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Kinetics of the pearlite to austenite reversion transformation Riehm, Derek J. 1990

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K I N E T I C S O F T H E P E A R L I T E T O A U S T E N I T E R E V E R S I O N T R A N S F O R M A T I O N By Derek J. Riehm B. Sc. Metallurgical Engineering, Queen's University, 1983 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S M E T A L S A N D M A T E R I A L S E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1990 © Derek J. Riehm , 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Metals and Materials Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, British Columbia Canada V6T 1W5 Date: 2 Z S ^ W l ^ r Abstract The pearlite-to-austenite reversion transformation kinetics under isothermal and contin-uous heating conditions in a eutectoid plain-carbon steel have been measured, using a dilatometric technique on a Gleeble 1500 Thermomechanical Simulator. The isothermal data was characterized in terms of the transformation start time at temperature for the onset of the P—• 7 transformation, and in terms of the Avrami parameters n and b. Under the assumption that the P—• 7 transformation was additive, the Scheil equation was applied to the measured isothermal transformation start data to predict the onset of the transformation on continuous heating, and the isothermal phase transformation kinetics were used to predict the continuous heating kinetics. It was found that the kinetic model significantly underpredicted the transformation start time during continuous heating. This was attributed to the large experimental error inherent in the estimation of the isothermal transformation start time, t0. The model's continuous heating kinetic predictions were excellent at low heating rates, but it tended to overpredict the kinetics at higher heating rates. The problem was traced to an observed difference between the measured temperature and the programmed tem-perature during the high heating rate tests. When the model was modified to incorporate the actual temperature profile, its prediction of the kinetics was considerably improved. Thus the austenite reversion transformation was concluded to be experimentally additive. A n average Avrami n value of 2.2 suggested that austenite was nucleating on pearlite colony corners and edges. This conclusion was verified with optical and scanning electron microscopy. 11 Previously published data, which indicated that the pearlite-to-austenite transforma-tion is isokinetic, was found to be based on questionable assumptions. Metallographic information suggests, however, that the nucleation sites are saturated early in the re-action. Furthermore, the isothermal austenite formation data generated in this work was found to meet the effective site saturation criterion for additivity, implying that the austenitization process would be expected to be additive. The effect of starting microstructure was evaluated by performing isothermal and continuous heating tests on two different pearlitic microstructures. It was found that, in agreement with published results, the transformation rate varied in inverse proportion with the pearlite spacing and colony size. in Table of Contents Abs t rac t i i L is t of Tables v i i L is t of Figures ix Lis t of Symbols x i i Acknowledgement x v Dedica t ion x v i 1 In t roduct ion 1 1.1 Isothermal Transformation Kinetics 2 1.2 Continuous Heating and Cooling Transformations 9 1.3 Kinetics of Austenitization 14 1.4 Microstructural Aspects of Austenitization 16 1.4.1 Nucleation of Austenite 17 1.4.2 Growth of Austenite 20 1.4.3 Equilibration 23 2 Scope and Objectives 25 3 Expe r imen ta l Procedure 27 3.1 Characterization of the Kinetics 27 iv 3.1.1 Test Apparatus 27 3.1.2 Sample Preparation 28 3.1.3 Isothermal Tests 30 3.1.4 Continuous Heating Tests 31 3.2 Assessment of the Additivity Principle 31 3.3 Characterization of Eutectoid Steel #634044 34 3.3.1 Characterization of Pearlitic Microstructures 34 3.3.2 Austenite Grain Size 36 3.3.3 Calculation of A i and A 3 Temperatures 37 4 Results and Discussion 39 4.1 Data Analysis 39 4.1.1 Isothermal Data 39 4.1.2 Continuous Heating Data 43 4.2 Prehminary Testwork 44 4.3 Modifications to Test Equipment 49 4.4 Microstructural Characterization 53 4.4.1 Pearlite Spacing and Colony Size 53 4.4.2 Austenite Grain Size 57 4.4.3 A i and A 3 Temperatures 59 4.5 Isothermal Test Results 60 4.5.1 Effect of Starting Microstructure on Isothermal Kinetics 71 4.5.2 Apparent Activation Energy 74 4.5.3 Partially Transformed Samples 76 4.6 Continuous Heating Test Results 82 v 4.7 Austenitization as an Additive Process 83 4.7.1 Prediction of Tstart 83 4.7.2 Prediction of Continuous Heating Kinetics 86 4.7.3 The Applicability of the Additivity Principle 95 5 Conclusions 103 6 Recommendations 105 References 107 A Statistical Techniques 110 A.l Linear Modelling Il l A.2 Nonlinear Modelling 112 B Source Code: Kinetics Model 115 vi List of Tables 3.1 Composition of eutectoid steel, Stelco heat #634044 (wt. %) 27 4.2 Isothermal test results obtained during preliminary testwork 48 4.3 Microstructural characterization of eutectoid steel #634044 54 4.4 7 —>P transformation range during continuous cooling of QC samples at 5°C/s showing the measured recalescence 56 4.5 Measured prior austenite grain sizes for the F C and QC austenitization treatments 59 4.6 A i and A 3 temperature calculations for eutectoid plain-carbon steel (heat #634044) 60 4.7 Summary of isothermal kinetics, sample groups C and D 64 4.8 Estimated values of the transformation start time, to, f ° r the isothermal pearlite-to-austenite transformation 66 4.9 Effect of small variations in Do on the transformation start time, as calcu-lated by the linearization method, for an isothermal test performed at 736°C. 66 4.10 Comparison of measured isothermal kinetics with calculated kinetics based on isothermal austenitization diagrams proposed by Roosz et al. [20]. . . 73 4.11 Apparent activation energies calculated from isothermal data for the F C and QC microstructures, and from data presented by Roosz et al. [20]. . 74 4.12 Summary of continuous heating test kinetics, sample groups C and D. . . 82 4.13 Summary of Tstart predictions made by the austenite reversion kinetics model on the basis of the application of the Scheil Equation to the isother-mal austenitization data 84 4.14 Measured slopes of pearlite and austenite portions of continuous heating dilatometer traces 92 vii 4.15 Application of Kuban's effective site saturation criterion [13] to the isother-mal P—> 7 transformation V l l l List of Figures 1.1 Theoretical transformation curves, fraction transformed versus time. . . . 4 1.2 Illustration of the use of the additivity principle for the prediction of con-tinuous cooling phase transformation kinetics, showing a series of isother-mal transformation curves and their relationship to a stepped transforma-tion event 11 1.3 Illustration of the approximation of continuous cooling kinetics as a sum of a series of short duration isothermal increments 12 1.4 Potential austenite nucleation sites in a pearlitic microstructure: inter-lamellar (site A), colony boundary (site B), and colony edge (site C). . . 18 1.5 Schematic diagram of growth of austenite into a pearlitic microstructure. 21 3.6 Schematic diagram of heating and temperature/dilation measurement on the Gleeble 1500 29 3.7 Flowchart for austenite reversion kinetics model 33 3.8 Printout of spreadsheet showing calculation of Ae3 temperature for eutec-toid steel #634044, following the method of Kirkaldy and Baganis [43]. . 38 4.9 Measured temperature and dilatometer signal traces for an isothermal test conducted at 738°C 40 4.10 Temperature and dilatometer traces for an isothermal test performed at 735°C, showing a spike in the dilatometer signal 42 4.11 Measured temperature and dilatometer signal traces for a continuous heat-ing test conducted at l°C/s 45 4.12 Preliminary testwork: dilatometer traces for several isothermal tests con-ducted at temperatures between 727 and 729°C 46 4.13 Variation of In (6*) with temperature, for n = 0.84 50 ix 4.14 Comparison of measured and predicted continuous heating transformation kinetics for n = 0.84 51 4.15 Typical scanning electron micrograph used in pearlite spacing measure-ment work. F C microstructure, mag. 2000 x 55 4.16 Typical optical photomicrograph used for prior austenite grain size mea-surement (FC microstructure). Mag. 400x 58 4.17 Isothermal transformation curves, QC microstructure 61 4.18 Isothermal transformation curves, F C microstructure 62 4.19 Variation of ln(6*) with temperature, for n = 2.19 63 4.20 Comparison of linear and nonlinear modelling techniques for the isother-mal pearlite to austenite transformation at 740°C 65 4.21 Dilatometer trace for isothermal test conducted at 736°C, illustrating range of DQ values considered for transformation start time analysis. . . 68 4.22 Hypothetical dilatometer curve for a double transformation event 70 4.23 T T T diagram for austenitization in eutectoid plain-carbon steel 72 4.24 Apparent activation energy plots for the F C and QC microstructures. . . 75 4.25 Temperature and dilatometer traces for interrupted isothermal test con-ducted at 727° C 77 4.26 Optical photomicrograph of an "interrupted and quenched" isothermal sample, showing martensite (white) and fine pearlite/bainite (dark) islands in untransformed coarse pearlite (light gray). Mag. 600x 79 4.27 Scanning electron micrograph of an "interrupted and quenched" isother-mal sample. Mag. 2000 x 80 4.28 Scanning electron micrograph showing austenite nodule growing from the region of four pearlite colonies, indicating nucleation from colony corners or edges. Mag. 5000x 81 4.29 Isothermal transformation start times for the F C and QC microstructures. 85 4.30 Measured and predicted continuous heating kinetics, QC microstructure. 88 x 4.31 Measured and predicted continuous heating kinetics, FC microstructure. 89 4.32 Dilatometer trace for continuous heating test performed with FC mi-crostructure at 0.5°C/s 91 4.33 Temperature and dilatometer traces for a QC microstructure test per-formed at 5°C/s, showing temperature depression during the P—» 7 trans-formation 93 4.34 Improved kinetics prediction resulting from modified temperature-time relationship, for a continuous heating test performed at 5°C/s 94 4.35 Comparison of austenitization kinetics predictions made with and without the Scheil Equation, for a continuous heating test performed at 1.0°C/s. 96 4.36 Nucleation and growth rates calculated from data presented by Roosz et al., showing proportionality between N and G 99 xi List of Symbols Transformation Theory V Total volume of transforming material V^, Va Volumes of 3 and a phases, respectively k Rate constant t Transformation time T Transformation temperature X Volume fraction transformed X Volume fraction transformation rate T Incubation time I Nucleation rate per unit volume of material Y, G Growth rate of newly formed phase vT Volume of new phase which nucleated at time r Vg Extended volume of 3 phase N Number of nucleation sites available at time t N Nucleation rate iVo Number of nucleation sites available at time t = 0 v Frequency at which individual nucleation sites become nuclei n Avrami equation time exponent navg Avrami equation time exponent, average value b Avrami equation rate constant b* Avrami equation rate constant, back-calculated for n — navg xii S Grain boundary surface per unit volume L Grain edge length per unit volume C Number of grain corners per unit volume n' Cahn modified Avrami time exponent d Grain diameter m Grain size exponent, Umemoto modified Avrami equation t0 Incubation time XF Fraction ferrite formed ta (T) Isothermal time to reach specific fraction transformed H (T) General function of temperature G (X) General function of fraction transformed Quantitative Metallography F Pearlite structure factor GT Mean true interlamellar pearlite spacing dr Mean random pearlite spacing Sy^P Specific interface of pearlite colonies ap Average edge length of pearlite colonies approximated as space filling truncated octahedrons NL Number of intersections of pearlite lamellae or colony boundaries per unit length of circumfrence of inscribed circle M Magnification dc Circle diameter xiii Data Analysis Tstart Transformation start temperature to, tstart Transformation start time D m i m Dmax Minimum and maximum dilatometer signals reached during a test Do Starting dilatometer signal Df Dilatometer signal after transformation AD trans Change in dilatometer signal due to P—> 7 transformation io .02^0.98 Time for 2%-98% transformation during continuous heating Statistical Methods k Number of operating variables m Number of data points y Response variable r?, E (y) Predicted response value / Response approximation function {£} Vector of values of operating variables {9} Vector of model parameters {#} Vector of estimated model parameters r (1) Residual (2) Correlation coefficient S Sum of squares of residuals {x} Vector of operating variable parameters {Bf Vector of linear model parameters | / ? | Vector of estimated linear model parameters xiv Acknowledgement I would like to thank professor E .B . Hawbolt, who supervised the work described herein, and professor J .K. Brimacombe, for their advice and guidance during the course of this work. The assistance of Binh Chau, who operated the Gleeble 1500, is also appreciated. Bernardo Hernandez-Morales and Willie Fajber provided considerable assistance with word processing software and C programming, respectively. And Sanjay Chandra and Chris Parfeniuk were always available for a discussion over a,cup of coffee, which made a cramped office more livable. I am indebted to my parents for the guidance and support they have provided over the last 18 months. Lastly, I'd like to thank Suzanne Perry for her love, support, prodding, and occasional kicks in the rear. xv Dedication This thesis is dedicated to the memory of Mark Allan Latham, a young man of courage, optimism, and vision. May he continue to be a source of inspiration to the Queen's University Class of '83. xvi Chapter 1 Introduction As competing materials vie for new markets, steel producers are seeking to make steel products meet more demanding mechanical property specifications. The properties of a steel are heavily dependent on its microstructure, which is shaped by the conditions under which the steel was processed through such operations as casting, hot and cold rolling, and annealing. Thus, in order to be able to control and predict the properties of the final product, it is necessary to control many important processing variables, and to characterize their influence on microstructure. The term microstructural engineering has been coined for the procedure of applying the basic principles of heat and mass transfer, fluid flow, and physical metallurgy to the modelling of process operations. The resulting model is capable of predicting the influence of a wide range of process variables on microstructure and, therefore, physical properties. These ideas have been successfully applied in the Centre for Metallurgical Process Engineering (CMPE) at U .B .C . , most recently by Campbell [1] and Devadas [2]. A n important part of microstructural engineering is the characterization and predic-tion of phase transformation kinetics, which affect material properties through their influ-ence on microstructural parameters such as grain size and pearlite spacing. The kinetics of ferrous phase transformations are to a considerable degree controlled by internal heat transfer during processing, and the internal heat transfer is to no small degree influenced by the transformation heat associated with endothermic and exothermic phase changes. Thus, in order to predict microstructural evolution, phase transformation kinetics and 1 Chapter 1. Introduction 2 heat flow modelling must be considered together. Despite the interaction between phase transformations and heat flow, the two can be decoupled under controlled laboratory conditions, allowing the phase transformation kinetics to be characterized independently. The mathematical relationships so derived for a given grade of steel can then be used in any thermomechanical model seeking to predict microstructural evolution in the same steel. A major aim of this project has been to provide these relationships for austenitization in a eutectoid, plain-carbon steel. 1.1 Isothermal Transformation Kinetics The isothermal kinetics of nucleation and growth processes were considered first by John-son and Mehl [3] and Avrami [4,5,6]. Their fundamental research provides the basis of much of the phase transformation work which has been carried out over the past sev-eral decades, and serves as one of the foundations of the technique of microstructural engineering. The simplest case to consider is a homogeneous transformation, in which every small portion of untransformed material has the same probabihty of transforming in a given time. In this case, the rate of transformation will be proportional to the volume of material as yet untransformed, that is, for the transformation a — » (3, d-£ = Hv-v») ( i . i ) where V is the total volume of the material, V@ is the volume of the newly formed /3 phase, k is a rate constant, and t is the transformation time. Equation 1 . 1 can be rearranged and integrated to give (V -VP\ In — — ) = - * * (1.2) Chapter 1. Introduction 3 which can be rearranged into the more familiar form X = — = 1 -exp (-fa) (1.3) where X is the volume fraction transformed. A graph of X versus t shows the well known 1st order rate curve, an example of which is shown in Figure 1.1. The situation becomes considerably more complex when a nucleation and growth process is considered. In this case, an individual transformed region is formed after an incubation time r , after which it grows continuously. If the nucleation rate is / per unit volume of untransformed material, then the number of new nuclei formed in a time dr is IVadT. Assuming that the growth rate Y is isotropic, and that a spherical precipitate is formed, the volume of 8 at time t which originated at time r is vr = l*R3 = \*Y3 (t - r) 3 (1.4) During the early stages of the transformation, it may be assumed that V3 <C Va (that is, Va V); thus the 8 nuclei are widely spaced and do not impinge on one another. At time t, then, the volume transformed which originated from nuclei formed between time r and time r + dr is dV? = vTIVadr (1.5) Approximating Va ~ V, Equation 1.5 can be integrated to give the total volume trans-formed at time i , rVl3 A H = dV? = ^TCV / IY3 (t - r) 3 dr (1.6) Jo 3 Jo The integration may be carried out by making certain assumptions about the nucleation rate J . If, for instance, I is assumed to be constant, then V? = UIY3V I* It - r) 3 dr (1.7) 3 Jo Chapter 1. Introduction 4 . 