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Phase-field modeling of microstructure evolution in low-carbon steels during intercritical annealing Zhu, Benqiang 2015

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Phase-Field Modeling ofMicrostructure Evolution inLow-Carbon Steels duringIntercritical AnnealingbyBenqiang ZhuB.Sc., Beijing Institute of Technology, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2015© Benqiang Zhu 2015AbstractIntercritical annealing is used widely in the steel industry to produce ad-vanced high strength steels for automotive applications, e.g. dual-phasesteels. A phase-field model is develop to describe microstructure evolutionduring intercritical annealing of low-carbon steels. The phase-field modelconsists of individual sub-models for ferrite recrystallization, austenite for-mation and austenite to ferrite transformation. In particular, a Gibbs-energydissipation model is coupled to the phase-field model to describe the effectsof solutes on migration of austenite/ferrite interfaces. The model is ap-plied to a low-carbon steel with a cold-rolled pearlite/ferrite microstructuresuitable for industrial production of dual-phase steels (DP600 grade). Thesub-model parameters, e.g. nucleation parameters and interface mobilities,are tuned using experimental data. The interaction of concurrent ferrite re-crystallization and austenite formation is investigated using the developedmodel. The simulation results reveal that ferrite recrystallization can beinhibited by the pinning effect of austenite particles and concurrent ferriterecrystallization can lead to intragranular distribution of austenite in the fi-nal microstructure. The transition of austenite morphology from a networkstructure to a banded structure with increasing heating rates is replicated bythe phase-field model. The model is validated using a simulated industrialiiAbstractintercritical-annealing cycle. Moreover, the developed phase-field model isused to describe cyclic phase transformations in the intercritical region fora plain-carbon steel and a manganese-alloyed low-carbon steel. The con-sideration of Gibbs-energy dissipation in the phase-field model rationalizesthe existence of stagnant stages during cyclic phase transformations in themanganese-alloyed low-carbon steel. In summary, the developed model pro-vides a single tool that is able to describe various physical phenomena occur-ring in an entire intercritical-annealing cycle. Phase-field modeling can be apromising approach for developing process models for advanced steels in thefuture.iiiPrefaceThis dissertation is written based on the original work conducted by theauthor Benqiang Zhu in Department of Materials Engineering, The Univer-sity of British Columbia, Point Grey campus. I was the lead investigator,responsible for developing phase field models, writing computer codes, im-plementing simulations, analyzing results, and writing this manuscript. Mysupervisor, Dr. Matthias Militzer was involved throughout the project andprovided great guidance and assistance with the manuscript writing. Allexperimental data in Chapter 5, Chapter 6 and Chapter 7 were providedby Dr. Mykola Kulakov (previous PhD student in Department of MaterialsEngineering, The University of British Columbia). All experimental data inChapter 8 were provided by Dr. Hao Chen (previous Postdoc in Departmentof Materials Engineering, The University of British Columbia).In Chapter 1 and Chapter 2, Fig. 1.1, 2.2, 2.8, 2.9, 2.10, 2.12, 2.13, 2.14,2.15, 2.16, 2.17, 2.18, 2.19, 7.12, 2.20, 2.21 and 2.22 have been taken withpermission from the cited sources.A version of Chapter 5 was published in a journal paper: B. Zhu, M.Militzer, “3D phase field modeling of recrystallization in a low-carbon steel”,Modeling and Simulation in Materials Science and Engineering, Vol. 20,No.8, P. 085011, 2012. I was the investigator, responsible for model devel-ivPrefaceopment, performing simulations, data analysis and manuscript composition.Matthias Militzer was the supervisory author and was involved in conceptformation and manuscript edits.A version of Chapter 6 and Chapter 7 was published in a journal paper:“B. Zhu, M. Militzer, “Phase-Field Modeling for Intercritical Annealing ofa Dual-phase Steel”, Metallurgical and Materials Transaction A, vol. 46, pp1073-1084, Mar 2015. I was the investigator, responsible for model devel-opment, performing simulations, data analysis and manuscript composition.Matthias Militzer was the supervisory author and was involved in conceptformation and manuscript edits.A version of Section 6.3 and Section 6.4 was published as part of a con-ference proceedings: “M. Kulakov, B. Zhu, W. Poole and M. Militzer, “Mod-eling Microstructure Evolution during Intercritical Annealing”, Thermome-chanical Processing, Sheffield, 2012. This paper combined the experimentaldata of Mykola Kulakov with some of my phase-field simulations. WarrenPoole and Matthias Militzer were the supervisory authors and were involvedin concept formation and manuscript edits.A version of Chapter 8 was published in a journal paper: B. Zhu, H.Chen, M. Militzer, “Phase-Field Modeling of Cyclic Phase Transformationsin Low-carbon Steels, Computational Materials Science, In Press, Feb 2015”.I was the investigator, responsible for model development, performing sim-ulations, data analysis and manuscript composition. Hao Chen providedthe necessary experimental results and contributed to the manuscript edits.Matthias Militzer was the supervisory author and was involved in conceptformation and manuscript edits.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Phase-Field Approach . . . . . . . . . . . . . . . . . . . . . . 72.3 Recrystallization . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Recrystallization in Low-carbon Steels . . . . . . . . . 102.3.2 Modeling Recrystallization . . . . . . . . . . . . . . . 13viTable of Contents2.4 Phase-transformation Theories in Steels . . . . . . . . . . . . 152.5 Austenite Formation and Decomposition . . . . . . . . . . . . 252.5.1 Austenite Formation from Pearlite . . . . . . . . . . . 252.5.2 Austenite Formation from Ferrite/pearlite Structures . 282.5.3 Austenite Formation from Ferrite/Carbide Structures 372.5.4 Austenite Formation from Bainite and Martensite . . 382.5.5 Austenite-to-ferrite Transformation . . . . . . . . . . . 402.5.6 Intercritical Annealing . . . . . . . . . . . . . . . . . . 432.5.7 Cyclic Phase Transformation . . . . . . . . . . . . . . 432.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . 494 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Basics of the Multi-Phase-Field Model . . . . . . . . . . . . . 524.3 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . 564.4 Initial Microstructures . . . . . . . . . . . . . . . . . . . . . . 594.5 Ferrite Recrystallization . . . . . . . . . . . . . . . . . . . . . 604.6 Austenite Formation . . . . . . . . . . . . . . . . . . . . . . . 624.7 Gibbs-energy Dissipation Model . . . . . . . . . . . . . . . . 674.8 Austenite-to-ferrite Transformation . . . . . . . . . . . . . . . 734.9 Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 744.10 Model Implementation . . . . . . . . . . . . . . . . . . . . . . 755 Modeling Recrystallization . . . . . . . . . . . . . . . . . . . . 775.1 Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . 77viiTable of Contents5.2 Effects of Inhomogeneity on Recrystallization . . . . . . . . . 825.3 Application to the Cold-rolled Ferrite/Pearlite Microstructure 855.3.1 Stored Energy Distribution . . . . . . . . . . . . . . . 855.3.2 3D Simulation . . . . . . . . . . . . . . . . . . . . . . 875.3.3 2D Simulation . . . . . . . . . . . . . . . . . . . . . . 955.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1016 Modeling Austenite Formation . . . . . . . . . . . . . . . . . . 1036.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 1D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 2D Simulation of Step Heating . . . . . . . . . . . . . . . . . 1096.4 2D Simulation of Rapid Continuous Heating . . . . . . . . . . 1166.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 Modeling Intercritical Annealing . . . . . . . . . . . . . . . . 1227.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.2 Effects of Heating Rate . . . . . . . . . . . . . . . . . . . . . 1247.3 Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . 1277.4 Simulated Industrial Annealing Cycle . . . . . . . . . . . . . 1307.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378 Modeling Cyclic Phase Transformation . . . . . . . . . . . . 1388.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388.2 Fe-C Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.3 Fe-C-Mn Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153viiiTable of Contents9 Summary and Future Work . . . . . . . . . . . . . . . . . . . . 1549.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158ixList of Tables4.1 Key alloying elements in the investigated steel. . . . . . . . . . 524.2 Carbon-diffusion parameters. . . . . . . . . . . . . . . . . . . . 565.1 Phase-field model parameters used in the benchmarking sim-ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Values of adjustable parameters in the 3D recrystallizationmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Values of adjustable parameters in the 2D recrystallizationmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1 Parameters in the austenite-formation model. . . . . . . . . . 1116.2 Fitted values of the adjustable parameters in the austenite-formation model. . . . . . . . . . . . . . . . . . . . . . . . . . 115xList of Figures1.1 Dual-phase microstructure containing martensite and ferrite. . 31.2 An intercritical annealing cycle in a hot-dip galvanizing line. . 32.1 Schematic illustrating an diffuse interface in the phase-fieldmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Micrograph showing recrystallization nuclei distribution in alow-carbon steel (0.03wt%C-0.19wt%Mn). . . . . . . . . . . . 122.3 Illustration of an isothermal section of a phase diagram fora Fe-C-M alloy with a bulk concentration of x0M (M is anaustenite-stabilizing element): solid tie-lines are NPLEs anddash tie-lines are PLEs. . . . . . . . . . . . . . . . . . . . . . . 182.4 Interaction potential of solute M with a grain boundary inCahn’s model: E0 denotes the binding energy. . . . . . . . . . 212.5 Illustration of the solute drag pressure as a function of theinterface velocity described by Cahn’s model. . . . . . . . . . . 222.6 Interaction potential of solute M with a ferrite/austenite in-terface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Illustration of the solute drag force described by the Purdy-Brechet model. . . . . . . . . . . . . . . . . . . . . . . . . . . 24xiList of Figures2.8 Phase-field modeling of austenite formation in a pearlitic Fe-Csteel during isothermal holding at 750 °C. . . . . . . . . . . . 272.9 Austenite nuclei density in the C22 steel (0.21wt%C-0.51wt%Mn)and the C35 steel (0.36wt%C-0.66wt%Mn) during continuousheating at 10 K/min. . . . . . . . . . . . . . . . . . . . . . . . 292.10 Optical micrographs showing austenite formation in a fer-rite/pearlite steel (0.11wt%C-1.47wt%Mn-0.27wt%Si-0.03wt%Nb)for different heating rates: (a) 0.05 K/s, 786 °C; (b) 5 K/s,784 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.11 TEM image of deformed pearlite being spheroidized in a cold-rolled low-carbon steel (0.17wt%C-0.74wt%Mn) heated at 1K/s to 1003 K. . . . . . . . . . . . . . . . . . . . . . . . . . . 312.12 Schematic illustration of austenite formation in a cold-rolledsteel (0.06wt%C-1.86wt%Mn-0.155wt%Mo) with different heat-ing rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.13 Phase-field modeling of austenite formation in pearlite in ahypo-eutectoid plain-carbon steel during holding at 750 °C. . . 332.14 Phase-field simulation of austenite formation assuming differ-ent equilibrium modes in a low-carbon steel (0.1wt%C-1.65wt%Mn-0.55wt%Cr-0.24wt%Si). . . . . . . . . . . . . . . . . . . . . . . 352.15 Microstructures and the carbon distribution in a low-carbonsteel (0.1wt%C-1.65wt%Mn-0.55wt%Cr-0.24wt%Si) obtainedin a 3D phase-field simulation of austenite formation. . . . . . 362.16 Schematic of austenite formation in a ferrite/carbide microstruc-ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38xiiList of Figures2.17 Microstructure evolution in a plain-carbon steel (0.17wt%C)with a ferrite/cementite microstructure by a phase-field simu-lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.18 Simulated microstructures during the austenite-to-ferrite trans-formation in a 3D phase-field simulation. . . . . . . . . . . . . 422.19 2D microstructures obtained in 3D (a) and 2D (b) phase-fieldsimulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.20 Schematic of cyclic heat treatments: (a) V-type; (b) H-type. . 442.21 Dilatometry data for H-type (a) and V-type (b) cyclic heattreatments of a low-carbon steel (0.1wt%C-0.5wt%Mn): thestagnant stages are marked with gray lines. . . . . . . . . . . . 452.22 Mn concentration gradients formed in a steel (0.1wt%C-0.5wt%Mn)during cyclic heat treatments, obtained in a 1D simulation us-ing Dictra®. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1 Schematic of the grid structure in the computational domain. 574.2 Para-equilibrium phase diagram for DP600 (α: ferrite; γ: austen-ite; θ: cementite). . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Schematic of the calculation of driving pressure for ferrite toaustenite transformation. . . . . . . . . . . . . . . . . . . . . . 664.4 Illustration of the Mn concentration across a moving α/γ in-terface for the ferrite-to-austenite transformation: a negativespike is formed in front of the α/γ interface. . . . . . . . . . . 704.5 The effect of the binding energy of Mn on the profile of thedissipated Gibbs-energy as a function of interface migrationrate at 770 °C for a low-carbon steel (0.1wt%C-1.86wt%Mn). . 71xiiiList of Figures4.6 Flowchart of the model implementation. . . . . . . . . . . . . 765.1 Influence of interface thickness and time step on numericalaccuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 3D microstructures with different volume fractions of recrys-tallizing grains obtained in a phase-field simulation assuminghomogeneous nucleation and uniform stored energy. . . . . . . 815.3 Double-logarithmic representation of the simulated recrystal-lization kinetics: homogeneous nucleation and uniform storedenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Double-logarithmic representation of recrystallization kinetics:various nucleation scenarios and non-uniform stored energy. . 835.5 Grain size distributions after full recrystallization (500 grains):(a) uniform stored energy and random nucleation; (b) non-uniform stored energy and random nucleation; (c,d) non-uniformstored energy and non-random nucleation with Est∗=4.0Ö106J/m3 and 5.0Ö106 J/m3, respectively. . . . . . . . . . . . . . . 845.6 Distribution of stored energy in the as-deformed material cal-culated with the Taylor-factor approach. . . . . . . . . . . . . 885.7 Cold-rolled ferrite/pearlite microstructure with 15% pearliteconstructed using Voronoi tessellation. . . . . . . . . . . . . . 895.8 Comparison of simulated recrystallization kinetics with exper-imental data for different isothermal tests. . . . . . . . . . . . 905.9 Grain size distribution after full recrystallization in 3D simula-tions (194 grains) and experiments (500 grains) for isothermaltests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91xivList of Figures5.10 Comparisons of 3D-simulated recrystallization microstructureswith experimental micrographs for isothermal tests at 600 °C:(a-b) 20% recrystallized; (c-d) 60% recrystallized. . . . . . . . 925.11 Experimental and 3D-simulated recrystallization kinetics forcontinuous heating at 1 °C/s. . . . . . . . . . . . . . . . . . . 935.12 Comparison of 3D-simulated grain size distribution with ex-perimental measurements (720 °C). . . . . . . . . . . . . . . . 935.13 3D-Simulated microstructures for continuous heating at 1 °C/s:(a) 670 °C; (b) 680 °C; (c) 690 °C; (d) 700 °C; (e) 710 °C; (f)720 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.14 Cold-rolled ferrite/pearlite microstructure with 15% pearliteconstructed with Voronoi tessellation. . . . . . . . . . . . . . . 975.15 Comparisons of experimental data with 2D simulation resultsfor various Est∗ values: (a-b) 1.0Ö106 J/m3; (c-d) 1.7Ö106 J/m3. 985.16 2D-simulated recrystallization microstructures for isothermaltests at 600 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . 995.17 Experimental and 2D-simulated recrystallization kinetics forcontinuous heating at 1, 10 and 100 °C/s. . . . . . . . . . . . . 1005.18 Comparison of 2D-simulated grain size distribution with ex-perimental measurements (720 °C). . . . . . . . . . . . . . . . 1016.1 Heating scenarios used in the simulations of austenite formation.1046.2 Schematic of the 1D simulation domain. . . . . . . . . . . . . 1056.3 Influence of numerical parameters on simulation accuracy. . . 1066.4 Influence of the ferrite/austenite mobility on austenite forma-tion kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xvList of Figures6.5 Carbon-concentration profile in the 1D simulation (mobility is1.0× 10−13 m4/(J·s)). . . . . . . . . . . . . . . . . . . . . . . . 1076.6 Austenite formation kinetics for simulations with and withoutGibbs-energy dissipation (mαγ = 1.0×10−13 m4/(J·s), DMnint =2.0× 10−17 m2/s). . . . . . . . . . . . . . . . . . . . . . . . . . 1086.7 Fully recrystallized microstructure at 720 °C obtained in a 2Dsimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.8 Comparison of the experimentally measured austenite forma-tion kinetics with simulation results for step heating. . . . . . 1126.9 Comparison of micrographs (a,c) with simulated microstruc-tures (b,d) in Case A at 790 °C (a-b) and C at 825 °C (c-d). . 1136.10 Comparison of the micrograph with the simulated microstruc-ture after full austenitization at 870 °C in case A. . . . . . . . 1146.11 Comparisons of experimentally measured transformation ki-netics with simulation results in Case D and E. . . . . . . . . 1176.12 Simulated microstructures at 750 °C in Case D (a) and E (b). 1186.13 Comparisons of micrographs at 790 °C with simulated mi-crostructures in Case D and E. . . . . . . . . . . . . . . . . . 1197.1 Heating scenarios used in the simulations of intercritical an-nealing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2 Schematic of an industrial intercritical-annealing cycle. . . . . 1247.3 Comparison of simulated microstructures with micrographs forintercritical annealing at 770 °C with heating rates of (a-b) 1°C/s and (c-d) 100 °C/s . . . . . . . . . . . . . . . . . . . . . 125xviList of Figures7.4 Austenite formation kinetics at 770 °C after heating at differ-ent heating rates. . . . . . . . . . . . . . . . . . . . . . . . . . 1267.5 Simulated microstructures for intercritical annealing at 790 °Cafter heating at of (a) 1 °C/s and (b) 100 °C/s after 300 s . . . 1287.6 Simulated austenite formation kinetics during holding at 770°C and 790 °C after heating at 100 °C/s. . . . . . . . . . . . . 1297.7 Nucleation kinetics in the simulations of intercritical annealingat 770 °C and 790 °C after heating at 100 °C/s . . . . . . . . . 1297.8 Simulated ferrite-recrystallization kinetics in the industrial an-nealing cycle for the DP600 steel: full recrystallization at 735°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.9 Comparison of the experimentally measured transformationkinetics with simulation results for the simulated industrialannealing cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.10 Comparison of experimental micrographs with simulated mi-crostructures at 770 °C after holding (a, b) and at 465 °C aftercooling (c, d). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.11 Carbon concentration field and a line plot along the verticalwhite line at 465 °C obtained in the phase-field simulation. . . 1337.12 Micrograph of the investigated steel (0.1wt%C-1.86wt%Mn-0.34wt%Cr) after intercritical annealing at 770 °C and isother-mal holding at 465 °C. . . . . . . . . . . . . . . . . . . . . . . 1337.13 Simulated phase fractions for an intercritical-annealing cyclewith two heating rates: (a) 5 °C/s; (b) 20 °C/s. . . . . . . . . 135xviiList of Figures7.14 Simulated microstructures at 770 °C and 465 °C for an intercritical-annealing cycle with two heating rates: (a-b) 5 °C/s; (c-d) 20°C/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.1 Micrographs of the Fe-0.1wt%C alloy used as the initial mi-crostructures in 2D simulations. . . . . . . . . . . . . . . . . . 1408.2 Comparison of the experimentally measured phase transfor-mation kinetics with the simulation results for the Fe-0.1wt%Calloy: (a) H-type; (b) V-type. . . . . . . . . . . . . . . . . . . 1418.3 Comparison of the experimentally measured phase transfor-mation kinetics with the simulation results for the Fe-0.1wt%Calloy: (a) H-type; (b) V-type. . . . . . . . . . . . . . . . . . . 1448.4 Simulated microstructures at various stages of the H-type cyclicphase transformation for the Fe-0.1wt%C alloy. . . . . . . . . 1458.5 Micrograph of the Fe-0.1wt%C-0.5wt%Mn alloy (quenched from785 °C) used as the initial microstructure in 2D simulations. . 1468.6 Comparison of the experimentally measured phase transfor-mation kinetics with the simulation data for the V-type heattreatment of Fe-0.1wt%C-0.5wt%Mn. . . . . . . . . . . . . . . 1478.7 Simulated microstructures at various stages of the V-type cyclicphase transformation for the Fe-0.1wt%C-0.5wt%Mn alloy. . . 1508.8 Comparison of the experimentally measured phase transfor-mation kinetics with the simulation data for the H-type heattreatment of Fe-0.1wt%C-0.5wt%Mn. . . . . . . . . . . . . . . 151xviiiList of SymbolsCMi concentration of component M in phase iC∗γ para-equilibrium carbon concentration in austeniteC∗α para-equilibrium carbon concentration in ferriteDim dimensionality of simulationsDMi diffusivity of component M in phase iDMint trans-interface diffusivity of component M in the α/γinterfaceE interaction energy of solute atoms with a grain boundaryE0 binding energy of solute atoms with the interfaceEsti stored energy in grain iEst∗ critical stored energy, a paramter in the nucleation model forrecrystallizationF Helmholtz free energy of a systemGi Gibbs free nergy of phase iLi kinetic coefficient in the Allen-Cahn equationM Taylor factorN number density of nucleiNRex number density of recrystallization nucleixixList of SymbolsNn number density of potential nucleation sitesP pearliteQb activation energy of recrystallizationQmij activation energy of the i/j interface mobilityQMi activation energy of the diffusivity of component M in phaseiQG activation energy of austenite growth in pearliteQN activation energy of nucleation for austenite in pearliteR ideal gas constantT temperatureT0 temperature at which austenite and ferrite have the sameGibbs free energyTa lower temperature in a cyclic thermal pathTb upper temperature in a cyclic thermal pathV interface velocityZ Zeldovich factor in the classical nucleation modelb rate constant in the JMAK equationb0 pre-factor of the rate constant in the JMAK equationd inter-atomic spacingd∗n critical nucleus sizef free energy density functionalf intf interface energy density functional in the multi-phase-fieldmodelxxList of Symbolsf chem chemical free energy density functional in themulti-phase-field modelfX volume fraction of recrystallized grainsg crystallographic orientationkij slope of the equilibrium concentration line on a phasediagram for phase i with respect to phase jkB Boltzmann constantl diffusion distancemij mobility of the interface between constituent i and jm0ij pre-factor of the i/j interface mobilitym0Pγ,1°C/s pre-factor of the P/γ interface mobility when heating at 1°C/s before austenite formationn JMAK exponentnη number of grid points across a diffuse interface in thephase-field modeln normal to a boundary of a simulation domaint timeuMi u-fraction of comonent M in phase iuM0 u-fraction of component M in the bulkxMi molar fraction of comonent M in phase ixM0 molar fraction of component M in the bulk2Λ physical thickness of an interfaceΨ parameter related to the shape and interface energy of nucleiin the classical nucleation modelxxiList of Symbols2∆E difference between chemical potentials in ferrite andaustenite∆Gchemij chemical driving pressure for the i→ j phase transformation∆Gdrivij driving pressure for the i/j interface migration4Gdragij solute drag pressure for the i→ j phase transformation4Gdissij dissipated Gibbs-energy by solute diffusion across an i/jinterface4Gdissij,0 dissipated Gibbs-energy by solute diffusion across astationary i/j interface∆Geffij effective driving pressure for the i/j interface migrationduring the i→ j transformation∆Gtij total driving force for the i→ j transformation∆G∗ nucleation activation energy∆gV difference between Gibbs free energy per unit volume anucleus and the substrate∆tD time step used for solving the diffusion equations∆tP time step used for solving the phase-field equations∆x grid spacingΩ integral domain of a systemα ferriteγ austeniteη interface thickness in phase-field modelsθ cementitexxiiList of Symbolsλ adjustable parameter related to the atom-jump frequency inthe classic nucleation modelµji chemical potential of component j in phase iµSCi chemical potential of the effective substitutional componentin phase iνD Debye frequencyξ factor in the calculation of time stepσij i/j interface energyφi phase-field variable representing a grain or phase iχkij thermodynamic coefficient correlating the carbonconcentration to the chemical driving pressure for the i→ jtransformationxxiiiAcknowledgmentsI would like to thank my supervisor