition, i.e. Mn partitioning between α and γ does not occur during transformation. Although the MICRESS code allows the derivation of the driving pressure of the transformation directly through the coupling with the Thermo-calc software, in this thesis a linearisation of the phase diagram is used to derive the driving

Summary

148

pressure under the assumption that it is proportional to the local undercooling at the interface. The phase field formulation used in this thesis does not incorporate in the driving pressure calculation the solute drag effect due to the Mn segregation at the interface; therefore the pre-exponential factor of the interface mobility, which is used as a 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 incorporates the solute drag effect. This explains why we found for this parameter different values when different cooling rates were employed. In the phase field 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. In this thesis the nucleation data were derived by metallographic examination and dilatometry, and they are set as input data of the model. In Chapter 3, where the γ decomposition to α of a Fe-0.10 C, 0.49 Mn (wt%) steel during cooling at different cooling rates were investigated, all nuclei were set to form at a single temperature, estimated from the ferrite start temperature evaluated from the dilatometric curves. An acceptable agreement is obtained between the experimental and simulated ferrite fraction curves at low and medium cooling rate, setting a cooling-rate independent interface mobility and increasing the nucleus density as the cooling rate increases in agreement with microscopic analysis. At high cooling rate the fit has been optimised at the early stages of the transformation where the fit is very good. However, at higher fractions transformed the predictions underestimate the actual transformation. The simulated microstructures at different cooling rates also reproduce quite well the variety in microstructures observed experimentally. The ferrite grain size and morphology as well as the distribution of pearlite colonies (the remaining austenite regions in the model) observed experimentally are well simulated by the model. The derived carbon profiles in austenite show an increase in the austenite carbon content at the interface with progression of the transformation, most pronounced at the highest cooling rate. It thus provides insight in the nature of phase transformation with respect to the interface-controlled or diffusion-controlled mode. It is found that at the initial stages the transformation is always nearly interface-controlled; subsequently it evolves in different ways depending on the morphologies formed during cooling and on the way the equilibrium is approached.

Summary

149

Chapter 4 presents a phase field analysis of the effect of Nb 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 kinetic. 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. 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. A decrease of the γ/α interface mobility is found when the austenitisation temperature is decreased from 1373 to 1173 K as a consequence of the presence of NbC precipitates during austenitisation. The presence of NbC has a distinct effect on the nature of the phase transformation with respect to the interface-controlled or diffusion-controlled mode. The phase field analysis reported in the chapters 3 and 4 of this thesis are limited to 2D space, mainly due to the computational demands required for performing meaningful 3D simulations. Nevertheless, the incorporation of 3D aspects in the modelling of diffusional solid-state phase transformations seemed to be essential 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 cases for continuous cooling transformation kinetics in a C-Mn steel with distinctly different ferrite nucleation conditions 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, namely at triple lines. For the second case, with cooling at 10 K/s, also nucleation at grain surfaces has to been into account to get a substantial α grain refinement as observed in the final microstructure. Ferrite nuclei form over a nucleation temperature range of 18 K for each nucleation mode. 3D simulation 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.

Summary

150

In Chapter 6 the influence of the nucleation behaviour on phase field modelling results is studied. In order to do so, 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.

In the final chapter, Chapter 7, a detailed analysis of the mixed-mode character of the γ-to-α transformation with respect to the diffusion-controlled and interface-controlled mode is performed. 2D and 3D phase field simulations are performed under isothermal conditions. Different interface mobilities and nucleus densities are employed since the former has a direct effect on the interface velocity and the latter has an effect on the time development of the carbon distribution in austenite. The interface position is used, alternatively to the ferrite fraction, to describe the transformation kinetics. For all the conditions investigated the transformation kinetics is interface controlled in the initial stage of transformation and shifts towards a diffusion control as the ferrite grows. The rapidity of change from interface control to diffusion control depends on the interface mobility and also on soft impingement. Also the geometric conditions of the growing ferrite affect the transformation character, explaining the differences in transformation kinetics as derived by 2D and 3D simulations.

