M O D E L I N G T H E A U S T E N I T E D E C O M P O S I T I O N I N T O F E R R I T E AND BAINITE By FATEH FAZELI B . Sc., Tehran University, 1990 M . Sc., Tehran University, 1995 A THESIS S U B M I T T E D IN P A R T I A L F U L L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Materials Engineering) T H E U N I V E R S I T Y OF BRITISH C O L U M B I A February 2005 ©Fateh Fazeli, 2005 ABSTRACT Novel advanced high-strength steels such as dual-phase (DP) and transformation induced plasticity (TRIP) steels, are considered as promising materials for new generation o f lightweight vehicles. The superior mechanical properties o f these steels, compared to classical high strength steels, are associated with their complex micro structures. The desired phase configuration and morphology can only be achieved through well-controlled processing paths with rather tight processing windows. To implement such challenging processing stages into the current industrial facilities a significant amount o f development efforts, in terms o f m i l l trials, have to be performed. Alternatively, process models as predictive tools can be employed to aid the process development' and also to design new steel grades. Knowledge-based process models are developed by virtue o f the underlying physical phenomena occurring during the industrial processing and are validated with experimental data. The goal o f the present work is to develop an integrated microstructure model to adequately describe the kinetics o f austenite decomposition into polygonal ferrite and bainite, such that for complex thermal paths simulating those o f industrial practice, the final microstructure i n advanced high strength steels can reasonably be predicted. This is i n particular relevant to hot-rolled D P and T R I P steels, where the intercritical ferrite evolution due to its crucial influence on the onset and kinetics o f the subsequent bainite formation, has to be quantified precisely. The calculated fraction, size and spatial carbon distribution o f the intercritical austenite are employed as input to characterize adequately the kinetic o f the bainite reaction Pertinent to ferrite formation, a phenomenological, physically-based model was developed on the ground o f the mixed-mode approach. The model deals with the growth stage since nucleation site saturation at prior austenite grain boundaries is likely to be attained during the industrial treatments. The thermodynamic boundary conditions for the kinetic model were assessed with respect to paraequilibrium. The potential interaction between the alloying atoms and the moving ferrite-austenite interface, referred to as solute drag effect, was accounted for rigorously in the model. To quantify the solute drag pressure the PurdyBrechet approach was modified prior to its implementation into the model. The integrated model employs three main parameters, the intrinsic mobility o f the ferrite-austenite interface, n the binding energy o f the segregating solute to the interface and its diffusivity across the transformation interface. These parameters are clearly defined in terms o f their physical meaning and the potential ranges o f their values are well known. However, no direct characterization techniques are currently available to precisely measure them hence they are treated as adjustable parameters in the model. The model predicts successfully the overall kinetics o f ferrite formation in a number o f advanced steels. The bainite evolution in different TRIP steels was analyzed using three available approaches, i.e. Johnson-Mehl-Avrami-Kolmogorov ( J M A K ) equation, the Zener-Hillert formulation for diffusional growth and the displacive definition proposed by Bhadeshia. Overall, it turned out that the predictive capability o f the three methodologies is similar. Further, some o f the ensuing model parameters pertinent to each approach are difficult to interpret in terms o f the underlying physics, which implies that all three models are employed in a semi-empirical manner. Assuming diffusional transformation mechanism for bainite, the isothermal incubation time and the onset o f bainite formation during continuous cooling treatments were described adequately. Consistently, for the purpose o f process modeling, the diffusional description of bainite growth can potentially be employed. However, from an academic point o f interest a more precise quantification for the nucleation part is still missing. in TABLE OF CONTENTS ABSTRACT T A B L E OF CONTENTS LIST O F T A B L E S LIST OF F I G U R E S LIST OF S Y M B O L S ACKNOWLEDGMENT ii iv vii viii xiii xvi Chapter 1 : General Introduction 1.1 1.2 1 Advanced High-strength Steels Scope and Objective o f the Thesis 1 8 Chapter 2 : Austenite Decomposition into Ferrite 10 2.1 Literature Review 2.1.1 Thermodynamics o f phase transformation 2.1.1.1 Orthoequilibrium 2.1.1.2 Paraequilibrium 2.1.1.3 Negligible partition local equilibrium 2.1.2 Austenite to ferrite transformation kinetics 2.1.2.1 Nucleation 2.1.2.2 L o c a l equilibrium model for thickening o f ferrite i n F e - C 2.1.2.3 Growth o f ferrite in ternary and other multi-alloyed steels 2.1.2.4 Retarding effect o f solute atoms 2.1.2.5 Mixed-mode model for ferrite growth 2.1.2.6 Overall kinetics o f ferrite formation 10 10 13 15 16 18 18 24 27 31 35 36 2.2 Model Development 2.2.1 Characterizations o f the thermodynamics o f ferrite formation 2.2.1.1 Paraequilibrium treatment 2.2.1.2 Negligible partitioning local equilibrium boundaries 2.2.2 Modeling the growth kinetics of ferrite 2.2.2.1 Carbon diffusion controlled model 2.2.2.2 Mixed-mode model 2.2.2.3 Assessment o f solute drag effect 2.2.2.4 Modification to Purdy-Brechet solute drag model 2.2.3 Crucial parameters in the kinetic model 39 39 40 44 45 46 51 54 58 60 2.3 M o d e l Application 2.3.1 Analysis o f allotriomorph thickening in Fe-C alloys 2.3.1.1 Prediction o f carbon diffusion model 2.3.1.2 Prediction o f mixed-mode model 2.3.2 Analysis o f allotriomorph thickening in Fe-C-z systems 2.3.2.1 Comments on the experimental data 2.3.2.2 Suitable thermodynamic treatment o f F e - C - i alloys 65 65 66 69 76 76 79 iv 2.3.2.3 Prediction o f mixed-mode model 81 2.3.2.4 Implementing the solute drag effect 84 2.3.2.5 Resulting solute drag parameters: Fe-0.21wt%C-l .52wt%Mn 84 2.3.2.6 Solute drag parameters: 0.12C-3.1Mn and 0.12C-3.3Ni(wt%) steels 86 2.3.2.7 Discussion on the temperature dependence o f binding energy 89 2.3.3 Analysis o f the overall kinetics o f ferrite formation in A 3 6 steel 93 2.3.3.1 Experimental kinetics 93 2.3.3.2 Modeling the isothermal ferrite transformation 96 2.3.3.3 Implication o f the modified S D model to isothermal transformation 99 2.3.3.4 Modeling the ferrite evolution during continuous cooling 101 2.3.4 Analysis o f ferrite evolution in quaternary systems: D P and T R I P steels.... 102 2.4 Summary and Remarks (Ferrite Formation) Chapter 3 : Bainite Transformation 3.1 107 110 Introduction 110 3.2 Literature Review 3.2.1 Overview 3.2.2 Mechanism o f bainite transformation 3.2.3 Displacive mechanism o f bainite transformation 3.2.4 Carbide precipitation 3.2.5 Thermodynamic criteria for displacive nucleation and growth 3.2.6 Displacive model to predict the overall kinetics o f bainite 3.2.7 Diffusional theory o f bainite formation 3.2.8 Mathematical expression for growth kinetics o f a plate 3.2.9 Evaluation o f bainite transformation kinetics 111 Ill 112 113 115 117 119 123 126 130 3.3 Study o f Isothermal Bainite Formation in F e - 0 . 6 C - l . 5 M n - l . 5 S i 3.3.1 Experimental procedures 3.3.2 Exp erimental results 3.3.3 Modeling 3.3.3.1 Semi-empirical modeling approach 3.3.3.2 Modeling the incubation period 3.3.3.3 Diffusion model 3.3.3.4 Kinetic effect o f strain energy in diffusion model 3.3.3.5 Displacive model 3.3.4 Summary and remarks (0.6C T R I P steel) 133 133 135 139 139 142 145 150 153 155 3.4 Study o f Isothermal Bainite Formation in F e - 0 . 1 8 C - l . 5 5 M n - l . 7 S i 3.4.1 Experimental procedure and results 3.4.2 Modeling 157 157 162 3.5 Study o f Continuous Cooling Bainite Formation in 0.19C-1.5Mn-1.6Si-0.2Mo T R I P Steel 167 3.5.1 Analyzing the onset o f bainite reaction 170 v 3.5.1.1 3.5.1.2 3.6 M o d e l i n g bainite formation Diffusion model with plastic work Summary and Remarks (Bainite Formation) Chapter 4 : Overall Conclusion 4.1 4.2 172 177 180 182 Findings and Achievements Future W o r k 182 184 REFERENCES APPENDICES 186 190 VI LIST OF TABLES Table 1-1 Table 1-2 Table 1-3 Table 2-1 Table 2-2 Table 2-3 The annual production o f typical structural materials [1] 1 Car body components and their required properties [6] 3 Typical mechanical properties o f cold-rolled, high strength steels [7] 4 Nominal values for n and m parameters [76] 37 List of elements included in Fe2000 database in their maximum permitted level (wt%) 40 Equilibrium austenite/ferrite transformation temperatures and ranges o f measured transformation temperatures, T , o f the investigated ternary alloys 76 The required values for the interface mobility (in terms o f the ratio to Krielaart and van der Zwaag mobility, Mpe-Mn) to replicate the measured thickening rate o f Fe-0.12wt%C-3.3wt%Ni 83 The required values for the interface mobility (in terms o f the ratio to Krielaart and van der Zwaag mobility, M -Mn) to describe the overall kinetics o f isothermal ferrite formation in Fe-0.17wt%C-0.74wt%Mn 97 Reported values for adjustable parameters based on the best fit for F e - C M n - S i alloys [23] 122 Models o f plate-lengthening rate [129] 129 M a x i m u m fraction o f bainite measured experimentally using optical micrographs 137 Parameters used in Equation 3-22 to predict the incubation period 145 Values for adjustable parameters found for the 0.6C-1.5Mn-1.5Si steel in this work 154 Growth rate o f bainite plates for different transformation temperatures assessed using Zener-Hillert equation (0.41wt%C-1.55wt%Mn-1.7wt%Si austenite) 164 Parameters used in displacive approach to describe the measured kinetics o f bainite formation from an intercritically treated austenite with carbon content o f 0.41wt% 165 List o f experimental treatments and the ensuing measurements of 0.16wt%C-1.5wt%Mn-1.6wt%Si-0.2wt%Mo, M o - T R f P steel 168 m e a s Table 2-4 Table 2-5 Fe Table 3-1 Table 3-2 Table 3-3 Table 3-4 Table 3-5 Table 3-6 Table 3-7 Table 3-8 vu LIST OF FIGURES Figure Figure Figure Figure 1-1 1-2 2-1 2-2 Figure 2-3 Figure 2-4 Figure 2-5 Figure 2-6 Figure 2-7 Figure 2-8 Figure 2-9 Figure 2-10 Schematic representation of the processing stages o f D P and T R I P steels Thermal and thermomechanical paths to produce T R I P steels Free energy surfaces of y and a and its equilibrium tie line for X Iron reach corner, isothermal section of phase diagram for F e - C - M n at 700°C showing orthoequilibrium phase boundaries between ferrite and austenite, as well as the tie lines Tangent planes to the free energy surfaces of a and y in paraequilibrium condition F e - C - M n isotherm representing O E and N P - L E phase boundary [34] The vertical section of a coherent pillbox nucleus [40] Carbon concentration profile along a direction perpendicular to the oc/y interface Schematic illustration of the tetrakaidecahedron model for an austenite grain and the planar and the spherical geometry used in the diffusional model [51] Schematic isothermal section of F e - C - i system representing the interfacial concentration for ferrite growth under O E and N P L E conditions when the nominal composition of alloy is different The composition path of interfacial condition during isothermal growth [55] Section of the free energy surfaces of ferrite and austenite along the concentration ray o f carbon for a given X[/Xp ratio Isothermal section at 700°C for F e - C - M n system, (a) shows paraequilibrium and orthoequilibrium phase boundaries as well as O E tie lines, (b) shows the iron reach corner Paraequilibrium pseudo-binary diagram illustrating the calculation of the 0 e Figure 2-11 Figure 2-12 chemical driving pressure at the start of ferrite growth where X\ m Figure 2-13 Figure 2-14 Figure 2-15 Figure 2-16 Figure 2-17 Figure 2-18 =X 0 Isothermal section of phase diagram at 700°C for the F e - C - M n system comparing N P L E , P E and orthoequilibrium boundaries (a) Ferrite thin shell decorating austenite grain boundaries, ferrite formed partially at 735°C from austenite with grain size o f 96um in 0.17wt%C0.74wt%Mn steel, followed by quenching to room temperature, (b) schematic representation of the ferrite growth as a spherical shell toward the center o f austenite grain Node configuration for (a) planar and (b) spherical geometries Carbon profile for various growth times calculated by diffusion model. Ferrite growth with planar geometry at 745°C from austenite o f 50)j,m grain size i n Fe-0.23wt%C was assumed Carbon profile for various growth times calculated by the mixed-mode model for reaction at 745°C in Fe-0.23wt%C alloy assuming planar geometry and (a) M=0.05 urn mol/J s and (b) M=0.38 urn mol/Js Chemical potential of solute in the oc/y interphase boundary [59] vm 5 6 14 14 16 17 21 25 27 28 30 41 43 44 45 47 48 51 53 55 Figure 2-19 Figure 2-20 Figure 2-21 Figure 2-22 Figure 2-23 Figure 2-24 Segregation profile inside the interface region (a) for various normalized velocities and AE=0 and (b) for different values o f A E for a stationary interface (V=0) Comparison o f the energy dissipated by solute drag for Eo=T.5RT and AE=-4kJ/mol, adopting the Purdy-Brechet model and the modified solute drag approach Experimental parabolic rate constant i n F e - C alloys [92] Zener predictions for parabolic rate constant and the measured [92] data in Fe-0.11wt%C Comparison o f the measured [92] parabolic rate constant with the prediction o f the carbon diffusion model for (a) Fe-0.11wt%C, (b) Fe0.22wt%C and (c) Fe-0.43wt%C alloys Measured allotriomorphs thickness and the predictions o f mixed-mode model assuming M =5.8 and 1.27 cm mol/Js, as well as the calculation o f the carbon diffusion model ( M = o o ) for growth at 775°C in Fe0.23wt%C Experimental data [92] o f parabolic rate constant and prediction o f mixedmode model assuming mobility o f 5.8 and 0.75 cm mol/Js for (a) Fe0.11wt%C, (b) Fe-0.22wt%C and (c) Fe-0.43wt%C alloys. The predictions o f the diffusion model are included as well Experimental Measurements o f parabolic growth rate constant o f ferrite in C - M n and C - N i steels [34, 62] The measured thickness o f ferrite allotriomorph at different growth temperatures in Fe-0.12wt%C-3.3.wt%Ni alloy [94], the solid line represents linear regression Predictions o f the diffusion model using different thermodynamic conditions for ferrite thickening in (a) 1.52wt%Mn alloy at 730°C and (b) 3.3wt%Ni alloy at 665°C Predictions o f mixed-mode model and the experimental [94] growth o f ferrite allotriomorphs for 0.12wt%C-3.3wt%Ni steel, intrinsic mobility is employed as the only fit parameter Arrhenius plot o f fitted mobilities for 0.12wt%C-3.3wt%Ni resulting in an apparent activation energy o f - 2 7 1 k J / m o l shows mobility would have to be decreased with increasing temperature Effect o f temperature on solute drag parameters to describe the experimental data [34] o f ferrite plate thickening in Fe-0.21 wt pet C-1.52 wt pet M n Effect o f the selection o f the intrinsic mobility on solute drag parameters to describe the experimental data [59] o f ferrite allotriomorph thickening in Fe-0.21 w t p c t C-1.52 w t p c t M n Solute drag parameters required to replicate the experimental growth kinetics [62] o f ferrite at different temperatures in Fe-0.12wt%C3.1wt%Mn Required solute drag parameters to describe the growth kinetics o f ferrite measured [62] at different temperatures in Fe-0.12wt%C-3.3wt%Ni 57 59 66 67 70 0 D Figure 2-25 Figure 2-26 Figure 2-27 Figure 2-28 Figure 2-29 Figure 2-30 Figure 2-31 Figure 2-32 Figure 2-33 Figure 2-34 ix 72 75 77 78 80 82 83 85 86 88 88 Figure 2-35 Figure 2-36 Figure 2-37 Figure 2-38 Figure 2-39 Figure 2-40 Variation o f applied binding energy with temperature to describe the experimental growth rate o f ferrite [34,62] i n 0.12wt%C-3.1wt%Mn, 0.12wt%C-3.3wt%Ni and 0.21wt%C-1.52wt%Mn steels 89 Arrhenius plot for D assuming constant values for the binding energy in Fe-0.12wt%C-3.1 w t % M n 91 Arrhenius plot for Db assuming constant values for the binding energy in Fe-0.12wt%C-3.3wt%Ni 92 Measured kinetics o f isothermal ferrite formation in a 0.17wt%C0.74wt%Mn steel with an austenite o f grain size o f 96um 95 Measured and calculated ferrite fractions by optical metallography and the calculated ones assuming O E and P E 95 Microstructure of Fe-0.17wt%C-0.74wt%Mn specimens reacted isothermally at (a) 735 and (b) 765°C, d =96|am .96 Experimental (symbols) and calculated (solid lines) kinetics o f isothermal ferrite formation in the temperature range o f 735 to 765°C in Fe0.17wt%C-0.74wt%Mn with an initial austenite grain size o f 96|j,m 98 Variation o f thermodynamic driving pressure and solute drag pressure with time for growth o f ferrite at 735°C i n 0.17wt%C-0.74%Mn steel (a) using modified S D model, (b) adopting the original Purdy-Brechet quantification 100 Ferrite formation kinetics at 735°C in Fe-0.17 wt% C-0.74 w t % M n showing the predictions adopting the original Purdy-Brechet solute drag theory and its modification, respectively 101 Measurements [63] and predictions for ferrite formation during continuous cooling from an initial austenite grain size o f 18um in 0.17wt%C0.74wt%Mn steel 102 Measured [96] (symbols) and predicted (solid lines) kinetics o f austenite decomposition into ferrite during continuous cooling o f a dual-phase steel, Fe-0.06wt% C - l . 8 5 w t % M n - 0 . 1 6 w t % M o 104 Comparison o f observed data [97] and model prediction o f ferrite formation kinetics during continuous cooling o f T R I P steel, Fe-0.21C1.53Mn-1.54Si(wt%), with an austenite grain diameter o f 20um 105 Comparison o f manganese diffusivity across ferrite-austenite interface for A 3 6 , D P and T R I P steels with bulk diffusivity i n austenite, assuming 25 o f lnm 106 Variation of manganese binding energy to a/y interface with temperature for DP and TRIP steels 106 3D representation o f lath and plate [109] 112 The T line, (a) The free energy curves o f austenite and ferrite and their relation to the different phase boundaries in the F e - C diagram, (b) the relative location o f T in the Fe-C diagram [109] 115 Evolution o f three different bainite morphologies from the supersaturated plate [109] 116 The change i n volume fraction o f bainite during austempering at different temperatures. Solid and dotted curves show calculated data based on the diffusional and displacive model, respectively [131] 130 b v Figure 2-41 Figure 2-42 Figure 2-43 Figure 2-44 Figure 2-45 Figure 2-46 Figure 2-47 Figure 2-48 Figure 3-1 Figure 3-2 0 0 Figure 3-3 Figure 3-4 x Figure 3-5 Figure 3-6 Figure 3-7 Figure 3-8 Schematic representation o f the heat treatment cycle applied to isothermally form bainite at different temperatures from a single phase austenite featuring 13 and 40|jm volumetric grain size 134 Dilatometer responses recorded during isothermal bainite formation at different temperatures from austenite grain sizes o f (a) 13fj.m and (b) 40um 136 T T T diagram o f bainite formation from two different austenite grain sizes in F e - 0 . 6 C - l . 5 M n - l . 5 S i 137 Optical micrographs showing bainite (dark gray phase) surrounded by martensite/austenite matrix formed at (a) 450, (b) 400 and (c) 350°C from d =40um 138 S E M micrographs showing bainite surrounded by martensite/austenite matrix formed at different transformation temperatures from d =40um 138 Incubation times o f bainite formation as a function o f temperature for different austenite grain sizes, i.e. 13 and 40um 141 Temperature dependency o f the parameter b to provide optimal description o f experimental data using the J M A K equation with n=l 141 Comparison the experimental kinetics o f isothermal bainite formation and model predictions adopting the J M A K equation 142 M o d e l prediction (lines) and experimental incubation period (points) o f isothermal bainite formation from two prior austenite grain sizes, 13 and 40(^m, in 0.6wt%C-1.5wt%Mn-1.5wt%Si steel 145 Interfacial carbon concentration o f austenite in equilibrium with ferrite plate bearing a tip curvature o f p for different temperatures 147 The variation o f growth rate with the tip curvature o f a ferrite plate for different reaction temperatures assuming the Zener-Hillert model 148 Variation o f the maximum growth rate o f a bainite plate with ferriteaustenite interfacial energy, calculated for bainite formation at 400°C in 0 . 6 w t % C - l . 5 w t % M n - l .5wt%Si steel 148 Comparison o f the measured kinetics and the predictions o f diffusion model for isothermal bainite formation i n 0 . 6 w t % C - l . 5 w t % M n - l .5wt%Si steel 149 Decreasing trend o f the growth velocity o f a plate as bainite fraction increases, transformation at 400°C in 0.6wt%C-1.5wt%Mn-1.5wt%Si steel 151 Prediction (lines) o f diffusion model modified with the effect o f plastic work and the experimental kinetics (points) o f bainite formation in Fe0 . 6 w t % C - l . 5 w t % M n - l . 5 w t % S i alloy. 152 Displacive model predictions and the measured kinetics o f bainite reaction for d =40p.m 154 Comparison o f the experimental data for the time o f 50% transformed, t o%, and model predictions employing diffusion and displacive approaches 156 Thermal cycle designed to form bainite from intercritically treated specimens with 0.18wt%C-1.55wt%Mn-1.7wt%Si initial composition 158 y Figure 3-9 T Figure 3-10 Figure 3-11 Figure 3-12 Figure 3-13 Figure 3-14 Figure 3-15 Figure 3-16 Figure 3-17 Figure 3-18 Figure 3-19 Figure 3-20 y Figure 3-21 5 Figure 3-22 XI Figure 3-23 Figure 3-24 Figure 3-25 Figure 3-26 Figure 3-27 Figure 3-28 Figure 3-29 Figure 3-30 Figure 3-31 Figure 3-32 Figure 3-33 Figure 3-34 Figure 3-35 Figure 3-36 Optical micrographs o f intercritically treated samples transformed partially to bainite at 400 and 300°C. Ferrite matrix (white) estimated to be 58%, and the rest is a mixture o f martensite (light gray) and packets o f bainite (dark gray), 2% nital etched 159 S E M micrographs o f intercritically treated samples transformed partially to bainite at 400 and 300°C 160 Kinetics o f isothermal bainite evolution from intercritically treated samples featuring 42% austenite 161 Comparison o f model prediction adopting J M A K approach and the experimental fraction o f bainite formed after intercritical treatment in 0.18wt%C-l . 5 5 w t % M n - l . 7 w t % S i steel 163 M o d e l prediction adopting diffusion approach and the experimental kinetics o f isothermal bainite evolution from an intercritically treated austenite with carbon content o f 0.41wt% 164 Comparison o f the model prediction based on the displacive approach and the measured kinetic o f isothermal bainite formation from an intercritically treated austenite with carbon content o f 0.41wt% 166 Experimental data o f overall austenite decomposition during continuous cooling treatments at 5°C/s after reheating at 950°C, 8=0 i n 0.19wt%C1.5wt%Mn-1.6wt%Si-0.2wt%Mo T R I P steel, (a) fraction transformed and (b) the rate o f decomposition 169 Variation o f the calculated driving pressure o f bainitic ferrite formation at the transformation start temperature. The lower line shows G n function [113], which is located about 300J/mol lower than the line followed by the data points 171 Normalized fraction o f bainite formed during continuous cooling with C R = ° 5 C / s for reheating at 950°C in M o - T R I P steel 173 Variation o f the maximum growth rate o f a bainite plate with temperature and carbon concentration for M o - T R I P steel 174 Diffusion model prediction and the experimental kinetics o f bainite formation for reheating at 1050°C and various subsequent treatments in M o - T R I P steel 175 Comparison o f the predicted (using the diffusion model) temperature for 50% transformation and the measured value for all the experiments listed in Table 3-8 177 Variation o f the maximum growth rate with transformation temperature for different values o f bainite fraction, i.e. different values o f Gibbs free dissipation due to plastic work 178 Comparison o f the predicted (using the diffusion model and accounting for the plastic work) temperature for 50% transformation and the measured value for all the experiments listed in Table 3-8 179 xii LIST OF SYMBOLS a Lattice parameter a Average lattice parameter o f ferrite and austenite b Rate parameter in J M A K equation c Initial carbon concentration o f an alloy L L 0 c Equilibrium carbon concentration in phase j J eq c Carbon concentration o f node i c- Carbon concentration o f node i at time step j i cj Interfacial carbon concentration in phase j nl c s Initial concentration o f substitutional component s in an alloy 0 c (x) Concentration o f segregating solute inside interface region d Austenite grain size f Volume fraction o f phase i h* k 21 21 Height o f critical pillbox nucleus Boltzmann's constant Size o f the remaining austenite grain Initial size o f an austenite grain m n Austenite grain size exponent J M A K exponent n p Number o f atoms in contact with critical nucleus Peclet number r* s u v v v^ Radius o f the critical pillbox nucleus Thickness o f a ferrite allotriomorph Size o f a bainitic ferrite sub-unit Velocity o f ferrite-austenite interface Growth velocity o f a bainite plate M a x i m u m growth rate o f a bainite plate s y t 0 B x x z z, Length axis Length axis Geometrical factors depending on the assumed shape for critical nucleus A (for i=l to 5) Parameters describing concentration and temperature i dependency o f v ^ A ax Intercept o f the linear temperature dependency of E E 0 B Slope o f the linear temperature dependency o f E C Normalized concentration o f segregating solute E C,, C D 0 0 2 Parameters used to describe the isothermal incubation time o f bainite reaction Pre-exponential factor for solute diffusivity across interface xiii Diffusivity o f a segregating solute across transformation interface Di Diffusivity o f species i in phase j E Solute-interface binding energy E(x) Interaction potential profile o f solute inside interface AE Chemical potential difference o f the segregating solute across the interface G M o l a r Gibbs free energy o f phase j Gcritical Critical driving pressure required at the onset o f bainite formation 0 J magnetic Magnetic contribution to Gibbs free energy o f a phase Universal nucleation function, criterion for displacive nucleation o f bainite plastic Energy dissipation due to plastic accommodation o f transformation strain Activation energy for nucleation AG AG eff Effective driving pressure at ferrite-austenite interface A G ,inl Thermodynamic driving pressure at ferrite-austenite interface AG M o l a r Gibbs energy change due to formation k from phase j A G SD Energy dissipated by solute drag effect Gspike Energy dissipation due to solute diffusion inside a solute spike AGy Volumetric Gibbs energy change due to the formation o f a new phase AG ^ AG ^ Free energy change for ferrite formation with paraequilibrium carbon content Driving pressure for composition invariant transformation o f austenite to supersaturated ferrite K, Adjustable parameters relevant to the displacive model o f bainite formation r y f+a a Adjustable parameter to describe the overall kinetics o f bainite formation by the diffusion model J in ' J out Atomic flux o f carbon to or from the interface Net atomic flux o f carbon across a moving ferrite-austenite interface M M N p 0 1 SD Intrinsic mobility o f transformation interface Pre-exponential factor for interface mobility Density o f viable atomic nucleation sites Solute drag pressure Activations energy for interface mobility Q Q Activation energy for the diffusivity o f solute across interface R SC Gas constant A n imaginary component relevant to the substitutional sublattice i n paraequilibrium Absolute temperature b The highest temperature at which the displacive nucleation o f bainite occurs Normalized velocity o f ferrite-austenite interface M o l a r volume xiv Normalized distance inside ferrite-austenite interface Initial mole fraction o f carbon in an alloy M o l e fraction o f carbon Equilibrium carbon mole fraction in ferrite or austenite M o l e fraction o f iron or substitutional elements Initial mole fraction o f iron or substitutional elements in an alloy Interfacial mole fraction o f carbon in phase j M o l e fraction o f iron or substitutional elements in specific location k Paraequilibrium mole fraction o f carbon i n ferrite or austenite Carbon content at the austenite side o f the bainitic ferrite-austenite interface corrected by capillary effect Site fraction o f species i (in a sub-lattice) Zeldovich nonequilibrium factor Ferrite Nucleation frequency factor H a l f thickness o f interface Austenite Shear component o f transformation strain associated with bainite formation Activity coefficient o f component i Wagner's interaction coefficient M a x i m u m fraction o f bainite determined experimentally Ratio o f substitutional mole fraction to iron mole fraction Chemical potential o f i in standard state Chemical potential o f i in phase j Change in the chemical potential o f / during phase change j—>k Chemical potential o f an imaginary substitutional component SC in phase j relevant to paraequilibrium pseudo-binary section Tip curvature o f a bainite plate Critical tip curvature o f a plate Interfacial energy between ferrite and austenite Interfacial energy for austenite grain boundaries Incubation time Ratio o f temperature to Curie temperature Y i e l d strength o f austenite Driving pressure exponent i n Russell's equation Supersaturation in front o f a bainite plate xv ACKNOWLEDGMENT It was a great opportunity for me to pursue my Ph.D. at U B C , in a research group conducted by decent and talented scientists, "Praise be to God, who has guided us to this". I am forever indebted to my supervisor Dr. Matthias Militzer for his efficient guidance, professional support and constant encouragement throughout the course o f this work. This provided me the enthusiasm to efficiently overcome the challenges. A sincere appreciation is extended to Dr. Warren Poole for his constructive advices. I have been fortunate to benefit o f his expertise in various aspects o f this research. Special thanks to all the colleagues and officemates for providing a friendly environment that I was always pleased to work in. Financial support from the Natural Science and Engineering Research Council o f Canada, and also the Stelco Fellowship are gratefully acknowledged. The great sacrifice, wisdom and patience o f my wife are much appreciated. This thesis is dedicated to my precious parents. xvi General Introduction Chapter 1 Chapter 1 : General Introduction 1.1 Advanced High-strength Steels Among structural materials, steel is o f the leading production tonnage globally. A s indicated in Table 1-1 it exceeds 960 million tones in 2003 [1], about 60% o f which is made from recycled scrap. Further, compared to other metallic alloys, its production requires lower amount o f energy. This substantially contributes in maintaining a sound ecology. There is virtually no limit i n alloying strategies and there are currently more than 3500 different steel grades available. The full potential o f this extensive diversity in chemistry, and consequently numerous distinctive transformation behaviors, can be exploited by applying appropriate thermomechanical treatments, such that various phase configurations and morphologies are likely to evolve i n practice. The enormous multiplicity in the microstructures o f steel results in diverse combination o f mechanical properties which are suitable for a wide range o f applications. Table 1-1 The annual production of typical structural materials [1] Steel 960 million tons (2003) Aluminum 21 million tons (2002) Copper 14.9 million tons (2002) Portland cement 87.8 million tons (2000) Timber 300 million m (2001) Despite many financial i crises that infrastructural industries have been facing, steel production is still a profitable, active and dynamic industry. A proof to this argument is the fact that 75% o f the current steel grades have been developed in the past 20 years and in particular, 70% o f the steels used in automotive applications today did not exist 10 years ago 1 General Introduction Chapter 1 [1]. This also implies that much progress in composition and processing o f new steels were driven by the automotive industry which requires the most demanding functionality o f steels for vehicle components and bodies [2,3,4]. However, in order to improve fuel efficiency and reduce emission, special efforts were made to reduce vehicle weight not only by optimizing the body design but also in part by introducing plastic parts and employing light metals. For instance the weight fraction o f steel being used in a family car was decreased from 74% in 1978 to 67% in 1997. However, over the same period the incorporation o f high strength steels was increased from 4% to 9%, a rise that is unique compared to the growth o f any other material class in automotive applications [5]. Several groups o f cold- and hot-rolled, high strength steels have been developed to satisfy the aforementioned demand for automotive applications. Although weight reduction would be possible through using stronger materials, there is a common trend that the ductility deteriorates as strength increases. Depending on the function o f individual components, the required properties vary in detail, as summarized in Table 1-2 [6]. Examples o f typical mechanical properties o f commonly used cold-rolled formable steels are presented in Table 1-3 [7]. High Strength L o w A l l o y ( H S L A ) steels were developed by combining the benefits o f precipitation hardening o f microalloying elements with the advantages o f grain refinement. Controlled rolling and accelerated controlled cooling are the two common processing routes of H S L A steels [8]. A complex chemistry plus complicated thermomechanical processing have led to the introduction o f the most advanced steel in this class, X I 0 0 grade steels with a tensile strength o f more than 800MPa for pipeline applications [9]. The main disadvantages 2 General Introduction Chapter 1 of H S L A steels are their inferior formability and high deformation resistance during hot or cold rolling. Interstitial Free (IF) steels have a carbon and nitrogen content o f 20-40 ppm, which is removed from solution by adding Ti/Nb as carbide and nitride formers. These steels are the current choice for the most complex parts o f car bodies because o f their high r-value and superior formability. However the upper limit o f yield and tensile strength in IF steels is about 320 and 450MPa, respectively [10]. Table 1-2 Car body components and their required properties [6], Components Required properties Factors for thickness selection Body panels Structural members Undercarriage parts Deep drawability, stretchability, shape flexability, Panel stiffness, dent panel stiffness, dent resistance, corrosion resistance resistance Stretchability, bendability, shape flexability, Structural rigidity, structural rigidity, crashworthiness, fatigue strength, crashworthiness, fatigue corrosion resistance, weldability strength Stretchability, stretch flanging, shape flexability, Structural rigidity, fatigue structural rigidity, crashworthiness, fatigue strength, strength corrosion resistance, weldability Reinforcements Bendability, shape flexability, crashworthiness, Crashworthiness weldability Bake Hardenable ( B H ) ultra-low carbon steels where a few ppm carbon remain in solution are suitable for outer large panels. They offer high formability and high dent resistance simultaneously. Hardening occurs during the paint baking process and results in a strength 3 General Introduction Chapter 1 increase of approximately 4 0 M P a . This is indeed a strain aging process and aids the material to be potentially dent resistant. Table 1-3 Typical mechanical properties of cold-rolled, high strength steels [7]. Yield strength (MPa) UTS (MPa) El. (%) r-value n-value Microalloyed 320 440 28 1.3 0.17 Bake Hardening 210 320 40 1.6 0.22 Interstitial Free 220 390 37 1.9 0.21 Dual Phase 350 600 27 0.9 - TRIP 400 640 32 - - Super-ultralow Carbon 200 300 45 2.0 0.24 Steels Dual phase (DP) and transformation induced plasticity (TRIP) steels have been proposed as new materials for more demanding applications in the automotive industry. These steels are characterized by complex multiphase microstructures that provide an excellent combination of strength and formability. Both D P and TRIP steels are processed in the intercritical region either by cooling from the austenite region after hot rolling or by annealing after cold rolling. The goal o f this processing step is to form an austenite-ferrite mixture with a ferrite fraction as desired in the final microstructure. D P steels are then quickly cooled to room temperature such that the remaining austenite transforms to martensite whereas the situation is more complex for T R I P steels, as illustrated schematically in Figure 1-1. Dual phase steels with a ferrite-martensite microstructure represent an improved combination of strength and formability [11,12]. The Ultra Light Steel Auto Body ( U L S A B ) project suggested that by employing different grades o f D P steel (about 75% by weight) in the proposed advanced vehicle concept a total weight reduction o f 25% could be attained [3]. 4 General Introduction Chapter 1 The conventional chemistry range o f these D P steels is: 0.04-0.2 C , 1-1.5Mn and 0.2-0.4Si (wt%). In general, cold-rolled D P steels are directly quenched to room temperature after intercritical annealing i n ferrite+austenite region, such that the formation o f pearlite or bainite is suppressed [13]. The resulting ferrite-martensite structure provides attractive properties such as: absence o f stretcher strain, low yield/tensile strength ratio o f about 0.5, and an initial high work hardening rate that warrants good formability. The low yield point has been attributed to the presence o f mobile dislocations i n the ferrite matrix around the martensite islands [14,15]. Finely dispersed martensite, which acts as obstacles for further dislocation motion, interacts severely with dislocations resulting i n a high work hardening rate. The tensile strength o f D P steels is primarily controlled by martensite volume fraction in the structure. Carbon wt% Figure 1-1 Schematic representation of the processing stages of DP and TRIP steels. Adding more silicon or other alloying elements, which prevent carbide formation, to D P steel chemistries, about 1-2%, and introducing a bainite formation step after intercritical treatment as shown in Figure 1-1, a substantial amount o f austenite can be retained in the final structure [16,17,18]. Indeed, during partial bainite formation, austenite is further enriched with carbon such that it can be stabilized to room temperature, i.e. upon subsequent cooling the 5 General Introduction Chapter 1 transformation to martensite is unlikely to take place. The metastable austenite transforms to martensite during later deformation, e.g. in a stamping operation, thereby leading to the TRIP effect. The volume expansion associated with transformation results in a localized increase o f strain hardening during straining [15]. This postpones the onset o f necking that finally leads to very high uniform elongation and a superior formability index ( U T S x E l % ) . Moreover, the resulting martensite can also contribute to further strengthening the material. These TRIP-assisted multiphase steels are richer in alloying elements than conventional D P steels, i.e.: 0.15-0.4C, l - 2 M n and l - 2 S i (wt%) [10]. The two following principal ways have been proposed to develop TRIP multiphase steels, which are depicted in Figure 1-2: (i) Controlled rolling during hot rolling to obtain hot-rolled T R I P , (ii) The combination of intercritical annealing and isothermal holding in the bainitic reaction region, results in coldrolled TRIP steels. Due to high content o f Si in traditional T R I P steels to suppress carbide formation, the wettability o f the surface during the subsequent hot-galvanizing treatment is drastically degraded. This drawback triggered the development o f a new generation o f TRIP steels in which S i is partially or totally replaced by other elements such as A l or P [19]. Figure 1-2 Thermal and thermomechanicalpaths to produce TRIP steels. 