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Microstructure-property models for heat treatment of A356 aluminum alloy Colley, Leo John 2011

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 MICROSTRUCTURE-PROPERTY MODELS FOR HEAT TREATMENT OF A356 ALUMINUM ALLOY  by LEO JOHN COLLEY  M.A.Sc. The University of British Columbia, 2003 B.Eng. University of Wales, Swansea, 2000  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  The Faculty of Graduate Studies (MATERIALS ENGINEERING)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  March 2011 © Leo John Colley, 2011 ii  ABSTRACT  The evolution of microstructure and mechanical properties during heat treatment of an industrially-cast A356 aluminum alloy was studied in an extensive experimental investigation. The temperature ranges of interest were; solution treatment at 500-560°C, natural ageing at room temperature, and artificial ageing at 150-200°C.  The changes in dendritic composition and eutectic morphology due to solution treatment were quantified by microprobe and image analysis for a wide range of processing conditions.  Subsequently, a microstructure model for solution treatment was constructed using sub-models for; i) the dissolution of Mg2Si particles, ii) the fragmentation of eutectic fibres, and iii) the coarsening of the fragmented eutectic.  For the ageing investigations, characterisation of mechanical properties was done by hardness and tensile testing, and the kinetics of precipitation was determined by an isothermal calorimetry technique.  A model to predict the evolution of yield strength during artificial ageing was developed based on established physical theories.  A yield strength model for natural ageing was also proposed using data from isothermal calorimetry tests performed close to room temperature. Two model Al-Si-Mg alloys were investigated in order to extend both ageing models to include the effects of; i) alloy chemistry, ii) incomplete solution treatment and iii) natural ageing prior to artificial ageing.  The validity of the models was verified using independent experimental measurements and literature data, and they were subsequently used as a tool to identify potential optimisation strategies for industrial heat treatment processes.  The linkages between the models revealed details of processing challenges arising from the interdependence of the heat treatment stages, such as reduced strengthening during ageing due to incomplete solution treatment, and delayed strengthening during artificial ageing as a result of prior natural ageing. iii  TABLE OF CONTENTS  ABSTRACT ....................................................................................................................................ii TABLE OF CONTENTS .............................................................................................................. iii LIST OF TABLES ....................................................................................................................... viii LIST OF FIGURES ........................................................................................................................ ix LIST OF SYMBOLS ..................................................................................................................... xv ACKNOWLEDGEMENTS .......................................................................................................... xix CHAPTER 1 INTRODUCTION ..................................................................................................... 1 CHAPTER 2 LITERATURE REVIEW .......................................................................................... 3  2.1 Introduction ........................................................................................................................... 3  2.2 Overview of Al-Si-Mg Casting Alloys ................................................................................. 3  2.3 Heat Treatment of Al-Si-Mg Casting Alloys ........................................................................ 5   2.3.1 Solution Treatment .......................................................................................................... 5   2.3.2 Quenching ....................................................................................................................... 7   2.3.3 Ageing Processes ............................................................................................................ 7  2.4 Microstructure Evolution during Heat Treatment ................................................................. 9   2.4.1 Dissolution of Soluble Second Phase Particles ............................................................... 9   2.4.2 Homogenization of Solute Elements ............................................................................. 10   2.4.3 Morphological Change of Insoluble Second Phases ..................................................... 11    2.4.3.1 Fragmentation ..................................................................................................... 12    2.4.3.2 Coarsening .......................................................................................................... 14   2.4.4 Precipitation .................................................................................................................. 14  2.5 Strengthening Mechanisms in Aluminum Alloys ............................................................... 17   2.5.1 Precipitation Strengthening Mechanisms ...................................................................... 17    2.5.1.1 Obstacle Strength ................................................................................................ 19 iv     2.5.1.2 Contribution of Precipitate Hardening towards the Yield Strength.................... 20    2.5.1.3 Obstacle Spacing ................................................................................................ 21    2.5.1.4 Superposition of Effects ..................................................................................... 22   2.5.2 Strengthening due to Second Phase Particles ................................................................ 22  2.6 Precipitation Kinetics .......................................................................................................... 24   2.6.1 Effect of Natural Ageing on Precipitation Kinetics ...................................................... 25  2.7 Microstructure-Property Models for Ageing Processes ...................................................... 27  2.8 Summary ............................................................................................................................. 29 CHAPTER 3 SCOPE AND OBJECTIVES .................................................................................. 31 CHAPER 4 EXPERIMENTAL METHODOLOGY ..................................................................... 33  4.1 Introduction ......................................................................................................................... 33  4.2 Materials .............................................................................................................................. 33  4.3 Sample Preparation ............................................................................................................. 38  4.4 Heat Treatment Experiments ............................................................................................... 38   4.4.1 Solution Treatment ........................................................................................................ 39   4.4.2 Artificial Ageing ........................................................................................................... 40  4.5 Material Characterization .................................................................................................... 41   4.5.1 Sample Preparation ....................................................................................................... 41   4.5.2 Microscopy .................................................................................................................... 41   4.5.3 Electron Probe Microanalysis (EPMA) ....................................................................... 44   4.5.4 Tensile and Hardness Testing ....................................................................................... 45   4.5.5 Isothermal Calorimetry ................................................................................................. 46 CHAPTER 5 EXPERIMENTAL RESULTS ................................................................................ 49  5.1 A356 Alloy Behaviour during Solution Treatment ............................................................. 49   5.1.1 Qualitative Description of Microstructure Changes ..................................................... 49   5.1.2 Quantitative Description of Microstructure Changes ................................................... 52 v    5.1.3 Electron Probe Microanalysis ....................................................................................... 56   5.1.4 Mechanical Properties of Solution Treated Material .................................................... 58  5.2 A356 Alloy Behaviour during Ageing ................................................................................ 59   5.2.1 Natural Ageing Behaviour ............................................................................................ 59   5.2.2 Immediate Artificial Ageing ......................................................................................... 62   5.2.3 Artificial Ageing after Natural Ageing ......................................................................... 66  5.3 Heat Treatment Behaviour of Model Alloys ....................................................................... 70   5.3.1 Behaviour of Al-11.1Si-0.22Mg ................................................................................... 70    5.3.1.1 Solution Treatment Behaviour ............................................................................ 71    5.3.1.2 Natural Ageing Behaviour .................................................................................. 72    5.3.1.3 Artificial Ageing Behaviour ............................................................................... 72   5.3.2 Behaviour of Al-1.3Si-0.3Mg ....................................................................................... 76    5.3.2.1 Natural Ageing Behaviour .................................................................................. 76    5.3.2.2 Immediate Artificial Ageing ............................................................................... 77    5.3.2.3 Artificial Ageing Following Natural Ageing ...................................................... 78  5.4 Discussion of Experimental Results.................................................................................... 80   5.4.1 Metallurgical Behaviour of A356 during Solution Treatment ...................................... 80    5.4.1.1 Dissolution of Mg-rich Phases............................................................................ 80    5.4.1.2 Eutectic Silicon Fragmentation and Coarsening ................................................ 81    5.4.1.3 The Relationship between Solution Treatment and Yield Strength ................... 83   5.4.2 Comparison of Ageing Behaviour of A356 and Model Alloys .................................... 84  5.5 Concluding Remarks ........................................................................................................... 90 CHAPTER 6 MICROSTRUCTURE-PROPERTY MODELLING .............................................. 91  6.1 Overall Approach ................................................................................................................ 91  6.2 Solution Treatment .............................................................................................................. 93   6.2.1 Dissolution of Mg2Si .................................................................................................... 94 vi    6.2.2 Morphological Change of Eutectic Silicon ................................................................... 99    6.2.2.1 Fragmentation of Silicon Particles ..................................................................... 99    6.2.2.2 Coarsening of Spheroidal Silicon Particles ...................................................... 104   6.2.3 Solution Treatment Model Results and Summary ...................................................... 106  6.3 Artificial Ageing ............................................................................................................... 107   6.3.1 Modelling of Precipitation Strengthening ................................................................... 109    6.3.1.1 Obstacle Strength .............................................................................................. 110    6.3.1.2 Obstacle Spacing .............................................................................................. 111    6.3.1.3 Estimation of Precipitate Volume Fraction ...................................................... 112    6.3.1.4 Calculating σppt ................................................................................................. 115   6.3.2 Calibration of the Model ............................................................................................. 116   6.3.3 Model Results.............................................................................................................. 117  6.4 Natural Ageing .................................................................................................................. 120  6.5 Artificial Ageing after Natural Ageing ............................................................................. 124   6.5.1 Estimation of fr ............................................................................................................ 124   6.5.2 Evolution of Yield Strength ........................................................................................ 128   6.5.3 Model Calibration ....................................................................................................... 129   6.5.4 Model Predictions and Validation ............................................................................... 130  6.6 Modelling of Al-Si-Mg Alloys.......................................................................................... 133   6.6.1 Development of Extended Model ............................................................................... 133    6.6.1.1 Effect of Silicon Content .................................................................................. 133    6.6.1.2 Effect of Mg Content ........................................................................................ 135    6.6.1.3 Effect of Incomplete Solution Treatment ......................................................... 136   6.6.2 Assessment of Extended Model .................................................................................. 138    6.6.2.1 Effect of Alloy Content .................................................................................... 138    6.6.2.2 Effect of Incomplete Solution Treatment ......................................................... 140 vii     6.6.2.3 Assessment using Literature Data for Al-Si-Mg Casting Alloys ..................... 142   6.6.3 Summary ..................................................................................................................... 147  6.7 Application of the Model for Optimization of Industrial Processes ................................. 148   6.7.1 Optimization of Solution Treatment Process .............................................................. 148    6.7.1.1 Effect of Soak Temperature .............................................................................. 148    6.7.1.2 Effect of Heating Rate ...................................................................................... 150    6.7.1.3 Effect of Charge Temperature .......................................................................... 153    6.7.1.4 Effect of Simultaneous Changes to Solution Treatment Parameters ................ 154   6.7.2 Optimization of Artificial Ageing ............................................................................... 157    6.7.2.1 Effect of Natural Ageing .................................................................................. 157   6.7.3 Summary ..................................................................................................................... 159 CHAPTER 7 CONCLUSIONS AND FUTURE WORK ............................................................ 161  7.1 Conclusions ....................................................................................................................... 161  7.2 Future Work ...................................................................................................................... 164 REFERENCES ............................................................................................................................ 167 APPENDIX A CORRELATION BETWEEN YIELD STRENGTH AND VICKERS HARDNESS ................................................................................................................................ 175    viii  LIST OF TABLES  Table 4.1: Chemical composition (wt%) of the A356 aluminum alloy used in this study ......... 33  Table 4.2: Chemical compositions (wt%) of the Al-Si-Mg model alloys used in this study ...... 34  Table 4.3: Predicted Microstructure Characteristics of the A356 Alloy and Model Alloys in the Solution Treated Condition ........................................................................................ 36  Table 5.1: FactSage-predicted equilibrium values for phase and solute content in the three alloys at solution treatment temperatures investigated in the present work ............... 85  Table 5.2: Summary of the yield strength contributions from solid solution strengthening, solute clusters, precipitates and eutectic particles for the three alloys studied in the as-quenched, naturally aged and peak aged conditions ............................................. 89  Table 6.1: List of microstructure variables and internal state variables for the metallurgical processes occurring during solution treatment ........................................................... 94  Table 6.2: Physical constants and values used in the dissolution model .................................... 96  Table 6.3: List of adjustable parameters in the fragmentation model and their values ............. 101  Table 6.4: List of adjustable parameters in the coarsening model and their values .................. 105  Table 6.5: Calculated Avrami coefficients for the JMAK equation used to describe the evolution of relative volume fraction of precipitates during immediate artificial ageing of the A356 alloy .......................................................................................... 114  Table 6.6: List of calibration parameters for the A356 alloy artificial ageing model ............... 116  Table 6.7: List of calibration parameters for the A356 alloy natural ageing model ................. 122  Table 6.8: Calibration parameters for the artificial ageing model for 24hr naturally aged A356 ...   .................................................................................................................................. 130  Table 6.9: FactSage thermodynamic calculation results for eutectic silicon content assuming equilibrium conditions at the typical solution treatment temperature of 540°C ...... 134  Table 6.10: Model predictions for completion of each microstructure process during solution treatment with various heating profiles (illustrated in Figure 6.32) ......................... 152  Table 6.11: Solution treatment parameters used for the model investigation into the effect of simultaneous parameter changes on the total processing time ................................. 155  Table 6.12: Model predictions for completion of solution treatment with various heating profiles (details of heating profiles are in Table 6.11) .......................................................... 156  ix  LIST OF FIGURES  Figure 2.1: The Al-rich section of the Al-Si phase diagram [after Massalski et al (1998)] .......... 4  Figure 2.2: Microstructure of Al/Si eutectic phase in an as-cast A356 aluminum alloy; (a) Unmodified alloy, (b) Modified with an addition of 156ppm strontium [Nafisi et al (2006)] ........................................................................................................................ 4  Figure 2.3: Morphological evolution of eutectic silicon in A356 (Al-7Si-0.4Mg) aluminum alloy during solution treatment at 540ºC; (a) as-cast, (b) 2 hr, (c) 8 hr. [Apelian et al (1990)] ........................................................................................................................ 6  Figure 2.4: Schematic illustrating a rod-shaped eutectic particle that fragments into a series of spherical particles [after Ogris et al (2002)] ............................................................. 13  Figure 2.5: Model results for silicon spheroidisation model presented by Ogris et al (2002) .... 13  Figure 2.6: Schematic view of three stages of the dislocation cutting mechanism ..................... 17  Figure 2.7: Schematic view of three stages of the Orowan looping mechanism ........................ 18  Figure 2.8: Schematic representation of a glide dislocation moving though an array of point obstacles [after Ardell (1985)] .................................................................................. 19  Figure 2.9: Variation of hardness in A356 alloy artificially aged at 180ºC without natural ageing and after 24 hours of natural ageing [after Shivkumar et al (1989)] ............. 26  Figure 4.1: Example of an automotive wheel indicating the wheel rim area used for the present .   .................................................................................................................................. 34  Figure 4.2: The end-chill casting apparatus and as-cast ingot at Rio Tinto Alcan ..................... 35  Figure 4.3: As-cast microstructures of: a) Al-1.3Si-0.32Mg and b) Al-11Si-0.22Mg alloys ..... 37  Figure 4.4: An example of the measured sample temperature-time profile during solution treatment in the salt bath ........................................................................................... 39  Figure 4.5: Schematic representations of the heat treatment processes studied: a) natural ageing at room temperature, b) immediate artificial ageing between 150ºC-200ºC, c) a period of natural ageing followed by artificial ageing between 150ºC-200ºC ......... 40  Figure 4.6: The equivalent circle diameter (ECD) is the diameter of a circle with area equal to that of the projection of the particle at a plane (i.e. Areaparticle = Areacircle) ....... 43  Figure 4.7: The aspect ratio (AR) is the ratio between the major axis length and the minor axis length.  Major and Minor are the primary and secondary axis of the best fitting ellipse ........................................................................................................................ 43  x   Figure 4.8: The circularity is a function of the perimeter P and the area A that has values between 0 and 1, where a value of 1 indicates a perfect circle ................................. 44  Figure 5.1: Optical micrographs of A356 alloy after solution treatment at 540°C (x500) a) as cast, b) 2 minutes, c) 30 minutes, d) 240 minutes .................................................... 50  Figure 5.2: Deep etched micrographs of A356 alloy after solution treatment at 540°C (x4000) a) As-cast, b) 2 minutes, c) 15 minutes, d) 30 minutes, e) 240 minutes ................... 51  Figure 5.3: BSE maps showing distribution of Si and Mg in A356 in the: a) as-cast condition, b) solution treated condition (following 240 minutes at 540°C) ............................. 52  Figure 5.4: The distribution of eutectic particle shape characteristics following solution treatment at 540°C: a) equivalent circle diameter, b) aspect ratio, c) particle roundness .................................................................................................................. 53  Figure 5.5: Change in average eutectic particle characteristics during solution treatment at 540°C: a) equivalent circle diameter, b) aspect ratio, c) particle roundness ............ 55  Figure 5.6: The distribution of a) magnesium and b) silicon across secondary dendrite arms in the A356 alloy in the as-cast condition and during solution treatment at 540°C ..... 57  Figure 5.7: Mechanical property behaviour for A356 in the as-cast and as-quenched condition following solution treatment at 540°C ...................................................................... 58  Figure 5.8: The mechanical property behaviour of the A356 alloy in the as-quenched (AQ) condition, and after natural ageing at room temperature for 2 hours, 24 hours and 2400 hours ................................................................................................................ 60  Figure 5.9: Exothermic heat flow traces for natural ageing of A356 at 25°C, 40°C and 60°C .. 61  Figure 5.10: The evolution of heat during natural ageing of A356 at 25°C, 40°C and 60°C ....... 62  Figure 5.11: The mechanical property behaviour of the A356 alloy in the as-quenched (AQ) condition, and after artificial ageing at 180ºC for various times .............................. 63  Figure 5.12: Ageing curves based on Vickers hardness data for artificial ageing of the A356 alloy in the temperature range 150°C-200°C ........................................................... 64  Figure 5.13: Thermograms for the artificially aged A356 alloy at 150°C, 180°C and 200°C ...... 65  Figure 5.14: The evolution of the relative fraction of heat during artificial ageing of A356 at: a) 150°C, b) 180°C, c) 200°C ....................................................................................... 66  Figure 5.15: The evolution of yield strength during artificial ageing at a) 150°C and b) 180°C following various natural ageing histories ................................................................ 67   xi  Figure 5.16: Thermograms for isothermal calorimetry of A356 artificially aged at 165°C, 180°C and 200°C following 24 hour natural ageing at room temperature .......................... 68  Figure 5.17: Thermograms of exothermic heat flow measured in A356 alloy during artificial ageing at 180°C following various natural ageing histories ..................................... 69  Figure 5.18: The distribution of magnesium across secondary dendrite arms in the as cast and solution treated Al-11Si-0.22Mg alloy ..................................................................... 71  Figure 5.19: The evolution of yield strength during natural ageing of Al-11Si-0.22Mg alloy ..... 72  Figure 5.20: The evolution of yield strength during artificial ageing of Al-11Si-0.22Mg alloy in the temperature range 150°C-200°C......................................................................... 73  Figure 5.21: The evolution of yield strength during artificial ageing of the Al-11Si-0.22Mg alloy at 180°C following a) no natural ageing, b) 24 hours natural ageing at room temperature ............................................................................................................... 74  Figure 5.22: The exothermic heat flow curves for Al-11Si-0.22Mg at 180°C during immediate artificial ageing and artificial ageing after 24 hours natural ageing ......................... 75  Figure 5.23: Exothermic heat flow curves for the Al-11Si-0.22Mg alloy during artificial ageing at 165°C, 180°C and 200°C following 24 hours natural ageing ............................... 75  Figure 5.24: The evolution of yield strength during natural ageing of Al-1.3Si-0.32Mg alloy .... 76  Figure 5.25: The evolution of yield strength during ageing of homogenized Al-1.3Si-0.32Mg .. 77  Figure 5.26: The evolution of yield strength during artificial ageing of the Al-1.3Si-0.3Mg alloy at 180°C following a) no natural ageing, b) 24 hours natural ageing at room temperature ............................................................................................................... 78  Figure 5.27: The exothermic heat flow curves for the Al-1.3Si-0.3Mg alloy artificially aged at 180°C following: i) no natural ageing and ii) 24 hours natural ageing .................... 79  Figure 5.28: Thermograms for isothermal calorimetry of Al-1.3Si-0.3Mg alloy artificially aged at 165°C, 180°C and 200°C after 24 hour natural ageing at room temperature ....... 79  Figure 5.29: The average magnesium content in A356 and Al-11Si-0.22Mg alloy during solution treatment at 540°C, calculated from the microprobe data ........................................ 81  Figure 5.30: Results of a sensitivity analysis to determine an appropriate particle roundness limit as a criterion for fragmented eutectic silicon particles in the A356 alloy ................ 82  Figure 5.31: The evolution of yield strength during solution treatment at 540°C, and the time periods over which each of the various metallurgical processes occurring are dominant ................................................................................................................... 83   xii  Figure 5.32: The evolution of yield strength in the underaged condition during immediate artificial ageing of the three alloys studied at 180°C ................................................ 85  Figure 5.33: The evolution of yield strength during the first 24 hours natural ageing at room temperature for the three alloys studied.................................................................... 89  Figure 6.1: Schematic diagram showing the solute concentration around a dissolving particle ....   .................................................................................................................................. 95  Figure 6.2:  Schematic diagram of the solute and eutectic content around the dissolving particle .   .................................................................................................................................. 96  Figure 6.3: Dissolution model predictions of the relative volume fraction of an Mg2Si particle for solution treatment involving a heating at 40°C/s to soak temperatures of 500°C, 540°C & 560°C.   The symbols and lines represent measured data and model predictions respectively ............................................................................................ 98  Figure 6.4: Experimental data vs. model predicted values at: a) 500°C, b) 540°C and c) 560°C ..   .................................................................................................................................. 99  Figure 6.5: The Arrhenius relationship between the JMAK constant k and temperature ......... 101  Figure 6.6: Model predictions and experimental data for fragmentation of the eutectic silicon phase during solution treatment at 500°C, 540°C and 560°C. The symbols and lines represent the measured data and model predictions respectively ........................... 102  Figure 6.7: Schematic representation of the fragmentation process showing the geometries of a rod containing a perturbation and a sphere that is of equal volume to the perturbed rod ........................................................................................................................... 103  Figure 6.8: Coarsening model predictions (lines) compared with experimental data (symbols) for solution treatment temperatures at 500°C, 540°C and 560°C .......................... 105  Figure 6.9: Process model predictions (lines) and experimental results (symbols) for solution treatment at 540°C .................................................................................................. 106  Figure 6.10: ln ln (1/(1-fr)) vs. ln t for the range of volume fraction between 0.05 and 0.95..... 113  Figure 6.11: A comparison of the experimental data (solid lines) and model predictions (dashed lines) for the evolution of the relative volume fraction of precipitates in the A356 alloy during immediate artificial ageing ................................................................. 114  Figure 6.12: The JMAK coefficient k has an Arrhenius relationship with temperature ............. 115  Figure 6.13: Comparison of model predictions and experimental data for the evolution of yield strength during immediate artificial ageing of the A356 alloy at: a) 150°C, b) 180°C & c) 200°C.  A heat ramp of 5°C/s from 20°C to the soak temperature is included at the start of the model .............................................................................................. 118  xiii   Figure 6.14: Comparison of model predictions and experimental data for the evolution of yield strength during immediate artificial ageing of the A356 alloy at a) 190°C, b) 220°C. A heat ramp of 5°C/s from 20°C to the soak temperature is included at the start of the model. ............................................................................................................... 119  Figure 6.15: A comparison of the experimental data (thick lines) and model predictions (dashed lines) for the evolution of relative volume fraction of precipitates during natural ageing ...................................................................................................................... 