Dom Michael Stefanescu

Science and Engineering of Casting Solidification Second Edition

4ü Sprin g er

CONTENTS

Length-scale in solidification analysis 1 References 4 Equilibrium and non-equilibrium during solidification 5

2.1 Equilibrium 5 2.2 The undercooling requirement 6 2.3 Curvature undercooling 9 2.4 Thermal undercooling 11 2.5 Constitutional undercooling 12 2.6 Pressure undercooling 15 2.7 Kinetic undercooling 15 2.8 Departure from equilibrium 17

2.8.1 Local interface equilibrium 19 2.8.2 Interface non-equilibrium 20

2.9 Applications 23 References 23 Macro-scale phenomena - general equations 25

3.1 Relevant Transport Equations 25 3.2 Introduction to diffusive transport 29

3.2.1 Flux laws 29 3.2.2 The differential equation for macroscopic heat transport 30 References 31 Macro-mass transport 33

4.1 Solute diffusion controlled segregation 33 4.1.1 Equilibrium solidification 36 4.1.2 No diffusion in solid, complete diffusion in liquid

(the Gulliver-Scheil model) 38 4.1.3 No diffusion in solid, limited diffusion in liquid 39 4.1.4 Limited diffusion in solid, complete diffusion in liquid 41 4.1.5 Limited diffusion in solid and liquid 44 4.1.6 Partial mixing in liquid, no diffusion in solid 44 4.1.7 Zone melting 47

4.2 Fluid dynamics during mold filling 49 4.2.1 Fluidity of molten metals 49 4.2.2 Capillary flow 49 4.2.3 Gating systems for castings 51

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4.3 Fluid dynamics during solidification 54 4.3.1 Shrinkage flow 55 4.3.2 Natural convection 55 4.3.3 Surface tension driven (Marangoni) convection 58 4.3.4 Flow through the mushy zone 58

4.4 Macrosegregation 60 4.4.1 Fluid flow controlled segregation 61 4.4.2 Fluid flow /solute diffusion controlled segregation 62

4.5 Fluid dynamics during casting solidification - macro shrinkage formation 64 4.5.1 Metal shrinkage and feeding 65 4.5.2 Shrinkage defects 68

4.6 Applications 69 References 74 Macro-energy transport 75

5.1 Governing equation for energy transport 76 5.2 Boundary conditions 77 5.3 Analytical solutions for steady-state solidification of castings 79 5.4 Analytical solutions for non-steady-state solidification of castings 81

5.4.1 Resistance in the mold 84 5.4.2 Resistance at the mold/solid interface 86 5.4.3 The heat transfer coefficient 89 5.4.4 Resistance in the solid 92

5.5 Applications 93 References 96 Numerical Macro-modeling of solidification 97

6.1 Problem formulation 97 6.1.1 The Enthalpy Method 98 6.1.2 The Specific Heat Method 99 6.1.3 The Temperature Recovery Method 99

6.2 Discretization of governing equations 100 6.2.1 The Finite Difference Method - Explicit formulation 100 6.2.2 The Finite Difference Method - implicit formulation 105 6.2.3 The Finite Difference Method - general implicit and

explicit formulation 105 6.2.4 Control-volume formulation 106

6.3 Solution of the discretized equations 107 6.4 Macrosegregation modeling 107 6.5 Macroshrinkage modeling 111

6.5.1 Thermal models 112 6.5.2 Thermal/volume calculation models 114 6.5.3 Thermal/fluid flow models 115

6.6 Applications of macro-modeling of solidification 118 6.7 Applications 121

References 125 Micro-scale phenomena and interface dynamics 127

7.1 Nucleation 128

Contents xiii

7.1.1 Heterogeneous nucleation models 131 7.1.2 Dynamic nucleation models 135

7.2 Micro-solute redistribution in alloys and microsegregation 135 7.3 Interface stability 142

7.3.1 Thermal instability 143 7.3.2 Solutal instability 144 7.3.3 Thermal, solutal, and surface energy driven

morphological instability 148 7.3.4 Influence of convection on interface stability 153

