Fundamentals of Solidification [5 ed.]

Table of contents :
Fundamentals of Solidification 5th Edition
Foreword
Table of Contents
Chapter One: Introduction
Chapter Two: Atom Transfer at the Solid-Liquid Interface
Chapter Three: Morphological Instability of a Solid/Liquid Interface
Chapter Four: Solidification Microstructure: Cells and Dendrites
Chapter Five: Solidification Microstructure: Eutectic and Peritectic
Chapter Six: Solute Redistribution
Chapter Seven: Rapid Solidification Microstructures
Chapter Eight: Solidification Microstructure Selection Maps
Appendix 1: Mathematical Modelling of Solidification at the Macroscopic Scale
Appendix 2: Analytical Solute and Heat Flux Calculations Related to Microstructure Formation
Appendix 3: Local Equilibrium at the Solid/Liquid Interface
Appendix 4: Nucleation Kinetics in a Pure Substance
Appendix 5: Atomic Structure of the Solid/Liquid Interface
Appendix 6: Thermodynamics of Rapid Solidification
Appendix 7: Analysis of Morphological Interface Stability
Appendix 8: Diffusion at a Dendrite Tip
Appendix 9: Dendrite Tip Radius and Spacing
Appendix 10: Eutectic Growth
Appendix 11: Transients in Solute Diffusion
Appendix 12: Mass Balance Equations
Appendix 13: Homogenisation of Interdendritic Segregation in the Solid State
Appendix 14: Introduction to the Phase-Field Method
Appendix 15: Relevant Physical Properties for Solidification
Symbols
Keyword Index

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2023. Trans Tech Publications Ltd. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.

Fundamentals of Solidification 5th Fully Revised Edition

Wilfried Kurz David J. Fisher Michel Rappaz

Cover: This view of a dendrite was obtained by etching away the Co12Sm2 matrix of a directionally solidified Co-Sm-Cu peritectic alloy from the primary Co dendrites. The cut surfaces (light) in the foreground are those which are typically seen on the surface of a polished section of a solidified alloy, and are the only visible signs of the three-dimensional dendrite as seen in two dimensions.

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Fundamentals of Solidification 5th Fully Revised Edition

Wilfried Kurz Professor, Institute of Materials Ecole Polytechnique Fédérale de Lausanne EPFL Lausanne - Switzerland

David J. Fisher Dr, Associate Editor TTP c/o JCR, Trevithick Building, Cardiff University, The Parade, Cardiff CF24 3AA, UK and

Michel Rappaz Professor, Institute of Materials Ecole Polytechnique Fédérale de Lausanne EPFL Lausanne - Switzerland

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Copyright  2023 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Seestrasse 24c CH-8806 Baech Switzerland https://www.scientific.net

Volume 103 of Foundations of Materials Science and Engineering ISSN print 2297-8143 ISSN web 2297-816X

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Foreword of the 4th Revised Edition Solidification phenomena play an important role in many of the processes used in fields ranging from production engineering to solid-state physics. For instance, a metal is usually continuously cast or ingot cast before forming it into bars or sheets. The bars then often serve as input to a sand-, permanent mould-, or precision-casting operation while the sheet is often fabricated into useful items by welding; another solidification process. At the other extreme, silicon is usually first prepared in the form of an impure reduction product and then, for electronic applications, has to be purified by zone-refining (a solidification process) and pulled, as a single crystal, from its melt. This broad range of interest in solidification, from the large tonnages of continuously cast products, through the intermediate weight output of superalloy precision castings, to the relatively small quantities of high-purity crystals, means that a book such as the present one must cater for the requirements of a very wide range of readers. To begin with, there is the graduate or final-year undergraduate who may eventually find himself dealing with any problem in the above range, and must therefore be thoroughly conversant with the basic principles and mathematical theory of the subject. Then there is the post-graduate researcher who may need to produce metallic specimens having a well-defined microstructure and, in order to do this, must be able to bring to bear all of the current understanding of solidification mechanisms. Finally, there is the foundryman who would like to exert close control over a cast product, but must contend with so many variables and unknown quantities that his work takes on the aspects of an art. It is hoped that this book will be of value to all three groups and, at least, provide the student with an introduction to modern solidification theory; the researcher with the fundamentals of the more quantitative models for predicting solidification microstructures; and the foundryman and the welding engineer with a framework into which he can fit his diverse observations. The ground covered by the present introduction is essentially the same as that covered by textbooks such as Winegard's "Introduction to the Solidification of Metals", and Chalmers' "Principles of Solidification", both of which were published in 1964, and Flemings' "Solidification Processing", published in 1974. Many of the ‘loose ends' of solidification theory, whose interrelationships were unclear ten or twenty years ago, have now been drawn together and many of the qualitative arguments which were a feature of the latter books can now be largely replaced by more quantitative models. It is currently possible to present major parts of the theory as a coherent whole, and to fit solidification microstructures into a logical framework. That is not to say that the present-day solidification literature is immediately accessible to the newcomer. Much of the most useful information is buried within a mass of mathematical formulae and scattered among many journals. Therefore, the aim here has been to collect together the key results obtained by the present and other authors, and to derive simpler solutions whenever possible. The sources of the models used can be found in the references at the end of each chapter but, for easier reading, are often not referred to in the text. In order to obtain the maximum benefit from this book, the reader should note that it is based upon a hierarchical scheme within which the subject matter can be studied at 3 levels. Firstly, an initial feeling for the subject and for the breadth of coverage, can be obtained simply by reading the extensive figure captions. Secondly, the main text describes the principles in more detail, but usually without deriving the necessary equations. Thirdly, the appendices contain detailed derivations and some essential mathematical background. It is stressed that only those readers who are specialising in the subject would usually need to study the appendices in detail.

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Within the main text, an essentially self-contained guide to the subject is presented. After a general overview, the reader is introduced at first to the mechanisms of crystal nucleation and growth which occur at the atomic scale (chapter 2). Subsequently, it is shown how the form of an initially planar solid/liquid interface evolves (chapter 3), before the most important single-phase (cellular and dendritic; chapter 4) and multi-phase (eutectic and peritectic; chapter 5) solid/liquid interface morphologies are presented. The section on peritectic growth has been updated for this fourth edition. The effect which solidification has upon the redistribution of solute is then discussed (chapter 6). Finally, the behaviour which is to be expected at high solidification rates is introduced (chapter 7). It is now possible to describe fully the effects of both high and low growth rates upon the microstructure. One subject which is not covered, due to the difficulties of presenting analytical results, is convection in the melt and its interaction with solidification microstructures. Each chapter includes a bibliography of key references for further study, as well as exercises which are designed to test the reader's understanding of the contents of the preceding chapter. For certain exercises, it is advisable firstly to work through the corresponding appendices. The authors hope that, after reading this book, the newcomer will feel confident when delving further into solidification-related subjects, and that the experienced engineer will also find some thought-provoking points. W. Kurz, D.J. Fisher

Lausanne, January 1998

Foreword of the 5th Fully Revised Edition The previous edition of Fundamentals of Solidification appeared 25 years ago, at the end of the last century. With its more than 6,000 citations it has secured its place in the literature on Materials Science and Engineering. During the intervening years however, great progress has been made, both theoretically and experimentally, in the understanding and modelling of solidification microstructures. The authors reasoned that a new, completely revised edition, would be of interest to students, engineers and others who want to know more about the topic in its present scientific state. The new co-author, who was a reviewer of the first edition in 1984, could be persuaded to join the project, thus ensuring an authoritative treatment of the newer topics. The most paradigm-shifting additions are: ■ A grain-size model combining both inoculation and growth; ■ The diffuse interface and its anisotropy; ■ An introduction to the phase-field method; ■ A new primary dendrite spacing model; ■ The effects of melt-flow upon dendrite growth; ■ Conditions for coupled peritectic growth; ■ General approach to solidification microstructure selection; ■ Initiation of banding in rapid solidification; ■ Stable to metastable phase selection in rapid solidification processing; ■ Columnar-to-equiaxed transition; ■ Coupled zone of eutectics; ■ Application of solidification concepts to additive manufacturing; ■ In situ imaging of solidifying metals.

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Specific highlights of the 5th edition include two new contributions; appendix 14 on phase-field modelling and chapter 8 on microstructure-selection. Appendix 14 is an original and easy-to-follow introduction to the rather involved mathematics of the phase-field method, now the method-of-choice for most microstructure computations. In 3D it involves fairly heavy computations and is therefore at present essentially limited to singular solidification phenomena and cannot yet be applied to the complex interacting solidification phenomena taking place in an industrial casting, a weld or in an additive manufactured part with e.g. Marangoni flow and dendrite fragmentation controlling the columnar to equiaxed transition. Yet, the possibility of simulating the solidification microstructure of a whole component is of paramount importance in industrial processes. It is there where analytical models as developed in this book can accelerate substantially the computations. Chapter 8 is a concluding section which explains how to combine the concepts of the seven preceding chapters of the book so as to model analytically real microstructures that form during complex solidification processes. These and the numerous other additions made to the chapters, make this 5th edition an essentially new book on the fundamentals of solidification. The topics are presented in the same well-received manner as in previous editions, being readable at three levels: (i) an initial feeling for the content of the chapter is obtained by consulting the figures with their detailed captions; (ii) a deeper understanding of the underlying physics is found by working through the main text; and (iii) 15 appendices offer an in-depth analysis of the various theories, by providing detailed derivations of the relevant equations. Each chapter includes a list of key references, including recommended articles for further study, as well as exercises which are designed to test the reader's understanding of the contents of the chapter. The authors hope that this introductory book will offer its readership a good start in the field, and prepare them for tackling more involved treatments of solidification, such as the Dantzig and Rappaz book, Solidification.

W. Kurz, D.J. Fisher, M. Rappaz Lausanne, December 2022

Acknowledgements Thanks go to R. Trivedi who, over several decades, was a most inspiring colleague and friend of the first author. The authors express their special thanks to Alain Karma, who contributed generously to the book, especially to appendix 14 on the phase-field method. Furthermore, they wish to thank for their contributions to and comments concerning the manuscript: R. Abbaschian, M. Apel, C. Beckermann, B. Billia, T.W. Clyne, J. Dantzig, P. Galenko, A.L. Greer, D.M. Herlach, H. Jones, M. Lorenz, A. Ludwig, A. Mortensen, J.H. Perepezko, A. Philion, D.R. Poirier, P.R. Sahm, T. Sato, F. Spaepen, T. Umeda, and P. Voorhees. Thanks go also to former PhD students and postdocs who have contributed to the modelling of solidification microstructures; M. Carrard, S. Dobler, H. Esaka, S. Fukumoto, C.-A. Gandin, M. Gäumann, P. Gilgien, S. Gill, B. Giovanola, M. Gremaud, O. Hunziker, J. Lipton, P. Magnin, S. Mokadem, P. Thévoz, M. Vandyoussefi, M. Wolf and M. Zimmerman.

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Wilfried Kurz is professor emeritus at the Ecole Polytechnique Fédérale de Lausanne (EPFL). After his doctorate from the University of Leoben, Austria, and 8 years at the Battelle Memorial Institute at Geneva, he joined EPFL where he worked as head of the Physical Metallurgy Laboratory and the Laser Materials Processing Centre. His main research interests are on microstructure development during solidification of materials in advanced processing techniques. He was cofounder of the first Swiss university curriculum in Materials Science and Engineering at EPFL in 1974, member of the Board of Swiss and European Materials Societies and member of the Board of Governors of Acta Materialia Inc. as well as Corresponding Member of the Austrian Academy of Sciences. He has published 4 books, including the bestseller Fundamentals of Solidification, and 250 scientific papers and patents. A Highly Cited Researcher (ISI-Thomson Reuters, 2001), he is recipient of international awards from TMS and ASM-International (USA), SF2M (France), DGM (Germany), AIM (Italy), IOM3 (UK), ISIJ (Japan), and FEMS, the Federation of European Materials Societies. David Fisher started work, as a junior metallurgist at Rolls Royce Aeroengines, on improving alloys for the turbine blades of Concord while simultaneously earning a BSc in physical metallurgy from the University of Wales. He then became one of Professor Kurz’s first doctoral students and presented a thesis on the use of organic analogues; the better to guide both composite turbine blade creation via the directional solidification of eutectic alloys and improving the understanding of the microstructures of traditional casting alloys. This work partially inspired the writing of the first draft of Fundamentals of Solidification. He then moved into scientific publishing and acted as an editor of several journal and book series. During that time he also wrote or co-authored books on subjects ranging from hydrogen-storage and high-entropy alloys to an encyclopaedic work on silicon nitride. His other interests range from the Bayesian and meta analysis of experimental results to empirical approximations of physical laws. In recent years he has written nearly 20 monographs on state-of-the-art topics in physics and materials science, including dissimilar-metal joining via molten-zone propagation or reaction-diffusion. Michel Rappaz has made a PhD in solid state physics at the Ecole Polytechnique Fédérale de Lausanne (EPFL). After a post-doc at Oak Ridge National Laboratory and two years in an engineering company, he joined in 1984 the laboratory of Prof. Wilfried Kurz at the Institute of Materials of EPFL to start an activity focused on solidification modelling. His main interests are the coupling of macroscopic aspects of heat and mass transfer of casting and welding processes with microscopic aspects of microstructure and defect formation. Nominated Adjunct Professor in 1990 and Full Professor in 2003, he retired in 2015 and is now Emeritus Professor and independent consultant. Michel Rappaz has received several awards, in particular the Mathewson award of TMS (1997), the Grand Medal from the French Materials Society (2011), the Bruce Chalmers Award of TMS (2002), the FEMS European Materials Gold Medal (2013), the Brimacombe Prize of TMS (2015) and a title of Doctor of Science - honoris causa from McMaster Univ. (2019). He is a highly-cited author of ISI, a fellow of ASM and TMS, and has coauthored more than 200 publications and two books.

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Table of Contents Foreword Chapter One: Introduction Chapter Two: Atom Transfer at the Solid-Liquid Interface Chapter Three: Morphological Instability of a Solid/Liquid Interface Chapter Four: Solidification Microstructure: Cells and Dendrites Chapter Five: Solidification Microstructure: Eutectic and Peritectic Chapter Six: Solute Redistribution Chapter Seven: Rapid Solidification Microstructures Chapter Eight: Solidification Microstructure Selection Maps Appendix 1: Mathematical Modelling of Solidification at the Macroscopic Scale Appendix 2: Analytical Solute and Heat Flux Calculations Related to Microstructure Formation Appendix 3: Local Equilibrium at the Solid/Liquid Interface Appendix 4: Nucleation Kinetics in a Pure Substance Appendix 5: Atomic Structure of the Solid/Liquid Interface Appendix 6: Thermodynamics of Rapid Solidification Appendix 7: Analysis of Morphological Interface Stability Appendix 8: Diffusion at a Dendrite Tip Appendix 9: Dendrite Tip Radius and Spacing Appendix 10: Eutectic Growth Appendix 11: Transients in Solute Diffusion Appendix 12: Mass Balance Equations Appendix 13: Homogenisation of Interdendritic Segregation in the Solid State Appendix 14: Introduction to the Phase-Field Method Appendix 15: Relevant Physical Properties for Solidification Symbols

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1 17 51 67 109 133 149 173 213 223 241 251 255 259 265 277 285 297 309 313 321 325 341 343

Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 1-15 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER ONE

INTRODUCTION 1.1. The Importance of Solidification Solidification is a phase transformation which is familiar to everyone, even if the only acquaintance with it involves the making of ice cubes in a refrigerator. It is relatively little appreciated that the manufacture of almost every man-made object involves solidification at some stage. The sheet-metal required to make for automobile bodies for example is obtained by the continuous casting of slabs of steel or aluminium which are then rolled to their final dimensions. The scope of this book is restricted mainly to a presentation of the theory of solidification as it applies to the widely-used metallic alloys. Here, solidification is generally accompanied by the formation of crystals; an event which is rarer during the solidification of inorganic glasses or polymers. Solidification is of such importance simply because its major practical applications, namely casting, welding or additive manufacturing, are economical methods for forming a component if the melting point of the metal is not too high. Metal products can be economically melted and solidified nowadays from alloys having melting points as high as 1660°C (titanium). Between solid and liquid there exists an enormous difference in viscosity, of some twenty orders of magnitude, as illustrated in Fig. 1.1. Instead of expending energy against the typically high flow stress of a solid metal during mechanical forming, it is thus possible to contend now only with the essentially zero shear stress of a liquid that almost eagerly fills a mould of complex shape. If the properties of solidified products were easier to control, then solidification would be an even more important process. Solidification theory plays a vital role in this respect since it forms the basis for influencing the microstructure, controlling the development of defects and hence improving the quality of the product. The effect of solidification is most evident when casting is the final operation since the resultant properties can depend markedly upon the position in the casting (Fig. 1.2). Its influence is also seen in a finished product, even after heavy working, since a solidification structure and its associated defects are difficult to eliminate once they are created. Solidification defects tend to persist throughout *

Top image: Air cooled gas turbine blade produced by advanced solidification processing. The directionally cast dendritic single crystal consisting of a ten-component superalloy allows substantial increase in the efficiency of aircraft engines.

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Fundamentals of Solidification 5th Edition

subsequent operations (Fig. 1.3). Good control of the solidification process at the outset is therefore of utmost importance.

Figure 1.1 Dynamic Viscosity as a Function of Temperature The fundamental advantage of solidification, as a forming operation, is that it permits metal to be shaped with a minimum of effort since the liquid metal offers very little resistance to shear stresses. When the material solidifies due to a decrease in the temperature, 𝜏, its viscosity increases continuously (formation of the amorphous structure of glass) or discontinuously (crystallisation, red curve), by over 20 orders of magnitude, to yield a strong solid, the viscosity of which is defined arbitrarily to be greater than 1014 Pa s. (The reduced temperature, 𝜏, is used here since it leads to a single curve which is applicable to many substances. The suffix, f or v, indicates the melting (fusion) point or boiling point, respectively. (Turnbull, 1961†)).

Some processes which involve solidification are shown in Table 1. Simple cast objects (in copper) first appeared before about 4000BC and were, no doubt, a natural by-product of the potter's skill in handling the clay used in furnace- and mould-making. The production of the renowned and highly sophisticated bronze castings of China began in about 1600BC. It is probable however that the technique had originally been imported from elsewhere. The lost-wax process for example was developed in Mesopotamia as long ago as 3000BC. Iron-casting in China began in about 500BC but, in Europe, cast iron did not appear until the 16th century and achieved acceptance as a constructional material only in England in the 18th century, under the impetus of the industrial revolution. Much of the delay experienced in exploiting cast materials probably originated from the complete lack of understanding of the nature of solidification phenomena and consequently of the microstructures produced. Fracture surfaces were invariably taken to reflect the nature of the 'crystals' of which a casting was supposedly composed. In the absence of an adequate picture of the solidification process, casting was bound to remain a black art rather than a science.

†

The full references can be found at the end of each chapter.

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Foundations of Materials Science and Engineering Vol. 103

Figure 1.2 Alloy Properties as a Function of Position, 𝒅, in a Casting The use of casting as a production route poses its own problems. One of these is the local variation of the microstructure, illustrated for example by the dendrite arm spacing, 𝜆2 , measured as a function of the distance, 𝑑, from the surface of the casting. This can lead to a resultant variation in properties such as the ultimate tensile strength, σ, and the elongation, ε. Like the weakest link in a chain, the inferior regions of a casting may impair the integrity of the whole. It is therefore important to understand the factors which influence the microstructure. Finer microstructures generally have superior mechanical properties and finer structures, in turn, generally result from higher solidification rates. Such rates are found at small distances from the surface of a mould, in thin sections, or at laser-remelted surfaces (Flemings, 1974). Table 1 Solidification Processes Casting

continuous-, ingot-, form-, precision-, die-

Additive Manufacturing

laser powder deposition, selective powder sintering

Welding

arc-, resistance-, plasma-, electron beam-, laser-, friction-

Soldering/Brazing Rapid Solidification Processing

melt-spinning, planar-flow casting, atomisation, bulk undercooling, surface remelting

Directional Solidification

Bridgman, liquid metal cooling, single crystal casting, Czochralski single crystal growth, electroslag remelting

Wear

micro-welding mechanism

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Fundamentals of Solidification 5th Edition

Figure 1.3 Effect of Deformation upon a Cast Microstructure Casting is often not the final forming operation. Subsequent deformation is however not a very efficient method for modifying the as-cast microstructure since any initial heterogeneity exhibits a strong tendency to persist. During the rolling of this L-shaped profile, which contains a heavily segregated central region, the latter defect thus survives the many stages between the cast billet at the left and the final product of some 100 mm in width at the right. This example emphasises the fact that an effective control of product quality must be exercised during solidification. (Pokorny, 1966)

The crystallisation of certain pure substances is also of great importance. The preparation of semiconductor-grade silicon crystals by using the Czochralski method is, for example, an essential feature of modern solid-state physics and technology. The production of integrated circuits; the basis of electronic devices such as computers, mobile phones or watches, requires the preparation of large single crystals of very high perfection, containing a controlled amount of a uniformly distributed dopant. At the moment such a crystal can be produced only by growth from the melt. The requirements of semiconductor physics have thus enormously influenced solidification theory and practice. During the past century solidification evolved from being a purely technological empirical field to become a science. From now on the term “solidification processes” will be used for the techniques listed in Table 1 such as casting, welding or additive manufacturing. 1.2. Heat Extraction The various solidification processes which are referred to above necessitate the extraction of heat from the melt in a more or less controlled manner. Heat extraction changes the energy of the phases (solid and liquid) in two ways: 1. There is a decrease in the enthalpy of the liquid or solid, due to cooling, given by: Δℎ = ∫ 𝑐𝑑𝑇‡. 2. There is a decrease in enthalpy, due to the transformation from liquid to crystal, which is equal to the latent heat of fusion, Δℎ𝑓 . Heat extraction is achieved by applying a suitable means of cooling to the melt in order to create an outward heat flux, q (< 0). The resultant cooling rate, dT/dt, can be deduced from a simple heat balance if the metal is isothermal (high thermal conductivity, low cooling rate) and if the specific heats of the liquid and the solid are the same. Using the latent heat per unit volume, Δℎ𝑓 = Δ𝐻𝑓 /𝑣𝑚 (defined to be positive for solidification), together with the specific heat per unit volume, 𝑐, gives:

‡

A list of symbols is given at the end of the book.

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Foundations of Materials Science and Engineering Vol. 103

5

𝐴′ 𝑑𝑇 𝑑𝑓𝑠 𝑞 ( ) = 𝑐 ( ) − Δℎ𝑓 ( ) 𝑣 𝑑𝑡 𝑑𝑡 so that: 𝑇̇ =

𝑑𝑇 𝐴′ 𝑑𝑓𝑠 Δℎ𝑓 = 𝑞( ) + ( )( ) 𝑑𝑡 𝑣𝑐 𝑑𝑡 𝑐

[1.1]

The first term on the right-hand-side (RHS) of Eq. 1.1 reflects the effect of casting geometry (ratio of surface area of the casting, 𝐴′ , to its volume, v) upon the extraction of sensible heat, while the second term takes account of the continuing evolution of latent heat effusion during solidification. It can be seen from this equation that, during solidification, heating will occur if the second term on the RHS of Eq. 1.1 becomes greater than the first one. This phenomenon is known as recalescence. When solidification occurs over a range of temperatures, as in the case of an alloy, the variation of the fraction of solid as a function of time must be calculated from the relationship: 𝑑𝑓𝑠 𝑑𝑓𝑠 𝑑𝑇 = ( )( ) 𝑑𝑡 𝑑𝑇 𝑑𝑡 since 𝑓𝑠 is a function of temperature. In this case: 𝑇̇ =

𝐴′ 𝑞 (𝑣𝑐 ) Δℎ𝑓 𝑑𝑓 1 − ( 𝑐 ) ( 𝑠) 𝑑𝑇

[1.2]

It can be seen that solidification decreases the cooling rate, since 𝑑𝑓𝑠 /𝑑𝑇 is negative. Figure 1.4 illustrates two fundamentally different solidification processes. Between an initial and final transient (Fig. 1.4(a)), the heat is steadily extracted by moving the crucible at a fixed rate, V ', through the temperature profile that is imposed by the furnace. Such a process is typically used for single-crystal growth or directional solidification. It permits the growth rate of the solid, 𝑉 (that is not necessarily equal to the rate of crucible movement - see Exercise 1.9), and the temperature gradient, 𝐺, to be separately controlled. If 𝑉 ′ is not too high, both the heat flux and the solidification are unidirectional. The cooling rate at a given location and time is given by: ∂𝑇 ∂𝑇 ∂𝑧 ′ ̇𝑇𝑠+𝜀 = ( ) =( ′ ) = 𝐺 𝑉|𝑠+𝜀 ∂𝑡 𝑠+𝜀 ∂𝑧 ∂𝑡 𝑠+𝜀

[1.3]

where the time-dependent position of the solid/liquid interface is s = z' - z, and z' is the coordinate with respect to the system (the crucible), while z is the coordinate with respect to the moving s/l interface (Fig. 1.4; see also Fig. A2.1), and  is a small quantity with respect to s. Here, 𝑉 is the rate of movement of the s/l interface and 𝐺 is the thermal gradient in the liquid when 𝑧 ′ = 𝑠 + 𝜀 (also called 𝐺𝑙 ) or the thermal gradient in the solid when 𝑧 ′ = 𝑠 − 𝜀 (called 𝐺𝑠 ), due to differences in the conductivity of solid and liquid and to the evolution of latent heat at a moving interface, 𝐺𝑙 ≠ 𝐺𝑠 . For reasons of simplicity the temperature gradient in the liquid will be mostly used in this book and written as 𝐺 ≡ 𝐺𝑙 . One has to bear in mind, however, that under certain conditions (as explained in Chap. 3) the physically meaningful temperature gradient is the mean gradient, 𝐺̅ = (𝐺𝑠 𝜅𝑠 + 𝐺𝑙 𝜅𝑙 )/(𝜅𝑠 + 𝜅𝑙 ).

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Fundamentals of Solidification 5th Edition

Figure 1.4 Basic Methods of Controlled Solidification Without heat extraction there is no solidification. The liquid must be cooled to the solidification temperature and then the latent heat of solidification appearing at the growing solid/liquid interface must be extracted. There are several methods of heat extraction. In directional (Bridgman-type) solidification (a): the crucible is drawn downwards through a constant temperature gradient, 𝐺, at a uniform rate, 𝑉 ′ , and therefore the microstructure is highly uniform throughout the specimen. The method is restricted to small specimen diameters and is expensive because it is slow and, paradoxically, heat must be supplied during solidification in order to maintain the imposed positive temperature gradient. For these reasons it is employed only for research purposes and for the growth of single crystals. In directional casting (b), the benefits of directionality, such as a better control of the properties and an absence of detrimental macrosegregation, are retained but the microstructure is no longer uniform along the specimen because the growth rate, 𝑉, and the temperature gradient decrease as the distance from the chill increases. Both processes are used for the production of columnar or monocrystalline gas-turbine blades. Combined with proper alloy development, these turbine blades result in a higher efficiency and a longer life for the gas-turbines of aircraft. (Note that the vertically hatched solid may represent single phase or eutectic solids.)

Another directional casting process is illustrated by Fig. 1.4(b). Heat is extracted here via a chill and, as in Fig. 1.4(a), growth occurs in a direction which is antiparallel to the heat-flux direction. In this situation, the heat-flux decreases with time as do the coupled parameters, 𝐺 and 𝑉. Thus 𝑇̇ which is the product of 𝐺 and 𝑉 also varies. Heat flow in the mould/metal system leads to an expression for the position, 𝑠, of the solid/liquid interface, which is of the type (Appendix 1): 𝑠 = 𝐾𝑡1/2

[1.4]

This equation is exact only if the melt is not superheated, if the solid/liquid interface is planar and if the surface temperature of the casting at the chill drops immediately, at 𝑡 = 0, to a constant value. In a casting, columnar grains with a dendritic morphology grow into the liquid, as shown schematically in Fig. 1.5. The insert represents a volume element of the casting which contains an infinitesimal narrow volume element in the mushy zone, the region between fully liquid and fully solid. The narrow volume can be regarded as an element that corresponds to Fig. 1.4(b). Now solidification on a microscopic scale takes place directionally and perpendicular to the primary growth axis of the dendrites. This representation permits a simple estimation of interdendritic microsegregation (Chap. 6).

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Figure 1.5 Solidification in Conventional Castings and Ingots In the vast majority of castings, no directionality is imposed upon the overall structure, but the local situation can be seen to be equivalent to that existing in directional casting (Fig. 1.4(b)). This is true of the way in which the solid advances inwards from the mould wall to form a columnar zone. During the growth of the columnar zone, three regions can be distinguished. These are the liquid, the liquid plus solid (so-called mushy zone), and the solid region. The mushy zone is the region where all of the microstructural characteristics are determined, e.g. the shape, size, and distribution of concentration variations, precipitates, and pores. (The mushy zone in this figure is highly schematic.) An infinitesimally narrow volume element which is fixed in the mushy zone and is perpendicular to the overall growth direction permits a description of the microscopic solidification process and therefore of the scale and composition of the microstructure.

1.3. Solidification Microstructures Three zones of solidification behaviour can generally be distinguished within a casting, (Fig. 1.6). The cooling rate is at its highest at the mould/metal interface due to the initially low relative temperature of the mould. Many small grains having random orientations are consequently nucleated at the mould surface and an 'outer equiaxed' zone is formed. These grains rapidly become dendritic, and develop arms which grow along preferred crystallographic directions (001 in the case of cubic crystals). The dendrite growth depends upon the alloy composition, and upon 𝐺 and 𝑉 (Chap. 4). Competitive growth between the randomly-oriented outer equiaxed grains causes those which have a favourable growth direction (parallel and opposite to the direction of heat flow) to eliminate the others. This is because their higher growth rate allows them to dominate the solid/liquid interface morphology, thus leading to the formation of the characteristic columnar zone. It is often observed that another equiaxed zone forms in the centre of the casting, mainly as a result of the growth of detached dendrite arms within the remaining slightly-undercooled liquid. Figure 1.7 shows the temperature fields in the various cast structures which might be typically encountered. In pure materials these are crystals with a planar interface (columnar grains – a) or

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Fundamentals of Solidification 5th Edition

thermal dendrites (equiaxed grains – b) and, in alloys, solutal (constitutional) dendrites (c, d). It can be seen that columnar grains always grow from the mould (which is the heat sink) in a direction which is opposite to that of the heat flow, while equiaxed grains grow in a supercooled melt which acts as their heat sink. The growth direction and the heat-flow direction are therefore the same in equiaxed growth.

Figure 1.6 Structural Zone Formation in Castings During cooling of a casting crystal nuclei having different crystallographic orientations first appear in the liquid at, or close to, the mould wall. They increase in size for a short time and form the outer equiaxed zone. Those crystals (dendrites) of the outer equiaxed zone which can grow parallel and opposite to the heat-flow direction then advance most rapidly. Other orientations tend to be overgrown, due to mutual competition, leading to the formation of a columnar zone (a). Beyond a certain stage in the development of the columnar dendrites, branches which become detached from the latter can grow independently. These tend to take up an equiaxed shape because their latent heat is extracted radially through the undercooled melt. The solidified region which contains them is called the inner equiaxed zone (b). The transition from columnar to equiaxed growth varies with alloy concentration and solidification conditions and is highly dependent upon the degree of convection in the liquid. Electromagnetic stirring is often used in continuous casting equipment in order to promote this transition and thus lead to a superior soundness of the ingot centre.

The form of a solidification microstructure depends not only upon the cooling conditions, but also upon the alloy composition (Fig. 1.8). There are essentially two basic growth morphologies which can exist during alloy solidification. These are the dendritic and eutectic morphologies §. A mixture of both morphologies will usually be present. It is reassuring in the face of the apparent microstructural complexity to remember that it is necessary to understand only these two growth forms in order to interpret the solidification microstructure of almost any alloy. Figure 1.9 illustrates the various stages of equiaxed solidification — from nucleus to grain — for the two major growth morphologies: dendrites and eutectic. Each grain has one nucleus at its origin. (In the literature on cast iron, an eutectic grain is often called an eutectic cell. This definition will not be adopted because the term “cell” is reserved here for another morphology which can develop in columnar growth when transitioning from planar to dendritic.)

§

Peritectic alloys typically grow in a dendritic manner but can also form eutectic-like structures.

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The transformation of liquid into solid involves the creation of curved solid/liquid interfaces (leading to capillarity effects) and the microscopic flow of heat (and also solute in the case of alloys). In normal processing of metals atom attachment kinetics are negligeable.

Figure 1.7 Solid/Liquid Interface Morphology and Temperature Distribution For solidification from the mould wall (hatched region), columnar grains are shown in (a) and (c) without the outer equiaxed zone for clarity. In the case of a pure metal which is solidifying inwards from the mould, the columnar grains (a) possess an essentially planar interface, and grow in a direction which is antiparallel to that of the heat flow. Within the equiaxed region of a pure metal (b), the crystals are dendritic and grow radially in the same direction as the heat flow. When alloying elements or impurities are present, the morphology of the columnar crystals (c) is generally dendritic. The equiaxed morphology in alloys (d) is almost indistinguishable from that in pure metals, although a difference may exist in the relative scale of the dendrites. This is because the growth in pure metals is heat-flow controlled, while the growth in alloys is mainly solute-diffusion controlled (in metals solute diffusion is much slower than heat diffusion). Note that in columnar growth the hottest part of the system is the melt while, in equiaxed solidification, the crystals are the hottest part.

1.4. Capillarity Effects With any solid/liquid interface of area, 𝐴, there is associated an excess interface energy which is required for its creation. Heterogeneous systems which possess a high 𝐴/𝑣 ratio will therefore be in a state of higher energy and thus unstable with respect to a system of lower A/v ratio, hence resulting in a coarsening of the structure. The relative stability of a heterogeneous system can be expressed by the equilibrium temperature between the two phases: the melting point. As shown in Appendix 3, the change in the melting temperature due to this curvature effect, often called the curvature or Gibbs-Thomson undercooling, is given by: Δ𝑇𝑟 = 𝐾Γ

[1.5]

Note that the curvature, 𝐾, and the Gibbs-Thomson coefficient, Γ, are defined here so that a positive undercooling (decrease in equilibrium melting point) is associated with a portion of solid/liquid interface which is convex towards the liquid phase. The curvature can be expressed as (Appendix 3):

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Fundamentals of Solidification 5th Edition

Figure 1.8 Principal Alloy Types It is important to understand how the various microstructures are influenced by the alloy composition and by the solidification conditions. This can fortunately be reduced in general to the study of two basic morphological forms: dendritic and eutectic. It is thus possible to distinguish: a) pure substances, which solidify in a planar or dendritic manner, b) solid-solution dendrites (with or without interdendritic precipitates), c) dendrites plus interdendritic eutectic and d) eutectic. The latter group includes the familiar ‘cast iron' and ‘Al-Si' alloys. In general, the design of casting alloys is governed by the twin aims of obtaining the required properties and good castability (i.e. easy mould filling, low shrinkage, small hot tearing tendency, etc.). Castability is greatest for pure metals and alloys of eutectic composition. The diagram presents, as a typical example, the Al-Cu system between Al and the intermetallic (theta) phase, Al2Cu.

𝐾=

𝑑𝐴 1 1 = + 𝑑𝑣 𝑟1 𝑟2

[1.6]

where 𝑟1 and 𝑟2 are the principal radii of curvature**. Thus, the total curvature of a sphere is 2/r and that of a cylindrical surface is 1/r. The Gibbs-Thomson coefficient is given by: Γ = 𝜎⁄Δ𝑠𝑓

[1.7]

For most metals Γ is of the order of 10-7 Km (see Appendix 15). The effect of the solid/liquid interface energy, , therefore becomes important only for morphologies which have a radius that is less than about 10 m. These include nuclei, interface perturbations, dendrite tips, and eutectic phases (Fig. 1.10). 1.5. Solute Redistribution The creation of a crystal from an alloy melt causes a local change in the composition. This is due to the equilibrium condition for a binary system containing two phases (Appendix 3): 𝜇𝑙A = 𝜇𝑠A   ,  𝜇𝑙B = 𝜇𝑠B

[1.8]

** The two principal radii of curvature are the minimum and maximum values for a given surface. It can be shown that they lie in planes which are always perpendicular to one another (Gauss, 2005).

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The difference in composition at the growing interface, assuming that local (i.e. at the interface) equilibrium exists in metals under normal solidification conditions, can be described by the distribution coefficient under isothermal and isobaric conditions (Fig. 1.11): 𝑘 = ( 𝐶𝑠 ⁄𝐶𝑙 ) 𝑇,𝑃

[1.9]

Figure 1.9 Process of Equiaxed Solidification of Dendrites and Eutectic In the cases of dendritic or eutectic solidification, single-phase nuclei form initially. In pure metals or single-phase alloys (a), the nuclei then grow into spherical crystals which rapidly become unstable and dendritic in form. These dendrites grow freely in the melt and finally impinge on one another. In a pure metal following solidification, no trace of the dendrites themselves will remain, although their points of impingement will be visible as the grain boundaries. The dendrites will remain visible in an alloy, upon etching, due to local composition differences (microsegregation). In an eutectic alloy (b) a second phase will soon nucleate on the initial singlephase nucleus. The eutectic grains then continue to grow in an essentially spherical form. In a casting both growth forms, dendritic and eutectic, often develop together. Note that each grain originates from a single nucleus.

In most of the theoretical (analytical) treatments to be presented later, the solidus and liquidus lines will be assumed to be straight and to intersect at zero concentration. In this case, which is typical of dilute solutions, 𝑘, the distribution coefficient, and 𝑚, the liquidus slope, are constant. This violates the condition for thermodynamic equilibrium (Eq. 1.8) but often makes theoretical analyses more tractable. At high concentrations, where the liquidus and solidus lines do not intersect at zero

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Fundamentals of Solidification 5th Edition

concentration, the distribution coefficient has to be deduced from the ratio of the liquidus and solidus slopes and not from that of the concentrations (Trivedi and Kurz, 1990). That is: 𝑘 ∗ = 𝑚𝑙 /𝑚𝑠

[1.10]

If the variation in composition is large however due, for example, to a large variation in growth rate, this assumption will lead to incorrect results. The problem has then to be treated using numerical methods, preferably combined with a thermodynamics database.

Figure 1.10 Various Solid/Liquid Interface Morphologies Solidification morphologies, typically of the size of μm, are determined by the interplay of two effects acting at the solid/liquid interface. These are the diffusion of solute (or heat), which tends to minimise the diffusion path and therefore the scale of the morphology (maximising curvature), and capillarity effects which tend to minimise interface energy and therefore maximise the scale. The crystal morphologies which are actually observed are thus a compromise between these two opposing tendencies, and this will be shown with respect to nucleation (a), morphological interface instability (b), dendritic growth (c) and eutectic growth (d).

The term, 𝑚, is defined throughout this book so that the product, (𝑘 − 1)𝑚, is positive: i.e. 𝑚 is defined to be positive when 𝑘 is greater than unity and to be negative when 𝑘 is less than unity. Two other important parameters of an alloy system are shown in Fig. 1.11. These are the liquidussolidus temperature interval, Δ𝑇0, for an alloy of composition, 𝐶0 : Δ𝑇0 = −𝑚Δ𝐶0 = (𝑇𝑙 − 𝑇𝑠 )

[1.11]

and the concentration difference between the liquid and solid solute contents at the solidus temperature of the alloy:

Δ𝐶0 =

𝐶0 (1 − 𝑘) 𝑘

[1.12]

Under rapid solidification conditions, Equation 1.8 may no longer be satisfied, and 𝑘 then becomes a function of 𝑉. This so-called non-equilibrium solidification can lead to a highly supersaturated crystal (see Chap. 7). Furthermore, the attachment kinetics of atoms to the solid becomes important (see Chap. 2). It will be shown later how the above parameters influence the solidification microstructure. The starting point of solidification (nucleation) will be briefly considered next, together with an introduction to the mechanisms via which atoms in the melt become part of the growing crystal (attachment kinetics). Both phenomena typically occur at the atomic scale.

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Figure 1.11 Solid/Liquid Equilibrium In order to simplify the mathematical treatment of solidification processes, it is generally assumed that the liquidus and solidus lines of the phase diagram are straight and converge to 𝑇𝑓 at 𝐶0 = 0, i.e. the distribution coefficient, 𝑘 , and the liquidus slope, 𝑚, are constants. The characteristic properties of the system are defined in the text (Eqs 1.9 to 1.12).

Exercises 1.1

Discuss the shape of the upper surface of the ingot in Fig. 1.6. What would happen if the solidifying material was one of the following substances: water, Ge, Si, Bi?

1.2

From a consideration of the volume element in the mushy zone of Fig. 1.5 (upper part), define the local solidification time, tf, in terms of the dendrite growth rate, V, and the length, a, of the mushy zone.

1.3

Sketch 𝜂 − 𝜏 diagrams for the crystallisation of a pure metal and for an alloy, and comment on their significance in each case.

1.4

Equiaxed dendrites are developing freely in an undercooled melt. Discuss the direction of movement of the equiaxed dendrites in a quiescent melt. Where would most of them be found in solidifying melts of a) steel, b) Bi?

1.5

Sketch two different phase diagrams having a positive and a negative value of 𝑚. Show that the product, 𝑚(𝑘 − 1), is always positive.

1.6

Using data from phase-diagram compilations (e.g. Kubaschewski, 1982), estimate the distribution coefficient, 𝑘, of S in Fe at temperatures of between 1500 and 988°C, and of Cu in Ni at temperatures of between 1400 and 1300°C. Discuss the validity of the assumption that 𝑘 is constant in these systems.

1.7

A molten alloy, like any liquid which has local density variations, will tend to exhibit the motion known as natural convection. What is the origin of this convection in a) pure metals, b) alloys? Discuss your conclusions with regard to various alternative solidification processes such as upward (as opposed to downward) directional solidification (Bridgman - Fig. 1.4), and casting (Fig. 1.5).

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1.8

Give possible reasons for the good mould-filling characteristics which are exhibited by pure metals and eutectic alloys during the casting of small sections. Discuss them with regard to the interface morphology shown in Fig. 1.7(a) and (c).

1.9

Figure 1.4 illustrates two directional solidification processes which differ with respect to their heat transfer characteristics. In one case, a steady-state behaviour is established after some transient changes. In the other case, changes continue to occur with the passage of time. One process is not limited with regard to the length of the product, but is limited by its diameter. The other process is not affected by the diameter, but rather by the specimen length. Sketch heat flux lines, and 𝐺 and 𝑉 values as a function of 𝑡 for both cases and relate them to the described characteristics of the two processes. Note that, in directional solidification, the temperature gradient in the liquid at the solid/liquid interface must always be positive, as shown in Fig. 1.7(a,c).

1.10 Illustrate the changes in the temperature distribution of a casting as a function of time between the moment of pouring of a pure superheated melt and the establishment of the situation shown in Figs 1.7(a) and (b). Discuss the fundamental differences between (a) and (b). References and Further Reading Solidification, General ▪ H.Biloni, W.J.Boettinger, Solidification, in Physical Metallurgy, R.W.Cahn, P.Haasen (Eds), 4th Edition, North Holland, 1996, p.669. ▪ J.A.Dantzig, M.Rappaz, Solidification, 2nd Edition, EPFL Press, 2016. ▪ M.C.Flemings, Solidification Processing, McGraw-Hill, New York, 1974, p.328. ▪ M.E.Glicksman, Principles of Solidification, Springer, 2011. ▪ M.Hillert, in Lectures on the Theory of Phase Transformations, H.I.Aaronson (Ed.), TMS of AIME, New York, 1975, p.1. ▪ A.J. Pokorny, De Ferri Metallographia, Vol. III, Luxembourg, 1966, p.287. ▪ D.M.Stefanescu, Science and Engineering of Casting Solidification, 3rd Edition, Springer, 2015. ▪ D.Turnbull, The liquid state and the liquid-solid transition, Transactions of the Metallurgical Society of AIME, 221 (1961) 422. Analytical Solutions to Heat Flow Problems in Solidification ▪ H.S.Carslaw, J.C.Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, London, 1959. ▪ G.H.Geiger, D.R.Poirier, Transport Phenomena in Metallurgy, Addison-Wesley, 1973. ▪ J.Szekely, N.J.Themelis, Rate Phenomena in Process Metallurgy, Wiley - Interscience, New York, 1971. Thermodynamics of Solidification / Phase Diagrams ▪ J.C.Baker, J.W.Cahn, in Solidification, American Society for Metals, Metals Park, Ohio, 1971, p.23. ▪ W.J.Boettinger, The Solidification of Multicomponent Alloys, Journal of Phase Equilibria and Diffusion, 37 (2016) 4. ▪ D.R.Gaskell, Metallurgical Thermodynamics, in Physical Metallurgy, R.W.Cahn, P.Haasen (Eds), 4th Edition, North Holland, 1996, p.413. ▪ M.Hillert, Phase Equilibrium, Phase Diagrams and Phase Transformations: their Thermodynamic Basis, 2nd Edition, Cambridge University Press, Cambridge, 2007.

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▪ U.R.Kattner, The thermodynamic modeling of multicomponent phase equilibria, Journal of Metals, 12 (1977) 14. ▪ O.Kubaschewski, Iron-Binary Phase Diagrams, Springer, Berlin, 1982. ▪ T.B.Massalski et al. (Eds), Binary Alloy Phase Diagrams, ASM, Metals Park, Ohio, 1990. ▪ Metals Handbook - Volume 8, ASM, Metals Park, Ohio, 1985. ▪ Thermo-Calc Software AB, 16967 Solna, Sweden. ▪ R.Trivedi, W.Kurz, Modeling of solidification microstructures in concentrated solutions and intermetallic systems, Metallurgical Transactions A, 21 (1990) 1311. Capillarity Effects ▪ J.A.Dantzig, M.Rappaz, Solidification, 2nd Edition, EPFL Press, 2016. ▪ C.F.Gauss, General Investigations of Curved Surfaces, Dover, 2005. ▪ W.W.Mullins, in Metal Surfaces - Structure, Energetics, and Kinetics, American Society for Metals, Metals Park, Ohio, 1963, p.17. ▪ R.Trivedi, in Lectures on the Theory of Phase Transformations, H.I.Aaronson (Ed.), The Metallurgical Society of AIME, New York, 1975, p.51. Casting / Welding / Additive Manufacturing ▪ S.S.Babu, N.Raghavan, J.Raplee et al., Additive Manufacturing of Nickel Superalloys, Metallurgical and Materials Transactions A, 49 (2018) 3764. ▪ T.F.Bower, D.A.Granger, J.Keverian, in Solidification, American Society for Metals, Metals Park, Ohio, 1971, p.385. ▪ M.Dal, R.Fabbro, An overview of the state of art in laser welding simulation, Optics & Laser Technology, 78 (2016) 2. ▪ T.DebRoy, H.L.Wei, J.S.Zuback, T.Mukherjee, J.W.Elmer, J.O.Milewski, A.M. Beese, A.Wilson-Heid, A.Ded, W.Zhang, Additive manufacturing of metallic components – process, structure and properties, Progress in Materials Science, 92 (2018) 112. ▪ R.Flinn, in Techniques of Metals Research - Volume I, R.F.Bunshah (Ed.), Wiley, New York, 1968. ▪ C.Koerner, M. Markl, J.A.Koepf, Modeling and simulation of microstructure evolution for additive manufacturing of metals: a critical review, Metallurgical and Materials Transactions A, 51 (2020) 4970. ▪ K.Mehta, Advanced Joining and Welding Techniques: an Overview, in Advanced Manufacturing Technology, Materials Forming, Machining and Tribology, K.Gupta (Ed.), Springer 2017, p.101. ▪ F.L.Versnyder, M.E.Shank, The development of columnar grain and single crystal high temperature materials through directional solidification, Materials Science and Engineering, 6 (1970) 213.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 17-50 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER TWO

ATOM TRANSFER AT THE SOLID-LIQUID INTERFACE From a thermodynamic point of view solidification requires a heat flux from the system to the surroundings which changes the free energies of the phases, and therefore their relative thermodynamic stability. From the same point of view thermodynamically stable phases are more likely to be detected but the transformation of one phase into another requires rearrangement of the atoms, with intermediate metastable states often being observed. As in the case of a pure substance these changes involve extensive atomic short-range rearrangements to form a crystal having a new structure. In the case of alloys atomic motion may have to occur over much larger, but still microscopic, distances when solute-diffusion controls the transformation. Because of these atomic movements solidification will always require some irreversible departure from equilibrium to exist in order to drive the process. Like chemical reactions, phase transformations are produced by thermal fluctuations and can occur only when the probability of transfer of atoms from the parent phase to the product phase is higher than that for the reverse process. Before this stage is reached it is necessary however that some of the new phase, to which atoms of the parent phase can migrate, should be present. Random atomic fluctuations within liquid metals create minute regions of different crystal structures (clusters, embryos), even at temperatures which are greater than the melting point, but these will not be stable. They continue to be metastable below the melting point because the relatively large excess energy which is required for surface creation tends to weight the ‘energy balance’ against their survival when they are small. Once nucleation has occurred, i.e. when a critical size of the new phase has been attained, atom transfer to the new crystals ensures their growth. 2.1. Conditions for Nucleation As indicated by Fig. 2.1 nucleation begins during cooling at some degree of undercooling, Δ𝑇 = Δ𝑇𝑛 †, which is generally very small for metals under most practical conditions. The initially *

Top image: cross-section through a critical nucleus of a hard sphere alloy computed with the Monte Carlo technique (Ganagalla and Punnathanam, 2013). This nucleus contains some 180 atoms and would have a diameter in case of Al of 2 nm. † The undercooling, Δ𝑇, is usually defined as the temperature difference between the equilibrium temperature of a system and its actual temperature. The latter is lower than the equilibrium temperature when the melt is undercooled. In this case, T is positive. The term, supercooling, is often used interchangeably with undercooling in the literature.

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tiny grains which begin to grow do not appreciably modify the cooling rate which is imposed by the external heat flux, 𝑞 (< 0). Increasing the undercooling has the effect of markedly increasing the nucleation rate, 𝐼, and also the growth rate, 𝑉, of the grains. As presented in Chap. 1, the overall solidification rate approaches a maximum value when the product of the latent heat of fusion and the volume rate of transformation, 𝑓𝑠̇ (= 𝑑𝑓𝑠 /𝑑𝑡), is equal to the external flux in absolute value (𝑞) (Eq. 1.1). The maximum undercooling is attained here and 𝑇̇ = 0. During the first stage of equiaxed solidification, which is essentially nucleation-controlled, the volume fraction of solid is still small. The temperature of the system then rises and the second stage of solidification becomes growthcontrolled. The number of grains thus remains essentially constant and solidification first proceeds via the lengthening of dendrites, and then via dendrite-arm thickening when the growing grains come in contact‡. On the basis of this reasoning, it is possible to deduce that nucleation is the dominant process at the beginning of solidification and leads very rapidly to the establishment of the final grain population, with each nucleus forming one equiaxed grain of the type shown in Fig. 1.9(a) or (b). Note that, even in the case of columnar solidification, the very first solid to appear in a casting is always in the form of equiaxed grains (outer equiaxed zone in Fig. 1.6). The conditions which lead to nucleation are therefore of utmost importance in determining the characteristics of any solidification microstructure. It is inherently difficult to observe the process of nucleation because it generally involves minute clusters of atoms. Advances in understanding have been made by the careful comparison of theoretical models with experimental results, by high-resolution transmission electron microscopy and by atomistic modelling (e.g. molecular dynamics methods, Sect. 2.5). Solidification involves interface creation and cannot occur at any arbitrarily small undercooling. A small crystal perforce possesses a large curvature of the interface, and this markedly changes the equilibrium melting point (Appendix 3): the smaller the crystal, the lower is its melting point. This occurs because the curvature creates a pressure difference between the two phases, and this can be of the order of 100 MPa (1 kbar) for a crystal radius of 1 nm (assuming that macroscopic thermodynamics remains valid at these scales). The equilibrium melting point of the system is thus lowered by an amount, Δ𝑇𝑟 . The critical size, 𝑟 ○ , of a crystal, i.e. the size which allows equilibrium to exist between the curved crystal and its melt, can be easily calculated. For a sphere (Eq. 1.5, Appendix 3), it is given by: Δ𝑇𝑟 = 𝐾Γ =

2Γ 𝑟○

and

𝑟○ =

2Γ 2𝜎 = Δ𝑇𝑟 Δ𝑇𝑟 Δ𝑠𝑓

[2.1]

This relationship indicates that, the smaller the difference between the melting point and the temperature of the melt (i.e. undercooling), the larger will be the size of the equilibrium crystal. For nucleation of a spherical crystal of radius, 𝑟, to occur, a number of atoms, each of volume, 𝑣 ′ , given by: 4𝑟 3 π 𝑛≅ 3𝑣 ′

[2.2]

have to arrange themselves on the sites of the corresponding solid crystal lattice. It is evident that the probability of this process occurring is very small for large values of 𝑟 ○ , i.e. at small undercoolings (Eq. 2.1).

‡

If the nucleus-density is high, dendrites cannot form and the grains retain a more spherical morphology.

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Figure 2.1 Thermal History of Equiaxed Dendritic Solidification The above temperature-time curve (a) that was pictured in Fig. 1.9(a) is typical of the solidification sequence of a polycrystalline alloy. The usual cooling curve first begins to deviate slightly at the undercooling at which nucleation occurs, Δ𝑇𝑛 = 𝑇𝑙 − 𝑇𝑛 . The first fraction of solid, 𝑓𝑠 , appears (c,d) at this point and affects the cooling rate (b). During further cooling the nucleation rate, 𝐼, rapidly increases to a maximum value (e). The cooling curve (T − t) reaches a minimum when the latent heat that is generated by transformations occurring in the volume of the specimen is equal to the heat extracted from its surface, i.e. the external heat flux, multiplied by the surface area. During the subsequent increase in temperature spherical grains or dendrites grow which ultimately “touch” one another via their solute fields. Note that the maximum of the cooling curve can lie above the nucleation temperature and that the nucleation rate 𝐼 (dashed curve) is much more sensitive to temperature changes than is 𝑉 (e). While the very early moments of transformation are rich in microstructure development (Fig. 2.9), most of the solidification process which takes place following impingement of the grains involves dendrite-arm coarsening (increase in spacing and solid volume fraction, 𝑓𝑠 ). During this time interval the tip growth rate, 𝑉, equals zero and the number of grains, 𝑁, remains constant. An analytical model and further details are given in Chap. 8.

As shown in Fig. 2.2, the critical condition for nucleation is derived by summing the interface and volume terms for the Gibbs free energy: Δ𝐺 = Δ𝐺𝑖 + Δ𝐺𝑣 = 𝜎𝐴 + Δ𝑔𝑣

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[2.3]

20

Fundamentals of Solidification 5th Edition

Figure 2.2 Free Energy of a Crystal Cluster as a Function of its Radius The phenomenon of nucleation of a crystal from its melt depends mainly upon two processes: thermal fluctuations which lead to the creation of variously sized crystal embryos (clusters), and creation of an interface between the liquid and the solid. The free energy change, Δ𝐺𝑣 , which is associated with the first process is proportional to the volume transformed. That is, it is proportional to the cube of the cluster radius. The free energy change, Δ𝐺𝑖 , which is associated with the second process is proportional to the area of solid/liquid interface which is formed. It is therefore proportional to the square of the cluster radius. At temperatures, 𝑇, greater than the melting point (a), both the volume free energy (Δ𝐺𝑣 ) and the surface free energy (Δ𝐺𝑖 ) increase monotonically with increasing radius, 𝑟. The total free energy, Δ𝐺, which is their sum therefore also increases monotonically. At the melting point (b), the value of Δ𝐺𝑖 still increases monotonically because it is only slightly temperature-dependent. Because thermodynamic equilibrium exists between the solid and liquid at the melting point, by definition, the value of Δ𝐺𝑣 is zero. Therefore Δ𝐺 again increases monotonically with increasing radius. At a temperature below the equilibrium melting point (c), the sign of Δ𝐺𝑣 is reversed because the liquid is now metastable, while the behaviour of Δ𝐺𝑖 is still the same as in (a) and (b). However Δ𝐺𝑣 has a 3rdpower dependence upon the radius while Δ𝐺𝑖 has only a 2nd-power dependence. At small values of the radius the absolute value of Δ𝐺𝑣 is less than that of Δ𝐺𝑖 while, at large values of 𝑟, the cubic dependence of Δ𝐺𝑣 predominates. The value of Δ𝐺 therefore passes through a maximum at a critical radius, 𝑟 ○. Fluctuations may move the cluster backwards and forwards along the Δ𝐺 − 𝑟 curve (c) due to the effect of random additions or removals of atoms to or from the unstable nucleus (d). When a fluctuation causes the cluster to become larger than 𝑟 ○ , growth will occur due to the resultant decrease in the total free energy. An embryo or cluster (𝑟 < 𝑟 ○ ) thus becomes a nucleus (𝑟 = 𝑟 ○ ) and eventually a grain (𝑟 ≫ 𝑟 ○ ). This thermally activated phenomenon is a stochastic process.

where 𝜎 is the solid/liquid interface energy, 𝐴 is the interface area, Δ𝑔 is the Gibbs free energy difference between the liquid and solid per unit volume and 𝑣 is the volume of the crystal. Again, assuming the simplest form for the nucleus, which is a sphere having the minimum 𝐴/𝑣 ratio, leads to:

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Foundations of Materials Science and Engineering Vol. 103

4π𝑟 3 Δ𝐺 = 𝜎4𝜋𝑟 + Δ𝑔 3

21

[2.4]

2

where the Gibbs free energy per unit volume, Δ𝑔, is proportional to T (Appendix 3): [2.5]

Δ𝑔 = −Δ𝑠𝑓 Δ𝑇

The right-hand-side of Eq. 2.4 is composed of a quadratic and a cubic term in the nucleus radius. The value of 𝜎 is always positive whereas g depends upon Δ𝑇, and is negative if Δ𝑇 is positive. This behaviour leads to the occurrence of a maximum in the value of Δ𝐺 when the melt is undercooled, i.e. when 𝑇 < 𝑇𝑓 (Fig. 2.2(c)). This maximum value can be regarded as being the activation energy which has to be overcome in order to form a crystal nucleus which will continue to grow. The criterion for the maximum is that: 𝑑(Δ𝐺) =0 𝑑𝑟

[2.6]

and can be regarded as being a condition for equilibrium between a liquid, and a solid with a curvature such that the driving force for solidification is equal to that for melting. It is thus not surprising that setting the first derivative of Eq. 2.4 equal to zero should lead to Eq. 2.1. Figure 2.2(d) demonstrates how fluctuations in a melt, corresponding to the conditions of Fig. 2.2(c), will behave. At least one cluster which is as large as the critical nucleus (of radius, 𝑟 ○ ) must be formed before solidification can begin. The time which elapses before this occurs will be different (t1, t2, …) at different locations in the melt. In this case fluctuations spontaneously create a small crystalline volume in an otherwise homogeneous melt (containing no solid phase). This is referred to as homogeneous nucleation because the occurrence of nucleation transforms an initially homogeneous system (consisting only of atoms in the liquid state) into a heterogeneous system (crystals plus atoms in a liquid). Using Eqs 2.2 to 2.6, the critical parameters can be calculated and are given in Table 2.1 where Δ𝐺𝑛 is equivalent to Δ𝐺 in Eq. 2.3; except that n (the number of atoms in the nucleus, Eq. 2.2) rather than the radius, r, has been used to describe the nucleus size. Table 2.1 Critical Dimensions and Activation Energy for the Nucleation of a Spherical Nucleus in a Pure Melt (𝚫𝒈 = −𝚫𝒔𝒇 𝚫𝑻) Homogeneous Nucleation 𝑟○

−

𝑛○

−(

Δ𝐺𝑛○

(

Heterogeneous Nucleation

2𝜎 Δ𝑔

32π 𝜎 3 )( ) 3𝑣 ′ Δ𝑔

16π 𝜎3 ) ( 2) 3 Δ𝑔

− −( (

2𝜎 Δ𝑔

32π 𝜎 3 ) ( ) 𝑓(𝜃) 3𝑣 ′ Δ𝑔

16π 𝜎3 ) ( 2 ) 𝑓(𝜃) 3 Δ𝑔

Suppose as an example that an undercooling of 230 K is required in order to cause homogeneous nucleation in small copper droplets. From this value and the properties of the metal (Appendix 15) it can be estimated that 𝑟 ○ = 1.28 nm and 𝑛○ = 634. When the melt contains solid particles, or is in contact with a crystalline crucible or oxide layer, nucleation may be facilitated if the critical number of atoms is decreased, or the presence of an existing surface reduces the activation energy required for nucleation. This is known as heterogeneous nucleation (Fig. 2.3). A purely geometrical calculation shows that, when the solid/liquid interface of the substance is partly replaced by an area of low-energy solid-solid interface

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Fundamentals of Solidification 5th Edition

between the crystal and a foreign solid substrate, nucleation can be greatly facilitated. The magnitude of the effect can be calculated by using the result derived in Appendix 3 (Fig. 2.3(b)): (2 + cos 𝜃)(1 − cos 𝜃)2 𝑓(𝜃) = 4

[2.7]

where 𝜃 is the solid-solid wetting angle, in the presence of the melt, between a growing spherical cap of crystal α (nucleus) and a solid substrate, s (particle or mould wall) (Fig. 2.3(a)). Note that the n° and Δ𝐺𝑛○ values are decreased by small values of 𝜃 whereas the 𝑟 ○ value is not (Fig. 2.3(c)).

Figure 2.3 Heterogeneous Nucleation on a Particle (Substrate) and the Effect of Wetting Angle between a Crystal-Embryo and a Substrate A schematic drawing of the cap of a heterogeneous nucleus is shown (a) together with the evolution of 𝑓(𝜃) of Eq. 2.7 in the form of an S-curve (b). The change in the free energy of nucleation for homogeneous and heterogeneous nucleation is also shown (c). The wetting of the substrate by the crystal in the presence of the melt reduces the activation energy for nucleation without changing the radius of the nucleus.

Under conditions of good solid-solid wetting (small 𝜃) between the crystal nucleus and the foreign substrate in the melt, a large decrease in 𝑛○ and Δ𝐺 ○ can be expected. This can have a dramatic effect upon nucleation and is used daily in foundries in the form of inoculation (see Sect. 2.3). Substances are there added to the melt which form crystals during cooling (but also at temperatures greater than the melting point). The effect is usually time-dependent because the added substances tend to dissolve. Another source of foreign nuclei is fragmentation; dendrite branches detach from the trunk and begin to grow immediately when they reach the undercooled melt. The above arguments have been developed for pure metals with or without foreign particles. They can also be applied to alloys. In this case the Gibbs free energy is not only a function of nucleus size, but also of composition. To a first approximation, the critical size and composition would be deduced in this case from the conditions, 𝑑(Δ𝐺)/𝑑𝑛 = 0 and 𝑑(Δ𝐺)/𝑑𝐶 = 0; which define a saddlepoint.

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2.2. Rate of Nucleus Formation In order to calculate the number of crystals nucleated within a given melt volume and time period (called the nucleation rate), the simplest case will be considered. This is an ideal mixture of an ensemble of 𝑁𝑙 atoms of liquid and 𝑁𝑛 small crystal clusters, each of which contains 𝑛 atoms. The equilibrium distribution (solubility) of these clusters can be calculated (Appendix 4), leading to the result (when 𝑁𝑛 ≪ 𝑁𝑙 ): 𝑁𝑛 −Δ𝐺𝑛 = exp [ ] 𝑁𝑙 𝑘𝐵 𝑇

[2.8]

Figure 2.4 Dependence of Cluster-Size Distribution upon Temperature Here Δ𝐺𝑛 is the free energy of a cluster containing n atoms, at two temperatures (a,c), and 𝑁𝑛 /𝑁𝑙 (b,d) is the ratio of clusters containing 𝑛 atoms to 𝑁𝑙 the number of atoms in the liquid phase. There is an exponential relationship between Δ𝐺𝑛 and 𝑁𝑛 . Thermal fluctuations are always creating small crystalline regions in the liquid, even at temperatures greater than the melting point (a). The relative number of clusters, 𝑁𝑛 , will be much smaller for large clusters than for small ones (b). This variation in the distribution of cluster sizes is represented schematically (a) as a shaded grey level (dark = high density). At temperatures below 𝑇𝑓 (c), there will be a maximum in the free energy Δ𝐺𝑛○ of the fluctuating system, as also shown by Fig. 2.2(c). The clusters (nuclei) which reach this critical size will continue to grow. The corresponding cluster-concentration, 𝑁𝑛○ /𝑁𝑙 , and cluster size, 𝑛○ (truncated minimum in figure d), are sensitive functions of the undercooling. The nucleation rate will depend upon the number of clusters which have the critical size, 𝑁𝑛○ /𝑁𝑙 .

Equation 2.8 indicates that there are always crystal clusters present in a melt, even though they are not always stable. The number of clusters is shown schematically in Fig. 2.4, where the darkness of the shade of grey increases with the number of clusters, and that number increases with decreasing value of Δ𝐺𝑛 . If the melt is superheated, 𝑑(Δ𝐺)/𝑑𝑛 is always positive and the equilibrium concentration of crystal nuclei is zero. In an undercooled melt (Fig. 2.4(c)), a maximum in Δ𝐺𝑛 as a function of 𝑛 exists, over which clusters can ‘escape’ as a flux of nuclei, I. The maximum value, Δ𝐺𝑛○ (Table 2.1), varies with 1/Δ𝑇 2 . The value of 𝑁𝑛○ varies according to Eq. 2.8 and therefore:

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Fundamentals of Solidification 5th Edition

𝑁𝑛○ = 𝐾1 exp (−

𝐾2 ) 𝑇Δ𝑇 2

[2.9]

where 𝐾1 and 𝐾2 are constants. If it is assumed here that the rate of cluster formation is so high or I is so low that the equilibrium concentration of critical clusters, 𝑁𝑛○ /𝑁𝑙 , will not change i.e. that the source of nucleation will not be exhausted§, the steady-state nucleation rate is given by:

Figure 2.5 Nucleation Rate and Nucleation Time as a Function of Absolute Temperature The overall nucleation rate, 𝐼 (number of nuclei created per unit volume and time), is influenced both by the rate of cluster formation, which depends upon the nucleus concentration (𝑁𝑛○ ), and by the rate of atom transport to the nucleus. At low undercoolings the energy barrier to nucleus formation is very high and the nucleation rate is very low. As the undercooling increases, the nucleus formation rate increases before decreasing again (a). The decrease in the overall nucleation rate, at large departures from the equilibrium melting point, is due to the decrease in the rate of atomic migration (diffusion) with decreasing temperature. A maximum in the nucleation rate, 𝐼𝑚 , is the result. This information can be presented in the form of a TTT (timetemperature-transformation) diagram (b) which gives the time required for nucleation. This time is inversely proportional to the nucleation rate, and diagram (b) is therefore the inverse of diagram (a) for a given alloy volume. The diagram indicates that there is a minimum time for nucleation, 𝑡𝑚 (proportional to 1/𝐼𝑚 ). This minimum value can however be moved to higher temperatures and shorter times by decreasing the activation energy for nucleation, Δ𝐺𝑛○ [dash-dot line in (b)]. When liquid metals are cooled by normal means, the cooling curve will generally cross the nucleation curve (curve 1). Very high rates of heat removal (curve 2) can cause the cooling curve to miss the nucleation curve completely and an amorphous solid (hatched region, glass) is then formed via a continuous increase in viscosity (Fig. 1.1). Note that this figure relates to nucleation (start of transformation) only. The second curve of a TTT diagram which describes the end of the transformation, after growth has occurred, is not shown. §

This assumption is a crude but useful simplification. For more details, the reader is referred to Christian (1975).

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𝐼 = 𝐾3 𝑁𝑛○ 𝐼 = 𝐾3 𝑁𝑙 exp (−

[2.10]

Δ𝐺𝑛○ ) 𝑘𝐵 𝑇

where 𝐾3 is a constant. The formation of clusters will however require the transfer of atoms from the liquid to the crystal. An activation energy, Δ𝐺𝑑 , for transfer through the solid/liquid interface must therefore be added to Eq. 2.10, giving (Appendix 4): 𝐼 = 𝐼0 exp (−

Δ𝐺𝑛○ + Δ𝐺𝑑 ) 𝑘𝐵 𝑇

[2.11]

where 𝐼0 is a pre-exponential factor. This important equation contains two exponential terms. One of these varies as −1/𝑇Δ𝑇 2 (Eq. 2.9), while the other varies, like the diffusion coefficient, as −1/𝑇. An increase in Δ𝑇, giving more numerous and smaller nuclei of critical size, is accompanied by a decrease in 𝑇 and fewer atoms are transferred from the liquid to the nuclei. These opposing tendencies lead to a maximum in the nucleation rate at a critical temperature, 𝑇𝑐 , which is situated somewhere between the melting point (Δ𝑇 = 0) and the point at which there is no longer any thermal activation (𝑇 = 0 K). This is illustrated by Fig. 2.5(a). Note that I would exhibit a maximum value even in the absence of the diffusion term, Δ𝐺𝑑 . The presence of the latter term increases the temperature at which the maximum occurs. Since the reciprocal of the nucleation rate is time, for a unit volume of melt, the I-T diagram can be easily transformed into a TTT-diagram (Fig. 2.5(b)) where the curve represents the beginning of the liquid-to-solid transformation. The effect of decreasing the wetting angle, 𝜃, is felt mainly via its influence upon the equilibrium concentration of nuclei and a decrease in Δ𝑇, i.e. nucleation, occurs closer to the melting point. At very high cooling rates, such as those encountered in rapid solidification processing, there may be insufficient time for the formation of even one nucleus, and a glassy (amorphous) solid then results (cooling curve 2 in Fig. 2.5(b)). It is interesting to calculate the effect of a slight change in Δ𝐺𝑛○ , due perhaps to a change in 𝑓(𝜃), upon the nucleation rate. This can easily be done by approximating Eq. 2.11. At low values of Δ𝑇, the exp[−Δ𝐺𝑑 /𝑘𝐵 𝑇] term is approximately equal to 0.01 and I0 is approximately equal to 1041 m3 -1 s . The nucleation rate (in units of m-3s-1) therefore becomes: −Δ𝐺𝑛○ 𝐼 = 10 exp [ ] 𝑘𝐵 𝑇 39

with

Δ𝐺𝑛○ =

16π 𝜎3 𝑓(𝜃) 3 (Δ𝑠 Δ𝑇)2

[2.12]

𝑓

A nucleation rate of one nucleus per cm3 per second (106 /m3s) occurs when the value of (Δ𝐺𝑛○ /𝑘𝐵 𝑇) is about 76. Changing the exponential term by a factor of two, from 50 to 100 for example, decreases the nucleation rate by a factor of 1022. When Δ𝐺𝑛○ /𝑘𝐵 𝑇 is equal to 50, 108 nuclei per litre of melt per microsecond are formed. If the latter term is equal to 100, only one nucleus will be formed per litre of melt over a period of 3.2 years (Fig. 2.6). This example shows that very slight changes in the solid-solid interface energy can have striking effects. Upon calculating the undercooling for a constant value (l/cm3s) of 𝐼, as a function of 𝜃, another interesting result is obtained. This is illustrated by Table 2.2 which reveals the change, in the undercooling for heterogeneous nucleation, as a function of the nucleus/substrate contact angle. If the substrate is highly dispersed, as in inoculation, the active surface area of the inoculant must also be taken into account in the pre-exponential factor (𝐼0 ), of Eq. 2.11.

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Fundamentals of Solidification 5th Edition

Table 2.2 Absolute and Relative Undercoolings, Required to Produce One Nucleus per Second per cm3 as a Function of 𝜽 𝜽(𝐝𝐞𝐠. ) 180 90 60 40 20 10 5 0

𝚫𝑻/𝑻𝒇 0.33 0.23 0.13 0.064 0.017 0.004 0.001 0.0

𝚫𝑻 (𝑻𝒇 = 𝟏𝟓𝟎𝟎 𝐊) 495 345 195 96 25.5 6.5 1 0

Figure 2.6 Nucleation Rate as a Function of Activation Energy, 𝚫𝑮○𝒏 Variations in the value of the term, Δ𝐺𝑛○ /𝑘𝐵 𝑇, have a remarkable effect upon the rate of nucleation, I, due to the exponential relationship. If for an observable rate of 𝐼 = 1 /cm3s, Δ𝐺𝑛○ /𝑘𝐵 𝑇 is changed by a factor of two, the resultant change in the nucleation rate is of the order of 1022. Changing the temperature or changing the value of Δ𝐺𝑛○ can thus enormously increase or decrease the nucleation rate. The value of Δ𝐺𝑛○ can be decreased by adding crystalline foreign particles, which ‘wet’ the growing nucleus to the melt (inoculation), or by increasing the undercooling.

2.3. Inoculation, Free Growth Metals and alloys are formed from multiple grains that are joined at grain boundaries. Fine equiaxed grains lead, in most cases, to a more homogeneous solid having improved mechanical properties. Grain refinement is therefore of great practical relevance. Coarse grains or even a single grain are preferred however for high-temperature applications because the grain boundaries are mobile under those conditions and this facilitates creep-deformation. Grains are rarely initiated by homogeneous nucleation of the bulk liquid because the related undercooling is very high. They are generally initiated by adding foreign particles to the melt which then act as heterogeneous nucleation sites. Stirring of the melt can also promote grain refinement via dendrite fragmentation, and is frequently applied to certain alloys (e.g. steel or copper-based) (see

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Foundations of Materials Science and Engineering Vol. 103

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Sect. 4.7). High shear of the melt can also refine grains by breaking up and dissipating oxide layers that inoculate Mg-alloys (Fan et al., 2009). Much research has been directed at the development of grain refiners (inoculants) which can promote heterogeneous nucleation throughout the entire volume of the alloy**. Among the many technically useful metals and alloys, grain refining by inoculation works especially well for aluminium and its alloys. Commercial refiners for these materials are generally based upon the threecomponent system, Al-5wt%Ti-1wt%B, which is added to the melt just before casting. The mechanisms underlying grain refinement using this system could be elucidated thanks to the transmission electron microscopy (TEM) work of Schumacher and Greer (1994) and Fan et al. (2015). As it is difficult to observe liquids in a TEM, Schumacher and Greer rapidly solidified a complex aluminium alloy to which a small amount of Al-Ti-B was added. This produced a glassy aluminium matrix that retained the disordered atomic structure of the liquid. Crystallization of this glass was then observed by TEM; assuming that this process was similar to that of nucleation from the liquid state. Based upon these observations, a plausible mechanism of grain initiation could be proposed: that is, TiB2 precipitates which are present in the grain-refiner act as nuclei for α-aluminium grains. In order to be an effective inoculant with an associated small undercooling however, a thin intermediate Al3Ti layer has to form on the close-packed surfaces of the hexagonal TiB2 particles. The interfaces of the compounds and the aluminium have similar close-packed parallel crystal planes, orientations and lattice parameters, giving the heterogeneous epitaxial relationships: ▪

{0001}TiB2 || {112}Al3Ti || {111}Al

▪

11¯20TiB2 || 20¯1 or 1¯10Al3Ti || 1¯10Al

as shown in Fig. 2.7. In this way a semi-coherent low-energy interface is created between the TiB2 particles and the α-Al, thus substantially reducing the contact-angle and the nucleation-undercooling. (To ensure a low interface-energy and a small wetting angle, atomic interactions also play a role.) Free-Growth Model (Greer et al., 2000): If the heteroepitaxial relationship with respect to a foreign inoculant particle is “good”, the interfacial energy between the particle and the crystal is very small and so is the wetting angle (typically less than a few degrees). In this case the maximum height of the spherical cap (Fig. 2.3) becomes less than the thickness of one atomic layer and the model of heterogeneous nucleation, described in Sect. 2.1, breaks down (Cantor and Doherty, 1979). This is usual in the case of aluminium alloys which are inoculated with TiB2 particles. It has been shown that small islands of fcc aluminium form on the hexagonal planes of the particles at very low undercoolings. These islands, which all have the same orientation, then grow to form a monocrystalline layer; but only on the hexagonal plane of TiB2, and limited by its edges. In order for this layer to initiate a grain, the solid/liquid interface has to curve as shown in Fig. 2.8(a). During the initial growth stage (i.e. low undercooling) the crystal layer forms a small cap having a large radius of curvature, 𝑟. With increasing Δ𝑇 the solid/liquid interface increases its curvature, and so the crystal radius decreases until it reaches a minimum value, 𝑟𝑐𝑟𝑖𝑡 , which corresponds to half the particle diameter, 𝑑. This radius corresponds to a critical undercooling, Δ𝑇𝑓𝑔 . Beyond 𝑟𝑐𝑟𝑖𝑡 , the curvature decreases and the crystal is now free to grow (thus giving the model its free growth title). According to Eq. 2.1, the critical undercooling for free growth is then given by:

**

In order to produce the so-called “equiaxed turbine blades” used in aeronautics, heterogeneous nucleation is promoted at the internal surface of the ceramic mold by adding a first coating which decreases the wetting angle. The grain density at the surface of the casting is thereby increased, and the grains then grow inwards from the surface. On the external surface of the casting, the grain structure appears to be equiaxed but, seen in cross-section, the grains are clearly growing perpendicularly to the external surface.

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Fundamentals of Solidification 5th Edition

Δ𝑇𝑓𝑔 =

4Γ 𝑑

[2.13]

Figure 2.7 Atomic Structure of the TiB2 Inoculant with the Al3Ti Layer which has a Good Match with the Al Atoms on Top (not shown) Heterogeneous nucleation in a metallic melt is favoured when there is a good match between the positions of the atoms of the substrate and of the new grain. For the Al-Ti-B grain-refiner in aluminium alloys, detailed high-resolution transmission electron microscopy work and molecular modelling has revealed a good crystallographic matching between TiB2 particles and aluminium to exist when a very thin intermediate Al3Ti layer is also present. The 3D image shows the structure of the TiB2 crystal, with titanium atoms exposed to the Al3Ti layer. The aluminium atoms which are present on top of the Al3Ti are not shown, in order to improve visibility. The crystallographic planes which are parallel to one another are shown on top in the images. (Figure courtesy of Fan et al., 2015).

Figure 2.8 Model of Spontaneous Growth of an Aluminium Crystal on a Grain Refiner Particle (Substrate) with Low Crystal/Particle Interface Energy (Free Growth Model). At first a semi-coherent epitaxial Al3Ti layer (1) forms at the {0001} face of the TiB2 substrate (a). During growth, this layer that is pinned at the edges of the substrate develops a spherical cap (2). With increasing undercooling, the radius decreases (curvature increases). Once a critical minimum radius (𝑟𝑐𝑟𝑖𝑡 = 𝑑/2) is reached (3), free growth of a new grain on the substrate particle happens as the curvature is decreasing with further growth (4). With a distribution of substrate particles in an inoculated melt (b), the largest particles will be the first to initiate grain growth according to Eq. 2.13, thereby solidifying the melt before the smaller particles become effective (see Sect. 8.1). This explains why only a small amount of inoculant (order of 1% ) is active in the solidification process. The smaller particles are dormant and are incorporated without participating in grain initiation. (Greer et al., 2000).

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where Γ is the Gibbs-Thomson coefficient. According to this model the initiation of solidification on a given particle begins instantaneously when Δ𝑇𝑓𝑔 is reached during cooling of the melt. The various stages of initiation of grain growth are shown in Fig. 2.9. Considering now the distribution of TiB2 particles present in a melt of uniform temperature, the largest particles will be the first to initiate growth (Fig. 2.8(b)). For 𝑑 = 4 m the undercooling for the free growth of aluminium is 0.15 K; a very low value. Before this undercooling is reached the radius of the thin crystal is large and “dormant”. The initiation of solidification by free growth is not a stochastic nucleation phenomenon (a function of time as presented in Sect. 2.1 and 2.2) but is a deterministic event (depending only upon the undercooling). As in the case of the martensitic transformation, this mechanism is called “athermal nucleation”; meaning that it is not a thermally-activated process. This term is slightly confusing however as it still depends upon the temperature.

Figure 2.9 Stages of Initiation of Growth During Cooling of a Melt This figure shows the sequence of phenomena which occur during cooling of the melt (coolingcurve in red). At above the melting-point the atoms of liquid near to the particle already interact with the atoms of the substrate to form a locally more ordered melt (pre-nucleation). When the nucleation temperature is reached a film of the crystal starts to form, with a cap growing outwards. When the condition for free growth is reached, the crystal engulfs the particle and growth of a spherical grain begins. Above a certain size the spherical grains undergo morphological instabilities which finally lead to dendritic forms if the nuclei density is low.

Certain alloying elements can “poison” the action of inoculants by, for example, adsorption on the low-energy faces of the particles. This is typical of zirconium in aluminium alloys, where it can form an Al3Zr layer at the surface of TiB2 particles instead of the Al3Ti, which is essential for aluminium-grain formation. It is standard inoculation practice to avoid the presence of such elements. Under certain circumstances however a poisoning mechanism can be used to inactivate very potent nucleation sites (i.e. those active at low undercoolings) and thus promote heterogeneous nucleation on much less efficient inoculants at greater undercoolings. Recalescence (re-heating) of the melt is very fast in this case due to the rapid release of latent heat (explosive nucleation, Fan et al., 2020). If the less potent inoculants are very densely dispersed, and not poisoned of course, this method can lead to a very small grain size. In the absence of foreign particles which promote heterogeneous nucleation, as previously noted, very large undercoolings (hundreds of degrees) can be produced. In order to explain this

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phenomenon Frank (1952) conjectured that liquid metals are not totally disordered and can exhibit a significant degree of icosahedral short-range order (ISRO); in particular, an arrangement of 12 atoms surrounding a central atom and having 5-fold symmetry (Fig. 2.10). These local “clusters” of atoms are incompatible with the long-range close-packed arrangements of atoms in cubic or hexagonal crystals, thus possibly explaining large nucleation undercoolings. This conjecture was later confirmed by various observations and simulations. The X-ray scattering of liquids also revealed partial icosahedral ordering. Molecular dynamics simulations (see next section) showed that icosahedral clusters in a supercooled liquid gradually disaggregate while BCC clusters appear (Hou et al., 2010). Some intermetallic phases, notably in aluminium alloys, have large unit cells with several icosahedral motifs. The clearest proof of ISRO was however the discovery by Shechtman et al. (1984) of quasicrystals (QC)††, i.e. solids which exhibit 5-fold symmetry with no translational order. Because these solids have a structure which is very close to that of ISRO liquid, their interfacial energy when in contact with the liquid is at least an order of magnitude smaller than that of other crystalline solids. They can form at fairly low undercoolings, with a moderate cooling-rate. It has recently been shown that QC can also promote the formation of FCC grains. When this is the case, the FCC phase forms in such a manner that it has a heteroepitaxial relationship with respect to the QC; with the {111} planes and 110 directions of the FCC phase corresponding to the triangular facets and edges of the icosahedron (Fig. 2.10). Kurtuldu et al. (2015) proposed that this was another mechanism for grain initiation and was intermediate between homogeneous- and heterogeneous nucleation (calling it “ISRO-mediated nucleation”).

Figure 2.10 Icosahedral Arrangement of Atoms, a Short-Range Order Typical of Metallic Liquids with 3- (and 5-) fold Symmetry An arrangement of 12 atoms on the vertices of an icosahedron which surrounds a slightly smaller atom at its centre. It corresponds to a local dense packing of atoms, without any possibility of translational symmetry or extended periodicity (considered necessary for a crystal). The 6 axes which link the opposite vertices exhibit a 5-fold symmetry, while the 10 axes which pass through opposed triangular facets exhibit 3-fold symmetry. The arrangement of the 3 atoms of the equilateral triangular facets correspond to those of the {111} planes of a FCC structure. Such icosahedral motifs are present in the large unit cell of several intermetallic phases (e.g. Al3Cr) and are the basis of many quasicrystals.

2.4. Glass Formation As can be seen in Fig. 1.1 the solidification of a glass is accompanied by a continuous increase in viscosity; a value of 1014 Pa s being defined as that of the solid state. Metal (monoatomic) glasses possess a dense random packing of their atoms. As indicated above, liquids as well as amorphous solids exhibit short-range order. If crystallization is avoided, i.e. if cooling is so rapid that crystals cannot nucleate or grow, the melt freezes while retaining the same ††

The Nobel Prize in Chemistry was awarded in 2011 to Dan Shechtman for the discovery of quasicrystals.

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configuration and a glass (or amorphous phase) forms (Fig. 2.5(b)). If the mobility of the atoms or molecules is low, as in the case of silicates or polymers, the nucleation rate is low and the nose of the crystallization curve in Fig. 2.5(b) is not crossed; even at fairly low cooling rates. It is therefore easy to obtain a glassy phase in bulk quantities. The crystallization of metals however is a very rapid process and very high cooling rates (>106 K/s) are usually required in order to avoid the nose of the crystallization curve and thereby form a metallic glass (metglass). This obviously limits both the sample sizes in which a metglass can be produced, and the processes available for their production. Thin metglass foils (typically < 0.1 mm thick) can be produced, for example, by planar flow casting; a process in which liquid metal is projected onto a rapidly-rotating water-cooled copper drum. During recent decades glass formation in metallic alloys has been an extremely active area of research (Greer, 2009, and other references at the end of this chapter). According to the theory presented in Sect. 2.2, the mobility of atoms can be reduced by lowering the equilibrium liquidus temperature of the alloy. In the case of alloys which exhibit a deep eutectic, the crystallization curve of Fig. 2.5(b) is shifted to longer times. The cooling rates which avoid the nose of the crystallization curve can then be several orders of magnitude lower; thus, permitting the casting of Bulk Metallic Glasses (BMG) having a typical thickness of several mm. The avoidance of heterogeneous nuclei, crystallization, and thus grain boundaries, in BMG endows them with unique properties, e.g. a high hardness and a large elastic-energy storage capacity; albeit at the cost of a reduced toughness. 2.5. Atomistic Modelling of Nucleation Classical nucleation theory (CNT) as presented above possesses the merit of relating nucleation kinetics to thermodynamic properties. It reveals its limits however when it treats small clusters, comprising less than a few hundred atoms. The difficulty of studying nucleation is that it occurs at the atomistic scale and, furthermore, in a liquid. High-resolution TEM observations of crystallization in glasses (Sect. 2.3) have however yielded interesting results; as have numerical modelling and the use of colloids as analogues (Sect. 2.9). Various atomistic models, based upon numerical techniques, have helped to advance the understanding of nucleation phenomena: e.g., Monte Carlo (MC) simulation, Molecular Dynamics (MD) models and Density Functional Theory (DFT). Due to the enormous numbers of atoms which are present even in small volumes (a tiny 100 nm nanocrystal can already contain up to 30×106 atoms), such computations require large computer memories and high processor speeds. Such computations have been made tractable, not surprisingly, by considering only small nuclei forming at large undercoolings. Monte Carlo: In the stochastic MC approach, configurations arising from statistical ensembles are probed by using a random-walk algorithm and classical interaction potentials. This method permits the efficient computation of thermodynamic and structural properties of the system, but has some difficulty in providing a connection to the physical time-scale. Molecular Dynamics: In contrast to MC, these models are based upon a deterministic approach (aside from the random thermal fluctuations of the atoms) in which the interactions between the atoms are simulated at a realistic time-scale by using empirical interatomic potentials, e.g. Lennard-Jones (LJ)‡‡. For metals having some free electrons, a “delocalized” potential is added in order to pair the interactions. This is often achieved by using the Embedded-Atom Method (EAM, see Voter, 1994). MD provides thermodynamic, structural and dynamic data on condensed systems which are in stable, metastable or non-equilibrium states. When applied to nucleation it was found that critical nuclei are far from spherical (Fig. 2.11) and that a steep gradient of the order parameter exists in nanocrystals; which exhibit a diffuse interface (see next section). ‡‡

Named after the British mathematician, this potential is an approximation to the interatomic interaction, i.e. with attraction at large interatomic separations and with repulsion at small separations.

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Density Functional Theory: DFT is based upon quantum mechanics and considers fixed atoms in order to solve, to some approximation, the Schrödinger equation for the ensemble of electrons which is involved in the interactions. As it is the closest to a real system, and thus the most complex method, it is also the most limited in the number of atoms that can be treated. As well as treating thermodynamic properties it can be adapted to the nucleation of condensed phases and to the prediction of spatially-dependent structures and densities. One of the conclusions drawn from these calculations was (as in the case of MD simulations) that a critical nucleus can exhibit a metastable structure at its centre (e.g. BCC) and evolve into a stable crystal (e.g. FCC) during growth. The theory predicts furthermore that the first step in crystal formation involves structuring and, only later, a change in density. Such results are not predicted by classical nucleation theory.

Figure 2.11 Early Stages of Crystal Nucleation The nucleation dynamics of a platinum crystal as modelled by MD-EAM computations at four different times (about 30 ps between each figure). The atoms of the melt have been removed for clarity. The fluctuating non-spherical shape in the early stages of nucleation is clearly visible. The contour maps next to the nanocrystals show regions of differing order parameter (with light-grey to dark-grey corresponding to 0.3, 0.5 and 0.7, respectively). The order parameter indicates how many atomic sites in a set match the phase in question. A strong gradient of the order parameter, in going from the interior to the interface, can be seen. (Zhou et al., 2019).

These numerical simulations reveal that nuclei are not simply tiny spherical entities having the properties of the bulk, as is assumed by the CNT presented at the beginning of the chapter. The small structures probably have several transitional states, and might be poorly represented by a single critical nucleus. The crystal–melt interface of metals is diffuse moreover (see next section) and, for small clusters, their size is comparable to the thickness of the diffuse interface. This questions the validity of using an interfacial energy which is usually determined by performing macroscopic thermodynamic measurements. Nucleation often appears to occur in two distinct stages: the formation of a metastable cluster and the subsequent nucleation of the final crystal within the cluster via a structural transformation; an example of Ostwald’s step rule (Anwar and Zahn, 2010). By considering short-range order only for pure melts, Fang et al. (2018) showed by molecular dynamics simulation that crystalline substrates induce atomic ordering at the liquid/substrate interface; even at temperatures above the liquidus (prenucleation, Fig. 2.9). Such atomic ordering does indeed facilitate nucleation as the interfacial energy is reduced. On the experimental side, colloidal analogues have been used to track nucleation mechanisms via direct observation using optical microscopy. Colloids consist of small, nanometre to micrometre sized, particles which are suspended in a fluid; the particles being charged or uncharged polymer spheres. The “crystallization” of these particles is initiated by increasing their volume-fraction; a convenient control-parameter. Even if the analogy between monoatomic systems and colloids is not perfect, the latter provide an interesting resource for the study of crystallization and glass formation.

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2.6. Interface Structure When a stable crystal has formed via nucleation, grain growth then begins and such growth is controlled by: ▪ atom attachment at the interface, ▪ diffusion of heat and mass from the moving interface into the neighbouring phase, ▪ capillarity action. The relative importance of each of these factors depends upon the substance in question and upon the solidification conditions. This section will consider the atomic attachment kinetics, and interface energy and its anisotropy. The other processes will be treated in Chaps 3 to 5. Crystals exhibit various growth-forms that can be attributed to the nanoscopic structure of the interface. The latter can either be (i) diffuse i.e. atomically rough with little difference between liquid and solid or (ii) sharp i.e. atomically flat with a solid which is very different to the liquid. This is the basis of the classification of substances into non-faceted and faceted ones (Fig. 2.12).

Figure 2.12 Non-Faceted and Faceted Growth Morphologies After nucleation has occurred, further atoms must be added to the crystal so that growth can follow. During this process the solid/liquid interface takes on a specific structure at the atomic scale; either rough or flat. This nanoscopic morphology depends upon the crystal structure and the bonding of the solid when in contact with its liquid. During the solidification of a non-faceted material, such as a metal with its atomically rough (diffuse) solid/liquid interface, atoms can be easily added at any point on the interface and the crystal shape is dictated mainly by the interplay of diffusion (of heat and/or solute) and capillarity effects. Smooth curved morphologies at the microscopic scale are the result (a). In faceted materials such as non-metallics or compounds having atomically flat (sharp) interfaces, the atoms cannot attach easily to the interface and the crystal shape becomes bounded by the slowest-growing facets (b). The differing crystallisation behaviours of the two classes of material are signalled by the entropy of fusion; low values indicate a small difference between liquid and crystal while large entropies are characteristic of materials which possess complex crystal structures and non-metallic bonding.

Metals and a special class of molecular compounds (plastic crystals, Sect. 2.9), usually solidify from their melt with a nanoscopically rough, and macroscopically smooth, solid/liquid interface and do not form facets in spite of their crystalline nature. This behaviour reflects the small dependence of the atomic attachment kinetics upon the orientation of the crystal plane. Slight anisotropies of the interface energy (Sect. 2.7) are however the cause of anisotropic dendrite growth and define the operating point of its tip (Chap. 4). The overall result is the appearance of smooth, but crystallographically oriented, dendrite trunks and arms (Fig. 2.12(a)). On the other hand, substances which have complex crystal structures and directional bonding form crystals having planar angular surfaces (the facets familiarly exhibited by minerals). Note that a substance which forms non-faceted crystals when grown from the melt can nevertheless form faceted crystals when grown from vapour.

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In the present book, the classification will be applied exclusively to melt-grown crystals. This classification is of practical interest to metallurgists, among others, because of the importance of the intermetallic phases and compounds which appear in most alloys, and because of the large-scale industrial use of two eutectic alloys (Chap. 5) which have a faceted phase as one component (e.g. graphite in Fe-C and silicon in Al-Si alloys). An understanding of atomic-attachment kinetics also aids the correct choice of the transparent analogues which are often used to model metal solidification.

Figure 2.13 Bond Numbers at the Solid-liquid Interface of a Simple-Cubic Crystal In order to understand the two types of growth shown in Fig. 2.12, the various ways in which an atom can be adsorbed at the solid/liquid interface have to be considered. Growth is determined by the probability that an atom in the liquid will reach the interface by diffusion and, despite thermal activation, will remain attached there until it has been fully incorporated into the crystal. This probability increases with increasing number of nearest-neighbours in the crystal (the numbers in the figure correspond to a simple cubic crystal for which the coordination number is 6). Only one atom of the liquid phase is shown here, for clarity. Atoms on sites of type 4 or 5 (attached to a step vacancy or surface vacancy) will be bonded preferentially. A special role in the growth of faceted crystals is played by kink atoms (type 3 in red) because, having three bonds with the crystal, they can be considered to be situated half in the solid and half in the liquid. A likely growth sequence would be: addition of type 3 atoms until a row is complete; addition of a type 2 atom to start a new row; and so on until a layer is complete. Thereupon the nucleation of a new layer by the addition of a type-1 atom would be necessary. This is typical for a flat (faceted) crystal. It is an unfavourable process and requires a high undercooling. In the case of the atomically rough crystal surface of a metal there are many favourable attachment sites of types 3 to 5 and growth is therefore uniform and smooth, even under low driving-forces.

The equilibrium shape of crystals is determined by planes having the lowest interfacial energy, according to the Wulff construction. Growth modifies the equilibrium form due to the different growth rates in various directions. The growth rate of a crystal depends upon the net difference between the rates of attachment and detachment of atoms at the interface (Appendix 5). The rate of attachment depends upon the rate of diffusion in the liquid, while the rate of detachment depends upon the number of nearest-neighbours which bind the atom to the interface. The number of nearest neighbours depends upon the crystal face which is considered, i.e. upon the surface roughness at the atomic scale (number of unsaturated bonds). This is the simplest possible situation. In general, surface diffusion, reorientation of molecules in the melt or other mechanisms may be active. Consider an essentially flat interface of a simple cubic crystal. An atom in the bulk crystal here has six nearest neighbours represented by the six faces of a cube (Fig. 2.13). There are five different positions at the interface, characterised by the number of nearest neighbours (1 to 5). In an undercooled system, where the crystal has a lower free energy, it is evident that an atom in position 5 will have a very much higher probability of remaining in the crystal than will an atom in position

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1. In order to incorporate atom 1 into the crystal, a very large difference must exist between the force binding it to a single neighbour bond of the crystal under the action of the other neighbours in the liquid. In order to create such a difference, a large undercooling of the melt is required. Figure 2.14 presents the two interface morphologies at the microscopic (above) and the nanoscopic scale (below).

Figure 2.14 Form of Faceted (a) and Non-Faceted (b) Interfaces This represents the interface morphologies at two very different scales: microscopic above and nanoscopic below. The images seen during directional solidification under an optical microscope (above) can be obtained with the help of transparent organic substances (solidified between two glass slides in a temperature gradient, bottom cool, top hot). One can observe either of the growth forms shown in Fig. 2.12. It is important to note that, during growth, a faceted interface (a) is jagged and faceted at the microscopic scale (above) but flat at the atomic scale (lower left insert). On the other hand, a non-faceted interface (b) can be microscopically flat with some slight depressions due to grain boundaries (above) while, at the atomic scale, it is rough and uneven (below). This roughness causes the attachment of atoms to be easy and largely independent of the crystal orientation. Note also that the interface of a non-faceted material will grow at a temperature which is close to the melting point, 𝑇𝑓 , while the interface of a faceted material might have a very high local undercooling. Such a point (arrow) is a re-entrant corner (Fig. 2.16(b)) and is associated with an increased number of nearest neighbours. Growth will thus tend to spread from there.

It can be generally said that the greater the difference in structure and bonding between the solid and liquid phases, the narrower will be the transition region over which these differences have to be accommodated. A sharp transition (i.e. an atomically flat microscopically faceted interface) exhibits little tendency to incorporate newly arriving atoms into the crystal. Hence growth is more difficult and requires an additional (kinetic) undercooling. On the other hand, high-index crystallographic planes tend to be inherently rough and contain many steps that provoke a marked anisotropy in growth rates when compared with low-index planes which are atomically flat. This in turn leads to the disappearance of the higher-index planes, due to their more rapid growth, and leads to a characteristic crystal form which is bounded by the slowest-growing faces (Fig. 2.15). On the other hand, metals have a similar structure, density and bonding in both the liquid and solid states. The transition from one phase to the other at the interface is thus very gradual and the interface becomes rough and diffuse (i.e. it exposes many suitable growth steps to atoms arriving from the melt).

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Figure 2.15 Development of Faceted Crystal Growth Morphologies A cubic crystal beginning growth from its equilibrium shape bounded by {100} planes (left hand side) will change its shape to an octahedral form (bounded by {111}) when the {111} planes grow more slowly than the {100} planes (a). Impurities often change the growth behaviour of specific planes and this results in the appearance of differing growth forms for the same crystal structure. If the {110} planes are the slowest-growing, this will lead to a rhombohedral dodecahedron (b). The slowest-growing planes (usually of low-index type) always dictate the growth habit of the crystal. The resultant growth form (minimum growth rate) is not the same as the equilibrium form, which is governed by minimisation of the total surface energy.

Related to the interface energy is its anisotropy. This topic is treated in more detail in Sect. 2.7. Faceted crystals are characterized by a high anisotropy, while most metals have a very low anisotropy and the equilibrium shapes are close to spherical, with some convexities in regions of slightly higher energy. Since the anisotropy of the interfacial energy is poorly known and the growth morphology (crystal habit, Fig. 2.15) is also influenced by the attachment of atoms, as mentioned before, the classification of non-faceted and faceted solids is based upon the entropy of crystallisation (or, with opposite sign, the entropy of fusion or melting). Table 2.3 Growth Morphologies and Crystallisation Entropies Dimensionless Melting Entropy, 𝜶 = 𝚫𝑺𝒇 /𝑹𝒈

Substance

Supersaturated Phase

Morphology

~1

metals

melt

non-faceted

~1

‘plastic’ crystals

melt

non-faceted

2-3

semiconductors

solution

nf/faceted

2-3

semimetals

solution

nf/faceted

~6

molecular crystals

solution

faceted

~10 ~20

metals complex molecules

vapour melt

faceted faceted

~100

polymers

melt

faceted

The entropy of crystallisation is indeed related to the difference in ordering between the solid and liquid phases: a low molar entropy of fusion Δ𝑆𝑓 is characteristic of metals, since the structures of the liquid and solid are not very different. Values of Δ𝑆𝑓 , normalized by the gas constant 𝑅𝑔 , 𝛼 = Δ𝑆𝑓 /𝑅𝑔 , which are less than ~2 can be taken to imply a tendency to non-faceted crystal growth, while

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higher 𝛼-values imply that faceted growth forms will be produced. Table 2.3 lists various substances and compares their growth forms and dimensionless entropies for crystallisation from a variety of media. The fact that metals and plastic crystals fall into the same group has been of great value in the study of solidification (see end of this chapter). A high entropy of fusion increases the disparity in growth rates between the low-index planes and the faster-growing high-index planes. Table 2.4 presents some numerical estimates of growthrate coefficients, 𝐾ℎ𝑘𝑙 , as a function of the dimensionless crystallisation entropy and two crystallographic planes for a simple-cubic crystal (see also Appendix 5 where a 2D crystal is used as a rough approximation to the behaviour of a real three-dimensional crystal). At small undercoolings with respect to 𝑇𝑓 the actual growth rate of the faces can be obtained from the expression: [2.14]

𝑉ℎ𝑘𝑙 = 𝐾ℎ𝑘𝑙 Δ𝑇

This expression corresponds to a limited range of Eq. A5.7 and assumes that a simple growth mechanism is operating. It can already be seen from this example that a very marked anisotropy results in the case of high melting-entropy. That is, the ratio of 𝑉{111} to 𝑉{100} is increased and the absolute growth rates are markedly decreased by increasing crystallization entropy. As a result, a crystal of a substance having a large value of 𝛼 will be bounded by {100} planes. Table 2.4 Growth Rate Coefficient, 𝑲𝒉𝒌𝒍 , as a Function of Crystallisation Entropy for a Simple-Cubic Crystal (Jackson, 1968) Crystallisation Melting Entropy, 𝜶 = 𝚫𝑺𝒇 /𝑹𝒈

𝑲𝟏𝟎𝟎

𝑲𝟏𝟏𝟏

𝑲𝟏𝟏𝟏 𝑲𝟏𝟎𝟎

1

0.2

0.1

0.5

5

0.007

0.01

1.4

10

0.000005

0.0001

20.0

Due to the greater difficulties experienced in attaching atoms to the surfaces of substances possessing a high entropy of fusion, surface defects are of particular importance in this case. Such defects clearly do not greatly facilitate growth if the attachment of the atoms easily removes them. Therefore, only those defects which cannot be eliminated by growth are effective (Fig. 2.16). These include screw dislocations which emerge at the growth surface, twin boundaries and rotation boundaries. Each of these supplies re-entrant corners (steps) which locally increase the number of bonds that an attached atom makes with the crystal, thus reducing the kinetic undercooling and leading to highly anisotropic growth. As a result the morphology becomes sheet-like (planar defect in Fig. 2.16(b) and (c) or rod-like (line defect in Fig. 2.16(a)). This phenomenon has a marked effect upon the growth behaviour of the most widely used eutectic alloys, Fe-C and Al-Si (Chap. 5). The faceted phase (graphite in cast iron, Fe-C alloy) grows at a different rate in different directions. When 𝑉(0001) is less than 𝑉{1010} , plates delimited by (0001) planes appear and the morphology is referred to as 'flake graphite'. When 𝑉{1010} is less than 𝑉(0001) , hexagonal prisms form. This is often not immediately apparent because the prisms tend to grow side by side in the radial direction and the morphology is then called 'spherulitic'. Changes in the growth behaviour of graphite are obtained in practice by controlling the trace-element concentration. The presence of sulphur, for example, leads to the appearance of flake graphite while the presence of magnesium or cerium results in the formation of nodular cast iron. It should be noted finally that, under a sufficiently high driving force (e.g. high undercooling), the solid/liquid interface of even a high melting entropy phase tends to become rough. As a result, its growth behaviour then becomes more isotropic.

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Figure 2.16 Repeatable Growth Defects in Faceted Crystals As pointed out under Fig. 2.13, kinks and steps on a crystal-liquid interface are favoured growth sites but can be easily eliminated by the very growth which they promote. Several types of crystal defect have been shown to provide steps without being eliminated by growth. These are (a) the emergent screw dislocation (Frank, 1949) which leads to the establishment of a spiral ramp; (b) the twin plane re-entrant mechanism (Hamilton Seidensticker, 1960) which acts as a macroscopic step and (c) the twist (rotation) boundary which also provides effective steps. Depending upon the type of defect present, the faceted crystal can exhibit various morphologies: (a) needles in the case of line defects, (b) plates in the case of planar defects - e.g. silicon in Al-Si alloys or graphite in cast iron (c) (Minkoff, 1968). The latter two defects are important in understanding the growth of irregular eutectic microstructures (Fig. 5.11).

2.7. Interface Energy and Anisotropy As shown in the previous section, the interface which separates the liquid metal from its solid is a 3-dimensional volume having a small but non-zero thickness over which the structure changes continuously from the short-range ordered arrangement of atoms of the liquid to the long-range ordered structure of the crystal. This transition occurs by adjusting the interface atoms to the structure of the crystal via increased ordering. This minimizes the density-deficit and decreases the entropy; the latter being the main contributor to the excess interfacial free energy between the crystal and its melt. The solid/liquid interface energy is of mainly negentropic origin (Spaepen, 1975/76). The transition zone of the interface is therefore accompanied by an excess volumetric energy, Δ𝑔, of the atoms which defines the interface energy, 𝜎 (Fig. 2.17): 𝜎 = ∫ ∆𝑔 𝑑𝑧

[2.15]

While the interface between two disordered phases, e.g. between liquid and gas or between a liquid and a glassy phase, is isotropic, that between a disordered phase and a crystal is not: the interface energy depends upon the orientation of the crystal/liquid or crystal/gas interface. “The origin of the anisotropy [of a solid/liquid interface] can be attributed to the nature of the localization of the liquid near a crystal surface. Depending on the structure of the surface, the extent of the localization (roughly the thickness of the layer with entropy loss), and the specific configurations in that localized layer will differ, resulting in different amounts of entropy loss.” (Spaepen, 2021). It is then natural to use Miller indices to characterize a flat interface, i.e. an interfacial energy 𝜎ℎ𝑘𝑙 = 𝜎(𝑛⃗ℎ𝑘𝑙 ) corresponding to that interface; the normal of which, 𝑛⃗ℎ𝑘𝑙 , is perpendicular to the {hkl} planes of a cubic crystal. In the case of metals, one generally has a value of 𝜎111 which is slightly smaller (by a few percent) than 𝜎100 or 𝜎110 because the {111} planes are densely packed and closer to the dense random arrangement of atoms in the liquid. This anisotropy is of course much higher for non-metallics, such as ceramics, which are usually characterized by sharp interfaces because the atomic arrangement of atoms and the composition of the solid which is in contact with the liquid are different.

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Figure 2.17 Diffuse Solid/Liquid Interface of a Pure Substance at Equilibrium Gibbs Free Energy 𝒈 and Phase-field Parameter 𝝓 In the homogeneous liquid and solid phases, the Gibbs free energy (molar or volumetric) 𝑔𝑙 and 𝑔𝑠 are equal and uniform at the melting point. Within the diffuse interface, atoms have an excess volumetric Gibbs free energy Δ𝑔 which is mainly due to a reduced configurational entropy. The 𝑔(𝑧) function represents the slope of the phase-field parameter, 𝜙(𝑧), introduced in the next section. The integral of Δ𝑔 over the thickness of the diffuse interface gives the surface energy 𝜎.

Some care must be taken when the interface energy is not isotropic. Firstly, some confusion is often found in the literature between surface energy, which is a scalar quantity having units of J/m2, and the surface tension, which is a 2-dimensional tensor having units of N/m. This confusion should absolutely be avoided when one or the two phases involved in the interface are ordered. For isotropic interfaces, these two entities are equal and the surface stress has two equal components in the plane of the interface. This is no longer so in the case of anisotropic interfacial energy, i.e. when the interfacial energy 𝜎 depends on the orientation of the interface. Provided that the anisotropy of  is not too large§§, as in the case of most metals, the stress tensor components, τij , of the surface tension are given by: τij = 𝜎δ𝑖𝑗 +

𝑑𝜎 𝑑𝑒𝑖𝑗

[2.16]

where δ𝑖𝑗 = 1 if 𝑖 = 𝑗 and δ𝑖𝑗 = 0 if 𝑖 ≠ 𝑗 (Kronecker symbol) and the second term represents variations of the surface energy with respect to small variations 𝑑𝑒𝑖𝑗 (deformations) in the tangent plane. This has several consequences: firstly, the derivation of the surface energy  along one direction on the surface, n, is not necessarily equal to that along the perpendicular direction, n, and the surface stress no longer lies in the interface plane. Secondly, at a local point of the interface the two principal radii of curvature are no longer equal when  is anisotropic. The extension of the Gibbs-Thomson contribution to the undercooling (Eq. 1.5, see Appendix 3) of a curved interface having two different radii of curvature, 𝑟1 and 𝑟2 :

Note that the derivative of  with respect to 𝑛⃗ is no longer valid for an interfacial energy which exhibits cusps (i.e. for crystals having faceted equilibrium shapes) (for details see Cahn Hofmann, 1974). §§

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∆𝑇𝑟 = 𝐾

with

𝐾=

1 1 + 𝑟1 𝑟2

and

=

𝜎 ∆𝑠𝑓

[2.17]

is no longer valid, since Γ is no longer a constant. In the case of solidification, however, an even small anisotropy of the solid/liquid interface energy is essential since dendrites would not form if 𝜎 were isotropic (see Chap. 4). For a weakly anisotropic interfacial energy, Herring (1951) has shown that the undercooling is given by the relationship: 1 1 𝜕 2𝜎 1 𝜕 2𝜎 ∆𝑇𝑟 = { [𝜎 + 2 ] + [𝜎 + 2 ]} ∆𝑠𝑓 𝑟1 𝑟2 𝜕𝑛1 𝜕𝑛2

[2.18]

where the coordinates (𝑛1 , 𝑛2 ) are coordinates on the surface and are associated with the principal radii of curvature, 𝑟1and 𝑟2 . Note that, when 𝜎 is isotropic, this expression retrieves the Gibbs-Thomson extension mentioned above, but then 𝑟1should also be equal to 𝑟2 ! The expression: 1 𝜕 2𝜎 𝜕 2𝜎 𝑆 = 𝜎 + [ 2 + 2] 2 𝜕𝑛1 𝜕𝑛2

[2.19]

which averages the second derivatives of the interfacial energy is called the interface stiffness, 𝑆: the magnitude reflecting just how easy it is to “curve” the interface. In two dimensions, Equation 2.18 is much simpler and can be written as: ∆𝑇𝑟 =

1 1 𝜕 2𝜎 1 1 [𝜎 + ]= 𝑆 2 ∆𝑠𝑓 𝑟 𝜕𝜑 ∆𝑠𝑓 𝑟

[2.20]

where 𝜑 is the azimuthal angle with respect to a reference direction in the crystal and 𝑟 is the local radius of curvature. This latter entity must not be confused with the radial distance in polar coordinates, i.e. if the interface is given by a function, 𝑦(𝑥), the local radius of curvature, r, is given by 𝑟 −1 = 𝑦′′/(1 + 𝑦′2 )3/2, where 𝑦′ and 𝑦′′ are the 1st and 2nd derivatives of 𝑦(𝑥). The next problem is how to represent the anisotropy of the interface energy. Since it is dictated by the structure of the crystal, let it be assumed that it has cubic symmetry. In this case the development of 𝜎(𝜑) in two dimensions and 𝜎(𝜃, 𝜑) in three dimensions, must respect this symmetry. Setting the reference coordinate system relative to the 10 and 100 directions, respectively, the expression of 𝜎 for cubic crystals can be developed as follows: 𝜎(𝜑) = 𝜎0 [1 + 𝜀4 cos(4𝜑) + 𝜀8 cos(8𝜑)+. . . ]

in 2D

𝜎(𝜃, 𝜑) = 𝜎0 [1 + 𝜀4 𝐻4 (𝜃, 𝜑) + 𝜀6 𝐻6 (𝜃, 𝜑) + 𝜀8 𝐻8 (𝜃, 𝜑)+. . . ]

in 3D

[2.21]

where 𝜎0 is the mean value of the interface energy, 𝜃 and 𝜑 are the spherical angular coordinates and the 𝜀ℓ values are anisotropy parameters (of the order of some percent in the case of metals). The functions, 𝐻ℓ (𝜃, 𝜑), are cubic harmonics, i.e. combinations of spherical harmonics, 𝑌ℓ𝑚 (𝜃, 𝜑), which respect the cubic symmetry. As an example in two dimensions, the stiffness of the interface (Eq. 2.20) for cubic symmetry becomes: 𝜕 2𝜎 𝑆 = [𝜎 + ] = 𝜎0 [1 − 15𝜀4 cos(4𝜑) − 63𝜀8 cos(8𝜑)+. . . ] 𝜕𝜑 2

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[2.22]

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Two conclusions can be immediately drawn: firstly, higher-order terms in the development of 𝜎, although becoming smaller as the order increases, are amplified in the stiffness parameter. This has been shown to exert a possible influence on the dendrite growth directions (Haximali et al., 2006). Secondly, upon retaining only the first term in the development, 𝜀4 , maxima of the interface energy correspond to minima of the stiffness (see Fig. 2.18). This reflects the fact that portions of the interface which have the maximum interface energy should be minimized if the global interface is to be minimal.

a)

b)

c)

Figure 2.18 Interface Energy Anisotropy, Stiffness and Equilibrium Shape of a 2D Crystal In (a) the interface energy as a function of the angle, 𝜎(𝜑), is plotted for a 2D crystal. Its deviation from the circle is due to a typical anisotropy for metals (𝜀4 = 0.03). In (b) the corresponding interface stiffness, 𝑆, and in (c) the shape of the equilibrium crystal are shown. The 2D “cubic” crystal has a maximum in the surface energy and a minimum in the stiffness lying along the 10 directions. The equilibrium shape corresponding to a constant curvature-undercooling along the whole interface has been calculated using Eq. 2.20. The equilibrium shape is very similar to the 𝜎-plot (a) (but this is not always the case; especially when the crystal is faceted, or has corners). Equiaxed dendrites grow preferentially along orientations of highest curvature, i.e. 10 in this case. This minimizes the surface energy and the interface is bent in the regions where its stiffness is minimal.