Samenvatting

151

Samenvatting

De eigenschappen van staal hangen in hoge mate af van de samenstelling en de microstructuur. Voor een gegeven samenstelling kunnen verschillende microstructuren worden gevormd op basis van bepaalde warmtebehandelingen tijdens het walsen, het daarop volgend gecontroleerd afkoelen en het oprollen van de staalband. In C-Mn staal is de vlakken-gecentreerde kubische structuur (Engels: fcc) austeniet (γ) de stabiele fase op hoge temperatuur. De temperatuur waarboven deze fase stabiel is, hangt af van de samenstelling: voor de gebruikelijke staalsoorten is de austenietfase stabiel boven een temperatuurgebied tussen 1000 K en 1185 K. Tijdens afkoelen transformeert de γ-fase in verschillende stabiele of metastabiele fasen, afhankelijk van de staalsamenstelling en de afkoelcondities. Lange tijd is de γ−decompositie geanalyseerd met semi-empirische modellen, waarbij de experimentele fracties van de nieuwe fasen werden gefit met bijvoorbeeld de Johnson-Mehl-Avrami-Kolmogorov vergelijking met twee fitparameters. De beperking van dit model is dat de waarden van de fitparameters, die de groeikinetiek beschrijven van een specifieke nieuwe fase, alleen geldig zijn voor de condities waaronder de experimentele gegevens zijn verkregen. Voor een betere voorspelling van de microstructuur zijn meer fysisch gebaseerde modellen nodig, die niet beperkt zijn tot een specifieke combinatie van samenstelling en procescondities. De belangrijkste transformatie om te onderzoeken met een fysisch model, is de transformatie van de γ−fase naar de ruimtelijk gecentreerde kubische structuur (Engels: bcc) ferriet (α). De α−fase is het eerste reactieproduct dat gevormd wordt uit γ tijdens afkoelen en dus beïnvloedt deze de daarop volgende formatie van andere fasen in het staal. In Fe-C stalen met een koolstof gehalte hoger dan de maximale koolstofoplosbaarheid in de α−fase vinden tijdens de γ → α transformatie twee verschillende fenomenen tegelijker-tijd plaats: a) de koolstofherverdeling tussen α en γ, beïnvloed door de koolstofdiffusie in γ (koolstofdiffusie in α is veel sneller dan die in γ); b) de aanmaak van het bcc (α) rooster uit het fcc (γ) rooster. De kinetiek van de γ → α transformatie wordt in de literatuur vaak gemodelleerd onder de aanname dat de koolstofdiffusie in austeniet de snelheidsbepalende factor is, hetgeen betekent dat de grensvlakreactie, die het fcc rooster transformeert in een bcc rooster, snel genoeg is om the transformatiesnelheid niet te beïnvloeden. Onder deze

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omstandigheden zijn de koolstofconcentraties in α and γ aan het grensvlak op elk moment tijdens de transformatie gelijk aan de evenwichtsconcentraties. De transformatie wordt dan diffusie-gestuurd genoemd. Het andere uiterste is om aan te nemen dat de transformatiekinetiek alleen wordt bepaald door de snelheid van de grensvlakreactie (gekarakteriseerd door de mobiliteit) en dat de koolstofdiffusie in austeniet snel genoeg is om een homogene koolstofconcentratie in austeniet te behouden. De transformatie wordt dan grensvlak-gestuurd genoemd. Het grensvlak-gestuurde model introduceert het concept grensvlakmobiliteit als een evenredigheidsfactor tussen de grensvlaksnelheid en de drijvende kracht van de transformatie. Deze parameter, die een maat is voor de snelheid van de roostertransformatie, wordt verondersteld thermisch geactiveerd te zijn, en dus een Arrhenius relatie te hebben, met een pre-exponentiële – temperatuur-onafhankelijke - factor en een activeringsenergie voor atomen die het grensvlak oversteken. Zowel de lange-afstands koolstofdiffusie in γ als de roostertransformatie aan het grensvlak moet worden meegenomen om de γ → α transformatie te beschrijven, omdat de transformatie in werkelijkheid een gemengd (“mixed-mode”) karakter heeft. De koolstofconcentratie aan het grensvlak in de austeniet heeft een waarde tussen de gemiddelde koolstofconcentratie in γ en de evenwichts-koolstofconcentratie in deze fase, en is sterk afhankelijk van het karakter van de transformatie: diffusie-gestuurd of grensvlak-gestuurd. Naast het karakter van de transformatie is er een ander aspect dat moet worden meegenomen in het transformatieproces, namelijk de rol van substitutionele legerings-elementen, zoals Mn. Het toevoegen van Mn aan het binaire Fe-C systeem maakt het identificeren van de werkelijke grensvlakcondities niet triviaal, zelfs als wordt aangenomen dat de transformatiekinetiek diffusie-gestuurd is met lokaal evenwicht voor koolstof aan het grensvlak. Als gevolg van de grote verschillen tussen de diffusie-

