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Modelling the interaction between recovery, recrystallization and precipitation in AA6111 Go, Johnson 2006

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M O D E L L I N G THE INTERACTION B E T W E E N RECOVERY, RECRYSTALLIZATION A N D PRECIPITATION IN AA6111 by JOHNSON GO B . A . Sc., University of British Columbia, 1998 M . A . Sc., University of British Columbia, 2001 A THESIS 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 DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Materials Engineering) T H E U N I V E R S I T Y OF B R I T S H C O L U M B I A July 2006 © Johnson Go, 2006 ABSTRACT The present work investigates the evolution of microstructure during the annealing of an industrial precipitation hardened aluminum alloy A A 6 1 1 1 . Special emphasis is placed on understanding the interaction between recovery, recrystallization and precipitation through a combination o f experimental and modelling approaches. Experimentally, extensive heat treatments were carried out to study the effect of varying initial precipitate conditions on the isothermal annealing behaviour of the alloy in the temperature range of 250-445°C. A total of four prior aging conditions were considered: naturally aged (T4), peak aged (PA), overaged (OA) and severely overaged (SOA). It was found that recrystallization was severely retarded at the annealing temperature of 325°C irrespective of prior precipitate conditions. Microscopic evidence confirmed that the growth of recrystallizing grains is directly related to the non-uniform spatial distribution of precipitates. Subsequently, a new microstructure model was developed to link the changes in the microstructure to the mechanical properties of the alloys. The model which was developed based on the internal state variable approach is capable of translating quantitatively the interaction between recovery, subgrain growth, recrystallization and precipitation into a yield stress vs. time relationship. The effect o f non-uniform distribution of precipitates was considered explicitly by using a simple rule of mixtures. The validity of the model was verified by comparing the model calculations to the experimental data obtained from overaged 40% cold rolled A A 6 1 1 1 . i i TABLE OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS i i i LIST OF TABLES vi LIST OF FIGURES vii LIST OF S Y M B O L S xii ACKNOWLEDGEMENTS xiv Chapter 1 INTRODUCTION 1 Chapter 2 LITERATURE REVIEW. 4 2.1 Recovery 5 2.1.1 General Observations 5 2.1.2 Recovery Modell ing 6 2.2 Recrystallization 10 2.2.1 General Observations 10 2.2.2 Recrystallization Modell ing 10 2.3 Interaction Between Recovery and Recrystallization 14 2.4 Precipitation 15 2.4.1 Precipitation Hardening Behaviour o f A A 6 1 1 1 16 2.4.2 Precipitation Modell ing 20 2.5 Interaction Between Precipitation and Recovery 22 2.6 Interaction Between Precipitation and Recrystallization 24 2.6.1 Precipitate Pinning 25 2.6.2 Solute Drag 33 2.7 Critical Assessment of the Literature 38 Chapter 3 SCOPE A N D OBJECTIVES 40 Chapter 4 EXPERIMENTAL METHODOLOGY 42 4.1 Starting Materials 42 4.2 Sample Preparations 42 4.3 Heat-treatment Experiments 45 4.3.1 Solution Heat-treatments 45 4.3.2 Artif icial Aging 46 i i i 4.3.3 Isothermal Annealing 47 4.4 Sample Characterization 47 4.4.1 Sample Preparation 49 4.4.2 Optical Metallography 51 4.4.3 Scanning Electron Microscopy 52 4.4.4 Electron Back Scattered Diffraction 52 4.4.5 Transmission Electron Microscopy 53 4.4.6 Resistivity Measurements 53 4.4.7 Tensile Measurements 54 Chapter 5 EXPERIMENTAL RESULTS A N D DISCUSSION 56 5.1 Experimental Results 57 5.1.1 Solution Heat Treatments 57 5.1.2 Artificial Aging 60 5.1.3 The Deformed State 67 5.1.4 Isothermal Annealing at 325°C 76 5.1.4.1 Evolution of Y i e l d Stress 76 5.1.4.2 Evolution o f Resistivity 78 5.1.4.3 Evolution of Microstructure 78 5.1.5 The Effect o f Annealing Temperature on Overaged Samples 93 5.1.6 T E M Studies o f the Annealing Behaviour o f Overaged Samples.... 97 5.2 Discussion of Experimental Results 100 5.2.1 F low Stress Addit ion Law 105 5.2.2 Evolution o f Resistivity 106 5.2.3 The Effect o f Prior Aging Conditions 110 5.3 Concluding Remarks 114 Chapter 6 M O D E L L I N G OF MICROSTRUCTURE EVOLUTION IN O V E R A G E D AA6111 115 6.1 Model Development - The Internal State Variable Approach 116 6.2 Recovery 121 6.3 Subgrain Growth 125 6.4 Recrystallization 128 6.4.1 Nucleation 130 6.4.2 Growth Rate 132 6.5 Precipitation 135 6.5.1 Dissolution/Growth 135 6.5.2 Coarsening 138 6.6. Parameter Identification 140 6.6.1 Solubility Product for Q' Precipitates 140 6.6.2 Molar Volume of Q' Precipitates 141 6.7 Mode l Implementation 141 6.8 Comparison with Experimental Results 144 6.9 Discussion of Modell ing Results 151 iv 6.9.1 Interaction Between Recovery and Precipitation 151 6.9.2 Interaction Between Recovery, Precipitation and Recrystallization 155 6.9.3 Composite Microstructure 155 6.10 Summary and Limitations of the Overall Mode l 157 Chapter 7 S U M M A R Y A N D CONCLUSIONS 161 7.1 Future Work 164 REFERENCES 165 V LIST OF TABLES Table 2.1. The value of ideal J M A K exponent as a function of growth dimensionality 13 Table 2.2. The effect of prior precipitate conditions on the recrystallization completion time for a 70% cold rolled high purity A l - M g - S i alloy 30 Table 4.1. Chemical composition of AA6111 in wt% 43 Table 4.2. Four step procedure for grinding and polishing aluminum alloys using a Phoenix 4000 automatic polisher 50 Table 5.1. Comparison of the recrystallization behaviour of 40% cold rolled AA6111 with varied precipitate conditions after annealing for 2 weeks at 325°C 82 Table 5.2. Summary o f the flow stress contributions from precipitates and dislocations as a function of precipitation state 107 Table 5.3. Summary of the softening and recrystallization behaviour of 40% cold rolled AA6111 with various precipitate conditions at the annealing temperature o f 325°C I l l Table 6.1. List o f internal state variables for recovery, subgrain growth, recrystallization and precipitation considered in the present modelling approach 117 Table 6.2. Reported diffusion data for M g , C u and Si atoms and self diffusion in bulk aluminum 124 Table 6.3. Comparison between calculated and measured Rp from T E M 145 Table 6.4. List o f adjustable parameters and their optimized values 146 Table 6.5. List o f physical parameters which were fixed in the model 147 v i LIST OF FIGURES Fig. 2.1. Recovery kinetics o f industrial processed A A 6 111 7 Fig. 2.2. Sigmoidal plot of typical recrystallization kinetics during isothermal annealing. ... 11 Fig. 2.3. Evolution of yield stress vs. aging time of AA6111 in the temperature range of 160-220°C 17 Fig. 2.4. Bright field T E M image showing the precipitate structure in AA6111 after aging for 7 hours at 180°C. N l and N 2 denote B" needles seen end-on and edge-on respectively 19 Fig. 2.5. Schematic showing the effect of precipitate size, volume fraction and prior strain (s) on recrystallization kinetics and mechanism. 26 Fig. 2.6. Isothermal recrystallization kinetics showing the effect o f precipitation which occurs at various stages during recrystallization 28 Fig. 2.7. Schematic representation of grain boundary migration controlled by (a) the transformation o f B' to /?precipitates and (b) local coarsening o f B precipitates at the recrystallization front 32 Fig. 2.8. The effect o f copper concentration on the migration rate o f boundaries in aluminum at various temperature 35 Fig. 2.9. The effect o f M g contents on the time to achieve 50% recrystallization at 275°C of a 95% cold rolled A l - M g alloy 36 Fig . 4.1. Optical micrograph showing the elongated grain structure i n the as received hot rolled AA6111 ; 44 Fig . 4.2. Schematic of the heat treatment and rolling experiments 48 Fig. 4.3. The effect o f natural aging on annealed samples. Samples G l and G2 were overaged prior to 40% cold rolled 55 Fig. 5.1. (a) Optical micrograph showing the solution treated microstructure with average grain size o f - 4 5 um, (b) E B S D micrograph of the same sample with grain size o f ~42 um 58 Fig. 5.2. S E M micrograph showing the solution treated microstructure 59 v i i Fig. 5.3. Age hardening curve showing the as aged yield stress o f samples with varied precipitate conditions: 61 Fig. 5.4. Plastic portion of the stress strain curves for (a)supersaturated solid solution (SSS), naturally aged and peak aged samples and (b) overaged and severely overaged samples. Note the serrated flow in the SSS and S O A samples 62 Fig. 5.5. (a) Bright field T E M image showing the lath shaped Q' precipitates (Li and L 2 seen end on and edge on respectively) in the O A sample and (b) corresponding diffraction pattern taken along the [001] zone axis of aluminum 64 Fig. 5.6. Bright field T E M images showing (a) the lath shaped Q' precipitates and (b) large square shaped Mg2Si particles in the S O A sample 65 Fig. 5.7. Optical micrographs showing the microstructure o f the as quenched sample after aging for 7 days at 250°C (OA) and 325°C (SOA) . The average grain size in both micrographs is ~43 um 66 Fig. 5.8. Precipitation hardening curves of AA6111 at 300 and 325°C. The as solution heat treated yield stress is indicated at the intercept with the vertical axis 68 Fig. 5.9. The as aged and as cold rolled yield stress o f samples with varied precipitate conditions. Note the significant increase in yield stress in the T4 sample after 40% cold rolling 70 Fig . 5.10. Optical micrograph showing the deformed microstructure o f a 40% cold rolled O A sample 71 Fig. 5.11. S E M micrograph showing the deformed microstructure o f a 40% cold rolled O A sample 72 Fig. 5.12. Dislocation structures in (a) S O A and (b) O A samples after 40% cold deformation 73 Fig. 5.13. T E M micrographs showing segments of fractured Q' precipitates in 40% cold rolled S O A sample. Courtesy of Dr. X. Wang 74 Fig. 5.14 (a) Distribution of insoluble Fe-rich constituent particles in the deformed matrix o f a deformed T4 specimen, (b) close up view o f one o f the particles and (c) X-ray spectrum indicates the presence o f Fe, S i , M n , and C u in the particle shown in (b) 75 Fig. 5.15. Isothermal evolution of yield stress during annealing at 325°C for 40% cold rolled (a) T4 and P A samples and (b) O A and S O A samples 77 v i i i Fig. 5.16. Evolution of resistivity during annealing o f AA6111 with various precipitate conditions. The prior aging conditions are indicated in the inset and the resistivity o f the as cold rolled and as solution treated samples are the intercepts with vertical axis 79 Fig. 5.17. Optical micrographs showing the recovered microstructure of 40% cold rolled T4, P A and O A specimens annealed for 1 minute at 325°C 80 Fig. 5.18. E B S D maps showing the partially recrystallized microstructures o f 40% cold rolled AA6111 with varied precipitate conditions after annealing for 2 weeks at 325°C 83 Fig. 5.19. Recrystallized grain size distribution in partially recrystallized (a) T4, (b) P A and (c) O A samples after annealing for 2 weeks at 325°C 84 Fig. 5.20. E B S D map showing the colonies of large recrystallized grains in 40% cold rolled O A sample annealed for 2 weeks at 325°C 85 Fig . 5.21. E B S D maps showing the partially recrystallized microstructures o f 40% cold rolled AA6111 with varied precipitate conditions after annealing for 40 d a y s a t 3 2 5 ° C 87 Fig. 5.22. Evolution o f (a) fraction recrystallized (b) recrystallized grain size and (c) number of recrystallized grains per unit area during isothermal annealing a t 3 2 5 ° C 88 Fig. 5.23. Spatial distribution of precipitates in (a) T4 and (b) O A samples annealed for 2 weeks at 325°C 89 Fig. 5.24. (a) S E M micrograph showing the boundary between precipitate and precipitate free zones in 40% cold rolled O A sample annealed for 2 weeks at 325°C. (b) X-ray spectrum showing the presence of M g , Si and C u in one of the precipitates 91 Fig. 5.25. (a) S E M micrograph showing the partially recrystallized microstructure of 40% cold rolled O A sample after annealing for 2 weeks at 325°C. (b) E B S D band contrast map showing the same area of the microstructure 92 Fig. 5.26. S E M micrograph using back scattered electrons showing a recrystallized grain nucleated from a Fe-rich intermetallic particle surrounded by precipitate free zones in 40% cold rolled O A sample annealed for 2 weeks at 325°C 94 Fig. 5.27. The effect of annealing temperature on the softening behaviour of 40% cold rolled (a) O A sample and (b) S O A sample. The as cold rolled yield stress is indicated at the intercepts with the vertical axis 95 ix Fig. 5.28. Isothermal recrystallization kinetics of 40% cold rolled AA6111 at (a) 325°C and (b) 445°C with varied precipitate conditions 96 Fig. 5.29. Ful ly recrystallized microstructure of the 40% cold rolled (a) O A and (b) S O A samples annealed for 100 minutes at 445°C 98 Fig. 5.30. T E M micrograph showing the formation of cell structure in 40% cold rolled O A sample annealed for 2 minutes at 325°C. Courtesy of Dr. X. Wang 99 Fig. 5.31. T E M micrograph showing the pinning of dislocations by precipitates in 40% cold rolled O A sample annealed for 2 minutes at 325°C 101 Fig. 5.32. T E M micrograph showing the transformation o f cell structure into subgrains in 40%) cold rolled O A sample annealed for 7 days at 325°C 102 Fig. 5.33. T E M micrographs showing subgrains formation in 40% cold rolled S O A sample after annealing for 100 minutes at 325°C. Courtesy of Dr. X. Wang 103 Fig. 5.34. (a) T E M micrographs showing a recrystallized grain embedded in the deformed matrix o f the O A samples annealed for 7 days at 325°C. T E M micrographs showing segment o f the migrating grain boundaries are shown in(b)and(c) 104 Fig. 6.1. Schematic outline o f the overall model framework 142 Fig. 6.2. Comparison between model calculated and experimental (a) softening and (b) recrystallization curves for 40% cold rolled overaged AA6111 during isothermal annealing at 325 and 445°C 149 Fig . 6.3. Comparison between predicted and experimental (a) softening and (b) recrystallization kinetics for 40% cold rolled severely overaged AA6111 during isothermal annealing at 325 and 445°C 150 Fig. 6.4. Softening due to recovery and precipitate coarsening in the precipitate zones o f 40% cold rolled overaged AA6111 isothermally annealed at 325°C 152 Fig. 6.5. (a) Evolution of precipitate radius during isothermal annealing of 40% cold rolled overaged AA6111 at 325 and 445°C and (b) corresponding evolution of concentration and precipitate volume fraction at 325°C 154 Fig. 6.6. Plots showing the time evolution of Zener pinning pressure vs. the stored energy in the precipitate zones of 40% cold rolled overaged A6111 annealed at 325°C 156 x Fig. 6.7. Comparison of (a) softening kinetics and (bj recrystallization kinetics in the precipitate and precipitate free zones during the isothermal annealing of 40% cold rolled overaged AA6111 at 325°C 1 x i LIST OF SYMBOLS Symbols Definitions/Values Ac Area o f a critical recrystallization nucleus (m 2) a Lattice parameter for Q' precipitates, 1.04 nm b Magnitude o f Burger's vector, 0.286 nm c Lattice parameter for Q' precipitates, 0.405 nm CM Residual solute concentration in matrix (CMS+CCU+CSI) cP Solute concentration in Q' precipitates, 81 at.% rMg ^eq Equilibrium concentration of M g in solid solution, at. % rCu ^eq Equilibrium concentration of C u l n solid solution, at. % rSi ^eq Equilibrium concentration of Si in solid solution, at. % Q Solute concentration at precipitate-matrix interface, at. % Co Nominal solute concentration of in A A 6 1 1 1 , 1.76 at.% Nominal concentration of M g in A A 6 111, 0.88 at.% nCu C 0 Nominal concentration of C u in A A 6 1 1 1 , 0.30 at.% Nominal concentration of Si in A A 6 1 1 1 , 0.58 at.% CM% Instantaneous concentration of M g in solid solution, at. % Ccu Instantaneous concentration of C u in solid solution, at. % Csi Instantaneous concentration of Si in solid solution, at. % DeQ Effective diffusivity of solute atoms (m 2/s) Do Diffusion coefficient for solute atoms (m 2/s) d Grain diameter (m) ds Solution treated grain size (m) drex Recrystallized grain size (m) E Young's modulus, 70 M P a F Volume fraction of precipitate free materials Fc Coarsening fraction FD Precipitate volume fraction G Shear modulus o f aluminum, 26 GPa K, Constant used to determine the equilibrium concentrations o f solutes in solution, 2 .8x10 2 8 k Constant that determines the potency of nucleation sites for recrystallization kc Constant in the coarsening fraction equation la Activation length for dislocation motions during recovery M Taylor factor ~ 3.1 Mb Grain boundary mobility (m 3/N-s) M0 Pre-exponential factor for mobility (m 3/N-s) NA Avogadro's number, 6.023x10 2 3 mol" 1 ND Number of precipitate particles per unit volume (m"3) x i i Nats Number of dislocation nodes,, assumed ~ 0.5pl 5 Nrex Number o f recrystallized nuclei per unit volume (m"J) Nr Number o f recrystallized grains per unit area (m2) NFe Number of Fe-rich particles per unit volume ( m 3 ) n Number o f atoms per unit cell for Q' precipitates, 21 Pd Driving pressure from dislocations (stored energy per unit volume) (Pa) QD Activation energy for solutes diffusion (J/mol) Qrex Recrystallization activation energy (J/mol) Qo Recovery activation energy (J/mol) R Recrystallized grain radius (m) RFe Fe-rich particle radius (m) Gas constant (J/mol-K) RP Precipitate radius (m) Sv,gb Deformed grain boundary area per unit volume (m"1) Sv.Fe Fe-rich particles surface area per unit volume (m"1) T Temperature (K) t Time V Activation volume for recovery (bJ) vm Molar volume of Q' precipitates, 1.09xl0" 5 mVmol X Volume fraction of recrystallized grains x, Volume fraction of recrystallized grains in precipitate free regions X„ Volume fraction of recrystallized grains in precipitate regions AH0 Standard enthalpy o f Q' phase dissolution/precipitation reactions, 495 kJ/mol a Constant in the order of 0.3 Constant used to determine the intrinsic grain boundary mobility ap Effective pinning parameter as Shape factor (~1) 8Sb Average subgrain radius (m) K Geometrical constant of order of unity High angle grain boundary velocity (m/s) Debye frequency, 8.11 x 10 1 2 s"1 pd Dislocation density (m"2) Q Aluminum atomic volume (m 3) °i Flow stress of precipitate free regions (Pa) On Flow stress of precipitate regions (Pa) Odis Dislocation contributions to flow stress (Pa) Oi Intrinsic flow stress of aluminum alloy, 10 M P a Ob Precipitation hardening yield stress (Pa) Ore Instantaneous recovered yield stress (Pa) Orex Precipitate-free fully recrystallized flow stress, 40 M P a Yi Precipitate-matrix interfacial energy, 0.35 J /m2 Ysb rj 1,1 " •••• • ' " L o w angle grain boundary energy, 0.16 J/m Yzb High angle grain boundary energy, 0.324 J /m 2 x i i i ACKNOWLEDGEMENTS I can't begin to describe my deep gratitude to my supervisors, Professors Warren Poole and Matthias Mili tzer . Many o f the ideas in the model were conceived during many hours of stimulating discussion with Professor Poole. I am indebted to Professor Mil i tzer who first offered me the opportunity to work on a. project in recrystallization. I have benefited immensely on both personal and professional level under their guidance over the past few years. O f course, I cannot forget the advice provided by Dr . M a r y Wells who is always so generous in sharing her thoughts and knowledge. The financial support from Alcan and N S E R C is gratefully acknowledged. Special thanks are extended to Dr. Xiang Wang at McMaster University for carrying out the T E M work. I wish to acknowledge my roommates in A M P E L 260, especially Dr . Fateh Fazeli for being so patient with me on many occasions. I would also like to thank Babak Raeisinia and Sujay Sarkar for their help in printing this thesis. This thesis is dedicated to the memory o f my father. I want to thank my family for their love in particular my wife Lee Leng for her constant encouragement throughout the course o f my PhD studies. xiv Chapter 1 Introduction The drive to produce high quality advanced aluminum sheet metals for the automotive industry has been intensifying over the past two decades, primarily due to the fierce competition with the steel industry. The aluminum sheet metal producers are faced by a number of significant challenges which can be grouped into two major categories. First, there are increased demands from car manufacturers for improved strength and formability. Strength is required for structural applications where steel has been traditionally the dominant material. On the other hand, highly formable aluminum sheets are required for body-in-white applications which offer the greatest potential for weight reduction. Unfortunately, from a metallurgical point o f view, strength and formability are two incompatible properties; an increase in strength in the materials is usually associated with some degree of loss in formability or vice versa. The second major challenge facing the industry lies in the ability to manufacture high quality products consistently at minimum cost. The key to overcoming these challenges is through innovative product and process improvements. After years of intensified research, advanced aluminum alloys with good combination of strength and formability are now available specifically for automotive applications, for example, the AA5754 and AA6111 aluminum alloys. The heat treatable 6xxx series alloys were developed specifically for outer panel applications due to its high dent resistance. For this alloy, it is critical to control the development o f texture in the materials 1 Chap. 11ntroduction throughout the entire processing chain since the surface quality after painting is directly related to the distribution of texture near the sheet surface. Furthermore, the isotropy o f the materials in terms o f mechanical properties can be significantly enhanced by the presence o f favourable texture components. One of the most effective means to control texture is by modifying the recrystallization behaviour o f the materials. For example, a more random recrystallized texture can be obtained i f particle-stimulated-nucleation (PSN) is the dominant nucleation mechanism. Hence, in order to exploit the full potential of these advanced alloys through thermomechanical processing, sheet metal producers are required to have an increasingly sophisticated understanding of the linkage between processing conditions and the evolution of microstructure in the materials. Consequently, much of recent collaborative research between academia and industry have been directed towards developing quantitative models based on the concept of microstructure engineering. This modelling concept which has been developed since the 1970s focuses on applying fundamentally based mathematical models to predict the product properties as a function of processing parameters by modelling microstructure evolution. A n accurate microstructure model, once established and verified, can be employed to systematically manipulate all existing processing parameters with material properties as a result. Therefore it provides a valuable tool for the industry to tailor their forming and heat treatment operations to achieve the most desirable product properties without the need to carry out expensive and time-consuming plant trials which are often based on the trial and error approach. Comprehensive microstructure models are currently available for the hot rolling o f aluminum alloys [Sellars, 1997 ,Wells et al., 1998]. However, comparatively little attention has been "2 Chap. 11ntroduction given to the annealing of cold rolled alloys. This is somewhat surprising because the annealing process constitutes one of the most critical processing steps in the production of sheet metals. It is usually the final heat treatment step where the microstructure of the alloys can be modified before the products are shipped to customers. More recently, microstructure models capable of predicting softening behaviour and recrystallized grain size have been developed for the continuous annealing of non-heat-treatable aluminum alloys [Go et al., 2003]. However, modelling work o f a similar nature has not yet been extended to the annealing of precipitation-hardenable alloys. Hence, the goal of the present work is to develop a basic modelling framework based on the concept of microstructure engineering for the annealing of cold rolled precipitation strengthened alloys. The commercially significant A l - M g - S i - C u AA6111 aluminum alloys has been chosen as the subject o f the present investigation. The experimental data generated from the present study represents the first systematic set o f recovery and recrystallization data for cold rolled AA6111 with a wide range of precipitate conditions. Based on these data, a first generation microstructure evolution model is proposed that is focused on the interaction between recovery, recrystallization and precipitation. Texture evolution is beyond the scope of the present work. The microstructure model developed in the present work can be seen as an important step towards the development of a comprehensive process model for the annealing o f heat-treatable aluminum alloys. 