Table of contents for Science and engineering of casting solidification / Doru Michael Stefanescu.


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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








Library of Congress subject headings for this publication: Metal castings Mathematical models, Solidification Mathematical models