2.8. Diffuse Interface/ Phase-Field As mentioned above, the solid/liquid interface of metals stretches over several atomic distances. In order to model such an interface the simplest system comprises two condensed phases: liquid and crystal, both separated by a diffuse solid/liquid interface***. The energy of such a heterogeneous system is given by the sum of the volumetric contributions of the bulk phases and the contribution from the interface which separates the phases. The atomic structure of a crystal is characterised by long-range order, while that of a liquid has short-range order. For a pure system under equilibrium conditions (𝑇 = 𝑇𝑓 ), Figure 2.19 shows a macroscopically flat and diffuse solid/liquid interface parallel to a set of crystalline planes. Averaging the atomic density in planes parallel to the interface shows fluctuations in the crystal, between maxima at the positions of the planes and minima between them. As the positioning of atoms in the liquid is random, the plane-average density is constant. Within the thickness of the diffuse interface, the amplitude of the average atomic-density oscillations increases from the constant value in the melt to a maximum in the homogeneous crystal (Fig. 2.19). The envelope of the amplitude of the oscillations, labelled 𝜙, can be used as an indicator of the phase: with 𝜙 = 0 corresponding to liquid, 𝜙 = 1 corresponding to solid and 0 < 𝜙 < 1 corresponding to

***

For the sake of simplicity, the molar volumes of both condensed phases, solid and liquid, are considered equal, i.e. the term 𝑝𝑉 can be omitted in the thermodynamic treatment of the phase transformation.

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the transition region between the two phases. This parameter is used in the phase-field method (PFM) for the computation of microstructure evolution. It can be seen that the interface is not a (sur)face (as simplified in the Gibbs thermodynamic treatment) but is rather a volume that stretches over several atomic distances (typically 3 to 4 in the case of metals) where the atoms of the solid and liquid phases form a sort of structural mixture having a reduced configurational entropy (Spaepen, 1975).

Figure 2.19 Atomic Structure of the Crystal-Melt Interface of a Pure Substance under Equilibrium Conditions, and Corresponding Atomic Density Function The atoms close to a diffuse solid/liquid interface are presented here. The interface is macroscopically planar and parallel to a set of crystal planes, the axis z corresponding to the interface normal. Note that atoms in the liquid are highly mobile (red arrows) and can move over long distances whereas, in the solid, they vibrate mainly about equilibrium positions in the lattice. Within the diffuse interface, the movement of atoms gradually decreases in going from the liquid to the solid. The atom density averaged on planes perpendicular to 𝑧 is displayed. It is constant in the liquid due to the random positioning of atoms, but exhibits maxima and minima in the crystal that correspond to atomic plane and mid-plane positions. According to Mikeev Chernov (1991), the phase-field parameter, 𝜙, can be viewed as being the envelope of the amplitude of the atomicdensity oscillations.

During the last decades, the phase-field method (PFM) for solidification, proposed by Langer in 1978, has become the method of choice for the simulation of phase transformation and microstructure evolution in materials science. It has permitted an increasing number of researchers to examine and simulate many complex transformation phenomena, such as nucleation, dendritic-, eutectic- and peritectic-growth, morphological interface instability, microsegregation, nonequilibrium effects in rapid solidification, coalescence near to the end of solidification, etc. Before PFM became available, the modelling of solidification microstructures was achieved mainly by using front-tracking techniques which were based upon a sharp (essentially 2-dimensional) interface (Stefan problem – see Appendix 1). The physical properties of the phases at a sharp interface change discontinuously, and two conditions have to be applied. For a pure system the temperature is imposed to be that of the melting-point, as modified by capillarity, while a local thermal flux-balance involves the velocity 𝑣 at which the interface is moving. This method is very complex, especially in 3dimensional geometries, because nodal points have to be positioned on the sharp interface in order to apply the above conditions and tracking of the latter also requires re-meshing of the domain.

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In the case of the PFM, all of the properties are continuous and averaged by the phase parameter, 𝜙. This technique has the great advantage, over front-tracking, that it is based upon equations that can be solved over the entire domain by using a fixed grid; without having to re-mesh it as the interface moves. An equation which describes the evolution of the phase field, i.e. 𝜙(𝑧, 𝑡) in one spatial dimension or 𝜙(𝑥, 𝑦, 𝑧, 𝑡) in three spatial dimensions, permits tracking of the evolution of the microstructure. The method is presented in further details in Appendix 14 and Chap. 4. In this chapter, attention is concentrated on the concept of the interfacial energy of a diffuse interface. Considering a pure substance at equilibrium, under the assumption that 𝜌𝑠 = 𝜌𝑙 , the molar or volumetric Gibbs free energies of the bulk solid and liquid phases are equal. The atoms which are located within the diffuse interface have an excess volumetric energy, ∆𝑔, given that they have both solid and liquid neighbours. In a continuum approach the volumetric Gibbs free energy 𝑔(𝑧) has the shape drawn schematically in Fig. 2.17: it is equal to 𝑔𝑠 in the solid, to 𝑔𝑙 = 𝑔𝑠 in the liquid and has a Gaussian form within the diffuse interface. As shown in more detail in Appendix 14, the integral of the excess volumetric Gibbs free energy Δ𝑔 [J/m3] over the thickness of the diffuse interface is the surface energy, 𝜎 [J/m2]. 2.9. Observation of Microstructure Development Analogues: It has already been mentioned elsewhere in this book that the use of physical systems whose behaviour is analogous to that of the phenomenon that one wishes to study can provide invaluable information. One example of this would be the cited use of colloids to investigate nucleation (Herlach et al., 2016). With regard to the larger scale of microstructures the use of plastic crystals such as succinonitrile and pivalic acid has allowed a huge step forward to be made with the light microscope. These transparent substances permit the observations of solidification of cells, dendrites, eutectic or peritectic. This is possible because members of this very rare class of organic compounds behave just as metals do, as far as solidification is concerned. Plastic crystals are a special class of molecular substance (organic or inorganic) whose intrinsic molecular asymmetry is destroyed by rapid rotational motion. The molecules consequently behave like spheres having surfaces defined by the envelope of all of the possible arrangements of the molecule's extremities. Since these 'spheres' no longer exhibit asymmetry or directional bonding, they can arrange themselves into simple crystal structures, particularly cubic ones, and certain of their properties are analogous to those of metals. The name, 'plastic', for example refers to their high malleability. They were however first recognised as a distinct class due to their low entropies of fusion. This also indicates that their crystallisation behaviour is analogous to that of metals (Table 2.3). After eliminating many members of the class on the grounds of their inconvenient melting-point or their unpleasant properties†††, only a handful remains. A wide range of other organic compounds can be mixed with plastic crystals so as to model the effect of impurities upon solidification. They also form eutectic mixtures between themselves, so that eutectic growth as well as dendrite growth can be observed. If one of the eutectic components is a non-plastic crystal, it is then possible to model the behaviour of those real eutectic alloys in which a metal and non-metal solidify cooperatively. Such an analogous system is very relevant to the understanding of the solidification behaviour of the two most widely used casting alloys: cast iron and aluminium-silicon (Kurz and Fisher, 1979). For experimental purposes the plastic crystals are sandwiched between two glass microscope slides which are maintained a small distance apart. In many cases a spacing is chosen which offers just a single-layer 2-dimensional view of the growth morphology (Fig. 4.16). If the spacing of the glass slides is too small, they can interfere with the growth morphology due to surface-wetting and thermal effects. They can also be placed a relatively large distance apart, thus permitting ‘3D’ observations to be made of the interface (Fig. 3.6(a)).

†††

Although widely used, succinonitrile is a cyanide and should be used with care. It is harmful if swallowed, may cause skin, eye and respiratory irritation.

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In order to impose directional solidification, the glass-enclosed layer of plastic crystal is moved through a temperature-gradient which is controlled so that the point of the gradient which corresponds to the melting-point of the sample remains stationary with respect to the microscope objective. Again for convenience the observations are usually performed with the sample horizontal, and this can conceivably suppress the effects of convection, thus allowing the results to be compared with the outcome of purely diffusion-based models. The use of organic analogues is however open to a wide range of experimental innovations including for example a vertical orientation of the sample or the free-fall conditions of the International Space Station. More details of these techniques can be found in the review article by Akamatsu and Nguyen-Thi (2016). Making Metals Transparent: it would be nice to be able to monitor the growth of metals in the same direct manner by which one can monitor that of plastic crystals. A familiar means of seeing through opaque materials is the use of X-rays. Due to advances in technology the direct real-time microscopic observation of metal solidification by means of synchrotron X-ray imaging has been able to supplant the use of plastic-crystal analogues. Although X-ray lenses exist, and can be inserted into the beam-path, the basic magnification technique is actually that of projection. By using a suitable scintillator screen to convert the X-rays into visible radiation, the projected image can then be examined more closely by using a conventional optical microscope. Because the X-rays can penetrate relatively thick samples, it is even possible to employ some of the techniques of conventional medical radiography. A large number of 2D projection images of a sample in 500 to 2000 orientations can be combined so as to produce a high-resolution 3D tomogram. It is even possible to exploit some of the tricks which are used by optical microscopists. The X-ray image contrast weakens with increasing photon-energy, but high-energy photons are the very ones which are required in order to penetrate a metallic sample. The traditional optical Zernike phasecontrast technique can enhance the contrast by converting phase modulations in the image into detectable amplitude modulations. More details of these techniques can be found in Rakete et al. (2011), Nguyen-Thi et al. (2012), Mirihanage et al. (2014), Gibbs et al. (2015) and Shahani et al. (2020) and Neumann-Heyme et al. (2022). Exercises 2.1

Derive Eq. 2.1.

2.2

Compare the curvature (d𝐴/d𝑣 ratio) of spheres and cylinders.

2.3

What is the meaning of the expression, d(∆𝐺)/d𝑟 = 0, in terms of the forces acting upon the critical nucleus?

2.4

Develop an equation for ∆𝐺, as a function of 𝑛, which is analogous to Eq. 2.4, and calculate ∆𝐺𝑛○ and 𝑛○ .

2.5

Derive Eq. 2.5. What approximations are made? Why must caution be exercised when applying it to cases involving high undercooling?

2.6

For small undercoolings, the nucleation rate given by Eq. 2.11 can be written in the form 𝐼 = 𝐾1 exp (−

𝐾2 ) 𝑇∆𝑇 2

(a) Assuming heterogeneous nucleation, explain the origin of the term ∆𝑇 2 and give expressions for 𝐾1 and 𝐾2 ; (b) Determine by how much 𝐾1 and 𝐾2 must change in order to double the value

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of ∆𝑇required to produce one nucleus per cubic metre per second when 𝐾1 s initially equal to 1039 /m3s. 2.7

In their classic experiments, Turnbull and Cech (1950) divided a melt up into droplets which were only a few microns in diameter in order to measure the temperature at which homogeneous nucleation seemed to occur. Why did they use this method?

2.8

By measuring the undercooling required for homogeneous nucleation, it is possible to determine the solid/liquid interface energy, which is difficult to measure using other methods. Develop the equations needed for this, with the aid of Eq. 2.11. [In order to learn more about this technique, the reader should consult the papers of Turnbull (1950) and Perepezko et al. (1979)].

2.9

Show that 𝑛○ and ∆𝐺𝑛○ are functions of 𝜃 but 𝑟 ○ is not. Why is this so? (See Appendix 3, Fig. A3.7).

2.10 Determine the curvature and the melting point of a pure iron crystal in a wetted conical pore in the crucible surface (𝜃 = 30°), as illustrated below.

2.11 What will happen if one stirs 0.4wt% of Ti powder into a pure Al melt at 700°C just before casting? (Hint: consult the phase diagram). [For more details, see Clyne and Robert (1980)]. 2.12 TiB2 compound particles are a good inoculant for Al but only if they have transformed their (0001) surface into another compound – which one? 2.13 Why are most (more than 95%) of the numerous inoculant particles that float in the melt inefficient for triggering crystallisation? 2.14 Calculate the contact angle for a cusp height which corresponds to 1 atomic diameter of Al. 2.15 Show graphically in two dimensions why rapidly growing planes disappear during crystal growth and leave the slowest-growing ones. 2.16 For a given substance, what is the difference in the shape of an equilibrium crystal (𝑉 = 0), when 𝜎 has a pronounced minimum for {111}, and the shape arising from growth (𝑉 > 0) when 𝑉 is a minimum for 100 ? 2.17 Explain why an atomically smooth interface is usually microscopically rough when growing in a temperature gradient (Fig. 2.14(a) - top). 2.18 Which orientation of a face centred cubic crystal has the lowest interface energy with its melt? And which has the lowest stiffness? 2.19 Draw the local atomic density variation between a cubic crystal and its melt.

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References and Further Reading Liquid Structure and Ordering ▪ K.F.Kelton, A.L.Greer, D.M.Herlach, D.Holland-Moritz, The influence of order on the nucleation barrier, MRS Bulletin, 29 (2004) 940. ▪ H.Men, Z.Fan, Atomic ordering in liquid aluminium induced by substrates with misfits, Computational Materials Science, 85 (2014) 1. ▪ M.Rappaz, P.Jarry, G.Kurtuldu, J.Zollinger, Solidification of metallic alloys: does the structure of the liquid matter? Metallurgical and Materials Transactions A, 51 (2020) 2651. Nucleation / Free Growth / Inoculation ▪ R.Becker, W.Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Annalen der Physik, 24 (1935) 719. ▪ P.G.Boswell, G.A.Chadwick, Heterogeneous nucleation in entrained Sn droplets, Acta Metallurgica, 28 (1980) 209. ▪ B.Cantor, R.D.Doherty, Heterogeneous nucleation in solidifying alloys, Acta Metallurgica, 27 (1979) 33. ▪ J.W.Christian, Theory of Transformations of Metals and Alloys, 2nd Edition, Pergamon, Oxford, 1975. ▪ T.W.Clyne, M.H.Robert, Stability of intermetallic aluminides in liquid aluminium and implications for grain refinement, Metals Technology, 7 (1980) 177. ▪ M.A. Easton, M. Qian, A. Prasad, D.H. StJohn, Recent advances in grain refinement of light metals and alloys, Current Opinion in Solid State and Materials Science, 20 (2016) 13. ▪ Z.Fan, Y.Wang, M.Xia, S.Arumuganathar, Enhanced heterogeneous nucleation in AZ91D alloy by intensive melt shearing, Acta Materialia, 57 (2009) 4891. ▪ Z.Fan, Y.Wang, Y.Zhang, T.Qin, X.R.Zhou, G.E.Thompson, T.Pennycook, T.Hashimoto, Grain refining mechanism in the Al/Al–Ti–B system, Acta Materialia, 84 (2015) 292. ▪ Z.Fan, F.Gao, B.Jiang, Z.Que, Impeding nucleation for more significant grain refinement, Scientific reports, 10 (2020) 9448. ▪ Z.Fan, F.Gao, Y.Wang, H.Men, L.Zhou, Effect of solutes on grain refinement, Progress in Materials Science, 123 (2022) 100809. ▪ A.L.Greer, A.M.Bunn, A.Tronche, P.V.Evans, D.J.Bristow, Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al–Ti–B, Acta Materialia, 48 (2000) 2823. ▪ A.L.Greer, Overview: application of heterogeneous nucleation in grain-refining of metals, Journal of Chemical Physics, 145 (2016) 211704. ▪ M.K.Hoffmeyer, J.H.Perepezko, Evaluation of inoculant efficiency in Al grain refining alloys, Scripta Metallurgica, 23 (1989) 315. ▪ K.N.Ishihara, M.Maeda, P.H.Shingu, The nucleation of metastable phases from undercooled liquids, Acta Metallurgica, 33 (1985) 2113. ▪ K.Kelton, A.L.Greer, Nucleation in Condensed Matter, Elsevier, Amsterdam, 2010. ▪ I.Maxwell, A.Hellawell, A simple model for grain refinement during solidification, Acta Metallurgica, 23 (1975) 229. ▪ J.H.Perepezko, D.H.Rasmussen, I.E.Anderson, C.R.Loper, in Solidification and Casting of Metals, The Metals Society, London (1979) p.169. ▪ J.H.Perepezko, Nucleation in undercooled liquids, Materials Science and Engineering, 65 (1984) 125.

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▪ J.H.Perepezko, G.Wilde, Melt undercooling and nucleation kinetics, Current Opinion in Solid State and Materials Science, 20 (2016) 3. ▪ T.E.Quested, A.L.Greer, The effect of the size distribution of inoculant particles on as-cast grain size in aluminium alloys, Acta Materialia, 52 (2004) 3859. ▪ P.Schumacher, A.L.Greer, Enhanced heterogeneous nucleation of a-Al in amorphous aluminium Alloys, Materials Science and Engineering A, l81-182 (1994) 1335. ▪ D.H.StJohn, A.Prasad, M.A.Easton, M.Qian, The Contribution of Constitutional Supercooling to Nucleation and Grain Formation, Metallurgical and Materials Transactions A, 46 (2015) 4868. ▪ D.Turnbull, Formation of crystal nuclei in liquid metals, Journal of Applied Physics, 21 (1950a) 1022. ▪ D.Turnbull, Kinetics of heterogeneous nucleation, Journal of Chemical Physics, 18 (1950-b) 198. ▪ D.Turnbull, R.E.Cech, Microscopic observation of the solidification of small metal droplets, Journal of Applied Physics, 21 (1950) 804. ▪ D.Turnbull, Kinetics of solidification of supercooled liquid mercury droplets, Journal of Chemical Physics, 20 (1952) 411. ▪ C.V.Thompson, F.Spaepen, Homogeneous crystal nucleation in binary metallic melts, Acta Metallurgica, 31 (1983) 2021. ▪ M.Volmer, A.Weber, Keimbildung in übersättigten Gebilden, Zeitschrift für Physikalische Chemie, 119 (1926) 277. Molecular Simulation ▪ J.Anwar, D.Zahn, Uncovering molecular processes in crystal nucleation and growth by using molecular simulation, Angewandte Chemie - International Edition, 50 (2011) 1996. ▪ S.R.Ganagalla, S.N.Punnathanam, Free energy barriers for homogeneous crystal nucleation in a eutectic system of binary hard spheres, Journal Chemical Physics, 138 (2013) 174503. ▪ Z.Y.Hou, L.X.Liu, R.S.Liu, Z.A.Tian, J.G.Wang, Kinetic details of nucleation in supercooled liquid Na: A simulation tracing study, Chemical Physics Letters, 491 (2010) 172. ▪ A.F.Voter, The embedded atom method, in Intermetallic Compounds, Vol. 1: Principles. J.H.Westbrook, L.Fleischer (Eds), John Wiley & Sons Ltd, 1994, p.77. ▪ J.Zhou, Y.Yang, J.Zhou, Y.Yang, Y.Yang, D.S.Kim, A.Yuan, X.Tian, C.Ophus, F.Sun, A.K.Schmid, M.Nathanson, H.Heinz, Q.An, H.Zeng, P.Ercius, J.Miao, Observing crystal nucleation in four dimensions using atomic electron tomography, Nature, 570 (2019) 500. Glasses and Quasicrystals ▪ E.Axinte, Metallic glasses from ‘‘alchemy’’ to pure science: present and future of design, processing and applications of glassy metals, Materials and Design, 35 (2012) 518. ▪ F.C.Frank, Supercooling of liquids, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 215 (1952) 43. ▪ A.L.Greer, Metallic glasses...on the threshold, Materials Today, 12 (2009) 14. ▪ A.Inoue, F.L.Kong, S.L.Zhu, A.L.Greer, Multicomponent bulk metallic glasses with elevated temperature resistance, MRS Bulletin, 44 (2019) 867. ▪ M.Rappaz, G.Kurtuldu, Thermodynamic aspects of homogeneous nucleation enhanced by icosahedral short-range order in liquid FCC-type alloys, Journal of Metals, 67 (2015) 1812. ▪ D.S.Shechtman, I.Blech, D.Gratias, J.W.Cahn, A metallic phase with long-ranged orientational order and no translational symmetry, Physical Review Letters, 53 (1984) 1951.

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▪ W.Steurer, Quasicrystals: What do we know? What do we want to know? What can we know? Acta Crystallographica A, 74 (2018) 1. Solid/Liquid Interface Structure/ Energy ▪ J.Cahn, D.Hoffman, A vector thermodynamics for anisotropic surfaces - II. Curved and faceted surfaces, Acta Metallurgica, 22 (1974) 1205. ▪ D.Camel, G.Lesoult, N.Eustathopoulos, Metastable equilibrium states of solid-liquid interfaces in metallic binary alloys, Journal of Crystal Growth, 53 (1981) 327. ▪ M.Elwenspoek, Comment on the α-factor of Jackson for crystal growth from solution, Journal of Crystal Growth, 78 (1986) 353. ▪ C.Fang, Z.Fan, Prenucleation at the liquid-Al/α-Al2O3 and the liquid-Al/MgO interfaces, Computational Materials Science, 171 (2020) 109258. ▪ T.Haxhimali, A.Karma, F.Gonzales, M.Rappaz, Orientation selection in dendritic evolution, Nature Materials, 5 (2006) 660. ▪ C.Herring, in The Physics of Powder Metallurgy, W.E.Kingston (Ed.), McGraw-Hill, New York, 1951. ▪ J.J.Hoyt, M.Asta, T.Haxhimali, A.Karma, R.E.Napolitano, R.Trivedi, B.B.Laird, J.R.Morris, Crystal-melt interfaces and solidification morphologies in metals and alloys, MRS Bulletin, December (2004) 935. ▪ K.A.Jackson, in Liquid Metals and Solidification, American Society for Metals, Cleveland, 1958, p.174. ▪ K.A.Jackson, On the theory of crystal growth: growth of small crystals using periodic boundary conditions, Journal of Crystal Growth, 3-4 (1968) 507. ▪ J.S.Langer 1978, ‘Phase-Field Model’, Research Notes Written for Colleagues at Carnegie Mellon University, Pittsburgh, published in the appendix of W.Kurz, D.J.Fisher, R.Trivedi, International Materials Reviews, 64 (2019) 350. ▪ L.V.Mikheev, A.A.Chernov, Mobility of a diffuse simple crystal-melt interface, Journal of Crystal Growth, 112 (1991) 591. ▪ R.E.Napolitano, S.Liu, Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system, Physical Review B, 70 (2004) 214103. ▪ F.Spaepen, A structural model for the solid-liquid interface in monatomic systems, Acta Metallurgica, 23 (1975) 729. ▪ F.Spaepen and R.B.Meyer, The surface tension in a structural model for the solid-liquid interface, Scripta Metallurgica, 10 (1976) 257. ▪ F.Spaepen, private communication, 2021. ▪ J.F.van der Veen, H.Reichert, Structural ordering at the solid-liquid interface, MRS Bulletin, December (2004) 958. ▪ L.Wang, J.Hoyt, N.Wang, N.Provatats, C.W.Sinclair, Controlling solid-liquid interfacial energy anisotropy through the isotropic liquid, Nature Communications, 11 (2020) 72. Crystal Growth ▪ E.A.Brener, H.Müller-Krumbhaar, D.E.Temkin, Europhysics Letters, 17 (1992) 535. ▪ A.A. Chernov, Notes on interface growth kinetics 50 years after Burton, Cabrera and Frank, Journal of Crystal Growth, 264 (2004) 499. ▪ S.R.Coriell, D.Turnbull, Relative roles of heat transport and interface rearrangement rates in the rapid growth of crystals in undercooled melts, Acta Metallurgica, 30 (1982) 2135.

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▪ F.C.Frank, The influence of dislocations on crystal growth, Discussions of the Faraday Society, 5 (1949) 48. ▪ G.H.Gilmer, K.A.Jackson, in Crystal Growth and Materials, E.Kaldis, H.J.Scheel (Eds), NorthHolland, 1977. ▪ D.R.Hamilton, R.G.Seidensticker, Propagation mechanism of germanium dendrites, Journal of Applied Physics, 31 (1960) 1165. ▪ K.A.Jackson, G.H.Gilmer, D.E.Temkin, J.D.Weinberg, K.Beatty, Non-equilibrium phase transformations, Journal of Crystal Growth, 128 (1993) 127. ▪ W.Kossel, Extending the law of Bravais, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, (1927) 143. ▪ I.V.Markov, Crystal Growth for Beginners, 3rd ed., World Scientific, Singapore, 2017. ▪ I.Minkoff, in The Solidification of Metals, Iron and Steel Institute Publication 110, London, 1968. ▪ L.V.Mikheev, A.A.Chernov, Mobility of a diffuse simple crystal-melt interface, Journal of Crystal Growth, 112 (1991) 591. ▪ F.Spaepen, D.Turnbull, Crystallization Processes, in Laser Annealing of Semiconductors, J.M.Poate, J.W.Mayer (Eds), Academic Press, 1982, p.15. ▪ I.N.Stranski, Zur Theorie des Kristallwachstums, Zeitschrift für Physikalische Chemie, 136 (1928) 259. ▪ I.Sunagawa, Crystals - Growth, Morphology, & Perfection, Cambridge University Press, 2007. In Situ Observation of Solidification Phenomena Colloids ▪ D.M.Herlach, T.Palberg, I.Klassen, R.Kobold, Overview: experimental studies of crystal nucleation: metals and colloids, Journal of Chemical Physics, 145 (2016) 211703. ▪ H.Hwang, D.A.Weitz, F.Spaepen, Direct observation of crystallization and melting with colloids, Proceedings of the National Academy of Sciences, 116 (2019) 1180. Organics ▪ S.Akamatsu, H.Nguyen-Thi, In situ observation of solidification patterns in diffusive conditions, Acta Materialia, 108 (2016) 325. ▪ K.A.Jackson, Current concepts in crystal growth from the melt, Progress in Solid State Chemistry, 4 (1967) 53. ▪ W.Kurz, D.J.Fisher, Dendrite growth in eutectic alloys : the coupled zone, International Metals Reviews, 24 (1979) 177. ▪ J.P.Mogeritsch, M.Abdi, A.Ludwig, Investigation of Peritectic Solidification Morphologies by Using the Binary Organic Model System TRIS-NPG, Materials 13 (2020) 966. Synchroton X-ray imaging ▪ J.W.Gibbs, K.A.Mohan, E.B.Gulsoy, A.J.Shahani, X.Xiao, C.A.Bouman, M.De Graef, P.W.Voorhees, The Three-Dimensional Morphology of Growing Dendrites, Nature-Scientific Reports, 5 (2015) 11824. ▪ W.U.Mirihanage, K.V.Falch, I.Snigireva, A.Snigirev, Y.J.Li, L.Arnberg, R.H.Mathiesen, Retrieval of three-dimensional spatial information from fast in situ two-dimensional synchrotron radiography of solidification microstructure evolution, Acta materialia, 81 (2014) 241. ▪ H.Neumann-Heyme, N.Shevchenko, J.Grenzer, K.Eckert, C.Beckermann, S.Eckert, In-situ measurements of dendrite tip shape selection in a metallic alloy, Physical Review Materials, 6 (2022) 063401.

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▪ H.Nguyen-Thi, L.Salvo, R.H.Mathiesen, L.Arnberg, B.Billia, M.Suery, G.Reinhart, On the interest of synchrotron X-ray imaging for the study of solidification in metallic alloys, Comptes Rendus – Physique, 13 (2012) 237. ▪ C.Rakete, C.Baumbach, A.Goldschmidt, D.Samberg, C.G.Schroer, F.Breede, C.Stenzel, G.Zimmermann, C.Pickmann, Y.Houltz, C.Lockowandt, O.Svenonius, P.Wiklund, R.H.Mathiesen, Compact x-ray microradiograph for in situ imaging of solidification processes: Bringing in situ x-ray micro-imaging from the synchrotron to the laboratory, Review of Scientific Instruments, 82 (2011) 105108. ▪ A.J.Shahani, X.Xiao, E.M.Lauridsen, P.W.Voorhees, Characterization of metals in four dimensions, Materials Research Letters, 8 (2020) 462. ▪ I.Snigireva, V.Honkima, R.H.Mathiesen, In situ hard X-ray transmission microscopy for materials science, Journal of Materials Science, 52 (2017) 3497. ▪ Y.Wang, S.Jia, M.Wei, L.Peng, Y.Wu, X.Liu, Research progress on solidification structure of alloys by synchrotron X-ray radiography: a review, Journal of Magnesium and Alloys, 8 (2020) 396.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 51-65 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER THREE†

MORPHOLOGICAL INSTABILITY OF A SOLID/LIQUID INTERFACE Classical thermodynamic definitions of stability are inapplicable to the determination of the morphology of a growing interface, and extensions of equilibrium thermodynamics have not furnished a fully acceptable alternative. In order to proceed with theoretical analyses of growth morphologies, it has been found necessary in the past to use heuristically-based stability criteria, such as proposing that the morphology which actually appears is that one which possesses the maximum growth-rate or minimum undercooling. Those criteria have now been replaced by stability arguments which involve (mathematically) perturbing the growing solid/liquid interface morphology in order to determine whether it is likely to change into another one. The interface is then said to be morphologically unstable if the perturbation is amplified with the passage of time and to be morphologically stable if it is instead damped-out (Fig 3.1). 3.1 Interface Instability of Pure Substances The conditions which lead to instability can easily be understood in the case of a pure substance. Figure 3.2 illustrates, in a schematic manner, the development of a perturbation of a plane front in a superheated liquid (a), i.e. in a positive thermal gradient (see also Fig. 1.7(a)), and in an undercooled liquid having a temperature lower than the melting point of the pure substance (Fig. 1.7(b)). The first situation is typical of columnar growth, while the second corresponds typically to equiaxed solidification. During the columnar growth of a pure substance (Fig. 3.2(a)) the temperature profile, 𝑇𝑞 (𝑧), imposed by the heat flux, increases in the z-direction (i.e. G is positive): by convention, the z-axis is defined to be normal to the solid/liquid interface and to point into the liquid. The interface will be *

Top image: Time evolution of a morphologically unstable planar solid-liquid interface in directional solidification showing the development of interface perturbations (a, b) into cells (c). At time (d) two cells become morphologically unstable also on their sides. These instabilities form the initiation of secondary arms of columnar dendrites (solidification of a transparent organic under the microscope by Trivedi and Somboonsuk (1984), width of solid/liquid interface = 1.7mm).

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Figure 3.1 Initial Evolution of a Stable (a) or Unstable (b) Interface Such an interface might be seen during the microscopic observation of transparent non-faceted organic substances (plastic crystals, Table 2.3). Any interface will be subjected to random disturbances during growth, due to temperature fluctuations, grain boundaries or particles. A stable interface is distinguished from an unstable interface by its response to such disturbances. It is imagined here that the interface is initially slightly distorted by a spatially regular disturbance. The perturbations will be unfavourably situated for their further development in the case of a stable interface (a) and will disappear. If they find themselves in a more advantageous situation for growth they will increase in prominence and the distorted interface is then clearly unstable (b). The solid/liquid interface is usually unstable during the casting of alloys. A stable interface is obtained only in special cases, such as the columnar solidification of pure metals (Fig. 1.7(a)) or the directional solidification of alloys in a Bridgman-type furnace (Fig. 1.4(a)) within a sufficiently high temperature gradient, 𝐺. The scale of this figure is typical for alloys under normal casting conditions.

located at the isotherm where the temperature, Tq, that is imposed by the heat flux is very close to the equilibrium melting point, Tf ‡. If an interface perturbation is to remain at the melting point over its entire surface and curvature effects upon Tf are neglected the temperature profile, 𝑇𝑞 (𝑧), must be locally deformed: at the tip of a perturbation (along line A-A), the temperature gradient in the liquid increases while it decreases in the solid. According to Fourier's first law this means that the heat flux from the liquid to the tip increases, and that the flux in the solid decreases. More heat will therefore flow into the tip and less will flow out of it, thus tending to make it retreat to its original position on the planar interface. The reverse process occurs meanwhile with regard to the depressions (line B-B) thus making them advance back to their position on the planar front. The perturbation consequently tends to be damped out. The interface of a pure substance during columnar growth in a positive temperature gradient will therefore always be stable.

‡

Due to interface-attachment kinetics, a solid/liquid interface will generally require some driving force (undercooling below the equilibrium melting temperature) to grow (Eq. 2.14). This cause of undercooling can be appreciable (greater than 1K) for faceted substances. Its magnitude for non-faceted substances such as metals will be negligibly small in most cases, and its effect can be safely discounted for these materials under conventional (i.e. not too rapid) solidification conditions.

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Figure 3.2 Columnar (a) and Equiaxed (b) Solidification of a Pure Substance The interface stability of a pure substance depends upon the direction of heat-flow. Under directional solidification conditions such as those existing in the columnar zone of a casting, the liquid temperature always increases ahead of the interface (a). The heat-flow direction is therefore opposite to that of the solidification. When a perturbation of amplitude, , forms at an initially smooth interface the temperature gradient in the liquid close to a protuberance increases, while the gradient in the solid decreases (red curve A-A). Since the heat flux is proportional to its position in the gradient, more heat then flows into the tip of the perturbation, and less flows out of it into the solid. The reverse occurs near a valley of the perturbation (thermal profile along line B-B). The perturbation consequently melts back near to its peaks and solidifies faster within its valleys, i.e. the planar interface is stabilised. The opposite situation occurs during equiaxed solidification (b). The free crystals here grow into an undercooled melt (cross-hatched region) and the latent heat produced during growth also flows down the negative temperature gradient and into the liquid. A perturbation which forms on the sphere will make this gradient steeper (compare the full red curve to the dotted curve) and thus allow the tip to reject more heat. The local growth rate is thereby increased and the interface is always morphologically unstable.

Turning now to the equiaxed solidification of a pure metal the situation is quite different (Fig. 3.2(b)). Solidification does not start from the mould wall in this case but is instead initiated by nuclei existing within the volume of the melt. The melt is thus undercooled while the temperature of the solid/liquid interface is essentially at the melting point, 𝑇𝑓 . The temperature gradient in the liquid is therefore negative while the gradient within the small particle of solid is essentially zero. A perturbation will now experience a higher negative gradient at its tip, leading to an increased heat flux and a resultant increase in the growth-rate of the tip. The opposite situation occurs near to a valley of the perturbation. The interface of a pure substance (again assuming a negligible attachmentkinetic undercooling) will therefore always be unstable under equiaxed solidification conditions. The result is that equiaxed grains in a pure metal will adopt a dendritic morphology, assuming that their number density is not too high and that dendrites have enough space in which to develop (Fig. 1.9(a)). Because no solute segregation occurs during the solidification of a pure metal the dendritic growth form will be effectively undetectable in the final solid. However, one may measure the local crystallographic orientations of the equiaxed grains. This can be achieved nowadays by means of electron back-scattered diffraction (EBSD) in a scanning electron microscope (SEM) or, more simply, by means of surface oxidation and examination of the oxide using polarized-light optical

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microscopy§. The characteristics of dendritic growth can of course be very closely observed directly by means of the optical microscopy of pure organic substances or of water. The most recent advance has been the direct in situ observation of aluminium by means of synchrotron X-ray imaging (see Sect. 2.9). It can be concluded that the solid/liquid interface of a pure metal will always be stable if the temperature gradient is positive, and unstable if the gradient is negative. Perturbations of amplitude, , will grow if the temperature gradient in the liquid at the interface, given by the heat flux, 𝐺 = 𝑑𝑇𝑞 /𝑑𝑧 , is negative: 𝜀̇ =

𝑑𝜀 >0 𝑑𝑡

in pure metals if

𝐺=

𝑑𝑇𝑞 𝑉1 , are shown.

§

EBSD essentially measures the local crystallographic orientation of a point (typically 1m in size) by monitoring the diffraction of back-scattered electrons, whereas oxidation provides an indirect measurement of the crystallographic orientation when the surface oxidation-rate, and therefore the final thickness of the oxide layer, can be made to depend upon the crystallographic orientation of the exposed surface.

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Due to the slowness of material transport via diffusion the excess solute which is rejected from the solid will accumulate within an enriched boundary layer ahead of the interface. A fully developed diffusion boundary layer is illustrated by Fig. 3.3 for a reference-frame, z, which is attached to the interface and has its origin fixed at the interface position. It is established during an initial transient period before steady-state growth begins (Fig. 6.1). All of the concentrations are constant with respect to the above reference frame in this steady-state situation, and the solid forms at the solidus temperature of the alloy. The composition of the solid is therefore equal to 𝐶0 , the composition of the liquid far ahead of the interface; where the effect of the solute pile-up has not yet been felt. The solute concentration in the boundary layer decreases exponentially, from 𝐶0 /𝑘 to 𝐶0 , according to (Appendix 2, Eq. A2.21): 𝐶𝑙 = 𝐶0 + Δ𝐶0 exp (−

𝑉𝑧 ) 𝐷

[3.2]

The thickness, 𝛿c , of the boundary layer is mathematically infinite but, for practical purposes, it can be taken to be equal to the 'equivalent boundary layer' (Appendix 2, Fig. A2.4): 𝛿𝑐 =

2𝐷 𝑉

[3.3]

This thickness is equal to the base-length of a right-angled triangle having a height which is equal to the excess solute concentration at the interface, ∆𝐶0 = (𝐶0 /𝑘 − 𝐶0 ), and an area which is the same as that under the exponential curve. Equation 3.3 reveals that the equivalent boundary layer thickness is inversely proportional to the growth rate. A simple flux balance shows that an interface of area, A, rejects Js atoms per second: 𝐽𝑠 = 𝐴 (

𝑑𝑧 ) (𝐶𝑙∗ − 𝐶𝑠∗ ) 𝑑𝑡

[3.4]

where the term, 𝐴(𝑑𝑧/𝑑𝑡) = 𝐴𝑉, represents the volume of liquid which is transformed into solid per unit time, and the second term represents the difference in the solute concentrations existing in the liquid and the solid at the interface. The resultant flux of rejected solute under steady-state conditions has to be balanced by an equal flux which transports solute away from the interface via diffusion. The flux in the liquid for a given cross-section, A, is: 𝑑𝐶𝑙 𝐽𝑙 = −𝐴𝐷 ( ) 𝑑𝑧

[3.5]

Equating the fluxes and noting that, in the steady state, 𝐶𝑙∗ = 𝐶0 /𝑘, gives the flux balance: 𝐺𝑐 = (

𝑑𝐶𝑙 𝑉 𝑉 𝐶0 ) = − ( ) Δ𝐶0 = − ( ) (1 − 𝑘) 𝑑𝑧 𝑧=0 𝐷 𝐷 𝑘

[3.6]

3.3 Interface Instability of Alloys It is seen from the above that there is a substantial change in the concentration ahead of the interface during the solidification of an alloy. This change will affect the local equilibrium solidification temperature, 𝑇𝑙 (𝐶𝑙 ), of the liquid, which is in turn related to the composition by:

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𝑇𝑙 (𝐶0 ) − 𝑇𝑙 (𝐶𝑙 ) = 𝑚(𝐶0 − 𝐶𝑙 )

[3.7]

where 𝑇𝑙 (𝐶0 ) is the liquidus temperature corresponding to the initial alloy composition and 𝑚 is the slope of the liquidus. This relationship is shown in Fig. 3.4 and indicates that the solutal boundary layer can be converted, using the phase diagram, into a corresponding liquidus-temperature boundary layer. The liquidus temperature increases with increasing 𝑧 (k < 1 and 𝑚 < 0, Fig. 1.11). It represents the local equilibrium temperature for the solidification of a corresponding volume element of the melt.

Figure 3.4 Constitutional Undercooling in Alloys The steady-state diffusion boundary layer, shown in Fig. 3.3, is reproduced in (a) for a given growth rate. As the liquid concentration, 𝐶𝑙 , decreases with distance, z (perpendicular to the interface), the local equilibrium liquidus temperature, 𝑇𝑙 (i.e. the melting temperature), of the alloy will increase as indicated by the phase diagram (c). This means that if small volumes of liquid at various distances ahead of the solid/liquid interface were to be extracted by some means and solidified, their equilibrium freezing points would vary with sampling-position in the manner described by the red curve in (b). Each volume element finds itself however located at a temperature, 𝑇𝑞 , which is imposed by the temperature gradient that arises from the heat flow occurring within the system. At the solid/liquid interface (𝑧 = 0), 𝑇𝑞 must be slightly less than 𝑇𝑠 (𝐶0 ) in order to attach the atoms to the crystal. Attachment is very easy in slowly solidifying metals, so one can neglect this undercooling and set 𝑇𝑞 = 𝑇𝑠 . There exists a volume of liquid which is undercooled when the gradient of 𝑇𝑞 is lower than the gradient of 𝑇𝑙 at 𝑧 = 0. This region (cross-hatched) is called the zone of constitutional undercooling. There exists a driving force for the development of perturbations in this volume, as in the case of the cross-hatched region of Fig. 3.2(b).

In order to investigate morphological stability it is also necessary to determine the temperature, 𝑇𝑞 , which is imposed by the heat flux. The temperature, 𝑇𝑞, neglecting any attachment-kinetics undercooling, must be equal to the liquidus, 𝑇𝑙 (𝐶𝑙∗ ), at the interface. During the steady-state growth of a planar interface this will correspond to 𝑇𝑙 (𝐶0 /𝑘) = 𝑇𝑠 (𝐶0 ), i.e. the solidus temperature for the nominal composition of the alloy, 𝐶0 , as shown in Fig. 3.4. Depending upon the temperature gradient:

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Foundations of Materials Science and Engineering Vol. 103

𝐺=(

𝑑𝑇𝑞 ) 𝑑𝑧 𝑧=0

57

[3.8]

in the liquid at the solid/liquid interface (which is imposed by the external heat flux) there may, or may not, exist a zone of constitutional undercooling (Fig. 3.5). Constitutional undercooling is said to be present in the volume of the melt ahead of the interface if the actual local temperature, 𝑇𝑞 , is lower than the corresponding local equilibrium solidification temperature, 𝑇𝑙 (Fig. 3.5(b)). The melt in this zone is thus undercooled, i.e., is in a metastable state. The condition required for the existence of such a constitutionally undercooled zone is that the temperature gradient in the liquid, 𝐺, is lower than the gradient of liquidus-temperature change in the melt; both being evaluated at the solid/liquid interface. The latter gradient is obtained by multiplying the concentration gradient at the interface, 𝐺𝑐 , by the liquidus slope, 𝑚. The criterion for morphological instability of an alloy is constitutional undercooling:

𝐺 < 𝑚𝐺𝑐

[3.9]

Figure 3.5 Condition for Constitutional Undercooling at the Solid/liquid Interface, and the Resultant Structures When the temperature gradient due to the heat flux is greater than the liquidus temperature gradient at the solid/liquid interface, the latter is stable (a). It can be clearly seen however that a driving force for interface change will be present whenever the slope of the local melting point curve (liquidus temperature) at the interface is greater than the slope of the actual temperature distribution. This is easily understood since the undercooling encountered by the tip of a perturbation which is advancing into the melt then increases and a planar interface is therefore unstable (b). Note that the temperature profile in (b) is only hypothetical; after the dendritic microstructure shown in the lower figure has developed, the region of constitutional undercooling is largely erased by the growth of the dendrites. Only a much smaller constitutional undercooling remains at the tips of the dendrites, as explained in Chap. 4.

Considering as before the behaviour of a perturbation which arises during the directional solidification of an alloy, such a protuberance at the solid/liquid interface will increase the local temperature gradient in the melt. In the case of a pure melt (Fig. 3.2), this leads to the disappearance of the perturbation. In the case of an alloy melt (Fig. 3.4), however, the local concentration gradient will also become steeper and the local gradient of the liquidus temperature will consequently increase. The region of constitutional supercooling will thus tend to be preserved. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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In the introduction to this chapter, it was noted that equilibrium thermodynamics principles could not properly be applied to solidification. It is interesting nevertheless to see how far the use of such classical methods can be pursued in this non-equilibrium situation. The growth rates and undercoolings are generally found to be closely related by functions whose form depends upon the process which controls growth (atomic attachment, mass diffusion, thermal diffusion). The growth rate increases with increasing undercooling in each case. An interface perturbation can thus be imagined to experience a driving force, f, for accelerated growth which is given by the negative value of the first derivative of the Gibbs free energy with respect to distance: 𝑓=−

𝑑(Δ𝐺) 𝑑𝑧

[3.10]

For small undercoolings, Δ𝐺 = −Δ𝑆𝑓 Δ𝑇 and Δ𝑆𝑓 = constant and thus, 𝑑Δ𝑇 ) = Δ𝑆𝑓 𝜙 𝑓 = Δ𝑆𝑓 ( 𝑑𝑧

[3.11]

where Δ𝑇 = 𝑇𝑙 − 𝑇𝑞 , and 𝜙 is the difference between the liquidus-temperature gradient (𝑚𝐺𝑐 ) and the heat-flux-imposed temperature gradient at the interface (𝐺): 𝑑Δ𝑇 𝑑𝑇𝑙 𝑑𝑇𝑞 ) ) 𝜙=( =( − = 𝑚𝐺𝑐 − 𝐺 𝑑𝑧 𝑧=0 𝑑𝑧 𝑑𝑧 𝑧=0

[3.12]

A positive driving force will exist which causes any perturbation to grow when 𝜙 is positive, i.e. when a zone of constitutional undercooling exists ahead of the interface. If its value is negative, 𝐺 is greater than 𝑚𝐺𝑐 . The limiting condition for constitutional undercooling is therefore: [3.13]

𝜙 = 𝑚𝐺𝑐 − 𝐺 = 0

This pseudo-thermodynamic approach thus gives the same result as that deduced above by considering the zone of constitutional undercooling. Since the concentration gradient at the interface is known, it is then simple to derive the criterion for the existence of constitutional undercooling in another form. The interface will always become unstable if Eq. 3.9 is satisfied. Using Eq. 3.6, this can be written: 𝐺 0)

pure metal

unstable

stable

alloy

unstable

unstable (𝜙 > 0) /

stable (𝜙 < 0)

Table 3.2 Minimum Stabilising Temperature Gradient (K/mm) as a Function of Distribution Coefficient, 𝒌, and Alloy composition, 𝑪𝟎 , for 𝑫 = 𝟎. 𝟎𝟎𝟓 mm2/s, 𝑽 = 𝟎. 𝟎𝟏 mm/s, 𝒎 = −𝟏𝟎 K/wt% C0 (wt %) k

10

1

0.01

0.5 0.1

200 1800

20 180

0.2 1.8

Only the limit of stability has been estimated up to this point. Nothing has been said about the form and scale of the perturbations which will develop if the interface is unstable. Information concerning the dimensions of the initial perturbed morphology is very important because this will influence the scale of the resultant growth morphologies. It must be remembered however that the morphology which initially develops beyond the limit of stability is usually only a transient structure, and disappears when the steady-state cellular, dendritic or eutectic morphology is established. 3.4 Perturbation Analyses A drawback of the constitutional undercooling criterion is that it ignores the effect of the surface energy of the interface and therefore cannot predict the wavelength of the instabilities. It is reasonable to suppose that capillarity should have a marked influence upon curved perturbations. In order to investigate this possibility, and to learn more about the morphological changes occurring near to the limit of stability, one supposes that the planar interface, positioned at 𝑧 = 0 in the reference frame moving at velocity V, has already been slightly disturbed (Fig. 3.6(a)). The time-evolution of this perturbation is then investigated under the constraints of diffusion and capillarity. Mullins and Sekerka (1963/64) were the first to compute the growth or decay rate of a small-amplitude perturbation of a planar solid/liquid interface, of the form: 𝑧 ∗ (𝑦, 𝑡) = 𝜀(𝑡) sin(𝜔𝑦)

[3.16]

where y is the lateral coordinate fixed in the interface (Fig. 3.6(b)), 𝜀(𝑡) is the amplitude of a Fourier component of the interface shape, having a wavelength 𝜆 = 2π/𝜔 and 𝑧 ∗ (𝑦, 𝑡) corresponds to the position of the solid/liquid interface. The application of linear perturbation theory leads to a relationship between the rate of amplitude development 𝜀̇ = 𝑑𝜀/𝑑𝑡 and the wavelength, 𝜆. Further details of this procedure are given in Appendix 7. The present discussion is intended merely to point out some of the basic physical principles which are involved. The undulation of the interface first induces a sinusoidal perturbation of the concentration profile with respect to the steady-state compositional field of a planar front (Eq. 3.2); the intensity of

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which decreases with increasing distance from the interface and with increasing . This modifies the local liquidus temperature of the interface according to 𝑇𝑓 + 𝑚𝐶𝑙∗ , where 𝐶𝑙∗ (𝑦) is the liquid composition at the perturbed interface. The curvature of the perturbed interface also modifies the equilibrium temperature of the interface, 𝑇 ∗ . Again, neglecting any temperature difference due to atomic-attachment kinetics, one then can write: 𝑇 ∗ = 𝑇𝑓 + 𝑚𝐶𝑙∗ − Γ𝐾 ∗

[3.17]

Figure 3.6 Morphological Perturbations at a Solid/liquid Interface The existence of a zone of constitutional undercooling implies that a driving force for a change in the morphology is available, but gives no indication of the scale of the morphology which will appear. Figure (a) shows the morphological breakdown of a planar solid/liquid interface (∼200 m wide) at an early stage of development (Fisher, 1978). It is a microscope view of a layer of succinonitrile directionally solidifying between two glass slides; the interface being inclined with respect to the microscope, thus offering a ‘three-dimensional’ image. The growth of the instabilities depends upon the initial local deformation of the plane front; slight grooves at low angle (sub-)grain boundaries and deep grooves at high-angle grain boundaries favour instabilities. Experimental observations show that the initial form of the new morphology is periodic and may be approximated by a sinusoidal curve (b). Perturbation analysis permits the calculation of the wavelength of the instabilities which develop. The result is of importance in the theory of dendrite growth.

This equation states that the difference between the melting point of the pure system, 𝑇𝑓 , and the interface temperature, 𝑇 ∗ , is equal to the sum of the temperature differences due to the local interface composition, 𝐶𝑙∗ , and the local interface curvature, 𝐾 ∗ , at any point of the interface (Γ is the Gibbs-Thomson coefficient, Eq. 1.7). Taking just two points at the tips (t) and depressions (d) of the interface (Fig. 3.6(b)) and calculating the temperature difference gives (removing ‘*’ for the interface): 𝑇𝑡 − 𝑇𝑑 = 𝑚(𝐶𝑡 − 𝐶𝑑 ) − Γ(𝐾𝑡 − 𝐾𝑑 )

[3.18]

The curvatures at tips and depressions which are not too extreme can be determined from the second derivative of the function which describes the interface shape (Eq. 3.16) at 𝑦 = 𝜆 and 𝜆/2: 𝐾𝑡 = −𝐾𝑑 =

4π2 𝜀 𝜆2

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[3.19]

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(Note that 𝐾 has the opposite sign to that which arises from the mathematical definition — Eq. 1.6). While Appendix 7 derives the full expression for the rate, 𝜀̇, at which a small perturbation of wavelength, 𝜆, is amplified or damped over time, a simple approximation can be used to find the wavelength, 𝜆𝑖 , for which 𝜀̇ = 0. This is called the marginal stability wavelength since it corresponds to a critical perturbation which is neither amplified, nor damped. It matches both the thermal and solutal diffusion fields of the unperturbed interface and so one can write: 𝑇𝑡 − 𝑇𝑑 = 2𝜀𝐺

[3.20]

𝐶𝑡 − 𝐶𝑑 = 2𝜀𝐺𝑐

[3.21]

and

Substituting Eqs 3.19 to 3.21 into Eq. 3.18 leads to: 1/2 Γ 1/2 Γ ) 𝜆𝑖 = 2π ( ) = 2π ( 𝜙 𝑚𝐺𝑐 − 𝐺

[3.22]

where 𝜙 is the degree of constitutional undercooling as defined by Eq. 3.12. Although this perturbation, of wavelength 𝜆𝑖 , will be stationary with respect to the unperturbed interface, this does not mean that the interface itself is stable as other perturbations of wavelength 𝜆 ≠ 𝜆𝑖 can induce 𝜀̇/𝜀 > 0. Considering now the whole spectrum of wavelengths 𝜆 that can perturb the interface, one can show that, to first order, the amplitude of the corresponding perturbation 𝜀(𝜆) evolves over time according to (see Appendix 7 for details, Eq. A7.11)**: 𝜀̇ 𝑉 𝑉 𝑉2 ) ( (1 − 𝑘) − 𝑏) (Γ𝜔2 + 𝐺 − 𝑚𝐺𝑐 ) − (𝜆) = ( 𝑘 𝜀 𝑚𝐺𝑐 𝐷 𝐷 with:

2π 𝜔= 𝜆

[3.23] 1/2

and

𝑉 𝑉 2 𝑏= + [( ) + 𝜔2 ] 2𝐷 2𝐷

The function 𝜀̇/𝜀(𝜆) is plotted in Fig. 3.7 for an Al-Cu alloy and exhibits a characteristic maximum. At 𝜆 values below the maximum the perturbations develop less quickly or disappear (𝜀̇/𝜀 < 0) due to the capillary effect of the high curvature. At 𝜆 values above the maximum the perturbations develop less quickly due to diffusion limitations. The wavelength, 𝜆𝑖 , describes the perturbed morphology which is at the limit of stability (𝜀̇/𝜀 = 0) under conditions of constitutional undercooling. Assuming the velocity 𝑉 to be small or/and the solubility of the solid to be zero (𝑘 = 0), the last term in Eq. 3.23 can be neglected. (This is equivalent to setting 𝜉𝑐 = 1 in Eq. A7.12). The first of the three remaining terms on the right-hand-side of Eq. 3.23 is non-zero and the second term becomes zero upon substituting for b (see list of symbols) when: 𝜆=∞

**

The last term of Eq. 3.23 naturally appears in Mullins-Sekerka’s analysis. As shown in Appendix 7 (also Exercise 3.10), this equation is equivalent to Eq. A7.12 when the parameter 𝜉𝑐 = [V/D − b][𝑉/𝐷(1 − 𝑘) − 𝑏]−1 is introduced.

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Figure 3.7 Rate of Development of a Perturbation at a Constitutionally Undercooled Interface Here the ratio 𝜀̇/𝜀 describes the relative rate of development of the amplitude of a small sinusoidal perturbation in the case of a specific alloy (Al-2wt%Cu) under given growth conditions (𝑉 = 0.1 mm/s, 𝐺 = 10 K/mm). At very short wavelengths, the value of this parameter is negative due to curvature damping and the perturbation will tend to disappear (morphological stability - Fig. 3.1(a)). At wavelengths greater than 𝜆𝑖 and above, the sinusoidal shape will become more accentuated (instability – Fig. 3.l(b)). The wavelength having the highest rate of development is likely to become dominant. The reason for the tendency to stability at high -values is the difficulty of diffusional mass transfer over large distances. When the interface is completely stable, the curve will remain below the 𝜀̇/𝜀 = 0 line for all wavelengths. This implies the disappearance of perturbations having any of these wavelengths (appendix 7, Fig. A7.2).

The third term of Eq. 3.23 vanishes when: 𝜔2 Γ = 𝑚𝐺𝑐 − 𝐺 = 𝜙 or: Γ 1/2 𝜆𝑖 = 2π ( ) 𝜙 This is exactly the same as Eq. 3.22 which was derived above using a much simpler method. As the temperature gradients in the liquid and solid are not necessarily the same, 𝐺 should be expressed as a conductivity-weighted mean temperature gradient: 𝐺̅ = (𝐺𝑠 𝜅𝑠 + 𝐺𝑙 𝜅𝑙 )/(𝜅𝑠 + 𝜅𝑙 )

[3.24]

The term, Γ/𝜙, is the ratio of the capillarity force to the driving force for instability. When 𝜙 tends to zero (the limit of constitutional undercooling) the minimum unstable wavelength approaches infinity. This is to be expected because, in the limit, only the planar interface should be observed. Far from the limit of constitutional undercooling in the unstable regime however: 𝐺 ≪ 𝑚𝐺𝐶 =

Δ𝑇0 𝑉 𝐷

The wavelength thus becomes, for 𝑉 ≫ 𝐺𝐷/Δ𝑇0 V:

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[3.25]

Foundations of Materials Science and Engineering Vol. 103

𝐷Γ 1/2 ) 𝜆𝑖 = 2π ( 𝑉Δ𝑇0

63

[3.26]

The latter expression reveals that the wavelength of the unstable morphology is proportional to the geometric mean of a diffusion length (𝐷/𝑉) and a capillarity length (Γ/Δ𝑇0 ). Increasing D or Γ, and decreasing 𝑉 or Δ𝑇0 will increase the minimum unstable wavelength. Another important prediction of the stability analysis is the possible re-stabilisation of a plane front at very high growth rates. This so-called absolute stability will be treated in Chap. 7 and Appendix 7. Exercises 3.1

A solid/liquid interface becomes unstable when relation 3.1 (3.9) is obeyed in the case of a pure metal (alloy). Show that relation 3.1 is implied by Eq. 3.9. Discuss the differences.

3.2

Indicate why the constitutional undercooling criterion cannot yield the wavelength of the perturbed interface resulting from instability.

3.3

Discuss the advantage of the Bridgman method (Fig. 1.4(a)), over directional casting processes (Fig. 1.4(b)), with respect to the control of interface stability.

3.4

Determine the phase diagram from the information that the solute distribution ahead of the planar solid/liquid interface of an Al-Cu alloy under steady-state conditions has been found to be described by: 𝐶[𝑤𝑡%] = 2 (1 + [

0.86 −𝑉𝑧 ] exp [ ]) 0.14 𝐷

It has also been determined that the interface temperature is 624°C, and it is known that the melting point of Al is 660°C. Give the values of 𝑘, 𝑚, Δ𝑇0, 𝑇𝑙 and Δ𝐶0 . 3.5

What is the limit of stability, 𝐺/𝑉, of the above alloy if it is given that 𝐷 = 310−5 cm2/s? Use the constitutional undercooling criterion.

3.6

Calculate the heat-flux required to produce a value of 𝐺 which is sufficient to stabilise the planar front in Exercise 3.5 at a growth rate, 𝑉, of 10 cm/h.

3.7

By analogy with Fig. 3.4, draw a diagram for 𝑘 > 1 and discuss the constitutional undercooling criterion for this situation. Point out differences between this case and the case where 𝑘 < 1.

3.8

Calculate the minimum value of 𝐺 which is required to stabilise the planar interface of an Fe0.09wt%S alloy during directional growth at a rate of 0.01 mm/s. Calculate also the limit of absolute stability. (Assume that 𝐷 = 810−9 m2/s, use 𝑘 and 𝑚 from Exercise 1.6, and obtain other data from Appendix 15).

3.9

Using Eq. A7.10, show that the 0-th order term retrieves the boundary condition of a planar interface (Eq. A2.22), while the 1-st order terms in 𝑧 ∗ result in Eq. A7.11.

3.10 Show that Eq. A7.12 is equivalent to Eq. A7.11 when the parameter 𝜉𝑐 is introduced. 3.11 Calculate the ratio, 𝜆𝑚 /𝜆𝑖 , for conditions of high constitutional undercooling and constant 𝑉 and 𝐺 values. Here, 𝜆𝑚 is the wavelength which corresponds to the maximum of 𝜀̇/𝜀(𝜆) in Fig. 3.7. Note that 𝑏 in Eq. 3.23 is a function of 𝜔 (Appendix 7, Eq. A7.4). Under certain conditions, EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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this can be simplified by comparing the relative magnitudes of 𝑉/2𝐷 and 𝜔. Is that true of the case depicted in Fig. 3.7? Does the value of 𝜆𝑚 /𝜆𝑖 depend upon the composition of the alloy? 3.12 Imagine that an experiment is carried out, on the alloy of question 3.4, in which the specimen is solidified using the Bridgman method with a temperature gradient, 𝐺, of 10 K/mm and is maintained at the limit of constitutional undercooling until steady-state conditions are established. The growth rate is then doubled without changing the value of 𝐺. Calculate the minimum, and the most probable, wavelength which the resultant perturbation is likely to have. 3.13 For the case of pure Al at an undercooling of 1 K, determine the order of magnitude of the radius of a growing sphere (𝑅 ≈ 𝜆𝑖 ) at which it will become unstable when growing in its undercooled melt. For the purpose of calculating 𝐺, assume that the quasi steady-state solution for the temperature field around a sphere (analogous to the concentration field solution derived in Appendix 2) is applicable.

References and Further Reading Stability Theory ▪ S.R.Coriell, G.B.McFadden, P.W.Voorhees, R.F.Sekerka, Stability of a planar interface during solidification of a multicomponent system, Journal of Crystal Growth, 82 (1987) 295. ▪ S. R.Coriell, G.B.McFadden, in Handbook of Crystal Growth, Vol. 1a, D.T.J.Hurle (Ed.), Elsevier, Amsterdam, 1993, p. 785. ▪ S.H.Davis, Theory of Solidification, Cambridge University Press, Cambridge, 2001. ▪ D.A. Huntley, S.H. Davis, Effect of latent heat on oscillatory and cellular mode coupling in rapid directional solidification, Physical Review B, 53 (1996) 3132. ▪ G.P. Ivantsov, Diffusional supercooling during crystallization of a binary alloy (in Russian), Doklady Akademii Nauk SSSR, 81 (1951) 179. ▪ W.W.Mullins, R.F.Sekerka, Morphological stability of a particle growing by diffusion or heat flow, Journal of Applied Physics, 34 (1963) 323. ▪ W.W.Mullins, R.F.Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics, 35 (1964) 444. ▪ R.F.Sekerka, A stability function for explicit evaluation of the Mullins-Sekerka interface stability criterion, Journal of Applied Physics, 36 (1965) 264. ▪ D.E.Temkin, V.B.Polyakov, Stability of plane front on phase-transition in a one-component system, Kristallografiya, 26 (1976) 661. ▪ W.A.Tiller, K.A.Jackson, J.W.Rutter, B.Chalmers, The redistribution of solute atoms during the solidification of metals, Acta Metallurgica, 1 (1953) 428. ▪ R. Trivedi, W.Kurz, Morphological stability of a planar interface under rapid solidification conditions, Acta Metallurgica, 34 (1986) 1663. Interface Stability Experiments ▪ Y.Chen, A.A. Bogno, N.M. Xiao, B Billia, et al., Quantitatively comparing phase-field modeling with direct real time observation by synchrotron X-ray radiography of the initial transient during directional solidification of an Al–Cu alloy, Acta Materialia, 60 (2012) 199. ▪ S.R.Coriell, R.F.Sekerka, Oscillatory morphological instabilities due to non-equilibrium segregation, Journal of Crystal Growth, 61 (1983) 499. ▪ D.J.Fisher, unpublished result, EPFL, 1978.

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▪ S.C.Hardy, S.R.Coriell, Morphological stability of ice cylinders in aqueous solution, Journal of Crystal Growth, 7 (1970) 147. ▪ W.Losert, B.Q.Shi, H.Z.Cummins, Evolution of dendritic patterns during alloy solidification: Onset of the initial instability, Proceedings of the National Academy of Sciences of the USA, 95 (1998) 431. ▪ R.J.Schaefer, M.E.Glicksman, Initiation of dendrites by crystal imperfections, Metallurgical Transactions, 1 (1970) 1973. ▪ K.Shibata, T.Sato, G.Ohira, Morphological stabilities of planar solid-liquid interfaces during unidirectional solidification of dilute Al-Ti and Al-Cr alloys, Journal of Crystal Growth, 44 (1978) 419. ▪ R.Trivedi, K.Somboonsuk, Constrained dendritic growth and spacing, Materials Science and Engineering, 65 (1984) 65. ▪ R.Trivedi, H.Miyahara, P.Mazumder, E.Simsek, S.N.Tewari, Directional solidification microstructures in diffusive and convective regimes, Journal of Crystal Growth, 222 (2001) 365.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 67-108 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER FOUR

SOLIDIFICATION MICROSTRUCTURE: CELLS AND DENDRITES Nearly all of the solidification microstructures which can be exhibited by a pure metal or an alloy can be divided into two groups: single-phase primary crystals and polyphase structures. The most important growth form, and the one to be discussed in this chapter, is the tree-like primary crystal, i.e. the dendrite. This feature has been the subject of research since the 18th century, the history of which is recounted in two articles referenced at the end of this chapter. Another microstructural element is the polyphase structure, comprising features such as the eutectics and peritectics to be described in the next chapter. The growth of these morphologies can be described by analogous theoretical models which follow a well-developed and simple two-step plan. More advanced treatments of the problem have shown that appropriate solutions to free boundary-value problems can be obtained in more rigorous physical/mathematical ways, such as using the concept of Microscopic Solvability (Langer, 1986; Pelcé, 1988). In order to avoid the complex modelling of interface morphologies it is preferred here to use the simpler two-step approach which, in spite of its simplifying assumptions, leads to physically reasonable solutions. These steps are: 1.

derivation of an equation which describes the general relationship between the scale of the microstructure, the undercooling of the growth-front and the growth rate (transport solution);

2.

choice of a criterion which permits the establishment of a unique relationship between (a) the scale of the microstructure and the undercooling (in the case of equiaxed growth) or (b) the scale of the microstructure and the growth rate (in the case of directional growth).

With regard to step 1 it is necessary to assume an interface morphology (e.g. a paraboloidal tipshape for dendrites) in order to determine an expression for the heat and/or solute fields under the constraint of the effects of capillarity. For step 2 one uses one of the following growth criteria: *

Top image: View on the tip of a Ni-base superalloy dendrite which has been uncovered by shrinkage melt flow (image width 3.5 μm, photo Mokadem, 2004). .

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(i) growth at the extremum - i.e. at the maximum growth-rate, as first proposed by Zener (1946) or, equivalently, growth at the minimum undercooling. This was applied to dendrites for three decades but did not lead to a quantitative description of dendrite growth. On the other hand, it is still useful for treating eutectics as will be seen in Chap 5. (ii) another argument was proposed by Langer and Müller-Krumbhaar (1977); that the tip grows at the limit of morphological (marginal) stability. In the case of dendrite growth the use of this criterion led to satisfactory agreement between theory and experiment. This criterion will therefore be used for treating the models presented in this chapter. 4.1. Constrained and Unconstrained Growth

Figure 4.1 Columnar (left) and Equiaxed (right) Dendrite Morphologies and Corresponding Cooling Curves In alloys the columnar dendrites (a) grow in a temperature gradient and are driven by the unidirectional displacement at velocity, 𝑉, of the isotherms (b). Equiaxed dendrites (d), on the other hand, grow radially from nuclei in an undercooled melt (e). If they grow little-influenced by their neighbours their morphology is equiaxed. Due to interaction with their neighbours the equiaxed morphology may be deformed (d) but is still termed equiaxed. Their growth is controlled by the local undercooling, Δ𝑇 (e). If a thermocouple is placed at a fixed position in the melt and is overgrown by the dendrites, different cooling curves will be recorded for directional growth (c) and equiaxed growth (f). This difference is essentially due to nucleation in the case of equiaxed solidification. Due to microsegregation (Chap. 6) some eutectic will also usually form during the last stages of solidification (i.e. at 𝑇𝑒 , (f)). Note that in the case of directional growth the crystals are in contact with the mould and heat will be conducted through them in a direction which is parallel, and opposite, to that of their growth (b). The melt is therefore the hottest part of the system. In the case of equiaxed growth the heat which is produced by solidification must be transported through the melt (e). In this case the crystals are the hottest part and the heat flux, 𝑞, is radial and in the same direction as that of growth. The local solidification time, 𝑡𝑓 , is the elapsed time between the beginning and the end of solidification at a fixed point in the system, i.e. from the tip to the root of the dendrite (c,f).

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The situation in which the heat-flow direction is opposite to the growth direction (as in directional or columnar solidification - Fig. 4.1(a) is often referred to as constrained growth. That is, the rate of advance of the isotherms constrains the dendrites (which in this situation are found only for alloy solidification) to grow at a given velocity. This forces them to adopt the corresponding tipundercooling. The boundaries of the grains (the latter containing many dendrites) are parallel to the primary dendrite axes (trunks) and are continuous along the length of the solid. Many columnar dendrites form a single grain by repeated branching. A characteristic trunk spacing, 1, and a secondary arm spacing, 2 , can be defined in this case. When the heat flows from the crystal into the melt (equiaxed solidification or free growth - Fig. 4.1(d) the dendrites can grow, in pure materials and alloys, as rapidly as the local undercooling will permit. The dendrites grow radially in their 6 principal cubic-crystal directions until they impinge upon the solute fields of dendrites originating from neighbouring nuclei. The grain boundaries form a continuous network throughout the solid. Equiaxed dendrites do not have a primary (trunk) spacing but do have a secondary-arm spacing, 2 , which depends upon the local solidification time. A transition from equiaxed to columnar or columnar to equiaxed microstructure is often observed and is the result of competitive growth between grains of differing crystal orientations (Fig. 4.2). This transition is described in more details in Chap. 8. 4.2. Morphology and Crystallography of Dendrites The formation of a dendrite begins with the breakdown of an unstable solid/liquid interface (planar in directional growth and spherical in equiaxed growth). Perturbations are amplified until a marked difference in growth of the tips and depressions of the perturbed interface has occurred (Fig on first page of Chap. 3). Because the tip rejects solute in the lateral direction, it will tend to grow more rapidly than does a depression, which tends to accumulate the excess solute which is rejected by the tips. The form of the perturbation is therefore no longer sinusoidal but takes on the form of cells (Fig. 4.3).

Figure 4.2 Schematic of Columnar and Equiaxed Dendritic Microstructures in a Casting Initial competition of crystals nucleated at random at the mould wall, and their growth into the liquid, leads to grain selection (known as geometrical selection (Lemmlein, 1945). In cubic metals, columnar grains having an [001] axis close to the heat-flux direction advance further into the liquid and ultimately outgrow the others. Note that, in this section, differences in the rotation of the columnar grains about their growth axis are not visible. Large orientational differences are accommodated at high-angle grain boundaries. Branching of the dendrites leads to the creation of new trunks which are crystallographically related to the initial primary trunk and form a grain. The transition from columnar to equiaxed growth occurs when the melt has lost its superheat, thus becoming slightly undercooled, and detached dendrite arms growing in the melt form a barrier ahead of the columnar zone. (Compare with Fig. 1.6.)

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If the growth conditions are such as to lead to dendrite formation, the cells will rapidly change to dendrites, which exhibit secondary arms and crystallographically-governed growth directions. Cell and primary dendrite spacings are much larger than the wavelength of the original perturbation which initiates the corresponding growth morphology (Fig. 4.3 and the figure on the first page of Chap. 3). It is important to note that, under normal solidification conditions, cells can appear only during the directional growth of alloys, i.e. for 𝐺 > 0. Cells can grow for instance under conditions such as those indicated in Fig. 4.1(b). In other cases (Fig. 4.1(e)) only dendrites will be observed. Figure 4.4 summarises the differences between cells and dendrites. Cells are usually a branchless crystal morphology which grows anti-parallel to the heat-flux direction. They grow under conditions which are close to the limit of constitutional undercooling at low growth rates. Cells can also be observed at high growth rates, close to conditions of absolute stability (see Chap. 7). There are always many cells within one grain. On the other hand dendrites are crystalline forms which grow far from the limit of stability of the plane front and adopt an orientation which is as close as possible to the heat-flux direction, or opposite to it, but follow one of the preferred growth axes. These directions are crystallographically determined (Table 4.1). Columnar dendritic single crystals are used for high-performance aircraft-engine turbine blades. In this case all the dendrites making up the mono-grain are aligned (Fig. 4.5), leading to improved high-temperature properties.

Figure 4.3 Cells Formed Following Breakdown of a Planar Solid/Liquid Interface The development of perturbations at the constitutionally undercooled solid/liquid interface (lower part of figure) is only a transient phenomenon. The tips of the perturbations can readily reject solute while the depressed parts of the interface accumulate solute and advance much more slowly. The initial wavelength is too small for further rapid growth to occur, and the final result is the formation of a cellular structure. Note that the wavelength has approximately doubled between the initial perturbation and the final cells. The spacing between the cells is also not constant. The initial cellular morphology can adjust itself to give a more optimum growth form via the cessation of growth of some cells (B) in order to decrease their number, or by the division of cells in order to increase the number present. The division of cells is not shown here, but it resembles the change in the spacing of the initial perturbations at point (A), with two branches continuing to grow. The larger centre cells (C) have moreover slightly perturbed surfaces and this suggests that, in the intercellular liquid, some driving force remains for further morphological change which might possibly lead to dendrite formation.

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Figure 4.4 Cells and Dendrites During directional growth in a positive temperature gradient, cells can exist as a stable growth form (a). Dendrites will be particularly common in conventional castings and are characterised by growth of the trunks and branches along preferred crystallographic directions, such as [001] in cubic crystals (b). Due to the anisotropy of properties such as the solid/liquid interface energy and the growth kinetics, dendrites will grow in that preferred crystallographic direction which is closest to the heat-flow direction, 𝑞, whereas well-developed cells grow with their axes parallel to the heat-flow direction; without regard to the crystal orientation. Only at the growing tip (a) may remain some trace of crystal anisotropy. Between these two extremes there is a range of intermediate forms (dendritic cell, cellular dendrite). Table 4.1 Preferred Growth Directions of Dendrites of Various Materials Structure face-centred cubic body-centred cubic body-centred tetragonal hexagonal close-packed

Dendrite Orientation

Example

⟨100⟩ ⟨100⟩ ⟨110⟩ ⟨1010⟩ ⟨0001⟩

Al -Fe Sn H2O (snow) Co17Sm2(Cu)

If a single dendrite could be extracted from the columnar zone of a casting or a weld during growth, it would resemble the one depicted schematically in Fig. 4.6. Behind a short paraboloid tip region, which often constitutes less than 1% of the length of the whole dendrite, perturbations appear on the initially smooth needle, as in the case of the breakdown of a planar interface. These perturbations grow and form branches oriented in the four 001 directions which are perpendicular to the trunk. If the primary spacing is sufficiently large, these cell-like secondary branches will develop into dendritic-type branches and lead to the formation of tertiary and higher-order arms (see the cover of the book). When the tips of the branches encounter the diffusion field of the branches of the neighbouring dendrites, they will stop growing and begin to ripen and thicken. The final secondary spacing will thus be very different to the initial one. Compare the right and left parts of Fig. 4.6. The final value of 𝜆2 is largely determined by the contact time between the branches and the liquid. This period is known as the local solidification time, tf, and is given by the time required e.g. for a thermocouple placed at a fixed point to pass from the tip to the root of the growing dendrite (Fig. 4.1).

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Figure 4.5 Transverse Section of a Monocrystalline Dendritic Gas-Turbine Blade In an alloy the last (interdendritic) liquid to solidify always has a different composition as compared to that of the first crystals (dendrite trunks) to form. The dendritic structure can therefore be revealed in a casting, due to variations in etching behaviour. This is seen in the above transverse section of a directionally solidified single crystal turbine blade. The orientation of the dendrite trunks is parallel to the blade-axis (and perpendicular to the plane of this section). The absence of high-angle grain boundaries improves the high-temperature strength and fatigue behaviour of the superalloy castings used in aircraft gas turbines. The formation of such a monocrystal can be understood with the aid of Fig. 4.2. During the initial stages of growth, when a columnar zone has progressed to some distance from a chill plate, the grains enter a restriction placed in the path of the growing crystals. As a result only one grain can continue to grow and form the blade. This is the principle used in single-crystal alloy casting.

4.3. Diffusion Field at the Tip of a Needle-Like Crystal The tip of a dendrite – invisible in the solidified end-product – very much controls the final microstructure. The tip-undercooling and growth-rate, as well as the dendrite morphology or primary spacing, are all largely dependent upon the behaviour of the tip region (Fig. 4.1(a)). During the growth of the tip, either heat (in the case of pure metals) or heat and solute (in alloys) are rejected (Fig. 4.7). These diffusion processes are driven by gradients in the liquid and the latter are in turn due to differences in temperature (Δ𝑇𝑡 ) and concentration (Δ𝐶) ahead of the growing crystal. The concentration difference, Δ𝐶, can be converted into a liquidus-temperature difference, (Δ𝑇𝑐 ), via the phase diagram, as shown in Fig. A8.1. After adding the temperature difference, at the solid/liquid interface, which is caused by the curvature of the tip (Δ𝑇𝑟 ), the coupling condition for the total undercooling can be written: Δ𝑇 = Δ𝑇𝑐 + Δ𝑇𝑡 + Δ𝑇𝑟

[4.1]

In this equation the possibility of an additional kinetic undercooling due to atom-attachment has been neglected. As explained in the previous chapter, this is a reasonable assumption in the case of materials, such as metals, which exhibit a low entropy of fusion under normal solidification conditions. Equiaxed dendritic growth takes place in the undercooled melt; in pure substances it is controlled by the diffusion of heat alone (Fig. 4.7(b), Appendix 9). In equiaxed dendritic growth of alloys, coupled heat- and mass-transport controls the growth behaviour (Fig. 4.7(d), Appendix 9). In the case of the directional growth of alloy dendrites, the situation is simpler than equiaxed alloy growth because, due to the imposed temperature gradient, the latent heat is transported through the solid and, to a first approximation, does not affect tip growth while solute is rejected ahead of the tips (Fig. 4.7(c)). In this case only mass diffusion needs to be considered. This permits a simple

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solution to the problem of alloy dendrite growth in the columnar zone. This case is of great practical importance and will be considered here in greater detail. The rejection of solute influences the temperature of the solid/liquid interface at the tip (Fig. 4.8). The ratio of the change in liquid concentration at the tip, Δ𝐶, to the equilibrium concentration difference, Δ𝐶 ∗ (= 𝐶𝑙∗ [1 − 𝑘]), the length of the tie-line at the temperature of the tip, is known as the dimensionless supersaturation, Ω. This supersaturation (or the related undercooling, Δ𝑇𝑐 ) represents the driving force for the diffusion of solute at the dendrite tip in an alloy. With increasing supersaturation the growth rate of the solid increases. The solute rejection-rate and therefore the growth-rate is influenced by the shape of the tip and, at the same time, the shape of the tip is affected by the distribution of the rejected heat or solute. This interaction makes the development of an exact theory extremely complex. In order to keep the present treatment simple, and yet remain physically meaningful, it is assumed that the dendritic shape can be satisfactorily described as being a paraboloid of revolution, as first suggested by Papapetrou (1935). The mathematical solution of the diffusion problem for a paraboloid was derived by Ivantsov (1947), who deduced the following relationship between the supersaturation, Ω, the dendrite tip radius, 𝑅, and the growth rate, 𝑉: Ω = I(𝑃𝑐 )

[4.2]

where: I(𝑃𝑐 ) = 𝑃𝑐 exp (𝑃𝑐 ) E1 (𝑃𝑐 )

[4.3]

Figure 4.6 Growing Dendrite Tip and Dendrite Root in a Columnar Structure Depending upon the directional growth conditions the dendrite (from the Greek, dendron = tree) will develop arms of various orders. A dendritic form is usually characterised in terms of the primary (dendrite trunk) spacing, 𝜆1 , and the secondary (dendrite arm) spacing, 𝜆2 . Tertiary arms are also often observed close to the tip of the dendrite. It is important to note that the value of 𝜆1 which is measured in the solidified microstructure is the same as that existing during growth whereas the secondary spacing is enormously increased due to the long contact-time between the highly-curved branched structure and the melt. The ripening process not only modifies the initial wavelength of the secondary perturbations, 𝜆′2 , to give the spacing which is finally observed, 𝜆2 , but also often causes dissolution of the tertiary or higher-order arms. The two parts of the figure are drawn at the same scale, refer to the same dendrite and illustrate morphologies which exist at the same time but which are widely separated along the trunk length (by about 100 𝜆1 ).

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Figure 4.7 Concentration and Temperature Fields of Dendrites These diagrams illustrate the heat and mass diffusion fields which exist along the dendrite axis, and correspond to various possible cases (Fig. 1.7). In pure substances (a,b) there is no soluterejection and dendrites can form only in an undercooled melt. In the latter case (b) the heatrejection which occurs during growth sets up a negative temperature gradient ahead of the interface. This leads to the establishment of the necessary conditions for the instability of thermal dendrites. In the case of alloys (c,d) dendrites can form regardless of the temperature gradient, if the interface is constitutionally unstable. If 𝐺 is greater than zero (c) the latent heat is transported, together with the unidirectional heat flux, into the solid. To a first approximation therefore solute rejection alone needs to be considered in the case of directionally solidified (solutal) dendrites. Equiaxed dendrites in alloys (d) reject both solute and heat.

and the Péclet number for solute diffusion, 𝑃𝑐 = 𝑉𝑅/2𝐷 = 𝑅/𝛿𝑐 .† Here E1 (𝑃) is the exponential integral function (Appendix 8). The form of the lvantsov function, I(𝑃), is shown by the continuous red curve in Fig. A8.4. Equation 4.3 can be approximated by a continued fraction of the type (Abramowitz, 1965): 𝑃

I(𝑃) =

1

𝑃+

1

1+ 𝑃+

2 2 1 + 𝑃+. . .

If the first term only is taken (broken line I0 in Fig. A8.4) one obtains: I0 (𝑃) ≅ 𝑃

[4.4]

and insertion of this approximation into Eq. 4.2 gives:

†

When the tip radius is related to the thermal diffusion length, 𝛿𝑡 , the corresponding dimensionless group is the thermal Péclet number, 𝑃𝑡 = 𝑉𝑅/2𝑎 (see Appendix 8).

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Ω ≅ 𝑃𝑐

75

[4.5]

This is the simple solution obtained for the diffusion field existing around a hemispherical cap (Eq. A8.11) and will be used in the following discussion because it permits a clearer insight into the physics of dendrite growth at low Péclet numbers than does the more complicated relationship of Eq. 4.3. Note however that diffusion in spherical coordinates does not lead to a steady state. It is a gross simplification of the problem which allows immediate qualitative insight into the functioning of the equations. It should nevertheless be kept in mind that Eq. 4.3 or more complicated ones (e.g. by Trivedi, 1970) can easily be used in numerical calculations and this will then lead to more exact solutions. Equation 4.5 simply states that the response of the system characterised by the Péclet number is proportional to the driving force defined by the supersaturation. Note that 𝑃𝑐 is the ratio of the tip radius, 𝑅, to the diffusion length, 2𝐷/𝑉 (Fig. A2.4).

Figure 4.8 Solute Rejection and Constitutional Undercooling at the Tip of an Isolated Columnar Dendrite During directional solidification, where the isotherms move due to the imposed heat flux, a needle-like crystal can grow more quickly than can a flat interface due to the more efficient solute redistribution: B-atoms rejected at the interface of a thin needle can diffuse outwards into a large volume of liquid. The solutal diffusion boundary layer, 𝑐 , of the needle is thus smaller than that of a planar interface. Because the interface is not planar the solid which is formed also does not have the same composition as the original liquid (as it does in the case of steady-state plane-front growth - Fig. 3.4). When a positive gradient is imposed, as in directional solidification, heat is extracted through the solid. If moreover the thermal diffusion is rapid (as in metals) the form of the isotherms will be affected only slightly by the interface morphology. In the case of directionally solidifying dendrites solute diffusion alone will thus be the limiting factor. The growth temperature, 𝑇 ∗, of the tip will define a solute undercooling, Δ𝑇𝑐 , or, via the phase diagram, the degree of supersaturation, Ω = Δ𝐶/Δ𝐶 ∗ . The determination of Ω as a function of the other parameters requires the solution of the differential equation which describes the solute distribution. The simplest solution is obtained when the tip morphology is chosen to be hemispherical. The real form of the dendrite tip is instead closely represented by a paraboloid-ofrevolution.

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The solution to the diffusion problem, described by Eq. 4.5, is shown in Fig. 4.9 for an alloy dendrite (under directional solidification conditions) as a straight line at an angle of 45° in logarithmic coordinates. This indicates that, for a given Ω-value, 𝑅 and 𝑉 are not defined unambiguously. Solution of the diffusion problem thus does not indicate whether the dendrite will grow quickly with a small 𝑅, or slowly with a large 𝑅, but merely relates the sharpness of the tip to its rate of propagation in the diffusional regime. The diffusion boundary-layer around the tip is proportional to the tip radius (Eq. A2.26). Because the gradient of 𝐶 is inversely proportional to the boundary-layer thickness, a sharper tip has a steeper gradient. A sharper tip can grow more rapidly because it can reject solute (or heat in the case of a thermal dendrite) more efficiently; the flux being proportional to the gradient. There is a limit however to the possible sharpness of the dendrite tip, which is represented by the critical radiusof-nucleation, 𝑅 ○ = 𝑟 ○ (Table 2.1). At 𝑟 ○ the growth rate is zero and therefore all of the supersaturation is used to create curvature and none remains to drive diffusive processes (Fig. 4.9).

Figure 4.9 Growth Rate of a Hemispherical Needle Crystal as a Function of Tip Radius for Constant Supersaturation (a) and Radius of Marginal Stability (b) For a hemispherical needle crystal the solution of the diffusion equation shows that the supersaturation, Ω, is equal to the ratio of the tip radius to the characteristic diffusion length. This dimensionless ratio is known as the Péclet number, 𝑃𝑐 (= 𝑅𝑉/2𝐷). For a given supersaturation the product, 𝑅𝑉, is therefore constant and means either that a dendrite with a small radius will grow rapidly or one with a large radius will grow slowly (diagonal line in (b)). At small 𝑅-values the diffusion limit is cut by the capillarity limit. The minimum radius, 𝑅 ○, is given by the critical radius of nucleation, 𝑟 ○ (Table 2.1). A maximum value of 𝑉 therefore exists. Because it was reasoned that the fastest-growing dendrites would dominate steady-state growth, it was originally assumed that the radius chosen by the system would be the one which gave the highest growth rate (extremum value, 𝑅 = 𝑅𝑒 ). Experiment indicates however that the radius of curvature of the dendrite is approximately equal to the lowest-wavelength perturbation of the tip (b), which is close to 𝜆𝑖 , (Fig. 3.7). This is referred to as growth at the limit of stability (𝑅 = 𝑅𝑠 ).

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4.4. Operating Point of the Needle Crystal - Tip Radius The overall growth curve of a needle-like crystal, which reflects the sum of the capillarity and diffusion effects, follows the solid line in Fig. 4.9(a) and exhibits a maximum value close to 𝑅 ○ . In the past, this maximum, 𝑅𝑒 , was considered to be the radius at which the dendrite would actually grow. This so-called extremum criterion permitted the establishment of an unique solution, to the otherwise indeterminate growth problem, by setting the first derivative of the equation of growth equal to zero. Langer and Müller-Krumbhaar proposed in 1977/78 that a dendrite grows with a tip having a size which is at the limit of stability (marginal stability). One can then determine the expected tip radius by setting: [4.6]

𝑅𝑠 = 𝜆𝑖

where 𝜆𝑖 is the shortest-wavelength perturbation which would cause the dendrite tip to undergo morphological instability. To a first approximation the wavelength of the marginally stable perturbation at a planar interface can be used. It is described by Eq. 3.22 and, for the condition 𝐺 ≪ 𝑚𝐺𝑐 (which is true for dendrites), it is given by Eq. 3.26: 𝜆𝑖 = (𝛿𝑐 𝑠)1/2 𝜎 ∗−1/2

[4.7]

This wavelength is proportional to the geometric mean of the diffusion length, 𝛿𝑐 (∼ 𝐷/𝑉) and the capillary length, 𝑠 (∼ Γ/Δ𝑇0 ). The proportionality constant of Eq. 4.7 is defined by the stability parameter, 𝜎 ∗ , which is equal to 1/4π2 for plane-front instability (Fig. 4.9; see also Huang and Glicksman, 1981). Subsequent theoretical modelling has shown that the physical interpretation of marginal stability was not correct and that this criterion had to be replaced by a microscopic solvability condition. The important difference between the two models is that the stability parameter, 𝜎 ∗ , which has a constant value (1/4𝜋 2 ) in marginal stability theory is a function of both surface-energy anisotropy and tip Péclet-number in microscopic solvability theory (Barbieri and Langer, 1989). The dependence upon Péclet number is relatively weak, so that 𝜎 ∗ depends predominantly upon anisotropy in the small Péclet number limit which is characteristic of low to moderate solidification rates; according to the relationship: 7/4

𝜎 ∗ = 𝜎0 𝜀4

[4.8]

where 𝜎0 is a constant which is of the order of unity and 𝜀4 is a measure of the four-fold anisotropy of the interface energy for cubic crystals (see Sect. 2.7. Eq. 2.21). For normal solidification conditions the fundamental relationship, originally developed between the tip radius, the diffusion length and capillarity length (Eqs 4.6 and 4.7), therefore remains the same but the stability parameter now varies as a function of the interface-energy anisotropy. In view of the above facts the theoretically more accessible marginal stability argument will, as noted before, be used instead of microscopic solvability while assuming that 𝜎 ∗ is an adjustable parameter that depends upon the specific alloy (see for example Oguchi and Suzuki (2007), Pan and Zhu (2010)). In Fig. 4.9, the value, 𝑅𝑠 , at which 𝑅 is at the limit of marginal stability is also indicated. It can be seen that this operating point is situated some distance away from the extremum and leads to larger 𝑅-values. This result is consistent with experimental measurements and also with analytical and numerical modelling. Since 𝑅𝑠 is considerably greater than 𝑅𝑒 the effect of curvature upon the growth curve can be neglected. (As shown in Fig. 4.9 the point, 𝑅𝑠 , is situated on the fully diffusion-limited curve; therefore Δ𝑇𝑟 in Eq. 4.1 is almost zero.) This does not mean however that capillarity has no influence upon the dendrite tip. The curvature now exerts its influence through the 𝜆𝑖 -value, via the capillary EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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length. Figure 4.10 shows how one obtains a unique solution by using the extremum criterion (thin black line), or with the aid of stability arguments (red line). The final result for constrained growth (𝑉 = f ′ [𝑅𝑠 ]) is given in Fig. 4.11 for growth rates, 𝑉 > 𝑉𝑐 , i.e. above the limit of constitutional undercooling of the planar front.

Figure 4.10 Growth Rate as a Function of Tip Radius and Supersaturation This figure demonstrates how, by using various optimisation criteria, the observed 𝑉 − 𝑅 -values are obtained as a function of Ω. That is, a given value of 𝑅 is found for a given value of 𝑉. The non-optimised growth rate, 𝑉, of Fig. 4.9 is shown here as a function of the tip radius, 𝑅, for various supersaturations, Ω3 > Ω2 > Ω1 (broken curves). Two optimisation criteria are shown here. One requires that tip growth should occur at the extremum (dotted line), and the other that it should occur at the limit of stability (red solid line). The optimised curve which is deduced from the stability criterion (red solid line) corresponds to the less steep portion of the curve in Fig. 4.11. (Note that the coordinates are inverted there).

The tip radius of a dendrite which is growing under conditions of directional solidification‡ and low Péclet number will now be developed in more detail. This is the solutal case with an imposed temperature field, as in Fig. 4.8. The minimum wavelength of the instability of the tip is approximated by Eq. 3.22: Γ 1/2 𝜆𝑖 = 2π ( ) 𝜙 Using Eq. 4.6 and dropping the subscript for 𝑅 gives: Γ 1/2 𝑅 = 2π ( ) 𝜙

where

𝜙 = 𝑚𝐺𝑐 − 𝐺

[4.9]

In evaluating 𝜙 it is assumed that 𝐺𝑠 = 𝐺𝑙 = 𝐺. The value of the solute gradient at the tip, 𝐺𝑐 , can be deduced (in the steady state) from a flux balance, i.e.: 𝐺𝑐 = −𝑉𝐶𝑙∗ 𝑝/𝐷

‡

[4.10]

Note that directional solidification can be imposed by using a Bridgman-type furnace (𝐺 and 𝑉 independently controlled) or will exist during directional growth by chill casting, laser-deposition or welding (𝐺 and 𝑉 coupled) - see . Fig. 1.4 and Exercise 1.9.

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Figure 4.11 Optimised Dendrite Tip Radius as a Function of Growth Rate in Directional Growth If it is assumed that, in directional solidification, growth occurs with a tip radius which is equal to the minimum instability wavelength, 𝜆𝑖 , curves such as those above can be generated. They indicate the magnitude of the dendrite tip radius for a given growth rate and temperature gradient. Note the marked effect of the temperature gradient upon the radius of curvature at low growth rates (constrained growth regime or cellular (c) regime e.g. for 𝐺 = 0.1 K/mm between 𝑉 = 10−4 and 𝑉 = 10−5 mm/s). A sufficiently high gradient, or a sufficiently low growth rate (𝑉𝑐 = 𝐺𝐷/Δ𝑇0 ) will lead to the re-establishment of a planar (p) interface (i.e. a 'dendrite' with an infinite radius of curvature).

The unknown quantity here is the tip concentration in the liquid, 𝐶𝑙∗ , which can be obtained by combining the supersaturation, Ω (Eq. A8.1), and the transport solution (Eq. 4.2) leading to: 𝐶𝑙∗ =

𝐶0 𝐶0 = 1 − 𝑝Ω 1 − 𝑝 I(𝑃𝑐 )

[4.11]

which can also be written as: 𝐶𝑙∗ = A(𝑃𝑐 ) 𝐶0

[4.12]

where A(𝑃𝑐 ) = [1 − 𝑝 I(𝑃𝑐 )]−1 and 𝐺𝑐 = −𝑉𝑝𝐶0 A(𝑃𝑐 )/𝐷. Substituting this gradient into Eq. 4.9 leads to: Γ1/2

𝑅 = 2π

1/2

{−

𝑚𝑉𝑝𝐶0 𝐴(𝑃𝐶 ) − 𝐺} 𝐷

[4.13]

and using the definition of the Péclet number, (𝑅 = 2𝑃𝑐 𝐷/𝑉), to: 𝑉 2 𝐴′ + 𝑉𝐵 ′ + 𝐺 = 0

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[4.14]

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with: π2 Γ 𝐴 = 2 2 𝑃𝑐 𝐷 ′

𝐵′ =

and

𝑚𝐶0 (1 − 𝑘)A(𝑃𝑐 ) 𝐷

This quadratic equation can be easily solved and the value of 𝑅 obtained as a function of 𝑉 (Fig. 4.11)§. Noting that for the dendritic growth regime the imposed temperature gradient, 𝐺, has little effect, one can rewrite Eq. 4.14 as: −𝐵 ′ 𝑉= ′ 𝐴

Δ𝑇0 𝑘A(𝑃𝑐 )𝑃𝑐2 𝐷 𝑉= π2 Γ

or

[4.15]

with Δ𝑇0 𝑘 = −𝑚𝐶0 𝑝. As for small Péclet numbers A(𝑃𝑐 ) can be approximated by 1 + 𝑝 I(𝑃𝑐 ) which, in this case, is close to unity. One obtains as an approximate solution: Δ𝑇0 𝑘𝑃𝑐2 𝐷 𝑉= π2 Γ

4π2 𝐷Γ 𝑅 𝑉= Δ𝑇0 𝑘

[4.16]

2

or

using the definition of the solutal Péclet number 𝑃𝑐 = 𝑅𝑉/2𝐷 ≅ (𝐶 ∗ − 𝐶0 )/(𝐶0 (1 − 𝑘)) = Δ𝑇/(𝑘Δ𝑇0 ) allows to obtain the dendrite tip growth kinetics:

𝑉=

1 𝐷 Δ𝑇 2 π2 Γ𝑘Δ𝑇0

[4.17]

This result is the same as that which one obtains when using the hemispherical low Péclet number approximation (Kurz and Fisher, 1981) of Eq. 4.5. The final equations of the latter approach lead to simple relationships which are useful for obtaining order-of-magnitude estimates and are summarised in Table 4.2. Table 4.2 Dendrite Growth Equations According to the Hemispherical Approximation [(𝟏 − 𝒌)𝑷𝒄 < 𝟏] 𝐷𝛤 1/2 𝑅 ≅ 2π ( ) Δ𝑇0 𝑘𝑉

𝑉𝛤 1/2 𝑃𝑐 ≅ π ( ) 𝐷Δ𝑇0 𝑘

𝐶𝑙∗ 1 ≅ ≅ 1 + (1 − 𝑘)𝑃𝑐 𝐶0 1 − (1 − 𝑘)𝑃𝑐 𝑉≅

Δ𝑇 = Δ𝑇𝑐 = 𝑚𝐶0 (1 −

1 ) 1 − (1 − 𝑘)𝑃𝑐

1 𝐷 Δ𝑇 2 π2 Γ𝑘Δ𝑇0

Figure 4.12 shows the tip concentration and temperature for cells and dendrites. It is interesting to note that the composition (Fig. 4.12 (a)) is high in the cellular regime near to the plane front at low and high growth rates. In the steady state the latter grows with the composition, 𝐶0 /𝑘; leading to an homogeneous solid having the composition, 𝐶0 . The temperature is related to the concentration via the liquidus line and shows, for a negative slope (𝑘 < 1), an inverse behaviour (Fig. 4.12(b)).

§

When using Eq. 4.14 one has to make sure that one always chooses the smaller radius, corresponding to 𝜆𝑖 , of the two 𝜆 values which are predicted by stability theory.

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Figure 4.12 Liquid Concentration (a) and Temperature (b) of a Needle Crystal Tip in Directional Growth For columnar growth (𝐺 > 0) the figures display the growth ranges as a function of the velocity, 𝑉, of a plane front (p), cells (c) dendrites, and back to cells and a plane front at high velocity. The blunt cell-tips which exist close to the limit of constitutional undercooling, 𝑉𝑐 = 𝐺𝐷/Δ𝑇0, cannot easily dissipate the solute which is rejected there and the tip concentration will therefore be higher than that ahead of a sharp dendrite-tip. (If the temperature gradient is zero or negative this will not occur as no cells are formed in those cases.) At very high growth rates the interface concentration in the liquid will again increase to high values due to the increase in supersaturation necessary to drive the process. At very high growth rates, 𝑉 ≥ 𝑉a = Δ𝑇0 𝐷/𝑘Γ, a crystal with a supersaturation of unity will grow which has the same composition, 𝐶0 , as the alloy (the composition of the liquid at the interface is then equal to 𝐶0 /𝑘). Under these conditions a planar solid/liquid interface (p) will result; as in growth at low rates in a positive temperature gradient (grey regions). Note that the composition of the solid is related to 𝐶𝑙∗ via the distribution coefficient, 𝑘. Use of the phase diagram and the assumed existence of local equilibrium at the solid/liquid interface permit a calculation of the temperature which is associated with the tip concentration (Fig. 4.12(b)). At low and high growth rates the tip temperature reaches the solidus temperature. At growth rates of the order of 100 mm/s or above the distribution coefficient, k, approaches 1 which reduces absolute stability (see Chap. 7).

The tip radius would not be so important if it did not influence other morphological parameters of the dendrite. The theory presented here is useful in the case of isolated needles. Dendrites can be regarded as being isolated crystals, at least at their tips. This is not the case for cells since they always grow with their tips close together (𝑅 ≅ 𝜆) and their diffusion fields overlap. Their tips differ furthermore from the parabolic shape of dendrites (Fig. 4.4).

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A better approximation that uses a reasonable fit for the Ivantsov function for small Péclet numbers has been given by Dantzig and Rappaz (2016). These authors approximated the Ivantsov function for small growth rates (𝑃𝑐 < 0.1) by I(Pc) ≈ 1.5Pc0.8, and neglected Δ𝑇𝑟 and Δ𝑇𝑡 . Further for the value of 𝜎 ∗ mode 4 of a marginally stable sphere was applied. This leads to: 𝑉≅

1 𝐷 Δ𝑇 2.5 2 5.51 π Γ(𝑘Δ𝑇0 )1.5

[4.18]

or: Δ𝑇 ≅ (𝑉/𝐾𝑑 )1/𝑛 with 𝐾𝑑 = 𝐷/[5.51 π2 [𝐶0 𝑚(𝑘 − 1)]1.5 Γ], and 𝑛 = 2.5. 4.5. Primary Dendrite Spacing in Directional Growth The dendrite trunk spacing, also termed the primary spacing, is an important characteristic of columnar dendrites only and has an effect upon the properties. In order to model this spacing a simple model has been developed by assuming that the dendrite envelope, which represents the mean crossSect. of the trunk and the branches along the axis, can be described approximately by an ellipse (Fig. 4.13). The smaller radius-of-curvature of the ellipse is given by: 𝑏2 𝑅= 𝑎

[4.19]

The semi-axis, 𝑏, is proportional to 𝜆1 , where the proportionality constant depends upon the geometrical arrangement of the dendrites. An hexagonal arrangement is assumed in Fig. 4.13, where the last liquid is assumed to solidify at the centre-of-gravity of the equilateral triangle formed on a cross-section by three densely-packed dendrites. This assumption leads to 𝑏 = 0.58𝜆1 . The semi-axis length, 𝑎, is given by the thermal length; the difference between the tip-temperature, 𝑇 ∗ , and the roottemperature, 𝑇𝑠′ , divided by the mean temperature gradient in the mushy zone: 𝑎 = Δ𝑇 ′ /𝐺 = (𝑇 ∗ − 𝑇𝑠′ )/𝐺, where, due to microsegregation, 𝑇𝑠′ is lower than the alloy solidus temperature (Chap. 6) and is often equal to the eutectic temperature - if such a transformation exists in the alloy system. Neglecting the mass balance, it is assumed as a crude approximation that the tip radius and the length of the interdendritic liquid zone together determine the primary spacing, due to purely geometrical requirements. From Eq. 4.19, one has λ1 ∝ (𝑅𝑎)1/2, and: 1/2

3Δ𝑇 ′ 𝑅 𝜆1 = ( ) 𝐺

[4.20]

Knowing that Δ𝑇 ′ is dependent upon the tip temperature, 𝑇 ∗ , a sharp drop in 𝑇 ∗ at low and high 𝑉 should also cause 𝜆1 to decrease sharply, as shown in Fig. 4.14. Cells will follow a different relationship for 𝜆1 as compared to that of dendrites as, at low velocity, their tip undercooling varies approximately as 𝐺𝐷/𝑉, i.e. increasing with a reduction in V until the limit of constitutional undercooling, 𝑉𝑐 = 𝐺𝐷/𝛥𝑇0 , is reached. For the most important range of dendritic growth, at moderate growth rates, the following equation can be obtained by substituting the 𝑅 of Eq. 4.16 or Table 4.2 (which applies only to diffusional growth) into Eq. 4.19 and assuming that, in this range, Δ𝑇 ′ ≅ Δ𝑇0 : 𝜆1 = 𝐾1 (Δ𝑇0 𝐷Γ)0.25 𝑘 −0.25 𝑉 −0.25 𝐺 −0.5

[4.21]

where 𝐾1 = 4.3. This equation is similar to that of Hunt (1979) but with a different constant (see

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Figure 4.13 Modelling Primary Spacing in Directional Solidification In practical applications the dendrite-tip radius is not as visible a parameter as is the primary spacing, since it disappears when solidification is complete. The tip radius has a direct influence upon the primary spacing however. In order to estimate the primary spacing the dendrites can be imagined to be elliptical in shape. The length of the major half-axis, a, of the ellipse is equal to Δ𝑇 ′ /𝐺, the thermal length of the mushy zone where Δ𝑇 ′ is the difference between the tip temperature and the temperature of the last interdendritic liquid. The primary spacing, 𝜆1 , which is proportional to the minor half-axis, 𝑏, can be determined from simple geometrical considerations. The factor, 0.58, arises from the assumption that the dendrite trunk arrangement is close-packed hexagonal.

Figure 4.14 Morphology, Tip Radius and Spacing of Cells and Dendrites According to the dendrite model the tip radius decreases from very large values at the limit of constitutional undercooling, 𝑉𝑐 , to small values at high growth rates. Over the range of dendritic growth the primary spacing decreases approximately as the square-root of 𝑅 (Eq. 4.20). The corresponding interface structures are also shown and vary from planar at growth rates which are less than 𝑉𝑐 to cells and to dendrites which become finer and finer until they give rise to cellular structures again when close to the limit of absolute stability. At 𝑉 > 𝑉𝑎 , plane-front growth is stabilised. (At the high-velocity transition from cells to plane-front, another structure appears: banding. See Chap. 7 on rapid solidification.)

Exercise 4.15). Both results indicate that a variation in the growth rate has a smaller effect upon 𝜆1 than does a change in the temperature gradient. Both models are gross simplifications of columnar dendrite growth; Hunt used a hemisphere and the extremum criterion for the tip which leads to small EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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tip radii while the present authors, despite using marginal stability, modelled the interdendritic region by using an ellipse which neglects the appropriate mass balance. Trivedi (1984) has developed an interesting spacing-model which however is presented in the form of complicated functions. A simpler but still physical model has been proposed also by Trivedi (2021). It starts with the model in Appendix 9: 𝐺𝜆12 𝐷𝐺 = −5,66 [𝑚𝐶𝑙∗ (1 − 𝑘) + ] 𝑅 𝑉

[A9.45]

Substituting the flux balance, 𝐶𝑙∗ (1 − 𝑘) = −(𝐷/𝑉)𝐺𝑐 , leads to 𝐺𝜆12 = 5.66(𝐷/𝑉)𝑅[𝑚𝐺c − 𝐺]

[4.22]

If one replaces 𝑅[𝑚𝐺c − 𝐺] by (Γ/𝜎 ∗ )(1/𝑅) (Eq. A9.23), then uses Eq. A9.38: 1/2

Γ𝐷 𝐶0 𝑅 = ( ∗ ) 𝜎 𝑘Δ𝑇0 𝑉 𝐶𝑙∗

[A9.38]

and sets 𝐶𝑙∗ ≅ 𝐶0 for low to medium velocities, this leads finally to:** [4.23]

𝜆1 = 2.38 𝑘1/2 [(Δ𝑇0 ⁄𝐺 )𝑅]1/2

This equation shows that the primary spacing corresponds to the general relationship in diffusional solidification theory, i.e. 𝜆1 is proportional to the geometric mean of two length scales, in this case the thermal length of the mushy zone (𝑙 𝑀 𝑇 = Δ𝑇0 ⁄𝐺 ), and the tip radius (𝑅) (see Table 4.3 at the end of this chapter). The final result for the primary spacing is: 1/4

𝜆1 = [32/𝜎 ∗ ]1/4 [𝛤kD]1/4 ΔT0

V −1/4 G −1/2

[4.24]

With σ∗ = 1/4π2 , the first term in Eq. 4.24, [32/𝜎 ∗ ]1/4 , becomes ~6. A graphical representation of the behaviour of the three primary spacing models, together with experimental results, is given in Fig. 4.15. It is clear that Trivedi’s theory corresponds best to the experimental results, provided that fluid-flow is negligible. Note that for the low-velocity part of the red curve the assumption of 𝐶𝑙∗ ≅ 𝐶0 has to be relaxed. Primary spacing values under purely diffusional conditions can be obtained via microgravity experiments in space (Dupouy et al., 1992; Zimmermann and Weiss, 2005). Developments in computer hardware and programming techniques, such as adaptive remeshing and massive parallelisation nowadays permit the phase-field computation of large dendritic arrays (Takaki et al., 2016). When modelling the diffusional columnar growth of hundreds of interacting dendrites of an aluminium alloy, a predominance of hexagonal arrays was found. The mean spacings obey Eq. 4.24, which shows that the geometrical mean of the tip radius and the thermal length of the dendrite is a reasonable approximation to the morphological scale when fluid flow is absent. Once the primary spacing is established it will not change during, or after, solidification. This is not true of the secondary arms, which undergo a ripening process.

**

The authors are grateful to R. Trivedi (2021) for communicating his unpublished results.

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Figure 4.15 Comparison of Three Models for Diffusional Primary Spacings of Dendrites and Cells in Succinonitrile – 5.5 mol% Acetone Comparison of experiment (dots) and three theories of primary spacing for succinonitrile – 5.5 mol% acetone with 𝐺 = 6.7 K/mm. The black curves are for diffusional spacing models by Kurz and Fisher (1981) and Hunt (1979). Trivedi’s (1984) model (red curve) comes closest to the experimental results (dots), including the transition from cells to dendrites.

Figure 4.16 Primary Spacing Selection in Directional Solidification This figure shows the spacing-selection mechanisms of dendrites during directional solidification of a transparent alloy (succinonitrile-7wt% acetone). (i) Trunk insertion: In the centre the branching at a divergent grain boundary is the origin of new dendrite trunks. When the spacing between two grains increases to twice the minimum spacing during directional growth, tertiary branches are no longer blocked by secondaries ahead of them (as can be seen in the centre-left of the figure). In this case the tertiaries catch up with the solidification front and become primaries. In this way, the primary spacing can vary between 1 and 2 the average spacing. (ii) Trunk elimination: In the case of a convergent grain boundary (lower part of the figure) a reduction in the interdendritic spacing leads to the elimination of a trunk. The figure is from a film by Esaka et al. (1985).

At grain boundaries the mechanisms of spacing-selection are clearly visible. At convergent grain boundaries the diffusion field of the dendrite tips increasingly overlap with time, leading to the elimination of a dendrite (bottom of Fig. 4.16). At a divergent boundary, on the other hand, the dendrites enjoy increasing space for the development of secondary arms … and even to the growth

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of tertiary arms on the secondaries. These tertiary arms compete with the secondaries ahead of them. If a tertiary arm can evade all of the preceding secondaries it can become a new trunk (Fig. 4.16, centre). This phenomenon occurs at a double spacing between neighbouring trunks. During directional growth the cooling rate is 𝑇̇ = −𝐺𝑉. According to the above model 𝜆1 will not obey simple relationships such as those often proposed in the literature: 𝜆1 ≠ 𝐾|𝑇̇|𝑛 As previously noted however nearly all experimental primary-spacing data are affected to various degrees by convection. It is therefore not surprising that there is a discrepancy with respect to the purely diffusional models developed above. Figure 4.17 presents the results of Bridgman-type experiments on succinonitrile-2.2wt%(d-)camphor alloy. For growth rates between 0.8 and 2 μm/s the measured dendrite trunk spacings, λ1, vary with the fluid flow velocity, 𝑉𝑙∞ : for low flow velocities the influence is negligible, while for 𝑉𝑙∞ > 20 μm/s the spacing first decreases slightly and then drops to some 40% of the diffusional value at higher fluid flow velocity.

Figure 4.17 Primary Dendrite Spacing of Succinonitrile-Camphor Alloy as a Function of Fluid Flow Velocity The dendrite trunk spacing, 𝜆1 , is influenced by the flow of the adjacent melt (see Sect. 4.7 below). This figure shows that during experiments on succinonitrile-2.2wt%(d-)camphor at growth rates, 𝑉, of the order of 1 μm/s, the primary spacing corresponds to the diffusional case for fluid velocities 𝑉𝑙∞ < 20 μm/s. For 𝑉𝑙∞ > 20 μm/s, the spacing drops sharply (Witusiewicz et al., 2021). By analogy one can say that, if the fluid-flow velocity in a Bridgman-type experiment varies around 𝑉𝑙∞ ≅ 100 μm/s, the primary spacings can vary by between 40 and 80%.

4.6. Secondary Spacing after Directional or Equiaxed Growth As seen from Fig. 4.1, the secondary arms start to develop very close to the tips. They initially appear as a sinusoidal perturbation of the paraboloid. As in the case of a planar solid/liquid interface which becomes unstable (Fig. 4.3), these perturbations grow, become cell-like, are sometimes eliminated by their neighbours (see the first 6 arms behind the tip in Fig. 4.6), and a number of them finally become full secondary dendrites growing perpendicularly to the primary trunk (in the case of a cubic crystal). These secondary arms, with their higher-order branches, grow and eliminate one another, provided that their length is less than 𝜆1 /2. When the diffusion fields of their tips come into contact with those of the branches growing from the neighbouring dendrites, they slow down and cease to grow. A ripening process causes the highly-branched structure to change, with time, into coarser less-branched and more widely-spaced ones (Fig. 4.18 and Fig. 6.7).

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Figure 4.18 Ripening of the Secondary Dendrite Arm Spacing in Equiaxed Solidification In contrast to the primary spacing the secondary dendrite arm spacing, as measured in the solidified metal, is largely determined by annealing processes which occur during growth of the dendrites (Fig. 4.6). Due to a ripening phenomenon smaller (higher-curvature) features disappear and 'feed' the growth of the already larger features. The upper figures illustrate the model used when calculating the effect of these changes while the lower photographs show equiaxed cyclohexane dendrites (a) just after solidification and (b) 20 min later. In these photographs the black areas correspond to the solid phase and the white areas to the liquid phase. Note that the primary spacing in an equiaxed structure is not defined. One could instead use the mean grain diameter. (Photographs from Jackson et al., 1966).

Careful inspection of the photographs in Fig. 4.18 suggests that one possible mechanism for the coarsening process is the melting of thinner secondary arms and an increase in the diameter of the thicker branches. This process is analogous to the Ostwald ripening of precipitates. The process is depicted schematically in the upper diagram of Fig. 4.18. Each time that a thin secondary arm melts the local spacing is doubled. The driving force for the ripening process is the difference in chemical potential of crystals with differing interfacial energies due to differing curvatures. As in the ripening of precipitates the spacing of the branches, 𝜆2 , is proportional to the cube root of time (Eq. A9.55). Following Kattamis and Flemings (1965) and Feurer and Wunderlin (1977) one can write: 1/3

[4.25]

𝜆2 = 5.5(𝑀𝑡𝑓 ) with:

𝑀=

𝐶𝑚 Γ𝐷 ln ( 𝐶𝑙 ) 0

𝑚(1 − 𝑘)(𝐶0 − 𝐶𝑙𝑚 )

[4.26]

where 𝐶𝑙𝑚 is often equal to 𝐶𝑒 . Depending on the parameters in Eq. 4.26, the value of 𝑀 can easily vary by an order of magnitude. Because its effect upon 𝜆2 is proportional only to the cube root the differences will however be relatively small when compared with the inevitable scatter which is to be expected in experimental data. Such results are presented in Fig. 4.19 for Al-4.5wt%Cu alloy. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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Measurements of the secondary spacing which are observed in directionally solidified or equiaxed structures give some indication of the local solidification conditions and can be helpful when cooling-curves are unavailable. In the case of directional solidification the local solidification time is given by: 𝑡𝑓 =

Δ𝑇 ′ Δ𝑇 ′ = |𝑇̇| |𝐺𝑉|

[4.27]

When there is no eutectic reaction present in the system the determination of Δ𝑇 ′ might be difficult. It is shown in Chap. 6 that one can obtain an approximate value in this situation. The change from morphologically unstable continuous interdendritic liquid films to isolated drops at 𝑓𝑠 ∼ 0.9 (Fig. 6.7) is important for the resultant mechanical properties; especially with regard to the hot-cracking tendency of the solidifying mushy-zone. A solid with 𝑓𝑠 ≲ 0.9 that is rather brittle in shear due to continuous liquid films between the dendrites and grain boundaries transforms at 𝑓𝑠 ≿ 0.9 into a solid supporting extensive plastic deformation due to isolated liquid drops.