122  Figure 6.16: Comparison of model predictions and experimental data for the evolution of yield strength during natural ageing of the A356 alloy at 20°C ...................................... 123  Figure 6.17: Deconvoluted dissolution and precipitation at 180°C after 24 hours natural ageing ...   ................................................................................................................................ 125  Figure 6.18: Arrhenius plot for k and B ...................................................................................... 128  Figure 6.19: Comparison of model predictions and experimental data for the evolution of yield strength during artificial ageing of the naturally aged A356 alloy at a) 150°C, b) 180°C, c) 200°C ...................................................................................................... 131  Figure 6.20: Comparison of model predictions and experimental data for the evolution of yield strength during artificial ageing of the naturally aged A356 alloy at 190°C .......... 132  Figure 6.21: Comparison of σeut values obtained from tensile test measurements with predicted values from Equation 6.16 ...................................................................................... 134  Figure 6.22: The effect of magnesium content, CMg, on strengthening due to precipitates (cppt) and solute clusters (ccluster) in the peak aged and naturally aged conditions in Al-Si- Mg alloys ................................................................................................................ 135  Figure 6.23: Model predictions for strengthening during immediate artificial ageing at 180°C for the A356 alloy and the two model alloys investigated ........................................... 139  Figure 6.24: Model predictions for strengthening during artificial ageing at 180°C following 24 hours natural ageing for the A356 and Al-11Si-0.22Mg alloys ............................. 139  Figure 6.25: Model predictions for strengthening of A356 alloy during immediate artificial ageing at 180°C following short solution treatments of 2 minutes and 5 minutes at 540°C, compared with full solution treatment of 30 minutes at 540°C ................. 141  Figure 6.26: Model predictions for Al-11Si-0.22Mg during immediate artificial ageing at 150°C, 180°C and 200°C following 30 minutes solution treatment at 540°C (fdis = 0.76) ......   ................................................................................................................................ 142   xiv  Figure 6.27: Model predictions for Al-7Si-Mg alloys with varying Mg content during artificial ageing at: a) 160°C and b) 180°C, compared with literature data from Moller et al (2008) ...................................................................................................................... 144  Figure 6.28: Model predictions for Al-7Si-Mg alloys with varying Mg content during natural ageing at 20°C compared with literature data from Moller et al (2008) ................ 145   Figure 6.29: Model predictions for Al-7Si-Mg alloys with varying Mg content during artificial ageing at: a) 160°C and b) 180°C after natural ageing, compared to data from Moller et al (2008) .................................................................................................. 146  Figure 6.30: Effect of soak temperature on the time required for microstructure changes during solution treatment. (Fixed process parameters: Dendrite Arm Spacing = 30µm, Initial silicon rod diameter = 0.5µm, Charge Temperature = 20°C, Solution treatment heating rate = 40°C/sec) ......................................................................... 149  Figure 6.31: Effect of heating temperature on microstructure changes during solution treatment. (Fixed process parameters: Dendrite Arm Spacing = 30µm, Initial silicon rod thickness = 0.5µm, Soak Temperature = 540°C, Charge Temperature = 20°C) .... 151  Figure 6.32: Modelled heating profiles, including the base case and two-step strategies .......... 152  Figure 6.33: Comparison of model predictions for solution treatment using a range of heating rates in the cases of cold charging (20°C) and hot charging (200°C) the as-cast component............................................................................................................... 153  Figure 6.34: Schematic illustrating heating profiles used to obtain model predictions, including the base case (0.2°C/sec to 540°C), and two-step strategies detailed in Table 6.11 ....   ................................................................................................................................ 154  Figure 6.35: Comparison of model predictions for the time to peak strength during immediate artificial ageing and after a 24 hour natural age at room temperature. Model predictions are given by solid lines and experimental data by symbols ................. 158  Figure A.1: Correlations and best fit equations between yield strength and Vickers hardness for the A356, Al-1.3Si-0.32Mg and Al-11Si-0.22Mg alloys used in this study .......... 176  Figure A.2: Correlations and best fit equation between yield strength and Vickers hardness for the Al-7Si-xMg alloys (where x= 0.28, 0.34 & 0.45) studied by Moller et al (2008) ..   ................................................................................................................................ 176   xv  LIST OF SYMBOLS   Appt  area under the heat flow curve for immediately artificially aged material Adis total heat absorbed due to the dissolution of natural ageing zones during subsequent artificial ageing  ANA  area under the heat flow curve for previously naturally aged material B temperature dependent parameter used in defining the dissolution kinetics of solute clusters B0 pre-exponential constant used in defining the temperature dependence of the dissolution kinetics parameter b  magnitude of the Burgers vector Ci solute concentration at the matrix-precipitate interface (wt%) Cp solute concentration in the precipitate (wt%) Ct solute concentration in the matrix at ageing time, t (wt%) C0 solute concentration in the matrix (wt%) CMg magnesium concentation in the matrix (wt%) CMg,ac solute magnesium concentration in the as-cast condition (wt%) CMg,st solute magnesium concentration in the solution treated condition (wt%) CSi silicon concentration in the matrix (wt%) c a proportionality constant ceutectic constant factor for contribution to the yield strength from eutectic phase cppt constant factor for contribution to the yield strength from precipitation cclusters constant factor for contribution to the yield strength from solute clusters D diffusion coefficient of the solute in the matrix (m2/s)  xvi  D0 proportionality constant for the diffusivity equation F obstacle strength, maximum obstacle-dislocation interaction force Fpeak average obstacle strength at the peak aged condition f volume fraction of precipitates feutectic volume fraction of eutectic phase ffrag volume fraction of fragmented eutectic silicon particles fpeak volume fraction of precipitates in the peak aged condition fr relative volume fraction of precipitates f0 initial volume fraction of precipitates fr,0 relative volume fraction of solute clusters at the beginning of artificial ageing k a constant (specific details for each use are described in the text) 0 fragk  proportionality constant for fragmentation 0 ck  proportionality constant for coarsening 0 pptk  proportionality constant for precipitation 0 clusterk  proportionality constant for solute clustering L effective obstacle spacing on the slip plane (m) M Taylor factor mppt constant factor relating the magnesium content to the magnitude of the precipitation contribution  to the yield strength (MPa wt%-1/2) mcluster constant factor relating the magnesium content to the magnitude of the solute cluster contribution  to the yield strength (Nm-5/2wt%-1) NA number of precipitates per unit area of the slip plane n a numerical exponent for the JMAK relationship  xvii  Q  an activation energy for a process (specific processes described in text) Qfrag apparent activation energy for the fragmentation of as-cast eutectic silicon rods (kJ/mol) QMg activation energy for diffusion of magnesium in aluminum (kJ/mol) QSi activation energy for diffusion of silicon in aluminum (kJ/mol) Qppt apparent activation energy for precipitation (kJ/mol) Qcluster apparent activation energy for clustering of solute (kJ/mol) Qdis apparent activation energy for dissolution of solute clusters (kJ/mol) Qd activation energy for diffusion of solutes in the matrix (kJ/mol) Qs enthalpy of solution for solute clusters (kJ/mol) R universal gas constant (J/mol-K) r radius of a spherical particle, or radius of the circular cross-sectional area of a precipitate on a slip plane (m) rcell  radius of a spherical matrix cell containing a single particle (m) rrod  radius of an as-cast eutectic silicon rod (m) r0 initial radius of a spherical particle (m) rpeak  average radius at the peak aged condition (m) S1, S2, etc internal state variable  T  temperature (K)  t  time  tpeak  time to peak-aged condition Vfrag volume of fragmented eutectic silicon particles (m3) Vtotal total volume of eutectic silicon particles (m3) α a dimensionless constant  xviii  η a parameter related to Ci, Cp, and C0 λ characteristic wavelength for fragmentation of eutectic silicon rods (m) σys  yield strength (MPa) σaq  as-quenched yield strength (MPa) σα-Al  contribution from α-Al phase to the yield strength (MPa) σclusters  contribution from solute clustering to the yield strength (MPa) σeutectic  contribution from eutectic phase to the yield strength (MPa) σint  intrinsic yield strength (i.e. frictional stress of the α-Al lattice (MPa) σppt  contribution from precipitation strengthening to the yield strength (MPa) σ'ppt contribution from a mixture of precipitates and solute clusters to the yield strength (MPa) σss  contribution from solid solution strengthening to the yield strength (MPa) σ0ss contribution from solid solution strengthening to the yield strength for the as-quenched material (MPa)      xix ACKNOWLEDGEMENTS  This thesis was made possible with the help of many colleagues, and I would like to take this opportunity to give special thanks to the following;  I would like to express my gratitude to my supervisors Dr. Mary Wells and Dr. Warren Poole, for their insight, and continuous support, guidance and encouragement throughout my studies.  Many thanks to Chris Hermesmann at Canadian Autoparts Toyota (CAPTIN) and Fred Major at Rio Tinto Alcan for supplying the materials that were used for this work.  Financial support from AUTO21 and the Rio Tinto Alcan Research Fellowship is also gratefully acknowledged.  I would like to thank Mati Raudsepp, Edith Czech, Mary Fletcher and Gary Lockhart for their help with the experimental work, Ross McLeod, Carl Ng and David Torok for their assistance with sample preparation, and Michelle Tierney, Mary Jansepar and Fiona Webster for all manner of administrative help.  Many thanks also to Michael Lin, Michael Mendenhall and Andrew Carne, who gave their time and effort as part of the Undergraduate Summer Research Assistant (USRA) program.  Special thanks to Jason Mitchell, Babak Raeisinia, Sujay Sarkar, Angela Kubiak, Qiang Du, Payman Babaghourbani, Jayant Jain, Hamid Azizi-Alizamini, Reza Rouminia and all other friends at UBC for stimulating discussions covering a wide range of topics.  Finally, I am forever indebted to my wife Lindsey, my son Nathan, and my parents Ingrid and John for their constant love, sacrifice, patience and support.  Thank you for everything.  1 CHAPTER 1 - Introduction  For many years, one of the largest potential markets for aluminum alloys has been the transportation sector, primarily due to their increasing use in automotive applications as a way of light-weighting cars.  The major driving forces for this increase have been the implementation of graduated government standards for vehicle fuel efficiency and recyclability, as well as the effects of higher fuel costs to the consumer.  As a consequence, automotive manufacturers have a strong incentive to reduce fuel consumption while maintaining product performance and cost levels.  One of the most cost effective ways of addressing these challenges has been to substitute lightweight materials such as aluminum alloys in existing automotive designs.  Cast components account for over 80% of aluminum alloy use in vehicles [Kaufman and Rooy (2005)].  These castings have replaced their steel counterparts on a part-by-part basis over a number of years, and include relatively large items such as engine blocks, transmission cases and wheels.  Often, these aluminum cast components are given a heat treatment after casting to improve their mechanical properties. An important aspect of the research efforts aiming to meet the demand for highly reliable cast automotive components is that an improved understanding of alloy behaviour during multi-stage heat treatment can allow the optimisation of the process from the standpoint of the material, resulting in lower variability in the properties of the heat treated component.  In addition there is a need to understand the heat treatment process from a metallurgical standpoint so that efficient heat treatments can be developed.  Although the benefit of heat treatment is undisputed, there exist several challenges for heat treatment operators, including market expectations of higher performance and reliability, lower production costs and energy use, as well as concern over environmental impacts.  Standard heat  2 treatment practises were established many years ago, and require re-examination as they remain unchanged despite product and process improvements occurring in the interim.  Developments in casting process technologies have led to changes in the scale of the as-cast microstructure, while current furnace designs produce higher heating rates and lower thermal variation within the charged components during heat treatment.  Consequently, there is scope for the optimisation of heat treatment processes by reducing the times and temperatures of each stage.  The main objective of this work is to develop a model framework that predicts the evolution of microstructure and mechanical properties during heat treatment and can be used as a tool to optimise industrial heat treatment operations and enable the production of components with consistently uniform microstructures and properties that are tailored to their specific application.  3 CHAPTER 2 - Literature Review  2.1 Introduction  The following sections provide a review of the heat treatment of Al-Si-Mg casting alloys, the theory of relevant metallurgical processes such as the precipitation and dissolution of soluble second phase particles, and the morphological change of insoluble particles, ending with a discussion of process models developed to predict the evolution of microstructure and mechanical properties during heat treatment.  2.2 Overview of Al-Si-Mg Casting Alloys  Hypoeuctectic Al-Si-Mg alloys are common non-ferrous foundry alloys due to their excellent castability, fluidity and corrosion resistance.  The as-cast microstructure consists of primary dendrites of α-Al containing magnesium and silicon in solution, surrounded by an Al/Si eutectic phase arising from the eutectic transformation shown in the Al-Si binary phase diagram presented in Figure 2.1.  The size and morphology of the eutectic silicon depends on the casting conditions, as well as the presence of chemical modifiers such as strontium, sodium or antimony. Without modification the eutectic silicon forms as coarse platelets, shown in Figure 2.2a, whereas a fine ‘fibrous’ or ‘coral-like’ structure occurs in modified alloys, Figure 2.2b.  Other phases found in as-cast Al-Si-Mg alloys include Mg2Si particles, which are taken into solid solution and precipitated during heat treatment, and minor Fe-rich phases including α-Al5SiFe, β-Al8Si2Fe and pi-Al8Mg3Si6Fe that arise from the presence of melt impurities.   4  α-Al Liquid α-Al +Si 577°C 660°C 12.2 Eutectic Point 0 5 10 15 20 400 Silicon Content (at%) 450 500 550 600 700 650 Te m pe ra tu re  (°C ) Al Si  Figure 2.1: The Al-rich section of the Al-Si binary phase diagram [after Massalski et al. (1998)].   α-Al Al/S i Eutectic a) b) 50µm50µm  Figure 2.2: Microstructure of Al/Si eutectic phase in an as-cast A356 aluminum alloy; (a) Unmodified alloy, (b) Modified with an addition of 156ppm strontium [Nafisi et al (2006)]†.  †  Reprinted from Materials Science and Engineering: A, 415, 1-2, S. Nafisi, R. Ghomashchi, Effect of modification during conventional and semi-solid metal processing of A356 Al-Si alloy, pp. 273-285, Copyright (2006), with permission from Elsevier.  5 2.3 Heat Treatment of Al-Si-Mg Casting Alloys  Controlled heat treatment of aluminum alloys can significantly influence properties such as strength, ductility, toughness, and corrosion resistance, as well as the formation of residual stresses and the thermal and dimensional stability of the component.  The main heat treatment process applied to cast Al-Si-Mg alloys is precipitation hardening, which is carried out to improve the mechanical strength of the casting and consists of three stages; solution treatment, quenching and ageing.  The ASTM standard procedure for T6 heat treatment of castings produced using the alloy studied in this work, A356 (Al-7Si-0.3Mg), involves [ASTM (2002)];  1) Solution treatment at 540°C for 4-12 hours, 2) Immediate water quench, 3) Natural age at RT for 8 hours, followed by artificial age at 155°C for 6-12 hours.  Each stage of this process is reviewed in the following section.  2.3.1  Solution treatment Solution treatment requires long soak times at high temperature in order to produce a homogeneous solid solution with maximum solute concentration.  The soak temperature is determined by alloy composition and solid solubility limit, and is typically close to the eutectic temperature.  Industrial Al-Si-Mg alloys are solution treated at a maximum temperature of 550°C to avoid localised melting.  Underheating can result in incomplete dissolution of particles, low solute concentrations, and inhomogeneous solute distributions in the matrix; all of which cause a reduction in the strengthening potential of the alloy.   6 The dissolution of non-equilibrium particles and homogenization of solute can take up to 24 hours depending on the soak temperature, the scale of the as-cast microstructure, the casting dimensions and furnace temperature.  However, studies by Closset et al (1986) and Shivkumar et al (1989) to examine the solution treatment behaviour of A356 alloys found that dissolution of Mg2Si particles and homogenization of solute were complete after 30 minutes at 550°C, and other researchers report faster rates during solution treatment of permanent mould castings [Zhang et al. (2002)].  Typically, small amounts of magnesium and silicon are also present in the as-cast alloy as components of Fe-rich phases that are slow to dissolve, such as pi-Al8Si6Mg3Fe; Gustafsson et al. (1986) found these particles dissolve after 4 hours at 520°C.  Another important metallurgical process during solution treatment is the change in shape of insoluble second phase particles.  In the case of Al-Si-Mg alloys this involves a change in the eutectic silicon phase from the as-cast structure to spheroidal globules.  The spheroidization process is illustrated in Figure 2.3, which shows (a) the as-cast eutectic silicon network, (b) its fragmentation, and (c) coarsening of fragmented particles during solution treatment.  20?mµ 20mm20µ Al Al-Si Eutectic 20? mµ20mm20µ m20µ Si Particles 20mm20µa) b) c)  Figure 2.3: Morphological evolution of eutectic silicon in A356 aluminum alloy during solution treatment at 540°C; (a) as-cast, (b) 2 hours, (c) 8 hours, [Apelian et al. (1990)]. ©1990 American Foundry Society, Schaumburg, Illinois, USA. Used with permission.  7 The use of fluidized beds for rapid heat transfer during solution treatment of Al-Si-Mg casting alloys has been studied in recent years.  Chaudhury et al [2006] found that the increased heating rate due to the fluidized bed resulted in faster spheroidisation times than conventional furnaces. This has been attributed to brittle fracture of eutectic silicon particles due to strains induced by the thermal expansion mismatch between the silicon and aluminum phases during heat up..  2.3.2 Quenching Following solution treatment the alloy must be rapidly cooled to produce a highly supersaturated solid solution containing large numbers of “quenched-in” vacancies.  The greatest benefit from the standpoint of alloy properties is achieved with the fastest quench as it ensures the maximum supersaturated solute concentration, but concerns regarding the development of residual stresses mean that industrial processes typically use a hot water quench.  Zhang and Zheng (1996) quenched specimens of an A356 alloy in various media, and compared their mechanical properties during artificial ageing, and found a significant decrease in yield strength and ultimate tensile strength when the quench rate is slower than 110°C/sec (i.e. in water above 60°C).  2.3.3 Ageing processes Ageing involves the controlled precipitation of a fine dispersion of second phase particles from a supersaturated solid solution.  This can be achieved by exposing the alloy to a suitable combination of temperature and time.  Typically, precipitation reactions involve the formation of intermediate phases prior to the equilibrium phase, each of which influences the overall strength.  Natural ageing refers to the decomposition of a supersaturated solid solution over time at room temperature following quenching.  Depending on the alloy, natural ageing occurs over a few  8 hours to several years, and results in an increase in strength from the as-quenched condition due to the formation of solute clusters or GP zones.  This condition is referred to as the T4 temper.  The decomposition of a supersaturated solid solution at an elevated temperature is commonly known as artificial ageing.  In this process, the solute elements form second phase precipitates that greatly increase the strength of the alloy.  Typical artificial ageing temperatures are in the range 150°C to 250°C, and artificial ageing times can be as long as 12 hours. Industrial artificial ageing strategies are designed to produce the optimum size, distribution, type and morphology of strengthening precipitate, and may involve one or more stages at different temperatures, a period of natural ageing prior to artificial ageing, or an intermediate ageing treatment at lower temperature (usually 60-120°C) known as “preageing” prior to artificial ageing.  The peak-aged condition (i.e. T6 temper) typically involves precipitate sizes in the range of 10nm, although many castings are “overaged” and contain larger precipitates to ensure dimensional stability during service at the expense of some strength (i.e. T5 and T7 temper).  In Al-Si-Mg casting alloys, natural ageing prior to artificial ageing is generally considered detrimental because the clustering of solute atoms during natural ageing reduces the driving force for precipitation and increases the time needed to reach the peak aged condition during artificial ageing.  Despite this, a period of natural ageing is included in the ASTM standard, mainly because it is difficult to avoid at least a short delay between quenching and artificial ageing during industrial processing.  This concludes a brief review of the heat treatment of Al-Si-Mg alloys.  The following sections contain reviews of the metallurgical processes occurring during solution treatment and ageing.   9 2.4 Microstructure Evolution During Heat Treatment  Several metallurgical processes operate during heat treatment to alter the microstructure of the material.  Phenomena occurring during solution treatment are: i) the dissolution of soluble second phase particles, ii) the homogenization of solute and iii) the morphological change of insoluble second phase particles.  Quenching involves rapid cooling to form a supersaturated solid solution although some precipitation is expected to occur at slower cooling rates.  The supersaturated solution decomposes during artificial ageing as the precipitation of a fine dispersion of second phase particles takes place.  Each process is discussed separately in the following sections.  2.4.1 Dissolution of soluble second phase particles A number of theoretical approaches have been applied to describe the diffusion-controlled dissolution of a second phase particle in an infinite matrix [Whelan (1969), Aaron et al. (1970)]. While these analyses consider a single particle in an infinite matrix, the presence of other particles cause overlapping of the diffusion fields and this has been taken into account by using a finite diffusion field approach [Tanzilli and Heckel (1968), Nolfi et al. (1969), Aaron and Kotler (1971)].  This approach is commonly known as the “cell concept” [Nolfi et al. (1969)], and assumes the particles are equidistant and of uniform size.  Reasonable descriptions of dissolution processes have resulted from the application of the cell concept to a number of alloy systems containing non-equilibrium particles [Aaron and Kotler (1971), Myhr and Grong (1991)].  In recent years numerical models have been developed to describe the dissolution kinetics of second phase particles [Tundal and Ryum (1992a, b), Vermolen et al. (1996), Vermolen et al. (1998a, b), Chen et al. (1999)].  Rometsch et al. (1999) presented a model simulating dissolution  10 of the Mg2Si phase and homogenization of magnesium in Al-Si-Mg alloys during solution treatment, and predicted complete dissolution and homogenization after 15 minutes, consistent with the experimental findings for the A356 alloy [Zhang et al. (2002)].  Subsequently, a numerical model based on a mass balance was developed to predict co-dissolution of Mg2Si and pi-Al8Si6Mg3Fe in A356 [Rometsch et al. (2001)].  They predicted that Mg2Si dissolves completely within 4 minutes at 540°C, whereas most pi-Al8Si6Mg3Fe particles dissolve within 30 minutes and complete dissolution occurs within 12 hours.  2.4.2 Homogenization of solute elements The homogenization of segregated solid solutions has been thoroughly reviewed previously [Martin (1980)].  It is known that the dendrite arm spacing of the as-cast alloy, the level of solute segregation within the dendrite, and the diffusion coefficient of the solute element in the matrix control homogenization kinetics.  Purdy and Kirkaldy (1971) have shown that the decay of a segregation profile can be represented by a cosine function with a time-dependent relaxation parameter, τ, as follows;      (2.1)  where;      (2.2)   In Equations 2.1 and 2.2, C0 is the bulk composition of the alloy, Ca is the amplitude of the segregation profile (both in wt%), d is the width of the dendrite arm and D is the temperature- ( ) 0 2cos expa x tC x C C d pi τ −    = +         2 2 4d D τ pi =  11 dependent diffusion coefficient for the solute element in the matrix (in m2/s).  A study by Ward (1965) showed that the value of τ controls the kinetics of homogenization, and that homogenization times close to 3τ are usually sufficient to remove solute concentration gradients.  2.4.3 Morphological change of insoluble second phases In Al-Si-Mg casting alloys, the eutectic silicon phase evolves during solution treatment from the as-cast morphology to spheroidal particles. Many authors [Parker et al. (1982), Rhines and Aballe (1986), Meyers (1986)] have analyzed this spheroidization process using the eutectic particle size, spacing and aspect ratio, and found that modified eutectic particles spheroidize faster than unmodified particles [Shivkumar et al. (1989), Apelian et al. (1990), Paray and Gruzleski (1994)] due to the refined structure’s larger interfacial area and driving force for morphological change.  In recent years the availability of 3-dimensional analytical methods led to renewed interest in this area.  Lasagni et al. (2007) combined focussed ion beam milling and energy dispersive spectroscopy (FIB-EDX) to obtain element maps for a sequentially milled strontium-modified Al-7Si alloy and reconstruct the three-dimensional silicon network.  Lower particle sphericities were reported using the 3-D reconstructions than equivalent 2-D element maps.  The spheroidization and coarsening of eutectic silicon particles is driven by the reduction of the surface energy associated with the Al/Si interface [Martin and Doherty (1980)].  At high temperature, the size and frequency of surface perturbations at the Al/Si interface increase, causing the breakdown of the eutectic into a series of near-spherical particles that subsequently coarsen to further reduce the interfacial area.  The processes of eutectic fragmentation and particle coarsening are discussed separately in the following two sections.  12  2.4.3.1 Fragmentation The shape instability problem was first addressed by Lord Rayleigh (1879) who considered the fragmentation of a continuous water jet into individual droplets.  This approach has been used to describe instabilities in several metallic systems that contain rods embedded in a matrix [Marich (1971), Nakagawa and Weatherly (1972), Walter and Cline (1973)].  In a later work, Stuwe and Kolednik (1988) developed an analytical model for the transformation of potassium cylinders into spheres within a tungsten matrix, and more relevantly Ogris et al. (2002) applied this approach to the disintegration of a eutectic silicon rod into spheres in a hypoeutectic Al-Si-Mg alloy, as shown in Figure 2.4.  The difference in surface area between a perfect cylinder and a cylinder with a perturbation having been expressed mathematically by Stuwe and Kolednik, Ogris considered the increase in surface energy when a single atomic layer diffuses from the neck to the bulge to increase the perturbation.  Subsequently they calculated the evolution of the perturbation assuming a constant driving force for the diffusion of silicon, and found the fluctuation wavelength that results in the maximum growth rate.  They estimated the time taken for a cylinder with a known initial radius to fragment, and the dependency of fragmentation time on initial silicon branch radius is shown in Figure 2.5.  The results of the work by Ogris indicate that the fragmentation time for eutectic silicon rods during solution treatment is highly dependent on the initial radius of the rod, and the solution treatment temperature.   13 Cylinder Radius Sphere Radius Fluctuation Wavelength Solution Treatment Time  Figure 2.4: Schematic illustrating a rod-shaped eutectic particle that fragments into a series of spherical particles [after Ogris et al. (2002)].  0 0.05 0.1 0.15 0.2 0.25 0.3 0 3 6 9 12 15 18 21 24 R ad iu s o f S ili co n  R o d (µ m ) Fragmentation Time (min) 400ºC 450ºC 500ºC 540ºC  Figure 2.5: Model results for silicon spheroidization model presented by Ogris et al. (2002).  Kovacevic (2008) used a phase field modelling approach to predict the spheroidization of a eutectic silicon plate 5µm long and 0.2µm thick in an Al-Si alloy.  The driving force for spheroidization is taken to be the minimization of surface energy at the Al/Si interface, assuming isotropic conditions.  The model forces the plate to spheroidise as a single particle and does not  14 allow for it’s fragmentation into several smaller spheroids, as would be expected when considering the large aspect ratio of the initial plate.  Furthermore, the initial size of the silicon plate is small when compared to typical as-cast silicon particles in unmodified alloys, although the model results were in reasonable agreement with experimental observations at solution treatment temperatures when an interfacial energy of 1J/m2 is assumed.  2.4.3.2 Coarsening The rate of coarsening of eutectic silicon particles in Al-Si-Mg alloys modified by additions of strontium can be described by a power law shown in Equation 2.3 [Parker et al. (1982), Rhines and Aballe (1986)] following the theory of diffusion-controlled growth proposed by Lifschitz and Slyozov (1961) and Wagner (1961), and known commonly as the LSW model.        (2.3)  In unmodified alloys, the coarsening of eutectic silicon was also found to behave according to the LSW model of diffusion-controlled growth after an initial delay during which time the fragmentation of the silicon plates occurs [Meyers (1985), Shivkumar (1989)].  2.4.4 Precipitation Precipitation hardening in aluminum alloys was discovered in the early years of the twentieth century when Wilm (1911) reported increasing strength with time at room temperature in Al-Cu alloys after quenching from high temperature.  Subsequently, Merica et al. (1920) suggested this behaviour is due to precipitation of a second phase not observed by optical microscopy. Attempts to uncover the mechanism of precipitation strengthening followed.  Mott and Nabarro 3 3 0r r kt− =  15 (1940) introduced the concept of a dislocation-precipitate interaction, and Orowan (1948) presented the first quantitative model for precipitation hardening.  