3 Chapter 2 Literature Review Prior to annealing, sheet metals undergo a cold rolling process where the strength of the material is greatly enhanced due to the accumulation of crystalline defects in the structure, mainly in the form of dislocations. During annealing, the deformed microstructure which is highly loaded with dislocations is eliminated through the processes of recovery and recrystallization. In materials o f medium or high stacking fault energy, significant subgrain growth may also occur prior to the onset of recrystallization. A l l these microstructural processes are significantly influenced by the precipitate condition in the deformed state. Depending on the annealing temperature, the pre-existing precipitates can either dissolve or coarsen concurrently with recovery and recrystallization. If the deformed matrix is supersaturated with solid solution, nucleation and growth of new precipitates becomes viable during annealing. Hence, annealing of cold rolled precipitation hardening alloys represents a complex process which may involve the interaction of various metallurgical phenomena. In this chapter, we w i l l focus on the three principle microstructural processes of recovery, recrystallization and precipitation. O f particular importance is the interaction between the various processes and their effect on the properties of the materials. The literature review is organized in the following order: First, the individual processes of recovery and recrystallization are reviewed in sections 2.1 and 2.2, respectively. The competing nature of recovery and recrystallization is delineated in section 2.3. Section 2.4 reviews the precipitation hardening behaviour of A l - M g - S i - C u AA6111 aluminum alloy (2.4.1) and 4 Chap. 2 Literature Review existing theoretical models for precipitation (2.4.2). The interaction between precipitation and recovery is discussed in section 2.5. In Section 2.6 the two most commonly observed interaction between precipitation and recrystallization are assessed. The chapter concludes with a critical assessment of the literature in an effort to accentuate the importance of integrated modelling and identify areas where improvement is most needed (section 2.7). 2.1 Recovery 2.1.1 General Observations Recovery refers to a process that involves the rearrangement and annihilation of dislocations into lower energy configurations. The palpable difference between recovery and recrystallization is that the former does not involve the sweeping of the deformed microstructure by migrating grain boundaries. The dislocation motions can be accomplished by either glide, climb or cross-slip [Nes, 1995, Kuhlmann-Wilsdorf, 2000]. In alloys of medium to high stacking fault energy, the rearrangement of dislocations often leads to the formation of subgrain structures with substantial size [Humphreys and Hatherly, 1995]. Several authors have suggested that subgrains that acquire a critical size eventually become the nuclei o f subsequent recrystallization [Doherty et al., 1997]. However, more research is still required to clarify the underlying mechanism for the transition to occur. Unlike recrystallization which can be quantified by measuring fraction transformed, recovery kinetics is usually followed by measuring the changes in the physical or mechanical properties of the materials, mainly because direct quantification of changes in dislocation density is difficult. 5 Chap. 2 Literature Review The recovery behaviour of industrial processed AA6111 has been investigated previously at U B C as part of a strategic project aimed at studying the continuous annealing behaviour of aluminum alloys [Go et al., 2001a]. The results are shown in Fig. 2.1 where the recovery kinetics is characterized by the softening in yield stress as a function o f annealing time. From Fig. 2.1a, it can be observed that the degree of softening due to recovery increases significantly with increasing annealing temperatures. The effect o f prestrain on softening is shown in Fig. 2.1b. It is evident that all the softening curves, irrespective of annealing temperatures or prior reduction, follow a logarithmic time decay. This logarithmic time dependence of softening is typical of the recovery process which has been observed in both steel [Mukunthan and Hawbolt, 1996] and aluminum alloys [Barioz et al., 1992, Burger et al., 1995, Verdier et al., 1996, G o et al., 2001b]. 2.1.2 Recovery Modelling During recovery, the annihilation and rearrangement o f dislocations into lower energy configurations can be accomplished by a combination of glide, climb and cross slip. Hence, in order to formulate a rate equation to describe the logarithmic time decay o f flow stress during recovery, the concern becomes to identify the correct rate controlling mechanism. In addition to the three basic mode of dislocation migration, solute drag becomes an alternative rate controlling mechanism in solute containing alloys. In the literature, a variety o f quantitative models based on the four different rate controlling mechanisms: glide, thermally activated climb and cross slip and solute drag have been developed [Friedel, 1964, Humphreys and Hatherly, 1995, Nes and Saeter, 1995, Kuhlmann-Wilsdorf, 2000]. However, many of the assumptions in these models are difficult to verify by experimental observations, mainly 6 Chap. 2 Literature Review 280 260 H 03 Q= 240 jg 220 co ? 200 > 03 Q_ CO 180 160 260 240 220 200 180 160 (a) • 175°C O 200°C • 225°C 250°C 60% cold rolled 101 <-*-f— 10 2 Time (s) (b) Annealing Temp. = 200°C 60% 40% 10 3 I i 1111 "" 101 I 1111 102 n 1 — n 104 105 103 Time (s) Fig. 2.1. Recovery kinetics of industrial processed A A 6 1 1 1 : (a) effect of annealing temperatures and (b) effect of prestrain. The as cold rolled yield stress is 230 and 253 M P a for 40% and 60% cold reduction, respectively [Go et al., 2001a]. .7 Chap. 2 Literature Review due to the difficulties in directly observing dislocation motion during recovery. Consequently, the models are often fitted to experimental results such as recovery softening curves by using a large number of adjustable parameters. A comprehensive review o f these modelling approaches can be found in a paper by Nes (1995). There are two dominant physically based models in the literature that have been developed to describe the characteristic logarithmic time dependence of recovery. The first approach is a single structural parameter model based on the overall dislocation density developed by Verdier et al. (1999). The mathematical formulation of the model follows the original theory proposed by Friedel (1964) in which recovery is assumed to occur by thermally activated dislocation motion. In this model, the instantaneous yield stress o f the material, cr, during recovery is given by a kinetic law of the form: da, 64 o \ 2 ( QQ\ . , (cjy\ dt 9M6al E { kTJ \kT ) In Equation 2.1, Q0 and V denote the activation energy and volume o f the elementary recovery event, respectively. vD is the Debye frequency, M is the Taylor factor and E is the Young's modulus. The parameter a is a constant of the order of 0.3 for aluminum. B y using Qo and V as fitting parameters, the model has been shown to be accurate in describing the recovery softening o f A l - M g alloys with different amounts o f prestrain [Verdier et al., 1999, Goer al, 2003]. 8 Chap. 2 Literature Review The second modelling approach is more rigorous in terms o f the number of structural parameters used. According to Nes (1995), recovery softening is best modelled by following the evolution of two parameters, namely, the cell/subgrain size, Ss and the dislocation density, Pd- The general form of the model is given by [Nes and Saeter, 1995]: where cr0 is the frictional stress of aluminum and b is the magnitude of the Burger's vector, ccj and 0(2 are constants with values around 0.3-0.5 and 2-3, respectively. This model has been successfully employed to interpret the recovery behaviour o f high purity iron, aluminum and A l - M g alloys [Nes, 1995]. Although it appears that the model proposed by Nes is more complete in describing the recovered microstructure, implementation of the model can become very complex since additional assumptions must be postulated in order to separate the flow stress contributions from dislocation density and subgrains. Consequently, additional adjustable parameters are required in order to fit the model to experimental data. In contrast to the argument by Nes, Verdier and coworkers have convincingly shown that recovery subgrain growth in commercial purity A l - M g alloys does not change the logarithmic time dependence o f yield stress [Verdier .et al., 1997a, 1997b, 1998]. Hence, they concluded that a single parameter approach based on dislocation density is sufficient to model recovery softening. The advantage o f the Verdier et al.'s model is that it can be easily integrated and applied to non-isothermal conditions with only two fitting parameters, the activation energy and volume, as demonstrated by Go et al., (2.2) (2003). 9 Chap. 2 Literature Review 2.2 Recrystallization 2.2.1 General Observations Recrystallization is accomplished by the migration o f high angle grain boundaries into the deformed or recovered matrix leaving behind a new set of dislocation free grains. Hence, the progress o f recrystallization can be followed directly by plotting the evolution of volume fraction recrystallized grains as a function of annealing time. Such a plot has the characteristic sigmoidal form o f Fig. 2.2. A s indicated in Fig. 2.2, an incubation time is usually observed prior to the start of recrystallization. Nucleation is followed by an increasing rate of recrystallization and then a decreasing recrystallization rate when recrystallizing grains start to impinge on their neighbours. Experimentally, the volume fraction o f recrystallized grains can be measured directly from optical micrographs. A comprehensive analysis of the quantitative metallography technique for recrystallization has been provided by Orsetti-Rossi and Sellars (1997). Due to the recent advancement in electron-backscattered diffraction (EBSD) , there has been a surge in using E B S D data to estimate fraction of recrystallized grains [Humphreys, 2001]. In addition to these microstructural techniques, recrystallization kinetics can also be quantified from softening measurements provided that the effect o f recovery is separated [Chen et al, 2002, Go et al, 2003]. 2.2.2 Recrystallization Modelling The most widely used analytical approach to describe the evolution o f volume fraction of recrystallized grains, X with respect to time, t follows the theory developed independently by 10 Chap. 2 Literature Review 1.0 H 3 0 8 N 1 0.6 o CD (£ 0.4 c o I 0.2 0.0 J Nucleation Incubation time Impingement of growing grains Log Time Fig. 2.2. Sigmoidal plot of typical recrystallization kinetics during isothermal annealing. 11 Chap. 2 Literature Review Kolmogorov (1937), Johnson and Meh l (1939) and Avrami (1940). The model is commonly known as the J M A K model. Assuming random distribution of nuclei, the standard mathematical form of the model is written as where B is a temperature dependent parameter and n is known as the J M A K exponent. The effect of nucleation and growth rates are embedded in the kinetic parameter B. Equation 2.3 was derived based on the assumption that growth rate is constant during recrystallization. There are two limiting cases in terms of nucleation: in the case of site saturation, i . e., all nucleation events effectively take place at the onset of recrystallization, then n = 3. If the nucleation rate is constant during recrystallization, then n = 4. The effect of growth dimensionality of the recrystallizing grains on the J M A K exponent is summarized in Table 2.1 [Humphreys and Hatherly, 1995]. The application of the J M A K approach to model recrystallization has been critically assessed in a recent review article by Vandermeer (2001). Overall, there are two main criticisms associated with the J M A K model. First, the experimentally determined J M A K exponent is often less than 3 even though the recrystallized grains are essentially equaixed (i.e., 3-d growth). Secondly, the assumption of constant growth rate is not expected to hold i f the driving force for recrystallization diminishes as recrystallization proceeds. B y reviewing a large body o f experimental data in the literature, Humphreys and Hatherly (1995) have concluded that the cause of these inconsistencies can be directly attributed to the heterogeneity (2.3) 12 Chap. 2 Literature Review Table 2.1. The value o f ideal J M A K exponent as a function o f growth dimensionality. Growth dimensionality Site saturation Constant nucleation rate 3-d 3 4 2-d 2 3 1-d 1 2 13 Chap. 2 Literature Review of the deformed structure. Inhomogeneous distribution of stored energy leads to variation of growth rate during recrystallization. Furthermore, recrystallization nucleation occurs first in regions with the highest stored energy and this contradicts the basic assumption of random distribution o f nuclei in the J M A K model. In view of these shortcomings, relaxed J M A K models in which the growth rate is a decreasing function of time were subsequently developed by various authors [Vandermeer and Rath, 1989, Furu et al., 1990, Rios, 1997]. However, the requirement of a random distribution of nuclei remains as a serious limitation of the J M A K model. 2.3 Interaction Between Recovery and Recrystallization During annealing, recovery and recrystallization compete for the same dislocation stored energy accumulated in the matrix during the cold working process. The classic early work by Vandermeer and Gordon provides the first direct experimental evidence to show that the driving force for recrystallization of aluminum alloys containing small amounts of copper is severely reduced by recovery [Vandermeer and Gordon, 1962]. In the literature, recovery and recrystallization are frequently treated as sequential processes but many aspects of the transition from recovery to recrystallization are still poorly understood. Several authors have suggested that recovery facilitates the nucleation of recrystallized grains by the mechanism of subgrain coalescence [Humphreys and Hatherly, 1995, Doherty et al., 1997]. However, careful analysis of experimental data has revealed that in most cases, recrystallization nucleation is close to being site saturated [Vandermeer and Rath 1989, Orsetti-Rossi and Sellars, 1999]. This is particularly the case for industrial alloys which have been severely cold rolled. In this case, the competing nature between recovery and recrystallization can be readily modelled by 14 Chap. 2 Literature Review using a time dependent growth rate, G without the need to consider in detail the nucleation mechanism [Furu et al, 1990, Zurob et al, 2002]: ^3 X = 1 - exp -nN\ \Gdt (2.4) G is directly proportional to the driving pressure, Pd exerted on the grain boundary of recrystallizing grains by the difference in dislocation density, pd across the recrystallization front: G = MPd = l-MGb2pd{t) (2.5) The constant of proportionality, M is defined as the mobility o f the grain boundary. The magnitude of the time dependent dislocation density in Equation 2.5 decreases as recovery proceeds. Therefore, recovery slows down recrystallization by lowering the driving pressure for the grain boundary to consume the deformed or recovered matrix. 2.4 Precipitation Precipitation has been the subject of numerous studies in the literature. In this work, the focus is on the effect o f precipitate conditions in the deformed microstructure on the recovery and recrystallization behaviour o f the alloys during subsequent annealing. O f particular interest is 15 Chap. 2 Literature Review the precipitation behaviour of the A l - M g - S i - C u A A 6 i l l alloy. The precipitation behaviour of this alloy has been extensively studied in the past in an effort to improve its age hardening response during the paint bake cycle. Some of these important observations are summarized in the next section. Existing precipitation models are examined in section 2.4.2. 2.4.1 Precipitation Hardening Behaviour of AA6111 In North America, the A l - M g - S i - C u AA6111 alloy is the main aluminum choice for automotive sheet skin applications, mainly due to its excellent paint bake response and high formability. Due to its significant commercial relevance, the precipitation hardening behaviour of the alloy in the intermediate temperature range of 160-220°C has been extensively characterized in many studies [Bryant, 1999, Quainoo et al., 2001, Esmaeili et al, 2003a, Wang et al., 2003]. More recently, a yield strength model specifically for the aging of AA6111 has been developed in the doctorate thesis of Esmaeili (2002). In Fig. 2.3, the age hardening curves of the material in the temperature range of 160-220°C is shown in terms of the evolution o f yield stress as a function of aging time. It can be observed that the peak strength obtained at different aging temperatures is similar (-340 MPa) but it took much longer for samples aged at lower temperatures to reach the peak strength: ~20 hours at 160°C vs. ~1 hour at 220°C. According to Lloyd et al., (2000), the corresponding precipitation sequence can be presented as SSS -> G P zones/clusters -» j3" + Q' -> equilibrium Q + M g 2 S i 16 Chap. 2 Literature Review 0.01 0.1 1 10 100 Fig. 2.3. Evolution of yield stress vs. aging time of AA6111 in the temperature range of 160-220°C [Esmaeili., 2002]. 17 Chap. 2 Literature Review where SSS represents the supersaturated solid solution state. Several authors have suggested that the precipitation sequence o f A l - M g - S i - C u alloys is dependent on the M g : S i ratio and the concentration of C u in solid solution [Miao and Laughlin, 2000, Murayama et al, 2001]. The formation of Guiner-Preston (GP) zones and solute clusters provides the increase in strength during aging at room temperature which is commonly known as the T4 condition. These clusters have been characterized by atom probe field ion microscopy and found to primarily contain co-clusters o f M g and Si [Murayama et al., 2001]. The strength of the alloy can be significantly enhanced by aging at elevated temperatures due to the precipitation of the two main hardening phases, B" and Q'. Quantitative T E M measurements have shown that approximately 80% o f the total volume fraction o f precipitates are B" while the remaining fraction belongs to the Q' phase after peak aging at 180°C for 7 hours [Wang et al., 2003]. The corresponding precipitate structure o f the sample is depicted in the bright field transmission electron micrograph shown in Fig. 2.4. It is observed that the two metastable phases J3" and Q' have a needle and lath shaped morphology, respectively and they both form with the long axis of the precipitate parallel to the <100> direction of the aluminum matrix [Chakrabarti and Laughlin, 2004]. The crystal structure of the B" and Q' precipitates have been identified as monoclinic and hexagonal, respectively [Perovic et al., 1999]. The Q' phase can be seen as a precursor for the equilibrium Q phase (some authors have used directly the notation Q instead o f Q' for the lath shaped precipitates, for example in Weatherly et al., 2001). Based on the T E M work of Wang and Embury (2002), the predominant precipitate after overaging for 7 days at 250°C is the Q' phase with lengths up to 350 nm. The length o f Q' precipitates increased significantly to -600 nm after overaging for 10 minutes at 315°C [Perovic et al, 1999]. 18 Chap. 2 Literature Review Fig. 2.4. Bright field T E M image showing the precipitate structure in AA6111 after aging for 7 hours at 180°C. N l and N 2 denote B" needles seen end-on and edge-on respectively. L I and L 2 corresponds to Q' laths seen end-on and edge-on respectively. [Esmaeili, 2002]. 19 Chap. 2 Literature Review During transformation from Q' to Q in the later stages or at higher temperatures, the precipitates maintain their lath shaped morphology and hexagonal crystal structure, only the size increases [Chakrabarti et al, 1998]. 2.4.2 Precipitation Modelling There are numerous kinetic models in the literature for the precipitation of second phase particles. In general, the precipitation process in a supersaturated solid solution can be divided into three distinct stages: nucleation, growth and coarsening. Theoretical models for each of these individual processes are long available in the literature [Martin et al., 1997]. The recent trend in modelling precipitation kinetics is to couple the nucleation, growth and coarsening processes within one comprehensive modelling framework. This approach has been successfully applied by several researchers to model the precipitation kinetics in aluminum alloys [Deschamps and Brechet, 1999, Myhr et al. 2001,] as wel l as in microalloyed steels [Dutta et al, 2001, Zurob et al, 2002]. In the model developed by Deschamps and Brechet (1999), the precipitation process is divided into two regimes. The first regime corresponds to the nucleation and growth of new precipitates. The growth and coarsening of these precipitates are considered in the second regime. A progressive transition from the first to second regime takes place and the transition occurs when the diminution o f the precipitate density, Np due to coarsening is larger than the increase in precipitate density due to nucleation, i . e.: 20 Chap. 2 Literature Review dNp > dNp (2.6) dt growth+coctrs dt Enucleation The mathematical formulation of the models is based on classical nucleation and growth theories [Martin et al., 1997]. In terms of precipitate coarsening, the standard L S W law is applied [Lifshitz and Slyozov, 1961, Wagner, 1961]. In the second regime, a coarsening fraction is used to integrate the individual contribution o f growth and coarsening to the overall average precipitate radius. The model was subsequently coupled with a yield stress model and applied to describe the precipitation hardening behaviour o f A l - Z n - M g alloys. There are a number of important assumptions in the model. First, the diffusion of Z n and M g in bulk aluminum is described by an equivalent diffusivity. However, no diffusion equation is given in the paper on how the equivalent diffusivity was calculated. It appears that the diffusivity at a given aging temperature is simply taken as a fitting parameter. Other fitting parameters in the precipitation model include the interfacial energy of the precipitate-matrix interface and the activation energy for precipitate nucleation. Secondly, the model does not consider directly the precipitation sequence, in other words, only one type of precipitate is considered. In spite of these simple assumptions, this model is considered as one o f the most comprehensive model frameworks for precipitation kinetics currently available in the literature. The model has also been extended to account for heterogeneous precipitation on dislocations and its interaction with the surrounding matrix [Deschamps and Brechet, 1999]. 21 Chap. 2 Literature Review 2.5 Interaction Between Precipitation and Recovery There have been few investigations into the interaction between recovery and precipitation. However, it is believed that fine and stable precipitates retard recovery in a similar fashion as retarding recrystallization during annealing [Humphreys and Hatherly, 1995]. In the case o f recovery, segments of dislocations may be pinned by precipitates thus render them immobile during recovery. O n the other hand, concurrent recovery lowers the dislocation density in the materials. This may in turn delay the progress of precipitation by reducing the number of available nucleation sites [Gomez-Ramirez and Pound, 1973]. Following Humphreys and Hatherly (1995), the pinning effect o f particles on dislocations can be treated based on a force balance approach. In this approach, the dislocation structure is assumed to be in the form of a three dimensional dislocation network o f mesh size R^. The dislocation velocity is proportional to the change in scale o f the network. The driving force, Fj for coarsening o f a 3-D dislocation network by dislocation migration can be approximated by r, Gb2 Fd=-T~ (2-7) In a matrix with a random dispersion of precipitate particles, the dislocation motion is opposed by a pinning force, Fp o f the order of (2.8) 22 Chap. 2 Literature Review where X is the spacing between the particles along the dislocation lines and c; is a constant. It should be noted that Equation 2.8 is based on the Orowan process i . e., dislocation lines pass through the particles by leaving behind dislocation loops (Orowan loops) around the particles. Humphreys and Hatherly (1995) have also considered the situation in which the recovery is dominated by subgrain growth. In this case, the pinning o f subboundaries by precipitates is treated as a special case of Zener pinning. The recovery is then expected to proceed at a rate controlled by the coarsening o f precipitates. A simplified approach to combine the pinning effect o f precipitates on dislocations with Verdier et al.'s recovery model (Equation 2.1) has recently been proposed by Zurob (2003b). In this model, the ratio of the net driving force to the total driving force is thought of as the unpinned fraction of the dislocation network, £ Using Equations 2.7 and 2.8, <f is given as e Fd~Fp . C^Rd t = — ^ J L = l-JTL (2-9) The mesh size Rd can be estimated from the dislocation density, Rd = (3/pd)05, and the average pinning spacing A, is equal to pd/Np where Np is the number of precipitates per unit volume. Combining these expressions with Equation 2.9, one gets Pd lydis where is the number of dislocation nodes and approximately equal to 0.5pd5. Finally, 23 Chap. 2 Literature Review Verdier et. al's recovery model is modified to account for the pinning effect, i.e.: - r L = Y 2 - ^ v D ^ P ~ s m h -hr (2-11) dt 9Mia2 E \ kT) \ kT ){ Ndis) Equation 2.11 assumes that the recovery kinetics is proportional to the fraction o f the network that is not available for recovery. It can be seen that recovery ceases when Np is equal to TV^. This simple approach has been shown to be sufficient in explaining the softening behaviour o f austenitic iron alloys containing various amounts of Nb and C during annealing [Zurob et al, 2003a]. •rf 2.6 Interaction Between Precipitation and Recrystallization Most of the existing studies on the interaction between precipitation and recrystallization focus on hot rolling of microalloyed steels [Jonas and Weiss, 1979, K w o n and DeArdo, 1991, Kang et al, 1997, Medina et al., 1999, Lee, 1999, Abad et al., 2001]. B y comparison, there has been very limited research, i f any, aimed at studying the precipitation-recrystallization interaction in commercial heat treatable aluminum alloys. This is unfortunate since Burger and co-workers have clearly shown that the microstructure and properties o f an industrial cold rolled annealed 6000 series aluminum alloys can be optimized by manipulating the thermomechanical history of the alloy (which determines the precipitate conditions in the material) prior to annealing [Burger et al., 1995, 1996]. In the following, the interaction between precipitation and recrystallization is examined in terms of precipitate pinning (section 2.6.1) and solute drag (section 2.6.2). 