Figure 4.19 Secondary Spacing as a Function of Solidification Time The best-fit curve to the experimental points for Al-4.5wt%Cu alloy over a wide range of solidification conditions shows that the secondary spacing varies approximately as the cube root of the local solidification time (Bower et al., 1966). The latter is defined as being the time during which each arm is in contact with liquid (Fig. 4.1) and is therefore a function of the growth rate, the temperature gradient and the alloy composition. The secondary spacing is important since, together with 𝜆1 , it determines the spacing of precipitates or porosity and thus has a considerable effect upon the mechanical properties of the as-solidified alloy (Fig. 1.2).

4.7. Fluid-Flow Effects In all of the theoretical developments presented so far, the only mode of matter transport that has been considered is the diffusion which is associated with the random movement of atoms. Such a situation is however almost never encountered on Earth and atoms instead tend to undergo coordinated bodily motions at the macroscopic scale; especially in liquids, where their mobility is high. This is termed fluid flow. Whatever the state of the other solidification conditions, there already has to exist a flow of liquid towards the solid in order to compensate for solidification shrinkage, i.e. to compensate for the fact that the solid has a density which is larger than that of the liquid in most cases††. This, however, is not the only cause of macroscopic transport of matter. If one considers the solid phase to be fixed, the liquid can flow for reasons such as: (i) natural convection (buoyancy) due to density gradients in the liquid which are associated with temperature and/or composition inhomogeneities, (ii) forced convection due to mould filling or mechanical/electromagnetic stirring, ††

Exceptions include water, silicon and germanium.

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(iii) surface-tension gradients induced by temperature or composition gradients at a free liquid surface (the so-called thermal/solutal Marangoni effect); particularly important in additive manufacturing and welding. Movement (or lack of movement) of the liquid can create several problems or defects such as: porosity; inhomogeneities of solute composition at the scale of the entire component (macrosegregation); partial melting and detachment of dendrite arms (fragmentation), thus promoting the formation of equiaxed grains; entrainment of equiaxed grains, etc. As shown above, the primary dendrite spacing is also influenced by melt flow (Fig. 4.17). Readers who are interested in these aspects of solidification are referred to the book by Dantzig and Rappaz (2016). This section presents just the influence that fluid flow can have upon the growth kinetics of dendrites as well as upon their possible fragmentation. Before treating these two topics the concept of the boundary layer, which is fundamental to all fluid-flow theories will be briefly introduced. Boundary layer The equation which governs the movement of a liquid, the so-called Navier-Stokes equation, is not fundamentally different to the equation which governs heat or solute diffusion. Consider a flow with a uniform velocity, 𝑉𝑙∞ , far from the edge of a fixed plate that is parallel to the flow (Fig. 4.20, 𝑦-axis). Because the atoms are assumed to “stick” to the plate (the so-called no-slip condition), the velocity 𝑉𝑙 (𝑧 = 0, 𝑦) must be zero all along the plate surface. At the edge of the plate, 𝑦 = 0, there is consequently a discontinuity since 𝑉𝑙 (𝑧 = 0, 𝑦 = 0) = 0, whereas 𝑉𝑙 (𝑧 > 0, 𝑦 = 0) = 𝑉𝑙∞ (the 𝑧axis is perpendicular to the plate surface). This singularity is smoothed-out (diffused) as the fluid moves along the 𝑦-axis of the plate, and a velocity-profile 𝑉𝑙 (𝑧, 𝑦𝑖 ) at a given position 𝑦𝑖 develops; from 0 at the surface of the plate to 𝑉𝑙∞ far from the plate. This occurs over a distance, 𝛿(𝑦𝑖 ), termed the boundary layer, that increases with 𝑦𝑖 . Two such velocity profiles are shown schematically in Fig. 4.20 for two positions, 𝑦1 and 𝑦2 . This propagation of the boundary condition 𝑉𝑙 (𝑧 = 0, 𝑦) = 0 into the bulk of the liquid is like a “diffusion” process; in the present case it is that of the density of momentum of the fluid, 𝜌𝑉𝑙 , where 𝜌 is its density. The process occurs with the kinematic viscosity, 𝜈 = 𝜂/𝜌 [m2/s], where 𝜂 is the dynamic viscosity [kg/(m s)]. Note that the kinematic viscosity, 𝜈, has the dimensions of a diffusion coefficient. The velocity boundary layer, 𝛿(𝑦), is given by the expression: 𝛿(𝑦) ≈ 5.0

𝑦 √Re

with

𝜌𝑉𝑙∞ 𝑦 𝑉𝑙∞ 𝑦 Re = = 𝜂 𝜈

[4.28]

in which Re is the Reynolds number (similar to a Péclet number). As expected in a diffusion-type problem, the distance, 𝛿(𝑦), over which the velocity profile develops, therefore varies as 𝑦1/2 (or as 𝑡1/2 since the time can be correlated to 𝑦/𝑉𝑙∞ ). If the temperature of the fluid far from the plate is given by 𝑇 ∞ while the plate temperature is maintained at 𝑇0 , the temperature of the fluid will also vary within a thermal boundary layer, 𝛿𝑇 (y). The thermal boundary layer, which follows that of the fluid momentum, is given by: 𝜈 𝜈 Pr = = with [4.29] 𝛿𝑇 (y) = 𝛿(𝑦) Pr −1/3 𝑎 𝜅/(𝜌𝑐𝑝 )

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Figure 4.20 Velocity and Solute Boundary Layers Above a Static Semi-Infinite Plate A horizontal flow of liquid is assumed to have a uniform velocity profile, 𝑉𝑙∞ , far from a fixed semi-infinite horizontal plate. Assuming a no-slip condition at the liquid/plate interface and laminar flow, the flow has a zero velocity along this surface. This creates a shear stress in the liquid which gradually makes the velocity profile evolve from 0 to 𝑉𝑙∞ over a typical distance, 𝛿(𝑦), that increases as one moves away from the edge of the plate. Such velocity profiles, 𝑉𝑙 (𝑧, 𝑦1 ) and 𝑉𝑙 (𝑧, 𝑦2 ), are shown schematically at two positions, 𝑦1 and 𝑦2 . If the temperature of the plate is different to that of the flow a temperature profile is also established, in a similar fashion to the boundary layer, over a thickness 𝛿𝑇 (𝑦) that also increases with the distance from the edge of the plate.

where Pr is the dimensionless Prandtl number, i.e. the ratio of the kinematic viscosity to the thermal diffusivity, a = 𝜅/𝑐, where 𝜅 is the thermal conductivity and 𝑐 is the specific heat of the fluid per unit volume. If the fluid is a pure substance while the surface of the plate has a “contaminant” of imposed concentration, the problem is identical to the thermal problem, with a solute boundary layer, 𝛿𝑐 (𝑦) = 𝛿(𝑦) Sc −1/3, where Sc is the dimensionless Schmidt number given by the ratio 𝜈/𝐷, with 𝐷 being the diffusion coefficient of the “contaminant” in the fluid. In the presence of convection during dendrite growth, similar boundary layers interact with thermal and/or solute diffusion fields in the liquid. Dendrite growth kinetics Two special situations are illustrated schematically in Fig. 4.21 for an isolated columnar dendrite growing in a vertical thermal gradient, 𝐺, with a velocity, 𝑉𝑇 , of the liquidus isotherm (directional solidification conditions). In the absence of any fluid-flow, the columnar dendrite with its preferred ⟨100⟩ crystallographic axis can be assumed to grow in perfect alignment with the heatflow direction. Under steady-state conditions, the velocity of the tip, 𝑉tip , is equal to 𝑉𝑇 and the undercooling, Δ𝑇, can be calculated as shown in the previous sections. Consider firstly case (a), where the flow is assumed to be opposite to the ⟨100⟩ growth direction of the dendrite trunk, with a characteristic velocity, 𝑉𝑙∞ , at some distance from the tip. This is the case for the flow induced by solidification-shrinkage, since, in this case, 𝑉𝑙∞ = −𝛽𝑉𝑇 , where 𝛽 = (𝜌𝑠 − 𝜌𝑙 )/𝜌𝑙 is the solidification shrinkage-factor, i.e. the normalized difference between solid and liquid densities. Shrinkage flow is not very significant in metals given that 𝛽 is here typically of the order of only a few percent and does not greatly modify the growth kinetics of the dendrite. The apparent velocity of the liquid, 𝑉𝑙∞ , can however be much larger as, for example, in the case of an EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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equiaxed grain settling in a liquid (Fig. 4.22(a)) or in some regions of a columnar dendrite front where the flow happens to be directed towards the tip‡‡. In Fig. 4.21(a), the antiparallel flow has a tendency to decrease the solute boundary layer, 𝛿𝑐 , ahead of the dendrite tip, i.e. it slightly compresses the isopleths of the solute composition profile. The flow is of course deviated by the solid dendrite while still respecting its axial symmetry. A simple approximation made by Cantor and Vogel (1977) for the case of Fig. 4.21(a) is to assume that the farfield composition of the liquid, 𝐶0 , is no longer imposed at  but rather at a confocal paraboloid which is located at a distance, 𝛿𝑐 , from the tip (red dashed curve). By using this approximate so-called “stagnant film” model, those authors have shown that the solutal supersaturation, Ω, given by Eqs 4.2 and 4.3, when there is no flow, is modified to give:

Figure 4.21 Effect of Fluid Flow on Dendrite Growth Schematic depiction of a single columnar dendrite growing in a thermal gradient, 𝐺, with an imposed velocity, 𝑉𝑇 , of the liquidus isotherm. In the case of fluid flow with a far-field velocity, 𝑉𝑙∞ , anti-parallel to the ⟨100⟩ growth direction of the dendrite trunk (a), the solute profile ahead of the tip (dashed curve) is “compressed” or, to be more precise, the diffusion layer thickness, 𝛿𝑐 , (red dashed curve) decreases with increasing value of 𝑉𝑙∞ . As the solute gradient, 𝐺𝑐 , is also increased the dendrite tip can more easily reject solute into the liquid. At a given undercooling the velocity of the tip, 𝑉tip , is therefore increased in the presence of this counterflow. Under Bridgman-type conditions where the velocity, 𝑉tip , is imposed by the speed of the liquidus, 𝑉𝑇 , the counterflow decreases the tip undercooling. In (b) the flow is assumed to be perpendicular to the thermal gradient, 𝐺, i.e. is parallel to the isotherms. In this case the solute profile on the upstream side of the tip is “compressed” but is “expanded” on the downstream side (solute boundary layer represented by the red dashed curve). As a result, the upstream side of the dendrite tip grows faster as compared with the downstream side and the tip no longer grows along the ⟨100⟩ crystallographic direction: It is instead tilted towards the incoming flow, and the inclination increases with 𝑉𝑙∞ . The stability parameter, 𝜎 ∗ , is unchanged by this type of flow. In the case of a columnar-dendrite front the flow rapidly decreases as it penetrates into the mushy zone. If the flow remains significant, as for a settling equiaxed grain (Fig. 4.22), the upstream secondary arms grow faster as compared with those on the downstream side; for the same reason as that given for the tip in (a).

‡‡

In order to conserve mass an incoming flow towards the columnar dendrite front must induce outgoing flows in other parts of the mushy zone (i.e. a convection loop).

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Ω = 𝑃𝑐 ex p(𝑃𝑐 ) {E1 [𝑃𝑐 ] − E1 [𝑃𝑐 (1 + 2

𝛿𝑐 )]} 𝑅

[4.30]

where 𝑃𝑐 = 𝑅𝑉𝑡𝑖𝑝 /2𝐷 is the solutal Péclet number which is associated with a dendrite tip of radius 𝑅 and E1 is the exponential integral, as defined for the purely diffusive regime. Note that in the absence of fluid flow the boundary layer extends to infinity, 𝛿𝑐 /𝑅 → ∞ (E1 (∞) = 0) and the second term within the curly braces vanishes, thus retrieving the expression which was derived for the purely diffusive regime (Eqs 4.2 and 4.3). When the flow is non-zero, the solute boundary layer, 𝛿𝑐 , is related to the boundary layer, 𝛿, as shown in the previous section. The object is however no longer a planar surface and 𝛿 is given in this case by 𝛿 = 2𝑅/Re1/2 , where the Reynolds number is now related to the curvature radius, 𝑅, of the tip and thus: −1/2

𝛿𝑐 𝛿 ν −1/3 𝑅𝑉𝑙∞ −1/3 −1/3 −1/2 = Sc = 2Sc Re = 2( ) ( ) 𝑅 𝑅 𝐷 ν

[4.31]

Cantor and Vogel concluded that, for rapid fluid flow, the dendrite grows at a higher velocity and with a smaller tip radius. The effect of fluid flow on constitutional dendrites is moreover greater than that on thermal dendrites.

a)

b)

Figure 4.22 Influence of Fluid Flow on a Growing Equiaxed Grain This figure shows (a) the growth of an equiaxed grain of NH4Cl falling into an undercooled NH4Cl-water solution (courtesy of Gerardin, 2002). As the heavier crystal falls with a velocity, 𝑉𝑠 , it experiences an apparent flow-velocity of the liquid: 𝑉𝑙∞ = −𝑉𝑠 . A primary arm that grows in the direction of fall is longer than an oppositely-directed primary arm, while secondary arms are much more fully developed in the direction of fall but are very sparse on the opposite site. Primary arms which grow in the direction perpendicular to the direction of fall are slightly tilted towards this direction. In (b) the left-hand side shows a 3-dimensional phase-field simulation of an equiaxed grain, with the left half governed by solute diffusion alone, while the right half shows a fixed equiaxed grain growing under identical conditions but with an upward flow-velocity. The grey level in the liquid shows the solute composition field, with the small upward directed arrows being the result of fluid-flow calculations (Phase-field simulation by Berger, 2018).

The present brief analytical presentation, for a columnar dendrite tip with an imposed counterflow, has been extended to the more complex situation of an equiaxed grain growing in an undercooled liquid (Melendez and Beckermann, 2012). In this case, no flow is imposed but instead results naturally from the rejection of solute and heat by the growing solid, i.e. thermo-solutal

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convection. By extending the analysis of Cantor and Vogel, these authors could reproduce the measured growth kinetics 𝑉tip − Δ𝑇 found for succinonitrile-acetone dendrites (Fig. 4.23).

Figure 4.23 Dendrite Tip Velocity 𝑽 as a Function of Undercooling 𝚫𝑻 The velocity, 𝑉, of dendrite tips of succinonitrile-0.1045mol% acetone alloy has been measured as a function of the undercooling, Δ𝑇. The red dots are the experimental data and the dashed curve corresponds to a purely diffusive regime (Ivantsov solution and marginal stability criterion) where both the thermal and solutal undercoolings are accounted for (LGK model by Lipton et al., 1987). The continuous red curve has been calculated using the LB model (Li and Beckermann, 2002), which accounts for thermo-solutal convection on the basis of a model which uses a stagnant-film approximation (Cantor and Vogel, 1977)). The open triangles correspond to the same theoretical model, but for a measured value of 𝜎 ∗ . It was found within experimental uncertainty that 𝜎 ∗ is independent of the composition but decreases slightly with undercooling (Melendez and Beckermann, 2012).

When the flow is not parallel to the growth direction of the dendrite the situation is much more complex and there is no simple analytical approach. In Fig. 4.21(b), the flow is assumed to be perpendicular to the heat-flow direction. In this case, the side of the dendrite tip which is in the upstream direction experiences a “compressed” solute field and can thus grow faster, for the same reason as that explained previously, while its downstream counterpart is characterized by a larger solute-layer thickness, 𝛿𝑐 , (red dashed curve) and thus grows slower. As a result, the dendrite axis which was growing along ⟨100⟩ in the absence of fluid-flow is now tilted towards the incoming flow, i.e. the dendrite axis corresponds to a ⟨ℎ𝑘𝑙⟩ direction of the crystal which is no longer ⟨100⟩. This effect is clearly seen in the primary arms of the falling NH4Cl crystal which grow in a direction that is perpendicular to its fall (Fig. 4.22(a)): they are not at right-angles to the primary arms that are aligned with the direction of fall and are instead tilted towards the direction of the apparent incoming flow of liquid. This is also the case for the equiaxed grain simulated by using the phase-field method (Fig. 4.22(b)). The secondary arms in Fig. 4.21(b) are therefore no longer symmetrical when they experience an incoming flow. The upstream arms are much more fully developed, for the same reason as that invoked for the primary trunk in Fig. 4.22(a), while the downstream secondary arms are much less developed. The same phenomenon is observed for the settling equiaxed grain in Fig. 4.22(a) or the

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grain simulated using the phase-field method in Fig. 4.22(b). Such an effect is much less pronounced for the secondary arms of a columnar zone because the fluid flow here does not penetrate far into the “forest” of columnar dendrites. It can be concluded from this brief coverage of a very complex topic that a detailed analysis of dendrite growth in the presence of fluid flow can be achieved only by performing numerical simulations. This can be done by coupling the phase-field method (Appendix 14), which already couples solute diffusion and possibly heat transfer, to the momentum-equation (Navier-Stokes) which governs liquid movement. Because the phase-field method solves equations over an entire domain it has to be modified so as to account for the fact that the velocity in the solid is zero; i.e. 𝑉(𝜙=1) = 0, while its component in the liquid, 𝑉(𝜙=0) = 𝑉𝑙 must be governed by the standard Navier-Stokes equation. To this end, a drag term function of 𝜙, similar to that which describes the movement (socalled Darcy flow) through a porous medium, is introduced. Further details can be found in the literature (Tong et al., 2000; Jeong et al., 2003). 4.8. Fragmentation of Dendrites As well as changing the growth kinetics of dendrites, fluid flow can have many other effects upon the formation of a microstructure given that it transports heat, solute, equiaxed grains and even suspended inoculant particles at the macroscopic scale. Solute-rich interdendritic liquid (for 𝑘 < 1) may be taken away to other regions of a component; for example, leading to what is termed macrosegregation. The latter is an inhomogeneous average composition at the scale of the component which is induced by microsegregation and relative movements of the solid and liquid phases. Readers who are interested in this topic will find further information in Chap. 14 of the book by Dantzig and Rappaz (2016). In this section will be treated the effect that fluid flow can have upon the fragmentation of dendrites, as this is an important process which promotes the formation of an equiaxed grain structure. The fragments which are carried away by convection are, if they do not re-melt in other regions of a casting, the perfect “nuclei” for forming new equiaxed grains given that there is clearly no nucleation barrier. Mechanical or electromagnetic stirring of a melt is in fact a common industrial method for obtaining a fine equiaxed grain structure, with more evenly and finely distributed secondary phases, possessing consequently improved properties. Figure 4.24 illustrates two “natural” mechanisms by which columnar dendrites can be fragmented due to the movement of fluid near to the tips of dendrites (mechanism labelled “1”) or deeper into the mushy zone (mechanism labelled “2”). In the first mechanism, two columnar dendrites are surrounded by the diffusion profile which is associated with their growth. If the solute element is heavier than the solvent (with 𝑘 < 1), the liquid layer which surrounds the tip of the dendrites is denser than it is at the mid-distance between two neighbours. This tends to induce two small fluid flow-loops (indicated by arrows); especially if the dendrite separation, 𝜆1 , is greater than “normal”, e.g. at grain boundaries. This increases the local composition of the liquid which surrounds the dendrite-arms that are located between them, thus decreasing the local liquidus temperature. As will be shown, the arms melt-through preferentially at the neck which is close to their attachment-point to the main dendrite, and the remaining part can detach (Fig. 4.26). If the solid is denser than the surrounding liquid, it stays in place and creates a grain having a slightly different orientation to that of the parent dendrite. On the other hand, still with 𝑘 < 1 and a solute element heavier than the solvent, the solid fragment can in fact be less dense than the liquid and its positive buoyancy makes it rise as indicated in Fig. 4.24 (1). This situation occurs in Al-Cu alloys having a nominal composition greater than about 8 wt% and is illustrated in Fig. 4.25(a) – (c).

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Figure 4.24 Two Mechanisms of Dendrite Fragmentation Schematic illustration of two fragmentation processes that can occur for columnar dendrites. In the first mechanism solute-rich liquid becomes denser than the bulk and two small loops (red arrows – 1) of natural convection are initiated in the tip region of two neighbouring dendrites where there is not yet complete mixing of solute in the liquid. This increases the composition of the liquid near to dendrite arms located in region “1” (for 𝑘 < 1) thus decreasing the local liquidus temperature and possibly inducing partial melting. Following detachment from the main stem the fragment, being leaner in solute, can be lighter than the liquid and thus rises. In the second mechanism the solvent is lighter and the interdendritic liquid is therefore less dense than the bulk liquid. The liquid tends to rise and local melting occurs if its velocity, 𝑉𝑙 , is greater than the speed, 𝑉𝑇 , of the isotherm. This creates what is known as a freckle, i.e. a chimney of liquid with an upward velocity at its centre which is fed by liquid arriving from its sides (red arrows – 2). Fragments of dendrites which have partially melted-through at their attachment-points to the columnar dendrite stems are entrained by the flow.

Figure 4.25 Observations of Fragmentation Three images (a-c) captured at 𝑡 = 0, 2.25 and 4.5 s of an in situ X-ray radiography video reveal the detachment of tertiary arms from one main columnar dendrite growing against gravity during the directional solidification of Al-20wt%Cu. The primary spacing between both columnar dendrites where this happens is about 400 m. The solid fragment which is leaner and thus less dense than the liquid rises towards the hotter part of the specimen (Ruvalcaba et al., 2007). (d) In a solution of NH4Cl–water solidified in a vertical thermal gradient, the NH4Cl dendrites (white) reject water. Because the water-rich interdendritic liquid is less dense than the liquid bulk (black), an upward flow of liquid (vertical arrows) has partially remelted the mushy zone and entrained a fragment. The upward flow, called freckle, can partially remelt secondary arms/fragments which are then entrained upwards and may become new equiaxed grains, as illustrated by the beautiful six-armed equiaxed grain which is visible above the columnar dendrites (Hansen et al., 1996).

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The second mechanism operates slightly deeper into the mushy zone and occurs when the solute is lighter than the solvent (again for 𝑘 < 1). As the interdendritic liquid is less dense than the bulk the fluid has a tendency to rise. If the velocity of the fluid in the mushy zone, 𝑉𝑙 , is greater than the speed of the isotherm, 𝑉𝑇 , local remelting occurs and forms what is known as a freckle. In this liquid channel, the remaining secondary arms can become detached and entrained upwards by the flow. Such a situation is shown in Fig. 4.24 (2) and Fig. 4.25(d) for a solution of NH4Cl-water (dark-field image) with an equiaxed grain visible above the columnar dendrites (with six 100-branches, 3D view). Fragmentation of dendrite arms naturally occurs near to their attachment-point to their stem as indicated schematically in Fig. 4.26(a). The geometry of a secondary dendrite arm at such a location is complex in that the two principal radii of curvature are unequal and can even be of opposite sign: i.e. convex along one principal radius and concave along the other. It can nevertheless be assumed to be almost cylindrical. As its radius of curvature at location A close to its attachment point to the

Figure 4.26 Schematic Representation of the Fragmentation Mechanism In Appendix 9 (Fig. A9.3) the exchange of solute between a thin secondary arm and a thicker arm is considered within the context of coarsening. A similar phenomenon can occur even for a single arm which is attached to a trunk at A (a). During growth it tends to develop a neck in region A as compared with its overall body form B. Neglecting the other radius of curvature (that in the plane of the figure) the radius, 𝑅 𝐴 (small circle), is smaller than 𝑅 𝐵 (large circle). Therefore the curvature undercooling Δ𝑇𝑟𝐴 = Γ𝐾 𝐴 > Δ𝑇𝑟𝐵 = Γ𝐾 𝐵 , where 𝐾 𝐴 = 1/𝑅 𝐴 and 𝐾 𝐵 = 1/𝑅𝐵 . Since the local temperature can be considered to be uniform at this local scale the curvature undercooling is accommodated by small variations in the solute composition (b). The liquid solute composition being such that 𝐶𝑙𝐴 < 𝐶𝑙𝐵 (for 𝑘 < 1) there is a small flux of solute from the main body of the arm towards the neck. The dendrite arm thus gets thinner and thinner at A until it detaches from the main stem. Entrained by the bulk of the liquid, it can lead to the growth of a new equiaxed grain as seen in Fig. 4.25. The coarsening of dendrite arms during isothermal holding with one arm at the centre that has shrunk to such an extent that it has become an isolated spherical fragment is shown in (c) (phase field simulation using the microstructure measured experimentally in Pb-Sn alloys, Cool and Voorhees, 2016, 2017).§§

Primary trunk is smaller than that of the main part of the arm B (Fig. 4.26(a)), the associated mean curvature is such that 𝐾 𝐴 > 𝐾 𝐵 and so are the associated curvature undercoolings Δ𝑇𝑟𝐴 = Γ𝐾 𝐴 > Δ𝑇𝑟𝐵 = Γ𝐾 𝐵 . Because heat diffuses much more rapidly than does solute, the temperature is uniform §§

Thanks to P. Voorhees for providing the images in (c).

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at this local scale, and the curvature undercooling is accommodated by small solute-composition differences (Fig. 4.26(b)). Given that the composition of the liquid at the neck level A, 𝐶𝑙𝐴 , is slightly smaller than 𝐶𝑙𝐵 (for 𝑘 < 1), there is a flux of solute from B to A (red arrows in Fig. 4.26(a)). This corresponds to a flux of solvent in the opposite direction and the neck thus tends to shrink while the radius of the main body of the secondary arm increases*** (small black arrows in (a)). As the neck becomes thinner, its curvature-undercooling increases and it therefore disappears at an increasing rate. This mechanism is similar to the coarsening mechanism, treated in Sect. 4.6, which involves two arms of differing size (see also Appendix 9, Fig. A9.3), and corresponds to what is known as a Rayleigh-Plateau instability: a small perturbation of an infinite cylinder leads to its fragmentation into a family of spherical droplets in order to minimize the total surface energy. This phenomenon can be observed daily in any laminar flow of water exiting a hose. Such a coarsening mechanism in Pb-Sn has been modelled using the phase-field method and is illustrated in Fig. 4.26(c). Although fragmentation can occur under almost any experimental conditions (temperature and/or solute-composition fluctuations), it is favoured when there is a sudden change in the cooling rate, e.g. due to the formation of an air gap at the interface between metal and mould, a temperature increase during recalescence…or when there is a flow of liquid near to the liquidus temperature (close to the dendrite front) or deeper in the mushy zone when it is parallel to the thermal gradient. 4.9. Phase-Field Analytical models for computation of the dendrite tip radius, 𝑅, primary trunk spacing, 𝜆1 , and secondary arm spacing, 𝜆2 , are very important as they give a quick and semi-quantitative description of how dendritic microstructures depend upon solidification parameters; mainly the thermal gradient, 𝐺, and the isotherm velocity, 𝑉 (or in some cases the cooling rate, 𝑇̇). Although established for steadystate conditions (e.g. Bridgman growth), such diffusion-based models can be used for the calculation of microstructural features, in casting, welding or additive manufacturing processes, based upon local values of 𝐺 and 𝑉 as deduced from heat- and mass-transfer calculations. In a similar manner to basic experimental investigations, research on numerical models for dendrite growth were carried out over many years; not only for the computation of the very complex morphologies of dendrites but also in order to answer fundamental questions such as: what is the effect of the solid/liquid interfacial-energy anisotropy upon the tip radius? How can one extend, to multicomponent analytical models, those usually developed for binary alloys? What is the effect of fluid flow? How do instabilities near to the dendrite tip initiate the formation of secondary arms? How does competition between dendrites occur at grain boundaries? For much of the time, various techniques based upon a sharp interface were developed in order to answer such questions; mainly in two dimensions. Finite-element or boundary integral methods are difficult to extend to 3-dimensions because the nodes must be precisely located at the sharp interface while it is moving. After being introduced by Langer in 1978, the phase-field method has revolutionized the field of microstructure modelling since the 1990s because it does not necessarily have to follow the solid/liquid interface and can use a fixed mesh. This method is briefly introduced in Sect. 2.7 and is explained in further detail in Appendix 14. The intent of this Appendix is to help the newcomer to grasp the essential points before deepening that understanding with the aid of the literature. Recapitulating what was introduced in Chap. 2, the phase-field method is based, as the name indicates, upon a phase parameter, 𝜙(𝑥, 𝑦, 𝑧, 𝑡), which is chosen here to be equal to 1 in the solid and 0 in the liquid and varies continuously and smoothly between these values across a diffuse interface of thickness, 𝛿 (see Fig. 2.17). As shown in Appendix 14 the phase-field equation which governs the formation of a dendrite is given by a diffusion-type relationship. For a pure substance which is

***

When combined with solidification this can lead to the coalescence of two dendrite arms.

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solidified at a fixed undercooling, Δ𝑇, this diffusion-type equation has, for one dimension, the form (Eq. A14.18): ∂𝜙 ∂2 𝜙 𝑑𝑔 𝑑𝑝 = 2𝜀𝑀𝜙 [ 2 ] − 𝑀𝜙 [ℎ + Δ𝑠𝑓 Δ𝑇 ] ∂𝑡 ∂𝑧 𝑑𝜙 𝑑𝜙

[4.32]

where the time-derivative of the phase variable, 𝜙 (left hand side), is given by a diffusive term (first term on the right-hand side) plus two additional terms: one related to the first derivative of a doublewell potential, (𝑔(𝜙)), of height ℎ (Fig. A14.2) and the other one being the driving force, Δ𝑠𝑓 Δ𝑇. The double-well potential, 𝑔(𝜙), is introduced in order to maintain the diffuse interface at a fixed thickness††† while 𝑝(𝜙) is a smooth interpolating function between the liquid, (𝑝(𝜙=0) = 0), and the solid, (𝑝(𝜙=1) = 1). The other parameters, 𝑀𝜙 and 𝜀, are related to the mobility of the interface and to the solid/liquid interfacial energy, respectively. Appendix 14 furnishes further details on how this equation is obtained and how it can be extended so as to handle alloys, non-uniform temperature fields and anisotropic interfacial energies. The phase-field method is fairly easy to implement as a computer program for the calculation of a complex dendrite… but there is also a “price to pay”; one which can be easily understood with the aid of Fig. 4.27. This figure shows how a columnar dendrite is situated between the scale of the columnar grains forming a directionally solidified superalloy turbine-blade, and the atomistic scale of a diffuse solid/liquid interface. That is, there are 3 orders-of-magnitude between the dendrite-tip radius, 𝑅, and the thickness, 𝛿𝑖 , of the diffuse interface while the tip-radius is about 2 orders-ofmagnitude smaller than the primary trunk spacing, 𝜆1 . It is therefore a real challenge to model dendrite formation in 3-dimensions by using a continuous variable, 𝜙(𝑥, 𝑦, 𝑧, 𝑡), which varies across the physical thickness, 𝛿𝑖 , of the diffuse interface. In fact 𝜙 is nearly constant in the solid and liquid phases and varies only across the thickness of the diffuse interface. The solution of the diffusion-type Eq. 4.32 requires having a minimum of about 10 points distributed across the thickness, 𝛿𝑖 , of the interface and would thus imply the use of a grid which was as fine as the size of the atoms! This might be practical in 1 dimension (although the morphology is then trivial!) but calculations capable of treating even a single dendrite within a domain of the order of mm in size would demand the use of 1014 mesh-points in 2 dimensions (or 1021 in 3 dimensions)! The phase-field method therefore uses a much thicker interface; typically 𝛿 = 102 − 103 𝛿𝑖 . For the calculation of a dendrite, this thickness should nevertheless satisfy other conditions: (i) 𝛿 should be some 5 to 10 times smaller than the smallest feature of the dendrite, i.e. the tip radius, 𝑅, because otherwise it will artificially enlarge this feature (due to so-called “numerical diffusion”); (ii) there should be 5 to 10 points of the mesh distributed across 𝛿. Increasing the thickness of the diffuse interface, from 𝛿𝑖 to 𝛿, has a direct impact upon solute-trapping (see Eq. A6.25). In order to compensate for this effect, the concept of an antitrapping current is used (Karma, 2001; Echebarria et al., 2004). In addition to increasing the thickness of the diffuse interface an alternative option for reducing the number of calculation points used in a phase-field simulation is to adopt an adaptative mesh which dynamically follows the region over which the variable, 𝜙, changes. This can be done by using finite elements which are refined within the region where 𝜙 varies, or coarse finite-difference meshes which are subdivided into smaller and smaller cubes (in 3-dimensions) within the region where the diffuse interface is located. †††

Because this double-well potential introduces an energy-penalty when 0 < 𝜙 < 1, it counterbalances the diffusive term which is proportional to 𝜀 and the thickness of the diffuse interface is thus given by δ = 2√ε/ℎ. The interfacial energy of the diffuse interface, 𝜎, includes two equal contributions, arising from the diffusive term and the double-well potential height, i.e. σ = √εh/3.

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Figure 4.27: Situating the Phase-Field Model of a Dendrite Between the Scale of a Component and the Diffuse Interface Thickness There is an order-of-magnitude difference between the scale of a casting, here a directionally solidified turbine blade with columnar grains (a), where a single grain might be made up of some 10 to 100 columnar dendrites. One of these columnar dendrites, modelled using the phase-field method, is shown in (b). The primary-trunk spacing, 𝜆1 , is of the order of a few hundred microns, the secondary-arm spacing, 𝜆2 , a few tens of microns and the dendrite tip radius, 𝑅, amounts to a few microns. Even this smallest feature of the dendrite is about 103 times bigger than the thickness, 𝛿, of the diffuse solid/liquid interface (c). (Columnar grains of a casting (a) modelled with a cellular automaton approach by Gandin et al.,1999).

Fig. 4.28: Phase-Field Result for an Equiaxed Dendrite Obtained by using the phase-field method, this figure shows an equiaxed dendrite growing freely into an infinite volume of undercooled liquid. The cubic crystal-symmetry exhibits a fourfold anisotropy of the interface energy, leading to the formation of 6 trunks. (Dantzig et al., 2013).

Whatever the technical details, the results of the phase-field method must firstly be validated on simpler cases by using, for example, solvability theory or by at least checking that the result has converged and is now independent of the mesh-size or the orientation of the dendrite with respect to the mesh. The phase-field method has been increasingly used over the past two or three decades to investigate various aspects of dendrite growth, such as (i) the influence of the anisotropy of the interfacial energy; in particular for dendrites exhibiting two different principal radii of curvature; (ii) the formation of twinned dendrites exhibiting a coherent twin plane at the trunk centre; (iii) the effect of convection; (iv) grain competition at grain boundaries; (v) coarsening; (vi) fragmentation of EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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dendrites; (vii) the growth of dendrites which interact with boundaries, e.g. in thin coatings; (viii) transitions from a planar front to cells and from cells to dendrites; (ix) dendrite selection under nonsteady thermal conditions; (x) extension of dendrites after passing a re-entrant corner; (xi) final-stage solidification and the coalescence of dendrites arms. This list is certainly non-exhaustive, and the reader is referred to the bibliography at the end of this chapter. 4.10. Dimensional Analysis of Microstructural Scales In concluding this chapter, a short review of the dimensional relationships between dendritic structural elements and the corresponding transitions will be presented (Trivedi and Kurz, 1994). In the absence of fluid-flow the microstructure depends mainly upon heat- and mass-diffusion phenomena on the one part, and upon capillary phenomena on the other part. Diffusion drives the system to adopt finer-scale morphologies (thereby accelerating local heat- and mass-transport) while capillary-action counterbalances this tendency of diffusion and coarsens the structures. The developing microstructure is always governed by at least these two phenomena, generally in the form of the geometrical means which can be found in Table 4.3 and in Fig. 4.29.

Figure 4.29: Characteristic Lengths, l, as a Function of Velocity (for Local Equilibrium). During the directional solidification of an alloy of constant composition in a fixed temperature gradient the thermal length of the dendritic structure, 𝑙 𝑀 𝑇 (= thermal length of mushy zone Δ𝑇𝑜 /𝐺), is constant. The capillary length, sc, is also constant while the log of the solute-diffusion length, 𝑙𝐷 , decreases linearly with log V. Low-velocity phenomena are located on the left, close to 𝑉𝑐 , while high-velocity phenomena group around absolute stability, 𝑉𝑎 . Here 𝑉𝑐 corresponds to 𝑙 𝑀 𝑇 = 𝑙𝐷 while 𝑉𝑎 corresponds to 𝑙𝐷 = 𝑘𝑠𝑐 (see Table 4.3).

The general form of the microstructural scaling equation can be written as: 𝐿𝑖 ∝ 𝑙𝐷𝑎 𝑙 𝑏𝑇 𝑠𝑐𝑐

or

𝑙𝐷 𝑎 𝑙 𝑇 𝑏 𝑠c 𝑐 [ ] [ ] [ ] = constant 𝐿𝑖 𝐿𝑖 𝐿𝑖

[4.33]

where 𝐿𝑖 is the characteristic length of the microstructure, and the exponents 𝑎, 𝑏 and 𝑐 are constant under the condition that: 𝑎 + 𝑏 + 𝑐 = 1.

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Table 4.3 Microstructural Scales and Transitions (Trivedi Kurz, 1994) Plane-front perturbation wavelength at 𝑉𝑐

𝜆 ∝ (𝑠𝑐 𝑙𝐷 𝑙 𝑇 )1/3

Dendrite tip radius

𝑅 ∝ (𝑠𝑐 𝑙𝐷 )1/2

Primary dendrite spacing (low 𝑉)

1/2 1/2 𝜆1 ∝ (𝑅 𝑙 𝑀 = (𝑙𝐷 )1/4 (𝑠𝑐 )1/4 (𝑙 𝑀 𝑇) 𝑇)

Secondary dendrite spacing (at tip)

𝜆𝑜2 ∝ (𝑠𝑐 𝑙𝐷 )1/2

Secondary dendrite spacing (final)

1/3 𝜆2 ∝ (𝑠𝑐 𝑙𝐷 𝑙 𝑀 𝑇)

Limit of plane-front stability 𝑉𝑐 (low 𝑉 )

𝑙𝐷 = 𝑙 𝑇

Limit of plane-front stability 𝑉𝑎 (high 𝑉 )

𝑙𝐷 = 𝑘𝑠𝑐

Cell-to-dendrite transition (low 𝑉)

𝑙𝐷 = 𝑘𝑙 𝑇

𝑠𝑐 = Γ/Δ𝑇0 capillary length of alloy; 𝑙𝐷 = 𝐷/𝑉 solute diffusion length in liquid; 𝑙 𝑇 = 𝑎/𝑉 thermal diffusion length for an undercooled pure and alloy melt; 𝑙 𝑀 𝑇 = Δ𝑇𝑜 /𝐺 thermal length of mushy zone in directional growth of an alloy; 𝑘 distribution coefficient; Δ𝑇0 = 𝑇𝑙 − 𝑇𝑠 equilibrium freezing range of alloy.

Exercises 4.1

What will happen when the angle of the dendrite trunk axis in Fig. 4.4 (b) is at exactly 45∘ to the heatflow direction? Sketch a portion of such a dendrite growing under these conditions.

4.2

A microstructure such as that in Fig. 4.3 is being formed by directional solidification. The material is cubic and the cube axis is 20∘ away from the axis of the cells. What will happen when the growth rate is markedly increased?

4.3

What does monocrystalline mean in the case of Fig. 4.5 (no concentration variation in the solid, absence of low-angle boundaries, absence of high-angle boundaries)? Compare with the columnar zone of a casting (Fig. 4.2).

4.4

Design a mould which is suitable for the production of a dendritic monocrystalline casting, e.g. a gas turbine blade.

4.5

Sketch a sequence of transverse sections of a dendrite which illustrates the region between the tip and the root of the dendrite in Fig. 4.6.

4.6

Derive the temperature gradient in liquid, 𝐺𝑙 , for a hemispherical dendrite tip of a pure substance in terms of Péclet number and unit undercooling.

4.7

Imagine that the radius of curvature of a dendrite tip is changed during growth under a given set of conditions. What will happen to the concentration field of a columnar dendrite (Fig. 4.8) as the tip becomes sharper and sharper at a fixed growth rate. Note that Δ𝑇 = Δ𝑇𝑐 + Δ𝑇𝑟 in directional solidification. Sketch the concentration profile for the case where (a) Δ𝑇 > Δ𝑇𝑟 , (b) Δ𝑇 = Δ𝑇𝑟 , (c) Δ𝑇 < Δ𝑇𝑟 . Which situation corresponds to the critical nucleation radius? Which dendrite grows and which melts?

4.8

Calculate the alloy undercooling for the case of Fig. 4.9 which represents an Al − 2wt%Cu alloy solidifying under directional solidification conditions. From the value of 𝑅 ○ , determine the Gibbs-Thomson parameter, , used.

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The unoptimised dendrite growth behaviour can be expressed as, 𝑉 = f(𝑅) (Fig. 4.10) or as Δ𝑇 = g(𝑅). Derive such an equation for the Δ𝑇 of a hemispherical needle crystal and indicate the range of 𝑅-values over which diffusion, or capillarity, is governing growth. Calculate the extremum values for 𝑅 and Δ𝑇𝑐 (begin with Eq. A8.3).

4.10 Develop graphically the evolution of unit undercooling for the dendrite tip as a function of velocity 4.11 When the results of more exact models for directionally solidifying dendrites are compared with those of the simple model given in Table 4.2, the discrepancy between them is large at low and high growth rates. At both extremes the Péclet number becomes large (P greater than unity). Discover which of the simplifications made is responsible for the unrealistic predictions of the simple model. 4.12 Evaluate the value of 𝜎 ∗ in Εq. 3.22. 4.13 Calculate the tip temperature (Table 4.2) of a dendritic growth front in Al − 2%Cu alloy when 𝑉 = 0.1 mm/s and 𝐺 = 10 K/mm. Determine 𝜆1 , 𝜆2 , and the length of the mushy zone, assuming that 𝐺 is constant in that region and that, due to microsegregation, the melting point of the last liquid is 𝑇𝑒 What is the value of the ratio, Δ𝑇 ′ /Δ𝑇0 ? 4.14 How much is the typical maximum primary spacing larger than the mean spacing? Arguments. 4.15 A comparison of the three diffusional primary spacing models shows the same (Γ𝑘𝐷)0.25 Δ𝑇00.25 𝑉 −0.25 𝐺 −0.5 dependence. Develop the factors that differentiate these models and discuss their significance. Upon assuming that 𝜎 ∗ = 1/4π2 and 𝑘 = 0.1, evaluate these factors. What are the main differences between these models? Which one is the most physical? Under which conditions do these models apply and under which do they fail? 4.16 Discuss qualitatively the influence of convection perpendicular to the dendrite trunk axis on primary spacing. 4.17 Does a transverse flow modify the dendrite growth direction? Does it change the crystallographic orientation? Arguments. 4.18 What is the major mechanism of fragmentation? 4.19 By using the lower limit, 𝐶0 /𝑘, instead of 𝐶𝑙𝑚 in equation 4.26, a simplification which is realistic for many systems, show how 𝜆2 varies with 𝐶0 in a given alloy system. 4.20 Measurements of temperature and microstructure for an Al − 5wt%Si alloy casting gave the results listed below. Compare these values with the theoretical ones and estimate the cooling rate (use the constants for Al-6wt%Si in Appendix 15). 𝑡𝑓 (s) 𝜆2 (μm)

43 41  3

330 81  13

615 93  3

4.21 In experiments involving strong uniform flow of the melt perpendicular to the direction of growth of the solid/liquid interface (e.g. during the electromagnetic stirring of steel during continuous casting), it is observed that columnar dendrites are inclined in a direction which is opposite to the flow direction. How would you explain this observation? Consider the manner in which the boundary layers around the dendrite tip are altered. Would the same effect occur in pure metals?

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4.22 Stirring of the melt during solidification is an efficient method for promoting the columnar-to equiaxed transition in a casting. The reason for this transition in stirred castings is that the melt becomes rapidly isothermal and, at the same time, many dendrite branches are detached from the mushy zone. With the aid of Fig. 4.1 indicate the two limiting temperatures which the melt must have in order to make this transition possible. 4.23 For reasonably rapid phase-field computations, the solid-liquid interface thickness is increased. Which phenomenon has to be taken into account in the model for a nevertheless realistic simulation? 4.24 Show that the solution of Eq. A14.11 is given by Eq. A14.12 with the thickness of the diffuse interface being given by Eq. A14.13. Indications: (i) the expression 2ϕ(1 − ϕ)(1 − 2ϕ) in Eq. A14.11 is coming from 𝑑𝑔/𝑑ϕ, where 𝑔 = ϕ2 (1 − ϕ)2; (ii) 𝑑𝑔/𝑑ϕ can be written as (𝑑𝑔/𝑑𝑧)(𝑑𝑧/𝑑ϕ); (iii) integrate to obtain 𝑧(ϕ). 4.25 Using the stationary solution ϕ(𝑧) (Exercise 4.24), show that the solid/liquid interfacial energy is given by Eq. A14.14. 4.26 Estimate the parameters ℎ and ε appearing in the phase field equation, knowing that the solid/liquid interfacial energy is 0.1 J/m2 and that the diffuse interface δ has been chosen to be 0.1 m. How does this last selected parameter compare with the “real” thickness of a diffuse interface? Why has it be chosen to this value? How does this value of δ compare with the microstructural parameters (𝑅, λ1 , λ2 ) of Exercise 4.13?

References and Further Reading History ▪ M.Hillert, Importance of Clarence Zener upon metallurgy, Journal of Applied Physics 60 (1986) 1868. ▪ W.Kurz, D.J.Fisher, R.Trivedi, Progress in modelling solidification microstructures in metals and alloys: dendrites and cells from 1700 to 2000, International Materials Reviews, 64 (2019) 311. ▪ W.Kurz, M.Rappaz, R.Trivedi, Progress in modelling solidification microstructures in metals and alloys: dendrites from 2001 to 2018, International Materials Reviews, 66 (2021) 30. In Situ Observations of Dendrites ▪ H. Esaka, J. Stramke, W. Kurz, Columnar Dendrite Growth in Succinonitrile–Acetone Alloys, Video, EPFL, 1985. ▪ S.Gerardin, Etude expérimentale de la croissance libre de cristaux équiaxes en mouvement, PhD thesis, Institut Jean Lamour, INPL, Nancy, France (2002). ▪ T.Gong, Y.Chen, S.Li, Y.Cao, L.Hou, D.Li, X.Q.Chen, G.Reinhart, H.Nguyen-Thi, Equiaxed dendritic growth in nearly isothermal conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation, Materials Today Communications, 28 (2021) 102467. ▪ G.Hansen, A.Hellawell, S.Z.Lu, R.S.Steube, Some consequences of thermosolutal convection: the grain structure of castings, Metallurgical and Materials Transactions A, 27 (1996) 569.

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▪ A.J.Melendez, C.Beckermann, Measurements of dendrite tip growth and side branching in succinonitrile-acetone alloys, Journal of Crystal Growth, 340 (2012) 175. ▪ H.Neumann-Heyme, N.Shevchenko, J.Grenzer, K.Eckert, C.Beckermann, S.Eckert, In-situ measurements of dendrite tip shape selection in a metallic alloy, Physical Review Materials, 6 (2022) 063401. ▪ D.Ruvalcaba, R.H.Mathiesen, D.G.Eskin, L.Arnberg, L.Katgerman, In situ observations of dendritic fragmentation due to local solute-enrichment during directional solidification of an aluminum alloy, Acta Materialia, 5 (2007) 4287. Dendrite Growth Theory ▪ A.Barbieri, J.S.Langer, Predictions of dendritic growth rates in the linearized solvability theory, Physical Review A, 39 (1989) 5314. ▪ E.A.Brener, H.Müller-Krumbhaar, D.E.Temkin, Kinetic phase diagram and scaling relations for stationary diffusional growth, Europhysics Letters, 17 (1992) 535. ▪ M.H.Burden, J.D.Hunt, Cellular and dendritic growth. II, Journal of Crystal Growth, 22 (1974) 109. ▪ B.Cantor, A.Vogel, Dendritic solidification and fluid flow, Journal of Crystal Growth, 41 (1977) 109. ▪ J.A.Dantzig, M.Rappaz, Solidification, EPFL Press, Lausanne, 2016. ▪ P.K.Galenko, D.A.Danilov, Model for free dendritic alloy growth under interfacial and bulk phase nonequilibrium conditions, Journal of Crystal Growth, 197 (1999) 992. ▪ C.A.Gandin, J.L.Desbiolles, M.Rappaz, P.Thévoz, A three-dimensional cellular automaton– finite element model for the prediction of solidification grain structures, Metallurgical and Materials Transactions A, 30 (1999) 3153. ▪ M.E.Glicksman, R.J.Schaeffer, J.D.Ayers, Dendritic growth-a test of theory, Metallurgical Transactions A, 7 (1976) 1747. ▪ M.E.Glicksman, N.B.Singh, M.Chopra, in Materials Processing in the Reduced Gravity Environment of Space (G.E.Rindone, Ed.), Elsevier, 1982, p.461. ▪ G.Horvay, J.W.Cahn, Dendritic and spheroidal growth, Acta Metallurgica, 9 (1961) 695. ▪ J.J.Hoyt, A.Karma, M.Asta, D.Y.Sun, From atoms to dendrites, Journal of Metals, April (2004) 49. ▪ S.C.Huang, M.E.Glicksman, Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth, Acta Metallurgica, 29 (1981) 701. ▪ S.C.Huang, M.E.Glicksman, Overview 12: Fundamentals of dendritic solidification—II. Development of sidebranch structure, Acta Metallurgica, 29 (1981) 717. ▪ J.D.Hunt, S.Z.Lu, Numerical modelling of cellular/dendritic array growth: Spacing and structure predictions, Metallurgical and Materials Transactions A, 27 (1996) 611. ▪ G.P.Ivantsov, Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in an undercooled melt, Doklady Akademii Nauk SSSR, 58 (1947) 567. ▪ G.P.Ivantsov, On a growth of spherical and needle-like crystals of a binary alloy, Doklady Akademii Nauk SSSR, 83 (1952) 573. ▪ W.Kurz, B.Giovanola, R.Trivedi, Theory of microstructural development during rapid solidification, Acta Metallurgica, 34 (1986) 823.

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▪ J.S.Langer H.Müller-Krumbhaar, Theory of dendritic growth—I. Elements of a stability analysis, Acta Metallurgica, 26 (1978) 1681 and 1697. ▪ J.S.Langer, Instabilities and pattern formation in crystal growth, Reviews of Modern Physics, 52 (1980) 1. ▪ J.S.Langer , Microscopic Solvability, Physical Review A, 33 (1986) 435. ▪ J.S.Lee, T.Suzuki, Numerical simulation of isothermal dendritic growth by phase-field model, Iron and Steel Institute of Japan International 39 (1999) 246. ▪ G.C.Lemmlein, The process of geometric selection in a growing aggregate of crystals, Doklady Akademii Nauk SSSR, 48 (1945) 177. ▪ Q.Li, C.Beckermann, Modeling of free dendritic growth of succinonitrile-acetone alloys with thermo-solutal melt convection, Journal of Crystal Growth, 236 (2002) 482. ▪ J.Lipton, M.E.Glicksman, W.Kurz, Equiaxed dendrite growth in alloys at small supercooling, Metallurgical Transactions A, 18 (1987) 341. ▪ J.Lipton, W.Kurz, R.Trivedi, Rapid dendrite growth in undercooled alloys, Acta Metallurgica, 35 (1987) 957. ▪ H.Müller-Krumbhaar, J.S.Langer, Theory of dendritic growth—III. Effects of surface tension, Acta Metallurgica, 26 (1978) 1697. ▪ K.Oguchi, T.Suzuki, Free dendrite growth of Fe-0.5 mass%C alloy – three-dimensional phasefield simulation and LKT Model, ISIJ International, 47 (2007) 1432. ▪ S.Pan, M.Zhu, A three-dimensional sharp interface model for the quantitative simulation of solutal dendritic growth, Acta Materialia, 58 (2010) 340. ▪ A.Papapetrou, Untersuchungen über dendritisches Wachstum von Kristallen, Zeitschrift für Kristallographie, 92 (1935) 89. ▪ P.Pelcé (Ed.), Dynamics of Curved Fronts, Academic Press, 1988. ▪ R.Trivedi, Growth of dendritic needles from a supercooled melt, Acta Metallurgica, 18 (1970) 287. ▪ R.Trivedi, The role of interfacial free energy and interface kinetics during the growth of precipitate plates and needles, Metallurgical Transactions, 1 (1970) 921. ▪ R.Trivedi, Theory of dendritic growth during the directional solidification of binary alloys, Journal of Crystal Growth, 49 (1980) 219. ▪ R.Trivedi, W.Kurz, Dendritic growth, International Materials Reviews, 39 (1994) 49. ▪ X.Tong, C.Beckermann, A.Karma, Velocity and shape selection of dendritic crystals in a forced flow, Physical Review E, 61 (2000) R49. ▪ A.Viardin, Y.Souhar, M.C.Fernández, M.Apel, M.Založnik, Mesoscopic modeling of equiaxed and columnar solidification microstructures under forced flow and buoyancy-driven flow in hypergravity: Envelope versus phase-field model, Acta Materialia, 199 (2020) 680. ▪ C.Zener, Kinetics of the decomposition of austenite, Transactions of the Metallurgical Society of AIME, 167 (1946) 550. Cells ▪ B.Billia, R.Trivedi, in Handbook of Crystal Growth Vol. IB, D.T.J.Hurle (Ed.), North Holland, Amsterdam, 1993, Ch.14, p.899.

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▪ T.F.Bower, H.D.Brody, M.C.Flemings, Measurements of solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (1966) 624. ▪ M.H.Burden, J.D.Hunt, Cellular and dendritic growth. II, Journal of Crystal Growth, 22 (1974) 109. Dendrite Spacing Models/Experiments ▪ T.Cool, Coarsening of dendrites in solid-liquid mixtures under microgravity: experiments and phase field simulations, Ph.D. thesis, Northwestern University, Evanston, USA, 2016. ▪ T.Cool, P.W.Voorhees, The evolution of dendrites during coarsening: Fragmentation and morphology, Acta Materialia, 127 (2017) 359. ▪ M.D.Dupouy, D.Camel, J.J.Favier, Natural convective effects in directional dendritic solidification of binary metallic alloys: dendritic array primary spacing, Acta Metallurgica et Materialia, 40 (1992) 1791. ▪ U.Feurer, R.Wunderlin, Einfluss der Zusammensetzung und der Erstarrungsbedingungen auf die Dendritenmorphologie binärer Al-Legierungen, Fachbericht Deutsche Gesellschaft für Materialkunde (DGM), 1977. ▪ J.D.Hunt, Cellular and primary dendrite spacings, in Solidification and Casting of Metals, The Metallurgical Society, London, 1979, p.3. ▪ K.A.Jackson, J.D.Hunt, D.R.Uhlmann, T.P.Seward, Transactions of the Metallurgical Society of AIME, 236 (1966) 149. ▪ T.Z.Kattamis, M.C.Flemings, Dendrite morphology microsegregation and homogenization of low-alloy steel, Transactions of the Metallurgical Society of AIME, 233 (1965) 992. ▪ T.Z.Kattamis, J.C.Coughlin, M.C.Flemings, Influence of coarsening on dendrite arm spacing of aluminum-copper alloys, Transactions of the Metallurgical Society of AIME, 239 (1967) 1504. ▪ W.Kurz, D.J.Fisher, Dendrite growth at the limit of stability: tip radius and spacing, Acta Metallurgica, 29 (1981) 11. ▪ S.P.Marsh, M.E.Glicksman, Overview of geometric effects on coarsening of mushy zones, Metallurgical and Materials Transactions A, 27 (1996) 557. ▪ I.Steinbach, Effect of interface anisotropy on spacing selection in constrained dendrite growth, Acta Materialia, 56 (2008) 4965. ▪ T.Takaki, S.Sakane, M.Ohno, Y.Shibuta, T.Shimokawabe, T.Aoki, Primary arm array during directional solidification of a single-crystal binary alloy: large-scale phase-field study, Acta Materialia, 118 (2016) 230. ▪ R.Trivedi, Interdendritic spacing: Part II. A comparison of theory and experiment, Metallurgical Transactions A, 15 (1984) 977. ▪ R.Trivedi, W.Kurz, Solidification microstructures: a conceptual approach, Acta Metallurgica et Materialia, 42 (1994) 15. ▪ R.Trivedi, Iowa State University, Ames Iowa, USA, private communication, 2021. ▪ A.Viardin, M.Založnik, Y.Souhar, M.Apel, H.Combeau, Mesoscopic modeling of spacing and grain selection in columnar dendritic solidification: Envelope versus phase-field model, Acta Materialia, 122 (2017) 386. ▪ P.W.Voorhees, M.E.Glicksman, Solution to the multi-particle diffusion problem with applications to Ostwald ripening - I. Theory, Acta Metallurgica, 32 (1984) 2001 and 2013.

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▪ J.A.Warren, J.S.Langer, Prediction of dendritic spacings in a directional-solidification experiment, Physical Review E, 47 (1993) 2702. ▪ V.T.Witusiewicz, L.Sturz, A.Viardin, C.Pickmann, G.Zimmermann, Effect of convection on directional solidification in transparent succinonitrile–2.2wt.%(d)camphor alloy, Acta Materialia, 216 (2021) 117086. ▪ G.Zimmermann, A.Weiss, Directional solidification of dendritic microstructures in microgravity and with forced melt flow, Microgravity Science and Technology, 16 (2005) 145. Dendrite/Cell Spacing Measurements ▪ H.Esaka, W.Kurz, Columnar dendrite growth: Experiments on tip growth, Journal of Crystal Growth, 72 (1985) 578. ▪ S.H.Han, R.Trivedi, Primary spacing selection in directionally solidified alloys, Acta Metallurgica et Materialia, 42 (1994) 25. ▪ C.M.Klaren, J.D.Verhoeven, R.Trivedi, Primary dendrite spacing of lead dendrites in Pb-Sn and Pb-Au alloys, Metallurgical Transactions A, 11 (1980) 1853. ▪ D.G.McCartney, J.D.Hunt, Measurements of cell and primary dendrite arm spacings in directionally solidified aluminium alloys, Acta Metallurgica, 29 (1981) 1851. ▪ T.Okamoto, K.Kishitake, I.Bessho, Dendritic structure in unidirectionally solidified cyclohexanol, Journal of Crystal Growth, 29 (1975) 131. ▪ K.Somboonsuk, J.T.Mason, R.Trivedi, Interdendritic spacing: Part I. Experimental studies, Metallurgical Transactions A, 15 (1984) 967. ▪ K.Somboonsuk, R.Trivedi, Dynamical studies of dendritic growth, Acta Metallurgica, 33 (1985) 1051. Additive Manufacturing ▪ S.Mokadem, Epitaxial laser treatment of single crystal nickel-base superalloys, Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne, 2004 Phase-Field Modelling ▪ R.Berger, Access e.V., Aachen, Germany, Dendrite model using MICRESS version 6.4, 2018. Thanks to Markus Apel, Access Aaachen, for providing the photograph. ▪ W.J.Boettinger, J.A.Warren, C.Beckermann, A.Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002) 163. ▪ J.A.Dantzig, P.Di Napoli, J.Friedli, M.Rappaz, Dendrite growth morphologies in Al-Zn alloys—Part II: Phase-field computations, Metallurgical and Materials Transactions A, 44 (2013) 5532. ▪ E.Dorari, K.Ji, G.Guillemot, C.A.Gandin, A.Karma, Growth competition between columnar dendritic grains - The role of microstructural length scales, Acta Materialia, 223 (2022) 117395. ▪ B.Echebarria, R.Folch, A.Karma, M.Plapp, Quantitative phase-field model of alloy solidification, Physical Review E, 70 (2004) 061604. ▪ J.H.Jeong, J.A.Dantzig, N.Goldenfeld, Dendritic growth with fluid flow in pure materials, Metallurgical and Materials Transactions A, 34 (2003) 459.

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▪ A.Karma, Phase-field formulation for quantitative modeling of alloy solidification, Physical Review Letters, 87 (2001) 115701. ▪ J.S.Langer, Phase-field model, unpublished notes, Pittsburgh, 1978. Published as appendix in W.Kurz, D.J.Fisher, R.Trivedi, International Materials Reviews, 64 (2019) 350. ▪ N.Moelans, B.Blanpain, P.Wollants, An introduction to phase-field modeling of microstructure evolution, Calphad, 32 (2008) 268. ▪ T.Takaki, S.Sakane, M.Ohno, Y.Shibuta, T.Shimokawabe, T.Aoki, Primary arm array during directional solidification of a single-crystal binary alloy: large-scale phase-field study, Acta Materialia, 118 (2016) 230. Mathematical Functions ▪ M.Abramowitz, I.A.Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 109-132 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER FIVE

SOLIDIFICATION MICROSTRUCTURE: EUTECTIC AND PERITECTIC The solid which forms during solidification can adopt various morphologies (Fig. 1.10), and each of them can have a wide range of sizes. Dendrites make up the bulk of the microstructures of most alloys but a number of important eutectic or hypoeutectic alloys are also used in practice, such as the Al-Si cast alloys which are much used in automobile engines and wheels and in aeronautical components. Eutectic morphologies are characterised by the simultaneous growth of two (or more) phases from the liquid. Due to their excellent casting behaviour, which is often similar to that of a pure metal, casting alloys are frequently of near-eutectic composition. The advantageous composite properties which are exhibited, when two finely distributed crystalline phases having differing properties solidify in close contact to one another, are an additional reason for their frequent use. One common example is that of cast iron where lamellae of a solid lubricant, graphite, are incorporated into a steel matrix. 5.1. Regular and Irregular Eutectics In addition to the fact that eutectics are composed of more than one phase, those phases can also exhibit a variety of geometrical arrangements. With regard to the number of phases present, as many as four phases have been observed to grow simultaneously (e.g. Fisher and Kurz, 1974; Oquab et al., 2019). The vast majority of technologically useful eutectic alloys are however composed of just two phases. For this reason, only the latter type of eutectic will be considered here (Fig. 5.1). At high volume fractions of both phases, a situation which is found in a symmetrical phase diagram (e.g. Pb-Sn), there is a preference for the formation of lamellar structures. On the other hand, if one phase is present in a small volume fraction, there is a tendency to the formation of fibres of that phase (e.g. W in Ni – see the figure above). As a rule of thumb, and using the argument of minimal solid/solid interface, one can suppose that when the volume fraction of one phase is such that 0 < 𝑓 < 0.28 the eutectic will probably be fibrous (especially if both phases are of non-faceted type and an anisotropic solid/solid interfacial energy is not controlling the transition). If 0.28 < 𝑓 < 0.50 the *

Top image: Directionally solidified Ni-W eutectic: W fibres (mean spacing 25 μm) in a deeply-etched nickel-matrix forming an in situ composite (Kurz and Lux, 1971).

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eutectic will tend to be lamellar (Exercise 5.2). If both phases possess a low entropy of fusion the eutectic will exhibit a regular morphology. Due to anisotropic growth kinetics the fibres will become faceted if one phase has a high entropy of fusion or if interfaces having a pronounced minimum energy exist between the two phases (Fig. 5.1). In this case the eutectic morphology often becomes irregular and this is particularly true of the two eutectic alloys of greatest practical importance: Fe-C (cast iron) and Al-Si (Silumin). These two eutectics predominantly form lamellae (graphite in Fe-C and silicon in Al-Si) even if the volume fractions of the second phases are much lower than 0.28. The marked anisotropy of growth of the faceted phase and of its solid/solid interface energy here play the dominant role†.

Figure 5.1 Types of Binary Eutectic Morphologies (Cross Section) Eutectic microstructures can be fibrous or lamellar, regular or irregular. The condition is that one phase (here the white -phase) must always have a low entropy of fusion, so that 𝛼𝛼 < 2. If both phases possess a low entropy of fusion their growth is easy in all crystallographic directions (Sect. 2.3) and the resultant structures are regular (non-faceted/non-faceted eutectic; upper part of the figure). When the low volume fraction phase possesses a high entropy of melting, as do semiconductors, graphite and intermetallic compounds for example, the eutectics are then of nonfaceted/faceted type and the microstructures are usually irregular. The important eutectic casting alloys, Fe-C and Al-Si, belong to the latter class (lower right of the figure). Fibres are usually the preferred growth-form when a small volume fraction of one phase is present, especially in the case of non-faceted/non-faceted eutectics. If the growth behaviour and the specific interface energy between the two solid phases is very anisotropic however, lamellae may also be formed at a low volume fraction, 𝑓 (as in Fe-C, where 𝑓C = 0.07).

The regularity of an eutectic has a marked effect upon its properties and this becomes especially important when it is required to control the orientation of the phases in order to obtain what are known as in situ composites. These are alloys where, by using a controlled heat flux to impose directional solidification (Fig. 1.4), the eutectic can be caused to grow in a well-aligned manner and a composite containing many kilometres of fibres per cm3 of material can be formed (Figure on first page of this chapter). †

Nodular graphite in ductile cast iron is an exception to the rule for non-faceted/faceted growth. In this case the faceted graphite phase develops spheres (formed of radial [0001] columns) which, in a second step, are surrounded by austenite shells and dendrites without forming a coupled growth front. Such a behaviour is called divorced growth. Also, if both phases are faceted the coupled (side-by-side) growth which is covered in the present chapter is not observed.

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Because eutectic alloys exhibit small interphase spacings which are typically of the order of the dendrite tip, i.e. 10 to 100 times smaller than primary dendrite spacings under similar conditions of growth, a large total interfacial area exists between the two solid phases. For a 1 cm3 cube, this area is typically of the order of 1 m2. The specific energy of the interface is moreover usually high and increases with increasing dissimilarity of the phases. As a result, there is a tendency for certain, lowest-energy, crystallographic orientations to develop between the phases and thereby minimise the interfacial energy. Extensive experimental studies have been made of the crystallography of eutectic alloys and the results have been reviewed by Hogan et al. (1971). Table 5.1 indicates the results of several systems. A list of some 300 directionally solidified eutectics can be found in Kurz and Sahm (1975). Table 5.1 Crystallography of Eutectic Alloys Eutectic

Growth Directions

Parallel Interfaces

Ag-Cu

[110]Ag || [110]Cu

(211)Ag || (211)Cu

Ni-NiMo

[11̅2]Ni || [001]NiMO

(110)Ni || (100)NiMo

Pb-Sn

[211]Pb || [211]Sn

(11̅1̅)Pb || (01̅1̅)Sn

Ni-NiBe

[112]Ni || [110]NiBe

(111)Ni || (110)NiBe

Al-AlSb

[110]Al || [211]AlSb

(111)𝐴𝑙 || (111)AlSb

5.2. Diffusion-Coupled Growth In order to model the growth behaviour of the two eutectic phases the simplest morphology will be assumed for the solid/liquid interface; i.e. that which exists during the growth of a regular lamellar eutectic with 𝑓 = 0.5. In this case the problem can be treated in two dimensions and, for reasons of symmetry (Appendix 10), only half of a lamella of each phase needs to be considered (Fig. 5.2). In this figure the alloy is imagined to be growing directionally in a crucible which is being

Figure 5.2 Phase Diagram and Regular Eutectic Structure in Directional Solidification The figure shows an eutectic phase diagram and a regular lamellar two-phase eutectic morphology of mean composition Ce growing unidirectionally in a positive temperature gradient. The - and - lamellae grow side-by-side and are perpendicular to the solid/liquid interface. The form of the trijunction where the three phases (, , liquid) meet is determined by the condition of mechanical equilibrium. In order to drive the growth front at a given rate, 𝑉, an undercooling, Δ𝑇, is necessary. Due to the perfection and symmetry of the regular structures only a small volume element of width, 𝜆/2, needs to be considered in order to characterise the behaviour of the whole interface under steady-state conditions.

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moved vertically downwards through a Bridgman apparatus at a rate, 𝑉′ (Fig. 1.4(a)). In a steadystate thermal environment this is equivalent to moving the solid/liquid interface upwards at a rate of 𝑉 = 𝑉′. The alloy of eutectic composition is growing with its interface at a temperature, 𝑇 ∗ = 𝑇𝑒 − Δ𝑇 below the equilibrium eutectic temperature. Due to the high thermal conductivity of metals and the small dimensions of the phases the interface is essentially isothermal. The /-phase interfaces are perpendicular to the solid/liquid interface and parallel to the growth direction. In order to proceed further, it is necessary to know more about the mass transport which is involved. It can be seen from the phase diagram that the concentrations of the two solid phases are very different while the melt concentration, 𝐶𝑒 , is intermediate in value. In the steady state the mean composition of the solid must obviously be equal to the composition of the melt. This makes it clear that eutectic growth is largely a question of diffusive mass transport. Imagine firstly that the two eutectic phases are growing separately from the eutectic melt with a planar solid/liquid interface (Fig. 5.3(a)). During growth the solid phases reject elements having a lower solubility into the liquid. The -phase will thus reject B-atoms into the melt while the -phase will reject A-atoms. Note here that when the concentration is expressed as a fraction, 𝐶B = (1 − 𝐶A ). When the phases are supposed to be growing separately with a planar front, solute transport must occur in the direction of growth. This involves long-range diffusion and, in the steady-state, the solute distribution is described by the exponential decay discussed in Chap. 3, with a boundary layer,

Figure 5.3 Eutectic Diffusion Field If it is imagined that the two eutectic phases are growing from a melt of eutectic concentration in separate adjacent containers (a) very large boundary layers, like that in Fig. 3.4, will be created. If the two containers are now brought together and the intervening wall is removed (b), extensive lateral mixing will take place because of the concentration jump at the / interface. The large boundary layers of the planar interfaces of (a) (approximately equal to 2𝐷/𝑉) are now replaced by a narrower layer whose thickness is of the order of magnitude of the phase separation, 𝜆. This marked change in the extent of the boundary layer is due to the diffusion flux which is established at, and parallel to, the eutectic solid/liquid interface and permits the rejection of solute by one phase to be balanced by incorporation of the solute into the other phase and vice versa (diffusion coupling). The interface concentration in the boundary layer varies periodically, by a small amount, about the eutectic concentration and the amplitude of the variation will decrease as 𝜆 decreases when 𝑉 is constant. The lateral concentration gradients create free-energy gradients which exert a 'compressive' force perpendicular to the /- interface and tend to decrease 𝜆. The corresponding phase diagram has been placed next to the solid/liquid interface in such a way that the local phase equilibria can be determined. It can be seen that the amplitude of the concentration 𝛽 variation, 𝐶𝑙 − 𝐶𝑙𝛼 , in the liquid at the solid/liquid interface is proportional to a maximum solute undercooling, Δ𝑇𝑐max . (In this figure, curvature effects at the interface are not considered).

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𝛿𝑐 = 2𝐷/𝑉. Such a long-range diffusion field will involve a very large solute or solvent build-up and a correspondingly low (much lower than 𝑇𝑒 ) growth temperature of the interface. During steady-state growth each phase would have the interface temperature indicated by the corresponding metastable solidus line when extended as far as the eutectic concentration (see Fig. 3.4). Imagine now that both phases are placed side-by-side and that the solid/liquid interfaces are at the same level (Fig. 5.3(b)). Lateral diffusion will immediately set in and decrease the high lateral concentration gradient. This situation is much more favourable than separate plane-front growth since the solute which is rejected by one phase is needed for the growth of the neighbouring phase. A periodic diffusion field will be established and it will couple the growth of both phases. Because the maximum concentration differences at the interface (compared to the eutectic concentration) are much smaller than in the case of single-phase growth, the temperature of the growing interface will be close to the equilibrium eutectic temperature. The proximity of the lamellae, while making diffusion easier, also causes however an increased undercooling due to capillarity effects (Fig. 5.4).

Figure 5.4 Curvature Effects at the Eutectic Interface The diffusion field causes the eutectic spacing, 𝜆, of the structure to be minimised and this leads to more rapid growth. There is an opposing effect which arises from the increased energy that is associated with the increased curvature (∝ 𝜆−1) of the solid/liquid interface as 𝜆 decreases. The latter can be expressed in terms of a curvature undercooling, Δ𝑇𝑟 , which depresses the liquidus lines of the equilibrium phase diagram as shown on the top right. The positive curvature of the solid phases in contact with the liquid arises from the condition of mechanical equilibrium of the interfacial forces at the trijunction (lower figure, see also Appendix 3). If the interface between the two eutectic phases is locked to specific crystallographic planes, in order to minimize the  interfacial energy, it might not grow parallel to the intended growth direction and a torque term then appears (Akamatsu et al., 2012).

Both effects, diffusion and capillarity, are considered together in Fig. 5.5. In (a) the flux-lines of the diffusion field of B atoms in the liquid close to the interface are shown schematically. These are most densely packed (higher flux) at points near to the interface. They rapidly become less significant as the distance from the interface increases. The characteristic decay distance for the lateral diffusion is of the order of half the interphase spacing, 𝜆. Note that the diffusion paths for the other species, passing in the opposite direction, are analogous. According to the phase diagram, the concentration variation at the solid/liquid interface (Fig. 5.5(b)) leads to a change in the liquidus

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temperature of the melt, 𝑇𝑙∗ , in contact with the phases (curves in Fig. 5.5(c)). The points at which the liquid concentration, 𝐶B∗ , is equal to 𝐶𝑒 are exactly at the eutectic temperature while those points of the -phase which are close to the / interface are at a higher liquidus temperature because the liquid in these regions has a lower content of B, as determined by the lateral diffusion field. On the other hand, the melt ahead of the -phase is always richer in A than is the equilibrium eutectic. Its liquidus temperature is therefore lower, as compared to the equilibrium eutectic temperature, and decreases with increasing values of 𝐶A (these relationships can be understood with the aid of the phase diagram – see Exercises 5.7 and 5.8). The capillarity undercooling, Δ𝑇𝑟 , also shown in Fig. 5.5(c) (hatched region), will be discussed below.

Figure 5.5 Eutectic Interface Concentration and Temperature During growth the diffusion field of component B will be as shown in diagram (a). The concentration in the liquid at the interface will vary as in diagram (b). Note that the eutectic concentration is not necessarily found at the junction of the two phases and that 𝐶B = 1 − 𝐶A . This sinusoidal concentration variation decays rapidly over one interphase spacing, in the direction perpendicular to the solid/liquid interface, as shown in Fig. 5.3(b). The balance between an ‘attractive’ force arising from the concentration gradient, and a ‘repulsive’ force between the three-phase junctions arising from capillarity effects at small 𝜆 determines the eutectic spacing. Under normal solidification conditions the growing interface can be regarded as being in a state of local thermodynamic equilibrium. This means that the measurable temperature, 𝑇𝑞∗ , of the interface which is constant along the solid/liquid interface (over 𝜆/2) corresponds to equilibrium at all points of the interface. The latter is a function of the local concentration and curvature (c). The sum of the solute (Δ𝑇𝑐 ) and curvature (Δ𝑇𝑟 ) undercoolings must therefore equal the interface undercooling, Δ𝑇. A negative curvature, as shown here at the centre of the  lamella, is required when the solute undercooling, Δ𝑇𝑐 is locally higher than Δ𝑇. The discontinuity in the solute undercooling, as the /-interface is crossed, is only a discontinuity in equilibrium temperature and is not a real temperature discontinuity.

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In order to determine the solute distribution, a flux balance (Appendix 2) must first be applied at the interface. In the present order-of-magnitude calculation it is assumed that the interface is planar, that both volume fractions are equal and that the interface concentration variation of Fig. 5.5(b) can be approximated by using a saw-tooth waveform (constant concentration gradient between the 𝛽 midpoints of the lamellae) with amplitude, Δ𝐶 = 𝐶𝑙𝛼 − 𝐶𝑙 , and a diffusion distance in the y-direction of 𝜆/2 (Fig. 5.2). In this way, the concentration gradient in the liquid at the solid/liquid interface is found to be: (𝑑𝐶/𝑑𝑦)z=0 = −Δ𝐶/(𝜆/2). Note that this concentration variation decays very quickly ahead the interface, i.e. the amplitude Δ𝐶 → 0 when 𝑧 ∼ 𝜆/2 and (𝑑𝐶/𝑑𝑦)𝑧=𝜆/2 → 0. The mean concentration gradient leading to lateral solute diffusion in a narrow liquid band with a thickness of about 𝜆/2, which is adjacent to the solid/liquid interface and perpendicular to the / interface, is therefore half that of the interface value given above. That is, 𝑑𝐶/𝑑𝑦 = −Δ𝐶/𝜆. The transverse flux, 𝐽𝑡 , within this boundary layer is then: 𝐽𝑡 = 𝐷

ΔC (ℎ𝜆/2) 𝜆

[5.1]

The flux is weighted here with the boundary-layer cross-section, ℎ𝜆/2, and has the dimensions [mole/s] if the concentration field, 𝐶, is given in mole/m3. This flux, which is perpendicular to the growth direction, is needed in order to redistribute the solute that is rejected during the crystallisation of solid having a lower concentration than that of the liquid. The rejected flux (in the 𝑧-direction), 𝐽𝑟 , for a symmetrical phase diagram and equal volume fractions of both solid phases, i.e. of one phase of half the width of the symmetry element and height, ℎ, is: 𝐽𝑟 = 𝑉𝐶𝑙∗ (1 − 𝑘) (ℎ𝜆/4) Assuming that under normal solidification conditions the interface concentration deviates only slightly from the eutectic concentration, i.e. Δ𝐶 is typically equal to a fraction of a percent and 𝐶𝑒 is of the order of 50%, one can assume that 𝐶𝑙∗ ≅ 𝐶𝑒 and the rejected flux becomes: 𝐽𝑟 = 𝑉𝐶𝑒 (1 − 𝑘) (ℎ𝜆/4)

[5.2]

Under steady-state conditions the flux balance, 𝐽𝑡 = 𝐽𝑟 , can then be written: Δ𝐶 𝜆𝑉 = 𝐶𝑒 (1 − 𝑘) 2𝐷

[5.3]

which is, in fact, entirely analogous to the previously presented equation for the diffusional growth of a hemispherical needle (Eq. 4.5). The left-hand-side of Eq. 5.3 corresponds to a supersaturation while the right-hand-side is the Péclet number for eutectic growth. One can therefore also write Eq. 5.3 in the form: Ωe = 𝑃𝑒

[5.4]

The concentration difference, Δ𝐶 (= Δ𝐶𝛼 + Δ𝐶𝛽 ) which is required in order to drive solute diffusion during eutectic growth, can be used to determine a temperature difference (undercooling) from the phase diagram (Fig. 5.6), via the liquidus slopes, Δ𝐶𝛼 = Δ𝑇𝑐 /(−𝑚𝛼 ), Δ𝐶𝛽 = Δ𝑇𝑐 /𝑚𝛽 and Δ𝐶 = Δ𝑇𝑐 [1/(−𝑚𝛼 ) + 1/𝑚𝛽 ], leading, via Eq. 5.3, to a relationship of the form:

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Δ𝑇𝑐 = 𝐾𝑐 𝜆𝑉

[5.5]

where 𝐾𝑐 is a constant (see Appendix 10). From Eqs 5.4 or 5.5, it can be seen that this problem is not completely solved because, as in the case of dendritic growth, the above equations apply equally well to a fine structure growing at high rates or a coarse eutectic growing at low rates.

Figure 5.6 Contributions to the Total Undercooling in Eutectic Growth Using Fig. 5.5(c), a mean solute undercooling and a mean curvature undercooling can be defined. Both undercoolings vary in opposite senses when the spacing is changed. (The situation of Fig. 5.6(a) is shown for an arbitrary spacing, 𝜆′). Here Δ𝑇𝑐 , which is proportional to Δ𝐶 (driving diffusion) increases while Δ𝑇𝑟 decreases with increasing spacing (b). Note that Δ𝑇 is measured downwards with respect to 𝑇𝑒 . The sum of the contributions exhibits a minimum in Δ𝑇 or a maximum in 𝑇 ∗ with respect to 𝜆 (red curve). At smaller spacings eutectic growth is controlled by capillarity effects (Δ𝑇𝑟 > Δ𝑇𝑐 ). At larger spacings diffusion is the limiting process. It is generally found in experiments, unlike the case of dendrites, that growth will occur at the extremum, 𝜆𝑒 . An increase in the growth rate increases the absolute value of the slope of the Δ𝑇𝑐 line, without influencing the Δ𝑇𝑟 curve, and displaces the maximum of the 𝑇 − 𝜆 curve to smaller spacings.

5.3. Capillarity Effects Returning to the periodic concentration-variation which exists ahead of the solid/liquid interface (Fig. 5.5), it can be seen that the corresponding liquidus temperature varies from values greater than 𝑇𝑒 , for certain regions of the -phase, to values below the actual interface temperature, 𝑇𝑞∗ for the central region of the -phase. The difference (hatched region in Fig. 5.5(c)) has to be compensated by the local curvature in order to maintain local equilibrium at the interface. Since 𝑇𝑞∗ is constant as presented above: Δ𝑇 = Δ𝑇𝑐 + Δ𝑇𝑟 = 𝑇𝑒 − 𝑇𝑞∗ = constant‡

[5.6]

A negative curvature (depression) may appear at the centre of a lamella in order to compensate for a high local solute-controlled interface undercooling which is often associated with a large spacing, 𝜆. At the //l three-phase junction, the / interface energy has to be balanced by the sum of components of the /l- and /l-interface energies (Fig. 5.4). The angles at the three-phase junction are thereby determined by considerations of mechanical equilibrium (Appendix 3).

‡

As in the case of directional dendrite growth, the thermal undercooling is zero during directional eutectic growth because heat is flowing into the solid (𝐺 > 0).

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The curvature of the /l- or /l-interface, which is necessary in order to match the angles at the trijunction (Fig. 5.4), changes the equilibrium temperature by an amount, Δ𝑇𝑟 , which is a function of 𝑦 (Fig. 5.5(c)). By calculating the mean curvature over the /l- and /l-interfaces (Appendix 10), the effect of capillarity can be related to a mean change in the liquidus temperature by (Eq. 1.5): Δ𝑇𝑟 = Γ𝐾 Because the curvature, 𝐾, is proportional to 1/𝜆: [5.7]

𝐾 = 𝐾𝑟 /𝜆

where 𝐾𝑟 is a constant. Use of Eqs 5.5 to 5.7 leads to the following relationships for the total solid/liquid interface undercooling: Δ𝑇 = 𝐾𝑐 λ𝑉 +

𝐾𝑟 λ

[5.8]

A relationship is thus obtained which exhibits a maximum in the growth temperature (equivalent to a minimum in Δ𝑇) as a function of 𝜆 (Fig. 5.6), where the growth rate is imposed and constant and Δ𝑇 is the dependent variable. Constant values of the growth rate are typical of directional solidification. On the other hand, the undercooling is imposed and constant in the initial stages of equiaxed solidification, and this results in a maximum in the (𝑉 − 𝜆) curve (Fig. 5.7).

Figure 5.7 Optimisation of the Eutectic Spacing A range of possible spacings exists, each of which satisfies local equilibrium requirements. This situation is described by the 𝑇 − 𝑉 − 𝜆 surface of this figure (linear coordinates) which forms a valley (thin dashed line) running from the left rear to the front right of the diagram. (The 𝑇 axis is reversed here with respect to the equivalent diagram in Fig. 5.6(b)). It is seen that if the growth rate is constrained (𝑉 = constant), as in directional solidification, a minimum in the 𝑇 − 𝜆 curve (e.g. at point A of the red dashed curve) is obtained, corresponding to the maximum in the 𝑇 − 𝜆 of Fig. 5.6(b). If 𝑇 is maintained constant, as in equiaxed (isothermal) growth, the 𝑉 − 𝜆 curve (continuous red curve) exhibits a maximum at point A. This curve for the plane-front growth of an eutectic is analogous to that in Fig. 4.9 for dendrites, where 𝑅 replaces 𝜆. The spacings which correspond to 𝑇𝑚𝑖𝑛 or 𝑉𝑚𝑎𝑥 are called 'extremum' or 'optimum' spacings and, in regular eutectics, correspond closely to experimentally determined values. (After Shingu, 1979).

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5.4. Operating Range of Eutectics Upon considering Eq. 5.8 it becomes clear that 𝑇 is not uniquely determined since it is a function of the product, 𝜆𝑉. Another equation is therefore required in order to determine the growth behaviour of an eutectic. This situation is analogous to that existing in dendrite growth. In the case of dendrites the assumption that growth at the limit of morphological stability is the operating point has been found to be consistent with experimental results and with microscopic solvability theory. In the case of eutectic growth, both the extremum point (for nf-nf eutectics), and also an operating point which is analogous to morphological stability (for nf-f systems), have been found to explain the experimental results for various alloys. Eutectic alloys which grow in a regular nf-nf fashion (e.g. PbSn) can be described well by using the extremum criterion (see also the discussion in Seetharaman and Trivedi, 1988). Under this assumption the first derivative of Eq. 5.8 is set equal to zero: 𝑑(Δ𝑇) =0 𝑑𝜆

[5.9]

Since Δ𝑇 = −Δ𝐺/Δ𝑆𝑓 the condition that Δ𝑇 should be a minimum implies that 𝑑(Δ𝐺)/𝑑𝜆 = 0 and means that the driving force for spacing changes is zero. Insertion of the corresponding value of the spacing leads to the final result for growth at the extremum: 𝜆2 𝑉 =

𝐾𝑟 𝐾𝑐

[5.10]

Δ𝑇 = 2(𝐾𝑟 𝐾𝑐 )1/2 𝑉 1/2

[5.11]

Δ𝑇𝜆 = 2𝐾𝑟

[5.12]

Figure 5.8 Eutectic and Eutectoid Mean Spacings as a Function of Growth Rate If the optimum (extremum) spacings (Fig. 5.7) are determined for a range of growth rates (projection of the thin dashed curve on the 𝜆 − 𝑉 surface in Fig. 5.7), straight lines are found in log-log plots, such as the red one above; the spacing versus growth rate relationship can be described by 𝜆2 𝑉 = constant. The eutectoid microstructure resembles that of an eutectic, as often does its growth law, but diffusion is slower because it occurs only through solid phases. There is thus a tendency to decrease the diffusion distances by decreasing the spacing, and the values of the latter are therefore smaller than those of regular eutectics. Irregular eutectics such as Fe-C and Al-Si do not appear to grow at the extremum but rather at larger values of 𝜆, and the larger spacings with respect to those of regular eutectics can be attributed to the branching difficulties of the faceted phase.

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Equation 5.10 is illustrated in Fig. 5.8 for two types of eutectics and for eutectoids. Note that the spacing is not uniquely defined; it is instead a relatively narrow distribution around a mean value which is determined by the above equations. It has been shown that the α/β/l trijunctions are not only free to move in the growth direction (as discussed above – dashed black arrows in Fig. 5.9) but can also move laterally (red arrows). This additional freedom of movement has a stabilising effect upon the squeezed β lamella in the centre of the figure, which also allows the interface to grow somewhat below the extremum where 𝑑𝑇/𝑑𝜆 > 0; an effect which is larger, the greater the value of 𝐺/𝑉.

Figure 5.9 Eutectic Interface Showing the Mobility of the α/β/l Trijunctions The mobility of the trijunctions determines the operating regime. The dashed black arrows correspond to the behaviour of the operating point in the above discussion. In addition, there is a lateral mobility of the trijunctions (red arrows) which can stabilise a lagging, and otherwise doomed, β-phase in a concave solid/liquid interface and extend the operating regime to below the extremum (Karma and Plapp, 2004).

The situation is more complex when irregular (nf-f) eutectics are considered. In this case a large variation of the spacings is observed; from a minimal spacing corresponding to a point close to the extremum value, to large spacings that can be explained by the difficulty which this class of eutectic experiences in branching. The mean spacing is therefore much larger than that for regular (nf-nf) eutectics (Fig. 5.8). Branching is an essential mechanism which permits the eutectic to adapt its scale to the local growth conditions and to approach the extremum point. If due to its atomic structure or planar defect-growth mechanism the facetted phase (e.g. C in Fe-alloys (Fig. 2.16c) or Si in Al-alloys (Fig. 2.16b)) cannot easily change its growth direction, some of the lamellae of this phase will diverge and increase its thickness until the point of branch-formation is reached. This behaviour can be understood with the aid of Fig. 5.10, which is a 2D approximation of the 3D reality. When two adjacent lamellae diverge the interface of the larger volume fraction phase will first become depressed because of the consequent increase in solute concentration at its centre (Fig. 5.5(a) and lower centre insert of Fig. 5.10). As the solute builds up more and more at the interface of the thickening diverging phases, the growth undercooling increases until the curvature can no longer compensate for the change and the interface becomes non-isothermal. The diverging phases will finally reach a spacing which is so large that even the low volume fraction phase will exhibit depressions at its solid/liquid interface (lower right insert). Under these conditions a single lamella may branch into two. When a new lamella has been created, it will usually diverge from its partner and tend to converge on neighbouring lamellae at the interface (Fig. 5.11). But because the spacing is now decreasing the interface temperature will increase, due to the decreasing solute build-up, and eventually reach the maximum in temperature (minimum undercooling). As the faceted phase cannot easily change its growth direction, its growth will decrease the local spacing to a value which is below the extremum value. Smaller spacing values will however appreciably decrease the temperature due to the steep capillarity-controlled slope of the curve. As a result, spacings which are smaller than the extremum value will tend to disappear (Fig. 5.10, lower left insert).

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Figure 5.10 A 2D Mechanism for Spacing Variation in Irregular Eutectics This figure illustrates the interface morphologies which may occur due to convergent and divergent growth of the minor faceted phase of an irregular eutectic. In this case the faceted phase can grow only along a planar growth defect (Fig. 2.16(b,c)), thus making changes in the growth direction of the faceted phase and therefore in the local spacing, very difficult. In convergentgrowth regions, the spacing will decrease during growth, thereby increasing the curvature of both phases. This will lead to the cessation of growth in that region because the interface temperature drops at the left of the extremum, 𝜆𝑒 , due to an increasing Δ𝑇𝑟 value. The mechanism of Fig. 5.9 allows the spacing of convergent lamellae to go slightly below the value of the extremum. In divergent-growth regions, the spacing becomes larger during growth, leading to increased solute pile-up ahead of both phases. This first leads to the formation of depressions in the major phase (Fig. 5.5), and later to depressions also in the minor phase. When the minor phase becomes depressed, branching (formation of two lamellae from one) is possible at 𝜆𝑏 , and the spacing will again decrease. This leads to zig-zag growth of the faceted phase between 𝜆𝑒 and 𝜆𝑏 . This is a simplified 2D-model. In reality the growth is 3D and branching can also occur out of the plane of the figure (Fig. 5.12).

The range of stable eutectic growth is thus confined between the extremum value, e, and the branching spacing, 𝜆𝑏 . Only those eutectics which experience branching difficulties will explore the entire range and this can explain their coarse spacings (Fig. 5.8), their correspondingly large undercoolings and their large spacing-variations (i.e. irregularity). It has been proposed (Jones and Kurz, 1981) to use, for microstructure-characterisation, the arithmetic mean of the minimum and maximum spacings. In theoretical work, one can define 𝜆mean = (𝜆𝑒 + 𝜆𝑏 )/2 and a ratio: φ=

λmean  λ𝑒

[5.13]

which allows the use of Eqs 5.10 and 5.11 as: 𝜆2 𝑉 = 𝜑

𝐾𝑟 𝐾𝑐

Δ𝑇 = 2(φ + 𝜑 −1 )(𝐾𝑟 𝐾𝑐 )1/2 𝑉 1/2

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[5.14] [5.15]

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Figure 5.11 2D Growth Model of Irregular Eutectics For nf-f eutectics the difficulty (stiffness) experienced in smoothly changing the growth direction of certain faceted phases results in the growth structure of the simple 2D model. The large spacings of divergent lamellae here lead to a non-isothermal nature of the solid/liquid interface, and the resultant microstructures are irregular. Common examples of this are the eutectics, Fe-C and Al-Si, upon which most commercial casting alloys are based. The magnified inset shows schematically the atomic processes which are occurring: on the left is the atomically diffuse easily-growing interface of the non-faceted -phase (Fig. 2.14(b)), and on the right is the defectconstrained growth of the faceted -phase along a specific crystal plane (e.g. Si in Al-Si eutectic, Fig. 2.16(b)). A planar growth defect is needed for the faceted phase to grow under a lower driving force and thus keep up with α at the trijunction at, however, the expense of a marked "stiffness".

Figure 5.12 Lateral Branching of Graphite Lamellae in the 3D-Growth of nf-f Eutectic Graphite lamellae in deeply etched Ni-C eutectic, showing branching in the 3rd dimension (instead of within the lamella’s width). Each branch bends slightly out of the plane of the micrograph of the lamella, thus forming a new lamella which adjusts itself to the appropriate spacing (Lux et al., 1969).

The branching-point can be calculated by using a stability analysis which is analogous to the criterion used in the case of dendritic growth (Magnin and Kurz, 1987). This model of irregular growth is applicable to thin-film growth between two glass slides (2D) where branching in the third

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dimension is not possible (Faivre and Mergy, 1992). Branching can in reality occur not only within the limited thickness of the minor phase, but also in the third dimension, the larger width of the lamellae (Fig. 5.12) thereby reducing the maximum spacing (a model to be developed). 5.5. Lamellae-Fibre Transition Another important phenomenon is the lamellae-fibre transition. This phenomenon has already been described by Mollard and Flemings (1967) for the Pb-Sn eutectic (Fig. 5.13). In order to obtain an initial estimate for this transition, one can compare the interface energies of both morphologies (as has been discussed above); the eutectic having the smaller interface energy will be formed. Considering at first only the interface area, 𝐴, at 𝜆 = constant the total interface energy that is attributed to the surfaces between the phases, decreases with decreasing volume fraction of cylindrical fibres according to 𝐴 ∝ f ½, while the interface area for lamellae remains constant. The interface area of fibres is thus lower than that for lamellae at volume fractions which are smaller than 0.28 and a transition from lamellae to fibres should occur at this volume fraction. This model neglects however the difference in the diffusion fields of the two morphologies (see Jackson and Hunt, 1966) and possible low-energy interface orientations (deep cusps in the solid/solid interface energy) which can stabilize lamellar morphologies down to low 𝑓𝛽 values.

Figure 5.13 Lamella to Fibre Transition in Sn-Pb Alloy Below the Pb-Sn phase diagram with its eutectic at 26.1 at% Pb are two transverse microstructures of alloys directionally solidified using a high 𝐺/𝑉 ratio; (a) 24.8 at% Pb (near-eutectic), (b) 12.6 at% Pb (hypo-eutectic). The effect of decreasing the volume fraction of Pb (black phase) upon the lamellae to fibre transition is clearly visible. The width of each micrograph is 280 μm. (According to Mollard and Flemings, 1967).

In the model developed by Liu et al. (2011), the fibre-to-lamellae transition depends upon the interface-energy parameter of the Jackson-Hunt equation. Writing Eq. 5.8 (or Eq. A10.28) for the eutectic growth undercooling of both morphologies, where the subscript 𝑖 = 𝐹 or 𝐿 represents fibrous or lamellar eutectic, one has: 𝐾𝑟𝑖 𝜆 Expressions for 𝐾𝑐𝑖 and 𝐾𝑟𝑖 are given by (Appendix 10): Δ𝑇𝑖 = 𝐾c𝑖 𝑉𝜆 +

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𝐾c𝑖 =

𝑚 ‾ 𝐶 ′ 𝑃𝑖′ 𝑓(1 − 𝑓)𝐷

𝑆𝑖 = [

Γ𝛼𝑖 sin (𝜃𝛼𝑖 ) Γ𝛽𝑖 sin (𝜃𝛽𝑖 ) + ] 𝑓|𝑚𝛼 | (1 − 𝑓)|𝑚𝛽 |

and

𝐾𝑟𝑖 = 2𝑚 ‾ 𝛿𝑖 𝑆𝑖

123

[5.16]

[5.17]

with 𝑚 ‾ = |𝑚𝛼 ||𝑚𝛽 |/(|𝑚𝛼 | + |𝑚𝛽 |) and 𝛿𝑖 =1 for lamellar eutectics and 𝛿𝑖 = 2𝑓 1/2 for fibrous eutectics (Liu et al., 2011). According to the convention of Jackson and Hunt, α represents a minor (fibrous) phase of volume fraction, f. The capillary terms, 𝑆𝑖 , largely control the transition from fibres to lamellae via the ratio: 𝑆 = 𝑆𝐹 ⁄𝑆𝐿

[5.18]

According to the analysis the variation of S as a function of the volume fraction of the minor phase has the form shown in Fig. 5.14. This figure indicates that, the higher the S ratio, the lower the transitional volume fraction. In addition to the volume fraction the transition therefore depends upon the Gibbs-Thomson parameters, the contact-angles and the liquidus slopes (Eq. 5.17). In the case of extremum growth, the transition is sharp (dashed curve in Fig. 5.14). Introducing a spacing range for both morphologies leads to a diffuse transition region which comprises a mixture of both morphologies (for details of the model, see Liu et al., 2011). This model does not include, however, any anisotropy of the solid/solid interface energy.

Figure 5.14 Fibre to Lamella Transition for Al-Al2Cu Eutectic This figure shows the transition from eutectic fibres to lamellae as a function of the volume fraction of the minor phase, f, and of the solid/liquid interface-energy ratio, 𝑆 (= 𝑆𝐹 /𝑆𝐿 ) (Liu et al., 2011). Here S is proportional to the ratio of the sum of Γ𝑝𝑖 sin(𝜃𝑝𝑖 ) /𝑚𝑝 for the phase p and the morphology i. The dashed curve is for extremum growth while the continuous curves delimit the range of mixed lamellar and fibrous structures.

5.6. Morphological Instability of Eutectics As shown in Fig. 5.15, binary eutectics can undergo several types of morphological instabilities; single-phase or two-phase. The latter is analogous to the breakdown of a planar single-phase interface (Chap. 3). Due to the discontinuity at the eutectic solid/solid interface analytical, solutions are very complex and phase-field modelling is usually indicated. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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It can be said in general that a third alloying element which is rejected by both solid phases will form a boundary layer ahead of the eutectic solidification front (in the 𝑧-direction) and, at a low 𝐺/𝑉 ratio, will form a constitutionally undercooled zone. As in the case of the morphological instability of a single-phase plane front, this leads to a two-phase instability and to the appearance of eutectic cells (Fig. 5.15(b)) or even two-phase eutectic dendrites. On the other hand, during the off-eutectic growth of a pure binary eutectic, one phase might become constitutionally undercooled and the resulting morphological single-phase instability could lead to the appearance of single-phase dendrites embedded in interdendritic eutectic (Fig. 5.15(a)). With an off-eutectic composition the alloy liquidus is always higher than the eutectic temperature and the corresponding primary phase will be more highly undercooled and tend to grow faster than the eutectic. If the 𝐺/𝑉 ratio is high enough however, primary single-phase dendrites will be replaced by plane-front two-phase growth as shown in Fig. 5.13. The range of purely eutectic microstructures is called the coupled zone (CZ). Outside of this zone a mixture of dendrites plus eutectic is observed. This case is of considerable importance because the properties of eutectic alloys can be appreciably impaired or enhanced when single-phase dendrites appear. More details concerning the selection of eutectic microstructures in a CZ can be found in Chap. 8. Beyond the scope of the present book, there are other types of morphological or oscillatory eutectic instabilities which can develop when, for example, the eutectic spacing has to adapt itself to a change in growth velocity. Oscillations of lamellae in 2-dimensions, or zig-zag lamellae in 3dimensions, have been well described by Karma and Sarkissian (1996), and Parisi and Plapp (2008), respectively.

Figure 5.15 Two Types of Eutectic Interface Instability The planar eutectic solid/liquid interface (left) can become morphologically unstable, just as in the case of a single-phase interface. There are then two different ways in which an instability can develop; instability of one phase (a), or instability of both phases (b). The former leads to the appearance of dendrites of one phase (plus interdendritic eutectic) and is seen mainly in offeutectic alloys in binary systems. A third (impurity) element, which is rejected by the two solid phases, may alternatively destabilise the morphology as a whole, because a long-range diffusion boundary layer of the third element leads to the establishment of a constitutionally undercooled zone ahead of the composite solid/liquid interface. (Recall that the eutectic tie-line of a binary system degenerates into an eutectic three-phase (l++) region in a ternary system). This can lead to the appearance of two-phase eutectic cells (b) or dendrites.

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5.7. Peritectic Growth Peritectic phase transitions are commonly found in metal and ceramic phase diagrams. Some technically important examples include steels (Fe-C, Fe-Ni, Fe-Cr-Ni), copper alloys (Cu-Sn, CuZn), aluminium (Al-Ti), permanent magnet materials (Fe-Nd-B) or high 𝑇𝑐 superconductors (such as Y-Ba-Cu-O). In a binary alloy, the peritectic is an invariant three-phase equilibrium of the form (l + α) = β (Fig. 5.16(a)). It is characterized by two interpenetrating l/s equilibria (l + α and l + β) with their liquidus slopes both having a negative sign; thus differentiating it from a eutectic. More on peritectics can be found in Kerr and Kurz (1996). Under normal solidification conditions a peritectic alloy, represented by a phase diagram of the form of Fig. 5.16(a) with 𝐶0 < 𝐶𝑝 , the α-phase crystallizes first as columnar or equiaxed grains. During cooling, the peritectic β-phase precipitates onto the α from the melt, and envelopes it (b). Because solid-state diffusion is very slow, the change from α to β during cooling is rarely completed and the typically non-equilibrium peritectic structure results (Fig. 2 in Kerr and Kurz, 1996). In accord with mechanical equilibrium of the interface energies, a trijunction between α, β and liquid forms which is similar to that for a eutectic (Fig. 5.16(c)). The (l + α) → β phase-change has been called a peritectic reaction. As α is not stable at low temperature, it is slowly replaced by β via solid-state transformation, termed peritectic transformation. Because of the slowness of solid-state diffusion one can generally differentiate three mechanisms of β-growth; (i) peritectic reaction (l + α → β), (ii) peritectic solid-state transformation (α → β ), and (iii) growth of β from the liquid (l → β).

Figure 5.16 A Typical Peritectic Equilibrium Phase Diagram and Several Corresponding Mechanisms of Peritectic 𝜷-Growth A peritectic equilibrium phase diagram, typical for steels, is displayed in (a). The dashed lines represent metastable liquidus and solidus lines. For concentrations below that of the peritectic liquid, 𝐶𝑝 , the stable α-phase generally nucleates first. During cooling this primary phase grows as dendrites that form a substrate for the nucleation and growth of the peritectic β-phase (shown in (b)). Simultaneously the dendrites are also transforming partially into the β-phase. Close to the trijunction β grows at the expense of α via the peritectic reaction (l + α → β)(shown in c). Mechanical equilibrium of the interface energies controls the form of the trijunction (Hillert, 1979).

In directional growth at high 𝐺/𝑉 ratio, α-dendrites are replaced by α-cells which grow jointly with, and slightly ahead of, intercellular β (Fig. 5.17(a)). The l/α/β trijunction of Fig. 5.16(c) is clearly visible in the figure. The fact that the liquidus slopes of both solid phases have the same sign means that both phases reject element B (Fig. 5.17(b)).

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a)

b)

Figure 5.17 The Quenched Solid/Liquid Interface of a Directionally Solidified Cellular Composite This figure shows a composite microstructure which is composed of cellular α–phase growing ahead of β–phase in a directionally solidified Fe-Ni alloy (in iron alloys α = ̂ δ and β = ̂ γ). The growth direction is upwards. The trijunction of Fig. 5.16(c) is visible in (a). In alloys with a phase diagram like that of Fig. 5.16(a), element-B is rejected by both solid phases. The corresponding diffusional fluxes of B are directed away from the solidification front in the growth direction (b). This is the reason why there is no diffusion-coupling like that observed in eutectics; thus making simultaneous two-phase growth very difficult.

Unlike eutectics, where element-B which is rejected by the α-phase is incorporated into the βphase via lateral diffusion (and vice versa for element-A which is rejected by β and incorporated into α), the fluxes from both solid phases are - in peritectics - both directed into the growth direction and produce a corresponding long-range solute boundary layer. As a result, the coupled growth which is typical of eutectics is difficult to sustain for peritectic alloys, even under high G-values, and was indeed thought to be impossible for many years.

Figure 5.18 Directionally Solidified Peritectic Composite in Fe – Ni (a) and Phase-Field Result for Oscillatory Instability at the Beginning of Cooperative Growth A (eutectic-like) peritectic growth-front produced at high 𝐺/𝑉 values in an Fe-4.22 at% Ni alloy is shown in (a) (Vandyoussefi et al., 2000). The specimen was directionally solidified with 𝐺 = 130 K/cm and 𝑉 = 10 m/s and then quenched (the dendrites which formed during quenching of the melt are visible above). Corresponding phase-field modelling (b) permitted the computation of similar structures, while showing the path towards the initiation of regular two-phase structures via oscillatory instability of the solid/liquid interface (Lo et al., 2003).

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Under conditions of large 𝐺/𝑉 ratio which are sufficient to counter constitutional undercooling, the plane-front cooperative (rather than coupled) growth of both solid phases can nevertheless be stabilized in peritectic alloys, and an eutectic-like structure can be observed. Figure 5.18(a) shows such a two-phase peritectic growth front with α (= δ) and β (= γ) lamellae growing with common α/β interfaces in a quenched Fe-Ni alloy. Dendrites which formed during quenching are visible above. Note the very large lamellar spacing which is typically associated with the very low growth velocity, 𝑉, which is required for the plane-front growth of both solid phases. In (b) the evolution of this simultaneous two-phase growth from an α-plane front, via morphological and oscillatory instabilities and branching, is demonstrated thanks to the phase-field computations of Lo et al., 2003. One can finally plot the occurrence of the various microstructures in a microstructure-selection map. Figure 5.19 shows such a map for Fe-Ni alloys and various 𝐺/𝑉 ratios. Dendrites prevail at the lower end but, with increasing 𝐺/𝑉 ratios above the morphological instability limit for the peritectic phase (γ in this case) but below the corresponding limit for the primary phase (δ), the cellular structure of Fig. 5.17 is found. At above the stability limit of both solid phases the simultaneous two-phase growth of either fibres or lamellae (depending upon the volume fraction) is found.

Figure 5.19 Peritectic Fe-Ni Phase Diagram and Corresponding Microstructure Selection Map for Directional Growth This map shows the various peritectic two-phase structures which can be observed in Fe-Ni alloys which are directionally solidified under high 𝐺/𝑉 ratios. (In Fe alloys, δ corresponds to α and γ corresponds to β.) At low 𝐺/𝑉 ratios the interface morphology is composed of δ-cells/dendrites with the intercellular precipitation of the peritectic γ-phase (Fig. 5.17). At high G/V ratios simultaneous or cooperative peritectic growth, similar to coupled eutectic solidification, is possible (Dobler et al., 2004). The condition for simultaneous growth is a G/V ratio which assures the morphological stability of both phases. Depending upon the volume fractions of the phases, fibres or lamellae can be the predominant morphology (Vandyoussefi et al., 2000).

While cooperative peritectic growth was initially observed in narrow melting-range alloys such as Fe-Ni, similar observations were made in Cu-Sn alloys, where both phases have a large value of

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Δ𝑇0 (Kohler et al., 2009). The velocity, 𝑉, in this case is however extremely low - typically less than 1 m/s - for the thermal gradients which can realistically be achieved in conventional directional growth experiments. Although steady-state cooperative growth with a diffusion layer ahead of the α/β front equal to 𝐷/𝑉 could not be attained§, the long transient in the cooperative growth regime of Cu-Sn clearly confirmed the oscillatory behaviour shown in Fig. 5.18(b) (Valloton et al., 2013). The possibility of growing oriented composite microstructures in peritectic alloys is certainly interesting in principle. The only drawback to obtaining these structures is that the solidification speed is limited by the melting range, Δ𝑇0. Even for Fe-Ni, which has a Δ𝑇0 value of just a few degrees, the maximum velocity is of the order of 10 m/s. Many peritectic alloys, such as the Cu-based ones, have however large melting ranges which would require 𝐺/𝑉 ratios that were much too high to be technologically achievable. Exercises 5.1

Calculate the equilibrium volume fraction of graphite at 𝑇𝑒 in an Fe-C alloy of eutectic composition. (Density of graphite: 2.15 × 103 kg/m3, density of C-saturated Fe: 7.2 × 103 kg/m3).

5.2

On purely geometrical grounds calculate the volume fraction at which fibrous or lamellar structures have the lower total / interface energy. Assume that the / interface energy is isotropic and that the phase separation, 𝜆, for a square fibre arrangement is equal and constant in both cases. Which assumption is in any case wrong?

5.3

In figure 5.5b the solute concentration in the liquid at the eutectic solid/liquid interface (𝑧 = 0) is given. Calculate the 𝐶(𝑦) function corresponding to 𝑧 = 𝜆 for an alloy of eutectic composition.

5.4

In order to demonstrate the potential value of producing 'in situ' composites via directional eutectic solidification, calculate the total length of fibres within a cube of such a composite which has an edge-length of 1 cm and a fibre-spacing, 𝜆, of 1 m.

5.5

A eutectic stores part of the transformational energy in the form of / interfaces (Fig. 5.4). By what amount, Δ𝑇, will the melting point of a lamellar eutectic with 𝜆 = 0.1 m and 𝜎𝛼𝛽 = 5 × 10−8 J/mm2 be lowered? As shown in Appendix 14 the magnitude of Δ𝑠𝑓 for metals is typically of the order of 106 J/m3K.

5.6

Find expressions for 𝐾𝑐 and 𝐾𝑟 (Eq. 5.8) for the simple case shown in the main text and derive the solutions for extremum growth.

5.7

Using the phase diagram explain the apparent discontinuity of the liquidus temperature, 𝑇𝑙∗ , at the /-junction shown in Fig. 5.5(c).

5.8

Draw analogous diagrams to those in Fig. 5.5 for the case where both phases have a positive curvature, and for the case where both phases have depressions at their centres.

5.9

It can be easily shown that, at least over one half-spacing of the eutectic (Fig. 5.2), the solid/liquid interface of the eutectic must be very close to isothermal. An enormous heat-flux in the y-direction would be required in order to change even slightly the interface temperature

At the onset of formation of the peritectic phase, , the diffusion layer ahead of the primary phase,  is equal to (𝐷/𝑉)Ω𝛼 (𝑇𝑝 ), where Ω𝛼 (𝑇𝑝 ) < 1 is the supersaturation of the primary phase,  at the peritectic temperature, 𝑇𝑝 (see Eq. 6.1). §

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of the two phases. Using the properties of aluminium (Appendix 15) calculate the lateral heat flux, assuming that Δ𝑇𝑦 = 0.1 K and that 𝜆 = 1 m. 5.10 Is the eutectic spacing at the extremum the lowest possible spacing? Argue why this is (is not) so. 5.11 Is an effect of 𝐺 upon the spacing of regular eutectic microstructures growing at a given rate to be expected? Explain your answer. 5.12 Experiments performed on eutectic Fe-C alloys reveal the following relationship for the mean lamellar spacing: 𝜆̅2 𝑉 = 4 × 10−7 mm3/s. Does this support the extremum criterion? Assume that 𝜃Fe = 20∘ and that 𝜃𝐶 = 80∘ . 5.13 Why is the 2D model of irregular eutectic growth developed in this book only a crude approximation for growth of the Ni-C(graphite) eutectic of Fig. 5.12 (the latter eutectic resembles Fe-C)? 5.14 Sketch the solid/liquid interface of a unidirectionally and dendritically solidifying (peritectic) steel which contains 0.2 wt% C. Note that due to the rapid solid-state diffusion of carbon in Fe and -Fe the liquid/solid and solid/solid transformations closely follow the behaviours which are to be expected on the basis of the equilibrium diagram. 5.15 During directional solidification at moderate 𝐺/𝑉 ratio of a peritectic alloy with 𝐶0 < 𝐶𝑝 , what is the morphology and the fraction of the primary phase α when reaching the peritectic temperature 𝑇𝑝 (assume Scheil-Gulliver, see Chap. 6)? 5.16 Below the peritectic temperature, after the β phase has formed around the primary phase α, draw the schematic composition profile in all three phases along a cut perpendicular to the α/β interface. 5.17 What is the driving force for the transformation of α into β and why, after solidification, this transformation is usually incomplete? 5.18 In Exercise 5.15 and assuming a high 𝐺/𝑉 ratio that stabilises a planar α-front, draw the profile of the liquid composition 𝐶𝑙 (𝑧) when reaching the peritectic temperature 𝑇𝑝 . What is the thickness of the diffusion layer in the liquid at that instant? 5.19 Using the profile 𝐶𝑙 (𝑧) drawn in the previous exercise, draw the associated profiles of the 𝛽 liquidus of the primary phase, 𝑇𝑙𝛼 (𝐶𝑙 (𝑧)), and of the peritectic phase, 𝑇𝑙 (𝐶𝑙 (𝑧)). Do the same for 𝑇 < 𝑇𝑝 , assuming that the β phase has not yet nucleated. What do you deduce ?

References and Further Reading General, Eutectic in situ Composites ▪ R.Elliott, Eutectic Solidification Processing - Crystalline and Glassy Alloys, Butterworth, London, 1983. ▪ L.M.Hogan, R.W.Kraft, F.D.Lemkey, in Advances in Materials Research, H.Hermann (Ed.), 1971, Vol. 5, p.83. ▪ W.Kurz, P.R.Sahm, Gerichtet erstarrte eutektische Werkstoffe, Springer, Berlin, 1975. ▪ E.R.Thompson, F.D.Lemkey, in Composite Materials, Volume 4, K.G.Kreider (Ed.), Academic Press, New York, 1974.