Direct evidence of dislocation-particle interactions became available with the introduction of transmission electron microscopy techniques [Kelly and Nicholson (1963)], and in recent years the development of novel microscopic and analytical techniques, such as high-resolution transmission electron microscopy [Andresen et al. (1998)] and 3-dimensional atom probe analysis [Edwards et al. (1998), Murayama et al. (1999)] have contributed towards an increased understanding of precipitation reactions and their associated strengthening mechanisms. Several reviews of precipitate hardening have been published [Brown and Ham (1971), Gerold (1979), Ardell (1985), Lloyd (1985), Nembach (1997), Martin (1998), Hornbogen (2001), Polmear (2004)], and an overview of precipitation in Al-Mg-Si alloys follows in this section, with the strengthening mechanisms and kinetics of precipitation discussed later.  Precipitation occurs as a result of a diffusional transformation in which thermally activated atomic movements control the nucleation, growth and coarsening of second phase particles forming within a supersaturated solid solution.  The precipitation reaction is expressed as follows;  α’  α + β  where α’ is the initial supersaturated solid solution, β is a stable or metastable precipitate, and α is a more stable solid solution with an equilibrium composition [Porter and Easterling (1992)]. As the precipitate has a different composition to the matrix, long-range diffusion of solute is required, and the precipitation rate is temperature-dependent.   16 Typically, a series of metastable phases precipitate during ageing prior to the equilibrium phase. Despite having a lower driving force for their precipitation, these transition phases have a lower activation energy barrier for nucleation and therefore the free energy of the system can be reduced more quickly through their formation in preference to the direct formation of the equilibrium phase.   Geisler and Hill (1948) and Guinier and Lambot (1948) observed needle- shaped ‘GP zones’ in artificially aged specimens of dilute Al-Mg-Si alloys as a precursor to the plate-like form of the equilibrium Mg2Si phase.  Although, there is still discussion regarding the exact details of the early stages of precipitation, it is now agreed that the precipitation sequence in Al-Mg-Si alloys is as follows [Dutta (1991), Edwards et al. (1998)]:  Supersaturated Solid Solution  Clusters of Mg and Clusters of Si  Co-clusters of Mg and Si  GP Zone I / Small Precipitates (equiaxed)  GP Zone II / Metastable β’’ (Mg5Si6 needles)  Metastable β’ (Mg2Si rods)  Equilibrium β (Mg2Si platelets)  The compositions of the early clusters and small precipitates remain relatively unknown, whereas the β″ precipitate, which is associated with the peak strength, is needle shaped and aligned along 100 Al [Pashley et al. (1966), and (1967)].  The β′ phase is rod shaped and aligned along 100 Al with a hexagonal crystal structure (a=7.05Å and c=4.05Å).  The equilibrium Mg2Si platelets lie in {100}Al planes with the face-centred cubic anti-fluorite structure (a=6.39Å) [Thomas (1961)].  Precipitation reactions result in significant strengthening in Al-Mg-Si alloys, and many other alloys can also be heat treated to improve properties via precipitation of second phase particles. This subject has been investigated in detail, and our fundamental knowledge of the strengthening mechanisms and kinetics of precipitation will be discussed in the following sections.  17 2.5 Strengthening Mechanisms in Aluminum Alloys  In this section, the strengthening mechanisms arising from the presence of precipitates as well as larger second phase particles are discussed, as these are both relevant to Al-Si-Mg casting alloys that contain strengthening precipitates and eutectic particles.  2.5.1 Precipitation Strengthening Mechanisms  The strengthening effect of precipitation is related to the interaction of glide dislocations with the precipitated particles, which act as obstacles to dislocation movement, and is dependent on several factors, including the particle characteristics (i.e. size, shape and volume fraction), their distribution within the matrix, and the nature of the particle-matrix interface.  Several strengthening mechanisms have been proposed to arise from dislocation-precipitate interactions. While a dislocation will move past the precipitate by the most energetically favourable method available, in general there are only two types of interaction; particle cutting in the case where the particle is shearable, and dislocation-looping around unshearable particles.  In the early stages of ageing, the precipitates are small and coherent or semi-coherent with the matrix and thus are shearable by dislocations, as illustrated in Figure 2.6.  a) c)b)  Figure 2.6:  Schematic view of three stages of the dislocation cutting mechanism.  18 The following strengthening mechanisms are considered to arise in the presence of coherent/semi-coherent particles and the dislocation cutting mechanism [Brown and Ham (1971), Gerold (1979), Ardell (1985), Lloyd (1985)]; • Coherency Strengthening - Coherent/semi-coherent particles increase the free energy of the system due to the elastic misfit between the particle and matrix. • Modulus Strengthening - If the particle has a higher elastic modulus than the matrix, a larger stress is required for the dislocation to continue moving forward. • Chemical Strengthening - There is an increase in interfacial energy when the particle is cut by the dislocation. • Atomic Order Strengthening - If an ordered particle is cut by a dislocation, the free energy increases due to creation of an anti-phase boundary. • Stacking Fault Strengthening - This arises from differences between the stacking fault energy of the matrix and precipitate.  As ageing continues the precipitate grows and becomes incoherent with respect to the matrix.  In this case, the dislocation bows between precipitates and forms loops in order to move forward, as shown in Figure 2.7.  Strengthening arises from bowing of the dislocation which is opposed by its line tension, as well as the formation of dislocation loops (Orowan loops) around precipitates.  a) c)b)  Figure 2.7:  Schematic view of three stages of the Orowan looping mechanism.  19 2.5.1.1 Obstacle Strength In general, it is useful to define precipitates and other obstacles as a ‘strong’ or a ‘weak’ obstacle depending on how far the dislocation must bow out before the obstacle is overcome. Considering a dislocation interacting with an array of obstacles within the matrix, as shown in Figure 2.8, the critical dislocation bowing angle at which the obstacle is overcome, ψc, is smaller when the obstacles are stronger. 2 cψ 2 cθ 2 cθ cR L Γ F Γ Direction of Dislocation Motion Dislocation Line Obstacle  Figure 2.8: Schematic representation of a glide dislocation moving though an array of point obstacles [after Ardell (1985)].  The maximum interaction force, F, between an obstacle and dislocation can be described using Figure 2.7 as:       (2.4)   where Γ is the dislocation line tension. c2 cos 2 F ψ = Γ      20 2.5.1.2 Contribution of precipitate hardening towards the yield strength The critical stress, τc, required to bow the dislocation to the critical angle, ψc, at which the dislocation overcomes the obstacle is given by [Gerold (1979), Martin (1998), Lloyd (1985)];    (2.5)  where b is the magnitude of the burgers vector and Rc is the radius of curvature of the dislocation.   Rc is also related to the effective obstacle spacing, L:   (2.6)   where θc/2 = pi/2 – ψc/2.  Using Equation 2.4 and 2.6 to replace for Γ and Rc respectively in Equation 2.5 gives:   (2.7)  Converting from shear stresses, τc, to normal stresses [Courtney (1990)], the contribution of precipitation strengthening, σppt, to the yield strength can thus be expressed as:   (2.8)  where M is the Taylor Factor. c cbR τ Γ = 2 sin 2 c c R Lθ  =    c F bL τ = ppt MF bL σ =  21 2.5.1.3 Obstacle spacing In the case of a dispersion of weak obstacles, the dislocation line is almost straight and the effective obstacle spacing, Ls, along the dislocation is larger than the mean obstacle spacing, L. The Friedel spacing [Martin (1998)] can be used to show that the stress required to overcome a weak obstacle is given by:   (2.9)   In the case of strong obstacles, a large amount of dislocation bowing occurs and the effective obstacle spacing approaches the mean obstacle spacing.  The critical stress required to overcome a strong obstacle with ψc<100° has been expressed by Brown and Ham (1971):   (2.10)   As the precipitates grow and coarsen, the mean obstacle spacing increases and a smaller force is required for a dislocation to overcome the obstacle, leading to a decrease in alloy strength.  In Equation 2.8 the maximum strength due to precipitation hardening occurs when a large force is required to overcome an obstacle and the effective obstacle spacing is small (i.e. F/L is at its maximum).  In the case of Al-Mg-Si alloys, it is now agreed that the dominant obstacle at peak strength is the β’’ precipitate [Edwards et al (1998)]. 3/ 2 cos 2 c c s Gb L ψ τ    =       0.8 cos 2 c c s Gb L ψ τ   =      22 2.5.1.4 Superposition of effects Precipitation in aluminum alloys typically results in precipitates of different strengths, either due to the formation of more than one type of precipitate or a distribution of precipitate sizes.  The overall strength of a material containing several different obstacle types simultaneously is generally given by a Pythagorean superposition rule [Koppenaal (1964), Brown and Ham (1971), Ardell (1985)]:  2 2 2 1 2τ τ τ= +                  (2.11)   In certain circumstances, including the combination of a small number of strong obstacles and a large number of weak obstacles, or the superposition of solid solution strengthening and precipitate strengthening, a linear law applies [Brown and Ham (1971), Ardell (1985)]:  1 2τ τ τ= +       (2.12)  2.5.2 Strengthening due to second phase particles Casting alloys based on the Al-Si system may be considered to be two-phase materials composed of hard silicon particles contained in the softer aluminum matrix.  The presence of a distribution of hard particles embedded in a metal matrix is known to result in an overall strengthening of the material, and there has been large interest in this area of research concerning metal matrix composites (MMC’s) [Clyne and Withers (1993)].  Typically, reinforcing particles are present in volume fractions between 5% and 20%, and are usually 10µm to 30µm in diameter, although they can range between 5µm and 250µm.  In Al-Si-Mg casting alloys, the silicon phase is  23 present up to a volume fraction of 11%, however the particles are usually smaller than 5µm in diameter except after prolonged solution treatment.  The strengthening effect of eutectic silicon particles in an Al-Si-Mg alloy can be explained by the transfer of load from the ductile aluminum matrix to the brittle silicon particle via the development of shear stresses at the particle-matrix interface. The load transfer process can be modelled by a number of approaches including the shear lag model, Eshelby models and finite element models based on continuum mechanics.  In general, if the second phase particles are strongly bonded to the matrix, and both particle and matrix behave elastically, a simple rule of mixtures equation can be used to predict the overall strength;      (2.13)  where Vf,particle is the volume fraction of the strengthening particle, and σtot, σparticle and σmatrix are the total stress acting on the material, the stress acting on the particle and the stress acting on the matrix respectively.  If the material is allowed to deform plastically, Equation 2.13 is modified to;      (2.14)  where; σ’matrix is the stress acting on the matrix at the particle fracture strain [Hertzberg (1996)].  The reader is directed to the review of metal matrix composites by Clyne and Withers (1993) for detailed information on the strengthening of a metal matrix by a dispersion of harder particles. ( ) , , . 1 .tot f particle particle f particle matrixV Vσ σ σ= + − ( ) , , . 1 . 'tot f particle particle f particles matrixV Vσ σ σ= + −  24 2.6 Precipitation Kinetics  The kinetics of an isothermal precipitation process involving nucleation and growth can be modeled by the Johnson-Mehl, Avrami, Kolomogorov (JMAK) model [Johnson and Mehl (1939), Avrami (1940), Kolmogorov (1937)].  According to this approach, the fraction of material transformed, fr, is given by;   (2.15)  where k is a function of the nucleation and growth rates and is sensitive to temperature. Provided the nucleation mechanism does not change, n is independent of temperature. Commonly referred to as the JMAK exponent, n typically varies between 1 and 4 [Christian (1975), Porter and Easterling (1992)].  In practice, n is not constant because the metastable phases in the precipitation sequence nucleate at different rates.  Overall, the JMAK kinetics model is valuable as a simple depiction of the kinetics of the overall transformation process, without providing a complete understanding of the nucleation and growth processes occurring during precipitation [Martin (1998)].  Several experimental methods have been used to investigate the kinetics of precipitation in aluminum alloys, including electrical resistivity measurements, differential scanning calorimetry, (DSC) and isothermal calorimetry [Starink et al. (1999), Sato et al. (2003)].  The results of these experimental investigations have been described using a modified JMAK kinetic model, whereas models that predict the nucleation, growth and coarsening processes separately have also been developed [Deschamps et al (1999), Myhr et al (2001)].  In both cases, the kinetic models have ( )1 exp nrf kt= − −  25 been incorporated into process models to predict the evolution of microstructure and mechanical properties of the alloy during non-isothermal ageing.  2.6.1 Effect of natural ageing on precipitation kinetics Pashley et al. (1966) studied the effect of various thermal histories on the microstructure and ageing response of wrought Al-Mg-Si alloys and observed an increased initial hardness and reduced hardening rate following natural ageing.  Later studies have shown a similar relationship between natural ageing and artificial ageing in Al-Si-Mg casting alloys, as shown in Figure 2.9 [Shivkumar et al. (1989)].  The detrimental effect of natural ageing on precipitation hardening in Al-Mg-Si alloys has been studied by several authors, and the clustering of solute atoms was found to be a key factor [Dutta et al. (1991), Murayama et al. (1999), Serizawa et al. (2006)].  TEM observations comparing specimens artificially aged directly after solution treatment and after natural ageing have shown that the strengthening β’’ precipitates are coarser and have a lower number density in the naturally aged alloy [Esmaeili et al. (2003a)].  Natural ageing appears to reduce the number density of nuclei available for subsequent artificial ageing, thereby decreasing the kinetics of precipitation.  The effect of natural ageing on precipitation can be justified as follows.  A high concentration of quenched in vacancies enhances the rate of solute clustering in the early stages of natural ageing, and this clustering of solute leads to a reduced supersaturation of solute in the matrix.  The solute clusters have a fine distribution within the matrix, and if they were to act as successful nuclei for the formation of β’’ during subsequent artificial ageing, a fine precipitate distribution would  26 result.  Evidently, this is not the case, a reason being that many of the clusters are below the critical size for stability at artificial ageing temperatures [Pashley (1966)].  Furthermore, a lower solute supersaturation is expected to reduce the kinetics of precipitation.  Thus, during artificial ageing, the dissolution of unstable clusters increase the solute concentration, while larger clusters that are stable remove solute by growing into GP zones that become nucleation sites for β’’. Therefore, the solute supersaturation is maintained at a relatively low level during artificial; ageing and the density of the β’’ is much lower than that occurring in alloys without natural ageing.  50 60 70 80 90 100 0.1 1 10 100 1000 Ha rd n es s Artificial Ageing Time (min) Without Natural Ageing With 24hr Natural Ageing  Figure 2.9: Variation of hardness in A356 alloy artificially aged at 180°C without natural ageing and after 24 hours of natural ageing [after Shivkumar et al. (1989)].     27 2.7  Microstructure-Property Models for Ageing Processes  Models that predict the response of a material to thermal processing have been the subject of significant interest in recent years, and the first application of a microstructure-property model for age hardening of aluminum alloys was made by Shercliff and Ashby (1990).  They defined their model as a mathematical relation between the process variables (e.g. alloy composition, heat treatment temperature and time), and the mechanical response of the alloy (e.g. yield strength, hardness), based on physical principles (e.g. thermodynamics, kinetics of precipitation, strengthening mechanisms etc.).  The internal state variable method, originally proposed by Richmond (1986) has been found to be well suited to model microstructure evolution under non- isothermal conditions, and has been combined with process modelling to make successful predictions of many non-isothermal transformations in different alloy systems [Bratland et al. (1997), Grong and Shercliff (2002)].  Depending on the problem, the microstructure evolution can often be described by one or two variables.  The microstructure of an age hardening alloy may be described by the evolution of two state variables with time; the volume fraction and the number density or particle radius [Shercliff et al. (1992)].  Microstructure-property models have addressed additional complexity in the precipitation process as knowledge of the evolution of precipitates and their relationship with the mechanical properties has increased.  These include the evolution of precipitates from shearable to non-shearable and the effect this has on strengthening [Poole et al. (2000)], and the effect of a precipitate size distribution on the overall strengthening behaviour [Deschamps et al. (1999)] in Al-Zn-Mg alloys, and also in Al-Mg-Si alloys [Myhr et al. (2001)].   28 Esmaeili et al. (2003a) have performed a comprehensive study of precipitation and age hardening in a wrought AA6111 alloy and developed a yield strength model using an internal state variable approach which incorporated the strengthening mechanisms discussed previously. This model was based on Equation 2.8 and described the development of the mean obstacle strength, F and average obstacle spacing, L, as functions of other microstructural variables that evolve with time.  A linear relationship was used to relate the mean obstacle strength to the mean precipitate radius prior to the peak-aged condition.  The effective precipitate spacing for strong and weak obstacles was estimated as functions of precipitate volume fraction and radius, taking into account the precipitate shape and orientation relationship with the matrix.  The model was further developed to predict the strength beyond peak-age by calculating the average obstacle strength from Equation 2.4, assuming a Gaussian size distribution of weak obstacles [Wang et al. (2003)], following an approach made by Deschamps and Brechet (1999).  The contribution of precipitate strengthening to the yield strength, σys, was then calculated using the equation;       (2.16)  where σppt and σss are the contribution of precipitation strengthening and solid solution strengthening to the yield strength respectively, and σi is the intrinsic strength of the pure aluminum matrix.  The yield strength model has since been developed to predict strengthening in an AA6111 alloy after natural ageing, by modeling concurrent cluster dissolution and precipitate formation during artificial ageing [Esmaeili et al. (2003b)].  In two further studies, a microstructure model of precipitate formation during preageing and artificial ageing was successfully incorporated into ys ppt ss iσ σ σ σ= + +  29 the yield strength model, and the evolution of specific precipitate morphologies and distributions during artificial ageing have also been considered [Esmaeili et al. (2005 a, b)].  As seen, process modelling using the internal state variable approach has been used successfully to predict microstructure and property evolution during age hardening of many aluminum alloys, and increasing complexity is included in these models as further work is carried out. However, the development of these models is focussed almost entirely on the heat treatment of wrought aluminum alloys, and only one study can be found in the literature concerning process models for the heat treatment of aluminum casting alloys [Rometsch and Schaffer (2002)].  In this model, the additional strengthening effects of the presence of eutectic silicon particles and Fe- rich intermetallics, as well as the precipitation of elemental silicon in addition to the Mg-Si precipitates is considered.  However, the model lacks much of the physical basis found in more recent process models for ageing wrought aluminum alloys, neglecting the influence of natural ageing and preaging on subsequent artificial ageing, and assuming the start material is in the fully solution treated condition, with the maximum concentration of supersaturated solute available for precipitation.  2.8 Summary  Although many previous investigations into the thermal processing of Al-Si-Mg casting alloys have been carried out, most focus on a single aspect of the overall process and a comprehensive experimental study considering all heat treatment stages is still required.  Furthermore, the development of process models for the prediction of microstructure and mechanical property changes in aluminum alloys has focussed on wrought alloys, while casting alloys that contain more complex microstructures have been overlooked and the evolution of the solution treated  30 microstructure and its influence on subsequent ageing behaviour has not been incorporated into the models.  Nevertheless, casting alloys are commercially used in large volumes and a process model for their microstructure and mechanical property evolution during multi-stage industrial heat treatments (i.e. heat up to solution treatment temperature, hold, quench, hold at room temperature and final age hardening) would be of significant benefit from the standpoint of process optimisation and the development of new heat treatment strategies.  31 CHAPTER 3 - Scope and Objectives  The present work aims to develop a comprehensive mathematical model of the microstructure and strength evolution in Al-Si-Mg casting alloys during heat treatment.  This will be achieved by performing a series of experimental investigations on an industrially-cast A356 alloy and laboratory-produced Al-Si-Mg alloys to enable the development of physically-based microstructure-strength models.  The heat treatment processes selected for investigation are;  • Solution treatment at temperatures in the range 500-560ºC, for times up to 24 hours • Natural ageing at near-ambient temperatures after solution treatment and quenching, • Immediate artificial ageing following solution treatment and quenching, • Artificial ageing following solution treatment, quenching and a period of natural ageing.  The experimental work required to characterise the behaviour of the material involves;  • An investigation into the microstructure changes that occur during solution treatment within the temperature range of interest, • A study to determine how changes in microstructure during solution treatment affects the strength of the material immediately following solution treatment, • An examination of the microstructure and strength changes during natural ageing following solution treatment, • A detailed investigation into the microstructure and strength evolution of the material during artificial ageing, in which the material has been subjected to a wide range of prior processing conditions (i.e solution treatment and natural ageing parameters).  32  The work required to develop the process model for heat treatment includes;  • Identifying established modelling approaches to describe the physical processes occurring in the material during heat treatment, • Developing an overall model and necessary microstructure sub-models, • Validating the model against independent experimental data, • Using the validated model to predict the behaviour of other Al-Si-Mg alloys, i.e. setting alloy composition boundaries under which the process model is applicable.  The model developed in this work is the first microstructure-strength model for heat treatment of aluminum casting alloys to incorporate the following;  • Strengthening arising from precipitation of second phase particles, • Strengthening due to large insoluble second phase particles and/or eutectic phases, • The effect of incomplete dissolution of particles due to short solution treatment times on the strengthening potential of the subsequent precipitation reactions, • The effect of natural ageing on strengthening during subsequent artificial ageing.  This model will be a significant contribution to the fields of processing-structure-property relationships and through-process modelling of Al-Si-Mg casting alloys, and will provide insight into the heat treatment process as well as acting as a tool to identify the most efficient heat treatment strategies to achieve certain property requirements.  33 CHAPTER 4 - Experimental Methodology  4.1 Introduction  This chapter outlines the materials, thermal treatment conditions and experimental methods used to characterize the microstructure and strength evolution in the material during heat treatment.  4.2 Materials  Materials studied in this investigation consisted of both industrial produced A356 (Al-Mg-Si) by Canadian Autoparts Toyota Inc. (CAPTIN) as well as model alloys that were laboratory cast at Rio Tinto Alcan. The industrial A356 aluminum alloy used in this investigation was provided by CAPTIN in the form of low-pressure die-cast automotive wheels in the as-cast condition.  The chemical composition of this material is given in Table 4.1, and an example of an as-cast wheel provided by CAPTIN is shown in Figure 4.1.  Table 4.1 – Chemical composition (wt%) of the A356 aluminum alloy used in this study Si Mg Fe Ti Sr Al 7.42 0.3 0.17 0.11 0.014 Bal.  The as-cast microstructure of the A356 alloy was found to consist of dendrites of primary α-Al, with an interdendritic Al/Si eutectic phase containing Mg-rich and Fe-rich intermetallic particles. Figure 4.1 shows an image of the as-cast microstructure.  The dendritic structure can be clearly seen as the lighter phase with the darker eutectic phase in the interdendritic regions.  The secondary dendrite arm spacing (SDAS) was measured at random locations across the wheel rim  34 and the average SDAS was calculated to be approximately 30 microns, with little variability throughout the rim.  The scale of the eutectic phase is observed to be very fine, and little information regarding the size and morphology of the eutectic silicon is available at these magnifications.  100µm  Figure 4.1: Example of an automotive wheel indicating the wheel rim area used for the present work and the typical through-thickness microstructure features observed.  A series of other Al-Si-Mg alloys were also studied as part of this investigation.  These alloys were obtained from Rio Tinto Alcan in the form of end-chill cast ingots with dimensions 223mm(l) x 30mm(w) x 255mm(h) in the as-cast condition. The chemical composition for each alloy is given in Table 4.2, and the ingot and end-chill casting apparatus is shown in Figure 4.2.  Table 4.2: Chemical compositions (wt%) of the Al-Si-Mg model alloys used in this study  Si Mg Fe Ti Sr Al Al-1.3Si-0.32Mg 1.33 0.32 0.15 0.014 0.00 Bal. Al-11Si-0.22Mg 11.1 0.22 0.19 0.16 0.15 Bal.  35   Figure 4.2: The end-chill casting apparatus and as-cast ingot at Rio Tinto Alcan  The rationale for selecting the Al-Si-Mg alloy chemistries outlined in Table 4.2 is that additions of silicon and magnesium to hypoeutectic Al-Si-Mg alloys have two specific effects on the microstructure of the alloy.  The silicon content controls the volume fraction of eutectic phase, whereas the magnesium content controls the maximum amount of solute magnesium that can be present during artificial ageing and therefore has a strong influence on the volume fraction of precipitated particles that form.  Thus, there are two main effects that can be studied when varying the silicon and magnesium content of the alloy;  1: A change in the volume fraction of the eutectic phase, controlled by Si additions 2: A change in the volume fraction of strengthening precipitates formed during ageing, controlled by Mg additions  36 The first alloy, Al-1.3Si-0.32Mg, was designed so the primary aluminum phase has a silicon content as close to the solubility limit as possible, but below the level required for the eutectic phase to be present in the solution treated condition.  The second alloy, Al-11Si-0.22Mg, was designed to contain a larger amount of silicon and therefore a higher volume fraction of eutectic phase, and a smaller amount of magnesium which reduces the amount of strengthening due to precipitation during ageing.  However, it should also be noted that the volume fraction of strengthening precipitates formed during artificial ageing is dependent on the complete dissolution of Mg-containing second phase particles during solution treatment.  The commercial thermodynamic database software FactSage has been used to predict the phase content in the A356 alloy and both of the model alloys.  Table 4.3 shows the predicted volume fraction of eutectic silicon and magnesium solute content in the two model alloys and the A356 alloy in the solution treated condition.  The predictions were made by calculating the equilibrium phase content of each alloy at 540°C, and assuming that the quench following solution treatment is rapid enough to prevent any significant changes in phase content.   Table 4.3: Predicted Microstructure Characteristics of the A356 Alloy and Model Alloys in the Solution Treated Condition  Al-1.3Si-0.32Mg A356 (Al-7Si-0.3Mg) Al-11Si-0.22Mg Volume fraction of Eutectic Silicon 0.03 6.19 10.48 Wt% Mg in Solution 0.32 0.3 0.22    37 The as-cast microstructures of the Al-1.3Si-0.32Mg and Al-11Si-0.22Mg alloys are shown in Figure 4.3.  The Al-1.3Si-0.32Mg alloy consists largely of primary α-Al, with small amounts of the Al/Si eutectic phase, Mg-rich and Fe-rich intermetallic particles in the intergranular regions, whereas the dominant phase in the Al-11Si-0.22Mg alloy is the Al/Si eutectic. Comparison with Table 4.3 illustrates that the trend in the observed proportion of eutectic phase in the as-cast alloys is similar to the thermodynamic predictions of eutectic silicon content in the solution treated condition.  The presence of the eutectic silicon phase in the Al-1.3Si-0.32Mg alloy micrograph in Figure 4.3a) arises from the casting conditions and this phase was entirely dissolved during homogenisation prior to testing.   a) α-Al Al/Si Eutectic   b) α-Al Al/Si Eutectic  Figure 4.3: As-cast microstructures of: a) Al-1.3Si-0.32Mg and b) Al-11Si-0.22Mg alloys.  38 4.3 Sample Preparation  Square coupons (25mm x 25mm x 4mm) were prepared to carry out the heat treatment investigation and subsequent metallographic analyses of each alloy.  The A356 alloy coupons were taken from the rim of the as-cast wheel, whereas coupons of the model alloys were taken from the as-cast ingot a short distance away from the chill surface. Additionally, cylindrical tensile samples with 16mm gauge length and 5.5mm gauge diameter were prepared for all three alloys.  All specimens were heat treated from the as-cast condition, except the Al-1.3Si-0.32Mg alloy, which was homogenised in order to remove any eutectic phase present in the as-cast condition.  Homogenisation was carried out in a conventional furnace at 560°C for 24hours, followed by cold-rolling to 50% reduction and recrystallisation at 560°C for 10 minutes to achieve a similar initial grain size to the other alloys being investigated.  4.4 Heat Treatment Experiments  The heat treatment experiments for all alloys involved solution treatment in a nitrate salt bath (60% potassium nitrate, 40% sodium nitrite), followed by quenching in water at room temperature.  Specimens were either examined immediately in the as-quenched condition, or subjected to further thermal treatments including natural ageing in air and artificial ageing at elevated temperatures using a silicon oil bath.  In order to minimise thermal gradients and heating times, the salt and oil baths were vigorously stirred for the duration of the experiments, and independent temperature readings were obtained from a second thermocouple immersed in the heating medium at regular intervals.  Longer heat treatments (i.e. greater than 24 hours) were carried out using a box furnace with air circulation and an aluminum blank instrumented with a thermocouple positioned next to the specimens to monitor the temperature.  39  4.4.1 Solution Treatment Solution treatment of the A356 alloy was carried out using a salt bath at temperatures in the range 500°C-560°C for times between 1 minute and 24 hours.  The range of solution treatment conditions investigated was chosen to produce specimens in a number of various partially solution treated conditions so that an investigation into the evolution of the solution treated microstructure could be made.  