7.4 Applications 154 References 155 Cellular and dendritic growth 157

8.1 Morphology of primary phases 157 8.2 Analytical tip velocity models 160

8.2.1 Solute diffusion controlled growth (isothermal growth) of the dendrite tip 160

8.2.2 Thermal diffusion controlled growth 163 8.2.3 Solutal, thermal, and capillary controlled growth 164 8.2.4 Interface anisotropy and the dendrite tip selection

parameter 171 8.2.5 Effect of fluid flow on dendrite tip velocity 172 8.2.6 Multicomponent alloys 174

8.3 Dendritic array models 175 8.4 Dendritic arm spacing and coarsening 177

8.4.1 Primary spacing 177 8.4.2 Secondary arm spacing 179

8.5 The columnar-to-equiaxed transition 183 8.6 Applications 188

References 193 Eutectic solidification 195

9.1 Classification of eutectics 195 9.2 Cooperative eutectics 197

9.2.1 Models for regular eutectic growth 199 9.2.2 Models for irregular eutectic growth 205 9.2.3 The unified eutectic growth model 207

9.3 Divorced eutectics 211 9.4 Interface stability of eutectics 214 9.5 Equiaxed eutectic grain growth 218 9.6 Solidification of cast iron 219

9.6.1 Nucleation and growth of austenite dendrites 219 9.6.2 Crystallization of graphite from the liquid 222 9.6.3 Eutectic solidification 226 9.6.4 The gray-to-white structural transition 231

9.7 Solidification of aluminum-silicon alloys 233 9.7.1 Nucleation and growth of primary aluminum dendrites 233 9.7.2 Eutectic solidification 233

9.8 Applications 240

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References 244 10 Peritectic solidification 247

10.1 Classification of peritectics 247 10.2 Peritectic microstructures and phase selection 249 10.3 Mechanism of peritectic solidification 254

10.3.1 The rate of the peritectic reaction 255 10.3.2 The rate of the peritectic transformation 257 10.3.3 Growth of banded (layered) peritectic structure 259

10.4 Applications 261 References 262

11 Monotectic solidification 265 11.1 Classification of monotectics 266 11.2 Mechanism of monotectic solidification 266

References 270 12 Microstructures obtained through rapid solidification 271

12.1 Rapidly solidified crystalline alloys 272 12.2 Metallic glasses 276

References 280 13 Solidification in the presence of a third phase 283

13.1 Interaction of solid inclusions with the solid/liquid interface 283 13.1.1 Particle interaction with a planar interface 285 13.1.2 Material properties models 287 13.1.3 Kinetic models 288 13.1.4 Mechanism of engulfment (planar S/L interface) 300 13.1.5 Particle interaction with a cellular/dendritic interface 3 01

13.2 Shrinkage porosity 303 13.2.1 The physics of shrinkage porosity formation 303 13.2.2 Analytical models including nucleation and growth

of gas pores 310 13.2.3 Analysis of shrinkage porosity models and defect prevention 312 References 313

14 Numerical micro-modeling of solidification 317 14.1 Deterministic models 318

14.1.1 Problem formulation 318 14.1.2 Coupling of MT and TK codes 322 14.1.3 Models for dendritic microstructures 323 14.1.4 Microporosity models 333

14.2 Stochastic models 341 14.2.1 Monte-Carlo models 342 14.2.2 Cellular automaton models 346

14.3 Phase field models 355 References 358

15 Atomic scale phenomena - Nucelation and growth 361 15.1 Nucleation 361

15.1.1 Steady-state nucleation - homogeneous nucleation 362 15.1.2 Steady-state nucleation - Heterogeneous Nucleation 368 15.1.3 Time-dependent (transient) nucleation 373

Contents xv

15.2 Growth Kinetics 15.2.1 15.2.2 15.2.3

Types of interfaces Continuous growth Lateral growth

15.3 Applications References

Appendix A Appendix B Appendix C

374 374 377 378 379 382

383 385 391