coëfficiënten van C en Mn in γ bij de gebruikte temperatuur ( 6/ 10Mn CD Dγ γ−≈ ), is het

normaal gesproken niet mogelijk om tegelijkertijd te voldoen aan lokaal evenwicht voor zowel C als Mn aan het grensvlak. Zelfs als er een grote afname van vrije energie zou plaatsvinden als gevolg van volledige Mn-verdeling volgens evenwicht (door de grotere Mn-oplosbaarheid in γ dan in α), bestaat de mogelijkheid dat de γ → α transformatie plaatsvindt zonder Mn-herverdeling. Daarom zijn er, als alternatief voor ortho-evenwicht (alle opgeloste atomen herverdelen volgens evenwicht tussen de originele en de nieuwe fase), verschillende aangepaste evenwichten gedefinieerd voor substitutionele elementen. Voor een eindige grensvlakmobiliteit moet het effect van Mn-ophoping bij het bewegende

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grensvlak op de drijvende kracht van de transformatie (solute drag) ook in rekening worden gebracht. De grensvlak-gestuurde, diffusie-gestuurde en mixed-mode modellen zijn fenomeno-logisch van aard en geven de mogelijkheid de getransformeerde fractie te voorspellen die optreden tijdens bepaalde warmtebehandelingen. Daarom zijn ze van toepassing op macroscopische lengteschaal. Deze fenomenologische benadering kan worden vertaald naar een mesoscopische schaal door het toepassen van front-tracking methoden. Als een alternatief voor deze modellen waarin een scherp grensvlak gedefinieerd wordt, hetgeen een aantal wiskundige complicaties met zich meebrengt, kan het phase-field model worden gebruikt voor het modelleren van fasetransformaties op mesoscopische schaal. Gebaseerd op de constructie van een Landau-Ginzburg vrije energie functional, behandelt het phase-field model op een integrale manier een multi-fase systeem, bestaande uit bulk- en

grensvlakgebieden. Eén of meerdere continue variabelen, ( ),i r tφ , ook wel phase-field of

order parameters genoemd, worden geïntroduceerd om op elk tijdstip t en elk punt r de verschillende domeinen in het systeem te beschrijven. In de berekeningen moeten tijd en ruimte gediscretiseerd worden, met tijdstappen ter lengte ∆t en een rooster met ruimtelijke elementen met afmeting ∆x. Dit representeert het materiaal op micrometer-schaal. Deze variabelen hebben een constante waarde in de bulkgebieden en hebben een continu verloop in een grensvlak met dikte η. In dit proefschrift is voor het analyseren van de austeniet naar ferriet transformatie in C-Mn staal gebruik gemaakt van het multiphase field model, dat ontwikkeld is door Steinbach et al. en geïmplementeerd is in de MICRESS (MICrostructure Evolution Simulation Software) code. De details van het multiphase field model worden, specifiek voor een poly-kristallijn systeem dat de α- en de γ-fase bevat, beschreven in hoofdstuk 2. De evolutie van een aantal continue veldvariabelen met de tijd, waarbij elke variabele een specifiek domein (korrel) van het systeem representeert, bepaalt de microstructurele veranderingen van het multi-domain (poly-kristalijne) systeem. Een stelsel van phase-field vergelijkingen wordt opgelost, rekening houdend met de vergelijking voor koolstofdiffusie, aannemende dat Mn niet diffundeert in de bulk van de α en γ fase tijdens de transformatie, vanwege de veel lagere diffusiecoëfficiënt van Mn-atomen in vergelijking met die van C-atomen. De grensvlakmobiliteiten, de grensvlakenergieën en de drijvende kracht voor de fasetransformatie zijn parameters van de phase-field vergelijkingen, en zij bepalen de kinetiek van de microstructurele evolutie. De drijvende