0 CD GO C o c o o D 0.5 0.0 s t •1 Order -Av ram i , n = 4 T ime F i g u r e 1.1: T h e o r e t i c a l t r a n s f o r m a t i o n cu rves , f r a c t i o n t r ans fo rmed versus t ime . Chapter 1. Introduction 5 which can be integrated and rearranged to give VP 7T X = Y = lIYH4 ( L 8 ) This equation predicts a rapidly rising rate of transformation in the early stages, before the growing regions of the newly formed B phase begin to impinge on one another. Clearly, however, the assumption of no impingement becomes increasingly less valid as the transformation proceeds. To deal with the problem of impingement, Johnson and Mehl [3] introduced what Avrami [4,5,6] later called an extended volume. The extended volume may be thought of as the volume transformed assuming nucleation is permitted in both the untransformed and the transformed regions. The extended volume resulting from nucleation events during time dr is therefore dVi = vTI (Va + Vfi) dr (1.9) which can be integrated to give a total extended volume of Vi = \T:V T IY3 (t - r ) 3 dr (1.10) 3 Jo Although Equation 1.10 is identical in appearance to Equation 1.6, it is important to realize that the restriction of V ~ Va has now been removed, since V = Va + V 3 ' . Therefore, Equation 1.10 should be valid not only in the early stages of a transformation, but also during the later stages. Of course, the extended volume is purely a theoretical construction arising out of kinetics considerations, and must therefore be related to the actual volume transformed, V@. The two volumes can be related to one another by understanding that as the trans-formation proceeds, on average, a fraction of the new volume formed will be in pre-viously untransformed regions, while the remainder will be in already transformed re-gions. Assuming that a new volume element dV^ is randomly located, this fraction is Chapter 1. Introduction 6 Va = (l - y-j. That is, dV* = dv£ ll - (1.11) This equation can be integrated to give r / V / 3 \ l (1.12) vi = -v The right side of Equation 1.12 can be substituted for the left side of Equation 1.10, giving In ( l - = In (1-X) = J* IY3 (t - r ) 3 dr (1.13) where X is the volume fraction transformed. Equation 1.13 can be integrated by making certain assumptions about J . The simplest assumption to make is that / is constant; this yields, after integrating and rearranging, X = 1 - exp ( - | / F 3 / ) (1.14) which is known as the Johnson-Mehl equation. There is, of course, little reason to believe that the nucleation rate J will remain constant throughout a transformation. Avrami [4,5,6] proposed instead that nucleation would occur only at certain sites, and that these sites would slowly become exhausted as the transformation proceeded. Assuming that there were A^ o sites to begin with, and assuming that v is the frequency at which individual sites become nuclei, he postulated that dN I=-^L = Nv (1.15) dt where N is the number of sites at any instant. This equation can be rearranged into the form -jf = -vdt (1.16) Chapter 1. Introduction 7 and integrated to yield N (t) = N0v exp (vt) (1.17) When Equation 1.17 is differentiated with respect to time, the result is 1= — = N0v exp (-vt) (1.18) Substituting this for I into Equation 1.13 and then integrating by parts yields X = 1 — exp 8KY3NO f , , „ vH2 v3t3' 1 exp (-vt) - 1 + vt + (1.19) v3 { r v ' 2 6 There are two limiting forms of this equation. The first corresponds to a very small value of vt, that is, to a constant nucleation rate. In this case Equation 1.19 reduces to a form comparable to that of the Johnson-Mehl equation. The second limiting form corresponds to a very large value of vt; in this case N —> 0, the nucleation sites are used up very quickly, and Equation 1.19 reduces to 4 -TTNOYH3 3 (1.20) X — 1 — exp Avrami went on to propose a general kinetic law, X = l- exp(-btn) (1.21) which is expected to hold for a variety of nucleation and growth processes. Equation 1.21 is generally known as the Avrami equation. Broadly speaking, b is a rate constant reflect-ing the nucleation and/or growth rate, while n is dependent on the nucleation conditions, the dimensionality of the transformation (ID, 2D, or 3D), and the time exponent of the rate controlling process. Christian [7] detailed this for interface controlled and diffusion controlled transformations. Cahn [8] discussed the dependence of n on the active nucleation site for the pearlite reaction. He differentiated between two conditions of very high and very low nucleation Chapter 1. Introduction 8 rates. In the first case, which he termed "site saturation" (see page 13), the nucleation rate is large enough that the nucleation sites are saturated early in the reaction. There-after, the nucleation rate disappears from the kinetic expression. For a site saturated transformation, he expressed the Avrami equation in three different ways, depending on the nature of the active growth site. For growth from grain boundary surfaces, for instance, he established that X = 1 - exp [-2SYt] (1.22a) where S is the grain boundary surface per unit volume. For growth from grain edges, X = 1 - exp [ - T T L Y 2 * 2 ] (1.22b) where L is the grain edge length per unit volume. For growth from grain corners, 47T X = 1 — exp -CYH3 3 (1.22c) where C is the number of grain corners per unit volume. Note that Equation 1.22c is identical to Equation 1.20. In the second case, where the nucleation rate is low enough that the nucleation sites are not used up early in the transformation, Cahn expressed the Avrami equation as X = 1 - exp [-fa 4 + n ' ] (1.23) where n' would be less than zero for a decreasing nucleation rate, zero for a constant nucleation rate, and positive for an increasing nucleation rate. For values of n greater than 1, the Avrami equation predicts a sigmoidally shaped fraction transformed versus time curve, an example of which is shown in Figure 1.1, for n = 4. Following an incubation period, the transformation starts slowly, as the second phase nucleates. The transformation rate X then increases, and eventually reaches a maximum. X then decreases due to grain impingement as the transformation approaches Chapter 1. Introduction 9 completion. Many solid state transformations are found to exhibit such behavior, and the Avrami equation is often used to characterize them. In order to consider the effect of grain size on transformation kinetics, Umemoto et al. [9] modified the Avrami equation as X — 1 — exp dm (1.24) where d is the starting grain size, k is a rate constant, and ra and n are empirical constants. They proposed that ra = 0 for random (homogeneous) nucleation, ra = 1 for nucleation on a grain surface, ra = 2 for nucleation on a grain edge, and ra = 3 for nucleation on grain corners. Umemoto et al. [10] later found that the Avrami equation was unable to characterize the kinetics of proeutectoid ferrite formation in a steel containing 0.2%C and in an Fe-0.43%C alloy. They therefore used the following equation to describe these kinetics: XF = 1 l + j-0.5k(T)(t-to(T)r -2 (1.25) where XF is the fraction of ferrite formed (relative to the final, or equilibrium, amount formed), k is the rate constant, to is the incubation time, d is the austenite grain size, and ra and n are empirical constants. 1.2 Continuous Heating and Cooling Transformations As noted by Christian [7], the treatment of non-isothermal transformations is complicated considerably by the fact that the rates of nucleation and growth usually vary indepen-dently with temperature. The problem can be simplified if the reaction is additive. The additivity principle was proposed originally by Scheil [11] to predict the incuba-tion time in a continuous cooling transformation. It has since been extended to describe Chapter 1. Introduction 10 nonisothermal transformations. As expressed by Cahn [12], a phase transformation is ad-ditive if the transformation rate can be expressed as a function only of the instantaneous temperature and the amount already transformed. That is, ^ = / (X,T) (1.26) It follows from this that a continuous heating or cooling transformation may be ap-proximated as a series of short isothermal segments, as illustrated in Figure 1.2. Essen-tially, the transformation is allowed to proceed isothermally, at Ti, for instance, for a short time interval, At. The temperature is then instantaneously changed, to T 2, and the transformation is continued again for a short time interval, and so on. At the start of each successive time step, a virtual time, tv, may be calculated. The virtual time is the time required to reach the current fraction transformed, X, if the entire transformation had been carried out isothermally at that temperature. The overall effect is that of a stepwise approximation to the continuous transformation, as shown in Figure 1.3. Cahn [12] and Christian [7] expressed the additivity principle generally as r* dt I-Jo ta (1.27) (T) in which the integral represents the summation of the fractional isothermal transforma-tion times, where ta (T) is the isothermal time to reach a specific fraction transformed. Christian further stated that the additivity condition is satisfied if ~df ~ G{X) ( ' where H(T) and G(X) are functions only of temperature, T, and fraction transformed, X, respectively. This has implications for the Avrami parameters n and b [13]. Solving Equation 1.21 for t yields t = (1.29) Chapter 1. Introduction 11 Figure 1.2: Illustration of the use of the additivity principle for the prediction of continu-ous cooling phase transformation kinetics, showing a series of isothermal transformation curves and their relationship to a stepped transformation event. Chapter 1. Introduction 12 Figure 1.3: Illustration of the approximation of continuous cooling kinetics as a sum of a series of short duration isothermal increments. Chapter 1. Introduction 13 The Avrami equation may be differentiated with respect to time, giving 1 dX ~dt (nbt"-1) exp (-btn) (1.30) Substituting Equation 1.29 into Equation 1.30 and rearranging gives dX nib) (1.31) dt n n-1 By examining Equation 1.31, it may be seen that Equation 1.28 is satisfied only if n is constant, and if b is a function only of temperature. Avrami [4,5,6] defined an isokinetic range, in which the nucleation rate is proportional to the growth rate, over which a reaction would be additive. This proportionality rarely holds, however [7]. Cahn [12] generalized the idea by defining an isokinetic reaction as one in which where h(T) is a single function of temperature. The consequence of Equation 1.32 is that a reaction in which the nucleation and growth rates vary independently is not, in general, expected to be additive. If, however, the nucleation sites are saturated early in the reaction, then the transformation subsequently depends only on the growth rate of pre-existing nuclei. Cahn termed this situation "site saturation", and stated that such a transformation would be additive. Kuban [13], in characterizing the 7 —»P transformation in a eutectoid, plain-carbon steel, noted that the reaction was experimentally additive even though it was not isoki-netic in the sense defined by Avrami, and though it did not meet Cahn's site saturation criterion. He therefore proposed a condition of "effective site saturation", which says xThe reader should note that Equation 1.30 presents a slightly different result of the differentiation of the Avrami equation than has been reported in the past[13]. Previous workers have calculated a negative value for dX/dt, which is clearly unrealistic. The correction changes the form of Equation 1.31 slightly, but does not affect the conclusion of the exercise. (1.32) Chapter 1. Introduction 14 that growth of pearlite nuclei formed early in the transformation event dominates the 7 —>P transformation, while nuclei formed later contribute very little. The effective site saturation criterion was originally expressed as [13] where £20 and tgo are the times to 20% and 90% transformation completion, respectively. In a later paper, Kuban et al. [14] proposed that for a transformation with a value of the Avrami time exponent of n — 2, the effective site saturation criterion would be expressed as which would ensure that nuclei formed before t2o would contribute at least 49% of the total volume transformed, at the time of 90% transformation completion. 1.3 Kinetics of Austenitization A number of workers [15]—[21] have measured isothermal austenitization kinetics. Typi-cally, their experimental technique involved immersing small test coupons into a molten salt, lead bath, or air furnace maintained at the desired temperature. Individual samples were removed from the bath at predetermined intervals and quenched to room temper-ature. Any portions of the sample which transformed to austenite in the bath then transformed to martensite. Metallographic determination of the fraction of martensite in each sample provided the austenite reversion kinetics, in the form of fraction transformed to 7 versus immersion time. The resulting kinetics, which exhibited a sigmoidally shaped fraction transformed versus time curve, have by some workers [17,20,21] been expressed in terms of the Avrami ^ > 0.38 (1.33a) ^ > 0.28 (1.33b) Chapter 1. Introduction 15 equation (1.21), which, when rearranged into the form ln ln 1 = ln b + n ln t (1.34) LI - X J is a straight line of slope n and intercept ln b. The Avrami parameters are thus estimated by performing a simple straight line least squares fit. Reported values of the time exponent, n, for austenite formation have varied between 0.65 and 4. For a hypereutectoid plain-carbon steel with a pearlitic microstructure, Speich and Szirmae [17] reported a value of n = 3, while Roosz et al. [20] reported a value of n = 4 for a pearlitic, eutectoid grade steel. Part of this variability may be attributed to the reproducibility of the experimental technique, but in fact it is the time base which is applied to the kinetics and the time taken for the sample to reach a uniform temperature through its thickness which is thought to account for most of the reported variability. In immersion tests, the transformation time is taken to be the immersion time. Yet salt immersion tests at U .B .C . [22] using test coupons instrumented with thermocouples have shown that a significant time (up to 6 seconds) is required for the surface of the coupon to reach the bath temperature. Additionally, the Avrami equation was derived to describe the transformation event (that is, the nucleation and growth of the 7 phase) but not the combination of incubation and transformation [23]. The use of the immersion time as the time base incorporates both the heating time and the incubation time with the transformation time, which can lead to significant errors, especially when the immersion time approaches that of the transformation event. Two workers [15,20] have discussed the effect of starting microstructure on the austen-ite formation kinetics in eutectoid plain carbon steel. Roberts and Mehl [15] concluded that the rates of nucleation and growth varied inversely with the pearlite spacing, and that the transformation rate was proportional to the a/Fe^C interfacial area. They further related the initial austenitic grain size to the rates of nucleation and growth, Chapter 1. Introduction 16 thereby connecting 7 grain size to pearlite spacing. They also examined spheroidized and martensitic starting microstructures, and found that in the martensitic steel, both the 7 nucleation and growth rates were considerably higher than in a pearlitic steel, while in the spheroidized steel, the nucleation rate was found to be very low, and the growth rate was similar to that observed in pearlite. Roosz et al. [20] introduced a so-called structure factor, which they calculated as F = ap(or)2, where dr is the mean true interlamellar pearlite spacing, and ap is the average pearlite colony edge length, calculated by approximating the pearlite colonies 2 367 as octahedrons, and equal to —pjp-, where Sy^P is the specific interface of the pearlite Sy colonies. The structure factor was used to make adjustments to an austenitization time read from a graph relating the fraction transformed to time at various temperatures. Re-garding the effect of microstructure on the rates of nucleation and growth, they concluded that N oc 4 (1.35a) and that G oc ^- (1.35b) OT Furthermore, they found that the initial 7 grain size was independent of the pearlite spacing in the starting microstructure. 1.4 Microstructural Aspects of Austenitization Microstructurally, austenitization is a complex and variable process, because of the great variety of possible alloys and starting microstructures. Nevertheless, it has been studied extensively over the past three decades, and so is fairly well understood mechanistically. The following discussion will center primarily upon plain-carbon steel. Broadly speaking, the process of austenitization can be divided into three stages: Chapter 1. Introduction 17 1. nucleation of austenite nodules at an interface between ferrite and iron carbide; 2. diffusion controlled growth of the newly formed 7 into the matrix, consuming ferrite and dissolving carbide; and 3. slow dissolution of any residual carbide, and elimination of carbon concentration gradients within the austenite. 1.4.1 Nucleation of Austenite Many workers have reported on the nucleation of austenite, from a variety of starting microstructures. Although the nucleation sites vary from steel to steel, it can generally be stated [17,19,24] that 7 nucleates at an interface between ferrite and iron carbide. Eutectoid and Hypereutectoid Steels In pearlitic steels, there is of course a large interfacial area between a and Fe3C. As Figure 1.4 illustrates, nucleation could conceivably occur either within a pearlite colony (site A), or at a colony boundary (site B), edge (site C), or corner (not shown). Because of the orientation relationship which exists between adjacent plates of a and FesC within a pearlite colony [25], this is a relatively low energy interface, whereas the boundary between two or more colonies has a relatively high surface energy. Generally, then, it is boundary sites which have been confirmed as being the effective nucleation sites [24]. Roosz et al. [20] examined the question of nucleation from pearlite when analyzing austen-itization kinetics tests performed with a eutectoid grade steel. They concluded that nu-cleation occurred at a/Fe3C interfaces on pearlite colony edges, and furthermore that the nucleation sites were not used up during the transformation. Speich and Szirmae [17] observed that nucleation occurred at colony boundaries, not necessarily colony edges. is Chapter 1. Introduction 19 In annealed samples, where the microstructure consists of spheroidized cementite in a ferrite matrix, the active nucleation site is usually at a cementite particle located at an a I a. grain boundary. Judd and Paxton [18] used a strain anneal technique to artificially enlarge the ferrite grains, and found that the nucleation rate at an Fe 3 C particle located within a single ferrite grain was from three to eight times less than the nucleation rate at particles residing on ferrite grain boundaries. Speich and Szirmae [17] also observed the predominance of ferrite grain boundary nucleation. Hypoeutectoid Steels The nucleation question is more complicated in hypoeutectoid steels, where the mi-crostructure can be a mixture of pearlite and grain boundary ferrite. Garcia and Dear-do [19] performed austenitization tests with four steels, all containing approximately 1.5% Mn , and varying in carbon content from 0.01% to 0.22%. They found that in these primarily ferritic samples, austenite nucleated at Fe 3 C particles located on ferrite grain boundaries. This was observed in both recrystallized and 70% cold worked samples. In samples containing a mixture of ferrite and pearlite, they observed that 7 nucleation took place on Fe 3 C particles which were located either on pearlite colony boundaries, or on boundaries between pearlite colonies and ferrite grains. Yang et al. [26] found that 7 nucleated first on boundaries between deformed and unrecrystallized ferrite grains, and then on spheroidized cementite particles in recrystal-lized ferrite. They also reported [27] that in a normalized ferritic-pearlitic microstructure, austenite nuclei generally form first at coarse carbide precipitates located at ferrite grain boundaries, but that austenite can also nucleate at carbide particles within a colony of spheroidized pearlite. Navara et al. [28] presented some contradictory observations in 1986, when they re-ported on work concerned with the intercritical austenitization of manganese partitioning Chapter 1. Introduction 20 dual phase steels. They observed that in conjunction with Mn diffusion induced grain boundary migration, manganese rich areas were formed along ctja. grain boundaries. Austenite nucleation was observed in these Mn rich areas, not necessarily at a grain boundary FeaC particle. They concluded that austenite nucleation in dual phase steels may depend more upon the Mn concentration along ferrite grain boundaries than upon the presence of grain boundary cementite particles, and that C diffusion along ferrite grain boundaries could provide the necessary carbon for the formation of austenite. The 7 nucleation discussion can be summarized briefly by saying that recent research has created a certain amount of controversy regarding nucleation along ferrite grain boundaries in hypoeutectoid steels. In pearlitic microstructures, however, it is well es-tablished that the active 7 nucleation site is an a/Fe^C interface located at a pearlite colony boundary, where excess surface energy exists to promote heterogeneous nucleation. 