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Theory of Eutectic Growth ▪ S.Akamatsu, S.Bottin-Rousseau, M.Serefoglu, G.Faivre, A theory of thin lamellar eutectic growth with anisotropic interphase boundaries, Acta Materialia, 60 (2012) 3199. ▪ L.F.Donaghey, W.A.Tiller, On the diffusion of solute during the eutectoid and eutectic transformations, part I, Materials Science and Engineering, 3 (1968/69) 231. ▪ D.J.Fisher, W.Kurz, A theory of branching limited growth of irregular eutectics, Acta Metallurgica, 28 (1980) 777. ▪ M.Hillert, The role of interfacial energy during solid-state phase transformations, Jernkontorets Annaler, 141 (1957) 757. ▪ K.A.Jackson, J.D.Hunt, Lamellar and rod eutectic growth, Transactions of the Metallurgical Society of AIME, 236 (1966) 1129. ▪ A.Karma, A.Sarkissian, Morphological instabilities of lamellar eutectics, Metallurgical and Materials Transactions A, 27 (1996) 635. ▪ A.Karma, M.Plapp, New insights into the morphological stability of eutectic and peritectic coupled growth, Journal of Metals, 56 (2004) 28. ▪ K.Kassner, C.Misbah, Spontaneous parity-breaking transition in directional growth of lamellar eutectic structures, Physical Review A, 44 (1991) 6533. ▪ S.Liu, J.H.Lee, R.Trivedi, Dynamic effects in the lamellar–rod eutectic transition, Acta Materialia, 59 (2011) 3102. ▪ P.Magnin, W.Kurz, An analytical model of irregular eutectic growth and its application to FeC, Acta Metallurgica, 35 (1987) 1119. ▪ A.Parisi, M.Plapp, Stability of lamellar eutectic growth, Acta Materialia, 56 (2008) 1348. ▪ T.Sato, Y.Sayama, Completely and partially co-operative growth of eutectics, Journal of Crystal Growth, 22 (1974) 259. ▪ P.H.Shingu, The extremum condition for the rate of cellular phase separation, Journal of Applied Physics, 50 (1979) 5743. ▪ W.A.Tiller, in Liquid Metals and Solidification, American Society for Metals, Cleveland, Ohio, 1958, p.276. ▪ R.Trivedi, P.Magnin, W.Kurz, Theory of eutectic growth under rapid solidification conditions, Acta Metallurgica, 35 (1987) 971. ▪ C.Zener, Kinetics of the decomposition of austenite, Transactions of the Metallurgical Society of AIME, 167 (1946) 550. Eutectic Growth Experiments ▪ G.Faivre, J.Mergy, Dynamical wavelength selection by tilt domains in thin-film lamellar eutectic growth, Physical Review A, 46 (1992) 963. ▪ D.J.Fisher, W.Kurz, Novel morphology of Pb/Sn/Cd/Zn quaternary eutectic, Metallurgical Transactions, 5 (1974) 1508. ▪ S.C.Gill, W.Kurz, Rapidly solidified Al-Cu alloys - II. Calculation of the microstructure selection map, Acta Metallurgica et Materialia, 43 (1995) 139. ▪ A.Hellawell, The growth and structure of eutectics with silicon and germanium, Progress in Materials Science, 15 (1970) 1. ▪ H.Jones, W.Kurz, Growth temperatures and the limits of coupled growth in unidirectional solidification of Fe-C eutectic alloys, Metallurgical Transactions A, 11 (1980) 1265. ▪ H.Jones, W.Kurz, Relation of interphase spacing and growth temperature to growth velocity in Fe-C and Fe-Fe3C eutectic alloys, Zeitschrift für Metallkunde, 72 (1981) 792.

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▪ R.M.Jordan, J.D.Hunt, Interface undercoolings during the growth of Pb-Sn eutectics, Metallurgical Transactions, 3 (1972) 1385. ▪ W.Kurz, B.Lux, On the Growth of the Ni-W Eutectic, Metallurgical Transactions, 2 (1971) 329. ▪ S.Liu, J.H.Lee, R.Trivedi, Dynamic effects in the lamellar–rod eutectic transition, Acta Materialia, 59 (2011) 3102. ▪ J.D.Livingston, H.E.Cline, E.F.Koch, R.R.Russell, High-speed solidification of several eutectic alloys, Acta Metallurgica, 18 (1970) 399. ▪ B.Lux, W.Kurz, M.Grages, The graphite in an eutectic Ni-C alloy seen in the scanning electron microscope, Praktische Metallographie, 6 (1969) 464. ▪ B.Lux, A.Vendl, H.Hahn, Über die Ausbildung eutektischer Gefüge in grau erstarrtem Gußeisen, Radex Rundschau, 1/2 (1980) 30. ▪ F.R.Mollard, M.C.Flemings, Growth of composites from the melt, Transactions of AIME, 239 (1967) 1534. ▪ S.Mohagheghi, S.Bottin-Rousseau, S.Akamatsu, M.Serefogglu, Decoupled versus coupled growth dynamics of an irregular eutectic alloy, Scripta Materialia, 189 (2020) 11. ▪ D.Oquab, C.Josse, A.Proietti, A.Pugliara, J.Lacaze, S.Steinbach, D.Ferdian, B.Viguier, Solidification sequence and four-phase eutectic in AlSi6Cu4Fe2 alloy, Materials Characterization, 156 (2019) 109846. ▪ Y.Sayama, T.Sato, G.Ohira, Eutectic growth of unidirectionally solidified iron-carbon alloy, Journal of Crystal Growth, 22 (1974) 272. ▪ V.Seetharaman, R.Trivedi, Eutectic growth: selection of interlamellar spacings, Metallurgical Transactions A, 19 (1988) 2955. ▪ H.A.H.Steen, A.Hellawell, Structure and properties of aluminium-silicon eutectic alloys, Acta Metallurgica, 20 (1972) 363. Cast Iron ▪ B.Lux, W.Kurz, M.Grages, The graphite in an eutectic Ni-C alloy seen in the scanning electron microscope, Praktische Metallographie, 6 (1969) 464. ▪ B.Lux, On the theory of nodular graphite formation in cast iron, Cast Metals Research Journal, 18 (1972) 25 and 49. ▪ P.Magnin, W.Kurz, Competitive growth of stable and metastable Fe-C-X eutectics, Metallurgical Transactions A, 19 (1988) 1955 and 1965. ▪ I.Minkoff, The Physical Metallurgy of Cast Iron, Wiley, New York, 1983. ▪ I.Minkoff, S.Myron, Rotation boundaries and crystal growth in the hexagonal system, Philosophical Magazine, 19 (1969) 379. Peritectics ▪ G.Azizi, B.G.Thomas, M.Asle Zaeem, Review of peritectic solidification mechanisms and effects in steel casting, Metallurgical and Materials Transactions B, 51 (2020) 1875. ▪ W.J.Boettinger, The structure of directionally solidified two-phase Sn-Cd peritectic alloys, Metallurgical Transactions, 5 (1974) 2023. ▪ S.Dobler, T.S.Lo, M.Plapp, A.Karma, W.Kurz, Peritectic coupled growth, Acta Materialia, 52 (2004) 2795. ▪ A.M.Figueredo, M.J.Cima, M.C.Flemings, J.S.Haggerty, Directional phase formation on melting via peritectic reaction, Metallurgical and Materials Transactions A, 25 (1994) 1747. ▪ M.Hillert, Eutectic and peritectic solidification, in Solidification and Casting of Metals, The Metals Society, London, 1979, p.81.

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▪ A.Karma, M.Plapp, New insights into the morphological stability of eutectic and peritectic coupled growth, Journal of Metals, 56 (2004) 28. ▪ A.Karma, W.J.Rappel, B.C.Fuh, R.Trivedi, Model of banding in diffusive and convective regimes during directional solidification of peritectic systems, Metallurgical and Materials Transactions A, 29 (1998) 1457. ▪ H.W.Kerr, W.Kurz, Solidification of peritectic alloys, International Materials Reviews, 41 (1996) 129. ▪ F.Kohler, L.Germond, J.D.Wagnière, M.Rappaz, Peritectic solidification of Cu–Sn alloys: microstructural competition at low speed, Acta Materialia, 57 (2009) 56. ▪ W.Kurz, R.Trivedi, Banded solidification microstructures, Metallurgical and Materials Transactions A, 27 (1996) 625. ▪ T.S. Lo, S. Dobler, M. Plapp, A. Karma, W. Kurz, Two-phase microstructure selection in peritectic solidification: from island banding to coupled growth, Acta Materialia, 51 (2003) 599. ▪ H.Nassar, H.Fredriksson, On Peritectic Reactions and Transformations in Low-Alloy Steels, Metallurgical and Materials Transactions A, 41 (2010) 2776. ▪ Y.Shiohara, A.Endo, Crystal growth of bulk high-Tc superconducting oxide materials, Materials Science and Engineering R-Reports, 19 (1997) 1. ▪ D.H.StJohn, L.M.Hogan, A simple prediction of the rate of the peritectic transformation, Acta Metallurgica, 35 (1987) 171. ▪ R.Trivedi, Theory of layered-structure formation in peritectic systems, Metallurgical and Materials Transactions A, 26 (1995) 1538. ▪ T.Umeda, T.Okane, W.Kurz, Phase selection during solidification of peritectic alloys, Acta Materialia, 44 (1996) 4209. ▪ M.Vandyoussefi, H.W.Kerr, W.Kurz, Two-phase growth in peritectic Fe–Ni alloys, Acta Materialia, 48 (2000) 2297. ▪ J.Valloton, J.A.Dantzig, M.Plapp, M.Rappaz, Modeling of peritectic coupled growth in Cu-Sn alloys, Acta Materialia, 61 (2013) 5549.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 133-148 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER SIX

SOLUTE REDISTRIBUTION It has been explained in some detail in Chap. 3 that the solid/liquid interface rejects solute into the liquid when the solubility of the solute in the solid is lower than that in the liquid. This leads to the creation of a boundary layer ahead of the solid/liquid interface. In this case the liquidus slope, 𝑚, is negative and the distribution coefficient, k, is less than 1. On the other hand, 𝑚 is positive and 𝑘 is greater than unity when the solubility is greater in the solid than in the liquid. In this case solute will diffuse from the liquid to the solid and a depleted zone forms ahead of the solid/liquid interface. As far as metals growing under normal solidification conditions are concerned, local equilibrium is assumed to hold at the solid/liquid interface. The solid concentration at the interface is then related to the liquid concentration by the equilibrium distribution coefficient (Eq. 1.9): 𝐶𝑠∗ = 𝑘𝐶𝑙∗ This compositional difference will always lead to concentration variations in the solidified alloy which are known as segregation. Note that the solute distribution in the liquid ahead of the solid/liquid interface leads to the appearance of the various growth morphologies, mainly dendrites, and that the latter in turn determine the solute distribution in the solid. The results are concentration variations occurring over microscopic distances, interdendritic precipitates, porosity and cracks. Because solute can be transported via diffusion and/or via convection, the segregation pattern will be quite different depending upon the processes involved. Convection can lead to the transport of mass over very large distances, as compared to those involved in diffusional processes, and may result in macrosegregation: that is, compositional differences which occur over distances that are a fraction of the size of a large casting. In this book attention will be limited to microsegregation†. It depends upon solute diffusion in the liquid and solid and is related to the dendrite shape and size. An **

Top image: Columnar dendrites with interdendritic enrichment of solute due to its rejection at the liquid/solid interface. Following solidification it persists as inhomogeneities in composition which are known as microsegregation. † For an introduction to convection effects in solidification the reader is referred to the bibliography at the end of the chapter: Flemings (1974), Geiger and Poirier (1973), Dantzig and Rappaz (2016).

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understanding of this phenomenon is key for the interpretation of the influence of solidification upon mechanical properties and the formation of defects of cast products or welds. Microsegregation reveals moreover the original microstructure of a solidified alloy due to differences in the etching behaviour of regions having differing local compositions. In order to understand the segregation occurring at the scale of the dendrites, with their complex morphology, it is useful to begin with a description of the solute distribution which exists during the directional solidification, of a rod of constant cross-section, having a planar solid/liquid interface (Fig. 1.4). In this special case, all the changes occur in only one dimension and are therefore easier to analyse. Once this case is fully understood, it will be possible to use the results to study more complicated cases in a qualitative manner by imagining that the changes which occur in small volume elements are the same as those which occur during directional solidification (Fig. 1.5). It is assumed throughout this chapter that convection is absent. 6.1. Mass-Balance in Directional Solidification Mass-balance was treated only with respect to steady-state conditions in Chap 3. In order to understand microsegregation, it is essential to consider the initial and final transients as well. The former is required in order to establish the steady-state boundary layer, and the latter arises from the interaction of the boundary-layer of the solid/liquid interface with the end of the specimen (Fig. 6.1). The diffusion boundary layer in the liquid ahead of a planar solid/liquid interface can be regarded as being a limited region of the system which transports the solute missing from the initial transient in the solid, where the concentration is below 𝐶0 : this satisfies the overall solute balance of the system. This moving boundary layer disappears at the end of solidification by 'depositing' its solute content into the final transient. The mean composition of the solid is therefore always the same as that of the liquid from which it formed. No account is taken here of reactions with the crucible or of vaporisation of some of the elements of the alloy. 6.2. The Initial Transient Figure 6.1 depicts in a schematic manner the mechanism which leads to the formation of the boundary layer and the initial transient. The process is shown for a bar of constant cross-section, 𝐴, (Appendix 11). When the first volume element, 𝐴𝑑𝑧′, has solidified (where 𝑑𝑧′ is vanishingly small) the solute which has not been incorporated into the solid is equal to the incremental volume of the solid, 𝐴𝑑𝑧′, multiplied by the difference in composition between the liquid and the solid, 𝐶𝑙∗ − 𝐶𝑠∗ , which at 𝑧 ′ = 0 is equal to 𝐶0 − 𝑘𝐶0 . This mass, divided by the time necessary for the advance of the interface by 𝑑𝑧′, leads to a solute flow at the interface where 𝑉 = 𝑑𝑧′/𝑑𝑡 (Eq. 3.4). One then has per unit of interface area: 𝐽1 = 𝑉𝐶𝑙∗ (1 − 𝑘) This flow leads to the creation of a pile-up at the interface and thence to the establishment of a concentration gradient and a diffusional flow into the liquid of: 𝐽2 = −𝐷𝐺𝑐 = 𝐷

𝐶𝑙∗ − 𝐶0 𝛿𝑐

where 𝛿𝑐 is the thickness of the boundary layer in the liquid ahead of the interface. Equating the two solute fluxes, 𝐽1 and 𝐽2 , shows this thickness to be: 𝐷 𝐶𝑙∗ − 𝐶0 𝐷 𝛿𝑐 = = Ωc ∗ (1 𝑉 𝐶𝑙 − 𝑘) 𝑉

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[6.1]

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Figure 6.1 Initial and Terminal Transients of a Plane Front During directional solidification with a planar solid/liquid interface in an apparatus such as the Bridgman furnace (Fig 1.4), the establishment of a steady-state boundary layer requires a distance of growth which corresponds to the length of the initial transient. This distance increases with decreasing growth rate. Within this transient (a) the concentration of the liquid at the interface increases from 𝐶0 to 𝐶0 /𝑘 (b). With respect to the phase diagram (assuming the existence of local equilibrium) this means that the first solid to freeze at 𝑇𝑙 has the composition, 𝑘𝐶0, and reaches the composition, 𝐶0 , and the interface temperature, 𝑇𝑠 , which corresponds to composition, 𝐶0 , in the steady state. At steady state, the flux of solute due to the interface advance and the difference in liquid and solid solubilities is equal to the diffusional flux induced by the concentration gradient at the solid/liquid boundary. In this case the exponential decay described in Chap. 3 is the exact solution. When the boundary layer becomes on the order of the length of the remaining liquid region, diffusion into the liquid phase is finally hindered by the system boundary (because the concentration gradient must clearly be zero at the end of the crucible). The concentration in the liquid at the solid/liquid interface thus begins to increase to a value which is greater than 𝐶0 /𝑘 and the solid concentration therefore becomes greater than 𝐶0 so that a terminal transient is created (c). To ensure mass conservation, the surface area of the grey region below 𝐶0 must be equal to the grey area above 𝐶0 . Note that the lengths of the initial and terminal transients are unequal. These concentration variations can be a major problem in solidification processes and are known as segregation. However, the same phenomenon is usefully exploited in 'zone-refining', where the initial solute-depleted (purer) part of the rod is used (see Exercise 6.12).

where Ω𝑐 is the supersaturation at any given time. During the transient both the thickness of the boundary layer, 𝛿𝑐 , and the liquid composition at the interface, 𝐶𝑙∗ , vary. The exact solution which gives the interface composition, 𝐶𝑙∗ (𝑧 ′ ), as a function of the solidification distance, 𝑧 ′ , can be found in Smith et al. (1955) (see also Mota et al., 2015, or Dantzig and Rappaz, 2016). Appendix 11 proposes an approximate solution for the evolution of 𝐶𝑙∗ , assuming that 𝛿𝑐 is constant. It is given by:

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𝐶0 𝑘𝑧 ′ 𝑉 𝐶𝑙∗ ≅ ( ) [1 − (1 − 𝑘) exp (− )] 𝑘 𝐷

[6.2]

As shown in Fig. 6.1, the initial transient exists until the solute boundary layer reaches its steady-state value, given by 𝛿𝑐 = 𝐷/𝑉, i.e. Ω𝑐 = 1. Figure 6.2 shows the evolution of the solid composition, 𝐶𝑠∗ = 𝑘𝐶𝑙∗ , given by Eq. 6.2 and plotted as a function of the dimensionless distance. This permits an estimation to be made of the length which must solidify before the steady state is reached. Note that in alloys with small values of 𝑘, solidifying at low rates, a large distance may be required. For example, if 𝑘 is equal to 0.1, 𝑉 is equal to 10-3 mm/s and 𝐷 is 5 × 10−3 mm2/s, the characteristic distance required to establish a steady-state planar interface with 82% of the theoretical concentration is of the order of 100 mm!

Figure 6.2 Length of the Initial Transient for a Planar Solid/Liquid Interface In many solidification experiments it is important to assess when the steady state has been reached. This diagram illustrates the length of the initial transient of Fig. 6.1 for typical distribution coefficients 𝑘 < 1 (𝑝 = 1 − 𝑘). A safe rule-of-thumb is that the distance of interface travel required for the establishment of the steady state is 4𝐷/𝑉𝑘.

This is an important factor to bear in mind when applying steady-state theory to experiments involving a planar interface. Equation 6.2 is only an approximate, but still useful, solution for the initial transient. 6.3. The Steady State The steady state is established when 𝑑𝐶𝑙∗ /𝑑𝑡 = 0 and 𝐶𝑙∗ = 𝐶0 /𝑘. Then: 𝛿𝑐 = 𝐷/𝑉

[6.3]

𝐶𝑙∗ − 𝐶0 = Ω𝑐 = 1 𝐶𝑙∗ (1 − 𝑘)

[6.4]

and:

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In other words, the planar solid/liquid interface grows, under steady-state conditions, at the solidus temperature of the alloy. The concentration of the melt there decreases exponentially as a function of the distance, 𝑧, from the interface (Eq. 3.2): 𝐶𝑙 (𝑧) − 𝐶0 = Δ𝐶0 exp (−

𝑉𝑧 ) 𝐷

[6.5]

The concentration gradient at the interface, 𝐺𝑐 , is equal to Δ𝐶0 /𝛿𝑐 = Δ𝐶0 𝑉/𝐷 (Fig. 3.4). The solute contained within the whole boundary layer, found by integrating Eq. 6.5 (or Eq. 3.2) from 0 to , is equal to Δ𝐶0 𝐷/𝑉. Replacing the exponential by a linear variation from 𝐶𝑙∗ to 𝐶0 defines an equivalent boundary layer: 𝛿𝑐′ = 2𝛿𝑐 = 2𝐷/𝑉. 6.4. The Final Transient When the boundary layer becomes comparable to the length of the remaining liquid zone, interaction with the boundary of the system occurs (Fig. 6.1). The end of the liquid zone can be regarded as being a perfectly impermeable wall which imposes a zero-flux condition at that point: (

𝑑𝐶 ) =0 𝑑𝑧 𝑧 ′ =𝐿

[6.6]

This is equivalent to saying that the flow into the liquid decreases and therefore, the concentration increases as shown in Fig. 6.1(c). For a mathematical description of this transient, it is assumed that Eq. 6.6 is satisfied if a mirror solute source of equal strength is placed beyond the end of the specimen at equal distance. A series of symmetrical sources are required to satisfy both Eq. 6.6 and the solute balance at the interface. This technique used by Smith et al. (1955) gives 𝐶𝑙∗ , as a function of the final liquid length, 𝑙 (Appendix 11). Note that, in this treatment, back diffusion into the solid which is specifically important during this transient is not taken into account. 6.5. Rapid Diffusion in the Liquid - Small Systems The analysis of the solute distribution becomes much simpler when it is assumed that diffusion in the liquid is sufficiently rapid to avoid the establishment of any concentration gradient ahead of the solid/liquid interface. This is a reasonable assumption for the case of very high diffusion coefficients in the liquid, strong convection and/or a very small system size, 𝐿, as compared to the boundary-layer thickness. Considering the case of rapid diffusion alone, no concentration gradient will exist in the liquid when the diffusion boundary-layer is much greater than the system size, 𝐿: 𝛿𝑐′ =

2𝐷 ≫𝐿 𝑉

[6.7]

The reason is that the insulating effect of the end of the specimen will then smooth out any concentration gradient. Under these conditions the most general treatment involves a combined approach (Appendix 12). Figure 6.3 shows the corresponding concentration profile. The boundary layer in the solid, 𝛿𝑠 , due to back-diffusion will here take a value between zero and infinity depending upon the value of the diffusion coefficient in the solid. At the relative interface position, 𝑠/𝐿 = 𝑓𝑠 = (1 − 𝑓𝑙 ), the mass balance, 𝐴1 = 𝐴2 + 𝐴3 , can be written to a first approximation as: (𝐶𝑙 − 𝐶𝑠∗ )𝑑𝑓𝑠 = 𝑓𝑙 𝑑𝐶𝑙 +

𝛿𝑠 𝑑𝐶 ∗ 2𝐿 𝑠

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[6.8]

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Figure 6.3 Segregation with Complete Liquid Mixing and Some Solid-State Diffusion When mass transport in the liquid is very rapid, e.g. due to the effect of convection, the excess solute will be redistributed evenly over the entire volume of liquid. In this case there will be an interaction with the far end of the crucible during the whole solidification process and the entire solute distribution will essentially be a long terminal transient beginning at the solid concentration, 𝐶0 𝑘. This behaviour is described by the 'Gulliver-Scheil' equation and predicts an infinite concentration at the end of solidification. In practice, eutectic solidification often intervenes and limits the maximum concentration. To obtain a more realistic description of the concentration profile at the end of solidification, solid-state back-diffusion must be taken into account. This can be done by using a simple mass balance. The solute rejected by the solid over the distance 𝑑𝑧 ′ (represented by surface 𝐴1 ) will partially increase the uniform liquid concentration by 𝑑𝐶𝑙 (surface 𝐴2 ), and this in turn will increase the interface concentration and the associated concentration gradient in the solid, and therefore the flux into the solid (surface 𝐴3 ). Mass conservation requires that the sum of the quantities described by the three surfaces must be zero. If diffusion in the solid is very rapid (e.g. carbon in -Fe), the boundary layer, 𝛿𝑠 , in the solid will be very large. In the limit, due to interaction with the initial boundary of the system (𝑧 ′ = 0), the concentration gradient in the solid will be decreased leading, for the extreme case of infinitely rapid back-diffusion, to equilibrium (lever-rule) solidification. In this case, a homogeneous solid with composition, 𝐶0 , will result after the completion of solidification. The extent of back-diffusion will depend upon a dimensionless parameter, 𝛼, which can be regarded as describing the ratio of the diffusion boundary layer in the solid, 𝛿𝑠 , to the size of the system, 𝐿.

where 𝐶𝑙 = 𝐶𝑙∗ . Assuming that the interface position, 𝑠, is a parabolic function of time‡, and integrating (Appendix 12) gives:

‡

This function gives a self-consistent solution to the problem. It is not the case when one assumes a linear relationship between s and t.

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𝐶𝑙 = (1 − 𝑢𝑓𝑠 )−𝑝/𝑢 𝐶0

139

[6.9]

where 𝑢 = 1 − 2𝛼 ′ 𝑘, 𝑝 = 1 − 𝑘, and 𝐶𝑠∗ can be obtained directly from 𝐶𝑠∗ = 𝑘𝐶𝑙 . The parameter, 𝛼 ′ , can be calculated from the interpolation function (Clyne and Kurz, 1981): 1 1 𝛼 ′ = 𝛼 [1 − exp (− )] − 0.5 exp (− ) 𝛼 2𝛼

[6.10]

where the dimensionless diffusion-time (Fourier number) is: 𝛼=

𝐷𝑠 𝑡𝑓 𝐿2

[6.11]

The behaviour of 𝛼 ′ is such that, at small 𝛼-values (less than 0.1), 𝛼 ′ = 𝛼, and at large 𝛼-values (greater than 50), 𝛼 ′ = 0.5 (Fig. A12.3).§ Substituting these limiting values of 𝛼 ′ into Eq. 6.9 leads to two well-known equations: ▪ When 𝛼 is equal to zero, 𝛼 ′ is also equal to zero (no solid-state diffusion takes place) and: 𝐶𝑙 1 1 = = 𝐶0 (1 − 𝑓𝑠 )𝑝 𝑓𝑙𝑝

[6.12]

This is known as the 'Gulliver-Scheil ' equation (Glicksman and Hills, 2001). ▪ On the other hand, when 𝛼 approaches infinity, 𝛼 ′ becomes equal to 0.5 and solidification occurs under equilibrium conditions, i.e. solid-state back-diffusion is so rapid (or 𝛿𝑠 ≫ 𝐿) that, as for the liquid, the zero-flux condition of the end of the solid (𝑧’ = 0) again smoothens out any concentration gradient in the solid. This case is described by the 'lever rule': 𝐶𝑙 1 = 𝐶0 1 − 𝑝𝑓𝑠

[6.13]

Figure 6.4 illustrates the behaviour of Eq. 6.9 for an Al-2wt%Cu alloy, including the limiting cases described by Eqs 6.12 and 6.13. It is evident that the Gulliver-Scheil equation is a very poor approximation with regard to the final liquid composition since the maximum liquid concentration, 𝐶𝑙𝑚 , is infinite. On the other hand, equilibrium solidification according to the lever-rule case, which leads to a final liquid concentration of 𝐶𝑙𝑚 = 𝐶0 /𝑘, is again unrealistic for most solutes because of their low solid-state diffusivity. There are very important exceptions however such as those of interstitial solutes, especially in open crystal structures, and small systems (e.g. interdendritic segregation of carbon in -Fe). The latter diffusion coefficient is so high that 𝛼-values greater than 100 are found. Knowledge of the value of 𝐶𝑙𝑚 also permits the temperature of the remaining liquid to be deduced from the phase diagram (Fig 6.5).

§

Due to the simplification made in deriving Eq. 6.9 it can be used only for k-values smaller than 1. For a more exact analysis of this problem see Ohnaka (1986) or Kobayashi (1988) (Appendix 12).

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Figure 6.4 Segregation Curves for Al-2 wt% Cu for Different 𝜶’ Values The composition of the liquid (assumed to be homogeneous as in Fig. 6.3) increases at the end of the specimen. Under lever-rule conditions the increase is from 𝐶0 to 𝐶0 /k, while the GulliverScheil equation predicts an increase from 𝐶0 to infinity. All the intermediate cases can be described by one relationship (Eq. 6.9) which contains a modified 𝛼-parameter, 𝛼 ′ , that can take values of between 0 and 0.5. Note that the curve represents the path of the interface concentration, 𝐶𝑙∗ , as a function of 𝑓𝑠 . The final solute distribution profile in the solid cannot be determined in this way because it changes with time, when 𝛼 > 0, due to back-diffusion. Only the endconcentration (𝑓𝑠 = 1) therefore represents a measurable value.

6.6. Microsegregation Solute distributions have been considered so far only for the case of relatively large systems undergoing directional solidification involving planar solid/liquid interfaces (Fig. 6.3). Because of the simplicity of such a unidirectional system and the approximations made, the equations which are derived are simple. The situation becomes extremely complex when the previously used approaches are applied to dendritic solidification. Further simplifications must therefore be made in order to make such problems analytically tractable. The case of dendritic (or better cellular) solidification in the columnar zone of a casting is first considered (Fig. 6.6). The mushy zone of length, 𝑎, is defined to be the region within which liquid and solid coexist at various temperatures, with various concentrations arising from solute redistribution. The zone length, 𝑎, is proportional to the non-equilibrium solidification range, Δ𝑇 ′ , which is usually larger than the equilibrium melting range, Δ𝑇0 (Fig. 6.5). In order to apply a mass balance, it is necessary to simplify the dendrite form in two respects. It is first assumed that there are no side-branches and, second, that the dendrite is plate-like rather than needle-like. It is finally assumed that 'directional solidification' is occurring in an infinitesimally narrow volume element between, and perpendicular to, two cells or dendrites (Fig. 6.6). All the relationships which were previously developed can now be applied in an approximate manner to the interdendritic region where the solidification time, 𝑡𝑓 , is: Δ𝑇 ′ 𝑡𝑓 = − 𝑇̇

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Figure 6.5 Relationship of the Segregate Freezing Point to the Phase Diagram An increasing concentration (for distribution coefficients of less than unity) is associated with a decreasing liquidus temperature since the slope, 𝑚, is then less than zero. Using the curves of Fig. 6.4, the temperature of the liquid as a function of volume-fraction solidified can be derived. The use of realistic diffusion coefficients shows that, for small systems (such as interdendritic regions – Fig. 6.6) interstitial carbon in - and -Fe will behave according to the lever-rule. The last liquid of a binary Fe-C melt will hence solidify at a temperature close to the solidus, while substitutional alloys, such as Al-Cu, which typically have much smaller solid-state diffusion coefficients will usually contain eutectic material in the last (interdendritic) regions to solidify; even when the overall composition is less than the solubility limit at 𝑇𝑒 .

The negative sign arises here due to the negative value of 𝑇̇, the cooling-rate. The length of the volume element is then, 𝐿 = 𝜆1 /2, and Δ𝑇 ′ = 𝑚(𝐶𝑙∗ − 𝐶𝑙𝑚 ), where 𝐶𝑙𝑚 corresponds to the composition of the last liquid. Substituting these values into Eq. 6.11 gives: 4𝐷𝑠 𝑚(𝐶𝑙∗ − 𝐶𝑙𝑚 ) 𝛼=− 𝜆12 𝑇̇

[6.14]

Under most conditions, the dendrite tip concentration, 𝐶𝑙∗ , is very close to 𝐶0 while 𝐶𝑙𝑚 depends markedly upon 𝛼 and can be obtained from Eq. 6.9 with 𝑓𝑠 = 1. Thus: 𝐶𝑙𝑚 = 𝐶0 (2𝛼 ′ 𝑘)−𝑝/𝑢

[6.15]

Considering now the dendrites of Fig. 6.7, the back-diffusion process, which is most marked at the end, will occur mainly between the secondary arms and not between the primary trunks. Except in the case of cellular growth, where 𝜆1 is the characteristic spacing, 𝜆2 would therefore be the most appropriate dimension for characterising dendritic solidified products.

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Figure 6.6 Characteristics of the Mushy Zone in Columnar Growth The results for macroscopic directional solidification (Fig. 6.3) can be applied to the cellular or dendritic structure of a casting by considering a small volume element solidifying at right-angles to the imposed growth direction of the alloy. An increase in the concentration or a decrease in the liquidus temperature of the remaining liquid shown before can thus also be applied qualitatively to the mushy zone of an as-solidified product. That is, 𝑓𝑠 is here now the local volume fraction of solid in the two-phase region with 𝑧 ′ = 0 at the centre of the cell trunk (corresponding to the position of the infinitely narrow volume element at the cell tip where 𝑓𝑠 = 0) and 𝑧′ = 𝐿 = 𝜆1 /2 at the last interdendritic liquid (corresponding to the position of the volume element at the cell root with 𝑓𝑠 = 1). In the case of dendrite growth with secondary branching, the characteristic back-diffusion distance is not 𝜆1 /2 as in cells, but instead the smaller distance, 𝜆2 /2.

Fig. 6.7 Evolution of Columnar Dendrites as a Function of Isothermal Holding Time. These figures show the ripening and coalescence of secondary branches of columnar dendrites in succinonitrile-acetone alloy as a function of time. The mean primary spacing is 130 μm. Close to complete solidification only drops of the last liquid remain. The morphological change from continuous interdendritic liquid films to isolated drops is found to occur at some 90% solid. This evolution of the solid fraction and its morphology has important consequences for interdendritic fluid flow, porosity formation and hot-cracking.

This dimension is also important in equiaxed solidification which, in many cases, is dendritic in nature. It is known that (Eq. 4.25): 𝜆2 = 5.5(𝑀𝑡𝑓 )1/3 so that the -value for dendrites is deduced to be: 𝛼 = 0.13𝐷𝑠 Δ𝑇 ′1/3 𝑀2/3 |𝑇̇|−1/3

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[6.16]

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where 𝑀 is defined by Eq. 4.26. Because Δ𝑇 ′ depends upon 𝛼, and this in turn depends upon Δ𝑇 ′ , the calculations have to be performed in an iterative manner, substituting for the initial Δ𝑇 ′ = ∆𝑇0 . To a first approximation one can also substitute for the final value of Δ𝑇 ′ = ∆𝑇0 leading to a constant value for the ripening parameter: 𝑀≅

−2Γ𝐷 ln( 𝑘) Δ𝑇0 𝑝

[6.17]

When 𝐶𝑙𝑚 is greater than 𝐶𝑒 precipitation of the eutectic will generally occur (Fig. 6.8) and its volume fraction, 𝑓𝑒 , can be calculated by using Eq. 6.9, knowing that 𝑓𝑒 = 1 − 𝑓𝑠 (for 𝐶𝑙 = 𝐶𝑒 ). This shows that: 1 𝐶0 𝑢/𝑝 𝑓𝑒 = ( ) [𝑢 − 1 + ( ) ] 𝑢 𝐶𝑒

[6.18]

Figure 6.8 Microsegregation For a solidification structure growing from a mould (in grey), the equilibrium melting range, Δ𝑇0 , does not (except for the lever-rule case) correspond to the range, Δ𝑇 ∗, over which the mushy-zone develops. The dendrite tips need a certain undercooling, determined by the growth kinetics of the tip. Due to non-equilibrium solidification the dendrite roots will usually have much higher concentrations than 𝐶0 /𝑘 (Fig. 6.4). This often leads to the interdendritic precipitation of eutectic phases of volume fraction, 𝑓𝑒 , even if the composition is not on the eutectic tie-line. In the columnar zone of a casting or a weld, as shown here, the volume fraction of solid, 𝑓𝑠 , will follow an S-shaped curve like that in the top diagram.

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Given the temperature gradient and the growth rate, the most important characteristics (𝐶𝑙𝑚 , 𝑓𝑒 , Δ𝑇 ′ , 𝜆1 and 𝜆2 ) of the solidified structure can therefore be obtained approximately. That is, the solidification microstructure as well as the microscopic inhomogeneities in chemical composition can be determined. In multicomponent systems, the 'path of solidification' is more complicated due to the greater number of variables. In Appendix 12 an example of such a situation is given for a ternary system (Eq. A12.29). The practically important case of post-solidification homogenisation of interdendritic segregation is treated further in Appendix 13 (only concentration variations without precipitations). These equations permit an estimation to be made of the degree of microsegregation which exists after cooling a solidified product to room temperature, and of the time required to reach a certain degree of homogenisation during a given heat-treatment process. Exercises 6.1

Write an equation for the 𝑇𝑙∗ − 𝑓𝑠 relationship in the lever-rule and Gulliver-Scheil equation cases.

6.2

Determine 𝛼 ′ values for Al-2wt%Cu and Fe-0.09wt%C alloys when 𝑡 = 10 s. Which system exhibits the greater tendency to segregate? First assume that 𝜆2 = 30 m for both alloys and then check whether this assumption is reasonable.

6.3

It is desired to purify part of a cylindrical metal ingot by directional solidification. What interface morphology (planar, cellular, dendritic) is required in order to accomplish this? What conditions are most favourable: a short initial transient, or Gulliver-Scheil-type solidification? Give the maximum growth rate which can be used.

6.4

Devise a method for estimating part of the phase diagram (𝑚, 𝑘 values) of a transparent organic alloy by solidifying it under planar interface conditions at the very limit of morphological stability (while observing it with a microscope). Indicate how one might perform the experiment. It is assumed that the values of 𝐶0 and 𝐷 are known and that 𝐺 can be determined during the experiment.

6.5

What will happen if the rod in Fig. 6.1 is solidified under conditions of strong convection? Sketch the solute profile and indicate which equation applies to this situation.

6.6

In what respect is Equation 6.2 an approximation to the initial transient? Examine the assumptions made concerning the boundary layer for the transient (see Appendix 11 and Fig. A2.3)

6.7

Describe a method for the production of a control sample of a given composition for use in microprobe measurements. Such a standard should present a composition to the electron beam which is homogeneous at a scale of the order of 1 m.

6.8

Write the equation, given in this chapter, which approximately describes the concentration variation along the curved interface of Fig. 6.6 under growth conditions under which the tip concentration in the liquid, 𝐶𝑙∗ , is roughly equal to 𝐶0 . In what region would one expect the concentration gradient in the liquid, perpendicular to the growth axis, to be (a) close to zero and (b) non-zero?

6.9

Indicate with the aid of Fig. 4.12 the growth rates for which, in an alloy, no intercellular nor interdendritic enrichment (segregation) will occur for (a) 𝐺 ≤ 0 and (b) G > 0.

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6.10 Why does Eq. 6.9 with 𝑢 = 1 − 2𝛼𝑘 produce increasingly incorrect results if 𝛼 increases beyond 0.1? What happens in this case? Sketch the solute profiles present in the solid and liquid under these conditions. Compare with Fig. A12.2. 6.11 Compare the fractions of eutectic predicted by the Gulliver-Scheil-equation, by the lever rule, and by Eq. 6.9, in the case of -Fe-0.6wt%C and Al-2wt%Cu. Discuss the differences. When calculating 𝛼 ′ use the same conditions as those assumed in Exercise 6.2. 6.12 Zone-melting is a process in which the tendency to segregation is exploited in order to produce a solid having a high purity. In practice this is done by causing a molten zone with planar solid/liquid interfaces to pass through the material many times (see figure below). The equation for the impurity distribution after the first pass, with strong mixing occurring in the liquid zone (𝐶𝑙 = constant), is: 𝐶𝑙 1 −𝑘𝑧 )] = [1 − (1 − 𝑘)exp ( 𝐶0 𝑘 𝐿 where 𝐿 is the zone-length. Discuss this equation with regard to the equation for the initial transient (equation 6.3).

6.13 In rapid solidification processing of Al-rich Al-Cu alloys it may happen that the stable θ-phase (Al2 Cu) is replaced by the metastable θ′-phase (also Al2 Cu). Assume that there is also a eutectic reaction l = α − Al + θ′. (a) Trace schematically the metastable phase diagram between Al and Al2 Cu. (b) Is 𝑘 𝛼 different for both phase diagrams?

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6.14 There are 3 classic microsegregation models (Appendix 12): i. Lever rule (𝐷𝑠 → ∞) ii. Gulliver-Scheil equation (𝐷𝑠 → 0) iii. Brody-Flemings equation (0 < 𝐷𝑠 < ∞) The classical models all assume that the liquid ahead of the solid-liquid interface is homogeneous and the differences appear in the solid-state diffusion only. a. Which phenomena can lead to 𝐶𝑙 = const. in space ? b. Evaluate the composition of the last liquid (𝑓𝑠 → 1) for Al-2wt% Cu with the lever rule and the Gulliver-Scheil models. c. What happens in the case of a large solid state diffusion coefficient as is the case for P in Fe, making α > 0.5 ? d. The solid composition at the interface as a function of 𝑓𝑠 can be obtained simply by multiplying the liquid composition with the partition coefficient. Does this curve correspond to the microsegregation profile observed in a casting? e. The initial condition for the concentration distribution in the three classical models (iiii) is given by 𝐶𝑙 = 𝐶0 . Why is it wrong in the case of the interdendritic composition profile? 6.15 Examining Fig. 6.7 estimate the range of volume fraction where easy transverse shear deformation is not possible. This limit is important for hot cracking tendency.

References and Further Reading Transients ▪ J.A.Dantzig, M.Rappaz, Solidification, EPFL Press, Lausanne 2016. ▪ F.L.Mota, N.Bergeon, D.Tourret, A.Karma, R.Trivedi, B.Billia, Initial transient behavior in directional solidification of a bulk transparent model alloy in a cylinder. Acta Materialia, 85 (2015) 362. ▪ V.G.Smith, W.A.Tiller, J.W.Rutter, A mathematical analysis of solute redistribution during solidification, Canadian Journal of Physics, 33 (1955) 723. ▪ Y.Song, D.Tourret, F.L.Mota, J.Pereda, B.Billia, N.Bergeon, R.Trivedi, A.Karma, Thermalfield effects on interface dynamics and microstructure selection during alloy directional solidification, Acta Materialia, 150 (2018) 139.

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Microsegregation ▪ T.F.Bower, H.D.Brody, M.C.Flemings, Measurement of solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (1966) 624. ▪ H.D.Brody, M.C.Flemings, Solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (1966) 615. ▪ T.W.Clyne, W.Kurz, Solute redistribution during solidification with rapid solid-state diffusion, Metallurgical Transactions A, 12 (1981) 965. ▪ E.A.Feest, R.D.Doherty, Dendritic solidification of Cu-Ni alloys: part II. The influence of initial dendrite growth temperature on microsegregation, Metallurgical Transactions, 4 (1973) 125. ▪ M.E.Glicksman, R.N.Hills, Non-equilibrium segregation during alloy solidification, Philosophical Magazine A, 81 (2001) 153. ▪ G.H.Gulliver, Metallic Alloys, Griffin, London, 1922. ▪ D.H.Kirkwood, D.J.Evans, in The Solidification of Metals, Iron and Steel Institute, London, Publication 110, 1968. ▪ S.Kobayashi, Mathematical analysis of solute redistribution during solidification based on a columnar dendrite model, Transactions of the Iron and Steel Institute of Japan, 28 (1988) 728. ▪ I.Ohnaka, Mathematical analysis of solute redistribution during solidification with diffusion in solid phase, Transactions of the Iron and Steel Institute of Japan, 26 (1986) 1045. ▪ W.G.Pfann, Zone Melting, 2nd Edition, Wiley, New York, 1966. ▪ J.A.Sarrel, G.J.Abbaschian, Effect of solidification rate on microsegregation, Metallurgical Transactions A, 17 (1986) 2063. ▪ E.Scheil, Bemerkungen zur Schichtkristallbildung, Zeitschrift für Metallkunde, 34 (1942) 70. ▪ K.Schwerdtfeger, Influence of solidification rate on microsegregation and interdendritic precipitation of manganese sulphide inclusions in a steel containing manganese and carbon, Archiv für das Eisenhüttenwesen, 41 (1970) 923. ▪ S.N.Singh, B.P.Bardes, M.C.Flemings, Solution treatment of cast Al-4.5pct Cu alloy, Metallurgical Transactions, 1 (1970) 1383. ▪ W.A.Tiller, K.A.Jackson, J.W.Rutter, B.Chalmers, The redistribution of solute atoms during the solidification of metals, Acta Metallurgica, 1 (1953) 428. ▪ Y.Ueshima, S.Mizoguchi, T.Matsumiya, H.Kajioka, Analysis of solute distribution in dendrites of carbon steel with δ/γ transformation during solidification, Metallurgical Transactions B, 17 (1986). ▪ V.R.Voller, C.Beckermann, A Unified Model of Microsegregation and Coarsening, Metallurgical and Materials Transactions A, 30 (1999) 2183. ▪ J.A.Warren, W.J.Boettinger, Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Materialia, 43 (1995) 689. Modelling Heat Flow, Convection and Segregation ▪ T.W.Caldwell, A.J.Campagna, M.C.Flemings, R.Mehrabian, Refinement of dendrite arm spacings in aluminum ingots through heat flow control, Metallurgical Transactions B, 8 (1977) 261. ▪ T.W.Clyne, Numerical modelling of directional solidification of metallic alloys, Metal Science, 16 (1982) 441. ▪ J.A.Dantzig, M.Rappaz, Solidification, EPFL Press, Lausanne 2016. ▪ M.C.Flemings, Solidification Processing, McGraw Hill, New York, 1974

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▪ G.H.Geiger, D.R.Poirier, Transport Phenomena in Metallurgy, Addison Wesley, 1973. ▪ P.N.Hansen, in Solidification and Casting of Metals, The Metals Society, London, 1979. ▪ R.Mehrabian, M.C.Flemings, Macrosegregation in ternary alloys, Metallurgical Transactions, 1 (1970) 455. ▪ M.Rappaz, Modelling of microstructure formation in solidification processes, International Materials Reviews, 34 (1989) 93.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 149-171 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

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CHAPTER SEVEN

RAPID SOLIDIFICATION MICROSTRUCTURES Rapid solidification processing is an important topic in solidification research and has proven to be useful in various applications, such as welding and additive manufacturing. In the latter processes, various techniques are used for the flexible manufacture of complex parts. One such technique is the laser-deposition process, in which a localised heat-source moves through the workspace while simultaneously melting the substrate and an injected powder. Components of very complex form can be built up in this way. Rapid solidification processing is generally understood to mean the use of high cooling rates or large undercoolings to produce high rates-of-advance (typically 𝑉 > 1 cm/s) of the solidification front. Under such conditions, the low Péclet number approximations which have been developed in the preceding chapters, and which assume that the characteristic diffusion distance, 𝐷/𝑉, is larger than the scale of the microstructure, are no longer valid and more general solutions are required. These are the subject of this chapter. The equations which will be developed are useful at both low and high growth rates. Attention is here concentrated on growth because, provided that the cluster populations change sufficiently rapidly, the nucleation models which were presented in Chap. 2 are affected only slightly by the rapid solidification conditions. At very high cooling rates, however, the steady-state nucleation models over-estimate the nucleation rate (see Kelton and Greer, 1986/2010). Depending upon the nucleation or growth temperature, metastable phases can form. Knowledge of the metastable phase diagram is thus required for modelling the resultant microstructure development. Rapid growth can occur for one of two reasons: ▪ High undercooling of the melt, which can be achieved by slow cooling in the absence of efficient heterogeneous nucleants (bulk undercooling) or by the rapid quenching of droplets (powder fabrication).

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Top image: Additive manufacturing is an advanced rapid-solidification method for producing complex parts in small batches. The parts are built up by, for example, scanning a laser beam (power 𝑃, beam-diameter, 𝐷𝑏 , and velocity, 𝑉𝑏 ) which melts the injected powder (mass flow rate, 𝑚̇) which then solidifies over successive traces. The technique thus resembles that of ‘3D printing’. (See also Chap. 8).

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▪ Rapidly moving temperature fields, as observed during surface-treatment or welding involving high power-density sources such as lasers or electron beams (see the heading of this chapter). Melt-spinning and rotating chill-block casting also belong to this class of process. 7.1. Departure from Local Equilibrium This section describes what happens when the moving solid/liquid interface is no longer at equilibrium. As shown in Table 7.1, the concept of local equilibrium at the solid/liquid interface (with 𝑘 and 𝑚 corresponding to equilibrium conditions) is no longer valid when the growth rate becomes large. This situation will arise in an alloy when the rate of interface displacement, 𝑉, approaches the rate of chemical diffusion over an interatomic distance. If 𝑉 is of the same order-of-magnitude as the diffusion-rate, the newly-formed crystal will not have time to adapt its chemical potential by altering its composition, with the result that solidification becomes diffusionless (Fig. 7.1).

Figure 7. 1 Loss of Local Equilibrium at the Solid/liquid Interface When the interface moves at a low rate, the atomic movements will be rapid enough to permit local equilibrium to be established (equality of chemical potentials, i.e. 𝜇𝑠 = 𝜇𝑙 for both the solvent and solute elements) (a). This is made possible when 𝑉 is much smaller than the diffusion rate, 𝐷𝑖 /𝛿𝑖 , where 𝐷𝑖 is the diffusion coefficient of solute species within the thickness, 𝛿𝑖 , of the diffuse solid/liquid interface. The velocity-dependent distribution coefficient, 𝑘𝑣 , tends then towards 𝑘; the equilibrium value. At the other extreme (b), the growth rate is so high that the solute atoms are frozen into the solid with the same composition as they arrive at the interface (an effect known as solute-trapping). In this case, where 𝑉 is larger than 𝐷𝑖 /𝛿𝑖 , 𝑘𝑣 tends towards unity and the chemical potential of the solute element in the solid, 𝜇𝑠 , will be higher than that of the liquid, 𝜇𝑙 . The chemical potential gradient is the cause of the back-flux of solute atoms which, at very high rates, will be much smaller than the rate of incorporation of atoms.

The diffusion rate is given by the ratio, 𝐷𝑖 /𝛿𝑖 , where 𝐷𝑖 is the interface diffusion coefficient in the growth direction (which is smaller than the bulk liquid diffusion coefficient) and 𝛿𝑖 is the thickness of the diffuse solid/liquid interface, of interatomic dimensions, which characterises compositional

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rearrangement (Fig. 2.19). As an example, when 𝐷𝑖 ∼ 10−10 m2/s and 𝛿𝑖 ∼ 10−9 m, the critical growth rate will be of the order of 0.l m/s. Under these conditions, and following Baker and Cahn (1971) and Boettinger and Coriell (1984, 1986), one must express the interface temperature, 𝑇 ∗ , and the composition of the solid at the interface, 𝐶𝑠∗ , as: 𝑇 ∗ = 𝑇(𝑉, 𝐶𝑙∗ ) − Γ𝐾

[7.1]

𝐶𝑠∗ = 𝐶𝑙∗ 𝑘(𝑉, 𝐶𝑙∗ )

[7.2]

where 𝑇(𝑉, 𝐶𝑙∗ ) and 𝑘(𝑉, 𝐶𝑙∗ ) reflect the changes in phase equilibrium caused by the high growth rate. At equilibrium, 𝑉 = 0 and the two functions, Eqs 7.1 and 7.2, are simply related to the phase diagram. That is, 𝑇(0, 𝐶𝑙∗ ) represents the equilibrium liquidus line and 𝑘(0, 𝐶𝑙∗ ) is the equilibrium distribution coefficient. The effect of curvature upon 𝑘 will be neglected here. Table 7.1 Rapid Solidification Phenomena (after Boettinger and Coriell, 1986).

If the atoms have no time to rearrange themselves in the interface, so as to equalise the chemical potentials of the two phases, complete solute-trapping will result. A crystal can be formed from its melt, with no change in composition, under specific thermodynamic conditions (Baker and Cahn, 1971). It will occur only when the system can decrease its Gibbs free energy. For this to be true, the interface temperature must drop below the 𝑇0 -line of the corresponding phase diagram (Fig. 7.2). (𝑇0 is the temperature of equality of the Gibbs free energies of the phases.) To be thermodynamically consistent, there are now two conditions to be fulfilled: one permits the liquid and solid compositions at the interface to converge at high rates (𝑘𝑣 tends to unity, Fig. 7.3(a)) and the other demands that this can happen only if the interface is below the 𝑇0 -temperature of the corresponding composition (Fig. 7.3(b,c)). A simple relationship for the first phenomenon has been given by Aziz (1982) (see Appendix 6):

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𝑘𝑣 =

𝑘 + (𝛿𝑖 𝑉/𝐷𝑖 ) 1 + (𝛿𝑖 𝑉/𝐷𝑖 )

[7.3]

where the parameter, 𝛿𝑖 𝑉/𝐷𝑖 , can be regarded as being an interface Péclet number (𝑃𝑖 ). Figure 7.4 shows the behaviour of this expression as a function of (log 𝑉).

Figure 7.2 The Importance of 𝑻𝟎 for Diffusionless Transformation According to the principles of thermodynamics complete solute-trapping or diffusionless solidification (𝑘𝑣 = 1) can occur only if, during the transformation process, the Gibbs free energy of the system is reduced, i.e. if the interface temperature is below 𝑇0 . Therefore 𝑇0 is the highest temperature at which -crystals can form with the same composition as the melt. It is the temperature at which both phases have the same free energy. 𝑇0 is a function of composition and lies between the liquidus and the solidus temperatures. The grey region represents the range of thermodynamically allowed compositions of -crystals that can form when a liquid of composition, 𝐶𝑙∗ , solidifies.

The second condition to be satisfied ensures that 𝑇 ∗ ≤ 𝑇0 when 𝑘𝑣 → 1. If the attachment kinetics of the solvent are also included in the treatment, but the curvature undercooling is neglected for the moment, one obtains for Eq. 7.1 (Appendix 6): 𝑇 ∗ = 𝑇𝑓 + m′𝐶𝑙∗ −

𝑅𝑔 𝑇𝑓 𝑉 Δ𝑆𝑓 𝑉0

[7.4]

with the velocity-dependent liquidus slope, 𝑚′ : 𝑚′ = 𝑚[1 + 𝑓(𝑘)]

and

𝑓(𝑘) =

𝑘 − 𝑘𝑣 [1 − ln( 𝑘𝑣 /𝑘)] (1 − 𝑘)

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[7.5]

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Figure 7.3 Liquidus and Solidus Temperatures as a Function of Interface Velocity At low velocities the liquidus and solidus curves correspond to the equilibrium values. At higher velocities local equilibrium is lost, 𝑘𝑣 rises if k < 1 (a); the liquidus curve 𝑇𝑙 (𝑉) falls, the solidus curve 𝑇𝑠 (𝑉) rises and both temperatures approach one another at 𝑇0 . At high velocities, the attachment-kinetics contribution Δ𝑇𝑘 also becomes dominant, i.e. the corresponding undercooling increases (dashed curve in b) and pushes the solidus and liquidus curves to lower temperatures (c).

Equation 7.5 is the thermodynamic correction to the interface temperature for dilute solutions. In the third term of Eq. 7.4, the velocity 𝑉0 corresponds to the limit at which crystallisation, i.e. atom attachment to the crystal, can still occur. The upper limit to 𝑉0 is the collision-limit, the velocity of sound, which is of the order of some 1000 m/s for metals. It can be seen that, at low growth rates (𝑘𝑣 → 𝑘 and 𝑉 ≪ 𝑉0), 𝑓(𝑘) becomes equal to zero, i.e. 𝑚′ = 𝑚 and the third term on the RHS of Eq. 7.4 approaches zero, leading to Eq. 3.17 when 𝐾 ∗ (the curvature of the interface) is nil.

Figure 7.4 The Variation of the Distribution Coefficient with Growth Rate The non-equilibrium distribution coefficient, 𝑘𝑣 , changes, over a critical velocity range, from the equilibrium value, 𝑘, to unity. At high growth rates, diffusionless solidification therefore occurs and the interface temperature must be below 𝑇0 . The variation in 𝑘𝑣 has a marked effect upon the microstructures formed under rapid solidification conditions, since the degree of solute rejection will depend upon its magnitude.

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7.2. Absolute Stability As has already been shown in Fig. 4.12, a planar interface will also become stable at sufficiently high growth rates. This happens when the solute diffusion distance, 𝛿𝑐 (∼ 𝐷/𝑉), approaches the solute capillary length, 𝑠𝑐 (∼ Γ/Δ𝑇0 ) (Fig. 4.29). The critical growth rate for absolute stability, 𝑉𝑎 (Appendix 7) is: Δ𝑇0𝑣 𝐷 (𝑉𝑎 )𝑐 = 𝑘𝑣 Γ

[7.6]

The suffix, c, indicates that this limiting velocity is controlled by solute diffusion. The main physical reason for this, at first sight strange, phenomenon is that the diffusion distance narrows with increasing growth rate and diffusion becomes more and more localised (Fig. 7.5), thus reducing lateral diffusion. At high rates, capillarity also becomes a dominant feature of the process and acts against refinement of the microstructure. In other words, the diffusion-distance decreases with 𝑉 −1 , while the microstructure (e.g. the wavelength, 𝜆𝑖 , of Eq. 3.26) decreases as 𝑉 −1/2. At some critical velocity, the microstructure therefore becomes too coarse for lateral diffusion to occur, thus leading to stabilization of a flat interface.

Figure 7.5 Diffusion Field Far and Near to Absolute Stability At low growth-rates, the solute diffusion distance (𝐷/𝑉) is large (a), while at high rates (b), it becomes so small that capillarity phenomena become dominant, due to the fineness of the microstructure that forms. This phenomenon again stabilises a planar interface at high growth rates. The critical growth rate at which this occurs is called the rate of absolute stability, 𝑉𝑎 . The stabilisation of a planar interface can be anticipated from the fact that, at 𝑉 = 𝑉𝑎 , the concentration gradients ahead of the tips and depressions of a perturbation are the same. The difference in growth-rate between the tips and depressions, which is the fundamental reason for microstructure formation, therefore vanishes.

From the above one can now conclude that, for a given alloy and a positive temperature gradient, planar growth will always occur when the growth rate and/or the temperature gradient are sufficiently high. With increasing growth rate and not too high a temperature gradient, there will be a transition from plane front to cells, to dendrites, to cells again, and back to plane front (Fig. 7.6 and Fig. 4.14). Note that Equation 7.6 is valid only for temperature gradients such that 𝐺 ≪ 𝐺𝑐𝑟𝑖𝑡 ≈

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𝑘𝑣 Γ𝑉 2 /𝐷2 where 𝐺𝑐𝑟𝑖𝑡 corresponds to the condition 𝑉𝑐 = 𝑉𝑎 (Fig. 7.6); that is, under most conditions of laser surface treatment where 𝐺 ∼ 106 K/m and 𝑉 > 1 cm/s.

Figure 7.6 Ranges of Interface Morphologies in Directional Solidification When a positive temperature gradient, 𝐺 1 , is imposed upon an alloy of a given composition, 𝐶0 , there is, at growth-rates of around 𝑉𝑐 (= 𝐺𝐷⁄Δ𝑇0), a transition from a planar to a cellular morphology due to constitutional undercooling. With increasing 𝑉, cells transform into dendrites and then a reverse transition to cells is found. Finally, there is a transition from cells to plane front at absolute stability (𝑉𝑎 = Δ𝑇0𝑣 𝐷/𝑘𝑣 Γ). This transition is not direct but instead exhibits an oscillatory behaviour over a certain velocity-range. Grey bands of cells parallel to the growthfront alternate with white bands corresponding to plane fronts as shown in the micrograph on the right (band spacing = 450 nm, cell spacing = 50 nm). 𝑉𝑎 is essentially independent of the temperature gradient. Above a certain temperature gradient, 𝐺𝑐𝑟𝑖𝑡 , the plane front is always stable.

7.3. Interface Response Plane Front: Assume firstly that a single-phase planar front has reached a steady state and, for some reason, is morphologically stable over the whole range of interface velocities. At low velocities, local equilibrium maintains the plane-front interface temperature, 𝑇𝑝 , at the equilibrium solidus temperature, 𝑇𝑠 (see Chap. 3). With increasing growth-rate, two non-equilibrium effects change 𝑇𝑝 according to: 𝑇𝑝 = 𝑇𝑓 +

𝑚′𝐶0 𝑉 − 𝑘𝑣 𝜇𝑘

[7.7]

where 𝑇𝑓 is the melting point and 𝜇𝑘 is the atom-attachment kinetics coefficient (Eq. 7.4, 3rd term). This equation describes the evolution of the planar interface temperature with velocity, as shown by the dashed black 𝑇𝑠 -curve in Fig. 7.3(b). (It is assumed here that, for dilute solutions, the liquidus and solidus curves can be approximated by straight lines, both of which have their origin at the melting point of the pure substance). Following a low-𝑉 range of constant 𝑇𝑝 (= 𝑇𝑠 ), the planar interface temperature increases due to a changing 𝑘𝑣 (second term on the RHS of Eq. 7.7) and approaches 𝑇0 at the maximum velocity. With a continually increasing interface velocity, the plane-front temperature decreases due to the now-dominant attachment-kinetics term, Δ𝑇𝑘 (third term on RHS of Eq. 7.7). Over this wide range of interface velocities, the liquidus temperature, 𝑇𝑙 , which at first is constant, decreases at high 𝑉 and becomes equal to 𝑇𝑠 when 𝑘𝑣 = 1 (Fig. 7.3). The liquidus-solidus temperature-interval changes according to:

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Δ𝑇0 (V) = 𝐶0 m′

𝑘v − 1 𝑘v

[7.8]

Figure 7.7 shows the results of two different modelling approaches to plane-front and cell/dendrite growth: (a) according to the models developed in this chapter (Eqs 7.7 and 7.17), and (b) the result of a linear stability analysis of a plane front by Huntly and Davis (1993, 1996). Four critical velocities can be observed: 𝑉𝑐 , constitutional undercooling, 𝑉𝑘 , decreasing driving force with increasing growth rate, 𝑉𝑎 , absolute stability, and 𝑉𝑚 , the maximum interface temperature, due to the increasing influence of attachment kinetics.

Figure 7.7 Microstructures of Al-Fe Alloys across a Wide Range of Interface Velocities The figure represents the single-phase interface temperatures and stability ranges as a function of velocity for Al-Fe and 𝐺 = 5 × 106 K/m. In (a) two steady-state interface temperature functions for Al-1 at% Fe are shown: (i) the lower curve is the plane-front temperature (similar to the solidus curve of Fig. 7.3(b), stable at low and high 𝑉 (red) and unstable in-between (dashed black); (ii) the upper red curve between constitutional undercooling, 𝑉𝑐 , and absolute stability, 𝑉𝑎 , represents the tip-temperature of cells (close to the extremes, 𝑉𝑐 and 𝑉𝑎 ) and dendrites for intermediate conditions. The full red curve, the Interface Response, spanning 8 orders-of-magnitude of 𝑉 corresponds to the envelope of observable interface temperatures of all morphologies as a function of velocity: plane front-cells-dendrites-cells-bands-plane front. In (b) the result of a linear stability theory for a plane front by Huntly and Davis (1993) are indicated in the form of two instability domains: (i) morphological instability inside the continuous black curve and (ii) oscillatory instability inside the broken curve. The composition of the alloy (1 at% Fe in (a)), is indicated by the horizontal red line in (b). The intersections of this line with the instability-limits correspond to the limits in (a), morphological instability between 𝑉𝑐 and 𝑉𝑎 , and oscillatory instability between 𝑉𝑘 and 𝑉𝑚 . Experimental results on the oscillatory instabilities known as banding are found close to 𝑉𝑎 (Gremaud et al., 1991), and agree with the theory.

Upon comparing the results of the analytical steady-state models (Fig. 7.7(a)) with the quantitative linear stability analysis of Huntley and Davis (1993) (Fig. 7.7(b)) the agreement is found to be astonishingly good. Figure 7.7(b) contains two regions: (i) morphological (spatial) instabilities and (ii) oscillatory (temporal) instabilities. The first region encompasses constitutional undercooling

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and absolute stability while the second covers the range of positive slope, d𝑇/d𝑉, between 𝑉𝑘 and 𝑉𝑚 . The experimental data on banding confirm the predictions of the models. Cells and dendrites: When a planar interface becomes unstable, above 𝑉𝑐 , it firstly reorganises into cells which interact strongly with one another. The increasing cell tip-temperature beyond 𝑉𝑐 (decreasing undercooling, Fig. 7.7(a)) can be approximated by using the approach developed by Bower et al. (1966): Δ𝑇 =

𝐺𝐷 𝑉

[7.9]

which corresponds to constitutional undercooling, i.e. when Δ𝑇 = Δ𝑇0. At higher velocities, the cell-tips become morphologically unstable and develop side-arms. The tips simultaneously become sharper and take on the shape of a paraboloid-of-revolution. Because the solute-diffusion fields of neighbouring dendrites, unlike those of cells, do not interact strongly when the dendrites are well-developed, the dendritic arrays can often be approximated by treating isolated dendrite tips. To model tip-growth, the solute-field alone is taken into account to a first approximation, together with the marginal stability criterion (Chap. 4). The interface-response, indicated by the red curve in Fig. 7.7(a), is the envelope of all of the microstructures which develop as a function of 𝑉: plane-front, cells, dendrites, cells, bands and plane-front (Trivedi and Kurz, 1994). According to the marginal stability criterion, the wavelength of the marginally stable state, 𝜆𝑖 , permits the choice of the tip radius (Eq. 4.6). This value can be obtained from the generalized stability equation (compare this with the third bracket on the right-hand-side of Eq. 3.23): −Γ𝜔2 − 𝐺 ∗ + 𝑚′𝐺𝑐∗ = 0

[7.10]

where 𝜔 = 2π/𝜆, the effective temperature gradient, 𝐺 ∗ = 𝜅̄ 𝑙 𝐺𝑙 𝜉𝑙 + 𝜅̄ 𝑠 𝐺𝑠 𝜉𝑠 , 𝜅̄ 𝑙 = 𝜅𝑙 /(𝜅𝑠 + 𝜅𝑙 ), 𝜅̄ 𝑠 = 𝜅𝑠 /(𝜅𝑠 + 𝜅𝑙 ), 𝐺𝑙 and 𝐺𝑠 are the temperature gradients in the liquid and solid, respectively, at the interface and 𝐺𝑐∗ (= 𝐺𝑐 𝜉𝑐 ) is the effective concentration gradient in the liquid at the solid/liquid interface. The stability parameters, 𝜉𝑙 , 𝜉𝑠 , and 𝜉𝑐 , are functions of the corresponding Péclet numbers and are given in Appendix 7 (Eq. A7.36). From the above equation the critical wavelength for zero amplification of the instabilities is 𝜆𝑖 = {(1/𝜎 ∗ ) [

1/2 Γ ]} (𝑚′𝐺𝑐∗ − 𝐺 ∗ )

[7.11]

where 𝜎 ∗ is a stability constant which is of the order of 1/42. Equation 7.11 has the same form as that of Eq. 3.22, except that 𝜙 is here the difference in the effective gradients, 𝑚′𝐺𝑐∗ − 𝐺 ∗ , which in turn depends upon the stability parameters, 𝜉𝑙 , 𝜉𝑠 , and 𝜉𝑐 . The latter three parameters approach unity at small Péclet numbers, and Equation 7.11 approaches Eq. 3.22. Stability is again obtained when the denominator of Eq. 7.11 becomes equal to zero. This happens at low growth rates: ▪ when m𝐺𝑐∗ = 𝐺 ∗ , and at high growth rates: ▪ under conditions of directional solidification (𝐺 ∗ > 0) when the effective concentration gradient, 𝐺𝑐∗ , approaches zero (𝜉𝑐 → 0 when 𝑃𝑐 ≫ 1). Banding: At higher velocities, a cellular microstructure is again stabilized. Increasing the velocity tends to make the microstructure finer. Diffusion becomes localized, and capillarity predominates, so that plane-front growth is eventually stabilised again (Fig. 7.8). This occurs at around the limit of EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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absolute stability, given by Eq. 7.6. Solute-trapping is important within this growth-rate range and leads to an increasing interface temperature (Fig. 7.7(a)). The decreasing undercooling with increasing growth-rate drives oscillatory instabilities, i.e. the interface accelerates, decelerates, accelerates, etc., thus forming a banded microstructure (Kurz and Trivedi, 1996). Atom-attachment kinetics become predominant at still higher growth-rates, leading to a maximum in the interface response. From then on, the interface temperature decreases again with increasing growth rate and the plane front becomes absolutely stable.

Figure 7.8 Phase-Field Result of Cell Growth Close to Absolute Stability At high velocities columnar dendrites change into cells. Phase-field computations by Boettinger and Warren (1999) show the refinement of the cellular structure of a Ni-Cu alloy with growth rate, leading to plane front at absolute stability at 𝑉 = 4 cm/s.

7.4. Rapid Dendritic Growth The equations for equiaxed and columnar dendrites which were developed in Chap. 4 will here be extended to rapid solidification conditions. For simplicity, the kinetic undercooling due to atomattachment will be neglected at first, but then re-introduced into the equations later. The capillarity-corrected transport equation can then be written in terms of the undercooling as: Δ𝑇 = Δ𝑇𝑡 + Δ𝑇𝑐 + Δ𝑇𝑟

[7.12]

where the various undercoolings are defined by Eqs A9.17 - A9.19. As shown in Fig. A8.1, Δ𝑇𝑡 is the thermal undercooling due to latent-heat release at the tip, Δ𝑇𝑐 is the solutal undercooling due to solute rejection by the dendrite and Δ𝑇𝑟 is the curvature undercooling. As in Chap. 4, the second equation which is required to solve the problem is obtained by applying stability or, to be more precise, solvability arguments. This defines the tip radius as a function of the local effective gradients of composition, 𝐺𝑐∗ , and temperature, 𝐺 ∗ , via Eq. 7.11, assuming that 𝑅 = 𝜆𝑖 , 𝐺𝑐∗ and 𝐺 ∗ are evaluated at the dendrite tip. Substituting Eq. 7.11 (with 𝑅 replacing 𝜆𝑖 ) into Eq. 7.12 gives differing relationships for the two cases of columnar and equiaxed growth.

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Columnar growth (𝑮 > 0) As shown in Fig. 4.8, there is an imposed macroscopic heat flux, from the superheated liquid to the solid, which produces in this case a positive temperature gradient at the dendrite tips. If the thermal conductivities and diffusivities of the liquid and solid are assumed to be equal (𝑙 = 𝑠 =  and 𝑎𝑙 = 𝑎𝑠 = 𝑎) and if, for simplicity, the effect of latent heat is neglected, the temperature gradients in both phases at the tip are also equal, i.e. 𝐺𝑙 = 𝐺𝑠 = 𝐺. In this case Eq. 7.11, with 𝑅 = 𝜆𝑖 , reduces to (see Eq. A7.13): 𝑅 = {(1/𝜎

∗)

1/2 Γ [ ′ ]} (𝑚 𝐺𝑐 𝜉𝑐 − 𝐺)

(𝐺 > 0)

[7.13]

Note that in this case the increase in 𝜉𝑠 cancels-out the decrease in 𝜉𝑙 with changing Péclet number and the effective temperature gradient, 𝐺 ∗ (Eq. 7.10), becomes equal to the imposed temperature gradient, 𝐺 (see Appendix 7). Noting furthermore that the imposed temperature field does not create any thermal undercooling in the steady state (since the temperature field is imposed from the exterior, i.e. depends upon the heating/cooling system and not upon the dendrite tip), Equation 7.12 reduces to: [7.14]

Δ𝑇 = Δ𝑇𝑐 + Δ𝑇𝑟

By substituting Δ𝑇𝑐 and Δ𝑇𝑟 from Eqs A9.18 and A9.19 into Eq. 7.14 one obtains the general result for columnar growth (corresponding to Eq. 4.14 at low V): 𝑉 2 𝐴′ + 𝑉𝐵 ′ + 𝐺 = 0

[7.15]

where 𝜋2Γ 𝜃𝑐 𝜉𝑐 and 𝐵′ = 2 2 P𝑐 𝐷 𝐷 The unit solutal undercooling, 𝜃𝑐 , is here equal to (Δ𝑇0 𝑘𝑣 A(𝑃𝑐 )), where A(𝑃𝑐 ) = 𝐶𝑙∗ /𝐶0 (Eq. A9.18(a)), and the stability function, 𝜉𝑐 , is given by Eq. A7.36(c). The growth rate, 𝑉, and the tip radius, 𝑅, can be deduced from Eq. 7.15 for various Péclet numbers. This leads to the relationship between 𝑅 and 𝑉 which is shown in Fig. 7.9 for different 𝐶0 and 𝐺 values. Figure 7.9(b) is similar to Fig. 4.11, the important difference being that due to the decrease in 𝜉𝑐 at high Péclet numbers, 𝑅 increases with increasing 𝑉 before reaching the limit of absolute stability, 𝑉𝑎 . This increase in 𝑅 arises from localisation of the diffusion field and indicates a tendency firstly to cellular and then, beyond 𝑉𝑎 , to planar growth (banding is neglected here for the moment). To calculate the tip concentration, use is made of the definition of the supersaturation (Eq. A8.1), leading to: 𝐴′ =

𝐶𝑙∗ = 𝐶0 𝐴( 𝑃𝑐 )

and

𝐶𝑠∗ = 𝑘𝑣 𝐶𝑙∗

[7.16]

The tip temperature can finally be calculated by using Eq. 7.4, to which the curvature undercooling has been added to give: 𝑇 ∗ = 𝑇𝑓 −

𝑅𝑔 𝑇𝑓 V 2Γ + 𝐶𝑙∗ 𝑚[1 + f(𝑘)] − 𝑅 Δ𝑆𝑓 𝑉0

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[7.17]

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Figure 7.9 Columnar Dendritic Growth as a Function of Composition (a) and Temperature Gradient (b) Within the cellular/dendritic regime of Fig. 7.6, the typical variation of the tip radius as a function of growth rate is of the form shown. When close to the limit of constitutional undercooling, 𝑉𝑐 , (large negative slope of the curves) cells will form, then dendrites and finally (close to the limit of absolute stability, 𝑉𝑎 ) cells will again be observed. Thereafter 𝑅 increases with increasing 𝑉. A change in composition will modify the entire range of microstructures formed (a) while a change in 𝐺 will modify the cellular regime at low rates (b). Note that in these diagrams the regimes close to 𝑉𝑐 and 𝑉𝑎 are the predictions of dendrite-growth equations and do not well-represent cellular growth.

It becomes evident that, at high growth rates, the variation of 𝐶𝑙∗ , and the undercooling (𝑇𝑙 − 𝑇 ∗ ) can become large and that certain alloy parameters may drastically change. These parameters are the velocity-dependent distribution coefficient, 𝑘𝑣 , (Eq. 7.3 and Fig. 7.4) which is also a function of temperature via the equilibrium distribution coefficient and the temperature-dependent diffusion coefficient, 𝐷(𝑇). Figure 7.10 shows how these variations will affect the concentration versus growth rate curve. Figure 7.11 shows, superposed on the linearised phase diagram, how the liquid and solid compositions and the temperature of the tip will change as a function of the growth rate. Equiaxed growth in undercooled melts (𝑮𝒍 < 𝟎) In the case of growth into an undercooled melt, the solid formed is at the temperature of the tip, which is the result of the overlapping thermal diffusion fields. The temperature gradient in the solid, 𝐺𝑠 , is equal to zero, making the term, 𝜅̄ 𝑠 𝐺𝑠 𝜉𝑠 , in 𝐺 ∗ of Eq, 7.10 equal to zero. One thus obtains for the dendrite tip radius, assuming equal conductivities in liquid and solid: 1/2 Γ 𝑅 = {(1/𝜎 ) [ ]} (𝑚′𝐺𝑐 𝜉𝑐 − 0.5𝐺𝑙 𝜉𝑙 ) ∗

(𝐺 < 0)

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[7.18]

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Figure 7.10 Growth-Rate Dependent Composition of Columnar Dendrite Tips The composition of the dendrite tips which corresponds approximately to the minimum in concentration of the microstructure depends very much upon the completeness of the physical modelling. At high growth rates (and high tip undercoolings), the parameters which depend upon 𝑉 and 𝑇 (i.e. 𝑘𝑣 , 𝑘(𝑇), 𝐷(𝑇)) will have a marked effect upon the results of the calculations. The highest curve takes account of all the possible variations in the above parameters for the binary system, Ag-5%Cu (Kurz et al., 1986).

Figure 7.11 Interface Temperatures – Concentrations for Laser Resolidification of an Ag-5wt%Cu Alloy The liquid and solid compositions and the temperature of the dendrite tip, as a function of growth rate, evolve as indicated (the linearised Ag-Cu equilibrium phase diagram is also plotted for reference). With increasing solidification rate the tip composition-temperature locus remains slightly below the liquidus and solidus, due to curvature undercooling. However, above a certain rate (where 𝑘𝑣 deviates from 𝑘) the solid composition increases more rapidly and approaches 𝐶0 at 𝑉𝑎 .

The full transport equation (7.12) now has to be solved since, in undercooled alloy melts, the tip is a source of both heat and solute. Evaluating the gradients with the aid of the usual flux balance Eqs A9.24 - A9.26 one obtains: 𝑅=

(Γ/𝜎 ∗ ) (2𝑃𝑐 𝜃𝑐 𝜉𝑐 + 𝑃𝑡 𝜃𝑡 𝜉𝑡 )

[7.19]

where 𝜉𝑡 = 𝜉𝑙 (Eq. A7.36(a)) and 𝜉𝑐 is defined by Eq. A7.36(c). One eventually finds the undercooling, tip radius and growth rate as shown in Appendix 9. Due to the presence of the solutal

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and thermal fields around the dendrite tip, the behaviour is more complicated in undercooled alloys (Fig. 7.12). Above a critical undercooling, Δ𝑇 + (Fig. 7.13), the distribution coefficient approaches unity and the dendrite becomes a purely thermal one. The curve for 𝑉(Δ𝑇) then joins the curve for 𝐶0 = 0. This abrupt transition can also be seen in Fig. 7.14.

Figure 7.12 Dendritic Growth in Undercooled Melts (𝒌 = constant) Undercooled growth, which produces many equiaxed grains in a casting under normal conditions, will not necessarily do so during rapid solidification. In highly undercooled powder particles for example, one particle usually equates to just one grain. Growth will be controlled by the undercooling and the related thermal diffusion fields at the dendrite tips will modify the overall growth behaviour as compared with that of columnar growth. It is shown here how, for an alloy, the tip radius (a) and the growth rate (b) change with undercooling (𝑘 = constant). For the chosen composition, and at low undercoolings, the dendrites are essentially solute-diffusion controlled. When these dendrites reach an undercooling that results in a velocity corresponding to the limit of absolute stability for solute diffusion, (𝑉𝑎 )𝑐 , a transition from a mostly solutal to a purely thermal dendrite is observed. As the thermal diffusion coefficient, 𝑎, of metals is much higher than the solute diffusion coefficient, 𝐷, the tip radius suddenly increases. At even higher undercoolings, approaching the limit of absolute stability for thermal diffusion, (𝑉𝑎 )𝑡 , the tip radius again increases. Beyond this undercooling (Δ𝑇𝑚𝑎𝑥 = 𝜃𝑡 + Δ𝑇0 ), it is predicted that morphologically stable (planar or spherical) solidification will be preferred. Note however that an absence of microsegregation is predicted above Δ𝑇𝑎 (corresponding to (𝑉𝑎 )𝑐 ). At low Δ𝑇 the slope of the growth-rate variation indicates a relationship of the form, 𝑉 = Δ𝑇 𝑛 , where 𝑛 is between 2.5 and 3, depending upon the composition. This exponent increases sharply as (𝑉𝑎 )𝑡 is approached. The corresponding changes in the stability parameters, 𝜉𝑐 and 𝜉𝑡 , are shown in (c). In this figure 𝑅, 𝑉 and Δ𝑇 are dimensionless quantities as defined by Lipton et al. (1987).

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Figure 7.13 Effect of Solute-Trapping Upon Undercooled Dendrite Growth The high growth rates which can be obtained by imposing large melt undercoolings will make kv tend towards unity. If one takes this effect into account, the 𝑉(Δ𝑇) relationship shown in Fig. 7.12 will drastically change. Above a certain undercooling, Δ𝑇 +, where 𝑘𝑣 tends towards unity, the essentially solutal (alloy) dendrites will become pure thermal dendrites, i.e. they will behave as if their composition, 𝐶0 , is equal to zero. (𝑉 and Δ𝑇 are dimensionless quantities as in Fig. 7.12) (Trivedi, Lipton et al., 1987).

Figure 7.14 Dendrite-Tip Composition as a Function of Melt Undercooling The change in interface composition as a function of the total undercooling is shown (a) for the case of Fig. 7.13. Here 𝑘𝑣 changes abruptly at the critical undercooling, Δ𝑇 + (b), since 𝑉 increases sharply there. A homogeneous solid is formed above Δ𝑇 + (Trivedi, Lipton et al., 1987).

7.5. Rapid Eutectic Growth High solidification velocities, of the order of 10 – 100 cm/s, can be attained during the lasertreatment of metals. Eutectic microstructures having interphase spacings of 17 nm have been obtained† at such growth rates (Fig. 7.15(a)). At around the limit of eutectic growth (corresponding to absolute stability), a wavy eutectic structure, and finally banding, have been observed †

Such a spacing corresponds to the finest ones observed in eutectoid steels, which reach strengths of 2% of the elastic modulus. It is assumed therefore that similarly fine aluminium-based eutectics might attain correspondingly high strengths.

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(Zimmermann et al., 1989). This is analogous to the banding of single-phase alloys, where a transition from cells to bands is found. In eutectics forming complex intermetallic phases with low solubilities, the growth-temperature can fall to such low values that the glass-transition stops crystallisation and leads to the formation of a glass. The latter is depicted in Fig. 7.15(b) for a heavily undercooled Al-38wt% Sm melt. The AlAl1Sm3 eutectic exhibits the same minimum spacing as the Al-Al2Cu eutectic (a).

Figure 7.15 Eutectics at the Limit of Growth: (a) Nanostructure in Al-Cu Eutectic at 0.2m/s and (b) Grain of Quenched Al-Sm Eutectic (λ = 17nm) Surrounded by Glass In a thin solidified layer (a) produced by rapid laser-treatment (corresponding locally to directional solidification) the lower spacing-limit has been reached in Al-Al2Cu eutectic at 0.2 m/s (Zimmermann et al., 1989). During rapid eutectic grain growth in heavily undercooled Al-38wt% Sm melt (Al-Al1Sm3 eutectic) (b), growth stops when the glass transition is reached (Wang et al., 2011). The smallest spacing in both cases is 17 nm.

As presented in Chap. 5, eutectic growth is generally modelled using the theory of Jackson and Hunt (1966). Those authors assumed that 𝑃 = 𝑉𝜆/2𝐷 ≪ 2π (Eq. A10.3 becomes 𝑏 = 2π/𝜆). In the flux balance (Eq. A10.5) furthermore, the interface composition in the liquid is, 𝐶𝑙∗ ≈ 𝐶𝑒 . For example, Equation 5.10 (see also Eq. A10.29) indicates that: 𝜆2 𝑉 =

Γ𝛽 sin 𝜃𝛽 2𝐷 Γ𝛼 sin 𝜃𝛼 [𝑓 − (1 − 𝑓) ] ′ ′ 𝐶𝑃 𝑚𝛽 𝑚𝛼

(𝑃 < 1)

[7.20]

This is certainly a reasonable assumption for low undercoolings but is not so at high undercoolings. Relaxation of the two low Péclet number approximations mentioned above leads to a more general analytical solution valid for any Péclet number (Trivedi, Magnin et al., 1987). For the special case where 𝑘𝛼 = 𝑘𝛽 = 𝑘, the product, 𝐶 ′ 𝑃′ in Eq. 7.20 is replaced by (1 − 𝑘)𝑃′′ . It is seen that the essential difference at high Péclet numbers is the replacement of the series, 𝑃′ , which is a function only of the volume fraction of the minor phase 𝑓 (Eq. A10.31), by the series, 𝑃′′ , which is a function of 𝑓, 𝑘 and P (compare Eq. A10.31 with Eq. A10.45 and consult Figs 4 and 5 in Trivedi, Magnin et al., 1987). As in the case of dendritic growth, 𝜆2 𝑉 is therefore constant only at small Péclet numbers and increases steeply when 𝑃 ≫ 1 (Fig. 7.16). Note that the second term on RHS of Eq. A10.45 is related to ξc (Eq. A 7.36c) which modifies dendrite growth under rapid solidification conditions (Eq. 7.15). This behaviour again leads to a sort of absolute stability for eutectic growth - a limiting rate above which coupled two-phase growth is less favourable than planar single-phase growth. This occurs when the interface temperature reaches the solidus temperature of the - or -phase, whichever is the higher. At that limit, 𝜆 increases strongly, as shown in Fig. 7.17, for distribution coefficients close to unity. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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Figure 7.16 Eutectic Growth at High Péclet Numbers As in the case of dendrites, eutectics also exhibit an increase in the scale of their microstructures at high growth rates (high Péclet numbers). At low growth rates, 𝜆2 𝑉 = constant (as shown in Chap. 5) but its value increases markedly when 𝑃 = 𝜆𝑉/2𝐷 becomes large with respect to unity. It is therefore important to remember that the rule-of-thumb, "finer structures at higher growth rates", is no longer valid at very high growth rates (of the order of 1 m/s) (Trivedi, Magnin et al., 1987).

Figure 7.17 Eutectic Spacing - Growth Rate Relationship for Systems with Various Distribution Coefficients If this figure for a eutectic with large solubility, i.e. a phase distribution coefficient, 𝑘 > 0.5, is compared with Fig. 7.9, the analogy between 𝜆(𝑉) for eutectics and 𝑅(𝑉) for dendrites becomes obvious. That is, both 𝜆 and 𝑅 decrease with 𝑉, before again increasing upon approaching the limit of absolute stability. For low-solubility systems with small 𝑘-values, the behaviour is more complicated due to the increased eutectic-growth undercooling, which decreases 𝐷(𝑇) and bends the curve back before it possibly increases again.

As discussed previously for small solid solubilities of the phases, the eutectic growth undercooling can reach high values. In this case the temperature-dependent diffusion coefficient, 𝐷(𝑇), slows the transformation down, i.e. the 𝑇 − 𝑉 curve for small 𝑘 in Fig. 7.18 bends back. The transformation to an amorphous state can be reached by directional growth from the upper branch of the 𝑇(𝑉) curve (Fig. 7.18, 𝑘 = 0.1) if the glass-transition temperature is above the extremum. If the glass transition is below the extremum, the lower decelerating branch of the eutectic can be reached only from below, i.e. by highly undercooled growth (Fig. 7.15(b), Wang et al., 2011).

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In addition to these high-velocity effects, the thermodynamic condition for solute-trapping has to be taken into account. Figure 7.19 shows the evolution of a eutectic phase diagram for three velocities, with the final three 𝑇0 -curves which satisfy the necessary condition to reach complete solute-trapping. These modifications of the phase equilibria have important consequences for the final eutectic microstructure (Kurz and Trivedi, 1991).

Figure 7.18 Eutectic Interface Temperature - Growth Rate Relationships for Systems with Various Distribution Coefficients The growth undercooling Δ𝑇(𝑉) increases as the 𝑘-value of the eutectic phases decreases (𝑘 < 1). Since the diffusion coefficient decreases with decreasing temperature growth will however ultimately slow down, resulting in a typical transformation curve with its characteristic nose. Extremely fine eutectic structures can therefore be obtained at low temperatures and low growth rates if 𝑘 is sufficiently small, as is observed for example during crystallisation of highly undercooled melts or after solidification from the amorphous state.

Figure 7.19 High Velocity Effect upon a Eutectic Phase Diagram for Al - 𝛉 Al2Cu Figure 7.3 at the beginning of this chapter shows that due to solute-trapping the liquidus and solidus curves converge, at high velocities, to 𝑇 ≤ 𝑇0 . In a eutectic system, one can distinguish three 𝑇0 -curves (for l-α, l-β and α-β) that are reference temperatures for complete solute-trapping (𝑘 = 1). The liquidus and solidus converge to 𝑇0 , as does the eutectic point.

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7.6. Solute Redistribution Another phenomenon whose theoretical treatment has to be modified at high growth rates is the interdendritic solute redistribution. In Chap. 6, the case of microsegregation where solidification starts at the liquidus temperature, i.e. where the tip undercooling is close to zero, has been discussed. Upon further assuming complete mixing in the interdendritic liquid, various segregation equations have been developed (lever rule, Gulliver-Scheil (Glicksman and Hills, 2001), or Brody-Flemings (1966)). However, at the high tip-undercoolings which are observed under rapid solidification conditions, these equations can no longer be applied: a high tip-supersaturation modifies the initial part of the microsegregation curve (Fig. 7.20). An exact solution to this problem is extremely difficult to obtain as one has to take account of the transient regime of solute fields around the dendrite or cell tips (Fig. 7.21). The problem can however be solved approximately in two steps: (i) the initial transient of the composition at the solid/liquid interface is assumed to obey a second-order polynomial up to a certain volume fraction, 𝑓𝑥 ; (ii) above 𝑓𝑥 , the liquid is homogeneous and the Gulliver-Scheil equation can be applied. A detailed description of this model can be found in Giovanola and Kurz (1991). The result of such a calculation is shown in Fig. 7.20 and compares favourably with experimental measurements. Wang and Beckermann (1993) have developed a general approach to solute redistribution which accounts of the tips undercooling. To sum up this chapter, one now has a complete theory which has proved to be useful in interpreting the manifold phenomena and microstructures observed following rapid solidification processing. To render the picture truly complete, precise knowledge is required concerning the various metastable phases which might appear at high undercoolings. Note that all the equations which have been developed in this chapter for the rapid solidification case are general in nature (even if only semiqualitative) and can equally well describe steady-state solidification under normal growth conditions. The simplified equations which have been developed in Chaps 3 to 6 are however easier to use when P is less than unity.

Figure 7.20 Solute Distribution Profiles at High Velocity According to Various Microsegregation Models Under rapid solidification conditions, the interdendritic solute distribution deviates markedly from the Gulliver-Scheil and Brody-Flemings equations due to the large solute build-up at the dendrite tip. The composition of the dendrite trunk is thus much higher and makes the microsegregation less pronounced, with the material becoming more homogeneous as shown by the more realistic red curve.

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Figure 7.21 Intercellular Solute Concentration Profile In the transient region of the intercellular/interdendritic liquid composition (1-2-3), the concentration gradients at the interface are large. The assumption of a homogeneous liquid is therefore valid only for a high volume fraction of solid, 𝑓𝑥 (4). This forms the basis of the model used for Fig. 7.20.

Exercises 7.1

Describe two different mechanisms which lead to a microsegregation-free solid.

7.2

Develop a simple relationship for the estimation of the concentration range of various alloys at which either solute-trapping or plane-front growth with local equilibrium beyond 𝑉𝑎 will be the major supersaturation mechanism. How will the form of the phase diagram (𝑘 → 1 or 𝑘 → 0) influence the critical composition?

7.3

Develop a relationship for the 𝑇0 -curve. (Hint: use Eq. 7.4).

7.4

Show under what conditions the characteristic diffusion distance will be equal to the relevant microstructural length for interface instabilities, for dendrites/cells and for eutectic. Calculate 𝑉𝑎 approximately for dendrites by using the low Péclet number approximation. Sketch the structures and the diffusion fields for all three cases, for 𝑃 ≪ 1 as well as for 𝑃 ≫ 1.

7.5

Show that the limit of absolute stability, 𝑉𝑎 , at low alloy compositions increases with the square root of the alloy concentration, 𝐶0 .

7.6

What is the main effect of the temperature dependence of the diffusion coefficient on eutectic growth? Can you relate this effect to the form of TTT diagrams?

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7.7

In which systems will the effect of the temperature-dependent diffusion coefficient become important at high rates; in alloys with low or high solubility? Does your reasoning hold for dendrites, for eutectic or for both?

7.8

What is the maximum value of the undercooling during columnar dendritic growth? Which growth rate corresponds to that undercooling?

7.9

What are the conditions for banding?

7.10 Which assumptions in the original Jackson-Hunt treatment restrict their solution to low Péclet numbers? 7.11 In a eutectic system which eutectic phase will reach absolute stability first? 7.12 Under what conditions would you predict glass formation during laser resolidification? Note that, at the bottom of the trace, crystals will always grow first as the growth rate is zero there and increases to a maximum at the surface. 7.13 Evaluate the interfacial energy which is stored in the finest observed eutectics?

References and Further Reading General Rapid Solidification Processing and Thermodynamics ▪ J.C.Baker, J.W.Cahn, in Solidification, ASM, Metals Park, Ohio, 1971, p.23. ▪ W.J.Boettinger, D.Shechtman, R.J.Schaefer, F.S.Biancaniello, The effect of rapid solidification velocity on the microstructure of Ag-Cu alloys, Metallurgical Transactions A, 15 (1984) 55. ▪ D.M.Herlach, Non-equilibrium solidification of undercooled metallic melts, Materials Science and Engineering R, 12 (1994) 177. ▪ H.Jones, Rapid Solidification of Metals and Alloys, The Institution of Metallurgists, London, 1982. ▪ J.H.Perepezko, W.J.Boettinger, Use of metastable phase diagrams in rapid solidification, Materials Research Society Symposium Proceedings, 19 (1983) 223. Nucleation in Rapid Solidification Processing ▪ K.F.Kelton, A.L.Greer, Transient nucleation effects in glass formation, Journal of NonCrystalline Solids, 79 (1986) 295. ▪ K.F.Kelton, A.L.Greer, Nucleation in Condensed Matter: Applications in Materials and Biology, Pergamon, Oxford, 2010. ▪ J.H.Perepezko, M.J.Uttormark, Nucleation-controlled solidification kinetics, Metallurgical and Materials Transactions A, 27 (1996) 533. Kinetics of Rapid Solidification Processing ▪ M.J.Aziz, Model for solute redistribution during rapid solidification, Journal of Applied Physics, 53 (1982) 1158. ▪ M.J.Aziz, T.Kaplan, Continuous growth model for interface motion during alloy solidification, Acta Metallurgica, 36 (1988) 2335. ▪ M.J.Aziz, Interface attachment kinetics in alloy solidification, Metallurgical and Materials Transactions A, 27 (1996) 671.

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▪ W.J.Boettinger, S.R.Corriell, R.F.Sekerka, Mechanisms of microsegregation-free solidification, Materials Science and Engineering, 65 (1984) 27. ▪ W.J.Boettinger, S.R.Corriell, in Science and Technology of the Undercooled Melt, P.R.Sahm, H.Jones, C.M.Adam (Eds), Martinus Nijhoff, Dordrecht, 1986, p.81. ▪ W.J.Boettinger, M.J.Aziz, Theory for the trapping of disorder and solute in intermetallic phases by rapid solidification, Acta Metallurgica, 37 (1989) 3379. ▪ A.A.Chernov, Modern Crystallography III: Crystal Growth, Springer, Berlin, 1984. ▪ S.R.Coriell, D.Turnbull, Relative roles of heat transport and interface rearrangement rates in the rapid growth of crystals in undercooled melt, Acta Metallurgica, 30 (1982) 2135. ▪ P.K.Galenko, Extended thermodynamical analysis of a motion of the solid-liquid interface in a rapidly solidifying alloy, Physical Review B, 65 (2002) 144103. ▪ A.L.Greer, H.Assadi, Rapid solidification of intermetallic compounds, Materials Science and Engineering A, 226-228 (1997) 133. Interface Stability and Banding ▪ M. Carrard, M. Gremaud, M. Zimmermann, W. Kurz, About the banded structure in rapidly solidified dendritic and eutectic alloys, Acta Metallurgica et Materialia, 40 (1992) 983. ▪ S.H.Davis, Theory of Solidification, Cambridge University Press, Cambridge, 2001. ▪ M. Gremaud, M. Carrard, W. Kurz, Banding phenomena in Al-Fe alloys subjected to laser surface treatment, Acta Metallurgica et Materialia, 39 (1991) 1431. ▪ D.A.Huntley and S.H.Davis, Thermal effects in rapid directional solidification: Linear Theory, Acta Metallurgica et Materialia, 41 (1993) 2025. ▪ D.A.Huntley and S.H.Davis, Effect of latent heat on oscillatory and cellular mode coupling in rapid directional solidification, Physical Review B, 35 (1996) 3132. ▪ A.Karma, A.Sarkissian, Interface dynamics and banding in rapid solidification, Physical Review E, 47 (1993) 513. ▪ W.Kurz, R.Trivedi, Banded solidification microstructures, Metallurgical and Materials Transactions A, 27 (1996) 625. ▪ R.Trivedi, W.Kurz, Morphological stability of a planar interface under rapid solidification conditions, Acta Metallurgica, 34 (1986) 1663. Rapid Cell/Dendrite Growth ▪ W.J.Boettinger, S.R.Coriell, R.Trivedi, Application of dendritic growth theory to the interpretation of rapid solidification microstructures in Rapid Solidification Processing: Principles and Technologies, R.Mehrabian, P.A.Parrish (Eds), Claitor’s, Baton Rouge, 1988, p.13. ▪ W.J.Boettinger, J.A.Warren, Simulation of the cell to plane front transition during directional solidification at high velocity, Journal of Crystal Growth, 200 (1999) 583. ▪ D.Herlach, Solidification from undercooled melts, Materials Science and Engineering A, 226228 (1997) 348. ▪ D.M.Herlach, P.K.Galenko, Rapid solidification: in situ diagnostics and theoretical modelling, Materials Science and Engineering A, 449-451 (2007) 34. ▪ W.Kurz, B.Giovanola, R.Trivedi, Theory of microstructural development during rapid solidification, Acta Metallurgica, 34 (1986) 823. ▪ J.Lipton, W.Kurz, R.Trivedi, Rapid dendrite growth in undercooled alloys, Acta Metallurgica, 35 (1987) 957.

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▪ R.Trivedi, Theory of dendritic growth under rapid solidification conditions, Journal of Crystal Growth, 73 (1985) 289. ▪ R.Trivedi, J.Lipton, W.Kurz, Effect of growth rate dependent partition coefficient on the dendritic growth in undercooled melts, Acta Metallurgica, 35 (1987) 965. ▪ R.Trivedi, W.Kurz, Dendritic growth, International Materials Reviews, 39 (1994) 49. Rapid Eutectic Growth ▪ W.J.Boettinger in Rapidly Solidified Amorphous & Crystalline Alloys, B.H.Kear, B.C.Giessen (Eds), Elsevier North Holland, New York, 1982. ▪ K.A.Jackson, J.D.Hunt, Lamellar and rod eutectic growth, Transactions of the Metallurgical Society of AIME, 236 (1966) 1129. ▪ W.Kurz, R.Trivedi, Eutectic growth under rapid solidification conditions, Metallurgical Transactions A, 22 (1991) 3051. ▪ R.Trivedi, P.Magnin, W.Kurz, Theory of eutectic growth under rapid solidification conditions, Acta Metallurgica, 35 (1987) 971. ▪ N.Wang, Y.E.Kalay, R.Trivedi, Eutectic-to-metallic glass transition in the Al-Sm system, Acta Materialia, 59 (2011) 6604. ▪ M.Zimmermann, M.Carrard, W.Kurz, Rapid solidification of Al-Cu eutectic alloy by laser remelting, Acta Metallurgica, 37 (1989) 3305. Solute Distribution ▪ W.J.Boettinger, L.A.Bendersky, S.R.Coriell, R.J.Schaefer, F.S.Biancaniello, Microsegregation in rapidly solidified Ag-15wt%Cu alloys, Journal of Crystal Growth, 80 (1987) 17. ▪ T.F.Bower, H.D.Brody, M.C.Flemings, Measurements of solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (1966) 624. ▪ H.D.Brody, M.C.Flemings, Solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (1966) 615. ▪ B.Giovanola, W.Kurz, Microsegregation under rapid solidification conditions with solid state diffusion, International Journal of Materials Research, 82 (1991) 83. ▪ M.E.Glicksman, R.N.Hills, Non-equilibrium segregation during alloy solidification, Philosophical Magazine A, 81 (2001) 153. ▪ C.Y.Wang, C. Beckermann, A multiphase solute diffusion model for dendritic alloy solidification, Metallurgical Transactions A, 24 (1993) 2787.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 173-211 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

*

CHAPTER EIGHT

SOLIDIFICATION MICROSTRUCTURE SELECTION MAPS Solidification microstructures largely depend upon: 1. Alloy properties: composition ( 𝐶0 ), equilibrium phase diagram parameters (such as the distribution coefficient, 𝑘 , and the slope of the liquidus, 𝑚 ), thermo-physical properties (density, 𝜌, solute diffusion coefficients in liquid and solid, 𝐷, thermal diffusivity, 𝑎, GibbsThomson coefficient, Γ), and 2. Local processing conditions such as the local cooling rate, 𝑇̇, the thermal gradient, 𝐺, the speed of the isotherms, 𝑉 †, all of which are functions of the global conditions imposed by the process, e.g., heat-extraction at the surface by conduction (into another medium), convection (forced or natural and associated with air, water or any other fluid) and radiation. (The effects of mechanical stress and deformation, which are also part of the processing conditions, are not treated here but can be found in Dantzig and Rappaz, 2016). For a given alloy, the conditions (𝐺, 𝑉) which prevail during various directional solidification processes are shown in Figure 8.1. These parameters are used to construct Solidification Microstructure Selection (SMS) maps. They plot the resultant microstructures as a function of processing variables such as the composition, solidification velocity, temperature gradient, etc. An important characteristic of solidification microstructures is the columnar or equiaxed morphology which can develop in different environments. Columnar dendrites or eutectics are favoured by unidirectional heat flow with a positive temperature gradient, while equiaxed dendritic or eutectic grains tend to form preferentially in an undercooled melt which extracts the heat of transformation (Fig. 4.1). However, in the section dealing with the transition from columnar to equiaxed microstructures (a so-called CET transition), it will be shown that these basic guidelines are slightly over-simplified. The first sections of this chapter are devoted to the formation of grain structures. While the velocity and associated undercooling of columnar grains can be easily deduced from the speed of the * †

Top image: Solidification Microstructure Selection Map of Al-Al2Cu (Gill and Kurz, 1993). These three local variables are interdependent in the case of directional growth (𝐺 > 0) because 𝑇̇ = −𝐺𝑉.

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isotherms, equiaxed morphologies require some elaboration before analysing the CET. The competition between microstructures, e.g. between dendrites and eutectics, and between stable and metastable phases, will be analysed next. Because such competition obviously also depends upon nucleation (of new grains of the same phase or of a new stable or metastable phase), this analysis is complex and is a function of the grain morphology (columnar or equiaxed) itself. For this reason, the analysis which is presented in the last sections of this chapter will focus on microstructure competition and/or phase competition which occurs during columnar growth.

Figure 8.1 𝑮 − 𝑽 Map for Various Solidification Processes with Columnar Structures The processing parameters imposed by various techniques are represented here, with velocities of 1 μm/s < 𝑉 < 1 m/s and temperature gradients of 103 K/m < 𝐺 < 107 K/m. The large possible range of each of these variables results in a multitude of possible microstructures for different alloys: columnar or equiaxed dendrites, single phase or multiphase structures, single- or poly-crystals, stable or metastable phases, etc.

8.1. Equiaxed Growth of Eutectic Grains The final grain size of a solidified product initially depends upon nucleation of the primary phase which, in most practical situations, is heterogeneous in nature. The final density of grains following complete solidification does not however depend only upon the density of potent nucleation sites in the melt. As described in this section, the final grain density or its inverse, the average grain volume (or size), is the result of an interplay between nucleation and growth. To simplify the problem, eutectic grains will be considered first, because the eutectic composition is constant and equiaxed solidification proceeds with a simple spherical morphology. Consider a small volume, 𝑣, which contains a eutectic alloy of uniform temperature (Figure 8.2). A heat flux, 𝑞 (W/m2), leaves the surface, 𝐴, of this volume. Since the temperature is assumed to be uniform within this small volume‡ , a local heat extraction rate, ℎ̇ = 𝑞𝐴/𝑣 (W/m3), can be defined. For a fully liquid or fully solid volume element, the heat extraction rate induces a cooling rate of 𝑇̇ = ℎ̇/𝑐, where 𝑐 is the specific heat per unit volume (ℎ̇ < 0 when heat leaves the volume). When solidification occurs, the heat extraction rate, ℎ̇, corresponds to variations in temperature, 𝑇, and fraction of solid, 𝑓𝑠 , according to:

The temperature difference within the specimen is proportional to 𝑞𝑣/(𝐴) where  is the thermal conductivity of the metal. ‡

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Figure 8.2 Solidification of Spherical Grains in a Small Volume Element, 𝒗, of Uniform Temperature Within a small volume element of uniform temperature, equiaxed spherical grains grow from nucleation centres until they collide and form a polycrystal. Eutectic alloys often solidify in this way. For modelling this process, it is assumed that an outward heat flux, 𝑞, leaves the surface, 𝐴, of a small element of volume, 𝑣, which contains an alloy of eutectic composition. Above the eutectic equilibrium temperature, 𝑇𝑒 , the cooling curve has a (negative) slope which is equal to 𝑞𝐴/(𝑣𝑐). As it cools to below 𝑇𝑒 , the first nuclei appear at an undercooling of Δ𝑇𝑛 (1). The temperature continues to decrease because the latent heat which is released by their growth remains modest and further grains nucleate while the initial ones have increased in size (2). As this nucleation-growth of grains continues, the latent heat released by solidification becomes equal to the heat extracted at the surface of the specimen, and the cooling-curve arrives at a minimum, corresponding to a maximum undercooling of Δ𝑇𝑚𝑎𝑥 (3). At this point the slope of the cooling-curve is zero. As the latent-heat release continues to increase, due to the increase in the extent of the effective solid/liquid interface, the temperature rises: a phenomenon known as recalescence which is characterized by a negative enthalpy variation and a positive temperature change (4). The growing grains start to impinge on each other at some points and the extent of the effective solid/liquid interface starts to diminish, together with the latent-heat release. The 𝑇(𝑡) curve reaches a maximum and then starts to decrease. Some pockets of liquid are still present between the grains, but their solidification is rapid as the undercooling increases (5). Following complete solidification, the cooling curve has roughly the same slope as that measured above the eutectic temperature, provided that the specific heat of the solid is equal to that of the liquid and that the external heat flux, 𝑞, did not change.

ℎ̇ =

𝑑ℎ 𝑞𝐴 𝑑𝑇 𝑑𝑓𝑠 = =𝑐 − ∆ℎ𝑓 𝑑𝑡 𝑣 𝑑𝑡 𝑑𝑡

[8.1]

where ∆ℎ𝑓 is the volumetric latent heat of fusion. Since there are two variables, namely 𝑇 and 𝑓𝑠 , the relationship between them must be determined in order to predict the evolution of solidification within the volume. This is done by using a nucleation-growth model. For this purpose it is assumed that nucleation is heterogeneous and is described by the free growth model (see Chap. 2) while the velocity of the growing grains is deduced for a eutectic, i.e. 𝑉 = KeΔ𝑇 2 (Chap. 5). The fraction of solid at any given time is given by: t

𝑓𝑠 (𝑡) = ∑ 𝑛𝑖 𝑣𝑖 (t) = ∫ i

0

𝑑𝑛 𝑣(𝜏, 𝑡)𝑑𝜏 𝑑𝜏

[8.2]

The first expression is a discrete one in which the index, i, identifies a time-step during which 𝑛𝑖 new grains (per unit volume) nucleate in the liquid. Since these grains are observed at a later time

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𝑡 > 𝑡𝑖 , each of them occupies a volume, 𝑣𝑖 (𝑡). In Figure 8.2 for example, 5 grains have formed during step (1) and have grown up to step (2), while 5 new grains have nucleated, and so on. The second expression in Eq. 8.2 corresponds to a continuous approach in which 𝑑𝑛 is the increment in the number of grains per unit volume that have nucleated between 𝜏 and (𝜏 + 𝑑𝜏). The grains which nucleated at time, 𝜏, occupy a volume, 𝑣(𝜏, 𝑡), at time, 𝑡. Assuming a free growth model (Chap. 2) associated with a distribution of potent nucleation sites, 𝑑𝑛/𝑑(∆𝑇), the first term in the integral of Eq. 8.2, can be written as: 𝑑𝑛 𝑑𝑛 𝑑Δ𝑇 𝑑𝑛 𝑑𝜏 = dτ =   − 𝑇̇(𝜏)d𝜏 𝑑𝜏 𝑑Δ𝑇 𝑑𝜏 𝑑Δ𝑇

[8.3]

Assuming that the grains remain spherical, their volume, 𝑣(𝜏, 𝑡), is given by: 3

4π 3 4π 𝑡 𝑣(𝜏, 𝑡) = 𝑟 (𝜏, 𝑡) = [∫ 𝑉(∆𝑇())𝑑] 3 3 τ

[8.4]

where 𝑟(τ, 𝑡) is the radius at time, 𝑡, of a grain which nucleated at time, 𝜏. In this equation appears the growth kinetics of the eutectic, 𝑉(Δ𝑇) = 𝐾𝑒 Δ𝑇 2 (Chap. 5). Equations 8.2 to 8.4 permit the establishment of a kinetic relationship between the solid fraction, 𝑓𝑠 (𝑡), and the temperature, 𝑇(𝑡), that depends upon nucleation (Eq. 8.3), growth kinetics (Eq. 8.4) and cooling rate, 𝑇̇. At some point during growth, the grains impinge upon one another and no longer remain spherical. In a manner which is analogous to the TTT (or CCT) diagrams of solid-state transformations, the actual solid fraction, 𝑓𝑠 , with account being taken of impingement, is related to the so-called extended solid fraction, 𝑓𝑠𝑒 , given by Eqs 8.2 to 8.4 (i.e. for spherical grains without impingement) by: 𝑑𝑓𝑠 = (1 − 𝑓𝑠 )𝑑𝑓𝑠𝑒

or

𝑓𝑠 = 1 − exp(−𝑓𝑠𝑒 )

[8.5]

In other words, the effective solid/liquid interface extent of impinging spherical grains decreases as (1 − 𝑓𝑠 ). This approximation corresponds to what is called “hard impingement”, i.e. the growth of two neighbouring grains stops only when their solid/liquid interfaces “collide”. This is a reasonable assumption for equiaxed eutectic grains given that solute diffusion between lamellae occurs in the liquid over a distance of the order of 𝜆𝑒 , which is much smaller than the radius, 𝑟, of the grain. 8.2. Equiaxed Growth of Dendritic Grains Following nucleation of the primary phase under 𝐺 ≅ 0 , most alloys of non-eutectic composition solidify in the form of globules which, upon attaining a certain size, become dendritic. When compared with the case of the eutectics described in the previous section, two complications now arise when dealing with the modelling of equiaxed dendritic morphologies. The first one is that the equiaxed grains are no longer fully solid and the concept of an “internal volume fraction of solid”, 𝑓𝑖 , has to be introduced. The second one is that the diffusion layer surrounding equiaxed grains can no longer be neglected and affects both nucleation and growth. This is shown schematically in Fig. 8.3 for a population of dendritic grains that have nucleated on a bimodal distribution of potent heterogeneous nucleation sites that became active at Δ𝑇1 and Δ𝑇2 > Δ𝑇1. Internal solid fraction Equiaxed dendritic grains of cubic metals preferentially grow with 100 arms and therefore exhibit (in two dimensions) the cruciform shape depicted in Fig. 8.3 for the grains formed at the

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smallest undercooling, Δ𝑇1 . The second family of nucleation sites, which may become active when the undercooling reaches Δ𝑇2 , are indicated by dots (the colours of which will be explained below). The concept of a “dendritic envelope” was first proposed in the 1980’s: it is a smooth surface which touches the tips of the dendrite arms and within which the interdendritic liquid has a nearly uniform composition. In Fig. 8.3, the envelope has been drawn schematically as a circle passing through the dendrite tips; it would preferably be much like a square in a more realistic model.

Figure 8.3 Equiaxed Dendritic Grains with their Diffusion Boundary Layer and Two Different Sizes of Inoculant Particles This figure illustrates grain initiation by two types of inoculant particles of differing size which become active at two different undercoolings, Δ𝑇1 < Δ𝑇2 . The equiaxed grains which nucleated on the large particles (free-growth model), characterized by Δ𝑇1 , have initially grown with a spherical morphology until the solid/liquid interface became unstable. This instability resulted in the formation of equiaxed dendrites which grew along 100 directions (assuming cubic crystals). The envelope of the dendritic grains, within which the interdendritic liquid has an almost uniform composition, is shown schematically as a circle which passes through the primary dendrite tips. The solute boundary layer surrounding these grains has further been outlined using a dashed circle. When Δ𝑇 = Δ𝑇2 , particles located outside the diffusion layer of the dendritic grains initiate nucleation and the formation of new grains (red dots). The black dots/particles located within the boundary layer of a dendritic grain find themselves in liquid of higher composition (for 𝑘 < 1), i.e. with a reduced undercooling. If this reduced undercooling does not attain Δ𝑇2 during growth, these particles will never initiate a new grain. The effect of Solute Suppressed Nucleation (SSN) zones which surround dendritic grains is discussed further under Fig. 8.4. The same result will of course occur if these particles are engulfed by the growth of dendritic grains which formed at a lower undercooling. (The boundary layer shown is schematic; in reality it is narrower and less spherical.)

The use of a lever-rule approximation (Chap. 6) for the grain envelope and a simple solute balance permits to establish that the internal fraction of solid of well-developed dendritic grains, 𝑓𝑖 , is equal to the supersaturation, i.e. 𝑓𝑖 = Ω(Δ𝑇), where Δ𝑇 is the undercooling of the dendrite tips (Rappaz and Thévoz, 1987). Equation 8.2 has to be multiplied by 𝑓𝑖 in order to obtain the volume fraction of solid, 𝑓𝑠 , i.e. 𝑓𝑠 = 𝑓𝑔 𝑓𝑖 where 𝑓𝑔 = (1 − exp(−𝑓𝑔𝑒 )) is the actual fraction of grains. This relationship between 𝑓𝑔 and the extended fraction of grains, 𝑓𝑔𝑒 , as well as the derivation of 𝑓𝑠 performed in the case of equiaxed eutectic growth (Eqs 8.2−8.5) are the same, provided that the growth kinetics of the dendrite tips replaces that of eutectics. Improvements in this simple relationship have been suggested by several authors in order to account for the limited diffusion in the solid, nonspherical grain envelope and, even more importantly, for the morphological transition from globular

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grains, just after nucleation, to dendritic ones following destabilisation of the spherical solid/liquid interface (Appolaire et al., 2008; Dantzig and Rappaz, 2016; Diepers and Karma, 2004). Just after nucleation, the grain radius is indeed very small and the spherical interface remains stable: the internal volume fraction of solid, 𝑓𝑖 , in this case is equal to 1. During the transition from a globular to a dendritic morphology, 𝑓𝑖 decreases from 1 to Ω(Δ𝑇) (Appolaire et al., 2008). If the average density of grains, 𝑛𝑔 , is very high, i.e. if the final average grain radius, 𝑟̅𝑔 , is small, the globular-to-dendritic transition may not occur and the grains will remain essentially globular. This is typically the case for low solute-content Al alloys which are heavily inoculated with TiB2 particles. The critical final grain radius, 𝑟̅𝑔,c , below which grains remain globular has been calculated (Martinez et al., 2006; Dantzig and Rappaz, 2016) to be: 1/3

𝑟̅𝑔,c

𝐷 Γ 𝜃𝑡 = (−96 ) 𝑘 Δ𝑇0 𝑇̇

[8.6]

where 𝜃𝑡 = Δℎ𝑓 /𝑐 is the unit undercooling. Diffusion around equiaxed dendritic grains If the grains are globular, the surrounding solute layer in the liquid is of the order of the grain radius, i.e. 𝛿𝑐 ~𝑟𝑔 ; typically hundreds of microns. For well-developed equiaxed dendritic grains, 𝛿𝑐 is of the order of 𝐷/𝑉 (Rappaz and Thévoz, 1987) and is in the range of tens of microns (dashed circle surrounding the grain envelope in Fig. 8.3). This has two consequences: ▪ The first effect is soft impingement, which is particularly important for globular grains. When the diffusion layers surrounding the grains meet (Fig. 8.4), the composition at the mid-point (i.e. the future grain boundary following solidification), 𝐶𝑔𝑏 , increases (for 𝑘 < 1 ). The supersaturation, which is proportional to the difference between the compositions at the tip, 𝐶 ∗ , and 𝐶𝑔𝑏 , therefore decreases. This diminishes the velocity of the dendrite envelope before hard impingement occurs. This effect, which must be accounted for in the model, leads to an increase in microsegregation at grain boundaries as compared with that occurring within the interdendritic regions. When the extradendritic and intradendritic liquids arrive at about the same composition, the remainder of the solidification occurs according to a Gulliver-Scheil or back-diffusion model; modified so as to account for the initial period of recalescence. ▪ The second effect is the formation of a so-called Solute Suppressed Nucleation (SSN) zone (Shu et al., 2011). Because the solutal undercooling of the melt is defined by the difference between the liquidus temperature, 𝑇𝑙 (𝐶), and the actual temperature, 𝑇, the solute diffusion layer surrounding each grain 𝐶𝑙 (𝑥) induces an associated undercooling profile, ∆𝑇𝑙 (𝐶𝑙 (𝑥)) = [𝑇𝑙 (𝐶𝑙 (𝑥)) − 𝑇], around each of them. A potent nucleation site characterized by a nucleation undercooling of Δ𝑇𝑛 and located at a position, 𝑥, can initiate a new grain only if the local undercooling, ∆𝑇𝑙 (𝐶𝑙 (𝑥)), is equal to Δ𝑇𝑛 . Although this nucleation site is still in the liquid and the “apparent” undercooling of the specimen, (𝑇𝑙 (𝐶0 ) − 𝑇), may be greater than Δ𝑇𝑛 , the diffusion layers around the growing grains can prevent the formation of a new grain. The SSN phenomenon is illustrated in Fig. 8.4 for the bimodal distribution of nucleation sites which was introduced in Fig. 8.3. A potent nucleation site, characterized by Δ𝑇2 , is located at the mid-point between two globular grains formed previously at the smallest undercooling, Δ𝑇1. If the distance which separates them is large (Fig. 8.4(a)), the composition, 𝐶𝑔𝑏 , at the mid-point remains at the nominal value, 𝐶0 , for a longer period of time as compared with the situation illustrated in (b), where the two grains are closer. While the temperature of the specimen, measured by the liquidus temperature at the solid/liquid interface of the grains (curvature undercooling being neglected),

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decreases over time, the solutal undercooling at the mid-point can increase beyond Δ𝑇2 in situation (a), but not in situation (b). In case (a) a new grain will form (and the mid-point will no longer become a grain boundary following solidification) while in case (b), the particle at the mid-point will eventually be trapped at the grain boundary without having had the opportunity to form a new grain, due to the overlapping of the solute boundary layers (or SSN).