Dr. Matthias Militzer for giving me thechance to work on this project and for his continuous support and guidance.I have benefited a lot from my supervisor’s knowledge and feedback dur-ing the project meetings and discussions. The discussions with Dr. WarrenPoole during the project meetings are very constructive and appreciated. Iam also thankful to Dr. Mykola Kulakov and Dr. Hao Chen for providingthe necessary experimental data. I appreciate the help of Mr. Jingqi Chenfor the EBSD experiments. The contribution of my departmental committeemembers, Dr. Chad Sinclair and Dr. Rizhi Wang, is acknowledged withgratitude. I am also grateful to my university examiners, Dr. Daan Maijer(Department of Materials Engineering), Dr. Andrew MacFarlane (Depart-ment of Chemistry), and my external examiner, Dr. Carl Krill III (Instituteof Micro and Nanomaterials, Ulm University, Germany), for their criticalfeedback. I would like to acknowledge the funding from The Natural Sciencesand Engineering Research Council of Canada (NSERC) and ArcelorMittalDofasco.Above all, I would like to thank my beloved parents and sister. I amindebted to them for all their love, understanding and support.xxivChapter 1IntroductionToday, steel continues to be an attractive automotive material in terms ofgood mechanical properties, cost effectiveness and recyclability. New gradesof automotive steels have been continuously engineered ever since the adventof cars, to keep up with the ever-changing challenges of producing inexpen-sive, safe and fuel-efficient transportation. For example, before the 1990s,mild steels having a maximum tensile strength of 280 MPa, e.g. plain-carbon steels and interstitial-free (IF) steels, were the dominant materialsin car bodies. In the 1990s, high-strength low-alloy (HSLA) steels with atensile strength up to 800 MPa were commonly used in the automotive in-dustry. Nowadays, advanced high-strength steels (AHSS) with better forma-bility than HSLA at the same strength level are widely used in auto bodies.In particular, dual-phase (DP) steels are commercially produced and sub-stantially used in designing modern vehicles. They are frequently employedfor automotive body panels, wheels and bumpers.Compared with conventional high strength steels, dual-phase steels haveadvantages of lower yield to tensile strength ratio, higher initial strain hard-ening rates and better uniform elongation [1] and thus have a better combi-nation of strength and formability as well as higher crashworthiness. Com-pared with other advanced high strength steels, e.g. transformation-induced1Chapter 1. Introductionplasticity (TRIP) steels and twinning-induced plasticity (TWIP) steels, dual-phase steels have lower contents of alloying elements and thus lower cost andbetter weldability [2].The advantageous properties of dual-phase steels are attributed to a com-posite microstructure in which the hard martensitic phase (a body-centeredtetragonal (BCT) crystal structure) is embedded in a matrix of soft ferrite(a body-centered cubic (BCC) crystal structure) (Fig. 1.1). There are twoprincipal processing routes to produce DP steel sheets: (i) controlled coolingfrom the austenite phase (a face-centered cubic crystal structure) during hotrolling; (ii) intercritical annealing of cold-rolled steels in continuous annealingor hot dip galvanizing lines. In particular, the intercritical annealing route(Fig. 1.2) is of primary interest to produce sheets with the required surfacequality and thickness of 1 mm and below. In an intercritical-annealing cy-cle, recrystallization is the first physical process occurring in a cold-rolledsteel upon heating. After the temperature reaches the start temperatureof austenite formation (usually some superheat over the eutectoid tempera-ture), austenite grains nucleate and grow to form a mixture of austenite andferrite during holding in the intercritical temperature range. Once cooleddown, some austenite will transform back into ferrite and the remaining willtransform into hard phases, i.e. bainite or martensite, depending on the steelchemistry and cooling rates. Continuous annealing is usually integrated withhot-dip galvanizing, in which steel sheets are dipped into hot zinc baths fora few seconds. During galvanization, some austenite may decompose intobainite.The mechanical properties of DP steels are controlled by size, shape,2Chapter 1. IntroductionFigure 1.1: Dual-phase microstructure containing martensite and ferrite (M:martensite, F: ferrite). [3]Figure 1.2: An intercritical annealing cycle in a hot-dip galvanizing line. [4]3Chapter 1. Introductioncomposition, morphology, distribution and volume fraction of hard phases.Robust annealing routes have to be designed and tightly controlled to getdesired DP microstructures. In order to evaluate these annealing routes,it is imperative to develop microstructure evolution models for intercriticalannealing.A number of microstructure process models were developed for controlledrun-out-table cooling of hot-rolled DP steels. There is, however, a remarkablelack of such models for intercritical annealing. The challenges of modelingintercritical annealing are related to the following aspects. First, a varietyof initial microstructures have to be taken into account that may result fromhot and cold rolling. Second, various physical processes, e.g. recrystalliza-tion, austenite formation and austenite decomposition, have to be consideredwhich may occur in sequence or concurrently. In the case of different phys-ical processes occurring concurrently, it is challenging to develop suitablemodels that can describe the interactions. The well-known Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation that has been used to model kineticsof recrystallization and phase transformations, cannot be used to describethe potential interaction of recrystallization and austenite formation thatmay occur in fast-annealing cycles [4]. Furthermore, this category of modelscan only describe the kinetics of the evolution of phase fractions but can-not provide further information on microstructures, e.g. solute distributions,grain sizes and phase morphologies.Given the complexity of intercritical annealing, mesoscale modeling is es-sential to track microstructure evolution. The phase-field modeling (PFM)approach has emerged as the modeling tool of choice to describe microstruc-4Chapter 1. Introductionture evolution especially with complex morphologies. Moreover, it providesa single modeling tool for all physical processes occurring in steel process-ing from casting to annealing. Thus it is suitable for modeling intercriticalannealing in the present project.The main objective of the present work is to develop a stand-alone phasefield model describing microstructure evolution during intercritical annealingof low-carbon steels. First, a model for ferrite recrystallization is developedand calibrated with experimental data. A phase-field model is then devel-oped for austenite formation from both recrystallized microstructures anddeformed microstructures in which ferrite recrystallization and austenite for-mation occur concurrently. The austenite-to-ferrite transformation is mod-eled as the reverse process of the ferrite-to-austenite formation during austen-ite formation. By integrating models for ferrite recrystallization, austeniteformation and austenite to ferrite transformation, a stand-alone phase-fieldmodel is developed for intercritical annealing and validated with experimen-tal data. Moreover, the phase-field model is used to simulate cyclic phasetransformations in the intercritical region and the simulation results are com-pared with experimental data.This thesis is organized as follows: Chapter 2 provides a literature reviewon the phase-field approach, phase equilibria and solute drag effects in alloyedsteels, and metallurgical phenomena during intercritical annealing of cold-rolled low-carbon steels. In Chapter 3 the scope and objectives of this workare presented. Chapter 4 introduces the methodology of the work. In Chapter5, recrystallization is systematically studied using the phase-field model andexperimental data are used to calibrate the model. In Chapter 6, a phase-5Chapter 1. Introductionfield model is developed to describe austenite formation in a pearlite/ferritemicrostructure. In Chapter 7, a stand-alone phase-field model for intercriticalannealing is developed and applied to a simulated industrial annealing cycle.In Chapter 8, the developed phase-field model for intercritical annealing isused to study cyclic phase transformations in a plain-carbon steel and amanganese-alloyed low-carbon steel. In Chapter 9, a summary is presentedwith an outlook into future work.6Chapter 2Literature Review2.1 IntroductionThe principles of the phase-field approach are reviewed first. Recrystalliza-tion in low-carbon steels is discussed then, including both experimental andmodeling work. Phase-transformation theories in low-carbon steels are intro-duced with an emphasis on the behavior of substitutional elements duringphase transformations. The latter part of this chapter is devoted to a reviewof the experimental and mesoscale-modeling work on austenite formation anddecomposition, in which the intercritical-annealing process and cyclic phasetransformations are discussed.2.2 Phase-Field ApproachIn the past few decades, the phase-field approach has emerged as a compu-tational tool to model various metallurgical phenomena that include solidifi-cation [5–12], solid-state phase transformations [13–20], recrystallization andgrain growth [21–29]. The phase-field approach has a few pronounced ad-vantages over other popular mesoscale models, e.g. Cellular Automata (CA)models and Monte Carlo (MC) methods [30–40]. One merit of phase-field72.2. Phase-Field Approachmodels is that there is no need to track interfaces explicitly, which enablesit to model microstructures of complex morphology, e.g. dendrite formation.Moreover, phase-field models do not require scaling between numerical timeand physical time that is an issue for CA and MC methods.Using diffuse interfaces in a phase-field model distinguishes it from CAand MC methods that use sharp interfaces. In sharp-interface models, theinterface is tracked as a sudden change of certain properties such as crystallo-graphic orientation and lattice structure. In contrast, the diffuse interface inphase-field models has a finite thickness within which the properties changesmoothly from one side to the other. The interface thickness in phase-fieldmodels is a numerical parameter and usually orders of magnitude larger thanthe physical thickness to use a large grid size for acceptable computationcost. It should also be small enough to resolve the characteristic lengths ofthe microstructure of interest [25].The phase-field theory originates from a single-phase-field concept de-scribing interaction of two phases [41]. In the single-phase-field approach, twophases are described by different values of a phase-field variable φ. Later, thesingle-phase-field method was extended towards multi-phase-field models ca-pable of modeling a number of phases or grains [42]. In the multi-phase-fieldmodel, each microstructure constituent, e.g. a phase or grain, is described byone phase-field variable φi . Each variable has constant but different valuesinside and outside the relevant constituent (e.g. 1 and 0) with a continuousvariation across the diffuse interface (Fig. 2.1).In phase-field models, microstructure evolution is described by a temporalvariation of values of phase-field variables, which is governed by a set of82.2. Phase-Field ApproachFigure 2.1: Schematic illustrating an diffuse interface in the phase-fieldmodel.partial differential equations, i.e. phase-field equations. The formulation ofthese partial differential equations is based on the theory of irreversible ornon-equilibrium thermodynamics. The evolution of phase-field variables φiis described by the Allen-Cahn equation [25]:∂φi∂t= −LiδFδφi(2.1)where Li is a kinetic coefficient associated with how quickly the phase-fieldvariable evolves, and F is the Helmholtz free energy of a non-equilibriumsystem which may consist of bulk free energy, interface energy, elastic strainenergy, etc.The Helmholtz free energy of a system is formulated as an integral of thelocal free energy density [25]:F =∫Ωf ({φi} , {∇φi}) dΩ (2.2)where the free energy density f is given by:92.3. Recrystallizationf = f0 ({φi}) + f1 ({∇φi}) (2.3)Here f 0 denotes a function of the phase-field variables and f 1 denotes afunction of the gradients of the phase-field variables.There are various formulations of the free energy density functional f inEq. 2.3 so that various phase-field equations have been derived from theAllen-Cahn equation [10, 43, 44]. In this work, the multi-phase-field modeldeveloped by Steinbach et al. is used [42, 45]. The details of the model willbe described in Chapter 4.To simulate microstructure evolution using a phase-field model, a sim-ulation domain has to be defined with specific boundary conditions. Thephase-field equations are solved in the domain with numerical methods, e.g.finite difference or finite element methods. An appropriate value of interfacethickness has to be chosen to both guarantee computational efficiency andresolve the characteristic length in the microstructure of interest.2.3 Recrystallization2.3.1 Recrystallization in Low-carbon SteelsRecrystallization is a process by which deformed grains are replaced by anew set of undeformed grains that nucleate and grow until the original grainshave been entirely consumed. The driving force for recrystallization is strainenergy stored in the form of dislocations (so-called stored energy). Disloca-tions are not introduced uniformly to the material by deformation. Instead,102.3. Recrystallizationsubstructures, e.g. dislocation cells and subgrains, are usually formed dur-ing deformation. Furthermore, in polycrystalline microstructures, each grainresponds differently to the macroscopic strain due to their different crystal-lographic orientations. Therefore, stored energy is inhomogeneous spatially.In low-carbon steels, the non-uniformity of the stored energy is related tothe crystallographic orientation of grains [46, 47]. Generally, there are twomain groups of crystallographic orientations in low-carbon steels, i.e. thealpha fiber and gamma fiber [48, 49]. Subgrains in the gamma fiber oftenhave smaller size and higher boundary misorientation than those in the al-pha fiber [50]. Therefore, stored energy in the gamma fiber is generally higherthan that in the alpha fiber [51].Recovery occurs in materials with high stacking fault energy, e.g. alu-minum alloys and ferritic steels [52, 53]. During recovery, dislocations an-nihilate and rearrange by glide and climb such that dislocation cells withtangled dislocations eventually form subgrains. Subgrains can grow furtherby migration of their low-angle grain boundaries.The recrystallization nuclei are formed from the substructures, e.g. dis-location cells or subgrains, which are already present in the deformed mi-crostructure. There are a few recrystallization nucleation theories in the lit-erature, e.g. strain induced boundary migration (SIBM) [54], abnormal sub-grain coarsening [31, 55, 56] and particle stimulated nucleation (PSN) [57].Subgrain coarsening is the dominant nucleation mechanism in low-carbonsteels [51]. Some subgrains keep growing into surrounding regions and theirboundary misorientation is increasing. They will form nuclei once theirboundaries become high-angle misoriented. Experimental results [52] have112.3. Recrystallizationshown that recrystallization nucleation is inhomogeneous in low-carbon steels(Fig. 2.2) and occurs preferentially in regions with high stored energy.Figure 2.2: Micrograph showing recrystallization nuclei distribution in a low-carbon steel (0.03wt%C-0.19wt%Mn): nuclei with equiaxed shapes are en-veloped with red lines [52].After nuclei have formed, recrystallization proceeds as new grains sweepdeformed regions by migration of high-angle boundaries and replaces thedeformed structure with a dislocation-free microstructure. Because storedenergy is spatially inhomogeneous, migration rate of recrystallization frontsvaries with space and time. Solute atoms and second-phase particles canexert drag and pinning forces on recrystallization fronts and thus slow downor even prevent interface migration [58–60].122.3. Recrystallization2.3.2 Modeling RecrystallizationThe volume fraction of a recrystallized structure fX is used to characterize thekinetics of recrystallization. To model recrystallization kinetics, the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation is frequently employed, i.e.:fX = 1− exp (−btn) (2.4)where n is the JMAK exponent and b is a rate constant which can usuallybe expressed as:b = b0 exp(−Qb/RT ) (2.5)where b0 is a pre-factor and Qb is the activation energy of recrystallization.The JMAK equation is derived mathematically with some assumptions,i.e. random nucleation and a constant growth rate. The value of the JMAKexponent depends on both nucleation mode and growth dimensionality. Al-though the JMAK model is only valid in an ideal case under its assumptions,it has been widely used to describe the actual recrystallization process thatin general does not fulfill the JMAK assumptions. The JMAK exponents fitto experimental data are usually lower than the ideal values, i.e. 3 for site-saturated nucleation and 4 for continuous nucleation with a constant rate.Computer simulations have indicated that lower JMAK exponents may resultfrom non-uniform microstructural features, e.g. inhomogeneous nucleation,non-uniform stored energy and anisotropic interface mobility [61–63]. Inlow-carbon steels, the JMAK exponent usually ranges from 1.0-2.0, and theactivation energy is between 300-400 kJ/mol [4].132.3. RecrystallizationIn the past decades, mesoscale approaches have been used to describerecrystallization [38,58,64–74]. Compared with the JMAK model, mesoscalemodels are more physical because the assumptions of the JMAK model canbe relaxed. Jou and Lusk have benchmarked a two-dimensional phase-fieldmodel successfully with the JMAK equation [75]. Takaki et al. coupled aphase-field model with a crystal-plasticity theory, to describe recrystalliza-tion in a structure with non-uniform stored energy [76]. Wang et al. haveapplied a two-dimensional phase-field model successfully to recrystallizationof a magnesium alloy [77]. The simulated recrystallization kinetics is in goodagreement with experimental results. Suwa [78] simulated recrystallization asabnormal subgrain growth with a two-dimensional phase-field model, by us-ing a polycrystalline microstructure with many subgrains as a well recoveredstructure. The simulation replicated the strain induced boundary migration(SIBM) nucleation. How to construct a realistic substructure is a challengefor this method. Moreover, the computational cost is high for this methodbecause a small mesh size is necessary to resolve substructures.A realistic grain topology is necessary for mesoscale modeling. Graintopologies can be obtained using microscopy and electron back-scattereddiffraction (EBSD). But it is more challenging to obtain a spatial varia-tion of stored energy or dislocation density. Various indirect methods havebeen used with some assumptions and simplifications to estimate stored en-ergy, e.g. hardness measurements [62], line broadening of X-ray or neutrondiffraction [79], EBSD image quality [58] and subgrain analysis [61]. In themethod of subgrain analysis, it is assumed that all stored energy is in theform of subgrain boundary energy. Therefore, stored energy can be esti-142.4. Phase-transformation Theories in Steelsmated if average subgrain sizes and boundary energy are known. Anotherdirect method is counting dislocations under a transmission electron micro-scope [79]. However, it is not applicable to subgrain structures and evenif feasible, tremendous work is required to obtain a map of stored energydistribution in a polycrystalline microstructure.All the mesoscale models are virtually growth models requiring intro-duction of nuclei to the simulation domain, except those that treat recrys-tallization as subgrain growth [31]. The embryos of dislocation-free grainsfor recrystallization nucleation are formed from dislocation cells or subgrainsrather than from thermal fluctuation. Some empirical nucleation models forrecrystallization have been postulated. For example, Raabe et al. assumedsite-saturated nucleation in the recrystallization of an interstitial-free (IF)steel [58] that occurs only wherever stored energy exceeds a critical value.This model is consistent with the general observation that nucleation occurspreferentially in the regions with high stored energy in low-carbon steels.2.4 Phase-transformation Theories in SteelsExtensive studies have been done in the past century on developing theoriesto rationalize the austenite-to-ferrite transformation kinetics. One of themis the diffusion-controlled theory, in which it is assumed that the movinginterface is in local equilibrium. It is simple to apply this theory to the Fe-Cbinary system by solving diffusion equations. The situation becomes compli-cated once a third element is added, especially if the element is substitutionaland diffuses much slower than carbon. Two thermodynamic equilibria have152.4. Phase-transformation Theories in Steelsbeen proposed for the interfacial conditions for a Fe-C-M system where Mdenotes a substitutional element.The first one is local equilibrium (LE) in which it is assumed that theinterface fulfills the following conditions [80]:µCα = µCγ (2.6)µFeα = µFeγ (2.7)µMα = µMγ (2.8)where α denotes ferrite, γ denotes austenite, and µ is the chemical potentialwith the superscript denoting the component and the subscript denoting thephase. At a given temperature, there are a multiplicity of tie-lines for a Fe-C-M system (Fig. 2.3), i.e. the interfacial concentrations of C and M underLE may follow different tie-lines during an isothermal phase transformation.Local equilibrium (LE) is further classified into two types: negligible-partitioning local equilibrium (NPLE) and partitioning local equilibrium(PLE). The term “partitioning” means that the concentration of M in theproduct phase formed during the phase transformation is not equal to thatin the bulk of the parent phase x0M . In contrast, “negligible-partitioning”means the product phase is formed with the same alloy content of M as thatin the bulk of the parent phase x0M . Thus, NPLE corresponds to a specifictie-line that connects the product phase with a M concentration of x0M . Fig.2.3 illustrates the NPLE tie-lines for an austenite-stabilizing element. For162.4. Phase-transformation Theories in Steelsthe austenite-to-ferrite transformation (γ → α), the product phase is ferriteand the parent phase is austenite. Thus, the relevant NPLE tie-line connectsferrite with a concentration of the bulk x0M and austenite with a M-enrichedconcentration. The enriched concentration of M in austenite at the interfaceleads to a sharp gradient (spike) in front of the interface. For the reversetransformation, i.e. the ferrite-to-austenite transformation (α → γ), on thecontrary, the product phase is austenite and the parent phase is ferrite andthus the relevant NPLE tie-line is different from that for γ → α. The NPLEtie-line for α → γ connects austenite with a concentration of the bulk x0Mand ferrite with a M-depleted concentration. The lower concentration of Min ferrite at the interface leads to a “negative” spike in ferrite (Fig. 2.3).To summarize, the tie-line of NPLE always connects a concentration of thebulk x0M for the product phase and a positive or negative spike is formedin the parent phase near the interface. Therefore, NPLE should be definedwith a designation of the transformation direction, e.g. NPLE for γ → αand NPLE for α→ γ. When the interface is under NPLE, there is only localredistribution of M at the interface and thus long-range diffusion of M is notnecessary, i.e. the interface migration is controlled by carbon diffusion.In contrast, when the interface condition follows the tie-lines other thanNPLE, the concentration of M in the product phase formed during the phasetransformation is not equal to x0M , i.e. the interface is under partitioninglocal equilibrium (PLE). In this circumstance, long-range diffusion of M isrequired and the interface migration is controlled by the slower diffusion ofsubstitutional elements.The second one is based on a constrained equilibrium, known as para-172.4. Phase-transformation Theories in SteelsFigure 2.3: Illustration of an isothermal section of a phase diagram for aFe-C-M alloy with a bulk concentration of x0M (M is an austenite-stabilizingelement): solid tie-lines are NPLEs and dash tie-lines are PLEs.equilibrium (PE). It is assumed that the chemical potential of carbon isconstant across the austenite/ferrite interface. The alloying element M is,however, assumed to remain immobile, i.e.:uMαuFeα=uMγuFeγ=uM0uFe0(2.9)where uFe and uM are molar fractions of Fe and M with respect to sub-stitutional sites which are termed u-fractions, and uM0 denotes the averagefraction of solute M.The alloy is treated as a pseudo-binary system similar to the Fe-C systemand all substitutional elements compose an effective substitutional compo-182.4. Phase-transformation Theories in Steelsnent (SC), the chemical potential of which is given by:µSCγ = uFeγ µFeγ + uMγ µMγ (2.10)µSCα = uFeα µFeα + uMα µMα (2.11)The para-equilibrium is defined by Eq. 2.6 andµSCγ = µSCα (2.12)A pseudo-binary phase diagram similar to the Fe-C phase diagram can becalculated using thermodynamic databases, e.g. Thermo-Calc®.The diffusion-controlled theory assumes that interface migration is lim-ited by solute diffusion only and interface reaction, i.e. transition of crystalstructures at the interface, is too fast to limit the interface migration. Incontrast, a mixed-mode theory assumes that interface migration is limitedby both solute diffusion and interface reaction that is a function of interfacemobility and driving pressure [81–83].The migration rate of an interface inthe mixed-mode theory is given by:V = mij∆Gdrivij (2.13)where mij is the i/j interface mobility and ∆Gdrivij is the driving pressurethat is a function of temperature and non-equilibrium solute concentrationsat the interface. Thus, the migration rate is affected by both the interfacemobility and solute concentrations. The mixed-mode phase transformation192.4. Phase-transformation Theories in Steelsis of significance for low-carbon steels [17].The solute-drag concept emerged when it was found that impurities couldretard recrystallization and grain growth significantly [84]. Then it was sug-gested that impurities tend to segregate to grain boundaries because theelastic stress induced by foreign atoms is less at interfaces than in the in-terior of a crystal. Once the interface starts moving, foreign atoms will beleft behind and exert a drag pressure on the interface 4Gdragij , such that theeffective driving pressure for interface migration ∆Geffij is given by:∆Geffij = ∆Gdrivij −∆Gdragij =Vmij(2.14)Cahn developed a solute drag model for grain boundaries, in which the solutedrag pressure is given by [85]:4Gdragij = −1Vm+Λ∫−Λ(x (y)− x0)dE (y)dydy (2.15)where V m is the molar volume, the grain-boundary width is denoted by 2Λ,x0 is the average molar fraction of solute atoms in the alloy, x (y) is themolar fraction of solute atoms across the grain boundary and E (y) is thefree energy of interaction of solute atoms with a grain boundary. A wedge-shaped profile of the interaction energy across the grain boundary is assumed,as shown in Fig. 2.4 where E0 is the binding energy of solute atoms to thegrain boundary.The concentration profile x (y) across the boundary is determined byassuming steady-state diffusion of solute atoms across the interface (trans-interface diffusion) and no long-range diffusion in the bulk, i.e. [86]:202.4. Phase-transformation Theories in SteelsFigure 2.4: Interaction potential of solute M with a grain boundary in Cahn’smodel: E0 denotes the binding energy.