1.4.2 Growth of Austenite The growth of the newly formed grains of austenite is, like nucleation, dependent upon both microstructure and alloy content. At temperatures below 910°C, austenite forms and grows only if carbon is available. In general, austenite growth is understood to be rate controlled by the diffusion of carbon from FesC through the austenitic phase to the advancing front of the 7 grain [24] (see Figure 1.5). Some workers have proposed different mechanisms. Navara and Harrysson [29], for instance, concluded that during the intercritical annealing of a low C steel containing 1.5% Mn, the rate controlling step of austenite formation was the diffusion of Mn in ferrite, thus demonstrating the potential importance to austenite formation of the par-titioning of alloying elements between ferrite and cementite. Speich and Miller [30], working with a similar steel, made a similar conclusion when in order to make their ex-perimental results fit their calculations, they assumed that growth was controlled by Mn Figure 1.5: Schematic diagram of growth of austenite into a pearlitic microstructure. Chapter 1. Introduction 22 diffusion in ferrite at low intercritical annealing temperatures. In the case of starting microstructures containing both proeutectoid ferrite and pearl-ite, the growth of the austenite phase is sometimes divided into two or more substages [31, 32]. Souza et al., for instance, observed that the first substage involves the transformation of pearlitic regions to austenite, while the second consists of the consumption of a by 7. The two substages are not necessarily completely distinct from each other in time; in fact, the beginning of the a —> 7 substage may very well coincide with the end of the P —> 7 substage. Presumably, the transformation in pearlite occurs more quickly because the carbide lamellae provide a ready source of carbon, reducing the diffusion distance in the 7 phase. Souza et al. went on to describe a third stage of the transformation, which they called the equilibration of ferrite and austenite, and which is equivalent to stage 3, as described in Section 1.4.3. Speich et al. [33] made similar observations of mixed ferritic-pearlitic microstructures. For steels of commercial purity, diffusion control has proven to be difficult to quantify. Judd and Paxton [18], for instance, modelled the isothermal transformation kinetics under the assumption of carbon diffusion control. They found that their model agreed with experimental data for a zone refined Fe-C alloy, but that agreement was poor both for an alloy containing Mn, and for a commercial steel. The addition of alloying elements was found to decrease the 7 growth rate by a factor of up to ten near the eutectoid temperature, though the effect was considerably smaller at higher temperatures. The reason for the poor agreement was proposed to be the effect of Mn and other impurities on carbon activity. In a later paper, Speich and Szirmae [17] also noted that alloying elements decreased the 7 growth rate considerably. They also attempted to model the dissolution of pearlite from a diffusional point of view, by extending the work done by Brandt [34], who cal-culated the growth rate of pearlite by solving the differential equation applicable to the Chapter 1. Introduction 23 short range diffusion controlled growth of a lamellar structure. When they modified Brandt's solution to account for the hypereutectoid composition of their steel, and used diffusion data presented by Wells et al. [35], Speich and Szirmae found good agreement between predicted and observed 7 growth rates, over the temperature range between T 4 3 and 900°C. The major exception to the general rule of diffusion control appears to be pure iron. Speich and Szirmae [17] calculated a 7 growth rate of about 1.6 cm/s in high purity Fe at 950°C, which led them to characterize the transformation as massive. As the a —> 7 transformation is composition invariant in pure iron, this is not an unexpected observation. 1.4.3 Equilibration If any residual FesC exists after the austenite grains have stopped growing, it dissolves slowly, at a rate determined by the temperature and by the composition of the steel. Even in areas which initially are transformed fully to austenite, carbon concentration gradients remain; these are the source of the pearlite "ghosts" discussed by Roberts and Mehl [15]. They reported that the gradients could persist for up to 30 minutes at 800°C, at which temperature the combined processes of 7 nucleation and growth would be expected to take no more than a few seconds. Thus the third and final stage of austenitization can be the longest. This observation has implications for microstructural modelling. If the concentration gradients remaining after the completion of stages 1 and 2 are significant, and if they persist for long enough, they might be expected to affect subsequent continuous cooling transformations. Clearly, the characterization of the kinetics of austenitization in terms of the Avrami parameters n and b does not take into account any diffusion gradients retained in the austenite phase. It was considered beyond the scope of this work to be able to make any estimate of this effect, but this Chapter 1. Introduction 24 would be an interesting point to investigate in the future. Chapter 2 Scope and Objectives A current research emphasis in the Centre for Metallurgical Process Engineering is the development of a finite element mathematical model capable of predicting the microstruc-tural evolution of steel during heat treatment. As part of this work, there is a need to characterize the kinetics of austenitization. This report details a project which was un-dertaken to: 1. describe isothermal and continuous heating pearlite-to-austenite reversion kinetics in a eutectoid plain-carbon steel; 2. determine if the principle of additivity could be applied to this transformation in order to use isothermal phase transformation kinetics to predict continuous heating kinetics; and 3. examine the effect of starting microstructure on isothermal and continuous heating reversion kinetics. The experimental work was divided into three major phases: 1. characterization of the isothermal and continuous heating pearlite-to-austenite re-version kinetics, for a variety of temperatures and heating rates, and for two starting microstructures; 2. an assessment of the applicability of the principle of additivity in using the isother-mal kinetics to predict the continuous heating kinetics; and 25 Chapter 2. Scope and Objectives 26 3. optical and scanning electron microscopy, primarily to characterize the starting microstructure and to verify the microstructural aspects of the transformation. Chapter 3 Experimental Procedure A l l experimental work was conducted with a eutectoid, plain-carbon steel, the composi-tion of which is given in Table 3.1. 3.1 Characterization of the Kinetics 3.1.1 Test Apparatus The isothermal and continuous heating tests were performed on a Gleeble 1500 Ther-momechanical Simulator, which provides for the simulation of a wide variety of metal working operations. The Gleeble is equipped with a sealed chamber, inside which sam-ples of various shapes and sizes can be resistively heated, gas or water quenched, and hydraulically tested in compression or tension. The control system is highly automated, and a wide variety of heating and cooling rates, strain rates, and sampling rates are possible. Data from each test is digitized and stored in binary format on disk in an I B M P C . Kinetics of solid state phase transformations involving changes in specific volume are measured by continuously monitoring the diameter of a sample during its transformation. Table 3.1: Composition of eutectoid steel, Stelco heat #634044 (wt. %). C M n S P Si 0.766 0.83 0.017 0.004 0.148 27 Chapter 3. Experimental Procedure 28 The Gleeble is supplied with an L V D T diametral and longitudinal strain measuring de-vice, which provides relatively low sensitivity, and which is generally used in compression tests, where large changes in sample diameter are expected. The high sensitivity, quartz tipped diametral dilatometer developed at U .B .C . [23] has been incorporated into the temperature control and data acquisition system of the Gleeble 1500. A schematic of the experimental setup is shown in Figure 3.6. To control and measure the sample temperature during isothermal and continuous heating tests, a C r - A l thermocouple was spot welded to the sample's surface at the plane of the diametral measurement. During each test, the time, programmed temperature, measured temperature, and dilatometer output were measured and recorded continuously. Most of the early testwork was carried out using extrinsic thermocouples, and run under a relatively low vacuum of approximately 1 torr. These tests exhibited poor reproducibility (see Section 4.2), which necessitated changes in the test procedures. Later tests were run with intrinsic thermocouple junctions, under a higher vacuum of approximately 1 0 - 4 torr. 3.1.2 Sample Preparation Test material was obtained from a single piece of 10 mm diameter eutectoid plain carbon steel rod, grade 1078, heat number #634044, obtained from Stelco. The rod was cut into lengths of about 45 cm, and straightened by pulling it in an Instron tensile machine while heating it (to dull red) with a water cooled induction heater. The rod ends, which had not been straightened fully, were then discarded. Individual thin walled tubular samples, 8 mm OD and 20 mm long, with a 1 mm wall, were then machined from the remain-ing center portions of the rods. Tubular samples were used to minimize through wall temperature gradients during testing. Each sample was drilled out carefully to control wall thickness variations to ±0.01 mm, in order to minimize circumferential temperature variations. Chapter 3. Experimental Procedure 29 Tubular sample 20 mm long 8 mm OD 6 mm ID Plane of diametral and temperature measurements tainless steel grips (a) Side view (dilatometer and t/c not shown) (b) End view Figure 3.6: Schematic diagram of heating and temperature/dilation measurement on the Gleeble 1500. Chapter 3. Experimental Procedure 30 After machining, the samples were placed in quartz tubes, which were evacuated, sealed under vacuum, and then annealed for one hour in an air furnace at 900°C. After an hour, the furnace was shut off, and the samples were removed after approximately 24 hours, which resulted in an average cooling rate of approximately 2°C/min through the temperature range of pearlite formation [36]. The resulting microstructure, which consisted of large nodules of coarse pearlite, was termed the F C (furnace cooled) mi-crostructure. In all, four separate batches of samples (termed batches A , B , C, and D) were processed in this way. For the purpose of assessing the effect of starting microstructure on austenitization kinetics, many samples were treated in situ on the Gleeble in order to produce a second microstructure. Prior to isothermal or continuous heating tests, these samples were heated to 800°C, held for one minute, and then cooled at 5°C/s to 600°C. They were then held for one minute at 600°C, before being tested. This treatment resulted in a relatively fine pearlite, with a small colony size. This second microstructure was termed the QC (quick cooled) microstructure. Aside from the two different microstructural pre-treatments, the test procedures used for measuring the pearlite to austenite kinetics in the F C and QC samples were identical. 3.1.3 Isothermal Tests Isothermal reversion of pearlite to austenite (P—> 7) tests were performed at temper-atures varying between 720°C and 750°C. Isothermal samples were heated to 680°C and held there for approximately one minute, before being heated to the isothermal test temperature. In early tests, samples were heated from 680°C to the test temperature at 500°C/s, but this was found to create problems with temperature overshoot. The practice was therefore changed to a two stage heating, in which the sample was brought from 680°C to 710°C at 500°C/s, held at 710°C for 0.1 s, and then heated to the test Chapter 3. Experimental Procedure 31 temperature at 200°C/s. This minimized the overshoot, while heating samples to within 5°C of the test temperature in less than 0.25 s. After completion of the isothermal P—> 7 transformation, the sample was cooled to room temperature, removed from the sample chamber, and retained for metallographic examination. Cooling rates, while not central to the experiment, were varied in order to affect the final microstructure. Some samples were He gas quenched, which resulted in a largely martensitic microstructure, while most were simply allowed to air cool in the vacuum atmosphere, which resulted in a pearlitic structure. A number of samples were He quenched part way through the transformation, and then mounted and polished for metallographic examination. This was done for two pur-poses: first, to verify that the dilatometer was indeed responding to the P—> 7 transfor-mation event, and secondly, to provide microstructural detail on the reversion mechanism. 3.1.4 Continuous Heating Tests Continuous heating tests were performed at heating rates which varied between 0.1°C/s and 10.0°C/s. Samples were brought quickly to 680°C, held for approximately one minute, and then heated to 800°C at a constant rate. 3.2 Assessment of the Additivity Principle The assessment of the applicability of the additivity principle to the pearlite to austenite reversion transformation was carried out by characterizing the isothermal transformation kinetics in terms of the Avrami equation parameters n and 6, and using these results to predict the continuous heating kinetics, as outlined in Section 1.2. Isothermal data were analyzed in two ways. The linearization method is discussed in Section 1.3. Due to statistical concerns with linearization, and problems encountered Chapter 3. Experimental Procedure 32 in employing the linearization technique with the dilatometer data, a nonhnear fitting method was also used to determine the Avrami parameters. This issue is discussed in depth in Sections 4.1.1 and 4.5, and the statistical background is covered in Appendix A. The kinetics model was designed to predict the continuous heating P—> 7 reversion kinetics, given a constant heating rate. Under the assumption that the austenitization process is additive, the prediction is made on the basis of a constant (average) value of the Avrami equation time exponent, n, and on an empirical relationship expressing the Avrami equation rate constant, 6, as a function of temperature. The model can estimate the transformation start time, to, and temperature, Tstartt in either of two ways. First, the Scheil equation can be used with the measured isothermal transformation start times to predict the continuous heating transformation start under the assumption that additivity holds. Secondly, empirical relationships relating Tstart to heating rate, generated during the continuous heating tests, can be used as input to the model. Because the heating rate is set as an input to the model, the calculations performed are independent of heat flow. In a finite difference or finite element heat flow model, these calculations would be performed at each node. In order to reach a stable solution for nodal temperatures at each time step, an iterative method would be required, which would account for the influence which internal heat flow and the endothermic 7 reversion transformation have upon one another. The austenitization kinetics model model was written in the C language. Its flowchart is shown in Figure 3.7, and the source code is contained in Appendix B . Chapter 3. Experimental Procedure f START ^ Input heating rate Calculate start temperature Calculate b, time, and temperature Calculate virtual time at temp I Calculate X no Figure 3.7: Flowchart for austenite reversion kinetics model. Chapter 3. Experimental Procedure 34 3.3 Characterization of Eutectoid Steel #634044 Following Roosz et al. [20], the starting microstructure was characterized in terms of the mean true pearlite spacing, OT, and the pearlite colony specific surface area, Sy^P. In addition, the prior austenite grain size was measured for the F C and QC microstruc-tures. Lastly, the A i and A 3 temperatures were calculated using methods available in the literature. These measurements are discussed separately in the following sections. 