Samenvatting

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kracht voor de transformatie hangt af van de lokale C- en Mn-concentratie. In eerdere studies, opgenomen in de hoofdstukken 3 en 4, hebben we aangenomen dat zowel C als Mn zich herverdelen tussen α en γ, en bijdragen aan de drijvende krachtberekening. In de hoofdstukken 5, 6 en 7 is verondersteld dat het systeem in para-evenwicht is; de herverdeling van Mn tussen α and γ vindt niet plaats tijdens de transformatie. Hoewel de MICRESS-code toestaat de drijvende kracht voor de transformatie direct te bepalen door middel van een koppeling met ThermoCalc software, is in dit proefschrift een linearisatie van het fasediagram gebruikt om de drijvende kracht te bepalen, onder de aanname dat deze proportioneel is met de lokale onderkoeling aan het grensvlak. De phase-field methode die gebruikt wordt in dit proefschrift is gebaseerd op een drijvende kracht die berekend wordt zonder het effect van solute drag, t.g.v Mn-segregatie aan het grensvlak, mee te nemen. Daarom is de pre-exponentiële factor van de grensvlakmobiliteit, die gebruikt is als fitparameter, niet de intrinsieke mobiliteit, maar dient deze geïnterpreteerd te worden als een effectieve mobiliteit waarin het solute-drag effect is verdisconteerd. Dit verklaart waarom er verschillende waarden voor deze parameter gevonden kunnen worden als verschillende afkoelsnelheden werden gebruikt. In het phase-field model wordt nucleatie in rekenschap gebracht door nieuwe korrels op geselecteerde plaatsen te laten groeien bij een bepaalde onderkoeling. De initiële afmetingen van de nieuwe korrel zijn gelijk aan één enkel roosterelement. Het plaatsvinden van nucleatie volgt niet vanzelf uit de phase-field methode, maar is opgelegd door voorgeschreven nucleatiecriteria, die voortkomen uit nucleatietheorie of experimentele data. De nucleatiegegevens in dit proefschrift zijn bepaald uit metallografisch onderzoek en dilatometrie, en dienen als invoer voor het model. In hoofdstuk 3, waar de omzetting van austeniet naar ferriet wordt onderzocht in Fe-0.10C-0.49Mn (wt%) staal bij verschillende afkoelsnelheden, worden alle nuclei verondersteld gevormd te worden bij een specifieke temperatuur die is bepaald met behulp van dilatometrie. Een acceptabele overeenstemming is gevonden tussen de experimentele en berekende curven van de ferrietfractie las functie van temperatuur bij lage en gemiddelde afkoelsnelheid, waarbij verondersteld wordt dat de grensvlakmobiliteit onafhankelijk is van de afkoelsnelheid en de kiemdichtheid toeneemt met de afkoelsnelheid. In het geval van een hoge afkoelsnelheid is de fit goed in het eerste deel van de transformatie, maar voor het verdere verloop van de transformatie is de voorspelde fractie te laag vergeleken met de gemeten fractie. De gesimuleerde microstructuren, zowel de ferriet korrelgrootte als de distributie van perlietkolonies (dat in het model voorkomt als restausteniet), komen goed overeen met de optisch waargenomen structuren. De koolstofdiffusie-profielen in austeniet tonen aan