∂∂y[DMint∂x (y)∂y+DMintx (y)RT∂E (y)∂y+ V x (y)]= 0 (2.16)or ratherDMint∂x (y)∂y+DMintx (y)RT∂E (y)∂y+ V x (y) = V x0where R is the ideal gas constant, T is temperature, DMint is the trans-interfacediffusivity of component M and V the interface velocity.The solution to Eq. 2.15 can be approximated with the following math-ematical equation [85]:∆Gdragij =aV1 + bV 2(2.17)where the parameters a and b are given by:a =(RT )2 ΛDMintE0(sinh(E0RT)−E0RT)212.4. Phase-transformation Theories in Steelsb =√aRTΛ2DMintE20The solute drag pressure starts from zero at a stationary interface, increasesto a maximum and then decreases to zero again with increasing interfacevelocities (Fig. 2.5).Figure 2.5: Illustration of the solute drag pressure as a function of the inter-face velocity described by Cahn’s model.Purdy and Brechet extended Cahn’s model to phase transformations [86].The wedge-shaped interaction potential E(y) in Eq. 2.15 is modified asgiven in Fig. 2.6, where 2∆E denotes the difference between the chemicalpotentials in the bulk of two phases. Thus, Cahn’s model can be treated asa special case where the difference of chemical potentials 2∆E is 0.The solute drag pressure in the Purdy-Brechet model has a similar curveto that given by Cahn’s model (Fig. 2.7), except that the drag pressureexerted on a stationary interface is nonzero if the chemical potentials in the222.4. Phase-transformation Theories in SteelsFigure 2.6: Interaction potential of solute M with a ferrite/austenite inter-face.bulk of the two phases is different, i.e. 2∆E 6= 0.In contrast to Cahn’s model and the Purdy-Brechet model that are de-veloped from the perspective of force, Hillert developed a Gibbs-energy dis-sipation model to describe the behavior of solute atoms at an interface fromthe perspective of energy [84]. Hillert claimed that “ the work put into themovement of a boundary in order to overcome solute drag, must be dissipatedby the diffusion of solute taking place as a result of the boundary movement.It is thus possible to evaluate the drag force as the dissipation of free energydue to diffusion” [84]. The details of the Gibbs-energy dissipation model isdiscussed in Section 4.7.The Gibbs-energy dissipation model has been applied to the diffusion-controlled theory [87, 88], in which the interfacial carbon concentration is232.4. Phase-transformation Theories in SteelsFigure 2.7: Illustration of the solute drag force described by the Purdy-Brechet model.evaluated by assuming that the chemical driving pressure ∆Gdrivij is equal tothe dissipated Gibbs-energy ∆Gdissij , i.e.:∆Gdrivij = ∆Gdissij (2.18)There are two important parameters in the solute drag model, i.e. thetrans-interface diffusivity of solute DMint and the binding energy E0. Sinclairet al. found that the value of Nb trans-interface diffusivity at ferrite grainboundaries is 15 times that of the bulk diffusivity in ferrite by fitting Cahn’ssolute-drag model with recrystallization data [89]. Zurob et al. assumedthat the trans-interface diffusivity of substitutional elements, e.g. Mn andNi, across the ferrite/austenite interface are equal to the geometric averageof bulk diffusivities in ferrite and austenite [90]. It was suggested that thevalue of trans-interface diffusivity should be comparable to the bulk diffu-242.5. Austenite Formation and Decompositionsivity rather than the diffusivity along the interface [87]. Jia and Militzerobtained a value of 210 kJ/mol for the activation energy of the Mn trans-interface diffusivity across the ferrite/austenite interface by fitting a solutedrag model with experimental data [91]. Various methods are used to esti-mate the values of the binding energy in the literature. For example, Sinclairet al. calculated the binding energy of Nb to the ferrite grain boundary(28.9 kJ/mol) with the experimentally measured interface concentration fora stationary grain boundary [89]. Jia et al. fitted the binding energy of Mnto the α/γ interface in a sharp-interface model with the austenite-to-ferritetransformation kinetics for various cooling rates [91], and a range of 6-18kJ/mol was obtained for a Fe-Mn system with various Mn additions. Chenet al. fitted the binding energy of Mn to the α/γ interface (9.9 kJ/mol) withthe final bainite fraction in an isothermal experiment of bainite formationusing the Gibbs-energy dissipation model by Hillert [92].2.5 Austenite Formation and Decomposition2.5.1 Austenite Formation from PearliteIn a fully pearlitic microstructure (a lamellar structure composed of alter-nating layers of ferrite and cementite), austenite nucleation occurs mainly athigh-angle boundaries of pearlitic ferrite, even though nucleation at less favor-able ferrite/cementite interfaces within pearlite colonies is also reported [93].Austenite nuclei maintain a near Kurdjumov-Sachs (K-S) orientation rela-tionship (OR) with one or two adjacent pearlitic ferrite grains [94]. Such nu-clei preferentially grow into the adjacent ferrite that has no K-S orientation252.5. Austenite Formation and Decompositionrelationship, because the interface has a high mobility due to incoherency.Austenite prefers to grow along the direction of cementite lamellae [95].The growth rate can be controlled by either volume diffusion of carbon orboundary diffusion of substitutional elements. Austenite usually replacesferrite faster than dissolving cementite, leaving cementite in austenite thatmay dissolve later. Alloying elements can affect the kinetics of the pearlite-to-austenite transformation. For example, Li et al. [95] found that Mn can accel-erate the pearlite-to-austenite transition whereas silicon (Si) and chromium(Cr) can retard it, in comparison with the transformation kinetics in Fe-Calloys.Characteristic parameters of microstructures, e.g. inter-lamellar spac-ing, also affect the pearlite-to-austenite transformation rate. Smaller inter-lamellar spacing leads to higher transformation rates.Gaude-Fugarolas and Bhadeshia [96] derived an approximate expressionof the interface velocity V for pearlite to austenite transformation based ona carbon diffusion-controlled theory (para-equilibrium):V ≈DCγl(CC∗γθ − CC∗γαCC∗γα − CC∗αγ)(2.19)where DCγ is carbon diffusivity in austenite, l is the inter-lamellar spacing,CC∗γθ , CC∗γα and CC∗αγ are equilibrium carbon concentrations between austenite(γ) and cementite (θ) or ferrite (α).Azizi-Alizamini used a 2D phase-field model to simulate austenite for-mation in a pearlitic plain-carbon steel [3]. A small domain was used toresolve the lamellar structure with lamellar spacing of 0.4 µm. The domainconsists of three pearlite colonies with one triple point at which an austenite262.5. Austenite Formation and Decompositiongrain nucleates (Fig. 2.8). Growth of austenite into pearlite is accompaniedFigure 2.8: Phase-field modeling of austenite formation in a pearlitic Fe-C steel during isothermal holding at 750 °C (F: ferrite; A: austenite; C:cementite) [3].by cementite dissolution. The simulation predicted undissolved cementitein austenite that is in agreement with experimental observations. Anotherfinding is that austenite tends to grow faster along ferrite/cementite inter-faces. Moreover, it was found that an increase of temperature from 750 °Cto 830 °C raises the growth rate of austenite by almost two orders of mag-nitude which is in good agreement with experimental observations and Eq.2.19. Simulations with different inter-lamellar spacing were also carried out.As expected, transformation rate increases with a decrease in the lamellarspacing.272.5. Austenite Formation and Decomposition2.5.2 Austenite Formation from Ferrite/pearliteStructuresMost studies on austenite formation are based on the typical ferrite/pearlitemicrostructure of hypo-eutectoid steels [96–102]. Austenite formation startsin pearlite regions in such microstructures. Nucleation and growth are simi-lar to that in a fully pearlitic microstructure. After a fast transformation ofpearlite to austenite, austenite grains with high carbon contents grow intosurrounding ferrite, which is controlled by either carbon diffusion or substi-tutional diffusion [97].Austenite nucleation also occurs at ferrite grain boundaries. Nuclei havea near-Kurdjumov-Sachs (K-S) relationship with at least one adjacent ferritegrain [103, 104]. This type of nucleation is affected by heating rate. Slowheating rates favor nucleation at ferrite grain boundaries [4, 105]. Savranmeasured the density of austenite nuclei in two ferrite/pearlite microstruc-tures using an in-situ 3D X-ray diffraction (XRD) technique (Fig. 2.9) [102].In a microstructure with 21vol.% pearlite (C22), site-saturated nucleationat ferrite/pearlite interfaces is followed by continuous nucleation at ferriteboundaries. But in a microstructure with 56vol.% pearlite (C35), nucleationis site-saturated. Savran fitted the classical nucleation model to the measurednucleation rates at ferrite grain boundaries in the C22 steel, given by [102]:dNdt= NnZβn exp(−∆G∗kBT)(2.20)where kB is the Boltzmann constant; T is temperature in Kelvin; Z is Zel-dovich factor; Nn is the density of potential nucleation sites; βn is a factor282.5. Austenite Formation and DecompositionFigure 2.9: Austenite nuclei density in the C22 steel (0.21wt%C-0.51wt%Mn)and the C35 steel (0.36wt%C-0.66wt%Mn) during continuous heating at 10K/min [102].related to the jump frequency of atoms; ∆G∗ is the activation energy fornucleation, given by:∆G∗ =Ψ∆g2V(2.21)where Ψ contains information about the nucleus shape and the nucleus/matrixinterface energy; ∆gV is the driving force for nucleation, i.e. the differencein Gibbs free energy per unit volume between the matrix and nucleus.It is argued that the presence of cementite at ferrite grain boundariesfavors austenite formation [105]. But Jeong and Kim [100] claimed that theformation of thin-film type austenite at ferrite grain boundaries does not re-quire cementite and is possible when carbon atoms diffuse from pearlite areato grain boundaries satisfying the carbon content for austenite formation.Savran [102] found with in-situ XRD that some austenite nuclei are not close292.5. Austenite Formation and Decompositionto cementite particles and suggested that the carbon required by austenitenuclei is provided by diffusion from carbon-enriched regions through ferrite.This observation is consistent with another observation that austenite nu-cleation at ferrite grain boundaries is rare at high heating rates because thetime for diffusion is insufficient.San Martin et al. found experimentally that austenite morphology de-pends on heating rate [105]. For example, in Fig. 2.10, a network of austeniteFigure 2.10: Optical micrographs showing austenite formation (light:martensite; dark: ferrite) in a ferrite/pearlite steel (0.11wt%C-1.47wt%Mn-0.27wt%Si-0.03wt%Nb) for different heating rates: (a) 0.05 K/s, 786 °C; (b)5 K/s, 784 °C [105].along ferrite boundaries is formed at 0.05 K/s at an intermediate stage ofaustenite formation. But at a heating rate of 5 K/s, ferrite boundaries arealmost free of austenite and blocky austenite islands are formed in priorpearlite regions. Similar findings were reported by Kulakov in a cold-rolledbut recrystallized low-carbon steel [4].The use of cold-rolled materials makes austenite formation more com-plicated. Ferrite recrystallization takes place and pearlite spheroidization302.5. Austenite Formation and Decompositionis accelerated significantly [106–109]. Depending on heating rate and steelchemistry, these processes may occur before or concurrently with austeniteformation. For example, if the heating rate is sufficiently low, ferrite recrys-tallization may complete before austenite formation and the lamellar struc-ture of cementite in pearlite will spheroidize to clusters of cementite particles(Fig. 2.11). Spheroidization in deformed pearlite is much faster than thatFigure 2.11: TEM image of deformed pearlite being spheroidized in a cold-rolled low-carbon steel (0.17wt%C-0.74wt%Mn) heated at 1 K/s to 1003 K(white arrow pointing to a recrystallization front) [110].in non-deformed pearlite [3,111–114]. If the heating rate is sufficiently high,ferrite recrystallization may be delayed to the intercritical region and over-lap with austenite formation and pearlite also spheroidizes to a lesser extentbefore austenite formation due to the short time. With high heating rates,austenite tends to form a banded morphology. Huang et al. [115] suggestedthat austenite only nucleates in pearlite colonies because the migrating fer-312.5. Austenite Formation and Decompositionrite boundaries are unsuitable to become nucleation sites. Huang illustratedthe effects of heating rates on austenite morphology in a cold-rolled Fe-C-Mn-Mo alloy (Fig. 2.12). Not only the microstructural morphology but alsoFigure 2.12: Schematic illustration of austenite formation in a cold-rolled steel (0.06wt%C-1.86wt%Mn-0.155wt%Mo) with different heatingrates [115].the austenite formation kinetics is affected by prior deformation. In a non-deformed microstructure, increasing the heating rate shifts austenite forma-tion to higher temperatures, because both nucleation and growth of austeniteare thermal activation processes. In a cold-rolled steel, however, the differ-ence in start temperature of austenite formation for various heating rates isreduced, comparing with that in the non-deformed microstructure [3, 115].Wycliffe et al. used a 1D local equilibrium model to study the growth ofaustenite during intercritical annealing of a Fe-C-Mn alloy [116]. The modelshowed that austenite growth is first controlled by carbon diffusion with neg-ligible partitioning of Mn and later by slow Mn-diffusion-controlled growth,leading to non-uniform Mn distributions in both austenite and ferrite.322.5. Austenite Formation and DecompositionThere are a few reports on modeling austenite formation in a ferrite/pearlitemicrostructure with both phase-field and cellular automaton methods. Forexample, Azizi-Alizamini developed a two-dimensional phase-field model tostudy austenite formation in a 0.17wt%C steel (Fig. 2.13) [3]. In the sim-Figure 2.13: Phase-field modeling of austenite formation in pearlite in ahypo-eutectoid plain-carbon steel during holding at 750 °C (F: ferrite; A:austenite; C: cementite) [3].ulation, finger-type austenite growing along ferrite/cementite interface inpearlite is duplicated after nucleation at ferrite/pearlite interfaces, which isin agreement with experimental observations. Slow transformation of ferriteto austenite after pearlite dissolution is also replicated in his work. Savranfound in a phase-field simulation that austenite interfaces move almost tentimes faster in pearlite than in ferrite [117].In order to get statistically meaningful data, large simulation domainsare often used with coarse grid spacing, such that the fine lamellar pearlitecannot be resolved. Instead, pearlite is usually considered as one effective332.5. Austenite Formation and Decompositionphase with eutectoid composition and the motion of the pearlite/austeniteinterfaces is assumed to be interface controlled and the driving force dependson the temperature only [37,101,118,119].Thiessen et al. carried out a phase-field simulation of phase transforma-tion during welding of a low-carbon steel [119]. It was assumed that substi-tutional elements are immobile in the sublattice and long-range diffusion ofcarbon only was taken into account. Pearlite was modeled as a pseudo phasewith the eutectoid carbon composition of 0.7 wt%. A pseudo phase diagramwas used to describe the thermodynamic equilibrium between pearlite andaustenite. Nucleation was assumed to be site-saturated and the nuclei den-sity was evaluated using the the post-weld micrographs. An intrinsic mobilityof the austenite/ferrite interface was used, given by:µ =d4νDkTexp(−QmRT)(2.22)where d is inter-atomic spacing; νD is Debye frequency; Qm is the activationenergy.Rudnizki et al. simulated austenite formation as a phase transformationcontrolled by long-range diffusion of carbon [101]. Using the commercialphase-field package MICRESS®, they described austenite formation in twodimensions in a low-carbon steel (0.1wt%C-1.65wt%Mn) from an annealedpearlite/ferrite microstructure. Austenite nucleation was assumed to takeplace in pearlite regions only and ferrite grain boundaries were not treatedas nucleation sites. Modeling of the pearlite-to-austenite transformation andthe ferrite-to-austenite transformation was performed separately in two steps,i.e. quick growth of austenite into pearlite followed by growth of austenite342.5. Austenite Formation and Decompositioninto ferrite. Pearlite is treated as a stoichiometric phase with a eutectoidcarbon concentration. A high interface mobility was adopted for both theaustenite/pearlite and austenite/ferrite interfaces such that both transforma-tions proceed in a carbon diffusion-controlled mode. Here, two equilibriummodes, i.e. para-equilibrium (PE) and negligible-partitioning local equilib-rium (NPLE), were used to simulate the transformation kinetics. It wasfound that the NPLE assumption describes the transformation kinetics thatis consistent with the experimental observations for one selected heating sce-nario (Fig. 2.14). Moreover, a 3D simulation was implemented [120] suchFigure 2.14: Phase-field simulation of austenite formation assuming differentequilibrium modes in a low-carbon steel (0.1wt%C-1.65wt%Mn-0.55wt%Cr-0.24wt%Si) [120].that more realistic microstructures and carbon distributions were obtained(Fig. 2.15). However, the applicability of the model to different processingconditions was not reported.Recently, the interaction of austenite formation with concurrent ferriterecrystallization has drawn much attention as it affects the final dual-phase352.5. Austenite Formation and DecompositionFigure 2.15: Microstructures and the carbon distribution in a low-carbonsteel (0.1wt%C-1.65wt%Mn-0.55wt%Cr-0.24wt%Si) obtained in a 3D phase-field simulation of austenite formation (black: pearlite; gray: ferrite; white:austenite) [120].microstructures and it opens new avenues to generate dual-phase steels withimproved properties [4, 110, 115]. Zheng and Raabe developed a 2D cellu-lar automaton model to simulate the interaction of ferrite recrystallizationand austenite formation during intercritical annealing of a cold-rolled low-carbon steel (0.08wt%C-1.75wt%Mn) [37]. Pearlite was assumed to be apseudo phase with a uniform carbon content of 0.71 wt%. The classical nu-cleation theory was employed to describe austenite nucleation occurring bothin pearlite regions and at ferrite grain boundaries. It was assumed that thepearlite-to-austenite transformation is interface-controlled and the ferrite-to-austenite transformation is of mixed-mode character. It was assumed thatthere is carbon diffusion only and substitutional elements are immobile inthe sublattice. The simulation replicated a few experimental observations byAzizi-Alizamini et al. [110], e.g. austenite nucleation occurs both in pearliteand at ferrite grain boundaries. Moreover, some recrystallization fronts may362.5. Austenite Formation and Decompositionbe pinned by austenite grains, whereas others cannot be pinned such thatselected austenite grains can appear as islands in ferrite grains even thoughno intragranular nucleation mode is operative. However, the model was notcalibrated to replicate the experimentally measured transformation kinetics.2.5.3 Austenite Formation from Ferrite/CarbideStructuresIn a ferrite/carbide structure with randomly distributed carbide particles,nucleation of austenite takes place at the interface between ferrite and carbidelocated at ferrite grain boundaries [4]. After nucleation, austenite tends toenvelope carbide and forms a layer. Subsequently, carbide dissolves graduallyin austenite while austenite grows into ferrite. As schematically depicted inFig. 