24 Chap. 2 Literature Review 2.6.1 Precipitate Pinning During recrystallization, grain boundaries are attracted to precipitates because when a boundary intersects a particle, a region of boundary equal to the intersection area is effectively removed which leads to a reduction in the energy of the overall system. In general, precipitation has four important effects on recrystallization [Humphreys and Hatherly, 1995]: 1. The presence of precipitates during cold deformation may lead to an increase in the stored energy and hence the driving force for recrystallization. 2. Large precipitates may act as nucleation sites for recrystallization via particle-stimulated-nucleation (PSN). 3. Precipitation decreases the matrix solute content and therefore reduces the effect of solute drag on the mobility of grain boundaries. 4. Fine and closely spaced precipitates may exert a significant pinning force on high angle grain boundaries. The first three effects tend to promote recrystallization, whereas the last tends to retard recrystallization. The importance of these effects during annealing is dependent on the nature of the precipitates. In principle, three simple parameters can be considered: volume fraction, size and spacing of the precipitates. The effect o f the precipitate volume fraction and size on recrystallization kinetics is schematically illustrated in Fig. 2.5. The effect of prior strain is also indicated. The graph can be analyzed in terms of the ratio between precipitate volume fraction and radius, i . e. FPIRP. B y examining a number o f experimental investigations, Humphreys and Hatherly (1995) have concluded that to a first approximation, the retardation 25 Chap. 2 Literature Review Fig. 2.5. Schematic showing the effect of precipitate size, volume fraction and prestrain on recrystallization kinetics and mechanism [after Humphreys and Hatherly, 1995]. 26 Chap. 2 Literature Review of recrystallization is most likely to occur when FPIRP is greater than 0.2 urn"1. If the ratio is less than 0.2 um" 1, recrystallization is often accelerated in comparison with precipitate free materials. The increase in the rate o f recrystallization can be attributed to the increased driving force that arises from the additional dislocations generated by the precipitates during deformation and to particle stimulated recrystallization (PSN). The critical diameter for P S N to occur has been found to be in the range of 1 pm for steel and aluminum alloys [Leslie et al, 1961, Humphreys, 1977, L loyd , 1985]. Furthermore, because the solutes solubility varies with temperature, it is probable that precipitation may precede or accompany recrystallization during annealing. Fig. 2.6 shows the effect of precipitation occurring at various stages during recrystallization on the shape o f the isothermal recrystallization curves [Liu et al, 1996]. Type I and III occur during annealing at high and low temperatures, respectively. At high annealing temperatures, precipitation does not interfere with recrystallization (in this case, the retardation effect would then primarily be one of solute drag which w i l l be discussed later). A t low annealing temperatures (type III kinetics) where precipitation takes place before recrystallization, the onset of recrystallization is greatly delayed by the precipitates induced pinning force on grain boundaries. Recrystallization is only allowed to proceed after sufficient coarsening o f the precipitates has occurred. In type II kinetics, a halt in the recrystallization process is observed which coincides with the precipitation process. This momentary cessation of the recrystallization process appears as a plateau on the recrystallized fraction curve. Recrystallization resumes after precipitation is completed. It has been suggested by a number of authors that type II and III recrystallization kinetics are controlled by the local coarsening of precipitates at the recrystallization front. [Hansen et al., 1980, Lotter et al., 1980, Wilshynsky-Dresler et al, 1992, Lillywhite et al, 2000]. 27 Chap. 2 Literature Review Log Time Fig. 2.6. Isothermal recrystallization kinetics showing the effect of precipitation which occurs at various stages during recrystallization [after L i u et al, 1996]. 28 Chap. 2 Literature Review Mechanistically, this is believed to occur as the migration rate o f the recrystallization fronts are retarded by a pinning force due to fine precipitates situated on the grain boundaries. A s long as the pinning force is larger than the magnitude of the driving force for recrystallization, the grain boundaries w i l l be completely arrested. Once the precipitates begin to grow and coarsen due to Ostwald ripening, the pinning force w i l l start to decrease. A s soon as the pinning force becomes sufficiently weak, the high angle grain boundaries w i l l start to move and consume the neighbouring deformed grains, albeit at some reduced overall velocity. To date, the most systematic study on the influence of prior precipitation state on recrystallization behaviour o f aluminum alloys was conducted on a high purity A l - M g - S i alloy by Lillywhite et al. in 2000. [Lillywhite et al., 2000]. In this work, a series of heat treatments were carried out before cold rolling in order to produce samples with different precipitate conditions. The samples were cold rolled to 70% reduction in thickness prior to annealing. The corresponding initial precipitate conditions are summarized in Table 2.2 along with the completion time for recrystallization obtained from isothermal annealing at 300°C. The experimental results in Table 2.2 show that the samples with coarse and widely spaced J3 particles (solution treated then furnace cooled) displays the most rapid recrystallization kinetics promoted by particle stimulated nucleation. In the as-quenched samples, recrystallization took an intermediate time to complete. The slowest recrystallization rate is observed in samples that have been preaged to produce fine metastable /?' particles. These metastable f3' particles were broken up and made partially incoherent by the deformation process. Upon annealing, before the onset of recrystallization, the broken /?' particles transform to the stable /? phase. Based on the results o f an extensive microstructural analysis, 29 Chap. 2 Literature Review Table 2.2. The effect o f prior precipitate conditions on the recrystallization completion time for a 70% cold rolled high purity A l - M g - S i alloy [Lillywhite et al, 2000]. Heat treatment before cold rolling Initial precipitate conditions Recrystallization completion time Solution treated, water quenched Supersaturated solid solution 10 hours Solution treated, furnace cooled Stable spheres 0.2 hours Solution treated, water quenched and then preaged at 300°C for 1 hour Fine metastable B' rods 1600 hours 30 Chap. 2 Literature Review Lillywhite et al. concluded that the rate determining step for recrystallization in the as-quenched samples is the transformation of the metastable B' phase to the equilibrium B phase while recrystallization in the preaged samples is controlled by the local coarsening o f P particles at the recrystallization front. These two rate controlling mechanisms are schematically illustrated in Fig. 2.7. Quantitatively, the retarding effect of precipitates can be readily modelled by incorporating the retarding pressure, Pz that arises from precipitates into Equation 2.5: The term (Pd - Pz) represents the net pressure acting on the grain boundaries. Therefore, recrystallizing grains w i l l only grow i f the net pressure is positive. Pz is commonly known as Zener pinning pressure due to Zener's seminal contribution to this subject. The mathematical derivation of the pinning pressure arising from precipitates has been performed in many studies in the past but unfortunately not always in a totally correct way [Nes et al, 1985]. The complete derivation leads to an expression of the pinning pressure, Pz which is directly proportional to the volume fraction of precipitates and inversely proportional to the mean radius of the precipitates [Humphreys and Hatherly, 1995]: G = M{Pd-Pz) (2.12) 3FPrgb 2Rp (2.13) 31 Chap. 2 Literature Review Fig. 2.7. Schematic representation of grain boundary migration controlled by (a) the transformation of /?' (grey rectangles) to J3 (dark circles) precipitates and (b) local coarsening of P precipitates at the recrystallization front. Arrows indicate the direction of boundary migration [after Lil lywhite et al., 2000]. 32 Chap. 2 Literature Review where ygb refers to the grain boundary energy. Fp and Rp denote the volume fraction and radius of the precipitates, respectively. Equation 2.13 is derived for a planar boundary intersecting an array o f precipitates which are randomly distributed in the matrix. A n additional requirement for Equation 2.13 to be valid is that the grain size must be much larger than the average precipitate spacing. It should be noted that in the original paper, the pinning pressure was half that of Equation 2.13. The reason behind this discrepancy is unclear but Nes et al. have speculated that it is probably because Zener has assumed only the particles behind the boundary w i l l act against the forward motion of the boundary thus obtaining a lower pinning pressure [Nes et al., 1985]. A more rigorous treatment of the pinning pressure involves the incorporation of precipitate shape and non-random distribution o f precipitates, as shown by Nes et al. (1985). In a recent review article by Manohar et al. (1998), it has been shown that a more comprehensive treatment of the particle-grain boundary interaction leads to considerably higher Pz than original Zener's estimate. However, Humphreys and Hatherly (1995) have concluded that more sophisticated calculations do not result in relationships which differ significantly from Equation 2.13. The Zener equation remains the most widely used approach in the estimation of particle pinning pressure. 2.6.2 Solute Drag In addition to precipitate pinning, the presence of a significant amount o f solutes can retard recrystallization by reducing the grain boundary mobility and evidence o f this has been found in a number of studies [Gordon and Vandermeer, 1966, Humphreys and Hatherly, 1995, Cahn, 1996]. A t high solute concentrations, the grain boundary mobility is low and decreases with 33 Chap. 2 Literature Review increasing solute concentrations. This behaviour is illustrated in Fig. 2.8 where the reciprocal velocity o f boundaries is plotted against the copper concentration in aluminum. Assuming a constant driving force, it can be seen that the boundary velocity is inversely proportional to the solute concentration. Fig. 2.8 also shows that the retarding effect of solutes diminishes with increasing temperatures. The aforementioned effect of solute on recrystallization is not observed in all materials. A n example is given in Fig. 2.9 where the time to achieve 50% recrystallization, f50% of a 95% cold rolled A l - M g alloys at 275°C is shown as a function of M g content. It can be seen that the rate of recrystallization is significantly lower when the M g content is increased from 0.5 to 1 wt%. However, at M g contents above 1 wt%, recrystallization is remarkably accelerated with increasing M g content up to 5% [Koizumi et al., 2000]. The accelerated recrystallization behaviour can be explained by the presence o f excess solutes during deformation which strongly inhibit dynamic recovery. A s a consequence, higher stored energy is available to drive subsequent recrystallization. Apparently, the increase in stored energy is more than enough to. overcome the inhibiting effect of solutes. Modern quantitative theory o f grain boundary mobility in a dilute solid solution is largely based on that developed by Cahn (1962) and Liicke and Detert (1957). In this model, the grain boundary velocity is divided into two regimes. The first regime corresponds to boundaries with high velocity for large driving force. In this case, the solutes have little effect on the boundary mobility and it is thought that the boundary has escaped from its solute atmosphere [Humphreys and Hatherly, 1995]. In the low mobility regime for small driving force, an atmosphere o f solute atoms is associated with the grain boundary. Under this condition, the 34 Chap. 2 Literature Review (x10°) 7- / 125°C / 139°C •/v* 6 * / 155°C s / 170°C / 189°C o O 1 4-O / S Id * . / / / / o O H / / 1 • 0- <_. • M M •f — *T . . * ? . r : . n » » I I I 1 1 1 0 50 100 150 200 250 (xlO"6) Cu concentration (at. fraction) Fig. 2.8. The effect of copper concentration on the migration rate of boundaries in aluminum at various temperatures [after Gordon and Vandermeer, 1966]. 35 Chap. 2 Literature Review 14000 12000 -10000 -£ 8000 -J p 6000 H 4000 2000 Temp. = 275°C 0 1 2 3 4 5 6 Mg (wt%) Fig. 2.9. The effect o f M g contents on the time to achieve 50% recrystallization at 275°C of a 95% cold rolled A l - M g alloy [after Ko izumi et al, 2000]. 36 Chap. 2 Literature Review mobility of the grain boundary, M is inversely proportional to the concentration of the solutes, CM according to: M = \ J _ + ccmC (2.14) where Mo is the intrinsic mobility o f the grain boundary and ocm is a temperature dependent parameter. Both of these quantities are difficult to specify, in particular the constant am as it requires an accurate estimation of the solute-boundary binding energy. To evaluate Mpure, the grain boundary diffusion data is required since the mobility is controlled by the rate of diffusion of solute atoms in the boundary region [Zurob et al., 2002]. A n interesting analysis of the effect of solute drag on recrystallization kinetics has recently been provided by Brechet and Purdy (2003). Based on their analysis, it can be shown that the general assumption that recrystallization kinetics samples the high mobility regime may not always be correct. According to their theory, i f the driving force is low and the number of recrystallization nuclei is larger than a critical value, each grain might always stay in the low mobility regime before the condition of hard impingement can be reached. Finally, since the motion of high angle grain boundaries during recrystallization is thermally activated, the effect o f temperature on mobility is often found to obey an Arrhenius relationship of the form [Humphreys and Hatherly, 1995]: M = MQ exp Q (2.15) where Q is the activation energy for boundary migration. 37 Chap. 2 Literature Review 2.7 Critical Assessment of the Literature A s delineated in the beginning of this chapter, the microstructural evolution during annealing of a cold rolled precipitation hardened alloys is a result o f the complex interplays between recovery, recrystallization and precipitation. In the literature, precipitation is studied more extensively than the processes of recovery and recrystallization. One o f the main hurdles in studying recovery and recrystallization is the lack o f understanding in the deformed state. This has prohibited the development of a quantitative theory for the nucleation of recrystallized grains. Recovery, on the other hand, has only received cursory attention in the literature and as a result many aspects related to the dislocation reactions are still poorly understood. For example, it is still not yet possible to establish, by either experiments or modelling, the rate controlling mechanism during recovery. In terms of modelling, comprehensive mathematical models are available to couple the nucleation, growth and coarsening kinetics o f precipitates. For recrystallization modelling, the J M A K model is at present the most widely used analytical approach. This is simply due to the lack of alternative models in the literature. B y comparison, the development of quantitative theories for recovery is still in its infancy. A unified theory o f recovery and recrystallization based on the stability and growth of cellular microstructures has been proposed by Humphreys (1997, 1999). However, this concept still requires significant development before it can be applied to model industrial alloys. In order to develop a realistic model framework for the annealing o f precipitation hardened alloys, the interaction between recovery, recrystallization and precipitation must be captured 38 Chap. 2 Literature Review based on sound physical theories. The Zener approximation is available as a quantitative tool to describe the retarding effect of precipitates on recrystallization and it has been verified in many studies. But much less is known about the effect o f precipitation on recovery. The quantitative approach proposed by Zurob et al. (2002) is reasonable but verification of the model's assumption is difficult, mainly due to the lack o f direct experimental observations on the local interaction between subboundaries and precipitates. In the current literature, the most comprehensive model in coupling the effect of recovery, recrystallization and precipitation was developed by Zurob et al. (2002) for the hot deformation of microalloyed steels. The experimental results provided by Lil lywhite et al. (2000), clearly indicate that the preexisting precipitate conditions in the deformed structure have an enormous effect on subsequent recrystallization behaviour o f a cold rolled A l - M g - S i alloy during annealing. However, quantitative physically based microstructure models for the annealing of cold rolled precipitation hardened aluminum alloys is still lacking in the literature. The present work is carried out to address this deficiency. The scope and objective are delineated in the next chapter. 39 Chapter 3 Scope and Objectives The primary objective of the present work is to obtain a fundamental understanding of the effect of simultaneous recovery and precipitation on the recrystallization behaviour of cold rolled AA6111 through a combination of experimental investigation and microstructure modelling. Experimentally, the goal is achieved through: • artificial aging o f solution treated materials at 20, 180, 250 and 325°C to obtain specimens with varied precipitate conditions prior to cold rolling, • isothermal annealing of the cold worked materials in the temperature range, i.e. 250-445°C where recovery, dissolution/precipitation and recrystallization are expected to occur concurrently, • thorough characterization of the annealed specimens using appropriate metallography techniques to identify on a microscopic scale the interaction mechanisms between the various microstructural phenomena, • measuring the yield stress of the specimens at various time intervals during annealing in order to relate the evolution of microstructure to the mechanical properties of the materials. 40 Chap. 3 Scope and Objectives In terms of modelling, the goal is to develop a comprehensive microstructure model to translate a qualitative description o f the interaction between recovery, recrystallization and precipitation into a quantitative prediction in terms of softening in yield stress as a function o f annealing time. The overall model w i l l be constructed by adopting physically based models in the literature for the individual processes of recovery, recrystallization and precipitation. The individual submodels w i l l be coupled based on well established physical theories in order to provide a realistic description for the complex behaviour of industrial alloys. Throughout the modelling exercise, a minimum number of adjustable parameters is sought and all the parameters used have a transparent physical meaning. Validation o f the model w i l l be carried out by comparing the model output with the experimental data obtained from overaged 40% cold rolled A A 6 1 1 1 . The present work provides the first scientific approach to consider recovery, recrystallization and precipitation as well as their, interaction within a single model framework for the annealing of A A 6 1 1 1 . The knowledge acquired from the experiments as wel l as modelling work w i l l make significant contributions to the development o f a through process model for the production of heat treatable aluminum alloys. 41 Chapter 4 Experimental Methodology The primary objective of the experimental work is to generate a series of recovery and recrystallization data using deformed samples with varied precipitate conditions. The as-received materials were subjected to a series of thermal and rolling processes. The microstructures of the specimens were extensively examined using a variety of characterization tools. The details of these experiments are described in this chapter. 4.1 Starting Materials A l l the samples used in this investigation were obtained from industrially hot rolled sheets (coil #38752) supplied by Alcan. The as-received A l - M g - S i - C u AA6111 alloy was ingot cast, homogenized and hot rolled to a final thickness of 3.5 mm. Table 4.1 outlines the composition of the A l - M g - S i - C u alloy AA6111 in wt%. A n optical micrograph showing the microstructure of the as received hot rolled sheet is given in Fig. 4.1. The highly elongated grain structure clearly indicates that recrystallization did not take place during or after hot rolling. 4.2 Sample Preparation Two sets of samples were prepared for subsequent heat-treatment experiments. Small rectangular sheet specimens were sheared from the as-received hot rolled sheets to be used 42 Chap. 4 Experimental Methodology Table 4.1. Chemical composition of AA6111 in wt%. Mg Si Cu Fe Mn Cr Ti Al 0.75 0.63 0.75 0.25 0.2 0.05 0.06 bal. 43 Chap. 4 Experimental Methodology Chap. 4 Experimental Methodology primarily for metallography examinations. A second set o f rectangular strip samples measuring 105 mm in length and 19 mm in width were sheared with the longitudinal direction parallel to the rolling direction of the sheets. Tensile samples with 40 mm gauge length were subsequently punched out from these strips after cold rolling using a manual die. In addition to tensile and metallography samples, a small number o f square coupons (25 x 25 mm) were prepared for resistivity measurements. 4.3 Heat-treatment Experiments For each combination o f rolling and heat-treatment conditions, the tests were carried out once using two identical samples. Heat-treatments that were shorter than two days were carried out in low temperature molten salt baths (60% potassium nitrate + 40% sodium nitrite). A n o i l bath was used for a limited number of tests at 250°C. In order to minimize the heat up time, the oi l and salt baths were stirred vigorously throughout the duration of the experiments. For heat-treatments longer than two days, either a tube or box furnace was utilized. In this case, samples were placed as close as possible to the thermocouples in the furnace. Temperatures of the baths and furnaces were checked periodically using a Fluke K-type thermometer. Differences between the thermometer and controller readings were typically within ±4°C. 4.3.1 Solution Heat-treatments A l l heat-treatment experiments were started with a solution heat-treatment at 560°C for 10 minutes in a salt bath followed by quenching in water at room temperature. The combination 45 Chap. 4 Experimental Methodology of time and temperature was chosen to dissolve all the pre-existing precipitates formed during hot rolling and coiling processes and hence restore the maximum solid solution in the materials. The solution heat-treatment is also carried out to eliminate the elongated grain structure seen in Fig. 4.1. A previous study at U B C has shown that the elongated grain structure may complicate the cold rolling process and increase the driving force for subsequent recovery and recrystallization processes during annealing [Go et al., 2001]. 4.3.2 Artificial Aging After the solution heat treatment, the samples were artificially aged to produce four different precipitate conditions: (i) naturally aged (T4), (ii) peak aged (PA) , (iii) overaged (OA) and (iv) severely overaged (SOA) . The T4 conditions were achieved by aging the samples at room temperature for 8 days. The P A samples were obtained by aging the samples at 180°C for 7 hours. The overaged and severely overaged samples were obtained by aging for 7 days at 250 and 325°C, respectively. Precautions were taken to minimize the time gap (usually within minutes) between quenching (from solutionizing temperature) and artificial aging in order to prevent natural aging from occurring. The formation of solute clusters/GP zones prior to artificial aging has been found to have an adverse effect on subsequent precipitation behaviour [Poole etal, 1997]. In addition to aging the materials to specific precipitate conditions, two precipitation hardening curves were generated. The evolution of yield stress as a function of aging time was measured at 300 and 325°C for times up to 7 days. 46 Chap. 4 Experimental Methodology 4.3.3 Isothermal Annealing Prior to isothermal annealing, the T4, T6, O A and S O A specimens were rolled at room temperature to a reduction of 40% in thickness (from 3.50 to 2.09mm) using a laboratory scale rolling mi l l . Generally, a total of 4 passes (reverse rolling) were applied to achieve the required thickness for all the specimens. Annealing experiments were carried out immediately after cold rolling in order to prevent the occurrence of room temperature recovery. In order to study the effect o f prior aging condition, all the samples were annealed isothermally at 325°C. The effect of annealing temperature was investigated by annealing the O A samples at 250 and 445°C and the S O A samples at 445°C. The high annealing temperature o f 445°C was chosen specifically to study the effect of precipitate dissolution. The holding time was varied from 1 minute up to 40 days. A t the end of the annealing cycle, the samples were quenched in water at room temperature. A summary of the overall thermal and mechanical processing applied to the as-received materials is schematically illustrated in Fig. 4.2. 