6 General Introduction The design and development Chapter 1 o f the aforementioned high-strength steels have been accomplished by considering the physical metallurgy concepts that dictate the microstructure evolution in response to the processing route the steel is exposed to. A crucial stage in processing these advanced high strength steels is the austenite decomposition into ferrite and other, non-equilibrium transformation products. These transformations primarily take place on the run out table o f hot strip mills and/or occur during intercritical annealing treatments. Therefore, to obtain a final microstructure o f the required features in terms o f phase fraction, morphology and distribution, the appropriate control o f the austenite decomposition kinetics is crucial. Consequently, microstructure models have gained significant attention as predictive tools to optimize the processing parameters to aid the design o f novel steel grades. A n example o f these sophisticated microstructure models that have been implemented into the steel industry as a predictive tool, is the Hot Strip M i l l M o d e l ( H S M M ) [ 2 0 ] , developed at U B C . Here, to describe the transformation kinetics a semi-empirical approach is employed using the Johnson-Mehl-Avrami-Kolmogorov ( J M A K ) equation together with additivity rule [21]. Although the J M A K approach is a versatile modeling tool to describe the kinetics o f various transformation products, due to its entirely empirical nature it cannot provide any information regarding transformation mechanism. A s a further limitation o f this approach, the validity o f the adjustable parameters determined from experimental data, is confined to the cases that already have been examined in the laboratory. It is thus imperative to develop a more physically-based model that gives insight into the actual mechanism o f austenite decomposition. This constitutes the main motivation behind the present research and w i l l thoroughly be discussed in the following chapters. 7 General Introduction Chapter 1 1.2 Scope and Objective of the Thesis 1. The overall objective o f the present work is to develop a thorough model to describe precisely the microstructural evolution during austenite decomposition into ferrite and bainite, relevant to industrial processing o f hot-rolled advanced high-strength steels, such as D P and TRIP products. A s a unique feature, all components o f the proposed model w i l l be developed on a solid physical foundation such that the theoretical knowledge are integrated elegantly into an efficient predictive tool applicable to real industrial processing. Although some o f the individual model components had already been developed, they suffer from some conceptual flaws and artifacts that have to be first identified and corrected. Further, the novelty o f the present model development is the adequate couplings o f these model components and their application to complex cooling paths and steel chemistries. 2. Relevant to ferrite formation, the goal o f the first part o f this research is to develop a phenomenological physically-based model for the ferrite growth kinetics such that both thermodynamic and kinetic effects o f alloying elements are explicitly taken into account. The model is designed to describe non-isothermal ferrite formation under industrial processing conditions during which the reaction features early stage nucleation site saturation, polygonal morphology for the growing phase, and a rather fast growth rate preventing partitioning o f substitutional elements. 3. Regarding subsequent austenite decomposition into bainite, the second part o f the present research is devoted to experimental and theoretical characterization o f bainite evolution in materials featuring a composition similar to those adopted traditionally for T R I P steels. Available modeling approaches, e.g. diffusional [22] and displacive 8 General Introduction Chapter 1 [23], are examined and their predictive capabilities are identified. One o f the goals o f this section is also to define a potential criterion for transition from ferrite to bainite formation during continuous cooling treatments. Although the proposed approach is aimed to provide an overall austenite decomposition model that integrates ferrite and bainite formations, different modeling strategies are adopted for each transformation. Therefore they w i l l be discussed in two separate chapters. The thesis is organized as follows. Following the introduction, a brief literature review relevant to ferrite formation constitutes the first part o f the second chapter. Then the model development for ferrite is delineated. Next the experimental kinetics and its analysis using the proposed model is presented. The third chapter deals with the bainite reaction. Here, following the outline of the relevant literature, available modeling approaches are examined. In the last chapter, conclusions and recommendations are given for the overall austenite decomposition modeling. 9 2.1 Literature Review Chapter Chapter 2 2 : Austenite Decomposition into Ferrite 2.1 2.1.1 Literature Review Thermodynamics of phase transformation Characterization the thermodynamics o f a transformation is the first essential step in modeling the kinetics o f a reaction. Concerning the austenite decomposition to ferrite, thermodynamics provides information about ferrite-austenite phase boundaries and the driving pressure for the reaction. This is accomplished by modeling the composition- and temperature-dependence o f the Gibbs free energy o f both phases and subsequently by quantifying the chemical potential o f individual components in austenite and ferrite phases. Since the steels under investigation usually contain more than one alloying element, the interaction among individual components in solution has to be addressed adequately in the thermodynamic model. The mathematical description o f the Gibbs energy can be formulated in the framework o f the regular solution model in which the chemical potential o f species i is o f the following form: //,. ="//,. + RT In X + RT In y. (2-1) i where represents the standard energy o f the pure element /, X and y t i denote the mole fraction and activity coefficient o f component i, respectively, and RT has its usual meaning. The second term is related to the entropy assuming random mixing and the last part is included to show the deviation from ideality and normally is referred to as partial excess term. 10 2.1 Literature Review Chapter 2 The activity coefficient, y in multicomponent systems accounts for the interaction among h the different species. Usually, Wagner's interaction coefficients are employed which are defined as: .. 8 d In/,. — = (2-2) Using Taylor's expansion for I n l a n d adopting a standard state at infinite dilution for all the alloying additions, the appropriate expression for iron and solutes in either ferrite or austenite phases can be derived [24]: \ny =-XI2 J X * ^ Fe <') 23 i=C,Mn... \ny = k £s[X l (for k=C, Mn, ...) (2-4) i=C,Mn,... where X Fe constitutes and X { are the mole fraction o f iron and the alloying element, respectively. This the thermodynamic model based on which Kirkaldy and coworkers [25] developed a method to predict the phase equilibria in alloyed steels. It has to be noted that due to the lack o f experimental data, only the binary interaction is considered. In many cases the size, shape and electronegativity o f the constituents are not similar and the assumption o f random mixing is no longer valid. For instance, carbon occupies the interstitial lattice sites in ferrite and austenite, respectively. Thus, it is more realistic to consider two sublattices, i.e. interstitial and substitutional ones. The concept o f the regular sublattice model for ionic metals and interstitial solutions was first proposed by Hillert and Staffansson [26]. Their analysis dealt with the pair-wise mixing o f four components in two sublattices. In this approach iron and all the other substitutional atoms occupy one sublattice while carbon and vacant interstitials reside in the second sublattice. Further to express the concentration the site fraction is used instead o f mole fraction bearing i n mind that ferrite has three 11 2.1 Literature Review Chapter 2 interstitial sites per metal atom whereas austenite has only one. Then the site fraction for substitutional, 7,, and carbon, Yc, atoms is defined as: Y, = ~^ — (for i=Fe, M n , . . . ) J \-X (2-5) c c X Y =r\ ^— c '\-X (2-6) c where rj is 1 for austenite and 3 for ferrite considering octahedral interstitial sites. The description o f the Gibbs energy applicable to either ferrite or austenite containing only carbon and one alloying element is given in Appendix I. The theory o f regular sublattice was later extended to systems o f several sublattices and components by Sundman and Agren [27] with the goal o f developing a practical numerical recipe, which can be implemented easily into a computational thermodynamic software. The frequently used Thermo-Calc software employs this numerical procedure together with Gibbs energy minimization technique [28] to assess the phase equilibria in complex, multicomponents systems. It is worth to note that in case o f ferro-magnetism, i.e. for the ferrite phase, the magnetic contribution has to be accounted for in the mathematical description o f the Gibbs energy. The detailed analysis o f the magnetic effect by Hillert and Jarl [29] led to the following quantification for this extra term: G =RTg(T )\n(P +\) mngnetic where r m =T IT C m m (2-7) and TQ is the Curie temperature which is composition dependent and (3 m is the magnetic momentum. The function g(i )takes m Curie temperature (cf. Appendix I). 12 different forms above and below the 2.1 Literature Review Chapter 2 In ternary Fe-C-/ alloys and multi-alloyed steels containing one or more substitutional elements ferrite formation hardly occurs under complete equilibrium condition. This equilibrium state is referred to as so-called orthoequilibrium and can only be achieved in the whole system after sufficiently long time [30]. Therefore, due to kinetic restrictions the system is likely to adopt different meta-equilibrium states depending on partitioning behavior of alloying elements during austenite decomposition. Paraequilibrium (PE) [31, 32] and negligible partition local equilibrium ( N P L E ) [32,33,34] are the two main alternative descriptions that can be assumed to assess the thermodynamics o f the reaction to be examined. It has to be noted that these two states tend to establish at the migrating ferriteaustenite interface while orthoequilibrium implies a full equilibrium state in the whole system [30]. W e shall now provide a brief comment on these thermodynamic conditions. 2.1.1.1 Orthoequilibrium Orthoequilibrium is established when all the components partition completely between the parent and product phases. The orthoequilibrium phase boundaries are the locus o f equilibrium tie lines for various temperatures. A tie line connects two points between parent and product phases, for instance austenite and ferrite, respectively, at which the chemical potentials of each component in both phases are identical, i.e. [35]: $ ~ Mi (for i=Fe,C, Mn,...) (2-8) In binary alloys, the tie line is the common tangent to the free energy curves, h i a ternary system the tie line is located on the common tangent plane to the free energy surfaces and includes the projection o f alloy composition, X , as depicted in Figure 2-1 for the F e - C - M n 0 system. The lever rule can be applied to the tie line to assess the fraction o f each phase. 13 2.1 Literature Review Chapter 2 a y Mn Mn Figure 2-1 Free energy surfaces of yand a and its equilibrium tie line forX„. For a range o f alloy compositions, the projection o f various tie lines on the composition triangle, which defines the two-phase field region at a given temperature, is usually represented by a perpendicular-axis plot o f element A versus B , as illustrated in Figure 2-2 for the F e - C - M n system at 700°C. 0.08 c 0.06 o o n <c 0.04 o E e 5 0.02 0.00 0.00 0.01 0.02 0.03 0.04 C mole fraction 0.05 Figure 2-2 Iron reach corner, isothermal section of phase diagram for Fe-C-Mn at 700°C showing orthoequilibrium phase boundaries between ferrite and austenite, as well as the tie lines. 14 2.1 Literature Review 2.1.1.2 Chapter 2 Paraequilibrium During the conventional thermal treatment o f steels, transformation time is not sufficient to let the complete partitioning o f all components take place, i.e. the system cannot approach orthoequilibrium. However, for more mobile diffusing species, e.g. carbon, partitioning is kinetically possible. Paraequilibrium depicts the equilibrium state where only interstitial atoms are free to redistribute while the substitutional sublattice remains configurationally frozen during the transformation. Then the ratio o f mole fraction X/X Fe for each substitutional alloying elements i is the same in ferrite and austenite, respectively, and is given by its nominal bulk value: A Fe.o Fe A A Fe In contrast to carbon, the chemical potential o f substitutional atoms cannot be equal in ferrite and austenite, although the following expression can be derived in paraequilibrium conditions for substitutional elements: i whereas for carbon: Mc = Ml V-U) holds, Here, ju" and ju are the chemical potentials o f carbon in ferrite and austenite, r c respectively, whereas Aju ^" 1 and A/uf~* denote the chemical potential difference between a austenite and ferrite phases for Fe and substitutional elements i, respectively. In a ternary system, the above constraints imply that the tie lines have to be composition rays of the mobile species carbon [31]. This tie line is not identical to the orthoequilibrium tie line and does not fall on the common tangent plane to the two free energy surfaces. However, the 15 2.1 Literature Review Chapter 2 tie line is situated on the common tangent line to the free energy surfaces o f ferrite and austenite, which defines the carbon concentration i n each phases, i.e. X and X r PE PE as shown in Figure 2-3. A l o n g the component ray the mole fraction o f any component j can be found as a function o f carbon content based on the following mass balance equation: ^(l + 5>,-) + *c = 1 (2-12) The tie line i n paraequilibrium is actually the intersection o f two tangent planes, one to the free energy surface o f ferrite at X a PE and the other to the free energy surface o f austenite at X . Looking at the geometry o f this configuration, as shown i n Figure 2-3, it is evident that r PE conditions expressed by the above equations are satisfied. Figure 2-3 Tangent planes to the free energy surfaces of a and yin paraequilibrium condition. 2.1.1.3 Negligible partition local equilibrium Ortho- and paraequilibrium describe the two extremes regarding the partitioning o f substitutional elements. The intermediate state can be defined when local redistribution at 16 2.1 Literature Review Chapter 2 migrating ferrite-austenite interface tends to occur. The boundary o f negligible partition local equilibrium ( N P L E ) divides ternary isotherm into the two region o f l o w and high supersaturation [34]. In the latter region, slow-diffusing solute, e.g. M n , can only redistribute locally at the interface resulting i n a solute spike. The bulk concentration o f this element remains unchanged. However, the interfacial activity o f carbon is determined by this spike. Regarding N P L E condition, as also illustrated i n Figure 2-4, it is assumed that manganese bulk concentration o f ferrite and austenite are the same as its concentration i n the initial alloy, i.e. they all fall on the M n isoconcentration line [34]: Y ^ Mn,o — Y" — Y R ^ Mn.bulk (2-13) ^ Mn,bulk Carbon isoactivity v ; line XM xjNPLE^ \ X, M n isoconcentration line \ M n Spike x„ O Carbon Xint n 3O 3 Figure 2-4 Fe-C-Mn isotherm representing OE and NP-LE phase boundary [34], Therefore the carbon and manganese contents o f the interface at the ferrite side can be found simply. Further, local equilibrium o f M n at the interface necessitates that: /*Mn,mt ( at X Mn = X nMlk M ) = 17 MMnjnt ( at Mn X ~ Mn,int X ) (2-14) 2.1 Literature Review Chapter 2 The above equation gives point O E , i.e. the interfacial manganese concentration at the peak of the M n spike, which governs the activity o f carbon at the interface, such that the intersection o f the carbon isoactivity line with M n isoconcentration line determines the interfacial carbon concentration at the root o f the manganese spike, i.e. point N P L E . 2.1.2 Austenite to ferrite transformation kinetics 2.1.2.1 Nucleation The formation o f a new phase is initiated by nucleation, where due to thermal fluctuation o f atoms a number o f clusters in form o f the new phase emerge inside the parent phase, preferentially at heterogeneous sites. In general, these clusters are not necessarily o f the equilibrium crystal structure and composition o f the product phase and nucleation can primarily be facilitated by formation o f metastable lattices. However, in case o f ferrite nucleation no transition metastable phases have been observed. Those clusters that are larger than a critical size provide stable nuclei. Tracking the evolution o f nuclei and characterization o f their structure, in particular the interface as well as the shape, are a challenging task for available experimental techniques due to the very tiny scale o f a nucleus, e.g. about 10 to 1000 atoms. This constitutes a major challenge for the study o f nucleation phenomena and modeling its kinetics. However, the recent emergence o f X-ray synchrotrons capable o f 3D data analysis and high energy neutron beams promises in-situ probing o f dynamic and meso-scale atomic phenomena, such as nucleation in the solid state [36]. The pioneering attempt to characterize the nucleation o f individual grains using these new facilities by Offerman et al.[36] provides new insight into this challenging area, yet more 18 2.1 Literature Review Chapter 2 mature software to interpret the raw data and finer time resolution o f data acquisition are required. Lacking intragranular inoculants, e.g. intermetallic inclusion, shear bands and subgrain boundaries, ferrite nucleates heterogeneously on potential sites, i.e. austenite grain corners, grain edges and grain faces, respectively [37]. To minimize the interfacial energy a semi coherent ferrite-austenite interface tends to appear giving rise to preferred orientation relationships between parent austenite and ferrite nuclei, e.g. { l l l } / / { 1 1 0 } . Enomoto et al. y a [38] compared nucleation rates calculated for grain edges and faces with experimental data in binary and ternary Fe-C alloys and found that only at very low undercooling does nucleation on edges dominate. Therefore, the total nucleation rate correlates with faces for most transformation temperatures. They also showed that the amount o f undercooling at which face nucleation becomes predominant depends on prior austenite grain size. A t smaller grain size, the temperature range in which edge nucleation dominates should be larger. A s the austenite grain size decreases, the ratio o f the grain edges length to the grain faces area per unit volume o f austenite increases, this leads to higher probability o f nucleation at grain edges. The nucleation rate o f ferrite at the austenite grain boundaries can be expressed by the classical nucleation theory, i.e.: — AG* —x I = NP'Z e x p ( — — ) e x p ( — ) kT t (2-15) where TV is the density o f viable atomic nucleation sites, the frequency factor pt is the rate at which single atoms are added to the critical nucleus, factor, AG Z is the Zeldovich nonequilibrium is the activation energy for critical nucleus formation and x is the incubation 19 2.1 Literature Review Chapter 2 time. The time-independent portion o f this equation is referred to as the steady state nucleation rate, I , when s t»r. To apply classical nucleation theory to ferrite nucleation, the shape o f the critical nucleus should be determined. This shape has a pronounced effect on the calculated nucleation rate, thus the selected model should provide consistent results with experimental values. If the interphase boundary is assumed to be incoherent featuring isotropic interfacial energy, then the interface is likely to be evenly curved such that the critical nucleus tends to form at grain faces in the shape o f two spherical caps. Following the same argument, the newly formed particle at grain edges would be bounded by three equivalent spherical surfaces with similar dihedral angles. A t grain corners the nucleus would be encompassed by four such spherical surfaces. The geometries, in particular the spherical cap shape, were frequently used in early studies to analyze the nucleation kinetics. For instance, adopting the aforementioned models for the critical nucleus with incoherent interfaces, Clemm and Fisher [39] performed a detailed theoretical analysis regarding nucleation at austenite grain boundaries, which yielded the following expression for the activation energy, A G * : K r ,* AG 4 (z cr -z^ryf 2 = - -— r 27 IL (2-16) z AGy 3 where <j and cr^ denote interfacial energy for interphase and grain boundaries, respectively, ay z/ , Z2 and zj are the factors to be determined from the assumed geometry and the value considered for the ratio o f (Jaylcjyy, and AG represents the volumetric Gibbs energy change V for ferrite formation. For grain face nucleation the value reported for the geometrical factors zj, Z2 and z , are 3.7, 1.6 and 0.48, respectively, when the ratio o f 0-^/0^=0.7 was assumed. 3 20 2.1 Literature Review Chapter 2 Although the spherical cap model can be treated easily and assumes only a single interfacial energy for the entire ferrite-austenite interface, the classical nucleation theory assuming this nucleus shape fails to describe the experimental nucleation rates [36,40]. Therefore some modification, in terms o f the geometry and the coherency o f the embedded surfaces have been discussed. Among the efforts to quantify the nucleation rate, the measurement carried out by Lange et al. [40] offered a thorough series o f experimental data. The study was undertaken by means of electron and optical microscopy on high purity Fe-C alloys with three different carbon contents, i.e. 0.13, 0.32 and 0.63wt%C. The particular technique employed in their treatment allowed them to discriminate between corner and edge nucleated particles. Further by appropriate correction, the measured number o f nuclei in the plane o f polish was converted to the surface density o f particles in the unreacted austenite grain faces. In terms o f modeling, Lange et al. [40], adopted classical nucleation theory and incorporated both the pillbox and the spherical cap models to evaluate the steady state nucleation rate o f ferrite at the austenite grain boundaries. They found that only the pillbox nucleus shape with low interfacial energy can give satisfactory results compared to the experimental measurements for steady state nucleation. Figure 2-5 shows a vertical section o f a disc-shaped pillbox nucleus. ~r Y a Figure 2-5 The vertical section of a coherent pillbox nucleus [40]. Following classical heterogeneous nucleation theory, Lange et al. [40] developed an expression for the steady state nucleation rate using the pillbox coherent model: 21 2.1 Literature Review where, %= a c ay + <j° ay Chapter 2 - <7 , yr D , is the appropriate diffusivity o f the rate controlling species with an atomic fraction X i n the austenite matrix prior to transformation, V is the molar t volume o f ferrite, a L a is the average lattice parameter o f ferrite and austenite, AGv is the free energy change, W is the strain energy associated with the nucleus (assumed to be zero), cr, indicate the interfacial energies as shown i n Figure 2-5. The radius and the height o f the critical nucleus, r* and h*, are: r = - (2-18) — (2-19) y A G , h = AG V Applying the appropriate parameters to Equation (2-17) is a challenging task [41]. The unknown parameters that have to be quantified correctly are: N, D , x> a n o t - AGy. Starting with the binary Fe-C system, the appropriate diffusivity o f carbon i n austenite [42,43] can be used and AGv can be calculated assuming the critical nuclei has a composition of either equilibrium or the one that provides the maximum energy change [44,45]. Then on the assumption that the pillbox height is one lattice parameter [41], % is obtained from Equation (2-19). Further, the value for <j should fall in the range o f 20 to 40 m J / m in Fe-C 2 e alloys [41]. Then to reflect the experimental measurement o f i] with Equation (2-17), N w a s 8 • 2 estimated as 10 sites/cm for Fe-C alloys, which can also be used in ternary and quaternary systems [41]. This potential nucleation site density is substantially smaller than the number of available sites at austenite grain boundaries, i.e. 2 x l 0 22 1 5 /cm . 2 2.1 Literature Review Chapter 2 In ternary and quaternary systems, more complexity is involved in determining the above parameters. Three different alternatives can be considered for the nucleation process: • paraequilibrium i n which carbon volume diffusion in the austenite is rate controlling • orthoequilibrium with mass transport controlled by volume diffusion o f substitutional element i in austenite • orthoequilibrium limited by grain boundary diffusion o f i. Enomoto and Aaronson [46] found that orthoequilibrium with grain boundary diffusion o f i is the mechanism controlling ferrite nucleation in Fe-C-/ alloys. In a more delicate study, Tanaka et al. [41] analyzed the nucleation in F e - C - M n - / alloys and suggested either paraequilibrium or orthoequilibrium limited by grain boundary diffusion o f i to obtain a nucleation rate consistent with experimental measurements. They also proposed that the possible synergism o f i and M n in diminishing ferrite nucleation rates has to be considered. So far, the coherent pillbox model with a radius o f the critical nucleus size represents the most consistent way to analyze the nucleation kinetics o f ferrite allotriomorphs at austenite grain faces in F e - C , Fe-C-/ and F e - C - M n - / [41]. However, it has been developed based on a number o f controversial assumptions, which are difficult to verify. Moreover, Offerman et al.[36] reported that their measured nucleation rate could not be replicated by a pillbox model assuming interfacial parameters suggested in the literature [36] such that decreasing the activation energy by at least two orders o f magnitudes is required to provide a decent agreement between experiment and prediction. Clearly, to resolve this controversy more detailed nucleation studies appear to be required and are now feasible with state-of-the art characterization techniques. 23 2.1 Literature Review Chapter 2 2.1.2.2 Local equilibrium model for thickening offerrite in Fe-C Temporarily setting aside the effect o f interface reaction, growth kinetics o f ferrite allotriomorphs in Fe-C alloy is controlled by mass transport o f carbon. Since the diffusivity of carbon i n ferrite is comparatively high, the carbon concentration within growing ferrite is uniform, equal to c a and excess carbon is rejected into austenite through the a/y interface. Thereby carbon redistribution i n austenite is the rate-controlling process. During the early stage o f thickening, particularly i n coarse-grained specimens, the moving interfaces, which have an essentially disordered structure, can be described with reasonable accuracy as planar [47]. Considering a carbon mass balance i n the system and assuming local equilibrium at the a/y interface, a linear carbon profile in semi-infinite austenite ahead o f the interface and constant carbon diffusivity i n austenite, Zener generalized the Dube analysis [48] and derived an analytical solution for parabolic growth o f ferrite allotriomorphs given by the following equation: (2-20) where s is the thickness o f the ferrite plate, c is initial carbon content o f the alloy, and c" 0 q and c are equilibrium carbon content o f ferrite and austenite, respectively as depicted in y eq Figure 2-6. Although Zener's simplifications are not a solution o f the underlying diffusion problem, analytical solutions for steady state diffusion problems confirm parabolic growth rates before soft impingement takes place. 24 2.1 Literature Review Chapter 2 Figure 2-6 Carbon concentration profile along a direction perpendicular to the a/y interface. To describe the overall growth kinetics including overlapping diffusion fields and concentration dependence o f diffusivity, numerical solutions have been invoked [49,50,51]. The main underlying principles adopted by this kind o f growth analysis can be summarized as follows: • Early site saturation takes place even at low undercooling. • Local equilibrium prevails at the interface. That is, the carbon concentrations at both sides o f the a/y interface are given by the equilibrium tie line. This implies that the net flux o f carbon at the interface is zero, so that the incoming flux, /,„, due to carbon rejection from the growing ferrite is balanced by the outgoing flux, J ouh as result o f carbon redistribution in the remaining austenite, i.e.: J. =v(c "in V eq -c ) r J J = -D y out M =J nel a dc (2-22) dz -J =o in oul where v is the interface velocity. 25 (2-21) eq) (2-23) 2.1 Literature Review • Chapter 2 To account for soft impingement due to overlapping diffusion fields within the austenite grain, the center o f the austenite grain is a point o f zero mass transfer: dc — = 0 atz =i o (2-24) dz where 21 0 • is the austenite grain size. Concentration dependency o f carbon diffusivity is incorporated, for instance by using the carbon diffusivity proposed by Agren [43] or by adopting the Trivedi-Pound [52] approach o f a weighted average o f the diffusion coefficient D . Another critical factor that has to be selected is the geometry o f growth. Planar geometry can describe the diffusion situation only at the early stage o f allotriomorph growth. A t the later stage spherical geometry appears to be more appropriate where an outer shell o f ferrite grows inward. Planar and spherical geometry are simple but do not reflect actual geometries. Thus, a model for the spherical growth o f ferrite nucleated at the corner o f Tetrakaidecahedron austenite grains, has been developed recently [53]. However, the unfilled space at the end o f growth remains a controversial issue i n this approach. Figure 2-7 shows the spherical and planar model geometry. B y applying the above assumptions the calculated ferrite layer thickness and its growth rate agree well with experimental values [50,51]. Therefore, diffusional controlled growth o f ferrite in Fe-C alloy is reasonably well justified. Kamat et al. [51] compared planar and spherical geometry and suggested that spherical geometry is more suitable to describe the overall kinetics o f ferrite growth. 26 Figure 2-7 Schematic illustration of the tetrakaidecahedron model for an austenite grain and the planar and the spherical geometry used in the diffusional model [51], 2.1.2.3 Growth of ferrite in ternary and other multi-alloyed steels In Fe-C alloys the carbon content o f growing ferrite from parent austenite is clearly defined by the equilibrium phase diagram asserting that the long-range redistribution o f carbon occurs during ferrite evolution, no matter how quickly the reaction proceeds depending on the applied cooling rate or undercooling. This is associated with the high diffusivity of carbon, as an interstitial species, in the iron lattice. In the presence o f substitutional alloying ferrite can grow with or without equilibrium composition depending on the partitioning degree o f these slow diffusing elements between the parent and product phases. A t sufficiently high temperatures, i.e. close to A , the redistribution o f the substitutional atoms e3 is thermodynamically required. The growth kinetics is then governed by long-range diffusion of these species in both participating phases and the equilibrium tie line constitutes the interfacial composition [54]. O n the isothermal section o f a ternary steel, Fe-C-z as illustrated in Figure 2-8(a), this tie line is indicated by a/ and y t composition o f y . o for an alloy featuring nominal The growth rate is relatively slow and eventually ferrite evolves with 27 2.1 Literature Review Chapter 2 equilibrium content o f all components, i.e. orthoequilibrium would be established at the end of transformation. Including in the left side o f Figure 2-8(a) is also a typical composition profile o f solute i in ferrite and austenite. The orthoequilibrium formation o f ferrite is hardly encountered in practice, however during nucleation due to very short diffusion distances, one can also presume it to be established. (a) (b) Figure 2-8 Schematic isothermal section of Fe-C-i system representing the interfacial concentration for ferrite growth under OE and NPLE conditions when the nominal composition of alloy is different. For the alloys inside the N P L E region, e.g. as indicated by y' a in Figure 2-8(b), the supersaturation is high and the interface migrates faster than what substitutional controlled growth permits. In these circumstances ferrite tends to form with substitutional concentration close to the parent austenite, i.e. point «/ and local equilibrium might be held at the interface such that the interfacial composition is given by j2. This yields to the evolution o f a substitutional spike in front o f the migrating interface with a height o f y -y' x 0 . This spike is illustrated in the left side o f the plot in Figure 2-8(b). The kinetics o f ferrite formation behind this spike is then controlled by carbon diffusion away from the interface with carbon content as defined by the boundary o f N P L E [32,34]. The term quasi-paraequilibrium was also 28 2.1 Literature Review Chapter 2 suggested to ferrite formation under this condition [35,54], although N P L E seems to be more accepted in the literature [34]. The spike o f substitutional elements is likely to evolve with time and it is more realistic that ferrite first forms with paraequilibrium composition, i.e. this situation is characterized by partitioning o f carbon and a frozen concentration profile o f substitutional species. The growth under paraequilibrium condition occurs without any substitutional rearrangement even locally adjacent to the interface. However as the reaction further proceeds the growth rate tends to slow down, due to either exhausting the thermodynamic driving pressure and/or overlapping the carbon diffusion fields, then limited partitioning o f substitutional elements close to the interface can be facilitated. This gives rise to the gradual evolution o f a spike at the transformation front and constitutes a kinetic transition from paraequilibrium to N P L E condition [55,56]. Although it was addressed in early work [57], the potential transition between different growth modes has been an appealing subject for research worldwide recently. A review on deviation from local equilibrium and its rationale was provided by Hillert [54]. O i et al. [58] investigated experimentally the boundary o f kinetic transition from partitioning to no partitioning growth o f ferrite in different F e - C - M n and F e - C - N i alloys. Although this transition is defined theoretically by the N P L E boundary, they observed that the exact boundary lies well above the N P L E yet inside the paraequilibrium region. The solute drag effect o f substitutional elements, as quantified by Purdy and Brechet [59], was invoked to describe these findings such that in order to initiate the growth, the potential drag pressure has to be overcome by sufficient driving pressure i.e. sufficient undercooling below the P E boundary is required. 29 2.1 Literature Review Chapter 2 Odqvist et al. [55] have also analyzed the kinetic transition between P E and N P L E during isothermal ferrite formation theoretically in F e - C - N i alloys. Attempting to map the composition path taken by the interface during this transition, as shown i n Figure 2-9, they employ an energy balance at the interface which facilitates finding simultaneously the interfacial concentration and the growth rate. Their quantification suggested that at the start of the reaction the interface features P E concentrations, i.e. point P. However, as growth proceeds the interfacial condition can no longer be at P E and a solute spike tends to form as the system approaches gradually the N P - L E condition, i.e. point F which shows the height o f the solute spike. Apparently, depending on the initial composition o f the alloy different composition paths would be predicted. isoaciiviiy line for C in y 1 y OF /& A? k ).05 u-fraction C Figure 2-9 The composition path of interfacial condition during isothermal growth [55]. For a moving boundary, the interfacial energy balance asserts that the total energy dissipation has to be equal to the thermodynamic driving pressure available at the interface. The energy dissipation for moving boundaries arises from various irreversible processes. For example, energy is dissipated by diffusion to carry forward a segregation profile inside and a solute spike in front o f a migrating interface. Further, i n case o f assuming a finite mobility for the 30 2.1 Literature Review Chapter 2 interface a contribution due to interface friction shall be introduced as well. Although it was predicted how the interfacial concentration deviates from local equilibrium as growth rate increase, Odqvist et al. did not elaborate what parameters would have to be adjusted in their model for description o f experimental kinetics. Recently Hutchinson and Brechet [56] proposed a transition between P E and N P L E to describe isothermal growth o f ferrite i n F e - C - N i systems where they had performed experimental studies. The ternary steels containing N i were chosen by virtue o f insignificant solute-interface interaction. Adopting a carbon diffusion model, it was revealed that in the early stage o f growth the experimental kinetics agree well with the prediction using P E boundary condition while the subsequent ferrite formation can be described only by assuming N P L E interfacial concentration. This implies that for the alloy inside the N P L E region the growth would virtually stop, as the average carbon content o f the untransformed austenite approaches the N P L E phase boundary. It was also observed that in alloys which lie above the N P L E limit the growth o f ferrite occurs rather quickly at early stages and almost ceases at later stages. The maximum ferrite fraction was also detected to be different from P E prediction in this situation. 