2.16, austenite grows anisotropically, in particular at low temperatures.It prefers growing along ferrite grain boundaries as grain boundaries are fasterdiffusion paths. Therefore, austenite will form a network along ferrite grainboundaries in a partially austenitized microstructure.Azizi-Alizamini modeled austenite formation in an ultra-fine ferrite/carbidemicrostructure (Fig. 2.17) [3]. The simulation revealed a variety of possiblegrowth scenarios during austenite formation. For example, it replicated theenveloping process, namely, a layer of austenite quickly surrounding carbide.As time goes on, isolated austenite grains will coarsen with smaller grainsdisappearing.372.5. Austenite Formation and DecompositionFigure 2.16: Schematic of austenite formation in a ferrite/carbide microstruc-ture [3].2.5.4 Austenite Formation from Bainite andMartensiteIn non-deformed bainite and martensite, austenite predominantly nucleatesat prior austenite grain boundaries and lath boundaries [121]. In a marten-sitic stainless steel, austenite nucleation sites depends on temperature, i.e.lath boundaries at low temperatures and prior austenite grain boundariesat high temperatures which was argued to be attributed to the difference innucleation energy barriers [122]. Austenite formed at lath boundaries tendsto grow along the lath and have an acicular shape while austenite formed atprior austenite grain boundaries tends to have a granular shape. For somemicrostructures with retained austenite, if a sufficiently high heating rate382.5. Austenite Formation and DecompositionFigure 2.17: Microstructure evolution in a plain-carbon steel (0.17wt%C)with a ferrite/cementite microstructure by a phase-field simulation (F: ferrite;C: cementite; A: austenite) [3].is used, the retained austenite may remain to high temperature and growwithout nucleation. [123].Ogawa studied austenite formation from a deformed bainitic structure inwhich recrystallization occurs simultaneously with austenite formation [124].It was found that austenite can nucleate near carbide at both the interfacebetween recrystallized and unrecrystallized ferrite as well as the subgrainboundaries inside unrecrystallized ferrite.Nakada et al. carried out a series of experiments to study the effects of de-formation and heating rate on martensite-to-austenite transformation [125].With a great amount of deformation and slow heating rates, recrystalliza-tion of martensite occurs before austenite formation. Therefore, a networkof austenite grains formed and decorated ferrite grain boundaries, similar toaustenite formed in a ferrite/carbide structure. In contrast, acicular mor-phology of austenite formed in lightly deformed martensite at fast heatingrates.Thiessen et al. applied a 2D phase-field method to modeling austeniteformation in a martensite/ferrite dual phase steel [14]. In the simulation,392.5. Austenite Formation and Decompositionmartensite was assumed to be carbon-supersaturated ferrite. Thermody-namic data based on para-equilibrium were used to consider carbon diffusiononly. Site-saturated nucleation and continuous nucleation were tested withexperimental data. It was found that simulation with continuous nucleationcould better describe the transformation kinetics.2.5.5 Austenite-to-ferrite TransformationThe intercritical annealing controls the volume fraction and chemical compo-sition of ferrite and austenite prior to the cooling step. However, after inter-critical annealing the phase fractions might change even during rapid cooling.The formation of ferrite can take place such that the austenite grains shrinkby movement of existing austenite/ferrite phase boundaries. Such growth ofexisting ferrite without nucleation is known as epitaxial ferrite growth [126].The austenite-to-ferrite transformation has been studied widely [17,88,90,127–132]. Many efforts have been made in modeling the transformation ki-netics, for example including semi-empirical formulations e.g. JMAK theorywith the additivity rule and some more fundamental formulations based onthe interface kinetics e.g. mix-mode models [91,133] and diffusion-controlledmodels [90]. An important emphasis of these efforts is the quantitative de-scription of solute drag effects of substitutional alloying elements. But thesemodels cannot predict the microstructure morphologies and spatial distribu-tion of different phases. Another drawback is that a simple geometry has tobe assumed. For example, a spherical austenite grain with a layer of ferritenuclei growing inside was assumed to study austenite-to-ferrite transforma-tion in [91,133].402.5. Austenite Formation and DecompositionSo far, a number of important advances have been made in modelingthe austenite-to-ferrite transformation with the phase-field method. Earlywork [13,134] has successfully investigated interface conditions and the tran-sition between different transformation modes e.g. from diffusion-controlledto interface-controlled transformation. Growth of Widmansta¨tten ferrite alsocan be predicted by assuming a sufficiently high anisotropy of the interfaceenergy [135]. Mecozzi et al. conducted a series of 2D simulations to describecontinuous cooling transformation kinetics in low-carbon steels including aNb micro-alloyed grade [19, 131, 136, 137]. Huang et al. [43, 138] also used a2D phase-field model to simulate the austenite-to-ferrite transformation fordifferent cooling rates. The nucleation scenarios and the interface mobilityare often kept adjustable in these simulations.The phase-field approach was used to simulate the austenite-to-ferritephase transformation in steel for the first time in 3D space by Militzer etal. [17]. The simulation results have shown that 3D simulations lead to morerealistic microstructures than 2D simulations (Fig. 2.18). The comparisonbetween 2D and 3D results has shown a faster transformation rate and lessrealistic morphologies in 2D. The shapes of remaining austenite are islands inthe 2D cuts from the 3D simulation and long elongated channels with severalnarrow inlet-type features between ferrite grains in the 2D simulation (Fig.2.19).412.5. Austenite Formation and DecompositionFigure 2.18: Simulated microstructures during the austenite-to-ferrite trans-formation in a 3D phase-field simulation (A: austenite; F: ferrite) [17].Figure 2.19: 2D microstructures obtained in 3D (a) and 2D (b) phase-fieldsimulations (A: austenite; F: ferrite) [17].422.5. Austenite Formation and Decomposition2.5.6 Intercritical AnnealingBos et al. developed a 3D cellular automaton model describing the mi-crostructure evolution during intercritical annealing of a cold-rolled steel(Fe-0.1wt%C-1.5wt%Mn) with an initial pearlite/ferrite microstructure [118].The model consists of sub-models for ferrite recrystallization, austenite for-mation and the austenite-to-ferrite transformation. These microstructurephenomena are assumed to occur in sequence rather than concurrently, i.e.the potential interaction of ferrite recrystallization and austenite formationis not taken into account. Site-saturated austenite nucleation is assumed totake place only in the pearlite regions. Pearlite is assumed to be an effectivephase with the eutectoid carbon concentration. The pearlite-to-austenitetransformation is assumed to be interface controlled. It is also assumed thatthe ferrite-to-austenite transformation is interface controlled even though thetransformation is typically of mixed-mode character in a low-carbon steel, i.e.both interface reaction and long-range diffusion would have to be considered.The austenite-to-ferrite transition is described using a mixed-mode model.NPLE is used as the equilibrium mode to calculate the driving force. Themodel was not validated with experimental data.2.5.7 Cyclic Phase TransformationRecently, a study of phase transformation between austenite and ferrite in aseries of low-carbon steels during cyclic heat treatments in the intercriticalregion have been carried out by Chen and Van der Zwaag [92,139,140], whichis named cyclic phase transformation. A schematic of cyclic thermal cycles432.5. Austenite Formation and Decompositionis shown in Fig. 2.20. Two types of cyclic heat treatments were used, i.e.Figure 2.20: Schematic of cyclic heat treatments: (a) H-type; (b) V-type [92].V-type and H-type. For both types, a fully austenitized material is cooledrapidly to a temperature in the intercritical region (Ta) and held for a givenperiod. Cyclic heat treatments are subsequently followed between Ta anda higher temperature (Tb) that is also in the intercritical region. There areonly heating and cooling stages for the V-type heat treatment (Fig. 2.20(a)).For the H-type heat treatment, there is an additional holding stage betweenheating and cooling (Fig. 2.20(b)).During cyclic heat treatments in the intercritical region, the ferrite-to-austenite phase transformation and austenite-to-ferrite phase transforma-tion occur alternately in low-carbon steels, which can be observed from thedilatometry curves in Fig. 2.21. One phenomenon of interest is that there isa temperature range at both heating and cooling stages in which the phasetransformation is sluggish termed the “stagnant stage”. A stagnant stagecorresponds to a linear segment upon heating or cooling on the dilatometrycurve (Fig. 2.20). Chen found that stagnant stages exist only in the steelsalloyed with either austenite-stabilizing or ferrite-stabilizing elements, e.g.442.5. Austenite Formation and DecompositionFigure 2.21: Dilatometry data for H-type (a) and V-type (b) cyclic heattreatments of a low-carbon steel (0.1wt%C-0.5wt%Mn): the stagnant stagesare marked with gray lines and the points A-D corresponds to the points inFig. 2.20 [92].Mn and Ni, but not in plain-carbon or Co-alloyed steels [92, 139, 140]. Thestagnant stage is lengthened with an increasing amount of alloying elements.One-dimensional simulations with a diffusion-controlled model were per-formed using the commercial software Dictra® to study the cyclic phasetransformation. Both para-equilibrium (PE) and local equilibrium (LE) wereused. The LE model does not explicitly distinguish NPLE and PLE, sincethey are both local-equilibrium tie-lines. The equilibrium concentrations atthe interface following either the NPLE tie-line or other tie-lines, are deter-mined dynamically based on the mass balance of all elements at the interfaceas assumed in the diffusion-controlled theory [116, 141]. It was found thatthe LE model can replicate the stagnant stages well, while the PE modeldescribes either no or shorter stagnant stages. In the LE simulations it isfound that in each heat cycle a concentration spike will be left near the loca-452.5. Austenite Formation and Decompositiontion where the moving direction of an interface is reversed (Fig. 2.22). TheFigure 2.22: Mn concentration gradients formed in a steel (0.1wt%C-0.5wt%Mn) during cyclic heat treatments, obtained in a 1D simulation usingDictra® [92].concentration spikes formed in a few cycles could become barriers inhibitingthe pass of interface during final cooling to room temperature.By assuming immobile substitutional elements in the sublattice, Gamsja¨geret al. also obtained a reasonable description of the cyclic transformation ki-netics in a Fe-C-Mn alloy using a 1D mixed-mode model [142]. Effectivemobilities were used for both the ferrite-to-austenite (α→ γ) and austenite-to-ferrite (γ → α) transformations, given by:mαγ = m0αγ exp(1RT(−Qmαγ + a (T − T1)))(2.23)where m0αγ is the pre-factor, Qmαγ is the activation energy, a is a constant,and T1 is the lower temperature of cyclic heat treatments (Fig. 2.20). Thepre-factor m0αγ and the constant a are kept adjustable. Two sets of valuesare obtained, one by fitting the α → γ transformation kinetics and theother by fitting the γ → α transformation kinetics. By using two different462.6. Summarymobilities for the α → γ and γ → α transformations, the simulated cyclictransformation kinetics is in quantitative agreement with experimental data.In fact, Eq. 2.23 can be re-written as:mαγ = m0αγ exp( aR)exp(−Qmαγ + aT1RT)(2.24)where(Qmαγ + aT1)can be regarded as an effective activation energy, which is1500 kJ/mol for the α→ γ transformation and -1200 kJ/mol, for the γ → αtransformation. The negative activation energy of the interface mobilityfor the γ → α transformation during cooling is required to obtain a smallmobility at the stagnant stage and a large mobility at lower temperatures.The decrease of an effective mobility for the ferrite/austenite interface withtemperature was also reported for the austenite-to-ferrite transformation ina Nb-Mo alloyed low-carbon steel during continuous cooling [143], which isattributed to the solute drag effect of alloying elements. Thus in the presentcase, it is attributable to the effect of Mn on the cyclic phase transformations.2.6 SummaryMeso-scale models describing the individual microstructure phenomena thatoccur during intercritical annealing of dual-phase steels, e.g. ferrite recrystal-lization, austenite formation and austenite decomposition, have been widelyreported in the literature. However, few integrated models are reported thatcan simulate an entire intercritical-annealing cycle. In particular, the poten-tial interaction of austenite formation with concurrent ferrite recrystalliza-tion has not been investigated in depth yet. The solute drag effect of solute472.6. Summaryatoms on phase transformations has not been taken into account in mesoscalemodels in the past. Moreover, the validation of the models is limited andtheir applicability to a multiplicity of thermal conditions has not been ver-ified. Therefore, there is a need to develop an integrated meso-scale modelthat can describe both the experimentally measured transformation kineticsand microstructural features for a variety of intercritical annealing scenarios.The experimental data on cyclic phase transformations is very appropriatefor phase-field model validation because the challenging nucleation processis avoided during cyclic phase transformation in the intercritical region. Asa fundamental research on phase transformation, it is of significance to in-vestigate the stagnant stages during cyclic phase transformations with thephase-field approach.48Chapter 3Scope and ObjectivesThis work aims to develop a stand-alone phase-field model to describe mi-crostructure evolution during intercritical annealing of dual-phase steels. Inparticular, the model will be applied to a pearlite/ferrite initial microstruc-ture. To achieve the overall goal, the following sub-objectives have to bemet:1. Phase-field models of recrystallization in both 2D and 3D will be devel-oped. First, effects of inhomogeneity, e.g. non-uniform nucleation andinhomogeneous stored-energy, on recrystallization will be systemati-cally investigated. Then, the phase-field model will be calibrated andvalidated with experimental results so that it can accurately describerecrystallization for other thermal processing scenarios.2. A phase-field model of austenite formation in 2D will be developed thatis combined with carbon diffusion and a solute drag model. First, themodel will be calibrated with experimental data of austenite formationfrom a well annealed microstructure. Then it will be used to interpretthe effects of incomplete recrystallization on austenite formation bycoupling to the recrystallization model.3. The phase-field model for austenite formation will be extended to the49Chapter 3. Scope and Objectivesaustenite-to-ferrite transformation. All sub-models will be integratedto describe the overall microstructure evolution during an entire intercritical-annealing cycle. The model will be validated with experimental dataof a simulated industrial annealing cycle.4. The phase transformation model for intercritical annealing will be ap-plied to cyclic phase transformation in the intercritical region, to ra-tionalize the stagnant stage that has been discovered in cyclic experi-ments [92].50Chapter 4Methodology4.1 IntroductionThe multi-phase-field method developed by Steinbach et al. [42, 45] is em-ployed in this study to develop a stand-alone model for intercritical annealingof low-carbon steels. Sub-models for ferrite recrystallization, austenite forma-tion and austenite to ferrite transformation are developed. Neither bainiticnor martensitic transformation is taken into account in the present work.A parallel computer code is developed and implemented on computer clus-ters for phase-field simulations. The sub-models are calibrated and validatedwith data from a systematic experimental study on intercritical annealing ofa low-carbon steel by Kulakov [4]. The chemistry of the studied steel is listedin Table 4.1. The steel consists of 85% ferrite plus 15% pearlite in a cold-rolled state (50% thickness reduction). Moreover, the developed phase-fieldmodel is used to study cyclic phase transformations in a plain-carbon steel(0.1wt%C) and a Mn-alloyed low-carbon steel (0.1wt%C-0.5wt%Mn), whichis validated with experimental data provided by Chen [92].514.2. Basics of the Multi-Phase-Field ModelTable 4.1: Key alloying elements in the investigated steel (wt.%) [4].C Mn Cr Si0.105 1.858 0.340 0.1574.2 Basics of the Multi-Phase-Field ModelIn the multi-phase-field model [42, 45], each microstructural constituent i,e.g. a grain or a phase, is represented by a phase-field variable φi. The valueof the phase-field variable represents the local volume fraction of the relevantconstituent. Thus the diffuse interface is a region with multiple nonzerophase-field variables and the sum of all nonzero phase-field variables is 1,i.e.:∑φi = 1 (4.1)In the bulk of the constituent i, the value of the relevant phase-field variableφi is 1 and the values of all other phase-field variables φj(j 6= i) are 0.In the phase-field approach, the Helmholtz free energy of a system is anintegral of the local density functional over the domain Ω [45]:F =∫Ω(f intf + f chem)dΩ (4.2)where f intf is the density functional of interfacial free energy and f chem is thedensity functional of the bulk chemical free energy. It is noted that for solid-state transformations, the work induced by volumetric change is negligible.Thus the Helmholtz free energy is equivalent to Gibbs free energy. The524.2. Basics of the Multi-Phase-Field Modeldensity functional of interfacial free energy is given by [45]:f intf =Np∑i=1Np∑j=1,j 6=i4σijη(−η2pi2∇φi · ∇φj + φiφj)(4.3)where NP is the number of non-zero phase-field variables, σij is the interfaceenergy and η is the thickness of a diffuse interface. The density functional ofthe chemical free energy is given by [45]:f chem =Np∑i=1φifi({cj}i)(4.4)where fi ({cj}i) is the chemical free energy of the constituent i which is afunction of all solute concentrations {cj}i (the bracket denotes the concen-trations of various solutes in constituent i in a multi-component system).The phase-field equations are derived with an antisymmetric approxima-tion from a variant Allen-Cahn equation [42,144]:∂φi∂t=Np∑j=1,j 6=ipi28ηmij(δFδφj−δFδφi)(4.5)where mij is the interface mobility. The derived multi-phase-field equationsare given by [45]:∂φi∂t=Np∑j=1,j 6=imij{σijIij +piη√φiφj∆Gdrivij}(4.6)withIij =[(φj∇2φi − φi∇2φj)+pi22η2(φi − φj)](4.7)534.2. Basics of the Multi-Phase-Field Modelwhere Iij is a term accounting for the interface curvature and ∆Gdrivij is thedriving pressure that is given by [45]:∆Gdrivij =(∂∂φj−∂∂φi)f chem (4.8)The multi-phase-field model is used to simulate microstructure evolution ina multi-phase or multi-grain structure. For a bi-constituent structure, Eq.4.6 is reduced to a single-phase-field equation:∂φ∂t= m{σ(∇2φ+pi2η2(φ−12))+piη√φ (1− φ)∆Gdriv}(4.9)where φ = 1 represents one constituent (phase or grain) and φ = 0 representsthe other constituent.It should be noted that the phase-field equations (Eq. 4.6 and Eq. 4.9)are solved only within diffuse interfaces and the value of a phase-field variableφi is kept constant in the bulk.The mobilities are assumed to obey an Arrhenius relationship, i.e.:mij = m0ij · exp(QmijRT)(4.10)where R is the ideal gas constant, T is the temperature, m0ij is the pre-factorand Qmij is the activation energy.Solute diffusion occurs during austenite formation and the austenite-to-ferrite transformation in low-carbon steels. In the present study, it is as-sumed that there is only long-range diffusion of the interstitial, i.e. carbon,and the long-range diffusion of substitutional elements does not occur. The544.2. Basics of the Multi-Phase-Field Modelshort-range trans-interface diffusion of the main substitutional element, i.e.manganese (Mn), is considered using a Gibbs energy dissipation model asdiscussed in Section 4.7.In the diffuse interface, each unit volume element represents a mixture ofvarious constituents each with a carbon concentration denoted by CCi . Thecarbon concentration in each constituent is not independent but related toeach other with the equality of chemical potential [45], which is approximatedwith:CCi − C∗iCCj − C∗j=kjikij(4.11)where C∗i is the para-equilibrium carbon concentration, kij and kji are theslopes of the equilibrium lines for phase i and phase j on a para-equilibriumphase diagram, which are calculated with Thermo-Calc® (TCFE7 database).Eq. 4.11 describes the local redistribution of carbon between two phases atan interface. The local carbon concentration in a unit volume element CC isgiven by [45]:CC =Np∑i=1(φiCCi)(4.12)The diffusion of carbon is described by a generalized Fick’s law [45]:dCCdt=Np∑i=1(∇(DCi φi∇CCi))(4.13)where DCi is carbon diffusivity in constituent i. It is assumed that the carbondiffusivity DCi does not depend on the carbon concentration and the carbon554.3. Numerical Techniquesdiffusivity also obeys an Arrhenius relationship:DCi = DC0i exp(−QCi /RT)(4.14)where DC0i is the pre-factor and QCi is the activation energy (Table 4.2).Table 4.2: Carbon-diffusion parameters [17].Phase Pre-factor (m2/s) Activation energy (kJ/mol)Ferrite 2.2×10-4 122.5Austenite 0.15×10-4 142.14.3 Numerical TechniquesThe phase-field equations (Eq. 4.6) and carbon-diffusion equations (Eq.4.13) are non-linear partial differential equations (PDEs). The finite dif-ference method is used to solve the PDEs in a discretized computationaldomain. The computational domains used in this study are square in 2Dand cubic in 3D. Thus the domain is simply discretized into a Cartesian gridhaving the same grid spacing ∆x in all directions (Fig. 4.1).564.3. Numerical TechniquesFigure 4.1: Schematic of the grid structure in the computational domain.A sparse data structure is used to store variables on each grid point,including the index and value of each non-zero phase-field variable as well aslocal solute concentrations. The detailed description of the data structurecan be found elsewhere [25].Centered finite difference with the nearest neighbors (5-point stencil in 2Dand 7-point stencil in 3D) is used to discretize the first and second derivatives.For example, the gradient and Laplacian of φ at grid point (i, j, k) in a 3Ddomain are given by:∇φi,j,k =∂φi,j,k∂n(4.15)≈(φi+1,j,k − φi−1,j,k2∆x,φi,j+1,k − φi,j−1,k2∆x,φi,j,k+1 − φi,j,k−12∆x)and574.3. Numerical Techniques∇2φi,j,k =∂2φ∂x2+∂2φ∂y2+∂2φ∂z2(4.16)≈φi+1,j,k + φi−1,j,k + φi,j+1,k + φi,j−1,k + φi,j,k+1 + φi,j,k−1 − 6φi,j,k∆x2The explicit Euler method is used to implement the integration in thetime domain for both phase-field and diffusion equations, e.g.(∂φi∂t)t≈φt+∆ti − φti∆t(4.17)where ∆t is the time step, φti is the current value of φi, φt+∆ti is the valuefor the next time step. The explicit Euler method is conditionally stable andthe time step has to fulfill the following requirement:∆t < ∆tc = min(∆x22Dim ·max (mijσij),∆x22Dim ·max (DCi ))(4.18)where Dim is the domain dimensionality. For simplicity, the time step isrepresented by ∆t = ξ∆tc where the factor ξ is adjusted between 0 and 1 tocompromise between computational cost and numerical accuracy.In contrast with the explicit Euler method, the implicit Euler method isunconditionally stable and thus a larger time step can be used. But largetime steps are usually discarded due to numerical inaccuracy. Further, thecomputational cost for each time step is higher. Thus the implicit methodhas no significant advantages over the explicit method. In contrast, someadvanced data structures [25] and parallel algorithms can be used in the584.4. Initial Microstructuresexplicit method to improve the computational efficiency.Suitable values of the other numerical parameters, i.e. the grid spacing∆x and the interface thickness η represented by the number of grid points, i.e.nη∆x, have to be chosen based on both computational cost and accuracy.Convergence and accuracy analyses were carried out to select appropriatevalues for the three parameters, i.e. nη, ∆x and ξ. The details are discussedin Chapter 5 and Chapter 6.4.4 Initial MicrostructuresThe initial microstructures used in the phase-field simulations (Chapter 5-7) are constructed using a modified Voronoi tessellation [145, 146], based onthe experimental data, e.g. the average ferrite grain size, the rolling reduc-tion, the phase fractions and pearlite-band spacing. To perform a phase-fieldsimulation, boundary conditions of the simulation domain have to be spec-ified, e.g. periodic and insulated boundaries. Periodic boundary conditionsare usually applied to a simulation domain that is a representative volumeelement (RVE) to approximate a system of an infinite size [147]. With pe-riodic boundary conditions, the mass conservation of carbon in the domainis guaranteed such that the total amount of carbon in the domain remainsconstant and the equilibrium phase fractions in the domain is consistent withthat in the system of an infinite size. However, periodic boundary conditionsare applicable only to computer-constructed microstructures but not to ex-perimental microstructures, e.g. micrographs and EBSD maps. In this case,insulated (no-flux) boundary conditions are alternatives, i.e. ∂φi/∂n = 0 and594.5. Ferrite Recrystallization∂CCi /∂n = 0 where n denotes the normal to the boundary [147]. Similar toperiodic boundary conditions, insulated boundary conditions can also ensurethat the total amount of carbon in the domain remains constant. There-fore, periodic boundary conditions are used in the simulations with initialmicrostructures that are constructed with the Voronoi tessellation, whereasinsulated boundary conditions are used in the 2D simulations of cyclic phasetransformations (Chapter 8) with micrographs as the initial microstructures.4.5 Ferrite RecrystallizationThe recrystallization model considers ferrite recrystallization only and mi-crostructure evolution in pearlite, e.g. spheroidization, is not taken intoaccount at this stage. The stored energy mainly in the form of dislocations,as the driving force for recrystallization, is an input of the phase-field model.The stored energy in the present study is assumed to be an attribute ofgrains and uniform within each grain. The calculation of stored energy inthe investigated cold-rolled steel is discussed in Chapter 5.In this model, the physical process of nucleation from substructures isnot considered. Rather, the nuclei are introduced to the simulation domainas spherical grains represented by new phase-field parameters. The initialsize of nuclei is limited by two aspects. For numerical stability and accu-racy [25], it has to be larger than nη∆x where nη is the interface thickness.To grow rather than shrink due to the interface curvature, the size has tobe larger than 2σij/∆Gdrivij where σij is the interface energy and ∆Gdrivij isthe driving pressure for interface migration. Thus, the initial size is set to604.5. Ferrite Recrystallizationmax(nη∆x, 2σij/∆Gdrivij). Site-saturated nucleation is assumed such that allnucleation events occur effectively at the beginning of recrystallization. Thenuclei density is based on the average ferrite grain size after full recrystal-lization. The spatial distribution of nuclei is non-random and is defined bya criterion that all nuclei are distributed only in the grains with a storedenergy greater than a critical value Est∗, i.e.Est > Est∗ (4.19)where Est∗ is an adjustable parameter, the value of which is calibrated withthe experimentally measured recrystallization kinetics.The driving pressure ∆Gdrivij for recrystallization in Eq. 4.6 is given by:∆Gdrivij = Estj − Esti (4.20)where Esti and Estj are the stored energy in grain i and j, respectively. Thestored energy in recrystallized grains is assumed to be 0.All ferrite grain boundaries are assumed to be high-angle grain bound-aries. Thus, the interface energy and mobility are the same for all ferritegrain boundaries. The ferrite grain-boundary energy is set to 0.76 J/m2 [58].The grain-boundary mobility is kept adjustable and calibrated with the ex-perimentally measured recrystallization kinetics. The details of tuning theadjustable parameters are discussed in Chapter 5.Ferrite recovery which can reduce the stored energy is not taken intoaccount in the model. Moreover, the possible solute drag effect of the sub-stitutional elements, e.g. Mn, on ferrite recrystallization is not explicitly614.6. Austenite Formationconsidered such that the grain boundary mobility is treated as an effectivemobility.4.6 Austenite FormationIn this sub-model, pearlite is assumed to be a pseudo phase, because itis a great computational challenge to model the lamellar structure with acharacteristic length of 100 nm when considering microstructures on a scaleof 10-100 μm. The carbon concentrations in both pearlite and ferrite are setto the para-equilibrium eutectoid compositions, e.g. 0.68 wt% and 0.012 wt%for the DP600 steel (point A and B in Fig. 