A coupon with a thermocouple cemented into its centre via a drilled hole was used to acquire temperature-time profiles for specimens in the salt bath, an example of which is shown in Figure 4.4.  It was found that the centre of the coupon reached the test temperature within 20 seconds after the specimen was immersed in the molten salt, and quenching after solution treatment resulted in cooling rates of 170°C/sec.  0 100 200 300 400 500 0 10 20 30 40 50 Time (sec) Te m pe ra tu re  (C ) Heat Up Hold Quench Te m pe ra tu re  (C )  Figure 4.4: An example of the measured sample temperature-time profile during solution treatment in the salt bath.    40 4.4.2 Ageing Following solution treatment and quenching, coupons were aged in order to characterize the following behaviours; i) strengthening due to natural ageing at temperatures close to room temperature, ii) strengthening due to immediate artificial ageing between 150°C-200°C, and iii) the effect of a period of natural ageing on strengthening during subsequent artificial ageing. Figure 4.5 shows schematic representations of these ageing processes.  Solution Treatment Natural Ageing 20°C a) T t Solution Treatment Natural Ageing 20°C c) T t Artificial Ageing Solution Treatment 20°C b) T t Artificial Ageing  Figure 4.5: Schematic representations of the heat treatment processes studied: a) natural ageing at room temperature, b) immediate artificial ageing between 150°C-200°C, c) a period of natural ageing followed by artificial ageing between 150°C-200°C.  41 4.5 Material Characterisation  The microstructure evolution and strength development of the alloys was measured using a multi-faceted range of characterization and testing techniques. The choice of the technique used was dependant on both the heat treatment stage in question and the metallurgical process of interest.  The metallurgical behaviour during solution treatment was investigated by optical microscopy techniques, electron probe microanalysis (EPMA), and image analysis of eutectic silicon particles exposed by deep etching.  During ageing the evolution of precipitation was characterized using exothermic heat traces obtained by an isothermal calorimetry technique, as well as by electrical resistivity measurements.  The mechanical response of the material was measured by performing Vickers hardness and tensile tests.  Further details are given in the following sections.  4.5.1 Sample Preparation  All metallographic examinations were carried out on through-thickness sections of the coupons, which in both wheel and ingot castings expose the microstructure perpendicular to the solidification direction.  The specimens for metallographic examination were mounted using Buehler Epoxicure acrylic resin, then ground and polished by hand to a 0.06µm finish using a colloidal silica suspension.  4.5.2 Microscopy The polished samples were first etched using Keller and Dix etch (1% hydrofluoric acid, 1.5% hydrochloric acid and 2.5% nitric acid in water) to improve the contrast between the various phases contained in the alloy microstructure.  Afterwards, photomicrographs of the microstructures were taken using a Nikon EPIPHOT 300 series inverted metallurgical  42 microscope equipped with a digital camera.  Post-processing of the micrographs was carried out using Clemex Professional Imaging and Adobe Photoshop 7.0 software.  Several microstructural characteristics were quantified by analysing the 2-D micrographs.  In the as-cast alloys, the secondary dendrite arm spacing of the primary α-Al dendrites and the average size of the as-cast Mg2Si particles were measured using the Imagetool software package developed at the University of Texas Health Sciences Centre at San Antonio.  Polished specimens were deep etched to reveal the change in eutectic silicon morphology and size during solution treatment.  This was achieved by immersing mounted and polished specimens into Keller’s etch (10% hydrofluoric acid and 5% hydrochloric acid in water) for 45- 60 minutes in order to dissolve the aluminum matrix and expose the eutectic silicon phase.  Deep etched specimens were then examined using a Hitachi S-3000 electron microscope in backscatter electron (BSE) mode.  The accelerating voltage was varied between 5keV and 20keV, with the best edge resolution found at lower voltages.  Stereographic images of selected areas were produced by overlaying two micrographs taken before and after the specimen was moved through a 7° rotation.  The eutectic silicon particle characteristics in both as-cast and solution treated specimens were quantified using the Imagetool software package to characterize the change in silicon particle morphology during solution treatment.  In this case, thresholding of the images was achieved by outlining and colouring the exposed particles manually and scanning the resulting monochrome images.  The particle parameters chosen for the analysis included the equivalent circle diameter, aspect ratio, and circularity, which are described in Figures 4.6-4.8 and calculated using equations 4.1-4.3.  43  Figure 4.6: The equivalent circle diameter (ECD) is the diameter of a circle with area equal to that of the projection of the particle at a plane (i.e. Areaparticle = Areacircle).  4. . . . particleAreaE C D pi =       (4.1)   Figure 4.7: The aspect ratio (AR) is the ratio between the major axis length and the minor axis length.  Major and Minor are the primary and secondary axis of the best fitting ellipse.  . . . . Max LengthA R MaxWidth =       (4.2)   44 Area Perimeter  Figure 4.8: The circularity is a function of the perimeter P and the area A that has values between 0 and 1, where a value of 1 indicates a perfect circle.  2 4. . particle particle Area Circularity Perimeter pi =      (4.3)  4.5.3 Electron Probe Microanalysis (EPMA) Electron probe microanalyses of the alloy microstructure were done using a fully automated CAMECA SX-50 electron-probe microanalyser, operating in wavelength-dispersion mode at the Electron Microbeam/X-ray Diffraction Facility in the Department of Earth and Ocean Sciences at UBC.  Quantitative analyses were made for Al Si, Mg, Fe and Ti using standards of known composition and the following operation conditions were used; excitation voltage, 15kV; beam current, 20nA; peak count time, 60s (10s for Al); background count-time, 30s (5s for Al); spot diameter, 1µm.  The interaction volume was estimated to have an approximate diameter and depth of 2.5µm and 2µm, respectively.  Consequently measurements were made across the width of the dendrite arms at spacings of 4µm, as well as selected measurements in the eutectic regions and at the  45 location of large (i.e. greater than 1µm) second phase particles. These results allow the spatial solute content of Mg across the dendrite to be estimated so as to determine the evolution of the dissolution and homogenisation processes. Between four and six dendrites were examined for each experimental condition studied; representative results from these analyses are presented in Chapter 5 and have been used to develop the models presented in Chapter 6.  4.5.4 Tensile and Hardness Testing The mechanical response of the material as a function of thermal history was initially characterized by performing Vickers hardness measurements using a Vickers apparatus with a load of 5kgf.  The indentation size was measured manually using an optical shutter and converted to Vickers Hardness (HV) using the following relationship;  21.854 FHV d =      (4.4)  where F is the applied load (measured in kilograms-force) and d is the average length of the diagonal left by the indenter (measured in mm2).  The yield strength of the samples after various heat treatment conditions was determined by performing tensile tests on a modified MTS servo-hyrdaulic tensile testing machine with an INSTRON 8500 controller at a displacement rate of 3.2mm/min (corresponding to a strain rate of 0.002s-1).  The elongation of the specimen during straining was measured by attaching an extensometer with a gauge length of 12.5mm to the reduced section of the samples. The recorded data for load and displacement were converted into engineering stress and engineering strain and the yield strength was calculated using the standard 0.2% offset method.  As discussed in the  46 previous sections, a period of natural ageing can significantly affect the microstructure and mechanical properties of the heat-treated alloy.  In order to avoid this, Vickers hardness and tensile tests were performed immediately after quenching from the heat treatment temperature.  Although care was taken to ensure the identical tensile testing procedure was conducted for each specimen, the inhomogeneous nature of cast metal alloys ensures there will be some variability in the mechanical property measurements.  A number of tensile tests were carried out on the A356 alloy in the as-cast condition and the variation in the measured mechanical properties was examined.  The variability in the yield strength data was found to be within +/-5MPa, and this level of error has been assumed for all yield strength data presented in the present work.  4.5.5 Isothermal Calorimetry Isothermal calorimetry experiments were conducted using a SETARAM C80 calvet-type calorimeter in order to obtain the variation in heat flow with time arising from precipitation and/or clustering in the material during artificial ageing and natural ageing.  In a Calvet-type calorimeter, the sample container is surrounded by a secondary wall, and highly accurate measurements of the temperature difference between the internal and external surfaces of this wall are made using a ring of thermocouples.  The geometry, thickness and thermal conductivity of the wall are all known, allowing the heat flow to be evaluated using the balance there.  As a consequence of the design, nearly all of the heat evolved by the specimen is typically detected.  For each alloy, three sets of specimens were tested.  The first set of experiments was performed on specimens that had been solution treated, quenched in water and immediately inserted into the calorimeter vessel at temperatures in the range 150-200°C.  The final set of experiments were performed on specimens that were solution treated, quenched in water and immediately inserted  47 into the calorimeter vessel at temperatures in the range 25-60°C.  For both of these sets of experiments the maximum delay between quenching and insertion was 5 minutes.  The final set of experiments were performed on specimens that were solution treated, quenched in water and allowed to naturally age for 24 hours before inserting into the calorimeter vessel at temperatures in the range 150-200°C.  Sample dimensions were 9mm x 9mm x 3mm (approx 650mg), and care was taken to ensure the specimen orientation within the vessel was identical in each test, with the large face (i.e. 8mm x 8mm) in contact with the base of the calorimeter chamber in order to minimise the time required for the specimen to reach the test temperature.  Two calorimeter runs were performed for each experimental condition studied.  In the first run the test is carried out as described above – in this case the heat flow data being collected includes the initial endothermic effect of adding a sample at room temperature to the calorimeter vessel and it’s heating to the set temperature, as well as the effect of any precipitation and dissolution occurring within the material.  The first run is allowed to continue until precipitation is complete, at which point the heat flow reaches a small, constant level (3x10-5W/g), and the sample is removed and allowed to cool.  The calorimeter is then returned to the set temperature and the sample inserted into the vessel for the second run.  As precipitation and dissolution processes are completed in the first run, the only phenomenon observed in the second run is the initial transient endothermic effect of the sample insertion.  The heat flow due to precipitation and dissolution can be isolated by subtracting the heat flow trace obtained during the second test from that of the first.  The thermal instability arising from insertion of the sample interferes with the heat flow trace for the first 6-10 minutes and therefore no heat flow data is recorded for this time period. The total heat evolved due to the precipitation reaction is given by the area underneath the exothermic part of the heat flow curve which can be found by an integration technique.  It is also  48 useful to determine the evolution of the exothermic heat release, and the fraction of the exothermic heat evolved, fheat, at time, t, is given by;  0 0 f t heat t dQ dt dtf dQ dt dt = ∫ ∫       (4.5)  where 0 t dQ dt dt∫  is the total heat evolved up to time, t, and 0 ft dQ dt dt∫  is the total heat evolved over the entire exothermic time range. 49  CHAPTER 5 - Experimental Results  This chapter presents the experimental results of this work, including the results of the investigations into the microstructure changes of the A356 alloy during solution treatment, and the ageing behaviour of the A356 and model alloys.  The final section presents a short discussion of these results with respect to the metallurgical processes that control the microstructure and mechanical property evolution of the alloys during solution treatment and ageing.  5.1 A356 Alloy Behaviour During Solution Treatment  In this section, the results of the solution treatment investigation on the A356 alloy at temperatures between 500°C and 560°C are presented.  Particular attention is paid to the results obtained during solution treatment at 540°C.  Metallographic specimens have been examined to provide a qualitative description of the changes in microstructure during solution treatment, and a quantitative description of the changes in eutectic silicon particle morphology and solute distribution has been obtained by image analysis and microprobe testing.  Finally, the variation in the mechanical properties of the alloy during solution treatment were measured by Vickers hardness and tensile tests.  5.1.1 Qualitative Description of Microstructure Changes Figure 5.1 shows optical micrographs of the alloy in the as-cast condition and after solution treatment at 540°C for 2 minutes, 30 minutes and 240 minutes.  It is clear from these micrographs that the size of the as-cast eutectic silicon particles increases significantly during solution treatment, whereas the secondary dendrite arm spacing of the alloy is unaffected. 50  40µm 40µm 40µm 40µm α-Al Al/Si Eutectic c) d) a) b)  Figure 5.1: Optical micrographs of A356 alloy after solution treatment at 540°C (x500) a) as cast, b) 2 minutes, c) 30 minutes, d) 240 minutes  The fine scale of the eutectic phase in the as-cast condition and early stages of solution treatment make it difficult to obtain accurate measurements of silicon particle characteristics using optical metallography techniques.  To address this, specimens were deep etched to expose the silicon particles in three-dimensions and examined using an SEM.  Figure 5.2 shows SEM micrographs of these specimens in the as-cast condition as well as after solution treatment at 540°C for 2 minutes, 15 minutes, 30 minutes and 240 minutes.  The as-cast silicon particle morphology is fibrous and interconnected, with an approximate fibre diameter of 0.5µm.   During solution 51  treatment the interconnected fibres break down into smaller spheroidal particles, which then coarsen with further solution treatment time.   a) b) e) c) d) 10µm 10µm 10µm 10µm 10µm  Figure 5.2: Deep etched micrographs of A356 alloy after solution treatment at 540°C (x4000) a) As-cast, b) 2 minutes, c) 15 minutes, d) 30 minutes, e) 240 minutes. 52  The microstructure of the as-cast alloy also contains Mg2Si and α-AlFeSi particles.  The Mg2Si particles dissolve during solution treatment and the change in magnesium distribution can be seen by comparing element maps of the alloy in the as-cast and solution treated conditions, as shown in Figure 5.3.  The distribution of magnesium atoms changes significantly as the Mg2Si particles dissolve and release magnesium atoms into solution in the aluminum matrix.  50µm 50µm 50µm 50µm a) a) b) b) Figure 5.3: Element maps showing distribution of Si and Mg in A356 in the: a) as-cast condition, b) solution treated condition (following 240 minutes at 540°C).  5.1.2 Quantitative Description of Microstructure Changes A quantitative description of the changes in eutectic silicon particle morphology during solution treatment has been made by analysing stereographic micrographs of the deep etched specimens. Bar charts showing the distribution of silicon particle characteristics in the as-cast condition and after solution treatment at 540°C from 2 minutes to 240 minutes are shown in Figure 5.4a)-c). 53  a) 0 0.2 0.4 0.6 0.8 1 0-1 1-3 3-5 5+ Eq. Circle Diameter (microns) N u m be r Fr a ct io n AC 2min 15min 30min 240min  b) 0 0.2 0.4 0.6 0.8 1 1-2 2-3 3-4 4+ Aspect Ratio N u m be r Fr a ct io n AC 2min 15min 30min 240min  c) 0 0.2 0.4 0.6 0.8 1 0. 86 5+ (sp he ric a l) 0. 64 - 0. 86 5 (sp he ro id a l) 0. 44 4- 0. 64 (irr e gu la r) 0- 0. 44 4 (co m pl ex ) Particle Roundness N u m be r Fr a ct io n AC 2min 15min 30min 240min  Figure 5.4: The distribution of eutectic particle shape characteristics following solution treatment at 540°C: a) equivalent circle diameter, b) aspect ratio, c) particle roundness. 54  Several interesting details regarding the solution treatment behaviour of the silicon particles can be ascertained from the bar charts.  Almost all particles measured in the as-cast condition were very small, with an equivalent circle diameter smaller than 3µm.  However, Figure 5.4 b) and c) also show that most as-cast silicon particles have a high aspect ratio and low roundness, confirming that they are elongated and non-spheroidal.  During solution treatment, Figure 5.4a) shows that the number fraction of particles with equivalent circle diameter smaller that 1µm decreases and the fraction of particles between 1µm and 5µm increases.  Simultaneously, the number fraction of particles with aspect ratio lower than 3 increases and there is a large drop in particles with roundness factors below 0.444, indicating that coarsening of the fine eutectic silicon fibres is occurring.  Particles that are smaller than 1µm become progressively less frequent with solution treatment time, while the number fraction of large particles increases.  After 30 minutes of solution treatment, nearly all particles have an aspect ratio close to 1 and particle roundness above 0.64, and longer solution treatments have no significant effect on these values, which suggests that the silicon particles fragment and become spheroidal within the initial 30 minutes of solution treatment.  The variation in the average particle characteristic values during solution treatment are presented graphically in Figure 5.5 and provide further information regarding the fragmentation and coarsening processes.  The average equivalent circle diameter increases during solution treatment from 1.14µm in the as-cast condition to 4.19 µm after 240 hours at 540°C, however there is a discontinuity between 15 minutes and 30 minutes where the average diameter decreases from 2.10 µm to 1.64 µm.  The average aspect ratio decreases from 4.01 in the as-cast condition to 1.32 after 30 minutes at 540°C and does not change significantly on further solution 55  a) 0 1 2 3 4 0.01 0.1 1 10 100 1000 Time (hr) M e a n  D ia m e te r (m ic ro n s )  b) 1 2 3 4 0.01 0.1 1 10 100 1000 Time (hr) M e a n  A s pe c t R a tio  c) 0 0.2 0.4 0.6 0.8 1 0.01 0.1 1 10 100 1000 Time (hr) M e a n  Pa rt ic le  R o u n dn e s s  Figure 5.5: Change in average eutectic particle characteristics during solution treatment at 540°C: a) equivalent circle diameter, b) aspect ratio, c) particle roundness. 56  treatment.  Similarly, the average particle roundness increases from 0.45 to 0.84 and then does not change significantly.  Rapid changes in aspect ratio and particle roundness take place between 15 minutes and 30 minutes after which no large changes are observed.  The decrease in average equivalent circle diameter between 15 minutes and 30 minutes, combined with the simultaneous achievement of low aspect ratio and high roundness suggests the fragmentation process is rapid during this time interval and is more or less complete after 30 minutes.  Cross-referencing Figure 5.5 with the micrographs in Figure 5.2 allows some further observations.  The coarsening of silicon rods suggested in Figure 5.5a) at the beginning of solution treatment is apparent in Figures 5.2a) & b).  Furthermore, rapid changes in aspect ratio and particle roundness at the beginning of solution treatment in Figures 5.5b) and c) are related to coarsening, while changes between 15 minutes and 30 minutes are related to fragmentation.  5.1.3 Electron Probe Microanalysis Representative curves from the electron probe microanalysis investigation are presented in Figure 5.6, showing the distribution of magnesium and silicon in the as-cast condition and after solution treatment at 540°C for different times.  In Figure 5.6a), a segregated magnesium distribution is observed in the as-cast condition between 0.16wt% at the dendrite centre and 0.29wt% at the edge, which arises due to rejection of solute during solidification of the alloy. During solution treatment, two effects are observed.  Firstly, the segregation profile is eliminated as homogenisation occurs via diffusion of magnesium atoms.  Secondly, the total amount of magnesium in the dendrite increases as interdendritic Mg2Si particles dissolve and release solute magnesium atoms into the aluminum matrix. The magnesium distribution in the dendrite reaches a relatively uniform concentration of 0.35wt% after 15 minutes at 540°C, and does not change 57  significantly with further solution treatment.  Six dendrites were examined for each experimental condition and the reproducibility of the microprobe results was found to be good.  For the as-cast dendrites, the mean magnesium concentration at the dendrite centre was 0.153wt%, with a 95% confidence interval of +/-0.023.  In the fully solution treated condition, the mean magnesium content of all dendrites examined was 0.36wt%, with a 95% confidence interval of +/-0.012.  The microprobe results for silicon reveal both similarities and differences in behaviour compared to that of the magnesium atoms.  In the as-cast condition silicon is segregated, however the lower content of 1.5wt% is found at the edge of the dendrite and the highest content of 1.78wt% is at the dendrite centre. This “inverse segregation” phenomenon was discussed previously by Snugovsky et al. (2000) and is related to directional solid state diffusion of silicon away from the dendrite centre towards the eutectic during cooling after solidification.  Solution treatment results in an average decrease in the silicon content of the dendrite as inverse segregation is removed and homogenisation occurs.  The microprobe data reveals some silicon-rich areas in the  0.1 0.2 0.3 0.4 0.5 -20 0 20 w t% M g Distance from Dendrite Centre (µm) As Cast 2 min 15 min 30 min 240 min 1.1 1.3 1.5 1.7 1.9 2.1 -20 0 20 w t% Si Distance from Dendrite Centre (µm) As Cast 2 min 15 min 30 min 240 min Figure 5.6: The distribution of a) magnesium and b) silicon across secondary dendrite arms in the A356 alloy in the as-cast condition and during solution treatment at 540°C. 58  dendrite after short solution treatment times, likely due to the presence of silicon particles in the as-cast dendrite that formed during cooling after final solidification.  The silicon concentration becomes uniform after 15 minutes at 540°C at a level of approximately 1.3wt%, and does not change significantly with further solution treatment.  This amount of silicon is in fairly close agreement with the solid solubility limit for silicon in aluminum at 540°C, as determined from a thermodynamic investigation using the FactSage software program, details of which follow later in this chapter.  5.1.4 Mechanical Properties of Solution Treated Material Mechanical property data from tensile testing of solution treated specimens in the as-quenched condition are shown in Figure 5.7.  Solution treatment was found to have several effects on the mechanical properties.  There is an initial drop in yield strength from 98MPa in the as-cast condition to 78MPa after a short solution treatment of 2 minutes at 540°C.  The yield strength continues to decrease slowly with further solution treatment to 67MPa after 24 hours at 540°C, and the ultimate tensile strength decreases from a maximum of 217MPa to 190MPa.  The drop in  50 60 70 80 90 100 110 1 10 100 1000 10000 Solution Treatment Time (min) Yi el d St re n gt h (M Pa ) 180 190 200 210 220 230 1 10 100 1000 10000 Solution Treatment Time (min) Ul tim at e Te n si le  St re n gt h (M Pa ) 0.15 0.2 0.25 0.3 0.35 1 10 100 1000 10000 Solution Treatment Time (min) St ra in  to  Fa ilu re  Figure 5.7: Mechanical property behaviour for A356 in the as-cast and as-quenched condition following solution treatment at 540°C. 59  yield strength and increase in ductility between the as-cast and 2 minute solution treated conditions may be related to the dissolution of precipitates that form in the as-cast material during cooling after solidification.  The changes in mechanical behaviour at longer solution treatment times is likely related to the microstructure changes during solution treatment, including changes in the amount of second phase strengthening due to the fragmentation and coarsening of the eutectic silicon particles.  5.2 A356 Alloy Behaviour During Ageing  In this section, the experimental results of the ageing investigation on the A356 alloy are presented.   The results of material behaviour during natural ageing are presented first, followed by details of the response of the material to immediate artificial ageing following the quench, and the behaviour of the material during artificial ageing following a 24 hour natural age.  5.2.1 Natural Ageing Behaviour The natural ageing response of the solution treated material was investigated by performing tensile tests on specimens that were quenched after solution treatment and left for a fixed period of time at room temperature.  Isothermal calorimetry tests were also carried out to examine the behaviour of solute and precipitates during natural ageing.  Figure 5.8 shows mechanical property data from tensile testing of the solution treated alloy in the as-quenched condition and after natural ageing at room temperature for 2 hours, 24 hours and 2400 hours.  An increase in yield strength from 72 MPa in the as-quenched condition to 104 MPa after 2 hours is observed, followed by a further increase to 122 MPa after 24 hours, rising 60  to 132MPa after 2400 hours (100 days) at room temperature.  Conversely, the strain to failure decreases from 0.31 to 0.21 as the length of the natural age increases.  This behaviour is associated with the formation of solute clusters that act as obstacles to dislocation movement.  0.15 0.2 0.25 0.3 0.35 0.1 1 10 100 1000 10000 St ra in  to  Fa ilu re Natural Ageing Time (hr) 200 210 220 230 240 250 260 0.1 1 10 100 1000 10000 Ul tim at e Te n si le  St re n gt h (M Pa ) Natural Ageing Time (hr) 60 80 100 120 140 0.1 1 10 100 1000 10000 Yi el d St re n gt h (M Pa ) Natural Ageing Time (hr)AQ AQ AQ  Figure 5.8: The mechanical property behaviour of the A356 alloy in the as-quenched (AQ) condition, and after natural ageing at room temperature for 2 hours, 24 hours and 2400 hours.  The kinetics of solute clustering at natural ageing temperatures was studied using isothermal calorimetry at 25°C (i.e. 5°C higher than ambient), 40°C and 60°C.  As discussed previously the exothermic heat flow can only be seen after a delay due to a thermal instability arising from the insertion of the specimen, however, for the natural ageing tests the delay is only 3 minutes long due to the proximity of the specimen temperature to that of the chamber.  Figure 5.9 shows thermograms for natural ageing at 25°C, 40°C and 60°C after adjusting for the delay (i.e. the time of ageing is calculated by tageing = t – tdelay).  The exothermic heat flow increases rapidly to a maximum, then decays more slowly to a negligible level, at which point the rate of clustering reaches an insignificantly small value.  The heat flow curves in Figure 5.9 show that the evolution of exothermic heat is dependent on the ageing temperature.  At higher ageing temperatures the exothermic heat flow increases more rapidly in the early stages of ageing to a 61  greater maximum value.  After the peak, the exothermic heat flow decays more quickly, and a negligible exothermic heat flow is recorded at earlier ageing times when the ageing temperature is increased.  The total heat evolved during natural ageing can be calculated from the area under the exothermic heat flow curve as described in Chapter 4.5.5.  For natural ageing at 25°C, 40°C, and 60°C, the total heat evolved was calculated and found to be 2.5J/g (+/-0.1).  0 0.001 0.002 0.003 0.004 0 1 2 3 tageing (hr) H e a t F lo w  (W /g ) 25°C 40°C 60°C  Figure 5.9: Exothermic heat flow traces for natural ageing of A356 at 25°C, 40°C and 60°C.  The fraction of heat evolved at a given time during ageing, which represents the fraction of the exothermic clustering reactions completed (and also the relative volume fraction of clusters formed), can be calculated from the heat flow data by dividing the total heat evolved at the time of interest by that of the entire precipitation reaction.  The evolution of the relative volume fraction of clusters during natural ageing at 25°C, 40°C and 60°C was calculated in this way and is shown in Figure 5.10.  The natural ageing temperature strongly influences the evolution of solute clusters - the relative volume fraction of clusters formed after 0.1 hours is 0.07, 0.21 and 0.39 at 25°C, 40°C and 60°C respectively. 62   0 0.2 0.4 0.6 0.8 1 0.01 0.1 1 10 100 Fr ac tio n  He at  Fl ow tageing (hr) 25ºC 40ºC 60ºC  Figure 5.10: The evolution of heat during natural ageing of A356 at 25°C, 40°C and 60°C.  5.2.2 Immediate Artificial Ageing A combination of tensile, Vickers hardness, and isothermal calorimetry experiments were performed to characterise the artificial ageing behaviour at temperatures between 150°C and 200°C.  Figure 5.11 shows the mechanical property behaviour of the solution treated alloy in the as- quenched (AQ) condition, and after artificial ageing at 180°C for various times in the underaged, peak-aged, and overaged condition.  The as-quenched yield strength of 72MPa increases to 198MPa and 231MPa after 15 and 30 minutes respectively.  The peak-aged material has a yield strength of 256MPa and is achieved after 3 hours at 180°C.  The ultimate tensile strength exhibits similar increases, from 208MPa in the as-quenched condition to 324MPa at the peak age, and the strain to failure decreases as the material approaches the peak-aged condition, from 63  0.312 in the as-quenched condition to 0.128 at the peak age.   These patterns of behaviour are similar to those observed during natural ageing, only on a larger scale, which indicates the precipitates formed during artificial ageing have a greater strengthening effect than the naturally aged solute clusters.  Overageing of the alloy is observed in the material at artificial ageing times longer than 3 hours at 180ºC.  The decrease in yield strength takes place very slowly compared with the strengthening observed during underageing – most of the strengthening during underageing at 180°C is achieved within the first 30 minutes, whereas a yield strength of 210MPa was measured after artificial ageing for 60 hours.  The ductility of the alloy also appears to recover slightly during overageing, as precipitated particles become larger, weaker and less able to prevent dislocation movement.  50 100 150 200 250 0.01 0.1 1 10 100 Artificial Ageing Time (hr) Yi el d St re n gt h (M Pa ) 200 250 300 350 0.01 0.1 1 10 100 Artificial Ageing Time (hr) Ul tim at e Te n si le  St re n gt h (M Pa ) 0.1 0.15 0.2 0.25 0.3 0.35 0.01 0.1 1 10 100 Artificial Ageing Time (hr) St ra in  to  Fa ilu re AQ AQ AQ  Figure 5.11: The mechanical property behaviour of the A356 alloy in the as-quenched (AQ) condition, and after artificial ageing at 180ºC for various times.  Ageing curves were calculated from Vickers hardness measurements of the immediately artificially aged material at 150°C, 180°C and 200°C, and these are shown in Figure 5.12.  The procedure for correlation of yield strength and Vickers hardness data is described in Appendix A.  Several important features of the ageing curves are apparent, including the temperature 64  dependence of strengthening during artificial ageing and the rapid change in strength during underageing compared with overageing.  It is also clear that the value of the peak strength is relatively unaffected within the range of artificial ageing temperatures studied.  50 100 150 200 250 0.1 1 10 100 1000 Artificial Ageing Time (h) Yi e ld  St re n gt h (M Pa ) 150°C 180°C 200°C  Figure 5.12: Ageing curves based on Vickers hardness data for artificial ageing of the A356 alloy in the temperature range 150°C-200°C.  The kinetics of precipitation was studied by isothermal calorimetry in the temperature range 150°C to 200°C.  