2.1.2.4 Retarding effect of solute atoms In addition to accounting for the partitioning o f solute atoms, the mutual interaction o f the moving ferrite-austenite interface and alloying elements has to be considered in the kinetic model. Due to the presence o f a high density o f structural imperfections at grain and interphase boundaries, segregation o f impurity atoms is likely at these interfaces, even in relatively high purity alloys. In case o f moving boundaries, the segregation exerts a drag 31 2.1 Literature Review Chapter 2 force that yields to a drastic drop in the interface migration rate. This retarding effect of solutes on moving interfaces, which is referred to as Solute drag effect, was first discovered and analyzed in grain boundary migration during recrystallization and grain growth [60,61]. This idea was later extended to the migration o f interphase boundaries [62] to explain the apparent discrepancies observed between measured and predicted kinetics. For instance, to resolve the disagreement between the measured and predicted kinetics o f ferrite evolution, Militzer et al. [63] attempted to incorporate the solute drag effect into a diffusional model. They proposed a segregation model for F e - C - M n alloys where the solute drag o f manganese at the ot/y interface changes the interfacial carbon activity. Although, their approach offered a decent description o f the experimental results, it was a phenomenological treatment in which the velocity dependence o f solute segregation was not addressed. A detailed quantitative description o f solute-interface interaction for moving grain boundaries was originally proposed by Liicke and Detert [60], and Cahn [61]. The term drag force stems from their treatment, in which the binding force on the segregated atoms inside the migrating interface region is integrated into an overall solute drag force. Assuming a symmetrical wedge shape well for the interaction potential inside the grain boundary and solving the flux equation for solute in this region, the segregation profile, c (x), s can be calculated. For moving boundaries, due to an asymmetry in the segregation profile the net drag pressure is not zero and it can be determined by accounting for the contribution o f all segregated atoms over the entire interface region, i.e. from -8 to +5. According to Cahn's treatment, the solute drag pressure, P$d, which can be converted to the free energy dissipated per mole o f substitutional atoms [64], is calculated by: 32 2.1 Literature Review Chapter 2 f AG SD where V and c m =P V SD m = - \[c (x)-c ] -a s So Oh* — dx * (2-25) c are the molar volume and the bulk concentration o f substitutional atoms, So respectively, and OE I dx denotes the gradient o f the interaction profile i n grain boundary region. The term c s 0 was introduced into the above integral by Cahn and constitutes the main difference between his approach and that proposed by Liicke and Detert [60]. Introducing an asymmetrical interaction potential, Purdy and Brechet [59] extended Cahn's solute drag theory to moving interphase boundaries. Alternatively, the kinetic effect of solute segregation to moving interfaces has been treated as Gibbs energy dissipation by Hillert [65] and was later extended to interphase boundaries for partitionless phase transformations by Hillert and Sundman [66]. A l l these treatments assume the interaction o f a single species with the moving interface, however recently a number o f attempts to model co-segregation have been reported in the literature as well [67,68]. Adopting the Purdy-Brechet approach to quantify the drag effect o f M o , Purdy et al. [69] attempted to model the formation o f the distinctive bay i n F e - C - M o alloys. O i et al. [58] evaluated the maximum driving force required to overcome the solute drag force i n Fe-C-z systems, thereby indirectly estimating a solute-boundary binding energy o f &RT for M n and &0.5RT for N i . Enomoto [64] refined the Purdy-Brechet theory by including the effect o f carbon cosegregation with M n on the ferrite-austenite interphase, i.e. considering a non ideal solution and introducing the Wagner's description o f the interaction between carbon and substitutional solutes. However, Enomoto's approach to interpret the drag force in terms o f energy dissipation was criticized by Hillert [70] who believes that his description o f energy dissipation has to be discriminated from the drag force defined by Cahn. Only when the 33 2.1 Literature Review thermodynamic properties Chapter 2 inside the interface region are invariant with respect to composition and location [70], Calm's treatment provides similar results to those Hillert's approach suggest. It has to be noted that the treatment o f Purdy-Brechet fails to predict a zero drag pressure for stationary interfaces. This artifact is not significant in the calculation provided in the original paper, since the assumed binding energy was taken to be much larger than AE. Hillert and his coworkers [70] argued that neglecting the spike o f substitutional elements in front o f the moving interface yields the artifact in the Purdy-Brechet approach. A t steady state the height of this spike is related to AE, i.e. to the partition coefficient k =exp(-AE/RT), r/a and by assuming an ideal solution model for austenite, the energy consumed by pushing the spike ahead is given by [70]: = -X \nk-" +(l-X ) In a i0 i0 V X i '°~ l l (2-26) ' 1,0 According to Hillert and Sundman [66] the drag force calculated using Cahn's definition, i.e. Equation 2-25, includes as well the energy dissipation by the spike which is o f finite value for AE * 0 . Therefore to predict a zero drag pressure for a stationary interface the contribution from the spike has to be accounted for [70], i.e.: AG °r c led s =AG -G SD Spike (2-27) This correction might be sensible in the presence o f a spike, i.e. when N P L E prevails or the interface deviates from P E and is in transition towards N P L E . However, as discussed before depending on reaction temperature and supersaturation, the ferrite-austenite interfaces can move relatively fast such that the interfacial condition is likely to remain at paraequilibrium. In these situations a solute spike does not existed in front o f the interface and further, the 34 2.1 Literature Review Chapter 2 chemical potential difference o f substitutional species is not zero, i.e. AE * 0 . Therefore, for ferrite formation under paraequilibrium condition alternative means to correct the aforementioned artifact are required. This problem w i l l be revisited in section 2.2.2.4. 2.1.2.5 Mixed-mode model for ferrite growth Austenite decomposition to ferrite involves the reconstruction o f B C C from F C C crystal that is accomplished by the individual jumping o f substitutional atoms across the interface. Simultaneously, carbon partitioning has to take place, presumably at a rate much higher than this interface reaction. However, i f the intrinsic interfacial friction becomes appreciable, including the effect o f interface reaction on growth kinetics is required. According to the theory o f thermally activated growth [71], the interface velocity, v is proportional to the driving force for interface migration, as given by: v - M AG (2-28) where AG is the Gibbs free energy difference per mole across the interface and M is the interface mobility. In addition to O E , P E and N P L E assumptions i n determining the driving force for interface migration, this parameter can be calculated based on the local chemistry at the interface or according to the average global concentration predicted by a "mean field" approach. In contrast to the local equilibrium approach o f diffusion models, the interfacial carbon concentrations are not constant. Carbon builds up at the interface as growth proceeds, it changes from the initial concentration to approach the equilibrium carbon concentration o f austenite, e.g. from c to c , respectively. This change o f interfacial carbon content is related r 0 eq to non-zero carbon net flux at the interface and can be expressed by the following mass transport equation: 35 2.1 Literature Review Chapter 2 Ac AJ, net (2-29) v inl Carbon volume diffusion and interface reaction are coupled together by the above equation to give the momentary interfacial carbon concentration, i.e. growth kinetics is mixed-mode controlled. V a n der Zwaag and his co-workers describe the kinetics o f austenite decomposition in Fe-C, Fe-C-z and Fe-/ systems successfully with the mixed-mode approach [72,73,53]. Although the model predictions seem promising, the interface mobility is treated as a fitting parameter which i n some cases yields to unusual outcome, e.g. cooling rate dependency o f the fitted mobility. Further, the interaction o f manganese with the interface was neglected. Thus the proposed mobilities have to be considered as effective values. 2.1.2.6 Overall kinetics offerrite formation The overall kinetics o f isothermal decomposition o f austenite to ferrite, agrees well with the sigmoidal curve o f the Avrami-equation [74] and its later modified version proposed by Umemoto [75]: fa = 1 - e x P(-7^) (2-30) y Here / i s the normalized fraction o f ferrite, b, n and m are empirically determined constants. a In general b is a kinetic parameter that represents the combination o f nucleation and growth rates and then depends on temperature. The n and m values i n some cases are considered to be constant. The nominal values [76] for these two parameters expected for different nucleation situations, i.e. nucleation site saturation and nucleation and growth are outlined in Table 2-1. 36 2.1 Literature Review Chapter 2 Table 2-1 Nominal values for n and m parameters [76] Site saturation Nucleation and growth Nucleation sites n m n m Grain surface 1 1 4 1 Grain edge 2 2 4 2 Grain corner 3 3 4 3 Campbell et al. [77], have found an average value o f n equal to 1.16 (0.88<«<1.33) and Umemoto [78] reported values o f n i n the range o f 1.2-1.3. These values are less than that o f pearlite, which fall i n the range o f 1.9-2.7 [77], and suggest ferrite nucleation takes place i n the early stages o f transformation such that nucleation site saturation occurs. The m value reported by Umemoto [75], is between 1.2 and 1.3, which verifies nucleation occurs mainly on grain surfaces. A model for the overall kinetics o f ferrite formation during continuous cooling processes can be developed by adopting the rule o f additivity [79] and applying the A v r a m i approach. A transformation is additive i f the criterion o f an isokinetics reaction is fulfilled. Mathematical interpretation o f this criterion is that the rate o f transformation has to be expressed by two separate functions o f fraction transformed and temperature, such that: f = ^- = H(f)G(T) at (2-31) The differential form o f A v r a m i ' s equation, as given by the following expression, suggests that to satisfy the above condition n has to be constant and only b is a function o f temperature: ^ = b " { n ( l - / ) [ - ln(l - / ) ] ? } U 37 (2-32) 2.1 Literature Review Chapter 2 Assuming an average value for n o f about 0.9 and finding out the temperature dependency o f b, one can model the overall kinetics of ferrite formation during continuous cooling for lowcarbon steels, readily [21]. 38 2.2 Model Development Chapter 2 2.2 Model Development The purpose o f this.chapter is to delineate in depth the modeling methodology and strategy based on which the kinetic model for austenite to ferrite transformation has been developed. The prime goal o f the present microstructural model is to describe the ferrite formation under industrial processing conditions, where the transformation kinetics is solely dictated by ferrite growth, i.e. the transformation starts from an austenite-ferrite microstructure or the applied cooling paths are such that nucleation site saturation occurs at the early stages o f the reaction. Therefore, the current model deals with the growth kinetics o f ferrite and to accomplish this, the mixed-mode approach was adopted as the main framework. Moreover, the kinetic effect o f alloying elements, i.e. solute drag, was incorporated into the model in a rigorous way. Depending on the partitioning degree o f substitutional elements, appropriate thermodynamic description o f the system, i.e. P E or N P L E , can be selected for the simulations, however the transition between them is not considered here. Integrating all the aforementioned features, the overall model consists o f several modules, i.e. thermodynamic routine, carbon diffusion, mixed-mode module and solute drag part, which w i l l be elaborated in the following sections. 2.2.1 Characterizations of the thermodynamics of ferrite formation Thermodynamic information, which is an essential input into any kinetic model, is evaluated by means o f Thermo-Calc version N . Thermo-Calc is a commercial/academic software developed by the Royal Institute o f Technology in Stockholm. This integrated software utilizes the Fe2000 database to get the required data for low alloy steels. Table 2-2 indicates 39 2.2 Model Development Chapter 2 the list o f elements included in the Fe2000 database and the maximum permitted level for each element provided that the total alloying does not exceeds 50 weight percent. Table 2-2 List of elements included in Fe2000 database in their maximum permitted level (wt%). Al B C Co Cr Cu Mg Mn Mo N Nb Ni O P s Si Ti V W 5 t 2 15 30 1 t 20 10 1 5 20 t t t 5 2 5 15 t: trace The kinetic model i n this study has been developed under C O M P A Q visual F O R T R A N version 6.5. To link this application program to Thermo-Calc, the TQ-interface is incorporated. Thermo-Calc version N is capable o f evaluating the thermodynamic parameters only under complete equilibrium conditions, therefore for ferrite formation under orthoequilibrium assumption the phase boundaries, chemical potentials and driving pressure can be assessed readily. However, to quantify the thermodynamic information o f a multicomponent system under paraequilibrium and negligible-partition local equilibrium assumptions appropriate subroutines have to be developed. 2.2.1.1 Paraequilibrium treatment For the sake o f transparency, the procedure steps to calculate the P E thermodynamic parameters o f a ternary steel, i.e. Fe-C-z , are described here, although the approach can be extended simply to higher order alloying steels as well. Fortunately, the configuration o f Gibbs energy o f ferrite and austenite phases in P E condition can be presented as a pseudo binary diagram. Therefore, a section through the free energy surfaces o f austenite and ferrite along the concentration ray o f carbon is first made with the help o f Thermo-Calc, which is schematically shown i n Figure 2-10. A l o n g the composition axis, which is characterized by 40 2.2 Model Development Chapter 2 carbon and an imaginary substitutional component, i.e. C and S C , carbon mole fraction changes while the ratio o f the mole fraction o f substitutional elements to iron remains constant (cf. Equation (2-9). F o r a given carbon mole fraction Xc, the pseudo-chemical potential o f the substitutional component i n phase j, can be expressed i n terms o f the chemical potentials o f iron and alloying atoms, ju (X ) and p.?(X ), J Fe c c respectively, as given by: (2-33) Msc ( c ) x The paraequilibrium tie line, i.e. the carbon equilibrium composition o f ferrite, X , and PE austenite, X , is then determined by the common tangent line to the free energy curves, i.e.: PE 8G a G +(x; -x ) a a E PE G r dX (2-34) r dG dG dX dX a y r (2-35) r i_ c a) Q) </> n n CD " PE ^ PE Carbon mole fraction Figure 2-10 Section of the free energy surfaces offerrite and austenite along the concentration ray of carbon for a given X/X . ratio. F< 41 2.2 Model Development Chapter 2 In order to calculate the first derivative o f G and G with respect to carbon mole fraction a a 7 continuous function is required, therefore a S P L I N E curve was fitted to each Gibbs energy section of ferrite and austenite. The F O R T R A N subroutine developed for assessment o f P E phase boundary is included i n Appendix II. A s an example the paraequilibrium phase boundaries were calculated at 700°C for the F e - C M n system and the results together with those for orthoequilibrium, are illustrated as a ternary isothermal section in Figure 2-11. Apparently, the paraequilibrium boundaries fall inside the orthoequilibrium region, moreover its tie lines are almost parallel to the carbon concentration axis. The essential thermodynamic parameter that the mixed-mode kinetic model employs is the driving pressure, based on which the velocity o f the ferrite-austenite interface is evaluated. It should be kept in mind that equilibrium concentration cannot be attained at interfaces with finite mobility. Therefore, adopting the paraequilibrium treatment for the mixed-mode growth is referred to here as the case in which the interfacial driving pressure is determined with respect to paraequilibrium such that the system is forced to approach this constraint equilibrium when growth ceases. A similar concept could be applied i f other thermodynamic description, e.g. N P L E , were assumed. In the case o f paraequilibrium treatment, during the mixed-mode ferrite formation, the interfacial composition in the parent phase, X\ , nt changes from the initial carbon concentration o f the alloy, X , to the paraequilibrium carbon content o f austenite, X , r 0 PE as growth proceeds. The interfacial carbon concentration in ferrite, X* , is then chosen such nt that there is no difference in the chemical potential o f carbon across the interface. Thus, Xf nl changes in accordance with the value o f Xj . Because o f the higher diffusivities in ferrite the nt 42 2.2 Model Development Chapter 2 carbon redistribution is assumed to be instant, i.e. the carbon concentration i n ferrite is uniform and given by the interfacial value, Xf . nt 0.08 t 0.0000 , 0.0005 0.0010 0.0015 0.0020 C mole fraction (b) Figure 2-11 Isothermal section at 700°C for Fe-C-Mn system, (a) shows paraequilibrium and orthoequilibrium phase boundaries as well as OE tie lines, (b) shows the iron rich corner. The thermodynamic driving pressure, as depicted in Figure 2-12, is evaluated based on the local chemistry at the interface, i.e.: 43 2.2 Model Development Chapter 2 AG,,, = (1 - XI )[^ {XI) - ju (XI) J (2-36) a sc sc \ 7 1 CD c CD CD CD Ug,, ( W SI (Xo) i 0 I PE = Mc ( X L ) ! 1 O Carbon mole fraction Figure 2-12 Paraequilibrium pseudo-binary diagram illustrating the calculation of the chemical driving pressure at the start of ferrite growth where X y inl = X. 0 2.2.1.2 Negligible partitioning local equilibrium boundaries The thermodynamic procedure described in section 2.1.1.3 was adopted to calculate the N P L E boundary. A typical calculation o f N P L E phase boundary at 700°C for F e - C - M n together with paraequilibrium and orthoequilibrium boundaries, are plotted in Figure 2-13. The F O R T R A N code to perform the aforementioned procedure is given in Appendix III. The required data is retrieved from Thermo-Calc by calling appropriate subroutines defined in the T Q interface. 44 2.2 Model Development Chapter 2 0.08 T=700°C c o u 0.06 1 03 o 0.04 \ o E 0.02 \ 0.00 0.00 0.01 0.02 0.03 0.04 0.05 C mole fraction Figure 2-13 Isothermal section of phase diagram at 700°C for the Fe-C-Mn system comparing NPLE, PE and orthoequilibrium boundaries. 2.2.2 Modeling the growth kinetics of ferrite The two main approaches adopted in the present work to describe the growth kinetics o f ferrite are the local equilibrium carbon diffusion model and the mixed-mode concept. The former is used solely for the sake o f comparison and as w i l l be discussed in the next chapter, its application seems to be restricted to steels containing more than 0.2wt%C, where the role of interface reaction can be neglected. In both approaches the carbon concentration profile inside the remaining austenite has to be quantified first therefore identical numerical procedures are implemented i n this part o f the kinetic model. However, it is the velocity o f the ferrite-austenite interface that has to be calculated i n a different way. In the carbon diffusion model, the balance o f carbon flux locally at the interface aids to estimate the interface velocity while the mixed-mode methodology employs the intrinsic mobility and the driving pressure at the interface to characterize the growth rate o f ferrite. 45 2.2 Model Development Chapter 2 Prior to solving the required differential equation pertinent to carbon diffusion in austenite, an appropriate geometry for the growing ferrite and the parent austenite phases has to be defined. Semi-infinite slab featuring planar geometry for the ferrite-austenite interface was adopted to analyze the thickening o f ferrite allotriomorph, the morphology that tends to develop at the initial stage o f transformation. The slab thickness is assumed to be the average austenite grain size. To describe the overall kinetics o f ferrite evolution a spherical austenite grain is assumed such that ferrite constitutes its outer shell and grows toward the grain center. This configuration replicates adequately the growth o f ferrite after nucleation site saturation at austenite grain boundaries, which was experimentally confirmed as shown in Figure 2-14. The physical methodology taken to develop diffusion and mixed-mode controlled models, as well as the numerical implementation o f the outlined geometries, is described here. 2.2.2.1 Carbon diffusion controlled model To calculate the carbon concentration profile inside the austenite, F i c k ' s second law for moving boundary situation was employed. The change o f carbon content in a specific location is related to two carbon fluxes, one is due to the movement o f the axis frame and one arises from the concentration gradient, e.g. for planar geometry: dc d . — = _( dt vc dz + _ dc. £)—) dz (2-37) A n implicit finite difference method ( F D M ) [80] was employed to solve this differential equation. 46 2.2 Model Development Chapter 2 (a) Ferrite shell | Carbon profile in y (b) Figure 2-14 (a) Ferrite thin shell decorating austenite grain boundaries, ferrite formed partially at 735°Cfrom austenite with grain size of96/mt in 0.17wt%C-0.74wt%Mn steel, followed by quenching to room temperature, (b) schematic representation of the ferrite growth as a spherical shell toward the center of austenite grain. 47 2.2 Model Development Chapter 2 Ferrite formed at austenite grain boundaries grows with planar or spherical geometry toward the grain interior. In either case the austenite grain is discretize into several nodes as depicted in Figure 2-15a. The austenite grain with initial size o f 21 0 is composed o f n nodes, while the ferrite layer o f thickness I has uniform carbon concentration identical to its interfacial value (e.g. in the diffusion model c - c ), such that it can be represent by one mesh. a a eq I to < W\ j £ a r 1 2 L._.i i 3 4 n _J (a) (b) Figure 2-15 Node configuration for (a) planar and (b) spherical geometries. A s growth proceeds the only grid i n ferrite becomes larger at the expense o f the shrinkage in the parent austenite phase. To handle the problem o f the moving ferrite-austenite interface, Murray-Landis [81] variable-grid method, which had been adopted by Tanzilli and Heckel for diffusional solid state transformations [82], was employed. Then considering different geometries, the concentration change o f an internal node i n austenite is given by: dc, — L dc, dz KD t = —'-—!- dt dz j dt ; + dc ; '-^- z t dD dc, : dz + —'-—^ i dz t ^ +D dz T j dc 2 : j- (2-38) dz* where c is the carbon concentration, z is the distance from the origin, D represent the carbon diffusivity in austenite, K is a geometry factor, i.e. 0 for planar and 2 for spherical geometry, and i denotes the node number. 48 2.2 Model Development Chapter 2 The migration rate o f an internal point i n austenite is related to the interface velocity, v = dll dt, and its distance from the interface as for the assumed geometries are given by the following expressions: dz, l -z,dt Planar: —'- = -? '-— 0 dt £ ~£ dt (2-39) 0 dz z Spherical: —'- = dt dt '•—— £ -£dt (2-40) 0 Local equilibrium is assumed at the ferrite-austenite interface such that interfacial carbon concentration remains constant at its equilibrium value given by the appropriate tie line selected, e.g. ortho-, para- or negligible partition local-equilibrium. To fulfill this condition the net flux o f carbon at the interface has to be zero, so that the interface velocity can be evaluated based on the carbon mass balance at the ferrite-austenite interface: 1 dl AJ int = 0 =^> — = Dl (2-41) For the configuration as depicted i n Figure 2-15, the discrete form o f these equations i n austenite, as are provided for planar and spherical configuration i n Appendix I V , can be evaluated by the following substitutions [80]:( for i=2, 3, ...n, i=\ corresponds to the interface at which concentration is given by the equilibrium value) dc, c/ dt At dz 2Az : 8 2 -c/ + 1 c -2cj+cU (2-42) (2-43) J Ci M dzf Az 7 £ Az = - 2 -I n-\ 49 (2-44) (2-45) 2.2 Model Development Chapter 2 where j denotes the time step. The selected time step has to satisfy the stability criterion, i n terms o f diffusivity and grid size, i.e.: . At < 0.5Az 2 (2-46() D r The boundary conditions, which have to be considered in the carbon diffusion model, are as follows: • A t austenite side o f the interface, i.e. z = £ - £ for t > 0 • Zero mass transfer at the center o f austenite grain: at i=n for t > 0 o c{ = c . y nl eq c' n = c„ _,. ; +l According to the mass conservation principle no carbon leakage or addition is permitted i n the system therefore at each time step the overall mass balance is performed to check the accuracy o f the numerical solution, such that the fraction o f austenite can be assessed using the following equation: c =(\-f )c +f c a 0 where c and c y 0 r r r (2-47) denote initial and average carbon content o f austenite. If this fraction would be different from the one calculated from the above finite difference procedure, then the interface location would have to be adjusted to satisfy the mass balance. This potential disagreement could perhaps be due to error accumulation i n the numerical solution. In contrast to the planar case, more numerical steps are involved i n determining the average carbon concentration o f austenite for a spherical grain. The example o f the mathematical steps and the associated numerical implications are described in Appendix V . Figure 2-16 shows an example o f the model predictions for ferrite growth in Fe-0.23wt%C alloy. Calculations were performed for planar geometry and an initial austenite grain size o f 50 2.2 Model Development Chapter 2 50um at the temperature o f 745°C where the equilibrium interfacial concentrations c and a c y were calculated as 0.017 and 0.62wt%C, respectively. 0.1 V o.o 1 0 1 , 5 • 1 10 15 Interface position, 20 1 25 Figure 2-16 Carbon profile for various growth times calculated by diffusion model. Ferrite growth with planar geometry at 745°C from austenite of 50^on grain size in Fe-0.23wt%C was assumed. 2.2.2.2 Mixed-m ode m odel In this approach, a similar F D M and node configuration as i n the diffusional model was employed to evaluate the carbon concentration profile i n untransformed austenite. However, in contrast to the diffusion-controlled model, the interface velocity is evaluated based on its intrinsic mobility and the chemical driving pressure, as expressed by Equation 2-28, i.e.: v =- =MAG,, dt where M is the intrinsic mobility o f the interface and &G (2-48) inl denotes the thermodynamic driving pressure for the austenite to ferrite reaction which here is evaluated based on the local composition at the interface as depicted in Figure 2-12. 51 2.2 Model Development Chapter 2 A s a consequence of, and i n contrast to the carbon diffusion model, the interfacial carbon concentrations at both sides o f the interface are variable quantities under isothermal conditions, for instance the interfacial carbon o f austenite changes from the initial to the equilibrium carbon concentration o f austenite, i.e. from c to c , as growth proceeds. This is r 0 a consequence o f a non-zero carbon net flux, J neh at the interface (cf. Equation 2-29). The change o f the interface carbon concentration in time step At during which the interface advances Al = vAt is calculated using the carbon net flux at the interface. The interfacial carbon concentration o f austenite is then assessed in each step using the discrete form o f Equation 2-29, i.e. [83]: c l { t , n = cl(l\t) where t + [clXl,t)-cl-Dl At cl(t,t + Al Az )-c(2,t) ] (2-49) - 1 + Al , t =t+At and c(2,t) denotes the carbon content o f the adjacent node to the interface. To maintain the equality o f the chemical potential o f carbon across the interface the carbon composition o f the growing ferrite, c,",, has to be re-evaluated in each time step accordingly. Because o f higher diffusivities in ferrite it is assumed that the carbon concentration in ferrite is uniform and is given by its interfacial content. In order to prevent any numerical instability the criteria depicted in Equation 2-47 has to be fulfilled for At, hence the time step required to perform the evaluation o f the new interfacial carbon concentration and new composition profile inside the remaining austenite, is readjusted continuously during growth. In the present numerical approach, this is accomplished by setting Al to a value, which is smaller than the node size, e.g. about 0.01 Az. 52 2.2 Model Development Chapter 2 Employing the mixed-mode approach and assuming two different mobilities, i.e. M=0.38 and 0.05 um mol/Js, the carbon concentration profiles inside the remaining austenite for different growth times are shown in Figure 2-17. 0.1 h o.o i i 0 5 i i 10 15 Interface position, um 1 1 20 25 (b) Figure 2-17 Carbon profile for various growth times calculated by the mixed-mode model for reaction at 745°C in Fe-0.23wt%C alloy assuming planar geometry and (a) M=0.05 fjm mol/Js and (b) M=0.38 /um mol/Js. 53 2.2 Model Development Chapter 2 The simulations assumed ferrite formation i n Fe-0.23wt%C alloy at 745°C where the equilibrium carbon content o f the interface at austenite side is 0.62wt%. The interfacial carbon concentration increases with time at a rate, which is related to the value assumed for the interface mobility. It has to be noted that at the extreme case, i.e. M -> oo, the equilibrium carbon concentration is established at the interface right from the onset of ferrite growth so that the mixed-mode model prediction is consistent with that o f the diffusion model. 2.2.2.3 Assessment of solute drag effect The kinetic interaction o f the alloying elements with the moving a/y interface, i.e. solute drag effect, was incorporated into the mixed-mode model by introducing an effective driving pressure, such that the free energy dissipated by drag, AGSD reduces the available chemical driving pressure at the a/y interface, i.e.[84]: AG =AG -AG EFF INL SD (2-50) The migration rate o f the interface is then calculated using the effective driving-pressure. Here, to model solute drag the quantitative description o f Purdy and Brechet [59] serves as starting point, therefore we shall elaborate it in detail. Treating the interface as a continuum media, they assumed an asymmetrical wedge shaped well for the interaction potential o f the solute with the interface as delineated in Figure 2-18. Here, 2<5is the interface thickness, 2AE is the chemical potential difference for i in austenite and ferrite, i.e. juf - /J." , and E is the 0 binding energy o f / to the interface which is represented by the depth o f the potential well. 54 2.2 Model Development Chapter 2 dE__ AE-E dE__AE + E dx~ dx ~ 0 a 5 8 2 AE a -8 I E +AE +8 0 Figure 2-18 Chemical potential of solute in the a/y interphase boundary [59]. The segregation profile, c (x), o f the substitutional element inside the interface region is s predicted by a flux equation, which consists o f three terms reflecting the contributions due to dc dE the concentration gradient—-, the potential gradient — and the velocity o f the migrating dx dx interface. For a boundary moving with quasi-steady velocity v, the net flux is zero and the concentration profile is given by: dC dX C dE • + - RT dX + F(C-1) = 0 (2-51) where C, X a n d V are dimensionless parameters defined by C=cs/cs, , X - x/S and V=v5/Dt,, 0 respectively. Here, cs, is the bulk composition o f solute atoms and Db is their diffusivity 0 across the interface. For a given binding energy, the feature o f the predicted segregation profile depends on the velocity and AE, such that as the migration rate increases the profile tends to flatten out however the degree o f its asymmetry becomes more pronounced. Further, the asymmetry also depends on AE, for instance for a stationary interface only in the case o f AE=0 this intrinsic asymmetry is eliminated and the segregation profile would be similar to that for grain boundaries as considered by Cahn [61]. Solute atoms segregate at different sides o f the moving interface where dissimilar gradients in the interaction potential exist, therefore they exert unequal and opposite forces on the 55 2.2 Model Development Chapter 2 boundary. The resulting net drag pressure is not zero and can be calculated by Cahn's formulation. The intensity o f drag pressure for a given interface velocity depends primarily on the binding energy and AE, the latter is a function o f interfacial carbon concentration and is calculated from the assumed thermodynamic conditions. Therefore, AE has to be reevaluated at each time step before being employed in the solute drag module. Assuming a binding energy o f \.5RT and AE=0, the variation o f segregation profile with normalized velocity are presented in Figure 2-19(a). A s velocity increases the resulting concentration profile flattens out while the degree o f its asymmetry becomes more pronounced. To explore the influence o f AE, the segregation profiles formed inside a stationary interface (F=0) for various values o f AE are plotted in Figure 2-19(b) as well. Note that for a non-moving boundary the degree o f asymmetry in the concentration profile increases with the chemical potential difference, AE, o f substitutional constituents across the interface. Only in the case o f AE = 0 this intrinsic asymmetry is eliminated and the segregation profile is consistent with that for grain boundaries, as considered by Cahn [61]. According to Cahn's theory any asymmetric profile is per definition associated with a solute drag pressure. Then by accounting for the contribution o f all segregated atoms over the entire interface region, the energy dissipation by solute drag per mole o f substitutional atoms is determined from Equation 2-25. 56 2.2 Model Development Chapter 2 -1.0 -0.5 0.0 0.5 1.0 Normalized distance, x/5 (b) Figure 2-19 Segregation profile inside the interface region (a) for various normalized velocities and AE=0 and (b) for different values of AE for a stationary interface (V=0). 57 2.2 Model Development Chapter 2 2.2.2.4 Modification to Purdy-Brechet solute drag model It is expected that the drag pressure vanishes gradually as the normalized velocity approaches to zero, this physical concept is valid for all values o f assumed solute-interface binding energy or calculated AE. However, adopting the Purdy-Brechet approach to evaluate the drag pressure leads to an apparent artifact for stationary interfaces when AE ^ 0 is considered. The problem arises from the presence o f asymmetry in the interaction potential wedge and the resulting asymmetric segregation profile for zero interface velocity as evident in Figure 2-19(b). For example, i f the paraequilibrium description is employed i n mixed-mode growth, the chemical potential difference o f the substitutional solute, e.g. M n , does not approach zero when the growth ceases at the end o f the reaction. Accordingly, a non-zero drag pressure would be predicted i f the Purdy-Brechet quantification were adopted preventing the system from reaching equilibrium. Therefore, it is imperative to remove this artifact by accounting for the intrinsic asymmetry i n the interaction potential well. In the present work, it is proposed that the solute drag pressure is evaluated with new concentration and potential terms, which correct for the intrinsic asymmetry as a function o f AE, i.e.: C New AF = Cexp(—X) dE_ d ~5X New (2-52) A 77 E AE (2-53) dX Then the drag pressure, or i n other words the energy dissipated by solute drag, is calculated based on these newly introduced parameters as: &G SD =-c \(C -\) So New 58 dE\ dX dX New (2-54) 2.2 Model Development Chapter 2 Using the above adjustment the drag pressure vanishes completely when the interface is brought to halt for all values o f AE. Figure 2-20 compares the predictions o f the modified and the original version o f the Purdy-Brechet solute drag model, where the binding energy o f 1.5RT,-4kJ/mol for AE and Xs, =0.0l are assumed for the calculations. Apart from the drop o of drag pressure to zero for stationary interface, the modified version predicts less drag pressure for all velocities compared to the original approach and the peak o f solute drag is slightly shifted from V « 1.8 to V « 2 . 150 i , Normalized velocity Figure 2-20 Comparison of the energy dissipated by solute drag for £•„=/. 5RT and AE=-4kJ/mol, adopting the Purdy-Brechet model and the modified solute drag approach. It is imperative to emphasize that in the present analyses the modified solute drag approach is employed to evaluate all drag forces. The reason for this adjustment is the fact that the isothermal kinetics cannot be captured accurately at the final stages o f reaction. However, as reported previously [84], the original Purdy-Brechet description o f the solute drag effect can still be applied successfully to continuous cooling cases, in which the thermodynamic driving 59 2.2 Model Development Chapter 2 pressure for growth increases with undercooling while simultaneously the drag pressure tends to decrease. Then the artifacts o f the Purdy-Brechet approach do not affect its general predictive ability for continuous cooling cases. However, the interpretation o f the resulting quantities for Db and E may be o f even more limited value. 