Samenvatting

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dat het koolstofgehalte in austeniet aan het grensvlak toeneemt als de transformatie verloopt, en dat dit effect sterker is voor hogere afkoelsnelheden. Dit geeft inzicht in het mixed-mode karakter van de transformatie in verhouding tot de puur diffusie-gestuurde en de puur grensvlak-gestuurde modes. Het blijkt dat in het begin (direct na nucleatie) de transformatie altijd grensvlak-gestuurd is; vervolgens verandert het karakter van de transformatie richting diffusie-gestuurd op een wijze afhankelijk van de morfologie van ferriet en de afkoelcondities. Hoofdstuk 4 behandelt een phase-field analyse van het effect dat Nb heeft op de γ → α transformatiekinetiek tijdens afkoelen. Bij een specifieke afkoelsnelheid (0.3 K/s) en verschillende austeniteertemperaturen is het effect van zowel Nb in oplossing als geprecipiteerd NbC op the transformatiekinetiek onderzocht. Zoals in het vorige hoofdstuk zijn ook hier de austeniet-microstructuur en de kiemdichtheid bepaald aan de hand van experimentele data. Anders dan in het vorige hoofdstuk wordt in dit hoofdstuk aangenomen dat de ferrietkiemen continu gevormd worden in temperatuurinterval van 50 K onder de starttemperatuur van de transformatie, hetgeen in overeenstemming is met de experimentele resultaten van Offerman et al. Ook hier is de grensvlakmobiliteit gebruikt als fitparameter en deze wordt verondersteld representatief te zijn voor het effect van NbC op de transformatie. Een verlaging van de γ/α grensvlakmobiliteit is gevonden wanneer de austeniteertemperatuur daalt van 1373 tot 1173 K. Dit wordt veroorzaakt door de aanwezigheid van NbC-precipitaten in de austenietmatrix. De aanwezigheid van NbC heeft een duidelijk effect op de aard van de transformatie ten aanzien van de grensvlak-gestuurde of diffusie-gestuurde mode, namelijk een veschuiving in de richting van grensvlak-gestuurd. De phase-field simulaties beschreven in hoofdstuk 3 en 4 van dit proefschrift zijn uitgevoerd in een twee-dimensionale ruimte (2D) vanwege de lange rekentijd voor een 3D-ruimte. Toch blijkt het essentieel te zijn om 3D-aspecten mee te nemen in de modellering van diffusionele vaste-stof transformaties om realistische diffusieprofielen te verkrijgen en om een betere verdeling van de kiemen te kunnen invoeren (nucleatie op quadruple points, triple lines en grain surfaces, resp. punten, lijnen en vlakken waar 4, 3 en 2 korrels aan elkaar grenzen). Hoofdstuk 5 beschrijft een serie 3D phase-field simulaties van de γ → α transformatie m.b.v. de MICRESS code. Het onderzoek houdt zich bezig met dezelfde legering als gebruikt in hoofdstuk 3. Twee afkoelsnelheden zijn beschouwd om te onderzoeken hoe 3D-simulaties verschillen van 2D-simulaties met betrekking tot de verschillen in ferriet nucleatiecondities. Een afkoelsnelheid van 0.4 K/s betekent dat maar één nucleatiemode een rol speelt, namelijk nucleatie op triple lines. Voor het geval dat de

Samenvatting

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afkoelsnelheid 10 K/s is, treedt nucleatie op grain surfaces ook op, hetgeen leidt tot een substantiële verfijning van de α-korrels, zoals ook optisch wordt waargenomen. Ferrietnuclei vormen in een temperatuurinterval van 18 K voor beide nucleatiemodes. 3D-simulaties worden gepresenteerd voor beide afkoelcondities. Vervolgens worden de 3D-resultaten vergeleken met de 2D-simulaties, en op basis van deze analyse worden de mogelijke richtingen en uitdagingen voor toekomstig onderzoek in dit vakgebied uiteengezet. In Hoofdstuk 6 is de invloed van het nucleatiegedrag op de phase-field simulaties bestudeerd. Om dit te doen is het 3D phase-field model uit Hoofdstuk 5 gecombineerd met de beste schatting van het werkelijke nucleatiegedrag, verkregen uit experimentele data, om een beschrijving te geven van de transformatiekinetiek en de uiteindelijke microstructuur. Niet alleen de effectieve mobiliteit, maar ook het temperatuurinterval waarin nucleatie plaatsvindt is gebruikt als een te bepalen parameter. Een aantal combinaties van deze twee parameters blijken een even goede overeenstemming met de experimentele fractiecurve te geven. De mate waarin de gesimuleerde en de experimentele verdeling van ferriet korrelgroottes overeenkomen, is gebruikt als maatstaf om de meest realistische combinatie van temperatuurinterval voor nucleatie en grensvlakmobiliteit te vinden.