In this case, the delay arising from the thermal instability when the specimen is inserted into the calorimeter chamber is approximately 10 minutes.  Figure 5.13 shows thermograms for immediate artificial ageing at 150°C, 180°C and 200°C, in which the initial delay has been removed.  The exothermic heat flow increases rapidly to a maximum, then decays more slowly to a negligible level, at which point the precipitation reaction has either completed or becomes insignificant.  An increase in the artificial ageing temperature results in larger maximum exothermic heat flow values and faster decay of the heat flow to negligible levels. 65  0 0.001 0.002 0.003 0.004 0.005 0.006 0 1 2 3 4 5 tageing (hr) H e a t F lo w  (W /g ) 150°C 180°C 200°C  Figure 5.13: Thermograms for the artificially aged A356 alloy at 150°C, 180°C and 200°C.  The total heat generated during artificial ageing is larger than that calculated during natural ageing (4.3 J/g Vs. 2.5 J/g), and is similar for all ageing conditions, except at 200°C where the total heat evolved is approximately 25% lower than that calculated for the other artificial ageing temperatures.  In this case, considering that the evolution of yield strength during ageing at 200°C results in significant strengthening at very short ageing times it is likely that a significant proportion of the heat release due to precipitation occurs during the thermal instability at the beginning of the calorimetry test and is therefore not recorded.  The heat evolution during artificial ageing at 150°C, 180°C and 200°C is shown in Figure 5.14, indicating the strong temperature dependence of the exothermic heat release due to precipitation. The analysis method for this data is described in section 4.5.5.  For ageing at 180°C, the fraction of heat evolves to 0.9 after 1 hour at temperature.  There follows a slower evolution of heat until precipitation is complete after 2.75 hours. 66  0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Time (hr) Fr ac tio n  He at  Ev o lv ed 200ºC 180ºC 150ºC  Figure 5.14: The evolution of the relative fraction of heat during artificial ageing of A356 at: a) 150°C, b) 180°C, c) 200°C.  5.2.3 Artificial Ageing After Natural Ageing The artificial ageing response of the material following 24 hours natural ageing was investigated between 150°C and 200°C.  Again, tensile, Vickers hardness, and isothermal calorimetry tests were performed.  Figure 5.15 shows yield strength curves for artificial ageing at 150°C and 180°C following three different thermal histories; (i) no natural ageing, (ii) natural ageing for 24 hours and (iii) natural ageing for 168 hours (150°C only).  Several differences in artificial ageing behaviour were observed.  As expected, the initial yield strength of the alloy after natural ageing is higher than that of the as-quenched condition (122 MPa vs. 72 MPa).  However, the naturally aged material exhibits a delay in strengthening at the beginning of artificial ageing that is not observed in the immediately artificially aged material.  The inhibition of precipitation strengthening persists for longer when the duration of the natural age is increased, and the yield strength of the naturally aged material may decrease slightly initially.  Furthermore, 67  strengthening during artificial ageing occurs more slowly in the naturally aged material than the directly artificially aged material and it takes longer to achieve the peak-aged condition.  Finally, the peak strength is similar for artificially ageing at 150°C, whereas at 180°C the peak strength of the naturally aged material is lower than that of the immediately artificially aged material by approximately 10MPa.  a) 50 100 150 200 250 0.1 1 10 100 1000 Artificial Ageing Time (h) Yi e ld  St re n gt h (M Pa ) No Natural Age 24hrs 168hrs  b) 50 100 150 200 250 0.1 1 10 100 1000 Artificial Ageing Time (h) Yi e ld  St re n gt h (M Pa ) No Natural Age 24hrs  Figure 5.15: The evolution of yield strength during artificial ageing at a) 150°C and b) 180°C following various natural ageing histories. 68  The heat flow curves from isothermal calorimetry tests at 165°C, 180°C and 200°C are shown in Figure 5.16.  Again, there is a delay of approximately 10 minutes at the beginning of testing, and a similar pattern of behaviour is observed in the characteristics of the heat flow curves in that the rate of heat release increases to a maximum then decreases slowly to zero, with strong temperature dependence.  The start and finish times for the exothermic heat release occur later when ageing at lower temperatures, while the total heat evolved during the artificial ageing process does not show any strong trends and is reasonably constant for the temperature range studied.  It should be noted that the sum of the heat evolved during natural ageing at room temperature for 24 hours (2.5J/g) and the subsequent artificial ageing (2.4J/g) is close to the total heat released during immediate artificial ageing (4.3J/g).  Thus, cluster formation during natural ageing followed by cluster dissolution and precipitate formation during artificial ageing appears to result in the same total exothermic heat release as an immediately artificially aged material in which only precipitate formation occurs.  0 0.0005 0.001 0.0015 0.002 0 2 4 6 8 10 12 Ageing Time (h) H e a t F lo w  (W /g ) 165°C 180°C 200°C  Figure 5.16: Thermograms for isothermal calorimetry of A356 artificially aged at 165°C, 180°C and 200°C following 24 hour natural ageing at room temperature. 69  The effect of the length of the natural ageing period on the kinetics of precipitation during subsequent artificial ageing was also investigated by isothermal calorimetry.  Figure 5.17 shows the first 4.5 hours of exothermic heat flow for the material during artificial ageing at 180°C after three different prior treatments; i) no natural ageing, ii) 30 minutes natural ageing at room temperature, and  iii) 24 hours natural ageing at room temperature.  The duration of natural ageing has a strong effect on the heat flow characteristics.  The maximum exothermic heat flow is delayed and becomes smaller with natural ageing.  Specimens subjected to longer natural ageing prior to testing also took longer to reach a negligible heat flow.  It is likely that this behaviour is related to the amount and size of the solute clusters that form during natural ageing. Larger solute clusters require longer to dissolve at a given artificial ageing temperature and this delays the availability of these atoms for the exothermic precipitation reactions.  A model is required to analyse the concurrent precipitation and dissolution processes taking place in the material under these conditions, and this shall be returned to in the following chapter.  0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 0.0028 0 1 2 3 4 Time (hr) H e a t F lo w  (W /g ) Immediate Artificial Age 30min Natural Age 24h Natural Age  Figure 5.17: Thermograms of exothermic heat flow measured in A356 alloy during artificial ageing at 180°C following various natural ageing histories. 70   5.3 Heat Treatment Behaviour of Model Alloys  A series of experiments were conducted on two model Al-Si-Mg alloys to provide data that can be used to aid development of a microstructure-strength model capable of predicting the material response of a range of Al-Si-Mg alloy compositions to heat treatment.  As described in Chapter 4.2, the silicon and magnesium content of the model alloys have been chosen to control the volume fraction of eutectic and the maximum amount of solute magnesium available for precipitation.    The Al-11Si-0.22Mg alloy was designed to contain a larger amount of silicon and smaller amount of magnesium than the A356 alloy studied in sections 5.1 and 5.2.  As a result, this alloy contains a higher volume fraction of eutectic phase and a smaller precipitation strengthening effect.  The Al-1.3Si-0.32Mg alloy was designed so the silicon content is as close to the solubility limit as possible and with similar magnesium content to the A356 alloy.  As a result this alloy will not exhibit any strengthening due to the eutectic phase, and is expected to have a similar precipitation strengthening potential as the A356 alloy.  The results of experiments to characterise the behaviour of the two Al-Si-Mg model alloys are presented in the following section.  The experimental results for the Al-11Si-0.22Mg alloy are presented first, followed by the results for the Al-1.3Si-0.32Mg alloy.  5.3.1 Behaviour of Al-11Si-0.22Mg The solution treatment and ageing behaviour of the Al-11Si–0.22Mg alloy was studied by a combination of electron microprobe analysis, Vickers hardness, tensile and isothermal calorimetry testing.  71  5.3.1.1 Solution Treatment Behaviour Electron probe microanalysis was used to investigate the changes in distribution of alloying elements during solution treatment at 540°C.  Figure 5.18 shows representative curves for the distribution of magnesium across dendrite arms in the as-cast condition and after solution treatment for 30 minutes and 180 minutes.   The distribution of magnesium in the as-cast dendrite varies from 0.16wt% at the core to 0.23wt% at the edge.  During solution treatment, the segregation profile is eliminated and the total amount of magnesium in the dendrite arm increases.  The microprobe results indicate that, in contrast to the A356 alloy, the dissolution of Mg-rich phases and homogenisation of magnesium in solution in the Al-11Si-0.22Mg dendrite are incomplete after 30 minutes at 540°C.  However, after 180 minutes at 540°C, a relatively uniform magnesium distribution has been measured with a concentration close to 0.25wt% indicating that dissolution and homogenisation are complete after this longer time.  Optical metallographic examination confirmed that the coarse interdendritic Mg2Si particles had been fully dissolved after 180 minutes at 540°C.  0.1 0.15 0.2 0.25 0.3 -20 -10 0 10 20 Distance From Dendrite Centre (Microns) w t% M g As Cast 30min 180min  Figure 5.18: The distribution of magnesium across secondary dendrite arms in the as cast and solution treated Al-11Si-0.22Mg alloy. 72  5.3.1.2 Natural Ageing Behaviour The change in mechanical behaviour during natural ageing was investigated by Vickers hardness and tensile testing specimens that were solution treated at 540°C for 180 minutes.  Figure 5.19 shows the change in yield strength during natural ageing.  The yield strength increases from 81MPa to 107MPa after 2 hours natural ageing, and continues to 123MPa after 24 hours.  70 80 90 100 110 120 130 0 4 8 12 16 20 24 Time (hr) Yi el d St re n gt h (M Pa )  Figure 5.19: The evolution of yield strength during natural ageing of Al-11Si-0.22Mg alloy.  5.3.1.3 Artificial Ageing Behaviour Figure 5.20 shows the evolution of yield strength in the Al-11Si-0.22Mg alloy during immediate artificial ageing at 150°C, 180°C and 200°C.   The yield strength in the as-quenched condition is 81MPa, increasing to 210MPa in the peak-aged condition.  During artificial ageing at 180°C, the yield strength increases rapidly from the as-quenched value to 190MPa after 1 hour, and subsequently reaches the peak condition of 210MPa after 3 hours.  73  60 80 100 120 140 160 180 200 220 0.1 1 10 100 Yi e ld  St re n gt h (M Pa ) Time (hr) 150°C 180°C 200°C  Figure 5.20: The evolution of yield strength during artificial ageing of Al-11Si-0.22Mg alloy in the temperature range 150°C-200°C.  The artificial ageing response of the Al-11Si-0.22Mg alloy was also investigated between 150°C and 200°C after a 24 hour natural age.  Figure 5.21 compares the evolution of yield strength during immediate artificial ageing at 180°C and after natural ageing for 24 hours.  Similar patterns in ageing behaviour were observed as were reported for the A356 alloy.  The decrease in yield strength at early stages of artificial ageing in the previously naturally aged material is clearly observed. 74  60 80 100 120 140 160 180 200 220 0.1 1 10 100 Yi e ld  St re n gt h (M Pa ) Time (hr) No Natural Ageing 24hr Natural Aged  Figure 5.21: The evolution of yield strength during artificial ageing of the Al-11Si-0.22Mg alloy at 180°C following a) no natural ageing, b) 24 hours natural ageing at room temperature.  A comparison between the heat flow curves for artificial ageing of the Al-11Si-0.22Mg alloy at 180°C in Figure 5.22 shows that, as previously seen in the A356 alloy, the heat evolution in the immediately artificially aged specimen is higher than that of the specimen subjected to natural ageing prior to artificial ageing.  The isothermal calorimetry technique was also used to investigate the effect of natural ageing on the kinetics of precipitation in more detail.  Similarly to the results of the immediate artificial ageing investigation for this alloy, the total heat evolved is slightly smaller than the equivalent heat evolution for the A356 alloy (2.0J/g vs. 2.4J/g), and the heat flow curves of both alloys share general characteristics.  The exothermic heat flow curves for the tests at 165°C, 180°C and 200°C are shown in Figure 5.23.   75  0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 0.0028 0.0032 0 1 2 3 4 Time (hr) H e a t F lo w  (W /g ) No Natural Ageing 24hr Natural Ageing  Figure 5.22: The exothermic heat flow curves for Al-11Si-0.22Mg at 180°C during immediate artificial ageing and artificial ageing after 24 hours natural ageing.  0 0.0005 0.001 0.0015 0.002 0 2 4 6 8 Time (hr) He at  Fl o w  (W /g ) 165°C 180°C 200°C  Figure 5.23: Exothermic heat flow curves for the Al-11Si-0.22Mg alloy during artificial ageing at 165°C, 180°C and 200°C following 24 hours natural ageing.   76  5.3.2 Behaviour of Al-1.3Si-0.32Mg As a result of the presence of the eutectic silicon phase in the as-cast Al-1.3Si-0.32Mg alloy microstructure, which is expected to dissolve at solution treatment temperatures, this material was homogenised prior to testing and consequently only the experimental results for the ageing behaviour are presented.  5.3.2.1 Natural Ageing Behaviour A study of the natural ageing behaviour following complete homogenisation of the Al-1.3Si- 0.32Mg alloy was made by Vickers hardness and tensile testing. The evolution of yield strength during natural ageing is shown in Figure 5.24.  The as-quenched yield strength of 50MPa increases to 81MPa after 2 hours, followed by a further increase to 99MPa after 24 hours.  40 50 60 70 80 90 100 110 0 4 8 12 16 20 24 Time (hr) Fr ac tio n  He at  Ev o lv ed  Figure 5.24: The evolution of yield strength during natural ageing of Al-1.3Si-0.32Mg alloy.   77  5.3.2.2 Immediate Artificial Ageing The age hardening response of the homogenised Al-1.3Si-0.32Mg alloy was investigated using Vickers hardness, tensile testing and isothermal calorimetry experiments between the temperatures 150°C and 200°C.  Figure 5.25 shows the evolution of yield strength during artificial ageing at 150°C, 180°C and 200°C immediately after quenching.   The yield strength in the homogenised and as-quenched condition is 51 MPa, increasing to 230 MPa in the peak-aged condition.  During artificial ageing at 180°C, the yield strength increases rapidly from the as- quenched value to 200MPa after 1 hour, and subsequently reaches the peak condition after approximately 3 hours.  This material response is similar to the A356 and Al-11Si-0.22Mg alloys, in which almost 90% of the strengthening at this temperature occurs within the first hour of ageing.  50 100 150 200 250 0.1 1 10 100 1000 Time (hr) Yi e ld  St re n gt h (M Pa ) 200°C 180°C 150°C  Figure 5.25: The evolution of yield strength during ageing of homogenised Al-1.3Si-0.32Mg. 78  5.3.2.3 Artificial Ageing Following Natural Ageing The artificial ageing response of the naturally aged Al-1.3Si-0.32Mg alloy was investigated between 150°C and 200°C.  Figure 5.26 shows the evolution of yield strength during ageing at 180°C after no natural ageing and after natural ageing for 24 hours.  Similarities in the material response of this alloy and the A356 and Al-11Si-0.22Mg alloys are again apparent.  The yield strength in the naturally aged condition is significantly higher than that of the as-quenched material (93 MPa vs. 51 MPa), the naturally aged material does not strengthen immediately during artificial ageing, and strengthens more slowly than the immediately artificially aged material, although the peak strength of the material in both conditions is similar (230 MPa).  50 70 90 110 130 150 170 190 210 230 250 0.1 1 10 100 Yi e ld  St re n gt h (M Pa ) Time (hr) No Natural Ageing 24hr Natural Age  Figure 5.26: The evolution of yield strength during artificial ageing of the Al-1.3Si-0.32Mg alloy at 180°C following a) no natural ageing, b) 24 hours natural ageing at room temperature.  Figure 5.27 compares the exothermic heat flow curves for artificial ageing of the immediately artificially aged and 24 hour naturally aged cases at 180ºC, and exothermic heat flow curves for tests performed on naturally aged specimens at 165°C, 180°C and 200°C are shown in Figure 79  5.28.  Again, the heat flow characteristics for all tests are similar to the results obtained for the A356 and Al-11Si-0.22Mg alloys, although the peak heat flows are smaller in this alloy.  0 0.0004 0.0008 0.0012 0.0016 0.002 0 1 2 3 4 Time (hr) H e a t F lo w  (W /g ) No Natural Ageing 24hr Natural Ageing  Figure 5.27: The exothermic heat flow curves for the Al-1.3Si-0.32Mg alloy artificially aged at 180°C following: i) no natural ageing and ii) 24 hours natural ageing.  0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0 2 4 6 8 10 Time (hr) H ea t F lo w  (W /g ) 165°C 180°C 200°C  Figure 5.28: Thermograms for isothermal calorimetry of Al-1.3Si-0.32Mg alloy artificially aged at 165°C, 180°C and 200°C following 24 hour natural ageing at room temperature. 80  5.4 Discussion of Experimental Results  The experimental results are analysed in detail in the following section.  First, the metallurgical phenomena occurring in the A356 alloy during solution treatment are identified and their influence on the yield strength of the solution treated alloy is discussed.  Secondly, a comparison of the ageing behaviour of the A356 alloy and the two model alloys is made, and an additive rule is used to quantify the contribution of the solid solution, solute clusters, precipitates and eutectic particles to the yield strength of the solution treated, naturally aged and peak aged alloys.  5.4.1 Metallurgical Behaviour of A356 During Solution Treatment It is evident from the results of the investigation into the solution treatment behaviour of the A356 alloy that two main processes occur in the material.  Firstly, magnesium rich phases that are unstable at solution treatment temperatures dissolve, leading to an increase in the solute content of the aluminum matrix, and secondly the insoluble eutectic silicon phase fragments and coarsens.  These effects are considered separately in the following sections.  5.4.1.1 Dissolution of Mg-rich Phases The dissolution of Mg-rich phases can be observed directly in the backscattered electron maps presented in Figure 5.3, which shows the Mg-rich phases in the interdendritic regions of the as- cast alloy.  Over the same time period, the magnesium content of the aluminum dendrites, measured by microprobe and presented in Figure 5.6, increases significantly as magnesium atoms diffuse from the dissolving particle in the interdendritic region into the dendrite.  The change in dendritic magnesium content during solution treatment at 540°C has been calculated from the microprobe results for the A356 and Al-11Si-0.22Mg alloys, and is shown in Figure 81  5.29.  No change is observed in the A356 alloy after 15 minutes following a rapid increase in magnesium content in the first minutes of solution treatment.  In comparison, the magnesium content of the Al-11Si-0.22Mg alloy continues to increase after 30 minutes at 540°C, suggesting that the Mg-rich phases in this alloy are larger than those in the A356 alloy and require longer time at temperature to dissolve completely.  0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.1 1 10 100 1000 Time (min) Av e ra ge  M g Co n te n t ( w t% ) A356 Al-11Si-0.22Mg  Figure 5.29: The average magnesium content in the A356 and Al-11Si-0.22Mg alloys during solution treatment at 540°C, calculated from microprobe data.   5.4.1.2 Eutectic silicon fragmentation and coarsening The behaviour of the silicon particles during solution treatment can be divided into two regimes; initial fragmentation of rods into spheroids, followed by coarsening of the fragmented particles. In order to determine the evolution of fragmented particles a criterion to describe a fragmented particle must be defined.  In this study, a limiting value of particle roundness was chosen to determine whether a silicon particle is fragmented.  The volume fraction of fragmented particles, ffrag, has been estimated at each solution treatment condition by calculating the equivalent 82  spherical volume of each particle based on its measured area and taking the sum of the total volume of particles that are within the fragmentation criterion, Vfrag, as a fraction of the sum of the volume of all silicon particles considered in the analysis, Vtotal, using Equation 5.1.  frag frag total Vf V =        (5.1)  A sensitivity analysis was carried out to find a suitable value for the roundness criterion, and the change in the volume fraction of fragmented particles is presented in Figure 5.30 for roundness criteria between 0.51 and 0.82.  0 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1 10 Time (hr) Fr ac tio n  Fr a gm e n te d 0.51 0.59 0.69 0.77 0.82 Roundness Limit  Figure 5.30: Results of a sensitivity analysis performed to determine an appropriate particle roundness limit as a criterion for fragmented eutectic silicon particles in the A356 alloy.  The sensitivity analysis shows that a tight criterion (i.e. only particles with roundness greater than 0.82) will not allow the fragmentation process to approach unity, whereas a loose criterion 83  (i.e. only particles greater than 0.51) gives unrealistic values for short solution treatment times, and a limiting particle roundness value of 0.69 was settled on as the determining factor for particle fragmentation.  Using this criterion, it can be seen in Figure 5.30 that the evolution of fragmented particles occurs slowly in the initial minutes of solution treatment, but proceeds rapidly after 15 minutes, reaching near-completion after 30 minutes at 540°C.  5.4.1.3 The relationship between solution treatment and yield strength The results of the analysis presented above suggest that the various metallurgical phenomena dominate the solution treatment process over different time periods at 540°C.  Figure 5.31 shows the evolution of the as-quenched yield strength with solution treatment time at 540°C, indicating which phenomenon is dominant during the process.  50 60 70 80 90 100 1 10 100 1000 10000 Time (min) Yi e ld  St re n gt h (M Pa ) Dissolution of Soluble Phases Fragmentation of Eutectic Silicon Coarsening of Eutectic Silicon AC Yi e ld  St re n gt h (M Pa ) Yi e ld  St re n gt h (M Pa )  Figure 5.31: The evolution of yield strength from the as-cast (AC) condition during solution treatment at 540°C, showing the time periods in which each metallurgical process occurs.  84  In general, the change in as-quenched yield strength is small and no strong effects can be observed between it and the changes in microstructure during solution treatment.  The largest change in yield strength occurs at the beginning of solution treatment when fine precipitates dissolve during the initial heat up of the as-cast alloy, resulting in a decrease from 98MPa to 78MPa.  Subsequent dissolution of coarse interdendritic Mg-rich particles, and the resulting increase in solid solution content of magnesium does not have a measurable effect on the yield strength.  The fragmentation of eutectic silicon, which can be seen directly in Figure 5.2 b-d), has a weak effect as the yield strength decreases from 77MPa after 15 minutes solution treatment to 72MPa after 30 minutes at 540°C.  A further decrease in yield strength from 72MPa to 67MPa is also observed due to particle coarsening, shown in Figure 5.2 d) and e), and reported in Figure 5.5 as an increase in average particle diameter from 1.64µm after 30 minutes of solution treatment, to 2.62µm after 240 minutes.  5.4.2 Comparison of Ageing Behaviour of A356 and Model Alloys A comparison of the evolution of yield strength in the underaged condition during artificial ageing at 180°C is presented in Figure 5.32 for the three alloys studied.  The yield strength of each alloy in the as-quenched condition differs due to variation in the solution treated microstructure of each alloy.  In order to determine the differences in the as-quenched alloy microstructure for the three alloys studied, a thermodynamic investigation using the FactSage software package was carried out to determine the phase content of each alloy at 500°C, 540°C and 560°C.  The results of this investigation are presented in Table 5.1.  The main difference in phase content between the alloys is the volume fraction of eutectic following solution treatment; the Al-1.3Si-0.32Mg alloy having no eutectic phase, and the A356 and Al-11Si-0.22Mg alloys containing approximately 6% and 10% respectively. 85    40 80 120 160 200 240 280 0.1 1 10 Time (hr) Yi e ld  St re n gt h (M Pa ) A356 (Al-7Si-0.3Mg) Al-1.3Si-0.32Mg Al-11.1Si-0.22Mg  Figure 5.32: The evolution of yield strength in the underaged condition during immediate artificial ageing of the three alloys studied at 180°C.   Table 5.1: FactSage-predicted equilibrium values for eutectic silicon and solute content in the three model alloys at the solution treatment temperatures investigated in the present work.  Solution Treatment Temperature (°C) Al-1.3Si-0.32Mg Al-7Si-0.3Mg Al-11Si-0.22Mg wt% Eutectic Silicon 500 0.46 6.19 10.36 540 0.17 5.90 10.05 560 0.00 5.71 9.86 wt% Silicon (Solute) 500 0.82 0.82 0.82 540 1.17 1.17 1.17 560 1.33 1.37 1.37 wt% Magnesium (Solute) 500 0.23 0.23 0.23 540 0.32 0.32 0.24 560 0.32 0.32 0.24  86  Figure 5.32 also shows that the increase in yield strength between the as-quenched and peak aged condition is different for the three alloys.  This is due to differences in the volume fraction of precipitates that form during artificial ageing, which is related to the solute level in the α-Al phase in each alloy following solution treatment.  FactSage predicted solute levels in the α-Al phase of each alloy are shown in Table 5.1.  The A356 and Al-1.3Si-0.32Mg alloys contain a similar amount of magnesium solute (0.32wt%), which explains their identical strength increase during artificial ageing (178MPa), whereas the magnesium solute content of the Al-11Si-0.22Mg alloy is lower (0.24wt%), and this is reflected in its smaller strengthening increment (138MPa).  Previous researchers have described the yield strength of an alloy as the sum of a number of contributing factors including the contribution due to precipitates and solid solution strengthening [Shercliff and Ashby (1990), Esmaeili et al (2003a, b)].  intys ppt ssσ σ σ σ= + +       (5.2)  where σint is the intrinsic strength of aluminum.  The contribution of solid solution strengthening can be isolated in the case of the Al-1.3Si-0.32Mg alloy by considering the yield strength in the as-quenched condition (i.e. intss aqσ σ σ= − ) assuming no precipitates are present and taking 10MPa as an estimate of the intrinsic strength [Wang et al 2003].  The same approach is not possible in the other alloys because the presence of the eutectic phase modifies Equation 5.2 to;  intys ppt ss eutecticσ σ σ σ σ= + + +           (5.3)  where σeutectic is the contribution of the eutectic phase to the total yield strength. 87  In the case of the A356 alloy, the contribution of solid solution strengthening is expected to be almost identical to that of the Al-1.3Si-0.32Mg alloy in the as-quenched condition.  Hence, the contribution of the eutectic phase in the A356 alloy in the as-quenched condition can be estimated from Equation 5.3, assuming no precipitates are present (i.e. inteutectic ys ssσ σ σ σ= − − ).  Further information is required to determine the relative contributions of the solid solution and eutectic phase towards the overall yield strength in the Al-11Si-0.22Mg alloy.  Kishi and Sakakibara (1984) have reported that the strength of an Al-1.3Si alloy is approximately 30MPa, so the contribution of the silicon addition can be estimated as 20MPa when the intrinsic strength, σint,  is subtracted.  Furthermore, the total contribution due to solid solution strengthening, σss is a function of the average concentration of all solute atoms in the matrix. An approximate relationship between the contribution due to solid solution strengthening and the average concentration of solute atoms has been shown previously [Nabarro (1967), Labusch (1970), Shercliff and Ashby (1990)] to be of the form;  2/3 ss soluteaCσ =       (5.5)  where Csolute is the total solute concentration and a is a constant value.  Hence, an estimate for σss for the as-quenched Al-11Si-0.22Mg alloy can be found, assuming that the maximum solid solubility of silicon at the solution treatment temperature is identical for all three alloys, and that all magnesium is in solid solution in the as-quenched condition (these assumptions are supported by the thermodynamic data presented in Table 5.1).  The calculated value for  σss can then be substituted into Equation 5.4 to obtain an estimate for the eutectic contribution, σeutectic. 88  In the peak aged condition the precipitation hardening contribution can be determined, assuming negligible contributions from solid solution and no change in the contribution from the eutectic phase (i.e. t intppt peak euσ σ σ σ= − − ).  This assumption is valid if most of the solutes are precipitated out which is the case in the present analysis.  A similar approach can be used to quantify the effect of natural ageing on the strength of the alloys.  The natural ageing curves for the three alloys studied are shown in Figure 5.33.  In the case of natural ageing, the strengthening increment is related to the formation of solute clusters, which act as weak obstacles rather than the stronger precipitates that form during artificial ageing.  A modified equation to Equation 5.3 has been used to determine the contribution of solute clustering assuming all of the solute atoms come out of solution to form the clusters.  intys clusters ss eutecticσ σ σ σ σ= + + +     (5.6)  The results of the calculations for the three alloys studied in the as-quenched, natural aged and peak-aged conditions are presented in Table 5.2.  89  30 50 70 90 110 130 0.1 1 10 100 Time (hr) Yi e ld  St re n gt h (M Pa ) A356 (Al-7Si-0.3Mg) Al-1.3Si-0.32Mg Al-11.1Si-0.22Mg  Figure 5.33: The evolution of yield strength during the first 24 hours natural ageing at room temperature for the three alloys studied.  Table 5.2: Summary of the yield strength contributions from solid solution strengthening, solute clusters, precipitates and eutectic particles for the three alloys studied in the as-quenched, naturally aged and peak aged conditions.    σss (MPa)  σclus (MPa)  σppt (MPa)  σeut (MPa)  σI (MPa)  σtot (MPa) Al-1.3Si-0.32Mg AQ 42 0 0 0 10 52 NA 0 90 0 0 10 100 AA 0 0 220 0 10 230 Al-7Si-0.3Mg AQ 42 0 0 20 10 72 NA 0 90 0 20 10 120 AA 0 0 220 20 10 250 Al-11Si-0.22Mg AQ 40 0 0 32 10 82 NA 0 78 0 32 10 120 AA 0 0 168 32 10 210    90  5.5 Concluding remarks  The heat treatment of A356 alloy result in a wide range of microstructure conditions as dissolution of soluble particles and morphological changes of insoluble particles occur during solution treatment, followed by the formation of precipitates and solute clusters at the expense of the solute content of the alloy during natural and artificial ageing.  In the next chapter, microstructure models based on the internal state variable approach are developed for the solution treatment and ageing processes based on the knowledge acquired in this chapter.    91 CHAPTER 6 - Microstructure-Property Modelling  The work presented in this chapter aims to develop processing-structure-property relationships for Al-Si-Mg alloys using physical principles.  The objective is to develop a model that can predict the microstructure and yield strength of the alloy during heat treatment based on its thermal history.  The overall approach towards the model development is introduced in Chapter 6.1, and details of the solution treatment and ageing models are presented in Chapter 6.2 and 6.3. In Chapter 6.4, the range of applicability of the model has been examined and Chapter 6.5 tests the sensitivity of the models to some of the physical constants used.  Chapter 6.6 assesses how successful the model is when extended to predict the heat treatment behaviour of Al-Si-Mg alloys with differing alloy compositions to the A356 alloy, and Chapter 6.7 outlines some model predictions to aid the optimization of industrial heat treatment operations.  6.1 Overall Approach  The overall modelling effort has been divided into two sections, as a model for the prediction of microstructure and yield strength are sought for i) solution treatment and ii) ageing processes. These models have been developed using the internal state variable approach, which relates the evolution of material properties to the process variables of temperature and time via the changes in certain microstructure characteristics.  