0 2.2.3 Crucial parameters in the kinetic model The integrated growth model as outlined above employs parameters which are defined clearly in terms o f their physics. Diffusion controlled reaction, and i n part mixed-mode growth rely on the quantification o f carbon concentration in austenite, diffusivity o f carbon, D, y c therefore the is o f common significance i n both modeling approaches. Fortunately the diffusion coefficient o f carbon in austenite has been characterized accurately by many researchers [42,43,85], and since no discernible influence o f M n , S i and M o on carbon diffusivity was reported, here the expression proposed by Agren, which is a function of temperature and carbon concentration, is adopted for all calculations [43]: D £ ( / n V ) = 4.53xlO" where Yc = Xcl(l-Xc), 7 1+ 8339.9 Y (l-Y ) c 2.221x10"" )(17767-26436 Y) c c (2-55) represents carbon site fraction in the interstitial sublattice. Apparently the most crucial kinetic parameter that the mixed-mode growth depends on is the intrinsic mobility o f the ferrite-austenite interface, i.e. M. This physical quantity is related to the jump frequency o f substitutional species, including iron atoms, across the interface that leads to local reconstruction o f B C C crystal from F C C lattice. Conceptually it is expected that both local composition and structure o f the interface affect the mobility, however no conclusive theoretical description o f these dependencies has been provided yet. Among a few 60 2.2 Model Development Chapter 2 attempts to define the mobility i n terms o f the physical features o f the interface, Turnbull's model [86] for grain boundaries is presented here for instance, although it overpredicts the experimental data o f grain growth i n pure iron [87] and moreover it relies on parameters that are difficult to determine accurately, i.e.: SD V rR M = -f B b RT '" (2-56) where 8 is the interface thickness, b is the interatomic distance and DCB is the grain boundary diffusivity. Further, due to the fine scale and discontinues nature o f the reaction interface, there is, at least currently, no experimental means by which the mobility can be characterized quantitatively. Treating the interface migration as a thermally activated process, one can consider a simplified version o f the above equation i n the form o f an Arrhenius temperature dependence as given by: M=M exp(^) o (2-57) where M is the pre-exponential factor and Q represents the activation energy o f the rate 0 controlling process. A s an attempt to determine these two parameters, Hillert [87] analyzed the kinetic data pertinent to grain growth i n zone-refined iron and suggested the following mobility: M(cm J I T S r irv5 ,-\AAkJ I moL mol I Js) = 5 x 10 exp( — ) (2-58) This mobility also had been applied to massive austenite to ferrite transformation i n low carbon steels showing satisfactory agreement with the experimental data. A more recent quantification o f ferrite-austenite mobility for binary F e - M n alloys has been carried out by Krielaart and van der Zwaag [83]. They described the measured kinetics o f ferrite formation, for different M n contents and cooling rates, solely by means o f an interface-controlled 61 2.2 Model Development Chapter 2 growth model treating the mobility as an adjustable parameter. The overall mobility was proposed as: M(cmmolI -UOkJ/mol Js) = 5.8 exp(RT (2-59) ) in which the activation energy is almost identical with the one suggested by Hillert. However, the pre-exponential value reported by Hillert is about five orders o f magnitude larger than that o f Krielaart and van der Zwaag. Clearly, there is a great deal o f uncertainty about the actual value o f the intrinsic mobility. Assuming Hillert's mobility i n the mixedmode model would lead to an interface reaction that occurs so quickly to not play any kinetic role such that the model would predict a transformation entirely controlled by carbon diffusion. Consequently, in the present simulations the following assumptions are made regarding the intrinsic mobility. The activation energy o f 140kJ7mol is taken and M 0 is essentially an adjustable parameter. A s the starting point to perform the simulations, the value proposed by Krielaart and van der Zwaag [83] is adopted for M and re-adjusted when required to describe 0 the experimental kinetics. Incorporation o f the solute drag module into the mixed-mode model is associated with introducing additional parameters. In terms of drag pressure two main physical characteristics have to be defined, i.e. the interaction energy between the interface and the segregated solute, E , 0 and the diffusivity o f this solute across the interface region, D . h Unfortunately, neither o f these quantities can be measured directly by means o f available experimental techniques, however the physically expected range o f them can be derived as an order o f magnitude estimate. For instance, Auger electron microscopy on the fracture surface of austenite grains i n Fe-C-/ systems [88], revealed that among M o , M n and N i , the 62 2.2 Model Development Chapter 2 enrichment factor is the highest for M o , and decreases in the order o f M n and N i . Moreover, adopting the co-segregation model o f Guttmann [89], the binding energy o f solute to grain boundaries was estimated as 18, 9 and 5kJ/mol for the above elements, respectively. O i and his coworkers [58] adopted the methodology proposed by Purdy et al. [69], i.e. balancing the driving pressure with solute drag force, and determined the following binding energies to ferrite-austenite interfaces: RT for M n and 0.5RT for N i . These values are consistent with the findings o f Enomoto [64,88]. A binding energy i n the order o f RT for M n and N i is assumed here, however as w i l l be addressed in the following chapter, the analyses o f ferrite formation suggest that for adequate description o f the experimental kinetics a temperature dependency for E is required, i.e.: 0 A +B T E (2-60) E where the slope and intercept are treated as adjustable parameters. Thus, E is employed as 0 an effective binding energy. Definitely, the diffusivity o f segregating solute across the interface is the parameter with the least certainty regarding its estimation. The nature o f this atomic jump process seems to be quite different from that along grain boundaries, although it is expected to be a few orders o f magnitude larger than the bulk diffusivity i n austenite. For instance Dt, was assumed to be larger by a factor o f 1000 than the bulk diffusivity in some simulations [64,90] while a value close to the average o f ferrite and austenite bulk diffusivity was taken for Dt, as well [56]. In the present simulations an Arrhenius temperature variation is adopted for the normalizing factor o f the interface velocity, i.e. Dt/5, as given by: (2-61) 63 2.2 Model Development Chapter 2 where similar to the case o f solute migration along the grain boundaries, the activation energy, Q , is considered to be half o f the one for bulk diffusion in austenite. For example b this quantity for M n and N i is set as 132 and 137kJ/mol, respectively [91]. The diffusivity i n austenite is used as a reference point even though it is smaller than the diffusivity i n ferrite. However, below the Curie temperature (770 °C) ferromagnetic ordering decreases diffusion coefficients and the data available for diffusion i n paramagnetic ferrite cannot be extrapolated to the temperature range o f transformation. The pre-exponential factor, D /S, 0 constitutes then another fitting parameter. \ 64 2.3 Model Application Chapter 2 2.3 Model Application 2.3.1 Analysis of allotriomorph thickening in Fe-C alloys In this section the thickening o f ferrite allotriomorphs is analyzed for binary Fe-C systems. This analysis deals with a growth geometry where ferrite forms along austenite grain boundaries and features parabolic growth, both in lengthening and thickening, at the early stage o f the reaction, i.e. prior to the impingement o f carbon diffusion fields. Therefore, as also confirmed experimentally [92,93], the thickness, s, varies with the reaction time, t, as: s = a,4~t (2-62) where a is the parabolic rate constant. t The experimental measurement o f allotriomorph growth is a challenging task and comparatively few results have been reported. There is noticeable data scattering, such that substantial variations in thickness from one to another allotriomorph i n a given specimen has been observed [93]. This has been in part attributed to the presence o f facets on the ferriteaustenite interface o f ferrite allotriomorphs, which otherwise is expected to be disordered. Further, in many cases the thickness is measured in the plane that may not pass through the center o f the allotriomorph, i.e. it is not normal to the grain boundary, which gives rise to the so-called stereological error. Most o f the published data suffer from inaccuracy originating from these measuring complications. However, to improve the precision o f the measured growth kinetics, the experimental technique developed by Bradley et al. seems to provide more reliable data. A summary o f their measurements [92] o f parabolic rate constants in three different Fe-C alloys containing 0.11wt%, 0.23wt% and 0.42wt%C is replicated in Figure 2-21. The examined temperatures range from 740 to 815°C. A s expected, for each individual alloy, as reaction temperature decreases the rate o f growth increases. Further, the 65 2.3 Model Application Chapter 2 specimen with the highest carbon level, i.e. 0.42wt%, represents the smallest parabolic rate constant for all the investigated temperatures. • A • CO E =L Fe-0.11wt%C Fe-0.23wt%C Fe-0.42wt%C *J C CO -M CO c o o CD -*-» CO 5 o O 700 720 740 760 780 800 820 Temperature, °C Figure 2-21 Experimental parabolic rate constant in Fe-C alloys [92] 2.3.1.1 Prediction of carbon diffusion model The growth o f ferrite allotriomorphs has traditionally been assumed to be controlled by carbon diffusion in austenite. However, considerable discrepancies between the measured parabolic rate constant and the predictions from diffusion models can be frequently found for several binary F e - C systems in the literatures [93]. To examine the predictive capability o f the carbon diffusion model, an attempt has been made to replicate the experimental data presented in Figure 2-21. The natural starting point is to utilize the analytical solution for the thickening o f allotriomorphs as proposed by Zener [48]: 1 { C ; q C o a ) ^ D ^ = a, exp(a, 2 /AD ) r 0 c 2 > £ 66 (2-63) 2.3 Model Application where c and c" denote the equilibrium carbon content o f the interface in austenite and eq r Chapter 2 Q ferrite sides, respectively, c is the nominal carbon level in the alloy and D r 0 c is the carbon diffusivity in austenite. The analytical solution is applicable only for the growth period before soft impingement. Moreover, in this approach a concentration independent carbon diffusivity in austenite is assumed. However, in the final solution as expressed by Equation 2-63, D Y C can be evaluated based on various carbon concentrations, e.g. using an average carbon concentration in front o f the ferrite-austenite interface, e.g. (c + c )I2, r e taking the interfacial carbon content o f austenite, c , or considering initial carbon content o f alloy, c . r 0 Then, the calculated parabolic rate constant is quite sensitive to the value taken for the carbon diffusivity as illustrated i n Figure 2-22 for Fe-0.11wt%C, where the experimental data are also included for the purpose o f comparison. To evaluate the carbon diffusivity in austenite, the expression proposed by Agren [43] was adopted in all the calculations. W E * 3] C 2 in § 2 o O) -M to -C Experiment A: using C B: using (C +C )/2 C: using C A *J • T 5 o u CD eq T 0 cq 0 740 760 780 800 820 Temperature, °C Figure 2-22 Zener predictions for parabolic rate constant and the measured [92] data in Fe-0.11wt%C 67 2.3 Model Application Chapter 2 Independently from the selection o f the diffusion coefficient, it is evident that the analytical solution significantly overpredicts the measured growth rate in Fe-0.11wt%C for all examined temperatures. Although, adjusting the diffusivity led to smaller discrepancy with respect to the experimental data at the lowest reaction temperature, i.e. 745°C, no noticeable improvement was attained for the temperature o f 815°C. This can be described by the larger difference between the value o f (c + c ) / 2 and c r 0 Y eq eq (or c ) as the growth temperature 0 decreases. One of the main drawbacks o f the above analysis is that a variable diffusivity which is physically more sensible, can not be employed in the analytical solution. Further, it is obvious that none o f the above values for carbon concentration, i.e. average carbon concentration, the initial carbon content o f the alloy or interfacial carbon content, can represent the exact carbon composition profile ahead o f the interface. Therefore, it is required to utilize a more realistic treatment, i.e. a numerical solution, to overcome these simplified assumptions. The advantage o f employing it to predict the growth kinetics o f ferrite allotriomorph is its capability to account for the variable carbon diffusivity, e.g. Equation 2-55. This numerical approach was subsequently used to estimate the parabolic rate constant of the three Fe-C alloys. Both the model predictions and the experimental values are represented in Figure 2-23. For the purpose o f comparison the predictions o f Zener's analytical solution assuming average carbon concentration, i.e. (c + c j ) / 2 to assess D y Q c are also included. The first glance at the prediction for all the examined alloys revealed that there is a good agreement between the analytical solution (Zener's equation) and the numerical solution. Further, for the alloys containing 0.11 and 0.23wt%C, the predictions o f the numerical diffusion model are considerably above 68 the measured data outside the 2.3 Model Application Chapter 2 experimental error for all investigated temperatures. The degree o f this disagreement tends to diminish as the carbon level o f the investigated alloy increases, such that eventually the diffusion model can replicate the reported values o f the rate constant in the Fe-0.42wt%C system. 2.3.1.2 Prediction of mixed-mode model The failure o f the carbon diffusion model to predict adequately the experimental growth kinetics o f low-carbon alloys, i.e. with carbon contents o f 0.23wt% and lower, indicates that another type o f atomic phenomenon has to be operative. For the F e - C alloys it can be ruled out that inappropriate evaluation o f either D y c or the interfacial composition causes this issue, since in Fe-C alloys the quantitative description o f carbon diffusivity in austenite and ferriteaustenite boundaries i n the F e - C phase diagram are well established. Therefore, it is imperative in these circumstances to account for the kinetic role o f the interface reaction rigorously. In particular the contribution o f interface reaction tends to be more significant with decreasing the carbon content o f alloy, such that as carbon concentration approaches to zero the growth kinetics o f ferrite would solely be an interfaced-controlled one. These two atomic events, i.e. the reaction at the interface and the redistribution o f carbon atoms in the remaining austenite, are coupled together elegantly i n the framework o f the mixed-mode model as explained in section 2.1. The key parameter i n the mixed-mode model is the intrinsic mobility o f the ferrite-austenite interface, with a temperature dependence that can be described by an Arrhenius relationship, i.e. Equation 2-57. A s outlined earlier, this mobility is used as an adjustable parameter. 69 2.3 Model Chapter 2 Application 1.0 700 720 740 760 780 Temperature, °C Figure 2-23 Comparison of the measured [92] parabolic rate constant with the prediction of the carbon diffusion model for (a) Fe-0.11wt%C, (b) Fe-0.22wt%C and (c) Fe-0.43wt%C alloys. 70 2.3 Model Application Chapter 2 The parabolic growth is described with a constant, a , which can be partially expressed in t terms o f interfacial carbon concentrations that remain constant during the isothermal ferrite formation, provided that local equilibrium at the interface prevails. In the carbon diffusion model, the ferrite-austenite interface is in equilibrium with respect to carbon therefore the condition for the parabolic thickening o f allotriomorphs is certainly met. In contrast, in the mixed-mode model due to a non-zero net flux o f carbon across the ferrite-austenite interface the interfacial carbon contents vary as growth proceeds. Thus, by definition a, is no longer constant and per se some deviations from the parabolic regime are expected. This inherent effect is more pronounced at early stages o f ferrite growth, when a relatively quick carbon built up occurs at the interface. A s soon as the interfacial carbon level approaches its equilibrium value, i.e. c , the growth becomes carbon diffusion-controlled. r Thus, the mixed-mode approach provides a smooth transition from an interface- to a diffusion-controlled mode. The period, during which growth is mainly governed by interface reaction, is characterized by a marked deviation from the parabolic growth behavior. This period tends to be shorter when a larger value for the interface mobility is assumed. To elaborate the discussion and as an example, the predictions o f the mixed-mode approach are illustrated in Figure 2-24 for allotriomorph thickening in Fe-0.23wt%C at 775°C for two different mobilities M =5.8 and 1.27 cm mol/Js. Included in this figure is the prediction o f 0 the carbon diffusion model as well as the experimental data points [92]. It is seen that neither the diffusion model nor the mixed-mode approach, assuming Af =5.8 cm mol/Js, can capture 0 the measured thickness. The calculation using M =1.27 cm mol/Js seems to provide an 0 adequate description o f the experimental kinetics, although it does not follow the parabolic 71 2.3 Model Application Chapter 2 growth. Besides, it is evident that the deviation from linearity is more significant for this mobility, i.e. M =\.21 0 cm mol/Js, compared to that for M =5.8 cm mol/Js. 0 0 2 4 time, s" ' 1 6 2 Figure 2-24 Measured allotriomorphs thickness and the predictions of mixed-mode model assuming M =5.i9 and 1.27 cm mol/Js, as well as the calculation of the carbon diffusion model (M 0 0 = <x>)for growth at 775°C in Fe-0.23wt%C. The above argument implies that in terms o f allotriomorph thickening, the prediction o f the mixed-mode model consists o f two growth modes: a initial non-parabolic part which is governed primarily by the interface reaction and the later stage where carbon diffusion is the main controlling step. The experimental growth kinetics reported in the literature for a given composition is usually expressed in terms o f a parabolic rate constant. Unfortunately, the exact data points similar to the ones depicted in Figure 2-24, are seldom reported in the literature. Thus two alternative approaches exist to analyze the experimental kinetics o f parabolic growth in the framework o f the mixed-mode model and to find the appropriate value for the mobility. The slope o f the second stage, i.e. the parabolic portion o f the model prediction, can be compared with the reported a or alternatively, the exact measured data h 72 2.3 Model Application Chapter 2 points, i f they are available, are used for the comparison with the calculated growth kinetics. Figure 2-24 shows that the model with M =1.27 cm mol/Js can capture the experimental data 0 reasonably and the slope o f its linear part, does agree well with the linear fit to the measured data points. A s is outlined in the following paragraphs, it was found that overall, the mixedmode model assuming M =0.78 cm mol/Js is capable o f predicting the thickening kinetics o f o ferrite at various temperatures in three Fe-C alloys featuring carbon content ranges between 0.11 to 0.4wt%. For the present case, i.e. allotriomorph thickening at 775°C in Fe-0.23wt%C, using this mobility predicts an overall slope o f 0.8 urn s" 1/2 in the s versus 4~t plot, which concurs with the experimental slope and lies within the scattering band o f the reported a , i.e. t 1II 0.6±0.2 urn s" for M 0 . Therefore, either o f the 1.27 or 0.78 (cm mol/Js) quantities can be selected in Fe-0.23wt%C. The implication is that the intrinsic mobility o f ferrite-austenite interface can be determined reasonably well from the analysis o f allotriomorphs thickening by mixed-mode model. The error o f the proposed mobility is likely to be less than one order of magnitude. Since the reported parabolic rate constants are the only data available for all three Fe-C alloys they are considered in the analysis o f allotriomorph thickening and compared with the parabolic part o f the mixed-mode model predictions. Since the total time during which the experimental measurements had been carried out is not available, the simulation time is assigned to be the time for the onset o f soft impingement in an austenite grain o f 200um size. This grain size is consistent with that used for allotriomorph thickness measurement [92], i.e. A S T M 1-2 . Attempts then have been made to replicate the experimental a o f Fe-C alloys 1 t presented in Figure 2-21 by changing the value o f the mobility term. The slope predicted by 1 N(number of grains per unit of area, inch square, in 100X magnification)=2 "' n 73 2.3 Model Application Chapter 2 the mixed-mode model was determined from the stage in which the growth is parabolic. The results including the prediction using the Krielaart and van der Zwaag mobility, i.e. M =5.8 0 cm mol/Js, and the experimental values o f parabolic rate constant for ferrite formation in three Fe-C alloys are illustrated in Figure 2-25. For the comparison purpose the calculations of the diffusion model are also included. It is quite obvious that the calculations using M o f 5.8 (cm mol/Js) overpredict the measured 0 kinetics o f the 0.11 and 0.23wt%C alloys, with the discrepancy being larger for the lower carbon content. However, for the Fe-0.42wt%C alloy, this assumed mobility provides rather good agreement with the experiment. A s anticipated from the concept o f mixed-mode modeling approach, the role that reconstruction o f B C C from F C C plays in overall kinetics tends to be more profound when the carbon content o f parent austenite decreases, such that eventually the overall kinetics becomes solely interface-controlled in pure iron. Attempting to find a unique mobility that would be able to replicate the experimental a for t all the investigated Fe-C alloys, gives a value o f 0.75 cm mol/Js for M . A s shown in Figure 0 2-25, the model predictions using this mobility show good agreement for all carbon levels and the investigated temperature range. This corresponds to a mobility which is 0.13 times smaller than the value obtained from the studies in the F e - M n systems by Krielaart and van der Zwaag. Therefore, it is concluded that although accounting for the interface mobility is quite essential, no universal intrinsic mobility can be proposed for all chemistries. This constitute a main challenge o f using the mixed-mode model, i.e. the model relies on the unknown parameters that first here to be determined for each alloying system by means o f fitting practices. 74 2.3 Model Application Chapter 2 (a) Fe-0.11wt%C in E 1 2 CO •«-» M =0.75 in c o u CD 1 5 • ? Experiment Mixed-mode model (M in cm mol/Js) Diffusion model o O o 740 760 780 800 820 Temperature, °C (b) Fe-0.23wt%C m E c CO .+-» in c o u M =0.75 3 CO u Experiment % Mixed-mode model (M in cm mol/Js) o o Diffusion model L. o 700 720 740 760 780 Temperature, °C 1.0 in (c) Fe-0.42wt%C M =5.8 E a. *s c CO in O 0.5 O M =0.75 cu CO .c % o O Experiment Mixed-mode model (M in cm mol/Js) Diffusion model 0 0.0 700 720 740 760 780 Temperature, °C Figure 2-25 Experimental data [92] ofparabolic rate constant and prediction of mixed-mode model assuming mobility of 5.8 and 0.75 cm mol/Js for (a) Fe-0.11wt%C, (b) Fe-0.22wt%C and (c) Fe0.43wt%C alloys. The predictions of the diffusion model are included as well. 75 2.3 Model 2.3.2 Application Chapter 2 Analysis of allotriomorph thickening in Fe-C-/ systems 2.3.2.1 Comments on the experimental data The major aim o f this section is to analyze the growth o f ferrite allotriomorphs in ternary FeC-i systems that contain either M n or N i as the only substitutional element. Primarily, the predictive capability o f the mixed-mode model to replicate the measured kinetics is going to be examined. Subsequently, the incorporation o f the solute drag effect into the mixed-mode model and the physical relevance o f the resulting interface parameters w i l l be discussed. The experimental thickening rates, against which the model is examined, were taken from published data for Fe-0.12wt%C-3.1wt%Mn, Fe-0.12wt%C-3.3wt%Ni [62] and Fe-0. 21wt%C-1.52wt%Mn [34]. The ranges o f investigated temperatures to quantify the allotriomorph thickness are presented in Table 2-3. Included in the table are also the theoretical Ae3 temperatures calculated based on the different thermodynamic assumptions, i.e. orthoequilibrium, paraequilibrium and negligible partitioning local equilibrium, by means of Thermo-Calc software. Table 2-3 Equilibrium austenite/ferrite transformation temperatures and ranges of measured transformation temperatures, T A l l o y composition, wt% OE T , C 0 A e 3 m e a s PE T , of the investigated ternary alloys. A e 3 ,°C NPLE T A e 3 , °C T °C Fe-0.21C-l.52Mn 792 772 733 725-735 Fe-0.12C-3.1Mn 771 732 588 550-650 Fe-0.12C-3.3Ni 772 742 670 650-720 Bradley and Aaronson [92,62] measured the ferrite thickness at austenite grain boundaries using room temperature, optical microscopy o f the specimens, which had been reacted 76 2.3 Model Application Chapter 2 isothermally at the transformation temperature for different times followed by quenching. To minimize structural and stereological errors o f the measurements, a particular austenitization technique was employed. This method tends to align the grain boundaries perpendicular to the intended plane o f polish. The obtained parabolic growth constant for 0.12wt%C3.1wt%Mn and 0.12wt%C-3.3wt%Ni steels are summarized in Figure 2-26. Moreover, to appreciate the quality o f the measured kinetics in terms o f parabolic growth, typical measurements o f thickness variation with the square root o f time for the N i steel are duplicated in Figure 2-27. Despite minor scattering, ferrite thickens linearly with Vt and the growth looks to be reasonably parabolic for this system. CNI 1.25 1 550 600 650 700 750 Temperature, °C Figure 2-26 Experimental Measurements ofparabolic growth rate constant of ferrite in C-Mn and C-Ni steels [34, 62]. 11 2.3 Model Application Chapter 2 (a) 665°C (b) 680°C E d_ E a. in in (0 0) c 3 o V c u 2 • *± 2 (1) u. Linear fit: y=0.85x-0.14 1 2 3 y=0.65x+0.23 A Time, s ' Time, s 1 2 (c) 700°C E E =1- 3 in m <o zl in in c o • (d) 715°C (0 c o • 2 *; 1 • • 0) Li. y=0.46x+0.15 4 y=0.25x-0.50 5 10 Time, s 1/2 12 14 Time, s Figure 2-27 The measured thickness offerrite allotriomorph at different growth temperatures in Fe0.12wt%C-3.3.wt%Ni alloy [94], the solid line represents linear regression. Alternatively, Purdy and coworkers [34] carried out the annealing o f diffusion couples to obtain the growth kinetics o f ferrite in F e - C - M n alloys. The diffusion couple was constituted of a pure iron layer deposited electrolytically onto a polished martensitic segment o f C - M n steel with the desired composition. The couple was upquenched to the single austenite field, where martensite reversion to austenite occurred rather promptly, then quenched down to the target growth temperature in the two-phase region. The growth o f the ferrite layer into 78 2.3 Model Application Chapter 2 austenite with an approximately planar interface was observed to follow the parabolic behavior until ferrite can precipitate later inside the austenite part. Although this technique reduces the stereological and to lesser extent the crystallographic errors, the narrow temperature range at which measurements can be performed, is its main practical limitation. The measured parabolic rate constant at three temperatures for 0.21wt%C-1.52wt%Mn steel by means o f this method are also represented in Figure 2-26. 2.3.2.2 Suitable thermodynamic treatment of Fe-C-i alloys To model the kinetics o f ferrite growth in ternary alloys, an appropriate thermodynamic description o f the system, which sets the major boundary conditions o f the simulation, first has to be selected. In contrast to binary Fe-C alloys, three different, well-quantified alternatives are available for alloyed steels, i.e. O E , P E and N P L E . Orthoequilibrium arises from the long-range redistribution o f substitutional atoms inside the parent austenite, the state that is unlikely to be attained at the investigated temperature ranges for the above alloys. A s also indicated in Table 2-3, the N P L E condition cannot be applied for all the growth temperatures, since the upper range o f those temperatures lies outside the N P L E boundary. However, the entire measured data fall within the P E limit, which suggests P E is the only potential choice for the thermodynamic treatment o f these systems. To further evaluate the merit o f either P E or N P L E as the thermodynamic treatment, particularly for the growth temperatures inside the N P L E limit, the predictions o f the carbon diffusion model assuming either o f these choices are compared with the experimental kinetics. Despite the fact that carbon contents o f the examined alloys are relatively low, the diffusion model is chosen solely because it provides the upper limit for the ferrite growth rate, which corresponds to the prediction o f the mixed-mode model i f an infinite mobility is 79 2.3 Model Application Chapter 2 assumed. The results o f the analysis for ferrite growth are depicted in Figure 2-28 for the alloys containing 3.3wt%Ni and 1.52wt%Mn at temperatures o f 665 and 730°C, respectively. 0 1 2 3 Time, s ' 1 4 2 Figure 2-28 Predictions of the diffusion model using different thermodynamic conditions for ferrite thickening in (a) 1.52wt%Mn alloy at 730°C and (b) 3.3wt%Ni alloy at 665°C. For the alloy containing M n , the prediction based on the full local equilibrium at the interface to determine the interfacial carbon concentration is included as well for the sake o f comparison only, since the associated local partitioning at the interface can be postulated, at 80 2.3 Model Application Chapter 2 least in a formalistic way. It is evident that N P L E treatment gives rise to the kinetics which underpredicts the measured thickness, thereby it is definitely not a justifiable assumption. O n the other hand, i f P E description is employed; the model significantly overestimates the experimental data. However, the predictions can be potentially brought into accord with the measurements i f a finite mobility for the interface is considered. Consequently, from the mentioned analyses it can be deduced that P E remains as the only viable choice for the thermodynamic treatment and is adopted in the following analysis. It has to be noted that the growth rate decreases as the transformation proceeds, therefore the partitioning degree o f substitutional atoms is expected to become more significant. However it is more relevant for the later stages o f reaction, where growth is no longer parabolic in nature. 2.3.2.3 Prediction of m ixed-m ode m odel Prior to considering the retarding effect o f alloying elements, i.e. solute drag effect, it is imperative to evaluate the mobility term by which, in the framework o f the mixed-mode model, the measured ferrite thickness in Fe-C-z alloys can be replicated. In other words, by employing the mobility as the only adjustable parameter, the predictions o f the mixed-mode model are brought into agreement with the experimental data. The 3.3wt%Ni steel is selected for this analysis, since less tendency for solute-interface interaction compared to manganese containing steels has been reported [58]. Therefore, neglecting the probable solute drag effect is expected to cause minimal misinterpretation o f the conclusion to be drawn. The calculations o f the mixed-mode model and the measured thickness o f allotriomorphs as function of time for growth temperatures o f 665, 680, 700 and 715°C are illustrated in Figure 2-29a to d, respectively. 81 2.3 Model Application Chapter 2 Figure 2-29 Predictions of mixed-mode model and the experimental [94] growth offerrite allotriomorphs for 0.12wt%C-3.3wt%Ni steel, intrinsic mobility is employed as the only fit parameter. It is imperative to remark that the linear fit to the experimental data points does not go through the origin i n some o f the cases shown in Figure 2-29, i.e. they do not follow parabolic growth entirely. In this regard the mixed-mode model prediction, which deviates from parabolic growth at early stages, provides an adequate description o f the reported kinetics. For each transformation temperature, the value for mobility was chosen such that 82 2.3 Model Application Chapter 2 the best reasonable agreement with experimental data points could be achieved. Table 2-4 summarizes the mobility values obtained in this way expressed in units o f the mobility proposed by Krielaart and van der Zwaag [72] for F e - M n alloys. In this scale the mobility varies from 2 to 0.11 when the growth temperature increases from 665 to 715°C indicating that the activation energy for mobility must be essentially different from that suggested for F e - M n systems. Presenting the mobility in an Arrhenius-plot as shown in Figure 2-30, the slope o f this curve suggests a value o f -271kJ/mol for the activation energy. Clearly a negative activation energy is inconsistent with the physical concept o f thermally activated processes, the intrinsic mobility must increase with temperature and not decrease. Table 2-4 The required values for the interface mobility (in terms of the ratio to Krielaart and van der Zwaag mobility, M . ,J Fe M M fitted/M Fe-Mn T, °C to replicate the measured thickening rate of Fe-0.12wt%C-3.3wt%Ni. 2 1.2 0.7 0.11 665 680 700 715 -15 -16 S c -17 -18 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1000/T, K" 1 Figure 2-30 Arrhenius plot offitted mobilities for 0.12wt%C-3.3wt%Ni resulting in an apparent activation energy of-271kJ/mol shows mobility would have to be decreased with increasing 83 temperature. 2.3 Model Application Chapter 2 2.3.2.4 Implementing the solute drag effect It is concluded from the preceding discussion that a simple mixed-mode model cannot provide a reasonable description o f the experimental kinetics i n Fe-C-z alloys, no matter how the problem is treated thermodynamically. Although to resolve this drawback, different remedies might be potentially employed, accounting for the retarding effect o f soluteinterface interaction is considered here. This so-called solute drag effect has been originally proposed as the prime reason for the observed discrepancies between the measured growth kinetics o f ferrite and diffusion model predictions [62]. The amount o f the required drag pressure to be employed in the mixed-mode model can be evaluated by means o f the optimal fit to the experimental growth data as presented in the following. 2.3.2.5 Resulting solute drag parameters: Fe-0.21wt%C-l. 52wt%Mn The analysis o f allotriomorph thickening i n the 0.21wt%C-1.52wt%Mn alloy is first dealt with. A s mentioned earlier, the drag pressure o f segregated solute, here manganese, on ferrite-austenite interface can be quantified in terms o f two prime parameters. For the diffusivity term an Arrhenius temperature dependency can be considered as is expressed by Equation 2-61. In this way the uncertainty associated with the assumed value for interface thickness is also embedded i n the diffusivity parameter, which leads to minimizing the number o f applied fitting parameters. In the present work the activation energy is taken to be half o f the one for austenite bulk diffusivity, i.e. 132kJ/mol and 137kJ/mol for M n and N i atoms [91], respectively. The preexponential factor, DJ8 is then adjusted such that Db to be at least one order o f magnitude larger than solute diffusivity i n austenite. For given values o f the diffusivity term, Di/S and 84 2.3 Model Application Chapter 2 mobility, M, the solute drag pressure which is required to replicate the experimentally observed kinetics can be found by employing an adequate binding energy E 0 as the third fitting parameter. Using the intrinsic mobility proposed by Krielaart and van der Zwaag, the solute drag parameters obtained from the optimal fit to the measured kinetics o f ferrite growth in Fe-0.21wt%C-1.52wt%Mn are illustrated in Figure 2-31 for the three examined temperatures, i.e. 725, 730 and 735°C. To perform this simulation the D /8ratio 0 was changed from 30 to 1400cm/s. B y making reasonable assumption for 5, i.e. 0.5 nm, the diffusivity across the interface can be expressed in units o f the M n diffusivity i n austenite, therefore the above range corresponds approximately to a D I D r b combinations o f solute drag parameters, i.e. E 0 Mn ratio o f 60 to 1700. Obviously, various and DJ5, can be selected to replicate the observed experimental kinetic data for each growth temperature. Further, the result suggests that stronger solute-interface interaction is required as transformation temperature increases. 4.0 3.5 3.0 I2.5 ill 2.0 1.5 1.0 0 300 600 900 1200 1500 D /8, cm/s 0 Figure 2-31 Effect of temperature on solute drag parameters to describe the experimental data [34] of ferrite plate thickening in Fe-0.21 wt pet C-l. 52 wt pet Mn. 85 2.3 Model Application Chapter 2 As a rational outcome, it is expected that the intensity o f the required solute drag effect does depend on the value selected for the interface mobility. In order to characterize the role o f the mobility selection, the value o f the Krielaart and van der Zwaag mobility was decreased and increased by a factor 0.3 and 3, respectively. The resulting solute drag parameters for the growth temperature o f 725°C are depicted in Figure 2-32. Employing the mobility greater than that o f Krielaart and van der Zwaag appears to have an insignificant influence on the required binding energy, while decreasing the mobility leads to a drastic drop in the resulting drag pressure. However, over the entire examined D ID y b Mn ratio and mobility range the binding energy is revealed to be within order o f RT, which is consistent with the values reported in the literature [58,64]. 3.5 3.0 2.5 ¥ 2.0 0 LU 1.5 1.0 0.5 10 2 3x10 2 10 3 3x10 3 Figure 2-32 Effect of the selection of the intrinsic mobility on solute drag parameters to describe the experimental data [59] of ferrite allotriomorph thickening in Fe-0.21 wtpet C-1.52 wtpet Mn. 2.3.2.6 Solute drag parameters: 0.12C-3.1Mn and 0.12C-3.3Ni(wt%) steels Using the prescribed temperature dependencies for mobility and diffusivity terms, the combinations o f the required solute drag parameters to describe the measured allotriomorph 86 2.3 Model Application Chapter 2 thickness in Fe-0.12C-3.1Mn and Fe-0.12C-3.3Ni(wt%) alloys for the range o f investigated growth temperatures are presented in Figure 2-33 and Figure 2-34, respectively. To perform the simulations M =5.8 and 17.4cm mol/Js were assumed for M n and N i containing system, 0 respectively. A s is evident in Figure 2-26 the growth o f ferrite in 0.12C-3.3Ni(wt%) steel is much faster than that for Fe-0.12C-3.1Mn alloy for the entire investigated temperatures. Therefore in the framework o f the mixed-mode approach, a larger M , i.e. 17.4cm mol/Js was 0 selected for the Ni-containing alloy. This is close to the minimum mobility that is required to replicated the thickening rate at 665°C, which corresponds to the temperature o f the fastest growth rate i n 0.12C-3.3Ni(wt%) steel (cf. Figure 2-26). Consistent with the previous discussion, different sets o f E and DJ8 values can be employed 0 to describe the experimental data adequately. Moreover, a strong temperature dependency o f the required binding energy is suggested. For example, assuming ZV(5=710cm/s, E 0 increases from 0.75 to 3JRT when temperature rises from 550 to 650°C in the 0.12wt%C-3.1wt%Mn steel. Taking the same value o f DJ8, the binding energy changes from 0.65 to 2.2RT in the N i containing alloy when the temperature increases from 665 to 715°C. A l s o it is noted that manganese tends to show stronger interaction with ferrite-austenite interface compared to nickel. 87 2.3 Model Application Chapter 2 5x10° 5x10 5x10 1 2 5x10 3 D / 5 , cm/s Figure 2-33 Solute drag parameters required to replicate the experimental growth kinetics [62] of ferrite at different temperatures in Fe-0.12wt%C-3.1wt%Mn. 715°C / 700°C / 6 8 0 ° C - ' 9x10 9x10 1 2 665°C 9x10 3 D /5, cm/s 0 Figure 2-34 Required solute drag parameters to describe the growth kinetics of ferrite measured [62] at different temperatures in Fe-0.12wt%C-3.3wt%Ni. 88 2.3 Model Application Chapter 2 2.3.2.7 Discussion on the temperature dependence of binding energy The present analysis o f ferrite growth in the manganese containing alloys indicates that a binding energy o f order o f RT is required for the model calculations to replicate the experimental kinetics. A binding energy o f RT is consistent with measurements o f M n segregation to austenite grain boundaries [88]. The alloy containing nickel shows less tendency for solute-interface interaction, this agrees with the expectation that N i has a lower binding energy which is not more than about half o f that o f M n [58,88]. However as an overall outcome it is apparent that stronger drag pressure o f the segregated solute is required at the higher growth temperatures. The variation o f binding energy with temperature, in terms o f undercooling below T 3 , is compared for all the investigated alloys using DJ8 = Ae 710 cm/s in Figure 2-35. 35 30 25 o E 20 15 • \ ± • • * \ 0.12C-3.1Mn 0.21C-1.52Mn 0.12C-3.3Ni • - \ v LU 10 • \^ 5 D /5 = 710 cm/s A c 0 20 i 50 80 < 110 i i 140 170 200 Figure 2-35 Variation of applied binding energy with temperature to describe the experimental growth rate of ferrite [34,62] in 0.12wt%C-3.1wt%Mn, 0.12wt%C-3.3wt%Ni and 0.21wt%C-1.52wt%Mn steels. 89 2.3 Model Application Chapter 2 The predicted temperature dependence o f the binding energy observed here is rather unexpected since according to the general concept o f the solute-interface interaction, a temperature independent binding energy would be expected. It could be argued that it is simply an artifact resulting from the inappropriate temperature dependency which were assumed for the intrinsic mobility as well as for the solute diffusivity across the interface. Therefore, to exclude any potential impact caused by these two parameters, their activation energies have to be revisited. First the role o f the mobility term is considered by changing the assumed activation energy within an acceptable range, i.e. 140±40 kJ/mol in 0.12wt%C-3.1wt%Mn steel. Reducing the activation energy, Q to lOOkJ/mol leads to an E that increases by 1.8i?rper each 100°C 0 increase in temperature. This temperature dependence o f the binding energy is 20% less than that obtained by assuming the nominal value (>=140kJ/mol. Definitely, the effect due to changing the activation energy for the mobility is insufficient to eliminate the temperature dependence o f binding energy. The diffusivity o f solute across the interface is the other unknown parameter for which, as first approximation, an activation energy, which is half o f that for bulk diffusivity in austenite was assumed. Alternatively, postulating a temperature independent binding energy the activation energy Q b for the jump across the interface can be treated as an adjustable parameter. This is simply accomplished by using the data represented in the format of Figure 2-33 and Figure 2-34, i.e. the activation energy can be obtained by assuming a certain value for the binding energy, E . For example taking E =18 and 28kJ/mol, respectively, the results 0 0 of this analysis are summarized with an Arrhenius plot i n Figure 2-36 for Fe-0.12wt%C3.1wt%Mn. A similar plot assuming E =10 and 15kJ/mol, respectively, for Fe-0.12wt%C0 90 2.3 Model Application Chapter 2 3.3wt%Ni alloy is illustrated i n Figure 2-37. The resulting negative apparent activation energies, i.e. approximately -lOOkJ/mol and -480kJ/mol for M n and N i containing alloys, respectively, indicate that this approach leads to an unreasonable behavior for the diffusivity of solute across the interface. Therefore, it can be ruled out that the increase o f binding energy with temperature is an artifact o f the selection o f activation energies for mobility and diffusivity, respectively. -26 •—• A Q=-80kJ/mol A -27 • -28 • Q=-130kJ/mol -29 • A • -30 1.06 1.09 1.12 E =18kJ/mol E =28kJ/mol 0 0 1.15 1.18 1000/T, 1/K Figure 2-36 Arrhenius plot for D assuming constant values for the binding energy in Fe-0.12wt%Cb 3.1wt%Mn. Alternatively, the temperature dependency o f E might be explained by a gradual transition 0 from paraequilibrium towards N P L E , i.e. the increase in the degree o f solute partitioning in the vicinity o f the interface as growth rate decreases at the later stages o f ferrite formation. A t the onset o f transformation, sufficiently large thermodynamic driving pressure is available which gives rise to rather fast growth, such that no appreciable redistribution o f substitutional elements is expected even locally at the ferrite-austenite interface. Therefore, the paraequilibrium assumption is a justifiable treatment at the beginning o f growth. It is worth 91 2.3 Model Application Chapter 2 to note that the deviation from paraequilibrium is likely to occur sooner at higher growth temperatures, where substitutional elements are more mobile and less driving pressure for allotriomorph thickening is available. Neglecting to account for this deviation from paraequilibrium could result in the overprediction o f the experimental growth rates, which has to be compensated by considering a stronger solute-interface interaction, i.e. a temperature-dependent binding energy. 