4.2). The other substitutionalelements are assumed to be uniformly distributed in the microstructure andthe possible segregation of Mn in the pearlite bands is not considered.Figure 4.2: Para-equilibrium phase diagram for DP600 (α: ferrite; γ: austen-ite; θ: cementite).624.6. Austenite FormationSavran et al. [102] conducted dedicated studies on austenite nucleationand observed continuous nucleation in a low-carbon steel. These results arein qualitative agreement with the observations by Kulakov [4] in the steelinvestigated here. Similar to previous observations, Kulakov found rapidpearlite-to-austenite transformation followed by a more gradual transforma-tion of ferrite to austenite. Further, austenite nucleation was observed tooccur at both pearlite/ferrite interfaces and ferrite grain boundaries. Thus,two separate nucleation models are postulated for the two nucleation sites.Site saturation is assumed for austenite nucleation at pearlite/ferrite in-terfaces with a nuclei density that is, based on the experimental observa-tion [148], approximately independent of heating scenarios. A nucleus den-sity of 0.024 μm-2 [148] is adopted and the nucleation temperature is setto 730 °C, in agreement with the measured start temperature of austeniteformation at the lowest investigated heating rate, i.e. 1°C/s [4].The classical nucleation theory is employed to describe the more gradualaustenite nucleation at ferrite grain boundaries:dNdt= λNn(t) exp(−Ψ∆g2V+QFeαkBT)(4.21)where kB is the Boltzmann constant, λ is a constant related to the atom jumpfrequency, Nn(t) is the number of potential nucleation sites represented bythe total length of ferrite grain boundaries in the 2D simulation domain (totalnumber of grid points within interfaces×grid spacing×grid spacing/interfacethickness) and decreases with time t as more and more nuclei occupy theferrite grain boundaries, Ψ is a constant related to the assumed nucleusshape and interface properties, QFeα is the activation energy of iron self-634.6. Austenite Formationdiffusion in ferrite, and ∆gV is the driving force for nucleation which dependson the temperature and the average carbon concentration in ferrite (thecarbon concentration gradient in ferrite is presumably negligible due to thehigh diffusivity and low solubility) which is calculated with Thermo-Calc®(TCFE7 database). The value of Ψ for ferrite nucleation at austenite grainboundaries during the austenite-to-ferrite transformation in Fe-C alloys (2.1×10−6 J3/m6) [149–151] was used for austenite formation, which was obtainedby fitting the classical nucleation model to the nuclei density measured withmetallography [149]. The value of QFeα is taken to be 3.9 × 10−19 J [102].The factor λ is kept adjustable and calibrated with the experimental data.In the case of concurrent ferrite recrystallization and austenite formation,it is assumed that the migrating recrystallization fronts are not potentialnucleation sites. Triple junctions are not distinguished from grain boundariesin the nucleation model.Further, it was observed that austenite grains formed at ferrite grainboundaries grow mainly into one of the two neighboring ferrite grains [103].This observation can be rationalized based on special crystallographic ori-entation relationships between the austenite grain and one of the adjacentferrite grains, e.g. the Kurdjumov-Sachs (K-S) relationship, making this por-tion of the α/γ interface semi-coherent and of low mobility. In the presentmodel, it is postulated that all the austenite nuclei at a boundary segmentbetween two ferrite grains α1 and α2, have a special orientation relationshipwith one specific ferrite grain (e.g. α1) but not with the other one (α2),such that the mobility of the α1/γ interface is 1% of that of the α2/γ inter-face, leading to dominant growth of austenite grains into α2. It is a random644.6. Austenite Formationselection in the model as to whether α1 or α2 is the grain with which austen-ite has a special orientation relationship. The introduction of nuclei to thesimulation domain is the same as that for ferrite recrystallization.Because pearlite is a pseudo phase, the following postulations are made:(1) no carbon diffusion is considered in pearlite; (2) pearlite interacts withaustenite only; (3) transformation of pearlite (P ) to austenite (γ) is assumedto be interface controlled with a driving force of:∆GdrivPγ (T ) = GP (T )−Gγ(T ) (4.22)where the Gibbs free energies of pearlite GP and austenite Gγ are evaluatedwith Thermo-Calc®. Because pearlite (P ) is not a phase but a mixture offerrite (α) and cementite (θ), its free energy is calculated from:GP = xαGα(T ) + (1− xα)Gθ (T ) (4.23)where xα is the molar fraction of ferrite in eutectoid pearlite; Gα and Gθare the molar Gibbs free energy for ferrite and cementite, respectively. TheGibbs free energy of austenite and ferrite in Eq. 4.22 and 4.23 is calculatedwith constant carbon concentrations of 0.68wt% and 0.012wt%, respectively.The chemical driving pressure for the ferrite-to-austenite transformationis described by:∆Gchemαγ = χαγ (T )(CCγ − C∗γ)(4.24)where CCγ is the interfacial carbon concentration in austenite, C∗γ is the para-equilibrium carbon concentration in austenite and χαγ (T ) is a proportional-654.6. Austenite FormationFigure 4.3: Schematic of the calculation of driving pressure for ferrite toaustenite transformation (SC denotes the effective substitutional compo-nent).ity factor that is calculated with Thermo-Calc®. For a given temperature,the value of χαγ (T ) is obtained by fitting Eq. 4.24 to a series of data points(CCγ ,∆Gchemαγ)that are calculated with Thermo-Calc®. The calculation ofthe chemical driving pressure ∆Gchemαγ is illustrated in Fig. 4.3. For a givencarbon concentration in austenite CCγ , there is a unique carbon concentrationin ferrite such that the chemical potential of carbon in ferrite is equal to thatin austenite (Fig. 4.3). The chemical driving pressure is then given by:∆Gchemαγ = µSCα − µSCγ (4.25)where µSCα is the chemical potential of the effective substitutional componentin ferrite (Eq. 2.10).Further, the solute drag effect of the main substitutional alloying element,664.7. Gibbs-energy Dissipation Modeli.e. Mn, is taken into account with a Gibbs-energy dissipation model [84] .The effective driving pressure for the ferrite/austenite interface migration∆Geffαγ in the phase-field equations (Eq. 4.6) is given by:∆Geffαγ = ∆Gchemαγ −∆Gdissαγ (4.26)where ∆Gdissαγ is the dissipated Gibbs-energy that is calculated with theGibbs-energy dissipation model as described in Section 4.7. The stored en-ergy in a deformed ferrite grain is included in the driving pressure for thedeformed-ferrite-to-austenite interface migration.The γ/P interface energy is 0.9 J/m2 [119] and the γ/α interface energyis 0.4 J/m2 [17]. The γ/P and γ/α interface mobilities are adjusted to fitthe experimentally measured transformation kinetics. The calibration of theadjustable parameters in the present sub-model is discussed in Chapter 6.4.7 Gibbs-energy Dissipation ModelIn the present study, the solute-drag effect of Mn on the ferrite-to-austenitetransformation is described with a Gibbs-energy dissipation model by Hillert[84]. As reviewed in Section 2.4, the dissipated Gibbs-energy in the model isequivalent to the solute drag pressure in the Purdy-Brechet model [86]. Theterm “Gibbs-energy dissipation” is used in the present work, because thecontroversial non-zero solute drag pressure on a stationary interface obtainedin the Purdy-Brechet model was rationalized in the Gibbs-energy dissipationmodel [152].In the Gibbs-energy dissipation model, it is assumed that the interface674.7. Gibbs-energy Dissipation Modelmovement during the ferrite-to-austenite transformation (α→ γ) consists oftwo processes [153], i.e. (i) the transition of a crystal structure from body-centered cubic (BCC) to face-centered cubic (FCC) with a concentration ofthe bulk xMn0 and (ii) local redistribution of Mn, i.e. trans-interface diffusionof Mn driven by its non-uniform chemical potential across the interface. Fromthe perspective of energy, Eq. 4.26 is an energy balance that the total freeenergy consumed by the two processes is equal to the released free energy ofthe system per unit volume of Fe, i.e. the chemical driving pressure. The freeenergy consumed by transition of the crystal structure is equivalent to theeffective driving pressure for interface migration ∆Geffαγ and the free energyconsumed by trans-interface diffusion is the dissipated Gibbs-energy ∆Gdissαγ .By assuming the system is an ideal solution and the trans-interface diffu-sion is steady-state, the Gibbs-energy dissipated by trans-interface diffusionof Mn per unit volume of Fe (J/m3) is given by [153]:∆Gdissαγ = −1Vm+Λ∫−Λ(xMn (y)− xMn0) d∆µMn (y)dydy (4.27)where the physical thickness of the interface is 2Λ, xMn0 is the molar fractionof Mn in the bulk, xMn (y) is the molar fraction of Mn across the interface and∆µMn (y) = µMn0 (y) − µFe0 (y) is the difference between the standard-statechemical potentials of Mn and Fe. ∆µMn (y) is equivalent to the free energyof interaction of the solute with an interface E(y) in the Purdy-Brechet model(Section 2.4).A wedge-shaped profile of ∆µMn (y) across an α/γ interface is assumed(Fig. 2.6). Based on the assumption of steady-state diffusion across an α/γ684.7. Gibbs-energy Dissipation Modelinterface (Eq. 2.16), the molar fraction of Mn across an interface movingwith a velocity of V is described by [86]:xMnxMn0=1 + a · exp (−V ′ (1 + a) (Y + 1))1 + aif −1 ≤ Y ≤ 0xMnxMn0=1 +(a(1+b)·exp(−V ′(1+a))1+a +b−a1+a)exp (−V ′ (1 + b)Y )1 + bif 0 ≤ Y ≤ 1 (4.28)with:Y =yΛ(4.29)V ′ =V ΛDMnint(4.30)a =(∆E − E0)RTV ′(4.31)b =(∆E + E0)RTV ′(4.32)where DMnint is the trans-interface diffusivity of Mn, E0 is the binding energyof Mn to an α/γ interface, 2∆E is the difference of ∆µMn in ferrite andaustenite, i.e. ∆µMnα − ∆µMnγ which is calculated with Thermo-Calc®. Aschematic of the Mn concentration profile across an α/γ interface is shownin Fig. 4.4.694.7. Gibbs-energy Dissipation ModelThe dissipated Gibbs-energy is calculated by substituting the Mn molarfraction (Eq. 4.28) into Eq. 4.27:∆Gdissαγ = −xMn0 RTVm[−a2V ′1 + a−b2V ′1 + b+a2 (1− exp (−V ′ (1 + a)))(1 + a)2+b2 (1− exp (−V ′ (1 + b)))(1 + b)2−ab (1− exp (−V ′ (1 + a))) (1− exp (−V ′ (1 + b)))(1 + a) (1 + b)](4.33)where Vm is the molar volume of the iron matrix.Figure 4.4: Illustration of the Mn concentration across a moving α/γ interfacefor the ferrite-to-austenite transformation: a negative spike is formed in frontof the α/γ interface.At a given temperature, the dissipated Gibbs-energy is a function of the704.7. Gibbs-energy Dissipation ModelFigure 4.5: The effect of the binding energy of Mn on the profile of thedissipated Gibbs-energy as a function of interface migration rate at 770 °Cfor a low-carbon steel (0.1wt%C-1.86wt%Mn).interface migration rate and the binding energy (Fig. 4.5). When the bindingenergy is 0, the wedge-shaped profile of ∆µMn (y) reduces to a straight lineand there is still trans-interface diffusion driven by the gradient of ∆µMn.The nonzero dissipated Gibbs-energy for an interface migration rate of zero isthe limit of the dissipated Gibbs-energy as an interface moves at an infinitelylow rate. Based on Eq. 4.33, the dissipated Gibbs-energy for a stationaryinterface ∆Gdissαγ,0 is:∆Gdissαγ,0 = −xMn0 RT/Vm(1−2∆ERT− exp(2∆ERT))(4.34)The non-zero dissipated Gibbs-energy for a stationary interface can alsobe interpreted as the free energy that the trans-interface diffusion wouldconsume if the stationary interface started moving. Therefore, to enable714.7. Gibbs-energy Dissipation Modela stationary interface to move, the chemical driving pressure has to belarger than the dissipated Gibbs-energy, i.e. ∆Gchemαγ > ∆Gdissαγ,0. Other-wise (∆Gchemαγ ≤ ∆Gdissαγ,0), the interface remains stationary. Based on Eq.4.26, the interfacial carbon concentration CCγ at a stationary interface underthe critical condition, i.e. ∆Gchemαγ = ∆Gdissαγ,0, is:CCγ = C∗γ +∆Gdissαγ,0χαγ (T )(4.35)where C∗γ is the para-equilibrium carbon concentration in austenite. Odqvistet al. proved that the carbon concentration given by Eq. 4.35 is the negligible-partitioning local equilibrium (NPLE) carbon concentration [87]. Therefore,the equilibrium limit in the Gibbs-energy dissipation model is NPLE. Basedon the present model assumptions, the para-equilibrium condition can existonly at a fast moving interface where Gibbs-energy dissipation is negligible.The value of Λ was set to 0.5 nm [91] and the binding energy E0 was takento be 10 kJ/mol [92]. The trans-interface diffusivity of Mn was assumed to bethe geometrical average of bulk diffusivities in ferrite and austenite [90,154],i.e.DMnint = 0.5× 10−4 exp(−247650RT)m2/s (4.36)In order to integrate the Gibbs-energy dissipation model with the phase-field model, the interface velocity in the Gibbs-energy dissipation model (Eq.4.33) is further expressed with the phase-field variable φi :V = |∂n∂t| =∂φi∂t/|∂φi∂n| (4.37)724.8. Austenite-to-ferrite Transformationwhere |∂φi∂n | is given by Eq. 4.15.It is possible to model solute segregation in a phase-field model by re-formulating the free energy density functional. For example, to simulatethe solute segregation on grain boundaries, Cha et al. formulated the den-sity functional of the chemical free energy in each grain f i (Eq. 4.4) as afunction of both the solute concentration ci and the phase-field variable φi,i.e. f i (ci, φi), such that an equal chemical potential corresponds to differentsolute concentrations in the bulk and at the interface:{∂fi (ci, 1) /∂ci}ci=c0 ={∂fi(ci,12)/∂ci}ci=cintf(4.38)where c0 is the solute concentration in the bulk (φi = 1) and cintf is thesegregated solute concentration (cintf > c0) in the center of an interface(φi = 1/2). The limitations of this approach are that a very thin interfacethickness (0.5-1 nm) is required to eliminate the artificial solute-trappingeffect (solute segregation in thick interfaces leads to significant depletion ofsolutes in the bulk) [155] and the slow diffusion of substitutional solutes hasto be taken into account, leading to high computational cost. Therefore,using an analytical solute-drag pressure is a pragmatic approach to considerthe solute drag effect in a phase-field model.4.8 Austenite-to-ferrite TransformationIn the cooling process of an intercritical annealing cycle, austenite will trans-form back to ferrite partially by reversing the moving direction of austen-ite/ferrite interfaces without ferrite nucleation, which is termed epitaxial fer-734.9. Grain Growthrite growth. The formulations proposed for austenite formation in Section4.6 and Section 4.7 are used for epitaxial ferrite growth, e.g. calculation ofthe driving pressure and solute drag pressure. The austenite/ferrite interfacemobility and energy are also kept the same.By integrating the models of ferrite recrystallization, austenite formationand austenite-to-ferrite transformation, a stand-alone phase-field model isdeveloped that can describe all aspects of microstructure evolution duringan entire intercritical-annealing cycle.The stand-alone model is also applied to modeling the cyclic phase trans-formations in a plain-carbon steel (0.1wt%C) and a Mn-alloyed steel (0.1wt%C-0.5wt%Mn) in Chapter 8. The thermodynamic parameters for each steel arecalculated with Thermo-Calc®. The parameters in the Gibbs-energy dissi-pation model is the same as that in Section 4.7. The interface properties,i.e. the γ/α interface energy and mobility are the same as that used for theDP600 steel.4.9 Grain GrowthGrain growth of both ferrite and austenite is included in the phase-fieldmodel. The interface energy and mobility of the ferrite grain boundary areset in the recrystallization sub-model (Section 4.5). The austenite-grain-boundary energy is taken to be 0.7 J/m2 [14]. The activation energy ofthe austenite-grain-boundary mobility is taken to be 185 kJ/mol [14] andthe pre-factor is adjusted such that the average austenite grain sizes afterfull austenitization for various heating scenarios are consistent with the ex-744.10. Model Implementationperimental measurements. The possible solute drag effect of substitutionalelements on grain growth is not taken into account.4.10 Model ImplementationBoth FORTRAN and C/C++ are used as the programming language totranslate the phase-field model into a computer program. The code is par-allelized using OpenMP and MPI (Message Passing Interface) libraries andalso accelerated with GPU (Graphics Processing Unit) computing. The im-plementation of the program is schematized in Fig. 4.6. First, input filescontaining the necessary data, e.g. domain size, phase diagram, interfaceproperties and thermal paths, are read by the program. Then an initialmicrostructure is constructed using Voronoi tessellation or input from a filealternatively. The main body of the program is a series of time loops inwhich nucleation models and solvers of phase-field equations and diffusionequations are implemented. In each loop, the values of model parameters areupdated, e.g. time, temperature and solute diffusivity. Selected data can bewritten to disk as specified by the user. Once the condition of terminationis met, the program stops.Both phase-field and carbon concentration field are written to files andvisualized with the software ParaView®. Other statistical information isalso exported by the program, e.g. recrystallized ferrite fraction, austenitefraction and grain sizes. The equivalent diameter is used as the grain size [4].In order to compare with the grain sizes measured using 2D micrographs,various 2D cuts of the 3D domain are used to calculate grain sizes.754.10. Model ImplementationFigure 4.6: Flowchart of the model implementation.76Chapter 5Modeling Recrystallization15.1 Benchmarking1D simulations of a moving interface driven by a constant driving pressure∆G were implemented to select appropriate values for the numerical parame-ters, i.e. the grid spacing ∆x, the number of grid points through an interfacenη and the time-step factor ξ = ∆t/∆tc (Eq. 4.18). In the simulations, theinterface energy σ was set to 1 J/m2 and the interface mobility m was set to1.0Ö10-12 m4/(J·s). The simulation domain length was set to 10 μm and thegrid spacing ∆x was set to 0.1 μm. Periodic boundary conditions were used.First, a series of simulations were performed with different values of nηand ξ. The driving pressure ∆G was set to 1.0Ö106 J/m3 such that theinterface migration rate is 1 μm/s in the sharp-interface limit. The simu-lated interface migration rates are shown in Fig. 5.1. For a given interfacethickness, the interface migration rate changes no more than 5% with theselected range of ξ. Therefore, a value of 1/4 was chosen for ξ. The num-ber of grid points used to discretize the interface has an influence on thenumerical stability and accuracy. The minimum value of nη for numerical1A paper based on this chapter was published: B. Zhu, M. Militzer, “3D phase fieldmodeling of recrystallization in a low-carbon steel”, Modeling and Simulation in MaterialsScience and Engineering, Vol. 20, No.8, P. 085011, 2012.775.1. BenchmarkingFigure 5.1: Influence of interface thickness and time step on numerical accu-racy.stability is 5 and larger values of nη lead to higher numerical accuracy. Forexample, More than 10 grid points are required for 95% accuracy whereas5 grid points lead to 85% accuracy. From the perspective of computationalcost, a small value of nη is desirable. Thus the value of nη was set to 5 for allphase-field simulations in this study. Because the interface mobilities are alladjustable parameters in the developed phase-field models, the consequenceof using a value of 5 for nη is that the fitted values of the mobilities will beabout 20% larger than the theoretical values. For example, as shown in Fig.5.1, to obtain an interface migration rate of 0.85 μm/s with a driving pres-sure of 1.0Ö106 J/m3, the mobility in the sharp-interface limit is 0.85Ö10-12m4/(J·s), whereas it is 1Ö10-12 m4/(J·s) in the present phase-field simulation.Shahandeh [25] found that the characteristic length of the studied microstruc-ture, e.g. the average grain size, should be at least 5 times larger than the785.1. Benchmarkinginterface thickness to attain 95% accuracy. The characteristic length in thestudied material is in the range of 3-10 μm. Thus the grid spacing was setto 0.1 μm such that the interface thickness is 0.5 μm. Whether the selectedvalues of these numerical parameters are appropriate for the simulations ofdiffusional phase transformations will be verified in Chapter 6.As the JMAK equation is frequently used to describe recrystallizationkinetics, it is critical to verify the capacity of the phase-field model to repli-cate the ideal JMAK recrystallization kinetics before it is applied to morecomplex situations. Thus a 3D simulation that follows the assumptions ofthe JMAK model was carried out.In the JMAK model, two assumptions are made: (1) random nucleationwith either site saturation or constant nucleation rate; (2) new grains growat a constant rate in all directions. In the phase-field simulation, only site-saturated nucleation was considered. A uniform stored energy was assumedin the simulation domain. Table 5.1 lists the relevant parameters used in theTable 5.1: Phase-field model parameters used in the benchmarking simula-tion.Domain size (μm3) 40Ö40Ö40Grid spacing ∆x (μm) 0.1Number of nuclei 500Stored energy (J/m3) 4.0Ö106Interface mobility (m4/(J·s)) 1.1Ö10-14Interface energy (J/m2) 0.1simulation. A small interface energy was used, for the purpose of reducing theeffect of boundary curvatures on interface migration because the curvature795.1. Benchmarkingeffect is not considered in the JMAK model.Fig. 5.2 shows the 3D microstructures of recrystallization at variousstages. New grains grow as spheres before they impinge on each other.When the recrystallization is finished, the simulation domain is occupiedcompletely by recrystallized grains in equiaxed and polyhedral shapes. Thedouble-logarithmic representation of the JMAK equation was used to obtainthe JMAK exponent n such that:ln (−ln (1− fX)) = ln(b) + nln(t) (5.1)The plot of ln (−ln (1− fX)) against ln(t) is ideally a straight line with aslope of n. As shown in Fig. 5.3, the double-logarithmic plot of the simulatedrecrystallization kinetics is also a straight line with a slope of 2.98. The slightdeviation of the fitted slope from the theoretical value of 3 may be due tothe interface curvature and the difference in the initial size of nuclei, i.e. 0in the JMAK model and 0.5 µm in the phase-field simulation.805.1. BenchmarkingFigure 5.2: 3D microstructures with different volume fractions of recrys-tallizing grains obtained in a phase-field simulation assuming homogeneousnucleation and uniform stored energy.815.2. Effects of Inhomogeneity on RecrystallizationFigure 5.3: Double-logarithmic representation of the simulated recrystalliza-tion kinetics: homogeneous nucleation and uniform stored energy.5.2 Effects of Inhomogeneity onRecrystallizationThe JMAK exponent n fitted with experimental data is usually smaller thanthe theoretical value of 3.0, indicating that some unrealistic assumptionsare made in the JMAK model. Random nucleation and uniform growthrates are not possible in reality because of the inhomogeneities in deformedmicrostructures. Therefore, the effects of non-uniform stored energy andnon-random nucleation on recrystallization are systematically investigatedby a series of 3D phase-field simulations.In order to mimic inhomogeneity of stored energy, a 50% cold-rolled poly-825.2. Effects of Inhomogeneity on Recrystallizationcrystalline microstructure consisting of 50 grains was used. Each grain wasassumed to have a specific value of stored energy. For simplicity, random val-ues between 2.0Ö106 J/m3 and 6.0Ö106 J/m3 were assigned to each grain. Tostudy the effects of non-random nucleation, the nucleation model described inSection 4.5 was used. It is obvious that, nuclei are limited in fewer deformedgrains as the value of Est∗ increases from the minimum stored energy to themaximum. Three different values of Est∗ were used in the simulations, i.e.0 J/m3, 4.0Ö106 J/m3 and 5.0Ö106 J/m3. The values of model parametersused in the simulations are listed in Table 5.1.There is a variation in migration rates of recrystallization fronts that isproportional to the variation of the assumed stored energy. The effect of anon-uniform stored energy distribution on the growth stage is investigatedby setting Est∗=0. As shown in the JMAK plot (Fig. 5.4), the change of theFigure 5.4: Double-logarithmic representation of recrystallization kinetics:various nucleation scenarios and non-uniform stored energy.835.2. Effects of Inhomogeneity on RecrystallizationJMAK exponent is modest in this case, decreasing from 2.98 to 2.85, which isconsistent with what is reported in the literature that a variation of growthrates of recrystallization fronts has little effect on the JMAK exponent whennucleation is random [156]. The final grain size distribution also changeslittle in comparison with the case of a uniform stored energy distribution(Fig. 5.5).Figure 5.5: Grain size distributions after full recrystallization (500 grains):(a) uniform stored energy and random nucleation; (b) non-uniform storedenergy and random nucleation; (c,d) non-uniform stored energy and non-random nucleation with Est∗=4.0Ö106 J/m3 and 5.0Ö106 J/m3, respectively.In contrast, the JMAK exponent changes significantly when nucleationis spatially non-random. The clustering of nuclei in some specific grains845.