This modelling approach was initially proposed by Richmond (1986) for deformation of aluminum alloys, and has since been applied to a wide range of other industrial processes involving a wide range of solid-state diffusion processes, as well as solidification and grain growth [Grong and Shercliff (2002)].   92 The first step in the development of the process-microstructure-property model is to identify relevant microstructure features that can be used as internal state variables to describe the metallurgical processes occurring in the alloy as shown in Equations 6.1.  1 2( , ,...)ys f S Sσ =      (6.1)  Following identification of the relevant internal state variables, a series of mathematical equations must be developed in order to track the evolution of these variables as a function of time at temperature.   It is typical to describe these variables in terms of their differential variation with time considering that each state variable may be a function of temperature as well as other state variables.  1 1 2( , , ,...) dS f T S S dt =      (6.2a) 2 1 2( , , ,...) dS f T S S dt =      (6.2b)  In the present work, evolution laws are used to determine the changes in the eutectic particle morphology and dissolution of Mg-rich phases during solution treatment, and the precipitation and dissolution of solute clusters and precipitates during ageing.  The final step in the development of an internal state variable model is to construct appropriate material response equations to connect the output of the microstructure evolution equations.  In the present approach the evolution of internal state variables is linked to the change in yield strength during solution treatment and ageing as a result of variation in solute content, and the  93 characteristics of insoluble second phase particles, precipitates and solute clusters.  The model is formulated for non-isothermal cases, and all figures and predictions following in this chapter include a heating period defined by a characteristic heating rate specified in the text.  As stated previously, the overall objective of this work is to develop a model for predicting the evolution of yield strength in an Al-Si-Mg alloy during the ageing process that can account for changes arising from different thermal processing histories.  The desired accuracy for this model is in the region of +/-10%, and R2 values† calculated from observed-predicted plots have been used to assess the goodness-of-fit between the model predictions and experimental data. Consistency of the physical constants and parameters used in the model is deemed critical, and as a result some deviation between model predictions and experimental data may be tolerated in some of the sub-models provided the overall objective is achieved.  6.2 Solution Treatment  The internal state variables used to describe the three important metallurgical processes occurring during solution treatment are shown in Table 6.1.  It is important to note that all internal state variables listed are physical quantities that can be directly measured.  The development of models to predict the evolution of the microstructure features for each process is discussed separately for dissolution, fragmentation and coarsening in the following sub-sections, and the construction of the appropriate material response equation for the evolution of yield strength during solution treatment is presented in the final subsection.  †  R2 is a commonly used statistic to evaluate model fit, typically ranging from 0 (poor fit) to 1 (perfect fit).  The R2 value is calculated from the sum of the squares of the distances between the experimental data and model predicted values, normalised to the sum of the square of the distances between the experimental data and a horizontal line through the experimental mean value.  [Montgomery (2000)].  94 Table 6.1: List of microstructure variables and internal state variables for the metallurgical processes occurring during solution treatment. Microstructural Process Microstructure Variables Internal State Variables Dissolution of Mg2Si Dendrite Arm Spacing Dendritic Magnesium Content Mg2Si Particle Radius, rMg2Si Fragmentation of Eutectic Si Initial Silicon Rod Diameter Fraction Fragmented, ffrag Coarsening of Eutectic Si Initial Silicon Rod Diameter Si Particle Radius, rSi  6.2.1 Dissolution of Mg2Si The dissolution of Mg2Si particles has been modelled under the assumption that it is controlled by the diffusion of magnesium atoms in the α-Al matrix.  The model is based on the dissolution of a single spherical particle of radius, r, in an infinite matrix, following the well-established approach by Whelan (1969):  dr kD Dk dt r tpi = − −       (6.3)  where k=(C0-Ci)/(Cp-C0) as depicted in Figure 6.1.  The 1/r term on the right hand side of Equation 6.3 arises from the steady-state diffusion field, whereas the 1/t1/2 term originates from the transient part.  In order to describe the evolution of the particle radius, r, in terms of its differential variation with time, the steady-state term has been isolated and the transient diffusion field has been neglected.    As a result, Equation 6.3 is simplified to;  dr kD dt r = −       (6.4)  95 Particle CMg Distance Mg-rich Particle Mg-depleted Dendrite C0 Matrix Diffusion of Solute Ci Cp  Figure 6.1: Schematic diagram showing the solute concentration around a dissolving particle.  In calculating k in Equation 6.4, the microprobe measured magnesium content at the centre of the as-cast dendrite is substituted for the solute concentration at the infinite point, Ci. Furthermore, the temperature dependences of C0 and D have been described using linear and Arrhenius relationships respectively.  The dissolving particle is assumed to have Mg2Si stoichiometry, and the initial particle size has been estimated using a mass balance approach in which the amount of magnesium contained in the particle is equated to the difference in the mass of magnesium measured in the as-cast and fully solution treated dendrite by the microprobe.  The calculated mass of magnesium is then converted into the radius of the Mg2Si particle assuming it is spherical and fully dense, with a density of 1.9kg/m3 [Mondolfo (1976)].  The magnesium content of the eutectic region is assumed to be uniform at the bulk concentration and unchanging during solution treatment.  For simplicity it is also assumed that no silicon or Fe-rich particles exist in the system.  Thus, the compositional schematic in Figure 6.1 is modified to reflect these changes and assumptions in Figure 6.2.  A list of the constants used is presented in Table 6.2.  96 α-Al Dendrite α-Al DendriteMg2Si Particle Al/Si Eutectic CMg Distance Mg-rich Particle Mg-depleted Dendrite C0  Figure 6.2: Schematic diagram of the solute and eutectic content around the dissolving particle.  Table 6.2: Physical constants and values used in the dissolution model Parameter Value Secondary Dendrite Arm Spacing  (µm) 30 Mg Concentration at Dendrite Core, Ci (wt%Mg) 0.16 Mg Concentration in Mg2Si Particle, Cp  (wt%Mg) 0.6336 Density of Mg2Si, ρMg2Si  (kg/m3) 1.9 Initial Mg2Si Particle Radius, 0r  (µm) 2.36 Activation Energy, Q (kJ/mol) 175 Diffusion Coefficient, D0 1.2x10-3   97 The model calculates the rate of change in particle radius with time and hence the evolution of particle size can be determined.  The ‘cell concept” introduced in Chapter 2.4 can be used to calculate the relative volume fraction of the Mg-rich particle as follows [Nolfi et al. (1969)];  3 3 cell rf r =       (6.5)  The relative volume fraction of the particle, fr, during dissolution can be calculated using a similar approach:  3 3 0 0 r f rf f r= =        (6.6)   where the instantaneous particle radius, r, is predicted at each time step.   The model predictions have been assessed in Figure 6.3 by comparing them against estimates of the fraction dissolved obtained from microprobe data of specimens in the partially solution treated condition.  The volume-averaged dendritic magnesium content is calculated from the microprobe data according to the approach presented in Figure 6.2, assuming the measured values from the microprobe are taken from a section that passes through the centre of a cylindrical dendrite, thereby capturing the minimum magnesium content at the dendrite core. Experimental estimations of the fraction dissolved can then be made by calculating the relative increase in magnesium content of the dendrite between the minimum and maximum cases (i.e. the as-cast and fully solution treated specimens) and assuming that the relative fraction increase in dendritic magnesium content is equal to the volume fraction of particle dissolved.  98  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 1000 10000 100000 Re la tiv e Vo lu m e Fr ac tio n Time (s) 500°C 540°C 560°C  Figure 6.3: The evolution of the relative volume fraction of an Mg2Si particle during solution treatment at 500°C, 540°C and 560°C.   The symbols and lines represent measured data and model predictions respectively.  The model predictions fit the experimental data very well during solution treatment, although they slightly under-predict the rate of dissolution at 560°C.  The goodness-of-fit of the dissolution model has been assessed by calculating R2 values for each temperature studied, shown in the diagnostic plots in Figure 6.4.  The R2 value for the model at 500°C, 540°C, and 560°C are all greater than 0.9, indicating the good fit.  These results are important with regard to the overall model because the predictions for the relative volume fraction dissolved are used in the ageing model to assist in calculating the amount of solute available for precipitation and the size of the strengthening increment due to precipitation.  99 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fr ac tio n  Di ss o lve d (P re di ct ed )       Fraction Dissolved (Experimental) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fr ac tio n  Di ss o lve d (P re di ct ed )       Fraction Dissolved (Experimental) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fr ac tio n  Di ss ol ve d (P re di ct ed )       Fraction Dissolved (Experimental) 500ºC R2 = 0.98 540ºC R2 = 0.95 560ºC R2 = 0.92 a) b) c)  Figure 6.4: Experimental data vs model predicted values at: a) 500°C, b) 540°C and c) 560°C.  6.2.2 Morphological Change of Eutectic Silicon  6.2.2.1 Fragmentation of Silicon Particles The isothermal silicon fragmentation kinetics were quantified using the evolution of the volume fraction of fragmented silicon with time.  A common expression used to mathematically describe the evolution of a volume fraction transformed with time at temperature (i.e. the volume fraction of fragmented particles, ffrag) is the JMAK equation;  1 exp( )nfragf kt= − −      (6.7)  100  which in its differential form is given as;  1 1 ( ln(1 )) (1 ) n frag n n frag frag df nk f f dt − = − − −     (6.8)  where k is a kinetic parameter and n is a time exponent.  Values for k and n have been determined empirically by performing experiments under isothermal conditions and doing a best fit to the equation.  The temperature dependency of k can be described by an Arrhenius type relationship shown in Figure 6.5 and Equation 6.9.  0 exp Qk k RT −  =          (6.9)  where k0 is a pre-exponential factor and Q is an activation energy.  The volume fraction fragmented was calculated from image analysis data from deep etched specimens as described in the previous chapter.  The criterion for a fragmented particle was defined as a minimum particle circularity of 0.69, and the relative volume fraction of fragmented particles was calculated from total spherical volume of the fragmented particles as a fraction of the total spherical volume of all particles considered in the analysis.  frag frag total Vf V =      (6.10)   101 The JMAK equation was fitted to image analysis data from specimens tested at 500°C, 540°C and 560°C by finding values for the three adjustable parameters 0fragk , Qfrag and n, shown in Table 6.3.  -9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 -5 0.00115 0.0012 0.00125 0.0013 0.00135 Temperature-1 (1/K) ln  k (k in  s- 1 )  Figure 6.5: The Arrhenius relationship between the JMAK constant k and temperature.    Table 6.3: List of adjustable parameters in the fragmentation model and their values. Parameter Value Rate coefficient, 0 fragk  (s-1) 7.5 x 106 Activation energy for fragmentation, Qfrag (kJ/mol) 150 Avrami exponent, n 1  Model predictions are presented in Figure 6.6 for solution treatment at 500°C, 540°C and 560°C, compared with the results of the experimental image analysis results.  There is good agreement between model predictions and experimental data over the range of temperatures studied.   R2  102 values were calculated by the same method as previously and were 0.97, 0.88 and 0.96 for fragmentation at 500°C, 540°C and 560°C, respectively.  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 1000 10000 100000 Fr a c tio n  Fr a gm e n te d Time (s) 500°C 540°C 560°C  Figure 6.6: Model predictions and experimental data for fragmentation of the eutectic silicon phase during solution treatment at 500°C, 540°C and 560°C. The symbols and lines represent the measured data and model predictions respectively.  An accurate estimate for the radius of the spheroidal particles immediately following fragmentation is necessary for successful modelling of the subsequent coarsening of these particles, as presented in the following section (i.e. Chapter 6.2.2.1.).  The approach taken in the present work to predict the initial fragmented particle radius is based on an analysis of the Plateau-Rayleigh instability [Rayleigh (1879)], introduced in Chapter 2.4 and involving the decomposition of a rod into a series of spheres, driven by a reduction in interfacial energy and initiated by a perturbation in the interface.  This process is illustrated schematically in Figure 6.7.   103 λ rsphere rrod Vcylinder = Vsphere  Figure 6.7: Schematic representation of the fragmentation process showing the geometries of a rod containing a perturbation and a sphere that is of equal volume to the perturbed rod.  Rayleigh’s analysis shows that the perturbations that cause the breakdown of the cylinder have a characteristic wavelength, λ, corresponding to the fastest growth rate which may be defined by Equation 6.11, in which rrod is the radius of the rod.  2 2 rodrλ pi=        (6.11)  In the present work, the fragmented silicon particle radius is calculated using the relationship given in Equation 6.11, assuming the as-cast rods and fragmented particles are perfect cylinders and spheres, and that conservation of volume laws are observed.  For a rod thickness of 0.636µm, the maximum decomposition rate occurs at a characteristic wavelength, λ, of 2.798µm which results in a mean fragmented silicon particle radius of 0.601µm.     104 6.2.2.2 Coarsening of Spheroidal Silicon Particles The growth kinetics of the spheroidal eutectic silicon particles during coarsening after fragmentation follow an r3 versus time law, where r is the particle radius, commonly known as cubic coarsening behaviour [Martin (1980)].  The governing differential equation used to model the progress of coarsening with time is;  2 dr k dt r =       (6.12)  Where r is the instantaneous particle radius and k is a kinetic constant that is dependent on the initial particle radius and temperature.  A suitable value for time at the beginning of the coarsening process (i.e. the time at which r = r0) must also be determined.  In the present approach the start time for coarsening has been taken as the time at which half of the total volume of eutectic silicon has become fragmented (ffrag = 0.5). While it should be noted that not all silicon particles will have fragmented and become spheroidal at this point, the silicon particle characteristic distributions presented in Figure 5.4 indicates that the majority of particles are small, with only a few, much larger, particles present throughout fragmentation.  Hence, the number fraction of fragmented particles is expected to be higher than 0.5 when the relative volume fraction of fragmented particles equals 0.5.  The temperature dependency of the coarsening rate coefficient is determined using an Arrhenius relationship of the form shown in Equation 6.9, and values for the activation energy for diffusion of silicon in aluminum, QSi and 0ck  were used as shown in Table 6.4.   The results of the coarsening model are shown in Figure 6.8, compared to the experimental measurements of  105 solution treated A356 alloy specimens presented in Chapter 5.1.2.  There is close agreement in the early stages of coarsening, which suggests that the estimates for the coarsening start time and initial particle radius are reasonable.  The fragmentation and coarsening models predict complete fragmentation of silicon rods into spheres with an average radius of 0.601µm after 10 minutes at 540°C, with subsequent coarsening to an average particle radius of 0.655µm after 15 minutes at 540°C.  This is an under-prediction of 5% when compared to the experimentally measured mean particle size of 0.690µm, and is within the 95% confidence interval (+/-0.06µm) for this data point.  Table 6.4: List of adjustable parameters in the coarsening model and their values. Parameter Value Coarsening rate coefficient, 0 ck  (s-1) 4.5x10-15 Activation energy for coarsening, QSi (kJ/mol) 136  0.1 1 10 100 1000 10000 100000 1000000 r (µm ) Time (s) 500°C 540°C 560°C  Figure 6.8: Coarsening model predictions (lines) compared with experimental data (symbols) for solution treatment temperatures at 500°C, 540°C and 560°C.  106 6.2.3 Solution Treatment Model Results and Summary  Each of the models presented in the previous sections have been combined and run simultaneously for a number of non-isothermal cases.  Initially, the model fit can be assessed by comparing the model predictions with experimental measurements for solution treatment at 540°C in a nitrate salt bath (equivalent to a heating rate of 40°C/s).  The predictions are shown in Figure 6.9, and indicate that the model fit is good, with dissolution of the Mg2Si particles progressing rapidly to completion within 10 minutes, while the fragmentation process is slightly slower and finishes within 15 minutes.  These observations are in general agreement with previous work on dissolution and fragmentation during solution treatment [Closset et al (1986), Shivkumar et al. (1989), Apelian et al. (1990), Zhang et al. (2002)].  0.1 1 10 0 0.2 0.4 0.6 0.8 1 1 10 100 1000 10000 100000 r (µm ) Re la tiv e Vo lu m e Fr ac tio n ,  f r an d f fra g Time (sec) ffrag fr r  Figure 6.9: Process model predictions (lines) and experimental results (symbols) for solution treatment at 540°C.  107 A microstructure model for solution treatment of A356 alloy has been developed based on a series of sub-models that predict the evolution of solute content and second phase particle characteristics throughout the process.  The change in microstructure during solution treatment is important because it influences the evolution of strength during ageing.  In contrast, the as- quenched yield strength results presented in Chapter 5.1.4 and discussed in Chapter 5.4.1.3 were found to be relatively insensitive to solution treatment.  As a consequence, the modelling work presented thus far has focussed on the successful prediction of microstructure evolution during solution treatment rather than the development of a yield strength model, and the overall measure of success will be determined by the accuracy of the microstructure model at predicting the strengthening behaviour of the alloy during subsequent ageing processes.  6.3 Artificial Ageing  The yield strength of a precipitation-hardened alloy contains contributions from the various strengthening mechanisms, including precipitation hardening, solid solution strengthening, and strengthening due to large second phase particles, as well as the intrinsic strength of the alloy. Each of these contributions is considered below.  The contribution due to precipitation is related to the microstructure variables that describe the precipitates in the alloy.  (i.e. radius, volume fraction, maximum interaction force between dislocation and precipitate, average spacing between precipitates, and the precipitate shape and orientation in the matrix.)  It has been shown previously that only the volume fraction of precipitates, f, and average precipitate radius, r, need to be considered to evaluate strengthening due to precipitates [Shercliff et al. (1992)].   108 The contribution due to solid solution strengthening, σss is a function of the average concentration of solute atoms in the matrix, Cx. An approximate relationship has been shown previously [Nabarro (1967), Labusch (1970), Shercliff and Ashby (1990)] to be of the form;  2/3 ss taCσ =       (6.13)  where Ct is the instantaneous solute concentration at a given artificial ageing time, t.  The change in the contribution due to solid solution strengthening can be related to the precipitation state of the alloy because solute is removed from the matrix as precipitation occurs and the strength contribution due to the solute decreases [Shercliff and Ashby (1990), Esmaeili et al. (2003a)].  2/3 0 (1 )ss ss rfσ σ α= −      (6.14)  The contribution towards the yield strength arising from the second phase particle content includes the presence of the eutectic phase.  This contribution can be estimated based on its dependency on the volume fraction of eutectic phase in the material.  Using values of 30MPa and 70MPa for the yield strength of Al-1.3wt%Si and Al-12.6wt%Si binary alloys [Kishi and Sakakibara (1984), Guiglionda and Poole (2002)], which correspond to feutectic values of 0 and 1 respectively, a simple linear relationship between σeutectic and feutectic can be derived as follows;   eutectic eutectic eutecticc fσ =     (6.15)  where the constant factor for the eutectic strengthening contribution to the total yield strength, ceutectic, is 40MPa.  109 The intrinsic strength of the alloy is the strength of the aluminum matrix, including the contribution due to Fe-rich intermetallics and any inclusions in the material. The grain size effect on strengthening has also been included in the intrinsic strength, and this is assumed to be a constant, small value.  The cast nature of the material results in significant amounts of porosity and in order to address the effect this has on the yield strength it was assumed that the amount of porosity and its strength contribution are both constant and included in the intrinsic strength.  After each of the strength contributions are calculated the total yield strength is obtained at each time step of the model using the appropriate additive rule introduced in the previous chapter.  intys ppt ss eutecticσ σ σ σ σ= + + +     (6.16)  The evolution of the volume fraction of precipitates during ageing has been determined from data obtained during isothermal calorimetry tests.  As discussed previously here and in other process models for yield strength evolution during ageing [Esmaeili (2002)], this method only allows the calculation of the total volume fraction of all types of precipitates formed during ageing.  As a consequence, the complex nature of the precipitation process, which involves the formation of a series of transitional phases, cannot be addressed by the model.  6.3.1 Modelling of Precipitation Strengthening As discussed in Chapter 2, the response equation obtained from considering the interaction between a glide dislocation and a point obstacle is;  ppt MF bL σ =       (6.17)  110 where M is the Taylor factor and b is the magnitude of the burgers vector.  The obstacle strength, F, given by the maximum interaction force between the obstacle and dislocation, and obstacle spacing, L are microstructure variables that evolve with ageing time, and are dependent on the ageing temperature as well as other microstructure variables.  A critical step towards the prediction of precipitation strengthening is the accurate determination of the evolution laws for F and L, and this will be discussed in the following sections.   6.3.1.1 Obstacle Strength The obstacle strength is dependent on the strengthening mechanisms involved.  In the case of shearable particles, most strengthening mechanisms indicate there is proportionality between the size of the obstacle and its strength [Gerold (1979), Lloyd (1985), Ardell (1985)].  A linear relationship between obstacle strength and size is usually assumed for coherency, atomic order and stacking fault strengthening mechanisms [Hirsch and Kelly (1965), Gerold and Haberkorn (1966), Gerold and Hartman (1968), Brown and Ham (1971), Kocks et al (1975), Melander and Persson (1978), Gerold (1979), Ardell, (1985), Lee and Park (1998)].  In contrast, power-law and exponential relationships are proposed for the modulus strengthening mechanism [Nembach (1983), Russell and Brown (1972)].  The linear relationship has been used by Deschamps and Brechet (1999), Myhr et al (2001) and Esmaeili et al (2003a) in models for precipitation strengthening of Al-Mg-Zn, Al-Mg-Si and Al-Mg-Si-Cu alloys, whereas Gerold (1979) considered the power-law relationship for general situations.  Irrespective of the exact relationship, it is clear that the obstacle strength increases with obstacle size to a peak value, after which the obstacle becomes non-shearable and the obstacle strength decreases.    111 In the A356 alloy, the precipitates remain shearable until after the peak aged condition. Therefore, the coherency strengthening mechanism is likely to dominate in the underaged and peak-aged conditions when large coherency strains are expected to surround the precipitates, and the obstacle strength – obstacle size relationship can be approximated by making F linearly proportional to the average obstacle radius, r.  An estimate for F can be obtained by normalising the obstacle strength – obstacle size relationship about the peak-aged condition;  peak peak rF F r =       (6.18)  6.3.1.2 Obstacle Spacing In the A356 alloy the strengthening β’’ precipitates form as rods along the <100> directions of the Al matrix, while the closest-packed planes are {111}; thus the orientation relationship between the precipitate and glide dislocation ensures the distribution of obstacles can be described as a triangular array.  Consequently the average obstacle spacing is calculated as;  2 3 A L N =       (6.19)  The obstacle spacing can be related to the volume fraction, f, as follows [Nie (1996)];  1 22L rf pi  =           (6.20)    112 6.3.1.3 Estimation of Precipitate Volume Fraction As precipitation is a nucleation and growth process, the JMAK equation, given previously in equation 6.8 can be used to model the evolution of the relative volume fraction of precipitates;  1 exp( )n r f kt= − −      (6.21)  If precipitation is complete when the material is in the peak-aged condition, it is possible to state;  r peak ff f=       (6.22)  In Equation 6.22, fpeak is the volume fraction of precipitates at the peak-aged condition.  The value of fr can be related to the exothermic heat flow curves obtained from the isothermal calorimetry experiments.  If the time to reach the peak-age condition corresponds to the end of the precipitation process, then;  0 0 0 peak t t r t ppt dQ dQdt dt dt dtf AdQ dt dt = = ∫ ∫ ∫      (6.23)  where dQ dt  is the rate of heat release derived from the heat flow curves and Appt is the total area under the heat flow curve.   113 An adjustment is required for the thermal instability at the beginning of the calorimetry test, during which time no data is collected.  This adjustment involves shifting the zero time to the end of the instability prior to the fitting procedure so that the evolution of the volume fraction of precipitates is calculated after the thermal instability has passed.  The JMAK coefficients are calculated in the usual manner, by calculating the term ln ln (1/(1-fr)) from the experimental data and plotting against ln t, and are shown in Figure 6.10.  The calculated JMAK parameters are shown in Table 6.5, and a comparison of experimental and predicted evolution of relative volume fraction of precipitates in Figure 6.11 demonstrates the good fit at 150°C and 180°C. The differences between experimental data and model predictions at 200°C are related to the duration of the thermal instability at the beginning of the isothermal calorimeter test, in which a small proportion of the total heat evolved at this temperature is not recorded.  Values obtained for the JMAK exponent, n, are all close to 1, which is consistent with theories of diffusion controlled growth of needle/lath shape precipitates [Christian (1975), Doherty (1996)].  -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) -4 -3 -2 -1 0 1 2 3 0 2 4 6 8 10 ln  ln  (1/ (1- Vf ) ln t (t in s) 150ºC 180ºC 200ºC  Figure 6.10: ln ln (1/(1-fr)) vs. ln t for the range of volume fraction between 0.05 and 0.95.  114  Table 6.5: Calculated Avrami coefficients for the JMAK equation used to describe the evolution of relative volume fraction of precipitates during immediate artificial ageing of the A356 alloy Temperature (°C) k (s-1) n 150 0.000225 1 180 0.00065 1 200 0.0016 1   0 0.2 0.4 0.6 0.8 1 0.01 0.1 1 10 Fr ac tio n He at  Ev ol ve d Time (hr) 150ºC 180ºC 200ºC  Figure 6.11: A comparison of the experimental data (solid lines) and model predictions (dashed lines) for the evolution of the relative volume fraction of precipitates in the A356 alloy during immediate artificial ageing.   115 Figure 6.12 shows the Arrhenius plot used to derive the kinetic parameters 0 pptk  and Q for the artificial ageing model, which results in an activation energy, Q, of 64kJ/mol and proportionality constant, 0 pptk , of 19000s-1.   It should be noted that the activation energy is the sum of positive and negative energy values for nucleation and growth mechanisms operating during precipitation rather than the activation energy of the precipitation process itself [Berkenpas et al. (1986), Luo et al. (1993)].  The values reported here are close to literature data for other Al-Mg-Si alloys (e.g. Esmaeili (2002) reported an activation energy of 58kJ/mol for precipitation in an AA6111 alloy).  -10 -9 -8 -7 -6 -5 -4 0.0021 0.0022 0.0023 0.0024 ln  k (k in  s- 1 ) 1/T (K-1) Q = 64kJ/mol k0 = 19040s -1  Figure 6.12: The JMAK coefficient k has an Arrhenius relationship with temperature.  6.3.1.4 Calculating σppt In the case of strong obstacles, σppt can be calculated by substituting the equations for F and L into the Equation 6.24.    116 ( ) 1/2 1 1/2 2 1/22 peak peak ppt r ppt r peak MF f f c f br σ pi = =     (6.24)  Equation 6.24 shows that σppt is dependent only on the precipitate volume fraction. Similar behaviour is found for coherency strengthening from spherical precipitates, order strengthening due to strong obstacles, and modulus strengthening [Ardell (1985)].  6.3.2 Calibration of the Model The model was calibrated by assuming the peak-aged values for the average precipitate radius, rpeak, volume fraction of precipitates, fpeak and obstacle strength, Fpeak, are independent of temperature over the range studied (150-220°C).  These assumptions are justified on the basis of the constant total heat evolved during isothermal calorimetry tests and constant peak strength measured during tensile testing for this temperature range.  The calibration parameters are presented Table 6.6.  Table 6.6: List of calibration parameters for the A356 alloy artificial ageing model Parameter Value As-Quenched Yield Strength, σaq  (MPa) 72 Intrinsic Strength of α-Al, σint (MPa) 10 Initial Contribution to Strength due to Solute, σ0ss  (MPa) 42 Coefficient of Precipitation Strengthening cppt  (MPa) 220 Precipitation Rate Coefficient, 0 pptk  (s-1) 19000 Activation Energy, Qppt (kJ/mol) 64   117 6.3.3 Model Results Figure 6.13 shows the modelling results and experimental data for the ageing process at 150°C, 180°C and 200°C.  The initial heat ramp at the beginning of artificial ageing has been simulated in the model by applying a constant heating rate of 5°C/s between the initial temperature (20°C) and the artificial ageing temperature.  The value for the heating rate was chosen to approximate the experimental conditions (i.e. the heating rate resulting from insertion into an oil bath).  The model predictions show very good agreement with the experimental data, with the largest error appearing in the early stages of artificial ageing.  R2 values were calculated to be greater than 0.9 for all three temperature conditions.  Validation tests have been done at 220°C and 190°C, and these are shown in Figure 6.14.  Again the good agreement between the model and data is apparent (R2 = 0.8 for both cases).  The ability of the ageing model to handle non-isothermal conditions is apparent in the validation test at 220°C, because the heating ramp comprises a significant proportion of the total artificial ageing time and is expected to have a significant influence over the evolution of yield strength at the shorter ageing times considered in this case.   118 0 50 100 150 200 250 0.01 0.1 1 10 100 Yi el d St re n gt h Co n tri bu tio n  (M Pa ) Time (hr) 0 50 100 150 200 250 0.01 0.1 1 10 Yi el d St re n gt h Co n tri bu tio n  (M Pa ) Time (hr) 0 50 100 150 200 250 0.01 0.1 1 10 Yi el d St re n gt h Co n tri bu tio n  (M Pa ) Time (hr) σexpt σtot σppt σss σeut + σint σexpt σtot σppt σss σeut + σint σexpt σtot σppt σss σeut + σint  Figure 6.13: Comparison of model predictions and experimental data for the evolution of yield strength during immediate artificial ageing of the A356 alloy at: a) 150°C, b) 180°C & c) 200°C. A heat ramp of 5°C/s from 20°C to the soak temperature is included at the start of the model.   