1.00 1.03 1.06 1.09 1000/T, 1/K Figure 2-37 Arrhenius plot for D assuming constant values for the binding energy in Fe-0.12wt%Cb 3.3wt%Ni. Apart from formation o f a solute spike in front o f the interface, at sufficiently high temperatures it is increasingly likely that the substitutional atoms may diffuse beyond the interface region. The extent o f this redistribution is simply expected to exceed more than several times o f the interface thickness. This is the concept that is proposed to be considered for rationalizing the aforementioned increase o f the apparent binding energy. 92 2.3 Model Application 2.3.3 Chapter 2 Analysis of the overall kinetics of ferrite formation in A36 steel Although the kinetic analysis o f allotriomorph thickening provides valuable insights into the underlying mechanisms o f ferrite reaction and particularly aids to characterize the interfacial and solute drag parameters, it is o f little industrial significance. In practical processing o f low-carbon steels, austenite decomposes entirely to ferrite and some other phases, such that the assumption o f planar geometry for the ferrite-austenite interface is no longer valid. Further, the applied cooling rates or undercooling is sufficiently large that nucleation site saturation at austenite grain boundaries is likely to occur promptly when less than 5 percent of ferrite forms. In these circumstances the overall kinetics can be accurately modeled by considering the subsequent growth o f these ferrite particles provided that an appropriate geometry for the growing phase is assumed. Here, a spherical shape is adopted for the parent austenite grain that has a diameter replicating the mean volumetric austenite grain size. Ferrite constitutes the outer shell having uniform thickness circumferentially and grows toward the center o f the spherical austenite grain. 2.3.3.1 Experimental kinetics The examined material was chosen to be a model ternary steel with nominal composition o f 0.17wt%C-0.74wt%Mn. The overall kinetics o f ferrite formation during continuous cooling cycles had already been quantified in previous work at U B C [63]. However for the sake o f completion, some selected isothermal and continuous cooling thermal treatments have also been conducted using the Gleeble 3500 thermomechanical simulator equipped with a dilatometer. Tubular samples o f 8 mm outer diameter having a wall thickness o f 1mm and a length o f 20mm were employed in the transformation tests. The volume change during transformation is measured in terms o f the change in diameter by means o f a dilatometer 93 2.3 Model Application Chapter 2 attached on the outer surface o f the sample right at its center plane, the same location where a thermocouple for temperature measurement is welded. The specimen is heated up employing resistance heating and cooled down with the help o f forced convection by blowing helium gas through inside the tube. A precise computer-aided controller facilitates applying any complex thermal cycles readily. Special care has to be taken for isothermal cycles where the sample is subjected to high cooling rates after the austenitization stage, such that undershooting below the intended transformation temperature is prevented. The recorded kinetics can be substantially influenced by the amount o f a potential temperature undershooting, even i f it takes only a short time period. The details o f the experimental procedure, as well as the practical method to interpret the acquired dilation data is delineated elsewhere [63,95]. The microstructure o f the specimens was characterized by optical microscopy after sample preparation using traditional metallography procedures. The results can confirm i f the employed thermal cycle yields to polygonal ferrite as the dominant structure. Further, the final ferrite fraction in either o f iso- or non-isothermally treated specimens is quantified from optical micrographs. Despite the more engineering relevance o f continuous cooling ferrite formation, first the kinetics o f isothermal austenite decomposition into ferrite is going to be analyzed here. The measured kinetics for transformation at four different temperatures 735, 745, 755 and 765°C from a prior austenite volumetric grain size o f 96um are illustrated in Figure 2-38. Using the lever law the dilation response recorded during isothermal ferrite formation was converted to ferrite fraction such that the plateau o f the dilation curve at the end o f reaction corresponds to the final fraction o f ferrite predicted by paraequilibrium. A s shown in Figure 2-39, the final fractions quantified from optical micrographs appear to be in an acceptable agreement with 94 2.3 Model Application Chapter 2 both orthoequilibrium and paraequilibrium calculations, which are quite close for the investigated cases. Therefore, the experimental ferrite fractions at the end o f reaction were normalized with respect to paraequilibrium predictions. 0 200 400 600 800 1000 1200 1400 Time, s Figure 2-38 Measured kinetics of isothermal ferrite formation in a 0.17wt%C-0.74wt%Mn steel with an austenite of grain size of 96pm. 0.4 710 720 730 740 750 760 770 780 Temperature, °C Figure 2-39 Measured and calculated ferrite fractions by optical metallography and the calculated ones assuming OE and PE. 95 2.3 Model Application Typical microstructures Chapter 2 at the end o f isothermal austenite decomposition followed by quenching to room temperature are represented in Figure 2-40 for growth temperature o f 735 and 765°C. These micrographs confirm that ferrite nucleates at austenite grain boundaries and grows with polygonal morphology. Figure 2-40 Microstructure of Fe-0.17wt%C-0.74wt%Mn specimens reacted isothermally at (a) 735 and (b) 765°C, d =96^tm. r 2.3.3.2 Modeling the isothermal ferrite transformation To replicate the above experimental growth kinetics, an attempt was first made to employ the mixed-mode model solely, without including the solute-interface interaction, i.e. the intrinsic mobility served as the only adjustable parameter for simulations. T o calculate the required thermodynamic information, the system is treated under paraequilibrium condition. The fitting practices predicted smaller values for the mobility than that suggested for F e - M n by Krielaart and van der Zwaag. The mobility varies from 0.5 to 0.1 in unit o f that for F e - M n when the transformation temperature increases from 735 to 765°C. The results are also 96 2.3 Model Application Chapter 2 depicted in Table 2-5. The Arrhenius plot o f this fitted mobility, i.e. the natural logarithm o f mobility versus the reciprocal o f temperature, suggests an apparent activation energy of 327kJ/mol, which is not a sensible physical value for the thermally activated jump o f atoms across the interface and indicates the mobility would have to decrease as temperature increases. This confirms again that in addition to the local reconstruction o f B C C from F C C lattice at the transformation interface, another atomic process has to be in operation. Therefore solute-interface interaction seems to be a justifiable postulation here. Table 2-5 The required values for the interface mobility (in terms of the ratio to Krielaart and van der Zwaag mobility, Mf . ) e Mn t o describe the overall kinetics of isothermal ferrite formation in Fe0.17wt%C-0.74wt%Mn. M fitted/M Fe-Mn T, °C 0.5 0.35 0.2 0.1 735 745 755 765 Thus, to model the isothermal kinetics o f ferrite reaction in Fe-0.17wt%C-0.74wt%Mn alloy, the mixed-mode approach incorporating solute drag was employed. The ensuing result shown in Table 2-5, which suggests a mobility smaller than that o f Krielaart and van der Zwaag would be sufficient to describe the isothermal ferrite kinetics for the transformation temperatures above 735°C. However, the analysis o f ferrite formation during continuous cooling conditions, as w i l l be discussed in the next section, revealed that a larger mobility is required for the lower transformation temperature down to 620°C, i.e. well below the temperature range studied i n the isothermal test series. Consequently, a mobility twice o f that suggested for F e - M n is adopted for the calculations. This mobility would overpredict the measured kinetics i f no retarding effect o f solute atoms is accounted for. Then DJ8 and the binding energy, E , serve as fitting parameters. It was 0 97 2.3 Model Application Chapter 2 found that for Do/S= 142cm/s the optimal descriptions o f the isothermal kinetics between 735 to 765°C can be achieved i f the following temperature dependence for the binding energy is taken: E (JI mol) = 4\RT - 3 . 3 x l O (2-64) 5 0 Adopting this binding energy implies that E is i n the order o f R T , such that it increases from 0 1.1RT at 735°C to 2.3RT at 765°C. Further, the assumed DJSof in terms o f the D ID y b Mn 142cm/s can be interpreted ratio by taking a l n m thickness for the interface, then D ID Y b Mn varies from 270 at 7 3 5 ° C to 170 at 765°C, which as intended, yields to solute diffusivity across the interface to be at least two orders o f magnitude larger than austenite bulk diffusivity. The model predictions using these adjustable parameters are plotted against the experimental overall kinetics in Figure 2-41. Apart from a slight overprediction at 765°C, the measured data are well captured using the present modeling methodology. 0.8 Time, s Figure 2-41 Experimental (symbols) and calculated (solid lines) kinetics of isothermal ferrite formation in the temperature range of 735 to 765°C in Fe-0.17wt%C-0.74wt%Mn with an initial austenite grain size of 96pm. 98 2.3 Model Application Chapter 2 2.3.3.3 Implication of the modified SD model to isothermal transformation To quantify the drag effect o f the segregated solute, the modified Purdy-Brechet theory was employed for all the simulations in the present work. The details o f the refinements made to the original solute drag approach are elaborated i n section 2.2.2.4. For ferrite formation at 735°C the variation o f the solute drag pressure and the chemical driving pressure at the interface, with isothermal reaction time are represented in Figure 2-42. In case o f employing the modified approach as shown in Figure 2-42a, both o f these two energy terms, i.e. and AG INT decrease monotonically and vanish entirely as transformation approaches to the AGSD, end. In contrast, using the original solute drag model, as shown i n Figure 2-42b, results i n a non-zero drag pressure at the end o f transformation, the point that is defined by a zero effective driving pressure at the interface. In analysing the overall kinetics o f isothermal ferrite evolution, the mentioned artifact enforces the growth to cease earlier, thereby the final fraction o f ferrite is predicted to be substantially smaller than what experimental observation indicates. F o r instance Figure 2-43 shows the measured kinetics at 735°C together with the predictions o f the mixed-mode approach using both the original and the modified solute drag model. Although incorporating the original Purdy-Brechet theory provides a decent agreement with experiments up to a ferrite fraction o f 70%, the final fraction is significantly underestimated whereas the modified theory can capture the kinetics to final transformation stages. Attempting to assume a linear temperature variation for E , the observed premature cessation o f growth is even 0 more pronounced at higher transformation temperature because o f the generally lower interface velocities. i 99 2.3 Model Application Chapter 2 100 (a) Modified SD Model Driving pressure Drag pressure o E CO c UJ 0 50 100 150 200 250 300 350 Time, s 80 (b) Original SD Model Driving pressure Drag pressure 0 50 100 150 200 250 300 350 Time, s Figure 2-42 Variation of thermodynamic driving pressure and solute drag pressure with time for growth offerrite at 735°C in 0.17wt%C-0.74%Mn steel (a) using modified SD model, (b) adopting the original Purdy-Brechet quantification. 100 2.3 Model Application Chapter 2 1 .uu 0.75 - / 0.50 If 0.25 / • 0.001 Experimental data - Model prediction (Purdy-Brechet SD) - Model prediction (Modified Purdy-Brechet SD) 1 0 100 1 1 200 300 i 400 500 Time, s Figure 2-43 Ferrite formation kinetics at 735°C in Fe-0.17' wt% C-0.74 wt%Mn showing the predictions adopting the original Purdy-Brechet solute drag theory and its modification, respectively. 2.3.3.4 Modeling the ferrite evolution during continuous cooling As the prime benefit o f developing a physically based model, it is expected that the model will be capable o f capturing the situations beyond the cases where it was validated experimentally. Driven by this concept, attempts were made to apply the interfacial parameters which had been determined from the analyses o f isothermal ferrite formation, to simulate the austenite decomposition into ferrite during non isothermal conditions in the same steel, i.e. Fe-0.17wt%C-0.74wt%Mn. The experimental kinetics for an initial austenite grain size o f 18um and the cooling rates of 1 and 16°C/s are examined, where polygonal ferrite was found to be the predominant microstructure. A s illustrated in Figure 2-44, the model prediction is i n good agreement with the measured kinetics. Capturing both isothermal and continuous cooling kinetics by employing a consistent set o f adjustable parameters confirms the versatile capability o f the proposed fundamental modeling approach. 101 2.3 Model Application Chapter 2 660 680 700 720 740 760 780 Temperature, °C Figure 2-44 Measurements [63] and predictions for ferrite formation during continuous cooling ft an initial austenite grain size of 18pm in 0.17wt%C-0.74wt%Mn steel. 2.3.4 Analysis of ferrite evolution in quaternary systems: DP and T R I P steels The chemistry o f advanced high strength steels is rather complex, containing more than one substitutional alloying element. This ensures the material to show the appropriate transformation characteristics during the industrial processing that it undergoes. In these multicomponent systems, the simultaneous segregation o f different species to the ferriteaustenite interface is likely to occur. In these circumstances the subject o f co-segregation becomes relevant and some attempts have been made to quantify this effect [67,68]. Undoubtedly, the observed apparent solute drag effect arises from the contribution o f all segregated elements, which would have to be appropriately accounted for in a detailed fundamental model. However, to characterize the individual effects at least two physical properties, i.e. diffusivity across the interface and binding energy to the interface, would have to be introduced into the calculation for each alloying element. Noting that none o f these 102 2.3 Model Application Chapter 2 parameters are independently known from experimental or theoretical investigations, it is impractical to propose an overall transformation model which would incorporate all these parameters. Thus, in order to maintain a minimum number o f unknown interfacial parameters, the drag effect o f manganese is treated in the model as an effective force resulting from all segregated solute atoms. This assumption is based on the fact that manganese is the major alloying elements in all advanced high strength steels, i.e. usually in the 1.5 to 2wt% range. Then the activation energy o f the diffusivity across the interface is taken as described for the F e - C - M n system and A E represent the chemical potential difference o f M n across the interface which is affected by the presence o f other substitutional elements, e.g. M o or S i . Driven by this philosophy, the continuous cooling kinetics o f ferrite formation in two novel dual-phase (DP) and T R I P steels, with composition o f Fe-0.06wt%C1.85wt%Mn-0.16wt%Mo and Fe-0.21wt%C-1.53wt%Mn-1.54wt%Si, respectively, have been analyzed. In both cases the simulation is restricted to the cooling conditions where polygonal ferrite is the predominant phase in the final microstructure. The comparison o f experimental data [96] with the model prediction is illustrated in Figure 2-45, for the D P steel with austenite grain sizes, d , of 16 and 24fam, respectively. The y adequate description o f observed transformation kinetics is accomplished for two cooling rates o f 1 and 18°C/s by selecting the mobility factor, M , as 10.5 cm mol/Js and setting D /8 0 0 to 100 cm/s, while the binding energy is kept in the order o f RT as given by: E (J/mol) o = \.5RT-3.\x\0~ 3 103 (2-65) 2.3 Model Application Chapter 2 0.8 k A ^ O, \\ ^ A \ A \ ~ ° A Q ) C A dT = 13um 600 I A\ i 620 ^l°C/s% 18°C/s °L A\ A\ A\ dY = 20 nm Model prediction 1 — 580 \ 640 I ^A\ Q \ \ \ \ i i 660 680 700 720 740 Temperature, °C Figure 2-45 Measured [96] (symbols) and predicted (solid lines) kinetics of austenite decomposition into ferrite during continuous cooling of a dual-phase steel, Fe-0.06wt% C1.85wt%Mn-0.16wt%Mo. For the hot-rolled T R I P steel with prior austenite grain diameter o f 20p.m, the experimental data for the cooling rates o f 1 and 5°C/s, where predominantly polygonal ferrite had been observed in the microstructure, were taken from literature [97]. Taking the mobility factor o f 5.8cm mol/Js and assuming DJ8 as 28.4 cm/s, the model can accurately capture the measured kinetics. F o r the simulation, a linear temperature dependency for the binding energy was adopted, i.e.: E (J/mol) o = lART-2Ax\0- 3 which is very similar to that obtained for the D P steel. 104 (2-66) 2.3 Model Chapter 2 Application 1.0 r0.8 - 600 650 700 750 800 Temperature, °C Figure 2-46 Comparison of observed data [97] and model prediction offerrite formation kinetics during continuous cooling of TRIP steel, Fe-0.21C-1.53Mn-1.54Si(wt%), with an austenite grain diameter of 20pm. Figure 2-47 and Figure 2-48 compare the solute drag parameters employed to perform the above simulations for D P and TRIP steels. Assuming an interface thickness, 28, o f l n m , manganese diffusivities across the interface are plotted for the entire temperature range o f transformation. For the sake o f comparison the diffusivity data considered for A 3 6 steel is also included i n Figure 2-47. Clearly these diffusivities are several orders o f magnitude larger than the manganese bulk diffusivity i n austenite. These results confirm that the diffusivity values used in the simulations are physically relevant and meaningful. Further, as illustrated i n Figure 2-48, the required binding energy o f solute to the interface is in the order o f RT for both steels i n the examined temperature range and increases slightly with temperature. This is consistent with the previous findings for the F e - C - M n systems and also with that suggested i n the literature regarding the interaction o f alloying elements with moving ferrite-austenite interfaces [59,58,64], 105 2.3 Model Chapter 2 Application io- ] io- : Dual- phase 1/1 CM^ E o 10 13 TRIP 10 14 io- Bulk D|v| in y Q=264kJ/mol : io- n : 0.9 1.0 1.1 1000/T, K" 1.2 1 Figure 2-47 Comparison of manganese diffusivity across ferrite-austenite interface for A36, DP and TRIP steels with bulk diffusivity in austenite, assuming 25of lnm. 590 630 670 710 750 790 Temperature, °C Figure 2-48 Variation of manganese binding energy to a/y interface with temperature for DP and TRIP steels. 106 Chapter 2 2.4 Summary and Remarks 2.4 Summary and Remarks (Ferrite Formation) Relevant in particular to industrial processing conditions, a physically based model in the framework o f the mixed-mode approach was developed to describe ferrite growth in multicomponent steels. Assuming paraequilibrium, the kinetic effect o f substitutional elements, which arises from the interaction o f solute atoms with the migrating ferrite-austenite interface, was quantified using the modified Purdy-Brechet description. However, the transition from paraequilibrium to local equilibrium was not considered in the model. The model employs three parameters, the intrinsic interface mobility, M, the binding energy of the major substitutional element to the interface E , and the diffusivity o f this solute across 0 the interface Db, thus, the model has the same number o f adjustable parameters as the traditional J M A K approach. In contrast to the J M A K methodology with entirely empirical parameters, the parameters employed in the proposed model are well defined in terms o f their physics and the potential ranges o f their values are well known. However, since the exact value of these parameters cannot be determined precisely by currently available experimental and/or modeling techniques, they have to be determined by the optimal description o f the measured transformation kinetics. Analyzing different steel grades is then expected to provide a base guideline for selecting these parameters for new chemistries to be examined. To verify the predictive capability o f the model and also to further examine the physical relevance and temperature dependence of the model parameters, several isothermal and continuous cooling kinetics o f ferrite formation were analyzed. The major findings and conclusions o f the model applications are summarized here: • The analysis o f the thickening o f ferrite allotriomorphs in three binary Fe-C alloys containing 0.11, 0.23 and 0.42wt%C revealed that the diffusion model overestimates 107 2.4 Summary and Remarks Chapter 2 the experimental parabolic growth constants for the lower carbon contents, i.e. less than in 0.23wt%C. Assuming an interface mobility close to what was proposed by Krielaart and van der Zwaag, most o f the measured thickening rates can reasonably be described by the mixed-mode model. • Analyzing the kinetics o f allotriomorph thickenings i n 0.21wt%C-1.52wt%Mn, 0.12wt%C-3.1wt%Mn and 0.12wt%C-3.3wt%Ni steels, showed that various combinations o f parameters, E and Dt>, can be employed to replicate the measured 0 growth rate. Further, the ensuing solute drag parameters were found to hinge on the value selected for the mobility term. • The overall kinetics o f isothermal ferrite formation in a 0.17wt%C-0.74wt%Mn steel were described accurately by the model. Further, using the same interfacial parameters determined from the isothermal data, the model replicated reasonably the kinetics o f ferrite evolution during continuous cooling treatments. • Relevant to advanced high strength steels with complex chemistry, the continuous cooling ferrite formation o f 0.06wt%C-1.85wt%Mn-0.16wt%Mo D P steel and 0.21wt%C-1.53wt%Mn-1.54wt%Si TRIP steel can also be described using the present model framework. Since for both cases M n constitutes the major alloying element, its drag pressure was solely accounted for i n the model. However, the ensuing values for solute drag parameters should be regarded as effective ones describing the apparent solute-interface-interaction o f all the segregating species. • Consistent with the literature, the determined binding energy was i n the order of RT, however, a linear temperature dependency for E was found here for all examined G cases. This can potentially be related to the increase o f the partitioning degree o f 108 2.4 Summary and Remarks substitutional atoms Chapter 2 at higher temperatures and phenomenon is required. 109 further investigation o f this 3.1 Introduction Chapter Chapter 3 3 :Bainite Transformation 3.1 Introduction Apart from the diverse characteristics o f bainite, in terms o f both kinetics and morphological appearance that renders this area o f research to be an active one for further explorations, bainite transformation has recently gained more attention due to its significance in processing of newly-developed advanced high-strength steels. In particular the bainite reaction is a crucial processing step for T R I P steels during which stabilization o f untransformed austenite is accomplished. Further, in processing of D P steels formation o f bainite upon cooling from intercritical region has to be prevented. This constitutes the motivation behind the research outlined in the present chapter. The underlying mechanism o f the bainite reaction has been a matter o f debate for many years [98, 99,100,101], In the case o f T R I P steels, the issue becomes even more complex since bainite formation occurs from very small austenite grains, i.e. l-3|j,m, encompassed by a ferrite matrix. Moreover, due to presence o f silicon, aluminum or other suitable alloying elements the formation o f carbides is delayed. A s a result, carbide-free bainite morphologies develop which adds further controversy to the subject [102,103]. The aim o f this chapter is to better understand the features o f bainite formation in the alloying system that was originally developed for classical T R I P steels. Moreover, the focus is to develop a model that is applicable to the bainite reaction in T R I P steels. A s a first step, the austenite-to-bainite transformation is investigated in a 0 . 6 C - 1 . 5 M n - l .5Si (wt%) steel, i.e. for the chemistry expected at the beginning o f the bainite treatment in a classical 0.2C1.5Mn-1.5Si T R I P steel (cf. points 2 and 3 in Figure 1-1). The isothermal transformation 110 3.1 Introduction Chapter 3 kinetics are analyzed using available modeling approaches, i.e. Johnson-Mehl-Avrami- Kolmogorov ( J M A K ) , as well as diffusional [104,22] and displacive [23] methodologies. The capabilities and limitations o f these approaches are examined i n regard o f capturing the experimental kinetics. In the second part, the kinetics and microstructural features o f bainite evolution after intercritical treatment o f a classical T R I P steel, i.e. 0.18C-1.55wt%Mn1.70wt%Si, is explored. Finally, regarding transformation treatments for a M o - T R I P during continuous cooling steel with 0.19wt%C-1.5wt%Mn-1.6wt%Si-0.2wt%Mo, the kinetics o f bainite evolution after ferrite formation is studied. 3.2 Literature Review 3.2.1 Overview Bainite is a product o f austenite decomposition i n the temperature gap between the region at which the products are formed b y a diffusional mechanism, e.g. pearlite or polygonal ferrite, and by a displacive mechanism, i.e. martensite region. The reason that at sufficiently low temperature below A e ] , bainite competes with pearlite formation is attributed to the asymmetry o f the phase diagram [105], i.e. the extrapolated A c m line is much steeper than the extrapolation o f A 3 line. This gives rise to the preference o f separate formation o f ferrite e plates rather than lamellar-cooperative growth o f ferrite and cementite. However as the temperature is lowered further, then enough driving pressure to overcome the strain energy associated with the diffusionless formation o f supersaturated bet ferrite, i.e. martensite, would be provided and austenite transforms athermally to martensite. The start temperature of bainite transformation depends on steel chemistry and can be represented by the following empirical equation [106]: 111 3.2 Literature Review Chapter 3 B (°C) = 830-2 70C-90Mn-3 7Ni- 70Cr-83Mo ( 3-1) s (For composition ranges of wt%C: 0.1-1.55. wt%Mn: 0.2-0.7. wt%Si: 0.1-0.35, wt%Mo: 0.0-1.0) From the microstructural point o f view, although bainite features quite diverse morphologies depending on the alloying system and reaction temperature [107,108] it is traditionally defined as a fine aggregate o f ferrite plates (or laths) and cementite particles precipitated between or within the ferritic counterpart. These two distinct types o f morphology are referred to as upper and lower bainite, respectively. The carbides that appear as coarse particles between two adjacent laths are called interlath carbides, and these precipitate in form of fine discs within each lath at the (112)BF plane are known as intralath. It is common to refer to the ferritic part o f bainite as bainitic ferrite (BF), which is quite different from polygonal ferrite. Bainitic ferrite has a very high dislocation density o f about 10 m* , and has 14 2 a lath or plate morphology. Figure 3-1 shows the three-dimensional view o f lath and plate, respectively. Figure 3-1 3D representation of lath and plate [109]. 3.2.2 Mechanism of bainite transformation Due to the complex characteristics o f bainite evolution, w h i c h has led to conflicting observations and interpretations, no conclusive description regarding the underlying physics of this reaction has been provided yet. Adopting shear mode formation o f bainite as 112 3.2 Literature Review Chapter 3 originally suggested by Zener [104], Bhadeshia has divided the decomposition o f austenite into two categories, "Displacive" and "Reconstructive" [109]. The former is characterized by invariant-plane strain shape change featuring a large shear component in which neither iron nor substitutional atoms can migrate individually. In case o f the reconstructive transformation, a new lattice is constructed by diffusion o f iron and substitutional atoms, as well as the interstitial species during both nucleation and growth events. In this approach, Widmanstatten ferrite, bainite and martensite fall in the displacive category, although different growth criteria have to be fulfilled for each to be evolved [109]. O n the other side o f this long lasting dispute, several researchers are arguing for diffusional edgewise growth o f bainite, which i n some cases is suggested to be controlled by a ledge mechanism [99], and regarded bainitic ferrite essentially to be the same as a Widmanstatten plate. A brief review of both approaches is provided in the following. 3.2.3 Displacive mechanism of bainite transformation According to the displacive mechanism o f bainite formation as described in detail by Bhadeshia [109], the transformation has a shear nature and resembles the martensite transformation. A t sufficiently high under cooling, when diffusional transformation o f ferrite or pearlite becomes impossible, sufficient driving pressure for diffusionless, shear mode transformation becomes available. A t this condition, bainitic ferrite laths (or platelets) nucleate at austenite grain boundaries and grow to a certain limited size, without any carbon diffusion. The change i n crystal structure o f the substitutional sublattice, which occurs during diffusionless growth, has a large shear component that gives rise to the observation o f a surface relief effect. I f the strain is elastically accommodated, then the strain energy o f 113 3.2 Literature Review Chapter 3 bainitic ferrite amounts to about 400Jmor'. Some o f the shear strain due to shape change is accommodated plastically by slip in both parent austenite and product ferrite. The dislocation debris in austenite, created by plastic accommodation (or plastic relaxation) stifles further growth o f bainitic ferrite laths. Thereby they grow to a limited size, much smaller than the austenite grain size. Transformation proceeds by successive nucleation o f new laths, face-toface near the tip o f already existing laths, and subsequent growth. The aggregate o f these parallel laths is known as a sheaf or packet o f bainite. The subunits within a sheaf have small relative misorientations but they all exhibit the same variant o f parent-matrix orientation relation. This sub-structural feature o f a bainite plate, i.e. consisting o f several subunits detected only by T E M technique, is an inherent characteristic that discriminates bainitic ferrite from Widmanstatten ferrite. The supersaturation o f bainitic ferrite with respect to carbon constitutes the central basis o f displacive growth, after which the excess carbon is soon partitioned into the adjacent residual austenite, encompassed by B F laths. The thickness o f interlath residual austenite is very small, so that their transformation carbon proceeds, enrichment carbon takes place only in few accumulates in untransformed milliseconds. A s austenite until the diffusionless growth o f newly nucleated laths becomes thermodynamically impossible. A t this stage (the so-called transformation stasis) further transformation ceases, which is the basis of the "incomplete reaction" theory of bainite. The maximum extent to which bainite reaction can proceed is determined by the T line, which represent the final limit o f austenite 0 enrichment with carbon before the diffusionless growth ceases. Actually, the T line is the 0 locus point where austenite and ferrite with the same chemistry have equal free energies. Figure 3-2(a) illustrates the free energy curve for austenite and ferrite and the definition of 114 3.2 Literature Review Chapter 3 the T line, and Figure 3-2(b) shows the location o f the T line in the F e - C diagram. To 0 0 account for the strain energy term the T line is modified to a T' line. 0 0 Carbon Concentration _ Carbon (b) (a) Figure 3-2 The T„ line, (a) The free energy curves of austenite and ferrite and their relation to the different phase boundaries in the Fe-C diagram, (b) the relative location of T in the Fe-C diagram 0 [109]. 3.2.4 Carbide precipitation When the initial carbon concentration is sufficiently high and elements such as A l or Si are not present, the conditions for carbide precipitation from enriched austenite, trapped between ferrite platelets may be satisfied. Therefore, i f the carbon concentration o f residual austenite, c , exceeds the amount which is given by extrapolation o f the y/(y+cementite) boundary in r the Fe-C phase diagram, the carbide precipitation from enriched austenite takes place. The shaded area in Figure 3-2(b) shows the region where the austenite is unstable with respect to cementite and its decomposition occurs. Note that below the critical temperature T , carbide c precipitation from austenite occurs simultaneously with the growth o f bainitic ferrite laths. At sufficiently low temperature, when due to low mobility the time to decarburize supersaturated B F platelet is larger than the time required to precipitate cementite within the 115 3.2 Literature Review Chapter 3 platelet, the latter process precedes the carbon rejection to adjacent austenite and intralath carbides would form. It has been reported that in plain Fe-C alloys, more than 0.3% carbon is necessary to fulfill this condition. A s a consequence o f carbide precipitation within platelets, some carbon is already tied up in the form o f cementite. Therefore, precipitation o f cementite from enriched austenite is reduced. If the initial carbon content is very low, e.g. ultra-low carbon bainitic ( U L C B ) steels, or graphite-former elements such as Si and A l are present, carbide formation can be inhibited or lags behind B F formation for a long period o f time. The resulting morphology is referred to as carbide-free bainite. In T R I P steels, carbide-free bainite is required to guarantee sufficient carbon enrichment in austenite in order to shift the martensite start temperature below room temperature. The effect o f silicon is associated with its low solubility i n cementite [110]. Therefore, the need for silicon to diffuse away from the cementite/ferrite interface could explain its retardation effect on cementite growth. Figure 3-3 shows the three different cases for bainite: carbide free bainite, upper bainite and lower bainite. Carbon supersaturated plate Carbon diffusion into austenite _ . , Carbon supersaturated, plate ^ S s % -.. . ^ ™ I ~ Carbide precipitation from austenite Carbon diffusion Imo ^ austenite and carbide precipitation in ferrite \ J * ' "^'^' ^^^^^^^^^^^ ^^^^^^^^ } Carbide-free bainite Upper Bainite Lower bainite Figure 3-3 Evolution of three different bainite morphologies from the supersaturated plate [109], 116 3.2 Literature Review 3.2.5 Chapter 3 Thermodynamic criteria for displacive nucleation and growth According to the displacive theory [109], a sheaf o f bainite consists o f many ferrite subunits formed by a shear mechanism and grow to a certain size. Autocatalytic nucleation, in which new subunits nucleate near the tip o f an already existing one, is a key factor i n formation o f a sheaf. The nucleation process is displacive [111] in which similar to martensite nucleation the activation energy o f nucleation varies linearly with the chemical driving pressure [112], in contrast to classical nucleation theory that predicts inverse proportionality to the square o f driving pressure. The linear dependence arises from the idea that many preexisting embryos are available in the undercooled austenite. Therefore, the activation energy is then a barrier to interface migration. It is suggested that glissile dislocation located i n the austenite grain boundaries can dissociate directly into an embryo o f ferrite at sufficiently high driving pressure. Nucleation takes place with carbon partitioning under paraequilibrium condition [111]. The minimum driving pressure required to initiate the displacive nucleation, GN, is found to be a linear function o f the transformation start temperature for bainite, T$ [113]. A universal nucleation function for GN has been proposed that predicts the highest temperature at which ferrite can nucleate by displacive mechanism, irrespective o f alloy chemistry. This universal function, which is a criterion for displacive nucleation, is given by [113]: G (Jlmol) N = 3.647; - 2 5 4 0 (3-2) This equation can be applied to either bainitic or Widmanstatten ferrite since their nucleation process is suggested to be similar in nature. However, different growth criteria have to be fulfilled that the nuclei can grow into a Widmanstatten or bainite plate as outlined below. 117 3.2 Literature Review Chapter 3 Displacive growth o f bainite plates occurs through an interface-controlled mechanism and involves full carbon supersaturation. Bainite subunits do not seem to grow in a mutually accommodating manner unlike the growth o f Widmanstatten plates, so that the resulting surface relief does not represent a tent shape [111]. The motion o f the interface is soon terminated by an accumulating friction stress. Indeed, the dislocation debris created by plastic accommodation stifles further growth o f ferrite plates. The proposed stored strain energy associated with the formation of bainitic ferrite is 400 J/mol. Therefore; the growth criterion can be expressed as [111]: AG ^ y where, AG ~* y < - 4 0 0 Jmol~ a (3-3) x is the driving pressure for composition invariant transformation o f austenite a to supersaturated ferrite, i.e. without compositional change. Graphical representation of this equation is the T' line i n the phase diagram by which the maximum carbon content o f 0 remaining austenite can be evaluated (see Figure 3-2). Similarly, for Widmanstatten ferrite the growth criterion is suggested as [113]: A G ^ where, AG ~* y y +a / M <-50 Jmol~ ] (3-4) indicates the free energy change accompanying the ferrite formation with paraequilibrium carbon content, i.e. with carbon partitioning. Although both Widmanstatten and bainitic ferrite are proposed to evolve with a displacive transformation mechanism [109], carbon partitioning is assumed for their nucleation stage. This seems to degrade the self-consistency of the displacive philosophy. 