In het laatste hoofdstuk, Hoofdstuk 7, wordt een uitgebreide analyse gegeven van het mixed-mode karakter van de γ-α transformatie. Zowel 2D als 3D phase-field simulaties zijn uitgevoerd onder isotherme condities. Verschillende waarden voor de grensvlak-mobiliteit en de kiemdichtheid zijn gebruikt omdat de eerstgenoemde een direct effect heeft op de grensvlaksnelheid, en de tweede een effect heeft op het verloop van koolstof-verdeling in austeniet. De grensvlakpositie is gebruikt om de transformatiekinetiek te beschrijven als alternatief voor de ferrietfractie. Voor alle bestudeerde condities is de transformatiekinetiek in het begin van de transformatie grensvlak-gestuurd en verschuift het karakter richting diffusie-gestuurd als de ferrietfractie toeneemt. De mate van verandering van grensvlak-gestuurd naar diffusie-gestuurd hangt af van de grensvlak-mobiliteit en soft impingement, het overlappen van de koolstofverrijkingsgebieden rond verschillende ferrietkorrels. Ook de geometrische condities van de ferriet korrelgroei beïnvloeden het karakter van de transformatie, hetgeen verklaart waarom 2D- en 3D-simulaties verschillende transformatiekinetiek voorspellen.

Acknowledgement

157

Acknowledgements

The work presented in this thesis began in 2002 when I started to work at the Microstructural Control in Metals (MCM) group of the department of Material Science at the Delft University of Technology within a fifth framework project: “Development of a physically-based research tool to conduct virtual experiments to solve problems in steel metallurgy (VESPISM)”. Thanks to the funding by the European Community I could work for two years on the phase field modelling of solid-state transformation in steels. Nevertheless, my PhD work was possible only thanks to the financial support of the Netherlands Institute for Metals Research (NIMR) that, extending my contract for two further years, gave me the opportunity to perform a PhD research. However, only the contribution of several people guaranteed the good outcome of the research activity reported in this thesis. Thus it is now time to thank them all. At first, I would like to thank Jilt Sietsma, my daily supervisor, not only for the big scientific contribution that he gave to my work, but also for the friendly relationship that he established with me during the past four years; a special thanks to him also for speaking to me in Italian and making me feel more at home. A special thanks also for my promoter, Sybrand van der Zwaag, for his continuous support during all the PhD work; the regular meetings with him and Jilt were always the occasion to learn new things through stimulating discussions which allowed me to move forward in my work. I would also like to thank Sybrand for supporting me in writing papers, which I consider the most difficult task of my PhD activity. Besides Jilt and Sybrand, Matthias Militzer gave a big contribution to my PhD work. It was a privilege to collaborate with him during the several months he spent in Delft to work on phase field modelling. I would like to express my sincere gratitude to him for his daily help in allowing me to understand the most difficult aspects of my research activity. I can say that without his contribution this thesis could not be as it is. During this research activity I had contact with several people from ACCESS (Aachen) where the phase field model used in this thesis was developed and implemented in the MICRESS code. I would like to take this occasion to thank Markus Apel, who was always

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willing to help me understand the most detailed aspects of the model, either by e-mail or by useful discussions I had the occasion to have with him in Aachen. His contribution in the modelling part was fundamental for getting the results reported in this thesis. I am also very thankful to Philippe Schaffnit who was always so effective in helping me any time I had problems running the MICRESS code. In the last year an important contribution to my research was given by Richard Huizenga. He developed the program for the visualisation of the 3D microstructure from the MICRESS output file, which made the analysis of the microstructure evolution during transformation easier than the 2D cuts of the 3D microstructure in the original MICRESS output file. I also enjoyed the discussions with him regarding several metallurgical topics. I would also like to thank all my collegues of the MCM group for the friendly realtionship I had with them. Special thanks are due to Erica Anselmino, for the support she gave me in difficult moments of my private and professional life. I also want to thank Nico Geerlofs for the nice conversations we had in the four years we shared an office and also for the help he gave me to perform dilatometric tests. My thanks also to Eric Peekstok for his help in metallographic measuments. I would express my sincere gratitude again to Richard Huizenga and to Stefan van Bohemen for writing the samenvatting of this thesis. Finally I thank the person that more than any other supported me in the past fours years, my husband Antonio. Without his help I could not have performed the PhD research reported in this thesis; in the difficult moments I considered interrupting my work, he always encouraged me to go forward, giving me also a continuous and increasing support in the duties of the family life. For this and also more I express to him all my love.

