Science and Engineering of Casting Solidification Second Edition

Science and Engineering of Casting Solidification Second Edition

Doru Michael Stefanescu Science and Engineering of Casting Solidification Second Edition

Doru Michael Stefanescu Department of Materials Science and Engineering The Ohio State University Columbus, OH USA ISBN 978-0-387-74609-8 e-isbn 978-0-387-74612-8 Library of Congress Control Number: 2008929617 2009 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

To my teachers: my mother my father Prof. Laurentie Sofroni Prof. Suzana Gâdea Prof. Carl Loper Max, my grandson my first and dearest my first metallurgy teacher my cast iron teacher my doctorate advisor my American teacher for recasting my understanding of this world

PREFACE "The book of nature is written in mathematical language" Galileo We come to know about the world in two distinctive ways: by direct perception and by application of rational reasoning which, in its highest form, is mathematical thinking. The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. In other words, we only know what we can describe through mathematical models. Casting of metals has evolved first as witchcraft, to gradually become an art, then a technology, and only recently a science. Many of the processes used in metal casting are still empirical in nature, but many others are deep-rooted in mathematics. In whatever form, casting of metals is an activity fundamental to the very existence of our world, as we know it today. Foundry reports indicate that solidification modeling is not only a costeffective investment but also a major technical asset. It helps foundries move into markets with more complex and technically demanding work. The ability to predict internal soundness allows foundries to improve quality and deliveries, and provides the information required to make key manufacturing decisions based on accurate cost estimates before pattern construction even begins. The acceptance of computational modeling of solidification by the industry is a direct result of the gigantic strides made by solidification science in the last two decades. Yet, solidification science is of paramount importance not only in understanding macro- and microscopic changes during the solidification of castings, but is also the basis of many new processes and materials such as semi-solid casting, laser melting, powder atomization, metal matrix composites, bulk metallic glasses. This book is the second attempt by the author to synthesize the information that can be used for engineering calculations pertinent to computational modeling of casting solidification. It includes additional material on the fundamentals of rapid solidification and bulk metallic glasses. This book is based on the author s more than forty years experience of teaching, research and industrial practice of solidification science as applied to casting processes. It is an attempt to describe solidification theory through the complex mathematical apparatus that includes partial differential equations and

viii Preface numerical analysis, required for a fundamental treatment of the problem. The mathematics is however restricted to the elements essential to attain a working knowledge in the field. This is in line with the main goal of the book, which is to educate the reader in the fast moving area of computational modeling of solidification processes. While the book is not intended to be a monograph, for the sake of completion, a special effort has been made to introduce the reader to the latest developments in solidification theory, even when they have no engineering applications at this time. In this respect this is a unique attempt to integrate the newest information in a text book format. The text is designed to be self-contained. The author s teaching experience demonstrates that some of the students interested in solidification science are not fully proficient in partial differential equations (PDE) and/or numerical analysis. Accordingly, elements of PDE and numerical analysis required to obtain a working knowledge of computational solidification modeling have been introduced in the text, while attempting to avoid the interruption of the fluency of the subject. Numerous modeling and calculation examples using the Excel spreadsheet as an engineering tool are provided. The book is addressed to graduate students and seniors interested in solidification science, as well as to industrial researchers that work in the field of solidification in general and casting modeling in particular. The book is divided in 15 major chapters. After introducing the length scale of solidification analysis in the first chapter, the reader is exposed to the basic concepts of driving force for solidification, undercooling, local equilibrium, and interface non-equilibrium from the thermodynamic perspective (Chapter 2). The following three chapters present a detailed analysis of the governing transport equations and their application at the macro-scale level to predict such features of interest in casting solidification as segregation, shrinkage cavity, solidification time and velocity, and temperature gradients. Numerical approximation methods with an emphasis on finite difference approximations are presented in Chapter 6 together with numerous examples of solidification modeling through analytical and numerical methods solved on the Excel spreadsheet. In this chapter, the reader is also introduced to the applications of macro modeling of solidification in today s casting technology. Chapters 7 through 11 extend the transport equations to the study of microscale phenomena and the formation of casting microstructure. Nucleation is discussed from the engineering standpoint that is emphasizing possible methodologies for quantification in solidification analysis of castings. A detailed analysis of existing models for dendritic, eutectic, peritectic and monotectic growth is provided. Again, the emphasis is on the use of this knowledge to build computational solidification models. To achieve this goal, each section of this chapter includes a comprehensive discussion of the applicability and limitations of transferring the information available from steady state analysis to continuous cooling solidification. Chapter 12 extends the concepts introduced earlier to the evolution of microstructure during rapid solidification. Rapidly solidified crystalline alloys and metallic glasses are briefly discussed. The solidification behavior in the presence of a third phase (gaseous or solid impurities) is covered in the 13 th chapter.