4.4 Sample Characterization The microstructures were characterized using a variety o f experimental, techniques including optical microscopy (OM) , scanning electron microscopy (SEM) , electron back-scattered diffraction ( E B S D ) , and resistivity measurements. Transmission electron microscopy ( T E M ) was also carried out in collaboration with researchers from the Brockhouse Institute of Materials Research at McMaster University. 47 Chap. 4 Experimental Methodology Fig. 4.2 Schematic of heat treatment and rolling experiments. 48 Chap. 4 Experimental Methodology A A A Sample Preparation A l l microstructural examinations were done on sections parallel to the rolling direction revealing the through thickness microstructures. To prepare the surface of interest for microstructural examination, specimens were cold mounted in an acrylic resin and polished to 0.05 pm finish using a Phoenix 4000 automatic polisher. Due to the ductile nature of aluminum alloys, deformation induced damage is a common problem during the grinding and polishing process. To overcome this problem, a four step grinding and polishing procedure developed by Buehler specifically for aluminum alloys was adopted. B y following the steps outlined in Table 4.2, a mirror like surface finish can be obtained consistently with minimal fine polishing scratches. Specimens for electron microscopy examination were further electropolished in order to remove the thin deformed layer on the surface caused by mechanical polishing. The composition o f the electropolishing solution was 100 m l perchloric acid (60%) + 500 ml denatured ethyl alcohol. The samples were made the anode by immersing in the solution for about 1 minute at a bath temperature of below -10°C. A steel cup which contains the solution was used as the cathode. Power is provided by a rectified power supply. The current density was controlled by adjusting the voltage until the ampere meter on the power supply reads 1 A . 49 Chap. 4 Experimental Methodology Table 4.2. Four step procedure for grinding and polishing aluminum alloys using a Phoenix 4000 automatic polisher. Surface Abrasive/Size Load Lb. (N)/Specimen Base Speed (rpm)/Direction * Time (min:sec) Abrasive disc 240 or 320 grit S i C with water cooled 5(22) 240-300 Comp. Unt i l plane Ultra-Pol cloth 6 pm diamond suspension 6(27) 120-150 Comp. 6:00 Trident cloth 1 pm diamond suspension 6(27) 120-150 Comp. 4:00 Micro cloth 0.05 pm colloidal silica suspension 6(27) 120-150 Contra. 2:00 *Comp. = platen and specimen holder both rotate in the same directions Contra. = platen and specimen holder rotate in opposite directions 50 Chap. 4 Experimental Methodology 4.4.2 Optical Microscopy In order to reveal the microstructures under optical microscope, the samples were first anodized using the Baker's reagent (200 ml distilled H 2 0 + 6 m l H B F 4 (48 wt%)). A thin layer of oxide film was deposited on the surface after anodization allowing the microstructures to be revealed under crossed-polarized illumination. Photomicrographs o f the microstructures were taken using a N i k o n E P I P H O T 300 series inverted metallurgical microscope equipped with a digital camera. Post processing of the micrographs was completed employing the Clemex Professional Imaging and Adobe Photoshop 6 software. The volume fraction of recrystallized grains was quantified using the ImageTool software developed by the researchers from The University of Texas Health Science Centre. Before analyzing the microstructures in ImageTool, recrystallized grain boundaries were outlined on transparency and the image scanned into a P C . Recrystallized grains in a partially recrystallized structure were selected primarily based on their shape. Highly elongated grains are identified as deformed matrix and grains with aspect ratio o f equal or less than 3 were considered recrystallized grains. The ImageTool software calculates the area of each individual recrystallized grain, Arex. The sum o f all the recrystallized grain areas divided by the total area of the micrograph gives the volume fraction of recrystallized grains. Typically, a minimum total area of 6.3x10" mm was measured for each sample. The recrystallized grain size, drex, was estimated by calculating the equivalent area diameter assuming spherical grains in 2-d [Orsetti-Rossi and Sellars, 1997], i.e., 51 Chap. 4 Experimental Methodology (4.1) The number of grains included in the analysis ranges from 200 to 700 depending on the degree of recrystallization in the sample. 4.4.3 Scanning Electron Microscopy (SEM) A l l S E M micrographs were taken on a Hitachi S-3000N electron microscope. The accelerating voltage was varied from 5 keV up to 20keV. L o w voltage is generally applied for taking back-scattered electron (BSE) images. In order to obtain the best quality B S E images, the contrast was increased to maximum and brightness adjusted to its lowest level. 4.4.4 Electron Back Scattered Diffraction (EBSD) E B S D scans were carried out on a Hitachi scanning electron microscope (SEM) S570 operating at 20 keV. After inserting the sample into the S E M chamber, the sample holder was tilted 70° towards the detector. The H K L Channel 5 suite o f programs was used to acquire and process the diffraction data. E B S D mapping of the grain structures was carried out at a step size of 2 um unless otherwise noted. The indexing quality ranges from a minimum of 85% up to as high as 95%. To reconstruct the microstructure from the E B S D data, it is convenient to divide grain boundaries into low and high angle grain boundaries. In this study, the transition from low to high angle grain boundaries is taken as 15° [Humphreys and Hatherly, 1995]. 52 Chap. 4 Experimental Methodology 4.4.5 Transmission Electron Microscopy (TEM) Initial characterization o f the as aged specimens was carried out at U B C . To prepare the T E M thin foils, small discs measuring 3 mm in diameter were cut from the sheet samples using electron discharging. The discs were then mechanical polished to reduce the thickness to approximately 100-120 urn. The ground discs were then electropolished to perforation in a Struers Tenupol-2 jet polishing unit using an electrolyte of 10% perchloric acid, 20% glycenol and 70%> methanol at around -20°C. The operating voltage o f the jet polisher was 20V. The microscope is a Hitachi 800 scanning transmission electron microscope operating at 200keV. Additional T E M observations were performed by Dr. X . Wang from the Brockhouse Institute of Materials Research at McMaster University using a conventional Philips C M - 1 2 electron microscope operating at 120keV. T E M thin foils were prepared by mechanically polishing the samples to approximately 100 um and then jet polishing in a solution o f perchloric acid and methanol at -35°C. The averaged dimensions of the precipitates were determined from bright field images. The measurements of cross-sectional area were converted to an equivalent diameter using an equation similar to Equation 4.1 assuming spherical precipitates. 4.4.6 Resistivity Measurements The resistivity of the specimens, pr were calculated from conductivity data measured using a portable Verimet M4900C conductivity tester. The tester was calibrated to measure the electrical conductivity o f a specimen and display the result in percent I A C S (International 53 Chap. 4 Experimental Methodology Annealed Copper Standard, 100% LACS copper = 58 M Q m ) . A l l measurements were carried out at room temperature in C O M P mode to compensate for any temperature effects. 4.4.7 Tensile Measurements The mechanical response of the samples at a given annealing temperature was followed by the evolution of yield stress with respect to annealing time. A l l tensile tests were carried out at a strain rate of 0.002 s"1 on a M T S servo-hydraulic tensile machine with the tensile axis parallel to the rolling direction of the sample. A n extensometer with gauge length of 40 mm was attached to the reduced section of the samples to measure the elongation during straining. The yield stress, <jy, was measured from engineering stress-strain curves employing the standard 0.2% offset method. Two tests were conducted for each experimental condition. The average values between the two measurements were taken. The difference between the two measurements is typically within ±5 M P a . Tensile tests were carried out immediately (within a few minutes) after annealing to prevent natural aging from occurring. This step is critical particularly at high annealing temperature, since significant amount of solutes can be dissolved during annealing and the structure becomes unstable upon quenching. A n example is given in Fig. 4.3 where the tensile curves o f two annealed samples were compared. The two overaged (OA) and cold rolled samples were annealed for 1.75 hours at 445°C. However, the yield stress o f sample G l was measured 2 days after the annealing was completed. It can be seen that the yield stress increased drastically (from 50 to 80 MPa) due to the effect o f natural aging. 54 Chap. 4 Experimental Methodology 03 co co cu CO 200 150 100 50 0 0.00 G1, YS = 80 MPa 0.06 Fig. 4.3. The effect of natural aging on annealed samples. Sample G l was tested immediately after annealing at 445°C for 1.75 hours while sample G2 was tested 2 days after the annealing heat treatment. Both samples were in their overaged conditions prior to 40% cold rolling. 55 Chapter 5 Experimental Results and Discussions In this chapter, the experimental results are presented by following the sequence of the experimental procedures illustrated in Fig. 4.2. Prior to cold rolling, the as received materials underwent a series o f heat treatments in order to achieve the desire starting microstructure. The effect of solution heat treatment and artificial aging is presented first in sections 5.1.1 and section 5.1.2, respectively. The deformed microstructure is analyzed in the next section (5.1.3). In section 5.1.4, the isothermal annealing behaviour of the deformed specimens with various precipitate conditions are presented in terms of the evolution of yield stress, resistivity and microstructure as a function of annealing time. The second part of this chapter is devoted to the discussion of the experimental results. In section 5.2.1, a simple nonlinear addition law is employed to examine the work hardening behaviour of the various samples. Then, in section 5.2.2, resistivity measurements are analyzed in terms of nucleation, growth and coarsening of precipitates. The effect of prior aging conditions on the isothermal annealing behaviour is discussed in section 5.2.3 with special attention given to the overaged samples. 56 Chap. 5 Experimental Results and Discussion 5.1 Experimental Results 5.1.1 Solution Heat Treatments The solution heat treatment step was carried out to recrystallize the elongated grain structure and dissolve all the preexisting precipitates in the as received hot band. Fig. 5.1a shows the through thickness microstructure of the as quenched sample after solution heat treatment (supersaturated solid solution). The fully recrystallized microstructure comprises grains which are slightly elongated in the rolling direction. Using E B S D mapping, the fully recrystallized microstructure is reproduced in Fig. 5.1b with the same magnification. In this micrograph, high angle grain boundaries with a minimum of 15° misorientation are represented by black lines and boundaries between 2-15° misorientation are represented by grey lines. It can be seen that the recrystallized grains are essentially free of internal substructures. The average grain size measured from the E B S D map (ds ~42pm) agrees very well with the grain size measured from the optical micrograph (ds ~45um). This is interpreted as a validation o f the use of 15° misorientation as the criterion for high angle grain boundaries. Fig. 5 .2 shows a S E M micrograph depicting the fully recrystallized microstructure at higher magnification. The most notable feature of this micrograph is the random dispersion of coarse and irregularly shaped intermetallic particles throughout the microstructure. The chemistry and distribution of these particles are analyzed in section 5.1.3. 57 Chap. 5 Experimental Results and Discussion Fig . 5.1. (a) Optical micrograph showing the solution treated microstructure with average grain size of -45 urn, (b) E B S D micrograph of the same sample with grain size of -42 um (black lines represent high angle grain boundaries > 15 c misorientation). 58 Chap. 5 Experimental Results and Discussion Fig. 5.2. S E M micrograph showing the solution treated microstructure. Irregularly shaped Fe-rich intermetallic particles as indicated in the micrograph are randomly distributed throughout the microstructure. 59 Chap. 5 Experimental Results and Discussion 5.1.2 Artificial Aging To prepare the solution treated specimens for cold rolling, the samples were subjected to a series of artificial aging processes as outlined in section 4.3.2. Four precipitate conditions were studied: (1) naturally aged, T4, (2) peak aged, P A , (3) overaged, O A and (4) severely overaged, S O A . The yield stress o f the as quenched samples with varied precipitate conditions can be compared conveniently by plotting the data on a typical age hardening curve. This is shown in Fig. 5.3 where the horizontal axis is used to represent the precipitate state o f the samples instead o f aging time. Starting from the supersaturated solid solution (SSS) condition, the yield stress increases from 58 to 139 M P a after 8 days o f natural aging (T4). The outstanding precipitation hardening potential of this alloy is illustrated by the P A sample where the solution treated yield stress was increased by nearly six fold (from 58 to 334 MPa) . The yield stress decreases progressively to 153 and 92 M P a as the aging temperature is increased to 250 and 325°C, respectively. The plastic portions o f the stress strain curves for the as aged samples with varied precipitate conditions are shown in Figs. 5.4. The stress strain curve of the supersaturated solid solution is included in Fig. 5.4a for comparison. In contrast with the smooth flow observed in the T4, P A and O A samples, the plastic stress strain curves for the SSS and S O A samples exhibit substantial evidence o f serrated flow. The serration is a well known effect in A l - M g alloys which is associated with the pinning of dislocations by M g atoms in solution [Lloyd, 1980, Inagaki and Komatsubara, 2000, Tian, 2003]. Therefore, the appearance of serration in the 60 Chap. 5 Experimental Results and Discussion 400 co 300 CL a> 200 O c <D CT 5 100 0 -334 -139 / \ 153 92 58 i s s s T4 PA OA SOA Precipitate Conditions Fig. 5.3. Age hardening curve showing the as aged yield stress o f samples with varied precipitate conditions: SSS: supersaturated solid solution, T4: naturally aged, P A : peak aged, O A : overaged and S O A : severely overaged. This corresponds to heat treatment of 10 minutes at 560°C, 8 days at room temperature, 7 hours at 180°C, 7 days at 250°C and 7 days at 325°C, respectively. The corresponding values of the as aged oy are indicated on the curve. 61 Chap. 5 Experimental Results and Discussion co CL CO Q_ 180 150 (b) Peak aged 0.00 0.03 0.06 0.09 0.12 0.15 0.18 Plastic strain Overaged Severely Overaged 0.00 0.03 0.06 0.09 0.12 Plastic strain 0.15 0.18 Fig. 5.4. Plastic portion of the stress strain curves for (a)supersaturated solid solution (SSS), naturally aged and peak aged samples and (b) overaged and severely overaged samples. Note the serrated flow in the SSS and S O A samples. 62 Chap. 5 Experimental Results and Discussion stress strain curve of the S O A sample suggests that significant amount of solute atoms remained in solid solution after aging for 7 days at 325°C. The precipitate structures of the T4 and P A samples have been extensively studied in the past and these results have been summarized in section 2.4.1. Hence, only the precipitate structures of the O A and S O A samples are characterized in the present work. Bright field T E M micrographs depicting the precipitate structure in the O A and S O A samples are shown in Figs. 5.5a and 5.6a, respectively. In contrast with the P A sample (Fig. 2.4) where 80% of the total volume fraction o f precipitates are B", the majority of the precipitates in the O A and S O A specimens are the lath shaped Q' precipitates (indicated as L i and L 2 in Fig. 5.5a). The corresponding diffraction pattern for the Q' precipitates is shown in Fig. 5.5b. In addition to the fine Q' precipitates, square shaped M g 2 S i particles with size in the order of several microns are also observed in the S O A sample (Fig. 5.6b). N o M g 2 S i particles were found in the O A sample. From the T E M micrographs, the average equivalent diameter of the Q' precipitates are estimated as 13 nm and 35 nm for the O A and S O A samples, respectively. Not surprisingly, the dimensions of the precipitates increased significantly after aging at higher temperature leading to a coarser precipitate spacing. The corresponding optical micrographs of the as aged O A and S O A samples are shown in Fig. 5.7a and 5.7b. The average grain size is measured as ~43 pm in both samples. This is similar to the recrystallized grain size found in the as solution treated microstructure shown in Fig. 5.1 thus confirming that no grain growth has occurred in the aging process. 63 Chap. 5 Experimental Results and Discussion (a) Fig. 5.5. (a) Bright field T E M image showing the lath shaped Q' precipitates (Li and L 2 seen end on and edge on respectively) in the O A sample and (b) corresponding diffraction pattern taken along the [001] zone axis of aluminum. 64 Chap. 5 Experimental Results and Discussion 400 nm 3.8 um . 5.6. Bright field T E M images showing (a) the lath shaped Q' precipitates and (b) large square shaped M g 2 S i particles in the S O A sample. Courtesy of Dr. X. Wang. 65 Chap. 5 Experimental Results and Discussion Fig. 5.7. Optical micrographs showing the microstructure of the as quenched sample after aging for 7 days at 250°C (OA) and 325°C (SOA) . The average grain size in both micrographs is -43 pm. 66 Chap. 5 Experimental Results and Discussion In addition to aging the specimens to specific precipitate conditions, the age hardening behaviour of the solution treated samples was investigated at 300 and 325°C. The results are shown in Fig. 5.8 where the age hardening response of the material is characterized by measuring the evolution of yield stress with respect to aging time. It can be seen that the yield stress increases rapidly and attains its peak value in less than 1 minute at 325°C. A t the lower aging temperature of 300°C, the peak strength was reached after about 200 seconds. Beside the difference in precipitation kinetics, the magnitudes of the peak strength are also different. The initial yield stress which represents the yield stress o f supersaturated solid solution was more than doubled from 58 to -150 M P a at 325°C. The peak strength obtained at 300°C is even higher, -170 M P a . This behaviour conforms to the typical age hardening response of a heat-treatable alloy where higher peak strength is usually observed at lower aging temperature at the expense o f longer time. After the peak strength is reached, the precipitates entered their coarsening phase which leads to the gradual decrease in yield stress shown in the aging curves. Precipitate coarsening occurs at a much slower rate than the nucleation and growth o f precipitates. This behaviour is evident in the time it took for the samples to soften: it took nearly 44 and more than 65 hours for the samples to give up 90% of the gain in yield stress at 325 and 300°C, respectively. 5.1.3 The Deformed State It is important to examine the deformed state because it sets the stage for subsequent recovery and recrystallization processes. Following aging heat treatment, the specimens with varied precipitate states were cold rolled to a reduction of 40% in thickness. This corresponds to an 67 Chap. 5 Experimental Results and Discussion 03 CL 250 200 150 b ^ 100 Ag ing T e m p . - • - 300°C - • - 325°C 50 0 " j " " I • I I • 11 I I • I I 1111 I T V I ' T T T f p I I I I I 11 If " - ^ | | | | 1 1 | | | I I I I I I I 0 101 102 103 104 105 106 Time (s) Fig . 5.8. Precipitation hardening curves of AA6111 at 300 and 325°C. The as solution heat treated yield stress is indicated at the intercept with the vertical axis. 68 Chap. 5 Experimental Results and Discussion equivalent strain o f 0.58. The yield stress o f the various as deformed samples are given in Fig. 5.9 along with the as aged yield stress. It can be seen that the yield stress o f the T4 sample was markedly increased by more than 200 M P a while the increase in yield stress for the P A , O A and S O A samples is in the order o f - 1 0 0 M P a . This work hardening behaviour w i l l be further analyzed in section 5.2.1. A n optical micrograph showing the as deformed microstructure o f the O A specimen is given in Fig. 5.10. A closer view o f the deformed grains is shown in the S E M micrograph appearing in Fig. 5.11. The major microstructural change is that the slightly elongated recrystallized grains seen in Figs. 5.7 become more elongated in the rolling direction after deformation. Macroscopic shear bands which extend across several grains are also observed (Fig. 5.10). The dislocation structures after cold rolling are shown in the T E M micrographs of Fig. 5.12a and 5.12b for the as cold rolled O A and S O A samples, respectively. The lath shaped Q' precipitates were fractured into small segments during deformation. This is shown in Fig. 5.13 where the precipitate structure o f the as cold rolled S O A sample is given as an example. Another important aspect of the deformed microstructure is the distribution of insoluble Fe-rich particles which act as potential nucleation sites for recrystallized grains. Fig. 5.14a shows the typical spatial distribution of these particles in the deformed matrix o f a T4 specimen. One of the larger particles is shown at higher magnification in Fig. 5.14b. The X-ray spectrum (Fig. 5.14c) indicates that significant amounts o f S i , C u and M n are present in the Fe rich particle. Based on quantitative metallography, the average equivalent diameter of the particles was determined as 2.5 ± 0.05 um and the number density is approximately 820 mm" 3. 69 Chap. 5 Experimental Results and Discussion OL 500 400 A 300 A ^ 200 100 0 T4 PA OA Precipitate Conditions SOA Fig. 5.9. The as aged and as cold rolled yield stress of samples with varied precipitate conditions. Note the significant increase in yield stress in the T4 sample after 40% cold rolling. 70 Chap. 5 Experimental Results and Discussion 200 um Fig. 5.10. Optical micrograph showing the deformed microstructure of a 40% cold rolled O A sample. The slightly elongated grains shown in F i g . 5.11 become more elongated in the rolling direction after deformation. 7 1 Chap. 5 Experimental Results and Discussion Fig. 5.11. S E M micrograph showing the deformed microstructure of a 40% cold rolled O A sample. 72 Chap. 5 Experimental Results and Discussion Chap. 5 Experimental Results and Discussion 1 um Fig . 5.13. T E M micrographs showing segments of fractured Q' precipitates in 40% cold rolled S O A sample. Courtesy of Dr. X. Wang. 74 Chap. 5 Experimental Results and Discussion 0 2 4 6 8 k«V Fig. 5.14. (a) Distribution of insoluble Fe-rich constituent particles in the deformed matrix of a deformed T4 specimen, (b) close up view of one of the particles and (c) X-ray spectrum indicates the presence of Fe, S i , M n , and C u in the particle shown in (b). 75 Chap. 5 Experimental Results and Discussion 5.1.4 Isothermal Annealing at 325°C After cold rolling, all the deformed samples with varied precipitate conditions were isothermally annealed at 325°C. The results of these experiments are presented in three parts: First, the material response to the annealing heat treatment is characterized by following the softening in yield stress as a function of annealing time. The second part compares the evolution of resistivity in the deformed samples during annealing with the evolution o f resistivity in the solution treated sample during artificial aging. In the third part, the isothermal recrystallization behaviour is described by comparing the partially recrystallized microstructures of the T4, P A and O A samples. 5.1.4.1 Evolution of Yield Stress Fig. 5.15 shows the softening behaviour o f the samples with varied precipitate conditions after annealing at 325°C for time up to 40 days. The uncertainty associated with these measurements is estimated to be in the order o f 5-10 M P a . It can be seen that the yield stress decreased rapidly in the initial stage of annealing irrespective o f prior aging condition. The initial softening is particularly severe for the P A sample which lost about 60-70% of its as deformed yield stress after only 1 minute of annealing. The rate of decrease in yield stress slows down considerably after the first few minutes o f annealing. This can be seen in the softening curve of the S O A sample where the yield stress decreases by only 15 M P a (from 120 to 105 MPa) between 5 minutes and 48 hours of annealing time. The softening kinetics o f the T4 and P A samples is basically the same with both curves eventually reaching a plateau at - 70 M P a after 2 weeks of annealing (Fig. 5.15a). 