Figure 8.4 Effect upon Nucleation and Growth of the Solute Boundary Layer Surrounding Two Neighbourly Equiaxed Grains In the upper figures two neighbouring equiaxed globular grains of a primary phase are shown at three different instants (1 to 3) of their growth. They have formed at an undercooling, Δ𝑇1 , while a potent nucleation site of characteristic nucleation undercooling, Δ𝑇2 (> Δ𝑇1 ), is located at the mid-point. The diffusion profile, 𝐶𝑙 (𝑥), is depicted schematically as a function of the distance, 𝑥, which measures the separation between the grains. By neglecting any curvature-undercooling the interface composition, 𝐶𝑙∗ , can be directly converted into a liquidus temperature, 𝑇𝑙 (𝐶𝑙∗ ) (lower figures). If the temperature, 𝑇, of the specimen is assumed to be uniform, the line, 𝑇 = 𝑇𝑙 (𝐶𝑙∗ ), in the lower figure shows the specimen temperature at these three instants, while the three profiles, 𝑇𝑙 (𝐶𝑙 (𝑥)), permit to deduce the solutal undercooling at the mid-point between the grains, Δ𝑇 = 𝑇𝑙 (𝐶𝑔𝑏 ) − 𝑇𝑙 (𝐶𝑙∗ ). This undercooling governs the speed of the spherical grain or of the dendrite tips which are growing towards the future grain boundary, and thus the soft impingement of the grains via their overlapping diffusion fields. When the distance between the two grains is fairly large (a), the composition, 𝐶𝑔𝑏 , in the liquid at the position of the future grain boundary starts to deviate from the nominal composition, 𝐶0 , only in stage 2. At that time. the solutal undercooling which is indicated by Δ𝑇2 is large enough to allow nucleation on the foreign particle. In (b), the distance between the two grains is smaller and the diffusion layers overlap earlier. The same potent nucleation site will here never become active, because Δ𝑇2 is never reached, i.e. the centre is located within the Solute Suppressed Nucleation zone.

The solute suppressed nucleation zone is therefore a zone of loss of constitutional undercooling (and not the “zone of constitutional undercooling” as often described in the literature). While this phenomenon has been illustrated for a continuously decreasing temperature of the specimen, it is of course “amplified” by the temperature increase when recalescence occurs, i.e. (𝑇𝑙 (𝐶0 ) − 𝑇) decreases. In a free growth model (Chap. 2), the nucleation sites characterized by Δ𝑇𝑛
Δ𝑇2 . The final grain density will consequently be higher and the final grain size smaller. Unlike eutectics, where the composition is fixed, the final grain size of equiaxed globular or dendritic grains is a function of the nominal composition, 𝐶0 . Indeed, the growth rate, 𝑉(Δ𝑇), of a dendritic grain is proportional to the ratio of the solute diffusivity, 𝐷 , and to the undercooling characterising the solute rejection which, at low growth rates, corresponds to 𝐶0 (𝑘 − 1)𝑚 = Δ𝑇0 𝑘 (Table 4.2). The temperature difference, 𝑘Δ𝑇0 = 𝑚𝐶0 (𝑘 − 1), associated with the solute rejection at the liquidus (or nominal) composition, 𝐶𝑙∗ = 𝐶0 , is also known as 𝑄, the growth restriction factor: It is

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proportional to the solute concentration, 𝐶0 , and to the phase diagram equilibrium, via (𝑘 − 1)𝑚. When the nominal composition, 𝐶0 , of the alloy is increased, this lowers the growth rate of the dendrite tips or of the spherical interface of globular grains. Under identical inoculation conditions, the slowing of grain growth due to an increase in 𝑄 decreases the latent-heat release and thus allows larger undercoolings to be explored. More nuclei then become active and the final grain size is reduced. Some authors use Δ𝑇0 , rather than 𝑘Δ𝑇0 , to define a growth restriction factor 𝑃 (which should not be confused with the Péclet number): this is equivalent of using the liquidus composition, 𝐶0 /𝑘, rather than 𝐶0 as a reference point for the rejection of solute, thus corresponding better to high undercoolings. Others include the diffusion coefficient, via 𝑚𝐶0 (𝑘 − 1)/𝐷 , for alloys having differing diffusion properties. Figure 8.6(a) shows the final grain size as a function of the cooling rate (in absolute value) for a commercially pure Al alloy (Greer et al., 2000). The filled symbols are experimental results while the open symbols correspond to simulations performed using the free growth model of nucleation and the growth kinetics of spherical grains. This model, which was initially proposed by Maxwell and Hellawell (1975), assumes that the radius of the spherical grain is proportional to the square root of 𝐷𝑡. Details of the experiments and models can be found in the references. In Fig. 8.6(b) the final grain size of various binary Al alloys is plotted as a function of the diffusivity-weighted growth restriction factor§. The points are experimental results and the curve is the best fit.

Figure 8.6 Dependence of the Equiaxed Grain Size of a Primary Phase as a Function of Cooling Rate and Growth Restriction Factor The final grain size of a commercially-pure Al alloy (i.e. 0.001 to 0.1wt% of various elements) is represented as a function of the cooling rate in (a) (Greer et al., 2000). The filled symbols are experimental results while the red curve connecting the open symbols correspond to simulations performed using the free growth model of nucleation and the growth kinetics of spherical grains. In (b) the final grain size is presented as a function of the growth restriction factor, 𝑄, for various binary Al alloys (Qi et al., 2015). Here Q is a measure of the solute rejection of the alloy and is proportional to the liquidus-solidus difference via 𝑘Δ𝑇0 . The points are the experimental results of Spittle and Sadli (1995) and the curve in red is a best fit obtained using a modified growth restriction factor accounting for differences in the diffusion coefficients of the various solute species. Comparison of these figures with the results of the analytical model (Fig. 8.7) confirms their qualitatively similar behaviours.

This trend towards smaller grain sizes with increasing cooling rate (in absolute value) is emphasized when the SSN effect is considered. A higher cooling rate indeed means a deeper undercooling, a smaller grain radius for globular grains and a higher velocity, 𝑉(Δ𝑇), of the dendrite §

To account for the diffusion coefficient, 𝐷𝑖 , of the various solute elements, a diffusivity-weighted growth restriction factor, 𝑄𝑖′ , can be defined as (𝐷ref /𝐷𝑖 )𝑄𝑖 , where 𝐷ref is a reference diffusion coefficient (Ti in the case of Fig. 8.6(b)).

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tips. In both situations, the solute layer around globular or dendritic grains is reduced and SSN is less pronounced, i.e. fewer potent nucleation sites are rendered inactive by the solute layers surrounding existing growing grains. An extension to multiple-component alloys can be made, to a first approximation, by summing over the contributions made by the various solute elements: 𝑄 = ∑ 𝑚𝑖 𝐶0𝑖 (1 − 𝑘𝑖 )

𝑄=∑

or

𝑖

𝑖

𝑚𝑖 𝐶0𝑖 (1 − 𝑘𝑖 ) 𝐷𝑖

[8.7]

where the values, 𝑚𝑖 and 𝑘𝑖 , can be deduced from the binary phase diagrams and the binary diffusion coefficients, 𝐷𝑖 , ignore cross-diffusion, i.e. any change in the diffusivity in one species, due to the solute gradients of other species, is ignored. A detailed analysis of the correct methods for simulating dendritic microstructures of multicomponent alloys, using analytical models, can be found in Guillemot et al. (2022). 8.4. A Simple Grain Size Model Which Combines Heat Flow, Nucleation and Growth Kinetics To achieve a deeper insight into the physics governing the dependence of the final size of equiaxed grains upon the nucleation kinetics, the growth kinetics and the cooling rate, a simple analytical model is presented here. The following assumptions are made (see Exercise 8.1): ▪ The cooling rate, 𝑇̇ = 𝑇̇0 , is constant and corresponds to the rate when the system-temperature passes the melting point. It neglects a smooth variation from the initial 𝑇̇0 to 𝑇̇0 = 0 at the maximum undercooling, Δ𝑇𝑚𝑎𝑥 , before recalescence sets in. Setting 𝑡 = 0 as the time at which the equilibrium temperature is crossed one has: Δ𝑇(𝑡) = −𝑇̇0 𝑡. ▪ A free growth model for grain nucleation is adopted with a flat distribution, 𝐵 = 𝑑𝑛/𝑑(Δ𝑇) (m-3K-1). Thus the nucleation rate is given by 𝑛̇ = 𝐼 = −𝐵𝑇̇0 [m-3s-1] for Δ𝑇 > 0, where 𝐵 is a constant. The growth kinetics for dendrites is given by: 𝑉 = 𝐾𝑑 Δ𝑇 𝑛

[8.8]

with 𝐾𝑑 = 𝛽𝐷/(Γ𝑄) and 𝑛 = 2 (Table 4.2). The parameter, 𝛽, is related to the operating point of the dendrite tip. It varies according to the criterion which is used: for the extremum condition, 𝛽 = 0.1, and for the marginal stability condition, 𝛽 = 0.025. ▪ The grains have the internal volume fraction of solid, 𝑓𝑖 = 1, and no solute boundary-layer (hard impingement). With these simplifications, Equation 8.2 becomes (without considering grain impingement): t

𝑓𝑠𝑒 (𝑡) = ∫ 0

t 𝑑𝑛 4π 3 𝑣(τ, 𝑡)𝑑τ = −𝐵𝑇̇0 ∫ 𝑟 (𝜏, 𝑡)𝑑𝜏   𝑑τ 0 3

[8.9]

where the radius, 𝑟(τ, 𝑡), of the grains which nucleated at time, τ, and are observed at time, 𝑡, is given by Eq. 8.4: 𝑡

𝑡

𝑟(𝜏, 𝑡) = ∫ 𝐾𝑑 Δ𝑇 2 (𝜉)𝑑𝜉 = 𝐾𝑑 𝑇̇02 ∫ 𝜉 2 𝑑𝜉 = 𝜏

𝜏

𝐾𝑑 𝑇̇02 3 (𝑡 − 𝜏 3 ) 3

Inserting Eq. 8.10 into Eq. 8.9 and integrating gives:

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[8.10]

Foundations of Materials Science and Engineering Vol. 103

3 3 ̇ 0 )7 ̇02 π𝐾 𝐵(𝑇 4π 𝐾 𝑇 𝑑 𝑑 (𝑡 3 − 𝜏 3 )] 𝑑𝜏 = − 𝑓𝑠𝑒 (𝑡) = 𝐵𝑇̇0 ∫ [ 𝑡10 3 3 35 0

183

𝑡

[8.11]

At some point, the latent heat released by the growth of these grains, i.e. Δℎ𝑓 𝑓𝑠̇ 𝑒 , becomes equal to the specific heat variation in absolute value, i.e. −𝑐𝑇̇0 ; at which point 𝑇̇ should be zero and not equal to 𝑇̇0 . This is the time, 𝑡𝑟𝑒𝑐 , at which recalescence starts (see Exercise 8.1): −1/9 𝑡𝑟𝑒𝑐 = (2/7π𝐾𝑑3 𝐵𝑇̇06 𝜃𝑡 )

[8.12]

from which the maximum undercooling, Δ𝑇𝑚𝑎𝑥 = Δ𝑇(𝑡𝑟𝑒𝑐 ) = −𝑇̇0 𝑡𝑟𝑒𝑐 , can be deduced together 3 with the final grain density, n, and the grain size, d, of the specimen: 𝑑 = √6/(π𝑛): −1/9 𝑛 = 𝐵Δ𝑇(𝑡𝑟𝑒𝑐 ) = −𝐵𝑇̇0 𝑡𝑟𝑒𝑐 = −𝐵𝑇̇0 (2/7π𝐾𝑑3  𝐵𝑇̇06 𝜃𝑡 )

[8.13]

1/9 𝑑 ∝ (𝐷/−𝑇̇0 𝑄) 𝐵 −8/27

[8.14]

Figure 8.7 Modelling Results of the Influence of Cooling Rate, Nucleation (B) and Growth Kinetics (𝑲𝒅 ), upon the Final Size of Globular Grains The simple analytical nucleation-growth-model for globular grains assumes the existence of a uniform distribution of nucleation sites, 𝑑𝑛/𝑑Δ𝑇 = 𝐵 (free growth model) and growth kinetics such that 𝑉Δ𝑇 −2 = 𝐾𝑑 . If the smooth evolution of the cooling rate, from 𝑇̇0 above the equilibrium temperature to 𝑇̇ = 0 at the onset of recalescence, is replaced by a constant cooling rate 𝑇̇0 , the final grain size can be calculated (Eq. 8.14). In (a) the grain size is represented as a function of −𝑇̇0 for a reference case (red) while the curves below and above reveal the influence of the nucleation kinetics (parameter 𝐵) and growth kinetics (parameter 𝐾𝑑 ), respectively. The inverse of the parameter 𝐾𝑑 is proportional to the growth restriction parameter, 𝑄, and the final grain size in (b) has been plotted against Q for 𝑇̇0 = −1 K/s and 𝐵 = 2 × 109 m-3K-1. The unit undercooling is 𝜃𝑡 = 350 K in each case.

Using this result, Figure 8.7(a) thus shows the influence of the nucleation and growth kinetics via the variations in the parameters 𝐵 and 𝐾𝑑 , respectively. Increasing the density of potent nucleation sites (factor 𝐵) by an order of magnitude reduces the final grain size by a factor of about 2, according to 𝐵 −8/27 , while increasing the growth kinetics by an order of magnitude increases the grain size by about 30%, according to 𝐾𝑑 1/9. This simple model thus shows that the nucleation kinetics is more important with regard to grain-size control than the growth kinetics.

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The growth parameter, 𝐾𝑑 , linking the velocity to the undercooling, is related to the growth restriction parameter, 𝑄**, if the equiaxed globular grains of a primary phase are considered. Although the above simple analytical model does not consider the variation in the cooling rate and ignores the effects associated with the boundary layer that surrounds equiaxed grains, it qualitatively explains the decrease in grain size with increasing cooling rate, |𝑇̇0 |, and growth restriction factor, 𝑄 ∝ 1/𝐾𝑑 (compare Fig. 8.7 with Fig. 8.6). Although the nucleation-growth approach presented in these sections constitutes a good starting point for understanding equiaxed grain growth formation, it should be kept in mind that the effects of convection can dramatically change the situation. As mentioned in Chap. 4, fluid flow can even fragment dendrites and promote the formation of new grains, regardless of the inoculation conditions. Armed with this basic understanding of equiaxed grain formation, the next section describes the competition that can occur between columnar and equiaxed structures, while still neglecting convection. 8.5. Competition between Columnar and Equiaxed Microstructures (CET) Equiaxed growth which involves both nucleation and growth kinetics is fairly complex, as described in the previous sections. Columnar growth is much simpler in comparison because the undercooling of dendrite tips (or of a eutectic front) can be calculated on the basis of a growth kinetics model and the velocity of the isotherm. The analytical modelling of both types of microstructures within a single unified approach, in particular the prediction of the Columnar-to-Equiaxed Transition (CET), is complex. The more suitable tool for modelling this phenomenon is the numerical Cellular Automaton (CA) method briefly presented at the end of this section. We use here instead the simple analytical CET-model developed by Hunt (1984). Ahead of a columnar dendritic front, the solute rejected by the dendrites creates a solute boundary-layer within which the melt is constitutionally undercooled (Fig. 8.8). If the undercooling of the columnar dendrite tips is greater than the nucleation undercooling, equiaxed grains can form and grow in that region. If the volume fraction of the equiaxed grains which form before they are trapped by the columnar front is sufficiently large, the columnar dendrites are halted and a CET occurs. A first appreciation of the main parameters which control the CET can be obtained by the following simple approach. Assuming zero nucleation undercooling (typical of dendrite fragments), a CET criterion can be deduced by equating two length scales: (i) a characteristic distance for equiaxed grain growth (equal to the final grain radius) which is proportional to the inverse of the cube root of the nuclei density, 𝑛0 , (times a factor close to 1) and (ii) the distance over which equiaxed growth can occur, i.e. the length of the undercooled zone, Δ𝑇/𝐺, where 𝐺 is the thermal gradient and Δ𝑇 = 𝑇𝑙 – 𝑇 ∗ the tip undercooling of the columnar dendrites. The criterion for a transition from columnar to equiaxed morphologies is then simply: Δ𝑇 −1/3 ≃ 𝑛0 𝐺

[8.15]

Using the simple dendrite relationship, 𝑉 = 𝐾𝑑 Δ𝑇 2 (Eq. 8.8), already used for the equiaxed growth model with 𝑉 being the velocity of the columnar front, a CET occurs if: 2/3

𝐺 2 𝑛0 ≃ 𝑉 𝐾𝑑

[8.16]

which corresponds to Eq. 8.19 if Δ𝑇n = 0.

**

Note that the growth restriction factor, 𝑄 = 𝑘Δ𝑇0 , enters into the growth kinetics coefficient of dendrites 𝐾𝑑 (Table 4.2), but does not directly give their velocity, equal to 𝐾𝑑 Δ𝑇 𝑛 in the present model.

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Figure 8.8 Equiaxed Growth within the Constitutionally Undercooled Liquid Ahead of Columnar Dendrites During the growth of columnar dendrites, heat is extracted by the solid (𝐺 > 0) while the isotherms move at a velocity, 𝑉. Under steady-state conditions and assuming that the primary trunks grow parallel to the thermal gradient, this velocity is also that of the dendrite tips. Neglecting the small capillarity effect, the Constitutionally Undercooled Zone (CUZ) ahead of the columnar dendrites is a region within which crystals can nucleate and grow if the nucleation undercooling, Δ𝑇𝑛 , is smaller than Δ𝑇. Assuming that equiaxed grains which form in the CUZ grow according to the same kinetic law as that of the columnar dendrites, their volume fraction at the location of the columnar front can be simply calculated. If their volume fraction is sufficiently large, they can block the columnar dendrites and a Columnar-to-Equiaxed Transition (CET) occurs. The critical length-scales for the CET are (i) the growth radius, i.e. half the final grain −1/3 size of the equiaxed crystals, ≃ 𝑛0 , and (ii) the distance over which equiaxed grains can grow, ≃ (Δ𝑇– Δ𝑇𝑛 )/𝐺.

In his more detailed analysis, Hunt (1984) assumed that the same growth kinetics governs columnar and equiaxed dendrites and that all of the equiaxed grains nucleate at the same critical undercooling, Δ𝑇n . The radius of the equiaxed grains is then given by integrating their growth velocity from the start of nucleation up to the time at which the equiaxed grains begin to interfere with the columnar front. Assuming a steady state, 𝑑(Δ𝑇)/d𝑡 = 𝑉𝐺, the radius, 𝑟, of these grains when they are at the location of the columnar front is given by: Δ𝑇

Δ𝑇

𝑟 = ∫ 𝐾𝑑 Δ𝑇 2 dt = ∫ Δ𝑇n

Δ𝑇n

𝐾𝑑 Δ𝑇 2

dt Δ𝑇 3 − Δ𝑇n3 d(Δ𝑇) = 𝐾𝑑 d(Δ𝑇) 3𝑉𝐺

[8.17]

Assuming a more general form for the growth kinetics (Eq. 8.8 with 𝑛 ≠ 2) and using the relationship between the fraction of grains, 𝑓𝑔 , and the extended fraction of grains, 𝑓𝑔𝑒 , (Eq. 8.5) it is left as an exercise (Exercise 8.3) to show that Equation 8.17 becomes:

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1/3

1 −4𝜋𝑛0 𝐺= [ ] 𝑛 + 1 3ln(1 − 𝑓𝑔 )

[1 −

Δ𝑇𝑛𝑛+1 ] Δ𝑇 Δ𝑇 𝑛+1

[8.18]

In the case of zero nucleation undercooling, which is a good approximation when fragmentation occurs, or for rapid solidification processing (𝑉 > 1 cm/s and 𝐺 = 106 K/m), this reduces to: 1/3

𝐺𝑛 −4𝜋𝑛0 = 𝐾𝑑−1 ([ ] 𝑉 3ln(1 − 𝑓𝑔 )

1 ) 𝑛+1

𝑛

[8.19]

This equation is formally equivalent to Eq. 8.16 when 𝑛 = 2 and grain impingement is accounted for, i.e. the actual fraction of grains is 𝑓𝑔 = 1 − exp(−𝑛0 4π𝑟 3 /3). Three microstructures can develop: (1) fully columnar, (2) mixed columnar and equiaxed and (3) equiaxed grains alone. Fully equiaxed growth is considered to occur if the extended volume fraction of equiaxed grains is 𝑓𝑔𝑒 ≥ 0.66, corresponding to an effective volume fraction of equiaxed grains of 𝑓𝑔 = 1 − exp(−𝑓𝑔𝑒 ) = 0.49. The structure is fully columnar if 𝑓𝑔 ≈ 𝑓𝑔𝑒 ≤ 0.0066. Again setting 𝑛 = 2, the final condition for a CET as related to the thermal gradient, 𝐺, and the inter-nuclei −1/3 distance, 𝑛0 , is: 1/3 𝑛0 Δ𝑇 (1

Δ𝑇n3 − 3) Δ𝑇

[8.20a]

Δ𝑇n3 ) Δ𝑇 3

[8.20b]

Fully equiaxed:

𝑓𝑔 > 0.49

𝐺 < 0.617

Fully columnar:

𝑓𝑔 < 0.0066

𝐺 > 2.86 𝑛0 Δ𝑇 (1 −

1/3

Hunt’s CET model is often used as a benchmark for other models, but one has to be aware however that Hunt used several simplifications for the tip growth kinetics. The square-root relationship, Δ𝑇 ∝ 𝑉 1/2 , is a crude model for dendrite growth. A physically better solution is obtained by using the Ivantsov transport solution together with the solvability (marginal stability) condition (see Chap. 4 and Appendix 9). Gäumann et al. (1997) for example used the Ivantsovmarginal stability model to deduce the influence of various parameters upon the CET in binary Al-Cu alloys (Fig. 8.9). The difference with respect to the Hunt model is the steeper slope shown in Fig. 8.9(a), which is of particular importance under additive manufacturing conditions. The Ivantsov model indicates moreover that a higher concentration, 𝐶0 , (Fig. 8.9(b)) and a higher nuclei density, 𝑛0 , (Fig. 8.9(d)) require higher 𝐺 𝑛 /𝑉 ratios in order to avoid equiaxed growth. Nucleation undercooling is effective only at low velocities and in low temperature-gradients (Fig. 8.9(c)). The use of the latter model in iterative microstructure computations for larger volumes is rather time-consuming. An approximation to the Ivantsov function is therefore useful. Some of these approximations for low Péclet numbers can be found in Appendix 8. A good approximation is the model by Dantzig and Rappaz (2016) (Eq. 4.18). Figure 8.10 summarises the main points considered in this chapter. For any given alloy, 𝐺 and 𝑉 are the main variables which determine the morphology and scale of the microstructures formed during solidification. Specific 𝐺/𝑉 values (the lines and bands running from the bottom left to the upper right) are associated with a constancy of microstructure (planar, cellular, columnar or equiaxed dendritic). On the other hand, the various 𝐺𝑉(= |𝑇˙|) values (the lines running from the upper left to the bottom right) are associated with a constancy of the scale of those structures (e.g. 𝜆2 ). Fine or coarse dendrites can thus be produced when 𝐺 and 𝑉 can be varied independently (see the insets in Fig. 8.10). Herein lies the value of the directional solidification (DS) method. That is, structures can be tailored to some extent so as to possess optimum properties, as in the case of a turbine blade (Fig. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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4.5). In a traditional casting, 𝐺 and 𝑉 tend to be interrelated via the heat flux and the thermal properties of the metal. In such a casting, only those conditions which are close to the arrow at the top in Fig. 8.10 can therefore be exploited. When the growth rate is below the limit of constitutional undercooling, 𝑉𝑐 (Eq. 3.15 ), plane-front solidification occurs and no solidification microstructure will develop. Note however that solidification with a planar interface does not alone ensure that a monocrystal will be obtained. To achieve that aim, the elimination of all but one grain is necessary.

Figure 8.9 Microstructure Maps for CET Showing the Effect of the Important Variables This figure shows the upper limit of the CET for fully equiaxed growth (𝑓g = 0.49). It summarises the effects of the variables: (a) growth kinetics, (b) concentration 𝐶0 , (c) nucleation undercooling Δ𝑇n and (d) nuclei density 𝑛0 . It shows that under conditions of slow solidification (small 𝑉 and 𝐺) the nucleation undercooling is dominant while under rapid solidification conditions (high 𝑉 and 𝐺) the nucleation density and the composition are the controlling variables. An important finding is the effect of the chosen dendrite model upon the limit. Hunt used the simplified hemispherical tip with extremum criterion relationship, while Gäumann chose the Ivantsovstability model. The difference visible in (a) is particularly significant for rapid solidification processes such as additive manufacturing.

Another important phenomenon occurs within the melt of a casting: This is the convective flow, which can markedly influence the transition from columnar to equiaxed dendritic growth. The presence of strong convection will generally decrease the length of the columnar dendritic zone and reduce the primary spacing (Chap. 4). Increased equiaxed grain formation is moreover the result of the melting-off of parts of dendrites (fragmentation) and of a decreased value of 𝐺 due to convective heat transfer. The promotion of an equiaxed microstructure can have a beneficial effect upon the internal quality of castings. In the case of the continuous casting of steel, the use of electromagnetic stirring has become a technologically important means for controlling a predominantly equiaxed solidification structure and thus producing a more homogeneous product. On the other hand, in the case of a single crystal part, fluid flow should rather be limited.

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Figure 8.10 Columnar and Equiaxed Structures in 𝑮 − 𝑽 Space for Various Solidification Processes This schematic diagram summarises the columnar and equiaxed microstructures which can be produced in a typical alloy when the imposed temperature gradient, 𝐺, or the growth rate, 𝑉, are varied. Provided that a unidirectional heat flow is imposed, the product, 𝐺𝑉, is equivalent to the cooling rate, 𝑇̇, which controls the scale of the microstructure formed. Moving from the lower left to the upper right along the lines at 45∘ leads to a refinement of the structure for a given morphology. The ratio, 𝐺/𝑉, i.e. lines perpendicular to those representing 𝑇̇ = 𝐺𝑉, largely determines the type of growth morphology. Passing from the lower right to the upper left leads to changes in morphology (from planar, to cellular to dendritic growth). Superimposed on the diagram are typical conditions pertaining to three solidification processes: traditional casting, directional solidification (DS) and additive manufacturing by laser deposition (AM). The conditions required to produce single-crystal turbine blades (Fig. 4.5) are those at the upper end of the vertical bar marked DS. Processes producing perfect (homogeneous) single crystals under conditions of plane-front growth, such as those required for semiconductor devices (silicon), are found at the bottom of the same vertical line. The columnar to equiaxed transition (red curve) is sluggish and indicates the start of CET for an alloy with a typical nucleant density and nucleation undercooling (Gäumann et al., 1997). The red curve may take a different form and position on the diagram depending upon the alloy and the nucleation mechanism.

This section on grain structure could hardly be concluded without mentioning the development of the Cellular Automaton (CA) method introduced in the mid-90’s (Gandin and Rappaz, 1994). Its basic premise is fairly simple: the volume undergoing solidification is subdivided into very small regular cells, typically 100 m in width. Potent nucleation sites are randomly distributed within these cells according to a distribution of nucleation undercoolings given by a free growth model (Chap. 2). Similar nucleation sites can be defined for specific cells located at the surface of the volume, i.e. heterogeneous nucleation at the surface of the container. The temperature within this volume is EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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predicted using a thermal model, which is discretised at the nodes of Finite Elements (FE). Interpolation between the FE nodes and CA cells permits the deduction of the temperature at each cell location (CAFE model). If a cell contains a nucleant particle, and the local undercooling is greater than its prescribed nucleation undercooling, a new grain having a random orientation will appear. It then grows, with a dendritic morphology, along 100 directions with a prescribed 𝑉(Δ𝑇) kinetics such as developed in Chap. 4. At some point, the envelope of the grain can capture neighbouring cells thus passing-on its crystallographic orientation. The grains thus propagate from cell to cell, permitting the modelling of competition not only between columnar and equiaxed grains but also among columnar grains of various orientations. The evolution of the fraction of solid within each cell is then interpolated back to the FE nodes at each time step in order to calculate the next temperature field (see the reference of Gandin and Rappaz (1994) for details). Figure 8.11 compares the grain structure observed on a longitudinal section of a directionally solidified Al-7wt%Si ingot with a CAFE simulation. In both images, the grains are represented by using various shades of grey. Such a model not only permits visualization of the grains, it also reveals the evolution of the crystallographic texture within the columnar zone. It shows moreover that, in a thermal gradient, the transition from columnar to truly equiaxed grains is not abrupt but occurs via the intermediate formation of elongated grains nucleating ahead of the columnar front and growing in a thermal gradient. The shape of these elongated grains has been analysed in the case of eutectics (Rappaz et al., 1994) and provides another criterion for predicting a gradual (rather than abrupt) CET.

a) b) Figure 8.11 Columnar-to-Equiaxed Transition in Directional Growth of an Al-Si Ingot A longitudinal section of a directionally-solidified Al-7wt%Si ingot is shown in (a). The sidewalls were insulated while heat was extracted at the bottom by using a copper chill. From the fine grains which nucleated at the bottom surface (referred to as an “outer equiaxed zone”), columnar grains emerge and grow up to the mid-height of the ingot. Grains within the transition region at the centre clearly formed ahead of the columnar zone, but have an elongated shape because they grew in a thermal gradient. This type of elongated “equiaxed” grain cannot be accounted for by using the simple analytical approach of Hunt. Near to the top of the ingot, these grains become truly equiaxed and coarser as the upper surface is approached. In (b) is shown the result of a simulation performed by using a 3D version of the so-called CAFE model developed by Gandin et al. (1994). The model represents the microstructure in (a) astonishingly well.

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8.6. Basic Considerations of Phase and Microstructure Selection in Directional Solidification The transition from a columnar to an equiaxed grain structure within the same primary phase has been treated in the previous section. The competition that can occur among different phases and microstructures with a columnar morphology which are growing under constrained conditions (𝐺 > 0) will now be considered. This includes in particular: ▪ Transition from stable α to metastable β dendrites or vice versa, ▪ Transition from α dendrites to α + β eutectic, the so-called coupled zone (CZ), ▪ Transition from stable (α + β) to metastable (α + β′ ) eutectic, ▪ Microstructures in Additive Manufacturing (AM) processes. All of these transitions involve of course the nucleation of a new phase at some point and, if this is not possible, that transition will consequently not occur. Assuming that nucleation can occur, the transition will still be observed only if the new phase or microstructure is favoured, i.e. if it enjoys a growth advantage over the phase or microstructure that it replaces. The type and scale of the microstructure can be evaluated by using several simulation tools. One of the most useful theories by far is currently the phase-field (PF) method (Appendix 14). The drawback of this method, however, is its computational complexity, which necessitates a fairly massive computer capacity. It also requires careful specification of all of the physical parameters that enter into such a model, e.g. the interfacial energy and anisotropy, which are not always precisely known. At present, it is still difficult to use PF when it comes to complex solidification phenomena such as, for example, the modelling of the skewed coupled zone of technically important alloys. A different approach, requiring only a little computation, is the methodology presented here. Although it is semiquantitative at best, it is very useful for the planning of experiments, for understanding experimental results and for the analysis of transformation products. This general approach will be described in the present section before going on to consider the above transitions. The analytical models presented in the various chapters of this book permit the approximate computation under steady-state conditions of the growth of dendrites, cells, eutectics or mixtures of all of the foregoing. Under steady-state constrained growth at a fixed velocity, 𝑉, and thermal gradient, 𝐺, a general methodology for their selection is: (i) evaluation of the growth temperatures of the various single-phase and multiphase structures, i.e. their interface responses (IR); and (ii) determination of the phase(s) and/or morphology growing with the highest interface or tip temperature for a given composition (Hunt and Jackson, 1967; Trivedi and Kurz, 1994). This extremum criterion or competitive growth criterion can be used as a strong, although approximate, indicator of the structure that will be selected. It neglects the interaction between competing growth forms at the transition limit and assumes that nucleation of the various phases can actually occur. Despite its simplicity, this approach is of great help in determining microstructure maps for more rational alloy development. Interface Response (IR). The basic concept of microstructure-modelling in columnar growth is that the temperature of the growing interface under steady-state conditions is a function of the interface velocity, 𝑉, the thermal gradient, 𝐺, and the nominal composition, 𝐶0 . When 𝐺 > 0, the interface morphology at low and very high growth rates is always planar (Eq. 7.7). Above the limit of constitutional undercooling, 𝑉𝑐 , a single-phase interface assumes a cellular or dendritic morphology and the growth temperature (𝑇 ∗ ) ranges between solidus and liquidus (Fig. 8.12). The low-velocity cell-tip undercooling can be approximated by using the approach developed by Bower et al. (1966) (Eq. 7.9). At higher velocities, the cell-tips become morphologically unstable, develop side-arms and grow along preferred crystallographic directions. The tips simultaneously become sharper and take on a form close to that of a paraboloid of revolution. Because the solute-diffusion fields of neighbouring dendrites do not

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(unlike cells) interact strongly when the dendrites are well-developed, the growth of dendritic arrays can often be approximated by treating just isolated dendrite tips (Eq. 7.17).

Figure 8.12 Single-Phase Interface Response (IR in Red) and Concentration Distributions at the Interface for Various Growth Velocities in Directional Growth (𝑮 > 𝟎). As indicated on the phase diagram (a), with its straight equilibrium liquidus and solidus lines and typical non-equilibrium (broken) lines, the upper and lower black curves in (b) correspond to velocity-dependent liquidus, 𝑇𝑙 , and solidus, 𝑇𝑠 , for alloy 𝐶0 . With increasing velocity both curves approach one another, which is a result of the distribution coefficient, 𝑘𝑣 , approaching unity at a high 𝑉. This effect can also be seen in the concentration-jumps at the interface (c) for five velocity ranges. In the unstable regime of plane-front growth between 𝑉𝑐 and 𝑉𝑎 , cells and dendrites are prevalent and their IR usually goes through a maximum. Oscillatory interface instabilities develop when close to 𝑉𝑎 due to a lowering of the undercooling, with increasing 𝑉, leading to banded structures (see text). Beyond the second maximum in the IR, attachment-kinetics takes over and plane-front growth prevails.

Using the analyses developed in Chaps 4 and 7, the interface response (IR) for the common planar, cellular and dendritic structures can be summarised by three expressions: Low-𝑉 plane-front:

𝑇 ∗ = 𝑇𝑓 − Δ𝑇0

Cells/dendrites:

𝑇 ∗ = 𝑇𝑓 −

𝐺𝐷 2Γ 𝑅𝑔 𝑇𝑓 𝑉 + 𝐶𝑙∗ 𝑚′ − − 𝑉 𝑅 Δ𝑆𝑓 𝑉0

High-V plane-front: 𝑇 ∗ = 𝑇𝑓 + 𝐶0 𝑚′

𝑘𝑣 − 1 𝑉 − 𝑘𝑣 𝜇𝑘

𝑉 < 𝑉𝑐

[8.21a]

𝑉𝑐 < 𝑉 < 𝑉𝑎

[8.21b]

V > 𝑉𝑎

[8.21c]

where 𝐶𝑙∗ = 𝐶0 𝐴(𝑃𝑐 ) (Eq. A9.18(a)), 𝑉𝑐 = 𝐺𝐷/Δ𝑇0 and 𝑉𝑎 ≅ Δ𝑇0𝑣 𝐷/(𝑘𝑣 Γ). Note that the temperature of the plane-front at a low velocity, i.e. below constitutional undercooling, corresponds to the equilibrium solidus temperature. Equations 8.21(a-c) together produce the red curve (IR) plotted in Fig. 8.12. The cells and dendrites always remain below the liquidus, while the plane-front

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generally follows the solidus temperature but is unstable between 𝑉𝑐 and 𝑉𝑎 . The two maxima in the IR curve characterise the cell-to-dendrite transition, and the combined effect of the velocitydependent distribution coefficient and attachment-kinetics undercooling, respectively. A detailed description of the full model can be found for example in Trivedi and Kurz (1994), Kurz (2001) or Mohammadpour et al. (2020). Figure 8.13 illustrates the principle of selecting the observed microstructure or phase on the basis of the maximum interface temperature criterion. As pointed out earlier, these results are computed for steady-state growth conditions and their comparison does not permit the specification of transition phenomena. When steady-state growth has been established, however, the prediction should be a reasonable approximation to reality. Assuming a fixed temperature gradient and hypoeutectic alloy composition, Figure 8.13 graphically compares the IR function, shown in Fig. 8.12 for a single phase, α, with the function, 𝑇(𝑉), calculated for an (α + β) plane-front eutectic (broken curve). On the basis of the maximum-temperature criterion it is easy to see that the order of microstructures which appear with increasing 𝑉 is: plane-front (pf) two-phase eutectic – single-phase cells – single-phase dendrites – pf eutectic - banding – single-phase pf. The observed and “dormant” (i.e. inoperative) morphologies at V1 are shown on the right-hand side in the form of a Bridgman experiment.

Figure 8.13 Microstructure-Selection According to the Criterion of Steady-State Maximum Growth Temperature in a Hypoeutectic Alloy (𝑮 > 𝟎) This simple criterion for microstructure selection involves firstly the calculation of the temperature of the solidification front of each of the various competing microstructural elements during steady-state growth at a fixed temperature gradient, 𝐺, as a function of the velocity, 𝑉. This has been done here for a single phase, α, (see the IR shown in Fig. 8.12) and for the planefront (α + β) eutectic. The criterion is then based upon selecting the structure having the highest growth temperature as being the one which will actually be observed: at velocity, 𝑉1 , α-phase dendrites precede both plane-front eutectic and plane-front α-phase (see interfaces on the RHS of the figure). The α-dendrites are therefore predicted to be the predominant morphology. Although the eutectic will probably appear as a segregation product in the interdendritic spaces at this velocity, it is not the leading morphology. At low and high velocities plane-front eutectic predominates, with no formation of α-cells or dendrites ahead of the front. Plane-front growth of α is observed beyond absolute stability in the form of bands and, beyond the maximum temperature, as a single-phase planar front.

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Close to absolute stability, the IR goes through a minimum. Beyond this minimum, the rising tip-temperature of fine cells at high 𝑉 is due to increasing solute-trapping. As a consequence of the decreasing undercooling with V, the interface develops an oscillatory (temporal) instability (see Chap. 7), which shows up, after solidification, as bands lying parallel to the solid/liquid interface. These alternating bands consist of plane-front and cells and have a spacing proportional to the geometric mean of the solutal and thermal lengths (Fig. 7.6). The “explosive” initiation of this oscillation when close to 𝑉𝑎 has been computed using a phase-field model and is shown in Fig. 8.14.

Figure 8.14 Initiation of a Plane-Front Morphology at Absolute Stability This phase-field modelling result by Ji et al. (2023) shows the sequence of stages that terminates the steady-state cellular-dendritic branch of an Al-3wt% Cu alloy (𝐺 = 5 × 106 K/m) when the growth velocity reaches absolute stability at 𝑉𝑎 ≈ 0.88 m/s. High-𝑉 cells start to sharpen and oscillate (a), then almost explosively expand their tips (b,c), consume the remaining liquid by backwards growth (d,e) and finally transit to a planar morphology with residual liquid which starts rebuilding a diffusion boundary layer and initiates banding. The velocity evolution with a 7-fold peak during a fraction of a microsecond is displayed in (f).††

8.7. Phase Selection in Dendritic Growth – Stable-to-Metastable Single-Phase Transition In the case of a peritectic system such as Fe-C or Fe-Ni, the IR approach is useful for estimating the conditions under which the stable or metastable phase appears. Figure 8.15 shows a typical peritectic phase diagram with its stable and metastable liquidus and solidus lines. For an alloy of composition, 𝐶0 , the IR of both the primary phase, δ, and of the peritectic phase, γ, are shown as a function of the growth rate, 𝑉. Because 𝑘γ > 𝑘𝛿 and |𝑚γ | < |𝑚δ |, the solidification interval of the peritectic phase, γ, is much smaller than that of the primary phase, δ, and the IR of δ at low speed is much more pronounced than that of γ. When 𝑉 < 𝑉𝑐 , plane-front growth is the stable morphology under steady-state conditions and occurs at the solidus temperature. The metastable γ-phase with its higher solidus at 𝐶 = 𝐶0 therefore appears first. When 𝑉 > 𝑉𝑐 , the δ-cells/dendrites have a tip temperature which is higher than that of γ-dendrites and are thus the ones which are observed, as expected on the basis of the phase diagram. However, as the velocity is further increased, the metastable γ-phase is again stabilized up to high velocities due to its lower solute-rejection rate. ††

The authors are indebted to A.Karma for sharing his results before publishing.

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The second cross-over point for this dendrite growth transition, which occurs at medium velocities, is signalled by equality of the growth temperatures of the two phases (Umeda et al., 1996). Using the square-root relationship for the dendrite undercooling of each phase, the transition-velocity becomes: 2

𝛿𝛾

Δ𝑇𝑓 + 𝐶0 Δ𝑚𝛿𝛾

𝛿𝛾

𝑉𝑡𝑟 ≅ 𝜎 ∗ 𝐷 (

1/2 (Δ𝑇0𝛿 𝑘 𝛿 Γ 𝛿 )

−

1/2 𝛾 (Δ𝑇0 𝑘 𝛾 Γ 𝛾 )

)

[8.22]

Figure 8.15 Phase/Microstructure Selection in Binary Peritectic Fe-Ni Alloys Fe-Ni alloys exhibit a peritectic phase equilibrium at 4.33 at% Ni (a). For low Ni concentrations, δ-iron is prevalent and, at higher concentrations, it is γ-iron. The corresponding IR for δ and γ are shown in (b). The microstructure selection map (c) shows fields in 𝑉 − 𝐶0 space for different phases and structures that form during directional solidification with 𝐺 = 105 K/m. Solidification at very low rates (𝑉 < 𝑉𝑐 ) of a 4.33at% Ni alloy occurs with a γ-plane-front. With increasing speed, γ-plane-front transforms into δ-dendrites which, at still higher rate, change into γ-dendrites, and finally into absolutely stable plane-front of γ. In the microstructure map (c), the transition limit from δ to γ is in the form of an S-curve. The form of this limit has important consequences in technical alloys with respect to hot cracking (Umeda et al., 1996). At very low 𝑉 with 3.8 < 𝐶0 < 4.3at% Ni (horizontally hatched region), two-phase (δ-γ) growth of peritectic occurs (see Chap. 5). 𝛿𝛾

The velocity, 𝑉𝑡𝑟 , at which the δ − γ transition occurs is controlled by thermodynamic 𝛿𝛾 equilibrium and by the growth kinetics. Thus, Δ𝑇𝑓 is the melting-point difference between δ and γ, and Δ𝑚𝛿𝛾 is the difference in the liquidus slopes of δ and γ , while Δ𝑇0ν 𝑘 ν = 𝐶0 𝑚ν (𝑘 ν − 1), with EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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ν = δ or γ . The numerator within the large parentheses represents the difference in liquidus temperature between δ and γ for the composition, 𝐶0 , and reveals the effect of the phase equilibria 𝛿𝛾 𝛿𝛾 upon 𝑉𝑡𝑟 . Therefore, 𝑉𝑡𝑟 is zero when the numerator in this equation is zero, corresponding to the composition of the liquid at the peritectic temperature. The denominator meanwhile represents the difference in the growth kinetics of the dendrites of the two phases. 8.8. Dendrite-to-Eutectic Transition: the Coupled Zone (CZ) in Columnar Growth As shown by Fig. 5.15, binary eutectics can undergo two types of morphological instability: single-phase or two-phase. The latter is analogous to the morphological instability of a planar singlephase interface (Chap. 3) but, due to the two-phase structure and its discontinuity at the solid/solid interface, any quantitative analysis is complex. In general it can be said that a third alloying element which is similarly partitioned between both solid phases will lead to two-phase instability and the appearance of eutectic cells (Fig. 5.15(b)) or even eutectic dendrites. During the growth of a pure binary eutectic with an off-eutectic composition, 𝐶0 , single-phase instability can occur and result in the appearance of mixed structures; that is, dendrites of one phase plus interdendritic two-phase eutectic (Fig. 5.15(a)). The explanation of this structure is that, in an off-eutectic composition, the alloy liquidus is always higher than the eutectic temperature (Fig. 8.16). In this case, due to the long-range boundary layer built up ahead of the solid/liquid interface, the corresponding primary phase is more highly undercooled and accelerates to reach a steady state, i.e. to adopt an undercooling which results in the same growth rate as that of the eutectic. A critical discussion of the stability of off-eutectic growth can be found in Karma and Sarkissian (1996). Off-eutectic growth is of considerable importance to industry because the properties of a casting can be appreciably impaired or enhanced when single-phase dendrites appear in solidified eutectic alloys. The coupled zone (CZ) is defined as the region in a kinetic 𝐶 − Δ𝑇(𝑉) space below Te wherein the α − β eutectic grows alone without the preceding formation of single-phase dendrites. It is most often represented as a diagram centred on the eutectic composition, with the equilibrium phase diagram above, and a kinetic map below the eutectic temperature, with the vertical coordinate being the growth temperature (equal to 𝑇𝑒 − Δ𝑇𝑒 (𝑉). Such a diagram is shown in Fig. 8.16. The CZ of a given alloy may take two different forms depending upon the value of the temperature gradient. When 𝐺 → 0, the cell branch of 𝑇(𝑉) for the hypereutectic alloy is absent (red curve in (b)) and the CZ converges to the eutectic concentration as the undercooling approaches zero (red in (a)). For 𝐺 > 0, cells form at low velocities. The temperature-velocity curve for cells and dendrites exhibits a maximum (black curve in Fig. 8.16(b)), while the pf eutectic curve (which is unaffected by 𝐺) exhibits a monotonic decrease in 𝑇 as 𝑉 increases. The cell/dendrite curve intersects the eutectic curve at low and high velocities (black and red dots): between these two velocities, βcells/dendrites form with interdendritic eutectic, while outside of this region the eutectic alone forms. This explains the anvil-like shape of the CZ for 𝐺 > 0. Another important factor in eutectic solidification is the symmetry of the CZ. For nf-nf eutectics, the CZ is generally symmetrical (Fig. 8.17(a)). This is due to the similarity of the (𝑇 − 𝑉) curves of both phases, α- and β- cells/dendrites. However, if the growth behaviour of the two phases is different and involves the formation of a nf-f eutectic with an irregular structure, the CZ is skewed towards the faceted phase. A comparison of Figs 8.17(a) and 8.17(b) explains that the much higher undercooling of the faceted β-dendrites in the form of plates (i.e. 2D-dendrites) and also of the irregular eutectic leads to displacement of the CZ to higher compositions in B-element. This explains the appearance of a microstructure consisting of α-dendrites plus eutectic when 𝐶B = 𝐶𝑒 , and even in hypereutectic alloys (𝐶0 in Fig. 8.17(b)) at very high growth rates! This seems strange at first but can be easily explained by comparing the corresponding growth-kinetics curves of the two primary phases and of the eutectic.

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Figure 8.16 Coupled Zones for Two Temperature Gradients in Directional Growth The coupled zone (CZ) of a binary alloy (a) is the range of compositions and temperatures (below the eutectic equilibrium temperature) within which only eutectic multiphase structures are observed. The CZ is delimited on both sides by the appearance of single-phase growth, mostly in the form of dendrites or cells together with eutectic. A semi-quantitative relationship between the growth kinetics and the microstructures can be easily obtained only for directional growth ( 𝐺 > 0) (a). (For equiaxed grains, competition between microstructures/phases depends upon both nucleation and growth kinetics, see Sect. 8.4.). For 𝐺 > 0 but small (red CZ limits in (a)), the CZ converges to the eutectic concentration (Eq. 8.25(b)) and there is only one cross-over point of the IR of both microstructures (red dot). When 𝐺 ≫ 0, the CZ has an anvil-like shape with two crossover points (black and red) (Eq. 8.25(a)). The reason for this behaviour can be found in the growth curves (b). The upper part of the anvil-shaped CZ in (a) is due to the low-velocity cellular branch of the growth curve, a branch which does not exist when 𝐺 → 0.

The most important practical effect of this is that a fully eutectic microstructure might well not be obtained when an alloy of eutectic composition is rapidly solidified. Because the zone is skewed towards any faceted phase which may experience some “difficulty” in growth (e.g. Si in Al-Si), it is necessary to use a starting composition which is richer in the element associated with the faceted phase if a eutectic microstructure free from α-dendrites is desired. The limits of the CZ represent growth kinetics and are not metastable extensions of the liquidus lines. Eutectic growth under normal solidification conditions can be well-modelled by using the Jackson-Hunt equation (Jackson and Hunt, 1966, see Chap. 5 and Appendix 10), i.e.: Δ𝑇𝑒 = (𝑉/𝐾𝑒 )1/n where 𝑛 = 2. To simplify the CZ computation, one often uses an approximate dendrite model having the same exponent, 𝑛 = 2, as that of the eutectic: Δ𝑇𝑑ν ≃ (𝑉/𝐾𝑑ν )1/n where 𝐾𝑑ν is the growth constant for the corresponding phase, ν (Eq. 8.8). To include also cellular growth at low velocity in a positive temperature gradient, the growth kinetics expression is modified to give (also setting 𝑛 = 2 in the growth kinetics of α as an example):

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Figure 8.17 Symmetric and Skewed Coupled Zones in Binary Eutectics During Directional Growth Symmetrical coupled zones (a) are associated with regular eutectics which involve easy-growing nf–nf phases (such as metals) and exhibit similar growth undercoolings for both single-phase cells/dendrites. Plane-front eutectic growth is found in the hatched region. Outside this range, dendrites are leading the interdendritic eutectic front. In the case of nf–f eutectics (b), the associated high undercoolings at high growth rates of the irregular eutectic and of the faceted primary β-phase as compared to that of the non-faceted α-phase, lead to the existence of a skewed coupled zone. In the latter case, due to the anisotropic growth mechanism (Fig. 2.16), the primary β-phase often grows with a 2D plate-like morphology (Fig. 2.12) and requires a higher growth undercooling than do 3D needles. In order to model a plate morphology, the diffusion-field solution for a parabolic cylinder (2D dendrite) is often used (Eq. A8.16).

Δ𝑇𝑑𝛼 ≅

𝐺𝐷 + (𝑉/𝐾𝑑α )1/2 𝑉

[8.23]

By linearizing the phase diagram around the eutectic concentration, the equality of the eutectic front and the primary phase temperature can be written: Δ𝑇𝑑𝛼 − Δ𝑇e = 𝑚𝛼 (𝐶𝑒 − 𝐶0 )

[8.24]

The CZ boundaries are therefore described by: −1/2

1 𝐺𝐷𝐾𝑒−1 𝐾𝑑𝛼 − 𝐾e −1/2 (𝑇e − 𝑇 ∗ )] 𝐶0 = 𝐶e − [ + −1/2 𝑚𝛼 (𝑇e − 𝑇 ∗ )2 𝐾e

[8.25]

At a constant temperature gradient, the interface temperature, 𝑇 ∗ , as a function of 𝐶0 , describes the CZ boundary between eutectic and α-dendrites+eutectic. The first term in the square bracket of

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Eq. 8.25 represents the eutectic versus α-cell competition at small undercoolings, (𝑇e − 𝑇 ∗ ), when 𝐺 > 0. Upon ignoring the second term in the square bracket it can be approximated by: 𝐶e − 𝐶0 ≃

𝐺𝐷𝐾𝑒−1 𝑚𝛼 (𝑇e − 𝑇 ∗ )2

[8.25a]

The second term in the square bracket of Eq. 8.25 corresponds to dendrite-eutectic competition at large undercoolings and is approximately given by: 𝐶e − 𝐶0 ≃

𝐾𝑑𝛼

−1/2

− 𝐾e −1/2

𝑚𝛼 𝐾e

−1/2

(𝑇e − 𝑇 ∗ )

[8.25b]

It can be seen that, at small undercoolings (temperatures close to the eutectic temperature), 𝐺 is the parameter which governs the CZ while, at higher undercoolings, it is the difference between −1/2 the growth kinetics of dendrites and eutectic, (𝐾𝑑𝛼 − 𝐾e −1/2 ) which governs. When the coupled zone is symmetrical (Fig. 8.17(a)), a eutectic morphology will obviously be observed for the eutectic composition at any growth rate (undercooling). Skewed coupled zones are usually associated with eutectics in which one of the phases exhibits an anisotropic growth behaviour. This results in irregular eutectics which require higher undercoolings (e.g. Al-Si), while the β-phase, with an anisotropic crystal morphology, grows with a plate-like morphology (e.g. Si with its twinplane re-entrant growth mechanism, Chap. 2). The growth behaviour of the latter can be approximated by using a parabolic cylinder (a 2D dendrite, see Appendix 8, Eq. A8.16), which has also a higher undercooling versus growth-rate relationship. Important: Equation 8.23 and Figure 8.17 explain why the limits of the CZ are not metastable liquidus lines which have been extended to below the eutectic temperature (an error that can be found in the literature). They are instead a result of the growth undercoolings of the competing phases/morphologies. The competition between dendrites and eutectic is of importance to many industrial alloys that are of hypo- or hyper-eutectic composition, especially those of Al-Si type. It has also become of increased interest with the advent of additive manufacturing (see below). For a more detailed coverage of CZ theory, see Trivedi and Kurz (1988). 8.9. Transition From Stable to Metastable Eutectic Stable and metastable phases can generally form in alloys at different equilibrium temperatures. Dilute Al-Fe alloys for example exhibit two eutectics: a stable one with Al3Fe and another, metastable one, with Al6Fe (Fig. 8.18(a)). The phases and their associated microstructures in this system are: Al3Fe plate-dendrites, Al6Fe needle-dendrites, Al-Al3Fe eutectic, Al-Al6Fe eutectic and Al needledendrites. Modelling of these structures and morphologies in the manner explained previously permits the construction of the microstructure selection map of Fig. 8.18(b). This diagram agrees reasonably well with the experimentally determined map (Hughes and Jones, 1976). Another more-complex phenomenon has been observed in cast irons where superposition of a metastable-to-stable transition (white to grey iron) competed with a columnar-to-equiaxed transition of both eutectics (Jacot et al., 2000).

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Figure 8.18 Microstructure Selection Map of Stable and Metastable Eutectic Al-Fe Alloys In (a) the stable and metastable eutectics of dilute Al-Fe alloys are shown. The corresponding microstructure map (b) for both stable (black) and metastable (grey) eutectics under conditions of directional solidification is typical of a skewed CZ. Note that the velocity scale can be converted into an undercooling scale, and the latter can be flipped over and superimposed on to the phase diagram as shown in Fig. 8.17. Experimental (black and grey surfaces, Hughes and Jones, 1976) and theoretical data (dashed curves, Kurz and Gilgien, 1994) are superimposed here, showing that they agree reasonably well.

8.10.

Microstructures in Additive Manufacturing

In recent years, Additive Manufacturing (AM) has become an ever-more popular method for producing limited numbers of complex parts. A localised heat-source, generally a laser, is used for the melting of a powder either deposited as a bed (Powder Bed Fusion (PBF) or Selective Laser Melting (SLM)), or directly blown into the melt-pool (Direct Metal Laser Melting (DMLM) or Laser Metal Forming (LMF)). In either case, the solidified powder forms a more or less complex component. These processes are conceptually similar to the laser surface treatment which has been extensively studied since the 1980’s. The latter will therefore be used here as a model for all laserbased additive manufacturing processes. In the process of surface-melting, a focused high-power laser beam is moved at a velocity, 𝑉𝑏 , over a surface (Fig. 8.19). Under steady-state conditions the melt-pool that forms below the focal point moves at the same velocity, 𝑉𝑏 , through the material. Neglecting for the moment any undercooling, the fully-liquid region (in red) is bordered by the liquidus (or eutectic) isotherm for a primary phase (or eutectic). The points on this surface that have a normal perpendicular to 𝑉𝑏 define a line that marks the transition between melting (ahead of the line) and solidification (behind the line). The projection of this line onto a surface perpendicular to the travel-direction of the laser defines the re-melted trace. Surrounding the weld trace is a region where partial melting has occurred and the solid might have been affected by solid-state transformations (Heat Affected Zone or HAZ in Fig. 8.19(a)). Although melting can occur very rapidly, the rate of solidification behind the laser-beam is dictated mainly by the rate of heat-extraction from the bulk material. As 𝑉𝑏 is increased, the isotherms ahead of the laser becomes denser and denser, thus increasing the thermal gradient, 𝐺, but the trailing liquid pool simultaneously becomes more elongated. The actual velocity, V, of the liquidus (for dendrites) or of the eutectic temperature (for eutectics) at any point along this trail is given by:

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Figure 8.19 Melt-Pool Produced by a Moving Laser Beam A focused laser beam permits the rapid melting and solidification of the surface of a component (a). The phenomena occurring during such a treatment are quite similar to those encountered during Additive Manufacturing (AM). In this figure, the interface is represented by two isotherms corresponding to the liquidus and solidus temperatures. Neglecting any undercooling, the region within the liquidus isotherm has been fully re-melted, while that between the two isotherms corresponds to partial melting. Together with a zone where solid-state transformations may have occurred, the overall region is known as the Heat Affected Zone (HAZ). The liquid pool forming under the focused laser-beam is characterized by the occurrence of rapidly changing conditions from bottom to top, or across a transverse section of the trace. The bottom of the laser trace marks the transition from melting (ahead of the beam) to solidification (behind the beam), and therefore 𝑉 = 0 at that point (line). However, this is also the region where the isotherms are densest, as compared with those at the back of the melt pool; i.e., the thermal gradient, 𝐺, is highest here. At steady state, the velocity any point of the liquidus surface, 𝑉, is equal to 𝑉𝑏 cos 𝜃, where 𝑉𝑏 is the travel velocity of the beam, and 𝜃 the angle between 𝑉𝑏 and the normal to the liquidus surface (b). The growth velocity, across the pool, of dendrites having a preferred ℎ𝑘𝑙 growth direction depends not only on 𝑉 (= 𝑉𝑏 cos 𝜃), but also on the angle between the interface normal and that ℎ𝑘𝑙 orientation which is closest to the heat-flow direction (Rappaz et al., 1990).

𝑉 = 𝑉𝑏 cos 𝜃

[8.26]

where 𝜃 is the angle between the travel-speed of the laser, 𝑉𝑏 , and the normal to the liquidus/eutectic surface. For dendrites growing along preferred ℎ𝑘𝑙 directions, it is also necessary to consider the angle, 𝜓ℎ𝑘𝑙 , between their primary trunks and the velocity of the isotherm in order to deduce their velocity (Rappaz et al., 1990): 𝑉ℎ𝑘𝑙 =

𝑉 cos 𝜃 = 𝑉𝑏 cos 𝜓ℎ𝑘𝑙  cos 𝜓ℎ𝑘𝑙

[8.27]

In Fig. 8.20, the relevant processing variables for a laser-remelted Al-Cu alloy are plotted as a function of melt-depth below the surface for two beam velocities, 𝑉𝑏 , the variables being the solidification velocity, 𝑉, the temperature gradient at the solid/liquid interface, 𝐺, the cooling-rate, Ṫ, and the 𝐺 /𝑉 ratio. As can be seen, the growth-rate and the cooling-rate are zero at the bottom of the melt pool, while the temperature gradient and 𝐺 /𝑉 ratio are highest at that point.

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Figure 8.20 Solidification Conditions during Laser Treatment of an Al-Cu Alloy For a given beam-intensity and two beam-velocities, 𝑉𝑏 = 0.2 m/s and 𝑉𝑏 = 2 m/s, the pooldepth is 154 m and 31 m, respectively. The local solidification conditions in the symmetryplane of the re-melted trace are displayed. In (a) the solidification-velocity, 𝑉, is shown to increase rapidly and approach the laser-beam velocity, 𝑉𝑏 ; (b) the temperature gradient at the solid/liquid interface, 𝐺, decreases from bottom to top; (c) the cooling-rate, |𝑇̇|, behaves in a similar manner to that of 𝑉; and (d) the 𝐺/𝑉 ratio, which characterises constitutional undercooling, is infinite at the bottom of the trace (Frenk and Kurz 1992).

The values shown are typical of such high-energy processes, but the exact details of the processvariables will differ from case to case, especially for different laser-power densities and for different alloys. The solidification rates are generally of the order of m/s, and the cooling rates are of the order of 106 K/s, thus widening the range of solidification conditions far beyond those normally encountered in traditional casting or in Bridgman-type experiments (see Fig. 8.1). These conditions permit the creation of unique products such as nanostructures, highly supersaturated crystals, metastable phases or single crystals. The possibility of creating such structures, combined with the flexible production of complex parts, is the reason for the great interest in AM. The small size of the melt-pool and the high imposed temperature gradients together create a strong Marangoni flow‡‡, with the free liquid surface then subject to oscillations and instabilities which often degrade the surface quality of the finished components and are frequently the cause of defects. Because the defect structure is closely related to the microstructure, close control of the process is a prerequisite for guaranteeing the quality and safety of the components produced by using this method. In Fig. 8.21(a), a microstructure map for Al-4wt% Cu (already shown in Fig. 7.6) is reproduced together with superimposed 𝐺(𝑉) curves, calculated along the mid-plane of the laser trace, for the two beam velocities of Fig. 8.20 (the two dashed curves in Fig. 8.21(a)). The local solidification conditions traverse the entire gamut of microstructures on the map, from planar front at the bottom of the pool (𝑉 = 0 and maximum 𝐺 ), to cellular and dendritic morphologies, and finally to the ‡‡

The Marangoni effect is associated with a variation in the surface tension between two fluids: in the case of AM these are the liquid and the surrounding atmosphere. The surface tension usually increases with decreasing temperature (thermal Marangoni effect) thus pulling liquid away from the hot centre of the molten pool and towards its edges. Shearing of a surface can also be induced by solutal gradients, i.e. the solutal Marangoni effect.

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banding regions near to the top of the melt pool where the velocity, 𝑉, is at its maximum. The corresponding structural sequence is sketched in Fig. 8.21(b) for an Al-Fe alloy. Cell-growth changes abruptly to a band structure at 𝑉 = 0.5 m/s. The finest cell-spacing before the appearance of bands is 40 nm.

Figure 8.21 Processing Conditions and Microstructure in Laser Treatment of Al Alloys The microstructural regimes and processing conditions for laser traces of Al − 4wt% Cu alloy at two beam velocities 𝑉𝑏 (2 and 0.2 m/s), leading to two different treatment depths: 𝑑 = 31 (red) and 154 𝜇m (black) (see also Fig. 8.20) are shown in (a) (Kurz and Trivedi, 1992). The dashed 𝐺 − 𝑉 curves for both laser traces represent solidification conditions with time (from left, the bottom - to right, the surface). In (b) the observed evolution of the microstructure across the pool depth of the trace is schematically drawn for Al-2wt% Fe from 𝑉 = 0 at bottom of the trace and 𝑉 = 𝑉𝑚𝑎𝑥 at the surface (Gremaud et al., 1991). Despite the differences in composition of both alloys, the diagram in (a) gives a good indication of the solidification conditions leading to the cellular and banded microstructure observed in (b).

The SMS maps are useful when selecting AM process-parameters for the design of products on the basis of the control of solidification microstructures. With this aim in mind, the microstructure map of Al-Si-0.5 wt% Mg has been modelled using the concepts presented in this book for the velocity range of 10−4 to 102 m/s (Fig. 8.22). In the case of 8 wt% Si, the alloy is anticipated to follow the sequence: eutectic – Al-dendrites – eutectic – Al-dendrites – bands and Al-plane-front (Fig. 8.22(a)). The SMS map shown in Fig. 8.22(b) for a range of velocity 𝑉 permits to anticipate the changes in microstructure which will occur as the Si composition of the alloy is changed. As stated before, AM is similar to welding and so the methodology of SMS maps has also been applied to the welding of stainless steels (Fukumoto and Kurz, 1999). In these steels, Cr is a stabilising element for ferrite (δ) while Ni favours the formation of austenite (γ). The δ-γ transition is relevant to the quality of the weld because the δ-iron phase is much less prone to hot-cracking than the γphase. To ensure optimum rapid solidification processing, a higher Cr/Ni ratio is therefore required in order to avoid cracking. The AM of superalloys has been analysed by Babu et al. (2021). These authors specifically examined the cracking tendency, which is a major problem for these brittle high-temperature alloys. One way of avoiding the cracking problem is to eliminate the high-angle grain boundaries at which the cracks tend to form preferentially. Single-crystal (SX) components are much less cracking-prone. The repair and forming, by using AM, of the costly SX blades, which work under high-

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temperature/high-pressure (HTHP) conditions in aircraft or land-based gas turbines, was developed in the 1990’s. It requires a tight control of the processing parameters and aims to avoid the formation of stray and equiaxed grains, with their high-angle grain boundaries, a strategy which is based upon avoiding the CET (Gäumann et al., 2001).

Figure 8.22 SMS Map for the Al-Si-(𝟎. 𝟓 wt. % Mg) System The strategy to estimate the microstructures to be formed in AM of an Al-Si-0.5wt%Mg alloy has been applied by Mohammadpour et al. (2020). The IR functions of the phases and microstructures involved can be seen in (a) for 8wt%Si and 𝐺 = 106 K/m. In (b), the SMS map is plotted for Si concentrations from 4 to 16wt% and solidification velocities in the range 10-4 to 102 m/s, with indication of typical velocities encountered in additive manufacturing. The CZ of this eutectic alloy has a complex form: at first, it becomes narrower with velocity and then widens before it bends over to higher Si concentrations. The high-V behaviour is due to the strong variation of the distribution coefficient. These results are theoretical and have to be confirmed by experimental work. They form however an excellent frame for the experimental exploitation of microstructure selection.

According to the modelling introduced in Sect. 8.4, the CET can be avoided, regardless of the nucleation undercooling Δ𝑇𝑛 , by using high thermal gradients, 𝐺, which are of the order of 106 K/m and velocities, 𝑉 ≅ 0.1 m/s (upper right-hand corner of the 𝐺 − 𝑉 diagram in Fig. 8.10). In this case, the nuclei density, 𝑛0 , is the predominant factor and Δ𝑇𝑛 can be set equal to 0. Equation 8.19 is useful in this case, with an 𝑛-exponent corresponding to rapid solidification conditions. Correlation of Δ𝑇(𝑉) with the marginal stability dendrite model for 𝑉 > 2 × 10−2 m/s (for laser treatment of the superalloy CMSX4 (Fig. 8.23(a)) leads to 𝑛 = 3.4 in Eq. 8.8. With 𝑓𝑔 = 0.0066, the critical condition for a complete absence of equiaxed grains in this alloy has been determined (Gäumann et al., 2001) to be: 𝐺 3.4 3.4/3 ∝ 𝑛0 𝑉

[8.28]

This condition is specific to CMSX4 alone and has to be re-evaluated for other alloys. It is valid only if the nucleation undercooling can be neglected and is a useful criterion for processes such as welding or laser treatment. Furthermore, the dendrites of cubic crystals generally select the ⟨001⟩ growth direction which is closest to the heat-flow direction (see Eq. 8.27). The corresponding local

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growth conditions (𝑉, 𝐺) change along the curved pool surface, while the angle between 𝑉 and the ⟨ℎ𝑘𝑙⟩ growth direction depends locally upon the orientation of the considered grain (Fig. 8.19). This effect has been evaluated by Rappaz et al. (1990) for single-crystal welds made by using an electron beam, and was taken into account by Anderson et al. (2010) and Wang et al. (2015). The columnarto-equiaxed transition is sluggish (Fig. 8.11) and proceeds through multiple misoriented elongated grains competing over a certain distance (Mokadem et al., 2007; Chih-Hung Chen et al., 2021).

Figure 8.23 Microstructure Selection Map (MSM) for Two Processing Conditions and Resulting Microstructure of Single Crystal Superalloy CMSX-4§§ To illustrate the use of the concepts of the CET presented in Sect. 8.5, the figure shows (a) a microstructure selection map with two 𝐺 − 𝑉 curves A and B representing the solidification conditions during laser remelting (arrow). The CET transition has been calculated through a thermodynamic database and using 𝑛0 = 2 × 1015 m-3. The dotted line is a representation of Eq. 8.28. In (b), two microstructures obtained with the conditions A (melt-depth 400 μm) and B (meltdepth 550 μm) are shown (at different scales to display the whole width of the trace). Condition B leads to a fully equiaxed microstructure while condition A just reaches the CET at the very end of solidification, as shown by the few equiaxed grains at the surface of the trace. Otherwise, the remelted region in A is the single crystal grown epitaxially on the substrate (Gäumann et al., 2001).

The solidification conditions which have been presented above still have to be translated into processing conditions for the purposes of laser treatment. Figure 8.24 displays the range of microstructures in a computed processing map and indicates the effects of laser-power vs laser-beam velocity for two preheating temperatures. Too high a power leads to equiaxed structures. A combination of relatively low power and medium velocity is best for producing and re-shaping superalloy single-crystal components. As a concrete example of the use of such process-microstructure modelling, the epitaxial repair of a single-crystal HTHP turbine blade is shown in Fig. 8.25. During service of an aircraft gas-turbine, cracks can form due to thermomechanical exposure, e.g. in the platform of a single-crystal blade. If the crack exceeds a certain length or forms in the main part of the blade, the blade has to be rejected. In view of the high price of such complex monocrystalline castings, a repair is indicated if the safety of the component can be assured. In order to demonstrate the potential value of laser AM, the cracked region of the platform has first been machined away and then rebuilt using successive layers that were added by means of epitaxial laser metal forming. The coloured figure shows the section through a repaired platform in the form of a perfect superalloy single-crystal, with only a thin superficial layer of equiaxed grains forming during deposition of the last layer. This layer can be easily eliminated by §§

Epitaxial LMF or epitaxial single crystal (SX) Laser Metal Forming is a process which has been developed at EPFL in the 1990’s (Gremaud et al., 1996; Frenk et al., 1997). It allows the repair of SX gas turbine blades. It is a typical (early) additive manufacturing process.

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machining the part to its final dimensions. This is an example where complex microstructural conditions have to be satisfied for the successful additive manufacturing or repair of high-quality components. It is a good example of a successful epitaxial AM technique.

Fig. 8.24 Processing - Microstructure Map for Epitaxial Single Crystal Deposition The map shows, for a constant beam diameter, the laser power – beam velocity relationship required for columnar growth of CMSX4 for two preheating temperatures of the substrate. Too high power tends to form equiaxed structures and too high velocity leads to absence of melting. The map has been obtained by a combination of the CET equation (Eq. 8.28) and thermal modelling of the laser source with a given beam diameter and absorption coefficient. In order to speed-up the computation time, the thermal field of the 3D plate geometry has been obtained by a modified semi-infinite Rosenthal solution (Gäumann et al., 2001).

Figure 8.25 Successful Epitaxial-Laser Single-Crystal Repair of a CMSX-4 Gas-Turbine Blade The costly high-temperature-high-pressure SX turbine blade was repaired by first machining the cracked platform (a) of the blade and then rebuilding it by successively depositing epitaxial layers using the Laser Metal Forming(or DMLM) process (i.e. the powder is directly blown into the laser beam) (b). An electron back-scattering diffraction (EBSD) image of the transverse section of the deposit (c) proves, by its uniform blue crystal-orientation colour, that the successive layers have solidified while preserving the single-crystal nature of the component. The thin layer of equiaxed grains (in red and yellow) that have formed at the surface of the final deposition layer have been eliminated by machining the component to its final dimensions (Wagnière, 2000).

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Exercises 8.1. Using Eqs 8.9 to 8.11 and the definition of the extended volume fraction of solid, establish the time, 𝑡𝑟𝑒𝑐 , at which recalescence begins (Eq. 8.12). By using this time, demonstrate that the final grain density and grain size are given by Eqs 8.13 and 8.14, respectively. 8.2. Analyse the dependence of the grain size as a function of the cooling rate, 𝑇̇0 , nucleation kinetics (parameter 𝐵 ) and growth kinetics (parameter 𝐾𝑑 ) and, alternatively, the growth restriction parameter, 𝑄. 8.3. Using the general growth kinetics of Eq. 8.8, firstly calculate the radius, 𝑟, of a dendritic grain that has nucleated at an undercooling, Δ𝑇𝑛 , and grows up to the undercooling, Δ𝑇, of the columnar front. The thermal gradient, 𝐺, and velocity, 𝑉, of the isotherm are fixed (Bridgman conditions). Assuming that 𝑛0 grains per unit volume have formed under such conditions, the fraction of equiaxed grains at the level of the columnar front is given by 𝑓𝑔 = 1 − exp(−4π𝑟 3 /3𝑛0 ). This permits the deduction of the relationship between the thermal gradient, the density of grains and the velocity of the isotherm. 8.4. Why is the nucleation undercooling of equiaxed grains negligeable for the CET under rapid solidification conditions? 8.5. In a single-crystal turbine blade solidified under Bridgman conditions (𝐺 > 0) such as the one in Fig. 8.25, nucleation of a new misoriented (stray) grain can frequently form at the corner of the platform, leading to the rejection of the component. Mention 3 possible reasons which can explain that. How could the formation of stray grains be avoided? 8.6. Hunt’s analysis of the CET considers either columnar and or truly equiaxed grains. However, under 𝐺 > 0 , one can have a transition region where the newly formed grains ahead of columnar ones are elongated in the direction of the thermal gradient and are thus not truly equiaxed (see the grain structure at mid-height of the ingot of Fig. 8.11). Why is it so? Make a sketch of such a situation for a single “equiaxed” grain and consider the local undercooling of the various primary dendrite tips. Why is such a grain finally more elongated in the direction of G rather than on the opposite direction (as measured from the centre of nucleation)? 8.7. The transition between sedimentation or floatation of equiaxed grains occurs at around 8wt% Cu in Al-Cu. Why is it so? Describe the consequences this can have on the CET (considering that Δ𝑇𝑛 < Δ𝑇𝑐 and neglecting the influence of the moving grains on the thermal field). 8.8. In Fig. 8.25, why do equiaxed grains form only at the last added layer? Do they form in underneath added layers and if they do, what is the condition for not observing them at the end of the AM process? 8.9. What is the reason for the start of banding? This phenomenon happens at high velocity. Give an estimate of this velocity. 8.10. Considering Fig. 7.18, the key factor which determines the creation of ultra-fine eutectic microstructures is a large undercooling (far-from-equilibrium) and not a high velocity (rapid solidification). Why is this so? Which eutectic will form the finer spacing, a system with 𝑘values close to 1 or close to 0? Sketch both types of equilibrium diagrams. 8.11. What is the reason for having a transition between the primary phase and a planar front eutectic in a hypoeutectic alloy at low velocity? Does it occur for any thermal gradient? Which morphology of the primary phase would you expect?

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8.12. Al − 7wt%Cu base alloys have a wide use in castings, e.g. for car wheels. Under constrained growth conditions (𝐺 > 0) at low growth rates, eutectic growth in these alloys with no sign of dendrites is possible at off-eutectic compositions. Calculate the limit of stability of such an alloy at which α-Al dendrites plus eutectic will appear. (Hint: use the simple constitutional undercooling criterion and replace Δ𝑇0 by 𝑇𝑙 − 𝑇𝑒 ). Show qualitatively to what point in Fig. 8.16 this situation corresponds. What will happen if 𝐺 is doubled? 8.13. From the growth equations for dendrites (when 𝐺 → 0 ) and eutectic, Δ𝑇𝑑 = 𝐾𝑑 𝑉 1/2 and Δ𝑇𝑒 = 𝐾𝑒 𝑉 1/2, determine the limiting growth rate and temperature of the coupled zone as a function of the composition. Assume that the constants, 𝐾𝑑 and 𝐾𝑒 , are independent of the composition. 8.14. Can the extrapolation of the liquidus lines into the metastable region be taken as the CZ limits when 𝐺 → 0? Explain. 8.15. You observe in the microscope α dendrites + eutectic in an alloy of hyper-eutectic composition. What is your conclusion: About the alloy? About the growth conditions? 8.16. Consider the peritectic phase diagram shown in Fig. 8.15(a) and solidification conditions 𝐺 > 0 and 𝑉 < 𝑉𝑐δ . If there is no nucleation barrier for the formation of either phases, δ or γ, why a stable steady-state planar front of either δ or γ is impossible for a hypo-peritectic alloy, i.e. with a composition 𝐶δ < 𝐶0 < 𝐶𝛾 ? To which region of Fig. 8.15(c) does this situation correspond? 8.17. The Rosenthal solution, frequently used in AM or welding, assumes a power 𝑃 of a punctual heat source moving at velocity 𝑉𝑏 over a semi-infinite planar surface in the direction 𝑦, no convection, constant thermal properties, no thermal exchanges with the surrounding air and neglects latent heat release. The temperature field in this case is given by: T(𝑥, 𝑦, 𝑧) − 𝑇0 =

𝑃 −𝑉𝑏 (𝑟 + 𝑦) exp ( ) 2πkr 𝑎

where 𝑇0 is the temperature of the base material, 𝑘 the thermal conductivity, 𝑎 the thermal diffusivity and 𝑟 = (𝑥 2 + 𝑦 2 + 𝑧 2 )(1/2) . In the mid-section of the trace ( 𝑥 = 0 ), draw schematically (or with a small program) the shape of the isotherms. What is the slope of the isotherms at the surface 𝑧 = 0? If the base material is a single crystal (such as the turbine blade of Fig. 8.25), with 100 directions parallel to the (𝑥, 𝑦, 𝑧) coordinate system, draw schematically the shape of columnar dendrites in the mid-section of the melted trace, using Eq. 8.27. 8.18. In laser surface treatment the highest G/V ratios are found at which position in the trace? Indicate its value.

References and Further Reading Inoculation, Grain Size ▪ T.W.Clyne, M.H.Robert, Stability of intermetallic alumimides in liquid aluminium and implications for grain refinement, Metals Technology, 7 (1980) 177. ▪ M.A.Easton, M.Qian, A.Prasad, D.H.StJohn, Recent advances in grain refinement of light metals and alloys, Current Opinion in Solid State and Materials Science, 20 (2016) 13. ▪ Z.Fan, F.Gao, B.Jiang, Z.Que, Impeding nucleation for more significant grain refinement, Scientific Reports, 10 (2020) 1. ▪ Z.Fan, F.Gao, Y.Wang, H.Men, L.Zhou, Effect of solutes on grain refinement, Progress in Materials Science, 123 (2022) 100809.

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▪ A.L.Greer, A.M.Bunn, A.Tronche, P.V.Evans, D.J.Bristow, Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al-Ti-B, Acta Materialia, 48 (2000) 2823. ▪ A.L.Greer, Overview: application of heterogeneous nucleation in grain-refining of metals, Journal of Chemical Physics, 145 (2016) 211704. ▪ I.Maxwell, A.Hellawell, A simple model for grain refinement during solidification, Acta Metallurgica, 23 (1975) 229. ▪ X.B.Qi, Y.Chen, X.H.Kang, D.Z.Li, Q.Du, An analytical approach for predicting as-cast grain size of inoculated aluminium alloys, Acta Materialia, 99 (2015) 337. ▪ T.E.Quested, A.L.Greer, The effect of the size distribution of inoculant particles on as-cast grain size in aluminium alloys, Acta Materialia, 52 (2004) 3859. ▪ P.Schumacher, A.L.Greer, J.Worth., P.V.Evans, M.A.Kearns, P.Fisher, A.H.Green, New studies of nucleation mechanisms in aluminium alloys: implications for grain refinement practice, Materials Science and Technology, 14 (1998) 394. ▪ G.Shao, P.Tsakiropoulos, Prediction of phase selection in rapid solidification using time dependent nucleation theory, Acta Metallurgica et Materialia, 42 (1994) 2937. ▪ D.Shu, B.Sun, J.Mi, P.S.Grant, A quantitative study of solute diffusion field effects on heterogeneous nucleation and the grain size of alloys, Acta Materialia, 59 (2011) 2135. ▪ J.A.Spittle, S.Sadli, Effect of alloy variables on grain refinement of binary aluminium alloys with Al-TiB, Materials Science and Technology, 11 (1995) 533-537. ▪ A.Zambon, B.Badan, K.Eckler, F.Gärtner, B.Badan, K.Eckler, F.Gärtner, A.F.Norman, A.L.Greer, D.M.Herlach, E.Ramous, Microstructure and phase selection in containerless processing of Fe-Ni droplets, Acta Materialia, 46 (1998) 4657. Equiaxed Grain Growth ▪ B.Appolaire, H.Combeau, G.Lesoult, Modeling of equiaxed growth in multicomponent alloys accounting for convection and for the globular/dendritic morphological transition, Materials Science and Engineering, 487 (2008) 33. ▪ H.J.Diepers, A.Karma, Globular-dendritic transition on equiaxed alloy solidification, in: Solidification Processes and Microstructures, M. Rappaz, C. Beckermann, R.Trivedi (Eds), TMS Publication, Warrendale, Pennsylania, USA, (2004), p.369. ▪ M.T.Rad, M.Založnik, H.Combeau, C.Beckermann, Upscaling mesoscopic simulation results to develop constitutive relations for macroscopic modeling of equiaxed dendritic solidification, Materialia, 5 (2019) 100231. ▪ J.A.Dantzig, M.Rappaz, Solidification, 2nd Edition, EPFL-Press, Lausanne, 2016. ▪ R.A.Martinez, A.Karma, M.C.Flemings, Spheroidal particle stability in semisolid processing, Metallurgical and Materials Transactions A, 37 (2006) 2807. ▪ W.W.Mullins, R.F.Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics, 35 (1964) 444. ▪ M.Rappaz, P.Thévoz, Solute diffusion model for equiaxed dendritic growth: analytical solution, Acta Metallurgica, 35 (1987) 2929. ▪ P.Thévoz, Modélisation de la solidification dendritique équiaxe, PhD Thesis, EPFL, 1988. ▪ C.Y.Wang, C.Beckermann, A multiphase solute diffusion model for dendritic alloy solidification, Metallurgical Transactions A, 24 (1993) 2787. Interface Response, Banding, 𝑺𝑴𝑺-Maps ▪ S.Fukumoto, W.Kurz, Solidification phase and microstructure selection maps for Fe-Cr-Ni alloys, ISIJ International, 39 (1999) 1270. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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▪ S.C.Gill, W.Kurz, Rapidly solidified Al-Cu alloys - I. Experimental determination of the microstructure selection map, Acta Metallurgica et Materialia, 41 (1993) 3563. ▪ M.Gremaud, M.Carrard, W.Kurz, Banding phenomena in Al-Fe alloys subjected to laser surface treatment, Acta Metallurgica et Materialia, 39 (1991) 1431. ▪ D.A.Huntley, S.H.Davis, Thermal effects in rapid directional solidification: linear theory, Acta Metallurgica et Materialia, 41 (1993) 2025. See also appendix in A.Ludwig, W.Kurz, Acta Materialia, 44 (1996) 3643. ▪ K.Ji, E.Dorari, A.J.Clarke, A.Karma, Microstructural pattern formation during far-fromequilibrium alloy solidification, 23rd September 2022, Physical Review Letters 130 (2023) 026203. ▪ A.Karma, A.Sarkissian, Interface dynamics and banding in rapid solidification, Physical Review E, 47 (1993) 513. ▪ W.Kurz, Solidification microstructure-processing maps: theory and application, Advanced Engineer. Materials, 3 (2001) 443. ▪ P.Mohammadpour, A.Plotkowski, A.B.Phillion, Revisiting solidification microstructure selection maps in the frame of additive manufacturing, Additive Manufacturing, 31 (2020) 100936. ▪ R.Trivedi, W.Kurz, Modeling of solidification microstructures in concentrated solutions and intermetallic systems, Metallurgical Transactions A, 21 (1990) 1311. ▪ R.Trivedi, W.Kurz, Dendritic growth, International Materials Reviews, 39 (1994) 49. ▪ T.Umeda, T.Okane, W.Kurz, Phase selection during solidification of peritectic alloys, Acta Materialia, 44 (1996) 4209. ▪ M.Vandyoussefi, H.W.Kerr, W.Kurz, in Solidification Processing 1997, J.Beech, H.Jones (Eds), University of Sheffield, Sheffield, 1997, p.564. Cell/Dendrite growth, CET ▪ T.F.Bower, H.D.Brody, M.C.Flemings, Measurements of solute redistribution in dendritic solidification, Transactions of the AIME, 236 (1966) 624. ▪ M.H.Burden, J.D.Hunt, Cellular and dendritic growth, Journal of Crystal Growth, 22 (1974) 109. ▪ C.H.Chen, A.M.Tabrizi, P.A.Geslin, A.Karma, Dendritic needle network modeling of the columnar-to-equiaxed transition. Part II: three-dimensional formulation, implementation and comparison with experiments, Acta Materialia, 202 (2021) 463. ▪ J.A.Dantzig, M.Rappaz, Solidification, 2nd Edition, EPFL-Press, Lausanne, 2016. ▪ C.A.Gandin, M.Rappaz, A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes, Acta Metallurgica et Materialia, 42 (1994) 2233. ▪ M.Gäumann, R.Trivedi, W.Kurz, Nucleation ahead of the advancing interface in directional solidification, Materials Science and Engineering A, 226-228 (1997) 763. ▪ G.Guillemot, O.Senninger, C.A.Hareland, P.W.Voorhees, C.A.Gandin, Thermodynamic coupling in the computation of dendrite growth kinetics for multicomponent alloys, CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry, 77 (2022) 102429. ▪ D.M.Herlach, Non-equilibrium solidification of undercooled metallic melts, Materials Science and Engineering R, 12 (1994) 177. ▪ J.D.Hunt, Steady state columnar and equiaxed growth of dendrites and eutectic, Materials Science and Engineering, 65 (1984) 75. ▪ D.J.McCartney, J.D.Hunt, R.M.Jordan, The structures expected in a simple ternary eutectic system: part 1Metallurgical Transactions A, 11 (1980) 1243. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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▪ M.Rappaz, C.Charbon, R.Sasikumar, About the shape of eutectic grains solidifying in a thermal gradient, Acta Metallurgica et Materialia, 42 (1995) 2365. ▪ R.Trivedi, W.Kurz, Dendritic growth, International Materials Reviews, 39 (1994) 49. Eutectic Growth and Coupled Zone ▪ I.R.Hughes, H.Jones, Coupled eutectic growth in AI-Fe alloys. Part 1, Effects of high growth velocity, Journal of Materials Science, 11 (1976) 1781. ▪ J.D.Hunt, K.A.Jackson, The dendrite-eutectic transition, Transactions of the Metallurgical Society of AIME, 239 (1967) 864. ▪ K.A.Jackson, J.D.Hunt, Lamellar and rod eutectic growth, Transactions of the Metallurgical Society of AIME, 236 (1966) 1129. ▪ A.Jacot, D.Maijer, S.L.Cockcroft, A two-dimensional model for the description of the columnar-to-equiaxed transition in competing gray and white iron eutectics and its application to calendar rolls, Metallurgical and Materials Transactions A, 31 (2000) 2059. ▪ A.Karma, A.Sarkissian, Morphological instabilities of lamellar eutectics, Metallurgical and Materials Transactions A, 27 (1996) 635. ▪ W.Kurz, D.J.Fisher, Dendrite growth in eutectic alloys: the coupled zone, International Metals Reviews, 24 (1979) 177. ▪ W.Kurz, P.Gilgien, Selection of microstructures in rapid solidification processing, Materials Science and Engineering A, 178 (1994) 171. ▪ D.J.McCartney, J.D.Hunt, R.M.Jordan, The structures expected in a simple ternary eutectic system: part 1. Theory, Metallurgical Transactions A, 11 (1980) 1243. ▪ R.Trivedi, P.Magnin, W.Kurz, Theory of eutectic growth under rapid solidification conditions, Acta Metallurgica, 35 (1987) 971. ▪ R.Trivedi, W.Kurz, Microstructure selection in eutectic alloy systems, in: Solidification Processing of Eutectic Alloys, D.M.Stefanescu, G.J.Abbaschian, R.J.Bayuzick (Eds), The Metallurgical Society, 1988, pp.3. ▪ N.Wang, Y.E.Kalay, R.Trivedi, Eutectic-to-metallic glass transition in the Al-Sm system, Acta Materialia, 59 (2011) 6604. Additive Manufacturing ▪ T.D.Anderson, J.N.Dupont, T.Debroy, Stray grain formation in welds of single-crystal Ni base superalloy CMSX-4, Metallurgical and Materials Transactions A, 41 (2010) 181. ▪ S.S.Babu, N.Raghavan, J.Raplee, S.J.Foster, C.Frederick, M.Haines, R.Dinwiddie, M.K.Kirka, A.Plotkowski, Y.Lee, R.R.Dehoff, Additive manufacturing of nickel superalloys: opportunities for innovation and challenges related to qualification, Metallurgical and Materials Transactions A, 49 (2018) 3764. ▪ A.Frenk, W.Kurz, Microstructure formation in laser materials processing, Lasers in Engineering, 1 (1992) 193. ▪ A.Frenk, M.Vandyoussefi, J.D.Wagnière, A.Zryd, W.Kurz, Analysis of Laser-CladdingProcess for Stellite on Steel, Metallurgical and Materials Transactions, 28B (1997) 501. ▪ S.Fukumoto, W.Kurz, Solidification phase and microstructure selection maps for Fe-Cr-Ni alloys, ISIJ International, 39 (1999) 1270. ▪ M.Gäumann, C.Bezençon, P.Canalis, W.Kurz, Single-crystal laser deposition of superalloys: processing microstructure maps. Acta Materialia, 49 (2001) 1051. ▪ M.Gremaud, M.Carrard, W.Kurz, Banding phenomena in Al-Fe alloys subjected to laser surface treatment, Acta Metallurgica et Materialia, 39 (1991) 1431.

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▪ M.Gremaud, J.D.Wagnière, A.Zryd, W.Kurz, Laser Metals Forming : Process Fundamentals, Surface Engineering, 12 (1996) 251. ▪ W.Kurz, R.Trivedi, Microstructure and phase selection in laser treatment of materials, Transactions of the ASME, 114 (1992) 450. ▪ P.Mohammadpour, A.Plotkowski, A.B.Phillion, Revisiting solidification microstructure selection maps in the frame of additive manufacturing, Additive Manufacturing, 31 (2020) 100936. ▪ S.Mokadem, C.Bezencon, A.Hauert, A.Jacot, W.Kurz, Laser repair of superalloy single crystals with varying substrate orientations, Metallurgical and Materials Transactions A, 38 (2007) 1500. ▪ N.Raghavan, S.Simunovic, R.Dehoff, A.Plotkowski, J.Turner, M.Kirka, S.Babu, Localized melt-scan strategy for site specific control of grain size and primary dendrite arm spacing in electron beam additive manufacturing, Acta Materialia, 140 (2017) 375. ▪ M.Rappaz, S.A.David, J.M.Vitek, L.A.Boatner, Analysis of solidification microstructures in Fe-Ni-Cr single-crystal welds, Metallurgical Transactions A, 21 (1990) 1767. ▪ K.Sisco, A.Plotkowski, Y.Yang, D.Leonard, B.Stump, P.Nandwana, R.R.Dehoff, S.S.Babu, Microstructure and properties of additively manufactured Al-Ce-Mg alloys, Scientific Reports, 11 (2021) 6953. ▪ J.D.Wagnière, 2000, unpublished report, Institute of Materials, EPFL, Lausanne. ▪ L.Wang, N.Wang, W.J.Yao, Y.P.Zheng, Effect of substrate orientation on the columnar-toequiaxed transition in laser surface remelted single crystal superalloys, Acta Materialia, 88 (2015) 283.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 213-221 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 1 MATHEMATICAL MODELLING OF SOLIDIFICATION AT THE MACROSCOPIC SCALE HEAT DIFFUSION EQUATION Solidification of a material with a melting point which is above the ambient temperature will occur spontaneously once some of the solid phase has formed. The heat-flux from the hot melt to the surroundings allows the liquid to cool, to transform to solid and that solid to cool further to the temperature of the surrounding medium. Transformation to the solid can mean the formation of crystals, as in the case of most metals (crystallisation), or the creation of an amorphous state (glass formation). Latent heat is evolved at the solid/liquid interface in the case of crystallisation. It is this heat source, together with an often highly complicated interface morphology, which makes the solution of the heat diffusion equation difficult. Although the phase-field method (Appendix 14) has proven to be the best tool for the calculation of complex microstructures, it involves fairly heavy computation, especially in 3D. It is therefore at present essentially limited to singular solidification phenomena and cannot be applied to the complex situation of a whole industrial casting or an additive manufactured part. Yet, the prediction of the solidification of a whole component is of paramount importance in industrial processes to determine cold shuts, hot spots, porosity formation, hot cracking and other solidification defects. To achieve such a goal, dedicated industrial software have been developed and are now routinely used in foundries. Hidden behind user-friendly interfaces are many of the analytical concepts presented in the present book, such as dendrite growth or eutectic kinetics (Chaps 4 and 5), solidification path (Chap. 6) or micro-macroscopic modelling of equiaxed grain morphologies (Chap. 8). The coupling of macroscopic aspects of solidification (essentially heat- and mass-transfer) with microscopic models of solidification (nucleation and growth kinetics) was a topic of intense developments since the 1980’s (Rappaz, 1989; Wang and Beckermann, 1993). The emergence of Cellular Automata coupled with Finite Elements (CAFE approach, Gandin and Rappaz, 1994) in the mid-90’s allowed to consider simultaneously columnar and equiaxed grain structures, - and thus to predict the Columnar-to-Equiaxed Transition (CET) -, and to visualize them on a computer. These approximate models require to solve the heat-flow equation at the scale of a casting, with the account of the very strong non-linearity induced by the latent heat release (the unit undercooling Δℎ𝑓𝑓 /𝑐𝑐 is several hundreds of degrees for most alloys). Although this is not the main scope of the present book, this appendix gives the basic equations that are necessary to understand the heat exchanges that are involved during solidification. Front-Tracking Method (Two-Domain Method, Stefan Problem) In these methods, the overall volume within which the heat-flow equation is to be solved is subdivided into 2 domains (liquid and solid) with a solid/liquid interface (see Fig. A1.1). One thus has two diffusion equations to solve, each for a single-phase region. For a two-dimensional problem these are:

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𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 �𝜅𝜅𝑙𝑙 � + �𝜅𝜅𝑙𝑙 � = 𝑐𝑐𝑙𝑙 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

[A1.1]

𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 �𝜅𝜅𝑠𝑠 � + �𝜅𝜅𝑠𝑠 � = 𝑐𝑐𝑠𝑠 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

[A1.2]

𝑇𝑇𝑙𝑙∗ = 𝑇𝑇𝑠𝑠∗ = 𝑇𝑇𝑓𝑓 − Γ𝐾𝐾

[A1.3]

where κ𝑙𝑙 and κ𝑠𝑠 are the thermal conductivities of the liquid and solid, and 𝑐𝑐𝑙𝑙 and 𝑐𝑐𝑠𝑠 are the volumetric specific heats of the liquid and solid, respectively. Appropriate boundary conditions (e.g. equality of heat flux) are then applied at the fixed boundaries, while two boundary conditions have to be satisfied at the free (moving) solid/liquid interface:  continuity of temperature (if atom-attachment kinetics can be neglected):

 continuity of heat flow: 𝜅𝜅𝑠𝑠

∂𝑇𝑇 ∂𝑇𝑇 � − 𝜅𝜅𝑙𝑙 � = Δℎ𝑓𝑓 𝑉𝑉 ∂𝑛𝑛 𝑠𝑠 ∂𝑛𝑛 𝑙𝑙

[A1.4]

Figure A1.1

where the temperature gradients are those normal to the local solid/liquid interface. This situation (without the curvature term, Γ𝐾𝐾, in A.1.3) is known as the Stefan problem. Analytical solutions are available in some cases but, in most situations, one has to use numerical methods involving a fronttracking procedure, i.e. a method which can follow the solid/liquid interface with time. This approach, which is quite complicated, is primarily used: i. in non-stationary calculations of the solidification of pure metals (Viskanta and Beckermann, 1987; Crank, 1984), ii. in calculations of steady-state microstructures (Coriell et al., 1985; Saito et al., 1988). In such cases solute diffusion within the two media (solid and liquid) also have to be considered as well as appropriate interface boundary conditions for both the concentration and the solute flow. Averaging Method (One-Domain Method) Since front-tracking methods are very difficult to apply to general problems of solidification, especially when dealing with alloys which have a mushy zone, a simpler approach is commonly used (see Fig. A 1.2). Here one defines a local solid fraction, 𝑓𝑓𝑠𝑠 , at a given "point" or, to be more precise, EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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within a volume element which is large with respect to the microstructure and small with respect to the temperature inhomogeneities. With the enthalpy ℎ defined as: 𝑇𝑇

[A1.5]

ℎ(𝑇𝑇) = � 𝑐𝑐(𝜃𝜃)𝑑𝑑𝑑𝑑 + Δℎ𝑓𝑓 (1 − 𝑓𝑓𝑠𝑠 ) 0

where Δℎ𝑓𝑓 is assumed to be independent of temperature, one has: 𝑑𝑑ℎ �⎯⎯� div(𝜅𝜅 grad 𝑇𝑇) = 𝑑𝑑𝑑𝑑

[A1.6]

The change in enthalpy during heating or cooling of the material is, according to Eq. A1.5: 𝜕𝜕𝜕𝜕 𝑑𝑑ℎ 𝜕𝜕𝑓𝑓𝑠𝑠 = 𝑐𝑐(𝑇𝑇) − 𝛥𝛥ℎ𝑓𝑓 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑑𝑑𝑑𝑑 Equation A1.6 can be written in terms of a single spatial dimension (𝑧𝑧 or 𝑟𝑟) as: ∂ ∂𝑇𝑇 𝑑𝑑ℎ �𝜅𝜅 � = ∂𝑧𝑧 ∂𝑧𝑧 𝑑𝑑𝑑𝑑

1 ∂ ∂𝑇𝑇 𝑑𝑑ℎ �𝜅𝜅𝜅𝜅 � = 𝑟𝑟 ∂𝑟𝑟 ∂𝑟𝑟 𝑑𝑑𝑑𝑑

1 ∂ ∂𝑇𝑇 𝑑𝑑ℎ �𝜅𝜅𝑟𝑟 2 � = 2 𝑟𝑟 ∂𝑟𝑟 ∂𝑟𝑟 𝑑𝑑𝑑𝑑

(plate)

[A1.7]

[A1.8]

(cylinder)

[A1.9]

(sphere)

[A1.10]

Figure A1.2

The relationship between enthalpy and temperature (Eq. A1.7) is highly non-linear within the solidification interval of alloys (Chap. 6) and even discontinuous for pure metals or eutectics. Very few analytical solutions to Eq. A1.6 therefore exist. Numerical solutions are therefore usually required, especially in the case of multidimensional problems. It is beyond the scope of this textbook to provide a full introduction to the subject. Some basic notions are given here while more details can be found in the references listed at the end of this appendix, in particular the book of Dantzig and Rappaz (2016).

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ANALYTICAL SOLUTIONS FOR SEMI-INFINITE SINGLE-PHASE SYSTEMS Neglecting heat generation for the moment, and keeping 𝑎𝑎 (= κ/𝑐𝑐) constant, Eqs A1.1 or Al.6/Al.7 can be written for a unidirectional heat flux as (Carslaw and Jaeger, 1959; Geiger and Poirier, 1973; Szekely and Themelis, 1971; Dantzig and Rappaz, 2016): 𝑎𝑎

∂2 𝑇𝑇 ∂𝑇𝑇 = ∂𝑧𝑧 2 ∂𝑡𝑡

[A1.11]

0 ≤ 𝑧𝑧 ≤ ∞

One possible general solution of Eq. A1.11 is:

[A1.12]

𝑇𝑇(𝑧𝑧, 𝑡𝑡) = 𝐴𝐴 + 𝐵𝐵erf (𝑍𝑍)

where 𝐴𝐴 and 𝐵𝐵 are constants and 𝑍𝑍 =

𝑧𝑧 2(𝑎𝑎𝑎𝑎)1/2

[A1.13]

The error function, erf, (and its complement, erfc, which can be a better choice for problems involving heating), have the properties (Fig. A1.3) *: erf (0) = 0 ; erf (∞) = 1 ; erf (−𝑍𝑍) = −erf (𝑍𝑍) ; erfc (𝑍𝑍) = 1 − erf (𝑍𝑍)

Noting that:

erf (𝑍𝑍) =

2

𝑍𝑍

[A1.14]

�   exp (−𝑍𝑍 ′2 )𝑑𝑑𝑍𝑍 ′ π1/2 0

the first derivative is:

2 𝑑𝑑(erf [𝑍𝑍]) = 1/2 exp [−𝑍𝑍 2 ] 𝑑𝑑𝑑𝑑 π

[A1.15]

The error function, for 0 ≤ 𝑍𝑍 ≤ ∞ can be easily determined by using the approximation (Abramowitz and Stegun,1965): erf (𝑍𝑍) = 1 − (𝑎𝑎1 𝑡𝑡 + 𝑎𝑎2 𝑡𝑡 2 + 𝑎𝑎3 𝑡𝑡 3 + 𝑎𝑎4 𝑡𝑡 4 + 𝑎𝑎5 𝑡𝑡 5 )exp (−𝑍𝑍 2 ) with

1 𝑡𝑡 = 1 + 𝑎𝑎0 𝑍𝑍

a0 = 0.3275911 a2 = −0.284496736 a4 = −1.453152027

and

Some useful approximations are:

[A1.16] a1 = 0.254829592 a3 = 1.421413741 a5 = 1.061405429

1/2

4𝑍𝑍 2 erf (𝑍𝑍) ≅ �1 − exp �− �� π erf (𝑍𝑍) ≅ 1

*

𝑍𝑍 > 2

;

erf (𝑍𝑍) ≅

2𝑍𝑍 π1/2

𝑍𝑍 < 0.2

The exponential integral, E1, which is also plotted in Fig. A1.3 will be used in Appendix 8 (Eq. A8.17)

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Figure A1.3

A good approximation, at least over the range 0 < 𝑍𝑍 < 0.25, is provided by the simple Padétype expression: erf (𝑍𝑍) ≅

2𝑍𝑍 3 π1/2 (𝑍𝑍 2 + 3)

[A1.17]

In order to demonstrate that Eq. A1.12 is a general solution of Eq. A1.11, one can use Eq. A1.15, so that:

and

∂𝑇𝑇 𝐵𝐵 = exp [−𝑍𝑍 2 ] ∂𝑧𝑧 (π𝑎𝑎𝑎𝑎)1/2 ∂2 𝑇𝑇 𝐵𝐵𝐵𝐵 = − exp [−𝑍𝑍 2 ] ∂𝑧𝑧 2 2(π𝑎𝑎3 𝑡𝑡 3 )1/2

[A1.18]

[A1.19]

The time derivative is:

∂𝑇𝑇 𝐵𝐵𝐵𝐵 =− exp [−𝑍𝑍 2 ] ∂𝑡𝑡 2(π𝑎𝑎𝑡𝑡 3 )1/2

[A1.20]

Substituting the latter two expressions into the differential Eq. A1.11 shows that the error function is indeed a solution. Other solutions might be 𝑇𝑇 = 𝐴𝐴 + 𝐵𝐵erfc (𝑍𝑍) or 𝑇𝑇 = �1/ 𝑡𝑡1/2 �exp (−𝑍𝑍 2 ). The choice of the solution (assuming that one exists) will depend upon the boundary conditions.

THE MOVING BOUNDARY PROBLEM As Figure A1.4 indicates, several regions exist in a casting, each of which exhibits a specific thermal behaviour. These include the mould, the air-gap, the solid, the mushy (solid-plus-liquid) zone and the liquid. An analytical solution of the general problem is not possible. For this reason, and to permit the reader to make a quick rule-of-thumb estimate, attention will be restricted here to just one simple case.

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Simple Analytical Solution Under the assumptions of one-dimensional heat flow, semi-infinite system, highly-cooled mould at Tm, no air-gap, planar solid/liquid interface and zero superheating of the melt, the boundary conditions (Fig. A1.5) are: 𝑇𝑇𝑚𝑚 = 𝑇𝑇0 = constant

∞−< 𝑧𝑧 < 0

𝑇𝑇𝑙𝑙 = 𝑇𝑇𝑓𝑓 = constant

𝑧𝑧 > 𝑠𝑠

where the interface position on the 𝑧𝑧-axis is given by the function, 𝑠𝑠(𝑡𝑡). The temperature in the solid can be determined with the aid of Eqs A1.12 and A1.13, giving: 𝑇𝑇𝑠𝑠 = 𝐴𝐴𝑠𝑠 + 𝐵𝐵𝑠𝑠 erf �

𝑧𝑧 � 2(𝑎𝑎𝑠𝑠 𝑡𝑡)1/2

[A1.21]

applying the boundary conditions: 𝑧𝑧 = 0: 𝐴𝐴𝑆𝑆 = 𝑇𝑇0

𝑧𝑧 = 𝑠𝑠: 𝐵𝐵𝑠𝑠 =

𝑇𝑇𝑓𝑓 − 𝑇𝑇0 erf [Φ]

Φ=

𝑠𝑠 2(𝑎𝑎𝑠𝑠 𝑡𝑡)1/2

Since under the above assumptions, 𝑇𝑇𝑓𝑓 and 𝑇𝑇0 are constant (Fig. A1.5), Φ is constant and: 𝑠𝑠 = 2Φ(𝑎𝑎𝑠𝑠 𝑡𝑡)1/2

or

𝑡𝑡 =

𝑠𝑠 2 4𝑎𝑎𝑆𝑆 Φ2

Figure A1.4

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[A1.22]

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219

In order to determine the value of Φ, it is necessary to consider the (Neumann) boundary condition which reflects the heat flow at the solid/liquid interface (see also Appendix 2). The latent heat which is generated during interface advance (defined in this book to be positive for solidification) must be conducted away through the solid, giving the flux balance (for zero superheat in the liquid): Δℎ𝑓𝑓

𝑑𝑑𝑑𝑑 ∂𝑇𝑇𝑠𝑠 = 𝜅𝜅𝑠𝑠 � � 𝑑𝑑𝑑𝑑 ∂𝑧𝑧 𝑧𝑧=𝑠𝑠

[A1.23]

Figure A1.5

From Eq. A1.22: 𝑑𝑑𝑑𝑑 𝑎𝑎𝑠𝑠 1/2 =� � Φ 𝑑𝑑𝑑𝑑 𝑡𝑡 while, from Eq. A1.18 at 𝑧𝑧 = 𝑠𝑠, 𝑉𝑉 =

∂𝑇𝑇𝑠𝑠 𝐵𝐵𝑠𝑠 exp [−Φ2 ] 𝐺𝐺𝑆𝑆 = � � = (π𝑎𝑎𝑆𝑆 𝑡𝑡)1/2 ∂𝑧𝑧 𝑧𝑧=𝑠𝑠

[A1.24a]

[A1.24b]

Substituting 𝑉𝑉 and 𝐺𝐺𝑆𝑆 from Eq. A1.24 into Eq. A1.23, with 𝐵𝐵𝑠𝑠 given by Eq. A1.21, gives (with as = κs/cs): Δ𝑇𝑇𝑠𝑠o = π1/2 Φ erf (Φ) exp (Φ2 )

[A1.25]

Δ𝑇𝑇𝑠𝑠o ≅ 2Φ2 (Φ2 + 1)

[A1.26]

where the dimensionless temperature, Δ𝑇𝑇𝑠𝑠o , is equal to �𝑇𝑇𝑓𝑓− 𝑇𝑇0 �𝑐𝑐𝑠𝑠 /Δℎ𝑓𝑓 . Evaluation of Δ𝑇𝑇𝑠𝑠o permits the calculation of Φ by iteration and, using Eq. A1.22, the position of the interface as a function of time can be found to obey a square-root law. For typical values of Δ𝑇𝑇𝑠𝑠o , ranging irom 0 to 4, Equation A1.25 can be approximated quite well by (Fig. A1.6): A more general solution, which furnishes a better description of the real situation shown in Fig. A1.4(b), has been given by Garcia et al. (1979) and Clyne and Garcia (1980). An analytical treatment which takes account of the mushy zone has been developed by Lipton et al. (1982). For 2- and 3-dimensional problems and complex casting geometry, the solution of Eq. A1.6 can only be obtained with the help of numerical methods (Finite Differences-, Finite Volumes- or Finite Elements-methods). It is beyond the scope of the present appendix to describe in details these methods

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applied to solidification problems The interested reader is referred to other sources (e.g. Croft and Lilley, 1977; Patankar, 1980; Rappaz et al., 2010).

Figure A1.6

Bibliography for Further Reading  M.Abramowitz, I.A.Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.  H.S.Carslaw, J.C.Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, London, 1959.  T.W.Clyne, A.Garcia, Assessment of a new model for heat flow during unidirectional solidification of metals, International Journal of Heat & Mass Transfer, 23 (1980) 773.  T.W.Clyne, The use of heat flow modeling to explore solidification phenomena, Metallurgical Transactions B, 13 (1982) 471.  S.R.Coriell, G.B.McFadden, R.F.Sekerka, Cellular growth during directional solidification, Annual Review of Materials Science, 15 (1985) 119.  J.Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.  D.R.Croft, D.G.Lilley, Heat Transfer Calculations using Finite Difference Equations, Applied Science Publications, 1977.  J.A.Dantzig, S.C.Lu, J.W.Wiese, Modeling of heat flow in sand castings, Metallurgical Transactions B, 16 (1985) 195, 203.

 J.A.Dantzig, M.Rappaz, Solidification, 2nd Edition, EPFL-Press, Lausanne, Switzerland, 2016.  J.L.Desbiolles, J.J.Droux, J.Rappaz, M.Rappaz, Simulation of solidification of alloys by the finite element method, Computer Physics Reports, 6 (1987) 371.  C.A.Gandin, M.Rappaz, A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes, Acta Metallurgica et Materialia, 42 (1994) 2233.  A.Garcia, T.W.Clyne, M.Prates, Mathematical model for the unidirectional solidification of metals: II. Massive molds, Metallurgical Transactions B, 10 (1979) 85.  G.H.Geiger, D.R.Poirier, Transport Phenomena in Metallurgy, Addison Wesley, Reading, 1973.  J.Lipton, A.Garcia, W.Heinemann, An analytical solution of directional solidification with mushy zone, Archiv für das Eisenhüttenwesen, 53 (1982) 469.