3.3.1 Characterization of Pearlitic Microstructures Both optical and scanning electron microscopy were applied for the metallographic eval-uation of pearlitic test samples. For the optical microscopy, 2% nital was used as an etchant. This was found to etch samples more deeply than required for S E M observa-tion, and so S E M samples were etched lightly in 1% nital. Pearlite Spacing The interlamellar spacing of pearlite has long been understood to be an important mi-crostructural parameter. Over the past several decades, numerous methods have been developed to measure it. Russ [37] provided a general summary of the stereological as-pects of measuring interlamellar spacing, while VanderVoort and Roosz [38] summarized several methods which have been developed for pearlite. They discussed the advantages and disadvantages of the various techniques, and concluded that the simplest and most unbiased method is the so-called random circles technique. As recommended by Under-wood [39], this technique calculates the mean random pearlite spacing, ay, from which the mean true pearlite spacing, &T, is determined as [40] oT = 0.5crr (3.36) Chapter 3. Experimental Procedure 35 In the random circles technique, photomicrographs are taken of random locations on a polished and etched sample. The magnification is maintained at the minimum required in order that individual lamellae are resolvable on the photograph; this yields the largest possible sample size (i.e. number of colonies) per photograph. A circular test grid is then overlaid on each photograph, and the number of intersections between the circle and the carbide lamellae are counted. The mean random spacing, crr, is calculated as the reciprocal of the number of intersections per unit length of circumfrence (NL), that is [38], 1 irdc ar = — = —± 3.37 NL nM y ' where n is the total number of intersections, M is the magnification, and dc is the circle diameter. The main advantage of the random circles technique is that it ensures a random and unbiased measurement. It is thus repeatable, relatively operator independent, and easy to automate. Its main disadvantage is that like any random measurement technique, it requires a large sample size. For characterizing the FC and QC microstructures in terms of or, at least 10 photomi-crographs were taken of each sample, using a Hitachi SEM. The magnifications adopted were such that each pearlite spacing photomicrograph contained, on average, approxi-mately 10 pearlite colonies. Thus, each spacing measurement represented an estimate drawn from approximately 100 colonies. Specific Surface The specific surface of the pearlite colonies, Sy^P, was measured in order to provide an estimate of the relative size of the pearlite colonies in the FC and QC microstructures. The random circles technique was followed for this measurement. SEM photomicrographs Chapter 3. Experimental Procedure 36 were taken at random locations within a sample, the magnification being controlled so that individual colonies were resolvable. The colony boundaries in each photograph were then outlined with a felt tip marker before the circular grid was applied. The specific surface, in mm 2 /mm 3 , was then calculated as [20] Sr = ^  (3.38) TCdc At least 10 photomicrographs were included for each measurement, representing 150-200 pearlite colonies. 3.3.2 Austenite Grain Size In order to more fully characterize the QC and F C microstructures, the austenite grain sizes resulting from the QC and F C austenitization treatments were measured. Austeniti-zation of a QC sample was performed in situ on the Gleeble, as described in Section 3.1.2. The sample was helium gas quenched, which resulted in a martensitic microstructure. The F C furnace treatment was simulated by heating another sample to 900° C and hold-ing for 10 minutes, before quenching to martensite. This is of course a rough simulation of the furnace treatment which was used for the test samples, in that the test sample was at temperature for a considerably greater time. This was not viewed as a serious problem, as the austenite grain size is generally a function more of temperature than of time at temperature. Nevertheless, the measured 7 grain size for the F C microstructure should be viewed as conservative. After being mounted and polished, the 7 grain size samples were etched according to a procedure developed in house [41]. This procedure, which combines two established methods for outlining prior austenite grain boundaries in martensite, is outlined below: 1. Boil sample for 10 minutes in freshly prepared and filtered mixture of 2 g picric acid and 25 g NaOH in 150 ml water. Chapter 3. Experimental Procedure 37 2. Etch very lightly with 2% nital (w 1 second). 3. Etch 1 minute in mixture of 2 ml sodium tridecylbenzene, 6 drops teepol, and 100 ml saturated (filtered) picric acid. The etched samples were examined and photographed under an optical microscope, and the austenite grain size was estimated from the photomicrographs. 3.3.3 Calculation of A i and A 3 Temperatures The A i and A 3 temperatures were estimated following the methods of Andrews [42], whose empirical relationships expressed the Ae 3 , A c 3 , and Aci temperatures in terms of the steel composition. These relationships are summarized below in Equation 3.39, with the temperatures in °C, and the alloy compositions in wt. %. For the A e 3 temperature, A e 3 = 910 - 25Mn - l l C r - 20Cu + 60Si + 60Mo + 40W + 100V + 700P + 3 - (250A1 + 120As + 400Ti) + / (C, Ni) (3.39a) where the "3" represents sulfur, which was assumed to give a constant rise in the A e 3 temperature, and / ( C , N i ) is a tabulated function of the C and Ni levels. The A c 3 temperature was estimated as A c 3 = 910 - 2 0 3 ^ - 15.2Ni + 44.7Si + 104V + 31.5Mo + 13.1W (3.39b) and the Aci was given by Aci = 723 - 10.7Mn - 16.9Ni + 29-lSi + 16.9Cr + 290As + 6.38W (3.39c) The A e 3 was also estimated following the more rigorous thermodynamic method pre-sented by Kirkaldy and Baganis [43]. A spreadsheet was employed for this purpose. Figure 3.8 shows a sample A e 3 calculation, for steel #634044. The results of the temperature calculations are presented in Section 4.4. Chapter 3. Experimental Procedure 38 Ae3 Temperature Calculation Ref: Kirkaldy + Baganis, Met. Trans. A, 9A, A p r i l 1978, 495-501 Using eq. (10) - l i n e a r analytic formula St e e l : 1078 #634044 Date: 14.08.90 Composition: wt. '/. g/mol at. */. dGk(To) Elk Intermediate Sums c 0. .765 12, .01 3. .455 -31952, .7 8. ,100 Mn 0. .830 54. .94 0, .820 -9036. .2 -4. ,933 -0.00477 S i 0. . 148 28. .09 0. .286 836. .8 12. ,050 0.00064 Ni 0. .005 58. .71 0, .005 -5489. .0 4. .733 -2.7E-05 Cr 0. .021 52, .00 0, .022 -2033, .3 -12. ,533 -2.1E-05 Mo 0. .002 95, .94 0, .001 2994, .0 -10. ,400 3.66E-06 Cu 0. .005 63, .54 0, .004 -5381, .4 3. .800 -2.4E-05 (Fe) 98. .224 55, .85 95, .408 Total: -0.00419 To = dGo(To) dHo(To) dHl(To) 1000.0 K 338.0 3890.6 -64111.0 Numerator: Denominator: d e l t a T = -34875.5 3830.4 -9.10 K Ae3 temp: 990.9 K 717.9 deg C Figure 3.8: Printout of spreadsheet showing calculation of Ae3 temperature for eutectoid steel #634044, following the method of Kirkaldy and Baganis [43]. Chapter 4 Results and Discussion 4.1 Data Analysis 4.1.1 Isothermal Data T w o methods were employed to characterize the isothermal test data: 1. l inearization of the data, followed by a least squares straight line fit yielding the A v r a m i parameters n and b; and 2. a nonhnear least squares fit applied directly to the A v r a m i equation. The more common method, linearization, was performed by estimating the transfor-mation start time, t 0 , the starting diameter, Do, and the final diameter, Df from the graph of dilatometer signal versus t ime, as shown i n Figure 4.9. Do was taken as the m a x i m u m value of D, while Df was taken as the value to which the dilatometer signal converged as t —> oo. The transformation start time, to, was estimated init ia l ly as the time elapsed when the dilatometer signal was at its m a x i m u m level; subsequent analysis improved upon this estimate, as outlined below. The fraction transformed at each point along the dilatometer curve was then calculated as V V 0 ) <4-4o> Vo — Df The resulting graph of X versus (t — to) was linearized into the form of Equat ion 1.34, and a straight line least squares fit was used to estimate the A v r a m i parameters n and b. 39 Chapter 4. Results and Discussion 40 F i g u r e 4.9: M e a s u r e d t e m p e r a t u r e a n d d i l a t ome te r s i gna l t races for a n i s o t h e r m a l test c o n d u c t e d at 738° C . Chapter 4. Results and Discussion 41 Linearization was found to present problems quite aside from the statistical concerns discussed in Appendix A . In particular, it proved difficult in many cases to visually estimate values for to, Do, and Df. There were several reasons for this. First, it was often observed that there were spikes in the dilatometer signal shortly after isothermal samples were heated to the test temperature. These spikes were generally associated with slight (PS1-2°C) overshooting of sample temperatures beyond the programmed test temperatures (see Figure 4.10). Depending on the size of the spike, this introduced uncertainty into the estimates of Do and t0. Second, problems were encountered in many tests with the stability of the dilatometer signal (see Section 4.2). This made it difficult to estimate Df. Third, the control system of the Gleeble tended to slow the rate of change in sample temperature as it approached the test temperature. This resulted in a rounded off appearance in the T versus t and D versus t curves at the beginning of the isothermal hold (see Figure 4.9). Especially at higher temperatures, it is possible that the transformation had already begun by the time the dilatometer signal reached its maximum. Thus, the visual method of estimating t0 and DQ was not considered valid. One solution to the problem of estimating to is to perform repeated straight line least squares fits on the linearized data, while varying the value of to by small increments. The correlation coefficient is calculated for each fit, and the correct value of t0 is considered to be that value which yields the highest value of the correlation coefficient, r. In practice, this technique works quite well, however it can be a time consuming exercise. In addi-tion, it generally involves a series of comparisons of r values differing slightly from each other. Because of this, it is possible that the chosen value of to could result in a poor characterization of the early stage of the transformation. The statistical implications of this determination of t0 are outlined in Appendix A . Chapter 4. Results and Discussion 42 CD ^_ ZJ o i _ CD C L CD 7 5 0 7 4 0 7 3 0 7 2 0 h 7 1 0 7 0 0 n 1 r Di la tomete r Tempera tu re 0 10 20 30 40 5 0 60 Time (s) 7.5 7.0 6.5 o c CO CD E o _D Q 6.0 5.5 70 80 9 0 100 Figure 4.10: Temperature and dilatometer traces for an isothermal test performed at 735°C, showing a spike in the dilatometer signal. Chapter 4. Results and Discussion 43 The second method of data analysis involved fitting a five parameter nonhnear equa-tion of the form D (t) = D0 - (Do - Df) [1 - exp (-b(t - t0)n)] (4.41) directly to the dilatometer data, which yielded least squares estimates of Do, Df, to, n, and b. The Marquardt-Levenberg algorithm (see Appendix A) was used to determine the least squares estimates of the five parameters. After values of n and b were obtained for several temperatures, the mean value of n (termed navg) was calculated. New values of b (termed b*) at each test temperature In bx, which essentially L TlT J r 1 i forces the two Avrami curves to intersect at In In were then back calculated according to the relation In bT •  .l-Xl The value of In b* was found to vary linearly with temperature, and so the variation of = 0 (that is, at X « 0.632). b* with temperature was characterized as b* = exp (#, + PiT) (4.42) where j30 and fii are the y intercept and slope, respectively, of the In b* versus T least squares line. The value navg and the relationship in Equation 4.42 were then used as inputs to the kinetics model for the prediction of the continuous heating kinetics. Estimates of the transformation start time, to, were also used for model input. The Scheil equation was used to estimate the transformation start temperature, Tstart, during continuous heating (see Section 4.7.1). Tstart was also calculated empirically from the continuous heating data; see below. 4.1.2 Continuous Heating Data Continuous heating kinetics were analyzed as illustrated in Figure 4.11. Straight lines were fit to the pearlite and austenite regions of the D versus t curve. These were extended Chapter 4. Results and Discussion 44 into the transformation region of the curve, and the fraction transformed was calculated as The transformation start and finish temperatures were estimated by determining the temperatures at which the dilatometer signal began to deviate significantly from the least squares lines at either end of the transformation. The empirical relationship between Tstart and heating rate was used as input to the austenitization kinetics model. 4.2 Preliminary Testwork The first set of isothermal and continuous heating tests was carried out using extrinsic thermocouple junctions, in a low vacuum (wl torr) environment. Nominal test tem-peratures varied between 721°C and 736°C; actual (measured) test temperatures varied between 723°C and 737°C. The results of these tests were suspect for several reasons. First, the observed kinetics were highly variable. Figure 4.12 illustrates this point. It contains dilatometer traces of five isothermal tests, all of which were run at measured temperatures of between 727°C and 729°C. It is readily apparent that there is nearly an order of magnitude difference between the time to completion of the transformation for the fastest and slowest tests. Note also that two of the traces show trends which are completely unexpected in the eutectoid P—+ 7 transformation. In test IT727-2, the signal rises after the transformation is apparently complete, and in test IT727-3, the signal shows an initial steep decline, followed by a levelling off, before it drops again and finally levels off. Another problem which was observed with these tests was that the dilatometer signal did not remain constant while the sample was being held at a constant temperature. This can be seen in Figure 4.12, in the initial part of each of the signal traces, and also in the Chapter 4. Results and Discussion 45 Figure 4.11: Measured temperature and dilatometer signal traces for a continuous heating test conducted at l °C/ s . Chapter 4. Results and Discussion 46 2 1 h 0 - 1 h - 2 Test -IT726— 1 - I T 7 2 7 - 1 - I T 7 2 7 - 2 - I T 7 2 7 - 3 - I T 7 2 7 - 4 Note : d i l a t ome te r s igna ls z e r o e d at t ime t = Os. 0 50 100 T ime (s) 1 50 2 0 0 Figure 4.12: Preliminary testwork: dilatometer traces for several isothermal tests con-ducted at temperatures between 727 and 729°C. Chapter 4. Results and Discussion 47 post transformation regions of tests IT727-2, IT727-4, and IT726-1. Without exception, all traces exhibited a decreasing dilatometer signal during the the initial isothermal hold at 680°C, and in most cases, the rate of decrease rose considerably after the sample was heated to the test temperature. This was considered to be a serious problem, largely because the observed changes in the dilatometer signal were not necessarily due only to the P—> 7 transformation. This interfered with the estimation of the Avrami parameters n and b. Finally, it was observed that the magnitude of the change in the dilatometer signal due to the transformation was not constant. Some variation was expected, because the signal amplification varied from test to test. Even so, however, the ratio between the signal change due to the transformation event and the signal change due to heating to the isothermal test temperature should vary only in proportion to the difference between 680°C, the pre-test isothermal hold temperature, and the test temperature. That is, referring to Figure 4.9, is expected to be constant for all tests conducted at a given temperature. This was not observed, as can be seen quite clearly in Figure 4.12. In fact, the ratio was found to vary widely, between approximately 1 and 3. Continuous heating testwork was performed in parallel with the isothermal tests. Dur-ing tests with sample groups A and B , the same problems of temperature measurement and dilatometer drift were encountered, although they manifested themselves differently in the continuous heating tests. The temperature measurement irregularities showed up as highly variable transformation start temperatures, in repeat tests at single heating rates. The dilatometer drift was less noticeable, largely because the samples were not transformation (4.44) AD, temperature Chapter 4. Results and Discussion 48 Table 4.2: Isothermal test results obtained during preliminary testwork. Test Temp (°C) n Hb) 1724-2 723 0.83 -2.83 -2.85 1726-1 727 0.58 -1.26 -1.82 1727-4 729 0.75 -2.12 -2.36 1730-2 731 1.04 -2.98 -2.41 1730-3 731 0.86 -2.28 -2.23 1733-1 734 0.92 -1.88 -1.71 1733-2 733 0.58 -1.26 -1.82 1736-1 737 1.20 -2.60 -1.81 1736-2 736 0.79 -1.41 -1.70 -- 0.84 held at a constant temperature during the tests. It was found that the slope of the D ver-sus t curve in the single phase pearlite and austenite regions was not constant; rather, it decreased continuously. This interfered with the analysis of the data, because it was difficult to extrapolate the curves into the transformation region in order to calculate the fraction transformed. The isothermal test data was analyzed in terms of the Avrami parameters n and 6, using the linearization method outlined in Sections 1.3 and 4.1.1. Table 4.2 summarizes the results. The average value of n was calculated to be 0.84. Such a low value suggests an initial very high rate of transformation, which decreases as the transformation proceeds. Referring again to Figure 4.12, it can be seen that several of the transformation curves (note especially tests IT726-1 and IT727-4) exhibit such behavior. Values of b* were back calculated using navg = 0.84. Figure 4.13 shows the variation of ln(6*) with temperature. Although there is an apparent linear relationship between the two, the correlation coefficient r was only 0.77; this relatively low value may be attributed to the lack of repeatibility discussed above. When the values of navg and b* were used to predict the continuous heating kinetics, it was found that there was Chapter 4. Results and Discussion 49 very poor agreement between the observed and predicted kinetics. This is illustrated in Figure 4.14, which compares the measured and predicted continuous heating kinetics, for a heating rate of 1.0°C/s. Without exception, observed CHT kinetics curves showed an initial low rate of transformation, which increased to a maximum, and then decreased as X—> 1.0. Irrespective of the relationship of b* with temperature, the kinetics model was unable to reproduce this form of curve, largely because the low value of n dictated that, for any practical heating rate, the transformation rate would decrease continuously from an initially high value. The lack of agreement between the measured and predicted continuous heating kinet-ics raised the possibility that the P—> 7 reversion transformation was not experimentally additive. 