76 Chap. 5 Experimental Results and Discussion 500 400-1 b ^ 200 -I 100 A Prior Aging Conditions • T4 (8 days, Room Temp.) • PA (7 hrs, 180°C) 0 A~""i * • i II i i l ium i i n i | i i n 0 101 102 103 104 10s 10s 107 108 300 250 200 150 •{ 100 50 H (b) Prior Aging Conditions • - OA (7 days, 250°C) • - S O A (7 days, 325°C) 0 T" ™l I— "I— i| | | — i i i mill—i i | 0 ; 101 102 103 104 105 106 107 108 Time (s) Fig. 5.15. Isothermal evolution of yield stress during annealing at 325°C for 40% cold rolled (a) T4 and P A samples and (b) O A and S O A samples. The prior aging conditions are indicated in the inset and the as cold rolled yield stress is indicated at the intercepts with vertical axis. 77 Chap. 5 Experimental Results and Discussion The final yield stress o f the O A and S O A samples annealed for 40 days is 73 and 41 M P a respectively (Fig. 5.15b). This represents an overall softening of 285, 359, 147 and 153 M P a for the T4, P A , O A and S O A samples, respectively. The onset of recrystallization (determined by following the changes in microstructure) for the respective samples is indicated on the softening curves. 5.1.4.2 Evolution of Resistivity Fig. 5.16 shows the evolution of resistivity in the T4, P A and O A specimens with respect to annealing time at 325°C. The resistivity curve of an artificially aged sample at 325°C is included for comparison. It can be observed that prior aging conditions did not affect significantly the evolution of resistivity during annealing. A l l the curves seem to collapse onto one another in the early stage of annealing and eventually follow closely the change in resistivity of the solution treated sample during artificial aging. The implication of these results is important since resistivity is directly related to the solutes concentration in solid solution. This aspect w i l l be further discussed in section 5.2.2. 5.1.4.3 Evolution of Microstructure Optical micrographs showing the recovered microstructures o f the T4, P A and O A samples annealed for 1 minute at 325°C are shown in Fig. 5.17. The recovered microstructures are essentially the same as the as deformed structure (Fig. 5.10) thus confirming that no recrystallization took place in the initial stage of annealing in all o f the samples. B y following 78 Chap. 5 Experimental Results and Discussion 50 45 a f 40 H w "</> 35 T Annealing temp. = 325°C \ \ \ \ Prior Aging Conditions • A - T4 (8 days, Room Temp.) • • - OA (7 days, 250°C) e - PA (7 hrs, 180°C) Q - Artificially aged 325°C \ \ -a 30 lSS ' • I I I I I I '""I I I I I I I I 0 101 102 103 104 105 106 107 Time (s) Fig. 5.16. Evolution of resistivity during annealing of AA6111 with various precipitate conditions. The prior aging conditions are indicated in the inset and the resistivity of the as cold rolled and as solution treated samples are the intercepts with vertical axis (time = 0). 79 Chap. 5 Experimental Results and Discussion Fig. 5.17. Optical micrographs showing the recovered microstructure of 40% cold rolled T4, P A and O A specimens annealed for 1 minute at 325°C. 80 Chap. 5 Experimental Results and Discussion the change in the grain structure with respect to annealing time, the onset o f recrystallization was determined to occur after approximately 12 hours in the T4 and P A samples and 48 hours in the O A sample. This is summarized in Table 5.1 along with several key microstructural parameters describing the partially recrystallized microstructures obtained after annealing for 2 weeks at 325°C. These parameters include the average recrystallized grain size drex, density of recrystallized grains per unit area and fraction of recrystallized grains. The corresponding E B S D maps of the partially recrystallized microstructures are shown in Fig. 5.18a to 5.18c. It is difficult to identify the preferential nucleation sites for recrystallization. However, it can be observed that some o f the recrystallized grains are located on the boundaries o f deformed grains indicating that grain boundary nucleation is a possible mechanism. It w i l l be shown later that in addition to grain boundary nucleation, evidence of particle-stimulated-nucleation can also be found in the microstructure. Based on quantitative microscopy, the mean recrystallized grain size drex is measured as 14.6, 11.5, and 18.6 urn for the T4, P A and O A samples, respectively. The bar charts of Fig. 5.19 show the distribution of recrystallized grain size in the various samples normalized by the averaged grain size. The distribution of normalized grain size in the O A sample is slightly wider than in the T4 and P A specimens which show relatively narrow distributions. It is also interesting to compare the density of recrystallized grains per unit area listed in Table 5.1. While the number of recrystallized grains in the O A sample is less than one half of the number o f recrystallized grains i n the T4 and P A samples, these grains have grown to a larger size. These relatively large recrystallized grains are distributed heterogeneously in the microstructure. This is shown in Fig. 5.20 where colonies o f large recrystallized grains (on the order o f 100 um in diameter) can be seen to congregate in the centre of the micrograph. 81 Chap. 5 Experimental Results and Discussion Table 5.1. Comparison o f the recrystallization behaviour of 40% cold rolled AA6111 with varied precipitate conditions after annealing for 2 weeks at 325°C. Prior aging conditions Onset of recrystallization at 325 V (hours) Fraction recrystallized Average recrystallized grain size, drex No. of recrystallized grains per mm2 T4 12 0.37 14.4 pm 1387 P A 12 0.32 11.5 pm 1946 O A 48 0.30 18.6 pm 617 82 Chap. 5 Experimental Results and Discussion 100pm Fig. 5.18. E B S D maps showing the partially recrystallized microstructures of 40% cold rolled AA6111 with varied precipitate conditions after annealing for 2 weeks at 325°C. The corresponding prior aging conditions are (a) T4, (b) P A and (c) O A . Black lines represent grain boundaries which are >15° and grey lines represent boundaries between 2-15°. 83 Chap. 5 Experimental Results and Discussion 30 25 20. 15 £ 10 0 ( 30 25 ^- 20 ( T (U LL 10 5 0 ( 30 25 5- 20 5 1 5 cr <u £ 10 5; 0: 0.1 0.1 0.1 (a) 10 (b) 10 (c) 10 drex / ^rect Fig. 5.19. Recrystallized grain size distribution in partially recrystallized (a) T4, (b) P A and (c) O A samples after annealing for 2 weeks at 325°C. 84 Chap. 5 Experimental Results and Discussion 100 um Fig. 5.20. E B S D map showing the colonies of large recrystallized grains in 40% cold rolled O A sample annealed for 2 weeks at 325°C. Black lines represent grain boundaries which are >15° and grey lines represent boundaries between 2-15°. 85 Chap. 5 Experimental Results and Discussion Lastly, the average internal misorientation of the substructure in the deformed grains is found to be in the range o f 5-6° for al l the samples based on E B S D measurements. This value is in agreement with the measurements by Vatne et al. (1996) in hot deformed aluminum alloys. Generally, recrystallization proceeds at a very sluggish rate at 325°C irrespective of prior aging conditions. The recrystallization kinetics of the T4 and P A specimen are similar: It took 2 weeks of annealing for the recrystallized grains to consume - 3 7 % of the deformed microstructure (Figs 5.18a and b). The recrystallization kinetics i n the O A sample is by far the slowest with only - 3 0 % of the microstructure recrystallized after 2 weeks of annealing (Fig. 5.18c). In an attempt to fully recrystallize the deformed microstructure, the annealing time for the T4 and O A samples was subsequently increased to 40 days. The resulting partially recrystallized microstructures are shown in Figs. 5.21a and 5.21b for the O A and T4 samples respectively. The volume fraction of recrystallized grains in the T4 samples increased to - 4 8 % while only - 3 7 % o f the microstructure in the O A samples were recrystallized after 40 days of annealing. Fig. 5.22a compares the recrystallization kinetics o f the two samples in terms of the fraction recrystallized vs. annealing time. The evolution o f recrystallized grain size and number of recrystallized grains per unit area are shown in Figs. 5.22b and 5.22c, respectively. The magnitude o f the error bars shown in all the figures represents the standard deviations between multiple fields of measurements obtained from different areas of the microstructure. There is a considerable scatter in the recrystallized fraction data indicating that the recrystallization occurred heterogeneously in the microstructure. The spatial distribution o f precipitates was examined i n S E M and the micrographs are shown i n Fig. 5.23a and 5.23b for the T4 and O A samples, respectively. Both samples were obtained after annealing for 2 weeks 86 Chap. 5 Experimental Results and Discussion 100 um Fig. 5.21. E B S D maps showing the partially recrystallized microstructures of 40% cold rolled AA6111 with varied precipitate conditions after annealing for 40 days at 325°C. The corresponding prior aging conditions are (a) O A and (b) T4. Black lines represent grain boundaries which are >15° and grey lines represent boundaries between 2-15°. 87 Chap. 5 Experimental Results and Discussion 0 200 400 600 800 1000 1200 Time (hrs) o o • 30 • E a. 5 25 • aged 20 • Avei 15 10 (b) Annealing Temp. = 325°C OA J -l-— -i-— JU 200 400 600 800 Time (hrs) 1000 1200. E E CD L_ D) T3 CD N o 0) JUUU • (c) 2500 • Annealing Temp. = 325°C 2000 • 1500 - T4 j 1000 -500 -I OA 0 • • •III—•—i—i—• i i -I—1 1 1 I I I ! 1 • • • • • • 1 1 1 0 200 400 600 800 1000 1200 Time (hrs) Fig. 5.22. Evolution of (a) fraction recrystallized (b) recrystallized grain size and (c) number of recrystallized grains per unit area during isothermal annealing at 325°C. 88 Chap. 5 Experimental Results and Discussion Fig. 5.23. Spatial distribution of precipitates in (a) T4 and (b) O A samples annealed for 2 weeks at 325 °C. Note the absence of precipitate clusters in the T4 sample. 89 Chap. 5 Experimental Results and Discussion at 325°C. Precipitates can be found along grain boundaries as wel l as in the matrix. The main difference between the two samples is the spatial distribution o f the precipitates. In the case o f the T4 sample, relatively coarse particles are homogeneously distributed throughout the microstructure resulting in a more uniform distribution of particle spacing (Fig. 5.23a). On the other hand, localized clusters o f relatively fine and closely spaced precipitates are observed in the O A sample, as shown in Fig. 5.23b. These precipitated clusters are not distributed uniformly throughout the microstructure. Precipitate free zones can be found in the matrix adjacent to some of the precipitate clusters. In the example shown in Fig. 5.23b, precipitate free zones comprise approximately 15-25% o f the microstructure. A clearer view o f the boundary between precipitate and precipitate free zones is shown in Fig. 5.24a. The X-ray spectrum confirms the presence of M g , Si and C u in the precipitates in the precipitate zones (Fig. 5.24b). In order to assess the effect o f the spatial distribution o f precipitates in the O A sample, specifically the precipitate free zones on the recrystallization process, it is necessary to link the precipitate structure seen in the S E M micrographs to the partially recrystallized microstructures as observed in E B S D (Fig. 5.23b vs. Fig. 5.18c). This is illustrated in Fig. 5.25 where the same area o f the O A sample annealed for 2 weeks are shown employing the two different experimental techniques. The E B S D micrograph in Fig. 5.25b is obtained by utilizing the band contrast function which shows the indexing quality in terms of grayscale: the darker the grains the lower the indexing quality. Recrystallized grains which are free of internal substructures are shown in white and indicated by the letters A , B and C. B y using a hardness indent as a marker, the precipitate structure of the same area is then examined in 90 Chap. 5 Experimental Results and Discussion 0 2 4 6 8 keV Fig. 5.24. (a) S E M micrograph showing the boundary between precipitate and precipitate free zones in 40% cold rolled O A sample annealed for 2 weeks at 325°C. (b) X-ray spectrum showing the presence of M g , Si and C u in one of the precipitates. 91 Chap. 5 Experimental Results and Discussion Fig. 5.25. (a) S E M micrograph showing the partially recrystallized microstructure of 40% cold rolled O A sample after annealing for 2 weeks at 325°C. (b) E B S D band contrast map showing the same area of the microstructure. The recrystallized grains A , B and C in the E B S D map are associated with precipitate free zones in the S E M micrograph. The precipitate clusters are marked by D , E , F and associated with deformed grains in the E B S D map. 92 Chap. 5 Experimental Results and Discussion S E M employing the back-scattered electron mode as shown in Fig. 5.25a. It can be seen that the recrystallized grains are clearly associated with precipitate free zones while the deformed grains (marked by the letters D , E and F) are related to the precipitate clusters in the S E M micrograph. Further evidence of this is shown in Fig. 5.26 which shows a growing recrystallized grain nucleated in the vicinity of a Fe-rich intermetallic particle surrounded by precipitate free zones. 5.1.5 The Effect of Annealing Temperature on Overaged Samples The softening behaviour o f the O A sample at 250, 325 and 445°C is illustrated in Fig. 5.27a. Fig. 5.27b shows the softening curves for the S O A specimens obtained at the annealing temperatures o f 325 and 445°C. It can be observed that the yield stress o f the S O A sample decreases rapidly to -50 M P a after only 1 minute of annealing at 445°C and remains constant after that. The decrease in yield stress for the O A sample is slower with the yield stress reaching 50 M P a in about 30 minutes. A t the lower annealing temperature o f 250°C, the decrease in yield stress of the O A sample appears to reach a plateau at around 180 M P a after 5 minute before decreasing to 160 M P a after 3 hours o f annealing. The effect of annealing temperature on the recrystallization kinetics is shown in Fig. 5.28. It is observed that recrystallization is faster in the S O A sample for the two temperatures investigated. The main difference is observed at the annealing temperature of 325°C where recrystallized grains were detected after -10 hours o f annealing i n the S O A sample compared to 48 hours in the O A sample. On increasing the annealing time to 40 days, 80% o f the 93 Chap. 5 Experimental Results and Discussion Fig. 5.26. S E M micrograph using back scattered electrons showing a recrystallized grain nucleated from a Fe-rich intermetallic particle surrounded by precipitate free zones in 40% cold rolled O A sample annealed for 2 weeks at 325°C. 94 Chap. 5 Experimental Results and Discussion 250 o io1 1 0 2 1 0 3 io4 io5 : 1 0 6 1 0 7 io8 Time (s) Time (s) Fig. 5.27. The effect of annealing temperature on the softening behaviour o f 40% cold rolled (a) O A sample and (b) S O A sample. The as cold rolled yield stress is indicated at the intercepts with the vertical axis. 95 Chap. 5 Experimental Results and Discussion 1.0 -0.8 -0.6 -0.4 0.2 H 0.0 (a) Annealing Temp. = 325°C 104 X 1.0 -0.8 -0.6 -0.4 -0.2 0.0 (b) r 1 / / SOA / / / / / O A m / / / / / • i i i i i 11 i i i i i i i 11 105 106 I I I I I I I I 107 Time (s) SOA ! OA TI I Annealing Temp. = 445°C i i i 102 103 • 111 i I I 104 101   105 Time (s) Fig. 5.28. Isothermal recrystallization kinetics of 40% cold rolled AA6111 at (a) 325°C and (b) 445°C with varied precipitate conditions. 96 Chap. 5 Experimental Results and Discussion deformed microstructure in the S O A sample was consumed by recrystallized grains but only 30% of fraction recrystallized was achieved in the O A sample. The recrystallization kinetics in both of the samples increases considerably as the annealing temperature was increased to 445°C. In the S O A sample, 80%> of the microstructure was recrystallized within 10 minutes of annealing while it took 30 minutes for recrystallization to reach the same fraction in the O A sample. Fig. 5.29 shows the fully recrystallized microstructure of the two samples after annealing for 100 min at 445°C. The average recrystallized grain size was determined to be 57 and 52 um for the O A and S O A samples, respectively. Finally, it is noted that no recrystallization was observed in the O A samples annealed at 250°C. 5.1.6 TEM Studies of the Annealing Behaviour of Overaged Samples The objective of the T E M experiments was to gain additional insight on the annealing behaviour o f the overaged materials, in particular the mechanism o f interaction between recovery, recrystallization and precipitation. The experiments focused on examining selected O A and S O A samples annealed at 325°C. A l l the T E M work presented i n the following was carried out in collaboration with Dr. X . Wang from the Brockhouse Institute of Materials Research at McMaster University. The T E M micrograph o f Fig. 5.30 shows the recovered microstructure o f the O A sample after annealing for 2 minutes at 325°C. It can be seen that cell structures consisting of diffuse cell walls with high density of tangled dislocations begin to form in the early stage of annealing. Precipitates are observed to pin low angle grain boundaries during recovery as shown in Fig. 97 Chap. 5 Experimental Results and Discussion Fig. 5.29. Fully recrystallized microstructure of the 40% cold rolled (a) OA and (b) SOA samples annealed for 100 minutes at 445°C. 98 Chap. 5 Experimental Results and Discussion Chap. 5 Experimental Results and Discussion 5.31. Fig. 5.32 shows that the tangles o f dislocations in cell walls transformed into subgrain boundaries as the annealing time was increased to 7 days. The formation o f subgrains is more prominent in the S O A sample where well defined subgrains with boundaries made up o f multiple sets o f dislocations are observed after 100 mins o f annealing, as shown in Fig. 5.33. A series o f T E M images is shown in Fig. 5.34 to illustrate the interaction between precipitation and recrystallization in the O A sample annealed for 7 days at 325°C. Fig. 5.34a shows a growing recrystallized grain embedded in the deformed matrix. In Fig. 5.34b, a T E M image is shown across the recrystallization front for the growing grain. Precipitates are observed to pin the migrating grain boundaries thus causing segments o f the boundaries to curve forward (Fig. 5.34c). 5.2 Discussion of Experimental Results The experimental results are analyzed in detail in the next three subsections: First, a nonlinear addition law is employed to quantify the contributions o f dislocations and precipitates to the as cold rolled yield stress. Then, the evolution o f resistivity i n the T4, P A and O A samples during isothermal annealing is interpreted in terms of the nucleation, growth and coarsening of the precipitates. In the end, the effect o f prior aging condition on the isothermal annealing behaviour is analyzed with special attention given to the O A sample. 100 Chap. 5 Experimental Results and Discussion 101 Chap. 5 Experimental Results and Discussion 102 Chap. 5 Experimental Results and Discussion Chap. 5 Experimental Results and Discussion (a) Fig . 5.34. (a) T E M micrographs showing a recrystallized grain embedded in the deformed matrix of the O A samples annealed for 7 days at 325°C. T E M micrographs showing segment of the migrating grain boundaries are shown in (b) and (c). Courtesy of Dr. X. Wang. 104 Chap. 5 Experimental Results and Discussion 5.2.1 Flow Stress Addition Law Previous studies on AA6111 have shown that the work hardening behaviour of AA6111 is strongly influenced by the precipitation states in the material [Cheng et al., 2003]. This is evident in the data presented in Fig. 5.9 which shows the remarkable increase in yield stress in the naturally aged materials after 40% cold rolling. On a macroscopic level, one can quantify this effect by carefully considering the manner in which the contributions from precipitates and dislocations are summed. In general, the summation of the solid solution strengthening component, crss precipitate contribution, <jp and dislocation hardening contribution, cr^ can be described by using a nonlinear addition law with the following form [Cheng et al., 2003]: \_ <jy = CTi+cjss +(ap" +crdis"Y (5.1) where cr, is the intrinsic strength of aluminum and n is a constant. Taking 10 M P a as a reasonable estimate for the intrinsic strength of aluminum [Wang et al., 2003] and assuming negligible contributions from solid solution for the aging conditions examined, the precipitation hardening contributions can be calculated from the as aged yield stress, i. e., crp = <?as-aged - oj. The latter assumption is valid i f most of solutes are precipitated out which is the case in the present analysis. The results of this simple calculation are tabulated in Table 5.2 for each of the precipitate conditions. Furthermore, by using an appropriate value for the constant n in Equation 5.1, the contribution from dislocation hardening to the flow stress can be calculated. The value of n depends on the strength and density o f the obstacles. Typically, n 105 Chap. 5 Experimental Results and Discussion varies from 1 for weak obstacles (e.g. solid solution or clusters) to a value of 2 for strong obstacles (e.g. large precipitates). For intermediate cases, the value o f n is expected to fall between these extremes. In the present case, the strengths o f the precipitates in the T4 and P A samples can be classified as weak (n - 1) and moderate (n = 1.5) obstacles respectively while the precipitates in the O A and S O A samples can be considered as strong obstacles (n = 2). Based on these assumptions, the magnitudes of cfe can be calculated and the results are summarized in Table 5 .2 . B y comparison, the T4 sample work hardened to a much larger extent than the rest o f the samples. The magnitude o f Orf,s which is directly responsible for the stored energy for subsequent recovery and recrystallization processes is considerably higher in the T4 and P A samples as compared to the O A and S O A samples. It should, however, be noted that the current interpretation of the as deformed yield stress represents a greatly simplified approach as the work hardening process is expected to be strongly dependent on whether the precipitates can be sheared by dislocations. Additional complications may arise due to the fracture o f large precipitates in the O A and S O A samples during the cold working process. Improvements are possible by adopting a more complicated work hardening model [Sevillano, 1993]. However, with the current modelling objective in mind, this does not appear to be justified. 5.2.2 Evolution of Resistivity A n y factors that tend to distort the regularity of the crystal lattice w i l l increase the resistivity of the materials, pr by scattering the conducting electrons [Stanley, 1963]. In the present case, there are three relevant factors that may contribute to the increase of resistivity in the 106 Chap. 5 Experimental Results and Discussion Table 5.2. Summary of the flow stress contributions from precipitates and dislocations as a function of precipitation state. Initial precipitation state n ^As-aged ^ As-cold rolled 0~dis T4 1 139 356 129 217 P A 1.5 334 430 324 197 O A 2 153 220 143 154 S O A 2 92 194 82 165 107 Chap. 5 Experimental Results and Discussion materials: i) dislocation density, ii) solute atoms in solid solution and iii) precipitates. The contribution of these three factors can be summed linearly according to Matthiessen's law: Pr=P(T) + Ti PiCi + Pppt + Pdis (5 -2) i In Equation 5.2, p(T) represents the resistivity o f pure A l which is a constant at a given measuring temperature, C, is the concentration of a solute element and pt is the corresponding resistivity coefficient. pppt and pdiS are the resistivity contributions due to precipitate and dislocations respectively. In cold worked materials, the magnitude o f pdiS is expected in the order of -0.3 nQm. Furthermore, it is not far from reality to assume that the term ppp, diminishes rapidly upon annealing at elevated temperature due to the significant increase in both precipitate size and spacing. Therefore, based on Equation 5.2, the variation in resistivity can be directly related to the variation in the concentration o f solute in solid solution. This can be verified by careful examination of the resistivity curves shown in Fig. 5.16 in terms of nucleation, growth and coarsening of precipitates: 1) During the artificial aging of solution treated samples, the nucleation and growth of fine scale precipitates which significantly deplete the matrix o f solute atoms caused the resistivity to decrease sharply in the early stage of annealing. This corresponds to the initial rise in the age hardening curve to peak strength at 325°C (Fig. 5.8). The depletion o f solute atoms appears to continue slowly until -3000 seconds at which point the value of resistivity becomes identical to the O A sample. A t this annealing time, the precipitates are well into their coarsening stage. During coarsening, the 108 Chap. 5 Experimental Results and Discussion variation o f solute concentration is not significant and this is reflected in the resistivity curve which shows very small variations. 2) The evolution of resistivity of the T4 and P A samples can be explained in a similar way as the artificially aged samples. The initial rapid nucleation and growth of the Q' phase caused the resistivity to drop by drawing solutes from solid solution. The effect o f this nucleation and growth processes is again short lived (up to ~3000s). After that the precipitation process is dominated by the coarsening of precipitates. 