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureleads to a marked decrease of the JMAK exponent, as low as 1.7 in thepresent simulations. Clustering of nuclei makes recrystallized grains impingewith each other earlier and because these recrystallized grains are located inregions with high stored energy, the growth rates of recrystallization frontsgenerally decrease with time as they migrate into regions of low stored energy.Luo et al. derived an equation for the JMAK exponent, i.e. n = 3(1− r), byassuming that the decrease of growth rates can be described by Ct−r where Cand r are positive constants [157]. Therefore, the geometrical constraint onthe growth of recrystallization fronts due to early impingement of clusterednuclei and decreasing growth rates with time, lead to smaller values of theJMAK exponent. Moreover, a larger variance of the grain size distributionis obtained in the case of a larger Est∗. For example, the standard deviation(√1NN∑i=1(di − 1NN∑i=1di)2where di is the grain size) in the case of Est∗ equalto 5.0Ö106 J/m3 (Fig. 5.5(d)) is 0.48, in comparison with 0.37 in the case ofrandom nucleation (Fig. 5.5(a)). It is concluded that spatially non-randomnucleation is the main cause of small values of the JMAK exponent, whichare frequently reported for recrystallization.5.3 Application to the Cold-rolledFerrite/Pearlite Microstructure5.3.1 Stored Energy DistributionThe volumetric distribution of stored energy in the form of dislocations wasestimated with a Taylor-factor analysis. Experimental studies demonstrated855.3. Application to the Cold-rolled Ferrite/Pearlite Microstructurethat the stored energy of each grain Est generally increases with the mag-nitude of the Taylor factor M that is associated with its crystallographicorientation g [46, 63]. Therefore, as a first-order approximation, it was as-sumed that the stored energy in a grain with a crystallographic orientationg is given by:Est(g) = κM(g) (5.2)where the constant κ was calculated with the average Taylor factor and theaverage stored energy. The average stored energy in the investigated steelwas estimated by Kulakov with hardness tests [4], assuming that the changeof yield stress before and after recrystallization (in MPa) ∆σY is equal toone third of the Vickers hardness drop (in MPa). The change of dislocationdensity before and after recrystallization ∆ρdislocation is given by [4]:∆σY = 0.5Gshear ~|b|√∆ρdislocation (5.3)where ~b is Burger’s vector and Gshear is the shear modulus of ferrite. Theaverage stored energy was estimated with the dislocation density, given by [4]:Estavg = 0.5Gshear~|b|2∆ρdislocation (5.4)The calculated average stored energy is 1.8Ö106 J/m3 [4]. The drawback ofthis method is that it neglects the effect of other factors on hardness, e.g.changes of grain size and morphology before and after recrystallization.Based on the assumption that the plastic strain of all grains is the sameand equal to the macroscopic plastic strain, the Taylor factor M is given865.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureby [158,159]:M =∑dlldε(5.5)where dl is the absolute value of shear strain contributed by a specific slipsystem and dε is the incremental macroscopic strain. In the calculation ofthe Taylor factor, it was assumed that the cold rolling is uniform plane-straincompression and the active slip systems are {110} 〈111〉 and {112} 〈111〉 withthe same critical resolved shear stress (CRSS). The distribution of crystal-lographic orientation or the orientation distribution function (ODF) of thedeformed microstructure was measured with the electron back-scatter diffrac-tion (EBSD) technique 2. The calculated average Taylor factor is 3.2 and thusκ is 0.56Ö106 J/m3. The obtained distribution of stored energy is shown inFig. 5.6.5.3.2 3D SimulationA 3D domain with a size of 56Ö56Ö56 μm3 was used with the grid spacingof 0.1 μm. The initial microstructure consists of 160 ferrite grains such thatan average grain size of 9.6 μm is obtained (Fig. 5.7). The pearlite-bandspacing was selected to be consistent with the experimental measurement(14 μm on average). The stored energy in each grain was adjusted such thatthe actual distribution of stored energy in the simulation domain matchesthe calculated distribution in Fig. 5.6.2The EBSD experiment was conducted with the assistance of Jingqi Chen at TheUniversity of British Columbia. A similar measurement procedure can be found elsewhere[160].875.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.6: Distribution of stored energy in the as-deformed material calcu-lated with the Taylor-factor approach.Simulations of isothermal recrystallization were carried out for three tem-peratures (600 °C, 625 °C and 650 °C) to determine the adjustable parame-ters, i.e. the temperature-independent nucleation parameter Est∗, the nucleidensity NRex and the interface mobility mαα. As concluded in Section 5.2,the JMAK exponent n and the final grain size distribution are determined bya unique value of Est∗, and the interface mobility mαα only affects the timescale of the recrystallization kinetics. Thus the fitting procedure is as follows.First, the nucleation parameter Est∗ was adjusted such that the JMAK expo-nents calculated from simulation results and experimental data match eachother. Meanwhile, the nuclei density NRex was adjusted together with Est∗ tokeep the average grain size consistent with the experimental value (7.7± 0.5885.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.7: Cold-rolled ferrite/pearlite microstructure with 15% pearlite con-structed using Voronoi tessellation (black: pearlite; gray: deformed ferrite).µm [4]). Once these two nucleation parameters were determined, suitablevalues of the interface mobility were chosen to match the measured recrys-tallization kinetics for the three temperatures, based on which the pre-factorand activation energy are calculated.Fig. 5.8 and Fig. 5.9 compare the recrystallization kinetics and grainsize distributions that are experimentally measured with that obtained inthe simulation. The fitted values of the adjustable parameters are listed inTable 5.2. It is noted that the simulated grain size distribution is identicalfor various temperatures because nucleation parameters are assumed to betemperature-independent. As shown in Fig. 5.8, the simulated recrystallizedferrite fractions are consistent with the experimental measurements, exceptone data point at 650 °C. The grain size distribution obtained in the sim-895.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.8: Comparison of simulated recrystallization kinetics with experi-mental data [4] for different isothermal tests.Table 5.2: Values of adjustable parameters in the 3D recrystallization model.Nuclei density NRex (1/μm3) 1.1× 10−3Critical stored energy for nucleation Est∗ (J/m3) 2.1Ö106MobilityPre-factor (m4/(J·s)) 2.4Ö105Activation energy (kJ/mol) 325ulation has an average of 7.8 μm and a standard deviation of 3.6 μm, alsoin good agreement with the experimental measurements (7.7±0.5 μm and3.4±0.3 μm). The final grain sizes basically follow a log-normal distribution,with the maximum grain size more than twice the average. The activationenergy of the effective mobility (325 kJ/mol) is much higher than that of theself-diffusion in pure iron, i.e. 250 kJ/mol [161], which may be attributableto a solute drag effect of the alloying elements, e.g. manganese.The 2D cuts of the microstructure in the simulation are compared with905.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.9: Grain size distribution after full recrystallization in 3D simula-tions (194 grains) and experiments (500 grains) [148] for isothermal tests (thesimulated grain size distribution is the same for various temperatures).the experimental micrographs (Fig. 5.10). The recrystallization nucleation isnon-random but clustered at the early stage (20% recrystallized). At a laterstage, pearlite bands can constrain recrystallizing fronts to grow only in thedirection parallel to the bands after they impinge the bands (Fig. 5.10(b)).Because the PFM employs a diffuse interface, there are some artifacts inthe 2D cuts, e.g. extended grain boundaries are displayed when the grainboundaries are approximately parallel to the 2D-cut plane.The 3D model is applied to the thermal condition of continuous heating at1 °C/s. The simulated recrystallization kinetics fits the experimental resultwell such that all simulation data are within the experimental measurementerrors (Fig. 5.11). Moreover, the mean (7.8 μm) and standard deviation(3.6 μm) of the simulated ferrite grain size distribution (Fig. 5.12) are both915.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureconsistent with the experimental results (8.0 μm and 3.7 μm). 3D views ofthe simulated microstructures at different stages are shown in Fig. 5.13.Figure 5.10: Comparisons of 3D simulated recrystallization microstructures(black: pearlite; gray: deformed ferrite; white: recrystallized ferrite) withexperimental micrographs [148] for isothermal tests at 600 °C (recrystallizedgrains are outlined in (a) and non-recrystallized grains are outlined in (c)):(a-b) 20% recrystallized; (c-d) 60% recrystallized.925.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.11: Experimental [4] and simulated (3D) recrystallization kineticsfor continuous heating at 1 °C/s.Figure 5.12: Comparison of 3D-simulated grain size distribution with exper-imental measurements (720 °C).935.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFigure 5.13: Simulated 3D microstructures for continuous heating at 1 °C/s(black: pearlite; gray: deformed ferrite; white: recrystallized ferrite): (a) 670°C; (b) 680 °C; (c) 690 °C; (d) 700 °C; (e) 710 °C; (f) 720 °C.945.3. Application to the Cold-rolled Ferrite/Pearlite Microstructure5.3.3 2D SimulationStrictly speaking, 3D experimental data are required to construct the initialmicrostructures and to calibrate the model parameters in the 3D simulations,though 2D experimental data were used in Section 5.3.2. The computationalcost of 3D simulation is high particularly for diffusional phase transformationsuch as austenite formation. Therefore, most mesoscale simulations in theliterature were performed in 2D, though 2D simulations are less physical than3D simulations. A 2D simulation describes microstructure evolution on a 2Dplane, whereas the realistic microstructure evolution occurs in 3D.A few studies comparing 2D and 3D phase-field models were reported inthe literature. Toloui et al. replicated the experimentally measured graingrowth kinetics (average grain size) in both 2D and 3D phase-field simu-lations where the fitted grain boundary mobility in 3D is 70% of that in2D [162, 163]. Militzer et al. compared the austenite-to-ferrite transfor-mation kinetics and microstructural morphology obtained from 2D and 3Dphase-field simulations for a low-carbon steel [17]. The overall transforma-tion kinetics was similar for 2D and 3D simulations by using a lower interfacemobility in 2D, but the microstructural morphology was less realistic in 2Dsimulations. Rudnizki et al. replicated the experimental recrystallization ki-netics and recrystallized-grain size distributions for a low-carbon steel in both2D and 3D phase-field simulations, by using the same mobility but differentnucleation scenarios, i.e. site saturation in 3D and continuous nucleation in2D [120]. They replicated experimentally measured transformation kinet-ics and final ferrite grain size distributions in both 2D and 3D simulationsfor austenite-ferrite transformation during intercritical annealing of a low-955.3. Application to the Cold-rolled Ferrite/Pearlite Microstructurecarbon steel with the same nucleation conditions (site-saturated in pearlite)and interface mobilities [120]. Vaithyanathan et al. compared precipitatecoarsening in 2D and 3D phase-field simulations with the same thermody-namic and kinetic parameters [164]. Similar precipitate size distributions andthe linear relationship between the cube of average precipitate size and timewere described in both 2D and 3D simulations, but the coarsening kineticsis faster in 3D simulations.These studies indicate that experimental results can be replicated in 2Dsimulations by choosing appropriate nucleation scenarios and interface mo-bilities. But to some extent, the fitted nucleation scenarios and interfacemobilities in 2D simulations are not physically meaningful. For example, toobtain a JMAK exponent of 3 for recrystallization in a deformed microstruc-ture with a uniform stored energy and site-saturated nucleation, an unreal-istic nucleation mode, i.e. continuous nucleation with a constant nucleationrate, has to be used in a 2D simulation. Further, the growth geometries ofgrains are intrinsically different in 2D and 3D simulations. 2D growth canbe regarded as cylindrical growth from the perspective of 3D, whereas 3Dgrowth is spherical in a first approximation. Due to the difference in growthgeometries, different mobilities have to be used in 2D and 3D simulationsto match the same experimental data. Therefore, the fitted nucleation pa-rameters and interface mobilities in 2D simulations have to be considered aseffective parameters that may not provide proper insight into the underlyingphysics.Nevertheless, a 2D phase-field model was developed for ferrite recrystal-lization in the present section and was coupled to the 2D models for austen-965.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureite formation and decomposition in the following chapters. A domain sizeof 100Ö100 μm2 was used with grid spacing ∆x of 0.1 μm (Fig. 5.14). Thesame model settings as in 3D simulations were used, e.g. the stored-energydistribution, the nucleation model and the grain boundary energy. The sameprocedure for parameter fitting was used as in 3D simulations.Figure 5.14: Cold-rolled ferrite/pearlite microstructure with 15% pearliteconstructed with Voronoi tessellation (black: pearlite; gray: deformed fer-rite).Four simulations with different values of the nucleation parameter Est∗were carried out, i.e. 1.0Ö106, 1.4Ö106, 1.7Ö106 and 2.0Ö106 J/m3. Basedon the study in Section 5.2, it is known that the nucleation parameter Est∗affects both the recrystallization kinetics and the final grain size distribution.As shown in Fig. 5.15(a)-(b), the value of Est∗ leading to the best fit ofrecrystallization kinetics (1.0Ö106 J/m3) does not lead to a good fit of thegrain size distribution. By making a compromise between the best fits ofrecrystallization kinetics and grain size distribution, the value of Est∗ was975.3. Application to the Cold-rolled Ferrite/Pearlite Microstructurechosen to be 1.7Ö106 J/m3. The fitted values of the other model parametersare shown in Table 5.3.Figure 5.15: Comparisons of experimental data [4] with 2D simulation resultsfor various Est∗ values (the simulated grain size distribution is the same forvarious temperatures): (a-b) 1.0Ö106 J/m3; (c-d) 1.7Ö106 J/m3.The simulated microstructures at intermediate stages are shown in Fig.5.16. It can be seen that at the early stage (Fig. 5.16(a)), the recrystallizedgrains in the simulation are smaller than those in the 3D simulation. Similarto the 3D simulation, the growth of some recrystallized grains is constrained985.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureTable 5.3: Values of adjustable parameters in the 2D recrystallization model.Nuclei density NRex (1/μm2) 1.7× 10−2Critical stored energy for nucleation Est∗ (J/m3) 1.7× 106Mobility (m4/(J·s))Pre-factor (m4/(J·s)) 1.2× 105Activation energy (kJ/mol) 325by pearlite bands at a later stage (Fig. 5.16(b)).Figure 5.16: 2D-simulated recrystallization microstructures for isothermaltests at 600 °C: (a) 20% recrystallized; (b) 60% recrystallized (black: pearlite;gray: deformed ferrite; white: recrystallized ferrite).An additional simulation with a domain size of 200Ö200 μm2 was per-formed with the estimated model parameters at 650 °C. There is negligibledifference between the simulated recrystallization kinetics for the domain sizeof 200Ö200 μm2 and that for the domain size of 100Ö100 μm2. Thus, a do-main size of 100Ö100 μm2 was sufficient to obtain statistically meaningfulresults.Simulations for continuous heating at rates of 1, 10 and 100 °C/s were995.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureperformed. The same mobility and nucleation scenario as determined fromthe isothermal tests are adopted in the simulation. The simulation resultsfor the heating rate of 1 °C/s are compared with experimental data. Thesimulated recrystallization kinetics matches the experimental data within themeasurement uncertainty (Fig. 5.17). Both the experimental and simulatedresults show that the recrystallized ferrite fraction reaches a measurable level(about 10%) at approximately 670 °C and is finished around 720 °C belowthe start temperature of austenite formation (730 °C [4]). Fig. 5.18 showsthe ferrite grain size distribution after full recrystallization at 720 °C. Theaverage grain size after complete recrystallization (7.6 μm) is consistent withthe experimental measurements (8.0 μm). The standard deviation of the finalgrain size distribution in the simulation is 3.1 μm, a little smaller than thatin the experiments (3.7 μm).Figure 5.17: Experimental [4] and 2D-simulated recrystallization kinetics forcontinuous heating at 1, 10 and 100 °C/s.1005.3. Application to the Cold-rolled Ferrite/Pearlite MicrostructureFerrite recrystallization is finished at 780 °C for the heating rate of 10 °C/sand 850 °C for the heating rate of 100 °C/s. Therefore, ferrite recrystallizationtakes place concurrently with austenite formation for the two heating rates.Figure 5.18: Comparison of 2D-simulated grain size distribution with exper-imental measurements (720 °C).5.3.4 Summary3D simulations provide a realistic description of recrystallization in spiteof the high computational cost, duplicating the recrystallization kinetics,microstructures at intermediate stages and ferrite grain size distributionsafter complete recrystallization in the studied material.2D simulations as a means to reduce the computational cost, also providea consistent description of ferrite recrystallization in the studied material by1015.3. Application to the Cold-rolled Ferrite/Pearlite Microstructureselecting appropriate model parameters. The fitted value of the nucleationparameter is, however, different from those obtained in the 3D simulation,implying that different nucleus distributions are used to fit the recrystalliza-tion kinetics and grain size distributions. Nucleation takes place in 60% ofgrains (area fraction) in 2D simulations and in 10% of grains (volume frac-tion) in 3D simulations. That is, less clustered nucleation is required to fitthe recrystallization kinetics with a JMAK exponent of 1.7 in 2D simulationsthan that in 3D simulations, due to the growth dimensionality. Thus, thefitted parameters in 2D simulations must be interpreted as “effective” valuesto translate the actual 3D recrystallization process into 2D simulations. Inparticular, the fitted nucleation scenarios and interface mobility cannot beused as reliable references for other problems. They are intended only for thepresent 2D phase-field model to consistently describe the ferrite recrystalliza-tion in the investigated steel for various processing scenarios. The fully orpartially recrystallized microstructures predicted by the 2D phase-field modelwill be used as the initial microstructures for austenite formation.102Chapter 6Modeling Austenite Formation36.1 IntroductionThe ferrite-to-austenite transformation has not been investigated as widely asthe austenite-to-ferrite transformation with phase-field approaches. Thus, inorder to have a straightforward understanding of the characteristics of austen-ite formation and the differences from the austenite-to-ferrite transformation,a series of 1D simulations without considering austenite nucleation were car-ried out. First, a convergence analysis for the selected values of grid spacing,interface thickness and time step in Chapter 5 is implemented for the diffu-sional austenite/ferrite phase transformation. The effect of the magnitude ofmobility on austenite formation kinetics is investigated, too. Moreover, theeffect of Gibbs-energy dissipation by the trans-interface diffusion of Mn oninterface migration is investigated.A 2D phase-field model considering both nucleation and growth of austen-ite is then developed and integrated with the ferrite recrystallization model.Five heating scenarios are used to study austenite formation in the investi-gated steel (Fig. 6.1). They are categorized into two groups: step heating3A journal paper based on this chapter was published: B. Zhu, M. Militzer, “Phase FieldModeling for Intercritical Annealing of a Dual-phase Steel”, Metallurgical and MaterialsTransaction A, vol. 46, pp 1073-1084, Mar 2015.1036.2. 1D SimulationFigure 6.1: Heating scenarios used in the simulations of austenite formation.(A-C) and continuous heating (D-E). A slow heating rate (1 °C/s) below 720°C is used in the step-heating group to ensure completion of ferrite recrys-tallization before austenite formation starting at 730 °C, for the purpose ofinvestigating the effects of heating rates on austenite formation in a well-annealed microstructure. Continuous-heating simulations were carried outto study the interaction of concurrent ferrite recrystallization with austeniteformation.6.2 1D SimulationThe simulations were carried out under an isothermal condition at 770 °C .The domain length is 14 μm with periodic boundary conditions (Fig. 6.2), i.e.equal to the spacing of pearlite bands in the studied steel. It is assumed thatthe initial microstructure consists of 14% austenite with a uniform carbon1046.2. 1D SimulationFigure 6.2: Schematic of the 1D simulation domain (dark: austenite; light:ferrite).concentration of 0.68 wt% and 86% ferrite with a uniform carbon concentra-tion of 0.012 wt% .The convergence analysis was implemented first. In Chapter 5, it wasfound that grid spacing of 0.1 μm and a time-step factor of ξ = ∆t/∆tc = 1/4(Eq. 4.18) lead to converged results for simulations of recrystallization. Butfor simulations of diffusional phase transformation, e.g. austenite formation,it is necessary to test whether the numerical solutions of the carbon diffusionequation are converged with the selected values of numerical parameters. Thenumber of grid points through the diffuse interface nη was set to 5. A smallertime-step factor ξ = 1/16 and smaller grid spacing ∆x = 0.02 μm were usedin simulations for comparison. The α/γ interface energy and mobility wereset to be 0.4 J/m2 and 1.0× 10−12 m4/(J·s), respectively. Fig. 6.3 shows therepeated transformation kinetics for four combinations of ∆x and ξ, provingthat the selected values of ∆x = 0.1 μm and ξ = 1/4 lead to converged resultsin simulations of austenite formation, and thus were used in the following 1Dand 2D simulations. Moreover, the convergence analysis was implementedwith the other values of the interface mobility, i.e. 1.0×10−13 and 1.0×10−11m4/(J·s) and the selected values of ∆x = 0.1 μm and ξ = 1/4 also lead toconverged results.1056.2. 1D SimulationFigure 6.3: Influence of numerical parameters on simulation accuracy.Simulations with three values of the α/γ mobility, i.e. 1.0 × 10−13,1.0 × 10−12 and 1.0 × 10−11 m4/(J·s) were performed without consideringtrans-interface diffusion of Mn and Gibbs-energy dissipation. The simula-tion results (Fig. 6.4) demonstrate that the transformation kinetics becomesgradually independent of the α/γ interface mobility as it increases beyondthe magnitude of 1.0 × 10−12 m4/(J·s). In all cases, the austenite fractionapproaches the para-equilibrium fraction (about 0.64).Fig. 6.5 shows the carbon concentration profiles at various stages. Theaustenite with a high carbon concentration tends to grow into surroundingferrite as carbon atoms diffuse to the α/γ interface, which is different from theaustenite-to-ferrite transformation where carbon atoms diffuse from the α/γinterface to the austenite interior. As austenite grows, the average carbonconcentration becomes lower and the concentration gradient also gets smaller,leading to slower transformation rates.1066.2. 1D SimulationFigure 6.4: Influence of the ferrite/austenite mobility on austenite formationkinetics.Figure 6.5: Carbon-concentration profile in the 1D simulation (mobility is1.0× 10−13 m4/(J·s)).Simulations considering the Gibbs-energy dissipation by trans-interfacediffusion of Mn were carried out with various values of the binding energy1076.2. 1D SimulationE0, i.e. 0, 8 and 10 kJ/mol, and the simulation results are compared withthe austenite formation kinetics for the simulation without consideration ofGibbs-energy dissipation in Fig. 6.6.Figure 6.6: Austenite formation kinetics for simulations with and withoutGibbs-energy dissipation (mαγ = 1.0 × 10−13 m4/(J·s), DMnint = 2.0 × 10−17m2/s).A mobility of 1.0 × 10−12 m4/(J·s) was used in all simulations. Whenno Gibbs-energy dissipation is taken into account, i.e. no trans-interface dif-fusion of Mn, the final austenite fraction approaches the para-equilibriumfraction (0.64). On the other hand, when Gibbs-energy dissipation is con-sidered, regardless of the magnitude of binding energy, the austenite fractionapproaches the negligible-partitioning local equilibrium fraction (0.37). Agreater binding energy only reduces the transformation rate at the interme-diate stage of phase transformation.1086.3. 2D Simulation of Step Heating6.3 2D Simulation of Step Heating2D Simulations of step heating scenarios were carried out first to study theeffect of heating rate on austenite formation in a well-annealed microstruc-ture where ferrite recrystallization is complete before the onset of austeniteformation. A simulation domain with the same size used for ferrite recrys-tallization (Section 5.3.3), i.e. 100 × 100 μm2, was used with grid spacingof 0.1 μm. The fully recrystallized microstructure at 720 °C is the initialmicrostructure for austenite formation in the case of step heating (Fig. 6.7).Figure 6.7: Fully recrystallized microstructure at 720 °C obtained in a 2Dsimulation (black: pearlite; white: recrystallized ferrite; blue: interface).A list of the parameters in the austenite formation model is given inTable 6.1, with five unknown parameters to be determined based on experi-mental data. The fitting procedure of their values are as follows. First, it is1096.3. 