Solidification

Solidification is one of the oldest processes known for producing complex shapes for applications ranging from art to industry, and today it still remains one of the most important commercial technologies for many materials. Since the 1980s, numerous fundamental developments in the understanding of solidification processes and microstructure formation have been derived from both analytical theories and the application of computational techniques using commonly available powerful computers. This book integrates these developments in a comprehensive volume that also presents and places them in the context of more classical theories. Divided into three sections, the text evolves from fundamentals to applications, giving professional engineers and students a firm understanding that they can readily apply. The first part, Fundamentals and Macroscale Phenomena, presents the thermodynamics of solutions and then builds on that subject to motivate and describe equilibrium phase diagrams. Transport phenomena are discussed next, focusing on the issues of most importance to liquid-solid phase transformations, then moving on to describe in detail both analytical and numerical approaches to solving such problems. The second part, Microstructure, employs these fundamental concepts for the treatment of nucleation, dendritic growth, microsegregation, eutectic and peritectic solidication, and microstructure competition. This part concludes with a chapter describing the coupling of macro- and microscopic phenomena in microstructure development. Defects, the third and final section describes various types of defects that may occur — with emphasis on porosity, hot tearing, and macrosegregation — presented using the modeling tools and microstructure descriptions developed earlier in the text. Users of this book can find software, figures, movies, and other supporting materials at the author's website by clicking on the downloads tab above.

Table of Contents:
Each chapter starts with an introduction and ends with a Summary, Exercises, and References. NOMENCLATURE AND DIMENSIONLESS GROUPS 1 OVERVIEW 1.1.1 Organization of the text 1.2 Solidification processes 1.2.1 Shape casting 1.2.2 Continuous and semi-continuous casting 1.2.3 Crystal growth processes 1.2.4 Welding 1.3 Summary 1.4 References I Fundamentals and Macroscale Phenomena 2 THERMODYNAMICS 2.1 Introduction 2.2 Thermodynamics of unary systems 2.2.1 Single phase systems 2.2.2 Equilibrium of phases 2.3 Binary alloys 2.3.1 Thermodynamics of a single phase solution 2.3.2 Ideal and regular solutions 2.3.3 Equilibrium of two phases 2.3.4 Multi-component alloys and Gibbs’ phase rule 2.4 Departure from equilibrium 2.4.1 Interfacial equilibrium 2.4.2 True departure from equilibrium 3 PHASE DIAGRAMS 3.1 Motivation 3.2 Binary systems 3.2.1 Isomorphous systems: preliminary concepts 3.2.2 Isomorphous systems: construction from Gibbs free energy curves 3.2.3 Eutectic systems 3.2.4 Peritectic systems 3.2.5 Other binary systems 3.2.6 Calculation of binary alloy phase diagrams 3.3 Ternary systems 3.3.1 Ternary isomorphous systems 3.3.2 Ternary three-phase equilibrium 3.3.3 Ternary four-phase equilibrium: ternary eutectic 4 BALANCE EQUATIONS 4.1.1 Reference frames and definitions 4.1.2 Control volumes 4.2 Mass balance 4.3 Momentum balance 4.3.1 Linear elastic solids 4.3.2 Plastic deformation 4.3.3 Newtonian fluids 4.3.4 Average form of the momentum balance 4.4 Energy balance 4.5 Solute balance in multicomponent systems 4.6 Scaling 4.7 Summary 4.8 Exercises 4.9 References 5 ANALYTICAL SOLUTIONS FOR SOLIDIFICATION 5.1 Introduction 5.2 Solidification in a superheated melt 5.2.1 Pure materials 5.2.2 Planar front solidification of a binary alloy 5.2.3 Transient solidification of a binary alloy at constant velocity 5.3 Solidification in an undercooled melt 5.3.1 Planar front growth 5.3.2 Solidification of a paraboloid 5.4 The effect of curvature 5.4.1 Solidification of a sphere in an undercooled melt 5.5 Summary and conclusions 6 NUMERICAL METHODS FOR SOLIDIFICATION 6.1 Introduction 6.2 Heat conduction without phase change 6.2.1 Finite difference method 6.2.2 Finite volume method 6.2.3 Finite element method 6.3 Heat conduction with phase change 6.3.1 Fixed grid: Enthalpy methods 6.3.2 Fixed