4.3 Modifications to Test Equipment The observed inconsistencies in the data, discussed above, suggested that there was a problem with the operation of the Gleeble. This was investigated, and two major trou-bles were found: first, the samples were oxidizing heavily, and second, the temperature readings were questionable. A battery of tests was conducted, in which the oxidation, temperature measurement, and dilatometer signal stability problems were investigated. The tests involved three major variables: 1. sample material — plain-carbon steel versus stainless steel; 2. thermocouple junction — extrinsic versus intrinsic; and 3. chamber atmosphere — low vacuum, high vacuum, and backfilled with argon. Chapter 4. Results and Discussion Figure 4.13: Variation of In (&*) with temperature, for n = 0.84. Chapter 4. Results and Discussion 51 TD CO E o H— ( / } c o c o 1.0 0.5 0 .0 0 • • • • • • • • • Expe r imen ta l Da ta P r e d i c t e d , n = 0 .84 8 10 12 Time (s) 14 1 6 18 20 Figure 4.14: Comparison of measured and predicted continuous heating transformation kinetics for n = 0.84. Chapter 4. Results and Discussion 52 As a result of the testwork, changes were made in the way the Gleeble was operated. A short summary of the tests, observations, and modifications follows. First, it was found that the dilatometer drift problem was considerably reduced when stainless steel samples were used. It was not (and still is not) completely understood why this was so. Furthermore, it was not known whether this was related to the temperature measurement observations discussed below. Oxidation of the samples was found to be due to a badly leaking vacuum chamber, and to a nonfunctioning diffusion pump. Both of these problems were rectified, permitting the chamber to be pumped down to, and held at, a total pressure of approximately 1 0 - 4 torr. This vacuum was found in subsequent tests to be sufficient to prevent noticeable oxidation and decarburization in samples heated to temperatures as high as 1000°C. Changing the thermocouple junction had a very noticeable effect on temperature measurement. It was found that the intrinsic junction always sensed a higher temperature than the extrinsic junction. This difference was found to be related to the chamber atmosphere and to the sample material. In general, the difference was greatest (variable, but up to 5% of indicated reading in °C, between 600°C and 1000°C) when a mild steel sample was used in a low vacuum atmosphere. This observed difference is believed to be a radiation effect. When a mild steel sample oxidizes heavily, as happens in a low vacuum atmosphere, its emissivity increases considerably, from about 0.2 to 0.8 [44]. This increase may considerably raise the radiative heat loss in the region of the thermocouple junction. Furthermore, the heat loss from the junction itself may also increase if the thermocouple wires oxidize. When this happens, the temperature of the junction may be depressed, and the reading may be artificially low. Extrinsic thermocouple junctions are not only more susceptible to radiation effects than are intrinsic junctions, but are also more prone to variability. This is because in an Chapter 4. Results and Discussion 53 extrinsic junction, the contact between the two thermoucouple wires is not at the sample surface; rather, it is above the surface, by a distance of approximately one wire diameter. This, in addition to the fact that the thermal mass in the immediate vicinity of the junction is greater than in an intrinsic junction, may lead to questionable temperature readings. It is believed that this is an important consideration for all work which is performed on the Gleeble. Obviously, the potential for a temperature measurement error of up to 50°C at 1000°C has serious implications for any experimental work. The importance of preventing sample oxidation is well understood by many who use the Gleeble in their research work [45], and the issue of thermocouple junction type and chamber atmosphere is discussed in some detail by Duffers Scientific (the manufacturer of the Gleeble) in reference [46]. Overall, the most consistent temperature measurements, and least variable dilatome-ter signal, were found when an intrinsic thermoucouple junction was used under high vacuum. The test procedure for all subsequent tests was changed to incorporate this new practice. As will be demonstrated in Section 4.5, this greatly improved the reliability of the experimental results. 4.4 Microstructural Characterization 4.4.1 Pearlite Spacing and Colony Size Sample groups A and B (samples involved in the preliminary testwork described in Sec-tion 4.2) were characterized only for pearlite spacing, in the FC condition. For group C the pearlite spacing and colony size were measured for both the FC and QC microstruc-tures, and for group D, the spacing and size measurements were made only for the FC microstructure, because these samples were not tested in the QC condition. Chapter 4. Results and Discussion 54 Table 4.3: Microstructural characterization of eutectoid steel #634044. Sample Group Micro-structure (pm) (mm 2 /mm 3 ) (x lO F ~ 9 mm 3) X s X s X s A F C 0.45 0.09 n.a. — — — B F C 0.39 0.08 n.a. — — — C F C 0.42 0.09 I l l 11 3.7 1.6 c QC 0.23 0.04 251 51 0.55 0.22 D F C 0.53 0.10 120 26 5.8 2.4 A typical scanning electron micrograph used in the pearlite spacing measurements is shown in Figure 4.15. Table 4.3 shows the results of the starting microstructural characterization, and lists calculated values of the pearlite structure factor F proposed by Roosz et al. [20] for sample groups C and D. In the table, x refers to the mean measurement for each parameter, while s refers to the estimated standard deviation. Clearly, the errors inherent in measuring the pearlite spacing and colony specific surface are large enough that there is considerable uncertainty in the calculated values of the structure factor. The effect of the two different sample treatments can be seen in the microstructural measurements. The higher cooling rate of the QC treatment results in a finer pearlite spacing and smaller colony size than in the F C samples. In order to verify the experimental measurements of the pearlite spacing, the under-cooling during the 7 —»P transformation was estimated for a number of QC samples, from the temperature data logged by the Gleeble. The results are presented in Table 4.4, which Chapter 4. Results and Discussion 55 Chapter 4. Results and Discussion 56 Table 4.4: 7 —+P transformation range during continuous cooling of QC samples at 5°C/s showing the measured recalescence. Temperature Recalescence Test Range (°C) (°C) RHC05-1 658 ->673 15 RHC1-1 660 ->675 15 RHC1-2 662 ->676 14 RHC5-1 658 -+675 17 RH727-1 662 - » 6 7 3 11 RH730-1 660 -> 673 13 RH730-2 658 ->670 12 RH740-2 660 ->674 14 catalogues the temperature range, in °C, over which the exothermic pearlite formation took place. Clearly, there was little sample-to-sample variability in pearlite formation in the QC samples, which suggests that the test-to-test microstructural variability should be small. On average, there was about 15°C of recalescence in each sample, which suggests that there was some spread in the pearlite spacing within each QC sample. The data in Table 4.4 was used to estimate an average pearlite formation temperature of 668°C. Given an A i temperature of approximately 715°C (see Section 4.4.3), this yields an average undercooling below TA1 of 45-50°C. A value of 50°C was used to compare the measured values of the mean true pearlite spacing, cry, against those of Campbell [1], who presented a comparison of his measurements with those of Pellisier et al. [47]. For an undercooling of 50°C, the least squares line calculated from Pellesier et a/.'s data would indicate a pearlite spacing of 0.31 /xm. The QC values calculated in the present work (0.235 fxva) are in the order of 25% lower than this (and somewhat higher than Campbell's measurements), but within one standard deviation of the least squares line. It was concluded that the pearlite spacing measurements were valid. The colony size measurements were compared to those of Roosz et al. [20]. For a Chapter 4. Results and Discussion 57 furnace cooled microstructure, they reported values of the pearlite specific surface, SP^P, of 172 and 177 mm2/mm3, or about 50% higher than the 111 mm2/mm3 reported in Table 4.3. For air cooled samples, they reported values of 240 and 320 mm2/mm3, as compared to 250 mm2/mm3 which was measured for the QC microstructure in the present work. These values are comparable, though Roosz et al. reported no cooling rate for their air cool. Given the inherent uncertainties associated with the stereoscopic measurement of the specific surface, and given that the values being compared were generated from different steels which had been given different heat treatments, it was concluded that the results reported in the present work are valid, at least for the purposes of comparing the QC and FC microstructures. 4.4.2 Austenite Grain Size Figure 4.16 shows an optical micrograph of a martensitic sample which was given the simulated FC austenitization treatment on the Gleeble, quenched, and then mounted, polished, and etched as outlined in Section 3.3.2. The prior austenite grain boundaries are clearly visible. The results of the austenite grain size measurements are presented in Table 4.5. The measurements indicate that the FC austenitization treatment resulted in a larger austenite grain size than the QC austenitization, a trend expected given the difference in austenitization temperature between the two treatments. As mentioned in Section 3.3.2, the FC sample was treated in situ on the Gleeble in order to facilitate quenching, a treatment which resulted in a shorter time at temperature than the FC test samples experienced. Thus the FC 7 grain size reported in Table 4.5 underestimates the actual grain size of the FC test samples. The extent to which the FC prior austenite grain size was underestimated was gauged by examining data collected by Chau and Hawbolt [48], and summarized by Devadas [2]. Chapter 4. Results and Discussion 58 Figure 4.16: Typical optical photomicrograph used for prior austenite grain size mea-surement (FC microstructure). Mag. 400x. Chapter 4. Results and Discussion 59 Table 4.5: Measured prior austenite grain sizes for the F C and QC austenitization treat-ments. Micro- A S T M Grain structure Grain Size Diameter (fim.) F C 7.8 24 QC 8.5 19 These data indicate that for a eutectoid plain-carbon steel austenitized at 900° C, the average austenite grain diameter would be expected to increase by less than 10% between 10 minutes and 30 minutes holding time. Because the rate of grain growth is so slow after 30 minutes at temperature, it is estimated that the F C 7 grain diameter reported in the present work is approximately 10% below the actual grain diameter reached after 60 minutes at temperature. 4.4.3 Ai and A 3 Temperatures The results of the A i and A 3 temperature calculations are presented in Table 4.6. Gen-erally, the calculations of Andrews [42] are expected to be within ±10°C of the true temperatures; judging from the observed transformation start temperatures and from the A e 3 temperature calculation according to Kirkaldy and Baganis [43] (see Figure 3.8), it appears that Andrew's calculations are high by a few degrees for the eutectoid plain-carbon steel used in this work. Given an A e 3 temperature of approximately 718°C and a steel which is slightly hy-poeutectoid, this would suggest an A e i temperature of approximately 715°C. Chapter 4. Results and Discussion 60 Table 4.6: A i and A 3 temperature calculations for eutectoid plain-carbon steel (heat #634044). Source Temperature Estimated Value (°C) Andrews [42] A c i 719 A c 3 739 A e 3 724 Kirkaldy [43] A e 3 718 4.5 Isothermal Test Results Following the changes in Gleeble operating practice, isothermal tests were performed on samples from groups C and D. Figures 4.17 and 4.18 show the fraction transformed versus time at temperature curves for the QC and F C microstructures, respectively. The results of the kinetics calculations are summarized in Table 4.7 and in Figure 4.19, which shows the variation of In b* with temperature, for a value of navg = 2.2. As Table 4.7 indicates, the isothermal data from these tests was analyzed by both the linearization method and by the nonhnear Marquardt-Levenberg algorithm. Generally, there is no large difference in the values of n and b calculated by the two methods, and the two values of navg are not statistically different from one another. Nevertheless, the nonhnear technique is the preferred method of analysis, for at least three reasons: first, as noted in Appendix A , there are several statistical concerns with the linearization method, all of which are avoided with a nonhnear technique; second, nonhnear modelling yields additional least squares estimates of t0, D 0 ) and Df] and third, the nonhnear model was observed to yield a better fit to the data than the linearization method. This is illustrated in Figure 4.20, which compares the experimental D versus t curve with the predicted kinetics using both the linear and nonhnear Avrami coefficients, for an isothermal test conducted at 740°C. The plot of the residuals (that is, the difference Chapter 4. Results and Discussion Time (s) Figure 4.17: Isothermal transformation curves, QC microstructure. Chapter 4. Results and Discussion 180 T ime (s) Figure 4.18: Isothermal transformation curves, F C microstructure. Chapter 4. Results and Discussion - 8 1 I 1 1 i , i i 1 1 1 i • QC m i c r o s t r u c t u r e . A FC m i c r o s t r u c t u r e -- -- A - - " ' " --Note : va lues c a l c u l a t e d for of b* n = 2'.19 i A I i i i i i 7 3 0 7 3 5 7 4 0 7 4 5 7 5 0 T e m p e r a t u r e (°C) Figure 4.19: Variation of ln(6*) with temperature, for n = 2.19. Chapter 4. Results and Discussion 64 Table 4.7: Summary of isothermal kinetics, sample groups C and D. Micro-structure Test Temp. (°C) Nonlinear n ln b ln b* Linearization n ln b ln b* QC RH730-2 RH730-1 RH735-1 RH740-2 RH745-2 733.0 734.0 738.0 743.5 747.5 1.86 -5.79 -6.81 2.47 -7.17 -6.35 2.46 -5.73 -5.09 2.29 -3.18 -3.03 2.51 -1.62 -1.41 .1.98 -6.17 -6.71 2.29 -6.55 -6.17 2.16 -5.10 -5.10 2.44 -3.63 -3.21 2.34 -1.44 -1.33 F C 730CM-1 733CM-1 735CM-1 740CM-5 745CM-1 731.0 735.0 736.0 742.0 746.0 1.97 -8.60 -9,58 2.07 -8.78 -9.30 2.00 -7.37 -8.07 2.02 -6.82 -7.38 2.24 -6.07 -5.94 2.06 -9.12 -9.55 2.09 -8.90 -9.20 2.02 -7.44 -7.96 1.85 -6.12 -7.14 2.34 -6.33 -5.82 2.19 2.16 between the measured and predicted data values) indicates clearly that the nonHnear model fits the raw data more closely than the linear model which was generated from linearized data. To reiterate what was said in Section 4.1.1, three different methods were employed to estimate the transformation start time, to: 1. visual examination, with to = t (Dmax); 2. repeated linearizations and least squares fits to Equation 1.34, with to taken to be that value yielding the highest r value; and 3. a nonHnear least squares fit to Equation 4.41, which yields to directly. The results of the transformation start time estimation are catalogued in Table 4.8, for all three of these methods. As Table 4.8 indicates, the visual method yields to values Chapter 4. Results and Discussion 65 5.5 3.0 Expe r imen ta l Data Non l inear R e g r e s s i o n L inear R e g r e s s i o n Test : 7 4 0 C M - 5 10 20 30 4 0 50 60 70 80 Time (s) (a) Linear and nonlinear predictions compared with experimental data. o 3 7 3 in QJ CC 0 .10 0 .05 0 .00 •0.05 -0 .10 r'\ A-(b) Magnitude of the difference between the experimental and predicted dilatometer signals, for linear and nonhnear Avrami characterizations. Figure 4.20: Comparison of linear and nonhnear modelling techniques for the isothermal pearlite to austenite transformation at 740°C. Chapter 4. Results and Discussion 66 Table 4.8: Estimated values of the transformation start time, to, for the isothermal pearlite-to-austenite transformation. Micro- Transformation Start Time (s) structure Test Temp. (°C) Visual Linearization Nonlinear RH730-2 733.0 1.2 4.3 4.2 RH730-1 734.0 2.1 2.7 2.0 QC RH735-1 736.0 2.1 1.5 1.7 RH740-2 743.5 1.6 0.6 1.0 RH745-2 747.5 0.7 0.4 0.4 730CM-1 731.0 2.1 4.0 7.2 733CM-1 735.0 1.6 5.0 5.2 F C 735CM-1 736.0 1.7 5.5 5.7 740CM-5 742.0 2.4 1.0 1.1 745CM-1 746.0 1.9 1.4 1.5 Table 4.9: Effect of small variations in Do on the transformation start time, as calculated by the linearization method, for an isothermal test performed at 736°C. Do 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 to (s) 11.2 10.2 9.6 8.2 6.6 5.5 3.8 2.4 0.9 0.0 0.0 which exhibit no readily apparent relationship with the test temperature, while the lin-earization and nonhnear methods both suggest that the transformation start time drops as the test temperature rises. The latter two methods yielded similar estimates for most temperatures. Making an accurate estimate of the transformation start time is difficult, not only because of the factors discussed in Section 4.1.1, but also because the final value of to is statistically dependent on the value of Do, the initial dilatometer signal. The magnitude of the dependence may be determined by calculating to as a function of Do- Table 4.9 shows the results of such a calculation, which was performed for an isothermal test conducted at 736°C. Chapter 4. Results and Discussion 67 The starting dilatometer signal shown in bold in Table 4.9 is the optimum value for Do, as determined by the nonhnear method. The transformation start time which corresponds to it (also in bold) was calculated by the linearization method; this value also appears in Table 4.8, for test 735CM-1. The remaining transformation start times were determined by making slight variations to the value for Do, and then using the linearization method and repeated straight line least squares fits to estimate to- The strong dependence of t0 on Do is readily evident. Table 4.9 leaves unanswered the question of how large these changes in D 0 are. Fig-ure 4.21 put's this into perspective. Consider that the change in dilatometer signal due to the P—> 7 transformation for this test was found to be ADtrans — 1.6. Thus the maximum variation of ±0.05 in Table 4.9 represents a change of approximately ± 3 % in terms of A D f r a n s . In this case, it would reasonably have been possible to estimate Do to within ±0.01 of the "correct" (i.e. least squares) value of 7.20. This suggests an error of roughly ±20% in the estimate of to, a figure which is regarded as a conservative estimate of the error inherent in the calculated values of to-For experimental data, such as that shown in Figure 4.9, it was possible to visually estimate Do and Df such that the error expected in the estimate of A D t r a n s was less than 1%. This small error introduces significant uncertainty into the value of to calculated by the linearization method, as Table 4.9 indicates. For less well behaved data (Figure 4.10), where the error in the visual estimate of A D t r a n s could be more like ±2%, the resulting value of the transformation start time is even less certain. It must be concluded that a reasonable estimate of t 0 is possible only if a valid estimate of the starting dilatometer signal can be made. It would certainly be possible to estimate Do by doing repeated linearizations, as is done for to- But as Table 4.9 demonstrates, the two parameters are strongly interdepen-dent, and therefore cannot be estimated independently of each other. Linearization thus Chapter 4. Results and Discussion 68 0 20 40 60 8 0 100 Time (s) Figure 4.21: Dilatometer trace for isothermal test conducted at 736°C, illustrating range of DQ values considered for transformation start time analysis. Chapter 4. Results and Discussion 69 becomes a very tedious and time consuming exercise. Nonlinear modelling, on the other hand, yields both parameters directly, and so is the preferred method of analysis. Another advantage of the nonHnear curve fitting is its potential to analyze double transformation events. Austenite decomposition kinetics of a hypoeutectoid steel have been analyzed in terms of the 7 —• a + P transformation [49], but the analysis can be difficult and time consuming to handle with the linearization method. If, for instance, the austenitization of a mixed pearlitic-ferritic