3) The resistivity of the O A sample remains virtually unchanged in the initial stage of annealing. This behaviour is expected since the stable precipitates are well into the coarsening stage prior to cold rolling. In other words, there is no supersaturation in the solution to drive the nucleation and growth of new precipitates upon annealing. The flat resistivity curve also confirms that the reduction o f dislocation density due to recovery did not influence the resistivity measurements. It is less clear why all the resistivity curves slowly decrease in the later stage o f annealing. One possible explanation for this is that the precipitation o f equilibrium phases reduces slightly the concentration o f solute in the matrix after long annealing time. Finally, it should be noted that while the resistivity measurements indicate that the precipitation process during isothermal annealing at 325°C is mostly dominated by the coarsening o f precipitates, the kinetics of the process might be different for different initial precipitate conditions. Furthermore, the spatial distribution of precipitates is expected to play an influential role in the recrystallization process. 109 Chap. 5 Experimental Results and Discussion 5.2.3 The Effect of Prior Aging Conditions The effect of varying the initial precipitate conditions on the isothermal annealing behaviour at 325°C is summarized in Table 5.3. Two important observations are noted: i) the overall recrystallization kinetics is sluggish; the fastest recrystallization is observed in the S O A sample while the O A sample recrystallized at the slowest rate, ii) the majority of softening is due to the simultaneous occurrence of recovery and precipitate coarsening, especially in the T4 and P A samples. These observations can be explained by examining the combined effect of recovery, recrystallization and precipitation. T4 (naturally aged) It is expected that the clusters and G P zones formed during the natural aging process started to dissolve rapidly during heat up to the annealing temperature. Some o f the clusters may act as potential nucleation sites for the subsequent precipitation o f Q' phase. A significant amount of this precipitation took place preferentially at the elongated grain boundaries and substructures (Fig. 5.23a), thereby pinning these boundaries and also pinning any new recrystallized grains that may form. The inhibiting effect diminishes as precipitates coarsen which ultimately leads to the onset of recrystallization. P A (peak aged): The isothermal softening kinetics of the T4 and P A samples is essentially identical as shown in Fig. 5.15a. The dramatic drop in yield stress of the P A sample can be attributed to the rapid transformation of the fine fi" precipitates to the coarser Q' phase. The fraction of 110 Chap. 5 Experimental Results and Discussion Table 5.3. Summary of the softening and recrystallization behaviour of 40% cold rolled AA6111 with various precipitate conditions at the annealing temperature of 325°C. Initial heat treatment Initial precipitate phases Softening Recrystallization As deformed ay (MPa) Final ay (MPa) j Onset time Fraction recrystallized1 T4 (8 days at room temperature) Solute clusters and G P zones 356 71 12 hours ~ 48% P A (7 hours at 180°C) Fine P" (80% by volume fraction) + Q' 430 73 after 2 weeks 12 hours ~ 35% after 2 weeks O A (7 days at 250°C) Lath shaped Q' 220 73 48 hours ~ 37% S O A (7 days at 325°C) Lath shaped Q' + some square shaped M g 2 S i 194 41 10 hours ~ 90% After 40 days of annealing unless otherwise noted 111 Chap. 5 Experimental Results and Discussion recrystallized grains in the two samples is also found to.be similar after 2 weeks of annealing at 325°C (see Figs. 5.18a and 5.18b). O A (overaged) and S O A (severely overaged) The precipitates in the O A and S O A samples are already in their stable configuration prior to annealing. These relatively large precipitates were fractured into small segments in the cold working process and as a result the number density o f precipitates was significantly increased (Fig. 5.13). Upon annealing, coarsening resumes but the high density of the precipitates pins grain boundaries making recrystallization a difficult process. However, the recrystallization rate was greatly enhanced by raising the annealing temperature to 445°C as shown in Fig. 5.28b. A n important consideration with regard to raising the annealing temperature is the effect o f precipitate dissolution. Based on the studies by Burger et al. (1996) on AA6111 , significant dissolution of precipitates was observed to occur at temperatures above 430°C. Hence, it is l ikely that the number density of precipitates quickly diminishes upon annealing at 445°C. This coupled with the fact that recrystallization being a thermally activated process results in the rapid recrystallization seen in Fig. 5.28b. Lastly, it should be mentioned that recrystallization is found to occur at two commonly observed sites in aluminum alloys: i) deformed grain boundaries (Fig. 5.18) and ii) deformation zones around large intermetallic particles (Fig. 5.26). It is interesting to further compare the recrystallization behaviour of the O A and T4 samples based on the data presented in Fig. 5.22. There are a number o f notable differences in the development of recrystallizing microstructures in the two samples. First, it is clear that 112 Chap. 5 Experimental Results and Discussion recrystallization is faster in the T4 samples. One possible reason behind this is the difference in the stored energy accumulated during cold rolling. It has been shown in section 5.2.1 that samples with T4 condition work hardened to a much larger extent than the O A samples. O n increasing the annealing time to 2 weeks, the resultant average recrystallized grain size in the O A sample is larger than the grain size found in the T4 sample (20 vs. 15 um). Colonies of large recrystallized grains with a diameter as large as 100 urn can be observed in the O A sample (Fig. 5.20) whereas smaller but more evenly distributed recrystallized grains are found in the T4 sample (Fig. 5.18a). This difference can be explained by comparing the spatial distribution o f precipitates in the two samples since the limiting grain size is directly related to the spacing between precipitates. The large precipitate free zones in the O A sample has been shown to correlate with recrystallizing grains (Fig. 5.25). In the case of the T4 sample, re-precipitation of Q' phase occurred in the matrix and along grain boundaries during annealing. These precipitates then coarsen resulting in a much finer distribution o f precipitate spacing (Fig. 5.23a). Additional insight on the recrystallization process can be gained by carefully examining the T E M micrographs presented in Fig. 5.34. It is apparent from the curvature of the migrating front, the boundary is subjected to two opposing pressures. The subgrains surrounding the recrystallization front exert a forward pressure in order to consume the recovered microstructure. But segments of the grain boundary are immobile due to the pinning pressure exerted by precipitates. The consequence o f these competing pressures is that the segment of the migrating front which is in contact with the subgrains bow forward and where the boundary is pinned by the precipitates, it is bowed back (Fig. 5.34c). 113 Chap. 5 Experimental Results and Discussion 5.3 Concluding Remarks The aging heat treatments carried out prior to cold rolling encompass a wide range of possible precipitate conditions in A A 6 1 1 1 . In the next chapter, a microstructure model based on the internal state variable approach is developed for the overaged samples based on the knowledge acquired in this chapter. 114 Chapter 6 Modelling of Microstructure Evolution for Overaged AA6111 In this chapter, the internal state modelling approach is adopted to develop a comprehensive microstructure model for the annealing of overaged A A 6 1 1 1 . Special emphasis is placed on integrating existing physically based models for the various elementary microstructural reactions: recovery, subgrain growth, recrystallization and precipitation. The chapter is organized in the following order: in the first part of this chapter, section 6.1, the internal state variable modelling approach is introduced along with the internal state variables that have been selected to represent the individual processes of recovery, subgrain growth, precipitation and recrystallization. In the formulation of material response equations, a modelling strategy based on the rule of mixtures is employed to account for the effect o f heterogeneous spatial distribution of precipitates in the deformed microstructure. From sections 6.2 to 6.5, the respective evolution laws for recovery, subgrain growth, recrystallization and precipitation are derived and coupled using physically sound theories in the literature. Material specific parameters which are required as input to the model are calculated in section 6.6. Section 6.7 explains the implementation of the model and gives a schematic outline of the overall model framework. Section 6.8 is devoted to compare the model predictions with available experimental data in terms of softening and recrystallization kinetics. In section 6.9 the model is employed to analyze quantitatively the interaction between the various microstructural processes. Finally, this chapter concludes with a brief discussion in section 6.10 on the limitations of the present model. 115 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 6.1 Model development - The Internal State Variable Modelling Approach The concept of microstructural modelling based on internal state variables was originally proposed by Richmond (1986) and applied to deformation processes. Since then, this modelling method has been expanded and widely applied to other industrial processes of commercial alloys which includes casting, cooling after hot forming, aging heat-treatment and welding [Grong and Shercliff, 2000]. The internal state variable modelling approach is particularly suited for modelling industrial thermomechanical processes because it is capable of relating changes in product properties to the changes in microstructure resulting from changes in processing parameters. Hence, an accurate internal state variable model developed based on physically sound mathematical equations is extremely useful in process design where operational parameters are optimized to achieve desired product properties. The first step in the development of an internal state variable model is to identify appropriate state variables to represent the most significant aspect o f the microstructure. In general, a microstructural evolution phenomenon can be well described by one variable. It is unusual to consider more than three variables for a given microstructure process [Grong and Shercliff, 2000]. These internal state variables are, in principle, measurable physical quantities although some of them are measured indirectly in practice; for example, dislocation density is often quantified based on flow stress measurements. Table 6.1 lists the internal state variables that have been selected to represent the four principal metallurgical reactions considered in the present model. 116 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Table 6.1. List o f internal state variables for recovery, subgrain growth, recrystallization and precipitation considered in the present modelling approach. Microstructural Process Internal State Variables Recovery Dislocation density, pdis* Subgrain growth Subgrain size, Ssb Recrystallization Volume fraction o f recrystallized grains, X Precipitation Precipitate radius, Rp and density, Followed indirectly by the evolution of flow stress 117 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The next step involves the formulation o f a series of mathematical equations to capture the evolution of internal state variables as a function of reaction time at a given reaction temperature. Mathematically, it is most convenient to formulate the evolution laws based on a system of coupled, in general non-linear first-order differential equations: ^ L = gl(T,ShS2 .S^ (6.1) where T is the instantaneous temperature, and Si, S2,---Si are the instantaneous values of the internal state variables. Each internal state variable, Si, evolves with increasing time and may be a function of other state variables for a given set of processing conditions such as temperature and amount o f prior cold reduction. Equation 6.1 is strictly applicable to a thermally controlled process where the evolution of each internal state variable in the next time increment is uniquely defined by these instantaneous values and the current temperature. Hence, introduction of some initial conditions, i.e., the condition at time, t = 0 is required. These evolution laws can be integrated over the time and temperature history of the heat treatment cycle using an appropriate numerical method thereby determining the internal state variables o f the resulting microstructure. In the present work, established models in the literature are adopted to devise the evolution laws for the individual processes of recovery, subgrain growth, recrystallization and precipitation. The final step i n the development o f an internal state variable model is to construct appropriate material response equations to link the output of the evolution laws, i.e., the resulting microstructure parameters to the physical or mechanical properties of the materials. In the 118 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 present modelling approach, the evolution of internal state variables is linked to the softening in flow stress as a function o f isothermal annealing time. One simple approach is to model the microstructure as consisting o f a mixture o f precipitate and precipitate-free zones. Based on this composite model, the overall flow stress of the material, <yy at any time during annealing is obtained by a simple rule of mixtures: The subscripts / and II indicate the quantities of interest in the precipitate-free and precipitate zones, respectively. A s a first approximation, Equation 6.2 is assumed to be valid since the volume fraction of the material that is free of precipitates, F, is relatively small (F - 0.15 estimated from S E M micrographs) and the microstructure is dominated by the precipitate zones. Furthermore, the strength levels of the softer (precipitate free zones ) and the harder (precipitate zones) fractions are expected to be within one order of magnitude. Following Equation 6.2, the overall volume fraction recrystallized grains is also given by the rule o f mixtures: It is straightforward to determine the flow stress of the precipitate free zones, o/. It contains a contribution from the flow stress of the fully recrystallized fraction, crrex (-40 MPa) and a contribution from the recovered flow stress, o-rc/ in the unrecrystallized fraction. Again using the rule of mixtures, 07 is calculated as follows: c r y = c r I F + * I I ( l - F ) (6.2) X = XIF + X I I ( \ - F ) (6.3) a-1 = a , + crrexXj + a r c I ( l - X j ) (6.4) 119 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 where at is the intrinsic strength of A l (-10 MPa) . crrex is larger than cr, by 30MPa due to residual contributions from precipitation hardening and at the highest annealing temperature of 445°C there is also an additional contribution from solid solution strengthening ass • However, changes in ass during annealing heat treatment is considered o f minor importance in the present analysis since ass scales with the solute concentration, C in a non-linear manner, i.e., &ss ~ with q < 1 [Esmaeili, 2002]. The determination of the flow stress of the precipitates region, an is more complicated. It contains an additional contribution from precipitates, ap. This can be determined by combining a series o f non-linear addition laws with the rule o f mixtures: ° 7 / = °7 + Vrex-matrixXll + GUnrex-matrix 0 ~ X l l ) (6-5a) °rrex-matrix ~ \Grex +<Tp (6.5b) I 2 2~ GUnrex-matrix= \GrcIl +<Tp (6.5c) The initial flow stress is assumed to be the same for precipitate and precipitate free zones, i.e., aj=an at t - 0. In Equations 6.4 and 6.5c, the recovered flow stress, arcj and arcn are obtained directly by numerical integration of the evolution law for recovery over the time and temperature history o f the annealing heat-treatment. Given that the precipitates are well into their overaged conditions, particle bypass is expected to be the dominant precipitate-dislocation interaction mechanism. Based on the work by Esmaeili et al. (2003b) on A A 6 1 1 1 , the precipitation contributions to the flow stress in the precipitate zones, ap, under this condition is given by: 120 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 MGbFpl (6.6) (2x)l/2Rp where M is the Taylor factor, G is the shear modulus and b the magnitude of the Burgers vector. The quantities Rp and Fp represent the average radius and volume fraction of the precipitates respectively. 6.2 Recovery It is assumed that the dislocation density, pd is directly related to the flow stress of the material according to the classical forest work hardening theory [Taylor, 1934]: where or is a constant in the order of 0.3. In the present modelling treatment, the recovery model developed by Verdier et al. (1999) is adopted to describe the recovery kinetics. This model is favored for two reasons: Firstly, the model relates the reduction in flow stress to the lowering of the average dislocation density and thus fits well with the internal state variable approach. Secondly, the model allows only for two adjustable parameters, namely the activation energy and activation volume, thus in line with the current modelling objective in keeping the unknown parameters to a minimum. (6.7) 121 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The mathematical formulation of the Verdier et a/.'s model for the precipitate free zones is shown in Equation 6.8a. In the precipitate zones, the retarding effect o f precipitates on recovery is captured by combining the model with the approach proposed by Zurob et al. (2002), as shown in Equation 6.8b. Precipitate free zones: d{crrcI-crrex) _ 64(arcI-arex)2yD dt 9M 3a 2E exp sink NA(°rd-°rex)V RgT (6.8a) Precipitate zones: di^rd! -^rex) dt 9M 3a 2E exp Qo sink NA(<7rcII-<7rex)V^ a » N ^ RgT 1 — Ndis J (6.8b) In Equations 6.8a and 6.8b, Rg denotes the gas constant and T is the temperature in Kelv in . The effective pinning parameter, ap which has a value o f less than 1, is introduced in Equation 6.8b to estimate the fraction of precipitates that are available to pin the dislocation networks. The magnitude o f the parameter ap is determined by fitting the model calculation to experimental data. Complete retardation of the recovery process, i.e., darc/dt = 0, is expected when cCpNp = NdiS. The number o f dislocation nodes, Ndis is approximated as 0.5p></'5 [Dutta et al, 2001, Zurob et al, 2002]. 122 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The key parameters in Equations 6.8a and 6.8b are the two activation parameters, i.e. the recovery activation energy, Q0 and activation volume, V. Estimation of the magnitude of these two parameters is problematic simply because Verdier et a/.'s recovery model does not discriminate between thermally activated cross slip, climb or solute drag as the rate determining process. However, the activation energy is l ikely to be approximated by the activation energy for self diffusion (-142 kJ/mol for A l [Smithells and Brandes, 1976]) i f dislocation climb is the rate controlling process since it involves the formation and movement of vacancies. On the other hand, i f solute drag is the rate controlling mechanism, then Qo is expected to lie within the range o f the activation energy for solute diffusion in aluminum, i.e. 130-140 kJ/mol. The reported values of the activation energy o f diffusion of the various species in aluminum are listed in Table 6.2. The values for the self diffusion o f aluminum is included for comparison. The second activation parameter, i.e. the activation volume is a quantity which is difficult to specify except for being o f the order o f b3. In the case o f thermally activated mechanism, the activation volume can be written as V - b2la where la is an activation length associated with the dislocation motion. In the case of solute drag as rate controlling mechanism, the activation length is a physical parameter defined as the distance between solute pinning points along the dislocation lines. Hence, /„ is inversely proportional to the concentration of solutes in the solution, CM [Nes, 1995]: 123 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Table 6.2. Reported diffusion data for M g , C u and Si atoms and self diffusion in bulk aluminum. Elements Diffusivity in Aluminum M g D0 = 2.2 x 10"4 m 2/s, QD = 130 kJ/mol M y h r and Grong, (2000), Myhr et al, (2001) Si D0 = 2.0 x 10"4 m 2/s, QD = 137 kJ/mol Burachynsky and Cahoon, (1997), Fujikawa et al, (1978) C u Do = 6.5 x 10"5 m 2/s, QD =136.1 kJ/mol Burachynsky and Cahoon, (1997), Fujikawa and Hirano, (1989) A l Do = 1.7 x 10"4 m 2/s, &> = 142 kJ/mol Smithells and Brandes, (1976) 124 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 where K is a geometrical constant of order of unity and Q is the atomic volume of aluminum (1.65xl0" 2 9 m 3 ) . In A A 6 1 1 1 , the concentration of residual solutes typically ranges from 0.26-0.74 atomic% after overaging at temperatures in the range of 200 to 400°C. Therefore, based on Equation 6.9, the activation volume is expected to be in the order o f 6-\2b3. In the present modelling approach, the activation volume is taken as a constant and set equal to an average value of 9b . This consideration amounts to the fact that bulk solute concentrations do not change significantly during annealing of overaged alloys. In summary, the two adjustable parameters in the recovery model are the activation energy (allowed to vary between 130-142 kJ/mol) and the effective pinning ccp (0 < ccp < 1). 6.3. Subgrain Growth The formation o f a wel l defined subgrain structure such as the one shown in Fig. 5.34 is promoted by the delay in the onset of recrystallization. A s annealing time increases, these subgrains may grow and coarsen in order to lower the stored energy o f the recovered structure via the reduction of grain boundary area. Following 0rsund and Nes (1989), i f the substructure can be approximated by an array of subgrains of radius Ssb then the driving pressure for subgrain growth, Psb may be estimated from P s b = ^ (6-10) 125 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 where a} is a geometrical factor in the order of 1.5 and ysb is the energy of the low angle subgrain boundaries. The magnitude of ysb is dependent on the misorientation of the boundary and may be estimated from the Read-Shockley equation [Vatne et al, 1996, Humphreys and Hatherly, 1995, Furu et al, 1999]: where v is the Poisson ratio (-0.33) and 6C is the critical misorientation angle for a high angle grain boundary (commonly taken as 15° [Humphreys and Hatherly, 1995]). Hence, by taking an average subgrain misorientation of 5° (obtained from E B S D measurements of the internal substructure of the deformed grains), the value of ysb is calculated as 0.16 J/m 2 . The rate o f subgrain growth is assumed to be proportional to the driving pressure, P S B , as shown in Equation 6.12a in the precipitate free zones. The constant of proportionality being the mobility of the subgrain boundaries, M'. In the precipitate zones, the retarding effect o f fine particles on the rate of growth of subgrains is considered based on a simple approach proposed by Humphreys and Hatherly (1995). In this approach, the driving pressure for subgrain growth is modified by a pinning term as shown in Equation 6.12b. Precipitate free zones: (6.11) W = M , a\Ysb (6.12a) d t Ssb,l (0 126 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Precipitate zones: dS, 'sb,II dt 3^(0 <W/(0 2RP(t)j (6.12b) If the motion of the subgrain boundaries is thermally activated, then the mobility, M' can be assumed to vary with temperature in accordance with an Arrhenius relationship of the form: M' - M'Q exp ( \ Q * } K R 8 T J (6.13) where QSb is the activation energy for subgrain growth and M'Q is the intrinsic mobility of the subgrain boundary. Admittedly, this calculation is not rigorous since a comprehensive approach in modelling subgrain growth should include the effect o f misorientations on the mobility of the boundaries [Furu et al., 1995, Humphreys and Hatherly, 1995]. Nonetheless, this simplified approach allows us to maintain the simplicity of the model and yet obtain a reasonable description of the underlying microstructure evolution. In the present model, both the parameters, Qsb and M'Q are treated as fitting parameters. Lastly, it is important to note that the contribution of subgrain size is not considered in the flow stress equations (Equations 6.4 and 6.5). In doing so, we have attributed the recovery softening during annealing solely to the annealing out of dislocations described in section 6.2. This is consistent with the approach proposed by Verdier and co-workers (1999) who showed that recovery subgrain growth does not effect the logarithmic time decay o f yield stress in A l -127 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 M g alloys. A single internal state variable based on dislocation density is sufficient to describe the softening curve. This is in contrast with the approach by Nes and Saeter (1995) who consider the flow stress of the material at any time during recovery is a function of both dislocation density, pd and subgrain size, 8sb- However, it should be noted that the relative importance of these two internal state variables in determining the flow stress is still a question of much debate in the literature [Nes, 1995]. 