Pina Mecozzi Oegstgeest, November 2006

List of publications

159

List of publications

Journal paper

1. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, M. Apel, P. Schaffnit, I. Steinbach, “Analysis of the γ → α transformation in a C-Mn steel by phase field modelling”, Metallurgical and Materials Transactions A, 36A (2005), 2327.

2. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, “Phase field modelling of the interfacial condition at the moving interphase during the γ → α transformation in a Nb microalloyed C-Mn steel”, Computational Material Science 34, (2005), 290.

3. M.G. Mecozzi, J. Sietsma, S. van der Zwaag “Analysis of the γ → α transformation in a Nb microalloyed C-Mn steel by phase field modelling ” Acta Mater. 53 (2005) 1431

4. M. Militzer, M.G. Mecozzi, J. Sietsma, S. van der Zwaag, “3D phase field modelling of the γ → α transformation” Acta Mater, 54 (2006) 3961

Conference papers

1. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, M. Apel, P. Schaffnit, I. Steinbach, “Analysis of γ → α Transformation in a C-Mn Steel by Dilatometry, Laser Scanning Confocal Microscopy and Phase Field model”, Material Science & Technology 2003, Austenite Formation and Decomposition, Chicago, (2003), 353

2. M.G. Mecozzi, J. Sietsma, S. van der Zwaag, “Phase field modelling of the interfacial condition at the moving interphase during the γ → α transformation in a Nb microalloyed C-Mn steel”, Material Science & Technology Conference 2004, Sept. 26-28, New Orleans

List of publications

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3. M.G. Mecozzi, M. Militzer, J. Sietsma, S. van der Zwaag, “Analysis of nucleation and interfacial mobility for polygonal ferrite formation using phase field simulations”, Proceeding of TMS 2006 Annual Meeting 12-16 March, San Antonio, TMS letters, 3 (2006), 15

Curriculum vitae

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CURRICULUM VITAE

Maria Giuseppina Mecozzi

Born on 16 July 1964 in Viterbo, Italy

Education 1978-1983 Diploma di Maturità scientifica (High school degree) − Liceo Scientifico

Statale “Tullio Levi Civita,” Rome, Italy, grade 58/60 1984-1989 Laurea in Fisica 1989 − University of Rome “La Sapienza,” grade

110/110. Thesis title: “Photopyroelectric simultaneous monitoring of specific heat and thermal conductivity: application to the analysis of phase transition of high critical temperature superconductors.”

Tutor: Prof. F. Scudieri 1990-1992 Master Science (Engineering) in Ferrous Metallurgy: University of

Leeds (UK) Thesis title: “Effect of Alloying elements and heat treatments on microstructure and properties of 12 % Cr ferrite martensite stainless steels.”

Tutor: Prof. T. Gladman 2002-2006 PhD in Material Science: Technical University of Delft Thesis title: “Phase Field Modelling of Austenite to Ferrite Transformation

in Steel” Promotor: Prof. S. van der Zwaag Work experience 1990-2001 Researcher in the Metallurgy and Corrosion Department of the Centro

Sviluppo Materiali, Roma, Italy

Appendix

1-A

Appendix

Figure 3.1

Figure 3.2

a b

a b

Appendix

2-A

Figure 3.3

1040 K

1010 K

1025 K

950 K

a b

7 µm 7 µm

7 µm 7 µm

Figure 4.11

Appendix

3-A

1040 K

1010 K

1025 K

950 K

Figure 4.12

7 µm

7 µm 7 µm

7 µm

A

A

Appendix

4-A

1085 K 1080 K

1075 K 1060 K Figure 5.3

Appendix

5-A

1067 K 1060 K

1017 K 840 K

Figure 5.6

Appendix

6-A

Figure 6.1

a

c

b

d

Figure 6.5

Appendix

7-A

900 950 1000 1050 11000.00.10.20.30.40.50.60.70.80.91.0

Ferri

te fr

actio

n

Temperature (K)

homogeneus nuclei distribution nuclei clusters

. Figure 6.9

a

b

c

Appendix

8-A

Figure 7.1

a

b

a

d

b

c

A A

B B

Figure 7.5