microstructure were being studied, the dilatometer data could be analyzed by fitting a 9 parameter nonHnear model of the form V = D0 + (D1- Do) {1 - exp [-W(t - t0)ni}} + (D2 - D0) {1 - exp [-b2(t - t0)n2}} (4.45) directly to the data. This is illustrated in Figure 4.22. This would likely be a difficult analysis, and its success would depend on a more stable dilatometer signal than was available for the present work. Existing plans call for the purchase of a second dilatometer for the Gleeble, and so this may very weU be a realistic possibility in the future. The forgoing discussion of isothermal data analysis can be summarized by saying that although the changes in Gleeble operating procedure discussed in Section 4.3 did consid-erably improve the reHabiHty of the data which was collected, they did not completely alleviate the signal drift problem. Slight downward drift in the signal tended to obscure the end of the transformation, where the signal should be levelling off to its final value of Df. This trouble was most noticeable near completion of tests performed at low tem-peratures (fts 730°C), where the transformation rate remains relatively low throughout the transformation. Chapter 4. Results and Discussion Time Figure 4.22: Hypothetical dilatometer curve for a double transformation event. Chapter 4. Results and Discussion 71 4.5.1 Effect of Starting Microstructure on Isothermal Kinetics Referring again to Figure 4.19, the effect of microstructure on isothermal transformation kinetics can be seen clearly. The F C microstructure, with its coarse pearlite spacing and large colony size, exhibits much lower values of b (that is, slower transformation kinetics) than the QC microstructure. This trend may also be observed by comparing Figures 4.17 and 4.18. This observation is in general agreement with the results of Roosz et al. [20] (see Section 1.3). It is also consistent with past observations that austenitization is a short range diffusion controlled process. The smaller interlamellar spacing in the QC microstructure reduces the effective carbon diffusion distance in the growing austenite grains, permitting a higher transformation rate. The isothermal kinetics were used to construct a TTT-type diagram for austenite formation in the eutectoid plain-carbon steel used in the present work. The diagram is shown in Figure 4.23. The effect of microstructure on the isothermal kinetics is clearly visible on the diagram; the finer pearlite spacing of the QC microstructure decreases the incubation and transformation times. In order to compare the isothermal austenitization kinetics measured in the present work with those reported in the literature, the structure factors tabulated in Table 4.3 for the F C and QC microstructures were applied to the isothermal austenitization dia-gram presented for eutectoid plain carbon steel presented by Roosz et al. As Table 4.10 indicates, the austenitization times calculated in this manner are much higher than those observed in the experimental portion of this project. There are at least two possible reasons for this discrepancy. First, the compositions of the steels were different (the eutectoid steel used by Roosz et al. was higher in M n and lower in Si). More importantly, the austenitization times predicted by Roosz et a/.'s diagram are markedly sensitive to the values of the pearlite spacing, <ry, and colony size Chapter 4. Results and Discussion 7 6 0 7 5 0 7 4 0 7 3 0 7 2 0 0.02 *"0.98" t0.02~ ^0.98' QC FC \ s A3 7 1 0 7 0 0 0.1 10 T ime (s) 100 1 0 0 0 F i g u r e 4 .23: T T T d i a g r a m for aus ten i t i za t i on i n eu tec to id p l a i n - c a r b o n steel Chapter 4. Results and Discussion 73 Table 4.10: Comparison of measured isothermal kinetics with calculated kinetics based on isothermal austenitization diagrams proposed by Roosz et al. [20]. Micro- Temp. 0^.99 ( s) structure (°C) Experimental Calculated * Calculated * 730 80 300 123 QC 740 16 23 10 750 3 12 5 730 231 2000 110 FC 740 62 170 75 750 16 75 40 * Structure factors from Table 4.3 * Reduced structure factors as represented by the specific surface, SP^P. Any errors associated with these measure-ments would be expected to have a significant effect on the kinetics predicted by Roosz's austenitization diagram. Recalling the estimated standard deviations of the structure factor calculations presented in Table 4.3, it is easily understood that there is consider-able uncertainty associated with the comparison of test results by different workers. To investigate the magnitude of this effect, the values of <fx and SP^P tabulated in Table 4.3 were adjusted so as to reduce both the pearlite spacing and colony size, which would tend to decrease the predicted austenitization time. The fifth column in Table 4.10 contains austenitization times calculated on the basis of a pearlite spacing decreased by 20% and a specific surface increased by 40%. These changes caused the structure factor, F, to decrease by slightly over 50% for the FC and QC microstructures. This decrease clearly had a large effect (up to an order of magnitude) on the predicted values of £o.99> which were much closer to the observed values. Thus a large portion of the difference between the observed and predicted isothermal kinetics are probably due to errors inherent in the measurement of microstructural parameters. Chapter 4. Results and Discussion 74 Table 4.11: Apparent activation energies calculated from isothermal data for the F C and QC microstructures, and from data presented by Roosz et al. [20]. Apparent Activation Steel Energy (kcal/mol) F C 220 QC 340 A* 320 B* 480 C* 460 D* 430 * Work reported by Roosz et al. for four different microstructures. 4.5.2 Apparent Activation Energy The apparent activation energy for the formation of austenite from pearlite was calculated according to Shewmon [25], using the relationship1 d l n t o 5 = 9. (446) dl/T R ^ ' where t0_5 is the time to half completion of the transformation, T is the temperature, Q is the apparent activation energy for the nucleation and growth process, and R is the gas constant. Q/R was taken as the slope of a plot of l n £ 0 . 5 versus 1/T. The apparent activation energy was calculated for the F C and QC microstructures, and for comparison purposes, the data reported by Roosz et al. [20] was used to calculate apparent activation energies for their work, for the four microstructures they studied. The results of the calculations are shown in Table 4.11. Figure 4.24 shows the plots of lni 0.5 versus 1/T for the F C and QC microstrucures. The values calculated from the isothermal data of the present work and from the data presented by Roosz et al. are considerably larger than would be expected on the basis 1This equation neglects the effect of temperature on the driving force of the transformation, hence the term "apparent activation energy". Chapter 4. Results and Discussion 4 3 2 -1 1 1 I -- -— - b -- ^« -- -- • QC m i c r o s t r u c t u r e _ o FC m i c r o s t r u c t u r e i i i i 10 .00 9 .95 9 .90 9 . 8 5 9 .80 9 .75 1 /T X 1 0 4 ( K ~ 1 ) Figure 4.24: Apparent activation energy plots for the F C and QC microstructures. Chapter 4. Results and Discussion 76 of carbon diffusion control. This reflects the large temperature dependence of the P—• 7 reaction. Clearly, the increase in the carbon diffusion coefficient at higher temperatures is expected to influence the rate of growth of austenite into pearlite if carbon diffusion in austenite is the rate controlling step. Additionally, however, the equilibrium carbon concentrations at the 7 / a and 7 /Fe 3 C interfaces change as the temperature increases, such that the effective carbon diffusion gradient in austenite rises with temperature. This may account for the very high calculated Q-values. 4.5.3 Partially Transformed Samples In order to verify the microstructural aspects of the pearlite-to-austenite transformation in eutectoid steel #634044, a number of samples were allowed to transform partially to austenite before being quenched, mounted and polished, and examined metallograph-ically. The time at temperature was varied so as to allow examination of samples at various stages of the transformation. Figure 4.25 shows the temperature and dilatometer traces for one such isothermal test, in which a sample in the F C condition was brought to 727°C, held for 8 seconds, and then helium gas quenched. The dilatometer trace shows a slight decrease from Dmax after the 8 seconds, indicating that the P—• 7 transformation had just begun by the time the sample was quenched. The temperature and dilatometer traces show slight inflections during the quench, which corresponds to an expansion of the sample caused by the formation of pearlite from the small amount of 7 formed at 727°C. The dilatometer trace shows no evidence of any expansion due to martensite formation. When the sample was examined met allograph! cally after the test, its microstruc-ture was found to consist largely of coarse, untransformed pearlite from the original F C microstructure. There were regions of fine pearlite and/or bainite, which were formed during cooling, and very small isolated regions of martensite. Presumably the amount Chapter 4. Results and Discussion 77 8 0 0 6 0 0 CJ CD CD CL E CD 4 0 0 2 0 0 0 ,/ i -' V; l'\ \\ \\ \\ \'' \ T e m p e r a t u r e D i l a tome te r S igna l 0 Test : 8 H 7 2 7 - 1 10 2 0 T ime (s) 30 D C 4 Cn CO 3 CD CD 5 0 - 1 40 Figure 4.25: Temperature and dilatometer traces for interrupted isothermal test con-ducted at 727°C. Chapter 4. Results and Discussion 78 of martensite formed was too small to register a noticeable effect on the dilatometer signal. Optical and scanning electron micrographs are shown in Figures 4.26 and 4.27, respectively. One of the main reasons for examining the interrupted isothermal samples was to attempt to determine the active 7 nucleation sites. This is difficult, largely because conclusions about a three dimensional transformation event are being deduced on the basis of an observation in two dimensions. Metallographic observation showed that no 7 nuclei formed within pearlite colonies, confirming numerous observations in the literature (see Chapter 1). The majority of nuclei were observed to be growing from the vicinity of three or more colonies. Figure 4.28 shows a sample S E M micrograph of an "interrupted and quenched" sam-ple. The martensitic region at the center of the photograph formed from an austenite nucleus which developed before the quench. It can be seen to have been growing from a region common to four pearlite colonies. This typical observation is a strong indication of nucleation occurring at pearlite colony corners or edges. Examination of partially transformed samples which had been transformed for various periods of time indicated that the P—• 7 transformation proceeded by the growth of existing nuclei more than by the generation of new ones; that is, the apparent number of 7 nodules did not change appreciably with time after an early stage in the reaction. This suggests that the austenite formation reaction may meet Cahn's early site saturation criterion. Figure 4.26: Optical photomicrograph of an "interrupted and quenched" isothermal sam-ple, showing martensite (white) and fine pearlite/bainite (dark) islands in untransformed coarse pearlite (light gray). Mag. 600x. Figure 4.27: Scanning electron micrograph of an "interrupted and quenched" isothermal sample. Mag. 2000 x . Figure 4.28: Scanning electron micrograph showing austenite nodule growing from the region of four pearlite colonies, indicating nucleation from colony corners or edges. Mag. 5000 x. Chapter 4. Results and Discussion 82 Table 4.12: Summary of continuous heating test kinetics, sample groups C and D. Micro- Heating Data Predicted structure rate (°C/s) Tgtart io.02-+0.98 (s) *0.02-+0.98 (s) 0.25 731 27.5 25.8 0.5 733 17.3 14.7 QC 1.0 738 8.6 8.2 1.0 736 8.9 8.2 2.5 739 3.8 3.7 5.0 742 2.3 2.1 0.25 736 38.0 35.2 0.5 738 17.7 21.0 F C 1.0 740 11.1 12.1 2.5 743 6.8 5.5 5.0 748 3.2 3.0 4.6 Continuous Heating Test Results The kinetics measured in the continuous heating tests are summarized in Table 4.12. The effect of starting microstructure on continuous heating kinetics, while less notice-able than observed for the isothermal kinetics, was manifested in two ways. First, the transformation start temperature at a given heating rate was generally higher for the FC microstructure (coarse pearlite) than for the QC microstructure (fine pearlite). Second, the time to completion for F C samples was longer than for QC samples. Both of these trends are apparent in Table 4.12, and both are consistent with the observed differences in isothermal kinetics between the QC and F C microstructures, as shown in Table 4.7. Chapter 4. Results and Discussion 83 4.7 Austenitization as an Additive Process 4.7.1 Prediction of Tstart The additivity principle (as outlined by Scheil [11] and Cahn [12]) was applied in an attempt to predict the transformation start temperature during continuous heating. The code to perform this calculation was included in the austenitization kinetics model (see Appendix B). Equation 1.27 was applied to the isothermal transformation start time data (as estimated by the nonHnear Marquardt-Levenberg algorithm) presented in Table 4.8. The program worked by approximating Equation 1.27 as tstart /\-f E ^ T ) - 1 ( 4 - 4 7 ) The transformation start time tstart and temperature Tstart were taken as the time and temperature, respectively, at which the summation in Equation 4.47 reached unity. The program was successful in predicting that for a given heating rate, Tstart will be higher for the F C microstructure than for the QC microstructure, in agreement with the experimental observations shown in Table 4.12. It is clear from Table 4.13, however, that the program underestimated Tstart in all cases. It is interesting to note that a past attempt to apply the additivity principle to predict the transformation start on continuous cooling was also unsuccessful [23], although in that case, the transformation start time was overpredicted. The question of why the transformation start temperature was poorly predicted can perhaps best be answered by examining Figure 4.29, which shows the isothermal transfor-mation start time data from Table 4.8, graphed as a function of isothermal temperature. The graph iUustrates the considerable scatter in the calculated values of the transforma-tion start time, and also shows the lack of data in the region between the Ae3 temperature Chapter 4. Results and Discussion 84 Table 4.13: Summary of Tstart predictions made by the austenite reversion kinetics model on the basis of the application of the Scheil Equation to the isothermal austenitization data. Micro- Heating Tgtart (°C) structure Rate (°C/s) Data Predicted * Predicted * 0.25 731 719.5 726.2 0.5 733 720.9 728.3 QC 1.0 737 723.5 731.0 2.5 739 730.0 735.6 5.0 742 737.3 740.2 0.25 736 721.0 728.0 0.5 738 723.7 730.6 F C 1.0 740 728.2 733.8 2.5 743 737.3 739.5 5.0 748 744.0 745.1 * Linear back-extrapolation. ' Parabolic back-extrapolation. (718°C) and 730°C 2 . The transformation start time data was handled in two ways in the Tstart algorithm. First, least squares straight lines of the form to = 30 + fa (T - TA3) (4.48) were fit to the F C and QC data over the range of isothermal test temperatures. To cover the region between the A 3 temperature and 730°C, these were simply back extrapolated to 718°C. This behavior would not be expected, either on the basis of T T T diagrams or on the basis of isothermal austenitization diagrams as proposed by Roosz et al. [20] and Lenel [50]. The linear extrapolation was done because it was the simplest way to 2The aforementioned problems with dilatometer signal stability made it very difficult to characterize the austenitization kinetics at temperatures below 730°C. This had no adverse effect on the prediction of continuous heating kinetics, because the observed continuous heating transformation start temperatures were above 730°C. Clearly, however, it did have a detrimental effect on the attempt to characterize the transformation start time at temperatures only slightly above the Ae 3 temperature. Chapter 4. Results and Discussion 85 2 5 2 0 A FC m i c r o s t r u c t u r e • QC m i c r o s t r u c t u r e L inear back—ex t rapo la t i on P a r a b o l i c back —ext rapo la t ion 15 10 5 T. , = 71 8°C A 3 " •V . A 0 7 1 5 7 2 0 7 2 5 A -7 3 0 7 3 5 7 4 0 T e m p e r a t u r e (°C) 7 4 5 7 5 0 Figure 4.29: Isothermal transformation start times for the F C and QC microstructures. Chapter 4. Results and Discussion 86 estimate the start time. The transformation start time versus test temperature relationship was also approx-imated by a least squares fit of the form ( ° = ( O T ( 4 ' 4 9 ) which yields an estimate of t0 quite similar to that of Equation 4.48 over the temperature range actually covered by the isothermal tests, but with a steeply increasing estimate near the A 3 temperature. As Table 4.13 shows, the model underestimated Tstart irrespective of which back-extrapolation was used. The parabolic back-extrapolation of Equation 4.49 clearly made a better estimate at all temperatures, however. The effect of this on the continuous heating kinetics predictions is discussed in the next section. It should be recalled that the A e 3 temperature of 718° C was estimated following the thermodynamic method of Kirkaldy and Baganis [43]. They estimate an error of approximately ±5°C for their method. If the A e 3 temperature were underestimated by this much, this would be expected to have a significant influence on the prediction of the transformation start time, under the assumption of additivity. 