6.4 Recrystallization The present description of the evolution of volume fraction recrystallized grains, X with respect to annealing time is derived based on the J M A K approach. Assuming that the recrystallized grains are distributed randomly throughout the volume, then there is a simple mathematical equation to relate the real volume fraction recrystallized to the extended volume fraction recrystallized, Xext: dX dX"=J-x) ( 6 1 4 ) Differentiating Equation 6.14 with respect to time and rearranging gives: dX dX , /, x „ \ 128 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Assuming isotropic growth of spherical grains, the extended volume fraction is related to the volume of the growing grains and number of recrystallization nuclei per unit volume, Nrex by * . = f * * X , (6-16) where R is the average radius of recrystallizing grains in the absence o f hard impingement. Equation 6.16 can be differentiated with respect to time to obtain the term dXext/dt in Equation 6.15: Finally, the evolution of volume fraction recrystallized grains as a function of time is obtained by combining Equations 6.15 and 6.17: § = 4 ^ § N j l - X ) (6.18) dt dt Equation 6.18 allows recrystallizing grains to grow isotropically in three dimensions until impingement by neighbouring grains. This hard impingement effect is taken into account by the term (l-X). Therefore, as recrystallization proceeds to completion (X-> 1), dX/dt w i l l tend to become 0. It can be seen from Equation 6.18 that the evolution of volume fraction recrystallized grains depends on three factors: (a) the instantaneous grain radius, R (b) the recrystallized grain growth rate, dRJdt and (c) the number density o f recrystallization nuclei, 129 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Nrex. The growth rate depends critically on the stored energy in the deformed structure and therefore is a function of both the recovery and precipitate processes. The number density of recrystallized nuclei can be calculated from the amount o f rolling reduction applied to the as aged materials. These two quantities are evaluated in separate sections below. 6.4.1 Nucleation It is difficult to formulate a physical description o f the nucleation process for recrystallization primarily due to lack of understanding of the deformed structure. The first major problem one encounters is the definition of recrystallized "nuclei" because recrystallized grains do not "nucleate" as totally new grains via the classical random atomic fluctuation mechanism proposed for phase transformations [Cahn, 1950, Humphreys and Hatherly, 1995, Doherty et al, 1997]. However, it is now widely accepted that new recrystallized grains originate from pre-existing sites in the deformed microstructure. A common feature among these pre-existing sites is high local misorientations, for examples, highly misoriented deformation zones around large particles and deformed grain boundaries [Doherty et al, 1997]. Another important consideration is the role of recovery in facilitating the nucleation of recrystallization [Ray et al., 1975, Bay and Hansen, 1979, Humphreys and Hatherly, 1995]. Unfortunately, a theoretical model is presently not available for recrystallization nucleation. Hence, under the prevailing circumstances, it is assumed that all nucleation events effectively take place at the onset of recrystallization, i.e., site saturated nucleation. This assumption has been shown to be satisfactory for industrial alloys with large prior deformation [Furu et al., 1990, Humphreys and Hatherly, 1995]. The number of recrystallization nuclei per unit volume, Nrex, is assumed 130 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 to be proportional to the sum of the grain boundary area and Fe-containing particles surface area: A where Ac is the area o f the critical nucleus (~RC2)- The basic assumption o f Equation 6.19 is that both grain boundary area and area in the vicinity o f Fe-containing particles have the same potency for the nucleation o f recrystallized grains. The critical nuclei radius, Rc is assumed to be in the order o f 0.5 urn based on estimation o f subgrain size from T E M micrographs. From Equation 6.19, it can be seen that the maximum number of possible nuclei is given by the ratio between the total surface area per unit volume and the critical nucleus area. The nucleation parameter k (0 < k < 1) determines the potency o f the nucleation sites which in the present modelling treatment needs to be treated as a fitting parameter. In Equation 6.19, two possible nucleation sites are considered: 1. Area adjacent to the Fe-rich constituent particles: The total surface area per unit volume, SVwFe, can be calculated from the number density, NViFe and the averaged radius of the Fe-rich particles, Rpe,'-SvFe=A7tRFe2NFe (6.20) 131 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The values for A V e and RFe are determined from experimental measurements (see section 5.1.3). 2. Deformed grain boundary area: The total deformed grain boundary area per unit volume, Sv,gb after cold rolling can be estimated from the solutionized grain size, ds (~ 42pm). B y assuming that the shape of the solutionized grains resembles a regular tetrakaidecahedron, Sv>gb is obtained as follows [Chen et al., 2002]: — a + 3 f l J l + -^ -+3 < 2ds[ i a2 I a 4 2 + 2a' a (6.21) where a = exp(sT). The true strain, sT is related to the amount o f prestrain, r (in % reduction) according to £j = ln f 1 ^ v l - r / 1 0 0 , (6.22) 6.4.2 Growth rate The growth rate of the recrystallizing grains is given by the wel l known relationship of ~d- = MPd (6-23) 132 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 where M is the mobility of the high angle grain boundary and Pd is the driving pressure for recrystallization. The calculation of the driving pressure is somewhat complicated because the recovered microstructure is composed of well defined subgrain structure and deformed structure. The subgrain boundaries are made up of multiple sets of well defined dislocation networks while the interior o f the subgrains are relatively free o f dislocations (Fig. 5.33). Outside the subgrains in the deformed regions, more or less uniformly distributed dislocations can be identified (Fig. 5.34). Following Vatne et al. (1996), the average stored energy per unit volume of such a composite structure can be defined as follows: Pd=\pdGb2+^L ( 6 > 2 4 ) 2 °sb The growth rate o f the recrystallizing grain in the precipitate free zones is obtained by combining Equations 6.23 and 6.24, as shown in Equation 6.25a. The dislocation density is time dependent due to the effect o f concurrent recovery. In the precipitate zones, the driving pressure for recrystallization is modified by a Zener retarding term under the condition of evolving precipitate size and density. This is shown in Equation 6.25b. Precipitate free zones: dt \ P d J { t ) G b 2 + - a ^ 2 ^ W <W(0. (6.25a) 133 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Precipitate zones: dRn dt .2 , a\Ysb IrgbFpi*) $sb,n{t) 2Rp(t) J (6.25b) In Equation 6.25b, ygb denotes the high angle grain boundary energy (-0.324 J/m 2 [Murr, 1975]). The high angle grain boundary mobility, M i s a complex function of the concentration of the solutes due to the solute drag effect as discussed in section 2.6.2. Since recrystallization is a thermally activated process, a simplified approach is to assume that the mobility obeys an Arrhenius relationship, i.e.: Here, Qrex denotes the activation energy for recrystallization and M0 is a pre-exponential factor. Equation 6.26 has been found to work generally wel l with many materials including both steel and aluminum alloys [Huang and Humphreys, 1999, Humphreys and Hatherly, 1995]. A s a first approximation, the magnitude of Qrex is set equal to 200 kJ/mol based on the recrystallization data from Vatne (1995). The value for Mo is found by fitting the model to experimental data. M = MQ exp -V (6.26) 134 6.5 Precipitation Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 In the precipitate zones, an evolution law is required to follow the change in the precipitate size and density during annealing. In the present work, the evolution laws for dissolution and precipitation are mainly derived from the models developed by Cheng et al. (2000) and Deschamps and Brechet (1999). Since the model is only applicable for overaged materials, the nucleation and growth o f new precipitates are not considered. The present precipitation model primarily deals with concurrent dissolution/growth and coarsening o f Q' precipitates. The complexity o f the precipitation sequence involving various intermediate phases is not accounted for in the model. For mathematical simplicity, the progress of precipitates dissolution/growth and coarsening are monitored by following the changes in the average values of the precipitate density, Np and radius, Rp. Furthermore, it is assumed that individual segments of fractured Q' precipitates (Fig. 5.13) can be represented by spherical precipitates. This greatly simplifies the mathematical formulation o f the evolution equations for dissolution/growth and coarsening. 6.5.1 Dissolution/Growth In the first stage o f annealing, pre-existing precipitates in the deformed matrix can either grow or dissolve. Whether the precipitates may dissolve or grow depends on its initial size and the annealing temperature. The rate at which this occurs can be expressed by a growth law for spherical particles [Cheng et al., 2000]: 135 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 dRp CM ~ Q Deq I Deq Cp-Q Rp y nt x l / 2 dt (6.27) \d/g where CP and C, denotes the solute concentration in the precipitate and at the precipitate-matrix interface, respectively. CM represents the average solute concentration in the matrix and is the equivalent diffusivity. Following Deschamps and Brechet (1999), it is assumed that the precipitate composition is a fixed combination of A L j C ^ M g g S i ? and that one can describe the diffusion kinetics for this combination of species using an equivalent diffusivity. The temperature dependence o f the diffusion process is expressed by an Arrhenius type of equation: where QD is the activation energy for diffusion and Do is a pre-exponential factor. A s a first approximation, the magnitude of QD is set equal to the activation energy of self diffusion aluminum, i.e., QD = 142 kJ/mol (Table 6.2). The magnitude o f the pre-exponential factor It can be seen from Equation 6.27 that the precipitates w i l l dissolve i f C, > CM and grow i f C, < CM- The effect o f curvature on dissolution and growth kinetics enters Equation 6.27 via the Gibbs-Thompson equation [Martin et al, 1997]: f Deq = D0 exP - (6.28) (between ~10"4 to 10"5 m 2/s) is found by fitting the model to experimental data. 136 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Cj — Ceq exp *0_ Rr, l-C C -C ^eq J (6.29) with R,=^ff (6.30) RgT In Equation 6.30, y denotes the precipitate-matrix interfacial energy and Vm is the molar volume of the Q' precipitates. Ceq is the equilibrium solute concentration at the annealing temperature. The magnitude o f the precipitate-matrix interfacial energy, y is a function of the coherency between the precipitates and the matrix. Incoherent precipitates are expected to have higher interfacial energy due to larger structural distortions at the interface. However, experimental determination of the interfacial energy is a difficult problem. A s a result, y is usually taken as a fitting parameter in most modelling approaches. For example, in the age hardening model of M y h r et al. (2000, 2001), the Mg2Si-matrix interfacial energy was varied from 0.2 to 0.26 J /m 2 in order to fit the model calculations to experimental data obtained from A l - M g - S i alloys. In another precipitation model developed by Deschamps and Brechet (1999) for A l - Z n - M g alloy, value as high as 0.3 J /m 2 was used for the interface between r/' precipitates and the matrix. In the present modelling exercise, yt is taken as a constant in the order of 0.3 J/m 2 . 137 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 6.5.2 Coarsening It is expected that the majority of the precipitates entered the coarsening regime in the early stage of annealing since the precipitates are well into the coarsening stage prior to cold rolling. Following Deschamps and Brechet (1999), the standard Lifshitz-Slyozor-Wagner (LSW) coarsening law is used to describe the evolution of average precipitate radius [Lifshitz and Slyozor, 1961, Wagner, 1961]: dRp_ _ 4 ^ ( i - Q ^ J V ^ dt 27 (r r \ 2 R  2 coars \Cp-Cegj *p In the coarsening regime, precipitates that are larger than the average radius w i l l grow and those that are below w i l l dissolve. The corresponding average solute concentration is given by replacing Q with CM in Equation 6.29. Differentiating CM with respect to time, one gets dCM _ RpCM dt R 2 1 - C eq C -C dR, dt (6.32) Subsequently, the evolution equation for the number density o f precipitate is obtained by differentiating the mass balance equation: C M l-^NnRn3 ) = C n - - n N „ R 3 C 'P"P P^p ^p (6.33) 138 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 and combining it with Equations 6.31 and 6.32 which leads to dNr dt coars 4 Ceq ( l ~ Ceq ) Rd^eff "(cP-cetlf v RQ C M { \ - C M ) RP (cP-cM) K4nRp J~NP •3Nr (6.34) Finally, under the conditions of simultaneous dissolution/growth and coarsening, the contribution from the respective processes to the overall precipitate distribution is weighted by a coarsening fraction, Fc [Deschamps and Brechet, 1999]: dRp dRv —E- = (I_F )—^ dt V 0 1 dt dRp + F, p d/g dt coars (6.35a) with dNK = f d N R dt dt coars Fc=\-erf • ( K A h CM -1 r K eq J (6.35b) (6.36) where kc = -0.1 i f Q > CM and kc = 0.1 i f C, < CM. Equation 6.36 provides a smooth transition from the dissolution/growth stage to the coarsening stage. Deschamps and Brechet (1999) 139 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 have shown that the exact form of the coarsening function is not critical to the overall coarsening kinetics as long as FC approaches 1 when the matrix solutes concentration approaches the equilibrium condition, i.e., CM -> CEQ-6.6 Parameter Identification A n inherent requirement in all microstructure modelling work is the need to identify a wide spectrum of physical parameters. The values of some of the known parameters such as the Young and shear modulus are given in the list of symbols on page x i i . However, there are two material specific parameters which are required as model input, namely, the solubility product and molar volume for Q' precipitates. These two quantities are evaluated below. 6.6.1 Solubility Product for Q' Precipitates A n important assumption in the precipitation model is that all particles are spherical with uniform thermodynamic properties and identical chemical composition. In addition, the model does not consider directly the complex precipitation sequence. A l l the pre-existing precipitates are assumed to be in Q' phase which has a H C P crystal structure with a fixed chemical composition o f ALjC^MggSiy . The solubility product for the Q' phase within the aluminum rich corner of the quaternary A l - M g - S i - C u phase diagram can be expressed as [Raeisinia, et al, 2004]: ^eq C; Si eq C. Cu eq = K\ exp AHo V , (6.37) 140 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 where C^8 , and are the equilibrium concentrations (in at. %) of M g , C u and Si in solution, respectively. AHo is the standard enthalpy o f the reversible Q' phase dissolution/precipitation reaction (-495 kJ/mol) and Kj is a constant (~2.8xl0 2 8 ) . The nominal solute concentration, Co in AA6111 is given by the sum of the nominal concentrations of the elements in the alloy, c{f8, CQU and C§'. 6.6.2 Molar Volume of Q' Precipitates The molar volume, Vm o f the Q' precipitates is obtained from the lattice parameters reported by Chakrabarti et al. (1998). For H C P crystal structure, the volume of a unit cell, Vc is given by a2c(sin!20°) where a and c are the lattice parameters. Us ing a = 1.04 nm and c = 0.405 nm, the molar volume is computed as follows [Gladman, 1997]: Vm=^J^ = \.09x\0-5— (6.38) n mol where n is the number o f atoms per unit cell (n - 21). 6.7 Model Implementation The overall model framework is constructed by connecting a series of differential equations, as shown in Fig. 6.1. The internal state variables that were used to link the various submodels are highlighted to emphasize the interaction between the various microstructural processes. 141 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Precipitate Free Zones (I) Recovery I. = exp, -— — Subgrain growth I: - j ' M ^7(7) 4 Recrystallization I: Precipitate Zones (II) I Subgrain growth II: —=f^  = A T » ' ^bjl(') 2Rf{t)j \dR, Precipitate radius: dt dR, Cu-C, V-g CP~C> dt 4 Cqjl-Cg) XyDtf Precipitation II 27 Cp-Ceqf *t Overall: - j f - O - S ) -dt dR„ dt Precipitate density: $ r« 4 Cjl-CeylRoDtf 27 RQCM{1-C^)\ 3 • !> ^* F " — E r Recrystallization II: Fp,RP Fig . 6.1. Schematic outline of the overall model framework. 142 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The numerical integration of these equations is performed in twelve stages using the Runge-Kutta method: at each calculation time step, the rate o f variation o f the five internal state variables (Rp, Np, <jrc, Ssb and X) is calculated as a function o f the current value o f all parameters. A mass balance is also carried out at each time step to evaluate the residual solute concentration in the matrix. The outcome of the integrations is then fed into the appropriate material response equations (Equations 6.2 to 6.6) to obtain the instantaneous flow stress and volume fraction of recrystallized grains. Before the model can be implemented, the conditions of the as deformed material must be specified. In the precipitate zones, the respective contributions of precipitates and dislocation to the as deformed yield stress were determined accordingly using Equations 6.5 and 6.6. The precipitate size at t = 0 is determined by assuming that after 7 days o f aging at 250 and 325°C, the precipitate conditions are reasonably close to the equilibrium conditions. In this case, the volume fractions o f precipitate in the materials can be related to the residual solute concentrations (in atomic%) in the matrix according to [Raeisinia et al., 2006]: CMg = 0.9-0.35F p CSi=0.6-0.3lFp CCu=0.3-0.09Fp (6.39a) (6.39b) (6.39c) Equations 6.39a to 6.39c were derived using a mass balance between each solute in the precipitate and in the matrix and the stoichiometry o f the precipitate. The volume fraction of precipitate at a given temperature is obtained by inserting Equations 6.39 into the solubility 143 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 product of Q' (Equation 6.37) and solving for Fp. Using Fp and the experimentally determined as aged yield stress, <Jas-aged, the precipitate radius, Rp at t = 0 is estimated from Equation 6.6. Following that, the initial number density of precipitates is given by Np = 3Fp/4nR/. Table 6.3 compares the calculated precipitate radius after 7 days of aging at 250 and 325°C with the measurements from T E M . It can be seen that the values of calculated Rp are reasonable considering that T E M measurements are regarded as only crude estimations o f the true values. It should be noted that no fitting parameter is used to obtain the calculated Rp. These results confirmed the validity of using Equation 6.6 to calculate the precipitate contributions to flow stress. 6.8 Comparison with Experimental Results The model is evaluated in terms o f the time evolution o f yield stress and fraction recrystallized. First, the model is fitted to the experimental data (both softening and recrystallization curves) o f the O A samples by adjusting the values o f seven parameters. These adjustable parameters are physically motivated and therefore are allowed to vary between a prescribed range, for instance, the activation energy for recovery is allowed to vary between 130-142 kJ/mol. The optimized values of these parameters are listed in Table 6.4. Table 6.5 lists other physical parameters which were fixed in the model. These parameters were obtained either from literature or estimated from experimental data. The quality of the fit was determined by a least square iteration method by minimizing the difference between the model output and the softening and recrystallization data of the O A 144 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Table 6.3. Comparison between calculated and measured Rp from T E M . Aging conditions Calculated Rp Measured (TEM) Rp 7 days at 250°C 8.6 nm 6.6 nm 7 days at 325°C 14.2 nm 17.5 nm *Equivalent radius 145 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Table 6.4. List o f adjustable parameters and their optimized values. Model Parameter Value Recovery Effective pinning parameter, ctp 9 x 10"2 Recovery Activation energy, Q0 140 kJ/mol Subgrain growth L o w angle grain boundary mobility, MQ 5 x 10"6 m 3 /N-s Subgrain growth Activation energy, QSB 155kJ/mol Precipitation Diffusion coefficient, D0 5.5 x 10 _ 5 m 2 /s Recrystallization Nucleation parameter, k 5 x 10"8 Recrystallization High angle grain boundary mobility, M0 1 x 10 2 m 3 /N-s 146 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Table 6.5. List of physical parameters which are fixed in the model. Model Parameter Value Source Recovery Activation volume, V 9b3 Calculated (Equation. 6.9) Subgrain growth Critical misorientation angle, 0C 15° Humphreys and Hatherly, 1995 Subgrain growth L o w angle grain boundary energy, ysb 0.16 J /m 2 Calculated (Equation. 6.11) Recrystallization Activation energy, Qrex 200 kJ/mol Vatne, 1995 Recrystallization High angle grain boundary energy, ygb 0.324 J /m 2 Murr, 1975 Recrystallization Critical nuclei radius, Rc 0.5 pm Estimated from T E M subgrain size Precipitation Interfacial energy, y 0.3 J /m 2 Deschamps and Brechet, 1999 147 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 samples. The results are shown in Figs. 6.2a and 6.2b. In terms o f softening, it can be seen that good agreement between the model calculations and experimental data is obtained at the annealing temperatures o f 250 and 445°C. The slow down in recovery rate at 250°C due to precipitate pinning indicated by the plateau in the softening curve between ~10 2 to 10 4 seconds is well captured by the model. A t the annealing temperature of 325°C, the model appears to have overestimated the decrease in yield stress in the later stage o f annealing. Deviation between model and experimental results is also observed in the recrystallization curves in Fig. 6.2b after ~5 x 10 6 seconds o f annealing. Nevertheless, the model gives good predictions in the initial stage of annealing. The magnitude of the activation energy for recovery, 140 kJ/mol, is fully in the possible range of solute diffusion in aluminum (Table 6.2). But the activation energy for subgrain growth, 155 kJ/mol, is somewhat higher than the values given in Table 6.2. The values of the diffusion parameters utilized in the precipitation model, i.e., D0 - 5.5 x 10"5 J /m 2 and QD - 142 k/mol are reasonably close to the values for the self diffusion of aluminum reported in the literature. The magnitude of the effective pinning parameter is small indicating that only a small fraction o f the precipitates are available to pin dislocations during recovery. Further validation of the model is carried out by applying the model to predict the softening curves of the S O A samples using the same set of parameters listed in Table 6.4. The results are shown in Figs. 6.3a and 6.3b. It is obvious that the model is capable o f predicting the softening curves at 325°C accurately. However, at the higher annealing temperature of 445°C, the model predictions are less satisfactory, especially in the early stage o f annealing. 148 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 03 CL CO CO CO T3 g> >-250 200 H 150 H 100 10-3 10"2 10"1 10° 101 102 103 104 105 106 107 108 Time (s) 101 102 103 104 105 106 107 108 Time (s) 109 Fig. 6.2. Comparison between model calculated and experimental (a) softening and (b) recrystallization curves for 40% cold rolled overaged AA6111 during isothermal annealing at 325 and 445°C. 149 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 Time (s) Fig. 6.3. Comparison between predicted and experimental (a) softening and (b) recrystallization kinetics for 40% cold rolled severely overaged AA6111 during isothermal annealing at 325 and 445°C. 150 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 It is believed that the rapid transformation of Q' precipitates to the relatively coarse Mg2Si particles may have contributed the initial drop in yield stress in the S O A samples which is not accounted for in the present model. This postulation is reasonable because significant number of Mg2Si particles were detected in the S O A samples even before annealing (see Fig. 5.6b). It is also possible that the recrystallization rate is enhanced due to the presence of these particles which are not effective in pinning grain boundaries. 6.9 Discussion of Modelling Results The most significant aspect of the present model is that it is capable of translating a qualitative description o f the interaction between recovery, recrystallization, subgrain growth and precipitation into a quantitative prediction in terms o f softening in yield stress vs. time relationships. In the ensuing discussion, the isothermal annealing behaviour o f the overaged samples at 325°C as predicted by the model is used as an example to illustrate the efficacy of the present model. 