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 S.V.Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, DC, 1980.  Q.T.Pham, The use of lumped capacitance in the finite-element solution of heat conduction problems with phase change, International Journal of Heat & Mass Transfer, 29 (1986) 285.  M.Rappaz, Modelling of microstructure formation in solidification processes, International Materials Reviews, 34 (1989) 93.  M.Rappaz, M.Bellet, M.Deville, Numerical modeling in materials science and engineering, Springer, Berlin, 2010.  Y.Saito, G.Goldbeck-Wood, H.Müller-Krumbhaar, Numerical simulation of dendritic growth, Physical Review A, 38 (1988) 2148.  J.Szekely, N.J.Themelis, Rate Phenomena in Process Metallurgy, Wiley Interscience, New York, 1971.  R.Viskanta, C.Beckermann, Mathematical Modelling of Solidification, Symposium on Interdisciplinary Issues in Materials Processing and Manufacturing, American Society of Mechanical Engineers, Annual Meeting, Boston, USA, 1987.  C.Y.Wang, C.Beckermann, Equiaxed dendritic solidification with convection, Metallurgical Transactions A, 27 (1993) 2754, 2765, 2784.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 223-239 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 2 ANALYTICAL SOLUTE AND HEAT FLUX CALCULATIONS RELATED TO MICROSTRUCTURE FORMATION The literature of solidification microstructure theory is complex. Various mathematical methods have been used in developing the models for morphological stability of interfaces, for dendritic and eutectic growth. Traditionally, the differential equations governing heat- and masstransfer during growth of these microstructures are solved analytically. Since the 1980’s, the phasefield theory, which is presented in Appendix 14, has become the method of choice. Here, we introduce the elements of the analytical solutions, the latter being helpful in understanding the role of key parameters in solidification mechanisms and models. The various published analyses differ only with respect to the approximations which are made, and to the weight which is given to various aspects of the problem in question. The most common approximation which is made is that solidification is occurring under steadystate conditions and that, therefore, the concentrations and solid/liquid interface morphology are independent of time. The principal disadvantage of this assumption is that no evolution of the interface shape can occur. The result of this constraint is that the solution to the basic diffusion problem is indeterminate and a whole range of morphologies is permissible from the mathematical point of view. In order to identify the solution which is the most likely to correspond to reality, it is necessary to find some additional criteria. Examination of the stability of a slightly perturbed growth form is probably the most reasonable way to treat this situation. One aim of the present appendix is to supply the reader with mathematical techniques which are sufficient to attack the problems of microscopic heat- and mass-transfer which are treated elsewhere in the book. Another aim is to provide the reader with a general and systematic method for approaching steady-state solidification problems. The general features of a solidification problem can be described as follows: a solid/liquid interface whose form is defined by a given mathematical function containing one or more variable parameters, is assumed to be advancing without change into the melt. As it advances, heat and/or solute evolve at each point of the interface and diffuse into the solid and the melt. The diffusing solute will build up ahead of the interface when 𝑘𝑘 < 1 and form a boundary layer, while a uniform level of the solute, 𝐶𝐶0 , is supposed to exist at a sufficiently large distance from the interface. The boundary layer can be characterised by the ratio of the diffusion coefficient to the growth rate. Typical orders of magnitude of the equivalent boundary layers for a planar interface (characteristic diffusion distance) are shown in Table A2.1. It shows that the boundary layer thicknesses for mass transfer of Table A2.1 Equivalent Boundary Layers for a Planar Interface Type of Diffusion

solute: 𝛿𝛿𝑐𝑐 = 2D/V heat: 𝛿𝛿𝑡𝑡 = 2𝑎𝑎/𝑉𝑉

Diffusing Species

Matrix at Tf

interstitial atom

crystal

substitutional atom either heat

crystal

liquid crystal/ liquid

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Layer Thickness (mm) at 𝑉𝑉 = 0.01 mm/s

10-4 10-1 100 >102

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substitutional and interstitial elements in the solid are small, while the boundary layer thicknesses for heat transfer (at low growth rates) in the solid or liquid are comparable to, or even larger than the scale of most castings. Convection must also be considered since the presence of a hydrodynamic boundary layer will reduce the thickness of the thermal boundary layer (Sect. 4.7).

DIFFERENTIAL EQUATION FOR DIFFUSION In view of the previous arguments, attention will be restricted here to solute diffusion occurring in the liquid. For simplicity, the suffix, 𝑙𝑙, will be dropped from the concentrations since these always refer to the liquid in the present appendix. That is, 𝐶𝐶𝑙𝑙 ≡ 𝐶𝐶. The equation governing any diffusion process is: ∂2 𝐶𝐶 ∂2 𝐶𝐶 ∂2 𝐶𝐶 1 ∂𝐶𝐶 + + = ∂𝑥𝑥 2 ∂𝑦𝑦 2 ∂𝑧𝑧 2 𝐷𝐷 ∂𝑡𝑡

[A2.1]

In physics texts this is usually written: 1 𝜕𝜕𝜕𝜕 𝐷𝐷 𝜕𝜕𝜕𝜕 while, in mathematical texts, it is often written in the suffix notation: 1 𝐶𝐶𝑥𝑥𝑥𝑥 + 𝐶𝐶𝑦𝑦𝑦𝑦 + 𝐶𝐶𝑧𝑧𝑧𝑧 = 𝐶𝐶𝑡𝑡 𝐷𝐷 ∇2 𝐶𝐶 =

Equation A2.1, which is analogous to Eq. A1.1, applies to three-dimensional space. However, the problems which are treated in the present book usually require the consideration of no more than two spatial dimensions: ∂2 𝐶𝐶 ∂2 𝐶𝐶 1 ∂𝐶𝐶 [A2.2] + = ∂𝑦𝑦 2 ∂𝑧𝑧 2 𝐷𝐷 ∂𝑡𝑡 Solutions can often be simplified by using a suitable coordinate system. For example, in spherical polar coordinates * Equation A2.1 becomes (Carslaw and Jaeger, 1959; Crank, 1956; Moon and Spencer, 1961): 2 1 cot 𝜃𝜃 1 1 𝐶𝐶𝑟𝑟𝑟𝑟 + 𝐶𝐶𝑟𝑟 + 2 𝐶𝐶𝜃𝜃𝜃𝜃 + 2 𝐶𝐶𝜃𝜃 + 2 2 𝐶𝐶𝜓𝜓𝜓𝜓 = 𝐶𝐶𝑡𝑡 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 sin 𝜃𝜃 𝐷𝐷

where 𝜃𝜃, 𝜓𝜓, and 𝑟𝑟 are the equatorial and azimuthal angles, and radial distance, respectively, and the suffix notation has been used for clarity. When the diffusional behaviour is independent of the angular orientations, 𝜃𝜃 and 𝜓𝜓, this equation becomes: ∂𝐶𝐶 ∂2 𝐶𝐶 2 ∂𝐶𝐶 = 𝐷𝐷 � 2 + � ∂𝑡𝑡 ∂𝑟𝑟 𝑟𝑟 ∂𝑟𝑟

More generally, one can write: ∂𝐶𝐶 ∂2 𝐶𝐶 𝑛𝑛 ∂𝐶𝐶 = 𝐷𝐷 � 2 + � ∂𝑡𝑡 ∂𝑟𝑟 𝑟𝑟 ∂𝑟𝑟

where 𝑛𝑛 = 2. Further reductions to the equation for a cylindrical or plate geometry can be obtained by replacing 𝑛𝑛 in the above expression by unity or zero, respectively. The steady-state growth of a sphere is thus governed by: For other coordinate systems, see Moon and Spencer (1961). In principle, any interface shape can be given a suitable system of coordinates. This however would be pointless unless the resultant transformed differential equation could be separated. This can be achieved only in some eleven systems, including the cylindrical, spherical, and parabolic. The latter system is very useful for treating dendrite tip problems.

*

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𝑑𝑑2 𝐶𝐶 2 𝑑𝑑𝑑𝑑 + =0 𝑑𝑑𝑟𝑟 2 𝑟𝑟 𝑑𝑑𝑑𝑑

The principal characteristic of the diffusion equation is its conservative nature, i.e. it acts so as to even out any irregularities. This can be seen firstly by considering the one-dimensional equation for rectangular coordinates: ∂2 𝐶𝐶 1 ∂𝐶𝐶 = ∂𝑧𝑧 2 𝐷𝐷 ∂𝑡𝑡

[A2.3]

Note that the left-hand-side is the expression which defines the sense of the curvature of a function. When the second derivative is positive it denotes a concave-upwards part of a function. In the present case it would correspond to a local minimum in the concentration distribution. From Eq. A2.3, the local change in concentration with time is also positive and therefore the depression in the concentration distribution will tend to be removed. The reverse is true for negative values of the left-hand side.

DIRECTIONAL GROWTH EQUATION Equation A2.3, which is known as Fick's second law, would be of little use in treating moving boundary problems because the interface movement would have to be accounted for. An equation which is expressed in a coordinate system that is moving with the interface can be used instead. In Fig. A2.1, the coordinates of the point, 𝑃𝑃, with respect to axes moving with the interface are (𝑦𝑦, 𝑧𝑧). Its coordinates with respect to a stationary observer are (𝑦𝑦 ′ , 𝑧𝑧 ′ ). From the diagram it is evident that: [A2.4]

𝑧𝑧 = 𝑧𝑧 ′ − 𝑉𝑉𝑉𝑉

Since ∂𝑧𝑧/ ∂𝑧𝑧 ′ = 1, ∂𝐶𝐶/ ∂𝑧𝑧 ′ = (∂𝐶𝐶/ ∂𝑧𝑧)(∂𝑧𝑧/ ∂𝑧𝑧 ′ ) = ∂𝐶𝐶/ ∂𝑧𝑧, and similarly, ∂2 𝐶𝐶/ ∂𝑧𝑧 ′2 = ∂ 𝐶𝐶/ ∂𝑧𝑧 2 , the left-hand side of Eq. A2.3 is unchanged. The concentration is a function of 𝑧𝑧 ′ and 𝑡𝑡, but must be transformed to become a function of 𝑧𝑧(𝑡𝑡) and 𝑡𝑡. In the moving reference frame the local variation in concentration as a function of time, ∂𝐶𝐶/ ∂𝑡𝑡, becomes: ∂𝐶𝐶 ∂𝐶𝐶 ∂𝐶𝐶 → − 𝑉𝑉 ∂𝑡𝑡 ∂𝑡𝑡 ∂𝑧𝑧 2

Figure A2.1

As the frame is moving in the z-direction one therefore has: 𝐷𝐷

∂2 𝐶𝐶 ∂𝐶𝐶 ∂𝐶𝐶 = −𝑉𝑉 + 2 ∂𝑧𝑧 ∂𝑧𝑧 ∂𝑡𝑡

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[A2.5]

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Re-introducing a second spatial coordinate (which is unaffected by the above transformation of coordinates) in order to describe lateral diffusion, and rearranging, gives: ∂2 𝐶𝐶 ∂2 𝐶𝐶 𝑉𝑉 ∂𝐶𝐶 1 ∂𝐶𝐶 + + = ∂𝑦𝑦 2 ∂𝑧𝑧 2 𝐷𝐷 ∂𝑧𝑧 𝐷𝐷 ∂𝑡𝑡

or in its time-independent form (steady-state growth): ∂2 𝐶𝐶 ∂2 𝐶𝐶 𝑉𝑉 ∂𝐶𝐶 + + =0 ∂𝑦𝑦 2 ∂𝑧𝑧 2 𝐷𝐷 ∂𝑧𝑧

[A2.6]

This equation, which is known as the directional growth equation, will be used to solve most of the problems in this book.

SOLUTIONS OF THE DIRECTIONAL GROWTH EQUATION The first step in solving a directional growth problem is to discover what functions satisfy Eq. A2.6. These functions can then be used as the starting point in solving any problem (in rectangular coordinates), and an exact solution would be obtained if all the boundary conditions could be satisfied everywhere. It is assumed firstly that the solution of Eq. A2.6 can be expressed as the product of separate functions of 𝑦𝑦 and 𝑧𝑧 alone (Carslaw and Jaeger, 1959; Crank, 1956; Moon and Spencer, 1961): 𝐶𝐶(𝑦𝑦, 𝑧𝑧) = 𝑌𝑌(𝑦𝑦)𝑍𝑍(𝑧𝑧)

[A2.7]

Inserting the relevant derivatives of this expression into Eq. A2.6 gives: 𝑑𝑑 2 𝑍𝑍 𝑉𝑉 𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑌𝑌 𝑍𝑍 + 𝑌𝑌 + 𝑌𝑌 = 0 2 2 𝐷𝐷 𝑑𝑑𝑑𝑑 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧

and dividing throughout by 𝐶𝐶(𝑦𝑦, 𝑧𝑧) gives: 1 𝑑𝑑2 𝑌𝑌 1 𝑑𝑑 2 𝑍𝑍 𝑉𝑉 1 𝑑𝑑𝑑𝑑 + + =0 𝑌𝑌 𝑑𝑑𝑦𝑦 2 𝑍𝑍 𝑑𝑑𝑧𝑧 2 𝐷𝐷 𝑍𝑍 𝑑𝑑𝑑𝑑

[A2.8]

Each term which involves 𝑦𝑦 or 𝑧𝑧 alone must be equal to a constant known as the separation constant. This can be seen by considering the term in 𝑍𝑍 for instance: either it is a constant or it is a function of 𝑧𝑧. In the latter case the other terms in Eq. A2.8 must be functions of 𝑧𝑧 in order to satisfy the equation. This however contradicts the assumption that the functions each depend upon only one variable. Each term is therefore equal to a constant, a, and the sum of the constants must be zero (from Eq. A2.8). The sign of the separation constant is determined by inspection after considering the properties which the solution must have in order to reflect the characteristics of the physical situation. Thus one can write:

or:

1 𝑑𝑑2 𝑍𝑍 𝑉𝑉 1 𝑑𝑑𝑑𝑑 + = 𝑎𝑎 𝑍𝑍 𝑑𝑑𝑧𝑧 2 𝐷𝐷 𝑍𝑍 𝑑𝑑𝑑𝑑

𝑑𝑑2 𝑍𝑍 𝑉𝑉 𝑑𝑑𝑑𝑑 + − 𝑎𝑎𝑍𝑍 = 0 𝑑𝑑𝑧𝑧 2 𝐷𝐷 𝑑𝑑𝑑𝑑

[A2.9]

In the overall direction of advance of the interface (𝑧𝑧 axis), one expects the existence of a boundary layer which is theoretically of infinite extent. This fact together with the form of Eq. A2.9 (i.e. a weighted sum of successive differentials) makes the exponential function a likely candidate. Therefore setting:

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[A2.10]

𝑍𝑍 = exp [𝑏𝑏𝑏𝑏]

and performing the differentiations indicated by Eq. A2.9 gives: 𝑉𝑉𝑉𝑉 exp [𝑏𝑏𝑏𝑏] − 𝑎𝑎exp [𝑏𝑏𝑏𝑏] = 0 𝐷𝐷

𝑏𝑏 2 exp [𝑏𝑏𝑏𝑏] +

The factor, exp [𝑏𝑏𝑏𝑏], cancels to leave: 𝑏𝑏 2 +

𝑉𝑉𝑉𝑉 − 𝑎𝑎 = 0 𝐷𝐷

This quadratic algebraic equation is solved by elementary means to give: 1/2

𝑉𝑉 𝑉𝑉 2 𝑏𝑏 = − − �� � + 𝑎𝑎� 2𝐷𝐷 2𝐷𝐷

The positive root is not considered because the solution to the problem might then predict infinite values of concentration in the liquid far from the interface. The general solution to Eq. A2.9 is thus: 1/2

𝑉𝑉 𝑉𝑉 2 𝑍𝑍(𝑧𝑧) = exp ��− − �� � + 𝑎𝑎� 2𝐷𝐷 2𝐷𝐷

[A2.11]

� 𝑧𝑧�

Considering now the term in 𝑌𝑌 (Eq. A2.8), one can set: or:

1 𝑑𝑑2 𝑌𝑌 = −𝑎𝑎 𝑌𝑌 𝑑𝑑𝑦𝑦 2

𝑑𝑑2 𝑌𝑌 + 𝑎𝑎𝑎𝑎 = 0 𝑑𝑑𝑦𝑦 2

[A2.12]

The form of this equation suggests that the function to be substituted should be such that its second derivative is of the same form as the original function, but of opposite sign. As the second derivative of an exponential function has the same sign as the original function, a circular function is a more likely candidate. Thus: or

𝑌𝑌 = cos [𝑐𝑐𝑐𝑐]

𝑌𝑌 = sin [𝑐𝑐𝑐𝑐]

[A2.13]

𝑌𝑌 = sin [𝑎𝑎1/2 𝑦𝑦]

[A2.14]

Substitution of either expression into Eq. A2.12 gives:

and

𝑐𝑐 = 𝑎𝑎1/2 𝑌𝑌 = cos [𝑎𝑎1/2 𝑦𝑦]

or

Substituting Eqs A2.14 and A2.11 into Eq. A2.7 finally gives:

or

𝐶𝐶 = cos �𝑎𝑎 𝐶𝐶 = sin �𝑎𝑎

1/2

1/2

1/2

𝑉𝑉 𝑉𝑉 2 𝑦𝑦�exp ��− − �� � + 𝑎𝑎� 2𝐷𝐷 2𝐷𝐷

1/2

𝑉𝑉 𝑉𝑉 2 𝑦𝑦�exp ��− − �� � + 𝑎𝑎� 2𝐷𝐷 2𝐷𝐷

� 𝑧𝑧�

[A2.15a]

� 𝑧𝑧�

[A2.15b]

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Examination of the form of these equations shows that, overall, the lateral distribution will be cyclic in form and that the concentration will decrease exponentially away from the interface in the growth direction (Fig. A2.2). Using the theory of Fourier series, any number of cosine (or sine) functions can be added together in order to satisfy the boundary conditions. Note that when the constant, a, is very large it will dominate the exponential decrease of the boundary layer. When it is very small, the boundary layer will be the same as that for a uniform plane interface. These two cases correspond to a frequently varying lateral concentration, and to a slowly varying lateral concentration respectively, since the value of a is inversely proportional to the wavelength of the interface morphology. This is seen in the case of eutectic growth at normal speeds where the rate of exponential decrease is dominated by the wavelength, and the thickness of the boundary layer becomes proportional to the eutectic spacing (Appendix 10). When the scale of the interface morphology can vary over a wide range or the boundary layer is reduced, the full solution (Eq. A2.15) must be used. These situations arise when carrying out stability analyses (Appendix 7) or studies of eutectic growth at high rates (Appendix 10).

BOUNDARY CONDITIONS Three types of mathematical boundary condition are generally imposed on the solution of a differential equation. These are the Dirichlet condition, which defines the absolute value of the solution at a boundary point, the Neumann condition, which defines the normal gradient of the solution at the boundary and the Robin (mixed) condition which establishes a relationship between the absolute value and the gradient of the solution at the boundary. The latter condition is the main source of difficulty in solidification problems because the interface concentrations and their gradients are usually not given explicitly but must be found as part of the solution. Since it arises from the balance between solute rejection and diffusion at the interface, this condition will be called the flux condition. The other conditions will also be given names which reflect their significance in solidification problems (Fig. A2.2). Flux Condition As a solid/liquid interface advances at the local normal growth rate, 𝑉𝑉𝑛𝑛 , with an interface concentration in the liquid, 𝐶𝐶 ∗ , and solid concentration, 𝑘𝑘𝐶𝐶 ∗ , the quantity of solute rejected per unit time will be 𝑉𝑉𝑛𝑛 (1 − 𝑘𝑘)𝐶𝐶 ∗ . This must be balanced by the creation of a concentration gradient in the liquid, normal to the isoconcentration contours, which permits solute removal at the same rate via diffusion. Thus: 𝑉𝑉𝑛𝑛 (1 − 𝑘𝑘)𝐶𝐶 ∗ = −𝐷𝐷

∂𝐶𝐶 ∂𝑚𝑚

[A2.16]

One complication is that the normal to the interface, 𝑛𝑛, along which the interface advances locally, is not generally the same as the normal, 𝑚𝑚, to the isoconcentration contours in the liquid. In order to simplify the problem, the interface should thus be assumed to be of uniform concentration (or isothermal). The flux condition can otherwise be easily applied only on an axis of symmetry, where the two normals are bound to be aligned. Such a point would be the tip (or trough) of the perturbation shown in Fig. A2.2.

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Figure A2.2

Far-Field Condition A Dirichlet condition can be imposed far ahead of the interface because here the original composition is expected to be unaffected by the advance of the interface, i.e.: 𝐶𝐶 = 𝐶𝐶0

Symmetry Condition

𝑧𝑧 = ∞

[A2.17]

Most interface morphologies consist of arrays of similar shapes. Advantage can be taken of this fact by studying just one half-period of the shape along the y-axis. If the shapes are presumed to be identical there can be no mass transfer between them. Zero concentration gradient (Neumann) conditions can thus be imposed at the boundaries of a typical interface ‘motif’, i.e.: ∂𝐶𝐶 =0 ∂𝑦𝑦

where 𝑛𝑛 = 0, 1, 2, …

𝑦𝑦 =

𝑛𝑛𝑛𝑛 2

[A2.18]

Coupling Condition

Under normal solidification conditions for metals (𝑉𝑉 < 100 mm/s), each location along the interface will have a local freezing point which is a function of the local concentration and of the

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local curvature. In steady-state growth each point of the interface must lie on the corresponding isotherm of the temperature field: 𝑇𝑇 ∗ = 𝑇𝑇𝑓𝑓 + 𝑚𝑚Δ𝐶𝐶 − Γ𝐾𝐾

[A2.19]

SATISFACTION OF BOUNDARY CONDITIONS

In a previous section, a general solution to the Laplace equation was obtained in terms of elementary functions. It would be overly optimistic to expect any real situation to involve boundary conditions permitting such a solution to be used without any further effort. The greater part of the problem unfortunately still lies ahead. The requirement that the basic solution (or its derivatives) should have certain values at the boundaries of the region studied indeed accounts for much of the effort expended by applied mathematicians. Their research works over the past two hundred years have produced an enormous range of methods for attacking the problem (Crank, 1956). Several methods which are used elsewhere in the text will be described below. Two cases will first be considered in which the boundary conditions can be satisfied exactly. Steady-State Diffusion Field Ahead of a Moving Planar Interface As was shown earlier in this appendix, the unidirectional diffusion equation A2.5 in a coordinate system with its origin fixed at the solid/liquid interface takes, after a transit (Fig. A2.3(a)), the steady-state form (Fig. A2.3(b)): ∂2 𝐶𝐶 𝑉𝑉 ∂𝐶𝐶 + =0 ∂𝑧𝑧 2 𝐷𝐷 ∂𝑧𝑧

Figure A2.3

Noting the similarity of the above expression to Eq. A2.9, setting the separation constant, a, equal to zero, and repeating the steps following Eq. A2.10, gives: 𝐷𝐷𝑏𝑏 2 + 𝑉𝑉𝑉𝑉 = 0

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The solutions of this so-called 'auxiliary' equation are 𝑏𝑏 = 0 and 𝑏𝑏 = −𝑉𝑉/𝐷𝐷. The general solution is therefore: 𝐶𝐶 = 𝐴𝐴 + 𝐵𝐵 exp �−

𝑉𝑉𝑉𝑉 � 𝐷𝐷

[A2.20]

Again following the principles described above, a far-field condition is applied. That is, far from the interface the concentration must be equal to the original composition, 𝐶𝐶0 . Letting 𝐶𝐶 = 𝐶𝐶0 when 𝑧𝑧 tends to infinity shows that 𝐴𝐴 = 𝐶𝐶0 . Therefore: 𝐶𝐶 = 𝐶𝐶0 + 𝐵𝐵 exp �−

𝑉𝑉𝑉𝑉 � 𝐷𝐷

One can next apply the flux (Robin) condition at the solid/liquid interface. The rate of solute rejection must here be equal to the diffusional flux in the liquid at the interface: 𝑑𝑑𝑑𝑑 𝐶𝐶 ∗ (1 − 𝑘𝑘)𝑉𝑉 = −𝐷𝐷 � � 𝑑𝑑𝑑𝑑 𝑧𝑧=0

Therefore when 𝑧𝑧 = 0: and

𝐶𝐶 ∗ = 𝐶𝐶0 + 𝐵𝐵

𝑉𝑉𝑉𝑉 𝑑𝑑𝑑𝑑 � � =− 𝑑𝑑𝑑𝑑 𝑧𝑧=0 𝐷𝐷

Substituting these expressions into the above shows that: 𝐵𝐵 = 𝐶𝐶0

1 − 𝑘𝑘 = Δ𝐶𝐶0 𝑘𝑘

The complete solution for the solute distribution ahead of a planar solid/liquid interface advancing under steady-state conditions is therefore (Tiller et al., 1953): 𝐶𝐶0 𝑉𝑉𝑉𝑉 𝐶𝐶 = 𝐶𝐶0 + � − 𝐶𝐶0 � exp �− � 𝑘𝑘 𝐷𝐷

[A2.21]

The boundary layer shown in Fig. A2.4 is of infinite extent. In order to obtain a convenient practical estimate of its thickness an equivalent boundary layer, 𝛿𝛿𝑐𝑐 , is often defined This equivalent layer is chosen so as to contain the same total solute content as the infinite layer, and has a constant concentration gradient across its thickness. The area of the triangle, OMN, must thus be equal to the area of the grey surface. That is: ∞ Δ𝐶𝐶0 𝛿𝛿𝑐𝑐 𝑉𝑉𝑉𝑉 = Δ𝐶𝐶0 � exp �− � 𝑑𝑑𝑑𝑑 2 𝐷𝐷 0

giving:

𝛿𝛿𝑐𝑐 =

2𝐷𝐷 𝑉𝑉

Differentiating Eq. A2.21 at 𝑧𝑧 = 0 gives: 𝐺𝐺𝑐𝑐 = �

𝑑𝑑𝑑𝑑 Δ𝐶𝐶0 𝑉𝑉 � =− 𝑑𝑑𝑑𝑑 𝑧𝑧=0 𝐷𝐷

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[A2.22]

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Figure A2.4

These developments show that the absolute value of the concentration gradient at the interface is equal to twice the absolute mean concentration gradient of the equivalent boundary layer. Such relationships are very useful in understanding the constitutional undercooling criterion (Chap. 3). Diffusion Field Around a Growing Sphere When no tangential diffusion is occurring, the diffusion equation can be written in terms of the radial coordinate alone (Zener, 1949; Parker, 1970): ∂2 𝐶𝐶 2 ∂𝐶𝐶 ∂𝐶𝐶 = 𝐷𝐷 � 2 + � 𝑟𝑟 ∂𝑟𝑟 ∂𝑡𝑡 ∂𝑟𝑟

[A2.23]

where 𝑟𝑟 is the radius. Under conditions where the Péclet number of the growing sphere, 𝑃𝑃 = 𝑟𝑟𝑟𝑟/𝐷𝐷, is small, the system reaches a quasi steady-state and one can write, 𝑑𝑑 2 𝑑𝑑𝑑𝑑 �𝑟𝑟 �=0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

The general solution of this equation is: 𝐶𝐶 = 𝐴𝐴 +

𝐵𝐵 𝑟𝑟

[A2.24]

For the situation described in Fig. A2.5, the boundary conditions are: 𝐶𝐶 = 𝐶𝐶0

𝐶𝐶 = 𝐶𝐶 ∗

𝑟𝑟 = ∞ 𝑟𝑟 = 𝑅𝑅

Satisfaction of these conditions shows that 𝐴𝐴 = 𝐶𝐶0 and 𝐵𝐵 = 𝑅𝑅(𝐶𝐶 ∗ − 𝐶𝐶0 ). Equation A2.24 thus becomes: 𝐶𝐶 = 𝐶𝐶0 +

𝑅𝑅 ∗ (𝐶𝐶 − 𝐶𝐶0 ) 𝑟𝑟

The concentration gradient in the liquid becomes: 𝑑𝑑𝑑𝑑 𝑅𝑅 = − 2 (𝐶𝐶 ∗ − 𝐶𝐶0 ) 𝑑𝑑𝑑𝑑 𝑟𝑟 EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A2.25]

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and at the interface: 𝑑𝑑𝑑𝑑 𝐶𝐶 ∗ − 𝐶𝐶0 � � =− 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑅𝑅

[A2.26]

Figure A2.5

This shows that, to a first approximation, the thickness of the boundary layer around a growing sphere is equal to the radius and increases with increasing size of the sphere. For other solutions see Frank (1950) and Engberg et al. (1975). Some methods which are available for the treatment of more difficult cases can now be considered.

CLASSICAL METHOD When using this method, advantage is taken of the linearity of Eq. A2.6. That is, the sum of any series of terms having the same form as the basic solution (such as the sine and cosine functions found previously) will also be a solution. This means that although the basic solution (often called an eigenfunction) is unlikely to satisfy the boundary conditions, 'adjustment' using terms of the same form will allow the conditions to be approximated more and more closely. For example, if the basic solution is of the form, cos [2π𝑦𝑦/𝜆𝜆], one can consider adding terms such as cos [4π𝑦𝑦/𝜆𝜆], etc., e.g.: 𝐶𝐶(𝑦𝑦) = 𝐴𝐴1 cos �

2π𝑦𝑦 4π𝑦𝑦 2π𝑖𝑖𝑖𝑖 � + 𝐴𝐴2 cos � � + ⋯ + 𝐴𝐴𝑖𝑖 cos � � 𝜆𝜆 𝜆𝜆 𝜆𝜆

[A2.27]

where the 𝐴𝐴𝑖𝑖 are constants. Because the oscillation of the cosine functions increases in frequency with increasing value of 𝑖𝑖, finer and finer adjustments can be made to the basic solution. In order to carry this out in practice, the constants before each term in a series such as the one above have to be suitably chosen. Many methods have been devised in order to find the required values of these constants. The best method, although rarely feasible, is to take advantage of the orthogonality of functions such as the circular ones (sine, cosine). In this method, each term in the above series would be multiplied by cos [2π𝑗𝑗𝑗𝑗/𝜆𝜆]: EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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𝐶𝐶(𝑦𝑦)cos �

2π𝑗𝑗𝑗𝑗 2π𝑦𝑦 2π𝑗𝑗𝑗𝑗 4π𝑦𝑦 2π𝑗𝑗𝑗𝑗 � = 𝐴𝐴1 cos � � cos � � + 𝐴𝐴2 cos � � cos � � + .. 𝜆𝜆 𝜆𝜆 𝜆𝜆 𝜆𝜆 𝜆𝜆 . . . + 𝐴𝐴𝑖𝑖 cos �

2π𝑖𝑖𝑖𝑖 2π𝑗𝑗𝑗𝑗 � cos � � 𝜆𝜆 𝜆𝜆

When both sides are integrated over one wavelength: 𝜆𝜆

𝜆𝜆 2π𝑗𝑗𝑗𝑗 2π𝑦𝑦 2π𝑗𝑗𝑗𝑗 �   𝐶𝐶(𝑦𝑦)cos � � 𝑑𝑑𝑑𝑑 = 𝐴𝐴1 �   cos � � cos � � 𝑑𝑑𝑑𝑑 + 𝜆𝜆 𝜆𝜆 𝜆𝜆 0 0 𝜆𝜆

+𝐴𝐴2 � cos � 0

𝜆𝜆

4π𝑦𝑦 2π𝑗𝑗𝑗𝑗 � cos � � 𝑑𝑑𝑑𝑑+. . . 𝜆𝜆 𝜆𝜆

. . . + 𝐴𝐴𝑖𝑖 � cos � 0

2π𝑖𝑖𝑖𝑖 2π𝑗𝑗𝑗𝑗 � cos � � 𝑑𝑑𝑑𝑑 𝜆𝜆 𝜆𝜆

[A2.28]

all of the terms on the RHS will disappear unless 𝑗𝑗 = 𝑖𝑖. Therefore by setting 𝑗𝑗 equal successively to 1, 2, 3, etc., the value of any 𝐴𝐴𝑖𝑖 can be 'singled out'. A general expression is usually obtained for all the adjustable constants of the series. The reader should satisfy himself that, if this 'trick' were not available, the 𝐴𝐴-values (Fourier coefficients) could be determined exactly only by solving an infinite set of simultaneous algebraic equations. This is unfortunately usually the case since the above method can be employed only when the boundary coincides with a coordinate line over which the functions are also orthogonal. In response to this common difficulty, approximate methods have been developed and will be described in the next section. Meanwhile a relatively little-known technique for obtaining the Fourier coefficients without integration will be described. This technique is however applicable only when the boundary conditions are discontinuous in some way. For example Fig. A13.1, where the first and higher derivatives of the concentration distribution are discontinuous at regular intervals. In the case of a cosine series the Fourier coefficients are given directly by: 𝑚𝑚

𝑚𝑚

𝑚𝑚

𝑠𝑠=1

𝑠𝑠=1

𝑠𝑠=1

1 1 1 𝐴𝐴𝑖𝑖 = �− �   𝐽𝐽𝑠𝑠 sin(𝑖𝑖𝑦𝑦𝑠𝑠 ) − �   𝐽𝐽𝑠𝑠′ cos(𝑖𝑖𝑦𝑦𝑠𝑠 ) + ⋯ + 2 � 𝐽𝐽𝑠𝑠″ sin(𝑖𝑖𝑦𝑦𝑠𝑠 ) +. . . � 𝑖𝑖π 𝑖𝑖 𝑖𝑖

[A2.29]

where the 𝐽𝐽, 𝐽𝐽′ , 𝐽𝐽′′ , etc are 'jumps' in the function, first derivative, second derivative, etc. The definition of such a jump will be given in later appendices where the technique is applied to various problems. The derivation of Eq. A2.29 is quite simple (Kreyszig, 1968) but is beyond the scope of the present book.

METHOD OF WEIGHTED RESIDUALS This technique (Finlayson, 1972) sets out in effect to satisfy the boundary conditions of the problem in the same way as the classical method does. Thus a series consisting of functions having adjustable multiplying constants is used. In this case however the functions are rarely orthogonal and the constants can be found only by solving a set of simultaneous algebraic equations. The number of terms in the series is chosen so as to be equal to the number of adjustable constants. The method is approximate in nature and the higher-order approximations can be handled only by using numerical analysis techniques and a computer. Surprisingly, accurate analytical results can nevertheless often be obtained by using just a few terms; and sometimes only one (Aris, 1978). Such an analytical solution has the advantage that it will reveal the influence of the various experimental and physical EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

Foundations of Materials Science and Engineering Vol. 103

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parameters directly, whereas a more accurate but numerical method will not. The method is extremely flexible and can be applied to any problem. As an example it will be used to find a solution to the dendritic growth problem in two dimensions. Simple dendrite analysis A function will be chosen which satisfies the boundary conditions. This is known as the external method. The trial solution can also be such that, alone, it does not satisfy either the differential equation or the boundary conditions. The reader will doubtless appreciate the flexibility of the method but on the other hand he must also have a firm grasp of the physics of the real situation in order to be able to use it properly. In the present case a Dirichlet condition will be applied at the interface, assuming that the concentration is constant over the parabolic surface of the dendrite. Thus: 𝐶𝐶 = 𝐶𝐶 ∗

𝑧𝑧 = −𝛽𝛽𝑦𝑦 2

Applying also the far-field condition, 𝑧𝑧 = ∞

𝐶𝐶 = 𝐶𝐶0

one can easily construct an expression which satisfies these conditions. Take firstly the basic solution for a planar interface which was derived above: 𝐶𝐶 = 𝐶𝐶0 + (𝐶𝐶 ∗ − 𝐶𝐶0 ) exp[ − 𝑏𝑏𝑏𝑏]

[A2.30]

𝐶𝐶(𝑦𝑦, 𝑧𝑧) = 𝐶𝐶0 + (𝐶𝐶 ∗ − 𝐶𝐶0 ) exp[ − 𝑏𝑏(𝛽𝛽𝑦𝑦 2 + 𝑧𝑧)]

[A2.31]

This describes a solute distribution in which the value at infinity is 𝐶𝐶0 . The far-field condition is thus satisfied. It is also required that the concentration be equal to 𝐶𝐶 ∗ when the value of the exponential term is unity. In the case of the planar interface this occurs only when 𝑧𝑧 is equal to zero. It is therefore necessary to replace 𝑧𝑧 by an expression which is equal to zero whenever a coordinate pair, (𝑦𝑦, 𝑧𝑧), corresponds to the surface of the parabolic plate dendrite. This is true when 𝑧𝑧 = −𝛽𝛽𝑦𝑦 2 , so that the required expression is: The reader should prove for himself that this satisfies the boundary conditions and that the concentration decreases exponentially with 𝑦𝑦 2 and 𝑧𝑧. The value of 𝑏𝑏 can be determined by means of the flux condition, applied at the tip: ∂𝐶𝐶 𝑉𝑉(𝑘𝑘 − 1)𝐶𝐶 ∗ = 𝐷𝐷 � � ∂𝑧𝑧 𝑦𝑦=𝑧𝑧=0

[A2.32]

Substituting the 𝑧𝑧-derivative of Eq. A2.31 into Eq. A2.32 gives: 𝑏𝑏 =

so that:

𝑉𝑉 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝐷𝐷 𝐶𝐶0 − 𝐶𝐶 ∗

𝑉𝑉(𝑘𝑘 − 1)𝐶𝐶 ∗ (𝛽𝛽𝑦𝑦 2 + 𝑧𝑧) 𝐶𝐶 = 𝐶𝐶0 + (𝐶𝐶 − 𝐶𝐶0 ) exp �− � 𝐷𝐷(𝐶𝐶0 − 𝐶𝐶 ∗ ) ∗

The differential equation (Eq. A2.6) is:

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[A2.33]

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Fundamentals of Solidification 5th Edition

𝜕𝜕 2 𝐶𝐶 𝜕𝜕 2 𝐶𝐶 𝑉𝑉 𝜕𝜕𝜕𝜕 + + =0 𝜕𝜕𝑦𝑦 2 𝜕𝜕𝑧𝑧 2 𝐷𝐷 𝜕𝜕𝜕𝜕

Substituting Eq. A2.33 into Eq. A2.6 and simplifying gives: 𝑉𝑉 4𝑦𝑦 2 𝛽𝛽 2 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝑉𝑉 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝑉𝑉 − 2𝛽𝛽 + − =0 𝐷𝐷 𝐷𝐷 (𝐶𝐶0 − 𝐶𝐶 ∗ ) 𝐷𝐷 (𝐶𝐶0 − 𝐶𝐶 ∗ )

The LHS of the above equation is the 'residual' which gives the method its name. The equation must be satisfied in order to solve the original problem. The first term can be eliminated by considering only the point of the dendrite. It will be equal to zero when 𝑦𝑦 = 0, so that: 𝑉𝑉 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝑉𝑉 = + 2𝛽𝛽 𝐷𝐷 𝐶𝐶0 − 𝐶𝐶 ∗ 𝐷𝐷

[A2.34]

The curvature at the tip of the parabola, 𝑧𝑧 = −𝛽𝛽𝑦𝑦 2 , is given by the second derivative of 𝑧𝑧 with respect to 𝑦𝑦. Thus, the curvature is 2𝛽𝛽 and the radius of curvature is: 𝑅𝑅 =

1 2𝛽𝛽

[A2.35]

Substituting this value into Eq. A2.34 gives:

and:

𝑉𝑉 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝑉𝑉 1 = + 𝐷𝐷 𝐶𝐶0 − 𝐶𝐶 ∗ 𝐷𝐷 𝑅𝑅 (𝑘𝑘 − 1)𝐶𝐶 ∗ 𝐷𝐷 =1+ ∗ 𝐶𝐶0 − 𝐶𝐶 𝑉𝑉𝑉𝑉

but 𝑉𝑉𝑉𝑉/2𝐷𝐷 is the Péclet number, so that: 1 (𝑘𝑘 − 1)𝐶𝐶 ∗ =1+ ∗ 𝐶𝐶0 − 𝐶𝐶 2𝑃𝑃

But (𝐶𝐶 ∗ − 𝐶𝐶0 )/(𝐶𝐶 ∗ − 𝑘𝑘𝐶𝐶 ∗ ) is the dimensionless supersaturation, 𝛺𝛺, so: Ω=

2𝑃𝑃 2𝑃𝑃 + 1

[A2.36]

which is the same as the Zener-Hillert (Engberg et al., 1975) solution (Appendix 8).

PERTURBATION METHOD If the geometry of the problem is such that the interface form corresponds closely to some simple shape, any exact solution which is available for the simple shape can be assumed to be similar to that for the slightly different morphology. This principle will be illustrated by treating an almost planar solid/liquid interface growing under steady-state conditions. A related problem will be studied in Appendix 7.

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Foundations of Materials Science and Engineering Vol. 103

237

Slightly-Perturbed Interface The form of the planar solid/liquid interface, described by the equation, 𝑧𝑧 = 0, is assumed to be changed so that it is then represented by the expression: [A2.37]

𝑧𝑧 = 𝜀𝜀 sin[ 𝜔𝜔𝜔𝜔]

where 𝜀𝜀 is assumed to be a very small amplitude and 𝜔𝜔 = 2π/𝜆𝜆 is the wave number of the perturbation. Recall that the exact solution for a planar solid/liquid interface under steady-state conditions (Eq. A2.21) is: 𝐶𝐶0 𝑉𝑉𝑉𝑉 𝐶𝐶 = 𝐶𝐶0 + � − 𝐶𝐶0 � exp �− � 𝑘𝑘 𝐷𝐷

where 𝐶𝐶0 /𝑘𝑘 is the concentration in the liquid at the interface and 𝐶𝐶0 is the original composition. Using the perturbation technique, a term having the same form as the perturbation (Eq. A2.37) is now added to the exact solution for the unperturbed interface, i.e.: 𝐶𝐶0 𝑉𝑉𝑉𝑉 𝐶𝐶 = 𝐶𝐶0 + � − 𝐶𝐶0 � exp �− � + 𝐴𝐴𝐴𝐴sin [𝜔𝜔𝜔𝜔]exp [−𝑏𝑏𝑏𝑏] 𝑘𝑘 𝐷𝐷

[A2.38]

where 𝑏𝑏 has to be equal to (𝑉𝑉/2𝐷𝐷) + [(𝑉𝑉/2𝐷𝐷)2 + 𝜔𝜔2 )]1/2 in order that the added term should satisfy Eq. A2.6 (see Eq. A2.11), and 𝐴𝐴 is a constant whose value is to be determined by forcing Eq. A2.38 to satisfy the boundary conditions. For 𝑧𝑧 = 𝜀𝜀sin [𝜔𝜔y] these are: 𝐶𝐶 = 𝐶𝐶 ∗

[A2.39]

𝑉𝑉(1 − 𝑘𝑘)𝐶𝐶 ∗ = −𝐷𝐷

𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

[A2.40]

Note that Eq. A2.38 already satisfies the far-field condition, 𝐶𝐶 = 𝐶𝐶0 when 𝑧𝑧 = ∞, since the unknown constant, 𝐴𝐴, disappears. Rather more work is required in order to make it satisfy the other boundary conditions (e.g. Eq. A2.39). The first step is to substitute 𝐶𝐶 ∗ for 𝐶𝐶 and 𝜀𝜀sin [𝜔𝜔𝜔𝜔] for 𝑧𝑧: 𝐶𝐶0 𝑉𝑉𝑉𝑉 𝐶𝐶 ∗ = 𝐶𝐶0 + � − 𝐶𝐶0 � exp �− � + 𝐴𝐴𝐴𝐴 exp[ − 𝑏𝑏𝑏𝑏] 𝑘𝑘 𝐷𝐷

[A2.41]

where, for clarity, 𝑆𝑆 has been used to represent 𝜀𝜀sin [𝜔𝜔𝜔𝜔]. The above expression can be evaluated only because of the assumption that 𝜀𝜀 (and 𝑆𝑆) are small. In this case an exponential function, exp [−𝑥𝑥], can be approximated by 1 − 𝑥𝑥. Again because 𝜀𝜀 is small, terms involving 𝜀𝜀 2 (and 𝑆𝑆 2 ) can be neglected. This leads to: 𝐶𝐶0 𝑉𝑉𝑉𝑉 𝐶𝐶 ∗ = 𝐶𝐶0 + � − 𝐶𝐶0 � �1 − � + 𝐴𝐴𝐴𝐴(1 − 𝑏𝑏𝑏𝑏) 𝑘𝑘 𝐷𝐷

By differentiating Eq. A2.21 with respect to 𝑧𝑧 and then setting 𝑧𝑧 equal to zero, the concentration gradient, 𝐺𝐺𝑐𝑐 , at the plane interface is found to be (Eq. A2.22): 𝐺𝐺𝑐𝑐 = −

𝑉𝑉 𝐶𝐶0 � − 𝐶𝐶0 � 𝐷𝐷 𝑘𝑘

Substituting this expression into the previous equation and dropping terms in 𝑆𝑆 2 gives: EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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Fundamentals of Solidification 5th Edition

𝐶𝐶 ∗ =

𝐶𝐶0 + 𝐺𝐺𝑐𝑐 𝑆𝑆 + 𝐴𝐴𝐴𝐴 𝑘𝑘

[A2.42]

Thus far no real progress has been made since the value of 𝐶𝐶 ∗ is also unknown. It is necessary to find another equation which links 𝐶𝐶 ∗ and 𝐴𝐴, i.e. the flux condition (Eq. A2.40). Differentiating Eq. A2.38 thus gives: 𝑑𝑑𝑑𝑑 𝑉𝑉 𝐶𝐶0 𝑉𝑉𝑉𝑉 = − � − 𝐶𝐶0 � exp �− � − 𝑏𝑏𝐴𝐴𝐴𝐴exp [−𝑏𝑏𝑏𝑏] 𝑑𝑑𝑑𝑑 𝐷𝐷 𝑘𝑘 𝐷𝐷

At the interface this becomes:

𝑑𝑑𝑑𝑑 𝑉𝑉 𝐶𝐶0 𝑉𝑉𝑉𝑉 � � = − � − 𝐶𝐶0 � �1 − � − 𝑏𝑏𝑏𝑏𝑏𝑏(1 − 𝑏𝑏𝑏𝑏) 𝑑𝑑𝑑𝑑 𝑧𝑧=𝑆𝑆 𝐷𝐷 𝑘𝑘 𝐷𝐷

Substituting 𝐺𝐺𝑐𝑐 for −(𝑉𝑉/𝐷𝐷)(𝐶𝐶0 /𝑘𝑘 − 𝐶𝐶0 ) gives: 𝑑𝑑𝑑𝑑 𝑉𝑉𝑉𝑉 � � = 𝐺𝐺𝑐𝑐 �1 − � − 𝑏𝑏𝑏𝑏𝑏𝑏 𝑑𝑑𝑑𝑑 𝑧𝑧=𝑆𝑆 𝐷𝐷

[A2.43]

Substituting Eqs A2.42 and A2.43 into Eq. A2.40 and cancelling 𝑆𝑆 throughout leads to: 𝐴𝐴 =

𝑘𝑘𝑘𝑘𝐺𝐺𝑐𝑐 𝑉𝑉𝑉𝑉 − 𝐷𝐷𝐷𝐷

where 𝑝𝑝 = 1 − 𝑘𝑘. Thus, the original expression becomes:

𝐶𝐶0 𝑉𝑉𝑉𝑉 𝑘𝑘𝑘𝑘𝐺𝐺𝑐𝑐 𝜀𝜀 sin[ 𝜔𝜔𝜔𝜔] � exp[ − 𝑏𝑏𝑏𝑏] 𝐶𝐶 = 𝐶𝐶0 + � − 𝐶𝐶0 � exp �− � + � 𝑘𝑘 𝐷𝐷 𝑉𝑉𝑉𝑉 − 𝐷𝐷𝐷𝐷

[A2.44]

This expression describes the solute distribution ahead of the slightly perturbed solid/liquid interface. Use is made of perturbation analysis in Appendix 7, where the stability of a planar solid/liquid interface is considered. References and Further Reading  R.Aris, Mathematical Modelling Techniques, Pitman, London, 1978.  H.S.Carslaw, J.C.Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, London, 1959.  J.Crank, The Mathematics of Diffusion, Oxford University Press, London, 1956.  G.Engberg, M.Hillert, A.Oden, Estimation of the rate of diffusion-controlled growth by means of a quasi-stationary model, Scandinavian Journal of Metallurgy, 4 (1975) 93.  B.A.Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.  F.C.Frank, Radially symmetric phase growth controlled by diffusion, Proceedings of the Royal Society of London A, 201 (1950) 586.  E.Kreyszig, Advanced Engineering Mathematics, Wiley, New York, 1968.  P.Moon, D.E.Spencer, Field Theory Handbook, Springer-Verlag, Heidelberg, 1961.

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 R.L.Parker, Crystal growth mechanisms: energetics, kinetics, and transport, Solid State Physics, 25 (1970) 151.  L.F.Richardson, A freehand graphic way of determining stream lines and equipotentials, Philosophical Magazine, 15 (1908) 237.  W.A.Tiller, K.A.Jackson, J.W.Rutter, B.Chalmers, The redistribution of solute atoms during the solidification of metals, Acta Metallurgica, 1 (1953) 428.  C.Zener, Theory of growth of spherical precipitates from solid solution, Journal of Applied Physics, 20 (1949) 950.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 241-250 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 3 LOCAL EQUILIBRIUM AT THE SOLID/LIQUID INTERFACE THE PHASE DIAGRAM In most analyses of alloy solidification it is assumed that the solid/liquid interface behaves locally as if it were in a state of equilibrium. This means that the reaction rates, in the small volume which makes up the very thin but finite interface layer, are expected to be rapid in comparison with the rate of interface advance. As a result, the transfer of atoms and changes in their arrangement, which are required in order to maintain the constancy of the chemical potentials in both phases, are relatively rapid and can therefore be neglected. Such a simplification is permissible in the case of metals which are solidifying at the rates encountered in normal casting and welding operations. * The assumption of local equilibrium means that, if the interface temperature is known, one can obtain the liquid and solid compositions at the interface by reference to the equilibrium phase diagram for a binary alloy. This does not mean that the system as a whole is at equilibrium, as gradients of temperature and composition are present. In order to avoid non-essential variables which would complicate the analysis without revealing any new principles, it is usually assumed that both the liquidus and the solidus lines of the relevant part of the phase diagram are straight (see Fig. 1.11). During solidification, the liquidus line is the more important one and represents the point where the liquid 'first' could transform to solid. The constant slope, 𝑚𝑚, of the liquidus which is defined as: 𝑚𝑚 =

𝑑𝑑𝑇𝑇𝑙𝑙 𝑇𝑇𝑙𝑙 (𝐶𝐶) − 𝑇𝑇𝑓𝑓 = 𝑑𝑑𝑑𝑑 𝐶𝐶0

[A3.1]

is thus used in calculations. The solidus composition can be determined at any time from the definition of the distribution coefficient: 𝐶𝐶𝑠𝑠 𝑘𝑘 = � � 𝐶𝐶𝑙𝑙 𝑇𝑇,𝑃𝑃

[A3.2]

The constants, 𝑚𝑚 and 𝑘𝑘, are always defined in the present book in such a way that the product, 𝑚𝑚(𝑘𝑘 −1) is positive. In general 𝑚𝑚 can be positive or negative and 𝑘𝑘 can be greater or less than unity, respectively. Two important properties characterise the range of coexistence of solid and liquid for a given alloy (Fig. 1.11): Δ𝑇𝑇0 = −𝑚𝑚Δ𝐶𝐶0

[A3.3]

At very high growth rates (𝑉𝑉 > 100 mm/s), such as those which occur during rapid solidification processing, conditions of local equilibrium no longer exist. Therefore the value of 𝑘𝑘 changes (Aziz, 1994). See Appendix 6.

*

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Fundamentals of Solidification 5th Edition

Δ𝐶𝐶0 =

[A3.4]

𝐶𝐶0 (1 − 𝑘𝑘) 𝑘𝑘

Figure A3.1

All of these properties depend upon the Gibbs free energy (free enthalpy) of the alloy system, as shown by Fig. A3.1. The latter relates a free-enthalpy versus concentration diagram and a freeenthalpy versus temperature diagram to a temperature versus concentration (phase) diagram. The form of the curves in the Δ𝐺𝐺 − 𝐶𝐶 diagram can be described by using the regular solution model (Gaskell, 1981; Hillert, 1975) which shows that, for 𝑋𝑋, the mole fraction of solute B: 𝐺𝐺𝑚𝑚 = (1 − 𝑋𝑋)𝐺𝐺𝐴𝐴• + 𝑋𝑋𝐺𝐺𝐵𝐵• + Ω𝑋𝑋(1 − 𝑋𝑋) + 𝑅𝑅𝑔𝑔 𝑇𝑇[𝑋𝑋 ln( 𝑋𝑋) + (1 − 𝑋𝑋) ln( 1 − 𝑋𝑋)]

[A3.5]

where Ω is the interaction parameter. The curves in the 𝐺𝐺 − 𝑇𝑇 diagram depend upon the standard values for the pure components: 𝐺𝐺 • = 𝐻𝐻 • − 𝑇𝑇𝑆𝑆 •

where the upper index “•” indicate values for the pure component. The free enthalpy (Gibbs free energy) difference which exists between the pure liquid and pure solid is: Δ𝐺𝐺 • = Δ𝐻𝐻 • − 𝑇𝑇Δ𝑆𝑆 •

where

𝑇𝑇𝑓𝑓

Δ𝐻𝐻 • = Δ𝐻𝐻𝑓𝑓• − � Δ𝑐𝑐 •𝑚𝑚 𝑑𝑑𝑑𝑑 𝑇𝑇

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[A3.6]

Foundations of Materials Science and Engineering Vol. 103

Δ𝑆𝑆 • = Δ𝑆𝑆𝑓𝑓• − �

𝑇𝑇𝑓𝑓

𝑇𝑇

243

Δ𝑐𝑐 •𝑚𝑚 � � 𝑑𝑑𝑑𝑑 𝑇𝑇

and Δ𝑐𝑐 •𝑚𝑚 = 𝑐𝑐 𝑙𝑙𝑙𝑙 − 𝑐𝑐 𝑠𝑠𝑠𝑠 is the difference of the molar specific heat between pure liquid and solid. The quantity, Δ𝐺𝐺 • , in Eq. A3.6 is very important for nucleation and growth processes and can be evaluated when the temperature dependence of Δ𝑐𝑐 •𝑚𝑚 is known. At high temperatures, the difference in specific heat of the liquid and solid can be described by: [A3.7]

Δ𝑐𝑐 •𝑚𝑚 = 𝐾𝐾1 𝑇𝑇 + 𝐾𝐾2

For an undercooled melt, Δ𝑇𝑇 = 𝑇𝑇𝑓𝑓 − 𝑇𝑇 (Thompson and Spaepen, 1979): Δ𝐻𝐻𝑓𝑓• Δ𝑇𝑇 𝐾𝐾1 𝐾𝐾2 |Δ𝐺𝐺 | = − Δ𝑇𝑇 2 � + � 𝑇𝑇𝑓𝑓 2 𝑇𝑇𝑓𝑓 + 𝑇𝑇 •

[A3.8]

If the value of Δ𝑐𝑐 •𝑚𝑚 is unknown, the simplest assumption (and one which is quite reasonable for metals) is that of Δ𝑐𝑐 •𝑚𝑚 = 0. This leads to: |Δ𝐺𝐺 • | =

Δ𝐻𝐻𝑓𝑓• Δ𝑇𝑇 = Δ𝑆𝑆𝑓𝑓• Δ𝑇𝑇 𝑇𝑇𝑓𝑓

[A3.9]

The quantity, Δ𝑆𝑆𝑓𝑓• , is the difference in slope of the 𝐺𝐺 − 𝑇𝑇 function of two phases (Fig. A3.1). For a more detailed discussion of this problem the reader is referred to Dubey and Ramachandrarao (1984). From Eqs A3.5 and A3.9, one can now see which parameters influence the magnitudes of 𝑚𝑚 and 𝑘𝑘. These are the interaction parameter, Ω, for the atoms of both species in the alloy, which determines the form of the 𝐺𝐺𝑚𝑚 − 𝑋𝑋 curves, and the melting entropy, Δ𝑆𝑆𝑓𝑓• , which separates the origins of the Δ𝐺𝐺𝑚𝑚 − 𝑋𝑋 curves. A description of further relationships between 𝑘𝑘, 𝑚𝑚, and the thermodynamic properties is given by Flemings (1974). For small concentrations of an alloying element, and assuming ideal solution behaviour, a useful relationship between the melting entropy of the solvent, (A) and the liquidus slope and distribution coefficient of the solute (B) can be obtained. Under these assumptions, the chemical potential of the solvent is: •

𝜇𝜇𝐴𝐴𝑠𝑠 = 𝜇𝜇𝐴𝐴𝑠𝑠 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln( 1 − 𝑋𝑋𝑠𝑠 ) •

𝜇𝜇𝐴𝐴𝑙𝑙 = 𝜇𝜇𝐴𝐴𝑙𝑙 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln( 1 − 𝑋𝑋𝑙𝑙 )

[A3.10]

where 𝑋𝑋𝑠𝑠 and 𝑋𝑋𝑙𝑙 are the mole fractions of B atoms in the solid and liquid, respectively. At equilibrium (𝜇𝜇𝐴𝐴𝑠𝑠 = 𝜇𝜇𝐴𝐴𝑙𝑙 ): •

•

Δ𝐺𝐺𝐴𝐴• = 𝜇𝜇𝐴𝐴𝑙𝑙 − 𝜇𝜇𝐴𝐴𝑠𝑠 𝑅𝑅𝑔𝑔 𝑇𝑇 ln �

From Eq. A3.9, one obtains:

1 − 𝑋𝑋𝑠𝑠 � 1 − 𝑋𝑋𝑙𝑙

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[A3.11]

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Fundamentals of Solidification 5th Edition

Δ𝑆𝑆𝑓𝑓• Δ𝑇𝑇 1 − 𝑋𝑋𝑠𝑠 = ln � � 𝑅𝑅𝑔𝑔 𝑇𝑇 1 − 𝑋𝑋𝑙𝑙

[A3.12]

For small solute concentrations and noting that, for 𝑧𝑧 → 1, ln (𝑧𝑧) is approximately equal to 𝑧𝑧 − 1, it can be seen that the RHS of Eq. A3.12 becomes approximately equal to 𝑋𝑋𝑙𝑙 − 𝑋𝑋𝑠𝑠 = 𝑋𝑋𝑙𝑙 (1 − 𝑘𝑘) when 𝑋𝑋 approaches unity. Therefore: Δ𝑆𝑆𝑓𝑓• Δ𝑇𝑇 = 1 − 𝑘𝑘 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑋𝑋𝑙𝑙

and substituting for Δ𝑇𝑇/𝑋𝑋𝑙𝑙 = 𝑚𝑚𝑎𝑎 , the liquidus slope expressed as degrees per atomic fraction, one obtains: 1 − 𝑘𝑘 Δ𝑆𝑆𝑓𝑓• = 𝑚𝑚𝑎𝑎 𝑅𝑅𝑔𝑔 𝑇𝑇

[A3.13]

Or as 𝑋𝑋𝑙𝑙 tends to zero and 𝑇𝑇 tends to 𝑇𝑇𝑓𝑓 : Δ𝑆𝑆𝑓𝑓• 1 − 𝑘𝑘 = 𝑚𝑚𝑎𝑎 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓

[A3.14]

Here Δ𝑆𝑆𝑓𝑓• is the melting entropy of the pure solvent with melting point, 𝑇𝑇𝑓𝑓 , and both 𝑚𝑚 and 𝑘𝑘 have to be defined in terms of mole (atom) fractions.

Figure A3.2

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Foundations of Materials Science and Engineering Vol. 103

245

CAPILLARITY EFFECTS (Mullins, 1963; Trivedi, 1975) The total Gibbs free energy of a small solid particle in a melt is inversely proportional to its size. Because the free enthalpy of the solid increases with decreasing diameter, while the free enthalpy of the liquid remains constant (if the amount of liquid is much greater than the amount of solid), the melting point decreases for the pure solid as well as for the alloy (Fig. A3.2). The increase in the free enthalpy of the particle, due to its curved surface (of radius, r), can be regarded as being an increase in internal pressure (the dot suffix will be dropped for the remainder of this appendix): Δ𝐺𝐺𝑟𝑟 = 𝑣𝑣𝑚𝑚 Δ𝑃𝑃

[A3.15]

Δ𝑃𝑃 = 𝜎𝜎𝜎𝜎

[A3.16]

𝑇𝑇𝑓𝑓 − 𝑇𝑇𝑓𝑓𝑟𝑟 = Δ𝑇𝑇𝑟𝑟 = Γ𝐾𝐾

[A3.17]

where 𝑣𝑣𝑚𝑚 is the molar volume (assumed to be constant) and ∆𝑃𝑃 is given by the specific interface energy 𝜎𝜎 and the mean curvature 𝐾𝐾: Combining Eqs A3.9, A3.15 and A3.16 leads to a relationship between the equilibrium temperature drop and the curvature:

and

Γ=

𝜎𝜎𝑣𝑣𝑚𝑚 𝜎𝜎 = Δ𝑆𝑆𝑓𝑓 Δ𝑠𝑠𝑓𝑓

[A3.18]

where Δ𝑆𝑆𝑓𝑓 is the absolute value of the molar freezing entropy and Δ𝑠𝑠𝑓𝑓 is the volumic freezing entropy. In order to use these equations the parameters 𝜎𝜎 and 𝐾𝐾 have to be defined:

Specific Interface Energy, 𝝈𝝈: it is assumed that the solid/liquid interface (in fact a volume) is a surface across which the properties change discontinuously. The specific surface energy is defined there as being the reversible work, 𝑑𝑑𝑑𝑑, which is required in order to create new surface area, 𝑑𝑑𝑑𝑑. Assuming the solid/liquid interface to be isotropic, the specific interface energy 𝜎𝜎 [J/m2] can be set equal to the interfacial tension, 𝛾𝛾 [N/m] †. According to Fig. A3.3, the work necessary to extend the surface by 𝑑𝑑𝑑𝑑 is: 𝑑𝑑𝑑𝑑 = 𝑓𝑓𝑓𝑓𝑓𝑓 = 𝛾𝛾𝛾𝛾𝛾𝛾𝛾𝛾 = 𝛾𝛾𝛾𝛾𝛾𝛾

Figure A3.3

from which one can obtain the definition for the isotropic case: In the anisotropic case the surface tension is no longer a scalar equal to 𝜎𝜎 multiplied by the identity matrix: it is instead a tensor (see Chap. 2). †

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246

Fundamentals of Solidification 5th Edition

𝑑𝑑𝑑𝑑 𝜎𝜎 = 𝛾𝛾 ≡ � � 𝑑𝑑𝑑𝑑 𝑇𝑇,𝑉𝑉,𝜇𝜇

[A3.19]

Curvature, K: in two dimensions the curvature of a function is defined as the change in the slope, 𝛿𝛿𝛿𝛿, of that function over a length of arc, 𝛿𝛿𝛿𝛿 (Fig. A3.4(a)): 𝐾𝐾 =

𝛿𝛿𝛿𝛿 𝛿𝛿𝛿𝛿

𝐾𝐾 =

1 𝑟𝑟

𝐾𝐾 =

𝑧𝑧 ″ [1 + 𝑧𝑧 ′2 ]3/2

and since 𝛿𝛿𝛿𝛿 = 𝑟𝑟𝑟𝑟𝑟𝑟, where 𝑟𝑟 is the local radius of curvature: More generally it can be shown that, for a curve 𝑧𝑧(𝑦𝑦), the local curvature at a point is given by:

[A3.20]

where 𝑧𝑧 ′ and 𝑧𝑧 ″ are the first and second derivatives, respectively, of the function, z(y).

Figure A3.4

The average curvature of an arbitrary line segment depends only upon the gradients of the curve at its end-points, and upon the distance between the latter (Fig. A3.5). In the case of surfaces in three dimensions the mean curvature, 𝐾𝐾, can be defined as the variation in surface area divided by the corresponding variation in volume (Fig. A3.4(b)): EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

Foundations of Materials Science and Engineering Vol. 103

𝐾𝐾 =

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

247

[A3.21]

Calling 𝑟𝑟1 and 𝑟𝑟2 the two principal radii of curvature (at 90o to each other), one has: 𝑙𝑙1 = 𝑟𝑟1 𝜃𝜃

and

𝑙𝑙2 = 𝑟𝑟2 𝜃𝜃

Increasing the radii by dr gives:

𝑑𝑑𝑑𝑑 = 𝑙𝑙1 𝑑𝑑𝑙𝑙2 + 𝑙𝑙2 𝑑𝑑𝑙𝑙1 = 𝑟𝑟1 𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃 + 𝑟𝑟2 𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃 𝑑𝑑𝑑𝑑 = (𝑟𝑟1 + 𝑟𝑟2 )𝑑𝑑𝑑𝑑𝜃𝜃 2

[A3.22]

𝑑𝑑𝑑𝑑 = 𝑟𝑟1 𝑟𝑟2 𝑑𝑑𝑑𝑑𝜃𝜃 2

[A3.23]

𝑑𝑑𝑑𝑑 = 𝑙𝑙1 𝑙𝑙2 𝑑𝑑𝑑𝑑 = (𝑟𝑟1 𝜃𝜃)(𝑟𝑟2 𝜃𝜃)𝑑𝑑𝑑𝑑 Therefore from Eqs A3.21, A3.22, and A3.23, 𝐾𝐾 = �

1 1 + � 𝑟𝑟1 𝑟𝑟2

[A3.24]

In the case of a sphere, 𝑟𝑟1 = 𝑟𝑟2 = 𝑟𝑟 so that 𝐾𝐾 = 2/𝑟𝑟. In the case of a cylinder, 𝑟𝑟1 is infinite and 𝑟𝑟2 = 𝑟𝑟 so that 𝐾𝐾 = 1/𝑟𝑟.

Figure A3.5

Using Eq. A3.17 the decrease in melting point due to the curvature of a spherical crystal in a melt can thus be written: Δ𝑇𝑇𝑟𝑟 =

2Γ 𝑟𝑟

[A3.25]

It is recalled that the value of 𝜎𝜎 is isotropic in the above calculations. The effect of anisotropy is treated briefly in Chap. 2, and in more detail by Aaronson (1975). EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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Fundamentals of Solidification 5th Edition

MECHANICAL EQUILIBRIUM AT THE THREE-PHASE JUNCTION A junction between two solid phases at the solid/liquid interface forms a groove. At this point, the surface forces tends to impose an equilibrium (minimum energy) morphology in which: ∑ 𝑓𝑓 = 0

Again assuming that all the interfaces are isotropic, their surface tension, 𝛾𝛾𝑖𝑖𝑖𝑖 , is tangential to their surface and is equal in modulus to their surface energy, 𝜎𝜎𝑖𝑖𝑖𝑖 . It is evident from Fig. A3.6 that this condition will be satisfied when: and

𝜎𝜎𝛼𝛼𝛼𝛼 = 𝜎𝜎𝛼𝛼𝛼𝛼 cos( 𝜃𝜃1 ) + 𝜎𝜎𝛽𝛽𝛽𝛽 cos( 𝜃𝜃2 ) 𝜎𝜎𝛼𝛼𝛼𝛼 sin( 𝜃𝜃1 ) = 𝜎𝜎𝛽𝛽𝛽𝛽 sin( 𝜃𝜃2 )

[A3.26]

In establishing mechanical equilibrium, it is important to consider the equilibrium of the moments acting on the junction (Eq. A3.26) since this affects the angle of eutectic solid/solid interface with respect to the solid/liquid interface. In most cases, the angle is expected to be close to 90°.

Figure A3.6

CALCULATION OF 𝒇𝒇(𝜽𝜽) FOR HETEROGENEOUS NUCLEATION (Kirkaldy, 1968)

The application of Eq. A3.26 to heterogeneous nucleation shows that true mechanical equilibrium (Fig. A3.7b) cannot be established, and that a surface stress will be set up in the case of Fig. A3.7(a). Due to the presence of foreign crystalline surfaces (crucible, surface oxide, inclusions) in a melt, nucleation may become much easier. The effect of these interfaces can be deduced from the energy balance: Δ𝐺𝐺𝑖𝑖 = (interface energy creation due to nucleation) – (interface energy gained due to the substrate)

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Foundations of Materials Science and Engineering Vol. 103

249

Figure A3.7

Δ𝐺𝐺𝑖𝑖 = (𝐴𝐴𝑙𝑙𝑙𝑙 𝜎𝜎𝑙𝑙𝑙𝑙 + 𝐴𝐴𝑐𝑐𝑐𝑐 𝜎𝜎𝑐𝑐𝑐𝑐 ) − 𝐴𝐴𝑐𝑐𝑐𝑐 𝜎𝜎𝑙𝑙𝑙𝑙

Δ𝐺𝐺𝑖𝑖 = 𝐴𝐴𝑙𝑙𝑙𝑙 𝜎𝜎𝑙𝑙𝑙𝑙 + 𝜋𝜋𝑅𝑅 2 (𝜎𝜎𝑐𝑐𝑐𝑐 − 𝜎𝜎𝑙𝑙𝑙𝑙 )

[A3.27]

Δ𝐺𝐺𝑖𝑖 = 𝐴𝐴𝑙𝑙𝑙𝑙 𝜎𝜎𝑙𝑙𝑙𝑙 − 𝜋𝜋𝑅𝑅 2 cos 𝜃𝜃 𝜎𝜎𝑙𝑙𝑙𝑙

[A3.28]

𝜎𝜎𝑙𝑙𝑙𝑙 = 𝜎𝜎𝑐𝑐𝑐𝑐 + 𝜎𝜎𝑙𝑙𝑙𝑙 cos 𝜃𝜃

Δ𝐺𝐺 = Δ𝐺𝐺𝑣𝑣 + Δ𝐺𝐺𝑖𝑖 = 𝑣𝑣𝑐𝑐 Δ𝑔𝑔 + [𝐴𝐴𝑙𝑙𝑙𝑙 − 𝜋𝜋𝑅𝑅 2 cos 𝜃𝜃]𝜎𝜎𝑙𝑙𝑙𝑙

𝑣𝑣𝑐𝑐 =

π𝑟𝑟 3 [2 − 3 cos 𝜃𝜃 + cos3 𝜃𝜃] 3

[A3.29]

𝐴𝐴𝑙𝑙𝑙𝑙 = 2π𝑟𝑟 2 [1 − cos 𝜃𝜃] 𝑅𝑅 = 𝑟𝑟 sin 𝜃𝜃

sin2 𝜃𝜃 = 1 − cos 2 𝜃𝜃

4π𝑟𝑟 3 Δ𝑔𝑔 2 − 3 cos 𝜃𝜃 + cos 3 𝜃𝜃 Δ𝐺𝐺 = � + 4π𝑟𝑟 2 𝜎𝜎𝑙𝑙𝑙𝑙 � � � 3 4

Δ𝐺𝐺ℎ𝑒𝑒𝑒𝑒 = Δ𝐺𝐺ℎ𝑜𝑜𝑜𝑜 𝑓𝑓(𝜃𝜃)

where:

𝑓𝑓(𝜃𝜃) =

2 − 3 cos 𝜃𝜃 + cos3 𝜃𝜃 [2 + cos 𝜃𝜃][1 − cos 𝜃𝜃]2 = 4 4

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[A3.30]

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Fundamentals of Solidification 5th Edition

References and Further Reading  H.I.Aaronson, in Lectures on the Theory of Phase Transformations, H.I.Aaronson (Ed.), Transactions of the Metallurgical Society of AIME, New York, 1975, p.158.  M.J.Aziz, On the transition from short-range diffusion-limited to collision-limited growth in alloy solidification, Acta Metallurgica et Materialia, 42 (1994) 527.  K.S.Dubey, P.Ramachandrarao, On the free energy change accompanying crystallisation of undercooled melts, Acta Metallurgica, 32 (1984) 91.  M.C.Flemings, Solidification Processing, McGraw-Hill, New York, 1974.  D.R.Gaskell, Introduction to Metallurgical Thermodynamics, McGraw-Hill, New York, 1981.  M.Hillert, in Lectures on the Theory of Phase Transformations, H.I.Aaronson (Ed.), Transactions of the Metallurgical Society of AIME, New York, 1975, p.1.  J.S.Kirkaldy, in Energetics in Metallurgical Phenomena, Volume IV, W.M.Mueller (Ed.), Gordon & Breach, New York, 1968, p.197.  W.W.Mullins, in Metal Surfaces - Structure, Energetics, Kinetics, American Society for Metals, Metals Park, 1963, p.17.  C.V.Thompson, F.Spaepen, On the approximation of the free energy change on crystallization, Acta Metallurgica, 27 (1979) 1855.  R.Trivedi, in Lectures on the Theory of Phase Transformations, H.I.Aaronson (Ed.), Transactions of the Metallurgical Society of AIME, New York, 1975, p.51.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 251-254 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 4 NUCLEATION KINETICS IN A PURE SUBSTANCE To determine the nucleation rate, it is necessary to determine the number of critical nuclei, and the rate of arrival of atoms necessary to make up these nuclei (Turnbull and Fisher, 1949; Uhlmann and Chalmers, 1965; Christian, 1975).

EQUILIBRIUM DISTRIBUTION OF NUCLEI IN AN UNDERCOOLED MELT The system shown in Fig. A4.1 represents a mixture of 𝑁𝑁𝑙𝑙 atoms in the liquid state and 𝑁𝑁𝑛𝑛 small crystal clusters, containing each n atoms. Its Gibbs free energy difference with one system containing only atoms and no crystal nuclei, is given by: [A4.1]

Δ𝐺𝐺 = 𝑁𝑁𝑛𝑛 Δ𝐺𝐺𝑛𝑛 − 𝑇𝑇Δ𝑆𝑆𝑛𝑛

Here, Δ𝐺𝐺𝑛𝑛 is the Gibbs free energy change due to the formation of one nucleus containing 𝑛𝑛 atoms and Δ𝑆𝑆𝑛𝑛 is the entropy of mixing 𝑁𝑁𝑛𝑛 clusters with 𝑁𝑁𝑙𝑙 atoms. (In this equation, the existence of an ideal mixture of crystal clusters and atoms of liquid is assumed. The mixing enthalpy, Δ𝐻𝐻𝑛𝑛 , is therefore equal to zero.) In the case of sub-critical clusters (known as embryos), Δ𝐺𝐺𝑛𝑛 is positive due to the work of interface creation. This is demonstrated in Fig. A4.2 and corresponds to Fig. 2.2 where 𝑛𝑛 and 𝑟𝑟, characterising the difference in size, are related by Eq. 2.2. The value of Δ𝐺𝐺𝑛𝑛 can be determined in an analogous manner by using Eq. 2.3, and this leads to the relationship: Δ𝐺𝐺𝑛𝑛 = 𝑛𝑛Δ𝐺𝐺 ′ + 𝐴𝐴𝑛𝑛 𝜎𝜎 𝐴𝐴𝑛𝑛 = 𝜂𝜂𝑛𝑛

[A4.2]

2/3

Figure A4.1

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Figure A4.2

where Δ𝐺𝐺 ′ is the atomic Gibbs free energy difference, 𝐴𝐴𝑛𝑛 is the interface area of the cluster, η is a form-factor which depends upon the shape of the cluster and 𝑛𝑛1/3 is proportional to the cluster diameter. The mixing entropy of Eq. A4.1 can be derived by using the well-known relationship: Δ𝑆𝑆𝑛𝑛 = 𝑘𝑘𝐵𝐵 ln �

and therefore:

(𝑁𝑁𝑙𝑙 + 𝑁𝑁𝑛𝑛 )! � 𝑁𝑁𝑛𝑛 ! 𝑁𝑁𝑙𝑙 !

Δ𝐺𝐺 = 𝑁𝑁𝑛𝑛 Δ𝐺𝐺𝑛𝑛 − 𝑘𝑘𝐵𝐵 𝑇𝑇 ln[ (𝑁𝑁𝑙𝑙 + 𝑁𝑁𝑛𝑛 )!] + 𝑘𝑘𝐵𝐵 𝑇𝑇 ln[ 𝑁𝑁𝑛𝑛 !] + 𝑘𝑘𝐵𝐵 𝑇𝑇 ln[ 𝑁𝑁𝑙𝑙 !]

[A4.3]

[A4.4]

where Δ𝐺𝐺𝑛𝑛 is always positive for critical nuclei. An increase in 𝑁𝑁𝑛𝑛 will first decrease the total free energy of the system, due to the mixing of clusters and atoms (Fig. A4.3), and then reach a minimum value corresponding to the equilibrium concentration of clusters for a given undercooling. Applying Stirling's approximation for large values of 𝑁𝑁, i.e. ln (𝑁𝑁!) = 𝑁𝑁ln (𝑁𝑁) − 𝑁𝑁, differentiating Eq. A4.4 with respect to 𝑁𝑁𝑛𝑛 , and setting the result equal to zero gives: Δ𝐺𝐺𝑛𝑛 − 𝑘𝑘𝐵𝐵 𝑇𝑇[ln( 𝑁𝑁𝑙𝑙 + 𝑁𝑁𝑛𝑛 ) − ln(𝑁𝑁𝑛𝑛 )] = 0

from which: 𝑁𝑁𝑛𝑛 Δ𝐺𝐺𝑛𝑛 = exp �− � 𝑁𝑁𝑙𝑙 + 𝑁𝑁𝑛𝑛 𝑘𝑘𝐵𝐵 𝑇𝑇

[A4.5]

Since 𝑁𝑁𝑙𝑙 ≫ 𝑁𝑁𝑛𝑛

𝑁𝑁𝑛𝑛 Δ𝐺𝐺𝑛𝑛 = exp �− � 𝑁𝑁𝑙𝑙 𝑘𝑘𝐵𝐵 𝑇𝑇

The number of nuclei (critical clusters) in equilibrium is therefore given by: 𝑁𝑁𝑛𝑛○ = 𝑁𝑁𝑙𝑙 exp �−

Δ𝐺𝐺𝑛𝑛○ � 𝑘𝑘𝐵𝐵 𝑇𝑇

where Δ𝐺𝐺𝑛𝑛○ is the energy barrier for nucleation as defined in Fig. A4.2. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A4.6]

Foundations of Materials Science and Engineering Vol. 103

253

Figure A4.3

RATE OF FORMATION OF STABLE NUCLEI The nucleation rate must be proportional to the number of crystals of critical size, 𝑁𝑁𝑛𝑛○ . In order for these crystals to grow, however, the addition of further atoms is required. The system containing clusters of size, 𝑛𝑛○ , is in an unstable state and can produce smaller or larger clusters (Fig. A4.2) in order to decrease its energy. It is therefore necessary to determine the rate of incorporation, dn/dt, of new atoms into the nuclei. The nucleation rate is then: 𝐼𝐼 = 𝑁𝑁𝑛𝑛○

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

[A4.7]

where the adsorption rate, dn/dt, is the product of an adsorption frequency, ν, and the density of sites at which atoms can be adsorbed by the critical nucleus, 𝑛𝑛𝑠𝑠○ : 𝑑𝑑𝑑𝑑 = ν𝑛𝑛𝑠𝑠○ 𝑑𝑑𝑑𝑑

[A4.8]

The adsorption frequency is: ν = ν0 exp �−

Δ𝐺𝐺𝑑𝑑 � 𝑝𝑝 𝑘𝑘𝐵𝐵 𝑇𝑇

where ν0 is the atomic vibration frequency, exp[−Δ𝐺𝐺𝑑𝑑 /𝑘𝑘𝐵𝐵 𝑇𝑇] is the fraction of atoms in the liquid which are sufficiently activated to surmount the interface-addition activation energy, Δ𝐺𝐺𝑑𝑑 , and 𝑝𝑝 is the adsorption probability. The site density is given by: [A4.9]

𝑛𝑛𝑠𝑠○ = 𝐴𝐴○𝑛𝑛 𝑛𝑛𝑐𝑐

where 𝐴𝐴○𝑛𝑛 is the surface area of the critical nucleus and 𝑛𝑛𝑐𝑐 is the capture-site density per unit area. The nucleation rate is therefore: 𝐼𝐼 = 𝐼𝐼0 exp �−

Δ𝐺𝐺𝑛𝑛○ + Δ𝐺𝐺𝑑𝑑 � 𝑘𝑘𝐵𝐵 𝑇𝑇

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[A4.10]

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and 𝐼𝐼0 = 𝑁𝑁𝑙𝑙 ν0 𝑝𝑝𝐴𝐴○𝑛𝑛 𝑛𝑛𝑐𝑐

Since the exponential term is extremely sensitive to small variations in the argument (Fig. 2.6), the exact value of 𝐼𝐼0 is relatively unimportant and, for metals, is often approximated by: 𝑘𝑘𝐵𝐵 𝑇𝑇 � = 𝑁𝑁𝑙𝑙 ν0 ≅ 1042 /𝑚𝑚3 𝑠𝑠 𝐼𝐼0 = 𝑁𝑁𝑙𝑙 � ℎ

[A4.11]

All these relationships assume the existence of a steady state and are not of general applicability. They nevertheless offer a good guide to the principles involved. For a more complete treatment of nucleation theory, the reader should consult Christian (1975) or Kelton and Greer (2010). The present approach has to be modified in the case of alloys (Thompson and Spaepen, 1983). References and Further Reading  J.W.Christian, The Theory of Transformations in Metals and Alloys, 2nd Edition, Pergamon Press, Oxford, 1975.  K.F.Kelton, A.L.Greer, Nucleation in Condensed Matter, Elsevier, 2010.  C.V.Thompson, F.Spaepen, Homogeneous crystal nucleation in binary metallic melts, Acta Metallurgica, 31 (1983) 2021.  D.Turnbull, J.C.Fisher, Rate of nucleation in condensed systems, Journal of Chemical Physics, 17 (1949) 71.  D.R.Uhlmann, B.Chalmers, The energetics of nucleation, Industrial and Engineering Chemistry, 57 (1965) 19.

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Online: 2023-02-20

APPENDIX 5 ATOMIC STRUCTURE OF THE SOLID/LIQUID INTERFACE To explain the principles involved, the simple case of a two-dimensional crystal consisting of 'square atoms' of only one type, which interact only with their nearest neighbours and form singlelayer interfaces, will be treated here. It is assumed that there are three different structural elements making up the surface structure (three-site model, Fig. A5.1). This situation has been treated in detail by Jackson (1968). There is a continuous interchange of sites due to the thermally activated adsorption or desorption of atoms. For example, an 𝑛𝑛 = 1 site will be transformed into an 𝑛𝑛 = 2 site by the adsorption of one neighbouring atom. It is assumed that, after a short time, the interface structure reaches a steady state, i.e. the overall density of the three types of sites does not change with time. The rate of adsorption of atoms is independent of the rate of departure. Both are activated processes similar to diffusion in the liquid, but the desorption of atoms is more difficult from an undercooled crystal, due to the gain in energy which occurs when an atom is added to the solid/liquid interface. The probability of adsorption will depend upon the roughness of the interface: the greater the density of steps (and therefore the greater the number of exposed atomic bonds presented to the liquid), the higher is the probability that an atom will be incorporated into the crystal, as a result of the stronger bonding involved.

Figure A5.1

The rate of atom arrival, 𝐽𝐽+ , is governed by: 𝐽𝐽+ = 𝐽𝐽0+ exp �−

𝑄𝑄 � 𝑅𝑅𝑔𝑔 𝑇𝑇

[A5.1]

where 𝑄𝑄 is the activation energy for diffusion in the liquid. The flux of atoms leaving is determined by: 2𝑛𝑛Δ𝐻𝐻𝑓𝑓 𝑄𝑄 𝐽𝐽− = 𝐽𝐽0− exp �− � exp �− � [A5.2] 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑧𝑧𝑅𝑅𝑔𝑔 𝑇𝑇 EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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where Δ𝐻𝐻𝑓𝑓 is the latent heat of fusion, 𝑛𝑛 is the number of bonds which have to be broken by an atom in leaving the crystal, and 𝑧𝑧 is the coordination number (4 in the case of a two-dimensional crystal). An atom making two bonds with the crystal (i.e. half of the maximum possible number of bonds) can be viewed as being half in the solid and half in the liquid. At the melting point, 𝑇𝑇𝑓𝑓 , the rates of arrival and departure from such sites (𝑛𝑛 = 2) should be equal. Thus, 𝐽𝐽0+ exp �−

Therefore:

Δ𝐻𝐻𝑓𝑓 𝑄𝑄 𝑄𝑄 � = 𝐽𝐽0− exp �− � exp �− � 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓

Δ𝐻𝐻𝑓𝑓 𝐽𝐽0− � + = exp � 𝐽𝐽0 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓

[A5.3]

[A5.4]

Combining Eqs A5.2 and A5.4, and rearranging the exponential terms: 𝐽𝐽− = 𝐽𝐽0+ exp �−

Δ𝐻𝐻𝑓𝑓 𝑛𝑛Δ𝐻𝐻𝑓𝑓 𝑄𝑄 � exp � − � 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 2𝑅𝑅𝑔𝑔 𝑇𝑇

[A5.5]

The second exponential term can be regarded as being a measure of the probability that an atom which has n neighbours will leave the site. That is: 𝑝𝑝𝑛𝑛 = exp �

Δ𝐻𝐻𝑓𝑓 𝑛𝑛Δ𝐻𝐻𝑓𝑓 − � 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 2𝑅𝑅𝑔𝑔 𝑇𝑇

The net flux is determined by the difference between Eqs A5.1 and A5.5: 𝐽𝐽 = 𝐽𝐽0+ exp �−

𝑄𝑄 � [1 − 𝑝𝑝𝑛𝑛 ] 𝑅𝑅𝑔𝑔 𝑇𝑇

[A5.6]

It is now possible to set up balance equations for the arrival and departure of atoms at various types of sites. The detailed calculations have been given by Jackson (1968). Carrying these through and realising that, if �Δ𝐻𝐻𝑓𝑓 /𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 �(Δ𝑇𝑇/𝑇𝑇) ≪ 1, exp �−

Δ𝐻𝐻𝑓𝑓 Δ𝑇𝑇 Δ𝐻𝐻𝑓𝑓 Δ𝑇𝑇 �≅1− 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 𝑇𝑇 𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 𝑇𝑇

the net growth rate, 𝑉𝑉, of the crystal becomes: 𝑉𝑉 = 𝐽𝐽𝑣𝑣 ′ = 𝑣𝑣 ′ 𝐽𝐽0+ exp �−

𝑄𝑄 Δ𝑇𝑇 � 𝛼𝛼 � � 𝑓𝑓[ℎ𝑘𝑘] 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑇𝑇

[A5.7]

where 𝛼𝛼 = Δ𝐻𝐻𝑓𝑓 /𝑅𝑅𝑔𝑔 𝑇𝑇𝑓𝑓 = Δ𝑆𝑆𝑓𝑓 /𝑅𝑅𝑔𝑔 (the dimensionless melting entropy) and 𝑣𝑣 ′ is the atomic volume.

To a first approximation, the growth rate is therefore a linear function of the undercooling. It also depends upon the magnitude of the crystallographic factor, 𝑓𝑓[ℎ𝑘𝑘], which is a function of the interface structure. It thus depends upon the value of 𝛼𝛼 and upon the Miller indices, [ℎ𝑘𝑘], of the crystallographic plane. When 𝛼𝛼 takes on high values (e.g. 10), the fraction of extra atoms or holes in the interface is proportional to exp(-α). For the (11)-faces of the present square crystal (Fig. A5.2), 𝑓𝑓 is independent of 𝛼𝛼, but this is not so for the (10)-face (Table A5.1). The reason is that a 45° edge exhibits many growth steps which cannot be removed by atom addition. When the value of 𝛼𝛼 is large, the (11)-edges will grow much more quickly and leave the crystal bounded by slow-growing (10)-edges (Fig. A5.3 - see also Fig. 2.15). Note also in Table A5.1 that, for small values of α, the growth is almost isotropic (𝑓𝑓[10] ∼ 𝑓𝑓[11]). This behaviour is typical of metals.

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Foundations of Materials Science and Engineering Vol. 103

257

Figure A5.2

Figure A5.3 Table A5.1 Crystallographic Factor, f[hk], of two Faces of a Square Crystal 𝛼𝛼 𝑓𝑓[10] 𝑓𝑓[11] 1 5 10

References and Further Reading

0.56 0.30 0.039

0.60 0.60 0.60

 Y.F.Gao, Y.Yang, D.Y.Sun, M.Asta, J.J.Hoyt, Molecular dynamics simulations of the crystal– melt interface mobility in HCP Mg and BCC Fe, Journal of Crystal Growth, 312 (2010) 3238.  J.J.Hoyt, M.Asta, A.Karma, Atomistic Simulation Methods for Computing the Kinetic Coefficient in Solid-Liquid Systems, Interface Science, 10 (2002) 181.  J.J.Hoyt, M.Asta, T.Haxhimali, A.Karma, R.E.Napolitano, R.Trivedi, B.B.Laird, J.R.Morris, Crystal–Melt Interfaces and Solidification Morphologies in Metals and Alloys, MRS Bulletin, Dec. (2004) 935.

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 K.A.Jackson, in Liquid Metals and Solidification, American Society for Metals, Cleveland, 1958, p.174.  K.A.Jackson, On the theory of crystal growth: Growth of small crystals using periodic boundary conditions, Journal of Crystal Growth, 3-4 (1968) 507.  I.V.Markov, Crystal Growth for Beginners, 3rd Edition, World Scientific, Singapore, 2007, p.181.

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Online: 2023-02-20

APPENDIX 6 THERMODYNAMICS OF RAPID SOLIDIFICATION INTERFACE KINETICS As explained in Appendix 3, under normal solidification conditions the non-faceted solid/liquid interface itself is close to equilibrium, even if concentration and temperature gradients exist in the adjacent phases. The assumption of local equilibrium is therefore useful when the short-range diffusional rate of atoms crossing the interface is much higher than the rate of arrangement of atoms into a configuration of low mobility (e.g. crystallisation). Consider the case of an alloy and assume now that the time required to freeze-in solute atoms is, due to rapid interface movement, much smaller than the time taken to diffuse through a characteristic distance, δ𝑖𝑖 , of the order of the interatomic distance (Fig. A6.1). As a result, solute is incorporated into the solid at a concentration which is different to the equilibrium concentration. Such solute-trapping will always occur, at least partially, if the following condition is satisfied: 𝜏𝜏 = 𝛿𝛿𝑖𝑖 /𝑉𝑉 ≤ 𝛿𝛿𝑖𝑖2 /𝐷𝐷𝑖𝑖

[A6.1]

where 𝐷𝐷𝑖𝑖 is the interface diffusion coefficient. From this relationship, one can also see that local equilibrium will be lost when the interface Péclet number, 𝑃𝑃𝑖𝑖 , satisfies: δ𝑖𝑖 𝑉𝑉 𝑃𝑃𝑖𝑖 = ≥1 [A6.2] 𝐷𝐷𝑖𝑖

Figure A6.1

At very high interface Péclet numbers, 𝑘𝑘𝑣𝑣 is equal to unity (see below). For this to occur, another condition must be satisfied: the change in Gibbs free energy must still be negative (at least slightly) when the liquid transforms to solid. This is best seen with the aid of free energy diagrams (Fig. A6.2).

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Figure A6.2

𝑻𝑻𝟎𝟎 - CONDITION

For a given interface temperature, 𝑇𝑇 ∗ = 𝑇𝑇′, between solid (α) and liquid (l), the equilibrium compositions are 𝑋𝑋𝛼𝛼𝑒𝑒 and 𝑋𝑋𝑙𝑙𝑒𝑒 . Now if the interface composition is 𝑋𝑋𝑙𝑙′ and 𝑘𝑘𝑣𝑣 = 1, solid 𝛼𝛼 of the same composition as the liquid, 𝑋𝑋𝑙𝑙′ , cannot form since its free energy is higher than that of the liquid phase. Solid of composition, 𝑋𝑋𝛼𝛼′ , can however form from liquid of composition, 𝑋𝑋𝑙𝑙′ , at 𝑇𝑇′ since its free energy will decrease. The range of compositions of α which can crystallise from liquid of composition, 𝑋𝑋𝑙𝑙′ , at 𝑇𝑇′ is given by the intersection of the tangent at 𝐺𝐺(𝑋𝑋𝑙𝑙 ) with the curve, 𝐺𝐺α . In order to trap solute completely (𝑘𝑘𝑣𝑣 = 1) at 𝑇𝑇 ∗ = 𝑇𝑇′, the composition must become equal to, or smaller than, 𝑋𝑋0. That is, the maximum solid composition at 𝑇𝑇′ corresponds to 𝑋𝑋0 (= 𝑋𝑋𝑙𝑙 = 𝑋𝑋𝛼𝛼 ) and 𝑇𝑇 ′ = 𝑇𝑇0. Following the approach of Baker and Cahn (1971) and Boettinger and Coriell (1986), one can now calculate the driving force for crystallisation as a function of 𝑘𝑘𝑣𝑣 . This tends towards unity when 𝑉𝑉 becomes very large. From the definition of the chemical potential one can write, for solute B: 𝜇𝜇𝐵𝐵𝛼𝛼 = 𝜇𝜇𝐵𝐵o𝛼𝛼 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝑎𝑎𝛼𝛼 ] 𝜇𝜇𝐵𝐵𝑙𝑙

=

𝜇𝜇𝐵𝐵o𝑙𝑙

+ 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝑎𝑎𝑙𝑙 ]

[A6.3]

where the activity, 𝑎𝑎, of the solute equals 𝛾𝛾𝛾𝛾. Since only dilute solutions will be considered here, for simplicity, one can assume that the activity coefficient, 𝛾𝛾, is constant. From the condition of equilibrium, one has: 𝜇𝜇𝐵𝐵o𝛼𝛼 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝛾𝛾𝛼𝛼 𝑋𝑋𝛼𝛼𝑒𝑒 ] = 𝜇𝜇𝐵𝐵o𝑙𝑙 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝛾𝛾𝑙𝑙 𝑋𝑋𝑙𝑙𝑒𝑒 ]

[A6.4]

𝜇𝜇𝐵𝐵o𝛼𝛼 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝛾𝛾𝛼𝛼 𝑋𝑋𝛼𝛼 ] = 𝜇𝜇𝐵𝐵o𝑙𝑙 + 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝛾𝛾𝑙𝑙 𝑋𝑋𝑙𝑙 ] + Δ𝜇𝜇𝐵𝐵

[A6.5]

and under non-equilibrium conditions

From Eqs A6.4 and A6.5 one deduces that:

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Foundations of Materials Science and Engineering Vol. 103

261

Δ𝜇𝜇𝐵𝐵 = 𝑅𝑅𝑔𝑔 𝑇𝑇 ln[ 𝑘𝑘𝑣𝑣 /𝑘𝑘]

[A6.6]

Δ𝐺𝐺 = (𝜇𝜇𝐴𝐴𝛼𝛼 − 𝜇𝜇𝐴𝐴𝑙𝑙 )(1 − 𝑋𝑋𝛼𝛼 ) + (𝜇𝜇𝐵𝐵𝛼𝛼 − 𝜇𝜇𝐵𝐵𝑙𝑙 )𝑋𝑋𝛼𝛼

[A6.7]

where the non-equilibrium distribution coefficient, 𝑘𝑘𝑣𝑣 , is equal to 𝑋𝑋𝛼𝛼 /𝑋𝑋𝑙𝑙 , and the equilibrium distribution coefficient, 𝑘𝑘, is equal to 𝑋𝑋𝛼𝛼𝑒𝑒 /𝑋𝑋𝑙𝑙𝑒𝑒 . The same calculation can be carried out for the solute, A, in a binary system of concentration, (1 − 𝑋𝑋). From the relationship: and 𝑋𝑋𝛼𝛼 = 𝑘𝑘𝑋𝑋𝑙𝑙 , it follows that: Δ𝐺𝐺 1 − 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 1 − 𝑋𝑋𝑙𝑙𝑒𝑒 𝑘𝑘𝑣𝑣 = [1 − 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ] ln � 𝑒𝑒 � + 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ln � � 𝑅𝑅𝑔𝑔 𝑇𝑇 1 − 𝑋𝑋𝑙𝑙 1 − 𝑘𝑘𝑋𝑋𝑙𝑙 𝑘𝑘

[A6.8]

In the case of dilute solutions (𝑋𝑋𝑙𝑙 ≪ 1), one obtains after approximating ln (𝑧𝑧) by (𝑧𝑧 − 1) when 𝑧𝑧 → 1: Δ𝐺𝐺 1 − 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 1 − 𝑋𝑋𝑙𝑙𝑒𝑒 𝑘𝑘𝑣𝑣 = (1 − 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ) � 𝑒𝑒 − 1� + 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ln � � 𝑅𝑅𝑔𝑔 𝑇𝑇 1 − 𝑋𝑋𝑙𝑙 1 − 𝑘𝑘𝑋𝑋𝑙𝑙 𝑘𝑘

[A6.9]

For dilute solutions, terms of higher order can be neglected, i.e. the prefactor in Eq. A6.9, (1 − 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ) ≈ 1 and only first-order terms are kept in the large parenthesis. This leads to: Δ𝐺𝐺 𝑘𝑘𝑣𝑣 = [𝑋𝑋𝑙𝑙 (1 − 𝑘𝑘𝑣𝑣 ) − 𝑋𝑋𝑙𝑙𝑒𝑒 (1 − 𝑘𝑘)] + 𝑘𝑘𝑣𝑣 𝑋𝑋𝑙𝑙 ln � � 𝑘𝑘 𝑅𝑅𝑔𝑔 𝑇𝑇 1 − 𝑘𝑘 1 − 𝑘𝑘𝑣𝑣 + 𝑘𝑘𝑣𝑣 ln( 𝑘𝑘𝑣𝑣 /𝑘𝑘) � − 𝑋𝑋𝑙𝑙𝑒𝑒 𝑚𝑚� = �𝑋𝑋𝑙𝑙 �𝑚𝑚 𝑚𝑚 1 − 𝑘𝑘

[A6.10]

On the other hand, the driving force Δ𝐺𝐺 for atom attachment at an interface growing at temperature, 𝑇𝑇 ∗ , can be related to its velocity 𝑉𝑉 via the relationship (Turnbull, 1962): 𝑉𝑉 = 𝑓𝑓𝑉𝑉0 [1 − exp( Δ𝐺𝐺/𝑅𝑅𝑔𝑔 𝑇𝑇 ∗ )]

[A6.11]

𝑉𝑉/𝑉𝑉0 = −Δ𝐺𝐺/𝑅𝑅𝑔𝑔 𝑇𝑇 ∗

[A6.12]

where 𝑓𝑓 is the fraction of interface sites which are growth sites (close to unity for metals), and 𝑉𝑉0 is the upper limiting rate of atom movement which, for pure metals, is of the order of the velocity of sound. For 𝑉𝑉 ≪ 𝑉𝑉0 , one can write: Combining Eqs A6.10 and A6.12 and setting, for the non-equilibrium slope: 1 − 𝑘𝑘𝑣𝑣 + 𝑘𝑘𝑣𝑣 ln( 𝑘𝑘𝑣𝑣 /𝑘𝑘) 𝑚𝑚′ = 𝑚𝑚 1 − 𝑘𝑘

[A6.13]

one obtains for the temperature of a planar interface: −

𝑉𝑉 1 − 𝑘𝑘 [𝑋𝑋𝑙𝑙 𝑚𝑚′ − 𝑋𝑋𝑙𝑙𝑒𝑒 𝑚𝑚] = 𝑉𝑉𝑜𝑜 𝑚𝑚

From the linearised phase diagram, one finds that 𝑇𝑇𝑓𝑓 − 𝑇𝑇 ∗ = −𝑚𝑚𝑋𝑋𝑙𝑙𝑒𝑒 and finally: 𝑇𝑇 ∗ = 𝑇𝑇𝑓𝑓 + 𝑚𝑚′ 𝑋𝑋𝑙𝑙 +

m 𝑉𝑉 1 − k 𝑉𝑉𝑜𝑜

where the asterisks represent interface values. For a dendrite tip of radius, 𝑅𝑅: EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A6.14]

[A6.15]

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Fundamentals of Solidification 5th Edition

𝑇𝑇 ∗ = 𝑇𝑇𝑓𝑓 −

2Γ 𝑚𝑚 𝑉𝑉 + 𝑚𝑚′ 𝑋𝑋𝑙𝑙∗ + 𝑅𝑅 1 − 𝑘𝑘 𝑉𝑉𝑜𝑜

𝑇𝑇0 = 𝑇𝑇𝑓𝑓 +

𝑋𝑋𝑙𝑙∗ 𝑚𝑚 ln( 𝑘𝑘) 𝑘𝑘 − 1

[A6.16]

where 𝑚𝑚′ is the non-equilibrium liquidus slope as defined in Eq. A6.13, and 𝑋𝑋𝑙𝑙∗ is the dendrite tip composition. Setting 𝑅𝑅 = ∞ in Eq. A6.16, 𝑉𝑉0 = ∞ (infinitely rapid interface kinetics) and 𝑘𝑘𝑣𝑣 = 1, one obtains the equation for the 𝑇𝑇0 -line: [A6.17]

Because of the simplifications made, this equation holds only for dilute solutions.

NON-EQUILIBRIUM DISTRIBUTION COEFFICIENT The last equation to be developed is the one describing the relationship between the nonequilibrium distribution coefficient and the rate of interface movement. This can be easily found by considering the two extreme cases shown in Fig. 7.1. At 𝑉𝑉 = 0 the interface is at equilibrium: 𝜇𝜇𝛼𝛼 = 𝜇𝜇𝑙𝑙 and 𝑘𝑘𝑣𝑣 = 𝑘𝑘. Here the net flux of atoms crossing the interface is zero. The other extreme case, 𝑉𝑉 = ∞, leads to complete solute trapping and 𝑘𝑘𝑣𝑣 = 1 (𝑋𝑋𝛼𝛼∗ = 𝑋𝑋𝑙𝑙∗ ). This sets up a gradient of chemical potential which tends to drive atoms from α to l. This flux is proportional to the chemical potential gradient: 𝐽𝐽𝑑𝑑 = −𝑋𝑋𝑀𝑀𝑖𝑖

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

[A6.18]

For dilute solutions the interface diffusion coefficient is related to the interface mobility, 𝑀𝑀𝑖𝑖 , via: 𝐷𝐷𝑖𝑖 = 𝑀𝑀𝑖𝑖 𝑅𝑅𝑔𝑔 𝑇𝑇

[A6.19]

giving:

𝐽𝐽𝑑𝑑 = −

𝐷𝐷𝑖𝑖 𝑋𝑋 𝑑𝑑𝑑𝑑 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑑𝑑𝑑𝑑

[A6.20]

Note that the interface diffusion coefficient in the growth direction can be substantially lower than the bulk liquid diffusion coefficient. The chemical potential gradient at the interface, in the case of dilute solutions (see Eq. A6.6), is: 𝑑𝑑μ Δ𝜇𝜇𝐵𝐵 ln( 𝑘𝑘𝑣𝑣 /𝑘𝑘) ≅ = 𝑅𝑅𝑔𝑔 𝑇𝑇 𝑑𝑑𝑑𝑑 𝛿𝛿𝑖𝑖 𝛿𝛿𝑖𝑖

[A6.21]

Setting the composition of the interface, 𝑋𝑋, equal to its mean value, (𝑋𝑋𝑙𝑙∗ + 𝑋𝑋𝛼𝛼∗ )/2 = 𝑋𝑋𝑙𝑙∗ (1 + 𝑘𝑘𝑣𝑣 )/2 leads to: 𝐽𝐽𝑑𝑑 = −

𝐷𝐷𝑖𝑖 𝑋𝑋l∗ (1 + 𝑘𝑘𝑣𝑣 )ln (𝑘𝑘𝑣𝑣 /𝑘𝑘) 2𝛿𝛿𝑖𝑖

[A6.22]

This flux in the growth direction, which arises from the chemical potential gradient, must be equal to a source term if steady-state conditions are to prevail at the interface. This source term is simply the solute rejected at the interface opposite to V: 𝐽𝐽𝑟𝑟 = −𝑉𝑉𝑋𝑋𝑙𝑙∗ (1 − 𝑘𝑘𝑣𝑣 )

Setting 𝐽𝐽𝑑𝑑 equal to 𝐽𝐽𝑟𝑟 , one obtains: EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A6.23]

Foundations of Materials Science and Engineering Vol. 103

𝑉𝑉𝛿𝛿𝑖𝑖 1 1 + 𝑘𝑘𝑣𝑣 𝑘𝑘𝑣𝑣 = ln � � 𝐷𝐷𝑖𝑖 2 1 − 𝑘𝑘𝑣𝑣 𝑘𝑘

263

[A6.24]

This is just one relationship between the interface Péclet number (𝑃𝑃𝑖𝑖 = 𝑉𝑉 δ𝑖𝑖 /𝐷𝐷𝑖𝑖 ) and the nonequilibrium distribution coefficient. Other relationships have been derived by Aziz (1982, 1994). For example, and again for dilute solutions: 𝑘𝑘𝑣𝑣 =

𝑘𝑘 + 𝑃𝑃𝑖𝑖 1 + 𝑃𝑃𝑖𝑖

[A6.25]

The general form of all these relationships is such that 𝑘𝑘𝑣𝑣 = 𝑘𝑘 when 𝑃𝑃𝑖𝑖 is much smaller than unity, and 𝑘𝑘𝑣𝑣 = 1 when 𝑃𝑃𝑖𝑖 is much greater than unity (Fig. 7.4). The critical growth rate at which 𝑘𝑘𝑣𝑣 changes markedly depends mainly upon the interface diffusion rate, 𝐷𝐷𝑖𝑖 /𝛿𝛿𝑖𝑖 . For metals, it is of the order of 0.1 to 1 m/s. For more details, see Galenko (2007). References and Further Reading  M.J.Aziz, Model for solute redistribution during rapid solidification, Journal of Applied Physics, 53 (1982) 1158.  M.J.Aziz, On the transition from short-range diffusion-limited to collision-limited growth in alloy solidification, Acta Metallurgica et Materialia, 42 (1994) 527.  J.C.Baker, J.W.Cahn, in Solidification, ASM, Metals Park, Ohio, 1971, p.23.  W.J.Boettinger, S.R.Coriell, in Science and Technology of the Undercooled Melt, P.R.Sahm, H.Jones, C.M.Adam (Eds), Martinus Nijhoff Publications, Dordrecht, 1986, p.81.  P.Galenko, Solute trapping and diffusionless solidification in a binary system, Physical Review E, 76 (2007) 031606.  D.Turnbull, On the relation between crystallization rate and liquid structure, Journal of Physical Chemistry, 66 (1962) 609.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 265-275 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 7 ANALYSIS OF MORPHOLOGICAL INTERFACE STABILITY The constitutional undercooling criterion provides a useful means for estimating whether a solid/liquid interface will be planar under directional solidification conditions. From a theoretical point of view it has several faults, however. Firstly, it does not take account of the effect of surface tension, which tends to inhibit the formation of perturbations. Secondly, it takes account only of the temperature gradient in the liquid. Thirdly, the constitutional undercooling theory does not give any indication of the scale of the perturbations which will develop if an interface becomes unstable. In order to deduce more information about the morphological instability of a plane interface than that provided by the constitutional undercooling theory, one must suppose that the interface has already been slightly perturbed and then ask whether the perturbation will grow or disappear (Chap. 3). To this end, it is sufficient to consider a sinusoidal interface form having a very small time-dependent amplitude (Fig. A7.1). (Provided that the amplitude is small, a sinusoidal form represents the most general disturbance possible; it can be supposed to be one term of the Fourier series describing any possible perturbation.) The object of the present calculations is to determine the conditions which govern the growth or decay of a perturbation at the solid/liquid interface (Fig. 3.1). Following the developments made by Mullins and Sekerka (1964), the stability analysis is first performed while assuming the thermal field to be fixed, i.e. fixed thermal gradient, 𝐺, and fixed velocity of the isotherm, 𝑉, regardless of the fluctuations of the solid/liquid interface. These fluctuations influence the solute field in the liquid, and the reader is advised to consult the section in Appendix 2 dealing with a slightly perturbed interface. For the results to be applicable to high rates of solidification, this first approach is then extended in the second part of this appendix for both solute and heat diffusion. In the case of solute diffusion, the solid phase can be neglected because the rate of diffusion in the solid is so slow. In the case of heat diffusion, both the liquid and solid phases have to be considered. Finally, the last section of this appendix focuses on an absolute stability of the solid/liquid interface that occurs at high speeds.

Figure A7.1

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266

Fundamentals of Solidification 5th Edition

STABILITY ANALYSIS FOR A FIXED THERMAL FIELD In Appendix 2 the steady-state solute profile in the liquid ahead of a planar solid/liquid interface moving at velocity, 𝑉, was determined (Eq. A2.21): 1 −𝑉𝑧 −𝑉𝑧 ) = 𝐶0 + ∆𝐶0 exp ( ) 𝐶̃𝑙 (𝑧) = 𝐶0 + 𝐶0 ( − 1) exp ( 𝑘 𝐷 𝐷

[A7.1]

where 𝐷 is the diffusion coefficient of the solute element in the liquid. The planar interface, positioned at 𝑧 ∗ = 0 in the moving reference frame, is now disturbed by a sinusoidal perturbation, 𝑧 ∗ = 𝜀sin⁡(𝜔𝑦), where 𝜀 ≪ 1. The steady-state equation that needs to be solved for the solute field in the liquid then becomes: 𝜕 2 𝐶𝑙 𝜕 2 𝐶𝑙 𝑉 𝜕𝐶𝑙 + + =0 𝜕𝑦 2 𝜕𝑧 2 𝐷 𝜕𝑧

[A7.2]

Following the procedure of separating variables described in Appendix 2, it is fairly straightforward to show that the perturbed solute field in the liquid is given by: 𝐶𝑙 (𝑦, 𝑧) = 𝐶̃𝑙 (𝑧) + 𝐴𝜀sin(𝜔𝑦)exp(−𝑏𝑧)

[A7.3]

where: 𝑏=

𝑉 𝑉 2 + √( ) + 𝜔 2 2𝐷 2𝐷

[A7.4]

The constant, 𝐴, which appears in this equation has now to be determined. It is given by the temperature condition at the perturbed solid/liquid interface. The temperature, 𝑇 ∗ , of the interface, which is equal to the solidus of the alloy 𝑇𝑠 (𝐶0 ) when it is planar, is now given by 𝑇𝑠 (𝐶0 ) + 𝐺𝑧 ∗ when it is perturbed. The temperature, 𝑇 ∗ , must be equal to the liquidus temperature associated with the local composition in the liquid at the interface, 𝐶𝑙 (𝑦, 𝑧 ∗ ), while also taking into account the local curvature-undercooling. The thermal condition at the interface is therefore given by: 𝑇 ∗ = 𝑇𝑠 (𝐶0 ) + 𝐺𝑧 ∗ = 𝑇𝑙 (𝐶𝑙 (𝑦, 𝑧 ∗ )) − Γ𝐾

[A7.5]

where 𝐾 is the local curvature of the interface and Γ is the Gibbs-Thomson coefficient. Assuming a small perturbation, the local curvature of the interface is approximated as follows: 𝐾=

−𝑧 ∗ ′′ 𝑑2 𝑧 ∗ ∗ ≅ −𝑧 ′′ = − = 𝜀𝜔2 sin(𝜔𝑦) = 𝜔2 𝑧 ∗ (1 + 𝑧 ∗ ′2 )3/2 𝑑𝑦 2

[A7.6]

since the first derivative of the interface is such that 𝑧 ∗ ′ ≪ 1. Assuming a linear phase diagram, the liquidus temperature, 𝑇𝑙 (𝐶𝑙 (𝑦, 𝑧 ∗ )), is given by: 𝑇𝑙 (𝐶𝑙 (𝑦, 𝑧 ∗ )) = 𝑇𝑓 + 𝑚[𝐶̃𝑙 (𝑧 ∗ ) + 𝐴𝜀sin(𝜔𝑦)exp(−𝑏𝑧 ∗ )] = = 𝑇𝑓 + 𝑚 [𝐶0 + Δ𝐶0 exp (

−𝑉𝑧 ∗ ) + 𝐴𝑧 ∗ exp(−𝑏𝑧 ∗ )] = 𝐷

𝑉𝑧 ∗ ) + 𝐴𝑧 ∗ (1 − 𝑏𝑧 ∗ )] = 𝑇𝑓 + 𝑚 [𝐶0 + Δ𝐶0 ⁡ (1 − 𝐷

[A7.7]

Note that use has been made here of the fact that exp(−𝑥) = 1 − 𝑥 for ⌊𝑥⌋ ≪ 1. One can now insert expressions A7.6 and A7.7 into Eq. A7.5, neglecting 2nd-order terms (e.g. 𝐴𝑏𝑧 ∗2 ) and keeping in mind that 𝑇𝑠 (𝐶0 ) = 𝑇𝑙 (𝐶0 /𝑘). One thus obtains:

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Foundations of Materials Science and Engineering Vol. 103

𝐺𝑧 ∗ = 𝑧 ∗ [−𝑚Δ𝐶0

𝑉 + 𝐴𝑚] − Γ𝜔2 𝑧 ∗ 𝐷

267

[A7.8]

The solute gradient of the unperturbed solution being 𝐺𝑐 = Δ𝐶0 /(−𝐷/𝑉), this equation allows one to find 𝐴: 𝐴=

1 {Γ𝜔2 + 𝐺 − 𝑚𝐺𝑐 } 𝑚

[A7.9]

Now that the solute profile is fully determined (Eqs A7.3 and A7.9) it is possible to find out whether the perturbation of wave-frequency, 𝜔, and amplitude, 𝜀, will decrease or increase with time. For that purpose one sets up the solute balance at the moving interface: −𝐷𝐺𝑐∗

∂𝐶𝑙 ∗ ≃ −𝐷 [ ] = 𝑉 ∗ 𝐶𝑙 (𝑦, 𝑧 ∗ )(1 − 𝑘) ∂𝑧

[A7.10]

where the velocity of the interface is given by: 𝑉∗ =

𝑑𝑧 ∗ 𝜀̇ = 𝑉 + 𝜀̇sin(𝜔𝑦) = 𝑉 + 𝑧 ∗ 𝑑𝑡 𝜀

Note that the solute gradient at the interface, 𝐺𝑐∗ , has been approximated by taking only the component along the z-axis: i.e. one is again assuming that the amplitude of the perturbation is small. It is left as an exercise (Exercise 3.9) to show that the 0th-order terms cancel, while the 1st-order terms in 𝑧 ∗ give the following condition: 𝜖̇ 𝑉 𝑉 𝑉2 2 ( (1 − 𝑘) − 𝑏) [Γ𝜔 + 𝐺 − 𝑚𝐺𝑐 ] − = 𝑘 𝜖 𝑚𝐺𝑐 𝐷 𝐷

[A7.11]

The last term on the RHS of this equation can be introduced into the square bracket. It is left as an exercise (Exercise 3.10) to show that Eq. A7.11 can also be written as: 𝜖̇ 𝑉 𝑉 ( (1 − 𝑘) − 𝑏) [Γ𝜔2 + 𝐺 − 𝑚𝐺𝑐 𝜉𝑐 ] = 𝜖 𝑚𝐺𝑐 𝐷

[A7.12]

providing the parameter 𝜉𝑐 is introduced: 𝑉 −𝑏 ξ𝑐 = 𝐷 with 𝑝 = 1 − 𝑘 𝑉𝑝 − 𝑏 𝐷 As discussed in Chap. 3 (Fig. 3.7), a mode of spatial frequency ω is unstable and the corresponding perturbation grows over time if 𝜖̇/𝜖(𝜔) > 0. On the contrary, if 𝜖̇/𝜖(𝜔) < 0, the mode is stable and the corresponding perturbation decays over time. In order to find the marginal stability mode, for which 𝜀̇/𝜀(𝜔) = 0, the term in the square bracket of Eq. A7.12 must be set to 0: Γ𝜔2 + 𝐺 − 𝑚𝐺𝑐 𝜉𝑐 = 0

[A7.13]

Keep in mind that 𝜔 also appears in the b-term (Eq. A7.4). Equation A7.13 is formally the same as Eq. A7.30, derived in the next section, where the effect of the perturbation on heat diffusion is also considered. Figure A7.2 shows the result for 𝜀̇/𝜀 as a function of the logarithm of the wavelength,⁡, with 𝜆 = 2π/𝜔, using Eq. A7.11 for two thermal gradients. The first one (𝐺 = 1 × 104 K/m) is representative of a typical Bridgman experiment, while the second one (𝐺 = 3 × 107 K/m) is unrealistically high, except perhaps in the case of laser re-melting. For the lowest thermal gradient, the range of wavelengths for which 𝜀̇/𝜀 > 0 begins when 𝜆 > 𝜆𝑖 and ends at 𝜆𝑐 (its crossing of the

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abscissa being not visible because of the scale used). That is, all the modes 𝜆𝑖 < 𝜆 < 𝜆𝑐 are unstable. The mode 𝜆 = 𝜆𝑖 , for which 𝜀̇/𝜀 = 0, corresponds to the marginably stable mode used in dendrite growth theory. For the highest thermal gradient, the condition, 𝐺⁡ > ⁡𝑚𝐺𝑐 , is satisfied and all the modes are stable (𝜀̇/𝜀 < 0), i.e. the solid/liquid planar interface is stable.