119  0 50 100 150 200 250 0.01 0.1 1 10 Yi el d St re n gt h Co n tri bu tio n (M Pa ) Time (hr) 0 50 100 150 200 250 0.01 0.1 1 10 Yi el d St re n gt h Co n tri bu tio n (M Pa ) Time (hr) σexpt σtot σppt σsss σeut + σint σexpt σtot σppt σsss σeut + σint a) b)  Figure 6.14: Comparison of model predictions and experimental data for the evolution of yield strength during immediate artificial ageing of the A356 alloy at a) 190°C, b) 220°C. A heat ramp of 5°C/s from 20°C to the soak temperature is included at the start of the model.        120 6.4 Natural Ageing  The yield strength of a naturally ageing alloy can be modelled using the same approach as the precipitation hardening case described above, with a few important modifications.  During natural ageing, precipitation does not occur; rather the solute atoms form clusters that increase the strength of the alloy by acting as obstacles in a similar manner to the precipitates.  As such, the additive equation can be modified from Equation 6.16 to reflect this change;  intys clusters ss eutecticσ σ σ σ σ= + + +     (6.25)  In the approach to the natural ageing model, the clusters formed during natural ageing are assumed to be weak obstacles to dislocation motion, and the governing equation is given as [Esmaeili et al. (2003b)];  3/2 1/2 1 1/2 1/2 1/2 2 1/2 1/2 3/2(2 3 ) peak peak clusters r clusters r peak MF f r f c r f b r σ pi = = Γ    (6.26)  The contribution to strengthening due to clusters is dependent on the radius of the clusters as well as the volume fraction.  This result is in agreement with previous analyses for coherency strengthening of very small particles, as well as atomic order and stacking fault strengthening mechanisms [Brown and Ham (1971), Ardell (1985), Embury et al. (1989)].  However, recent work by Serizawa et al. (2008) indicates that although the yield strength increases with natural ageing time, the energy required for the dislocation to cut the solute cluster remains the same. This observation, despite being indirectly inferred from tensile tests is consistent with an  121 increasing density of solute clusters of similar radius.  Thus, the dependence of σclusters on the cluster radius, r, in the present work has been eliminated, and Equation 6.26 simplified to;  1 2 cluster clusters rc fσ =      (6.27)  which is of the same form as Equation 6.24.  Calibration parameters for the natural ageing model are listed in Table 6.7.  A further modification to the model involves the proportion of solute that leaves solution to form clusters. In the artificial ageing model, all solute is assumed to precipitate from the matrix during ageing, whereas only a proportion of the available solute, given by the variable α in Equation 6.14 is assumed to form clusters during natural ageing.  3-D atom probe experiments carried out by Esmaeili et al. (2007) on naturally aged AA6111 specimens indicate that only a small fraction of the initial solute concentration is depleted in the matrix during natural ageing and as a consequence the value of α used in the present work is 0.1.  The JMAK equation was used to model the evolution of exothermic heat obtained experimentally from calorimetry experiments at 25°C, 40°C and 60°C, and an Arrhenius relationship was used to obtain the kinetic parameters 0 clustersk  and Qclusters for the clustering process, which are given in Table 6.6.  Figure 6.15 shows the heat evolution experiment and JMAK model prediction for natural ageing at 25°C, 40°C and 60°C, indicating that there is close agreement between model and experiment.    122 Table 6.7: List of calibration parameters for the A356 alloy natural ageing model. Parameter Value As-Quenched Yield Strength, σaq  (MPa) 72 Intrinsic Strength of α-Al, σint (MPa) 10 Initial Contribution to Strength due to Solute, σ0ss  (MPa) 42 Solute Clustering Coefficient, cclusters  (Nm-5/2) 63 Kinetic Rate Coefficient, 0 clustersk  (sec-1) 40 Activation Energy, Qclusters (kJ/mol) 24 Proportion of solute released, α 0.1  0 0.2 0.4 0.6 0.8 1 0.01 0.1 1 10 100 Fr a c tio n  H e a t F lo w tageing (hr) 25ºC 40ºC 60ºC  Figure 6.15: A comparison of the experimental data (thick lines) and model predictions (dashed lines) for the evolution of relative volume fraction of precipitates during natural ageing.  The natural ageing model is limited to the point at which the heat flow trace becomes indistinguishable from the reference test – in this case, around 24 hours.  At this point the model does not predict any further change in strength; however there is slow increase in strength with further natural ageing as the clusters coarsen.  Figure 6.16 shows the modelling results for  123 natural ageing at 20°C compared with the experimental data for natural ageing at room temperature.  The model results have a good agreement with the experimental data, with model predictions of the yield strength within 10% of the experimentally measured values from tensile tests.  0 50 100 150 0.01 0.1 1 10 100 Yi el d St re n gt h Co n tri bu tio n  (M Pa ) Time (hr) σexpt σtot σcluster σsss σeut + σint  Figure 6.16: Comparison of model predictions and experimental data for the evolution of yield strength during natural ageing of the A356 alloy at 20°C.           124 6.5 Artificial Ageing After Natural Ageing  The approach used in the previous sections to model strength evolution during immediate artificial ageing and natural ageing is applied here to model strengthening during artificial ageing that takes place following a period of natural ageing.  Necessary modifications have been made to the model to account for the presence of solute clusters at the beginning of artificial ageing that dissolve concurrently with precipitate formation during artificial ageing.  It should be noted that the model has been constructed based on a fixed naturally ageing period of 24 hours at room temperature, and as a consequence it is valid only for artificial ageing of a fully solution treated material that has been subjected to a 24 hour natural age at room temperature.  6.5.1 Estimation of fr The heat flow curve obtained from isothermal calorimetry of previously naturally aged materials shows the net heat flow occurring due to all reactions occurring in the material during artificial ageing.  These reactions include cluster dissolution (endothermic) and precipitate formation (exothermic), and successful modelling of these concurrent processes is dependent on the deconvolution of the heat flow curve to isolate each reaction.  The work presented here follows the approach of Esmaeili et al. (2002), who propose that the following equations can be used to describe the precipitation and dissolution counterparts of the overall heat flow equation;  Precipitation   ( )1 expppt n npptdQ A knt ktdt −= −      (6.28) Dissolution 1 12 23 1 2 clusters disdQ BA t Bt dt −   = −        (6.29)   125 where B is the temperature dependent function; 1 2 1 32 0 DB r η pi =       (6.30)  Thus, the net heat flow can be modelled by combining the two equations for dissolution and precipitation; ( ) 1 212 1 23exp 1 2 n n dis ppt BA tdQ A knt kt Bt dt − −   = − + −             (6.31)  where Appt is the area under the trace for the immediately artificially aged material.  It follows that the total heat of dissolution, Adis, is given by the difference between the total heat for immediately artificially aged (i.e. Appt) and previously naturally aged material.  In Equation 6.31 n=1, and therefore only two parameters, k and B, are unknown and can be found by curve fitting. Figure 6.17 shows the deconvoluted dissolution and precipitation heat flow curves for 180°C.  -0.002 -0.001 0 0.001 0.002 0 1 2 3 Time (hr) He at  Fl ow  (W /g ) Experiment Model (ppt) Model (dis) Model (net)  Figure 6.17: Deconvoluted dissolution and precipitation at 180°C after 24 hours natural ageing.  126  The usual procedure was used to find the temperature dependence of k and B from Arrhenius plots.  For the precipitation process the temperature dependence of k is given by the equation;  0 exp pptppt Qk k RT −  =          (6.32)  The temperature dependence of the dissolution coefficient, B, is modelled according to the temperature dependence of ηD1/2, in which the diffusion coefficient, D, can be described as;  0 exp d QD D RT −  =          (6.33)  where D0 is the proportionality constant and Qd is the activation energy for diffusion of solute atoms in the matrix.  A relationship between η and T can be developed based on the assumptions used in developing Equation 6.5.  If the precipitate–matrix interface is at equilibrium [Whelan (1969)] then Ci can be considered to be the metastable solvus boundary for the solute clusters. Furthermore, the composition of the particle/cluster remains constant throughout dissolution [Aaron and Kotler (1971)], and is much larger than the composition at the interface.  If we further assume that Ci >> C0, meaning that the matrix is significantly depleted in solute compared to the interface, we can simplify η to;  0 2 iC C η =      (6.34)   127 In Equation 6.34, Ci is dependent on temperature according to the equation; exp si QC c RT −  =          (6.35)  where c is the proportionality constant and Qs is the enthalpy of solution of the solute clusters formed during natural ageing.  Using Equations 6.30, 6.33, 6.34 and 6.35 the temperature dependence of B can be described by the equation;  0 exp dis QB B RT −  =          (6.36)  where B0 is the pre-exponential coefficient and Qdis = Qs + (Qd/2).  Following the approach, calculation of the temperature dependency of the kinetic parameters k and B for precipitation and dissolution is possible from Arrhenius plots shown in Figure 6.18. Calculated values for Qppt and Qdis are 102kJ/mol and 74kJ/mol respectively.  These results are in reasonably good agreement with a prior analysis by Esmaeili et al. (2002), who calculated Qppt and Qdis for precipitation and dissolution in AA6111 to be 95kJ/mol and 88kJ/mol respectively. The activation energy for diffusion of Mg and Si in Al are both approximately 130kJ/mol, and using this value, the enthalpy of solution of solute clusters, Qs, can be estimated at approximately 9kJ/mol based on the relationship between Qs, Qdis, and Qd. described above.  Given that the typical error reported for estimates of Qd and Qdis are +/-10kJ/mol, this value of Qs falls close to the calculated activation energy for solute clustering obtained from isothermal calorimetry experiments in the present work and described in the previous section (i.e Qcluster = 24kJ/mol), as well as the range of values presented by Esmaeili for Qs in AA6111 (i.e 10-30kJ/mol).  128  -12 -10 -8 -6 -4 -2 0 2 0.002 0.0021 0.0022 0.0023 0.0024 0.0025 ln t (t in sec) ln  ln  (1/ 1- f) B k Qppt = 102.2kJ/mol Qdis = 73.6kJ/mol  Figure 6.18: Arrhenius plot for k and B  6.5.2  Evolution of Yield Strength Esmaeili (2002) has shown that strengthening due to natural ageing clusters can be modelled by Equation 6.37, in which the strength contribution due to solute clusters is dependent on the initial and current relative volume fraction of clusters, given by fr,0 and fr respectively;  ( )12,0clusters clusters r rc f fσ =     (6.37)  For strengthening due to precipitates in the previously naturally aged material, the behaviour is expected to be the same as the immediately artificially aged material, so Equation 6.24 is used. Solid solution strengthening is also assumed to behave in a similar manner to the immediately artificially aged case, and therefore a modified version of Equation 6.14 is used, in which the  129 value of σ0ss is the contribution of solid solution strengthening at the end of the natural ageing period instead of the as-quenched value.  The overall strengthening behaviour of the previously naturally aged material during artificial ageing can be described by the usual additive equation, with a modification to account for the simultaneous presence of precipitates and clusters.  In equation 6.38, σ’ppt is the combined strength contribution due to precipitates and clusters.  ' intys ppt ss eutecticσ σ σ σ σ= + + +    (6.38)  In the case of σ’ppt, a superposition rule is used to obtain an estimate of the strength contribution when both types of obstacles are present.  The pythagorean superposition method is applied following the previous work of Esmaeili et al (2003b) and Brown and Ham (1971).  ( )1' 2 2 2ppt ppt clustersσ σ σ= +     (6.39)  6.5.3 Model Calibration The model for artificial ageing of previously naturally aged material requires estimation of the constant parameters cppt, cclusters and fr,0.  cppt has been estimated from the peak aged yield strength of the previously naturally aged material, in this case 250MPa.  Subtraction of the intrinsic and eutectic strength contributions results in a value for cppt of 210.  cclusters is estimated from the strength of the material naturally aged for 24 hours material, which is 122MPa.  Again, subtraction of the other contributing factors leads to a value for cclusters of 82MPa.  Finally, the exothermic heat flow curve for naturally ageing material at 25°C is zero from approximately 20  130 hours onwards, and consequently it is assumed that the volume fraction of clusters, fr,0 after 24 hours natural ageing is 1.    These calibration parameters are listed in Table 6.8.  6.5.4 Model Predictions and Validation Model predictions and experimental results are compared in Figure 6.19 for the naturally aged material during artificial ageing at 150°C, 180°C and 200°C.  There appears to be an excellent fit between the model and experimental values and the calculated R2 value for each temperature was greater than 0.9.  The relative contributions of σppt and σclusters are shown in Figure 6.19, indicating that the contribution due to clusters is important not only during the early stages of ageing, but also at ageing times closer to the peak-aged condition.  The comparison between σppt and σclusters also reveals a slight decrease in overall strength owing to the dissolution of the clusters during the early part of precipitation.  Table 6.8: Calibration parameters for the artificial ageing model for 24hr naturally aged A356. Parameter Value Coefficient of Precipitation Strengthening, cppt (MPa) 210 Coefficient of Solute Cluster Strengthening cclusters (MPa) 82 Relative Volume Fraction of Solute Clusters, fr,0 1   131 0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re ng th  Co n tri bu tio n  (M Pa ) Time (hr) 0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h Co n tri bu tio n (M Pa ) Time (hr) 0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h Co n tri bu tio n (M Pa ) Time (hr) σexpt σtot σppt σclusters σeut + σint σexpt σtot σppt σclusters σeut + σint σexpt σtot σppt σclusters σeut + σint  Figure 6.19: Comparison of model predictions and experimental data for the evolution of yield strength during artificial ageing of the naturally aged A356 alloy at a) 150°C, b) 180°C, c) 200°C.  132  Validation tests have been carried out at artificial ageing temperature of 190°C after natural ageing for 24 hours, and the comparison between model predictions and experimental results are shown in Figure 6.20.  Excellent agreement between the model predictions and experimental data can be clearly seen.  0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h Co n tri bu tio n  (M Pa ) Time (hr) σexpt σtot σppt σclusters σeut + σint  Figure 6.20: Comparison of model predictions and experimental data for the evolution of yield strength during artificial ageing of the naturally aged A356 alloy at 190°C.           133 6.6 Modelling of Al-Si-Mg Alloys  This section aims to extend the microstructure-property model developed for the A356 alloy to other alloy compositions within the Al-Si-Mg alloy system.  The development of the model extension is presented first, followed by an assessment of the extended model’s ability to predict the heat treatment behaviour for a range of Al-Si-Mg alloy compositions.  6.6.1 Development of Extended Model As an initial step, the important parameters that need to be incorporated into the extended microstructure-property model must be identified.  In this work, most attention is paid to the effect of changes in alloy content, and therefore an approach is required to include dependency on the magnesium and silicon content, CMg and CSi in the model.  It is known that a higher magnesium content increases the volume fraction of precipitates and solute clusters formed, thereby influencing the amount of strengthening arising from precipitation, σppt and solute clustering σcluster.  In contrast, higher silicon content increases the volume fraction of eutectic, and this raises the overall strength of the alloy via the eutectic strengthening contribution σeut.  6.6.1.1 Effect of Silicon Content A review of the results presented in Table 5.2 for the A356 and model alloys in the as-quenched, natural aged and peak-aged conditions shows that a relationship exists between the eutectic contribution to strengthening, σeut, and the silicon content, CSi.  In Equation 6.15 a linear relationship between σeut and feutectic was developed in agreement with theories of second phase particle strengthening and the rule of mixtures approach.  134 The thermodynamic software program, FactSage, was used to estimate the volume fraction of eutectic at the solution treatment temperature under equilibrium conditions for each alloy.  The results are shown in Table 6.9, and have been used to calculate the eutectic contribution to strengthening, σeut, based on Equation 6.15.  The predictions of σeut are compared to the values presented in Table 5.2 derived from tensile testing of the three model alloys in Figure 6.21. There is clear agreement between both sets of data.  Table 6.9: FactSage thermodynamic calculation results for eutectic silicon content assuming equilibrium conditions at the typical solution treatment temperature of 540°C. Alloy Composition Volume Fraction of Eutectic Phase, feutectic Al-1.3Si-0.32Mg 0 Al-7Si-0.3Mg 55 Al-11Si-0.22Mg 85  0 5 10 15 20 25 30 35 40 0 10 20 30 40 Pr ed ic te d σσ σσ eu te ct ic (M Pa ) Measured σeutectic (MPa)  Figure 6.21: Comparison of σeut values obtained from tensile test measurements with predicted values from Equation 6.15.  135 Equation 6.15 is limited to hypoeutectic compositions (i.e. a range of silicon content between approximately 1.3wt% and 12.6wt%, assuming the solvus for Si in Al at the solution treatment temperature is 1.3wt%Si and the eutectic composition is 12.6wt%Si) because the effect of primary silicon particles has not been assessed.  It is also important to note that the model has been developed from experiments performed on alloys with a modified eutectic microstructure, and therefore may not be applicable to an alloy containing an unmodified eutectic phase.  6.6.1.2 Effect of Mg Content Referring again to the results presented in Table 5.2 for the A356 and model alloys, and with reference to measurements by Esmaeili et al. (2007) to determine the fraction of solute atoms that remain in solution in the naturally aged condition, calculated coefficients for precipitate and cluster strengthening cppt and ccluster can be plotted as a function of the magnesium content, CMg, as shown in Figure 6.22.  0 20 40 60 80 100 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 Cl u st er  St re n gt he n in g,  c c lu st e r (N m - 5/ 2 ) Pr e cip ita te  St re n gt he n in g,  c p pt (M Pa ) Magnesium Composition, CMg (wt%) cppt cclusters  Figure 6.22: The effect of magnesium content, CMg, on strengthening due to precipitates (cppt) and solute clusters (ccluster) in the peak aged and naturally aged conditions in Al-Si-Mg alloys.   136 Curve fitting allows the development of empirical equations that define the effect of magnesium content on the precipitation and solute clustering coefficients, cppt and ccluster; 1/2 ppt ppt Mgc m C=       (6.40) clusters clusters Mgc m C=      (6.41)  where mppt and mclusters are constant values given by 380MPa/wt%1/2 and 190Nm-5/2/wt% respectively.  In Equations 6.40 and 6.41, the difference in the relationships between the magnesium content and the precipitation and clustering coefficients can be explained by considering the dislocation-obstacle interactions and assuming the precipitates and clusters act as strong and weak obstacles to dislocation motion respectively.  As discussed earlier in Chapter 2, the bowing angle of the dislocation changes as the strength of the obstacle increases, and this in turn changes the effective spacing of the obstacles.  In the case of strong obstacles, a high bowing angle means the effective spacing increases proportionately with λ1/2, where λ is the precipitate spacing, whereas the effective spacing varies linearly with λ in the case of weak obstacles that do not cause the dislocation to bow significantly.  The range of magnesium alloy content that this model can be used for is assessed in the upcoming section.  6.6.1.3 Effect of Incomplete Solution Treatment A benefit of incorporating compositional dependence in the yield strength model is the ability to quantify the influence of short solution treatments on strengthening during ageing by defining an effective concentration of magnesium available for precipitation, which is lower than the bulk concentration due to incomplete dissolution of as-cast Mg2Si particles.  Furthermore, the development of a relationship between the dissolution of as-cast particles and the effective solute magnesium content results in an important connection between the solution treatment and ageing  137 models.  It should be noted that previous models for strength evolution during ageing assume the material is fully solution treated and all of the alloying additions are available for precipitation. The modification to the model to include incomplete dissolution of as-cast particles can be described as follows. The total amount of magnesium available for precipitation, CMg, has been used in equations 6.40 and 6.41 to calculate the strengthening coefficients for precipitate and clusters.  This term has been modified to include its dependency on the volume fraction of Mg2Si particles during solution treatment, which is given by fr and is described in detail in section 6.2. In order to achieve this, the amount of magnesium in solution in the initial (i.e. as-cast) and final (i.e. fully solution treated) conditions are required, and these have been estimated from the microprobe data presented in Section 5.1.3.  These estimations assume that all magnesium measured in the as-cast dendrite is in solution, and that the magnesium solute content increases from the initial to final cases as fr evolves.  This relationship is described as follows;  ( )( )( ), , ,1Mg Mg ac r Mg st Mg acC C f C C= + − −    (6.42)  where CMg,ac and CMg,st are the solute magnesium concentration in the as-cast and fully solution treated condition.  Using equation 6.42 in conjunction with equations 6.40 and 6.41 to calculate cppt and cclusters means the coefficients for strengthening due to precipitate and cluster formation during ageing are dependent on the alloy composition and solution treatment processing history.  An important assumption of this model is that all of the magnesium measured by the microprobe is in solution.  This is unlikely in the case of the as-cast condition, where some precipitation of Mg2Si from the matrix is expected to occur during cooling following solidification.  As a result, the model presented here does not apply to direct ageing of the as-cast alloy, and the minimum  138 solution treatment required for valid model predictions involves heating to a temperature above the Mg2Si solvus sufficient for dissolution of any small precipitates in the as-cast dendrite.  6.6.2 Assessment of Extended Model  6.6.2.1 Effect of Alloy Content The accuracy of the model predictions of Al-Si-Mg alloy strengthening behaviour over a range of alloy chemistries has been assessed by comparing model predictions for the Al-1.3Si-0.32Mg and Al-11.1Si-0.22Mg alloys with the experimental data presented in Chapter 5.  The model predictions for immediate artificial ageing are presented in Figure 6.23 at an artificial ageing temperature of 180°C. The predictions are observed to be a good fit to the experimental data for all three alloys.  In particular, in the case of the Al-11Si-0.22Mg alloy the increased strength arising from the larger eutectic content in the early stages of ageing is demonstrated, as well as the smaller amount of strengthening during ageing arising from the lower magnesium content and reduced precipitate formation.  Figure 6.24 shows the model predictions for the A356 and Al-11Si-0.22Mg alloys during artificial ageing at 180°C, following complete solution treatment (i.e. 540°C for 30 minutes and 540°C for 180 minutes for the A356 and Al-11Si-0.22Mg alloys respectively) and 24 hours natural ageing.  For both alloys, model predictions are within 10% of the experimental data plotted in the figure, and R2 is greater than 0.97, suggesting that the clustering and precipitation processes are well approximated for different compositions.   139 0 50 100 150 200 250 0.01 0.1 1 10 Yi el d St re n gt h (M Pa ) Time (hr) Al-1.3Si-0.3Mg A356 Al-11Si-0.22Mg  Figure 6.23: Model predictions for strengthening during immediate artificial ageing at 180°C for the A356 alloy and the two model alloys investigated.   0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) Al-7Si-0.3Mg Al-11SI-0.2Mg  Figure 6.24: Model predictions for strengthening during artificial ageing at 180°C following 24 hours natural ageing for the A356 and Al-11Si-0.22Mg alloys.   140 6.6.2.2 Effect of incomplete solution treatment The ability of the process model to predict the strengthening behaviour of Al-Si-Mg alloys after partial dissolution of the as-cast phases has been investigated by comparing the model predictions for the A356 and Al-11.1Si-0.22Mg alloys with experimental yield strength measurements during artificial ageing after short solution treatments.  These comparisons are made in Figures 6.25 and 6.26 for the A356 and Al-11Si-0.22Mg alloys respectively.  The A356 was solution treated for 2 minutes and 5 minutes at 540°C to achieve the partially solution treated condition in which the as-cast particles are only partly dissolved.  Referring to Figure 6.3 these solution treatment conditions result in a fraction dissolved of 0.4 and 0.85, respectively.  The model predictions for immediate artificial ageing at 180°C following the short solution treatments are presented in Figure 6.25, and compared to the results of tensile tests carried out on specimens heat treated in the same manner.  The base case (i.e. fully solution treated alloy) is also represented by the model predictions and experimental measurements of the yield strength of the material following solution treatment for 30 minutes.  The comparison shows that the model is accurate at predicting the reduced value of the peak strength arising from the lower solute magnesium content; however predictions of the evolution of strength during artificial ageing are inaccurate for the shortest solution treatment time.  To illustrate this, R2 values were calculated for the 2 minute, 5 minute and 30 minute cases presented in Figure 6.25, and these were 0.19, 0.82 and 0.89 respectively, indicating the poor fit of the predictions for the 2 minute solution treatment.  The poor fit occurs because the kinetic variables used to model the precipitation reaction are based on the fully solution treated condition and cannot account for variation in the precipitation rate arising from lower solute concentrations and lower driving forces.  In effect, the model will only be applicable to a finite  141 range of magnesium solute contents, within which the kinetic parameters do not change significantly enough to affect the evolution of precipitation and strength in the model.  The extent of this range of magnesium composition will be examined in the following section.  0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 2 min 5 min 30 min  Figure 6.25: Model predictions for strengthening of A356 alloy during immediate artificial ageing at 180°C following short solution treatments of 2 minutes and 5 minutes at 540°C, compared with full solution treatment of 30 minutes at 540°C.   The ability of the model to simultaneously handle changes in alloy content, eutectic volume fraction and available solute magnesium has been investigated by solution treating the Al-11Si- 0.22Mg alloy for 30 minutes and 180 minutes at 540°C followed by immediate artificial ageing. In this case, the solution treatment model predictions, obtained by changing the model inputs to reflect the differing alloy content and as-cast particle size, predict 76% dissolution after 30 minutes and complete dissolution after 70 minutes.  Figure 6.26 shows a comparison between the model predictions for the incomplete and complete solution treated cases at artificial ageing  142 temperatures of 150°C, 180°C and 200°C.  It is evident that the model is able to predict the behaviour of the eutectic alloy over this range of temperature and solution treatment condition, although the goodness of fit is better at 150°C and 180°C (R2 = 0.8) than at 200°C (R2 = 0.5).  0 50 100 150 200 250 300 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 150°C 180°C 200°C  Figure 6.26: Model predictions for Al-11Si-0.22Mg during immediate artificial ageing at 150°C, 180°C and 200°C following 30 minutes solution treatment at 540°C (fdis = 0.76).  6.6.2.3 Assessment using Literature Data for Al-Si-Mg Casting Alloys An important aspect of validating the extended microstructure-property model is to compare model predictions with independent data obtained from the literature.  Recently, Moller et al (2008) published results of an experimental investigation into the artificial ageing behaviour of semi-solid manufactured (SSM) Al-7Si-Mg alloys.  Their study investigated the effect of natural ageing time, artificial ageing temperature and magnesium content on strengthening during artificial ageing.  The microhardness data presented by Moller et al has been converted to estimates of the yield strength according to the procedure outlined in Appendix A, and these data are compared to the predictions made by the model.  143  A comparison of the literature values and model predictions for the effect of artificial ageing temperature and magnesium content on strengthening are shown in Figure 6.27. The model predictions replicate the literature data accurately at 180°C; however, the predictions at 160°C are less successful.  While the evolution of strength is predicted fairly accurately at 160°C, the value of the peak strength does not match the experimentally measured values.  This may occur because Moller et al reported higher peak strength at 160°C compared to ageing at 180°C, whereas the experimental investigation here found constant peak strength throughout the temperature range of interest.  Despite these differences, all goodness-of-fit analyses for this data reported very good R2 values greater than 0.8, with 70% of the model predictions within 10% of the corresponding experimental yield strength measurement, and 90% of model predictions within 15% of the measured value.  Model predictions for strengthening during natural ageing have also been compared with the data of Moller et al in Figure 6.28.  The model predictions are very close to the experimental data in the first 20 hours of natural ageing, however, the experimental measurements show continued strengthening beyond 20 hours, whereas the model predicts no further change in yield strength.  This error arises from the initial calorimetry experiments used to develop the kinetic model which found no significant heat release after about 20 hours at room temperature.  For the 0.28wt%Mg, 0.34wt%Mg and 0.45wt%Mg alloys the R2 values obtained for the first 20 hours of natural ageing are 0.86, 0.94 and 0.91 respectively, which indicates that the model is a very good fit for the data over this range of natural ageing time.   144 a) 0 50 100 150 200 250 300 350 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 0.28Mg 0.34Mg 0.45Mg  b) 0 50 100 150 200 250 300 350 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 0.28Mg 0.34Mg 0.45Mg  Figure 6.27: Model predictions for Al-7Si-Mg alloys with varying Mg content during artificial ageing at: a) 160°C and b) 180°C, compared with literature data from Moller et al (2008)   145 0 50 100 150 200 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 0.28Mg 0.34Mg 0.45Mg  Figure 6.28: Model predictions for Al-7Si-Mg alloys with varying Mg content during natural ageing at 20°C compared with literature data from Moller et al (2008)  Finally, the validity of the yield strength model for artificial ageing after natural ageing has also been assessed using the data from Moller et al (2008).  Model predictions for artificial ageing at 160°C and 180°C are shown in Figure 6.29, compared to the literature data for the three Al-Si- Mg alloys.  It should be noted that the natural ageing time for the model is 24 hours, whereas the experimental measurements were carried out on specimens that had received a 20 hour natural age.  