118 3.2 Literature Review 3.2.6 Chapter 3 Displacive model to predict the overall kinetics of bainite Rees and Bhadeshia [23] proposed a nucleation-based model that transformation proceeds by displacive autocatalytic nucleation of ferrite subunits. The growth rate o f subunits is so high that at the moment o f nucleation they can grow to a certain size promptly. The incremental change of volume fraction o f bainite due to nucleation of ferrite subunits between t and dt is given by: df ={\-f )uidt B (3-5) B where / is the actual volume fraction of bainite, / is the nucleation rate per unit of volume and u is the volume o f a subunit. The main underlying assumptions considered in this model can be summarized as follows: ( I) Carbide precipitation is absent due to presence of silicon. ( II ) Activation energy o f displacive nucleation is directly proportional to driving pressure for transformation, consistent with martensite nucleation. Therefore, the volumetric nucleation rate at austenite grain boundaries, I is given by: 0 / =£,exp[^f(l-^)] RI r (3-6) 0 where, k/ is the nucleation rate at the Widmanstatten ferrite start temperature W , AG is the s M driving pressure for bainite formation, r and K2 are constants, and R and T have their usual meaning. The displacive nucleation rate at W is identical for all steels [113]. S The minimum driving pressure for bainite transformation at W is G^,, referred to the S universal nucleation function, i.e. Equation (3-2). Indeed, GN is the minimum driving pressure required for displacive nucleation o f ferrite at the bainite start temperature, B , s which is defined as the highest temperature where bainite formation can be detected 119 3.2 Literature Review Chapter 3 experimentally during continuous cooling of the examined steel. Alternatively, B S can be estimated from empirical expressions such as the Steven-Haynes equation [106]. (III) Carbon enrichment o f the remaining austenite that affects the nucleation rate is taken into account. Partitioning the excess carbon of ferrite subunits soon after their growth increases the carbon content o f adjacent parent austenite, which induces a drop in A G . If m the driving pressure is assumed to vary linearly with the extent o f the reaction, i.e. f , B between its initial value AG ^ " R R+ and its final value GN (since below GN no displacive nucleation is permitted), then: A G M = A G ^ ' + A - f B ( A G ^ A -G N ) (3-7) ( I V ) The effect o f autocatalysis, i.e. the increase in number o f nucleation sites as volume fraction o f bainite increases, is taken into account as: 7 = / ( l + / (3e ) 0 fi fl where / is the effective nucleation rate, 0^ is the maximum fraction o f bainite, and (3-8) is the autocatalysis factor. However, due to carbon built up at the a/y interface, a temporary local decrease in driving pressure takes place, hence the autocatalysis factor /?, should be related to the mean carbon concentration o f austenite x , as: (3 = X (1 - X x) x 2 (3-9) where, X\ and Xi are empirical constants. ( V ) Effect o f austenite grain size, d y on nucleation rate, is assumed to be proportional to k'd , where k' is an empirical constant. The final equation for transformation kinetics that includes all o f the above assumptions is given by: 120 3.2 Literature Review df J K b (l-f )dt B Chapter 3 - K - Q ( "i +' */ -"'-~ p e ) e x [RT --f(i AG ^ J y+a 1 V J R B CL f l P + ) + rf ] (3-io) B B where K = ul d k' and T is given by: ] ^ K (AG ~* r =— ^ y Y+a 2 -G ) — N RT (3-11) The solution o f Equation 3-10, gives the time t, taken to form a certain volume fraction o f b a i n i t e a t the given reaction temperature. It is worthwhile to delineate more the significance o f the adjustable parameters used in the displacive approach. The parameter K describes the role o f austenite grain size and the t volume o f subunit is included. Further, parameter K dictates to which degree the nucleation 2 rate is affected b y the thermodynamic driving pressure, for instance setting K to zero or a 2 sufficiently small value (<200J/mol) results in a nucleation rate w h i c h is independent o f the calculated value for the driving pressure. The values reported by Rees and Bhadeshia for the three adjustable parameters (i.e. Kj, K , j3 ) quantified from fitting to a large number o f 2 experimental data i n F e - C - M n - S i steels are given i n Table 3-1. B y using different sets o f experimental data, quite different parameter combinations were proposed. Using an initial set o f well defined data, parameter values were found which were, i n particular for K and /?, 2 significantly different than those for other steels analyzed. M o s t notably the value for K is 2 sufficiently small for this set o f bainite transformations i n F e - C - M n - S i that the nucleation rate becomes independent o f the driving pressure. Including additional data, the parameter combination is also for the F e - C - M n - S i system similar to that for other steels and K is now 2 sufficiently large that the model becomes sensitive to the actual driving pressure. In summary, reviewing the analysis o f Rees and Bhadeshia indicates that application o f the 121 3.2 Literature Review Chapter 3 model is very sensitive to the data base employed and even the order o f magnitude o f the adjustable parameters is still not very well established. Table 3-1 Reported values for adjustable parameters based on the best fit for Fe-C-Mn-Si alloys [23]. Kj, m ' V 1 Generic values reported for F e - C - M n - S i alloys (initial data set) 2.6 x l O " / Generic values reported for F e - C - M n - S i alloys (extended data set) 5.2 x l O " / K2, J/mol P 1.925 4.756 8 A 8 A 6.395xl0 3 9.696 In the aforementioned approach a number o f crucial physical events, such as autocatalytic nucleation and the carbon enrichment o f the untransformed austenite, were treated in a crude manner. More recently, some o f these drawbacks pertinent to the original model were addressed by Matsuda and Bhadeshia [114]. The main improvement discriminating between seems to grain boundary and autocatalytic nucleation through be explicit quantification o f each contribution. In the initial stage o f the reaction grain boundary nucleation is assumed to govern the overall kinetics, whereas the evolution o f sheaves contributes to the further increase o f the bainite fraction later on. The growth o f a bainite sheaf is intermittent in nature controlled by autocatalytic nucleation o f its constituents, i.e. subunits. B y adopting the concept o f extended area developed by Cahn [115], the transformed area on a certain test plane parallel to the grain boundary can be translated into the incremental bainite fraction. Then a more realistic description o f the rate equation for bainite evolution was defined. It has to be noted that the intersection o f growing sheaves nucleated at different time scales with a test plane constitutes the transformed area on the plane under consideration. 122 3.2 Literature Review Chapter 3 The temperature dependence o f bainite plate thickness is implemented in the revised model as well. Moreover, the carbon content in the untransformed austenite is updated at each time step. Overall, although a more complex numerical scheme is associated with the revised model, it benefits from more realistic physics compared to the original approach, yet its predictive capability has to be examined and still its applicability is confined to carbide-free bainite. 3.2.7 Diffusional theory of bainite formation This idea is originated on microscopic observations performed by Hultgren [116], who proposed that bainite forms by diffusion controlled edgewise growth o f a set o f parallel Widmanstatten plates o f ferrite followed by precipitation o f cementite in the inter-plate regions. Carbide precipitation increases the growth rate of ferrite plates by up to three times [117] compared to the cases where cementite formation is suppressed by the presence o f Si or A l . Bainitic ferrite grows at a constant rate and without carbon supersaturation, i.e. with a carbon content according to the extrapolation o f the a/oc+y phase boundaries. However, the capillary effect, interface reaction and interaction o f segregating solutes with the interface, can potentially alter the interfacial conditions under which ferrite plates evolve. The proponents o f this school of thought based on detailed T E M studies [118,119] are in firm believe that the interfacial structure o f the broad face between ferrite plates and the austenite matrix is incapable o f motion by glide, the feature that is essential for displacive type of growth. It has been observed that the interface consists o f a single set o f misfit dislocation and a set o f structural ledges [118], which are formed due to energy minimization and provide an optimized match between the associated phases. The height o f these structural 123 3.2 Literature Review Chapter 3 ledges was observed to be o f a few atomic distances and is expected to be completely sessile. Aaronson and Kinsman [98], who argued against displacive growth o f bainite, proposed that the tip o f bainite plates comprises an array o f growth ledges. Therefore, it is necessary to consider the plate lengthening in terms o f the diffusion controlled ledge mechanism in which no atom attachment is allowed on the trace o f a ledge and interface can only move by lateral migration o f ledges. From a kinetic point o f view, the experimental measurements o f edgewise growth [120] did not reveal any abrupt change o f lengthening rate as the reaction temperature was lowered from 700 to 380°C and even further down to 230°C [117], indicating that the growth mechanism o f Widmanstatten and bainitic ferrite are essentially similar [120]. The fundamental similarity in the growth mode o f Widmanstatten ferrite and bainitic ferrite was also supported b y Hillert [22], who modified an equation proposed b y Zener [104] for diffusion controlled edgewise growth o f a plate precipitate surrounded by a disordered interface. The modified equation, which utilized the thermodynamic data given by a simple linear extrapolation o f phase boundaries to lower temperatures, was successfully applied to the kinetics o f Widmanstatten and bainitic ferrite [22]. However, he suggested that the lower growth rates experimentally observed as compared to the model predictions could be attributed to some resistance i n interface motion that must be overcome. The displacive theory o f bainite formation i n part relies on some other experimental observations, e.g. high defect densities i n bainitic ferrite, surface relief effect and transformation stasis. These issues were tried to be addressed as well b y the proponents o f the diffusional theory. Bainitic ferrite has a high dislocation density that increases as transformation temperature decreases. These types o f defects 124 that are left behind the Chapter 3 3.2 Literature Review transformation interface are referred to as the trailing effect [119]. According to the diffusional thought, the occurrence o f dislocation substructure can be ascribed to a normal migration mode o f the ledges, which becomes possible at sufficiently high driving pressures [101]. This mode o f migration induces high defect densities in the product phase. The ledges are usually expected to grow through lateral migration at intermediate undercoolings. Further, it is argued [119] that i f bainite would form by a shear mechanism it is definitely expected to observe twins as another type o f trailing defects, which can potentially be formed in higher carbon austenite or when the matrix is subjected to hydrostatic pressure. The lack o f twin evolution is regarded as further support for the diffusional argument. The emergence o f the surface relief was originally examined by K o and Cottrell [121] in 1942 who discovered that acicular ferrite features a surface effect similar to that of martensite. According to this finding and with respect to not very fast lengthening rate, they concluded that ferrite plates grow at the rate controlled by carbon diffusion in austenite while the substitutional sublattice may transform by martensite-like shear mechanism. The issue o f transformation stasis, in the framework o f the diffusional approach, was attributed to the role o f plastic accommodation in austenite [103]. Assuming the shape change associated with formation o f bainitic ferrite has a pure dilatational nature, at least on the macro-scale, and is plastically accommodated solely in the untransformed austenite, a crude quantification for the dissipation of Gibbs energy due to this plastic work was provided. The onset o f transformation stasis can then be defined when the energy release due to the diffusional formation o f ferrite from supersaturated austenite becomes equal to the energy loss as result o f the aforementioned plastic accommodation. Employing this criterion, 125 3.2 Literature Review Chapter 3 the carbon content o f the untransformed austenite at the stasis was shown to decrease with growth temperature and increase with the initial carbon concentration o f the alloy. Further, it is shown that the transformation stasis is not a unique feature o f bainite formation in Si and A l containing steels, it also can be observed in alloys with strong carbide formers such as M o [122] and C r [123]. It was found that the measured m a x i m u m fraction o f bainite during isothermal holding depends on the alloying content o f initial austenite, the pronounced effect that cannot be explained solely by thermodynamic considerations in terms of shifting the equilibrium boundaries, for instance T line which is invoked to define the 0 limit of displacive growth [109]. In these circumstances, the premature growth cessation might be attributed to solute-interface interaction. For example Reynolds et al. [122] explained the stasis in terms o f both nucleation and growth o f bainite subunits based on the extensive experimental characterization o f the transformation stasis in a series of high purity F e - C - M o steels containing 0.06 to 0.27wt%C and 0.23 to 4.28wt%Mo. They proposed that the simultaneous suppression o f sympathetic nucleation and the restriction o f growth yield to this phenomenon. Further, the growth cessation was considered to be due to the drag effect of molybdenum on the ferrite-austenite interface. 3.2.8 Mathematical expression for growth kinetics of a plate The edgewise growth rate, v , B o f a growing plate, e.g. bainitic plate, with a radius p o f curvature which is controlled by carbon diffusion away from the advancing disordered tip was originally discussed by Zener [104]. Based on the dimensional argument, he proposed an equation for steady state growth velocity, which was later modified by Hillert [22] in order to be applicable to parent phases with high degree o f supersaturation, as given by: 126 3.2 Literature Review Chapter 3 v B i xi-x 2p X t (3-12) o where X is the initial carbon content o f parent austenite and X y 0 is the modified equilibrium carbon concentration along the tip o f a/y interface to account for the surface energy term V <7 Ip. m Plate shape is assumed to be cylindrical featuring circular tip and the concentration is constant along the tip. Originally, to assess the supersaturation term as a function o f tip curvature, i.e. X -X , y p matrix 0 Zener proposed to assume a dilute solution for the and to consider the following replacement using the equilibrium interfacial concentration: x;-x where p c = (x -x )(\-^) y 0 eq o (3-13) is the critical tip curvature at which the entire chemical driving pressure is exhausted by the interfacial energy, i.e. growth would stop. Horvay and Cahn [124] treated the plate as elliptical paraboloid and considered constant concentration along the plate interface. A n exact solution to the diffusion problem o f a growing cylindrical plate featuring a parabolic tip was first presented by Ivantsov [125], i.e.: {n r P exp(p)erfc(p ) 05 = no (3-14) where the left side o f the expression, I(p), is a function o f the Peclet number, p=(v p/2D) and Q 0 denotes the supersaturation with respect to the equilibrium concentrations: X a X eq 111 (3-15) 3.2 Literature Review Chapter 3 In Ivantsov's treatment the role o f tip curvature on the interfacial concentration o f solute is neglected, however adopting Zener's simplification one can simply multiply the equilibrium supersaturation, Q , by (l-p /p), 0 c to account for the capillary effect. Trivedi [126] adopted Ivantsov's solution and improved it to capture the effect o f capillarity and interface reaction kinetics on the local interface concentration for a plate with parabolic cylinder geometry. A simplified version o f this model was proposed by Bronze and Trivedi [127]. A s discussed recently by Hillert et al [128], Trivedi's description is only valid for a high amount o f equilibrium supersaturation, i.e. Q 0 > 0.5 and further it does not provide any solution below a certain radius o f curvature. Quidort and Brechet [117] have recently employed this simplified equation to model the growth kinetics o f upper bainite plates in low alloy steels. They measured the longest plate observed by optical/scanning electron microscope and found linear lengthening rate o f plates that was in accord with the model prediction. The slower growth rates observed are attributed to a solute drag effect, which is not incorporated into their model. Enomoto [129] presented an informative comparison o f these various models for edgewise lengthening o f ferrite plate, which is reproduced in Table 3-2 with some additional remarks. The kinetic model based on ledgewise growth, which has been developed by Atkinson [130], is also included in this comparison. 128 3.2 Literature Review Chapter 3 Table 3-2 Models of plate-lengthening rate [129]. Author(s) Year Plate Boundary Condition Remarks/drawbacks Constant concentration No capillary effect considered Cylindrical Constant concentration along Dilute solution for matrix, with circular the tip maximum growth rate always at Geometry Ivantsov 1947 Cylindrical with parabolic tip Zener- 1957 Hillert tip Horvay- 1960 Cahn Tip of Constant concentration elliptical paraboloid (p,/ :)=0 P Trivedi 1970 Cylindrical Concentration varies with Dilute solution approximation. with a curvature and interface reaction Valid for moderate to high parabolic tip kinetics supersaturation, p /p corresponds c to maximum growth rate changes with the degree of supersaturation Atkinson 1981 Ledge No concentration gradient along the riser, no flux on the terrace 129 3.2 Literature 3.2.9 Review Chapter 3 Evaluation of bainite transformation kinetics Adopting the Zener-Hillert [22] growth equation, Minote et al. [131] employed a diffusional model to predict the overall kinetics o f bainitic ferrite formation in a classical T R I P steel, i.e. F e - 0 . 2 C - l . 5 M n - l . 5 S i (wt%). In addition, they adopted the displacive approach o f bainite formation [23] for modeling the overall kinetics. They observed that the kinetics o f bainite transformation experimental above 350°C is in accord with the predictions o f the diffusional mechanism, while it is consistent with the calculations o f the displacive model below 350°C, as shown in Figure 3-4. Note that at 3 5 0 ° C , both models give similar predictions, which are in good agreement with the experiments. P 1 10 100 1000 10000 1 10 100 1000 Austempring time (s) 10000 1 10 100 1000 10000 Figure 3-4 The change in volume fraction of bainite during austempering at different temperatures. Solid and dotted curves show calculated data based on the diffusional and displacive model, respectively [131] The displacive approach is a nucleation controlled model and does not present any estimation of subunit growth rate and its size. Cessation o f bainite reaction is not quantified explicitly in this nucleation-based model. The diffusional approach considers only the growth rate o f one single plate but not that o f a subunit as a member o f a bainite sheaf. Autocatalytic nucleation of subunits, which plays a significant role in formation o f bainite sheaf and affects the overall kinetics of the bainite reaction, is not taken into account. 130 3.2 Literature Review Chapter 3 Common deficiencies associated with both approaches can be summarized as follows. Neither model can predict the width o f subunits, which is the characteristic length in controlling the strength o f bainitic structures. Strain energy accompanying the formation o f bainitic ferrite governs the size, aspect ratio and dislocation density o f subunits. A realistic model has to incorporate this factor as well. However, in the framework o f diffusion model the kinetic role o f the plastic accommodation in austenite was evaluated by Quidort and Bouaziz [103], which seems to be a phenomenological quantification. Substitutional elements seem to slow down the rate of bainitic ferrite formation by changing the thermodynamic boundary conditions, and by means o f interaction with the moving a/y interface, i.e. by a solute drag effect. Transformation stasis, i.e. the so called "incomplete reaction phenomenon", can also be related to this interaction. The solute drag effect has to be introduced rigorously into the kinetic model. The intrinsic mobility o f the a/y interface, which is a measure o f the rate o f lattice change from F C C to B C C , can be rate controlling at higher undercooling or lower carbon content. Thus it is crucial to include the nature o f the a/y interface and its mobility into the model. Current models are not capable o f handling the morphology transition o f ferrite from allotriomorphic ferrite to Widmanstatten and bainitic ferrite that takes place through decreasing the transformation temperature. Alternatively, in most steels, the overall kinetics o f bainite transformation can be described by an Avrami equation [74]. Umemoto [75], proposed a modified form o f this equation to include the effect o f prior austenite grain size on the kinetics o f phase transformation, as presented in section 2.1.2.6. The reported values for n and m are different from those o f ferrite and pearlite. Umemoto et al. [75], suggested 4 and 0.6 for n and m , respectively, for bainite formation between 300 and 450°C in SUJ2 steel, Fe-0.99C-0.24Si-0.29Mn-l.34Cr. 131 3.2 Literature Review Chapter 3 Bhadeshia [109] also reported a range o f 1.8-4 for n and a value o f 0.65 for m. These suggested values indicate that nucleation and growth take place simultaneously and support the autocatalysis nucleation nature o f bainite [109]. 132 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 3.3 Study of Isothermal Bainite Formation in Fe-0.6C-1.5Mn-l .5Si The goal o f this part is to experimentally characterize the kinetics and micro structural features o f bainite evolution from single phase austenite in F e - C - M n - S i alloying system. The material selected for this study was a laboratory steel with a nominal composition o f 0.6C1.5Mn-1.5Si (wt%). The steel was supplied as 10mm hot-rolled plate with a predominantly pearlitic initial microstructure. The A 3 temperature o f the investigated steel was determined e as 755°C using Thermo-Calc software with Fe2000 database. The examined composition resembles the chemistry o f the intercritically treated austenite in classical 0.2wt%C T R I P steel at the start of bainite reaction, where due to presence o f 40 to 60 volume percent o f ferrite, the remaining austenite is enriched with carbon to approximately 0.6wt%. 3.3.1 Experimental procedures A Gleeble 3500 thermomechanical simulator equipped with a contact dilatometer was employed to perform austenitization and transformation experiments. T o establish suitable austenitization conditions, resulting austenite grain sizes were first studied by heat treating rectangular samples o f 3x6x15mm. The selected austenitizing procedures were to heat the samples at a rate o f 5°C/s to two different temperatures, i.e. 800 and 1050°C and hold for l m i n resulting in volumetric austenite grain sizes, d , r o f 13 and 40Lim, respectively. Subsequent transformation tests were carried out using tubular specimens o f 7mm diameter with 1mm wall thickness. A type-K thermocouple was spot welded on the outer surface o f the tubular sample, right at the middle, where the volume change during the imposed thermal 133 3.3 Bainite Formation in Fe-0.6C-l.5Mn-1.5Si Chapter 3 cycle was measured by the attached dilatometer. Due to the unique capability o f the Gleeble 3500, temperature can be controlled within a couple o f degrees o f the target temperature along any complex heating-cooling pattern. After austenitization at 800 and 1050°C, respectively, the tubular specimens were cooled using H e gas at a rate o f approximately 80°C/s to selected bainite reaction temperatures, where the kinetics o f isothermal bainite formation was quantified by dilatometry measurements. The isothermal temperatures applied were 300, 350, 400 and 4 5 0 ° C for samples with the smaller austenite grains, and 350, 375, 400 and 4 5 0 ° C for specimens having the larger prior austenite grains. In a second test series, the bainite formation was interrupted at the times for 10, 50 and 90 percent transformation and the samples were quenched to room temperature for microstructure characterization. It has to be noted that tests were performed using a vacuum o f 1.3mPa (10" Torr) to prevent 6 any potential oxidation or decarburization. The thermal path that the test specimens were subjected to is schematically depicted in Figure 3-5. - 450°C 1 ± _ time Figure 3-5 Schematic representation of the heat treatment cycle applied to isothermally form bainite at different temperatures from a single phase austenite featuring 13 and 40pm volumetric grain size. 134 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 Surface preparation for metallography was accomplished by mechanical grinding o f the mounted sample cut at the location o f temperature and dilation measurements, followed by polishing using 0.05um silicon solution as the final step. Then the specimens were etched in 2% nital solution to reveal the bainite microstructures, whereas the attacking time was varied to achieve the optimum contrast o f the microstructural constituents. T o reveal the prior austenite grain boundaries, the as-quenched samples after austenitization had been tempered at 550°C for 24 hours to enhance segregation o f impurities on the boundaries, followed by quenching to room temperature and etching using a solution containing 50ml picric acid, 0.4g wetting agent and 0.3g copper chloride. The etchant had been warmed up to 60-70°C and the C u layer deposited on the examined surface was removed afterward using 10% hydroxide ammonium solution. To determine the E Q A D (equivalent area diameter) o f austenite grains Jeffries method outlined in A S T M standard E l 12-96 was adopted. One can convert the E Q A D to volumetric austenite grain size by multiplying with a factor o f 1.2 [132]. 3.3.2 Experimental results The dilatometer responses recorded during the bainite formation stage are summarized in Figure 3-6(a) and (b) for both austenitization conditions. In all tests a significant incubation period is observed thereby confirming that the selected cooling rate from austenitizing to bainite temperatures was sufficient to prevent any bainite nucleation along the preceding cooling path. The data illustrated in Figure 3-6a for the specimens with the prior austenite grain size o f 13um, indicate that both the duration o f the incubation time and the time to completion increase with decreasing reaction temperature. Moreover, the total volume expansion recorded at the end o f transformation increases as temperature is lowered and is 135 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 the largest for the lowest temperature, i.e. 300°C. Similarly, as shown in Figure 3-6b, more bainite tends to form as the reaction temperature drops when the initial austenite grain size is 40um. However, the temperature trend of the bainite reaction rate is quite different here with the highest rate being observed for the lowest temperature, i.e. 3 5 0 ° C , and little dependence on temperature for the higher transformation temperatures. 0 1000 2000 3000 4000 5000 0 2000 Time, s 4000 6000 8000 Time, s (a) (b) Figure 3-6 Dilatometer responses recorded during isothermal bainite formation at different temperatures from austenite grain sizes of (a) 13pm and (b) 40 pm. Using the lever law the isothermal dilation was translated into normalized fraction transformed, based on which the T T T diagrams o f bainite formation from the two different austenite grain sizes can be constructed. This diagram is depicted in Figure 3-7. It is clearly seen that larger austenite grain sizes lead to a delay o f the bainite reaction, as for example indicated by ts% for the start o f transformation. Moreover, it can be seen that the rate o f bainite formation from 40u.m austenite grains displays a minimum at approximately 375°C. 136 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 480 5% 95% 5% 95% 440 O O § 400 CO 1_ a | 360 QJ I- O 13um —•— 40nm 320 o 1 280 10 100 Time, min Figure 3-7 TTT diagram of bainite formation from two different austenite grain sizes in Fe-0.6C1.5Mn-1.5Si. Typical optical and S E M microstructures o f isothermally formed bainite in the temperature range o f 350 to 4 5 0 ° C are presented in Figure 3-8 and Figure 3-9, respectively, for the prior austenite grain size o f 40um. For those specimens the isothermal transformation interrupted after formation o f about 50% bainite, followed by quenching to was room temperature. The volume fraction o f bainite at the end o f each isothermal reaction were measured based on the optical micrographs. The maximum fraction 9B o f bainite formed in each condition is summarized in Table 3-3. Table 3-3 Maximum fraction of bainite measured experimentally using optical micrographs d -\3\im d =40um y r Transformation temperature, °C 450 400 350 300 450 400 375 350 0B 0.61 0.65 0.80 0.84 0.64 0.76 0.80 0.85 137 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 Figure 3-8 Optical micrographs showing bainite (dark gray phase) surrounded by martensite/austenite matrix formed at (a) 450, (b) 400 and (c) 350°C from d =40pm. r 450°C 400°C • i' !,VWD21.6i nmtVo'. OkV xl't'Ok, C 50um IS V • ' t * / / f * •/• v. '-Mv " ^ v V ' ' { J N H P .».«s ^ WD'21.7mm 20 . OkV, x l . Ok ." 50um '••WD22 3mm 20 .'OkVi'-xi. Ok 5'0um ' Figure 3-9 SEM micrographs showing bainite surrounded by martensite/austenite matrix formed at different transformation temperatures from d =40jum. r In the optical micrographs, Figure 3-8a to c, bainitic ferrite can be seen as dark constituent embedded in either a martensite or retained austenite matrix. The bainite phase consists o f subunits in the form o f discrete packets. The morphology o f the subunits at the highest reaction temperature, i.e. 4 5 0 ° C , resembles that of degenerate ferrite. Here, individual ferrite 138 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 grains can be detected while lath and plate morphologies tend to emerge i n samples reacted at 400 and 350°C, respectively. The width o f subunits decreases with temperature and their length is limited by the prior austenite grain diameter or the available gap between units already formed. Although, the primary nucleation sites were revealed to be mainly austenite grain boundaries, some evidence o f autocatalytic nucleation was also observed at higher magnification levels. 3.3.3 Modeling Modeling the bainite reaction is a matter o f controversy because different mechanisms, i.e. edgewise diffusional growth and displacive transformation have been proposed. Here, different kinetic models, including semi-empirical approaches are employed to evaluate their predictive capabilities i n the present case. This i n particular helps to explore the potential benefits and inefficiencies o f each examined methodology. 3.3.3.1 Semi-empirical modeling approach To develop a semi-empirical model, the traditional Johnson-Mehl-Avrami-Kolmogorov ( J M A K ) methodology can be adopted, i.e. [74]: / = l-exp[-^-r) J n (3-16) where / is the normalized fraction transformed, t is the time at transformation temperature, r denotes the incubation time; b and n are rate parameters. The parameter n is usually assumed to be temperature independent and can be determined by representing the experimental data in a plot o f In In (1/1-f) vs In t. However, the obtained quantity for n is quite sensitive to the value assigned for the incubation time. This may be one o f the reasons that a wide range o f n values, i.e. from 1 to 4, are reported in the literature [78,109,117]. The parameter b is then 139 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 used to account for the effects o f temperature and initial austenite grain size on the transformation kinetics such that [78]: b =^ (3-17) d; where b is a function o f temperature and m is the grain size exponent. 0 Applying the J M A K approach to the present experimental kinetics o f bainite formation, n=\ has been obtained employing the incubation times summarized i n Figure 3-10. Then, b has been determined from a best fit analyses o f the individual transformation curves. Figure 3-11 shows b as a function o f temperature for both initial austenite grain sizes. A linear temperature dependency o f In b has been proposed elsewhere for ferrite [21 ] and can here be applied for the transformations below 450°C for d 7 13um and above 3 5 0 ° C for d r 40um such that: In b = -5Ad° M r + 0.00487 (3-18) where the temperature is in °C and d in um. Assuming the above relationship for In b, the y predicted kinetics are observed to be in good agreement with experimental data for all transformation temperatures except for those at 450°C for d = 13p.m and at 350°C for d = y r 40p.m. Figure 3-12 illustrates the quality o f the J M A K fit for the bainite transformation at 300 and 400°C (d =13um), and 375 and 450°C (d =13um). However, as evident from r r Figure 3-11, a more complex temperature function would have to be considered for b in order to replicate the experimental kinetics for all conditions. It is obvious that more experimental data are required to establish a conclusive function for b. 140 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 Figure 3-10 Incubation times of bainite formation as a function of temperature for different austenite grain sizes, i.e. 13 and 40/Jtn. -8 ' 250 ' • 300 350 1 400 . 1 450 500 Transforamtion temperature, °C Figure 3-11 Temperature dependency of the parameter b to provide optimal description of experimental data using the JMAK equation with n=l. 141 3.3 Bainite Formation in Fe-0.6C-l.5Mn-1.5Si 0 1000 Chapter 3 2000 3000 4000 Time, s Figure 3-12 Comparison the experimental kinetics of isothermal bainite formation and model predictions adopting the JMAK equation. 3.3.3.2 Modeling the incubation period Incubation period o f a reaction can i n general be related to the nucleation kinetics and basically is referred to the time for establishing the steady-state condition i n nucleation. However, nucleation itself as an atomic event is more elusive and complex than subsequent growth. Particularly, the debated nature o f bainite, i.e. diffusive or displacive, constitutes a major challenge such that any conclusive description o f the nucleation stage is still lacking for the bainite reaction. Therefore, here a rather phenomenological treatment based on the classical nucleation theory is adopted as a first attempt to predict the incubation period o f bainite formation. Assuming that the time for removing an atom from a critical nucleus is essentially similar to that for adding an atom to a cluster o f subcritical size, i.e. the time for formation o f a nucleus, 142 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 Russell [133] treated grain boundary nucleation and developed an expression for the incubation time, i.e.: .2 r oc kT—— — t (3-19) r AG (3 where n* is the number o f atoms embedded inside the critical nucleus, AG* is the activation energy for the formation o f this nucleus and (3* is the frequency factor o f jumps toward or from the critical nucleus. This factor is defined i n terms o f effective diffusivity that relates to boundary or volume diffusion, and the number o f atoms i n contact with the critical nucleus, k*, such that: P*=^%- where a L (3-20) denotes the lattice parameter. Various types o f grain boundary nuclei featuring different geometries and coherency characteristics were analyzed by Russell to quantify the aforementioned parameters used i n Equation 3-19. A s a consequence, the incubation period can in general be represent by the following simplified expression: T x <x (3-21) ^ D e f f where the energy term is the driving pressure for nucleation and its exponent w takes either the value o f 2 or 3 depending on the coherency o f the forming nucleus. A s also indicated by Bhadeshia [134], the above equation can be rearranged in the following format: \n^p- = Q /RT D +C ] (3-22) in which Qo denotes the activation energy for the relevant diffusion process and C / is a constant embedding a l l the pertinent constants such as geometrical factor, interfacial energy 143 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 between matrix and precipitate, the wetting angle effect and the pre-exponential factor o f the diffusivity term. To include the effect o f prior austenite grain size, d , this constant is r modified here as: C, = C d ; (3-23) n 2 Using Equation 3-22, the experimental incubation time o f isothermal bainite formation, as already presented in Figure 3-10, was tried to be replicated. Table 3-4 shows the adjustable parameters obtained from the optimal fit to the measured data. The model predictions employing these values are also illustrated in Figure 3-13. It is evident that the model can capture most o f the data points adequately. However, the shortest incubation times, which are associated with bainite formation at 450 and 350°C from d o f 13 and 40um, respectively, r were noticeably overestimated. The inaccurate prediction o f the bainite kinetics for these two conditions has already been mentioned in the previous section and suggests that more experimental measurements have to be carried out to verify and characterize the kinetics in those temperature regions. B y virtue o f the presence o f a coherent interface between bainitic ferrite and the parent austenite, the parameter TIT was assumed to be 2 and an attempt was then made to find the corresponding values for other parameters. A s indicated i n Table 3-4 the value for the activation energy reported is about half o f that for carbon diffusion in bulk austenite. Therefore it is plausible to speculate that grain boundary diffusion o f carbon governs the nucleation kinetics. It has to be noted that even during displacive nucleation o f bainite the partitioning o f carbon is allowed, which constitutes one common conjecture adopted by the two different schools o f thought. However, since the above analysis is a crude treatment o f nucleation, and different sets o f values for those parameters can yield acceptable predictions 144 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 as well, one can hardly draw a definite fundamental conclusion out o f these data and its application is then solely confined for modeling purposes. Table 3-4 Parameters used in Equation 3-22 to predict the incubation period. QD, kJ/mol c 0 -0.3 1.64 55 2 m 2 200 400 600 800 Incubation time, s Figure 3-13 Model prediction (lines) and experimental incubation period (points) of isothermal bainite formation from two prior austenite grain sizes, 13 and 40pm, in 0.6wt%C-1.5wt%Mn-1.5wt%Si steel. 