grid: Temperature recovery methods 6.3.3 Front tracking methods 6.3.4 Level set methods 6.4 Fluid flow 6.4.1 Finite difference method on staggered grids 6.4.2 Finite element methods for CFD 6.4.3 Example: Melting of pure Ga 6.5 Optimization and inverse methods II Microstructure 7 NUCLEATION 7.2 Homogeneous nucleation 7.2.1 Embryos and nuclei 7.2.2 Nucleation rate 7.3 Heterogeneous nucleation 7.3.1 Motivation 7.3.2 Basic theory 7.3.3 Instantaneous or athermal nucleation 7.4 Mechanisms for grain refinement 8 DENDRITIC GROWTH 8.2 Free growth 8.2.1 General observations 8.2.2 Stability and scale selection for a freely growing sphere of a pure material 8.2.3 Extension to binary alloys 8.3 Constrained growth 8.3.1 General observations 8.3.2 Length scales and pattern selection in constrained growth 8.3.3 Stability of planar front growth in binary alloys 8.4 Growth of a needle crystal 8.4.1 General observations 8.4.2 Approximate models for growth at the dendrite tip 8.4.3 Primary dendrite arm spacing in constrained growth 8.4.4 Secondary dendrite arm spacing: Coarsening 8.5 Convection and dendritic growth 8.5.1 Convection and free growth 8.5.2 Convection and columnar growth 8.6 Phase-field methods 9 EUTECTICS, PERITECTICS AND MICROSTRUCTURE SELECTION 9.2 Eutectics 9.2.1 General considerations 9.2.2 Coupled eutectic growth morphologies 9.2.3 Jackson-Hunt analysis for regular eutectics 9.2.4 Operating point and stability of regular eutectic 9.2.5 Irregular eutectics 9.2.6 Other eutectic morphologies 9.3 Peritectics 9.3.1 General considerations 9.3.2 Nucleation 9.3.3 Solidification of peritectics at normal speed 9.3.4 Solidification of peritectics at low speed 9.4 Phase selection and coupled zone 9.4.1 Phase competition 9.4.2 Coupled zone 10 MICROSEGREGATION AND HOMOGENIZATION 10.2 1-D microsegregation models for binary alloys 10.2.1 Microsegregation with diffusion in the solid state 10.2.3 Volume averaged model 10.3 Homogenization and solution treatment . 10.3.1 Homogenization 10.3.2 Solution heat treatment 10.4 Multicomponent alloys 11 MACRO- AND MICROSTRUCTURES 11.2 Equiaxed grains growing in a uniform temperature field 11.2.1 Nucleation and growth of equiaxed eutectic grains 11.2.2 Transition from globular to dendritic grain morphologies 11.2.3 Nucleation and growth of equiaxed dendritic grains 11.3 Grains nucleating and growing in a thermal gradient 11.4 Columnar grains 11.5 Columnar-to-Equiaxed Transition 11.5.1 Hunt’s criterion 11.5.2 Microsegregation and cooling curves 11.6 Micro-macroscopic models 11.6.1 Thermal conditions 11.6.2 Analytical models of microstructure formation 11.6.3 Stochastic models of microstructure formation 11.6.4 Influence of convection III Defects 12 POROSITY 12.2 Governing equations 12.3 Interdendritic fluid flow and pressure drop 12.3.1 Darcy equation 12.3.2 Niyama criterion 12.4 Thermodynamic of gases in solution 12.5 Nucleation and growth of pores 12.5.1 Pore Nucleation 12.5.2 The role of curvature during growth 12.5.3 Contribution of gas diffusion during growth 12.5.4 Summary of the coupling between pressure and pore fraction 12.6 Boundary conditions 12.7 Application of the concepts 13 DEFORMATION DURING SOLIDIFICATION AND HOT TEARING 13.2 Thermomechanics of castings 13.2.1 Origins of thermal stresses 13.2.2 General formalism for a fully solid material 13.2.3 Examples 13.3 Deformation of the mushy zone 13.3.1 Rheological measurements on semi-solid alloys 13.3.2 Coherency 13.3.3 Two-phase approach 13.4 Hot tearing 13.4.1 Characteristics of hot tears 13.4.2 Hot tearing tests and hot tear sensitivity 13.5 Hot tearing criteria and models 14 MACROSEGREGATION 14.2 Macrosegregation during planar front solidification 14.2.1 Thermal convection in a pure material 14.2.2 Convection during directional solidification of a binary alloy 14.3 Composition field and governing equations 14.4 Macrosegregation induced by solidification shrinkage 14.4.1 Initial solidification at the mold surface 14.4.2 Steady state 14.4.3 Final transient 14.5 Macrosegragation induced by fluid flow 14.5.1 Analysis based on Flemings’ criterion 14.5.2 General approach 14.5.3 Freckle formation 14.6 Macrosegregation induced by solid movement 14.6.1 Macrosegregation induced by grain movement 14.6.2 Macrosegregation induced by solid deformation