Figure A7.2 The parameters used here to calculate 𝜖̇/𝜖 = 𝑑(ln⁡𝜖)/𝑑𝑡⁡were: 𝑉 = 1 × 10−4 m/s, 𝐷 = 3 × 10−9 m2/s, 𝑘 = 0.2, 𝑚 = −10 K/%, 𝐶0 = 2%,  = 1 × 10−7 Km.

STABILITY ANALYSIS INCLUDING SOLUTE AND HEAT DIFFUSION Assuming now that the thermal field is also affected by the perturbation, Equation A7.2 for the solute field in the liquid has to be complemented by the heat diffusion equations for the solid and the liquid phases: 𝜕 2 𝑇𝑙 𝜕 2 𝑇𝑙 𝑉 𝜕𝑇𝑙 + + =0 𝜕𝑦 2 𝜕𝑧 2 𝑎𝑙 𝜕𝑧 which describes the temperature distribution in the liquid and 𝜕 2 𝑇𝑠 𝜕 2 𝑇𝑠 𝑉 𝜕𝑇𝑠 + + =0 𝜕𝑦 2 𝜕𝑧 2 𝑎𝑠 𝜕𝑧

[A7.14a]

[A7.14b]

which describes the temperature distribution in the solid. Note that the indices, s and l, used here in the equations for the temperature fields refer to the domains within which they apply and not to solidus and liquidus temperatures. The 𝑎𝑙 and 𝑎𝑠 are here the thermal (liquid and solid) diffusivities, respectively. For a plane interface, suitable solutions to these equations are, respectively: 𝐶𝑙 = 𝐶0 −

𝐷𝐺𝑐 𝑉𝑧 exp [− ] 𝑉 𝐷

[A7.15a]

𝑇𝑙 = 𝑇0 +

𝐺𝑙 𝑎𝑙 𝑉𝑧 [1 − exp (− )] 𝑉 𝑎𝑙

[A7.15b]

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Foundations of Materials Science and Engineering Vol. 103

𝑇𝑠 = 𝑇0 +

𝐺𝑠 𝑎𝑠 𝑉𝑧 [1 − exp (− )] 𝑉 𝑎𝑠

269

[A7.15c]

where 𝐺𝑐 , 𝐺𝑙 and 𝐺𝑠 are solute and temperature gradients (liquid and solid), respectively, at the plane interface. Note that 𝐶0 is the concentration at infinity, but 𝑇0 is the temperature at the interface. As in Appendix 2, the plane interface is assumed to be perturbed so that it has the form shown in Fig. A7.1. According to perturbation theory (Appendix 2), the solute and temperature distributions can be assumed to be of the form: 𝐶𝑙 = 𝐶0 −

𝐷𝐺𝑐 𝑉𝑧 exp (− ) + 𝐴𝜀 sin(𝜔𝑦) exp(−𝑏𝑐 𝑧) 𝑉 𝐷

[A7.16a]

𝑇𝑙 = 𝑇0 +

𝐺𝑙 𝑎𝑙 𝑉𝑧 [1 − exp (− )] + 𝐵𝜀 sin(𝜔𝑦) exp(−𝑏𝑙 𝑧) 𝑉 𝑎𝑙

[A7.16b]

𝑇𝑠 = 𝑇0 +

𝐺𝑠 𝑎𝑠 𝑉𝑧 [1 − exp (− )] + 𝑅𝜀 sin(𝜔𝑦) exp(−𝑏𝑠 𝑧) 𝑉 𝑎𝑠

[A7.16c]

where 𝐴, 𝐵, 𝑅, 𝑏𝑐 , 𝑏𝑙 and 𝑏𝑠 have to be determined by applying the various physical constraints imposed by the situation: As in the previous section, each of the 𝑏-values can be found by inserting the derivatives of the relevant equation into the corresponding differential Eqs, A7.2 and A.7.14. In this way (Appendix 2) it is found that: 1/2

𝑉 𝑉 2 𝑏𝑐 = + [( ) + 𝜔2 ] 2𝐷 2𝐷

[A7.17a] 1/2

𝑉 𝑉 2 𝑏𝑙 = + [( ) + 𝜔2 ] 2𝑎𝑙 2𝑎𝑙

[A7.17b] 1/2

𝑉 𝑉 2 ) + 𝜔2 ] 𝑏𝑠 = − + [( 2𝑎𝑠 2𝑎𝑠

[A7.17c]

The value of 𝑅 can be found by recalling that, at the perturbed interface, the temperatures in the solid and in the liquid must be the same. That is (putting 𝑆 = 𝜀sin(𝜔𝑦) = 𝑧 ∗ , Appendix 2): 𝑇0 +

𝐺𝑙 𝑎𝑙 𝑉𝑆 [1 − exp (− )] + 𝐵𝑆 exp(−𝑏𝑐 𝑆) = 𝑉 𝑎𝑙 = 𝑇0 +

𝐺𝑠 𝑎𝑠 𝑉𝑆 [1 − exp (− )] + 𝑅𝑆 exp(−𝑏𝑠 𝑆) 𝑉 𝑎𝑠

Since 𝜀 is very small one can write exp[−𝜀sin(𝜔𝑦)] ≈ 1 − 𝜀sin(𝜔𝑦), and terms in 𝜀 2 disappear. This leads to: 𝑅 = (𝐺𝑙 − 𝐺𝑠 ) + 𝐵

[A7.18]

A second condition is imposed by the coupling of the absolute temperatures and concentrations at the interface. That is, the actual temperature at the perturbed interface must correspond to that given by the sum of the constitutional supercooling and capillarity effects (if the attachment kinetics are very rapid). This can be written: 𝑇 ∗ = 𝑇𝑓 + 𝑚𝐶 ∗ − Γ𝜔2 𝑆

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[A7.19]

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where the index, “ ∗ ”, indicates a value determined at the perturbed interface and 𝑇𝑓 is the meltingpoint of the pure substance. The last term on the RHS involves only the second differential of the perturbing function rather than the full curvature expression, 𝐾 = −𝑧 ′′ /(1 + 𝑧 ′2 )3/2 ≈ −𝑧 ′′ . This is permissible because of the smallness of 𝜀. Thus: 𝑇0 +

𝐺𝑙 𝑎𝑙 𝑉𝑆 [1 − exp (− )] + 𝐵𝑆 exp(−𝑏𝑙 𝑆) = 𝑉 𝑎𝑙 = 𝑇𝑓 + 𝑚 [𝐶0 −

𝐷𝐺𝑐 𝑉𝑆 exp (− ) + 𝐴𝑆 − exp(−𝑏𝑐 𝑆)] − Γ𝜔2 𝑆 𝑉 𝐷

That is: 𝑇0 + 𝐺𝑙 𝑆 + 𝐵𝑆 = 𝑇𝑓 + 𝑚𝐶0 −

𝑚𝐷𝐺𝑐 + 𝑚𝐺𝑐 𝑆 + 𝑚𝐴𝑆 − Γ𝜔2 𝑆 𝑉

Since for the unperturbed interface, 𝑇0 = 𝑇𝑓 + 𝑚𝐶0 − 𝑚𝐷𝐺𝑐 /𝑉, one can write (1st-order terms): 𝐺𝑙 𝑆 + 𝐵𝑆 = 𝑚𝐺𝑐 𝑆 + 𝑚𝐴𝑆 − Γ𝜔2 𝑆 and cancelling 𝑆 throughout leads to: 𝐵 = 𝑚𝐺𝑐 + 𝑚𝐴 − Γ𝜔2 − 𝐺𝑙

[A7.20]

One now has 𝐵 in terms of 𝐴. In order to eliminate this last remaining unknown, one imposes the condition that the gradients of solute and temperature must satisfy solute and heat conservation at the perturbed interface for the same growth rate. This can be written: 𝜅𝑠 𝐺𝑠∗ − 𝜅𝑙 𝐺𝑙∗ 𝐷𝐺𝑐∗ = Δℎ𝑓 (𝑘 − 1)𝐶 ∗

[A7.21]

where 𝜅𝑠 and 𝜅𝑙 are the conductivities of the solid and liquid, respectively, and Δℎ𝑓 is the volumic heat of fusion. The quantities marked “ * ”, must be evaluated at the perturbed interface. That is: 𝐺𝑠∗ = 𝐺𝑠 −

𝐺𝑠 𝑉𝑆 − 𝑏𝑠 𝑅𝑆 𝑎𝑠

[A7.22a]

𝐺𝑙∗ = 𝐺𝑙 −

𝐺𝑙 𝑉𝑆 − 𝑏𝑙 𝐵𝑆 𝑎𝑙

[A7.22b]

𝐺𝑐∗ = 𝐺𝑐 −

𝐺𝑐 𝑉𝑆 − 𝑏𝑐 𝐴𝑆 𝐷

[A7.22c]

𝐶 ∗ = 𝐶0 −

𝐷𝐺𝑐 + 𝐺𝑐 𝑆 + 𝐴𝑆 𝑉

[A7.22d]

Inserting Eq. A7.22 into Eq. A7.21 gives: 𝑛1 + 𝑛2 𝑉𝑆 + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)𝑆 =

Δℎ𝑓 𝐷𝐺𝑐 − 𝐺𝑐 𝑉𝑆 − 𝐷𝑏𝑐 𝐴𝑆 (𝑘 − 1) 𝐶 ∗ + 𝐺𝑐 𝑆 + 𝐴𝑆

where: 𝑛1 = (𝜅𝑠 𝐺𝑠 − 𝜅𝑙 𝐺𝑙 )

and

𝜅𝑙 𝐺𝑙 𝜅𝑠 𝐺𝑠 ) 𝑛2 = ( − 𝑎𝑙 𝑎𝑠

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Foundations of Materials Science and Engineering Vol. 103

271

and 𝐶 ∗ is the liquid composition at a planar solid/liquid interface. Multiplying throughout by the denominator on the RHS, and remembering that terms in 𝑆 2 can be neglected, gives: 𝑛1 𝐶 ∗ + 𝑛1 𝐺𝑐 𝑆 + 𝑛1 𝐴𝑆 + 𝑛2 𝑉𝐶 ∗ 𝑆 + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)𝐶 ∗ 𝑆 = =

Δℎ𝑓 (𝐷𝐺𝑐 − 𝐺𝑐 𝑉𝑆 − 𝐷𝑏𝑐 𝐴𝑆) 𝑘−1

But, from Eq. A7.21, it follows that: 𝑛1 𝐶 ∗ =

Δℎ𝑓 𝐷𝐺 𝑘−1 𝑐

After dividing throughout by S, one therefore obtains: 𝑛1 𝐺𝑐 + 𝑛1 𝐴 + 𝑛2 𝑉𝐶 ∗ + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)𝐶 ∗ =

Δℎ𝑓 (𝐺𝑐 𝑉 + 𝐷𝑏𝑐 𝐴) 𝑝

where 𝑝 = 1 − 𝑘. By substituting for 𝑅 from Eq. A7.18, and for 𝐵 from Eq. A7.20, one can obtain the value of the last unknown: 𝑉𝐶 ∗ −𝑛1 ( 𝐷 + 𝐺𝑐 ) − 𝑛2 𝑉𝐶 ∗ − 𝑛3 𝐶 ∗ (𝜙 − 𝜔2 Γ) + 𝜅𝑠 𝑏𝑠 (𝐺𝑙 − 𝐺𝑠 )𝐶 ∗ 𝐴= [A7.23] 𝐶∗ 𝑛1 + 𝑛3 𝑚𝐶 ∗ + 𝑛1 (𝐺 ) 𝑏𝑐 𝑐 where 𝑛3 = 𝜅𝑙 𝑏𝑙 − 𝜅𝑠 𝑏𝑠 and 𝜙 = 𝑚𝐺𝑐 − 𝐺𝑙 (the degree of constitutional supercooling). In order to study the time-dependence, it is assumed that the local velocity of the perturbed interface of the form: [A7.24]

𝐴𝑉 + 𝜀̇ sin(𝜔𝑦)

must be equal to the heat conservation term on the LHS of Eq. A7.21. By inserting the values from Eqs A7.22a and A7.22b, one obtains: 𝑉 + 𝜀̇ sin(𝜔𝑦) = 𝜅𝐺 𝜅𝐺 (𝜅𝑠 𝐺𝑠 − 𝜅𝑙 𝐺𝑙 ) + ( 𝑎𝑙 𝑙 − 𝑎𝑠 𝑠 ) 𝑉𝜀 sin(𝜔𝑦) + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)⁡𝜀 sin(𝜔𝑦) 𝑙 𝑠 = Δℎ𝑓 or: 𝑉 + 𝜀̇ sin(𝜔𝑦) =

𝑛1 + 𝑛2 𝑉𝜀 sin(𝜔𝑦) + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)𝜀 sin(𝜔𝑦) Δℎ𝑓

[A7.25]

The above equation expresses the identity of two polynomials (in 𝜀 sin(𝜔𝑦)). The equivalent coefficients on the two sides must therefore be equal. Thus: 𝑉=

𝜅𝑠 𝐺𝑠 − 𝜅𝑙 𝐺𝑙 Δℎ𝑓

[A7.26a]

and: 𝜀̇ sin(𝜔𝑦) =

𝑛2 𝑉𝜀 sin 𝜔𝑦 + (𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅)𝜀 sin 𝜔𝑦 Δℎ𝑓

[A7.26b]

That is: 𝜀̇ 𝑛2 𝑉 + 𝜅𝑙 𝑏𝑙 𝐵 − 𝜅𝑠 𝑏𝑠 𝑅 = 𝜀 Δℎ𝑓

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[A7.27]

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Fundamentals of Solidification 5th Edition

The sign of 𝜀̇/𝜀⁡determines the stability of the interface. If it is positive for any value of 𝜔, then perturbations with that wave-number will be amplified. In order to study the stability behaviour in detail, it is necessary to use Eqs A7.18 and A7.20 to substitute for 𝑅 and 𝐵, respectively, in Eq. A7.27. That is: 𝜀̇ 𝑛2 𝑉 − 𝜅𝑠 𝑏𝑠 (𝐺𝑙 − 𝐺𝑠 ) + (𝜅𝑙 𝑏𝑙 − 𝜅𝑠 𝑏𝑠 )(𝜙 − 𝜔2 Γ + 𝑚𝐴) = 𝜀 Δℎ𝑓

[A7.28]

The denominator of Eq. A7.28, after substituting for A, will be the same as the denominator of Eq. A7.23. Examination of the denominator shows that it is always positive and therefore cannot affect the stability of the interface. Attention will now therefore be restricted to the numerator. This can be written as: 𝜀̇ 1 𝑏𝑐 𝑉 𝐺𝑐 = [𝑛1 𝑛2 𝑉 − 𝑛1 𝑛4 + 𝑛1 𝑛3 (𝜙 − 𝜔2 Γ)] ( ∗ + ) − 𝑛1 𝑛3 ( + ∗ ) 𝑚 𝜀 𝐶 𝐺𝑐 𝐷 𝐶 where 𝑛4 = 𝜅𝑆 𝑏𝑆 (𝐺𝑙 − 𝐺𝑆 ). Further rearrangement gives the final result for marginal stability, i.e. for 𝜀̇/𝜀 = 0: −Γ𝜔2 − [𝜅̄ 𝑙 𝐺𝑙 𝜉𝑙 + 𝜅̄ 𝑠 𝐺𝑠 𝜉𝑠 ] + 𝑚𝐺𝑐 𝜉𝑐 = 0

[A7.29]

where ξ𝑙 =

𝑏𝑙 − (𝑉/𝑎𝑙 ) 𝜅̄ 𝑙 𝑏𝑙 + 𝜅̄ 𝑠 𝑏𝑠

𝜅̄ 𝑙 = 𝜅𝑙 /(𝜅𝑙 + 𝜅𝑠 ) 𝜉𝑐 =

;

ξ𝑠 =

𝑏𝑠 − (𝑉/𝑎𝑠 ) 𝜅̄ 𝑙 𝑏𝑙 + 𝜅̄ 𝑠 𝑏𝑠

; 𝜅̄ 𝑠 = 𝜅𝑠 /(𝜅𝑙 + 𝜅𝑠 )

𝑏𝑐 − 𝑉/𝐷 𝑏𝑐 − 𝑉𝑝/𝐷

In order to understand the implications of this very important result, it is helpful to assume, for the moment, that 𝜅𝑠 = 𝜅𝑙 , 𝑎𝑠 = 𝑎𝑙 , 𝐺𝑠 = 𝐺𝑙 = 𝐺, and 𝜉𝑠 = 𝜉𝑙 . The latter is true when 𝜔 ≫ 𝑉/𝑎. (It is also true for the case of constrained growth with 𝐺 > 0 when 𝐺𝑆 = 𝐺𝑙 , a situation that was analysed in the previous section). This corresponds to the small thermal Péclet number solution of Mullins and Sekerka (1964) but allows for large solutal Péclet numbers. Under these conditions, Equtation A7.29 becomes: 𝑉𝑝 𝑉𝑝 𝑉 [A7.30] ) − 𝐺 (𝑏𝑐 − ) + 𝑚𝐺𝑐 (𝑏𝑐 − ) = 0 𝐷 𝐷 𝐷 which is identical to Eq. A7.12, that was derived in the previous section for the case of a fixed thermal profile. −𝜔2 Γ (𝑏𝑐 −

UPPER LIMIT OF STABILITY (ABSOLUTE STABILITY) Writing Eq. A7.30 in the form: −Γ𝜔2 − 𝐺 + 𝑚𝐺𝑐 𝜉𝑐 = 0 and assuming that the Péclet number is much larger than unity, i.e. 𝜉𝑐 → π2 /𝑘𝑃𝑐2 , one obtains: −Γ𝜔2 − 𝐺 + 𝑚𝐺𝑐 π2 /𝑘𝑃𝑐2 = 0 It can be assumed in most cases that 𝐺 can be neglected at very high growth rates, so that:

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Foundations of Materials Science and Engineering Vol. 103

273

−Γ𝜔2 + 𝑚𝐺𝑐 π2 /𝑘𝑃𝑐2 = 0 Substituting for 𝜔 and 𝑃𝑐 ⁡gives: −4𝜋 2 Γ/𝜆2 + 𝑚𝐺𝑐 (π2 /𝑘)(4𝐷2 /𝑉 2 𝜆2 ) = 0 Because it is assumed throughout that the long-range diffusion field is the same as that for the plane front, one can substitute Δ𝑇0 𝑉/𝐷 for 𝑚𝐺𝑐 . Thus: 𝑘𝑉Γ =1 Δ𝑇0 𝐷 When this condition is satisfied, or exceeded, the interface will be stable regardless of the value of 𝐺, if 𝐺 < 𝐺𝑐𝑟𝑖𝑡 (Fig. 7.6). This is referred to as 'absolute stability'. One can define a critical velocity for absolute stability: (𝑉𝑎 )𝑐 ≥

Δ𝑇0 𝐷 𝑘Γ

[A7.31]

Using typical values for metals: Δ𝑇0 = 10⁡K, D = 5 × 10−3 mm2/s 𝑘 = 0.5, and Γ = 10−4 ⁡Kmm, the limit of absolute stability should be attained when 𝑉 is greater than 1 m/s. Such rates are very high but can be attained in rapid solidification processes such as laser surface melting. At these rates, account must be taken of the value of the distribution coefficient, which is a function of the growth rate and may approach unity (see Appendix 6)*. The change in 𝑘 will decrease (𝑉𝑎 )𝑐 via the decreased Δ𝑇0 and increased 𝑘 values. Although the gradient, 𝐺𝑙 , is considered to have a negligible effect in the above case, it is still positive. One can also consider the case where the temperature gradient in the solid is equal to zero and the gradient, 𝐺𝑙 , in the liquid is negative, as in the case of equiaxed grain-growth. The growth rate is still assumed to be very high. Equation A7.29, in the case where 𝐺𝑆 = 0 and 𝜅𝑠 = 𝜅𝑙 , becomes: −Γ𝜔2 − 0.5𝐺𝑙 𝜉𝑙 + 𝑚𝐺𝑐 𝜉𝑐 = 0 Substituting the high Péclet number approximations, 𝜉𝑙 = 2𝜋 2 /𝑃𝑙2 and 𝜉𝑐 = π2 /𝑘𝑃𝑐2 , gives: −Γ𝜔2 − 𝐺𝑙 (𝜋 2 /𝑃𝑙2 ) + 𝑚𝐺𝑐 (π2 /𝑘𝑃𝑐2 ) = 0 This expression determines the form (values of 𝜔) which the perturbed interface must assume in order to satisfy all of the conditions of the problem. From the signs of the terms in Eq. A7.29 one can immediately deduce the effects of the parameters involved. For example, increasing Γ𝜔2 (curvature effect) or 𝐺 (imposed temperature gradient) tends to decrease the value of 𝜀̇/𝜀 and thus increases the stability, whereas 𝑚𝐺𝑐 (liquidus temperature gradient) is always positive (since 𝑚 and 𝐺𝑐 always have the same sign) and therefore decreases stability. If Equation A7.29 is further simplified by making the assumption that 𝑘 is equal to zero (that is, 𝑝 = 1), one can immediately obtain the result: 𝜔2 =

𝜙 Γ

*

[A7.32]

Note that the limit in Eq. A7.31 can also be derived by assuming that, in Eq. A7.30, the value of 𝐺 becomes irrelevant when the de-stabilizing term, 𝑚𝐺𝑐 , is balanced by the term V 2 kΓ/𝐷2 . Equating the two terms again gives Eq.A7.31.

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274

Fundamentals of Solidification 5th Edition

which, as pointed out in Sect. 3.4 (Eq. 3.22), is the same as the result derived there by using very simple arguments. Another relatively simple step one can take is to replace the exact expression for 𝑏𝑐 (which leads to an intractable cubic equation) by 𝑏𝑐 ∼ 𝑉/𝐷 + (𝐷/𝑉)𝜔2 (which implies 𝜔 ≪ 𝑉/2𝐷). This leads to the equation: 𝐷Γ 4 𝐷 𝑉𝑘 𝑉𝑘 𝜔 − (𝜙 − Γ ) 𝜔2 + 𝐺 =0 𝑉 𝑉 𝐷 𝐷

[A7.33]

Using Descartes' rule of signs, if the equation is to have no positive roots (implying stability) then the coefficient of 𝜔2 must be negative. This leads to the stability condition: 𝐺 > 𝑚𝐺𝑐 −

𝑉 2 𝑘Γ 𝐷2

[A7.34]

Returning now to consider the full form of Eq. A7.28, one question of interest is how 𝜉𝑙 , 𝜉𝑠 and 𝜉𝑐 vary as the parameters change. This is most easily seen by again assuming that 𝑎𝑙 = 𝑎𝑠 and 𝜅𝑠 = 𝜅𝑙 , which is true, for example, in the case of organic materials such as succinonitrile. This then leads to the equations: 𝜉𝑙 = 1 −

𝑉/2𝑎𝑡 [(𝑉/2𝑎𝑙 )2 + 𝜔 2 ]1/2

[A7.35a]

𝜉𝑠 = 1 +

𝑉/2𝑎𝑠 [(𝑉/2𝑎𝑠 )2 + 𝜔 2 ]1/2

[A7.35b]

𝜉𝑐 =

1 − (1 + 4𝐷 2 𝜔2 /𝑉 2 )1/2 1 − 2𝑘 − (1 + 4𝐷2 𝜔 2 /𝑉 2 )1/2

[A7.35c]

By substituting the relevant Péclet numbers, these equations can be written in the corresponding forms: 𝜉𝑙 = 1 −

1 [1 + (2π/𝑃𝑙 )2 ]1/2

[A7.36a]

𝜉𝑠 = 1 +

1 [1 + (2π/𝑃𝑠 )2 ]1/2

[A7.36b]

𝜉𝑐 = 1 −

2𝑘 [1 + (2π/𝑃𝑐 )2 ]1/2 − 1 + 2𝑘

[A7.36c]

As in Fig. A7.3, all three are equal to unity at low Péclet number, and the first and last tend towards zero as the Péclet number tends towards infinity. The second of the three tends towards 2 as the Péclet number approaches infinity. When the Péclet number is high but not infinite, the expressions can be further simplified since 1/[1 + (2π/𝑃)2 ]1/2 is approximately equal to 1 − 2π2 /𝑃2 under these conditions. This gives: 𝜉𝑙 ≅ 2π2 /𝑃𝑙2 𝜉𝑠 ≅ 2 − 2π2 /𝑃𝑠2

𝑃≫1

𝜉𝑐 ≅ π2 /𝑘𝑃𝑐2 Replacing 𝑚𝐺𝑐 , 𝑃𝑙 , 𝜔 and 𝑃𝑐 by their respective definitions gives: −𝐺𝑙

𝑎𝑙2 Δ𝑇0 𝐷 + =𝑉 Γ𝑉 2 Γ𝑘

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Foundations of Materials Science and Engineering Vol. 103

275

Figure A7.3

For growth in undercooled melts, with 𝐺𝑠 = 0, the heat flux balance is 𝑉Δℎ𝑓 = −𝜅𝑙 𝐺. Therefore:

or

Δℎ𝑓 𝑎𝑙2 Δ𝑇0 𝐷 + =𝑉 𝜅𝑙 Γ Γ𝑘

𝐷Δ𝑇0 𝑎𝑙 𝜃𝑡 + Γ𝑘 Γ where 𝜃𝑡 is the unit thermal undercooling. This condition can be written (Trivedi and Kurz, 1986): 𝑉=

𝑉𝑎 = (𝑉𝑎 )𝑐 + (𝑉𝑎 )𝑡

[A7.37]

This equation shows that absolute stability is a general phenomenon and will always be observed when the growth rate of the interface is higher than that given by Eq. A7.37. The second term on the RHS of this equation is much larger than the first one (for metals where 𝑎 ≫ 𝐷). Therefore, 𝑉𝑎 in an undercooled melt (equiaxed growth) is larger than in the case of directional growth, where the temperature gradients are positive and (𝑉𝑎 )𝑡 = 0. For a more complete analysis of interface stability, including oscillatory behaviour, the reader is referred to Coriell and Sekerka (1983), Huntley and Davis (1993, 1996) and Karma and Sarkissian (1993). References an Further Reading ▪ S.R.Coriell, R.F.Sekerka, Oscillatory morphological instabilities due to non-equilibrium segregation, Journal of Crystal Growth, 61 (1983) 499. ▪ D.A.Huntley, S.H.Davis, Thermal effects in rapid directional solidification: weakly-nonlinear analysis of oscillatory instabilities, Journal of Crystal Growth, 132 (1993) 141. ▪ D.A.Huntley, S.H.Davis, Effect of latent heat on oscillatory and cellular mode coupling in rapid directional solidification, Physical Review B, 53 (1996) 3132. ▪ A.Karma, A.Sarkissian, Interface dynamics and banding in rapid solidification, Physical Review E, 47 (1993) 513. ▪ W.W.Mullins, R.F.Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics, 35 (1964) 444. ▪ R.Trivedi, W.Kurz, Morphological stability of a planar interface under rapid solidification conditions, Acta Metallurgica, 34 (1986) 1663.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 277-283 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 8 DIFFUSION AT A DENDRITE TIP There are three different types of dendrites: i. equiaxed dendrites of pure substances - freely growing and governed by thermal diffusion (Fig. 4.7(b)); ii. equiaxed dendrites of alloys - freely growing and governed by solute and thermal diffusion (Fig. 4.7(d)); iii. columnar alloy dendrites - constrained in their growth by a positive temperature gradient and controlled by solute diffusion (Fig. 4.7(c)). The first two types of growth form are similar and both lead to the creation of an equiaxed polycrystal in which each randomly arranged grain is made up of six orthogonal primary trunks (in the case of a cubic crystal). The space remaining between the trunks is filled with secondary and possibly higher-order branches (Fig. 4.18). These crystals grow in an undercooled melt, and this makes them inherently unstable. No steady-state cellular morphologies can therefore be observed and the spacing, 𝜆𝜆1 , between the trunks corresponds approximately to the grain diameter. The growth of equiaxed dendrites of pure metals occurs under conditions where only heat flows from the interface to the surrounding liquid. That is, the temperature gradient is negative at the interface (left-hand side of Fig. A8.1) and a thermal undercooling, Δ𝑇𝑇𝑡𝑡 , exists. In the case of equiaxed alloy growth, there exists not only a negative temperature gradient but also a solute build-up (if 𝑘𝑘 is less than unity) ahead of the dendrite tip. This changes the local liquidus temperature (RHS of Fig. A8.1). When an equiaxed grain of an alloy of composition, 𝐶𝐶0 , is growing, it experiences an undercooling, Δ𝑇𝑇, which is the sum of a solute undercooling, Δ𝑇𝑇𝑐𝑐 , a thermal undercooling, Δ𝑇𝑇𝑡𝑡 , and a curvature undercooling, Δ𝑇𝑇𝑟𝑟 . In general, it can be said that, due to the large dendrite tip radius predicted by the stability criterion (Fig. 4.9 and Appendix 9), the curvature undercooling is small in comparison to the other contributions and can often be neglected to a first approximation under normal solidification conditions. At the tip, one can therefore write: 𝐶𝐶𝑙𝑙∗ (𝑟𝑟) ≅ 𝐶𝐶𝑙𝑙∗ . One can then evaluate the solutal and thermal undercoolings, Δ𝑇𝑇𝑐𝑐 and Δ𝑇𝑇𝑡𝑡 . From the definition of solutal supersaturation (Fig. A8.2): Ω=

𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 𝐶𝐶𝑙𝑙∗ 𝑝𝑝

𝐶𝐶𝑙𝑙∗ =

𝐶𝐶0 1 − Ω𝑝𝑝

Δ𝑇𝑇𝑐𝑐 = 𝑚𝑚(𝐶𝐶0 − 𝐶𝐶𝑙𝑙∗ ) = 𝑚𝑚𝐶𝐶0 �1 −

[A8.1]

1 � 1 − Ω𝑝𝑝

where 𝑝𝑝 = 1 − 𝑘𝑘. At low supersaturations, where Ω is much smaller than unity: Δ𝑇𝑇𝑐𝑐 ≅ −𝑚𝑚𝐶𝐶0 Ω𝑝𝑝

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[A8.2]

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Fundamentals of Solidification 5th Edition

Figure A8.1

Figure A8.2

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Foundations of Materials Science and Engineering Vol. 103

Substituting −𝑚𝑚𝐶𝐶0 𝑝𝑝 = Δ𝑇𝑇0 𝑘𝑘, one obtains:

Δ𝑇𝑇𝑐𝑐 ≅ ΩΔ𝑇𝑇0 𝑘𝑘

Ω≪1

279

[A8.3]

At small supersaturations, the solutal supersaturation, Ω, can be defined as the ratio of two temperature differences: Ω=

Δ𝑇𝑇𝑐𝑐 Δ𝑇𝑇0 𝑘𝑘

[A8.4]

(When, at high velocity, Ω approaches unity, the approximation to obtain Eq. A8.3 from Eq. A8.2 is no longer valid: the reduced undercooling Δ𝑇𝑇0 𝑘𝑘 in Eq. A8.4 must be replaced by Δ𝑇𝑇0). The thermal supersaturation, Ω𝑡𝑡 , can be similarly defined as the ratio of the thermal undercooling to the unit undercooling (Δℎ𝑓𝑓 /𝑐𝑐): Ω𝑡𝑡 =

Δ𝑇𝑇𝑡𝑡 Δℎ𝑓𝑓 /𝑐𝑐

[A8.5]

from which is obtained an equation that is analogous to Eq. A8.3: Δ𝑇𝑇𝑡𝑡 = Ω𝑡𝑡

Δℎ𝑓𝑓 𝑐𝑐

[A8.6]

HEMISPHERICAL NEEDLE APPROXIMATION

A cylinder with a hemispherical tip, growing along its axis, is the simplest approximation which can be made to the problem of dendrite tip growth (Fisher, 1966). It permits a rapid assimilation of the physical factors which are important in dendrite growth. The cross-section of the cylinder, 𝐴𝐴 = π𝑅𝑅 2 , determines the volume which grows within a given time, 𝑑𝑑𝑑𝑑, and which is responsible for the rejection of solute (Fig. A8.3(a)). The surface area of the hemispherical cap, 𝐴𝐴ℎ = 2π𝑅𝑅 2 , determines the amount of radial solute diffusion. Thus a flux due to solute rejection, 𝐽𝐽1 , and one due to diffusion in the liquid ahead of the tip, 𝐽𝐽2 , can be identifed: 𝐽𝐽1 = 𝐴𝐴𝐴𝐴(𝐶𝐶𝑙𝑙∗ − 𝐶𝐶𝑠𝑠∗ )

[A8.7]

𝑑𝑑𝑑𝑑 𝐽𝐽2 = −𝐷𝐷𝐴𝐴ℎ � � 𝑑𝑑𝑑𝑑 𝑟𝑟=𝑅𝑅

[A8.8]

Figure A8.3

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280

Fundamentals of Solidification 5th Edition

Under steady-state conditions the fluxes must be equal, leading to the relationship: 𝑑𝑑𝑑𝑑 𝑉𝑉𝐶𝐶𝑙𝑙∗ (1 − 𝑘𝑘) = −2𝐷𝐷 � � 𝑑𝑑𝑑𝑑 𝑅𝑅

[A8.9]

The concentration gradient at the tip can be approximated by the value which was found for a growing sphere (Eq. A2.26) *: 𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 𝑑𝑑𝑑𝑑 � =− 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑅𝑅 The diffusion equation therefore reduces to: 𝑉𝑉𝑉𝑉 𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 = 2𝐷𝐷 𝐶𝐶𝑙𝑙∗ (1 − 𝑘𝑘) �

[A8.10]

which is generally written in the abbreviated form: [A8.11]

𝑃𝑃𝑐𝑐 = Ω

Here, 𝑃𝑃𝑐𝑐 (= 𝑉𝑉𝑉𝑉/2𝐷𝐷) is the solute Péclet number (the ratio of a characteristic dimension, 𝑅𝑅, of the system to the solute diffusion distance, 2𝐷𝐷/𝑉𝑉). In Fig. 4.9 this expression is represented by the straight line which runs from the upper left to the lower right. In the case of thermal diffusion-limited dendrites, a similar flux balance to that above can be made and leads to the same relationship as that of Eq. A8.11, where the solute Péclet number is replaced by a thermal Péclet number: 𝑃𝑃𝑡𝑡 =

𝑉𝑉𝑉𝑉 2𝑎𝑎

[A8.12]

and the solute supersaturation is replaced by the thermal supersaturation, as defined by Eq. A8.5. The thermal case is therefore defined by: 𝑃𝑃𝑡𝑡 = Ω𝑡𝑡

[A8.13]

PARABOLOID OF REVOLUTION

As originally proposed by Papapetrou (1935), the dendrite tip has a parabolic shape. Ivantsov (1947) was the first to develop a mathematical analysis for this shape, and his analysis has since been generalised by Horvay and Cahn (1961). They found that: Ω = I( 𝑃𝑃)

[A8.14]

I(𝑃𝑃) = 𝑃𝑃 exp(𝑃𝑃) E1 (𝑃𝑃)

[A8.15]

I ′(𝑃𝑃) = (𝜋𝜋𝜋𝜋)1/2 exp(𝑃𝑃) erfc�𝑃𝑃1/2 �

[A8.16]

where the Ivantsov function for a paraboloid of revolution (needle crystal, 3D dendrite, Fig. A8.3(b)) is given by:

and for a parabolic cylinder (plate, 2D dendrite, Fig. A8.3(c)) is given by:

Here, E1 is the exponential integral function defined by: This case corresponds to the translation of a hemisphere with constant radius, different from the non-steady growth of a sphere. *

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Foundations of Materials Science and Engineering Vol. 103

E1 (𝑃𝑃) = �

∞

𝑃𝑃

exp(−𝑧𝑧) 𝑑𝑑𝑑𝑑 = − Ei(−𝑃𝑃) 𝑧𝑧

281

[A8.17]

Its value (Fig. A1.3) can be determined from the series (Abramowitz and Stegun, 1965): ∞

E1 (𝑃𝑃) = −0.5772157 − ln(𝑃𝑃) − � ≅ −0.577 − ln(𝑃𝑃) +

𝑛𝑛=1

4𝑃𝑃 𝑃𝑃 + 4

(−1)𝑛𝑛 𝑃𝑃𝑛𝑛 ≅ 𝑛𝑛 ⋅ 𝑛𝑛!

[A8.18]

for intermediate values of 𝑃𝑃. For numerical calculations, a good approximation to E1 (𝑃𝑃) is, for 0 ≤ 𝑃𝑃 ≤ 1: [A8.19]

E1 (𝑃𝑃) = 𝑎𝑎0 + 𝑎𝑎1 𝑃𝑃 + 𝑎𝑎2 𝑃𝑃2 + 𝑎𝑎3 𝑃𝑃3 + 𝑎𝑎4 𝑃𝑃4 + 𝑎𝑎5 𝑃𝑃5 − ln(𝑃𝑃)

where

𝑎𝑎0 = −0.57721566 𝑎𝑎2 = −0.24991055 𝑎𝑎4 = −0.00976004

and for 1 ≤ 𝑃𝑃 ≤ ∞

I(𝑃𝑃) = 𝑃𝑃 exp(𝑃𝑃) E1 (𝑃𝑃) =

where

𝑎𝑎1 = −0.99999193 𝑎𝑎3 = −0.05519968 𝑎𝑎5 = −0.00107857

𝑃𝑃4 + 𝑎𝑎1 𝑃𝑃3 + 𝑎𝑎2 𝑃𝑃2 + 𝑎𝑎3 𝑃𝑃 + 𝑎𝑎4 𝑃𝑃4 + 𝑏𝑏1 𝑃𝑃3 + 𝑏𝑏2 𝑃𝑃2 + 𝑏𝑏3 𝑃𝑃 + 𝑏𝑏4

𝑎𝑎1 = 8.5733287401 𝑎𝑎2 = 18.0590169730 𝑎𝑎3 = 8.6347608925 𝑎𝑎4 = 0.2677737343

[A8.20]

𝑏𝑏1 = 9.5733223454 𝑏𝑏2 = 25.6329561486 𝑏𝑏3 = 21.0996530827 𝑏𝑏4 = 3.9584969228

The Ivantsov function, I(𝑃𝑃), can also be written as a continued fraction: 𝑃𝑃 I(𝑃𝑃) = 1 𝑃𝑃 + 1 1+ 2 𝑃𝑃 + 2 1 + 𝑃𝑃+. . .

[A8.21]

Truncating the continued fraction at the zeroth, first, and higher terms will lead to various approximations: I0 = 𝑃𝑃

I1 =

I∞ = I(𝑃𝑃) = 𝑃𝑃 exp(𝑃𝑃) E1(𝑃𝑃)

𝑃𝑃 𝑃𝑃 + 1

𝐼𝐼2 =

2𝑃𝑃 2𝑃𝑃 + 1

These approximations are shown graphically in Fig. A8.4. Substituting the zeroth approximation into Eq. A8.14 gives the solution obtained previously for the case of a hemispherical tip (Eq. A8.11). It is interesting furthermore to note that the modification of Zener's analysis, made by Hillert (1957, 1975) for high supersaturations, leads to:

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Fundamentals of Solidification 5th Edition

𝑉𝑉𝑉𝑉 𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 Ω = = ∗ 𝐷𝐷 𝐶𝐶0 − 𝐶𝐶𝑠𝑠 1 − Ω

[A8.22]

which is equivalent to 2𝑃𝑃 = Ω/(1 − Ω) or Ω = 2𝑃𝑃/(2𝑃𝑃 + 1) (see also Eq. A2.36). This is the second approximation which arises from the continued fraction representation when it is substituted into Eq. A8.14. The paraboloid of revolution, for which the solution is given by Eq. A8.15, represents an isothermal/isoconcentrate dendrite, and closely approximates the experimentally observed form. From Fig. A8.4, it can be seen that I1 is a much better approximation than is I0 because the latter cannot be used when 𝑃𝑃 is greater than unity. As the series (Eq. A8.21) converges very slowly, it is usually preferable to use the polynomial expressions (Eqs A8.19 and A8.20). From Eq. A8.22 one sees that another relationship can be obtained for the inverse Ivantsov solution (Pelcé and Clavin,1987). This is a good approximation to Eq. A8.15 (Fig. A8.4): 𝑃𝑃 =

Ω − ln( Ω)

[A8.23]

Another simple but quite accurate approximation to the Ivantsov function for small Péclet numbers (Dantzig and Rappaz, 2016, Fig. 8.24 of this reference) is: [A8.24]

𝐼𝐼(𝑃𝑃) ≅ 1.5 𝑃𝑃0.8

More exact, non-isothermal, solutions have been developed in two articles by Trivedi (1970).

Figure A8.4

Bibliography for Further Reading  M.Abramowitz, I.A.Stegun (Eds), Handbook of Mathematical Functions, Dover, New York, 1965.  J.A.Dantzig, M.Rappaz, Solidification, 2nd edition, EPFL-Press, Lausanne, Switzerland, 2016.  J.C.Fisher, referred to by B.Chalmers in Principles of Solidification, Wiley, New York, 1966, p.105.  M.Hillert, The role of interfacial energy during solid-state phase transformations, Jernkontorets Annaler, 141 (1957) 757.  M.Hillert, Diffusion and interface control of reactions in alloys, Metallurgical Transactions A, 6 (1975) 5.  G.Horvay, J.W.Cahn, Dendritic and spheroidal growth, Acta Metallurgica, 9 (1961) 695. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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 G.P.Ivantsov, Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in an undercooled melt, Doklady Akademii Nauk SSSR, 58 (1947) 567.  G.P.Ivantsov, About the growth of spherical and needle crystals in a binary melt. Dokl. Akad. Nauk SSSR, 83 (1952) 573.  A.Papapetrou, Untersuchungen über dendritisches Wachstum von Kristallen, Zeitschrift für Kristallographie, 92 (1935) 89.  P.Pelcé, P.Clavin, The stability of curved fronts, Europhysics Letters, 3 (1987) 907.  R.Trivedi, Growth of dendritic needles from a supercooled melt, Acta Metallurgica, 18 (1970) 287.  R.Trivedi, The role of interfacial free energy and interface kinetics during the growth of precipitate plates and needles, Metallurgical Transactions, 1 (1970) 921.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 285-295 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 9 DENDRITE TIP RADIUS AND SPACING GROWTH AT THE EXTREMUM The relationships which define solutal (or thermal) diffusion at the hemispherical tip of a needle-like crystal (Appendix 8): Ω=𝑃

[A9.1]

or a paraboloid of revolution Ω = I( 𝑃)

[A9.2]

do not specify a unique functional dependence of the tip radius upon the growth conditions (supersaturation, undercooling). They relate only the product, 𝑉𝑅, to the supersaturation as shown in Fig. 4.9. An additional equation is therefore required in order to link the variables. For many years this was done by adding a capillarity term to the diffusion equation and then determining the extremum of the corresponding function. This approach will be illustrated first by means of the simple solution (Eq. A9.1): Ω=𝑃+

2𝑠 𝑅

[A9.3]

where 𝑠 is the capillarity length (for the solutal case at low Ω-values, 𝑠 = 𝑠𝑐 = Γ/Δ𝑇0 𝑘, and for the thermal case, 𝑠 = 𝑠𝑡 = Γ𝑐/Δℎ𝑓 ). Relationships such as Eq. A9.3 are presented for various models in Fig. A9.1, in terms of dimensionless variables, for free thermal dendrite growth and for a constant undercooling of Δ𝑇𝑡 = 0.05Δℎ𝑓 /𝑐 (Langer and Müller-Krumbhaar, 1977). These curves indicate that the addition of a capillarity term to Eq. A9.1 cuts off the diffusion solution at some tip radius (see also Fig. 4.9). The radius at which the cut-off occurs corresponds to the critical radius for nucleation. This can be easily verified by noting that this radius is the one using up the total supersaturation, indicated by Eq. A9.3, in satisfying the curvature. As a result, 𝑉 (and therefore 𝑃) must be equal to zero at that point. Therefore: 2𝑠 (𝑉 = 0) 𝑅○ where the critical radius, 𝑅 ○ , for dendrite tip growth is equal to the critical radius, 𝑟 ○ , for nucleation. Substituting Δ𝑇 for Ω from Eqs A8.3 or A8.6 leads to: 2𝑠 2𝛤 [A9.4] 𝑅○ = = = 𝑟○ Ω Δ𝑇 Ω=

Examination of Eq. A9.3 for the thermal or solutal case reveals that: 𝑉𝑅 Δℎ𝑓 2Γ ( )+ 2𝑎 𝑐 𝑅 𝑉𝑅 2Γ Δ𝑇𝑐 = (Δ𝑇0 𝑘) + 2𝐷 𝑅 Δ𝑇𝑡 =

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[A9.5] [A9.6]

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Figure A9.1

The extremum (maximum) value of 𝑉 for a fixed undercooling was once thought to define the tip radius. It is shown in Chap. 5 (Fig. 5.7) for eutectics that the maximum value of 𝑉 in an isothermal environment corresponds to a minimum in Δ𝑇 for constant-velocity growth. This applies also to dendrites. Minimising Δ𝑇 in Eqs A9.5 or A9.6 will therefore give the extremum radius and undercooling. This leads to: 1/2

𝑅𝑒𝑡

Γ𝑎𝑐 = 2( ) Δℎ𝑓

1 𝑉 1/2

ΓΔℎ𝑓 1/2 1/2 ) 𝑉 Δ𝑇𝑡 = 2 ( 𝑎𝑐 for thermal dendrites and to: 𝑅𝑒𝑐 = 2 (

Γ𝐷 1/2 1 ) Δ𝑇0 𝑘 𝑉 1/2

ΓΔ𝑇0 𝑘 1/2 1/2 ) 𝑉 Δ𝑇𝑐 = 2 ( 𝐷

[A9.7] [A9.8]

[A9.9] [A9.10]

for solutal dendrites when both are growing at the extremum. Equations A9.8 and A9.10 reflect the well-known square-root-of-velocity relationships observed for free dendrite growth in a constantsupersaturation environment. Other relationships can be found in the review paper by Glicksman et al. (1976). Instead of using the extremum criterion which resulted only in qualitative agreement between experiment and theory, Langer and Müller-Krumbhaar (1977, 1978) proposed a marginal stability criterion for the growth of the dendrite tip (see Sect. 4.4 for more details). This criterion works with the stability of a planar solid/liquid interface (Fig. 4.9). Some simple cases will be considered here in order to demonstrate the essential points of the more complete treatments. Despite the fact that marginal stability leads to agreement with experiment, it does not account of the anisotropy of the solid-liquid interfacial energy, which influences the tip radius, via the factor σ∗ , and dictates the preferential growth directions of dendrites. The more physical theory of microscopic solvability, however, is beyond the scope of the present book.

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GROWTH AT MARGINAL STABILITY Solution for Dendrite Growth in Undercooled Alloy Melts The essentials of such an analysis can be rapidly demonstrated in the case of a thermal dendrite, for example. Because the stability limit of the dendrite tip lies at much larger tip radii than the extremum value does, the capillarity term in Eq. A9.5 can be neglected and one instead obtains (for a hemispherical needle at low Péclet numbers): Δ𝑇𝑡 = 𝑃𝑡 𝜃𝑡

[A9.11]

where 𝜃𝑡 is the unit thermal undercooling (= Δℎ𝑓 /𝑐). With 𝐺𝑐 = 0 for a pure metal, the critical tip radius becomes: 1/2

Γ 𝑅=[ ∗ ] 𝜎 (−𝐺̄ )

[A9.12]

where 𝜎 ∗ = 1/4π2 (Eq. 4.7) and the mean gradient at the tip, 𝐺̄ , for the thermal (undercooled) dendrite can be obtained from 𝐺𝑠 = 0 (isothermal tip, Fig. A8.1) and 𝐺𝑙 = −2𝑃𝑡 𝜃𝑡 /𝑅 (see Exercise 4.6) via the relationship: 𝐺̄ =

𝜅𝑠 𝐺𝑠 + 𝜅𝑙 𝐺𝑙 𝜅𝑠 + 𝜅𝑙

[A9.13]

When 𝜅𝑠 = 𝜅𝑙 : 𝐺̄ = 𝐺𝑙 /2 = −

𝑃𝑡 𝜃𝑡 𝑅

[A9.14]

and from Eq. A9.12: 2 1/2 𝑎Γ 1/2 1 𝑅𝑡 = ( ∗ ) ( ) 𝜎 𝜃𝑡 𝑉 1/2

[A9.15]

Substituting Eq. A9.15 into Eq. A9.11 leads to: 1 1/2 𝜃𝑡 Γ 1/2 1/2 Δ𝑇𝑡 = ( ∗ ) ( ) 𝑉 2𝜎 𝑎

[A9.16]

Comparison of Eqs A9.15 and A9.16 with Eqs A9.7 and A9.8 shows that both the extremum and the stability arguments lead to qualitatively the same result (for low Péclet numbers), but with differing numerical constants. The tip radius for the marginally stable hemispherical dendrite is thus 4.4 times larger, and the undercooling is 2.2 times larger, than the corresponding extremum values. After considering this very simple case, a more realistic model will be developed, which is useful for the interpretation of both the low- and high-Péclet number cases, that is, dendrite growth under slow and rapid solidification conditions. In the case of alloy growth from the undercooled melt the coupled (solute and heat diffusion) transport problem has to be solved. The total undercooling, Δ𝑇 (= 𝑇𝑙 − 𝑇∞ ), is made up of three contributions when attachment kinetics are neglected (Fig. A8.1). These are: Δ𝑇𝑡 , the thermal undercooling (𝑇 ∗ − 𝑇∞ ); Δ𝑇𝑐 , the solutal undercooling (𝑇𝑙𝑟 − 𝑇 ∗ ) and Δ𝑇𝑟 , the curvature undercooling (𝑇𝑙 − 𝑇𝑙𝑟 ). Two of these undercoolings can be found from the Ivantsov solution for the case of a paraboloidal tip: Δ𝑇𝑡 = 𝜃𝑡 I( 𝑃𝑡 ) where:

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[A9.17]

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Δ𝑇𝑐 = 𝑚𝐶0 [1 − 𝐴( 𝑃𝑐 )]

[A9.18]

with: [A9.18a]

𝐴( 𝑃𝑐 ) = 𝐶𝑙∗ /𝐶0 = [1 − 𝑝 I( 𝑃𝑐 )]−1

The third contribution to Δ𝑇 is the curvature undercooling, which is determined by the relationship: Δ𝑇𝑟 = 2Γ/𝑅

[A9.19]

The sum of the three contributions to Δ𝑇 gives the total undercooling: Δ𝑇 = Δ𝑇𝑡 + Δ𝑇𝑐 + Δ𝑇𝑟

[A9.20]

It has been noted previously that this expression is not in itself sufficient to solve the problem. In order to find a unique solution to Eq. A9.20, one can advantageously use the marginal stability approach developed by Langer and Müller-Krumbhaar (1977). Use of the criterion gives results which are close to those found experimentally. See for example Fig. A9.1 where the open circle refers to marginal stability and the closed circle is the experimental result of Glicksman et al. (1976). This criterion supposes that the dendrite tip grows at a constant value of the stability constant, 𝜎 ∗ (≡ 𝛿𝑡 𝑠𝑡 /𝑅 2 : product of the diffusion length, 𝛿, and the capillarity length, 𝑠; each rendered dimensionless by dividing by the tip radius, 𝑅) which is also given by the relationship, 𝜎 = (𝜆𝑖 /2𝜋𝑅)2 . Taking the operating value of the stability constant, 𝜎 ∗ , to be about 1/4𝜋 2 , as has been observed experimentally, it follows that: 𝑅 = 𝜆𝑖

[A9.21]

where 𝜆𝑖 is the lower limiting wavelength for a perturbation which can grow at the solid/liquid interface (Fig. A7.2). Using this value for the wavelength at a planar interface (Eq. 3.22) as a zeroth approximation gives (Kurz and Fisher, 1981): Γ 1/2 𝜆𝑖 = 2π ( ) 𝜙

[A9.22]

Thus at low Péclet numbers: 1/2

Γ 𝑅=[ ∗ ] 𝜎 (𝑚𝐺𝑐 − 𝐺̄ )

(𝑃𝑐 , 𝑃𝑡 ≪ 1)

[A9.23]

where 𝜎 ∗ = 1/4π2 . The concentration gradient ahead of an advancing dendrite can be found from a simple flux balance, 𝑉𝐶𝑙∗ (1 − 𝑘) = −𝐷𝐺𝑐 : 𝐺𝑐 = −

2𝑃𝑐 𝐶𝑙∗ 𝑝 𝑅

[A9.24]

Substituting for 𝐶𝑙∗ from Eq. A8.1 and using Ivantsov's solution gives: 𝐺𝑐 = −2𝑃𝑐 𝑝𝐶0 𝐴(𝑃𝑐 )/𝑅

[A9.25]

In a similar manner, the temperature gradient, 𝐺𝑙 , in the liquid ahead of a steadily growing dendrite is found to be:

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Foundations of Materials Science and Engineering Vol. 103

𝐺𝑙 =

−2𝑃𝑡 Δℎ𝑓 𝑐𝑅

289

[A9.26]

In contrast to the solute diffusion field, which is evaluated only for the liquid phase, the temperature field has to be calculated for both phases, because of their similar thermal diffusivities. One therefore has to introduce a conductivity-weighted average temperature gradient. Assuming that 𝜅𝑠 = 𝜅𝑙 and that the Ivantsov dendrite is isothermal, one finds that 𝐺̄ at the tip is equal to 𝐺𝑙 /2 (Eq. A9.14). It is this value which has to be substituted into Eq. A9.23. Substituting Eq. A9.23 into Eq. A9.19 then yields an expression for the tip radius: 𝑅=

1 Γ ∗ 𝜎 𝜃𝑡 𝑃𝑡 − 2𝑃𝑐 𝑚𝐶0 𝑝 𝐴( 𝑃𝑐 )

[A9.27]

where 𝐴(𝑃𝑐 ) = 1/[1 − (1 − 𝑘)I(𝑃𝑐 )]. The capillarity undercooling at the tip is therefore: Δ𝑇𝑟 = 2𝜎 ∗ [𝜃𝑡 𝑃𝑡 − 2𝑃𝑐 𝑚𝐶0 𝑝 𝐴( 𝑃𝑐 )]

[A9.28]

The solutal Péclet number is simply related to the thermal Péclet number by 𝑃𝑐 = 𝑃𝑡 (𝑎/𝐷). The only unknown quantity in Eqs A9.17 to A9.20 is therefore the product, 𝑉𝑅 (or the Péclet number). From the definition of 𝑃𝑡 , for example, one can finally obtain the growth rate: 𝑉=

2𝑎𝑃𝑡 𝑅

[A9.29]

At high Péclet numbers, this solution has to be generalised by using a modified stability criterion (Eq A7.36). One can thus write, instead of Eq. A9.23 (Trivedi et al., 1987; Lipton, Kurz et al., 1987): 1/2 Γ 𝑅=[ ∗ ] 𝜎 (𝑚𝐺𝑐∗ − 𝐺 ∗ )

[A9.30]

where 𝜎 ∗ is again approximately equal to 1/4π2 and 𝐺𝑐∗ and 𝐺 ∗ are the effective concentration and temperature gradients as defined in Eq. 7.10. Note that because the temperature gradient in the solid at the tip, 𝐺𝑠 , is zero, the product, 𝜅̄ 𝑠 𝐺𝑠 𝜉𝑠 , is equal to zero in Eq. A7.29. From now on, one can therefore set 𝜉𝑙 = 𝜉𝑡 (for thermal diffusion in undercooled melts). Equation A9.27 then becomes, in its general form: 𝑅=

1 Γ ∗ 𝜎 𝜃𝑡 𝑃𝑡 𝜉𝑡 + 2𝑃𝑐 𝜃𝑐 𝜉𝑐

[A9.31]

where 𝜃𝑐 = Δ𝑇0 𝑘𝐴(𝑃𝑐 ) and 𝜉𝑡 = 𝜉𝑙 and 𝜉𝑐 are defined by Eq. A7.36. Substituting Eq. A9.31 into the curvature undercooling term (Eq. A9.19) and using the other undercooling relationships (Eqs A9.17 and A9.18) in combination with Eq. A9.20 gives the general solution for slow or rapid dendrite growth in undercooled alloy melts. Lee and Suzuki (1999) compared the Ivantsov-Marginal-Stability approach (Trivedi et al., 1987) with 2D phase-field computations and found reasonable agreement between both theories. The main difference came from the unknown of σ∗ value for delta-iron dendrites in an undercooled Fe0.15wt% C alloy. Figure A9.2 shows the variation in the dimensionless growth rate, 𝑉̄ = 𝑉𝑠𝑡 /2𝑎𝑙 (a), and dimensionless tip radius, 𝑅̄ = 𝑅/𝑠𝑡 (b), with dimensionless concentration 𝐶̄0 = 𝐶0 |𝑚|𝜃𝑡 for succinonitrile-acetone mixtures at an undercooling of 0.5K (Lipton, Glicksman et al., 1987). The red curve gives the predictions of the present model and the broken line gives those of a more complicated model due to Karma and Langer (1984). The points are experimental results from Chopra (1984). For

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a given (small) Δ𝑇, the solution of these equations reveals a maximum in 𝑉 and a minimum in 𝑅 occurring with increasing solute content in a given alloy system. For a detailed discussion of the behaviour of this model for small undercoolings see Lipton, Glicksman et al. (1987) and, for rapid solidification conditions (large undercoolings), see Trivedi et al. (1987) and Kurz et al. (1986, 1988). More recent results on this topic can be found in Melendez and Beckerman (2012).

Figure A9.2

Constrained (Alloy) Dendrite Growth During the directional growth of dendrites, the heat generated at the growing interface does not depend upon the tip radius. The latent heat flows into the solid, due to the imposed temperature gradient. The moving isotherms force the tips to grow at a given rate into the liquid. The resultant growth behaviour (e.g. the tip undercooling) is therefore determined by the solute flux at the tip. In order to treat this case, one again starts with Eq. A9.20 but neglects the thermal contribution to the tip diffusion fields (Δ𝑇𝑡 = 0). For the transport solution one thus has: Δ𝑇 = Δ𝑇𝑐 + Δ𝑇𝑟

(𝐺 > 0)

[A9.32]

The second equation is deduced from the general stability relationship (Eq. A9.30) by using the appropriate temperature gradient. Assuming that, at the very tip of the dendrite, the imposed temperature gradient is the same in both liquid and solid (𝐺 = 𝐺𝑙 = 𝐺𝑠 ) and that the thermal

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conductivities of the solid and the liquid are equal (𝜅 = 𝜅𝑠 = 𝜅𝑙 ), one obtains for the effective temperature gradient: 𝐺 ∗ = (𝐺/2)(𝜉𝑙 + 𝜉𝑠 )

[A9.33]

From Eq. A7.36, one sees that 𝜉𝑙 + 𝜉𝑠 is equal to 2 and the effective gradient is thus equal to the temperature gradient imposed by the furnace for example. That is: 𝐺∗ = 𝐺

[A9.34]

The case of columnar dendritic growth is therefore simpler than that of growth from an undercooled melt. Using the stability expression finally gives: 1/2 Γ 𝑅=[ ∗ ] 𝜎 (𝑚𝐺𝑐 𝜉𝑐 − 𝐺)

[A9.35]

Evaluating the tip concentration gradient (Eq. A1.25), and replacing it in Eq. A9.35 together with 𝑅 = 2𝐷𝑃𝑐 /𝑉, one finally obtains the result: 𝑉 2 𝐴′ + 𝑉𝐵 ′ + 𝐺 = 0

[A9.36]

where 𝐴′ = π2 𝛤/(𝑃𝑐2 𝐷2 ) and 𝐵 ′ = 𝜃𝑐 𝜉𝑐 /𝐷. This corresponds to Eq. 7.15. Since the temperature gradient has little effect upon the 𝑉 − 𝑅 relationship of Eq. A9.36 at high growth rates it can be neglected. This further simplifies the relationship to: 𝑅2𝑉 =

𝐷Γ

[A9.37]

𝜎 ∗ 𝜃𝑐 𝜉𝑐

Note that, at small 𝑃𝑐 values, 𝜉𝑐 tends towards unity, I(𝑃𝑐 ) tends towards zero and 𝜃𝑐 = Δ𝑇0 𝑘/[1 − 𝑝I(𝑃𝑐 )] tends towards Δ𝑇0 𝑘. Therefore, setting σ∗ = 1/4π2 : 𝑅 2 V = 4π2 DΓ/Δ𝑇0 k

(𝑃𝑐 ≪ 1)

[A9.38]

which is exactly the same as Eq. 4.16. The undercooling can be obtained from Eq. 7.17 by using the rate-dependent distribution coefficient of Eq. 7.3 and the temperature-dependent liquid-state diffusion coefficient. Figures 7. 9 to 7.11 show some results derived by using Eq. A9.36. For more details see Kurz et al. (1986).

PRIMARY SPACING IN CONSTRAINED GROWTH It has been assumed in Kurz and Fisher (1981) that the shape of a fully developed dendrite, including the mean volume of its branches, can be approximated by an ellipsoid-of-revolution (Fig. 4.13). The radius of an ellipse is given by its semi-axes, 𝑎 and 𝑏: 𝑏2 𝑅= 𝑎

[A9.39]

In the case of a hexagonal array, 𝑏 = 𝜆1 /√3 and 𝑎 = Δ𝑇 ′ /𝐺, where Δ𝑇 ′ is the non-equilibrium solidification range. Therefore: 1/2

3Δ𝑇 ′ 𝑅 𝜆1 = ( ) 𝐺

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[A9.40]

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Fundamentals of Solidification 5th Edition

Substituting Eq. A9.38 into Eq. A9.40 and replacing, to a first approximation, T ' by T0 gives a relationship for the primary trunk spacing: 𝐷ΓΔ𝑇0 1/4 −1/4 −1/2 ) 𝑉 𝜆1 = 4.3 ( 𝐺 𝑘

[A9.41]

This relationship is very similar to those derived by Hunt (1979) and by Trivedi (1984). The latter extended Hunt's treatment by using the marginally stable tip radius for low Péclet numbers. Hunt and Trivedi consider the flux balance for a small interdendritic volume element (Fig. 6.6), where the cell or dendrite axis is the 𝑧-axis, and 𝑟 is the radial distance. Considering that diffusion of solute species occurs only in the liquid along the longitudinal axis, an average solute balance at the scale of a small volume element gives: 𝜕 𝜕 ∂𝐶𝑙 (𝑓𝑙 𝐶𝑙 + 𝑓𝑠 𝐶𝑠 ) − (𝑓𝑙 𝐷 )=0 𝜕𝑡 𝜕𝑧 ∂𝑧 Neglecting the variation of the solid composition (Scheil-Gulliver) and since 𝑓𝑠 + 𝑓𝑙 = 1, this finally gives: 𝑓𝑙

𝜕𝐶𝑙 𝜕𝑓𝑠 𝜕 𝜕𝐶𝑙 = 𝐶𝑙 (1 − 𝑘) + [𝑓𝑙 𝐷 ] 𝜕𝑡 𝜕𝑡 𝜕𝑧 𝜕𝑧

[A9.42]

Setting 𝐺 = 𝑚(∂𝐶𝑙 / ∂𝑧), and assuming the existence of a constant interdendritic concentration gradient (∂2 𝐶𝑙 / ∂𝑧 2 → 0) leads, since ∂𝑓𝑙 / ∂𝑧 = −(1/𝑉)(∂𝑓𝑙 / ∂𝑡) under steady-state conditions, to: 𝑓𝑙 𝜕𝐶𝑙 = [𝐶𝑙 (𝑘 − 1) − (𝐷𝐺/𝑚𝑉)]𝜕𝑓𝑙

[A9.43]

Integrating from 𝑓𝑙 = 1 to 𝑓𝑙 and from 𝐶𝑙 = 𝐶𝑙∗ to 𝐶𝑙 gives: 𝑓𝑙 (k − 1) =

𝐶𝑙 (k − 1) − (𝐷𝐺/𝑚𝑉) 𝐶𝑙∗ (𝑘 − 1) − (𝐷𝐺/𝑚𝑉)

[A9.44]

For a cylindrical geometry one has 𝑓𝑙 = [1 − 𝑟 2 /(𝜆1 /2)2 ]. Substituting this expression for 𝑓𝑙 , and fitting a spherical tip to the 'Scheil-cell', finally leads to (Hunt, 1979): 𝑅=

−𝐺𝜆12 5.66[𝑚𝐶𝑙∗ (1 − 𝑘) + (𝐷𝐺/𝑉)]

[A9.45]

In Sect. 4.5, details of the path from Eq. A9.45 to the final equation for 𝜆1 can be found. Using for the stability coefficient 𝜎 ∗ = 1/4π2 , the relationship for diffusional columnar solidification becomes: 1/4

𝜆1 = 6[Γ𝑘𝐷]1/4 Δ𝑇0

𝑉 −1/4 𝐺 −1/2

[A9.46]

In summary, it should be pointed out that none of these models represent the mechanisms of the primary spacing selection as described by Esaka et al. (1988). However, it was found in phasefield modelling of large columnar dendrite arrays with a petaflop’s computer (Takaki et al., 2016) that the mean primary spacing comes close to Trivedi’s solution.

SECONDARY SPACING IN CONSTRAINED OR UNCONSTRAINED GROWTH To simplify the model, it is assumed that only two dendrite arms of differing diameters need to be considered. In reality, a distribution of arms having various thicknesses will exist. Following Kattamis and Flemings (1965) and Feurer and Wunderlin (1977), the situation illustrated in Fig. A9.3 will now be analysed in a very approximate manner. (More detailed

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information can be found in the literature, e.g. Sun et al. (2018), Neumann-Heyme et al. (2018)). Two arms of radius, 𝑅 and 𝑟, are placed in a locally isothermal melt. Because local equilibrium will be established very rapidly at the interface between the solid and the liquid, the concentration along the surface of the cylindrical arms will differ, with the thinner arms being situated in liquid of lower solute concentration. That is: 𝑇 ′ = 𝑇𝑓 + 𝑚𝐶𝑙𝑅 −

Γ 𝑅

𝑇 ′ = 𝑇𝑓 + 𝑚𝐶𝑙𝑟 −

Γ 𝑟

For an isothermal system one therefore obtains: 1 1 𝑚(𝐶𝑙𝑅 − 𝐶𝑙𝑟 ) = Γ ( − ) 𝑅 𝑟

[A9.47]

Solute will diffuse along the concentration gradient, from thick to thin arms, while the solvent will diffuse from thin to thick arms. The thin arms will therefore tend to dissolve while the thicker arms will tend to thicken. It is assumed for simplicity that the concentration gradient between two arms is constant and that the diffusion is unidirectional. If 𝑅 is now assumed to be much greater than 𝑟, 𝑑𝑅/𝑑𝑡 can be neglected with respect to 𝑑𝑟/𝑑𝑡, and the two fluxes existing between the arms are: 𝐽=𝐷

(𝐶𝑙𝑅 − 𝐶𝑙𝑟 ) 𝑑

𝐽 = −𝐶𝑙𝑟 (1 − 𝑘)

[A9.48]

𝑑𝑟 𝑑𝑡

[A9.49]

Combining Eqs A9.47 to A9.49: 𝑑𝑟 Γ𝐷 1 1 ( − ) = 𝑟 𝑑𝑡 𝑚𝐶𝑙 (1 − 𝑘)𝑑 𝑟 𝑅

[A9.50]

For small compositional differences in the liquid between the arms, 𝐶𝑙𝑟 is approximately equal to 𝐶𝑙 , the interdendritic concentration due to segregation. It is further assumed that 𝑑 is approximately equal to the arm spacing, leading to:

Figure A9.3

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Fundamentals of Solidification 5th Edition

𝑑𝑟 Γ𝐷 1 1 = o( − ) 𝑑𝑡 𝑚𝐶𝑙 (1 − 𝑘)𝜆2 𝑟 𝑅

[A9.51]

where 𝜆o2 is the arm spacing which existed before ripening. All the parameters, except for 𝐶𝑙 on the RHS of Eq. A9.51, are assumed to be constant. It is furthermore assumed that 𝑅/𝜆o2 and 𝑟0 /𝑅 are constant and that the interdendritic liquid concentration is a linear function of time, starting with the alloy concentration, 𝐶0 , and ending (due to segregation) at the composition, 𝐶𝑙𝑚 : 𝐶𝑙 = 𝐶0 + (𝐶𝑙𝑚 − 𝐶0 )

𝑡 𝑡𝑓

[A9.52]

where t is the time which has elapsed since the start of solidification, and 𝑡𝑓 is the local solidification time. If 𝐶𝑙𝑚 = 𝐶𝑒 , 𝑡𝑓 is approximately equal to (𝑇𝑙 − 𝑇𝑒 )/𝑇̇. Rearranging Eq. A9.51 and integrating from 𝑡 = 0 to 𝑡 = 𝑡𝑓 and from 𝑟 = 𝑟0 to 𝑟 = 0 gives: 𝜆o2 𝑅2 [

𝑟0 𝑟0 + ln( 1 − )] = 𝑀𝑡𝑓 𝑅 𝑅

[A9.53]

where: 𝑀=

−Γ𝐷 ln( 𝐶𝑙𝑚 /𝐶0 ) 𝑚(1 − 𝑘)(𝐶𝑙𝑚 − 𝐶0 )

[A9.54]

Feurer and Wunderlin (1977) then assume that 𝑅/𝜆2 ≅ 0.5 and that 𝑟0 /𝑅 ≅ 0.5, which gives 𝑀𝑡𝑓 = 0.1(𝜆𝑜2 )3 . They suppose moreover that, when the arms have melted, 𝜆2 = 2𝜆o2 and therefore: 𝜆2 = 5.5(𝑀𝑡𝑓 )1/3

[A9.55]

Due to the extreme simplification of the ripening phenomena as described here, the constant factor of 5.5 in Eq. A9.55 should not be accorded too much significance. This applies both to its value and to its constancy. The coarsening parameter, 𝑀, has nevertheless been shown to be of some use in estimating 𝜆2 -values in aluminium alloys (Feurer and Wunderlin, 1977). It should however be kept in mind that the coarsening described here relates to secondary-branch spacings next to the primary trunk. This is so because, when one measures 𝜆2 at points far from the trunk, one misses out the finer dissolving branches. Because the real situation is here again very complicated, numerical and in-situ X-ray studies of the behaviour of distributions of arms or particles as a function of time can be very useful in obtaining a better understanding of coarsening in general (Kammer and Vorhees, 2006; Wang et al., 2021). Bibliography for Further Reading ▪ M.Chopra, Ph.D. Thesis, Rensselear Polytechnic Institute, Troy, NY, 1984. ▪ H.Esaka, W.Kurz, R.Trivedi, in Solidification Processing 1987, The Institute of Metals, London, 1988, p.198. ▪ U.Feurer, R.Wunderlin, Einfluss der Zusammensetzung und der Erstarrungsbedingungen auf die Dendritenmorphologie binärer Al-Legierungen, Fachbericht Deutsche Gesellschaft für Materialkunde (DGM), 1977. ▪ M.E.Glicksman, R.J.Schaefer, J.D.Ayers, Dendritic growth - a test of theory, Metallurgical Transactions A, 7 (1976) 1747. ▪ S.C.Huang, M.E.Glicksman, Overview 12: Fundamentals of dendritic solidification—II. Development of sidebranch structure, Acta Metallurgica, 29 (1981) 717.

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▪ J.D.Hunt, Cellular and primary dendrite spacings, in Solidification and Casting of Metals, The Metals Society, Book 192, London, 1979, p.3. ▪ D.Kammer, P.W.Voorhees, The morphological evolution of dendritic microstructures during coarsening, Acta Materialia, 54 (2006) 1549. ▪ A.Karma, J.S.Langer, Impurity effects in dendritic solidification, Physical Review A, 30 (1984) 3147. ▪ T.Z.Kattamis, M.C.Flemings, Dendrite morphology, microsegregation and homogenization of lowalloy steel, Transactions of the Metallurgical Society of AIME, 233 (1965) 992. ▪ W.Kurz, D.J.Fisher, Dendrite growth at the limit of stability: tip radius and spacing, Acta Metallurgica, 29 (1981) 11. ▪ W.Kurz, B.Giovanola, R.Trivedi, Theory of microstructural development during rapid solidification, Acta Metallurgica, 34 (1986) 823. ▪ W.Kurz, B.Giovanola, R Trivedi, Microsegregation in rapidly solidified Ag − 15wt%Cu, Journal of Crystal Growth, 91 (1988) 123. ▪ J.S.Langer, H.Müller-Krumbhaar, Stability effects in dendritic crystal growth, Journal of Crystal Growth, 42 (1977) 11. ▪ J.S.Langer, H.Müller-Krumbhaar, Theory of dendritic growth, Acta Metallurgica, 26 (1978) 1681,1689 and 1697. ▪ J.S.Lee, T.Suzuki, Numerical simulation of isothermal dendritic growth by phase-field model, Iron and Steel Institute of Japan International, 39 (1999) 246. ▪ J.Lipton, M.E.Glicksman, W.Kurz, Equiaxed dendrite growth in alloys at small supercooling, Metallurgical Transactions A, 18 (1987) 341. ▪ J.Lipton, W.Kurz, R Trivedi, Rapid dendrite growth in undercooled alloys, Acta Metallurgica, 35 (1987) 957. ▪ A.J.Melendez, C.Beckermann, Measurements of dendrite tip growth and side branching in succinonitrile-acetone alloys, Journal of Crystal Growth, 340 (2012) 175. ▪ H.Neumann-Heyme, N.Shevchenko, Z.Lei, K.Eckert, O.Keplinger, J.Grenzer, C.Beckermann, S.Eckert, Coarsening evolution of dendritic sidearms: From synchrotron experiments to quantitative modeling, Acta Materialia, 146 (2018) 176. ▪ W.Oldfield, Computer model studies of dendritic growth, Materials Science and Engineering, 11 (1973) 211. ▪ Y.Sun, W.B.Andrews, K.Thornton, P.W.Voorhees, Self-Similarity and the Dynamics of Coarsening in Materials, Scientific Reports, 8 (2018) 17940. ▪ T.Takaki, S.Sakane, M.Ohno, Y.Shibuta, T.Shimokawabe, T.Aoki, Primary arm array during directional solidification of a single-crystal binary alloy: large-scale phase-field study, Acta Materialia, 118 (2016) 230. ▪ R.Trivedi, Interdendritic spacing: part II. A comparison of theory and experiment, Metallurgical Transactions A, 15 (1984) 977. ▪ R.Trivedi, J.Lipton, W.Kurz, Effect of growth rate dependent partition coefficient on the dendritic growth in undercooled melts, Acta Metallurgica, 35 (1987) 965. ▪ S.Wang, Z.Guo, J.Kang, M.Zou, X.Li, A.Zhang, W.Du, W.Zhang, T.L.Lee, S.Xiong, J.Mi, 3D Phase Field Modeling of Multi-Dendrites Evolution in Solidification and Validation by Synchrotron X-ray Tomography, Materials, 14 (2021) 520.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 297-307 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 10 EUTECTIC GROWTH LOW PÉCLET NUMBER SOLUTION The first part of this appendix follows the treatment of Jackson and Hunt (1966) but is a simplified version of that paper. Figure 5.2 shows the corresponding phase diagram and interface geometry. By using symmetry arguments (Appendix 2), the analysis of a solidifying lamellar (2D) eutectic interface can first be reduced to the consideration of just a pair of half-lamellae. No net mass transport can occur between this pair and another pair under steady-state conditions because this would cause changes in the morphology and thus violate the assumption of steady-state behaviour. The concentration gradient in the y-direction is thus equal to zero at the mid-point of each lamella (Fig. A10.1). The differential equation describing the solute distribution in the melt ahead of a steadily advancing interface is: 𝜕𝜕 2 𝐶𝐶 𝜕𝜕 2 𝐶𝐶 𝑉𝑉 𝜕𝜕𝜕𝜕 + + =0 𝜕𝜕𝑦𝑦 2 𝜕𝜕𝑧𝑧 2 𝐷𝐷 𝜕𝜕𝜕𝜕

[A2.6]

and it has been shown (Appendix 2) that the general solution which satisfies this equation consists of products of circular and exponential functions. It is logical to associate the exponential function with the 𝑧𝑧-variation of the solution because it is known that the boundary layer has this form for a singlephase interface. It is equally logical to associate the circular functions with the alternating pattern of eutectic phases. The distribution of component B can thus be deduced, merely by inspection, to be of the form: 𝐶𝐶 = 𝐶𝐶𝑒𝑒 + 𝐴𝐴exp �−

𝑉𝑉𝑉𝑉 � + 𝐵𝐵exp (−𝑏𝑏𝑏𝑏)cos (𝑔𝑔𝑔𝑔) 𝐷𝐷

[A10.1]

This approximate solution * can be seen to be made up of an exponential term reflecting the planarity of the interface as a whole, and another one reflecting the alternating pattern of the two phases. The cosine, rather than sine, function is chosen because the expression describing the gradient of the interface concentration in the y-direction must be able to take zero values at the origin of the region considered and at 𝜆𝜆/2 (Fig. A10.1). Another way of looking at the choice of the function in Eq. A10.1 is to imagine that it reflects the interface solute distributions depicted in Fig. 5.3. Thus Eq. A10.1, without the third term on the RHS and with a suitable choice of value for 𝐴𝐴, could describe the solute distribution ahead of either of the single phases (Fig. 5.3(a)). The re-introduction of the third term can then be looked upon as being the expected solution when the originally single-phase interface is 'perturbed' by the addition of the second phase (Fig. 5.3(b)).

The exact solute profile must be described by using a summed set of circular functions rather than by just one term. The present choice however simplifies the treatment without losing sight of the important physical phenomena.

*

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Fundamentals of Solidification 5th Edition

Figure A10.1

It is now necessary to determine the constants, 𝐴𝐴, 𝐵𝐵, 𝑏𝑏 and 𝑔𝑔. The value of 𝑔𝑔 can be immediately written down because, again due to symmetry considerations, the derivative of the solution with respect to 𝑦𝑦, ∂𝐶𝐶/ ∂𝑦𝑦, must be equal to zero when 𝑦𝑦 = 𝜆𝜆/2. Thus: 𝐶𝐶 = 𝐶𝐶𝑒𝑒 + 𝐴𝐴exp �−

𝑉𝑉𝑉𝑉 2π𝑦𝑦 � + 𝐵𝐵exp (−𝑏𝑏𝑏𝑏)cos � � 𝐷𝐷 𝜆𝜆

The derivation of 𝑏𝑏 has already been discussed (Eq. A2.10). It is equal to: 1/2

𝑉𝑉 𝑉𝑉 2 2π 2 𝑏𝑏 = + �� � + � � � 2𝐷𝐷 2𝐷𝐷 𝜆𝜆

[A10.2]

[A10.3]

This expression can be simplified on the basis of experimental observations. It is noted firstly that the term, 𝑉𝑉/2𝐷𝐷, is the inverse of the equivalent solute boundary layer of a planar interface. At low growth rates its value will be relatively small, whereas 2π/𝜆𝜆, is the wave-number of the eutectic and will be large due to the small values of λ usually encountered. Therefore: 4π2 𝑉𝑉 2 ≫ 𝜆𝜆2 4𝐷𝐷2

and Eq. A10.3 can be simplified to: 𝑏𝑏 ≅

2π 𝜆𝜆

(𝑃𝑃 ≪ 1)

by neglecting terms in 𝑉𝑉/2𝐷𝐷 (𝑃𝑃 = 𝑉𝑉𝑉𝑉/2𝐷𝐷). Equation A10.2 can now therefore be written: 𝐶𝐶 = 𝐶𝐶𝑒𝑒 + 𝐴𝐴exp �−

𝑉𝑉𝑉𝑉 2π𝑧𝑧 2π𝑦𝑦 � + 𝐵𝐵exp �− � cos � � 𝐷𝐷 𝜆𝜆 𝜆𝜆

[A10.4]

The values of the constants, 𝐴𝐴, 𝐵𝐵, can be found by forcing the above solution to satisfy the flux condition at the interface. The fluxes can be deduced from the phase diagram. The general flux boundary condition is:

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∂𝐶𝐶 𝑉𝑉(𝑘𝑘 − 1)𝐶𝐶 ∗ = 𝐷𝐷 � � ∂𝑧𝑧 𝑧𝑧=0

299

[A10.5]

but, because the variation of the interface concentration with respect to 𝐶𝐶𝑒𝑒 is relatively small in cases other than rapid solidification, it can be supposed that 𝐶𝐶 ∗ ≅ 𝐶𝐶𝑒𝑒 . The flux conditions for the two eutectic phases can thus be written in terms of the weight fraction of element 𝐵𝐵 as: α-phase: β-phase:

𝑉𝑉(𝑘𝑘𝛼𝛼 − 1)𝐶𝐶𝑒𝑒 = 𝐷𝐷 �

∂𝐶𝐶 � ∂𝑧𝑧 𝑧𝑧=0

−𝑉𝑉�𝑘𝑘𝛽𝛽 − 1�(1 − 𝐶𝐶𝑒𝑒 ) = 𝐷𝐷 �

[A10.6]

∂𝐶𝐶 � ∂𝑧𝑧 𝑧𝑧=0

[A10.7]

A negative sign appears in the case of the β-phase because the interface gradient of 𝐶𝐶𝐵𝐵 in the growth direction is expected to be positive. The concentration gradient at the solid/liquid interface can be found from Eq. A10.4: �

∂𝐶𝐶 𝑉𝑉 2π 2π𝑦𝑦 � � = − 𝐴𝐴 − 𝐵𝐵cos � ∂𝑧𝑧 𝑧𝑧=0 𝐷𝐷 𝜆𝜆 𝜆𝜆

[A10.8]

The α-phase extends from the origin to the point, 𝑓𝑓𝑓𝑓/2, where 𝑓𝑓 is the volume fraction of the α-phase (as defined in Jackson and Hunt, 1966). Using the method of 'weighted residuals' (Appendix 2), the values of 𝐴𝐴 and 𝐵𝐵 (the 'weights') can be found by satisfying the flux boundary conditions (Eqs A10.6 and A10.7) in an average fashion over the α/l and β/l interfaces, i.e.: 𝑓𝑓𝑓𝑓 2

𝑓𝑓𝑓𝑓 2

�   𝑉𝑉(𝑘𝑘𝛼𝛼 − 1)𝐶𝐶𝑒𝑒 𝑑𝑑𝑑𝑑 = �   𝐷𝐷 � 0

0

𝜆𝜆 2

∂𝐶𝐶 � 𝑑𝑑𝑑𝑑 ∂𝑧𝑧 𝑧𝑧=0 𝜆𝜆 2

[A10.9]

∂𝐶𝐶 �   − 𝑉𝑉�𝑘𝑘𝛽𝛽 − 1�(1 − 𝐶𝐶𝑒𝑒 )𝑑𝑑𝑑𝑑 = �   𝐷𝐷 � � 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓 ∂𝑧𝑧 𝑧𝑧=0

[A10.10]

𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 2𝐷𝐷sin (π𝑓𝑓)𝐵𝐵 = (1 − 𝑘𝑘𝛼𝛼 )𝐶𝐶𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓

[A10.11]

2

2

Inserting the value of (∂𝐶𝐶/ ∂𝑧𝑧)𝑧𝑧=0 from Eq. A10.8, and carrying out the integrations yields the simultaneous algebraic equations: (1 − 𝑓𝑓)𝑉𝑉𝑉𝑉𝑉𝑉 − 2𝐷𝐷sin (π𝑓𝑓)𝐵𝐵 = �𝑘𝑘𝛽𝛽 − 1�(1 − 𝐶𝐶𝑒𝑒 )(1 − 𝑓𝑓)𝑉𝑉𝑉𝑉

[A.10.12]

𝐴𝐴 = 𝑓𝑓𝐶𝐶 ′ − 𝐶𝐶𝛽𝛽

[A10.13]

and the values of A and B are then found to be:

𝐵𝐵 =

𝑓𝑓(1 − 𝑓𝑓)𝑉𝑉𝑉𝑉𝐶𝐶 ′ 2𝐷𝐷sin (π𝑓𝑓)

[A.10.14]

Here, 𝐶𝐶 ′ is the difference in composition between the ends of the eutectic tie-line and 𝐶𝐶𝛽𝛽 is the difference in composition between the eutectic and the maximum solid solubility in the β-phase (Fig. A10.2). Note that 𝐴𝐴 will be very small in general and will be equal to zero if the phases have the same density.

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Figure A10.2

A solution has thus been obtained which satisfies most of the boundary conditions. The final condition to be satisfied is the coupling condition, which relates the local melting point to the imposed temperature distribution. This has already been depicted in a qualitative fashion in Fig. 5.5. If account were to be taken of the detailed shape of the eutectic interface, the problem would rapidly become intractable with analytical methods because of the need to satisfy the coupling condition at each point †. The eutectic interface is instead assumed to be perfectly planar, in order to make a tractable solution possible, and the quantities involved are treated in an average fashion. The satisfaction of the coupling condition can then be achieved in three steps. These involve calculating the sum of the average solute undercooling and the average curvature undercooling for each phase and then equating the two total undercoolings. The average concentration differences of the liquid, Δ𝐶𝐶 = 𝐶𝐶 − 𝐶𝐶𝑒𝑒 , at the α/l and β/l interfaces, relative to the eutectic composition, will first be found. From Eq. A10.4, they can be shown to be, for 𝑧𝑧 = 0: 𝑓𝑓𝑓𝑓

2 2 2π𝑦𝑦 �� 𝑑𝑑𝑑𝑑 𝛥𝛥𝐶𝐶𝛼𝛼 = � �𝐴𝐴 + 𝐵𝐵 cos � 𝑓𝑓𝑓𝑓 0 𝜆𝜆 𝜆𝜆

2 2 2π𝑦𝑦 �� 𝑑𝑑𝑑𝑑 𝛥𝛥𝐶𝐶𝛽𝛽 = � �𝐴𝐴 + 𝐵𝐵 cos � (1 − 𝑓𝑓)𝜆𝜆 𝑓𝑓𝑓𝑓 𝜆𝜆

giving:

𝛥𝛥𝐶𝐶𝛼𝛼 = 𝐴𝐴 + 𝐵𝐵 𝛥𝛥𝐶𝐶𝛽𝛽 = 𝐴𝐴 − 𝐵𝐵

2

sin( π𝑓𝑓) π𝑓𝑓

sin( π𝑓𝑓) π(1 − 𝑓𝑓)

[A10.15] [A10.16]

Since 𝐴𝐴 is usually negligible and 𝐵𝐵 is positive, the average concentration difference in component B at the α/l interface is positive while that at the β/l interface is negative. The mean solute undercoolings, with respect to the eutectic temperature (Fig. 5.5), can be found by multiplying the above concentration differences by the absolute liquidus slopes, |mα | and �𝑚𝑚β �. The solute undercoolings are therefore:

† Eutectic solidification can be realistically modelled using the phase-field technique, providing 3 phase parameters, 𝜙𝜙𝛼𝛼 , 𝜙𝜙𝛽𝛽 and 𝜙𝜙𝑙𝑙 , are introduced with the condition 𝜙𝜙𝛼𝛼 + 𝜙𝜙𝛽𝛽 + 𝜙𝜙𝑙𝑙 = 1. For further details, see Folch and Plapp (2005). EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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Figure A10.3

Δ𝑇𝑇𝑐𝑐𝛼𝛼 = |𝑚𝑚𝛼𝛼 | �𝐴𝐴 + 𝐵𝐵 𝛽𝛽

sin (π𝑓𝑓) � 𝜋𝜋𝜋𝜋

Δ𝑇𝑇𝑐𝑐 = −�𝑚𝑚𝛽𝛽 � �𝐴𝐴 − 𝐵𝐵

sin (π𝑓𝑓) � 𝜋𝜋(1 − 𝑓𝑓)

[A10.17] [A10.18]

since both are positive. The second source of undercooling to be considered is the average undercooling due to the curvature of the interface. Note firstly that the average curvature of a line between two points which are a distance, 𝐿𝐿, apart can be found by integrating the general expression for the curvature (Appendix 3), where 𝑧𝑧’ and 𝑧𝑧’’ are the first and second derivatives, respectively, of the interface 𝑧𝑧(𝑦𝑦) shown in Fig. A10.3: Thus:

𝐾𝐾 =

𝑧𝑧 ″ (1 + 𝑧𝑧 ′ 2 )3/2

[A10.19]

1 𝐿𝐿 𝑧𝑧 ″ 𝐾𝐾 = � 𝑑𝑑𝑑𝑑 𝐿𝐿 0 (1 + 𝑧𝑧 ′ 2 )3/2

Making the successive substitutions, 𝑍𝑍 = 𝑧𝑧 ′ and tan (𝜃𝜃) = 𝑍𝑍 (Fig. A10.3) leads to: 1 𝐾𝐾 = [sin( 𝜃𝜃)]𝐿𝐿0 𝐿𝐿

Again using the above identities leads to: 1 � = sin [arctan(𝑧𝑧 ′ )]𝐿𝐿0 𝐾𝐾 𝐿𝐿

[A10.20]

The form of the eutectic interface is assumed to be as shown in Fig. A10.3. The average curvature of the β/l interface can thus be found by substituting the slopes at 𝑦𝑦 = 𝑓𝑓𝑓𝑓/2 and 𝜆𝜆/2 into Eq. A10.20: 𝐾𝐾𝛼𝛼 = 𝐾𝐾𝛽𝛽 =

2 sin( 𝜃𝜃𝛼𝛼 ) 𝑓𝑓𝑓𝑓

2 sin( 𝜃𝜃𝛽𝛽 ) (1 − 𝑓𝑓)𝜆𝜆

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The curvature has been defined to be positive if the solid projects into the melt. Mathematically speaking, it is actually negative. The curvature is defined differently here so as to give a positive undercooling (for a solid projection) when combined with the (positive) Gibbs-Thomson coefficient. By using the associated Gibbs-Thomson coefficients the curvature undercoolings thus become: Δ𝑇𝑇𝑟𝑟𝛼𝛼 = 𝛽𝛽

Δ𝑇𝑇𝑟𝑟 =

2Γ𝛼𝛼 sin( 𝜃𝜃𝛼𝛼 ) 𝑓𝑓𝑓𝑓

[A10.21]

2Γ𝛽𝛽 sin( 𝜃𝜃𝛽𝛽 ) (1 − 𝑓𝑓)𝜆𝜆

[A10.22]

Combining Eqs A10.17 or A10.18 with Eqs A10.21 or A10.22 gives the total undercooling for the two phases: Δ𝑇𝑇𝛼𝛼 = |𝑚𝑚𝛼𝛼 | �𝐴𝐴 + 𝐵𝐵

sin( π𝑓𝑓) 2Γ𝛼𝛼 sin( 𝜃𝜃𝛼𝛼 ) �+ π𝑓𝑓 𝑓𝑓𝑓𝑓

Δ𝑇𝑇𝛽𝛽 = −|𝑚𝑚𝛽𝛽 | �𝐴𝐴 − 𝐵𝐵

[A10.23]

2Γ𝛽𝛽 sin( 𝜃𝜃𝛽𝛽 ) sin( π𝑓𝑓) �+ π(1 − 𝑓𝑓) (1 − 𝑓𝑓)𝜆𝜆

[A10.24]

Multiplying Eq. A10.23 by �𝑚𝑚𝛽𝛽 � and Eq. A10.24 by |𝑚𝑚𝛼𝛼 |, and adding gives: �𝑚𝑚𝛽𝛽 �Δ𝑇𝑇𝛼𝛼 + |𝑚𝑚𝛼𝛼 |Δ𝑇𝑇𝛽𝛽 = |𝑚𝑚𝛼𝛼 |�𝑚𝑚𝛽𝛽 �𝐵𝐵 +

sin (π𝑓𝑓) + π𝑓𝑓(1 − 𝑓𝑓)

2�𝑚𝑚𝛽𝛽 �Γ𝛼𝛼 sin (𝜃𝜃𝛼𝛼 ) 2|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin �𝜃𝜃𝛽𝛽 � + (1 − 𝑓𝑓)𝜆𝜆 𝑓𝑓𝑓𝑓

[A10.25]

The coupling condition (Appendix 2) requires however that the α/l and β/l interfaces should lie on the same isotherm. Due to the very small distances between the phases (typically of the order of a few microns) and their high thermal conductivities, no substantial temperature difference can exist between the α and β phases. Therefore Δ𝑇𝑇𝛼𝛼 = Δ𝑇𝑇𝛽𝛽 = Δ𝑇𝑇 (Fig. 5.5): Δ𝑇𝑇 =

|𝑚𝑚𝛼𝛼 |�𝑚𝑚𝛽𝛽 �

|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 �

𝐵𝐵

sin (π𝑓𝑓) + 𝜋𝜋𝜋𝜋(1 − 𝑓𝑓) +

2�𝑚𝑚𝛽𝛽 �Γ𝛼𝛼 sin (𝜃𝜃𝛼𝛼 )

𝑓𝑓𝑓𝑓�|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 ��

Substituting for 𝐵𝐵 from Eq. A10.14 gives: 𝐶𝐶 ′ Δ𝑇𝑇 = 𝑉𝑉𝑉𝑉 + |𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 � 2π𝐷𝐷 |𝑚𝑚𝛼𝛼 |�𝑚𝑚𝛽𝛽 �

+

𝐾𝐾𝑟𝑟 𝜆𝜆

2|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin �𝜃𝜃𝛽𝛽 �

(1 − 𝑓𝑓)𝜆𝜆�|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 ��

[A10.26]

2(1 − 𝑓𝑓)�𝑚𝑚𝛽𝛽 �Γ𝛼𝛼 sin (𝜃𝜃𝛼𝛼 ) + 2𝑓𝑓|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin �𝜃𝜃𝛽𝛽 �

This can be written (see also Eq. 5.8):o Δ𝑇𝑇 = 𝐾𝐾𝑐𝑐 𝑉𝑉𝑉𝑉 +

+

𝑓𝑓(1 − 𝑓𝑓)𝜆𝜆�|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 ��

where the physical constants of the alloy are, for the diffusional part: 𝐾𝐾𝑐𝑐 =

|𝑚𝑚𝛼𝛼 ||𝑚𝑚𝛽𝛽 | 𝐶𝐶 ′ |𝑚𝑚𝛼𝛼 | + |𝑚𝑚𝛽𝛽 | 2𝜋𝜋𝜋𝜋

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[A10.27a]

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303

and for the capillarity contribution: 2(1 − 𝑓𝑓)�𝑚𝑚𝛽𝛽 �Γ𝛼𝛼 sin (𝜃𝜃𝛼𝛼 ) + 2𝑓𝑓|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin �𝜃𝜃𝛽𝛽 �

𝐾𝐾𝑟𝑟 =

𝑓𝑓(1 − 𝑓𝑓)�|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 ��

[A10.27b]

Assuming that the 𝜆𝜆-value chosen by the eutectic is the one which makes Δ𝑇𝑇 a minimum, i.e.: 𝑑𝑑(Δ𝑇𝑇) 𝐾𝐾𝑟𝑟 = 𝐾𝐾𝑐𝑐 𝑉𝑉 − 2 = 0 𝑑𝑑𝑑𝑑 𝜆𝜆

it is found that the final result of the simple eutectic growth model is: 𝜆𝜆2 𝑉𝑉 =

𝑓𝑓|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin�𝜃𝜃𝛽𝛽 � + (1 − 𝑓𝑓)�𝑚𝑚𝛽𝛽 �Γ𝛼𝛼 sin(𝜃𝜃𝛼𝛼 ) 4π𝐷𝐷 𝑓𝑓(1 − 𝑓𝑓)𝐶𝐶 ′ |𝑚𝑚𝛼𝛼 |�𝑚𝑚𝛽𝛽 �

[A10.28]

Equation A10.28 differs from the more exact solution only with respect to the first multiplying term on the RHS which 𝑖𝑖𝑖𝑖 2𝐷𝐷/𝐶𝐶’𝑃𝑃’ in Jackson and Hunt (1966). As mentioned in Chap. 5, the relationships for the extremum (Eqs 5.10 to 5.12) are: 𝜆𝜆2 𝑉𝑉 = 𝐾𝐾𝑟𝑟 /𝐾𝐾𝑐𝑐

Δ𝑇𝑇/√𝑉𝑉 = 2�𝐾𝐾𝑟𝑟 𝐾𝐾𝑐𝑐 Δ𝑇𝑇𝑇𝑇 = 2𝐾𝐾𝑟𝑟

The more exact derivation for the 𝐾𝐾-parameters by Jackson and Hunt (1966) for a lamellar eutectic in a compact form gives: the equivalent expression for f(1-f)/2π in the simple model is P’: 𝑚𝑚 ‾ 𝐶𝐶 ′ 𝑃𝑃′ 𝐷𝐷𝐷𝐷(1 − 𝑓𝑓)

𝐾𝐾𝑐𝑐 =

𝐾𝐾𝑟𝑟 = 2𝛿𝛿𝑚𝑚 ‾ �    𝑚𝑚 ‾ =

𝑖𝑖=𝛼𝛼,𝛽𝛽

|𝑚𝑚𝛼𝛼 |�𝑚𝑚𝛽𝛽 �

Γ𝑖𝑖 sin 𝜃𝜃𝑖𝑖 𝑚𝑚𝑖𝑖 𝑓𝑓𝑖𝑖

[A10.29]

|𝑚𝑚𝛼𝛼 | + �𝑚𝑚𝛽𝛽 �

where 𝛿𝛿 is a parameter characterising both eutectic morphologies ‡. For lamellae: 𝛿𝛿(𝐿𝐿) = 1 and the exact/approximate 𝑃𝑃′ (𝐿𝐿) functions are (Trivedi and Kurz, 1988): 𝑃𝑃

′ (𝐿𝐿)

∞

1 = �    � 3 3 � sin2 (𝑛𝑛π𝑓𝑓) 𝑛𝑛 π 𝑛𝑛=1

𝑃𝑃′ (𝐿𝐿) ≅ 0.335 [𝑓𝑓(1 − 𝑓𝑓)]1.65

[A10.30a]

[A10.30b]

while for fibres: 𝛿𝛿(𝐹𝐹) = 2�𝑓𝑓 (f is the volume fraction of the minor phase α) and the exact/approximate 𝑃𝑃′ (𝐹𝐹) functions are: ‡

The series P’ in Jackson and Hunt replaces 𝑓𝑓(1 − 𝑓𝑓)/2π of the simple model.

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∞

1 𝐽𝐽12 �𝛾𝛾𝑛𝑛 �𝑓𝑓� 𝑃𝑃 (𝐹𝐹) = �   � 3 � 2 𝐽𝐽0 (𝛾𝛾𝑛𝑛 ) 𝛾𝛾𝑛𝑛 ′

[A10.31a]

𝑛𝑛=1

𝑃𝑃′ (𝐹𝐹) ≅ 0.167 [𝑓𝑓(1 − 𝑓𝑓)]1.25 ,

[A10.31b]

0 < 𝑓𝑓 ≤ 0.3

where Jm are the Bessel functions of first kind order m, and 𝛾𝛾𝑛𝑛 are the roots of 𝐽𝐽1 (𝑥𝑥) = 0 (𝛾𝛾𝑛𝑛 ≅ 𝑛𝑛π). To help making quick calculations using Eqs A10.30 and A10.31, a few 𝑃𝑃’-values for lamellar and fibrous eutectics are given in Table A10.1. The 𝑃𝑃’(𝐿𝐿) values are symmetric around 𝑓𝑓 = 0.5 while the P’(F) values are asymmetric and show a maximum at 𝑓𝑓 = 0.17. 𝑓𝑓

𝑃𝑃′ (𝐿𝐿)

0.00

0.00000

0.06

0.00284

0.03 0.09 0.12 0.15

0.00091 0.00534 0.00819 0.01121

Table A10.1 Values of the 𝑷𝑷′ Functions for Lamellae (L) and Fibres (F) (J.-H. Model)

𝑃𝑃′ (𝐹𝐹)

𝑓𝑓

0.00000

0.18

0.03555

0.24

0.02848 0.03916 0.04103 0.04182

0.21 0.27 0.30 0.33

𝑃𝑃′ (𝐿𝐿)

𝑃𝑃′ (𝐹𝐹)

0.01734

0.04140

0.01429 0.02026 0.02301 0.02553 0.02777

𝑓𝑓

0.04188

0.36

0.04053

0.42

0.03935 0.03793 0.03633

0.39 0.45 0.48 0.50

𝑃𝑃′ (𝐿𝐿)

𝑃𝑃′ (𝐹𝐹)

0.03129

0.03272

0.02970 0.03252

0.03337 0.03383 0.03392

0.03458 0.03078 0.02878 0.02673 0.02536

EUTECTIC GROWTH AT HIGH RATES The main difference between the analyses of eutectic growth at high and low rates is that, in the former case, one cannot make some of the simplifying assumptions which were made earlier in this appendix. One can no longer assume that: (i) 4π2 /𝜆𝜆2 ≫ 𝑉𝑉 2 /4𝐷𝐷2 as 𝑉𝑉 is now very large, and (ii) the interface composition in the liquid is close to that of the eutectic. The interface gradients are therefore functions of the interface concentrations, and the exact solution of the problem becomes much more difficult. A solution has in fact been developed for lamellar eutectics by making the assumption that 𝑘𝑘𝛼𝛼 = 𝑘𝑘𝛽𝛽 (Trivedi et al.,1987). In the approximate derivation which follows, it is not necessary to make this simplification, but it will be retained anyway to facilitate comparison with the more exact result (Trivedi et al., 1987). Starting with Eq. A10.2, one again finds an expression for the concentration gradient in the liquid at the interface: ∂𝐶𝐶 𝑉𝑉 = − A − 𝑏𝑏𝑏𝑏cos(2π𝑦𝑦/𝜆𝜆) ∂𝑧𝑧 𝐷𝐷

[A10.32]

Note that one cannot assume here that 𝑏𝑏 = 2π/𝜆𝜆. One must later insert this expression for the gradient into Eq. A10.5 giving, since one has to assume that the liquid composition at the interface, 𝐶𝐶𝑙𝑙∗ , is not equal to 𝐶𝐶𝑒𝑒 : α-phase: β-phase:

𝑉𝑉(𝑘𝑘 − 1)𝐶𝐶𝑙𝑙∗ = 𝐷𝐷 �

∂𝐶𝐶 � ∂𝑧𝑧 𝑧𝑧=0

∂𝐶𝐶 −𝑉𝑉(𝑘𝑘 − 1)(1 − 𝐶𝐶𝑙𝑙∗ ) = 𝐷𝐷 � � ∂𝑧𝑧 𝑧𝑧=0

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[A10.33] [A10.34]

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305

(Recall that it is assumed that 𝑘𝑘𝛼𝛼 = 𝑘𝑘𝛽𝛽 = 𝑘𝑘). The composition at the interface is: 𝐶𝐶𝑙𝑙∗ = 𝐶𝐶𝑒𝑒 + 𝐴𝐴 + 𝐵𝐵 cos( 2π𝑦𝑦/𝜆𝜆)

[A10.35]

It is now necessary to set up equations which are equivalent to Eqs A10.9 and A10.10. This gives, upon substituting from Eqs A10.32 and A10.35: α-phase:

𝑉𝑉(1 − 𝑘𝑘) �

𝑓𝑓𝑓𝑓 2

0

β-phase:

�𝐶𝐶𝑒𝑒 + 𝐴𝐴 + 𝐵𝐵 cos

𝜆𝜆 2

= 𝐷𝐷 �

𝑓𝑓𝑓𝑓 2

0

2π𝑦𝑦 � 𝑑𝑑𝑑𝑑 = 𝜆𝜆

𝑉𝑉 2π𝑦𝑦 � 𝐴𝐴 + 𝑏𝑏𝑏𝑏 cos � 𝑑𝑑𝑑𝑑 𝐷𝐷 𝜆𝜆

𝑉𝑉(1 − 𝑘𝑘) � �1 − 𝐶𝐶𝑒𝑒 − 𝐴𝐴 − 𝐵𝐵 cos 𝑓𝑓𝑓𝑓 2

𝜆𝜆 2

= 𝐷𝐷 � �− 𝑓𝑓𝑓𝑓 2

2π𝑦𝑦 � 𝑑𝑑𝑑𝑑 = 𝜆𝜆

𝑉𝑉 2π𝑦𝑦 � 𝑑𝑑𝑑𝑑 𝐴𝐴 − 𝑏𝑏𝑏𝑏 cos 𝐷𝐷 𝜆𝜆

[A10.36]

[A10.37]

This leads to two simultaneous algebraic equations which are equivalent to Eqs A10.11 and A10.12: α-phase: β-phase:

𝑃𝑃(1 − 𝑘𝑘) �(𝐶𝐶𝑒𝑒 + 𝐴𝐴)𝑓𝑓 +

𝐵𝐵 𝜆𝜆 sin(π𝑓𝑓)� = 𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑏𝑏𝑏𝑏 sin(π𝑓𝑓) π 2π

𝐵𝐵 sin(π𝑓𝑓)�] = π 𝜆𝜆 = 𝑃𝑃𝑃𝑃(1 − 𝑓𝑓) − 𝑏𝑏𝑏𝑏 sin(π𝑓𝑓) 2π

𝑃𝑃(1 − 𝑘𝑘) �(𝐶𝐶𝑒𝑒 + 𝐴𝐴 − 1)(1 − 𝑓𝑓) −

[A10.38]

[A10.39]

where 𝑃𝑃 is the Péclet number (= 𝑉𝑉λ/2𝐷𝐷). Adding the two equations and simplifying gives: 𝐴𝐴 =

1 − 𝑘𝑘 [𝑓𝑓𝐶𝐶𝑒𝑒 − (1 − 𝑓𝑓)(1 − 𝐶𝐶𝑒𝑒 )] 𝑘𝑘

[A10.40]

This is the equivalent equation to Eq. A10.13, and 𝐴𝐴 is identical to 𝐵𝐵0 in Eq. 15(a) of Trivedi et al. (1987). Multiplying Eq. A10.38 by (1 − 𝑓𝑓) and Eq. A10.39 by 𝑓𝑓, and subtracting one from the other leads, after substituting for 𝑏𝑏 from Eq. A10.3, to: 𝐵𝐵 = 𝑃𝑃

(1 − 𝑘𝑘)(𝑓𝑓 − 𝑓𝑓 2 )(2π/𝑃𝑃) sin(π𝑓𝑓) [(1 + (2π/𝑃𝑃)2 )1/2 − 1 + 2𝑘𝑘)]

[A10.41]

The calculations can now be carried forward in exactly the same way as for the low-velocity case, but using the new values of A and B. For instance, the average concentration difference ahead of the α phase is now: (𝑓𝑓 − 𝑓𝑓 2 ) 2π (1 − 𝑘𝑘) 𝑃𝑃 1 − 𝑘𝑘 𝜆𝜆𝜆𝜆 2π [A10.42] 𝛥𝛥𝐶𝐶𝛼𝛼 = [𝑓𝑓𝐶𝐶𝑒𝑒 − (1 − 𝑓𝑓)(1 − 𝐶𝐶𝑒𝑒 )] + � � 𝑘𝑘 𝑓𝑓𝑓𝑓 (1 + (2π/𝑃𝑃)2 )1/2 − 1 + 2𝑘𝑘 The expression in the large bracket on the RHS of Eq. A10.42 is equivalent to the exact function, 𝑃𝑃(𝑓𝑓, 𝑝𝑝, 𝑘𝑘), to be found in Trivedi et al. (1987). It has been written in the present form so as to reveal the similarity. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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The overall effect of high growth rates can be seen by replacing B, in Eq. A10.25 by the expression in Eq. A10.41. This gives: 𝑉𝑉𝑉𝑉 |𝑚𝑚𝛼𝛼 ||𝑚𝑚𝛽𝛽 | 2π𝐷𝐷 (1 − 𝑘𝑘)2π/𝑃𝑃 Δ𝑇𝑇 = + |𝑚𝑚𝛼𝛼 | + |𝑚𝑚𝛽𝛽 | [1 + (2π/𝑃𝑃)2 ]1/2 + 2𝑘𝑘 − 1 +

That is:

2(1 − 𝑓𝑓)|𝑚𝑚𝛽𝛽 |Γ𝛼𝛼 sin(𝜃𝜃𝛼𝛼 ) + 2𝑓𝑓|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin�𝜃𝜃𝛽𝛽 � 𝑓𝑓(1 − 𝑓𝑓)𝜆𝜆(|𝑚𝑚𝛼𝛼 | + |𝑚𝑚𝛽𝛽 |)

[A10.43]

𝛥𝛥𝛥𝛥 = 𝐾𝐾𝑐𝑐 𝑉𝑉𝑉𝑉 + 𝐾𝐾𝑟𝑟 /𝜆𝜆

where:

𝐾𝐾𝑐𝑐 =

|𝑚𝑚𝛼𝛼 ||𝑚𝑚𝛽𝛽 | (1 − 𝑘𝑘) (2π/𝑃𝑃) |𝑚𝑚𝛼𝛼 | + |𝑚𝑚𝛽𝛽 | 2𝜋𝜋𝜋𝜋 [1 + (2π/𝑃𝑃)2 ]1/2 − 1 + 2𝑘𝑘

[A10.44]

Comparison of this equation with Eq. A10.27(a) indicates that it contains a third fraction on the RHS, a function of P and k to account for rapid solidification, and (1 - k) replaces 𝐶𝐶′. 𝐾𝐾𝑟𝑟 is still given by Eq. A10.27(b). Hence, after optimisation the volume rate of a rapidly growing lamellar eutectic in this simplified treatment is given by: 𝜆𝜆2 𝑉𝑉 =

4π𝐷𝐷 [1 + (2π/𝑃𝑃)2 ]1/2 − 1 + 2𝑘𝑘 ⋅ × 𝑓𝑓(1 − 𝑓𝑓) (2π/𝑃𝑃) ×

𝑓𝑓|𝑚𝑚𝛼𝛼 |Γ𝛽𝛽 sin�𝜃𝜃𝛽𝛽 � + (1 − 𝑓𝑓)|𝑚𝑚𝛽𝛽 |Γ𝛼𝛼 sin(𝜃𝜃𝛼𝛼 ) |𝑚𝑚𝛼𝛼 ||𝑚𝑚𝛽𝛽 |(1 − 𝑘𝑘)

[A10.45]

It is of interest to see how the results of this high growth rate (high 𝑃𝑃) solution differ from the low Péclet number solution. This is done by separating out the 𝑃𝑃-dependent terms. Thus: 𝜆𝜆2 𝑉𝑉 = 𝐷𝐷Θ

(𝑃𝑃2 + 4π2 )1/2 − 𝑃𝑃 + 2𝑘𝑘𝑘𝑘 2π(1 − 𝑘𝑘)

[A10.46]

where Θ is a constant which groups all the fixed parameters that appear in Eq. A10.45. When 𝑃𝑃 tends to zero, the RHS of Eq. A10.46 tends to: 𝜆𝜆2 𝑉𝑉 = 𝐷𝐷Θ/(1 − 𝑘𝑘)

This indicates that 𝜆𝜆2 𝑉𝑉 is constant, independent of the growth rate. It corresponds to the low𝑃𝑃 solution of Jackson-Hunt. As 𝑃𝑃 tends to infinity, Equation A10.46 tends to: 𝜆𝜆2 𝑉𝑉 = 𝐷𝐷Θ

𝑘𝑘𝑘𝑘 π(1 − 𝑘𝑘)

Thus 𝜆𝜆2 𝑉𝑉 is no longer constant but increases with 𝑃𝑃 at a rate which depends upon 𝑘𝑘. By differentiating Eq. A10.46 with respect to 𝑃𝑃 and equating the result to zero, one finds that the 𝜆𝜆2 𝑉𝑉 versus 𝑃𝑃 curve has a minimum when the Peclet number becomes: 𝑃𝑃 =

π(1 − 2𝑘𝑘) [𝑘𝑘(1 − 𝑘𝑘)]1/2

[A10.47]

The minimum moves towards infinity as 𝑘𝑘 tends to zero. If 𝑘𝑘 is greater than 0.5 the minimum disappears (Fig. 7.16). The exact solution for the high velocity series 𝑃𝑃′′, a function of 𝑓𝑓, 𝑃𝑃 and 𝑘𝑘, has been developed by Trivedi et al. (1987): EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

Foundations of Materials Science and Engineering Vol. 103

𝑃𝑃

′′ (𝑓𝑓,

∞

1 3 𝑝𝑝𝑛𝑛 𝑃𝑃, 𝑘𝑘) = � � � sin2 (𝑛𝑛π𝑓𝑓)   𝑛𝑛π �1 + 𝑝𝑝𝑛𝑛2 − 1 + 2𝑘𝑘

𝑝𝑝𝑛𝑛 = 2nπ/𝑃𝑃

𝑛𝑛=1

Equation A10.48 can be approximated by:

𝑃𝑃′′ (𝑓𝑓, 𝑃𝑃, 𝑘𝑘) ≅ 0.335(𝑓𝑓(1 − 𝑓𝑓))1.65 𝜉𝜉𝑒𝑒 with:

𝜉𝜉𝑒𝑒 =

2.5π/𝑃𝑃

�1 + (2.5π/𝑃𝑃)2 − 1 + 2𝑘𝑘

307

[A10.48]

[A10.49]

Comparison of Eq. A10.49 with Eq. A10.30 shows that eutectic growth models express rapid solidification through a 𝜉𝜉 – function (similar to dendrites, e.g. Eq. 7.13). The rapid eutectic growth of fibrous eutectics has been treated in more detail by Trivedi and Wang (2012) and the effect of solute trapping by Kurz and Trivedi (1991). For further treatments of eutectic growth the reader is referred to Kassner and Misbah (1991) and Karma and Sarkissian (1996). Bibliography for Further Reading  R.Folch, M.Plapp, Quantitative phase-field modeling of two-phase growth, Physical Review E, 72 (2005) 011602.  K.A.Jackson, J.D.Hunt, Lamellar and rod eutectic growth, Transactions of the Metallurgical Society of AIME, 236 (1966) 1129.  A.Karma, A.Sarkissian, Morphological instabilities of lamellar eutectics, Metallurgical and Materials Transactions A, 27 (1996) 635.  K.Kassner, C.Misbah, Spontaneous parity-breaking transition in directional growth of lamellar eutectic structures, Physical Review A, 44 (1991) 6533.  W.Kurz, R.Trivedi, Eutectic Growth under Rapid Solidification Conditions, Metallurgical Transactions A, 22 (1991) 3051.  R.Trivedi, P.Magnin, W.Kurz, Theory of eutectic growth under rapid solidification conditions, Acta Materialia, 35 (1987) 971.  R.Trivedi, W.Kurz, Microstructure selection in eutectic alloy systems, in Solidification Processing of Eutectic Alloys, D.M.Stefanescu, G.J.Abbaschian, R.J.Bayuzick (Eds), The Metallurgical Society, 1988, p.3.  R.Trivedi, N.Wang, Theory of rod eutectic growth under far-from-equilibrium conditions: Nanoscale spacing and transition to glass, Acta Materialia, 60 (2012) 3140.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 309-312 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 11 TRANSIENTS IN SOLUTE DIFFUSION INITIAL TRANSIENT As shown previously (Chap. 3), the build-up of solute ahead of a planar solid/liquid interface takes some time - the time required for the saturation of the diffusion boundary layer. It is assumed to a first approximation that diffusion is rapid enough to ensure, during this transient, the establishment of a quasi-steady state solute distribution. In this case, the concentration decreases exponentially from the interface and into the liquid, with the equivalent boundary layer always having the same thickness (Appendix 2): 2𝐷𝐷 𝑉𝑉 Assuming that the growth rate, 𝑉𝑉, increases instantaneously to a constant value at the start of solidification, the flux at an interface of unit area, due to the differing solubilities of the solute in the two phases, will be: 𝛿𝛿𝑐𝑐 =

𝐽𝐽1 = 𝑉𝑉𝐶𝐶𝑙𝑙∗ (1 − 𝑘𝑘)

[A11.1]

𝐽𝐽2 = −𝐷𝐷𝐺𝐺𝑐𝑐

[A11.2]

This flux is initially greater than the flux created by the concentration gradient in the liquid:

The difference between the two fluxes will be used to 'fill' the diffusion boundary layer, of thickness 𝛿𝛿𝑐𝑐 , up to a mean concentration, 𝐶𝐶̄𝑙𝑙 : 𝐽𝐽1 − 𝐽𝐽2 =

𝑑𝑑𝐶𝐶̄𝑙𝑙 𝛿𝛿 𝑑𝑑𝑑𝑑 𝑐𝑐

[A11.3]

Recalling that a quasi-steady state profile is assumed to exist, the concentration gradient at the solid/liquid interface is given directly by the first derivative (at 𝑧𝑧 = 0) of Eq. 3.2, with Δ𝐶𝐶0 in this transient state being replaced by Δ𝐶𝐶 = (𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 ) (Fig. A11.1): 𝐺𝐺𝑐𝑐 = (𝐶𝐶0 − 𝐶𝐶𝑙𝑙∗ )

𝑉𝑉 𝐷𝐷

[A11.4]

For unit area therefore:

𝐽𝐽2 = (𝐶𝐶𝑙𝑙∗ − 𝐶𝐶0 )𝑉𝑉

and from Eq. A11.3 2𝐷𝐷 𝑑𝑑𝐶𝐶̄𝑙𝑙 = 𝑉𝑉(𝐶𝐶0 − 𝑘𝑘𝐶𝐶𝑙𝑙∗ ) 𝑉𝑉 𝑑𝑑𝑑𝑑 EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A11.5]

310

Fundamentals of Solidification 5th Edition

Noting that, for the triangular equivalent boundary layer (Fig. A11.1(b)), the variation in the mean concentration is related to the variation in interface concentration, e.g.: 𝐶𝐶̄𝑙𝑙 =

𝑑𝑑𝐶𝐶𝑙𝑙∗ 2

with 𝑉𝑉 = 𝑑𝑑𝑧𝑧 ′ /𝑑𝑑𝑑𝑑V:

𝑑𝑑𝐶𝐶𝑙𝑙∗ 𝑉𝑉 𝑑𝑑𝑑𝑑 ∗ = 𝐶𝐶0 − 𝑘𝑘𝐶𝐶𝑙𝑙 𝐷𝐷

[A11.6]

Figure A11.1

Integrating: 𝐶𝐶𝑙𝑙∗

gives:

′

𝑑𝑑𝐶𝐶𝑙𝑙∗ ′ 𝑉𝑉 𝑧𝑧 � � 𝑑𝑑𝑑𝑑 ∗ = 𝐷𝐷 0 𝐶𝐶0 𝐶𝐶0 − 𝑘𝑘𝐶𝐶𝑙𝑙 ′ 1/𝑘𝑘

or

𝐶𝐶0 (1 − 𝑘𝑘) ln � � 𝐶𝐶0 − 𝑘𝑘𝐶𝐶𝑙𝑙∗ 𝐶𝐶𝑙𝑙∗ =

=

𝑉𝑉𝑧𝑧 ′ 𝐷𝐷

𝐶𝐶0 𝑘𝑘𝑧𝑧 ′ 𝑉𝑉 �1 − (1 − 𝑘𝑘) exp �− �� 𝑘𝑘 𝐷𝐷

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[A11.7]

Foundations of Materials Science and Engineering Vol. 103

311

Figure 6.2 shows the behaviour of this expression. It is a good approximation to the exact solution derived by Smith et al. (1955). More detailed and exact results on the initial transient can be found in Warren and Langer (1993) and in Mota et al. (2015).