4.7.2 Prediction of Continuous Heating Kinetics As discussed in Section 4.2, the kinetics model was unable to predict the continuous heat-ing transformation kinetics when a value of n — 0.84, determined during the preliminary testwork, was used. The model's predictions improved greatly when an average value of n = 2.2, resulting from the later isothermal tests, was input. Table 4.12 compares the predicted and measured times for the P—> 7 transformation (for 0.02 < X < 0.98), while Figures 4.30 and 4.31 compare the predicted and measured continuous heating Chapter 4. Results and Discussion 87 results, for the QC and FC microstructures, respectively, assuming an additive transfor-mation. Agreement between the model predictions and the experimental data is seen to be generally very good, except at high heating rates. Several points are worth noting. First, referring to Table 4.12, it can be seen that the model predictions for io.02-fO.9s generally agree with the experimental data, to within approximately 10%. There are three exceptions: QC microstructure, 0.5°C/s; FC microstructure, 0.5°C/s; and FC mi-crostructure, 2.5°C/s. The reasons for these discrepancies were investigated, and the causes are discussed in the following paragraphs. In the case of the QC microstructure tested at 0.5°C/s, the difference between the predicted and observed kinetics is believed to be due to the difficulties inherent in es-timating the continuous heating transformation start and finish times. As Figure 4.30 shows, the model predicts the continuous heating kinetics for this heating rate very well. The experimental data appears to level off at approximately X = 0.95, and converges to X = 1 more slowly than in most other tests. This is simply an artifact of the method of data analysis which was discussed in Section 4.1.2; in the case of this test, the least squares line for the austenitic region of the D versus t curve diverged slightly from the curve at X ~ 1.0. This is not viewed as a serious problem, because it does not signifi-cantly affect the bulk of the X versus t curve. It does, however, introduce uncertainty into the curve near its endpoint, and also into the calculated value of io.o2->o.98-For the FC microstructure test performed at 0.5°C/s, as Figure 4.31 illustrates, the model appears to significantly underpredict the continuous heating kinetics. This is believed to be caused by irregularities in the test data. As Figure 4.32 shows, the austenite portion of the D versus t curve exhibits a significant amount of curvature, of the same sort which was found to be due to the drifting dilatometer signal (see Section 4.6). This makes makes is difficult to extend the austenite portion of the curve into the transformation region, thereby introducing uncertainty into the calculation of X over the entire range 0 < Time (s) Figure 4.30: Measured and predicted continuous heating kinetics, QC microstructure. Chapter 4. Results and Discussion 89 Time (s) Figure 4.31: Measured and predicted continuous heating kinetics, F C microstructure. Chapter 4. Results and Discussion 90 X < 1. In practice, this problem is dealt with by fitting a straight line only to the initial portion of the 7 portion of the dilatometer trace, immediately following the completion of the transformation. This may minimize the error, but there remains a question of the validity of back-extrapolation of the line into the transformation region of the curve. This problem is illustrated in Table 4.14, which lists the measured slopes of the pearlite and austenite regions of the continuous heating dilatometer traces. Because the heating rate and signal amplification vary from test to test, the slopes themselves cannot be compared from test to test. The ratio between the two slopes, however, should be invariant, if the dilatometer is responding only to change in the test sample diameter. This ratio is shown in the fifth column. There is some scatter, and the ratio tends to increase as the heating rate increases. On average, though, it remains reasonably constant for the QC microstructure, but not for the F C microstructure. The major exception is the F C microstructure test conducted at 0.5°C/s. In this case, the initial slope of the 7 portion of the curve is much higher than would be expected. This suggests that the dilatometer is responding not only to the changing sample diameter. In general, a drifting dilatometer signal indicates a possible temperature measurement problem (see Section 4.3), which casts doubt on the X versus t curve. It was concluded, therefore, that the inability of the model to predict the continuous heating kinetics for the F C microstructure test conducted at 0.5°C/s was due to dilatometer drift during the test. In the F C test conducted at 2.5°C/s, the model significantly overpredicted the ob-served kinetics. This was observed to at least some degree in all of the tests done at heating rates in excess of 1.0°C/s, a trend better illustrated in Figures 4.30 and 4.31 than in Table 4.12. The reason for this was traced to the control system of the Glee-ble. It was found that during high heating rate tests, in which the endothermic heat Chapter 4. Results and Discussion 91 Figure 4.32: Dilatometer trace for continuous heating test performed with F C microstruc-ture at 0.5°C/s. Chapter 4. Results and Discussion 92 Table 4.14: Measured slopes of pearlite and austenite portions of continuous heating dilatometer traces. Micro- Heating Pearlite Austenite Ratio structure Rate (°C/s) Slope Slope (7/P) 0.25 0.0106 0.0146 1.372 0.5 0.0150 0.0202 1.341 QC 1.0 0.0281 0.0421 1.496 1.0 0.0295 0.0428 1.451 2.5 0.0722 0.1111 1.539 5.0 0.1297 0.2202 1.697 0.25 0.0067 0.0117 1.755 0.5 0.0110 0.0341 3.100 F C 1.0 0.0254 0.0465 1.827 2.5 0.0611 0.1347 2.205 5.0 0.1242 0.2385 1.920 of transformation is absorbed quickly, the Gleeble's control system was unable to com-pensate completely for the absorption of the heat, resulting in a temperature depression during the transformation, as illustrated in Figure 4.33. In the case of the high heating rate tests, this depression was significant enough to affect the observed transformation kinetics, with the result that the model appeared to overpredict the kinetics. To test this hypothesis, the model was modified in order to incorporate the actual temperature variation as a function of time, rather than the programmed continuous heating temperature. When this was done, it was found that the model's predictions were considerably improved, as illustrated in Figure 4.34, which compares the unmodified and modified model predictions with the experimental data for test RHC5-1 , which was run with a heating rate of 5.0°C/s, on a sample with the QC microstructure. It was concluded that the temperature fluctuations described above were the reason for the model's overprediction of the continuous heating kinetics at high heating rates. In order to determine the effect of the error in the estimation of Tstart on the kinetics Chapter 4. Results and Discussion 93 Figure 4.33: Temperature and dilatometer traces for a QC microstructure test performed at 5°C/s, showing temperature depression during the P—> 7 transformation. Chapter 4. Results and Discussion 94 1 .0 0.5 , fj e - B -6 P 13 0.0 CB • Da ta , test R H C 5 - 1 Model Model with mod i f i ed t e m p e r a t u r e prof i le 0.0 1.0 2 .0 T ime (s) 3.0 Figure 4.34: Improved kinetics prediction resulting from modified temperature-time re-lationship, for a continuous heating test performed at 5°C/s. Chapter 4. Results and Discussion 95 model's prediction, the model was run using the Scheil equation, employing both back-extrapolation methods discussed in Section 4.7.1. These predictions were compared to the model predictions which were based on the empirical data for T s t a r t collected during the continuous heating tests. It was found that the use of the Scheil equation introduced significant errors into the predictions of the continuous heating kinetics, as illustrated in Figure 4.35, which compares the predictions made for a test conducted with the QC microstructure, at a heating rate of 1.0°C/s. The parabolic back-extrapolation yielded better results than the linear back-extrapolation, possibly because t0 was better estimated in the region between 718°C and 730°C. Overall, it was concluded that the data available simply did not permit the use of the Scheil equation in the austenite reversion kinetics model. The empirically determined values for Tatart, obtained during the continuous heating tests, were used in the model runs reported in Figures 4.30 and 4.31, and Table 4.12. 4.7.3 The Applicability of the Additivity Principle The austenitization kinetics model has shown the P—• 7 transformation to be experimen-tally additive in a eutectoid, plain-carbon steel. This section discusses the mathematical and metallurgical implications of this. As discussed in Section 4.5, the experimental results of the isothermal testwork demonstrated that the value of the Avrami time exponent, n, is constant, over the tem-perature range covered in the tests. Furthermore, the Avrami rate constant, b, has been shown to be a simple function of temperature. The mathematical requirements for the additivity principle to be apphcable to the formation of austenite from pearlite (see Section 1.2) have thus been satisfied, as the requirements imposed on n and b by Equation 1.28 have been met. Note, however, that the requirements of Equation 1.32 have not necessarily been met, because b has not been shown to be a single function of Chapter 4. Results and Discussion 96 Figure 4.35: Comparison of austenitization kinetics predictions made with and without the Scheil Equation, for a continuous heating test performed at 1.0°C/s. Chapter 4. Results and Discussion 97 temperature. The isothermal tests yielded values of n ranging between 1.9 and 2.5, with an aver-age of 2.2. Christian [7] catalogued the expected value of n for a variety of nucleation and growth conditions, for diffusion controlled reactions (i.e. those obeying a parabolic growth law) and for interface controlled growth, eutectoid reactions, and discontinuous precipitation (i.e. those obeying a linear growth law). For a diffusion controlled reaction, a value of n « 2 reflects nuclei of any shape growing from small dimensions, with a decreasing nucleation rate. For an interface or short range diffusion controlled reaction, a value of around two indicates nucleation on grain edges following saturation of grain corner sites. Christian also noted that a simple determination of n is not necessarily sufficient to permit conclusions to be made about the active nucleation sites, because a variety of nucleation sites are usually active. Microscopic observation of the "interrupted and quenched" isothermal samples (see Section 4.5.3) suggested that the effective nucleation sites for austenite formation in a eutectoid plain-carbon steel are pearlite colony corners and/or edges. This is in agreement with the implications of the average value of the Avrami time exponent, n, for a reaction obeying a linear growth law, as proposed by Christian [7]. The isokinetic condition for the applicability of the principle of additivity to a given reaction, proposed by Avrami [4,5,6], was discussed in Section 1.2. In essence, an isoki-netic reaction is understood to be one in which the isothermal nucleation and growth rates are proportional over a certain temperature range (that is, the two rates vary identically with temperature). Such a reaction is expected to be additive. Isothermal nucleation and growth rates were not measured in the present work, and so it is not possible to directly check the isokinetic condition. Instead, the results of calculations presented by Roosz et al. [20] concerning the nucleation and growth rates for austenitization in a eutectoid, plain-carbon steel were used to verify whether the isokinetic condition held in their work. Chapter 4. Results and Discussion 98 Their results indicated that log N oc 1 (4.50a) AT and that log G oc 1 (4.50b) AT where AT = T — TACY • This clearly implies that N oc G. Results of calculations made from the nucleation and growth rates presented in the Roosz paper are summarized in Figure 4.36. As the graph illustrates, their results indi-cated that the rates of nucleation and growth were proportional; therefore, the isokinetic condition should be satisfied, and so the P—> 7 transformation would be expected to be additive. Despite this, Roosz's implied conclusion that N oc G should be viewed with some skepticism. To understand this, one must consider the method used by Roosz et al. to calculate the rates of nucleation and growth for the P—> 7 transformation. Nucleation and growth rates are most commonly measured metallographically [13]. Typically, N is estimated by measuring the number of nodules of product phase per unit volume of sample, Ny, for different times at temperature. The slope of the Ny versus time curve then gives the nucleation rate. The growth rate can be estimated by measuring the size of the largest visible product nodule as a function of time at temperature. The growth rate is then taken as the slope of the size versus time curve. Roosz et al. did not measure their nucleation and growth rates in this manner. Instead, they measured the volume fraction transformed to austenite, X, and the specific interface between the growing 7 phase and the parent pearlite, Sy^. They then used formulae derived by Gokhale et al. [51] to calculate N and G. Gokhale et a/.'s work concerned recrystallization in an Fe-Si alloy. Their derivation assumed that nucleation was taking place at grain edges, that the nucleation sites were Chapter 4. Results and Discussion 99 Figure 4.36: Nucleation and growth rates calculated from data presented by Roosz et al, showing proportionality between N and G. Chapter 4. Results and Discussion 100 saturating locally early in the transformation, and that subsequent growth was therefore occuring in a two dimensional manner, outward from grain edges. Yet Roosz's conclusions concerning the nucleation of 7 from pearlite were that grain corners acted as the effective nucleation sites, and that site saturation did not occur during the transformation, a conclusion which was based primarily on the observed value of 4 for the Avrami time exponent, n. If this were true, then it is unlikely that austenite growth would occur two dimensionally; thus their calculations are very likely in error. Indeed, Gokhale et al. noted that in general, measurement of the volume fraction transformed and the specific interface between parent and product is not considered to be sufficient information from which to derive both N and G. It is believed, therefore, that Roosz's conclusion that N oc G is unrehable. It was not possible to test Calm's early site saturation criterion directly, because nucleation and growth rates were not measured in the present work. As discussed in Sec-tion 4.5.3, however, there is metallographic evidence which suggests that site saturation takes place during the P—> 7 transformation. Kuban's effective site saturation criterion, as expressed for the 7 —>P transformation with n = 2 (Equation 1.33b), was checked by calculating the fraction ^ 2 0 / ^ 9 0 according to the measured isothermal austenitization kinetics. The results are shown in Table 4.15. It is seen that the criterion (t20 > 0.28igo) is met in all cases. It is to be noted that the limit of 0.28 was determined empirically from isothermal 7 —>P data reported in reference [23], and that this criterion would not necessarily apply to the reverse transformation. Never-theless, because the pearlite formation and dissolution kinetics curves exhibited similar shapes (that is, similar values of navg), it is expected that the effective site saturation criterion would imply similar contributions of early nuclei to the overall transformation. It is interesting to examine the effective site saturation criterion in light of kinetics which are expressed in terms of the Avrami equation. Consider the criterion as originally Chapter 4. Results and Discussion 101 Table 4.15: Application of Kuban's effective site saturation criterion [13] to the isothermal P—• 7 transformation. Micro- Temp. structure (°C) ^2o/^90 733.0 0.29 734.0 0.39 QC 738.0 0.39 743.5 0.36 747.5 0.39 731.0 0.31 735.0 0.32 FC 736.0 0.31 742.0 0.31 746.0 0.35 expressed by Kuban [13] (Equation 1.33a). Substitution of Equation 1.29 into the equality ho/Uo = 0.38 yields ln(l-0.2)l n — - — L = 0.38 (4.51) ln(l-0.9)l n -b J The only unknown in Equation 4.51 is n, because b cancels out on the left side. This equation may therefore be solved for a "critical" value of n, here termed recr,-t, which is approximately equal to 2.4. Thus for transformations whose kinetics are expressed in terms of the Avrami equation parameters n and b, those with n > ncrn will meet the effective site saturation criterion, whereas those with n < ncrn will not. Mathematically, this makes a certain amount of sense, when one considers that the shape of an Avrami transformation curve (and therefore the fraction £20/^90) is determined solely by the value of n. It must be noted that the effective site saturation criterion is considered to be a sufficient, not necessary, condition for additivity. It would be incorrect, therefore, to conclude from the above analysis that transformations with high values of n are additive, Chapter 4. Results and Discussion 102 while those with low values of n are not. Campbell's work [1], after all, showed that the 7 —> a transformation was experimentally additive in low and medium plain-carbon steels, with n « 1.0. Metallurgically, it seems reasonable that transformations exhibiting low values of n would be less likely to meet the effective site saturation criterion than those with higher values of n. Consider two transformations, A and B, with = 1 and ng = 2. According to Christian [7], transformation A is expected to exhibit grain boundary nucleation following saturation of corner and edge sites, while B should exhibit nucleation on grain edges following saturation. Clearly, there are a great many more potential sites on grain boundaries than on grain edges. Therefore, saturation of edge sites would be expected to occur earlier (i.e. at a lower value of X) than saturation of boundary sites. Thus the nodules nucleated before t^o in transformation A would be expected to contribute less to the total volume transformed than the corresponding nodules in transformation B. This point is recognized implicitly by Kuban et al. [14]; they note that the effective site saturation criterion is met at a lower value of <2o/^90 for a transformation with n = 2 than for a transformation with n = 4. This discussion can be summarized by saying that the experimentally observed addi-tivity of the P—• 7 transformation in a eutectoid, plain-carbon steel would be expected on the basis of the characterization of the process by the Avrami equation. The isoki-netic condition has been shown to hold in work done by Roosz et al. with a similar steel, though there is evidence that their nucleation and growth rate calculations were made on the basis of faulty assumptions. With an average Avrami time exponent value of n = 2.2, the fraction <2o/*90 is s e e n to meet the effective site saturation criterion as proposed by Kuban et al. [14]. Chapter 5 Conclusions The following conclusions summarize the results and discussion of the experimental work concerned with the kinetics of formation of austenite from pearlite in a eutectoid, plain-carbon steel. 1. The pearlite-to-austenite reversion transformation in a eutectoid, plain-carbon steel was determined to be experimentally additive, as continuous heating transformation kinetics were successfully predicted on the basis of measured isothermal kinetics. 2. The Scheil equation did not predict the onset of the pearlite to austenite trans-formation on continuous heating. This may have been due to uncertainties in the estimates of the isothermal transformation start times. 3. Isothermal transformation kinetics were characterized in terms of the Avrami pa-rameters n and b. An average value of n = 2.2 was taken as an indication, not inconsistent with metallographic observations, of saturation of pearlite colony cor-ner sites and subsequent nucleation on colony edges. 4. Nucleation and growth rate data published for a steel similar to the one used in the present work suggested that austenitization is an isokinetic reaction. It was found, though, that the calculated rates were based upon questionable assumptions, and therefore the P—> 7 reaction cannot necessarily be called isokinetic. The austenite formation data generated in the present work was found to meet the effective site saturation criterion for additivity. 103 Chapter 5. Conclusions 104 5. A nonlinear curve fitting algorithm was found to be an appropriate technique for the analysis of ferrous phase transformation data logged by a dilatometer. 