Preface ix Chapter 14 is dedicated to the fast moving field of numerical modeling of solidification at the micro-scale. Deterministic and cellular automaton models are covered in detail, while phase field modeling is briefly summarized. The analysis of nucleation and growth at the atomic scale level, required for a complete understanding of solidification and the associated phenomena is presented in chapter 15. Since the current level of understanding does not permit the use of this information directly in computational modeling of solidification, the emphasis is on the physics rather than on engineering.

CONTENTS 1 Length-scale in solidification analysis 1 References 4 2 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 3 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 4 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

xii Contents 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 - macroshrinkage formation 64 4.5.1 Metal shrinkage and feeding 65 4.5.2 Shrinkage defects 68 4.6 Applications 69 References 74 5 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 6 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 7 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 8 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 9 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

xiv Contents 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 301 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 374 15.2.1 Types of interfaces 374 15.2.2 Continuous growth 377 15.2.3 Lateral growth 378 15.3 Applications 379 References 382 Appendix A 383 Appendix B 385 Appendix C 391

NOMENCLATURE C, C o alloy composition p probability * CS interface composition in the solid q diffusion flux C L * interface composition in the liquid r radius (m) D species diffusivity (m 2 s -1 ) t time (s) E internal energy (J mole -1 or J m -3 ) v volume (m 3 ) F G Helmholtz free energy (J mole -1 or J m -3 ) Gibbs free energy (J mole -1 or J m -3 ) gradient v a atomic volume (m 3 atom -1 ) v m molar volume (m 3 mole -1 ) C o H enthalpy (J mole -1, J m -3, J kg -1 ) G v concentration difference between liquid and solid at the solidus temperature change in volumetric free energy (J m -3 ) I intensity of nucleation (m -3 ) H change in volumetric enthalpy (J m -3 ) J mass flux H f K curvature (m -1 ) S f permeability of porous medium (m 2 ) T latent heat of fusion (J mol -1, J kg -1, J m -3 ) entropy of fusion (J mol -1 K -1 or J m -3 K -1 ) undercooling (K) equilibrium constant (Sievert s law) T c constitutional undercooling (K) P pressure (Pa) T k kinetic undercooling (K) Péclet number T o liquidus-solidus interval (K)

xviii Nomenclature Q volumetric flow rate (m 3 s) T r curvature undercooling (K) R gas constant (J mol -1 K -1 ) Γ general diffusion coefficient T temperature (K or C) Gibbs-Thomson coefficient (m K) T L liquidus temperature (K) Φ phase quantity T S solidus temperature (K) α thermal diffusivity (m 2 s -1 ) S entropy (J mol -1 K -1 or J m -3 K -1 ) dimensionless back-diffusion coefficient V velocity (m s -1 ) β T thermal expansion coefficient (K -1 ) V o speed of sound (m s -1 ) β c solutal expansion coefficient (wt% -1 ) c specific heat (J m -3 K -1 ) γ surface energy (J m -2 ) f mass fraction of phase δ boundary layer, disregistry g volume fraction of phase ν kinematic viscosity(m 2 s) g, g gravitational acceleration (m s -2 ) vibration frequency h heat transfer coefficient (J m -2 K -1 s - 1 ) ρ density (kg m -3 ) k solute partition coefficient λ interphase spacing (m) thermal conductivity (W m -1 K -1 ) µ growth constant k B Boltzman constant chemical potential (J mole -1 ) l length (m) dynamic viscosity (N m -2 s) m slope of the liquidus line (K wt% -1 ) θ contact angle mass (kg) τ momentum flux n number of atoms (moles) superscripts subscripts het heterogeneous cr critical hom homogeneous e equilibrium m molar eut eutectic r property related to radius of curvature f fusion * interface g glass het hom heterogeneous homogeneous

Nomenclature xix subscripts subscripts E equivalent, eutectic i component, interface G gas k kinetic L liquid met metastable P particle, pressure n atoms per unit volume S solid r property related to radius of curvature T thermal s surface, stability c constitutional, solutal st stable v property related to volume