3.3.3.3 Diffusion model The diffusion model adopted here is based on the Zener-Hillert equation for edgewise growth of a plate in form o f a circular cylinder with a tip curvature o f p, as described in detail in section 3.2.8 using Equation (3-12). This equation, in which the supersaturation term is corrected to account for the Gibbs-Thomson effect, indicates that the growth rate depends on 145 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 the tip curvature. Usually, for a given value of the austenite-ferrite interfacial energy, a , ay a plate adopts the tip curvature that corresponds to the maximum growth rate, v ^ . To a x evaluate the maximum growth rate the derivative o f Equation (3-12) with respect to p has to be calculated for each transformation temperature. For this purpose a numerical methodology was developed here, the detail o f which is going to be described i n the following paragraph. The carbon diffusivity i n austenite was quantified adopting Agren's equation [43] in which for the carbon concentration the equilibrium value in austenite at the examined temperature was used. The interfacial carbon concentration, X , which can be quantified i n terms o f radius o f the y tip curvature and the austenite-ferrite interfacial energy, a , ay varies from X to X] , the 0 equilibrium carbon content i n austenite, when p increases from the critical radius for ferrite nucleation to infinity. This variation can be quantified by including the Gibbs-Thomson effect, o V ay m Ip, in the Gibbs free energy o f ferrite and finding the corresponding carbon content o f austenite i n equilibrium with such a ferrite particle. For this purpose a F O R T R A N program was designed to retrieve the required thermodynamic data from Thermo-Calc, construct the Gibbs energy curves o f ferrite and austenite under paraequilibrium assumption and finally determine X y for each growth temperature adopting the approach outlined above. The resulting calculation is represented i n Figure 3-14. In the present calculation the value of a was assumed to be 0.2J/m [135]. The start o f each curve corresponds to p , the critical 2 ay c tip curvature at which the Gibbs-Thomson effect would be equal to the driving pressure for nucleation from the parent austenite o f X 0 composition. Therefore all the available driving pressure would be exhausted by the surface effect and no growth could be expected. 146 3.3 Bainite Formation in 0.02 Chapter 3 Fe-0.6C-l.5Mn-l.5Si -J . < 1 1 ' 0 2 4 6 8 10 p, nm Figure 3-14 Interfacial carbon concentration of austenite in equilibrium with ferrite plate bearing a tip curvature of pfor different temperatures. Consequently as p changes and so does X , y the ratio X y p I p , based on which the growth rate is defined, experiences a maximum. In fact, the growth rate tends to be zero at the critical tip curvature, p , passes through a maximum and afterward asymptotically approaches zero. This c growth feature o f a ferrite plate is illustrated in Figure 3-15 for the analyzed reaction temperatures. After finding an expression for X y as a function o f p and a , the derivative o f Equation a y (3-12) was solved numerically to determine the radius at which the growth rate becomes a maximum. It was found that v^ ax is essentially a linear function o f the austenite-ferrite interfacial energy. This is shown in Figure 3-16 for bainite growth at 4 0 0 ° C , when the interfacial energy varies from 0 to U / m . 2 147 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 40 p, nm Figure 3-15 The variation of growth rate with the tip curvature of a ferrite plate for different reaction temperatures assuming the Zener-Hillert model. 140 0 i 0.0 1 0.2 . . , , 1 0.4 0.6 0.8 1.0 1.2 a/y interfacial energy, J / m 2 Figure 3-16 Variation of the maximum growth rate of a bainite plate with ferrite-austenite interfacial energy, calculated for bainite formation at 400°C in 0.6wt%C-1.5wt%Mn-1.5wt%Si steel. The isothermal fraction o f bainite can then be calculated by means o f the following equation for the transformation rate, i.e. [131,136]: 148 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si dt Chapter 3 = K v (l-f ) B D nm (3-24) B where KD is a temperature dependent parameter which is associated w i t h nucleation but has to be determined from experimental data. A s proposed by Suehiro et al. [136], an Arrhenius relationship is employed for K with: D „ , _,. 2 . 1 8 X 1 0 ' o( )= —J d? K 6 m e x 12 P ^2770" (3-25) T being obtained in the present case. Assuming that the diffusional growth starts after the incubation period has elapsed, the quality o f fitting experimental kinetics is similar to that obtained with the semi-empirical approach. Again, a more complex function for K D would be required to achieve a satisfactory agreement at 450°C for d = 13um and at 350°C for d = r r 40um, respectively. A s an example, typical predictions o f the diffusion model using the above KD are shown in Figure 3-17. 1.00 o u 0.75 CO L. © 0.50 N o • o E 2 0.25 A 400°C, 13um 300°C, 13nm 450°C, 40nm 375°C, 40nm model prediction 0.00 1000 2000 3000 4000 Time, s Figure 3-17 Comparison of the measured kinetics and the predictions of diffusion model for isothermal bainite formation in 0.6wt%C-1.5wt%Mn-1.5wt%Si steel. 149 3.3 Bainite Formation in 3.3.3.4 Fe-0.6C-l.5Mn-l.5Si Chapter 3 Kinetic effect of strain energy in diffusion model The formation o f bainitic ferrite is associated with both types o f dilatational and shear stresses due to shape change. The former arises from the specific volume difference between austenite and ferrite and the latter yields to the so-called surface relief effect. The elastic strain energy o f bainite formation is estimated to be about 400 J/mol and is considered as criterion for displacive growth as discussed in section 3.2.5. If the amount o f stress at the interface exceeds the yield strength o f austenite then some plastic accommodation is likely to occur. On this basis the kinetic effect of strain energy, in terms o f energy dissipation due to plastic accommodation, has been implemented into the diffusion model and is adopted here to verify i f the predictive capability o f the diffusion model can be improved. Assuming that the entire transformation shear is accommodated plastically in the untransformed austenite, Quidort and Bouaziz [103] estimated the amount o f this plastic work as: dG =T Y ^ -V r plastk where z Y (3-26) 1 y B m is the yield strength o f austenite that deforms as an ideal plastic medium, y B shear component o f transformation strain considered to be 0.22, and f B is the denotes the fraction of bainite. The energy dissipation due to the plastic work increases as the reaction proceeds. A s a consequence the effective driving pressure, AG im - G , plastic that remains to render the diffusional growth decreases correspondingly. This implies that the m a x i m u m growth rate o f a bainite plate, i.e. v * , is no longer constant at a given reaction temperature and it tends to a x drop as the transformation carries on. This is in contrast with the case that plastic 150 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 accommodation is neglected during the diffusional growth o f a bainite plate, i.e. bainite plates grow featuring constant velocity. To quantify the variation o f with the fraction o f bainite, first the dissipation o f energy as a function o f fraction was evaluated using Equation 3-26. The expression proposed by Bouaziz et al. [137] had been adopted to account for the temperature dependency o f austenite yield strength, as given by (T i n °C): x\(T) = x\(20) exp[-0.0011(7 - 20)] (3-27) For each bainite fraction, after removing G iastic from the chemical driving pressure, the p maximum growth rate was determined following the methodology described in section 3.3.3.3. A s an example, the variation o f v* ax with bainite fraction for growth at 400°C is illustrated here in Figure 3-18. The dashed line shows the constant growth rate when no plastic accommodation is considered. 30 constant \f mgx 5 0.0 0.2 (no plastic accomodation) 0.4 0.6 0.8 Bainite fraction Figure 3-18 Decreasing trend of the growth velocity of a plate as bainite fraction increases, transformation at 400°Cin 0.6wt%C-1.5wt%Mn-1.5wt%Sisteel. 151 3.3 Bainite Formation in Chapter 3 Fe-0.6C-l.5Mn-l.5Si Accounting for the energy dissipation by plastic work simply results i n slowing down the rate o f bainite formation i n particular as the reaction approaches to its completion. The fraction o f bainite can then be calculated using the rate equation as given by Equation (3-24) and considering the fraction dependency o f v* optimum description o f the for each transformation temperature. From the ax experimental kinetics o f bainite formation in 0.6wt%C- 1.5wt%Mn-l .5wt%Si steel, the following temperature dependency was found for KD, i.e.: v ( 2 - 7 8 x l 3170 (3-28) A typical prediction o f diffusion model modified by the effect o f plastic accommodation is illustrated in Figure 3-19 for isothermal bainite evolution at different temperatures from d of r 13 and 40 urn. 1.00 o u 0.75 "8 0.50 *3 T CO N TO E | 0.25 0.00 0 1000 2000 3000 4000 Time, s Figure 3-19 Prediction (lines) of diffusion model modified with the effect ofplastic work and the experimental kinetics (points) of bainite formation in Fe-0.6wt%C-1.5wt%Mn-1.5wt%Si alloy. 152 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 A s it is evident the overall agreement between model calculation and the measured kinetics is fairly good. However, for the examined alloy it appears that accounting for plastic work in the diffusion model does not lead to a substantial improvement i n its predictive capability, in particular for the highest reaction rates corresponding to bainite formation at 450 and 350°C for dy of 13 and 40 urn, respectively. Moreover, compared with the case when plastic work is neglected, i.e. Figure 3-17, it appears that for the bainite fractions above 0.7 the experimental data is being somewhat underpredicted when plastic work is taken into account. 3.3.3.5 Displacive model In the displacive theory o f bainite formation growth does not contribute to the overall kinetics, i.e. the rate o f reaction is assumed to be controlled solely by nucleation. Moreover, in contrast to a diffusional-type o f nucleation, the nucleation rate is a linear function o f driving pressure. A s described in section 3.2.6 the rate equation for isothermal bainite formation with displacive manner can be formulated by Equation 3-10, w h i c h is used here to describe the isothermal bainite formation in Fe-0.6wt%C-1.5wt%Mn-1.5wt%Si. The variation o f driving pressure, AG, with temperature for the investigated steel was quantified by means of thermodynamic data provided by Thermo-Calc assuming paraequilibrium condition. From the Thermo-Calc results, a simplified expression can be given for the temperature range o f 300 to 500°C, i.e.: AG(J/mol) = 7.017-4653.5 (3-29) when 7 is in °C. A s natural starting point, first the reported values for the three adjustable parameters, i.e. K h K2 and /? as depicted i n Table 3-1, were employed to predict the bainite evolution kinetics. Using the two sets o f fit parameters, no satisfactory description o f the present results in the 153 3.3 Bainite Formation in Chapter 3 Fe-0.6C-l.5Mn-l.5Si 0. 6C-1.5Mn-1.5Si steel were obtained. Thus, the model parameters were adjusted to provide the best fit possible with the eight experimental curves. Given the limited database it was possible to use different fit procedures with similar fit quality. F o r example, setting Kj to zero (i.e. eliminating the role o f driving pressure) or assuming (3- 0 yield quite similar results i n terms o f fit quality which cannot be improved by also varying this third parameter, 1. e. j3 and K2, respectively. A s an example, the parameter set with (3= 0 obtained for the present 0.6C-1.5Mn-1.5Si steel is shown i n Table 3-5. A typical comparison o f model predictions with the measured kinetics for the bainite formation at various temperatures i n specimens with d =40um is illustrated i n Figure 3-20. r Table 3-5 Values for adjustable parameters found for the 0.6C-1.5Mn-1.5Si steel in this work Kj, m V 2.6 x l O " " / K.2, J/mol P 553 0.0 8 Values determined for 0.6C-1.5Mn-1.5Si steel / 154 /d Y 3.3 Bainite Formation in Fe-0.6C-1.5Mn-l. 5Si Chapter 3 Reasonable agreement was found for all transformation conditions, except for those at 450°C for d = 13um and at 3 5 0 ° C for d = 40um. These are the conditions were also the J M A K y r approach and the diffusion model in conjunction with simplified temperature functions for the parameters b and K , respectively, failed to describe experimental results adequately. D 3.3.4 Summary and remarks (0.6C TRIP steel) Isothermal austenite-to-bainite transformation kinetics were analyzed for a F e - 0 . 6 C - l . 5 M n 1.5Si steel as a first step to develop a bainite transformation model for T R I P steels. Existing modeling approaches, i.e. the J M A K equation as well as a diffusional theory and the displacive model o f Rees and Bhadeshia, have been evaluated. U s i n g the general temperature trends o f the respective model parameters, as proposed in the literature, the bainite formation can be equally well described with all three approaches. To further illustrate these findings, the calculations o f displacive model and diffusion approach are summarized i n Figure 3-21, in which a comparison between the predicted and the measured transformation time for 50% bainite formation, i.e. 150% , are provided for all the examined transformation temperatures and austenite grain sizes. Clearly, the description o f tso% is acceptable except for the specimens with the highest transformation rate for each austenitizing condition, i.e. at 450°C for d o f 13um and at 350°C for J o f 4 0 u m , respectively. y r For all the examined approaches significant deviations o f predicted and observed kinetics have to be registered for the highest transformation rate at each austenitizing condition. Thus, additional careful experimental studies are required to develop new temperature dependencies o f these model parameters. Further, it was shown that Russell's treatment, which is consistent with the diffusional mechanism, can serve as the basis for developing a 155 3.3 Bainite Formation in Fe-0.6C-l.5Mn-l.5Si Chapter 3 submodel for the incubation time. This submodel appears to be a critical component o f any model approach for bainite formation. 1800 / • u £ 1200 /m / a /m 1/ A o • • • 0 -I 0 600 Diffusion, 13um Displacive, 13um Diffusion, 40um Displacive, 40um 1200 1800 Measured time, s Figure 3-21 Comparison of the experimental data for the time of 50% transformed, r predictions employing diffusion and displacive approaches. 156 5fl% , and model 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 3.4 Study of Isothermal Bainite Formation in Fe-0.18C-l.55Mn-l.7Si In this part o f research an attempt has been made to explore the kinetics and micro structural aspects o f isothermal bainite formation after intercritical treatment, i.e. from a mixture o f ferrite and austenite. Although bainite evolution in T R I P steels has received comprehensive experimental characterization [138,139], the major goal o f the present study is to gain more insight pertinent to the critical aspects that are particularly significant for modeling purposes. 3.4.1 Experimental procedure and results The examined material was a lab cast steel with a nominal composition o f 0.18wt%C1.55wt%Mn-1.7wt%Si, provided as hot forged blocks by Dofasco Inc. The as received microstructure consisting o f ferrite + pearlite mixture was revealed to be extensively banded. Therefore, rods o f 12mm diameter and 15cm length were machined, encapsulated in a quartz tube under rough vacuum and homogenized for 7 days at 1200°C. To remove the resulting coarse microstructure, this treatment was later followed by reheating the specimen to 980°C, holding for 2min and controlled cooling at rate o f l ° C / s down to 4 5 0 ° C , which results in appropriate fine aggregates o f ferrite + pearlite polygonal phases for the subsequent experiment. In a similar manner as outlined earlier, a Gleeble 3500 thermo-mechanical simulator was employed to conduct a number o f predetermined thermal cycles i n this study. Tubular specimens were ramp heated to the austenitizing temperature, i.e. 9 8 0 ° C , held for l m i n followed by cooling to the intercritical region where holding for 5min at 700°C yielded to a two phase microstructure consisting o f 58% ferrite and 42% austenite. The samples were subsequently cooled at about 100°C/s to the bainitic temperature region, i.e. between 300 to 157 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 400°C, and held at predefined times to transform isothermally. The aforementioned thermal pattern is schematically represented in Figure 3-22. I L _ time Figure 3-22 Thermal cycle designed to form bainite from intercritically treated specimens with 0.18wt%C-1.55wt%Mn-l. 7wt%Si initial composition. The optical microstructures o f specimens transformed partially to bainite at 400 and 300°C are illustrated in Figure 3-23. Here, the bainite reaction was interrupted after 50% progress by helium quenching to ambient temperature. The matrix consists o f polygonal ferrite and the prior intercritical austenite has partially been transformed isothermally to bainite (dark gray phase) and potentially to martensite (light gray phase) upon quenching to room temperature. Well-defined bainite pockets consisting o f many plates/laths are evident. A t lower growth temperature the bainite tends to adopt lenticular plate morphology with smaller thickness whereas at higher temperature, i.e. 400°C bainite plates are coarser and represent the same thickness along their entire section. In both cases the growth o f plates is impeded by the prior 158 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 austenite-ferrite boundaries, i.e. the length o f the largest plate is the same as the austenite grain size. Figure 3-23 Optical micrographs of intercritically treated samples transformed partially to bainite at 400 and 300°C. Ferrite matrix (white) estimated to be 58%, and the rest is a mixture of martensite (light gray) and packets of bainite (dark gray), 2% nital etched The characterization o f microstructure by S E M technique, shown in Figure 3-24, confirms the previous information obtained from optical micrographs. However, it reveals many plates intersecting with each other and shows some indications o f nucleation at pocket boundaries. 159 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 Figure 3-24 SEM micrographs of intercritically treated samples transformed partially to bainite at 400 and300°C. The dilation measurements recorded during isothermal holding at different temperatures are illustrated in Figure 3-25. The total volume change corresponding to the end o f the reaction increases as growth temperature was lowered, indicating more bainite forms at higher undercoolings. Further the overall rate o f reaction increases with decreasing temperature as well. It was found even after prolong holding, beyond the period that is shown in Figure 160 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 3-25, the dilation curves do not drop and follow a flat plateau. This rules out any significant precipitation o f carbides that potentially might be expected at longer isothermal holding time and would be associated with an abrupt drop in dilation curve. It is also worth remarking that in spite o f very fast cooling from intercritical to bainite region, about 100°C/s, no appreciable incubation period (less than a few seconds) was recorded. Therefore it is assumed that the incubation period is negligible and bainite transformation starts once the specimen reaches the target temperature. 16 Time, s Figure 3-25 Kinetics of isothermal bainite evolution from intercritically treated samples featuring 42% austenite. To further investigate the transformation behavior o f an intercritically treated austenite embedded in a ferrite matrix, a number o f specimens were cooled down at different rates from 700°C to ambient temperature. Those samples had been processed i n the two-phase field similar to the previous cases, i.e. reheating to 980°C and holding 5min at 700°C. B y analyzing the dilatometry data measured along the cooling path, the start temperature o f the transformation was determined for each cooling rate. It turned out that for a relatively large 161 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 range o f cooling rates, i.e. from 24 to 155°C/s, the observed transformation start temperature remains rather constant around 275°C. This temperature is w e l l below the bainite formation region examined i n the previous experiments. Considering 58% ferrite with a nominal carbon content o f about 0.02wt%, the carbon enrichment o f the intercritical austenite is estimated to be about 0.41wt%C. U s i n g Steven-Haynes [106] equation a martensite start temperature o f about 295°C is then predicted for this intercritical austenite, which is i n reasonable agreement with the experimental findings. The determination o f the critical cooling rate above which the pearlite/bainite formation can be suppressed is crucial for advanced high strength steels. In D P steels, the remaining austenite is expected to transform entirely to martensite. In T R I P steels having this information aids to design the appropriate thermal cycle parameters, i.e. the imposed cooling rate from intercritical region that prevents acicular/pearlite formation, and the temperature range o f isothermal holding i n the bainite region. 3.4.2 Modeling The measured kinetics o f isothermal bainite formation from a mixture o f ferrite-42%austenite was analyzed using available modeling approaches, similar to the strategy adopted in section 3.3.3. For this purpose the dilation data represented in Figure 3-25 was first translated into normalized fraction. A s starting point, the semi empirical methodology, i.e. J M A K equation as expressed by Equation (3-16), was employed. The optimal description o f the experimental kinetics was accomplished by assuming n =1 and considering the following temperature dependence for the b parameter: In 6 = -8.7 + 0.00977 162 1 (3-30) 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 The predictions o f the J M A K equation using these values for n and b are compared with the measured fraction i n Figure 3-26. Apart from a slight overestimation o f the fraction at final stages o f the reaction at 4 0 0 ° C , the overall fit quality is satisfactory. 1.00 750 1500 100 700 1400 100 700 1400 Time, s Figure 3-26 Comparison of model prediction adopting JMAK approach and the experimental fraction of bainite formed after intercritical treatment in 0.18wt%C-1.55wt%Mn-l. 7wt%Si steel. On the second attempt the kinetics o f bainite evolution was simulated assuming diffusioncontrolled edgewise growth o f plates at constant rate as delineated i n section 3.3.3.3. It has to be noted that the parent austenite, due to being treated in the intercritical region, has a carbon concentration o f about 0.41wt%. Then assuming an interfacial energy between bainitic ferrite and austenite to be 0.2 J/m the maximum growth rate for each transformation temperature was evaluated. Table 3-6 shows these growth velocities w h i c h were used to predict the isothermal kinetics o f bainite formation by means o f Equation (3-24). In the present analysis it was found that the experimental data can be accurately described, when KD is taken to be: K (m~') = 18.5 exp D 163 13000 (3-31) 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. 7Si Chapter 3 For comparison purpose the results o f this simulation are presented i n Figure 3-27 together with the measured kinetics o f isothermal bainite formation. Table 3-6 Growth rate of bainite plates for different transformation temperatures assessed using ZenerHillert equation (0.41wt%C-1.55wt%Mn-l. 7wt%Si austenite). Temperature, °C 0.00-1 0 , 750 300 350 400 9.9 22.2 39.5 • 1500 1 -L, 100 , 700 , J J_, 1400 100 , 700 J 1400 Time, s Figure 3-27 Model prediction adopting diffusion approach and the experimental kinetics of isothermal bainite evolution from an intercritically treated austenite with carbon content of0.41wt%. Finally, employing the methodology as discussed in section 3.3.3.5, the displacive model was adopted to simulate the kinetics. B y means o f Thermo-Calc the driving pressure, AG, for ferrite nucleation from 0.41wt%C-1.55wt%Mn-1.7wt%Si austenite was quantified to be linear for the temperature range o f 300 to 400°C, i.e.: 164 3.4 Bainite Formation in Fe-0.18C-1.55Mn-l. AG(Jfmol) 7Si Chapter 3 = 7.82T-5003.7 The main adjustable parameters i n the displacive model, i.e. K (3-32) h K and (3, were determined 2 from the best fit to the experimental kinetics. Table 3-7 shows the model parameters for this case. The parameter dealing with autocatalytic nucleation, /?, was found to be zero. Further, K 2 features the same order o f magnitude as that reported by Bhadeshia [23] based on the extended data set (see Table 3-5). For the present analysis, Kj appeared to be about two to three orders o f magnitude larger than what was found before for bainite formation from single austenite phase i n Fe-0.6wt%C-1.5wt%Mn-1.5wt%Si with d r o f 13 and 40um, respectively. This can be attributed to the faster rate o f bainite formation from an austeniteferrite aggregate where bainitic ferrite nucleates or perhaps continues to grow from austeniteferrite interphase boundaries. The lack o f incubation period prior to the onset o f bainite growth, as shown in Figure 3-25, can also be related to this conjecture. The predictions o f the displacive model using the parameters reported in Table 3-7 are illustrated in Figure 3-28 along with the experimental fraction o f bainite formation. The overall agreement between the calculation using the displacive approach and the measured data is, while still satisfactory, not as good as for example when the diffusion model is employed. Table 3-7 Parameters used in displacive approach to describe the measured kinetics of bainite formation from an intercritically treated austenite with carbon content of0.41wt%. K,, inY 1.6 1 K , J/mol 2 5xl0 j 165 P 0 3.4 Bainite Formation 0 in Fe-0.18C-1.55Mn-l. 750 1500 7Si 100 700 Chapter 3 1400 100 700 1400 Time, s Figure 3-28 Comparison of the model prediction based on the displacive approach and the measured kinetic of isothermal bainite formation from an intercritically treated austenite with carbon content of 0.41wt% 166 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 3.5 Study of Continuous Cooling Bainite Formation in 0.19C-1.5Mn1.6Si-0.2Mo TRIP Steel In this section the kinetics o f bainite formation during continuous cooling treatments is dealt with. This is relevant to the industrial processing o f hot-rolled T R I P steel, where hot band is cooled slowly through the intercritical region to form the desired fraction o f polygonal ferrite followed by coiling, such that the untransformed austenite decomposes partially into bainite. The material selected for this study was a M o - T R I P steel containing 0.19wt%C-1.5wt%Mn1.6wt%Si-0.2wt%Mo. The kinetics o f austenite decomposition along different cooling paths had been measured earlier [140 ] by means o f dilatometry using the Gleeble 3500 thermomechanical simulator. During the austenitization stage the test specimens were reheated at the rate o f 5°C/s to different temperatures, i.e. 950, 1000 and 1100°C, held for 2min followed by cooling to deformation temperature 850°C, where the samples were subjected to a strain s =0, 0.3 and 0.6 at a strain rate o f Is' . A s a result the parent austenite has different features, 1 in terms o f the grain size, i.e. 24, 34 and 52um, and the degree o f work hardening. The list o f experimental treatments that the examined material had been exposed to is shown in Table 3-8. The transformation start and finish temperatures o f each product, i.e. ferrite and bainite are also included i n the table. The true fraction o f ferrite and bainite i n the final microstructures, quantified using standard metallographic procedures, are outlined in the table as well. Based on the amount o f ferrite formed preceding to bainite reaction, the degree of carbon enrichment i n the untransformed austenite can be estimated as is indicated by l c unions • Assuming a spherical austenite grain with an outer shell o f ferrite, it is also possible to assess the grain diameter o f the remaining austenite at the end o f ferrite formation stage, which is listed by d yUntrms in the table. Therefore, both the carbon content and the size o f the 167 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 starting austenite for bainite reaction might be different for each thermal path and have to be considered explicitly i n the modeling analysis. The carbon enrichment i n austenite is expected to be the more significant effect for the subsequent bainite formation. Table 3-8 List of experimental treatments and the ensuing measurements of 0.16wt%C-1.5wt%Mn1.6wt%Si-0.2wt%Mo, Mo-TRIP steel. Prior Austenite Reheating 950°C, 24um 1050°C, 34um 1100°C, 52um Strain, Ferrite £ CR, °C /s T, °C 0 0 0 0 0.3 0.3 0 0 0 0.3 0.3 0.3 0.6 0.6 0 0 0 1 5 7.5 10 5 10 1 2.5 5 1 5 10 1 10 1 5 10 755 725 676 650 754 746 740 690 740 776 752 720 787 772 750 720 662 s fa; % 55 25 10 7 37 22 49 17 5 57 33 8 58 24 33 0 0 Untransformed Austenite o Untrans. y ^ f.Untrans , wt% , um 0.4 0.25 0.21 0.2 0.29 0.24 0.35 0.22 0.2 0.42 0.27 0.21 0.43 0.24 0.27 0.19 0.19 18 22 23 24 20 21 28 32 33 26 30 33 24 28 45 50 51 Bainite T, °C T °C /B, 580 607 619 590 575 600 590 598 620 555 565 575 512 580 590 610 600 364 380 400 394 381 400 365 354 404 360 400 417 350 351 364 360 403 31 52 63 54 30 27 37 63 61 15 32 36 10 21 52 68 61 s F) % Driving pressure, J/mol 706 620 575 723 773 658 670 673 574 842 834 801 1080 761 699 625 675 A typical measurement o f the overall decomposition kinetics o f austenite, in terms o f fraction transformed versus temperature, for the case where the initial austenite o f 24 um grain size (reheating at 950°C) was cooled constantly at the rate o f 5°C/s to ambient temperature, is illustrated in Figure 3-29a. It can clearly be seen that the overall decomposition is sequential which is characterized by the two (or three when martensite forms) distinct sigmoidal subcurves observed along the measured kinetics. This feature turned out to be common for most o f the thermal treatments applied to the examined M o - T R I P steel. Figure 3-29b shows the 168 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 variation o f the transformation rate with temperature evaluated based on the derivative o f the data presented i n Figure 3-29a. It is evident that for each transformation product a corresponding separate peak i n the rate can be detected, which was also used to confirm the transformation start and finish temperatures o f ferrite and bainite reactions. 300 400 500 600 700 800 Temperature, °C 40 0 I 300 1 400 1 1 500 600 ! 700 1 800 Temperature, °C Figure 3-29 Experimental data of overall austenite decomposition during continuous cooling treatments at 5°C/s after reheating at 950°C, £=0 in 0.19wt%C-1.5wt%Mn-1.6wt%Si-0.2wt%Mo TRIP steel, (a) fraction transformed and (b) the rate of decomposition. 169 3.5 Bainite Formation in Fe-0.19C-1.5Mn-1.6Si-0.2Mo Chapter 3 Considering the aforementioned data, in the following sections attempts w i l l be made to model two major aspects o f austenite to bainite transformation during continuous cooling paths, i.e. a criterion for the onset o f bainite formation after ferrite reaction, and also the kinetics o f bainite evolution in these circumstances. 3.5.1 Analyzing the onset of bainite reaction A s indicated i n Figure 3-29b, at a certain point along the cooling path the rate o f ferrite evolution appears to decline and shortly afterward further decomposition of the untransformed austenite is accomplished via bainite formation. This transition, i.e. the temperature at which bainite reaction comes into the picture, is o f crucial significance in modeling the overall kinetics. To analyze this transition point one could simply think o f the relative growth rate o f polygonal ferrite compared to that for a bainitic ferrite plate. Then naturally it is expected that bainite might form when it outgrows the polygonal ferrite, in terms o f the growth velocity. The ratio o f the two growth rates was quantified at the point o f experimental transition for all the conditions reported in Table 3-8, however, no consistent trend was found to be followed. Alternatively, the driving pressure o f bainitic ferrite formation at the experimentally observed transition point was calculated. For this purpose the carbon enrichment o f the untransformed austenite was accounted for and paraequilibrium treatment was employed. The calculated driving pressure, as also outlined i n the last column o f Table 3-8, is plotted versus the transformation start temperature o f bainite i n Figure 3-30. It appears that regardless o f the carbon content, grain size and degree o f work hardening o f the parent austenite, all the data points fall on the straight line showing a negligible scattering. This, at least as a first approximation, can constitute a phenomenological criterion for the onset o f 170 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 bainite formation, i.e. along the cooling path following by ferrite evolution, bainite tends to emerge when a critical driving pressure is attained. For the examined M o - T R I P steel this critical value is estimated as: G {J CrUical I mol) = 3.647TC) - 2240 (3-33) 900 O • • 800 950C 1050C 1100C 700 600 500 a 400 300 560 580 600 620 Temperautre,C Figure 3-30 Variation of the calculated driving pressure of bainitic ferrite formation at the transformation start temperature. The lower line shows G function [113], which is located about N 300J/mol lower than the line followed by the data points. Surprisingly, this linear dependence is o f the same slope as the universal nucleation function GN proposed by Bhadeshia (cf. section 3.2.5). However, the intercept given above, 2240J/mol, appears to be 300 J/mol larger than that indicated by GN- Although, the concept of the critical driving pressure, in terms o f GN, was originally proposed as a criterion for displacive nucleation o f bainite subunits, it is calculated by virtue o f carbon partitioning during the nucleation event. In other words it is the maximum thermodynamic driving 171 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 pressure for nucleation o f ferrite from the parent austenite i n paraequilibrium condition. The composition o f the critical ferrite nucleus corresponds to one that provides the maximum driving pressure. Regarding the displacive nucleation, it would be o f more physical sense i f the driving pressure is calculated assuming the same carbon content for bainitic ferrite and austenite. This implies a self-inconsistency o f the displacive model in providing a conclusive description o f the nucleation stage. Therefore, since the driving pressure in Equation 3-33 is evaluated assuming carbon partitioning, then approaching a critical value for the driving pressure at the onset o f bainite reaction, can potentially be considered as a criterion for diffusional formation o f bainitic ferrite. 3.5.1.1 Modeling bainite formation The experimental data points required for modeling the kinetics o f bainite evolution i n the M o - T R I P steel were taken from the measured overall kinetics, such that between the transformation start and finish temperatures o f the bainite stage the sub-curve was cut and then converted to normalized values using the maximum experimental fraction o f bainite. A s an example and w i t h reference to the overall kinetics represented i n Figure 3-29, the corresponding bainite stage, in terms o f normalized fraction versus transformation temperature, is shown i n Figure 3-31. In the right side o f the plot, the second scale is also included to show the true fraction o f bainite, which had been determined from dilatometry analysis and verified by metallographic measurements. To analyze the measured kinetics o f bainite the diffusion model as described in section 3.3.3.3, is adopted here. This is in part due to the fact that the criterion for bainite start, as expressed by Equation 3-33, assumes carbon partitioning to calculated the driving pressure. 172 3.5 Bainite Formation in Fe-0.19C-1.5Mn-1.6Si-0.2Mo Chapter 3 Further, bainite forms in the examined M o - T R I P steel at relatively high temperatures, i.e. mostly between 600 to 350°C, the temperature range within which bainite is expected to form by a diffusional mechanism [131]. reheating at 950°C in Mo-TRIP steel. Regarding the bainite evolution from austenite o f different carbon content during continuous cooling paths, the first step to model the kinetics is to quantify the maximum growth rate o f a plate, v * , as a function o f carbon concentration and temperatures. This allows accounting ax explicitly for the impact o f gradual carbon enrichment o f austenite on the growth kinetics, as the bainite reaction proceeds during a non-isothermal treatment. For the investigated temperature range and carbon concentrations, the evaluated v ^ , using the methodology as a x described in section 3.3.3.3, is illustrated in Figure 3-32. For the sake o f simplicity relevant to the numerical implementation o f v ^ , the following expression was fitted to the a x calculated values: 173 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo \ V 0.5 L {\mls) = A exp l V 2 A J Chapter 3 2 + wt%C-A< (3-34) where, the parameters Aj to A were found to be: 5 A i, pm/s 9.25xlO J A , °C 113.6 A , wt%C 0.74 2 A , °C 525.4 3 A, 4 5 wt%C -1.9 350 l O v 1 ^„ 550 ^ 500 450 oc Figure 3-32 Variation of the maximum growth rate of a bainite plate with temperature and carbon concentration for Mo-TRIP steel. Having quantified the maximum growth rate o f a plate, one can calculate the kinetics o f bainite formation during non-isothermal conditions by integrating the rate equation [cf. Equation (3-24)] along a given cooling path. Typical model predictions for the case o f reheating at 1050°C, corresponding to a prior austenite grain size o f 34u.m, are illustrated i n Figure 3-33. For the purpose o f comparison the corresponding measured data points are also included in each plot. 174 3.5 Bainite Formation 300 350 in 400 Fe-0.19C-l.5Mn-l.6Si-0.2Mo 450 500 550 350 Temperature, °C Chapter 3 400 450 500 550 Temperature, °C Figure 3-33 Diffusion model prediction and the experimental kinetics of bainite formation for reheating at 1050°C and various subsequent treatments in Mo-TRIP steel. 175 J . J Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 In these simulations the following temperature dependence were used for the parameter Kp, which was determined based on the optimum description o f all the experimental kinetics (listed in Table 3-8), i.e.: „ , o (m K 5xl(T )= 0.35 6 e x ,12000, P(—^T-) <- > 3 35 To summarize the predictions o f the diffusion model for bainite formation i n M o - T R I P steel regarding all the reheating conditions and the subsequent thermo-mechanical treatments, the calculated temperatures for 50% transformation are compared with the corresponding experimental values i n Figure 3-34. Overall, it is seen that the diffusion model can reasonably capture most o f the experimental kinetics. However, it is quite obvious that common to all reheating conditions, the measured data points for a cooling rate o f l ° C / s are overpredicted. Although a realistic justification for this discrepancy cannot be provided, one might argue that it may be associated with the potential interaction between substitutional solute and the moving ferrite-austenite boundaries. Basically for the slow cooling rates, e.g. l°C/s, the sample spends more time at higher temperatures, therefore the transformation interface can readily be loaded with solute atoms that essentially slow down the apparent growth rate. In particular, the examined material in addition to manganese has about 0.2wt% M o , which is known to have a strong tendency for segregation to ferrite-austenite interfaces. Alternatively, the discrepancy between the model prediction and the experimental kinetics for slow cooling rates can be attributed to the pinning effect o f carbide particles. In the investigated M o - T R I P steel bainite starts to form at relatively high temperature, i.e. around 600°C, during continuous cooling treatment. This temperature is much higher than the one is traditionally practiced for isothermal bainite formation i n T R I P steels, i.e. 300 to 450°C, during which the carbide formation is delayed due to the presence o f S i or A l . However, the 176 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 rate o f carbide formation is expected to accelerate as temperature increases by virtue o f the required mass transportation o f carbon (and solute) in the lattice. Further, the high thermodynamic affinity o f molybdenum for carbon atoms decreases the retarding effect of silicon on carbide formation. Therefore, at sufficiently high temperatures where atoms are o f high mobility and provided that sufficient time is available, i.e. low cooling rates, carbide formation is expected. 