Nevertheless, the model predictions in Figure 6.29 are very close to the experimental data. As discussed previously, the model over-predicts the decrease in strength in the early stages of artificial ageing arising from cluster dissolution, and under-predicts the final peak-aged strength at 160°C.  However, the evolution of strength during concurrent cluster dissolution and precipitate formation is accurately reproduced.   146 a) 0 50 100 150 200 250 300 350 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 0.28Mg 0.34Mg 0.45Mg  b) 0 50 100 150 200 250 300 350 0.01 0.1 1 10 100 Yi el d St re n gt h (M Pa ) Time (hr) 0.28Mg 0.34Mg 0.45Mg  Figure 6.29: Model predictions for Al-7Si-Mg alloys with varying Mg content during artificial ageing at: a) 160°C and b) 180°C after natural ageing, compared to data from Moller et al (2008)    147 6.6.3 Summary  The microstructure-property model developed through Chapters 6.2 – 6.5 has been extended to include their dependency on the alloy chemistry (i.e. silicon and magnesium content), eutectic volume fraction, and solution treatment condition.  An assessment of the ability of the extended model to predict the behaviour of Al-Si-Mg alloys over a range of compositions and heat treatment conditions has been made and a summary of the conclusions is presented here.  The models are capable of handling a variation in silicon content within the hypo-eutectic range (i.e between the solvus and eutectic point) and also provide good predictions of strengthening behaviour for alloys over a range of magnesium content between 0.2wt% and 0.45wt%. Discrepancies between model predictions and experimental measurements of strengthening during artificial ageing after short solution treatments indicate that the model predictions become inaccurate when the magnesium content is below 0.2wt% because of the slower kinetics of the precipitation reaction arising from the lower solute concentration in the matrix.  For the case of the A356 alloy studied in the present work, this results in a limiting case of solution treatment for at least 180 seconds at 540°C.          148 6.7 Application of the Model for Optimisation of Industrial Processes  As with any model for an industrial process, its ability to provide knowledge to industry about the process and to enable optimisation of the process should be examined.  This section considers model predictions from an industrial standpoint, with a view to identifying aspects of the heat treatment process that can be improved to increase productivity.  6.7.1 Optimisation of Solution Treatment Process The solution treatment process is a high-energy, high-cost operation that typically involves heating components of significant size to high temperatures for long periods of time.  As stated in section 2.3, the ASTM Standard for T6 heat treatment of A356 castings states that solution treatment should be carried out 540°C for 4-12 hours.  Although many heat treatment facilities operate at the lower end of this time scale, costs are still high and there is pressure to further shorten the processing time.  The microstructure-property model developed here has been used to examine three different approaches that can be used to reduce the processing time further; increased heating rates, higher soak temperatures, and preheating of the charged component.  6.7.1.1 Effect of Soak Temperature An increase in soak temperature results in faster solution treatment because of the temperature dependency of the physical processes that control the microstructure changes occurring in the material. These include the dissolution of as-cast particles as well as the fragmentation and coarsening of the eutectic silicon phase.  The effect of soak temperature on these microstructure changes during solution treatment has been examined by using the model to predict completion times for each process when the soak temperature is varied in isolation (i.e. initial temperature and heating rate are kept constant).  Model predictions of the time required for 98% completion  149 of Mg2Si dissolution and eutectic silicon fragmentation are presented along with the time required for the fragmented eutectic silicon particles to coarsen to an average radius of 1µm. The initial as-cast microstructure characteristics are known to affect the rate of microstructure changes during solution treatment so these were kept constant at a secondary dendrite arm spacing of 30µm and initial eutectic silicon rod diameter of 0.636µm.  The results are presented in Figure 6.30 for soak temperatures from 500°C up to 560°C, above which the integrity of the casting is likely to be compromised by incipient melting in the interdendritic regions.  500 510 520 530 540 550 560 0 60 120 180 240 300 360 420 480 So lu tio n  Tr ea tm en t T em pe ra tu re  (°C ) Solution Treatment Time (min) Dissolution Complete Fragmentation Complete Silicon Radius =1micron  Figure 6.30: Effect of soak temperature on the time required for microstructure changes during solution treatment. (Fixed process parameters: Dendrite Arm Spacing = 30 µm, Initial silicon rod diameter = 0.5µm, Charge Temperature = 20°C, Solution treatment heating rate = 40°C/sec)  It is evident that the soak temperature has a large influence on all three processes occurring during solution treatment and the time to achieve a suitable solution treated microstructure increases from around 90 minutes to 420 minutes as the soak temperature is decreased from  150 560°C to 500°C.  These predictions indicate that the current ASTM standard overestimates the required time for solution treatment, except in cases where an average eutectic silicon particle radius greater than 1 micron is required for greater fatigue and fracture resistance.  The predictions also indicate that energy savings achieved by reducing the soak temperature would be negated by an unreasonably large increase in the process time (i.e. a reduction of 40°C causes a greater than threefold increase in the processing time).  It should be noted that the dissolution process is thermodynamically unfavourable below a critical temperature, at which point the Mg2Si particles will not fully dissolve.  The minimum temperature for complete dissolution to occur is strongly dependent on the composition of the alloy.  The solution treatment model presented in the present work is unable to determine this critical temperature limit and hence it is recommended that plots of the form shown in Figure 6.30 are developed in conjunction with thermodynamic software (e.g FactSage, Thermocalc) that can be used to determine the process window for dissolution.  6.7.1.2 Effect of Heating Rate The second method investigated to achieve a decrease the solution treatment time is an increase in heating rate to the soak temperature.  The effect of heating rate on the microstructure changes during solution treatment has been examined in a similar manner by using the model to predict completion times for each process as the heating rate is varied in isolation (i.e. initial temperature and soak temperature are kept constant).  Model predictions of the time required for dissolution and fragmentation to reach 98% completion, and the time for the fragmented silicon particles to coarsen to an average radius of 1µm are presented in Figure 6.31 for heating rates between 0.1°C/sec and 10°C/sec.  The model predicts a large change in the time to achieve the appropriate solution treated microstructure as heating rate is varied.  When the heating rate is  151 increased from 0.1°C/sec to 1°C/sec the completion time decreases from 315 minutes to 130 minutes, although further increases in heating rate have a smaller effect.  It should be noted that high heating rates increase the likelihood of incipient melting due to non-equilibrium second phase particles in the as-cast microstructure, as well as thermal overshoots during processing, and as a result it may not be desirable to use very high heating rates.  0.1 1 10 0 60 120 180 240 300 360 420 480 He at in g Ra te  (°C /s ) Solution Treatment Time (min) Dissolution Complete Fragmentation Complete Particle Radius = 1micron  Figure 6.31: Effect of heating temperature on microstructure changes during solution treatment. (Fixed process parameters: Dendrite Arm Spacing = 30µm, Initial silicon rod thickness = 0.5µm, Soak Temperature = 540°C, Charge Temperature = 20°C)  A two step heating rate is common in solution treatment of wrought aluminum alloys, involving rapid initial heating to an intermediate temperature above the solvus of the precipitating phases, followed by slower heating to the soak temperature to avoid thermal overshoots and incipient melting.  This heating strategy has been examined to determine its suitability for optimising the solution treatment process of cast components.  A series of two-step heating profiles are  152 presented in Figure 6.32 compared to the single-step base case, and model predictions of solution treatment behaviour under these scenarios are presented in Table 6.10.  0 100 200 300 400 500 600 0 15 30 45 60 Te m pe ra tu re  (°C ) Time (min) 0.2°C/sec to 540°C 0.5°C/sec to 500°C, then 0.1°C/sec to 540°C 1°C/sec to 500°C, then 0.1°C/sec to 540°C 2°C/sec to 500°C, then 0.1°C/sec to 540°C  Figure 6.32: Modelled heating profiles, including the base case and two-step strategies.  Table 6.10: Model predictions for completion of each microstructure process during solution treatment with various heating profiles (illustrated in Figure 6.32) Heating Schedule Dissolution Time (sec) Fragmentation Time (sec) Coarsening Time (sec) Total Time (min) 0.2°C/sec to 540°C 3060 5580 16800 280 0.5°C/sec to 500°C, & 0.1°C/sec to 540°C 1710 4320 12480 208 1°C/sec to 500°C, & 0.1°C/sec to 540°C 1228 3840 10680 178 2°C/sec to 500°C, & 0.1°C/sec to 540°C 991 3660 9720 162   153 The results show that a two-step strategy may be a useful approach for improving solution treatment times.  The change from single-step heating to a rapid initial heating rate followed by slower achievement of the soak temperature may reduce the total solution treatment time from 280 minutes to around 162 minutes, a saving of over 40%.  6.7.1.3 Effect of Charge Temperature The final solution treatment variable investigated is the charge temperature of the component as it is inserted in to the furnace.  This has been done by comparing the completion time for the dissolution, fragmentation and coarsening processes for one component cold charged at room temperature and another hot charged at 200°C at a range of heating rates to a soak temperature of 540°C.  The comparison is shown in Figure 6.33, in which the completion of each microstructure process is represented by the full lines for the cold charge and the dashed lines for the hot charge.  The figure illustrates that a hot charge results in more rapid solution treatment when the heating rate is lower, while at high heating rates the use of a hot charge has little-to-no effect.  It appears that this finding may provide some scope for the heat treatment operator; if an increased heating rate is not possible, a preheated charge can help to reduce solution treatment times. 0.1 1 10 0 60 120 180 240 300 360 420 480 Solution Treatment Time (min) He at in g Ra te  (ºC /s ) Dissolution Complete Fragmentation Complete Particle Radius = 1micron Hot Charge (200ºC) Cold Charge (20ºC)  Figure 6.33: Comparison of model predictions for solution treatment using a range of heating rates in the cases of cold charging (20°C) and hot charging (200°C) the as-cast component.  154  6.7.1.4 Effect of Simultaneous Changes to Solution Treatment Parameters Each of the process parameters considered in the previous three sections have indicated that potential benefits exist in terms of reducing the total solution treatment time when they have been varied in isolation.  In this section the combined effect of simultaneous changes to all three parameters has been considered and an optimised solution treatment strategy has been determined.  Details of the process parameters used in this part of the investigation are given in Table 6.11.  The soak temperature was varied from 540°C to 550°C, while the heating rate was increased from the single-step base case to the two step strategy outlined in the section 6.7.1.2. The effect of hot charging was also considered by varying the initial temperature from 20°C to 200°C.   A diagram representing four of the heating profiles is given in Figure 6.34.  0 100 200 300 400 500 600 0 15 30 45 60 Te m pe ra tu re  (°C ) Time (min) HR1 HR2 HR7 HR12  Figure 6.34: Schematic illustrating heating profiles used to obtain model predictions, including the base case (0.2°C/sec to 540°C), and two-step strategies detailed in Table 6.11   155 Table 6.11: Solution Treatment Parameters Used for the Model Investigation into the Effect of Simultaneous Parameter Changes on the Total Processing Time I.D. Charge Temperature (°C) Heating Rate (°C/sec) Soak Temperature (°C) HR1 20 0.2 540 HR2 20 0.5 to 500°C, then 0.1 540 HR3 20 1 to 500°C, then 0.1 540 HR4 20 2 to 500°C, then 0.1 540 HR5 20 0.2 550 HR6 20 0.5 to 500°C, then 0.1 550 HR7 20 1 to 500°C, then 0.1 550 HR8 20 2 to 500°C, then 0.1 550 HR9 200 0.2 550 HR10 200 0.5 to 500°C, then 0.1 550 HR11 200 1 to 500°C, then 0.1 550 HR12 200 2 to 500°C, then 0.1 550  The model predictions for the twelve solution treatments scenarios considered are presented in Table 6.12.  The time to 98% completion of the Mg2Si dissolution and eutectic silicon fragmentation processes are presented with the time to achieve an average silicon particle radius of 1µm.  The total solution treatment time is estimated from these values and is also presented for comparison.  The predictions indicate that while changes to any one process parameter can have an effect to shorten the solution treatment time, the combined effect of several simultaneous changes to the process parameters leads to further improvements.  A comparison between the base case of heating to 540°C at a rate of 0.2°C/sec and HR12, involving heating a  156 preheated component (200°C) to 500°C at 2°C/sec followed by heating to 550°C at 0.1°C/sec, shows that the total solution treatment time is reduced from 280 minutes to 129 minutes, a time reduction of over 50%.  While there are additional processing costs associated with the modified thermal profile (preheating, higher heating rate and soak temperature), production time savings may generate a reduction in overall production costs and/or improved productivity.  Table 6.12: Model predictions for completion of each microstructure process during solution treatment with various heating profiles (details of heating profiles are in Table 6.11) I.D. Heating Time (sec) Dissolution Time (sec) Fragmentation Time (sec) Coarsening Time (sec) Total Time (min) HR1 2600 3060 5580 16800 280 HR2 1360 1710 4320 12480 208 HR3 880 1228 3840 10680 178 HR4 830 991 3660 9720 162 HR5 2650 2940 4800 14640 244 HR6 1460 1650 3540 10800 180 HR7 980 1163 3120 9060 151 HR8 740 928 2880 8100 135 HR9 1750 2040 3900 12000 200 HR10 1100 1272 3180 9480 158 HR11 800 977 2940 8340 139 HR12 650 833 2760 7740 129      157 6.7.2 Optimisation of Artificial Ageing Compared to solution treatment, the artificial ageing process is much less energy intensive; however the drive to optimise the process in terms of cost, production time and component quality remains.  The ASTM standard for T6 heat treatment (shown in section 2.3) recommends natural ageing for 8 hours at room temperature, followed by artificial ageing at 155°C for 6-12 hours.  However, many heat treatment operations do not allow natural ageing for as long as 8 hours, or the delay between the quench and artificial ageing may vary from component to component.  Furthermore, it is desirable to reduce the time of the artificial ageing process from the relatively long times given by the ASTM standard.  6.7.2.1 Effect of Natural Ageing An investigation using the model developed in the present work has been carried out to investigate the concerns surrounding the effect of natural ageing on production time during ageing.  The time to peak strength gives an estimate of the speed of the ageing process and the effect natural ageing has on this value allows us to assess the impact that delays prior to artificial ageing have on the overall production time.  Figure 6.35 shows the change in model predictions of the time to peak strength during immediate artificial ageing as well as following 24 hours natural ageing at different artificial ageing temperatures.   The predictions show that the delaying effect of natural ageing on the time to peak strength during artificial ageing is significant at artificial ageing temperatures close to the ASTM standard of 155°C, and that this effect decreases at higher artificial ageing temperatures until the difference between the two cases is minimal at an artificial ageing temperature of 220°C.   158 150 160 170 180 190 200 210 220 0 4 8 12 16 20 24 28 32 Artificial Ageing Time (hours) Ar tif ici al  Ag ei ng  Te m pe ra tu re  (ºC )    No Delay    24 Hour Delay Model Expt Ar tif ici al  Ag ei ng  Te m pe ra tu re  (ºC )  Figure 6.35: Comparison of model predictions for the time to peak strength during immediate artificial ageing and after a 24 hour natural age at room temperature. Model predictions are given by solid lines and experimental data by symbols.  The model predictions shown in Figure 6.35 can be considered in several ways.  The predictions quantify the delaying effect that natural ageing has on the time to peak strength during artificial ageing.  For example, the model predicts that during artificial ageing at 180°C, the time to peak strength is delayed by approximately 2 hours when a 24 hour natural age occurs immediately before.  Furthermore, the model also predicts that the naturally aged material will reach the peak strength after 3.5 hours during artificial ageing at 190°C.  This information is useful to industry when heat treating material that has been subjected to an unscheduled delay to determine the most cost-effective treatment for components that have been subjected to different processing histories.     159 6.7.3 Summary  The microstructure-strength model developed in the present work has been used to examine several different approaches to optimise heat treatment operations by reducing the processing time and energy consumption.  For the solution treatment process this has involved the investigation of the effect of changes to important processing parameters including the charge temperature, heating rate and soak temperature on the achievement of the desired solution treated microstructure, while for the artificial ageing process, strategies for overcoming the effect of a 24 hour delay (natural age) prior to the artificial age have been discussed.  The results of the solution treatment investigation indicate that changes to the charge temperature, heating rate and soak temperature can be combined to significantly reduce the total solution treatment time.  If a two-step heating strategy similar to that applied to many wrought alloys in industrial heat treatment operations is used, a small increase in soak temperature from 540°C to 550°C becomes possible as incipient melting becomes less of a concern.  Combination of the new heating strategy and soak temperature with preheating of the component to 200°C prior to charging in the furnace results in a predicted decrease in solution treatment time of approximately 50% compared to the base case of heating directly from room temperature to a soak temperature of 540°C.  Strategies identified for overcoming the detrimental effect of a 24 hour delay (natural age) on strengthening during subsequent artificial ageing include increasing the artificial ageing temperature and/or time.  The model has been used to make predictions that quantify the amount of temperature/time change required, and these results suggest that it is more efficient to raise the artificial ageing temperature to overcome the effects of the natural age.  It would be useful to  160 determines how long a delay can be tolerated before a change to the artificial ageing procedure becomes necessary, however, the ability to make these types of prediction lies beyond the scope of the current model because the influence of natural ageing on precipitation and dissolution kinetics during artificial ageing is dependent on the length of the natural ageing time and only the 24 hour naturally aged condition has been considered in the present work.  It would be beneficial to quantify the effect of natural ageing time on the artificial ageing kinetics such that this dependency can be integrated into the yield strength model.   161  CHAPTER 7 - Conclusions and Future Work  7.1 Conclusions  The objectives of the present work were to develop a comprehensive mathematical model of the microstructure and strength evolution in an A356 (Al-7Si-Mg) alloy during heat treatment (solution treatment, followed by natural and artificial aging).  This was achieved by performing experimental investigations to determine the effect that thermal processing has on the material behaviour and incorporating this knowledge into non-isothermal microstructure and yield strength models based on established physical theories.  The model predicts the microstructure evolution during solution treatment and the microstructure and yield strength evolution during natural ageing and artificial ageing.  Detailed conclusions are presented for each aspect of this work.  Solution Treatment Process The material response to solution treatment for various times within the temperature range 500- 560ºC was characterised experimentally by quantitative microprobe analysis and image analysis of eutectic particles exposed by a deep etching technique.  As expected, solute redistribution and particle morphology changes were slower at lower temperatures.  A microstructure model for solution treatment was developed based on sub-models for: i) dissolution of Mg2Si particles, ii) fragmentation of eutectic silicon particles and iii) coarsening of the fragmented eutectic particles. The model predictions are accurate with respect to the experimental data, although less satisfactory results were obtained for dissolution at 560ºC where the rate was significantly underpredicted.   Specific conclusions include: 162   • The modelling approach employed in this work (i.e. the use of three independent sub- models linked by solution treatment temperature and time) is able to accurately represent microstructure changes during solution treatment. • For the alloy chemistry, initial microstructure and process conditions studied, the complete dissolution of Mg2Si occurs rapidly within minutes, whereas the coarsening of fragmented eutectic silicon particles is the rate limiting parameter for solution treatment.  Ageing Processes Two process histories were investigated in the artificial ageing study: i) as-quenched and ii) quenched and naturally aged for 24 hours.  In addition, the material response during natural ageing was investigated.  Microhardness, tensile and isothermal calorimetry tests revealed that the peak strength and total heat evolved during artificial ageing were constant for the temperature range 150-200ºC.  Using the heat flow curves from isothermal calorimetry, the rate of precipitation was determined and used to develop yield strength models for each ageing process.  Model predictions and experimental data were in close agreement for the artificial ageing models, while the natural ageing model was accurate for 24 hours after quenching. Specific conclusions include:  • A model for yield strength evolution using a strong obstacle assumption to describe the interaction between particle and dislocation can be used to predict strengthening during artificial ageing of Al-Si-Mg casting alloys. • The yield strength evolution during the first 24 hours of natural ageing can be modelled, using a weak obstacle assumption for dislocation-particle interactions. 163  • The evolution of strength during artificial ageing of material subjected to prior natural ageing can be predicted based on kinetics of dissolution and precipitation obtained from mathematical deconvolution of the isothermal heat flow curve. • Natural ageing has a strong influence on the kinetics of precipitation during subsequent artificial ageing, and the point at which the natural ageing time becomes significant needs to be fully investigated.  An extended yield strength model is proposed that incorporates the dependence of strengthening on alloy composition and solution treatment history.  This novel approach provides linkages between the solution treatment and ageing models, allowing them to be applied to:  • A minimum range of Al-Si-Mg alloy compositions between 1.3<wt%Si<12.6 and 0.2<wt%Mg<0.45. • The prediction of strengthening following short/incomplete solution treatment. • The use of model alloys with simple microstructure features to predict the heat treatment behaviour of an alloy with more complex microstructure.  For example, data from wrought and eutectic alloys may be combined using a composite approach to predict strengthening in an A356 casting alloy.  Application of Models to Industrial Optimisation The model was used to identify process improvements leading to increased productivity.  In the solution treatment model, minimal improvements in productivity were predicted by an increase in soak temperature.  Significant improvements were predicted using increased charge temperatures, minimum heating rates of 1ºC/s and two-step heating strategies.  Solution 164  treatment processing time reductions in the region of 50% were predicted when a combination of parameter changes was applied.  The ageing model was used to quantify the effect of a delay prior to artificial ageing on the time to peak strength, thereby enabling strategies to minimise loss of productivity.  An increase in artificial ageing temperature to compensate for the reduction in the kinetics of precipitation was identified, and further work to fully characterise the effect of natural ageing time on precipitation kinetics during artificial ageing is required.  Specific conclusions include:  • The model can be used to identify process improvements in order to attain greater efficiencies.  Currently these improvements focus on increased productivity via reduction in processing time, however with suitable energy use calculations, the model may be easily adapted to optimise processes based on energy efficiency. • The model enables optimisation of the process based on alloy chemistry.  Small changes in alloy content may result in reduced operating costs while maintaining the desired material performance. • The model approach employed in this work is suitable for application to other heat- treatable aluminum casting alloys.  7.2 Future Work  The overall objective of the future work is the development of a through-process model for the entire heat treatment process.  The work proposed below aims to add complexity in areas that have not yet received sufficient attention in the current model:  165  • The inclusion of the relationship between the solidification conditions (e.g. the solidification rate, addition of chemical modifiers and presence of impurities), the as-cast microstructure (i.e. secondary dendrite arm spacing, eutectic silicon morphology and size, segregation of alloying elements), and the heat treatment behaviour of the material would be valuable to the model in terms of providing a linkage between casting and heat treatment operations.  Initially, this work would involve a study of the influence of the scale of the as-cast microstructure on the solution treatment response, with further examination to reveal the effect of different impurity levels and the influence on the material response during ageing. • The homogenisation of solute is not addressed in the present solution treatment model and this requires attention.  In the partially solution treated condition, segregation of solute may influence the material response during further processing, particularly in materials with larger alloying additions than those presented in the present work.  A sub- model for the evolution of the spatial distribution of solute is proposed as a means of quantifying its influence on the material response during ageing. • A model for the quenching process following solution treatment that can predict the amount of solute retained in solution during cooling would be a valuable addition to the overall model.  This would enable quantification of the reduced precipitation strengthening effect associated with slower quench rates. • The yield strength model for natural ageing could be extended in the following ways. The temperature dependence of cluster formation, if added to the model would allow the prediction of strengthening during pre-ageing at intermediate temperatures (e.g 60ºC). Secondly, the material response during natural ageing could be modelled beyond the current limit of 24 hours by considering coarsening of the solute clusters. 166  • An extension to the artificial ageing model to account for the effect of overageing processes at long ageing times could be made by taking into account the coarsening of precipitates and the shearable/non-shearable precipitate transition.  This would result in the ability to model the complete artificial ageing curve of the material. • The influence of natural ageing time on the kinetics of cluster dissolution and precipitation formation during subsequent artificial ageing could be further investigated. This would aid development of a comprehensive yield strength model capable of predicting the yield strength evolution for a wider range of non-isothermal and multi- stage ageing conditions.  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Journal of Light Metals, 2 (1), pp.27-36.  175  Appendix A - Correlation between Yield Strength and Hardness  Tabor (1947) & (1951) has shown that the strain associated with a Vickers indentation is approximately 8% for work-hardening alloys such as aluminum alloys, and Cahoon et al. (1971) have demonstrated that the tensile stress at a strain of 8% correlates with the Vickers hardness. However, the work hardening rate in Al-Mg-Si alloys is known to vary according to the precipitation state of the alloy [Cheng et al. (2003)], and therefore correlations between the yield strength and Vickers hardness of specimens heat treated to different conditions (i.e. as-quenched, naturally aged, underaged, peak aged, overaged) are not expected to  be linear.  Correlations between the yield strength and Vickers hardness of the A356 alloy and the two model alloys (Al-1.3Si-0.32Mg and Al-11Si-0.22Mg) were made in order to determine equations that best describe the relationship between the two properties for each alloy.  In all three cases, a power law equation was found to give the best fit to the data.  The correlations and resulting power law equations for the three alloys are given in Figure A.1.  The equations given in this figure were used to calculate the estimated yield strength values shown in Figures 5.12 and 5.15 for the A356 alloy, Figures 5.19, 5.20 and 5.21 for the Al-11Si-0.22Mg alloy, and Figures 5.24, 5.25 and 5.26 for the Al-1.3Si-0.32Mg alloy.  It should be clarified that the yield strength data presented for these alloys elsewhere in Chapters 5 and 6 are yield strength measurements from tensile testing, rather than estimates from the correlation equations.  Correlations were also made between the reported yield strength and Vickers hardness data for the Al-7Si-xMg alloys (where x = 0.28, 0.34 and 0.45) presented by Moller et al (2008) and used in this work to validate the extended microstructure-strength model.  In this case, due to the 176  limited number of correlated data points for each alloy, a single best fit equation was calculated for all three alloys.  The plot and best fit equation are shown in Figure A.2, and the equation has been used to calculate the estimated yield strength values in Figures 6.27, 6.28 and 6.29.  YS = 0.025VHN2.080 R² = 0.717 YS = 0.054VHN1.864 R² = 0.978 YS = 0.022VHN2.028 R² = 0.939 0 50 100 150 200 250 300 350 40 60 80 100 120 Yi el d St re n gt h (M Pa ) Vickers Hardness (kg/mm2) Al-11Si-0.22Mg Al-1.3Si-0.32Mg A356  Figure A.1: Correlations and best fit equations between yield strength and Vickers hardness for the A356, Al-1.3Si-0.32Mg and Al-11Si-0.22Mg alloys used in this study. YS = 0.324VHN1.428 R² = 0.994 0 50 100 150 200 250 300 350 40 60 80 100 120 Yi el d St re n gt h (M Pa ) Vickers Hardness (kg/mm2)  Figure A.2: Correlations and best fit equation between yield strength and Vickers hardness for the Al-7Si-xMg alloys (where x= 0.28, 0.34 & 0.45) studied by Moller et al (2008).  177  References Cheng, L.M., Poole, W.J., Embury, J.D., Lloyd, D.J., 2003. The influence of precipitation on the work-hardening behaviour of the aluminum alloys AA6111 and AA7030. Metallurgical and Materials Transactions A, 34 (11), pp.2473-2481.  Moller, H., Govender, G., and Stumpf, W.E., 2007. Natural and artificial aging response of semisolid metal processed AlSiMg alloy A356. International Journal of Cast Metals Research, 20 (6), pp.340-346.  Tabor, D., 1948. A simple theory of static and dynamic hardeness. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences. 192 (1029), pp.247-274.  Tabor, D., 1951. The Hardness of Metals. Oxford: Oxford University Press.  

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