6. In agreement with published results, rate of the P—> 7 transformation was found to vary inversely with the pearlite spacing and colony size in the starting microstruc-ture. Chapter 6 Recommendations Dilatometry has proven to be a useful technique for the study of ferrous phase transfor-mation kinetics, and the Gleeble 1500 is particularly well suited for this sort of work. The present work was hampered to some degree by equipment problems, however the changes in Gleeble operating practice described in this report have at least partially al-leviated the difficulties. More reliable results and higher productivity are sure to follow as our experience with the machine grows. Aside from the specific recommendations already made [52], it is further recommended that a Gleeble 1500 operating practice be drafted in-house, and that it be followed closely in all future research. Some topics to be included should be temperature measurement, prevention of sample oxidation, and periodic maintenance. A nonhnear curve fitting technique has been shown to be a practical method of analysis of dilatometer data. This technique should be considered for use in future phase transformation work. A combination of a more stable dilatometer signal with a nonlinear curve-fitting technique for data analysis could prove useful in providing a more accurate estimate of isothermal transformation start times in ferrous phase transformations. This would help to clarify the applicability of the principle of additivity to the prediction of the onset of nonisothermal transformations. The austenitization process has been shown to meet the effective site saturation cri-terion for additivity, but there remains uncertainty as to whether Cahn's early site sat-uration criterion is met, and as to whether the reaction is isokinetic. A future project 105 Chapter 6. Recommendations 106 could involve the measurement of isothermal nucleation and growth rates for 7 formation from pearlite, in order to more rigorously test these criteria. References [1] P.C. Campbell; Application of Microstructural Engineering to the Controlled Cooling of Steel Wire Rod, Ph.D. Thesis, University of British Columbia, 1989. [2] C. Devadas; Prediction of the Evolution of Microstructure During the Hot Rolling of Steel Strips, Ph.D. Thesis, University of British Columbia, 1989. [3] W.A. Johnson, R.F. Mehl; Trans. Met. Soc. AIME, 135, (1939) p. 416. [4] M. Avrami; J. Chem. Phys., 7, (1939) p. 1103. [5] M. Avrami; J. Chem. Phys., 8, (1940) p. 212. [6] M. Avrami; J. Chem. Phys., 9, (1941) p. 177. [7] J.W. Christian; The Theory of Transformations in Metals and Alloys, Permagon Press, Oxford, 1975. [8] J.W. Cahn; J. Metals, (1957) p. 140. [9] M. Umemoto, N. Nishioka, I. Tamura; J. Heat Treating, 1, [3], (1980) p. 57. [10] M. Umemoto, N. Nishioka, I. Tamura; Proc. Int. Congress on Heat Treatment of Materials, 7-11 Nov. 1983, Shanghai, p. 5.35. [11] E. Scheil; Archiv. fur Eissenhuttenvesen, 12, (1935) p. 565. [12] J.W. Cahn; Acta Metall, 4, [11], (1956) p. 572. [13] M.B. Kuban; Kinetics of Nucleation and Growth in a Eutectoid Plain Carbon Steel, M.A.Sc. Thesis, University of British Columbia, 1983. [14] M.B. Kuban, R. Jayaraman, E.B. Hawbolt, J.K. Brimacombe; Met. Trans. A, 17A, (1986) p. 1493. [15] G.A. Roberts, R.F. Mehl; Trans. ASM, 31, (1943) p. 613. [16] G. Molinder; Acta Metall, 4, [11], (1956) p. 565. [17] G.R. Speich, A. Szirmae; Trans. TMS AIME, 245, (1968) p. 1063. [18] R.R. Judd, H.W. Paxton; Trans. AIME, 242, (1968) p. 206. 107 References 108 [19] C.I. Garcia, A.J. Deardo; Met. Trans. A, 12A, (1981) p. 521. [20] A. Roosz, Z. Gacsi, E.G. Fuchs; Acta Metall, 31, [4], (1983) p. 509. [21] M. Hillert, K. Nilsson, L.-E. Torndahl; J. Iron & Steel Inst, January 1971, p. 49. [22] K.H. Magee; The Application of the Additivity Principle to Recrystallization, M.A.Sc. Thesis, University of British Columbia, 1986. [23] E.B. Hawbolt, B. Chau, J.K. Brimacombe; Met. Trans. A, 14A, (1983) p. 1803. [24] L. Samuels; Optical Microscopy of Carbon Steels, ASM, 1980. [25] P.G. Shewmon; Transformations in Metals, McGraw-Hill Book Company, New York, 1969. [26] D.Z. Yang, E.L. Brown, D.K. Matlock, G. Krauss; Met. Trans. A, 16A, (1985) p. 1385. [27] D.Z. Yang, E.L. Brown, D.K. Matlock, G. Krauss; Met. Trans. A, 16A, (1985) p. 1523. [28] E. Navara, B. Bengtsson, K.E. Easterling; Mat. Sci. & Technol, 2, (1986) p. 1196. [29] E. Navara, R. Harrysson; Scripta Metall, 18, (1984) p. 605. [30] G.R. Speich, R.L. Miller; Proc. Int. Conf. on Solid-Solid Phase Transformations, AIME, (1982) p. 843. [31] Woo Chang Jeong, Chong Hee Kim; J. Mat. Sci., 20, (1985) p. 4392. [32] M.M. Souza, J.R.C. Guimaraes, K.K. Chawla; Met. Trans. A, 13A, (April 1982) p. 575. [33] G.R. Speich, V.A. Demarest, R.L. Miller; Met. Trans. A, 12A, (1981) p. 1419. [34] W.H. Brandt; J. Appl Phys., 16, (March 1945) p. 139. [35] C. Wells, W. Batz, R.F. Mehl; Trans. AIME, 188, (March 1950) p. 553. [36] E.B. Hawbolt, private communication, 1990. [37] J.C. Russ; Practical Stereology, University of North Carolina, Raleigh, NC, 1985. [38] G.F. VanderVoort, A. Roosz; Metallography, 17, (1984) p. 1. [39] E.E. Underwood; Quantitative Stereology, Addison-Wesley, Reading, MA, 1970. References 109 [40] M. Gensarner; Trans. ASM, 30, (1942) p. 983. [41] B. Chau, U.B.C.; private communication, 1990. [42] K.W. Andrews; J. Iron & Steel Inst, (July 1965) p. 721. [43] J.S. Kirkaldy, E.A. Baganis; Met. Trans. A, 9A, (1978) p. 495. [44] G.H. Geiger, D.R. Poirier; Transport Phenomena in Metallurgy, Addison-Wesley Publishing Company, 1980. [45] J. Worobec, Dofasco Inc.; private communication, 1990. [46] Duffers Scientific, Inc.; Dynamic Thermal/Mechanical Metallurgy Using the Gleeble 1500, 2nd ed. [47] G.E. Pellisier, M.F. Hawkes, W.A. Johnson, R.F. Mehl; Trans. ASM, 30, (1942) p. 1049. [48] B. Chau, E.B. Hawbolt; unpublished research, 1989. [49] E.B. Hawbolt, B. Chau, J.K. Brimacombe; Met. Trans. A, 16A, [4], (1985) p. 565. [50] U.R. Lenel; Scripta Metall., 17, (1983) p. 471. [51] A.M. Gokhale, C.V. Iswaran, R.T. Dehoff; Met. Trans. A, 11A, [8], (August 1980) p. 1377. [52] D.J. Riehm; memo to Brimacombe, Chau, and Hawbolt, 25 January 1990. [53] D.W. Bacon; Collection and Interpretation of Industrial Data, Queen's University, Kingston, Ontario. [54] D.G. Berghaus; Experimental Mechanics, 17, (Jan. 1977) p. 14. [55] N.R. Draper, H. Smith; Applied Regression Analysis, 2nd ed., John Wiley Sz Sons, 1981. [56] K. Levenberg; Quart. Appl. Math., 2, (1944) p. 164. [57] D.W. Marquardt; J. Soc. Indust. Appl. Math., 11, [2], (June 1963) p. 431. [58] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling; Numerical Recipes in C, Cambridge University Press, 1988. Appendix A Statistical Techniques Statistical models are widely used in data analysis in most scientific and technical re-search. There is a wide range of techniques available, varying greatly in complexity and flexibility. This section outlines the basics of the linear and nonhnear modelling techniques which have been employed for data analysis in this project. A straight line least squares fit is a special case (the simplest) of statistical modelling. Consider the general case in which the values of k operating variables (or independent variables) are measured in ra tests, and it is desired to model their influence on a response variable y. A statistical model may be expressed in the form [53] where rj and E (y) are the predicted response value (that is, the expected value of y), / is the response approximation function, {£} is the vector of values of the operating variables, and {0} is the vector of parameters being fitted to the data. A residual ru is then defined as the difference between any measured response value yu and its predicted value riu. There are a number of criteria available for estimating the model parameters, all of which seek the "best possible fit" between the data and the model for the ra data points. The most common technique is the least squares method, in which the sum of the squares of the residuals V = E(y) = f({(},{6}) (A.52) m m s ({0}) = E (r«)2 = (Uh, W)]2 (A.53) 110 Appendix A. Statistical Techniques 111 is minimized by setting each of the derivatives "5/({a>w) _ — T i n . . i i t r r < r / i - i i d9i d9i (A.54) «=i to zero. This results in a set of so-called normal equations, which must be solved for the estimated parameters {#}. There are numerous methods for testing the statistical adequacy of a model. The simplest is probably the correlation coefficient, which measures the "goodness of fit". It is calculated as r N &.(*-«' ( ' where y is the arithmetic mean of the measured response values. The value of r as calculated in Equation A.55 varies as 0 < r < 1, with 1 representing a "perfect" fit between the data and the model, and 0 representing no relationship at all between y and the independent variable(s). The correlation coefficient is usually used for single operating variable models. More sophisticated techniques are available for assessing the adequacy of multivariable models [53]. A . l Linear Modelling A mathematical model is defined as linear if it is a linear function of the parameters to be estimated, that is, V = Po + fii^i + P2X2 + ••• + 6pxp (A.56) where xx, x2,..., xp are operating variable parameters which are functions only of the operating variables £1, £25 • • •, £fc> a n d Po, Pi, • • •, PP are the linear model parameters. It dri follows that in a linear model, each of the partial derivatives —- is a function only of one oBi or more of the operating variable parameters x,, and not of any of the parameters /?,-. The normal equations to be solved for j/^ }, then, are linear, and can be solved simultaneously using any of several standard techniques, such as Gaussian elimination. Appendix A. Statistical Techniques 112 A.2 Nonlinear Modelling In a nonlinear model, at least one of the partial derivatives dn Wi is a function of at least one of the model parameters 0,-. The normal equations which arise from setting the derivatives to zero cannot therefore be solved directly; instead, an iterative technique must be used to determine the least squares values of the model parameters. When the Avrami equation is used to characterize isothermal phase transformation kinetics, it is a good example of a nonhnear model of the form rj — 1 — exp (—9\i02), where 9\ = b and 62 = n. As discussed in Section 1.3, the most common method of estimating the Avrami parameters involves a linearization and a simple straight line fit. Linearization may, however, introduce statistical uncertainty into the parameter estimates. Bacon [53] and Berghaus [54] discuss several reasons for this. First, the least squares technique relies on several fundamental assumptions, one of which states that the variance of the random error is constant over the range of the operating variables (t in this case). While this is very often true for most experimental data, it is almost certainly not true for transformed data. Secondly, because linearization involves applying the least squares criterion to a transformed model function and transformed data, the sum of squares of the residuals between the raw data and its approximation are not minimized with respect to anything. Thirdly, the transformed model function may not approximate the transformed data as well as the untransformed approximation matches the untransformed data. Nonlinear fitting techniques are attractive for several reasons. First, they sidestep the above statistical concerns by fitting least squares parameters directly to the data being modelled. Secondly, they are the only method of fitting "intrinsically nonhnear" models, i.e. those for which no linearizing transformation exists. Lastly, specifically in the case of the Avrami equation, a nonhnear modelling technique can be used to solve for more Appendix A. Statistical Techniques 113 than just b and n. The five parameter model of Equation 4.41 employed in the present work, for instance, was of the general form where 8\ - D 0 , 82 = Df, #3 = b, 84 = to, and 85 = n. Numerous methods exist for determining the least squares estimates of nonhnear model parameters. For instance, the normal equations themselves can be solved itera-tively, using a method such as Newton-Raphson. Model parameters may also be esti-the minimum of S. Although both of these methods may work for simple models, it is very difficult to apply them to complex models with several parameters. Alternatively, a number of iterative nonhnear estimation algorithms have been devel-extensive background discussion of linear modelling techniques. The reader is referred to reference [55] for further information. A brief discussion of the three most common methods follows here. The linearization method (not to be confused with the linearizing transformation technique discussed above) assumes that the model is locally linear. This method involves expanding the model function in a Taylor series about some trial parameter estimates. Each successive iteration improves upon the previous estimate of {8}, and eventually the least squares solution is arrived at. This method is numerically efficient in the sense that it converges quickly, but it tends to be oscillitory. It can in many cases fail to converge at all, especially if the initial parameter guesses are far from their least squares values. The gradient method, on the other hand, employs the well-known optimization strat-egy of steepest descent to search the least squares surface for its minimum. This method is highly stable, but inefficient because of the large number of iterations required to reach (A.57) mated by generating values of S < 8 over a grid of values for 8Q, 8\,..., 9P, and locating oped [53,55]. A detailed discussion of each is not warranted here, as it would require an Appendix A. Statistical Techniques 114 the minimum. The so-called Marquardt-Levenberg algorithm has evolved into the standard nonHnear least squares technique. Proposed independently by Levenberg [56] and Marquardt [57], it amounts to a compromise between the Gauss method and the gradient method. It has been found to be stable and relatively insensitive to the "goodness" of the initial parameter guesses, and it converges quickly to the least squares parameter estimates. In the present work, an implementation of the Marquardt-Levenberg algorithm from reference [58] was employed. Appendix B Source Code: Kinetics Model /* ark.c * d j r 18.08.89 * * Numerically predicts Ausenite Reversion Kinetics on continuous heating * of eutectoid p l a i n carbon s t e e l , on the basis of isothermal transformation * data. */ /* Preprocessor #includes */ #include <stdio.h> #include <math.h> /* If using the Sch e i l equation, you must #define SCHEIL. If not, * simply comment i t out. */ /*#define SCHEIL*/ /* The following #define statement t e l l s the compiler which microstructure * i s being used. You must #define either "QC" or "FC", or the program * won't compile. Trust me! */ #define FC #if defined(FC) #define LNBSLOPE 0.24002 #define LNBYINT -185.189 #define T0SL0PE -0.4235 #define T0YINT 316.6129 #define TS025 736.0 #define TS05 738.0 #define TS1 740.0 #define TS25 743.0 #define TS5 748.0 #elif defined(qc) #define LNBSLOPE 0.367023 #define LNBYINT -275.844 #define TOSLOPE -0.20085 #define TOYINT 150.3311 #define TS025 731.0 #define TS05 733.0 #define TS1 737.0 #define TS25 739.0 #define TS5 742.0 #else 115 Appendix B. Source Code: Kinetics Model #error You didn't define a microstructure... #endif /* Other preprocessor #defines */ #define TRUE 1 #define FALSE 0 #define MINRATE 0.25 /* minimum allowable heating rate */ #define MAXRATE 5.0 /* maximum allowable heating rate */ #define COMPLETE 0.9999 /* transformation l i m i t */ #define MAXITS 30000 /* maximum allowable no. of i t e r a t i o n s */ #define BIGSTEP 0.1 /* time step f o r k i n e t i c s calcs */ #define SMALLSTEP 0.001 /* time step f o r Sch e i l equation */ #define NVALUE 2.190 /* value f o r Avrami n */ #define AE3TEMP 718.0 /* Ae3 temp, af t e r Kirkaldy */ #define FILENAME "ark.dat" /* output filename */ /* Function prototypes */ double Caleb (double); in t CalcStartTemp (double, double * ) ; void F a t a l (char * ) ; /* Global variables */ double b.Vt.X,Temp,Rate,trel; FILE *0 u t F i l e ; char ts[50]; /* * main() program * */ main(ac, av) in t ac; char **av; { in t i=0, done=FALSE; /* Get heating rate from command l i n e or from user */ i f (ac != 2) { p r i n t f ("\n\tEnter heating rate: " ) ; i f ((Rate=atof(gets(ts))) == 0) Fa t a l ("Invalid heating ra t e " ) ; > else i f ((Rate=atof(av[l])) == 0) Fa t a l ("Invalid argument"); i f (Rate < MINRATE I I Rate > MAXRATE) Fat a l ("Heating rate out of range"); /* Open output f i l e */ i f ((0utFile=fopen(FILENAME,"w")) == NULL) { s p r i n t f ( t s , "Error opening f i l e */.s" .FILENAME) ; F a t a l ( t s ) ; } Appendix B. Source Code: Kinetics Model /* I n i t i a l i z e variables */ i f (!CalcStartTemp (Rate, ftTemp)) { f c l o s e (OutFile); F a t a l ("Error i n CalcStartTempO: i t e r a t i o n l i m i t reached"); } p r i n t f ("\n\tTransf ormation start temp = */.5.11f deg C\n", Temp); Vt = t r e l = 0.0; X = 0.0; /* Write f i r s t data point to output f i l e */ f p r i n t f (OutFile, ' 7 . 1 f '/.If '/.If\n",trel,Temp,X); /* Calculate f r a c t i o n transformed f o r f i r s t time step, and p r i n t r e s u l t * out to ouput f i l e . */ i++; t r e l += BIGSTEP; b = Caleb (Temp); X = 1.0 - exp ( -b * pow (trel,NVALUE)); f p r i n t f (OutFile/"/.If '/.If '/.lf\n",trel,Temp,X); /* Now s t a r t looping. Keep going u n t i l the transformation i s * complete, or u n t i l the number of i t e r a t i o n s gets excessive. */ while (!done && ++i < MAXITS) { /* F i r s t , update time elapsed, temperature, and b */ t r e l += BIGSTEP; Temp += BIGSTEP * Rate; b = Caleb (Temp); /* Now calculate a v i r t u a l time taken at current temp to reach a * f r a c t i o n transformed of X. */ Vt = exp ( (log(log(1.0/(1.0-X)))-log(b)) / NVALUE); /* Increment the v i r t u a l time, calculate the new f r a c t i o n transformed, * p r i n t r e s u l t s , and fig u r e out i f we're done yet. */ Vt += BIGSTEP; X = 1.0 - exp ( -b * pow (Vt,NVALUE)); f p r i n t f (OutFile, "'/.If '/.If '/.If \n", t r e l , Temp, X) ; done = (X >= COMPLETE); } /* end while */ /* Wrap i t a l l up */ f c l o s e (OutFile); p r i n t f ("\n\n\t0utput i s i n f i l e '/.s\n\n" .FILENAME); i f ( i == MAXITS) { p r i n t f ("\n\n\tIteration l i m i t reached\n\n"); Appendix B. Source Code: Kinetics Model return OxFF; } else return 1 ; } /* end main() */ /* * CalcbO calculates b f o r the Avrami equation, using * data c o l l e c t e d during isothermal testwork. * */ double Caleb (double Temperature) •C return (exp (LNBYINT + LNBSLOPE * Temperature)); } /* * CalcStartTempO calculates the start of transformation temperature * based either on the Scheil equation or on empirical CHT data * */ i n t CalcStartTemp (HRate, STptr) double HRate, *STptr; •C #if defined(SCHEIL) double Sum, Degrees, IsoIncTime; i n t i ; /* I n i t i a l i z e variables. */ Sum = 0.0; Degrees = AE3TEMP; /* Loop, summing the b i t s as we go. */ while (++i < MAXITS) { IsoIncTime = TOYINT + T0SL0PE * Degrees; Sum += SMALLSTEP/IsoIncTime; i f (Sum > COMPLETE) break; Degrees += SMALLSTEP * HRate; > /* Now we're a l l done. Wrap i t up. */ i f ( i == MAXITS) { •STptr =0.0; return FALSE; } else { •STptr = Degrees; return TRUE; } 

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