460 480 500 520 540 560 580 Measured T „ , °C 50 /o Figure 3-34 Comparison of the predicted (using the diffusion model) temperature for 50% transformation and the measured value for all the experiments listed in Table 3-8. 3.5.1.2 Diffusion model with plastic work In the aforementioned analyses o f the role o f Gibbs energy dissipation, which occurs due to the plastic accommodation o f the transformation strain, was not be accounted for. A i m i n g at improving the quality o f the model predictions, in particular for the slowest cooling rate, i.e. l°C/s, the contribution o f the plastic accommodation is implemented in the diffusion approach for bainite formation in the M o - T P J P steel. A s delineated in section 3.3.3.4, the 177 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo Chapter 3 amount o f the Gibbs energy dissipation, and its retarding effect on growth rate, scales with the fraction transformed. Therefore, it is essential to quantify the decline o f v ^ r iax as a function o f bainite fraction for a given carbon concentration o f parent austenite and transformation temperature. A n example of this variation is shown i n Figure 3-35, which indicates that the growth rate o f a plate can potentially be reduced by about a factor o f two when the bainite fraction changes from zero to 0.75. 80 70 60 I 50 % 4 0 10 0 300 350 400 450 500 550 600 650 Temperature, °C Figure 3-35 Variation of the maximum growth rate with transformation temperature for different values of bainite fraction, i.e. different values of Gibbs free dissipation due to plastic work. The effect o f the plastic accommodation in the equation for m a x i m u m growth rate can be accounted for through modifying the parameters A] to As as a function o f bainite fraction, i.e. A = 0 ( / ) , where f t B B is the true fraction o f bainite with respect to the untransformed austenite remaining after the ferrite formation stage. Considering the modified diffusion model, the resulting predictions for the temperature for 50% transformation, T %, compared 50 with the measured values are illustrated in Figure 3-36. The parameter KD employed in the simulations is slightly different from the previous analysis, i.e.: 178 3.5 Bainite Formation in Fe-0.19C-l.5Mn-l.6Si-0.2Mo „ , D(™ Chapter 3 3.62xlO" 12000, TUT— e x p ( — — ) 5 )= K (3-36) Apparently, the model calculations agree reasonably well with most o f the measured data, however, similar to the prediction o f the diffusion model without accounting for the plastic work, the experimental kinetics for the slowest cooling rate, l C / s , are still significantly overestimated. This implies that implementation o f the plastic work into the diffusion model does not markedly improve its predictive capability for bainite formation i n the M o - T R I P steel, although from a conceptual prospect it adds more physical creditability to the diffusion model. 580 o 560 "-. 540 o in £ B 520 | 500 * 480 • A - o rr o °/ ' o a A d,=52|am d =34fim d =24|im 540 560 y v 460 460 480 500 520 Measured T 5 0 % ) 580 "C Figure 3-36 Comparison of the predicted (using the diffusion model and accounting for the plastic work) temperature for 50% transformation and the measured value for all the experiments listed in Table 3-8. 179 3.6 Summary and Remarks Chapter 3 3.6 Summary and Remarks (Bainite Formation) The significance o f bainite formation in processing o f T R I P steels i n particular, and D P steels to a lesser extent, is twofold. Firstly, it constitutes the essential stage during which the untransformed austenite enriches with carbon such that austenite is retained to room temperature i n T R I P steels. Secondly, it is o f crucial importance to determine the critical cooling rate above which bainite formation can be prevented. Therefore, to gain more insight regarding bainite transformation a series o f isothermal and continuous cooling treatments i n three different steels was carried out. The materials selected for isothermal bainite formation were a high carbon TRIP steel with nominal composition o f 0.6wt%C-1.5wt%Mn-1.5wt%Si and a classical TRIP steel containing 0.18wt%C- 1.55wt%Mn-1.7wt%Si. In the former bainite formation from single phase austenite was studied while for the latter an intercritical mixture o f ferrite-0.42%austenite was chosen as the starting microstructure for isothermal bainite reaction. Relevant to continuous cooling treatments, a M o - T R I P with 0.19wt%C-1.5wt%Mn-1.6wt%Si-0.2wt%Mo composition was considered, where ferrite reaction precedes the bainite formation. The ensuing results based on the modeling and analysis o f the experimental data are discussed as follows: • In the framework o f Russell's treatment based on classical nucleation theory, the incubation time associated with isothermal bainite formation i n 0.6wt%C-1.5wt%Mn1.5wt%Si steel was captured successfully for various temperatures and initial austenite grain sizes. • The isothermal kinetics o f bainite evolution i n T R I P steels can be described rather accurately by the J M A K equation, 180 the diffusion model and the displacive 3.6 Summary and Remarks Chapter 3 formulation. Therefore, based on these analyses a conclusive model, outperforming the other approaches in terms o f predictive capability, cannot be suggested. • The version o f displacive model analyzed here, as a purely nucleation approach to predict the overall kinetics, deals with the autocatalytic nucleation and sheaf formation i n a simplified manner. Its recent modification [114] has yet to be verified. • Analysis o f austenite decomposition during a non-isothermal treatment in Mo-TRTP steel revealed that following the formation o f polygonal ferrite, bainite transformation starts when a critical driving pressure is attained. This critical driving pressure varies linearly with bainite start temperature and is calculated assuming carbon partitioning in paraequilibrium condition. Hence it is regarded as a diffusional criterion. • Regarding bainite evolution during continuous cooling treatments o f a Mo-TRTP steel, the diffusion model can reasonably capture most o f the experimental kinetics. However, for the slowest cooling rate, i.e. l ° C / s , the measured rates were significantly overestimated. This might be related to the potential kinetic effects o f substitutional solutes, e.g. solute drag effect, which is not addressed rigorously in the model. The other potential reason for this discrepancy might be due to carbide formation and the resulting pinning effect, which slows down the growth rate o f bainite plates. These aspects call for more improvement and modification o f the model. • For the investigated isothermal and continuous cooling cases, it turned out that accounting for the effect o f the plastic work does not improve the overall predictive capability o f the diffusion model. 181 Overall Conclusion Chapter Chapter 4 4 : Overall Conclusion A thorough microstructure model to describe austenite decomposition into ferrite and bainite under industrial processing conditions was developed. This is in particular relevant to hotrolled advanced high-strength steels such as D P and TRIP products. To explicitly address all the undergoing microstructural events, the model elegantly integrates a number o f components, each developed on a solid physical ground, into a monolithic module which can easily be implemented into an industrial process model. Consequently, the employed model parameters have clear physical meaning. Therefore, the model application can potentially be extended to conditions that have not been considered in laboratory experiments, i.e. new steel grades. This constitutes a unique overall feature o f the proposed model. Specifically to the ferrite and bainite components the following characteristics are outlined. 4.1 Findings and Achievements • To replace the traditionally used J M A K methodology i n process modeling, a physically-based approach in the framework o f a mixed-mode model is proposed for polygonal ferrite formation. Assuming paraequilibrium the retarding effect o f alloying elements is accounted for by using a modified Purdy-Brechet solute drag model. It was identified that accounting for the solute-interface interaction in the mixed-mode approach is essential to adequately describe experimental kinetics. Moreover, the apparent artifact o f original Purdy-Brechet theory was corrected by accounting for the intrinsic asymmetry o f the segregation profile inside the interface. The proposed austenite-to-ferrite model proved that it is capable o f accurately describing the ferrite formation kinetics for different isothermal and continuous 182 Overall Conclusion cooling Chapter 4 situations. The physically well-defined model parameters, which are introduced to describe the interfacial properties o f the ferrite-austenite interface, were determined for a number o f examined cases. Further systematic experimental investigations would provide the basis to quantify these parameters as a function o f steel chemistry. This would then also increasingly provide better guidelines for the acceptable combination o f the employed parameters for each new steel which is being analyzed. In this transformation sense, the model provides a versatile tool to predict kinetics for industrial processing conditions where the laboratory simulations are limited. • Relevant to T R I P steels, a comprehensive experimental characterization o f bainite transformation kinetics for isothermal and continuous cooling treatments were carried out. In the sense o f describing the overall isothermal kinetics, the predictive capabilities o f the J M A K equation, the diffusional formulation and the displacive approach were revealed to be similar. Due to the nature o f the model parameters, which are determined entirely from fitting to experimental data, all approaches were employed in a semi-empirical manner. Therefore, proposing a conclusive model for bainite reaction remains a challenge and requires more investigation. Further, it was revealed that the plastic accommodation o f the transformation strain does not have a noticeable effect i n improving the prediction quality o f the diffusion model. • Assuming diffusional transformation mechanism, a criterion based on a critical driving pressure is presented to predict the onset o f bainite reaction along a given cooling path. Regarding isothermal bainite formation the incubation period can be predicted using Russell's theory. Relevant to the maximum growth rate o f a bainite 183 Overall Conclusion Chapter 4 plate in terms o f the Zener-Hillert equation, a numerical treatment was developed to determine its carbon and temperature dependencies. For the various conditions o f parent austenite, in terms o f grain size and carbon content, the diffusion model could reasonably describe the kinetics o f bainite evolution during continuous cooling treatments o f a M o - T R I P steel. The diffusion model was employed to bring more consistency into the overall model since both the isothermal incubation period and the criterion for the onset o f bainite formation have been quantified in the framework o f a diffusional transformation mechanism. 4.2 Future work • To further improve the physical credit o f the ferrite model, it is also intended to account for the potential deviation from paraequilibrium condition, which may also aid to remove the apparent temperature dependency o f the binding energy. A t higher transformation temperatures, the partitioning domain o f substitutional atoms is likely well extended beyond the interface region and the degree o f redistribution tends to be so pronounced that it cannot be described merely by a solute spike. Since during the industrial processing conditions paraequilibrium is expected to prevail, the aforementioned modification does not have a major significance for conventional process modeling and it is o f more academic interest. • Although, the diffusional approach was proven to be capable in capturing the major kinetic aspect o f bainite formation, to translate the growth rate into the overall kinetics an empirical parameter, KD, has to be introduced. This parameter provides a phenomenological description o f the nucleation event. A more sophisticated treatment 184 Overall Conclusion Chapter 4 of nucleation is required. In particular, the evolution o f bainite sheaves and autocatalytic (also called sympathetic) nucleation, which is an inherent feature o f the bainite reaction, has to be considered appropriately. Moreover, the assumption o f carbide-free bainite for bainite evolution in T R I P steels needs to be revisited. The potential effect o f carbide formation, in terms o f pinning the ferrite-austenite interface and removing some supersaturation from the untransformed austenite, has to be addressed in the diffusion model. Finally, a more rigorous approach, compared to the description o f Quidort and Bouaziz, to quantify the elastic and plastic strain energy of bainite reaction is required. • The recent version o f the displacive model [114], which relies on the extended area pertinent to nucleation on grain boundaries, and its general applicability have to be evaluated for bainite formation in TRIP steels. 185 REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 International Iron and Steel Institute website: www.worldsteel.org T. Senuma, ISIJInt, vol. 41, 2001, pp. 520-532. J. Shaw et al., SAE-SP 1685, New Steel Sheet and Steel Bar Product and Processing, 2002, pp. 63-71. J.G. Speer, D.K. Matlock, JOM, vol. 54, no. 7, 2002, pp. 19-24. R.A. Heimbuch, 40 MWSP Conference Proceedings, vol. X X X V I , ISS, Warrendale, 1998, pp. 3-10. ; K. Kishida, Nippon steels technical report, no. 81, 2000, pp. 12-16. W. Bleck, JOM, vol. 48, no. 7, 1996, pp. 26-30. A J . DeArdo, Microalloying'95 Conference Proceedings, Pittsburgh, USA, ISS, 1995, pp. 15-33. C. Ouchi, ISIJInt., vol. 41, 2001, pp. 542-553. A. Pichler et al., 40 MWSP.Conference Proceedings, vol. X X X V I , ISS, Warrendale, 1998, pp. 259-74. T. Matsuoka and K. Yamamori, Metall. Trans. A, vol. 6A, 1975, pp. 1613-22 S.Hayami and T. Furakawa, Proceedings of Micro Alloying '75, Washington, D.C., 1975, pp. 311-20. J. Vrieze et al., 41 ' MWSP Conference Proceedings, vol. X X X V I I , ISS, Warrendale, 1999, pp. 277-94. F. Hassani and S. Yue, 41 ' MWSP Conference Proceedings, vol. X X X V I I , ISS, Warrendale, 1999, pp. 493-98. Y. Sakuma, D. K. Matlock and G. Krauss, Metall. Trans. A, vol. 23A, 1992, pp. 1221-32. V.F. Zackay et al., Trans. ASM, vol. 60, 1967, pp. 252-59. O. Matsumura, Y. Sakuma and H. Takechi, ISIJInt., vol. 27, 1987, pp. 570-79 K. Sugimoto, M . Kobayashi and S.I. Hashimoto, Metall. Trans. A, vol. 23A, 1992, pp.3085-91 M . De Meyer, D. Vanderschueren and B.C. De Cooman, ISIJInt, vol. 39, 1999, pp. 813-822. R.A. Shulkosky et al., Modelling, Control and Optimization in Ferrous and non-Ferrous Industry, TMS, Warrendale, 2003, eds. F. Kongoli, B.G. Thomas and K. Sawamiphakdi, pp. 509-27. M . Militzer, E.B. Hawbolt and T.R. Meadowcroft, Metall. Mater. Trans., vol. 31A, 2000, pp. 1247-59. M. Hillert, Jernkont. Ann., vol. 141, 1957, pp. 757-64. G.I. Rees and H.K.D.H. Bhadeshia, Mater. Sci. Tech., vol. 8, 1992, pp. 985-93. J.S. Kirkaldy and G.R. Purdy, Can. J. Phys., vol. 40, 1962, p. 202-07. J.S. Kirkaldy, B. A. Thomson and E. Baganis, Hardenability Concept with Application to Steel, Eds. D.V. Doan and J.S. Kirkaldy, TMS-AEVIE, Warrendale, PA, 1978, pp. 82-125 M . Hillert and L.I. Staffansson, Acta Chem. Scand., vol. 24, 1970, pp. 3618-26. B. Sundamn and J. Agren, J. Phys. Chem. Solids, vol. 42, 1981, pp. 297-301. B. Sundman, B. Jansson and J.O. Andersson, Calphad, vol. 9, no. 2, 1985, pp. 153-190. M . Hillert and M . Jarl, Calphad, vol. 2, no. 3, 1978, pp. 227-238. M . Hillert and J. Agren, Scripta Materialia, vol. 50, 2004, pp. 697-699. J.B. Glimour, G.R. Purdy and J.S. Kirkaldy, Metall. Trans. A, vol. 3, 1972, pp. 1455-64. M . Hillert, Internal Report, Swedish Inst. Metals Res., 1953. J.S. Kirkaldy: Can. J. Phys., 1958, vol. 36, pp. 907-16. G.R. Purdy, D.H. Weichert and J.S. Kirkaldy, Trans. TMS-AIME, vol. 230, 1964, pp. 1025-34. M . Hillert, "Phase Equilibria, Phase Diagrams and Phase Transformations", Cambridge University Press, Cambridge, UK, 1998. S.E. Offerman et a l . Science, vol. 298, 2002, pp. 1003-05. ,h th s s 186 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 D.A. Porter and K.E. Easterlings, "Phase Transformations in Metals and Alloys ", 2 edition, Chapman and Hall, London, 1992. M. Enomoto, W. F. Lange, III, and H. I. Aaronson, Metall. Trans. A, vol.l7A, 1986, pp.13991404. P.J. Clemm and J.C. Fisher, Acta Met, vol. 3, 1955, pp. 70-73. W. F. Lange, III, M . Enomoto, and H.I. Aaronson, Metall. Trans. A, vol. 19A, 1988, pp. 427-40. T. Tanaka, H.I. Aaronson and M . Enomoto, Metall Trans A, vol. 26A, 1995, pp. 547-59. C. Wells, W. Batz and R.F. Mehl, Trans AIME, vol. 188, 1950, p. 553 J. Agren, Acta Metall, vol. 20, 1986, pp. 1507-10. M. Hillert, "Lectures on the Theory of Phase Transformations"', ed. H.I. Aaronson, 2 edition, TMS, Warrendale, PA, 1999, pp. 1-33. M. Enomoto and H.I Aaronson, Metall. Trans. A, vol. 17A, 1986, pp. 1381-84. M. Enomoto and H.I. Aaronson, Metall. Trans. A, vol. 17A, 1986, pp. 1385-97. H.I. Aaronson, "Decomposition of Austenite by Diffusional Processes", eds. V.F. Zackay and H.I Aaronson, Interscience Publishers, 1962, pp. 387-546. C. Zener, J. Appl. Phys., vol. 20, 1949, pp. 950-53. V.M.M. Silalahi, M. Onink and S. van der Zwaag, Steel Research, vol. 66, 1995, p. 482. R. A. Vandermeer, Acta Metal. Mater., vol. 38, 1990, pp. 2461-70. R.G. Kamat et al., Metall. Trans. A, vol. 23A, 1992, pp. 2469-80. R. Trivedi and G.M. Pound, J. Appl. Phys., vol. 38, 1967, pp. 3569-76. T.A. Kop et al., ISIJInt., vol. 40, 2000, p. 713-18. M. Hillert, Scrip. Mater., vol. 46, 2002, pp. 447-53. J. Odqvist, M. Hillert and J. Agren, Acta Mater., vol. 50, 2002, pp. 3211-25. C R . Hutchinson, A. Fuchmann and Y. Brechet, Metal. Mater. Trans. A., vol. 35A, 2004, pp. 1211-21. D.E. Coates, Metall. Trans., vol. 3, 1972, pp. 1203-12. K. Oi, C. Lux and G.R. Purdy: Acta Mater., vol. 48, 2000, pp. 2147-55. G.R. Purdy and Y.J.M Brechet, Acta Mater., vol. 43, 1995, pp. 3763-74. K. Luke and K. Detert, Acta Metall, vol. 5, 1957, pp. 628-37. J.W. Cahn, Acta Metall, vol. 10, 1962, pp. 789-98. J. R. Bradley and H. I. Aaronson, Metall. Trans. A, vol. 12A, 1981, pp. 1729-41. M. Militzer, R. Pandi and E.B. Hawbolt: Metall. Mater. Trans. A, vol. 27A, 1996, pp. 1547-56. M. Enomoto, Acta Mater., vol. 47, 1999, pp. 3533-40. M. Hillert, "The Mechanism of Phase Transformations in Crystalline Solids", Institute of Metals, London, 1969. M. Hillert and B. Sundman, Acta Metall, vol. 24, 1976, pp. 731-43. Y.J.M. Brechet and G.R. Purdy, Acta Mater., vol. 51, 2003, pp. 5587-92. ' M.I. Mendelev and D.J. Srolovitz, Interface Set, vol. 10, 2002, pp. 191-99. G.R. Purdy, E.T. Reynolds and H.I. Aaronson, Proceedings of International Conf: Solid-Solid Phase Transformations' 99, eds. M. Koiwa, K. Ousuka and T.Miyazaki, The Japan Institute of Metals Processing, vol. 12, 1999, pp. 1461-64. M . Hillert, J. Odqvist and J. Agren, Scripta Mater., vol. 45, 2001, pp. 221-27. J.W. Christian, "77ze Theory of Transformation in Metals and Alloys", 2 edition, Pergamon Press, Oxford, 1982. G.P. Krielaart, J. Seitsma and S. van der Zwaag, Mater. Sci. Eng., vol. A237, 1997, pp. 216-23. M. Onik, Ph. D. thesis, Delft University, 1995. M. Avrami, J. Chem. Phys., vol. 8, 1940, pp. 212-24. M. Umemoto, N. Komatsubara and I. Tamura, J. Heat Treating, vol.1, 1980, pp. 57-64. I. Tamura et al., "Thermomechanical Processing of High-strength Low-alloy Steels", Butterworths, 1988. nd nd 187 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 P.C. Campbell, E.B. Howbolt and J.K. Brimacombe, Metall. Trans. A, vol. 22A, 1991, pp. 277990. M . Umemoto, N. Komatsubara and I. Tamura, J. Heat Treat., vol. 1, no. 3, 1980, pp. 57-64. E. Scheil, Arch. Eisenhuttenw., vol. 12, 1935, pp. 656-67. J. Cranck, "The Mathematics of Diffusion", 2 edition, Oxford University Press, 1975. W.D. Murray and F. Landis: Trans. ASME, vol. 81, 1959, pp 106-12. R.A. Tanzilli and R.W. Heckel, Trans. TMS-AIME, vol. 242, 1986, pp. 2313-21. G. Krielaart, Ph. D. thesis, Delft University, 1995. F. Fazeli and M . Militzer, Steel Res., 2002, vol. 73, pp. 242-48. L. Kaufman, S.V. Radcliffe and M . Cohen, "Decomposition of Austenite by Diffusional Processes", Eds. V.F. Zackay and H.I. Aaronson, Interscience Publisher, 1962, pp. 313-67. D. Turnbull, Trans AIME, vol. 191, 1951, pp. 661-65. M. Hillert, Metall. Trans. A, vol. 6A, 1975, pp. 5-19. M . Enomoto, C.I. White and H.I. Aaronson, Metall. Trans. A, vol. 19A, 1988, pp. 1807-18. M. Guttmann and D. McLean, "Interfacial Segregation ", A S M , Metals Park, OH, 1979. M. Suehiro, Z.K. L i u and J. Agren, Acta Materialia, vol. 44, 1996, pp. 4241-51. H. Oikawa: Tetsu-to-Hagane, vol. 68, 1982, pp. 1489-97. J.R. Bradley, J.M. Rigsbee and H.I. Aaronson, Metall. Trans. A., vol. 8A, 1977, pp. 323-33. K.R. Kinsman and H.I. Aaronson, "Transformation and Hardenability in Steels", Climax Molybdenium C o , Ann Arbor, Michigan, 1967, pp. 39-54. J.R. Bradley, Ph.D. thesis, Carnegie Mellon University. E.B. Hawbolt, B. Chau and J.K. Brimacombe, Metall. Trans A., vol. 14A, 1983, pp. 1803-15. M . Militzer and F. Fazeli: Proceedings of International Conference on Thermomechanical Processing: Mechanics, Microstructure and Control, eds. E.J. Palmiere, M . Mahfouf and C. Pinna, 2002, Sheffield, UK, pp. 109-14. S.H. Park, H.N. Han, J.K. Lee and K.J. Lee: 40 Mechanical Working and Steel Processing Conference Proceedings, vol. X X X V I , ISS, Warrendale, PA, 1998, pp. 283-91. R.F. Hehemann, K.R. Kinsman and H.I. Aaronson, Metall. Trans., vol. 3, 1972, pp. 1077-94. H.I. Aaronson et a l , Metall. Trans. A., vol. 21A, 1990, pp. 1343-79. H.K.D.H. Bhadeshia and J.W. Christian, Metall. Trans. A, vol. 21 A, 1990, pp. 767-97. G.R. Purdy and M . Hillert, Acta Metall, vol. 32, 1984, pp. 823-28. M. Hillert and G.R. Purdy, Scripta Mater., vol. 43 , 2000, pp. 831-33. D. Quidort and Q. Bouaziz, Canadian Metall. Quart., vol. 43, 2004, pp. 25-33. C. Zener, Trans AIME, 1946, vol. 167, pp. 550-83. M . Hillert, ISIJInt., vol. 35, 1995, pp. 1134-40. W. Steven and A. G. Haynes, J. Iron Steel Inst., vol. 183, 1956, pp. 349-59. B.L. Bramfitt and J.G. Speer, Met. Trans. A, vol. 21 A, 1991, pp. 817-29. G. Spanos et a l . Met. Trans. A, vol. 21 A, 1991, pp. 1391-11. H.K.D.H. Bhadeshia, "Bainite in Steels", The Institute of Materials, London, 1992. W.S. Owen, ASM Trans. , vol. 11, 1954, pp. 812-29. H.K.D.H Bhadeshia, Acta Metal, vol. 29, 1981, pp. 1117-30. G.B. Olson and M.Cohen, Metall. Trans. A, vol. 7A, 1976, pp. 1915-23. A. A h and H.K.D.H. Bhadeshia, Mater. Sci. Tech., vol. 6, 1990, pp. 781-784. H. Matsuda and H.K.D.H. Bhadeshia, Proc. R. Soc. Lond. A., vol. 460, 2004, pp. 1707-22. J.W. Cahn, Acta Metall., vol. 4, 1956, pp. 449-59. A. Hultgren, J. Iron Steel Inst., vol. 114, 1926, p. 421-22. D. Quidort and Y. Brechet, Acta Materialia, vol. 49, 2001, pp. 4164-70. J.M. Rigsbee and H.I. Aaronson, Acta Metall., vol. 27, 1979, pp. 365-76. G.R. Purdy, Scripta Materialia, vol. 47, 2002, pp. 181-85. M . Hillert, Scripta Materialia, vol. 47, 2002, pp. 175-80. nd lh 188 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 T. K o and S.A. Cottrel, J. Iron Steel Inst., vol. 172, 1952, pp. 307-313. W.T. Reynolds et a l , Metall. Trans. A, vol.21A, 1990, pp. 1433-63. D. Quidort, Scripta Materialia, vol. 47, 2002, pp. 151-56. G. Horvay and J. W. Cahn, Acta Metall., vol. 9, 1961, pp. 695-705. G.P. Ivantsov, Dokl Nauk SSSR, vol. 58, 1947, p. 567-69. R. Trivedi, Metall. Trans., vol. 1, 1970, pp. 921-27. W. P. Bosze and R. Trivedi, Metall. Trans., vol. 5, 1974, pp. 511-12. M . Hillert, Acta Materialia, vol. 51, 2003, pp. 2089-95. M . Enomoto, Metall. Trans. A, vol. 25A, 1994, pp. 1947-55. C. Atkinson, Proc. R. Soc. Lond., vol. A378, 1981, pp. 351-68. T. Minote et a l , IJISInt., vol. 36, no. 2, 1996, pp. 201-07. M . Militzer et a l , Metall. Mater. Trans., vol. 27A, 1996, pp. 3399-09. K.C. Russell, Acta Metall, vol. 17, 1969, pp. 1123-31. H.K.D.H. Bhadeshia, Metal Science, vol. 16, 1982, pp. 159-65. L. Haufman and M . Cohen, Progress in Metals Physics, vol. 7, 1949, pp. 165-246. M . Suehiro, Tetsu-to-Hagane, vol. 73, 1987, pp. 1026-33. O. Bouaziz, D. Quidort and P. Maugis, Revue de Metallurgie-CIT, vol. 1, 2003, pp. 103-108. A. Zarei Hanzaki, P.D. Hodgson, and S. Yue, ISIJInt., 1995, vol. 35, No. 1, p. 79-85. P. Jacques et a l , Metall, Trans. A, 1998, V o l 29A, pp. 2383-93. D. Liu and M . Militzer, 2004, UBC, unpublished report. 189 APPENDIX ( I ) Sub-Lattice Model According to the concepts o f the sub-lattice model [26], the description o f the free energy o f either the ferrite or austenite phase for a ternary steel, e.g. F e - M n - C , is presented below. A l l the definitions presented below are consistent with the sub-lattice approach. The above system is defined in terms o f four compounds such that each pair o f them occupies one sub-lattice, i.e. substitutional sub-lattice which contains Fe and M n , and interstitial sub-lattice which is filled by C and interstitial vacancy. The ratio between the numbers o f atomic site on the substitutional lattice to interstitial one is called site ratio, n . This ratio is 1 for austenite and 3 for ferrite phase. The four components are defined by: Fe :C, Fe :Va , Mn : C and Mn : Va , by which the composition o f the system can be represented on a square, instead o f the well known Gibbs triangle for a ternary system. However, since the composition o f this system can be varied by only two ways, i.e. changing carbon (or vacancy) content in interstitial lattice and manganese (or iron) content in the substitutional lattice, then this four components aggregate resembles a ternary solution phase. The square o f composition is shown below: Mn: Va Fe:Va Yva Fe:C Mn:C Y M In To represent the concentration, the site fraction o f each component is defined as: 190 Y. = l-X for i=¥e and M n (A-1) c Y =r\ (A-2) c The Gibbs free energy o f either ferrite or austenite ( j=a or y ) can be represented by contribution o f different terms, standard free energy o f unmixed components, entropy o f mixing for an ideal solution, the excess energy due to non-ideality and the magnetic effect, i.e.: Qj o J _ = Q T S L b 1 + J G xcess + G L g n e H c ( A . 3 ) These contributions are quantified as: °G J = Y Y °GL Fe - TSJ deal c + Y Y °G , + Y Y °G j C Fe = RT(Y Va In Y + Y Fe Fe Gixcess ~ ^Fe^Mn^C^FeMn-C + J Fe Va Mn Mn c In Y ) + RTr\(Y Mn ^Fe^Mn^Va^Fe.MmVa + Y Y °G , j MmC Mn Mn Va In Y + Y In Y ) c c ^Fe^C^Va^Fe-.CVa + Va Va + Va ^Mii^C^Va^Mrr.C,Va (A- 4) (A- 5) ^) where the standard Gibbs energy term for each component, "G- , and the interaction term between two components, L\ , are normally defined i n terms o f the following temperature k dependency: °GJ orL\ = A + BT + CTInT + DT + ET + — + -^- + ^ 2 k 3 (A-7) and A through H denote the S G T E (Scientific Group Thermodata Europe) parameters, which are quantified and reported for different alloying systems in the literature. It has to be noted that to obtain the molar Gibbs free energy, the aforementioned three contributions have to be multiplied by (\-Xc). The magnetic contribution to Gibbs free energy is usually considered for ferrite and can be estimated using the following expression [29]: 191 magnetic = RTg(z )\n{f3 +\) m and (A- 8) m (A- 9) T/Tr. where /?,„ represents the magnetic momentum and for different temperature ranges, i.e. above and below the Curie temperature, Tc, the function g(T/T ) is given by: c For T>T : C g =Y and for 1 r 1.25 1.56 v 7 U V A 15 497y 135 (A- 10) • + - 600 T<TQ'- ( g = - 1.56 It has to be noted that both Tc and -s -15 Fe c i Fe l/ 2+ Mn \ (A- 11) 1500 depend on the alloy composition, i.e.: T =Y Y T +Y Y J Y Y T c -25 x. +• 10 315 X„ c 1 (A- 12) + YM„ Y Va YFC YMn Y J V P m = Y Y ft Ft c +Y YJ Fe v 2 + Y Y (3, + Mn where T, and /?, are parameters. 192 c Y YJ Mn v A 5 (A- 13) APPENDIX (II) Code for Paraequilibrium Calculation F O R T R A N subroutine to determine the paraequilibrium carbon contents o f ferrite and austenite at a given temperature for a quaternary F e - C - M n - S i steel. Definition o f the variables and parameters: IWSG (): Integer array for storage o f data inside T Q workspace. I_fcc, I_b.cc: Index for fee and bec phase. I_Fe, I M n , I S i , I_C: Index for the constituents o f alloy. Gm_fcc( ), Gm_bcc( ): Array containing C mole fraction and Gibbs free energy o f phases. SGM( ): Array containing C mole fraction, Gibbs energy and its derivative for fee and bec after fitting to S P L I N E function. NN: Number o f total moles in the system. T: Temperature i n K . P: Pressure i n Pa. X_alpha, X g a m m a : Equilibrium mole fraction o f C in ferrite and austenite (input parameters). X P a l p h a , XP_gamma: Paraequilibrium mole fraction o f C i n ferrite and austenite (output parameters). Tmn, Tsi: The ratio o f mole fraction o f M n (and S i ) to iron i n the parent austenite (input parameters). X X C , X X M N , XXSI: M o l e fractions o f C, M n and S i along component ray o f carbon. Kf, Kb: Pointer. Jf, Jb: Pointer. subroutine peq(T,x_alpha,x_gamma,Tmn,Tsi,xp_alpha,xp_gamma) implicit doubleprecision (a-h,o-z) common /tc/iwsg,i_fcc,i_bcc,ic,imn,isi,ife dimension iwsg(80000),gm_bcc(500,2),gm_fcc(5 00,2),sgm(3500,7), dimension xx(500),yy(500),yy2(500) d o u b l e p r e c i s i o n nn p=101325.d0 NN=1 c a l l tqsetc('P',-1,-1,P,num,iwsg) c a l l tqsetc('N ,-1,-l,NN,num,iwsg) c a l l tqsetc('T',-1,-1,T,num,iwsg) 1 kb=0 call call do tqcsp(i_fcc,'SUSPENDED',iwsg) tqcsp(i_bcc,'ENTERED',iwsg) XXc = 0', 3 * X _ a l p h a , X _ a l p h a / l 0 0 . kb=kb+l XXmn=(1-XXc)*Tmn/(1+Tmn+tsi) 193 XXsi=(1-XXc)*Tsi/(1+Tmn+Tsi) call call call tqsetc('X',-1,ic,XXc,num,iwsg) tqsetc('X ,-1,imn,XXmn,num,iwsg) tqsetc( X ,-1,isi,XXsi,num,iwsg) call call t q c e ( ' ',0,0,0.ODO,iwsg) tqgetl('GM',i_bcc,-1,gm_dummy,iwsg) 1 1 1 gm_bcc(kb,2)=gm_dummy gm_bcc(kb,1)=xxc Enddo kf = 0 c a l l t q c s p ( i _ f c c , 'ENTERED ,iwsg) c a l l tqcsp(i_bcc,'SUSPENDED',iwsg) 1 do XXc=0,1.5*X_gamma,X_gamma/2 00. kf=kf+1 XXmn=(1-XXc)*Tmn/(1+Tmn+tsi) XXsi=(1-XXc)*Tsi/(1+Tmn+Tsi) call call call tqsetc('X',-l,ic,XXc,num,iwsg) tqsetc( X ,-1,imn,XXmn,num,iwsg) tqsetc('X',-1,isi,XXsi,num,iwsg) call call tqce( ,0,0,0.ODO,iwsg) tqgetl( GM',i_fcc,-1,gm_dummy,iwsg) 1 1 1 1 1 gm_fcc(kf,2)=gm_dummy gm_fcc(kf,1)=xxc Enddo i i n=kb F i t t i n g t h e BCC d a t a t o a s p l i n e f u n c t i o n f S P L I N E s e e : N u m e r i c a l r e c i p e i n FORTRAN o r do i=l,n xx(i)=gm_bcc(i,1) yy(i)=gm_bcc(i,2) enddo ypl=(yy(2)-yy(1))/(xx(2)-xx(l)) ypn=(yy(n)-yy(n-1))/(xx(n)-xx(n-1)) call spline(xx,yy,n,ypl,ypn,yy2) jb=0 194 do x=xx (1) , x x (n) , (xx (n) - x x ( 1 ) ) /10 . / n jb=jb+l c a l l splint(xx,yy,yy2,n,x,y,yla) sgm(jb,1)=x s g m ( j b , 2) =y sgm(jb,3)=yla enddo ! Fitting t h e FCC d a t a to a spline function n=kf do i=l,n xx(i)=gm_fcc(i,1) yy(i)=gm_fcc(i,2) enddo ypl=(yy(2)-yy(1))/(xx(2)-xx(l)) ypn=(yy(n)-yy(n-1))/(xx(n)-xx(n-1)) call spline(xx,yy,n,ypl,ypn,yy2) jf =0 do x=xx (1) , x x (n) , (xx (n) -xx (1) ) /10 . / n jf=jf+l c a l l splint(xx,yy,yy2,n,x,y,yla) sgm(j f , 4 ) = x s g m ( j f , 5) =y sgm(j f , 6 ) = y l a enddo !-Finding I ferrite and a u s t e n i t e carbon contents i n p a r a e q u i l i b r i u m u s i n g common t a n g e n t l i n e m e t h o d check=0 do i=l,jb do j = l , j f dtl=sgm(i,3)-sgm(j,6) dt2=sgm(j,5)-sgm(i,2)-sgm(i,3)*(sgm(j,4)-sgm(i,1)) if(dtl.It.Id-5.and.dt2.It.Id-5)then Xp_alpha=sgm(i,1) Xp_gamma= s g m ( j , 4 ) 195 check=l exit endif enddo if(check.eq.1)exit enddo return end APPENDIX ( H I ) Code for N P - L E Calculation F O R T R A N subroutine to determine the N P L E carbon contents o f ferrite and austenite at given temperature for a quaternary F e - C - M n - S i steel. subroutine nple(t,xc,xmn,xsi,xn_alpha,xn_gamma xn) / i m p l i c i t d o u b l e p r e c i s i o n (a-h,o-z) common /tc/iwsg,i_fcc,i_bcc,ic,imn,isi,ife dimension iwsg(80000),xn(3,3) d o u b l e p r e c i s i o n NN PP=101325.dO NN=l.dO call call tgcsp(i_fcc,'ENTERED',iwsg) tqcsp(i_bcc,'ENTERED',iwsg) do 1=1,6 call enddo tqremc(i,iwsg) call call call call call call tqsetc('P',-1,-1,PP,num,iwsg) tqsetc('N',-1,-1,NN,num,iwsg) tqsetc('T',-l,-l,t,num,iwsg) tqsetc('X ,-1,ic,Xc,num4,iwsg) tqsetc('X',i_bcc,imn,Xmn,num5,iwsg) tqsetc('X',i_bcc,isi,Xsi,num6,iwsg) call tqce(' 1 ',0,0,0.0D0,iwsg) c a l l tqgetl('ac',-1,ic,AC_C,iwsg) !XN( call tqgetl('X',i_bcc,ic,xn_alpha,iwsg) call tqgetl('X',i_bcc,imn,xn(2,1),iwsg) call tqgetl('X',i_bcc,isi,xn(3,1),iwsg) !bcc !NPL call tqgetl('X',i_fcc,ic,xn(1,3),iwsg) c a l l tqgetl('X',i_fcc,imn,xn(2,3),iwsg) call tqgetl('X',i_fcc,isi,xn(3,3),iwsg) xn(1,1)=xn_alpha call call tqcsp(i_fcc,'ENTERED',iwsg) tqcsp(i_bcc,'SUSPENDED',iwsg) call call call tqsetc('ac',-l,ic,AC_C,num7,iwsg) tqsetc('X',-l,imn,Xmn,num,iwsg) tqsetc('X',-1,isi,Xsi,num,iwsg) call call call tqremc(num4,iwsg) tqremc(num5,iwsg) tqremc(num6,iwsg) call tqce(' call call tqgetl('X',-l,ic,xn(l,2),iwsg) tqgetl('X',i_fcc,imn,xn(2,2),iwsg) x x X ',0,0,0.0D0,iwsg) 197 !fcc ! x ! X ! X X X !) call call tqgetl('X',i_fcc,isi,xn(3,2),iwsg) tqremc(num7,iwsg) XN_gamma=xn(1,2) return end 198 APPENDIX ( I V ) Discrete Form of Fick's Equations The discrete form o f the F i c k ' s equations for a moving boundary configuration is outlined in this Appendix. Considering the node configuration depicted below for both planar and spherical geometries, the node size is given by: Az = £„-£ (A- 14) a ..L...1...1. 0 12 3 4 Then the following differential equation is employed to calculate the carbon profile in the remaining austenite: ( K is 0 for planar and 2 for spherical geometry) D J At ; 2Az c/ +1 "T Is. • V r• 2Az 2Az c J -c J 2Az (A- 15) - 2c/ + c Az 2 The equation in the discrete form can be expressed as: cr-c{ {cj -cU = ){n-i)d£ x At 2{£ -£) dt o + K DfcL- cUXn-1) 2 2{n-i){£ -£) 2 0 + (A-16) 4(£ -£) 2 o + D. (4 -2c/+< )(«-l) 1 1 where: 199 2 dl _ £>, (-3c/ + Acj - cj )(n - 1 ) 200 A P P E N D E X ( V ) Average Carbon Content of a Spherical Austenite The numerical approach to calculate the average carbon content o f the remaining austenite is depicted here. To fulfill the carbon mass balance in the system, the following equation has to be verified at each time step: c =Q-f )c"+f c (A-17) r 0 r r In case o f carbon leakage or built up, then the fraction o f the remaining austenite, containing the average carbon concentration o f c , must be corrected via changing the location of the T interface. Therefore, at each time step the assessment o f c r for a given carbon profile is crucial. For spherical geometry, a linear carbon profile between two adjacent nodes is first assumed: c T = C / _£tZ£LtL i Z ( z < _ z ) ( A . 1 8 ) M - Z the carbon content, Q , o f the austenite shell encompassed by these two nodes is given by: t z z 2 Qt = J4nz c dz 2 y z z, r= i Z = J4n z (T z + K)dz (A-19) 2 z, ' ~ C 2 C m and A = c, -°''~° z. M ~ M i Z Z ~ (A-20) M Z The average carbon o f austenite then is the sum o f Q over the volume o f austenite sphere, V : 7 t n-1 (A-21) which in the discrete format can be presented by this rather short solution, which is independent to the size o f the remaining austenite and node size Az : 201 0.25 ^ + (3c, + c , ) ( « - 0 /+ 202 3 - (c, + 3 c ) ( » - i - 1 ) /+1 (A- 22) 3
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Modeling the austenite decomposition into ferrite and...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Modeling the austenite decomposition into ferrite and bainite Fazeli, Fateh 2005
pdf
Page Metadata
Item Metadata
Title | Modeling the austenite decomposition into ferrite and bainite |
Creator |
Fazeli, Fateh |
Date Issued | 2005 |
Description | Novel advanced high-strength steels such as dual-phase (DP) and transformation induced plasticity (TRIP) steels, are considered as promising materials for new generation of lightweight vehicles. The superior mechanical properties of these steels, compared to classical high strength steels, are associated with their complex micro structures. The desired phase configuration and morphology can only be achieved through well-controlled processing paths with rather tight processing windows. To implement such challenging processing stages into the current industrial facilities a significant amount of development efforts, in terms of mill trials, have to be performed. Alternatively, process models as predictive tools can be employed to aid the process development' and also to design new steel grades. Knowledge-based process models are developed by virtue of the underlying physical phenomena occurring during the industrial processing and are validated with experimental data. The goal of the present work is to develop an integrated microstructure model to adequately describe the kinetics of austenite decomposition into polygonal ferrite and bainite, such that for complex thermal paths simulating those of industrial practice, the final microstructure in advanced high strength steels can reasonably be predicted. This is in particular relevant to hot-rolled DP and TRIP steels, where the intercritical ferrite evolution due to its crucial influence on the onset and kinetics of the subsequent bainite formation, has to be quantified precisely. The calculated fraction, size and spatial carbon distribution of the intercritical austenite are employed as input to characterize adequately the kinetic of the bainite reaction Pertinent to ferrite formation, a phenomenological, physically-based model was developed on the ground of the mixed-mode approach. The model deals with the growth stage since nucleation site saturation at prior austenite grain boundaries is likely to be attained during the industrial treatments. The thermodynamic boundary conditions for the kinetic model were assessed with respect to paraequilibrium. The potential interaction between the alloying atoms and the moving ferrite-austenite interface, referred to as solute drag effect, was accounted for rigorously in the model. To quantify the solute drag pressure the Purdy- Brechet approach was modified prior to its implementation into the model. The integrated model employs three main parameters, the intrinsic mobility o f the ferrite-austenite interface, the binding energy of the segregating solute to the interface and its diffusivity across the transformation interface. These parameters are clearly defined in terms of their physical meaning and the potential ranges of their values are well known. However, no direct characterization techniques are currently available to precisely measure them hence they are treated as adjustable parameters in the model. The model predicts successfully the overall kinetics of ferrite formation in a number of advanced steels. The bainite evolution in different TRIP steels was analyzed using three available approaches, i.e. Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation, the Zener-Hillert formulation for diffusional growth and the displacive definition proposed by Bhadeshia. Overall, it turned out that the predictive capability of the three methodologies is similar. Further, some of the ensuing model parameters pertinent to each approach are difficult to interpret in terms of the underlying physics, which implies that all three models are employed in a semi-empirical manner. Assuming diffusional transformation mechanism for bainite, the isothermal incubation time and the onset of bainite formation during continuous cooling treatments were described adequately. Consistently, for the purpose of process modeling, the diffusional description of bainite growth can potentially be employed. However, from an academic point of interest a more precise quantification for the nucleation part is still missing. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2009-12-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0078776 |
URI | http://hdl.handle.net/2429/17350 |
Degree |
Doctor of Philosophy - PhD |
Program |
Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2005-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_2005-994724.pdf [ 12.48MB ]
- Metadata
- JSON: 831-1.0078776.json
- JSON-LD: 831-1.0078776-ld.json
- RDF/XML (Pretty): 831-1.0078776-rdf.xml
- RDF/JSON: 831-1.0078776-rdf.json
- Turtle: 831-1.0078776-turtle.txt
- N-Triples: 831-1.0078776-rdf-ntriples.txt
- Original Record: 831-1.0078776-source.json
- Full Text
- 831-1.0078776-fulltext.txt
- Citation
- 831-1.0078776.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0078776/manifest