6.9.1 Interaction between Recovery and Precipitation In Fig. 6.4, the contributions from recovery, <jrc and precipitates, <TP to the overall yield stress is plotted as a function o f annealing time. Initially, recovery proceeds at a rate given by Equation 6.8b with little interference from the precipitates. A s recovery proceeds, the number of dislocation nodes, Ndts decreases and consequently the rate o f recovery is gradually reduced 151 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 200 150 H CO Q_ CO S 100 (/) T3 0) 50 H 0 Precipitate coarsening Recovery i r 10-2 10-1 10° 101 102 103 104 105 106 107 Time (s) Fig. 6.4. Softening due to recovery and precipitate coarsening in the precipitate zones of 40% cold rolled overaged AA6111 isothermally annealed at 325°C. 152 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 by the pinning term in Equation 6.8b. Recovery is completely halted when the number of dislocation nodes is equal to about 10% (ap ~ 0.09) of the total number of precipitates. This occurs at about 100 seconds into annealing at 325°C which corresponds very well with the T E M observations shown in Fig. 5.31. Complete retardation of the recovery process continues until the onset o f precipitate coarsening. In the precipitate coarsening regime, recovery proceeds at a rate dictated by the precipitate coarsening process. Hence, in the intermediate stage of annealing the overall softening of the material in the precipitate zones represents a convoluted effect o f recovery and precipitate coarsening. The evolution of precipitate radius with annealing time is shown in Fig. 6.5a for the annealing temperatures of 325 and 445°C. It can be seen that precipitate dissolution causes the average radius to decrease initially. The dissolution process is promoted by the increase in solubility during annealing since both the annealing temperatures are higher than the prior aging temperature of 250°C. Obviously, the effect of dissolution is more apparent at 445°C due to higher solubility. Nonetheless, in both cases precipitate coarsening quickly takes over as annealing progresses. The corresponding evolution o f matrix solute concentrations and volume fraction of precipitates are shown in Fig. 6.5b for the annealing temperature of 325°C. The onset o f precipitate coarsening can be identified by following the change in the matrix solute concentration as indicated in Fig. 6.5b. The initial small increase in solute concentration and a corresponding minute decrease in the precipitate volume fraction is the consequence of precipitate dissolution. 153 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 12 11 co , 3 2 9 <i) 03 I 8 CD o E o 7 6 1 1.0 0.8 0.6 -0.4 -/ / / /' i i i /325°C i =TZZ i-/ \ . ^ . . ^ . / 445°C 10-2 10-1 1 Q 0 1 Q 1 102 1 Q 3 1 0 4 1 Q 5 Time (s) 0.2 Coarsening Dissolution 'M 0.05 0.04 0.03 0.02 h 0.01 Annealing Temp. = 325°C 0.0 -I . . . , 1 . , 1- 0.00 10"1 10° 101 10 2 1 0 3 1 0 4 1 0 s 10 6 107 Time (s) Fig. 6.5. (a) Evolution of precipitate radius during isothermal annealing of 40% cold rolled overaged AA6111 at 325 and 445°C and (b) corresponding evolution of concentration and precipitate volume fraction at 325°C. 154 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 6.9.2 Interaction between Recovery, Precipitation and Recrystallization Both recovery and subgrain growth reduces the driving pressure for high angle grain boundaries during recrystallization according to Equation 6.25. O n the other hand, precipitate coarsening leads to lower pinning pressure thus promotes recrystallization. The best way to illustrate this delicate balance of pressure at the recrystallization front is to plot the stored energy per unit volume vs. the Zener retarding pressure. This is shown in Fig. 6.6. It can be seen that in the early stage of annealing, the dislocation density in the deformed microstructure is quickly consumed by the recovery process which leads to the initial drop in stored energy. A s annealing time increases, the decrease in stored energy slows down which corresponds to the pinning o f dislocations by precipitates. However, it can be observed that the Zener pressure quickly diminishes as precipitates coarsen thus allowing the driving pressure to overcome the pinning pressure. Upon further annealing, precipitate coarsening accelerates and releases the subgrain structure. The combination of subgrain growth and recrystallization cause the stored energy in the microstructure to rapidly decrease in the later stage o f annealing. In essence, the development of recrystallized microstructure is the consequence o f complex interplays between various microstructure phenomena. 6.9.3 Composite Microstructure A novel feature o f the present model is the description of the deformed microstructure as a composite model composed, of a mixture of precipitate and precipitate free zones. This assumption provides some interesting implications i n the description of microstructure evolution during annealing. The most important one being that different parts of the 155 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 1.2 10-1 1 0 o 1 0 1 102 103 104 105 106 107 108 Time (s) Fig. 6.6. Plots showing the time evolution of Zener pinning pressure vs. the stored energy in the precipitate zones of 40% cold rolled overaged A6111 annealed at 325°C. 156 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 microstructure w i l l assume different recovery and recrystallization rates during annealing. For instance, the recovery rate is expected to be faster in the precipitate free zones due to the absence of the pinning term (Equations 6.8a), leading to variation in stored energy for recrystallization in different parts of the microstructure. This behaviour is illustrated in Figs. 6.7a and 6.7b where the softening and recrystallization kinetics of the precipitate free zones are compared to those in the precipitate zones. It is obvious that deformed grains that are located in the precipitate free zones recover and recrystallize faster than the deformed grains in the precipitate zones. This type of inhomogeneous recovery and recrystallization behaviour is a commonly observed behaviour in many industrial alloys [Vandermeer and Gordon, 1962, Furu et al., 1990, Humphreys and Hatherly, 1995, Vandermeer, 2001]. However, this behaviour is often neglected in many previous work on the modelling o f recovery and recrystallization [Humphreys and Hatherly, 1995]. The present model provides a physical basis to relate the effect o f non-uniform spatial distribution of precipitates to the heterogeneous recovery and recrystallization behaviour o f cold deformed alloys. 6.10 Summary and Limitations of the Model It is important to recognize that the current modelling approach represents an attempt to describe a highly complex microstructure based on some greatly simplified assumptions. Having said that, the model is capable of addressing the challenge to quantitatively link the response of the material in terms of mechanical properties to the various microstructural changes that occur during isothermal annealing. The novel feature of the present modelling approach lies in its ability to explicitly take into account the spatial distribution of precipitate on recovery and recrystallization kinetics. The interaction between the various microstructure 157 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 1 ~ 1 1 1 1 1 1 1 1 1 1 10"2 10"1 10° 101 102 103 104 105 106 107 108 Time (s) Time (s) Fig. 6.7. Comparison of (a) softening kinetics and (b) recrystallization kinetics in the precipitate and precipitate free zones during the isothermal annealing of 40% cold rolled overaged AA6111 a t 3 2 5 ° C . Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 phenomena are considered based on physically sound principles and all the parameters utilized in the model have transparent physical meaning. The model framework which is constructed based on the internal state variable approach is versatile in a sense that additional microstructure phenomena, for example, grain growth, can be easily integrated, thus allowing the model to be easily adopted to other alloy system. Throughout the modelling exercise, attempts were made to keep the number of adjustable parameters to the minimum and eliminate any unknown parameters. However, in order to apply the model to a specific alloy system, experimental softening curves are required to calibrate the model in order to verify the magnitudes o f the adjustable parameters listed in Table 6.4. Two parameters which are expected to be highly alloy dependent are the effective pinning parameter, a in the recovery model and the nucleation parameter, k, in the recrystallization model. Furthermore, microstructure analysis needs to be carried out to estimate the volume fraction of precipitate free zones in the deformed microstructure. Te model suffers three main limitations: 1. Since the precipitation model considers only one type o f precipitate, i.e., the Q' phase, the model loses its validity at high annealing temperatures or after prolonged annealing time i f significant number of the Q' precipitates is transformed into the Mg2Si phase. 2. The yield stress model does not take into account the effect o f subgrain size. It is assumed that the annealing out o f dislocations in the deformed microstructure controls the recovery softening during annealing. 159 Chap. 6 Modelling of Microstructure Evolution for Overaged AA6111 The recrystallization model lacks a detailed nucleation model. Nucleation o f recrystallized grains is simply assumed to be site saturated and nucleation sites are limited to deformed grain boundary area and Fe-containing intermetallic particles. 160 Chapter 7 Summary and Conclusions The present work investigates the evolution of microstructure during the annealing of cold rolled precipitation hardened aluminum alloys. Through a combination of experimental and modeling approaches, the interaction between recovery, subgrain growth, recrystallization and precipitation and their effect on the mechanical properties of the materials were studied in detail. Experimentally, the isothermal recrystallization behaviour of 40% cold rolled aluminum alloy AA6111 was examined as a function of the precipitate conditions in the deformed state. A total of four prior aging conditions were included in the study: naturally aged (T4), peak aged (PA) , overaged (OA) and severely overaged (SOA) . The material response to annealing heat treatment in the temperature range of 250-445°C was quantified by following the softening in yield stress and the evolution of volume fraction of recrystallized grains with annealing time. Experimental results obtained at the annealing temperature of 325°C indicate that the recrystallization kinetics were extremely sluggish irrespective of the starting precipitation state, the slowest being the overaged samples. The T4 and P A samples displayed similar behaviour during annealing. The fastest recrystallization rates were observed in the S O A samples. Using a combination of S E M and E B S D techniques, it was shown that the development o f recrystallized microstructure in the overaged samples can be directly correlated to the heterogeneous spatial distribution of precipitates in the deformed structure. Significant subgrain growth was observed prior to the onset of recrystallization. In all cases, the combined effect o f recovery and precipitate 161 Chap. 7 Summary and Conclusions coarsening were found to be responsible for the majority o f the softening experienced by the samples during annealing. Increasing the annealing temperature to 445°C significantly enhances both the softening and recrystallization kinetics in the O A and S O A samples. A t 250°C, no recrystallization was observed in the overaged samples. Based on the knowledge gained in the experimental work, a comprehensive microstructure model was developed for the annealing of overaged alloys. The model which draws on the concept o f internal state variables adopts a simple rule o f mixtures to divide the microstructure into two parts: precipitate and precipitate free zones. This distinguish approach allows the model to describe the inhomogeneous recovery and recrystallization behaviour in the materials in a physical way. The overall model framework is constructed by coupling a series of submodels for recovery, subgrain growth, precipitate coarsening and recrystallization. The development of submodels relies on those already established in the literature. The linkages between various submodels are provided by following the evolution of internal state variables representing the respective microstructure phenomena, namely, dislocation density, subgrain size, precipitate size and density and volume fraction of recrystallized grains. It has been demonstrated that the model provides a unique tool to translate a qualitative description of the interaction between recovery, recrystallization, subgrain growth and precipitation into a quantitative prediction in terms o f softening in yield stress vs. time relationships. The model has been shown to be accurate in describing the softening and microstructure evolution during the annealing of 40% cold rolled overaged and severely overaged A A 6 1 1 1 . Less satisfactory results were obtained i f 162; Chap. 7 Summary and Conclusions significant precipitate phase transformations occur during annealing since the model only considers one type o f precipitates. This remains one of the limitations of the model. Lastly, the most significant contributions o f the present work are summarized in the following: • The experimental data gathered in the present study represents the first generation o f recovery and recrystallization data for cold rolled AA6111 with a wide range o f precipitate conditions. • The microstructure model developed in the present work represents a novel approach to relate directly the effect of inhomogeneous spatial distribution of precipitates to the recovery and recrystallization processes. • For the first time, quantitative descriptions o f the interaction between recovery, subgrain growth, precipitation and recrystallization are considered explicitly within a single model framework for cold rolled A A 6 1 1 1 . In conclusion, the present model can be seen as an important step towards the development of a comprehensive through process model for the industrial production o f precipitation hardenable alloys. The model framework is not limited to aluminum alloys. In principle, it can be applied to other alloy systems as well . Further advancement o f the model can be achieved in future work and these are discussed in the next section. 163 Chap. 7 Summary and Conclusions 7.1 Future Work The following future work is proposed: • A natural extension of the present model is to extend the precipitation model to take into account the entire sequence o f precipitation, i.e., nucleation, growth and coarsening o f precipitates. This w i l l greatly expand the applicability o f the model to cold rolled alloys with varied precipitate conditions. The model is currently limited to overaged alloys. • Another development of the present model which could be considered is to couple the microstructure model with a work hardening model which w i l l provide the deformed conditions of the materials. This w i l l significantly expand the model capability to study the effect of prestrain on the interaction between recovery, precipitation and recrystallization. The model is, in its present state, limited to 40% cold rolled materials. • The applicability of the model can be further expanded to study the effect of non-isothermal heat treatments on the interacting phenomena. These situations are extremely common in the industrial processing o f sheet alloys. Mathematically, the model which consists of a series o f differential equations can be readily integrated over the thermal histories o f the alloys. However, this should be carried out in a pragmatic manner in which a minimum number of adjustable parameters is sought. 164 REFERENCES Abad, R., Fernandez, A . I., Lopez, D . and Rodriguez-Ibabe, J. M . (2001), ISIJInt., 41, 1373-1382. Avrami, M . (1940), J. Chem. Phys., 8, 212-224. Barioz, C , Brechet, Y . , Legresy, J. M . , Cheynet, M . C , Courbon, J. , Guyot, P. and Ratnaud, G . M . (1992), in Proc. 3rd Int. Conf. On Aluminum Alloys, eds. Arnberg et al. Trondheim. 347-354. Bay, B . and Hansen, N . (1979), Metall. Trans., A10 , 279-288. Brechet, Y . J. M . and Purdy, G . R. (2003), Can. Metall. Quarterly, 42, 121-124. Bryant, J. D . (1999), Metall. Mater. Trans. A, (1999), 30A, 1999-2006. Burachynsky, V . and Cahoon, J. R. (1997), Metall. Mater. Trans. A, 28A, 563-582. Burger, G . B . , Gupta, A . K . , Jeffrey, P. W. and Lloyd , D . J. (1995), Mater. Characterization, 35, 25-39. Burger, G . B . , Gupta, A . K . , Sutak, L . and Lloyd, D . J. (1996), Mat. Sci. Forum, 217-222, 471-478. Cahn, J. W. (1962), Acta Metall, 10, 789-798. Cahn, R. W . (1996), in Physical Metallurgy, eds. Cahn and Haasen, North-Holland, New York, N Y , 2440-2448. Chakrabarti, D . J., Cheong, B . and Laughlin, D . E . (1998), in Automotive Alloys II, ed. Das, S. K . , T M S , 27-44. Chakrabarti, D . J. and Laughlin, D . E . (2004), Prog, in Mat. Sci., 49, 389-410. Chen, S. P., Hanlon, D . N . , Pei, Y . T. and Dehosson, H . Th. M . (2002), J. of Mat. Sci., 37, 989-995. Chen, S. P., Todd, I. and Van der Zwaag, S. (2002), Metall. Mater. Trans. A, 33A, 529-537. Cheng, L . M . , Hawbolt, E . B . and Meadowcroft, T. R. (2000), Metall. Mater. Trans. A, 31 A . 1907-1916. Cheng, L . M . , Poole, W . J. , Embury, J. D . and Lloyd , D . J. (2003), Metall. Mater. Trans. A, 34A, 2473-2481. 165 Deschamps, A . and Brechet, Y . (1999), Acta Mater., 47, 293-305. Doherty, R. D . , Hughes, D . A . , Humphreys, F. J., Jonas, J. J. , Jensen D . J., Kassner, M . E . , King , W . E . , McNel ley , T. R., McQueen, H . J. and Rollett, A . D . (1997), Mat. Sci. Eng. A, 238, 219-274. Dutta, B . , Palmiere, E . J. and Sellars, C. M . (2001), Acta Mater., 49, 785-794. Esmaeili, S. (2002), PhD Thesis, University of British Columbia, Vancouver, Canada. Esmaeili, S., Wang, X . , L loyd , D . J. and Poole, W . J. (2003a), Metall. Mater. Trans. A., 34A, 751-763. Esmaeili, S., L loyd , D . J. and Poole, W . J. (2003b), Acta Mater., 5J_, 2243-2257. Friedel, J. (1964), Dislocations, Pergamon Press, U K . Fujikawa, S., Hirano, K . and Fukushima, Y . (1978), Metall. Trans. A, 9A , 1811-1815. Fujikawa, S. and Hirano, K . (1989), Def. Diffus. Forum, 66-69, 447-452. Furu, T., Marthinsen, K . and Nes, E . (1990), Mat. Sci. and Technol, 6, 1093-1102. Furu, T., 0rsund, R. and Nes, E . (1995), Acta Mater., 43, 2209-2232. Furu, T., Shercliff, H . R., Baxter, G . J. and Sellars, C. M . (1999), Acta Mater., 47, 2377-2389. Gladman, T. (1997), The Physical Metallurgy of Microalloyed Steels, The Institute o f Materials, London, U K . Go, J., Poole, W . J., Mili tzer , M . and Wel l , M . A . (2001a), Unpublished work on AA6111, N S E R C Strategic Grant Project No. 101772, University o f Brit ish Columbia, Vancouver, Canada. Go, J., Mili tzer, M . , Wells, M . A . and Poole, W . J. (2001b), in Proc. of the Is' Int. Conf. on Recrystallization and Grain Growth, eds. Gottstein and Molodov, R W T H Aachen, Germany, 995-1000. Go, J., Poole, W . J., Mili tzer , M . and Wells, M . A . (2003), Mat. Sci. Technol., 19, 1361-1368. Gomez-Ramirez, R. and Pound, G . M . (1973), Metall. Trans., 4, 1563-1570. Gordon, P. and Vandermeer, R. (1966), in Recrystallization, Grain Growth and Textures, A S M , Metals Park, Ohio, 205-266. Grong, O and Shercliff, H . R. (2000), Prog, in Mat. Sci., 47,163-282. 166 Hansen, S. S., Vander Sande, J. B . and Cohen, M . (1980), Metall. Trans. A, 11 A . 387-402. Huang, Y . and Humphreys F. J. (1999), Acta Mater., 47, 2259-2268. Humphreys, F . J. (1977), Acta Metall, 25, 1323-1344. Humphreys, F . J. and Hatherly, M . (1995), Recrystallization and Related Annealing Phenomena, 1 s t edition, Pergamon Press, U K . Humphreys, F. J. (1997), Acta Mater., 45, 4231-4240. Humphreys, F. J. (1999), Mat. Sci. Technol, 15, 37-44. Humphreys, F. J. (2001), J. Mat. Sci., 36, 3833-3854. Inagaki, H . and Komatsubara, T. (2000), Mat. Sci. Forum, 331-337. 1303-1308. Johnson, W . A . and M e h l , R. F. (1939), AIME Trans., 135, 416-442. Jonas, J. J. and Weiss, I. (1979), Met. Sci., 13, 238-245. Kang, K . B . , Kwon , O., Lee, W . B . and Park, C. G . (1997), Scrip. Mater., 36, 1303-1308. Koizumi , M . , Kohara, S. and Inagaki, H . (2000), Z. Metallkd., 91, 460-467. Kolmogorov, A . N . (1937), Izv. Akad. Nauk. USSR. Ser. Mathmat., 1, 335. Kuhlmann-Wilsdorf, D . (2000), Mat. Sci. Forum, 331-337. 689-702. Kwon , O. and DeArdo, A . J. (1991), Acta Metall. Mater., 39, 529-538. Lee, K . (1999), Scrip. Mater., 40, 837-843. Leslie, W. C , Michalak, J. T. and A u l , F. W . (1961), in Proc. Conf. on Iron and its Dilute Solution, eds. Spencer and Werner, John Wiley and Sons, New York , 119-216. Lifshitz, I. M . and Slyozov, V . V . (1961), J. of Phys. Chem. Solids, 19, 35-50. Lillywhite, S. J., Pragnell, P. B . and Humphreys F. J. (2000), Mat. Sci. Technol, 16,1112-1120. L i u , C , Burghardt, J.C., Jacobs, T. H . and Scheffer, J. J. F. (1996), in Proc. 37th MWSP Conf, vol. X X X I I I , ISS, Warrendale, P A . Lloyd, D . J. (1980), Metall. Trans. A, UA, 1287-1294. 167 Lloyd, D . J. (1985), in Microstructural Control in Aluminum Alloys: Deformation, Recovery and Recrystallization, eds. Chia and McQueen, The Metallurgical Society of A I M E , New York, 45-66. Lloyd, D . J., Evans, D . R. and Gupta, G . K . (2000), Can. Met. Quarterly, 39, 475-482. Lotter, U . , Miischenborn W . and Thieman, E . (1980), in Recrystallization and Grain Growth of Multiphase and Particle Containing Materials, eds. Hansen et al., Ris0 National Laboratory, Roskilde, Denmark, 133-138. Liicke, K . and Detert, K . (1957), Acta Metall, 5, 628-637. Manohar, P. A . , Ferry, M . and Chandra, T. (1998), ISIJInt., 38, 913-924. Martin, J. W. , Doherty, R. D . and Cantor, B . (1997), Stability of Microstructure in Metallic Systems, Cambridge University Press, U K . Medina, S. F. , Quispe, A . , Valles, P. and Bahos, J. L . (1999), ISIJInt., 39, 913-922. Miao , W . F. and Laughlin D . E . (2000), Metall. Mater. Trans. A, 31A, 361-371. Mondolfo, L . F. (1979), Aluminum Alloys: Structure and Properties, Butterworth, London. Mukunthan, K . and Hawbolt, E . B . (1996), Metall. Mater. Trans. A. 27A, 3410-3423. Murr, L . E . (1975), Interfacial Phenomena in Metals and Alloys, Addison-Wesley, Reading, M A , 131. Murayama, M . , Hono, K . , Miao , W. F. and Laughlin D . E . (2001), Metall. Mater. Trans. A, 32A, 239-246. Myhr, O. R. and Grong, 0 . (2000), Acta Mater., 48, 1605-1615. Myhr, O. R. and Grong, 0 . and Andersen, A . J. (2001), Acta Mater., 49, 65-75. Nes, E . , Ryum, N . and Hunderi, O. (1985), Acta Metall, 33, 11-22. Nes, E . (1995), Acta Metall Mater., 43, 2189-2207. Nes, E . and Saeter, J. A . (1995), in Proc. of the 16th Ris0 Int. Symposium on Mat. Sci.: Microstructural and Crystallographic Aspects of Recrystallization, eds. Hansen et al, Ris0 National Laboratory, Denmark, 169-192. Orsetti-Rossi, P. L . and Sellars, C. M . (1997), Acta Mater., 45, 137-148. Orsetti-Rossi, P. L . and Sellars, C. M . (1999), Mat. Sci. Technol, 15, 185-192. 168 0rsund, R. and Nes, E . (1989), Scripta Metall, 23, 1187. Perovic, A . , Perovic, D . D . , Weatherly, G . C. and Lloyd , D . J. (1999), Scripta Mater., 41, 703-708. Poole, W . J., L loyd , D . J. and Embury, J. D . (1997), Mat. Sci. and Eng., A234-236, 306-309. Quainoo, G . K . , Yannacopoulos, S. and Gupta, A . K . (2001), Can. Metall. Quarterly, 40, 211-220. Raeisinia, B . , Poole, W . J. and Lloyd, D . J. (2006), Mat. Sci. Eng. A, 420, 245-249. Ray, R. K . , Hutchinson, W . B . and Duggan, B . J. (1975), Acta Metall, 23, 831-840. Richmond, O. (1986), J. of Met., 38, 16-18. Rios, P. R. (1997), Metall. Mater. Trans. A, 28A, 939-946. Sellars, M . C. (1997), in Thermec '97, eds. Chandra et al., T M S , Warrendale, 3-11. Sevillano, J. G . (1993), in Mat. Sci. and Technol, A Comprehensive Treatment, eds. Cahn et al., V C H , Weinheim, 6, 19-88. Smithells, C . J. and Brandes, E . A . (1976), Metals Reference Handbook, 5 t h edition, Butterworth, London. Stanley, J. K . (1963), Electrical and Magnetic Properties of Metals, A S M , Metals Park, Ohio. Suehiro, M . , L i u , Z . K . and Argen, J. (1998), Metall. Mater. Trans. A, 29A, 1029-1034. Taylor, G . I. (1934), Proc. Royal. Soc. A, 145, 362. Tian, B . (2003), Mat. Sci. Eng., A360, 330-338. Vandermeer, R. A . and Gordon, P. (1962), in Recovery and Recrystallization of Metals, ed. Himmel, L . , Interscience, N Y , 221-240. Vandermeer, R. A . and Rath, B . B . (1989), Metall. Trans. A, 20A, 391-401. Vandermeer, R. A . (2001), in Recrystallization and Grain Growth: Proc. of the First Joint Int. Conf, eds. Gottstein et al., Aachen, Germany, 645-657. Vatne, H . E . (1995), PhD Thesis, The Norwegian Inst, o f Technol., Trondheim, Norway. Vatne, H . E . , Furu, T., 0rsund, R. and Nes, E . (1996), Acta Mater., 44, 4463-4473. 169 Verdier, M . , Saeter, J. A . , Janecek, M . , Brechet, Y . , Guyot, P., Duly, D . and Nes, E . (1996), Mat. Sci. Forum, 217-222, 435-440. Verdier, M . , Groma, I., Flandin, L . , Lendvai, J., Brechet, Y . and Guyot, P. (1997a), Scripta Mater., 37, 449-454. Verdier, M . , Bley, F. , Janecek, Livet, F. , Simon, J. P. and Brechet, Y . (1997b), Mat. Sci. Eng., A234-236, 258-262. Verdier, M . , Janecek, M . , Brechet, Y . and Guyot, P. (1998), Mat. Sci. Eng., A248, 187-197. Verdier, M . , Brechet, Y . and Guyot, P. (1999), Acta Mater. 47, 127-134. Wagner, C. (1961), Z. Electrochem., 65, 581-591. Wang, X . and Embury, J. D . (2002), Unpublished TEM work on AA6111, McMaster University, Hamilton, Canada. Wang, X . , Poole, W . J . , Esmaeili, S., Lloyd, D . J. and Embury, J. D . (2003), Metall. Mater. Trans. A. 34A, 2913-2924. Weatherly, G . C , Perovic, A , Mukhopadhyay, N . K . , L loyd , D . J. and Perovic, D . D . (2001), Metall. Mater. Trans. A, 32A, 213-218. Wells, M . , L loyd , D . J., Samarasekera, I. V . , Brimacombe, J. K . and Hawbolt, E . B . (1998), Metall. Mater. Trans. B, 29B, 709-719. Wilshynsky-Dresler, D . O., Krauss, G . and Matlock, D . K . (1992), in Developments in the Annealing of Sheet Metals., eds. Pradhan and Gupta, T M S , Warrendale, P A , 189-218. Zurob, H . S., Hutchinson, C. R. , Brechet, Y . and Purdy, G . (2002), Acta Mater., 50, 3075-3092. Zurob, H . S., Hutchinson, C. R., Brechet, Y . and Purdy, G . (2003a), in Austenite Formation and Decomposition, eds. Damm and Merwin, T M S , Warrendale, P A , 121-138. Zurob, H . S. (2003b), PhD Thesis, McMaster University, Hamilton, Canada. 170 

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