2D Simulation of Step Heatingreasonable to assume that austenite grain growth is negligible at a heatingrate of 100 °C/s in Case C. Thus, the total nuclei density can be estimatedwith the average austenite grain size after full austenitization in Case C.Given the measured nuclei density of austenite in pearlite regions, the nucle-ation parameter λ (Eq. 4.21) that affects the nuclei density at ferrite grainboundaries was, therefore, adjusted such that the average austenite grain sizeafter full austenitization is consistent with the experimental measurement inCase C. With the calibrated nucleation scenario, the pre-factor of the mo-bility of austenite grain boundaries was fitted with the average austenitegrain sizes after full austenitization in Case A and B where austenite graingrowth is significant [4]. The pearlite/austenite interface mobility (both thepre-factor and activation energy) and the pre-factor of the ferrite/austeniteinterface mobility were determined with the austenite formation kinetics forthe three cases A, B and C. The former was adjusted such that the austeniteformation kinetics obtained in the simulations matches the experimentallymeasured transformation kinetics at the early stage of austenite formation(austenite fraction below 0.2) when the pearlite-to-austenite transformationis dominant. The latter was adjusted such that the austenite formation ki-netics obtained in the simulations is in agreement with the experimentallymeasured transformation kinetics at the later stage of austenite formation(austenite fraction above 0.2) when the ferrite-to-austenite transformationtakes place.Consistency of the simulated transformation kinetics and microstructureswith the experimental measurements were obtained as illustrated in Fig. 6.8,Fig. 6.9 and Fig. 6.10. In particular, the simulated microstructures at the1106.3. 2D Simulation of Step HeatingTable 6.1: Parameters in the austenite-formation model.Parameter Symbol ValueParameter related to the atom jumpfrequency in the classical nucleationmodel (s-1)λ Fitting parameterPearlite/austenite interface energy (J/m2) σPγ 0.9 [119]Ferrite/austenite interface energy (J/m2) σαγ 0.4 [17]Austenite grain boundary energy (J/m2) σγγ 0.7 [14]Ferrite grain boundary energy (J/m2) σαα 0.76 [58]Pre-factor of pearlite/austenite interfacemobility (m4/(J·s))m0Pγ Fitting parameterActivation energy of pearlite/austeniteinterface mobility (kJ/mol)QmPγ Fitting parameterPre-factor of ferrite/austenite interfacemobility (m4/(J·s))m0αγ Fitting parameterActivation energy of ferrite/austeniteinterface mobility (kJ/mol)Qmαγ 140 [17]Pre-factor of austenite grain boundarymobility (m4/(J·s))m0γγ Fitting parameterActivation energy of austenite grainboundary mobility (kJ/mol)Qmγγ 185 [14]Pre-factor of ferrite grain boundarymobility (m4/(J·s))m0αα 1.2× 105 (Table 5.3)Activation energy of ferrite grainboundary mobility (kJ/mol)Qmαα 325 (Table 5.3)Binding energy of Mn to theferrite/austenite interface (kJ/mol)E0 10 [92]Trans-interface diffusivity of Mn acrossthe ferrite/austenite interface (m2/s)DMnint Eq. 4.36Physical interface thickness in theGibbs-energy dissipation model (nm)2Λ 11116.3. 2D Simulation of Step HeatingFigure 6.8: Comparison of the experimentally measured austenite formationkinetics [4] with simulation results for step heating.intermediate stages of austenite formation are consistent with the metallo-graphic observations in terms of the austenite morphology (Fig. 6.9). Anetwork of austenite grains is formed and decorates the ferrite grain bound-aries in Case A with a heating rate of 1°C/s (Fig. 6.9(a-b)). In contrast,banded austenite morphology was obtained in Case C with a heating rateof 100 °C/s (Fig. 6.9(c-d)). The agreement of austenite morphology withexperiments in Case A, i.e. complete occupation of ferrite grain boundariesby austenite grains, validates the postulated nucleation model that is cali-brated using the experimental data of Case C. Site saturation of austenitenucleation at ferrite grain boundaries in Case A and rare nucleation in CaseC indicate a comparatively weak temperature dependence of austenite nucle-ation, in contrast to the experimental observation in the austenite-to-ferrite1126.3. 2D Simulation of Step HeatingFigure 6.9: Comparison of micrographs [148] with simulated microstructuresin Case A at 790 °C (a-b) and C at 825 °C (c-d) (A: austenite; F: ferrite; M:martensite).transformation during continuous cooling that fast cooling usually stimulatesmore nucleation than slow cooling.Values of the five adjustable parameters are listed in Table 6.2. The fittedactivation energy of the pearlite/austenite interface mobility is comparativelyhigh, reflecting a significant increase of the pearlite-to-austenite transforma-tion rate with temperature which is consistent with the findings of Speich etal. [97].The simulations reveal that Gibbs-energy dissipation by trans-interface1136.3. 2D Simulation of Step HeatingFigure 6.10: Comparison of the micrograph (left) [148] with the simulatedmicrostructure (right) after full austenitization at 870 °C in case A.diffusion has a great effect on the transformation kinetics for slow and mediumheating rates (1-10 °C/s) but not for the fast heating rate (100 °C/s). For thefast heating rate, the driving pressure is sufficiently high such that interfacemigration occurs in the high velocity regime with negligible Gibbs-energydissipation. The good description of transformation kinetics also indicatesthat the values of binding energy and trans-interface diffusivity of Mn usedin the model are reasonable. It is important to note that neglecting Gibbs-energy dissipation by trans-interface diffusion would have led to an interfacemobility with an unrealistic dependence on heating rate to replicate the ex-perimental austenite formation kinetics. Similarly, an α/γ interface mobilitythat depends on cooling rate was reported elsewhere for phase-field model-ing of continuous cooling austenite-to-ferrite transformations in low-carbonsteels [137].Further, the simulations demonstrate that austenite grain growth is sig-1146.3. 2D Simulation of Step HeatingTable 6.2: Fitted values of the adjustable parameters in the austenite-formation model.Parameter Symbol ValueParameter related to the atom jumpfrequency in the nucleation model (s-1)λ 2× 107Pre-factor of pearlite/austenite interfacemobility (m4/(J·s))m0Pγ 2× 107Activation energy of pearlite/austeniteinterface mobility (kJ/mol)QmPγ 398Pre-factor of ferrite/austenite interfacemobility (m4/(J·s))m0αγ 5× 10−6Pre-factor of austenite grain boundarymobility (m4/(J·s))m0γγ 5× 10−3nificant at lower heating rates. For example at 1 °C/s, the total number ofaustenite nuclei introduced into the domain is about 900, but only 150 grainsremain when full austenitization is attained. In particular, austenite grainsnucleated at pearlite regions coarsen predominantly to sweep those formedat ferrite grain boundaries. A comparison of the measured and simulatedfinal austenitic structures is shown in Fig. 6.10.An additional simulation with a domain of 200× 200 μm2 was performedfor Case A. There is negligible difference in austenite formation kinetics be-tween the present simulation and the previous simulation with a domain of100× 100 μm2, indicating that the employed domain size (100× 100 μm2) isconsidered to be sufficiently large such that the simulation results are statis-tically meaningful and representative.1156.4. 2D Simulation of Rapid Continuous Heating6.4 2D Simulation of Rapid ContinuousHeatingDuring rapid continuous heating scenarios, austenite formation may notstart from a well-annealed microstructure such that ferrite recrystallizationmay be incomplete and the pearlite morphology may be different [4]. Thespheroidization of pearlite is evident during step heating but not observableduring rapid continuous-heating, e.g. at a rate of 100 °C/s [4]. It was fur-ther observed that the pearlite-dissolution temperature decreases from 780°C in Case B to 760 °C in Case D, and from 800 °C in Case C to 760 °Cin Case E [4]. One cause of the faster transformation of pearlite to austen-ite during rapid continuous heating may thus be related to a lower degreeof spheroidization and the higher level of interface energy remaining in thepearlite colonies.In the present model, the pre-factor of the pearlite/austenite interfacemobility m0Pγ was adjusted to fit the accelerated pearlite-to-austenite trans-formation in Case D and Case E (Fig. 6.11). The fitted value of m0Pγ was4×107 m4/(J·s) in Case D and 1×108 m4/(J·s) in Case E, in comparison withthe value of 2× 107 m4/(J·s) in Cases A-C. Therefore, the pearlite/austenitemobility is phenomenologically a function of the heating rate below the starttemperature of austenite formation. To simulate other heating rates, thedependence of the pre-factor m0Pγ on heating rate dT/dt is described phe-nomenologically by:m0Pγ = m0Pγ,1°C/s (dT/dt)0.35 (6.1)1166.4. 2D Simulation of Rapid Continuous Heatingwhere m0Pγ,1°C/s is the value of the mobility pre-factor for a heating rate of1 °C/s (Table 6.2). Eq. 6.1 was developed for constant heating rates. If theFigure 6.11: Comparisons of experimentally measured transformation kinet-ics [4] with simulation results in Case D and E.heating rate dT/dt is not constant, a representative average heating rate maybe used. Further, Eq. 6.1 is intended only for a heating rate ranging from 1°C/s to 100 °C/s. The applicability of Eq. 6.1 for the heating rates out of therange is undetermined. It was found that ferrite recrystallization completes at770 °C in Case D, overlapping a little with austenite formation, while ferriterecrystallization proceeds concurrently with austenite formation until 830 °Cin Case E. The contribution of stored energy in non-recrystallized ferrite tothe transformation kinetics was found to be negligible in Case E. The mainreason is that the chemical driving force for ferrite/austenite transformationin Case E is relatively high (15-30 MJ/m3) in comparison to the averagestored energy in the non-recrystallized ferrite grains (about 2 MJ/m3).1176.4. 2D Simulation of Rapid Continuous HeatingIn the transformation of deformed austenite to ferrite, carbon diffusion indeformed austenite may be enhanced due to the presence of dislocations andother defects and, as a result, may contribute to the acceleration of trans-formation [165]. Whether or not enhanced carbon diffusion in ferrite couldaccelerate austenite formation was tested with the model by increasing thecarbon diffusivity in ferrite by a factor of 100. No acceleration of transfor-mation rates was found, which is reasonable because ferrite has a low carbonsolubility and carbon diffusion in ferrite is already much faster (about 100times) than that in austenite. Therefore, similar to the austenite-to-ferritetransformation, carbon diffusion in ferrite is not the rate-limiting factor forferrite-to-austenite transformation.Fig. 6.12 and Fig. 6.13 show a comparison of simulated and experimentalFigure 6.12: Simulated microstructures at 750 °C in Case D (a) and E (b) (P:pearlite; A: austenite; NF: non-recrystallized ferrite; F: recrystallized ferrite).micrographs at intermediate stages for continuous heating of 10 °C/s and 1001186.4. 2D Simulation of Rapid Continuous Heating°C/s.Figure 6.13: Comparisons of micrographs at 790 °C (a,c) [148] with simulatedmicrostructures (b,d) in Case D and E (A: austenite; NF: non-recrystallizedferrite; F: recrystallized ferrite; M: martensite).It is evident that ferrite recrystallization and austenite formation occursimultaneously in both cases. The overlapping period in Case D (10 °C/s) isshort, and ferrite recrystallization has already completed around 760 °C. In1196.5. SummaryCase E (100 °C/s), ferrite recrystallization is delayed to higher temperaturessuch that there is still a great amount of non-recrystallized ferrite at 790°C. In spite of some austenite grains nucleating at ferrite grain boundaries,austenite mainly forms in prior pearlite area, inheriting the banded morphol-ogy of pearlite. From the simulated microstructure, it can be observed thataustenite grains can act as barriers to inhibit migration of recrystallizationfronts. Another interesting observation is that some austenite grains are lo-cated inside recrystallized ferrite grains due to migration of recrystallizationfronts.6.5 SummaryThe 1D simulations demonstrate that carbon atoms diffuse from the centerof austenite grains to the α/γ interface, distinct from the austenite-to-ferritetransformation where carbon atoms diffuse from the interface to the centerof austenite grains. The consideration of Gibbs-energy dissipation by trans-interface diffusion of Mn changes the thermodynamic equilibrium mode frompara-equilibrium to negligible-partitioning local equilibrium (NPLE), suchthat the equilibrium austenite fraction at a given temperature is less thanthat for para-equilibrium.A 2D phase-field model considering both austenite nucleation and growthhas been developed to simulate austenite formation and its interaction withferrite recrystallization during heating of a cold-rolled ferrite-pearlite steel.A Gibbs-energy dissipation model is coupled to the phase-field model toenable the description of the austenite formation kinetics using a mobil-1206.5. Summaryity independent of heating rates. The interaction of ferrite recrystallizationand austenite formation is investigated, revealing that ferrite recrystalliza-tion is inhibited by the particle-pinning effect of austenite grains. Moreover,concurrent ferrite recrystallization makes some boundary-nucleated austenitegrains become intragranular in the final recrystallized microstructure. Thesimulations duplicate changes in microstructure morphology from a networkstructure to a banded structure with increasing heating rates.121Chapter 7Modeling IntercriticalAnnealing47.1 IntroductionIn Chapter 6, the phase-field model for austenite formation was developedand calibrated with the step-heating and continuous-heating experiments.By taking the austenite-to-ferrite transformation into account, an integratedmodel for intercritical annealing was developed, consisting of sub-models forferrite recrystallization, austenite formation and austenite to ferrite trans-formation, with all adjustable model parameters tuned in Chapter 5 andChapter 6. Bainitic and martensitic transformations were not considered inthe model.In this chapter, intercritical annealing was investigated with the inte-grated phase-field model. An intercritical-annealing cycle usually consists ofthree stages: heating, holding and cooling. The effect of heating rate at theheating stage and annealing temperature at the holding stage on the austen-ite formation were studied with simulations of four heating scenarios, i.e.4A journal paper based on this chapter was published: B. Zhu, M. Militzer, “Phase FieldModeling for Intercritical Annealing of a Dual-phase Steel”, Metallurgical and MaterialsTransaction A, vol. 46, pp 1073-1084, Mar 2015.1227.1. Introductioncombinations of two heating rates (1 °C/s and 100 °C/s) and two annealingtemperatures (770 °C and 790 °C) (Fig. 7.1). Furthermore, the integratedphase-field model was applied to a simulated industrial annealing cycle forvalidation (Fig. 7.2). In all simulations, the domain size and grid spacingare the same, i.e. 100× 100 μm2 and 0.1 μm.Figure 7.1: Heating scenarios used in the simulations of intercritical anneal-ing.1237.2. Effects of Heating RateFigure 7.2: Schematic of an industrial intercritical-annealing cycle.7.2 Effects of Heating RateFig. 7.3 compares the simulated microstructures with micrographs in the caseof annealing at 770 °C after heating at 1 °C/s and 100 °C/s. Similar to thecontinuous heating scenario, ferrite recrystallization overlaps with austeniteformation for the heating rate of 100 °C/s. However, ferrite recrystallizationcompletes within the first 10 s holding at 770 °C. For the heating rate of1 °C/s, ferrite recrystallization completes before austenite formation. Thetotal number of austenite nuclei in the simulation domain is about 900 inboth cases. However, the locations of austenite nuclei are different. Forthe heating rate of 1 °C/s, austenite nuclei occupy prior pearlite regionsand the recrystallized ferrite grain boundaries completely, forming a networkmorphology which is similar to the step-heating scenario. For the heating rate1247.2. Effects of Heating Rateof 100 °C/s, austenite nucleation first occurs at the pearlite/ferrite interfacesand deformed ferrite grain boundaries, leading to both intergranular andintragranular distribution of austenite grains. Intragranular austenite grainsare arranged in arrays parallel to the rolling direction. These austenite grainsare formed originally at the deformed ferrite grain boundaries. Later, as therecrystallization fronts sweep away the old grain boundaries, these austenitegrains are left inside the recrystallized ferrite grains.Figure 7.3: Comparison of simulated microstructures with micrographs forintercritical annealing at 770 °C with heating rates of (a) 1 °C/s and (b) 100°C/s (A: austenite; F: recrystallized ferrite; M: martensite).Fig. 7.4 compares the experimentally measured austenite formation kinet-1257.2. Effects of Heating RateFigure 7.4: Austenite formation kinetics at 770 °C after heating at differentheating rates (Experimental data are provided by Kulakov [4]).ics with the simulation results, which demonstrates some discrepancies. Firstof all, the experimental austenite fraction evaluated by the lever-rule analysisof dilatation data has an uncertainty of ±0.05 [4]. The lever rule analysisunderestimates the austenite fraction at the early stage of austenite forma-tion, because it does not consider the increased thermal expansion coefficientof austenite induced by its high carbon concentration (0.6wt%) [3, 4]. Thediscrepancy between metallographic measurements and lever rule analysis ofdilatation data can be as much as 0.1, e.g. 0.18 versus 0.08 [4]. Therefore,the actual amount of austenite formed in the experiment is expected to belarger than that measured with the dilatometry at the early stage. Takingthis uncertainty into account, the actual discrepancy between the simulatedaustenite fraction and the experimental austenite fraction can be less thanthe present discrepancy, i.e. 0.05-0.08. In the case of heating at 100 °C/s1267.3. Effect of Temperatureto 770 °C, the simulated austenite fraction does not increase beyond 130s (0.37), whereas the experimental austenite fraction keeps increasing (e.g.0.48 at 400 s) which is probably due to the enhanced long-range diffusionof Mn in deformed ferrite (fast diffusion along dislocations). Speich dividedaustenite formation in Fe-C-Mn steels into three stages in terms of solutediffusion: carbon diffusion in austenite first, followed by manganese diffu-sion in ferrite and finally manganese diffusion in austenite [97]. Because thepresent phase-field model only considers the first stage, i.e. carbon diffusion,the austenite formation involved with significant manganese diffusion at laterstages cannot be described in the present simulations.Another discrepancy between experimental data and simulation results isthat the austenite formation kinetics in the simulations does not exhibit sucha great dependence on the heating rate as the experimental measurements.The large fraction of austenite formed in the case of heating at 100 °C/s maybe caused by enhanced substitutional diffusion at the beginning of austeniteformation when ferrite recrystallization is incomplete. The accuracy of tem-perature control is another possible cause. Heating quickly, e.g. 100 °C/s,tends to overshoot the desired temperature.7.3 Effect of TemperatureFig. 7.5 shows the simulated microstructures for intercritical annealing at 790°C after heating at 1 °C/s and 100 °C/s. In comparison with Fig. 7.3, one sig-nificant difference is that the austenite fraction is increased with temperature,as the equilibrium austenite fraction increases with temperature according1277.3. Effect of Temperatureto the phase diagram. Fig. 7.6 shows the austenite fraction changes from0.37 at 770 °C to 0.57 at 790 °C. Comparing with the microstructures in thecases of annealing at 770 °C, the austenite bands formed in the prior pearlitebands grow thicker at 790 °C. Moreover, austenite grains occupy completelyferrite grain boundaries even in the case of 100 °C/s (Fig. 7.5(b)), forming amore developed network morphology than that at 770 °C (Fig. 7.3(b)).Fig. 7.7 shows the nucleation kinetics for the heating rate of 100 °C/s.The final nuclei density is similar for both holding temperatures, i.e. about1000 nuclei in the simulation domain. But the nucleation is faster at 790 °Cbecause it is a thermally activated process.Figure 7.5: Simulated microstructures for intercritical annealing at 790 °Cafter heating at of (a) 1 °C/s and (b) 100 °C/s after 300 s (A: austenite; F:recrystallized ferrite).1287.3. Effect of TemperatureFigure 7.6: Simulated austenite formation kinetics during holding at 770 °Cand 790 °C after heating at 100 °C/s.Figure 7.7: Nucleation kinetics in the simulations of intercritical annealingat 770 °C and 790 °C after heating at 100 °C/s (the dot lines separate theheating and holding stages).1297.4. Simulated Industrial Annealing Cycle7.4 Simulated Industrial Annealing CycleA selected industrial intercritical annealing cycle was simulated using theintegrated phase-field model (Fig. 7.2). The simulation starts from 600 °Cand stops at 465 °C before the isothermal galvanizing process when bainiteformation takes place.The ferrite recrystallization kinetics is shown in Fig. 7.8. The deformedferrite fraction is 0.05 at the start temperature of austenite formation, i.e. 730°C. Ferrite recrystallization is finished at 735 °C. Thus the overlap betweenferrite recrystallization and austenite formation is negligible. As austenitegrows into ferrite, the recrystallized ferrite fraction starts decreasing above740 °C.Figure 7.8: Simulated ferrite-recrystallization kinetics in the industrial an-nealing cycle for the DP600 steel: full recrystallization at 735 °C.The predicted transformation kinetics and the microstructures are com-1307.4. Simulated Industrial Annealing Cyclepared with the experimental results in Fig. 7.9 and 7.10, respectively. SimilarFigure 7.9: Comparison of the experimentally measured transformation ki-netics [4] with simulation results for the simulated industrial annealing cycle.to slow heating scenarios, many austenite grains decorate recrystallized fer-rite grain boundaries. Before fast cooling, the austenite fraction has reached0.4 at 770 °C, consistent with the experimental measurement (0.45 ± 0.05).During fast cooling, epitaxial ferrite grows into austenite, leading to anaustenite fraction of 0.17 at 465 °C in the simulation that is consistent withthe experimental measurement. But in the experimental micrograph (Fig.7.10(c)), some austenite has already transformed into bainite during coolingwhich is not considered in the present phase-field model. A sharp gradient ofthe carbon concentration with a thickness about 1 µm is found in a narrowregion of austenite in front of the α/γ interface due to insufficient diffusionof carbon in austenite (Fig. 7.11). The micrograph (Fig. 7.12) shows that ashell of martensite with a thickness of 0.5-1 µm is formed and envelops thebainite in the regions of prior austenite, which is consistent with the thick-1317.4. Simulated Industrial Annealing CycleFigure 7.10: Comparison of experimental micrographs (a,c) [4] with sim-ulated microstructures (b,d) at 770 °C after holding (a, b) and at 465 °Cafter cooling (c, d) (A: austenite; B: bainite; F: recrystallized ferrite; M:martensite).ness of the carbon-enriched layer in the simulation. Therefore, it is concludedthat the experimentally observed morphology of the bainite/martensite ag-gregates is due to the carbon enrichment near the ferrite/austenite interfacesduring the austenite-to-ferrite transformation.1327.4. Simulated Industrial Annealing CycleFigure 7.11: Carbon concentration field (left) and a line plot (right) alongthe vertical white line at 465 °C obtained in the phase-field simulation.Figure 7.12: Micrograph of the investigated steel (0.1wt%C-1.86wt%Mn-0.34wt%Cr) after intercritical annealing at 770 °C and isothermal holding at465 °C (F: ferrite, B: bainite, M: martensite) [4].The developed phase-field model was further evaluated for the industrialannealing cycle (Fig. 7.2) with the annealing time scaled by a factor of 0.4and 0.1, respectively, such that the heating rate between 600 °C and 780°C was changed to 5 °C/s and 20 °C/s, the holding time was shortened to1337.4. Simulated Industrial Annealing Cycle36 s and 9 s, and the cooling rate was increased to 55 °C/s and 220 °C/s,to mimic faster line speeds used in the steel industry [166]. The fractions ofrecrystallized ferrite, non-recrystallized ferrite and austenite are shown in Fig.7.13. For the heating rate of 5 °C/s, ferrite recrystallization is finished whenaustenite fraction is 0.15 at 765 °C during heating. In the case of heating at 20°C/s, ferrite recrystallization is finished when austenite fraction is 0.29 after11 s (9 s heating+2 s holding). Thus, ferrite recrystallization overlaps withaustenite formation in the two cases. Compared with the annealing cyclewith a heating rate of 2.1 °C/s, the shorter holding time leads to a smalleraustenite fraction before cooling (0.35 and 0.31). In the case of heating at2.1 °C/s, a network of austenite is formed at ferrite grain boundaries duringholding (Fig. 7.10(b)), whereas in the present cases, austenite grains donot