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1. Length-scale in solidification analysis 1 2. Thermodynamics of solidification 4 2.1 Equilibrium 4 2.2 The undercooling requirement 5 2.3 Kinetic undercooling 7 2.4 Curvature undercooling 9 2.5 Thermal undercooling 11 2.6 Constitutional undercooling 12 2.7 Pressure undercooling 14 2.8 Hierarchy of equilibrium 15 2.9 Local interface equilibrium 16 2.10 Interface non-equilibrium 16 2.11 References 19 3. Macro-scale phenomena - formation of macrostructure 20 3.1 Relevant transport equations 20 3.2 Introduction to the mathematics of diffusive transport 24 3.2.1 The differential equation for macroscopic heat transport 25 3.2.2 Solutions of the heat conduction equation 27 3.3 References 30 4. Macro-mass transport 31 4.1 Solute diffusion controlled segregation 31 4.2 Analysis of solute redistribution 34 4.2.1 Equilibrium solidification 35 4.2.2 No diffusion in solid, complete diffusion in liquid (the Gulliver-Scheil model) 36 4.2.3 No diffusion in solid, limited diffusion in liquid 38 4.2.4 Limited diffusion in solid, complete diffusion in liquid 40 4.2.5 Limited diffusion in liquid and solid 43 4.2.6 Partial mixing in liquid, no diffusion in solid 43 4.2.7 Zone melting 47 4.3 Fluid flow controlled segregation 49 4.3.1 Shrinkage flow 51 4.3.2 Natural convection 51 4.3.3 Flow through the mushy zone 54 4.3.4 Fluid flow /solute diffusion controlled segregation 55 4.3.5 Fluid dynamics during mold filling 63 4.3.6 Fluid dynamics during casting solidification - macro shrinkage formation 67 4.4 References 70 5. Macro-Energy Transport 72 5.1 Governing equation for energy transport 72 5.2 Boundary conditions 74 5.3 Analytical solutions for steady-state solidification of castings 75 5.4 Analytical solutions for non-steady-state solidification of castings 78 5.4.1 Resistance in the mold 81 5.4.2 Resistance at the mold/solid interface 85 5.4.3 The heat transfer coefficient 89 5.4.4 Resistance in the solid 91 5.4.5 Resistance in the solid and in the mold 92 5.5 References 93 6. Macro-modeling of solidification; Numerical approximation methods 94 6.1 Problem formulation 94 6.2 Discretization of governing equations 96 6.2.1 Taylor series and numerical differentiation 96 6.2.2 Finite difference discretization of the heat conduction equation 99 6.2.3 Control volume formulation 107 6.3 Solution of the discretized equations 108 6.4 Applications of macro-modeling of solidification 109 6.4.1 Model deliverables 110 6.4.2 Criteria functions 112 6.4.3 Scaling relationships for criteria functions 112 6.5 References 115 7. Micro-scale phenomena and interface dynamics 116 7.1. Problem formulation 117 7.2. Nucleation 119 7.2.1 Heterogeneous nucleation models 122 7.2.2 Dynamic nucleation models 126 7.3. Micro-solute redistribution in alloys and microsegregation 126 7.4. Interface stability 133 7.4.1 Thermal instability 133 7.4.2 Solutal instability 135 7.4.3 Thermal, solutal and surface energy driven morphological instability 138 7.4.4 Influence of convection on interface stability 143 7.5 References 143 8. Cellular and dendritic growth 145 8.1. Morphology of primary phases 145 8.2. Interface undercooling and growth velocity models for dendrites 148 8.3 Tip velocity models 149 8.3.1 Solute diffusion controlled growth (isothermal growth) 149 8.3.2 Thermal diffusion controlled growth 152 8.3.3 Solutal, thermal, and capillary controlled growth 153 8.3.4 Interface anisotropy and the dendrite tip selection parameter 158 8.3.5 Effect of fluid flow on dendrite tip velocity 160 8.3.6 Multicomponent alloys 161 8.4 Dendritic arrays models 162 8.5 Volume averaged dendrite models 165 8.6 Complex geometry models 173 8.7 Dendritic arm spacing and coarsening 174 8.7.1 Primary spacing 174 8.7.2 Secondary arm spacing 176 8.8 The columnar-to-equiaxed transition 180 8.9 References 184 9. Eutectic solidification 186 9.1 Types of eutectics 186 9.2 Cooperative eutectics 188 9.2.1 Model of eutectic growth assuming only diffusion parallel to the interface 189 9.2.2 Model of eutectic growth assuming diffusion parallel and perpendicular to the interface; the Jackson-Hunt model 191 9.3 Operating point of cooperative eutectics 195 9.3.1 The extremum criterion 195 9.3.2 Growth at the limit of morphological stability 196 9.4 Divorced eutectics 202 9.5 Modification of eutectics 203 9.6 Interface stability of eutectics 203 9.7 Competitive growth of eutectic and dendritic phases 204 9.8 Equiaxed eutectic grain growth 206 9.9 References 208 10. Peritectic and monotectic solidification 210 10.1 Peritectic solidification 210 10.1.1 The Peritectic Reaction 212 10.1.2 The Peritectic Transformation 213 10.2 Monotectic Solidification 214 10.3 References 218 11. Solidification in the presence of a third phase 219 11.1 Interaction of solid inclusions with the solid/liquid interface 219 11.1.1 Particle interaction with a planar interface 221 11.1.2 Thermodynamic models 222 11.1.3 Thermal properties criterion models 223 11.1.4 Kinetic models 223 11.1.5 Particle interaction with a cellular/dendritic interface 239 11.2 Microshrinkage 241 11.2.1 The physics of microshrinkage formation 241 11.2.2 Modeling of microshrinkage formation 243 11.2.3 Prevention of microshrinkage 250 11.3 References 250 12. Atomic scale phenomena: nucleation and growth 253 12.1 Homogeneous nucleation 253 12.2 Heterogeneous nucleation 259 12.3 Growth kinetics 265 12.3.1 Types of interfaces 265 12.3.2 Continuous growth 268 12.3.3 Lateral growth 269 12.4 References 270 13. Stochastic modeling of solidification 272 13.1 Monte-Carlo grain growth models 275 13.1.1 Dendrite envelope growth models 275 13.1.2 Eutectic S/L interface growth models 275 13.2 Cellular automaton models 277 13.2.1 Cellular automaton grain growth models 277 13.2.2 Cellular automaton dendrite growth models 279 13.3 References 283 14. Macro-micro modeling of solidification of some commercial alloys 284 14.1. Steel 285 13.1.1 Nucleation of primary dendrites 285 13.1.2 Growth of primary dendrites 285 13.1.3 Micro/macro-segregation 288 13.1.4 Inclusions 288 14.2. Cast iron 290 14.2.1 Nucleation and growth of austenite dendrites 290 14.2.2 Crystallization of graphite from the liquid 293 14.2.3 Eutectic solidification 297 14.2.4 Microsegregation 307 14.2.5 The gray-to-white structural transition 307 14.2.6 Room temperature microstructure and mechanical properties 311 14.3. Aluminum-silicon alloys 311 14.3.1 Nucleation and growth of primary aluminum dendrites 311 14.3.2 Eutectic solidification 312 14.4. Superalloys 317 14.5 References 320