occupy ferrite grain boundaries completely (Fig. 7.14). Furthermore,some austenite grains are located inside ferrite grains which is due to theoverlapping of ferrite recrystallization and austenite formation, as discussedin Section 7.2. The sizes of austenite grains at ferrite grain boundaries arealso smaller than that in the case of heating at 2.1 °C/s (Fig. 7.10(b)).After cooling to 465 °C, the austenite fractions decrease to 0.20 and 0.24(Fig. 7.13), respectively. The austenite grains both at ferrite grain bound-aries and inside ferrite grains shrink to a size of 0.5-1 µm such that thefinal dual-phase structures have banded morphologies, in comparison withthe network morphology obtained for the slow heating rate of 2.1 °C/s asshown in Fig. 7.10.The phase-field simulations predict that faster heating and cooling ratesand shorter holding time obtained by increasing the line speed increase1347.4. Simulated Industrial Annealing Cyclethe intercritical austenite fraction when reaching the zinc bath temperature(465 °C) and change the austenite morphology in the employed intercritical-annealing cycles.Figure 7.13: Simulated phase fractions for an intercritical-annealing cyclewith two heating rates: (a) 5 °C/s; (b) 20 °C/s.1357.4. Simulated Industrial Annealing CycleFigure 7.14: Simulated microstructures at 770 °C and 465 °C for anintercritical-annealing cycle with two heating rates: (a-b) 5 °C/s; (c-d) 20°C/s.1367.5. Summary7.5 SummaryThe developed phase-field model for intercritical annealing was applied tovarious heating scenarios in the intercritical region. Similar to continuousheating, the heating rate also has a significant influence on the austenitemorphology during intercritical annealing that consists of heating, holdingand cooling stages. When ferrite recrystallization is completed before austen-ite formation for low heating rates, austenite grains are all located in the priorpearlite regions and recrystallized ferrite grain boundaries, forming a networkof the austenite phase. When ferrite recrystallization is delayed and overlapswith austenite formation for high heating rates, some austenite grains arelocated in the interior of recrystallized ferrite grains in the final microstruc-ture.The phase-field model replicates the transformation kinetics and microstruc-ture morphologies reasonably for a simulated industrial annealing cycle, ex-cept that bainite formation is not taken into account in the model. The sharpgradient of carbon concentration in austenite near the α/γ interfaces explainsthe observed martensite shells around bainite in the final microstructure [4].137Chapter 8Modeling Cyclic PhaseTransformation58.1 IntroductionIn the present study, cyclic phase transformation in low-carbon steels refersto alternation of the austenite-to-ferrite and ferrite-to-austenite phase trans-formations during a cyclic heat treatment in the intercritical region (Fig.2.20). The stagnant stages (Section 2.5.7) existing during cyclic phase trans-formations for some alloyed steels are not only of great interest to funda-mental research but also provide the steel industry with insight to designinginnovative intercritical treatments. Thus the phase-field model was used toinvestigate the underlying physics of stagnant stages.In this chapter, the developed phase-field model coupled with the Gibbs-energy dissipation model as described in Section 4.6 was used to investigatecyclic phase transformations in two steels, i.e. a plain-carbon steel (0.1wt%C)and a low-carbon Mn-alloyed steel (0.1wt%C-0.5wt%Mn). Nucleation was5A version of this chapter was published in a journal: B. Zhu, H. Chen, M. Militzer,“Phase-Field Modeling of Cyclic Phase Transformations in Low-carbon Steels, Computa-tional Materials Science, In Press, 2015”. The experimental data used in this chapter iscited from the experimental work by Chen and Van der Zwaag [92].1388.2. Fe-C Alloyneglected in the simulations, because austenite or ferrite nucleation was notobserved during cyclic phase transformations in the investigated steels [92].2D simulations using micrographs as initial microstructures were performedand compared with experimental data.8.2 Fe-C AlloyThe plain-carbon steel (0.1wt%C) was used as a benchmark case to test thephase-field model. Two thermal paths, i.e. H-type and V-type, were used forthe cyclic phase transformations (Fig. 2.20). In both the H-type and V-typeexperiments by Chen et al. [92], the material was first fully austenitized at1000 °C and then cooled down to a temperature in the intercritical region(Ta as marked in Fig. 2.20) for 20 minutes to create a mixed ferrite-austenitemicrostructure. The temperature was then cycled between Ta and Tb in theintercritical region, with isothermal holding (H-type) or without isothermalholding (V-type) at Ta and Tb. For the H-type thermal path, Ta and Tb(Fig. 2.20(a)), are 830 °C and 860 °C, respectively [92]. For the V-typethermal path, Ta and Tb are 815 °C and 855 °C, respectively [92].The phase-field simulations start from point A as marked in Fig. 2.20. Aregion of 400× 400 μm2 was selected from the micrograph corresponding topoint A (Fig. 8.1) to provide the initial microstructure in each simulation.Ferrite and martensite (prior austenite) were distinguished by image contrastusing MATLAB®. All grain boundaries were neglected and the single-phase-field equation (Eq. 4.9) was used with φ = 1 representing austenite and φ = 0representing ferrite. The α/γ interface energy and mobility are the same as1398.2. Fe-C AlloyFigure 8.1: Micrographs of the Fe-0.1wt%C alloy used as the initial mi-crostructures in 2D simulations (light: ferrite; dark: martensite): (a) H-type;(b) V-type [167].that for the DP600 steel, i.e. 0.4 J/m2 [17] and 5 × 10−6 exp (140000/RT )m4/(J·s).The simulated transformation kinetics is compared with the transforma-tion kinetics calculated with dilatation data by Chen [92] (Fig. 8.2). Theequilibrium ferrite fraction as a function of temperature is also plotted as areference. First, during isothermal holding at 860 °C for the H-type path(Fig. 8.2(a)), the experimentally measured austenite fraction (0.83) is higherthan the equilibrium fraction (0.78), which is contradictory to the fact thatthe austenite formed during ferrite to austenite transformation should notexceed the equilibrium fraction at any temperature. Similarly, during heat-ing from 815 °C to 855 °C for the V-type heat treatment, the experimentallymeasured austenite fraction is larger than the equilibrium fraction. Suchdiscrepancies between the experimental data and the thermodynamic datamay be related to inaccuracy of either experimental measurements or the1408.2. Fe-C Alloyconversion of dilatation data to phase fractions using the lever rule.Figure 8.2: Comparison of the experimentally measured phase transforma-tion kinetics [167] with the simulation results for the Fe-0.1wt%C alloy: (a)H-type; (b) V-type.One of the experimental inaccuracies is the uncertainty of temperaturemeasurements and temperature gradients (±5 °C [92]). As shown in Fig.8.2, if the actual temperature in the specimens was higher than the mea-1418.2. Fe-C Alloysured value, the discrepancy between experimental data and thermodynamicequilibrium could be rationalized. For example, a temperature shift of 2 °Cin the H-type case will lead to the experimentally measured fractions afterholding at 860 °C equal to the thermodynamic equilibrium; a shift of 5 °Cin the V-type case will bring the experimentally measured ferrite fractionsinto a reasonable relationship with the thermodynamic equilibrium, i.e. fer-rite fractions during austenite formation at the heating stage are larger thanthe equilibrium fractions. According to the equilibrium lines in Fig. 8.2,the equilibrium phase fractions are more sensitive to temperature at highertemperatures. For example, an increase of 5 °C in temperature near 860 °Cleads to a decrease of 0.1 in the equilibrium ferrite fraction.Regardless of the inaccuracy of the experimental data, the simulated fer-rite fractions agree well with the experimental data at lower temperatures,i.e. below 850 °C in the H-type case (Fig. 8.2(a)). But there are some dis-crepancies (0.06-0.1) at higher temperatures, i.e. above 850 °C, where largerfractions of ferrite are obtained in the simulations. In the V-type case (Fig.8.2(b)), the simulated ferrite fractions are larger than the experimental datain the entire cycle. For example, the simulated ferrite fraction at 855 °Cis 0.40 in comparison with the experimental measurement of 0.24. Further-more, it was found that any further increase of the α/γ interface mobilityhas no effect on transformation rate in the present simulations, i.e. the phasetransformation is diffusion-controlled.Taking the inaccuracy of temperature measurements into account, sim-ulations with an increase of 2 °C and 5 °C in the overall temperature werecarried out for the H-type and V-type cases, respectively. The temperature1428.2. Fe-C Alloyrange [Ta, Tb] was changed to [832 °C, 862 °C] for the H-type case and[820 °C, 860 °C] for the V-type case. As shown in Fig. 8.3, the simulatedtransformation kinetics is in better agreement with the experimental data inboth cases. The simulation results indicate that the assumption of a carbondiffusion-controlled phase transformation is appropriate. The ferrite fractionkeeps varying during both heating and cooling in the H-type case, i.e. nostagnant stage. In the V-type case, upon cooling from 860 °C, austenitedoes not transform back to ferrite immediately. Instead, ferrite to austenitetransformation continues until cooling to 858 °C. The austenite formationduring cooling is due to the non-equilibrium condition in the material suchthat there is still a driving pressure for austenite formation. Below 858 °C,the inverse process, i.e. the austenite-to-ferrite transformation, takes place.Although the ferrite fractions at temperatures between 855 °C and 860 °Cchange insignificantly, it is not a stagnant stage. Based on the present model,there is no Gibbs-energy dissipation in the plain-carbon steel. Thus the ef-fective driving pressure for interface migration does not remain zero for anytemperature range during cooling. The heating stage has a similar situation.Therefore, there is no stagnant stage in either the H-type or the V-type case.The simulated microstructures at selected stages of the H-type heat treat-ment are shown in Fig. 8.4. The back-and-forth motion of α/γ interfaceshas been described in the phase-field simulation. At the heating stage, theisolated austenite grains grow into ferrite and some of them impinge eachother. At the cooling stage, the α/γ interfaces move in the reverse directionand austenite transforms back to ferrite.1438.2. Fe-C AlloyFigure 8.3: Comparison of the experimentally measured phase transforma-tion kinetics [167] with the simulation results for the Fe-0.1wt%C alloy withshifted temperatures (2 and 5 °C respectively) in the simulations for (a)H-type; (b) V-type.1448.3. Fe-C-Mn AlloyFigure 8.4: Microstructures (light: ferrite; dark: austenite) at various stagesof the H-type cyclic phase transformation for the Fe-0.1wt%C alloy (graphsa-d correspond to points A-D in Fig. 2.20) .8.3 Fe-C-Mn AlloyBoth the H-type and V-type thermal paths were used in the simulations.For each thermal path, two simulations were carried out, one with and onewithout an increase of 5 °C in temperature (upper bound of the temperatureuncertainty). The temperature ranges [Ta, Tb] in the simulations are [7801458.3. Fe-C-Mn Alloy°C, 835 °C] and [785 °C, 840 °C] for H-type, [785 °C, 842 °C] and [790 °C,847 °C] for V-type. The experimental microstructure obtained by quenchingfrom point A in Fig. 2.20 was used as the initial microstructure in both cases(Fig. 8.5).Figure 8.5: Micrograph of the Fe-0.1wt%C-0.5wt%Mn alloy (quenched from785 °C) used as the initial microstructure in 2D simulations (F: ferrite; M:martensite) [167].The numerical parameters, interface mobility and interface energy are thesame as in the simulations for the plain-carbon steel. Gibbs-energy dissipa-tion by trans-interface diffusion of Mn is taken into account. The values ofmodel parameters, i.e. the physical interface thickness, the binding energy ofMn and the trans-interface diffusivity of Mn, are given in Section 4.7. It wasfound that any increase of the interface mobility has a negligible influenceon the transformation kinetics, whereas a decrease below 1 × 10−6 m4/(J·s)leads to a slower transformation rate.1468.3. Fe-C-Mn AlloyThe simulated transformation kinetics for the V-type thermal path iscompared with the experimental data in Fig. 8.6.Figure 8.6: Comparison of the experimentally measured phase transforma-tion kinetics [167] with the simulation data for the V-type heat treatment ofFe-0.1wt%C-0.5wt%Mn: (a) the temperature range in the simulation is [785°C, 842 °C]; (b) the temperature range in the simulation is [790 °C, 847 °C]and the experimental curve is shifted up by 5 °C to [790 °C, 847 °C].There are various equilibria in a ternary system Fe-C-Mn, e.g. para-1478.3. Fe-C-Mn Alloyequilibrium, ortho-equilibrium and negligible-partitioning local equilibrium(NPLE). NPLE is selected to calculate the equilibrium ferrite fraction that isused as a reference in Fig. 8.6, because the present phase-field model mimicsNPLE at a stationary interface. NPLE corresponds to a specific tie-line wherethe Mn concentration in the growing phase is equal to the concentration inthe bulk. Thus, the NPLE tie-line for the austenite-to-ferrite transformation(γ → α) is different from the one for the ferrite-to-austenite transformation(α → γ) (shown in Fig. 2.3). Accordingly, there are two lines of NPLEferrite fractions in Fig. 8.6. If the fraction of a growing phase is larger thanthe relevant NPLE fraction, it implies that long-range diffusion of Mn hasoccurred.The initial microstructure used in the simulation was formed during theaustenite-to-ferrite transformation (γ → α) at 785 °C (Fig. 2.20(b)). Theagreement of the experimentally measured ferrite fraction with the NPLE(γ → α) ferrite fraction implies that the austenite-to-ferrite transformation(γ → α) has reached NPLE and long-range diffusion of Mn is negligible. Af-ter a full heating/cooling cycle, the ferrite fraction at 785 °C is smaller thanthe NPLE (γ → α) fraction in both experiments and simulations, indicat-ing a non-equilibrium condition in the material, i.e. the austenite-to-ferritetransformation (γ → α) is incomplete.In the case without a temperature increase, the simulated transformationkinetics is not in agreement with the experimental data at the upper temper-ature Tb (Fig. 8.6(a)). The ferrite fraction at 840 °C is 0.08 larger than theexperimental measurement. On the other hand, the simulation results arein better agreement with experimental data in the case with a temperature1488.3. Fe-C-Mn Alloyincrease of 5 °C. In particular, the phase fractions at both the lower andupper temperatures (Taand Tb) are consistent with the experimental data.Distinct from the plain-carbon steel, there is a stagnant stage for both theferrite-to-austenite transformation (α→ γ) during heating and the austenite-to-ferrite transformation (γ → α) during cooling. As shown in Fig. 8.6(a),the ferrite fraction does not decrease significantly upon heating until 802 °C,implying a sluggish transition of ferrite to austenite; the ferrite fraction doesnot increase significantly upon cooling until 820 °C, implying a sluggish tran-sition of austenite to ferrite. These stagnant stages have been quantitativelyreplicated in the phase-field simulation, with lengths of 18±3°C. The gooddescription of transformation kinetics by the phase-field model indicates thatlong-range diffusion of Mn is insignificant during the V-type heat treatmentand the assumption of carbon diffusion only in the phase-field model is ap-propriate. Fig. 8.7 shows the microstructures at selected stages of V-typecycles. The microstructure after a heating/cooling cycle generally resemblesthe one before the cycle.The simulated transformation kinetics for the H-type heat treatment iscompared with the experimental data in Fig. 8.8. The austenite-to-ferritetransformation (γ → α) takes place during cooling and holding at 780 °C,whereas the ferrite-to-austenite transformation (α → γ) takes place duringheating and holding at 835 °C. The model generally describes the trend ofcyclic phase transformations for the Fe-C-Mn alloy. In particular, the lengthsof stagnant stages for both heating and cooling are duplicated quantitativelyby the model, with a similar length of 18±3°C. The simulated ferrite fractionsare consistent with the experimental data at low temperatures, e.g. below1498.3. Fe-C-Mn AlloyFigure 8.7: Microstructures (dark: austenite; light: ferrite) at various stagesof the V-type cyclic phase transformation (a-d correspond to point A-D inFig. 2.20) for the Fe-0.1wt%C-0.5wt%Mn alloy.810 °C. Both the simulated and experimental measured ferrite fractions at780 °C after a full heating/cooling cycle are in agreement with the ferritefraction of NPLE for γ → α, implying that the long-range diffusion of Mnis negligible during the austenite-to-ferrite transformation (γ → α). Accord-ing to the local-equilibrium theory by Coates [141], the austenite-to-ferritetransformation (γ → α) is carbon diffusion-controlled first (NPLE) and Mndiffusion-controlled subsequently (partitioning local equilibrium, PLE).1508.3. Fe-C-Mn AlloyFigure 8.8: Comparison of the experimentally measured phase transforma-tion kinetics [167] with the simulation data for the H-type heat treatment ofFe-0.1wt%C-0.5wt%Mn: (a) the temperature range in the simulation is [780°C, 835 °C]; (b) the temperature range in the simulation is [785 °C, 840 °C]and the experimental curve is shifted up by 5 °C to [785 °C, 840 °C].If long-range diffusion of Mn occurs, the migration distance of a fer-1518.3. Fe-C-Mn Alloyrite/austenite interface is approximated with the Mn diffusion distance inthe parent phase, i.e. austenite. In the present case, holding at 780 °C for600 s leads to a diffusion length of about 14 nm (the diffusivity of Mn inaustenite is about 1.7 × 10−6 μm2/s at 780 °C [154]) which is negligible incomparison with an average ferrite grain size of 10-15 μm. Thus, long-rangediffusion of Mn is negligible and the assumption of carbon diffusion only isappropriate for the austenite-to-ferrite transformation (γ → α).The experimentally measured ferrite fraction at 835 °C (0.2-0.25) is, how-ever, smaller than the NPLE (α→ γ) ferrite fraction (0.3-0.4), or rather theexperimentally measured austenite fraction is larger than the NPLE (α→ γ)austenite fraction, implying that long-range diffusion of Mn occurs and in-duces further austenite formation. According to the local-equilibrium theoryby Wycliffe et al. [116], the ferrite-to-austenite transformation (α → γ) iscarbon diffusion-controlled first (NPLE) and Mn diffusion-controlled sub-sequently (PLE). If long-range diffusion of Mn occurs during the ferrite-to-austenite transformation (α→ γ), the migration distance of a ferrite/austeniteinterface that is estimated with the diffusion distance of Mn in the parentphase, i.e. ferrite, is about 1.3 μm after holding at 835 °C for 1200 s (the dif-fusivity of Mn in ferrite is about 1.5× 10−3 μm2/s in ferrite at 835 °C [154]).An increase of 1.3 μm in the equivalent radius of austenite grains with an av-erage size of 40-50 μm will contribute to a volume fraction of 0.12-0.16 in thepresent case, which is consistent with the discrepancy between the simulatedand the experimentally measured austenite fractions at 835 °C.Therefore, the present phase-field model considering carbon diffusion onlyis applicable to the V-type thermal path but not applicable to the H-type1528.4. Summarythermal path where long-time holding at the upper temperature Tb leads tosignificant long-range diffusion of Mn during the ferrite-to-austenite trans-formation (α→ γ).8.4 SummaryThe developed 2D phase-field model for intercritical annealing was appliedto the cyclic phase transformations in the intercritical region for both plain-carbon and Mn-alloyed steels. The phase-field model predicts that there isno stagnant stage in the plain-carbon steel, while stagnant stage exists inthe Mn-alloyed steel. Based on the model, the stagnant stage is whereverthe chemical driving pressure is smaller than the Gibbs-energy dissipationinduced by trans-interface diffusion of Mn or rather wherever long-rangediffusion of Mn is necessary for phase transformation. Compared with 1Dsimulation, the two-dimensional simulations can predict quantitatively thekinetics of the cyclic phase transformations by taking microstructural mor-phologies into account. The drawback of the model is that it cannot replicatethe phase transformation quantitatively if long-range diffusion of substitu-tional solutes is significant, e.g. for the H-type heat treatment of the Fe-C-Mnalloy.153Chapter 9Summary and Future Work9.1 SummaryA stand-alone phase-field model has been developed to simulate the mi-crostructure evolution during intercritical annealing of a cold-rolled low-carbon steel with a ferrite/pearlite microstructure. The model consists ofsub-models for ferrite recrystallization, austenite formation and austenite-to-ferrite transformation. The main modeling results and their implicationsare summarized below.Both 2D and 3D phase models have been developed to simulate recrys-tallization. The kinetics predicted by the JMAK theory was accurately du-plicated by the phase-field model. Based on the study of inhomogeneousdeformation and nucleation, it was concluded that the inhomogeneity of nu-cleation for recrystallization is the main reason that leads to the lower JMAKexponents (< 3) typically observed in experiments. Both nucleation scenar-ios and the grain boundary mobility in the model were estimated with theexperimental data. The model was validated by accurately predicting therecrystallization kinetics during continuous heating.Different values of the nucleation parameters and the ferrite grain bound-ary mobility were estimated in 2D and 3D simulations, based on the exper-1549.1. Summaryimentally measured recrystallization kinetics. In particular, nuclei distribu-tion in the 3D simulation is more inhomogeneous than in the 2D simulation.Despite of the less realistic description of microstructures in 2D simulations,the 2D model was calibrated successfully with the experimental data on re-crystallization kinetics and grain size distributions.A 2D phase-field model was developed to describe austenite formation.The model was integrated with the recrystallization model. The sequentialand concurrent ferrite recrystallization and austenite formation depend onthe heating scenarios, and are well replicated by the model. A Gibbs-energydissipation model was integrated with the phase-field model to consider theeffects of Mn trans-interface diffusion on the phase transformations. Themodel describes the variation of austenite morphologies with heating routes.Intercritical annealing was simulated to predict austenite formation as a func-tion of heating rate and annealing temperatures. The austenite-to-ferritetransformation is modeled by using the same interface mobility as for theferrite-to-austenite transformation, i.e. without any additional fitting, andby assuming no ferrite nucleation for decomposition of intercritical austenite.By integrating the models for ferrite recrystallization, austenite formationand austenite to ferrite transformation, a phase-field model for intercriticalannealing was developed. The model was validated with the experimentalresults of a simulated-industrial intercritical-annealing cycle.The model was also applied to describe the cyclic phase transformation inthe intercritical region. The existence of stagnant stages in a Mn-alloyed steelwas predicted by the model. It is the consideration of Gibbs-energy dissipa-tion by trans-interface diffusion of substitutional elements in the phase-field1559.2. Future Workmodel that leads to the prediction of stagnant stages.The work has demonstrated that phase-field modeling is a promising com-putational approach to study microstructure evolution in intercritical anneal-ing which is increasingly important in developing advanced high strengthsteels.9.2 Future WorkThe following extensions of the present study are suggested:ˆ A phase-field model for pearlite to austenite transformation that takesthe lamellar structure into account should be developed in the future.The model should be able to describe the accelerated spheroidization ofdeformed pearlite and its effect on the subsequent austenite formation.ˆ Long-range diffusion of substitutional elements, e.g. Mn, is not takeninto account in the present model. By using more advanced numericalmethods, e.g. adaptive meshing, it will be feasible to include the substi-tutional diffusion in modeling austenite formation. The multi-phase-field model used in the present study [45] is applicable to modelingmulti-component diffusion during phase transformation.ˆ A 3D phase-field model for intercritical annealing should be devel-oped in the future to provide a more realistic description of the mi-crostructure evolution by using the advanced numerical and comput-ing techniques, e.g. adaptive meshing, to reduce the computationaltime. Moreover, 3D experimental data, e.g. initial pearlite/ferrite mi-1569.2. Future Workcrostructures, are required for 3D simulations, which might be obtainedusing 3D EBSD.ˆ The current model only considers the austenite-to-ferrite transforma-tion. However, the austenite-to-bainite transformation is also of greatimportance especially in the hot-dip galvanizing process. Therefore,bainite formation should be taken into account in the future. So far,the mechanism of bainite formation is still under debate, i.e. eitherdisplacive or diffusive. 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