FINAL TRANSIENT While the initial transient is required to build up a boundary layer, the final transient is the result of a 'collision' of the boundary layer with the end of the specimen (Fig. A11.2). This problem has been solved by Smith et al. (1955) in the following way for the case where the system has reached a steady state. The origin of the z-axis is placed at the end of the specimen and the diffusion equation becomes, in terms of the new x-coordinate (which does not move with the solid/liquid interface): 𝑑𝑑2 𝐶𝐶𝑙𝑙 1 𝑑𝑑𝐶𝐶𝑙𝑙 = 𝑑𝑑𝑥𝑥 2 𝐷𝐷 𝑑𝑑𝑑𝑑

[A11.8]

Figure A11.2

The boundary conditions apply for all values of 𝑡𝑡. At 𝑥𝑥 = 0 (the end of the specimen) the diffusion flux must be equal to zero and: 𝑑𝑑𝐶𝐶𝑙𝑙 =0 𝑑𝑑𝑑𝑑

while at the solid/liquid interface, 𝑥𝑥 = 𝑥𝑥 ∗ : 𝑑𝑑𝐶𝐶𝑙𝑙 𝑉𝑉 ∗ = 𝐶𝐶 𝑝𝑝 𝑑𝑑𝑑𝑑 𝐷𝐷 𝑙𝑙

where 𝑥𝑥 ∗ is the length of the liquid zone. EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A11.9]

[A11.10]

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Fundamentals of Solidification 5th Edition

In order to meet the first boundary condition, an imaginary source placed in a symmetrical position with respect to the real solid/liquid interface is introduced. This is equivalent to a barrier to mass flux being placed at the end of the specimen (Fig. A11.2). The steady-state distributions at the two interfaces in an infinite specimen are: 𝑝𝑝 𝑉𝑉(𝑥𝑥 ∗ ± 𝑥𝑥) 𝐶𝐶𝑙𝑙 = 𝐶𝐶0 �1 + � exp �− � 𝐷𝐷 𝑘𝑘

[A11.11]

where 𝑧𝑧 = 𝑥𝑥 ∗ ± 𝑥𝑥. In order to cope with the second boundary condition (Eq. A11.10) when the sources are close enough to interact, further sources (interfaces) are introduced at distances, 𝑛𝑛𝑥𝑥 ∗ , where 𝑛𝑛 is an integer (Fig. A11.3). These sources travel at speeds 𝑛𝑛 times the speed of the real interface. This superposition permits the calculation of the necessary coefficients by constraining the interface to obey the flux balance (Eq. A11.10): ∞

𝐶𝐶𝑙𝑙 𝑉𝑉 𝑉𝑉 = 1 + � 𝐶𝐶𝑛𝑛 �exp �−𝑛𝑛 (𝑛𝑛𝑥𝑥 ∗ − 𝑥𝑥)� + exp �−𝑛𝑛 (𝑛𝑛𝑥𝑥 ∗ + 𝑥𝑥)�� 𝐶𝐶0 𝐷𝐷 𝐷𝐷

[A11.12]

𝑛𝑛=1

Figure A11.3

This procedure is given in more detail in Smith et al. (1955) and leads to: 𝐶𝐶𝑠𝑠 (𝑥𝑥) = 1 + 3

1 − 𝑘𝑘 2𝑉𝑉𝑉𝑉 (1 − 𝑘𝑘)(2 − 𝑘𝑘) 6𝑉𝑉𝑉𝑉 �+5 � +. .. exp �− exp �− (1 + 𝑘𝑘)(2 + 𝑘𝑘) 1 + 𝑘𝑘 𝐷𝐷 𝐷𝐷

. . . +(2𝑛𝑛 + 1)

(1 − 𝑘𝑘)(2 − 𝑘𝑘). . . (𝑛𝑛 − 𝑘𝑘) 𝑛𝑛(𝑛𝑛 + 1)𝑉𝑉𝑉𝑉 exp �− � (1 + 𝑘𝑘)(2 + 𝑘𝑘). . . (𝑛𝑛 + 𝑘𝑘) 𝐷𝐷

[A11.13]

The above equation describes the final transient in the solid, for the case in which a steady-state solute pile-up has been established at the solid/liquid interface, neglecting back diffusion. Bibliography for Further Reading  F.L.Mota, N.Bergeon, D.Tourret, A.Karma, R.Trivedi, B.Billia, Initial transient behavior in directional solidification of a bulk transparent model alloy in a cylinder, Acta Materialia, 85 (2015) 362.  V.G.Smith, W.A.Tiller, J.W.Rutter, A mathematical analysis of solute redistribution during solidification, Canadian Journal of Physics, 33 (1955) 723.  A.Warren, J.S.Langer, Prediction of dendritic spacings in a directional-solidification experiment, Physical Review E, 47 (1993) 2702.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 313-319 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 12 MASS BALANCE EQUATIONS The differing solubilities of a solute in the liquid and solid phases, together with differences in mobility, lead to the spatial concentration variations known as segregation. Provided that mass transport in the liquid is infinitely rapid (no concentration gradient in the liquid), the corresponding relationships can be easily deduced from a mass balance: 𝑓𝑓𝑠𝑠 𝐶𝐶𝑠𝑠 + 𝑓𝑓𝑙𝑙 𝐶𝐶𝑙𝑙 = 𝐶𝐶0

[A12.1]

𝑑𝑑(𝑓𝑓𝑠𝑠 𝐶𝐶𝑠𝑠 ) + 𝑑𝑑(𝑓𝑓𝑙𝑙 𝐶𝐶𝑙𝑙 ) = 0

These equations state that the solute which cannot be incorporated into the growing solid has to be incorporated into the liquid. No account is taken here the other possible phenomena which might occur in practical situations, e.g. solute vaporization or crucible reaction.

LEVER RULE The simplest case to which a mass balance can be applied is that of equilibrium solidification (no concentration gradient in the solid or liquid). This can be expressed by the relations: 𝐷𝐷𝑙𝑙 ≫ 𝐷𝐷𝑠𝑠 ≫ 𝐿𝐿𝐿𝐿

[A12.2]

𝐿𝐿 ≪ (𝐷𝐷𝑠𝑠 𝑡𝑡𝑓𝑓 )1/2

[A12.3]

Since 𝐿𝐿 is the length of the solidifying system (Fig. A12.1), Equation A12.2 states that the diffusion boundary layer, 𝛿𝛿𝑐𝑐 = 2𝐷𝐷/𝑉𝑉, is much larger than the maximum distance, 𝐿𝐿, over which either solid- or liquid-state diffusion can occur. It can similarly be supposed that if the growth rate, 𝑉𝑉, of the solid is constant and equal to 𝐿𝐿/𝑡𝑡𝑓𝑓 , the length of the specimen must be less than the characteristic diffusion length: Under these conditions: 𝜕𝜕𝐶𝐶𝑙𝑙 𝜕𝜕𝐶𝐶𝑠𝑠 = ′ ≅0 𝜕𝜕𝑧𝑧 ′ 𝜕𝜕𝑧𝑧

Taking the differential form of Eq. A12.1, 𝐶𝐶𝑠𝑠 = 𝑘𝑘𝐶𝐶𝑙𝑙 , 𝑑𝑑𝑑𝑑𝑠𝑠 = 𝑘𝑘𝑑𝑑𝑑𝑑𝑙𝑙 , 𝑓𝑓𝑠𝑠 = 1 − 𝑓𝑓𝑙𝑙 , and 𝑑𝑑𝑓𝑓𝑠𝑠 = −𝑑𝑑𝑓𝑓𝑙𝑙 can be substituted and integrations performed *: �

𝐶𝐶𝑙𝑙

𝐶𝐶0

𝑓𝑓𝑠𝑠 𝑑𝑑𝐶𝐶𝑙𝑙 ′ 𝑑𝑑𝑓𝑓𝑠𝑠 =� 𝑝𝑝𝐶𝐶𝑙𝑙 ′ 0 1 − 𝑓𝑓𝑠𝑠 𝑝𝑝

giving, for the lever rule:

*

𝐶𝐶𝑙𝑙 1 = 𝐶𝐶0 1 − 𝑝𝑝𝑓𝑓𝑠𝑠

The prime in 𝑑𝑑𝑑𝑑𝑙𝑙 ′ allows to distinguish from the upper bound 𝐶𝐶𝑙𝑙 .

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[A12.4]

314

Fundamentals of Solidification 5th Edition

Figure A12.1

SOLUTE DISTRIBUTION WITH BACK-DIFFUSION AND COMPLETE MIXING IN THE LIQUID The case of a bar having a constant cross-section, 𝐴𝐴, and a zero-concentration gradient, due to rapid mass transport in the liquid (𝐶𝐶𝑙𝑙 = 𝐶𝐶𝑙𝑙∗ ) and limited diffusion in the solid, is shown in Fig. 6.3. For this case, Brody and Flemings (1966) developed a flux balance which, in a slightly modified form (sum of redistributed mass represented by the surfaces, 𝐴𝐴1 + 𝐴𝐴2 + 𝐴𝐴3 = 0, in Fig. 6.3) is: (𝐶𝐶𝑙𝑙 − 𝐶𝐶𝑠𝑠∗ )𝐴𝐴𝐴𝐴𝐴𝐴 = (𝐿𝐿 − 𝑠𝑠)𝐴𝐴𝐴𝐴𝐶𝐶𝑙𝑙 + 𝑑𝑑𝐶𝐶𝑠𝑠∗ 𝐴𝐴

𝛿𝛿𝑠𝑠 2

[A12.5]

Here 𝐴𝐴3 represents the surface of the equivalent boundary layer in the solid (Appendix 2). Recognising that 𝑓𝑓𝑠𝑠 = 𝑠𝑠/𝐿𝐿, 𝑑𝑑𝑓𝑓𝑠𝑠 = 𝑑𝑑𝑑𝑑/𝐿𝐿, 𝛿𝛿𝑠𝑠 = 2𝐷𝐷𝑠𝑠 /𝑉𝑉 = 2𝐷𝐷𝑠𝑠 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, 𝐶𝐶𝑠𝑠∗ = 𝑘𝑘𝐶𝐶𝑙𝑙 , and 𝑑𝑑𝐶𝐶𝑠𝑠∗ = 𝑘𝑘𝑘𝑘𝐶𝐶𝑙𝑙 then: 𝐶𝐶𝑙𝑙 (1 − 𝑘𝑘)𝑑𝑑𝑓𝑓𝑠𝑠 𝐿𝐿 = 𝐿𝐿(1 − 𝑓𝑓𝑠𝑠 )𝑑𝑑𝐶𝐶𝑙𝑙 + 𝑑𝑑𝐶𝐶𝑙𝑙 𝑘𝑘𝐷𝐷𝑠𝑠

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

Dividing by 𝐿𝐿, and using a parabolic growth rate relationship (e.g. Eq. A1.22): 1/2

𝑠𝑠 𝑡𝑡 = 𝑓𝑓𝑠𝑠 = � � 𝐿𝐿 𝑡𝑡𝑓𝑓

[A12.6]

[A12.7]

with 𝑡𝑡𝑓𝑓 the solidification time. Evaluating 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, substituting the results into Eq. A12.6 and rearranging gives: 𝑑𝑑𝐶𝐶𝑙𝑙 𝑑𝑑𝑓𝑓𝑠𝑠 = 𝑝𝑝𝐶𝐶𝑙𝑙 (1 − 𝑓𝑓𝑠𝑠 ) + 2𝛼𝛼𝛼𝛼𝑓𝑓𝑠𝑠

[A12.8]

where 𝛼𝛼 is a dimensionless solid-state back-diffusion parameter (dimensionless time = Fourier number): 𝛼𝛼 =

𝐷𝐷𝑠𝑠 𝑡𝑡𝑓𝑓 𝐿𝐿2

(compare with Eq. A 12.3) from which: 𝑓𝑓𝑠𝑠 1 𝐶𝐶𝑙𝑙 𝑑𝑑𝐶𝐶𝑙𝑙 𝑑𝑑𝑓𝑓𝑠𝑠 � =� 𝑝𝑝 𝐶𝐶0 𝐶𝐶𝑙𝑙 0 1 − 𝑓𝑓𝑠𝑠 (1 − 2𝛼𝛼𝛼𝛼) EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

[A12.9]

Foundations of Materials Science and Engineering Vol. 103

315

and integration leads to: 𝑘𝑘−1 𝐶𝐶𝑙𝑙 = [1 − 𝑓𝑓𝑠𝑠 (1 − 2𝛼𝛼𝛼𝛼)]1−2𝛼𝛼𝛼𝛼 𝐶𝐶0

[A12.10]

Due to the simplifying assumptions made, this solution is limited to 2α𝑘𝑘-values which are smaller than 1, with α < 0.5. Apart from this limitation, Eq. A12.10 is an important one because it includes the two limiting cases (for ∂𝐶𝐶𝑙𝑙 / ∂𝑧𝑧 ′ = 0):  Lever rule:

 Gulliver-Scheil's equation: 𝐶𝐶𝑙𝑙 (𝑘𝑘−1) = 𝑓𝑓𝑙𝑙 𝐶𝐶0

(Eq. A12.4) when 𝛼𝛼 = 0.5

when 𝐷𝐷𝑠𝑠 = 0 (no solid-state diffusion) 𝛼𝛼 = 0. That is:

[A12.11]

According to Eq. A12.9, the case 𝛼𝛼 = 0.5 (lever rule) does not correspond to the physical characteristics of equilibrium solidification. There, 𝛼𝛼 should approach infinity (Eq. A12.3). A modified back-diffusion parameter, 𝛼𝛼′, was therefore proposed by Clyne and Kurz (1981). The basis for this calculation is that, in the original Brody-Flemings treatment for high 𝛼𝛼-values, solute was not conserved in the system. This can be easily understood with the aid of Fig. 6.3, where the solid diffusion boundary layer, 𝛿𝛿𝑠𝑠 , still has a small value. If this 𝛿𝛿𝑠𝑠 -value (proportional to 𝛼𝛼) becomes greater than the solidified length, s, there is no longer any mass conservation according to Eq. A12.5 because the end-effects are not considered. In reality, the initial specimen end (𝑧𝑧 ′ = 0) is an isolated system boundary and solute cannot leave the system at that point. In order to find a simple solution to this problem, it will be assumed that diffusion is semi-infinite and that the tail of the diffusion boundary layer at 𝑧𝑧 ′ -values less than zero (outside of the specimen) has to be taken into account. For the purpose of the analysis the coordinates are changed: 𝑦𝑦 = 0 is fixed at the solid/liquid interface and only the solid concentration profile shown in Fig. A12.2 is considered. The end of the specimen (𝑧𝑧 ′ = 0) then corresponds to the cut-off distance on the 𝑦𝑦-axis. The total amount of solute in the boundary layer in the solid is 𝐴𝐴̄𝑇𝑇 . The neglected portion of the solute, due to the small cut-off distance, is 𝐴𝐴̄𝐸𝐸 . These can be obtained from (Clyne and Kurz, 1981): ∞

𝐴𝐴̄ 𝑇𝑇 = � 𝐶𝐶 ′ 𝑑𝑑𝑑𝑑 = 𝐶𝐶0′ 0 ∞

𝐴𝐴̄𝐸𝐸 = � 𝐶𝐶 ′ 𝑑𝑑𝑑𝑑 = 𝐶𝐶0′ 𝑦𝑦𝑖𝑖

𝛿𝛿𝑠𝑠 2

𝛿𝛿𝑠𝑠 2𝑦𝑦𝑖𝑖 exp �− � 2 𝛿𝛿𝑠𝑠

[A12.12]

[A12.13]

It is assumed here, to a first approximation, that the back-diffusion can be described by an exponential function. To estimate the correction factor which is necessary to cope with the cut-off effect, a parameter, Σ, is defined: Σ(𝛼𝛼) =

1 𝑡𝑡𝑓𝑓 𝐴𝐴̄𝐸𝐸 � 𝑑𝑑𝑑𝑑 𝑡𝑡𝑓𝑓 0 𝐴𝐴̄ 𝑇𝑇

[A12.14]

It will be noted that, even if 𝐶𝐶0′ changes during solidification, the ratio contained in the integral will be independent of such changes. For parabolic growth laws (Eq. A12.7): 1/2

𝑡𝑡 𝑦𝑦𝑖𝑖 = 𝐿𝐿 � � 𝑡𝑡𝑓𝑓

Combining Eqs A12.12 to A12.15 gives:

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[A12.15]

316

Fundamentals of Solidification 5th Edition

1/2

1 𝑡𝑡𝑓𝑓 2𝐿𝐿 𝑡𝑡 Σ(𝛼𝛼) = � exp �− � � 𝑡𝑡𝑓𝑓 0 𝛿𝛿𝑠𝑠 𝑡𝑡𝑓𝑓

[A12.16]

� 𝑑𝑑𝑑𝑑

Figure A12.2

to:

Recognising that the exponential term in the integral of Eq. A12.16 is equal to (−1/2𝛼𝛼) leads Σ(𝛼𝛼) = exp �−

1 � 2𝛼𝛼

[A12.17]

Equation A12.17 approaches zero asymptotically at low 𝛼𝛼-values, and unity at high 𝛼𝛼-values. In the low 𝛼𝛼-range it therefore represents the deviation from 𝛼𝛼 and, in the high 𝛼𝛼-range, it represents the deviation from the limiting value, 𝛼𝛼 ′ = 0.5. A spline-like function which connects 𝛼𝛼 ′ and 𝛼𝛼 must therefore be found which satisfies the two boundary conditions: 𝛼𝛼 ′ → 𝛼𝛼 as Σ(𝛼𝛼) → 0 and 𝛼𝛼 ′ → 0.5 as Σ(𝛼𝛼) → 1. An expression which has the required form (Fig. A12.3) is: 1 1 1 𝛼𝛼 ′ = 𝛼𝛼 �1 − exp �− �� − exp �− � 𝛼𝛼 2 2𝛼𝛼

[A12.18]

Re-naming 𝛼𝛼 in Eq. A12.10 as 𝛼𝛼 ′ (Eq. A12.18) permits the calculation of any solute distribution when diffusion in the liquid is very rapid: that is, it is not necessary to decide whether the lever rule or Gulliver-Scheil's equation has to be used in a given situation. This is of great importance, particularly for alloys which contain interstitial and substitutional solutes. In Fig. A12.3, typical values of 𝛼𝛼 for phosphorus and carbon in δ-Fe are given. As can be seen, carbon obeys the lever rule while phosphorus is intermediate in behaviour. The most important application of Eq. A12.10 is related to the estimation of microsegregation, i.e. the modelling of the behaviour of the mushy zone. Replacing 𝐿𝐿 by the characteristic diffusion distance, 𝜆𝜆/2 (= 𝜆𝜆1 /2 for cells and 𝜆𝜆2 /2 for dendrites; Figs 6.6 and 6.7) leads to: 𝛼𝛼 =

4𝐷𝐷𝑠𝑠 𝑡𝑡𝑓𝑓 𝜆𝜆2

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[A12.19]

Foundations of Materials Science and Engineering Vol. 103

317

Rearranging Eq. A12.10, with 𝛼𝛼 re-named 𝛼𝛼 ′ , permits the solidified fraction to be calculated: 1−2𝛼𝛼′ 𝑘𝑘 𝑘𝑘−1

𝑇𝑇𝑓𝑓 − 𝑇𝑇 1 �1 𝑓𝑓𝑠𝑠 = − � � 1 − 2𝛼𝛼 ′ 𝑘𝑘 𝑇𝑇𝑓𝑓 − 𝑇𝑇𝑙𝑙

[A12.20]

�

Here, 𝑇𝑇𝑓𝑓 is the melting point of the pure element, and 𝐶𝐶 has been replaced by 𝑇𝑇, via the liquidus slope. The first derivative with respect to temperature is: ′

′

2𝛼𝛼 𝑘𝑘−1 2−2𝛼𝛼 𝑘𝑘−𝑘𝑘 1 𝑑𝑑𝑓𝑓𝑠𝑠 = �𝑇𝑇𝑓𝑓 − 𝑇𝑇𝑙𝑙 � 𝑘𝑘−1 �𝑇𝑇𝑓𝑓 − 𝑇𝑇� 𝑘𝑘−1 𝑑𝑑𝑑𝑑 𝑘𝑘 − 1

[A12.21]

This relationship can be used to make numerical calculations of the solidification microstructure of cast alloys.

Figure A12.3

Matsumiya et al. (1984) have compared previously developed models with numerical results. They showed that, in some cases, Equation A12.10 (with 𝛼𝛼 ′ from Eq. A12.18) did not agree with the more exact calculations. Ohnaka (1986) has proposed a more elegant approximation to microsegregation problems with back-diffusion: 𝐶𝐶𝑙𝑙 = (1 − 𝜓𝜓𝑓𝑓𝑠𝑠 )(𝑘𝑘−1)/𝜓𝜓 𝐶𝐶0 with:

[A12.22a]

𝜓𝜓 = 1 − 𝜂𝜂𝜂𝜂/(1 + 𝜂𝜂)

and

𝜂𝜂 = 4α (α see Eq. A12.19)

Finally, Kobayashi (1988) has developed an exact analytical solution to the microsegregation problem and has also provided some higher-order approximations. The following example comes very close to matching the rather complicated exact solution:

with

1 1 1 𝐶𝐶𝑠𝑠 = 𝑘𝑘𝐶𝐶0 𝜉𝜉 𝜂𝜂 �1 + 𝑈𝑈 � � 2 − 1� − 2 � − 1� − ln 𝜉𝜉�� 2 𝜉𝜉 𝜉𝜉 𝜉𝜉 = 1 − (1 − 𝛽𝛽𝛽𝛽)𝑓𝑓𝑠𝑠

𝛽𝛽 = 2𝛾𝛾/(1 + 2𝛾𝛾)

𝛾𝛾 = 8𝐷𝐷𝑡𝑡𝑓𝑓 /𝜆𝜆2

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[A12.22b]

318

Fundamentals of Solidification 5th Edition

𝜂𝜂 = (𝑘𝑘 − 1)/(1 − 𝛽𝛽𝛽𝛽)

𝑈𝑈 =

𝛽𝛽 3 𝑘𝑘(𝑘𝑘 − 1)[(1 + 𝛽𝛽)𝑘𝑘 − 2] 4𝛾𝛾(1 − 𝛽𝛽𝛽𝛽)3

A microsegregation model for binary alloys which accounts for diffusion in all phases and for nucleation undercooling and growth kinetics of peritectics and eutectics can be found in an article by Tourret and Gandin (2015).

SOLUTE DISTRIBUTION UNDER RAPID SOLIDIFICATION CONDITIONS In Chap. 7, it was shown that, under conditions of rapid solidification, the assumption of a homogeneous interdendritic liquid does not represent the observed microsegregation behaviour (Fig. 7.20). This is mainly due to the large solute pile-up around the rapidly growing dendrite tips. Once these boundary layers interact at higher volume fractions (𝑓𝑓𝑠𝑠 > 𝑓𝑓𝑥𝑥 ,), homogeneous interdendritic liquid can again be assumed to exist. One can treat these effects in the following way (Fig. 7.21):  for small solid fractions (𝑓𝑓𝑠𝑠 < 𝑓𝑓𝑥𝑥 ) the concentration is assumed to have a second-order polynomial form: [A12.23]

𝑓𝑓𝑠𝑠 = 𝑎𝑎1 𝐶𝐶𝑠𝑠∗2 + 𝑎𝑎2 𝐶𝐶𝑠𝑠∗ + 𝑎𝑎3

 Gulliver-Scheil's equation holds for large solid fractions (𝑓𝑓𝑠𝑠 > 𝑓𝑓𝑥𝑥 ), i.e.: 1

𝐶𝐶𝑠𝑠∗ 𝑘𝑘−1 𝑓𝑓𝑠𝑠 = 1 + (𝑓𝑓𝑥𝑥 − 1) + � � 𝐶𝐶𝑥𝑥

[A12.24]

There are five unknowns to be determined: 𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 , 𝑓𝑓𝑥𝑥 , 𝐶𝐶𝑥𝑥 . The corresponding five equations necessary for the solution can be obtained, as demonstrated by Giovanola et al. (1990). In this way, a realistic 𝐶𝐶𝑠𝑠∗ (𝑓𝑓𝑠𝑠 ) – function can be approximated.

PRECIPITATION IN A TERNARY SYSTEM, A-B-C, IN THE ABSENCE OF SOLID-STATE DIFFUSION

When the effect of back-diffusion is negligible, as in the case of a large volume of precipitating phase at the end of solidification (i.e. large concentration gradients close to 𝑓𝑓𝑠𝑠 = 1 are avoided), Gulliver-Scheil's Equation A12.11 can be used. For two solute elements, B and C, the corresponding solute profiles (trace of interface concentration as a function of 𝑓𝑓𝑠𝑠 ) are: 𝐶𝐶𝑙𝑙 � � = 𝑓𝑓𝑙𝑙 (𝑘𝑘𝐵𝐵 −1) 𝐶𝐶0 𝐵𝐵

[A12.25]

𝐶𝐶𝑙𝑙 � � = 𝑓𝑓𝑙𝑙 (𝑘𝑘𝐶𝐶 −1) 𝐶𝐶0 𝐶𝐶

When the solubility product of the phase, BxCy, is reached, precipitation will begin (if there is no difficulty in nucleation): 𝑦𝑦

(𝐶𝐶𝑙𝑙 )𝐵𝐵𝑥𝑥 (𝐶𝐶𝑙𝑙 )𝐶𝐶 = 𝐾𝐾𝐵𝐵𝑥𝑥 𝐶𝐶𝑦𝑦

From Eqs A12.25 and A12.26: 𝑦𝑦

𝐾𝐾𝐵𝐵𝑥𝑥 𝐶𝐶𝑦𝑦 = (𝐶𝐶0 )𝐵𝐵𝑥𝑥 (𝐶𝐶0 )𝐶𝐶 𝑓𝑓𝑙𝑙 (𝑥𝑥𝑘𝑘𝐵𝐵 +𝑦𝑦𝑘𝑘𝐶𝐶 −𝑥𝑥−𝑦𝑦)

When precipitation begins, 𝑓𝑓𝑙𝑙 = 𝑓𝑓𝑝𝑝 and the precipitate volume is:

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[A12.26]

Foundations of Materials Science and Engineering Vol. 103

𝑓𝑓𝑝𝑝 =

1 −𝑦𝑦 𝑥𝑥𝑘𝑘𝐵𝐵 +𝑦𝑦𝑘𝑘𝐶𝐶 −𝑥𝑥−𝑦𝑦 −𝑥𝑥 �𝐾𝐾𝐵𝐵𝑥𝑥 𝐶𝐶𝑦𝑦 (𝐶𝐶0 )𝐵𝐵 (𝐶𝐶0 )𝐶𝐶 �

319

[A12.27]

SOLIDIFICATION PATH IN TERNARY SYSTEMS The path of solidification (trace of liquid or solid composition as a function of 𝑓𝑓𝑠𝑠 ) can generally be obtained in the case of solidification by relating the composition to 𝑓𝑓𝑙𝑙 , which must be the same for all the elements). Using Eqs A12.10 and A12.18: 𝑝𝑝𝐵𝐵 𝐶𝐶𝑙𝑙 � � = (1 − uB fs )𝑢𝑢𝐵𝐵 𝐶𝐶0 𝐵𝐵

[A12.28]

𝑝𝑝𝐶𝐶 𝐶𝐶𝑙𝑙 � � = (1 − u𝐶𝐶 fs )𝑢𝑢𝐶𝐶 𝐶𝐶0 𝐶𝐶

where 𝑢𝑢𝑖𝑖 = 1 − 2𝛼𝛼𝑖𝑖′ 𝑘𝑘𝑖𝑖 and 𝑝𝑝𝑖𝑖 = 1 − 𝑘𝑘𝑖𝑖 . Eliminating 𝑓𝑓𝑠𝑠 : 𝑝𝑝𝐵𝐵 𝑢𝑢𝐶𝐶 −𝑢𝑢𝐵𝐵 𝑝𝑝𝐶𝐶

𝐶𝐶𝑙𝑙 𝑢𝑢𝐵𝐵 𝐶𝐶𝑙𝑙 � � = �1 − �1 − � � � � 𝐶𝐶0 𝐵𝐵 𝑢𝑢𝐶𝐶 𝐶𝐶0 𝐶𝐶

[A12.29]

The latter equation relates the composition of solute B to that of solute C in the ternary system, A-B-C, with constant 𝑘𝑘𝑖𝑖 and 𝛼𝛼𝑖𝑖′ .

Bibliography for Further Reading

 H.D.Brody, M.C.Flemings, Solute redistribution in dendritic solidification, Transactions of the Metallurgical Society of AIME, 236 (l966) 615.  T.W.Clyne, W.Kurz, Solute redistribution during solidification with rapid solid state diffusion, Metallurgical Transactions A, 12 (1981) 965.  B.Giovanola, W.Kurz, Modeling of microsegregation under rapid solidification conditions, Metallurgical Transactions A, 21 (1990) 260.  S.Kobayashi, Mathematical analysis of solute redistribution during solidification based on a columnar dendrite mode, Transactions of the Iron and Steel Institute of Japan, 28 (1988) 728.  T.Matsumiya, H.Kajioka, S.Mizoguchi, Y.Ueshima, H.Esaka, Mathematical analysis of segregations in continuously-cast slabs, Transactions of the Iron and Steel Institute of Japan, 24 (1984) 873.  I.Ohnaka, Mathematical analysis of solute redistribution during solidification with diffusion in solid phase, Transactions of the Iron and Steel Institute of Japan, 26 (1986) 1045.  D.Tourret, C.A.Gandin, A generalized segregation model for concurrent dendritic, peritectic and eutectic solidification, Acta Materialia, 57 (2009) 2066.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 321-323 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 13 HOMOGENISATION OF INTERDENDRITIC SEGREGATION IN THE SOLID STATE In order to determine the changes occurring during homogenisation of the cooling solid following solidification, only one dendrite arm has to be considered (due to symmetry - Appendix 2). In principle the form of the solute segregation can be described approximately by the Gulliver-Scheil equation (Fig. A13.1). It will be shown here that all the possible initial distributions can be related to one another by using dimensionless constants. The changes can be treated approximately by using the one-dimensional time-dependent diffusion equation: 𝜕𝜕 2 𝐶𝐶 𝜕𝜕𝜕𝜕 [A13.1] = 𝜕𝜕𝑥𝑥 2 𝜕𝜕𝜕𝜕 It is known (Appendix 2) that a likely solution to this equation involves circular and exponential functions. The exponential function is more likely to be associated with the time-dependence and a cosine or sine function is more likely to reflect the characteristics of spatial distributions such as those in Fig. A13.1. In practice the solute distribution may take any form at all, of course. It will be shown below however that analogous changes occur in the distribution, regardless of the initial state. One can thus suppose that: 𝐷𝐷𝑠𝑠

𝐶𝐶 = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿exp(𝑎𝑎𝑎𝑎)cos(𝑏𝑏𝑏𝑏)

[A13.2]

where 𝛿𝛿𝛿𝛿 is the initial amplitude of the concentration variation (Fig. A13.1). Substitution of the derivatives of Eq. A13.2 into Eq. A13.1 shows that: 𝑎𝑎 = −𝐷𝐷𝑠𝑠 𝑏𝑏 2

and therefore:

𝐶𝐶 = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿exp(−𝐷𝐷𝑠𝑠 𝑏𝑏 2 𝑡𝑡)cos(𝑏𝑏𝑏𝑏)

[A13.3]

The value of 𝑏𝑏 can be evaluated by using the boundary conditions. The gradient of the concentration at the origin must be zero at all times, 𝑡𝑡, since the origin is a point of symmetry. This can be assumed to be true even though the gradient of distributions such as those illustrated in Fig. A13.1 is not defined mathematically. The zero-gradient condition is already satisfied by Eq. A13.2 at 𝜆𝜆/2. The gradient of concentration must also be zero at 𝜆𝜆 (another point of symmetry) for all values of 𝑡𝑡. This is true only if: sin(𝑏𝑏𝑏𝑏) = 0

so that:

𝑏𝑏𝑏𝑏 = 𝑛𝑛π

where 𝑛𝑛 is an integer. The final general solution is: 𝐶𝐶 = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿exp �−𝐷𝐷𝑆𝑆

𝑛𝑛2 π2 𝑛𝑛π𝑥𝑥 𝑡𝑡� cos � � 2 𝜆𝜆 𝜆𝜆

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[A13.4]

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Figure A13.1

The final condition to be satisfied is that the concentration distribution at the start of homogenisation (Fig. A13.1) should be described by Eq. A13.4 (with 𝑡𝑡 equal to zero). Such a distribution can obviously not be described by the latter expression unless the initial distribution happens to be sinusoidal. Because the diffusion equation (Eq. A13.1) is linear however, any number of similar equations (with differing values of 𝑛𝑛) can be added together. This is simply the technique of Fourier analysis (Appendix 2) and leads to a solution, for any initial solute distribution described by 𝑓𝑓(𝑥𝑥), of the form: 𝜆𝜆 𝑛𝑛2 π2 2 𝑛𝑛π𝑥𝑥 𝑛𝑛π𝑥𝑥 𝐶𝐶 = 𝐶𝐶0 + � exp �−𝐷𝐷𝑠𝑠 2 𝑡𝑡� cos � � �   𝑓𝑓(𝑥𝑥)cos � � 𝑑𝑑𝑑𝑑 𝜆𝜆 𝜆𝜆 𝜆𝜆 𝜆𝜆 0

[A13.5]

𝑛𝑛

This solution can be used to determine the concentration distribution at any time. Recall that, using the method described in Appendix 2, the result of performing the above integration can be written down immediately when the 'jumps' in the function and its derivatives are known. Suppose that the measured distribution is parabolic (Fig. A13.1) with a maximum, C𝑚𝑚 , a minimum, 𝐶𝐶𝑚𝑚 , and an average, 𝐶𝐶0 . There is no jump in the function at 0, 𝜆𝜆, 2𝜆𝜆, … 𝑛𝑛, but there is a jump of (8/𝜆𝜆)(𝐶𝐶 𝑚𝑚 − 𝐶𝐶𝑚𝑚 ) in the first derivative at 0 and at 𝜆𝜆. There is no jump in the second and higher derivatives. Using Eq. A2.29 one can thus immediately find that the Fourier coefficients are given by: 16 (𝐶𝐶 𝑚𝑚 − 𝐶𝐶𝑚𝑚 ) 𝑛𝑛2 𝜆𝜆2 Use of this technique is considerably easier than integrating expressions of the type, 𝑥𝑥 2 cos(𝑛𝑛𝑛𝑛). Note that when one assumes an asymmetrical distribution such as the parabolic one, the maximum and minimum values of the distribution are not independent. They are instead related by the requirement that the average concentration should be equal to the original concentration, 𝐶𝐶0 . One can introduce the dimensionless value, 𝐼𝐼, which is defined to be the instantaneous value of the amplitude of the distribution, as compared with its original value; say 𝛿𝛿𝛿𝛿/(𝐶𝐶 𝑚𝑚 − 𝐶𝐶𝑚𝑚 ). Another dimensionless constant, 𝑛𝑛2 π2 𝐷𝐷𝑠𝑠 𝑡𝑡/𝜆𝜆2 (compare with Eq. A12.9), arises naturally in the above calculation. Using this constant, the relaxation time, τn, of that component can be defined: 𝐴𝐴𝑛𝑛 =

𝜏𝜏𝑛𝑛 =

𝜆𝜆2 𝑛𝑛2 π2 𝐷𝐷𝑠𝑠

[A13.6]

For the current purpose the simple sinusoidal concentration variation is quite useful because, as can be seen from Eq. A13.4, the higher-order terms (short wavelengths) decay much more rapidly

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than the longer ones do and the homogenisation process will therefore ultimately be determined by the relaxation time of the lowest-order term, i.e. by 𝜏𝜏1 = 𝜆𝜆2 /π2 𝐷𝐷𝑠𝑠 . If the initial concentration variation is given approximately by: π𝑥𝑥 𝐶𝐶(𝑥𝑥, 0) = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿cos � � 𝜆𝜆 the solution for the lowest-order term is: π𝑥𝑥 𝑡𝑡 [A13.7] 𝐶𝐶(𝑥𝑥, 𝑡𝑡) = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿cos � � exp �− � 𝜆𝜆 𝜏𝜏 where 𝜏𝜏 = 𝜏𝜏1. The maximum concentration at 𝑥𝑥 = 0, 𝐶𝐶 𝑚𝑚 , changes with time according to: 𝑡𝑡 [A13.8] 𝐶𝐶 𝑚𝑚 (𝑡𝑡) = 𝐶𝐶0 + 𝛿𝛿𝛿𝛿exp �− � 𝜏𝜏 giving, after a time, 𝑡𝑡 = 𝜏𝜏 (Eq. A13.6): 𝐶𝐶 𝑚𝑚 (𝜏𝜏) = 𝐶𝐶0 +

𝛿𝛿𝛿𝛿 = 𝐶𝐶0 + 0.37𝛿𝛿𝛿𝛿 𝑒𝑒

[A13.9]

or after a time, t = 3τ:

𝛿𝛿𝛿𝛿 [A13.10] = 𝐶𝐶0 + 0.05𝛿𝛿𝛿𝛿 𝑒𝑒 3 From Eq. A13.6, it can be seen that the secondary dendrite-arm spacing will have a significant effect upon the annealing time, given that the relaxation-time is proportional to 𝜆𝜆2. High solidification rates, which reduce 𝜆𝜆, will have a marked effect upon the reduction of the annealing time. For example, to reduce the amplitude of the concentration variation to 5% of its initial value, the necessary annealing time can be deduced by using Eqs A13.10 and A13.7: 𝐶𝐶 𝑚𝑚 (3𝜏𝜏) = 𝐶𝐶0 +

𝑡𝑡0.05 ≅ 0.3

𝜆𝜆2 𝐷𝐷𝑠𝑠

[A13.11]

Using Eq. A13.11, the annealing temperature required to homogenise (to less than a 5% variation) an alloy with a given dendrite-arm spacing within a given time can be calculated: 𝑄𝑄 [A13.12] 𝑡𝑡𝐷𝐷 𝑅𝑅 ln � 02 � 0.3𝜆𝜆 where 𝑄𝑄 and 𝐷𝐷0 are the activation energy and pre-exponential term, respectively, in the Arrhenius expression for the temperature-dependence of the diffusivity. 𝑇𝑇0.05 ≅

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 325-339 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 14 INTRODUCTION TO THE PHASE-FIELD METHOD* The volumetric energy of a heterogeneous system consisting of two phases is made up of two contributions: that of the bulk phases, and that associated with the interface separating the phases (Sect. 2.8). The atomic structure of a crystal is characterised by long-range order while, in a liquid, it is random … with possibly some short-range order (Fig. A14.1). The atomic density in the crystal, which corresponds to its lattice planes, is periodic while in the liquid it is fairly uniform due to the randomness of the atomic positions. In the interface, which is a transition region between the liquid and the crystal, the amplitude of the atomic density gradually increases up to its maximum value which characterizes the crystal lattice planes. In most metallic systems the interface is not a sharp surface (as simplified in Gibbs thermodynamic treatments) but is a volume of finite thickness 𝛿𝑖 .

Figure A14.1 *

The contribution of A. Karma to this Appendix is gratefully acknowledged.

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It is diffuse and stretches over several (typically 3 to 4) atomic distances where the atoms are partially ordered so as to form a sort of structural mixture having a reduced configurational entropy when compared with that of the liquid. In order to avoid having to track the interface, a phase-field variable, 𝜙, is introduced in order to describe the smooth transition between the solid (perfectly ordered) and the liquid (disordered). It can be regarded as being the envelope of the maxima of the local atomic density. Here, 𝜙 is defined as taking a value of 0 for the liquid phase and 1 for the solid phase. The volumetric free energy, 𝑔(𝑧), of a solid/liquid system in equilibrium at the melting point is characterised by taking equal values in the bulk phases, i.e. 𝑔𝑠 = 𝑔𝑙 , while an additional bell-shaped energy, ∆𝑔, is present over the diffuse interface. When integrated over the thickness, 𝛿𝑖 , of the diffuse interface, the excess volumetric energy (J/m3) associated with the interface corresponds to its surface energy, 𝜎 (J/m2). The 𝑔(𝑧)-function can be regarded as being the slope (gradient) of the phase-field variable, 𝜙, in absolute value: the slope is zero for the homogeneous bulk phases and goes through a maximum at the centre of the diffuse interface.

PHASE-FIELD FOR A PURE TWO-PHASE SYSTEM Free energy of the entire system For the sake of simplicity it is first assumed that the molar volume of atoms is the same in the solid and in the liquid, i.e. the densities of the two phases are equal. The Gibbs free energy per unit volume, 𝑔, is therefore such that 𝑔𝑠 = 𝑔𝑙 at the melting point of the pure element. This also simplifies the formalism when integrating over space to find the total energy of the system. Considering the solidifying system to be a continuum, one uses the phase-field variable, , to denote the liquid e.g. by 𝜙 = 0, and the solid by 𝜙 = 1. The diffuse solid/liquid interface corresponds to 0 < 𝜙 < 1, but those boundaries will be further discussed below. The variable, 𝜙, is therefore used as an additional parameter to the temperature and the composition, in order to describe the thermodynamic state at any location in the relevant domain. A one-dimensional situation will be considered in the following development, but the technique reveals its full power in two and three dimensions. For a pure system and a one-dimensional geometry the total energy (per unit area), 𝐹, can be written as: F = ∫ 𝑓 (𝑇, 𝜙,

∂𝜙 ∂ 2 𝜙 , , … ) dz ∂𝑧 ∂𝑧 2

[A14.1]

where the spatial 𝑧-coordinate is directed along the interface normal. The Gibbs free-energy density, 𝑓, is a function of the temperature, 𝑇, and of the phase-field variable, 𝜙, but also of its gradient, 𝜕𝜙/𝜕𝑧, and its higher-order derivatives. It will be shown that such an expansion permits the description of a diffuse interface having a certain width and steepness. Expanding the integrand, 𝑓, as a Taylor series in 𝜙†gives: ∂𝜙 ∂𝜙 2 ∂2 𝜙 𝑓 = 𝑓0 (𝜙, 𝑇) + 𝑘11 + 𝑘12 ( ) + 𝑘21 2 + ⋯ ∂𝑧 ∂𝑧 ∂𝑧

[A14.2]

Here 𝑓0 (𝜙, 𝑇) is the homogeneous part of the phase state. Upon discarding higher-order terms it is first noted that 𝑘11 must be equal to zero, on the grounds of symmetry. That is, because 𝑓 is a scalar function, it must be invariant with respect to the direction of the gradient, 𝜕𝜙/𝜕𝑧. For this

†

It could also be expanded like the other state variables, 𝑇 and 𝐶, but this makes the formalism more complicated and is not really necessary.

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reason, only terms in even powers of the derivative can appear. On the other hand the fourth term in the integrand of Eq. A14.2 can be integrated (assuming 𝑘21 to be constant): ∂2 𝜙 ∂ ∂𝜙 ∂𝜙 +∞ ∫ 𝑘21 2 dz = ∫ [ (𝑘21 )] 𝑑𝑧 = 𝑘21 | ∂𝑧 ∂𝑧 ∂𝑧 ∂𝑧 −∞

[A14.3]

Taking the phase-field, 𝜙, at the ends of the domain to be constant, one has ∂𝜙/ ∂z|+∞ −∞ = 0, and thus the second derivative of 𝜙 in Eq. A14.2 cancels out. Therefore only the square of the phasefield gradient is retained in the development and one has: ∂𝜙 2 F = ∫ (𝑓0 (𝜙, 𝑇) + ε ( ) ) 𝑑𝑧 ∂𝑧

[A14.4]

where 𝜀 = 𝑘12 . Note that in a three-dimensional problem, F is measured in Joules and thus 𝜀 has units of J/m. In summary the energy of the entire system is represented by the integral of Eq. A14.4, which contains a homogeneous part, 𝑓0 (𝜙, 𝑇), and the square of a gradient term in 𝜙 which is non-zero within the diffuse-interface thickness. The functional F is a number, the value of which depending on the functions appearing in the integral. Homogeneous and inhomogeneous contributions As a two-phase system tries to minimize its total energy, some care must be taken to express the volumetric contribution, 𝑓0 (𝜙, 𝑇). Indeed if 𝑓0 (𝜙, 𝑇) is simply written as a linear combination of the free energy of the two phases, i.e. 𝑓0 (𝜙, 𝑇) = 𝑓𝑠 (T)𝜙 + 𝑓𝑙 (T)(1 − 𝜙), the interface will be totally diffuse. A volume at the melting point of the pure substance, i.e. for which 𝑓𝑠 (𝑇𝑓 ) = 𝑓𝑙 (𝑇𝑓 ), having 50% of solid and 50% of liquid will minimize its energy by having 𝜙 = 0.5 everywhere, since the gradient term in Eq. A14.3 is zero. In order to counter this problem, a double-well potential term, 𝑓𝑑𝑤 (𝜙, 𝑇), is added for phases located within the diffuse interface and the homogeneous part of F in Eq. A14.4 is then given by the sum: [A14.5]

𝑓0 (𝜙, T) = 𝑓𝑑𝑤 (𝜙) + 𝑓𝑖𝑛 (𝜙, T)

where the double-well function, 𝑓𝑑𝑤 (𝜙), separates the homogeneous phases by a barrier of height, ℎ, and 𝑓𝑖𝑛 (𝜙, 𝑇) is a smooth interpolation of the energy of the bulk phases. Although there are various possible choices for the double-well potential, the most common one is defined by the even function: 𝑓𝑑𝑤 (𝜙) = ℎ𝑔(𝜙)

with

𝑔(𝜙) = 𝜙2 (1 − 𝜙)2

[A14.6]

It is to be noted that this double-well potential, within the interval 𝜙 = [0,1], has a shape similar to that of the squared gradient term, i.e. that of a bell. The function, 𝑔(𝜙), is shown in Fig. A14.2: it is zero for 𝜙 = 0 and 𝜙 = 1, and its maximum amplitude (1/16) occurs at 𝜙 = 0.5. This double-well potential introduces a penalty contribution which has an opposite effect to that of the squared gradient term in Eq. A14.4: it has a tendency to make the diffuse interface as thin as possible. As will be seen, the two combined terms result in a finite thickness of the diffuse interface. Note that ℎ has units of J/m3. Concerning the interpolation function, 𝑓𝑖𝑛 (𝜙, 𝑇), of the bulk energy densities, 𝑓𝑙 (𝑇) and 𝑓𝑠 (𝑇), a more general form than the linear expression above might be: 𝑓in (𝜙, 𝑇) = 𝑓𝑠 (𝑇)𝑝(𝜙) + 𝑓𝑙 (𝑇)[1 − 𝑝(𝜙)] = 𝑓𝑙 (𝑇) − Δ𝑓(𝑇)𝑝(𝜙)

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[A14.7]

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where Δ𝑓(𝑇) = 𝑓𝑙 (𝑇) − 𝑓𝑠 (𝑇) is the difference in the volumetric Gibbs free energies between the solid and liquid phases at a temperature, 𝑇. It is therefore given simply by Δ𝑠𝑓 ∆𝑇, where Δ𝑠𝑓 is the volumetric entropy of fusion and ∆𝑇 is the undercooling. At the equilibrium melting point of the pure system, 𝑇𝑓 , i.e. ∆𝑇 = 0, 𝑓𝑠 = 𝑓𝑙 and 𝑓𝑖𝑛 (𝜙, 𝑇𝑓 ) is therefore independent of 𝜙 and can be omitted from the formulation. To be compatible with the overall formalism of the phase-field, the function, 𝑝(𝜙), in Eq. A14.7 must satisfy several conditions. It must firstly satisfy 𝑝(𝜙 = 0) = 0 and 𝑝(𝜙 = 1) = 1, in order to describe properly the interpolation between the two phases. It must secondly satisfy the two conditions: 𝑑𝑝/𝑑𝜙 = 0 for 𝜙 = 0, 1, as does the double-well potential. The integral of the doublewell (Eq. A14.6) satisfies all these conditions (with a normalisation factor) and the interpolation function, 𝑝(𝜙), given by: 𝑝(𝜙) = 𝜙3 (6𝜙2 − 15𝜙 + 10)

[A14.8]

is also shown in Fig. A14.2.

Figure A14.2

In summary, a functional, F, of the free energy of a pure system (Eq. A14.4) has been constructed: it depends upon the distribution of the phase fraction, 𝜙, and temperature, 𝑇. It has three components: (i) an homogeneous part which interpolates the volumetric energies of the solid and liquid phases (Eq. A14.7); (ii) a double-well potential (Eq. A14.6) which takes account of the fact that atoms located within the diffuse interface make an additional energy contribution which tends to make the diffuse interface as sharp as possible; (iii) a gradient term, 𝜀(∂𝜙/ ∂𝑧)2 , which describes the spatial steepness of the phase field. It is the first term to be invariant in energy under rotation and spatial translation and has an opposite effect to that of the double-well potential: it tends to ‘blur’ the interface. The sum effect of these contributions to the free energy (Eq. A14.4) is thus to account for the different states of the liquid and solid, the separation of free energy between them and the surface energy of the diffuse interface. Stationary solid/liquid interface at the melting point As has been seen in the previous section, at 𝑇 = 𝑇𝑓 , Δ𝑓(𝑇𝑓 ) = 0, and in a one-dimensional problem, the free energy (Eqs A14.4 to A14.8) reverts to simply:

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Foundations of Materials Science and Engineering Vol. 103

F = ∫ (𝑓𝑑𝑤 (𝜙, 𝑇𝑓 ) + 𝜀 (

𝜕𝜙 2 ) ) 𝑑𝑧 𝜕𝑧

329

[A14.9]

with the integration bounds here being 𝑧 = ±𝐿/2, two points supposed to be far from the diffuse interface and 𝑓𝑑𝑤 (𝜙, 𝑇𝑓 ) = ℎ𝜙2 (1 − 𝜙)2. Because a system at equilibrium tries to minimize its energy the distribution of the phase, (𝑧), should be such that: 𝛿F(𝜙, 𝑇𝑓 ) F(𝜙 + 𝛿𝜙, 𝑇𝑓 ) − F(𝜙, 𝑇𝑓 ) = =0 𝛿𝜙 𝛿𝜙

[A14.10]

where this expression is here called a functional derivative. The increment, 𝛿𝜙, is a perturbation of the phase field, 𝜙(𝑧), e.g. a Dirac function located at any arbitrary point, z’. Without going into the details of this functional derivative, it can be shown that any function of 𝜙, such as the double-well potential, 𝑓𝑑𝑤 (𝜙), which enters into the functional, F, becomes simply: 𝑑𝑓𝑑𝑤 (𝜙)/𝑑𝜙 i.e. it is the familiar derivative of the function, but taken out of the integral. It is slightly more complicated to show that the gradient term squared in Eq. A14.4 becomes the second derivative of 𝜙, taken out of the integral and multiplied by (−2) (Boettinger et al., 2002). The functional derivative in Eq. A14.10 thus becomes: 𝑑𝑔 𝑑2𝜙 𝑑2𝜙 ℎ − 2𝜀 2 = 2ℎ𝜙(1 − 𝜙)(1 − 2𝜙) − 2𝜀 2 = 0 𝑑𝜙 𝑑𝑧 𝑑𝑧

[A14.11]

Integration of this equation yields the stationary solution, 𝜙(𝑧), of the phase field (see Fig. A14.3 and Exercise 4.24): 𝜀 1/2 𝜙 ) 𝑧 = − ( ) ln ( ℎ 1−𝜙

[A14.13]

𝛿 = 2√

𝜀 ℎ

with

[A14.12]

𝜙(𝑧) =

1 𝑧 (1 − tanh ) 2 𝛿

or

Figure A14.3

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The stationary solution of the phase-field equation, obtained for a pure system at 𝑇 = 𝑇𝑓 , has thus resulted in a diffuse interface, the thickness of which is proportional to the square root of the diffusive-term coefficient, 𝜀, and is inversely proportional to the square root of the double-well potential height, ℎ (Eq. A14.13). Because the hyperbolic tangent which appears in the solution (Eq. A14.12) has asymptotic limits of ±1 at ±∞, it is common to define the actual thickness of the diffuse interface as being the distance which separates 𝜙 = 0.05 and 𝜙 = 0.95. This thickness is then given by 3𝛿 = 6(𝜀/ℎ)1/2. The limit, 𝜀 = 0, makes the interface sharp (as does a double-well potential of infinite height, if 𝜀  0): the solution then becomes a step-function having a discontinuity centred at 𝑧 = 0. The excess energy of the diffuse interface or, in other words the surface energy 𝜎, can be obtained by simply integrating the functional F given by Eq. A14.4 with the solution found in Eq. A14.12, since F = 0 for both solid and liquid phases at 𝑇 = 𝑇𝑓 when there is no interface. It is fairly straightforward to show that the squared gradient term and the double-well potential term contribute equally to the surface energy, 𝜎 (see Exercise 4.25) and that integration of F from 𝑧 = − to + gives‡: 𝜀 ℎ 1/2 ℎ 𝜀 1/2 𝜎= ( ) + ( ) 6 𝜀 6 ℎ

or

𝜎=

(𝜀ℎ)1/2 ℎ = 𝛿 3 6

[J/m2]

[A14.14]

As can be seen, the double-well potential and the phase-field gradient term in Eq. A14.9 contribute equally to the surface energy, 𝜎, of the interface. This interface energy is also the product of the thickness, 𝛿, and the height, ℎ, of the double-well potential with a factor of 1/6. Interface kinetics at a fixed undercooling At small deviations from the equilibrium melting-point, 𝑇 ≲ 𝑇𝑓 , the velocity, 𝑉, of a sharp solid/liquid interface (with Miller indices (ℎ, 𝑘, 𝑙)) is linearly dependent upon the undercooling, Δ𝑇 = 𝑇𝑓 − 𝑇 (Eq. 2.14) 𝑉 = 𝐾ℎ𝑘𝑙 (𝑇𝑓 − 𝑇) = 𝐾ℎ𝑘𝑙 Δ𝑇

[A14.15]

In terms of the volumetric free energy, the driving force, found in Eq. A14.7 for the phase transformation at a temperature, 𝑇 ≲ 𝑇𝑓 , can be approximated by (assuming that the free energies of the solid and liquid are linear functions of T near to the melting point) §: 𝑓𝑖𝑛 (𝜙, 𝑇) = 𝑓𝑠 (𝑇)𝑝(𝜙) + 𝑓𝑙 (𝑇)[1 − 𝑝(𝜙)] ∼ −Δ𝑠𝑓 Δ𝑇𝑝(𝜙)

[A14.16]

where Δ𝑠𝑓 is the volumetric entropy of fusion. The volumetric energy given by this expression, together with the double-well potential, results in unequal energetic minima for the phases; as shown in Fig. A14.4 (the values which have been used to draw 𝑓𝑖𝑛 + 𝑓𝑑𝑤 correspond approximately to those for aluminium, with 𝛿 ≅ 30 nm (a factor about ‡

Note that this expression for 𝜎 can also be derived by treating the interface-thickness, 𝛿, as a variational parameter. Evaluating the excess free-energy of the interface (Eq. A14.9) by using Eq. A14.12 and with 𝛿 being treated as a free parameter yields: 𝜎 = ℎ𝛿 ⁄4 + 𝜀 ⁄𝛿 , where the first term on the RHS of this equation corresponds to the double-well potential, and the second term to the gradient-squared term. The double-well contribution increases with 𝛿 , since 𝜙 is not at its minimum over a wider interfacial region. On the other hand the gradient-squared term contribution decreases as 𝛿 increases because the gradient-squared term penalizes steep gradients for 𝜙. As a result of these two contributions the 𝜎versus-𝛿 relationship goes through a minimum at a particular value of 𝛿 that is obtained by setting 𝑑𝜎/𝑑𝛿 = 0; thus yielding the same value of 𝛿 that is given by Eq. A14.13 and corresponding to the equilibrium  profile. §

In phase-field computations the free energy can be shifted by an arbitrary constant, to retain only those terms responsible for the difference between the solid and liquid phase. In the case, T = Tf, a constant is neglected and, for 𝑇 ≲ 𝑇𝑓 , the energy is shifted so that the energy of the liquid (𝜙 = 0) remains fixed and that of the solid is shifted by Δ𝑠𝑓 Δ𝑇.

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10 as compared with 𝛿𝑖 ) and an undercooling of 5 K). The higher energy-minimum characterizes metastable liquid while the lower energy-minimum corresponds to the stable solid. The system will therefore solidify via propagation of the interface. The question remains as to how one should transform Eq. A14.10 to link the shape of a stationary diffuse interface to a kinetic equation for a moving interface. It is assumed that the temporal derivative of the phase-field, 𝜙, is proportional to the functional derivative of F. This at least permits the recovery of the stationary case previously described (𝛿F/𝛿𝜙 = 0) where the solution, , is independent of time. One thus writes: ∂𝜙 𝛿F = −𝑀𝜙 ∂𝑡 𝛿𝜙

[A14.17]

where 𝑀𝜙 is the mobility of the diffuse interface and will be related below to the physical parameter, 𝐾ℎ𝑘𝑙 , of Eq. A14.15. It describes just how rapidly the interface moves when the functional derivative of F is non-zero; the functional derivative here acting as a driving force (in units of J/m3) for interface movement. As described previously, the functional derivative of the squared gradient term leads to the second spatial derivative of 𝜙 (see Eq. A14.11). The above kinetic equation therefore leads to the following terms when 𝑇 ≲ 𝑇𝑓 : ∂𝜙 𝑑𝑔 𝑑𝑝 ∂2 𝜙 = −𝑀𝜙 [ℎ + Δ𝑠𝑓 Δ𝑇 ] + 2𝜀𝑀𝜙 2 ∂𝑡 𝑑𝜙 𝑑𝜙 ∂𝑧

[A14.18]

Figure A14.4 Plot of the function, ℎ𝑔(𝜙) − Δ𝑠𝑓 Δ𝑇𝑝(𝜙), with ℎ = 30 MJ/m3 and Δ𝑠𝑓 Δ𝑇 = 5 MJ/m3. These values are for aluminium, with Δ𝑇 ≅ 5 K and 𝛿 ≅ 3 × 10−8 m, i.e. an order of magnitude larger than the actual diffuse interface δ𝑖 .

The first term in parenthesis on the RHS of this equation corresponds to the driving force for interface movement when Δ𝑇 ≠ 0, but an interface which still remains constricted due to the doublewell potential, 𝑔(𝜙), while the last term corresponds to a diffusive term: the preceding coefficient, 2𝜀𝑀𝜙 , indeed having the units [m2/s] of a diffusion coefficient. That coefficient must be positive, thus justifying the minus sign in Eq. A14.17, in addition to physical considerations. The solution of Eq. A14.18 is no longer trivial, even for Δ𝑇 = constant, but it has a shape which is similar to that for the stationary solution which is found for Δ𝑇 = 0 (Eq. A14.12). It is

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precisely this equation which is used in numerical computations (and usually extended to 2 or 3 dimensions, see below), coupled to a heat-diffusion equation if the temperature is not fixed so as to describe the evolution of the solid/liquid interface in a pure system. Under steady-state conditions a solution of the form, 𝜙(𝑧 − 𝑉𝑡) = 𝜙(𝑢), is sought. The velocity, 𝑉, at which this (yet unknown) phase-field solution moves can be related to the parameter, 𝑀𝜙 , as follows. One takes the time-derivative of the functional, F. Since F is a function of the phasefield, 𝜙(𝑧, 𝑡), the derivation again involves the functional derivative of F, 𝛿F/𝛿𝜙, which has the units of (J/m3). The chain rule for the time-derivation thus becomes: 𝑑F δF ∂𝜙 =∫ 𝑑𝑧 𝑑𝑡 δ𝜙 ∂𝑡

[A14.19]

where the integration domain runs from - to +. This general expression reveals that the dissipation of energy, 𝑑F/𝑑𝑡, during the movement of the interface is nothing but the product of the driving force, 𝛿F/𝛿𝜙 (J/m3), and of the velocity of the phase-field evolution, ∂𝜙/ ∂𝑡 (s-1), as integrated over distance. Using Eq. A14.17, this expression becomes: 𝑑F δF ∂𝜙 1 ∂𝜙 ∂𝜙 1 ∂𝜙 2 =∫ 𝑑𝑧 = ∫ [− ] 𝑑𝑧 = − ∫ [ ] 𝑑𝑧 ≤ 0 𝑑𝑡 δ𝜙 ∂𝑡 𝑀𝜙 ∂𝑡 ∂𝑡 𝑀 ∂𝑡

[A14.20]

This energy-dissipation is of course negative. Assuming a steady state and using the approximate solution for the stationary case (Eq. A14.12), the integral in Eq. A14.20 is given by 𝑉 2 /3𝛿. So one finds finally that: 𝑑F V2 =− 𝑑𝑡 3𝛿𝑀𝜙

[A14.21]

On the other hand the energy-dissipation is given by the product of the velocity, 𝑉, and the volumetric change of the Gibbs free energy due to solidification, −Δ𝑠𝑓 Δ𝑇. The expression A14.21 is therefore also given by: 𝑑F V2 =− = −Δ𝑠𝑓 Δ𝑇𝑉 𝑑𝑡 3𝛿𝑀𝜙

[A14.22]

from which one can deduce the growth kinetics: 𝑉 = [3Δ𝑠𝑓 𝑀𝜙 𝛿]Δ𝑇 = 𝐾ℎ𝑘𝑙 Δ𝑇

[A14.23]

This equation is formally identical to Eq. A14.15 for a sharp interface, thus allowing one to relate the kinetic parameter, 𝑀𝜙 , in the dynamic phase-field (Eq. A14.17) to the physical law which was introduced for the sharp-interface model (Eq. (2.14), see Chap.2).

TECHNICAL ASPECTS OF THE PHASE-FIELD METHOD The previous section has presented the basic principle of the phase-field method for the case of a pure system; first for a stationary solid/liquid interface at the melting point and then for an interface that propagates under a small fixed undercooling. This section will briefly present some technical complications which arise when applying this technique to practical examples of microstructure prediction.

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Implementation in a numerical program and the thickness of the diffuse interface As has been shown, the phase-field Equation A14.18 is not that different to a (parabolic) diffusion equation in which the first time-derivative of 𝜙 is linked to its second spatial derivative. The only difference is that two additional terms appear: a double-well potential which constrains the width of the diffuse interface to a finite value, and a driving force which causes it to move. The solution of this equation requires the use of numerical techniques such as finite-difference or finite-element methods. As well as the conditions that would traditionally link time-steps, Δ𝑡, and grid-spacings, Δ𝑧, when using an explicit time-stepping scheme, the phase-field equation also imposes other constraints. In order to solve the phase-field equation properly, the grid-spacing, Δ𝑧, must in fact be about 10 times smaller than the thickness, 𝛿, of the diffuse interface. In one-dimensional problems, the numerical implementation of the phase-field method does not present any major difficulties because the grid can be made very fine and the interface remains planar. This is not the case for two- and three-dimensional problems. The physical thickness, 𝛿𝑖 , of a diffuse interface in metallic alloys is in fact in the nanometre range while microstructures typically develop over lengths of several millimetres (a typical grain size) and other features (dendrite tip radius, eutectic spacing, etc) are in the micrometer range. The physical interface thickness is therefore much too small when compared to the other lengths and with present-day computers will make the simulation at least impractical. The carrying-out of phase-field simulations at experimentally relevant length- and time-scales requires the choice of an enlarged interface thickness, 𝛿, which is about 1/10 of the characteristic length-scale of a solidification feature such as the dendrite tip radius, 𝑅. The thickness, 𝛿, of the diffuse interface of the model is thus about 2-3 orders-of-magnitude greater than the physical value of 𝛿𝑖 . The use of a diffuse interface having a thickness, 𝛿, which is much larger than 𝛿𝑖 has physical consequences, particularly with regard to solute-trapping. As explained in Chap. 7, solute-trapping occurs when the time required for solute to equilibrate across the spatially diffuse solid/liquid interface, ~ 𝛿 2 /𝐷𝑖 , where 𝐷𝑖 is the liquid diffusion coefficient within the interface, becomes comparable to the time, 𝛿/𝑉, required for the interface to advance by the distance, 𝛿**. At a low solidification-rate (𝑉 ≪ 𝐷𝑖 /𝛿), the degree of solute-trapping is negligibly small. But if 𝛿 is chosen to be 3 orders-of-magnitude greater than 𝛿𝑖 , a spurious degree of trapping occurs even at low 𝑉. The phase-field model must therefore be formulated in such a way as to eliminate solute-trapping, while nevertheless respecting the solute-conservation condition at the interface. The phase-field model for alloy solidification reduces to the standard sharp-interface solidification-model (involving just the Gibbs-Thomson condition, Eq. 1.5, and the solute fluxbalance at the moving interface, Eq. 3.6). In the so-called “sharp-interface limit” of vanishingly small 𝛿 (𝛿/𝑅 → 0), it reduces to a modified set of sharp-interface equations in the so-called “thin-interface limit” which is relevant to phase-field simulations when 𝛿/𝑅 is not vanishingly small (e.g. 𝛿/𝑅~ 1/10). This was first demonstrated for the solidification of a pure substance by Karma and Rappel (1998) and then for alloy solidification by Karma (2001) and Echebarria et al. (2004). In the alloy case, the corrections which are made to the sharp-interface equations include a jump in chemical potential at the solid/liquid interface which is related to solute-trapping, plus two corrections which are made to the solute-conservation condition at the interface (Stefan condition). This was accomplished by introducing an anti-trapping solute-current that, together with a thin-interface version of the phase-field model, makes it possible to eliminate excess solute-trapping while respecting the solute-conservation condition. This approach has permitted quantitative simulations to be made of isothermal dendritic solidification (Karma (2001)) and columnar growth (Echebarria et al. (2004)) that are independent of the choice of 𝛿 in the phase-field model.

**

Or equivalently when the Péclet number of the interface, 𝑉𝛿/𝐷𝑖 , becomes comparable to 1.

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Even when such corrections are made, the condition, 𝛿/𝑅~ 1/10, that is necessary in order to predict correctly the local curvature, 𝐾, of the microstructure is already a computational challenge in two dimensions, and even more so in three dimensions. The evolution of a dendritic grain within a domain of 1 mm with a tip dendrite tip radius 𝑅 of, say 5 m, requires for example a grid of 2000  2000 nodal points (8 × 109 points in three dimensions). In order to overcome this difficulty some authors have used a dynamic grid that is refined only near to the diffuse interface (Jeong et al., 2001; Burman and Picasso, 2003), thus reducing the total number of nodal points but rendering the program more complex. Two- and three-dimensional problems and anisotropy of the interfacial energy In one-dimensional problems, the phase-field method does not offer a clear advantage over, for example, a front-tracking technique given that the interface geometry remains planar. The beauty of the method lies however in computing the evolution of complex morphologies such as complete dendritic grains, which require dealing with two- and three-dimensional cases. In fact the basic equation remains essentially the same as that for any diffusion calculation and Equation A14.18 becomes, in three dimensions: ∂𝜙 𝑑𝑔 𝑑𝑝 ∂2 𝜙 ∂2 𝜙 ∂2 𝜙 = −𝑀𝜙 [ℎ + Δ𝑠𝑓 Δ𝑇 ] + 2𝜀𝑀𝜙 [ 2 + 2 + 2 ] ∂𝑡 𝑑𝜙 𝑑𝜙 ∂𝑥 ∂𝑦 ∂𝑧

[A14.24]

where the second derivative, ∂2 𝜙/ ∂𝑧 2 , has simply been replaced by the Laplacian of 𝜙. As seen in Chap. 3, the growth of a pure solid in an undercooled liquid is unstable and therefore, in two or three dimensions, the solid/liquid interface develops convexities and concavities. The phasefield method is entirely capable of predicting these morphological developments provided that account is taken of the constraints, mentioned in the previous section, regarding the fineness of the grid. A new challenge appears here however concerning the anisotropy of the solid/liquid interface. It is now well-understood that the formation of dendrites along preferential directions is associated with an anisotropy of the solid/liquid interface; in most cases that of the interfacial energy, 𝜎, (although that of the attachment kinetics can also play a role). In the absence of any anisotropy, the growth of a pure equiaxed grain in an undercooled liquid results in what is termed a sea-weed structure: there are no preferential growth directions and the situation, well predicted by Eq. A14.18, is analogous to pushing one liquid (or gas) into another fluid with which it is immiscible. The anisotropy of the solid/liquid interface energy, which is dictated by the crystallography of the solid, is therefore one of the important material parameters which has to be incorporated into the phase-field method (Caginalp and Fife, 1986; Wheeler and McFadden, 1996; Karma and Rappel, 1998). Without going into too much detail the surface energy, 𝜎, of a solid/liquid interface possesses an anisotropy which is dictated by the solid’s crystallography. It can be characterized (Chap. 2) by: 𝜎(𝜃, 𝜑) = 𝜎0 [1 + 𝜂ℓ𝑚 𝑌ℓ𝑚 (𝜃, 𝜑)]

[A14.25]

where 𝜎0 is the isotropic component of the interfacial energy, 𝜃 and 𝜑 are the two angles of the spherical coordinate system with respect to the crystallographic axes of the solid, 𝑌ℓ𝑚 (𝜃, ) are the spherical harmonics and the 𝑚 are anisotropy coefficients. The combination, 𝑌ℓ𝑚 (𝜃, ), must respect ℓ the symmetry of the crystal and summation over the indices, ℓ and m, is implicit. It is sufficient to consider only the first few terms of the series and, in most metallic systems, the coefficients, 𝑚 , do ℓ not exceed a few percent. They are nevertheless very important in “driving” the growth of the solid along certain directions. One can use an example often considered in phase-field computations to illustrate the results for a two-dimensional case of cubic symmetry. In this case, and retaining only the first term of the development, the interfacial energy can be expressed with respect to the [10] and [01] axes of the crystal by:

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[A14.26]

𝜎(𝜃) = 𝜎0 [1 + 𝜂4 cos (4𝜃)]

Because the interfacial energy, 𝜎, and the coefficient, 𝜀, in the phase-field equation are linked by Eq. A14.14, the energy gradient coefficient also becomes anisotropic: 𝜀(𝜃) = 𝜀0 [1 + 𝜂4 cos (4𝜃)]2 ≅ 𝜀0 [1 + 𝜀4 cos (4𝜃)]

[A14.27]

The phase-field equation (Eq. A14.24) then becomes similar to a diffusion coefficient having anisotropic diffusion coefficients (2𝜀𝑀𝜙 ) which depend upon the orientation. Further details can be found in Dantzig et al. (2013) for example. If the (𝑥, 𝑦) axes of the coordinate system which are used to compute the microstructure are not aligned with the [10] and [01] axes of the two-dimensional crystal, the expression A14.27 is modified simply by replacing 𝜃 with (𝜃 − 𝜃0 ), where 𝜃0 is the angle between the two coordinate systems. Figure A14.5 illustrates the influence of anisotropy upon the development of a Ni-Cu equiaxed grain (see below for the extension of the method to a binary alloy) as calculated using the phase-field method (Henry, 1999). The conditions, taken from Warren and Boettinger (1995), are identical for the two situations shown here, apart from the anisotropy. In the absence of anisotropy (top) destabilisation of the initially spherical nucleus leads to a seaweed structure, with the formation of several doublons (two solid tips growing with a liquid channel between them). With an anisotropy of only 2% (bottom) well-defined dendritic primary and secondary arms grow along 10-directions (directions of maxima in 𝜎 in this simple case or minima of the stiffness); these directions being rotated 9 deg. with respect to the (𝑥, 𝑦) axes.

Figure A14.5 (Henry, 1999)

A final point concerns the anisotropy which is introduced by expressing Eq. A14.24 in terms of finite differences (or finite elements). A numerical scheme which is based upon a regular grid tends to introduce what is known as a small “numerical anisotropy”: this anisotropy has axes which are aligned with grid axes and can be of the same order-of-magnitude as the weak physical anisotropy (a few percent) of most metals. If the axes of the physical anisotropy (e.g. of the 10 axes in cubic metals) coincide with those of the grid, this numerical anisotropy is not noticeable a priori but is already present. When the 10 axes are rotated by an angle, 𝜃0 , with respect to the grid mall differences are then noticed in the velocities and tip radii of the primary dendrite arms, as compared EBSCOhost - printed on 5/26/2023 7:16 AM via . All use subject to https://www.ebsco.com/terms-of-use

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with those found for 𝜃0 = 0. This numerical anisotropy must be compensated-for by subtracting a component of the principal axes, (𝑥, 𝑦), from the physical anisotropy of the axes, 10, in order to guarantee a result which is independent of 𝜃0 . Coupling with heat-transfer The analysis which was performed in the previous section assumed a constant temperature to prevail over the entire system, including the diffuse interface. The use of such an isothermal model permits one to define the equilibrium states or the kinetics of the diffuse interface for a pure system. In most practical cases the temperature is not fixed however and one has to distinguish two different situations. For constrained growth such as Bridgman solidification in which the isotherms are moving at a constant speed, 𝑉𝑇 , in a fixed temperature gradient, 𝐺, one can assume that the temperature field is “frozen”, i.e. that it is not influenced by the growth of the microstructure. This frozen-temperature approximation is of course only of interest for alloys (see next section) since a pure system exhibits a planar front in a positive temperature gradient. As the temperature field is given in this case by 𝑇(𝑧, 𝑡) = 𝑇(𝑧 − 𝑉𝑇 𝑡) = 𝑇(𝑧′), where the 𝑧-axis coincides with the thermal gradient direction, the calculation domain within which the phase-field equation is solved can be translated at the same velocity, 𝑉𝑇 , in order to remain at the location of the dendrite tips or eutectic front (Galilean transformation). The temperature in this reference frame which is moving at 𝑉𝑇 is given by 𝑇(𝑥, 𝑦, 𝑧′) = 𝑇𝑟𝑒𝑓 + 𝐺(𝑧𝑖 ′−𝑧𝑟𝑒𝑓 ) and the time-derivative which appears in the phase-field equation becomes 𝐷/𝐷𝑡 = 𝜕/𝜕𝑡+𝑉𝑇 𝜕/𝜕𝑧’. For equiaxed growth, it has been seen that the solid/liquid interface of a pure system always tends to develop instability due to the latent heat which is released. The approximation of fixedundercooling is usually not met in practice, e.g. recalescence may occur under the more realistic approximation of a fixed heat-extraction rate and in this case the phase-field equation (Eq. A14.24) has to be coupled to the heat-transfer equation. The non-uniform temperature field, 𝑇(𝑥, 𝑦, 𝑧), which appears (as an undercooling) in this equation is given by: 𝑐

∂𝑇 ∂ ∂𝑇 ∂ ∂𝑇 ∂ ∂𝑇 ∂𝜙 (𝜅(𝜙) ) + (𝜅(𝜙) ) + (𝜅(𝜙) ) + Δℎ𝑓 = ∂𝑡 ∂𝑥 ∂𝑥 ∂𝑦 ∂𝑦 ∂𝑧 ∂𝑧 ∂𝑡

[A14.28]

where Δℎ𝑓 is the volumetric enthalpy of fusion (= Δ𝑠𝑓 𝑇𝑓 ), 𝑐 is the volumetric specific heat and 𝜅 is the thermal conductivity (which may be a function of 𝜙 when the thermal conductivities of the solid and liquid are different). This last physical property might differ in the solid and liquid phases, in which case it depends upon 𝜙. The Equations A14.24 and A14.28 are therefore coupled, i.e. the evolution of the phase-field variable, 𝜙(𝑥, 𝑦, 𝑧), depends upon the temperature field, 𝑇(𝑥, 𝑦, 𝑧), in A14.24, and the temperature field, 𝑇(𝑥, 𝑦, 𝑧), is affected by the latent heat (final term of A14.28), which is given by the evolution of 𝜙(𝑥, 𝑦, 𝑧). In a time-stepping calculation, these two equations are usually solved consecutively, i.e. by using the latest calculated field of the other, and vice versa, within an explicit time-stepping scheme. This definitely simplifies the search for a solution to these complex and coupled equations, but with a condition imposed upon the time-step, t: once the grid size Δ𝑧 is fixed, this restricts the time step Δ𝑡: the numerical scheme in 3-dimensional problems is stable only if Δ𝑡 < Δ𝑧 2 /(6𝑎) for the diffusion equation and Δ𝑡 < Δ𝑧 2 /(12𝜀𝑀𝜙 ) for the phase-field equation. Solidification of alloys Most practical cases of solidification deal with alloys, and the phase-field method has been extended to this case by Warren and Boettinger (1995) under conditions involving a fixed undercooling. Because the diffusion of solute is generally much slower than that of heat, a “frozen” temperature assumption (uniform but not necessarily constant undercooling for equiaxed

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solidification, Bridgman conditions for columnar solidification) is reasonable. For the sake of simplicity, consider the equiaxed solidification at a fixed temperature of a binary alloy which is composed of species A and B with the composition, 𝐶, expressed in atomic percent. The solid and liquid phases are assumed moreover to be ideal solutions††. The solidification dynamics of this binary alloy is built upon the free energy developed for a pure system (Eq. A14.4), while taking account of the fact that that the free energy at the temperature considered is now a function of the phase field, 𝜙, and composition, 𝐶. In one dimension the free energy functional, F, becomes: F(𝜙, 𝑇, 𝐶) = ∫ (𝑓0 (𝜙, 𝑇, 𝐶) + ε (

∂𝜙 2 ) ) 𝑑𝑧 ∂𝑧

[A14.29]

The free energy density, 𝑓0 (𝜙, 𝐶), at a given temperature, 𝑇, is shown in Fig. A14.6. The free energies of the solid, 𝑓0 (𝜙 = 1, 𝐶), and liquid 𝑓0 (𝜙 = 0, 𝐶), phases are the parabolic-type curves which are visible in the corresponding planes, 𝜙 = 1 and 𝜙 = 0. A double-well potential is of course added to the interpolation between 𝑓0 (𝜙 = 1, 𝐶) and 𝑓0 (𝜙 = 0, 𝐶) in order to maintain the diffuse interface at a finite width.

Figure A14.6 Gibbs free energy, 𝑓0 (𝜙, 𝐶), of an ideal solution (grey surface) interpolating in (𝜙, 𝐶)-space the Gibbs free energies of the solid, 𝑓𝑠 (𝐶), and the liquid, 𝑓ℓ (𝐶). In this space the common-tangent construction appears as a plane which is parallel to the 𝜙-axis and intersects the 4 corners of the space at the corresponding chemical potential. (Redrawn from the original diagram of Warren and Boettinger (1995)).

The equilibrium which is found by equating the chemical potentials, 𝜇𝐴𝑠 = 𝜇𝐴𝑙 and 𝜇𝐵𝑠 = 𝜇𝐵𝑙 , corresponds to the well-known common-tangent construction: it gives for this temperature the equilibrium compositions, 𝐶𝑠 and 𝐶ℓ , of the solid and liquid phases, respectively. In the phase-field ††

Regular or sub-regular solutions, as well coupling to a thermodynamics database can, and have been, considered for phase-field computations (Böttger et al., 2008).

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Fundamentals of Solidification 5th Edition

formulation, the tangent construction corresponds to a plane which is parallel to the -axis and tangential to the two curves, 𝑓0 (𝜙 = 1, 𝐶) and 𝑓0 (𝜙 = 0, 𝐶). Without giving all of the details, which can be found elsewhere (Warren-Boettinger, 1995), the free energy which appears in the functional, F, at a given temperature is given by: 𝑓0 (𝜙, 𝑇, 𝐶) = 𝑓𝑑𝑤 (𝜙, 𝐶) + 𝑓𝑖𝑛 (𝜙, 𝑇, 𝐶) where: 𝑓𝑖𝑛 (𝜙, 𝑇, 𝐶) = 𝑝(𝜙)𝑓𝑠 (𝑇, 𝐶) + (1 − 𝑝(𝜙))𝑓𝑙 (𝑇, 𝐶) with

[A14.30] [A14.31]

𝑝(𝜙) = 𝜙3 (6𝜙2 − 15𝜙 + 10)

For the sake of simplicity, the height of the double-well potential, ℎ, has been taken to be constant so as to give a thickness, 𝛿, for the diffuse interface which is independent of the local composition‡‡ (see Eq. A14.13). Before detailing the free energy of the solid and liquid phases, 𝑓𝑠 and 𝑓𝑙 , one can derive the kinetic equation (Eq. A14.18) for this case: ∂𝜙 𝛿F ∂2 𝜙 = −𝑀𝜙 = −𝑀𝜙 [ℎ𝑔′ (𝜙) + (𝑓𝑠 (𝑇, 𝐶) − 𝑓𝑙 (𝑇, 𝐶))𝑝′ (𝜙) − 2𝜀 2 ] ∂𝑡 𝛿𝜙 ∂𝑧

[A14.32]

where 𝑔′ (𝜙) = 𝑑𝑔/𝑑𝜙 and 𝑝′ (𝜙) = 𝑑𝑝/𝑑𝜙 have been used to simplify the expression (note that 𝑝′(𝜙) = 30𝑔(𝜙)). Assuming an ideal solution for both the solid and liquid phases (and equal molar volumes of the A and B species) the difference, (𝑓𝑠 (𝑇, 𝐶) − 𝑓𝑙 (𝑇, 𝐶)), can be found in Warren and Boettinger (1995). It is given by: (𝑓𝑠 (𝑇, 𝐶) − 𝑓𝑙 (𝑇, 𝐶)) = Δ𝑠𝑓𝐴 (𝑇𝑓𝐴 − 𝑇)(1 − 𝐶) + Δ𝑠𝑓𝐵 (𝑇𝑓𝐵 − 𝑇)𝐶

[A14.33]

where the volumetric entropy of fusion and the melting-points of the pure elements appear. Because a fixed temperature has been assumed, this kinetic equation for 𝜙(𝑧, 𝑡) also involves the composition field, 𝐶(𝑧, 𝑡). As previously done for the temperature field, an equation which describes the evolution of 𝐶 has to be coupled to the phase-field equation. It is given by Fick’s law but it takes on a slightly more complicated form here for two reasons: firstly, the solid and liquid diffusion coefficients are extremely different (by nearly 3 orders-of-magnitude) and secondly, the local compositions of the solid and liquid phases are different. Thus the diffusion equation takes the form: ∂𝐶 ∂ ∂𝐶 ∂𝜙 = [𝐷(𝜙) ( + 𝑝′ (𝜙)𝜉(𝑇)𝐶(1 − C) )] ∂𝑡 ∂𝑧 ∂𝑧 ∂𝑧

[A14.34]

where

[A14.35]

𝐷(𝜙) = 𝑝(𝜙)𝐷𝑠 + (1 − 𝑝(𝜙))𝐷𝑙 𝜉(𝑇) =

Δ𝑆𝑓𝐵 𝑇𝑓𝐵 Δ𝑆𝑓𝐴 𝑇𝑓𝐴 ( − 1) − ( − 1) 𝑅𝑔 𝑇 𝑅𝑔 𝑇

[A14.36]

and where Δ𝑆𝑓𝐴 and Δ𝑆𝑓𝐵 are the molar entropies of fusion of pure A and B and 𝑇𝑓𝐴 and 𝑇𝑓𝐵 are their corresponding melting-points. This complex expression, in which the gradient of 𝜙 appears, reduces to the standard diffusion equation in the pure solid and liquid phases. This last equation, combined with the phase-field equation (Eq. A14.32), permits the tracking of the evolution of the phases and of the composition, as shown in Fig. A14.5 for Cu-Ni with various anisotropies.

‡‡

As mentioned by Warren and Boettinger (1995), the choice of the phase-field parameters is not totally free. Upon choosing a fixed height, ℎ, for the double-well potential its width, 𝛿, becomes independent of the composition because there is an unique value of 𝜀 (see Eq. A14.13). This choice also implies however that the surface energies, 𝜎𝐴 and 𝜎𝐵 , of pure A and B are also equal in Eq. A14.14), which is not generally true.

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Foundations of Materials Science and Engineering Vol. 103

339

Bibliography for Further Reading ▪ W.J.Boettinger, J.A.Warren, C.Beckermann, A.Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002) 163. ▪ B.Böttger, M.Apel, J.Eiken, P.Schaffnit, I.Steinbach, Phase-field simulation of solidification and solid-state transformations in multicomponent steels, Steel Research International, 79 (2008) 22. ▪ E. Burman, M. Picasso, Anisotropic, adaptive finite elements for the computation of a solutal dendrite, Interfaces and Free Boundaries, 5 (2003) 103. ▪ G.Caginalp, P.Fife, Higher-order phase field models and detailed anisotropy, Physical Review B, 34 (1986) 4940. ▪ J.A.Dantzig, P.Di Napoli, J.Friedli, M.Rappaz, Dendritic growth morphologies in Al-Zn alloys - part II: Phase-field computations, Metallurgical and Materials Transactions A, 44 (2013) 5532. ▪ B.Echebarria, R.Folch, A.Karma, M.Plapp, Quantitative phase-field model of alloy solidification, Physical Review E, 70 (2004) 061604. ▪ S.Henry, Etude de la germination et de la croissance maclées dans les alliages d'aluminium, PhD Thesis #1943, EPFL (1999) p.135. ▪ A.Karma, W.J.Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Physical Review E, 57 (1998) 4323. ▪ A.Karma, Phase-field formulation for quantitative modeling of alloy solidification, Physical Review Letters, 87 (2001) 115701. ▪ J.H.Jeong, N.Goldenfeld, J.A.Dantzig, Phase field model for three-dimensional dendritic growth with fluid flow, Physical Review E, 64 (2001) 041602. ▪ J.S.Langer, Models of pattern formation in first-order phase transitions, in Directions in Condensed Matter Physics, World Scientific, Singapore (1986) p.165. ▪ L.V.Mikheev, A.A.Chernov, Mobility of a diffuse simple crystal-melt interface, Journal of Crystal Growth, 112 (1991) 591. ▪ O.Penrose, P.C.Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990) 44. ▪ N.Provatas, K.R.Elder, Phase-Field Methods in Materials Science and Engineering. Weinheim: Wiley-VCH (2010). ▪ X.Tong, C.Beckermann, A.Karma, Q.Li, Phase-field simulations of dendritic crystal growth in a forced flow, Physical Review E, 63 (2001) 061601. ▪ S.L.Wang, R.F.Sekerka, A.A.Wheeler, B.T.Murray, S.R.Coriell, R.J.Braun, G.B.McFadden, Thermodynamically-consistent phase-field models for solidification, Physica D, 69 (1993) 189. ▪ J.A.Warren, W.J.Boettinger, Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metallurgica et Materialia, 43 (1995) 689. ▪ A.A.Wheeler, G.B.McFadden, A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics, European Journal of Applied Mathematics, 7 (1996) 367.

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Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 341-342 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

APPENDIX 15 RELEVANT PHYSICAL PROPERTIES FOR SOLIDIFICATION The properties listed here are intended for use in the exercises. They are therefore consistent with each other but should not be assumed to be the most accurate data available. Most of them have been taken, or deduced, from data in the 5th edition of Metals Reference Book (C.J.Smithells, Butterworths, London, 1976). (Volumetric properties such as Δℎ𝑓𝑓 have been calculated on the basis of the density at the melting point). Properties of Pure Materials at the Melting Point Property 𝑇𝑇𝑓𝑓 𝑇𝑇𝑓𝑓 Δℎ𝑓𝑓 Δ𝑠𝑠𝑓𝑓 𝜅𝜅𝑙𝑙 𝜅𝜅𝑙𝑙 𝑐𝑐𝑙𝑙 𝑐𝑐𝑠𝑠 𝜌𝜌𝑙𝑙 �𝑇𝑇𝑓𝑓 � 𝜌𝜌𝑠𝑠 �𝑇𝑇𝑓𝑓 � 𝑎𝑎𝑙𝑙 𝑎𝑎𝑠𝑠 𝑀𝑀 𝑣𝑣 𝑆𝑆,𝑚𝑚 Δℎ𝑓𝑓 /𝑐𝑐𝑙𝑙 𝜎𝜎 Γ 𝑠𝑠𝑇𝑇 Δ𝑣𝑣/𝑣𝑣

Units

°C K J/m3 J/m3K W/mK W/mK J/m3K J/m3K kg/m3 kg/m3 m2/s m2/s kg/mol m3/mol K J/m2 mK m -

Al

660.4 933.6 9.5 × 108 1.02 × 106 95 210 2.58 × 106 3.0 × 106 2.39 × 103 2.55 × 103 37 × 10-6 70 × 10-6 27 × 10-3 11 × 10-6 368 160 × 10-3 1.6 × 10-7 0.24 × 10-9 6.5 × 10-2 #

metastable,

Cu

1084.9 1358 1.62 × 109 1.2 × 106 166 244 3.96 × 106 3.63 × 106 8.0 × 103 8.35 × 103 42 × 10-6 67 × 10-6 63.5 × 10-3 8.3 × 10-6 409 177 × 10-3 1.5 × 10-7 0.37 × 10-9 4.2 × 10-2 ##

δ-Fe (γ-Fe#)

1538 (1526) 1811 (1799) 1.93 × 109 1.07 × 106 35 33 5.74 × 106 5.73 × 106 7.0 × 103 7.25 × 103 6.1 × 10-6 5.8 × 10-6 55.8 × 10-3 7.7 × 10-6 336 204 × 10-3 1.9 × 10-7 0.57 × 10-9 3.6(4.1) × 10-2

succinonitrile - CH2(CN)2CH2

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SCN##

58.08 331.23 4.6 × 107 1.4 × 105 0.223 0.225 2.0 × 106 2.0 × 106 0.988 × 103 1.05 × 103 0.116 × 10-6 0.112 × 10-6 80 × 10-3 76 × 10-6 23 9 × 10-3 0.64 × 10-7 2.78 × 10-9 6 × 10-2

342

Fundamentals of Solidification 5th Edition

Properties of Aluminium Alloys and Succinonitrile-Acetone Property 𝐶𝐶0 𝑇𝑇𝑙𝑙 Δ𝑇𝑇0 𝑇𝑇𝑒𝑒 𝐶𝐶𝑒𝑒 𝐶𝐶′ 𝑚𝑚𝛼𝛼

mβ

𝑘𝑘𝛼𝛼 𝑘𝑘𝛽𝛽 𝑓𝑓𝛽𝛽 𝐷𝐷𝐼𝐼 𝐷𝐷𝑠𝑠 Γ𝛼𝛼 Γ𝛽𝛽 𝑠𝑠𝑐𝑐 ### 𝑃𝑃′

Units

wt% °C(K) K °C(K) wt% wt% K/wt% K/wt% m2/s m2/s mK mK m -

Al-Cu

Al-Cu#

2 656 (929) 32 -2.6 0.14 3 × 10-9 3 × 10-13 2.4 × 10-7 54 × 10-9 -

#eutectic,

##

33.1 548 (821) 33.1 46.5 -4.9 3.3 0.18 0.05 0.46 3.4 × 10-9 2.4 × 10-7 0.55 × 10-7 3.36 × 10-2

metastable,

Al-Si

6 624 (897) 240## -6 0.13 3 × 10-9 1 × 10-12 2 × 10-7 6.4 × 10-9 -

###

Al-Si#

12.6 577 (850) 12.6 98.2 -7.5 17.5 0.13 2 × 10-4 0.127 5.5 × 10-9 1.96 × 10-7 1.7 × 10-7 8.9 × 10-3

𝑠𝑠𝑐𝑐 ≅ Γ/Δ𝑇𝑇0 𝑘𝑘 when Ω ≪ 1

SCN-ACE

1.3 54.25 (327.4) 32.8 -2.8 0.1 1.3 × 10-9 0.64 × 10-7 19.5 × 10-9 -

Properties of Iron Alloys Property 𝐶𝐶0 𝑇𝑇𝑙𝑙 Δ𝑇𝑇0 𝑇𝑇𝑒𝑒 𝑜𝑜𝑜𝑜 𝑇𝑇𝑝𝑝 𝐶𝐶𝑒𝑒 𝑜𝑜𝑜𝑜 𝐶𝐶𝑝𝑝 𝐶𝐶′ 𝑚𝑚𝛼𝛼

mβ

𝑘𝑘𝛼𝛼 𝑘𝑘𝛽𝛽 𝑓𝑓𝛽𝛽 𝐷𝐷𝐼𝐼 𝐷𝐷𝑠𝑠 Γ𝛼𝛼 Γ𝛽𝛽 𝑠𝑠𝑐𝑐 ### 𝑃𝑃′

Units

wt% °C(K) K °C(K) wt% wt% K/wt% K/wt% m2/s m2/s mK mK m #

δ Fe-C

0.09 1531 (1804) 36 1493 (1766)[p] 0.53 [p] -81 0.17 2 × 10-8 6 × 10-9 1.9 × 10-7 30 × 10-9 -

eutectic

γ Fe-C

0.6 1490 (1763) 72 1155 (1428)[e] 4.26 [e] -65 0.35 2 × 10-8 1 × 10-9 1.9 × 10-7 7.5 × 10-9 ###

γ Fe-C#

4.26 1155 (1428)[e] 4.26 [e] 97.9 -140 400 0.49 0.001 0.071 2 × 10-8 2 × 10-7 2 × 10-7 3.7 × 10-3

𝑠𝑠𝑐𝑐 ≅ Γ/Δ𝑇𝑇0 𝑘𝑘 when Ω ≪ 1

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γ Fe-Ni

10 1503 (1776) 6 -2.4 0.8 7.5 × 10-9 3 × 10-13 2 × 10-7 42 × 10-9 -

Foundations of Materials Science and Engineering ISSN: 2297-816X, Vol. 103, pp 343-348 © 2023 Trans Tech Publications Ltd, Switzerland

Online: 2023-02-20

Symbols For a better readability of the equations the authors avoid complex symbols. Therefore many symbols have several meanings which are however always defined when they are used in the text. Symbol

Meaning

Definition

Units

𝐴

surface or cross-sectional area

-

m2

𝐴

gradient term

𝑘𝑉𝐺𝑐 /[𝑉𝑝 − 𝐷𝑏]

%/m

𝐴′

surface area of casting

-

m2

𝐴(𝑃𝑐)

normalised dendrite tip composition, 𝐶𝑙∗ /𝐶0

[1 − (1 − 𝑘) I( 𝑃𝑐 )]−1

𝐵

constant

-

-

𝐶

concentration

-

at%, wt%

𝐶𝑒

eutectic composition

-

at%, wt%

𝐶′

length of eutectic tie-line

-

at%, wt%

concentration at the solid/liquid interface

-

at%, wt%

𝐶0 𝐶0̅

initial alloy concentration

-

at%, wt%

dimensionless alloy concentration

C0|m|𝛩𝑡

-

𝐷

diffusion coefficient in liquid

-

m2/s

𝐷0

pre-exponential term (diffusion)

-

m2/s

𝐷𝑖 𝐷𝑠,𝑙 𝐸

interface diffusion coefficient diffusion coefficient in solid, liquid energy

-

m2/s m2/s J

𝐸

internal energy

-

J/mol

𝐸1

exponential integral function

Appendix 8

-

𝐹

stability parameter

(𝜀̇/𝜀)(𝑚𝐺𝑐 /𝑉)

K/m2

F

free energy functional

Appendix 14

J

𝐺

Gibbs free energy

-

J/mol

𝐺 𝐺̄

interface temperature gradient

𝑑𝑇/𝑑𝑧

K/m

mean temperature gradient

𝜅𝑠 𝐺𝑠 + 𝜅𝑙 𝐺𝑙

𝐺𝑐

interface concentration gradient in liquid

(𝑑𝐶𝑙 /𝑑𝑧)𝑧=0

at%,wt%/m

𝐺𝑙

temperature gradient in liquid

(𝑑𝑇𝑙 /𝑑𝑧)𝑧=0

K/m

𝐺𝑠

temperature gradient in solid

K/m

𝐺∗

effective temperature gradient

(𝑑𝑇𝑠 /𝑑𝑧)𝑧=0 𝜅𝑙 𝐺𝑙 𝜉𝑙 + 𝜅𝑠 𝐺𝑠 𝜉𝑠 (𝜅𝑠 + 𝜅𝑙 )

𝐻

enthalpy (molar)

-

J/mol

𝐻𝑙

cubic harmonics (for surface energy anisotropy, 𝑙 = 4, 6, ..)

-

-

𝐼

nucleation rate

Appendix 2

/m3s

𝐽

mass flux

-

/m2s

𝐾

curvature

1/𝑟1 + 1/𝑟2

/m

𝐾

constant

-

-

𝐾ℎ𝑘𝑙

kinetic coefficient for an interface

𝑉[ℎ𝑘𝑙] /ΔT

m/s K

𝐶

∗

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K/m

344

Fundamentals of Solidification 5th Edition

𝐿

length

-

m

𝑀

atomic (molecular) weight

-

g/mol

𝑀𝜙

mobility of an interface

Appendix 14

m3/J s

𝑁

number

-

-

𝑁𝑛

number of clusters containing n atoms

-

23

𝑁𝐴

Avogadro's number

6.022x10

𝑃

pressure

-

𝑃

Péclet number

𝐿/𝛿

𝑃

Growth restriction factor (high ΔT)

𝑇0

K

𝑃′

series in Jackson-Hunt eutectic model

(Appendix 10)

-

𝑃𝑐

solute Péclet number

𝑉𝑅/2𝐷

-

𝑃𝑡

thermal Péclet number

𝑉𝑅/2𝑎

-

Pr

Prandtl number

𝑎/𝐷

-

𝑄

activation energy for diffusion

-

J/mol

𝑄

quantity of heat

-

J

𝑄

Growth restriction factor (low ΔT)

k𝑇0

K

𝑅 𝑅̅

radius

-

m

dimensionless radius

𝑅/𝑠𝑡

-

Re

Reynolds number

𝑅𝑉/ν

-

𝑅𝑔

gas constant

-

J/mol K

𝑆

Entropy (molar)

-

J/mol K

𝑆

perturbation term

𝜀 sin( 𝜔𝑦)

m

𝑆

stiffness of s/l interface

-

J/m2

Sc

Schmidt number

ν/𝐷

-

𝑇

temperature

-

K

T* 𝑇̇

interface/tip temperature

-

K

cooling rate

𝑑𝑇/𝑑𝑡

K/s

𝑇𝑓

melting point of pure substance

-

K

𝑇𝑙

liquidus temperature

-

K

𝑇𝑚

mould temperature

-

K

𝑇0

temperature of equal free energy of two phases

-

K

𝑇𝑝

plane front temperature

-

K

𝑇𝑞

measurable temperature

-

K

𝑇𝑠

solidus temperature

-

K

𝑇𝑠′

non-equilibrium solidus

-

K

𝑉

rate of interface movement

-

m/s

𝑉𝑎

critical growth rate for absolute stability

-

m/s

𝑉𝑏

travel velocity of a laser/electron beam source

-

m/s

𝑉𝑐

critical growth rate for constitutional undercooling

-

m/s

𝑉′

rate of crucible movement

-

m/s

𝑉(ℎ𝑘𝑙)

growth rate of a plane (hkl)

-

m/s

𝑉ℎ𝑘𝑙

growth rate of a dendrite along hkl direction

-

m/s

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/mol Pa

Foundations of Materials Science and Engineering Vol. 103

345

𝑉𝑙

fluid flow velocity

-

m/s

𝑉𝑙∞

fluid flow far from an obstacle

-

m/s

𝑉0 𝑉̅

limiting crystallisation velocity

~ sound velocity

m/s

dimensionless velocity

𝑉𝑠𝑡 /2𝑎𝑙

-

𝑋

mole fraction

-

-

𝑌

partial solution to differential equation

Appendix 2

-

𝑌𝑙𝑚

spherical harmonics

-

-

𝑍

partial solution to differential equation

Appendix 2

-

𝑎

thermal diffusivity

𝜅/𝑐

m2/s

𝑎

separation constant

Appendix 2

M

𝑎

half-axis of ellipsoid

-

m

𝑏

half-axis of ellipsoid

-m

𝑏

exponent in stability analysis

𝑐

volumetric specific heat

-

J/m3K

𝑐𝑝

specific heat

-

J/kgK

𝑐∗

effective specific heat

Appendix 1

J/m3K

𝑑

exponent

-

-

𝑑

distance

-

m

𝑑

grain size

-

m

𝑒

exponent

-

-

𝑓

force

-

N

𝑓

volume fraction of -phase in eutectic

-

-

𝑓[ℎ𝑘𝑙]

crystallographic factor

Appendix 5

-

𝑓𝑔

fraction of equiaxed grains

1 − exp(−𝑓𝑔𝑒 )

-

𝑓𝑔𝑒

extended fraction of equiaxed grains

-

-

M 2

2 0.5

(𝑉/2𝐷) + [(𝑉/2𝐷) + 𝜔 ]

∗

/m

𝑓𝑖

fraction of solid within equiaxed grains

= Ω(𝑇 )

-

𝑓𝑙

volume fraction of liquid

𝑣𝑙 /(𝑣𝑙 + 𝑣𝑠 )

-

𝑓𝑠

volume fraction of solid

-

-

𝑓𝑠𝑒

extended fraction of solid for equiaxed globular grains

𝑔

double-well potential

Appendix 14

-

ℎ

Planck's constant

6.6310-34

Js

ℎ

height of double well potential

Appendix 14

J/m3

ℎ

heat transfer coefficient

𝑞/𝛥𝑇𝑔

W/m2K

ℎ ℎ̇

enthalpy per unit volume

-

J/m3

heat extraction rate per unit volume

-

J/m3 s

𝑘

equilibrium distribution coefficient

𝐶𝑠 /𝐶𝑙

𝑘𝐵

Boltzmann's constant

1.3810

𝑘𝑣

non-equilibrium distribution coefficient

Appendix 6

-

𝑙

length

-

m

𝑙𝐷

diffusion length

𝐷/𝑉

m

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-

-23

J/K

346

Fundamentals of Solidification 5th Edition

𝑙𝑇

thermal diffusion length for an undercooled melt

𝑎/𝑉

m

𝑙𝑀 𝑇

thermal length of the mushy zone

Δ𝑇0 /𝐺

m

𝑚

liquidus slope

𝑑𝑇𝑙 /𝑑𝐶

K/at%, K/wt%

𝑚

mass

-

kg

𝑚

normal to isoconcentrates

-

-

m’ 𝑛

velocity dependent liquidus slope number

Eq. 7.5 -

𝑛

exponent

-

-

𝑛

interface normal

-

-

𝑛

grain number density

-

/m3

𝑛𝑠

adsorption site density

Appendix 4

-

𝑝

probability

-

-

𝑝

complementary distribution coeff.

(1 − 𝑘)

-

𝑝

interpolation function for the phase field

Appendix 14

-

𝑞

heat flux

-

W/m2

𝑟

radius

-

m

critical nucleation radius

-

m

𝑠

position of s/l interface

-

m

𝑠𝑐

solute capillarity length

Γ/𝑚𝐶0 (𝑘 − 1)

m

𝑠𝑡

thermal capillarity length

Γ𝑐/𝛥ℎ𝑓

m

𝑡

time

-

s

𝑡𝑓

local solidification time

-

s

𝑢

back-diffusion parameter

(1 − 2𝛼 ′ 𝑘)

-

𝑣

volume

-

m3

𝑣′

atomic volume

-

m3

𝑣𝑚

molar volume

-

m3/mol

𝑤

work

-

J

𝑥

coordinate in s/l interface

-

m

𝑦

coordinate in s/l interface

-

m

𝑧

coordinate normal to a planar s/l interface

-

m

𝑧′

system coordinate

-

m



dimensionless entropy of fusion

Δ𝑆𝑓 /𝑅𝑔

-



dimensionless coefficient for back-diffusion

𝐷𝑠 𝑡𝑓 /𝐿2

-

′

Dimensionless coefficient for back-diffusion

Appendix 12

-



Gibbs-Thomson coefficient

𝜎/Δ𝑠𝑓

Km

𝛿

boundary layer

-

m

𝑐′

solute boundary layer thickness

2𝐷/𝑉

m

𝑐 𝑖 𝑠

characteristic diffusion length

𝐷/𝑉

m

interatomic jump distance

-

m

solute boundary layer thickness in solid

2𝐷𝑠 /𝑉

m

𝑟

○

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K/at%, K/wt% -

Foundations of Materials Science and Engineering Vol. 103

347

𝑡

thermal boundary layer thickness

2𝑎/𝑉

m

δ𝑇

thermal boundary layer with fluid flow

Sect. 4.7

m

𝐶0

concentration difference between liquidus and solidus at solidus 𝐶0 (1 − 𝑘)/𝑘 temperature of alloy

at%, wt%

𝐺

total Gibbs free energy per mole

J/mol

𝐺𝑑

activation free energy for diffusion across s/l interface

𝐺 •

standard free energy

-

J/mol

𝐺

○

-

J/mol

activation energy for nucleation of critical cluster radius

-

J

Δ𝐺𝑛○

activation energy for critical cluster with n atoms

-

J

𝑔

Gibbs free energy per unit volume

Δ𝐺/𝑣𝑚

J/m3

𝐻 •

standard enthalpy

-

J/mol

𝐻𝑓

latent heat of fusion per mole (>0)

-

J/mol

ℎ𝑓

latent heat of fusion per unit volume

Δ𝐻𝑓 /𝑣𝑚

J/m3

𝐻𝑣

latent heat of vaporisation

-

J/mol

standard entropy

-

J/mol

𝑆𝑓

entropy of fusion per mole

Δ𝐻𝑓 /𝑇𝑓

J/mol K

𝑠𝑓

entropy of fusion per unit volume

Δℎ𝑓 /𝑇𝑓

J/m3K

𝑇

undercooling

𝑇𝑓 − 𝑇

K

Δ𝑇 ∗

dendrite tip undercooling

-

K

𝑇′

dendrite tip-to-root temperature difference

-

K

critical undercooling

Fig. 7.13

K

𝑇0

liquidus-solidus range at C0

𝑇𝑙 − 𝑇𝑠

K

𝑇𝑘

attachment kinetic undercooling

-

K

𝑇𝑟

curvature undercooling

Appendix 3

K

𝑇𝑐

solute undercooling

Appendix 8

K

𝑇𝑡 ̅̅̅̅ 𝑇

undercooling due to heat flow

Appendix 8

K

dimensionless undercooling

Δ𝑇/𝑡

-

  

surface tension

-

N/m

amplitude of perturbation

-

m

parameter for the phase-field gradient term

Appendix 14

J/m

𝜀̇

growth rate of amplitude

𝑑𝜀/𝑑𝑡

m/s

𝑖   𝑚 𝑙

coefficients for the surface energy anisotropy

-

-

dynamic viscosity

-

Pa s

shape factor

Appendix 4

-

coefficients of the surface energy anisotropy

Appendix 14

-

𝜃, 𝜑

angles (spherical coordinates system)

-



𝑐 𝑡

unit solutal undercooling

Δ𝑇0 𝑘 𝐴( 𝑃𝑐 )

K

unit thermal undercooling

Δℎ𝑓 /𝑐𝑙

K

 ̅  

thermal conductivity

-

W/Km

relative thermal conductivity

Appendix 7

W/Km

wavelength

-

m

spacing

-

m

𝑆

Δ𝑇

•

+

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348

Fundamentals of Solidification 5th Edition



chemical potential

𝜇𝑘

atom attachment kinetic coefficient

𝑣

adsorption frequency

Appendix 4

/s

𝑣

kinematic viscosity

-

m2/s

𝑣0

atomic frequency

-

/s

   0 ∗

stability parameter

Appendix 7

-

density

-

kg/m3

solid/liquid interface energy

-

J/m2

isotropic part of the surface energy

-

J/m2

stability constant

Appendix 9

-



correction term for back-diffusion

Appendix 12

-



reduced temperature

𝑅𝑔 𝑇/Δ𝐻𝑣

-



relaxation time

-

s

𝜙

degree of constitutional undercooling

𝑚𝐺𝑐 − 𝐺

K/m

𝜙

phase-field function

Appendix 14

-

 

reduced coordinate

Appendix 1

-

azimuthal angle

-



′

series in high Péclet number eutectic model

-

-

ψ𝑚 𝑙

spherical harmonics of surface energy anisotropy

Appendix 14

-

Ω

dimensionless solutal supersaturation

Appendix 8

-

Ω

interaction parameter

Appendix 3

-

Ω𝑡

dimensionless thermal supersaturation

Appendix 8

-



wave number

2𝜋/𝜆

/m

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-

J/mol 𝑅𝑔 𝑇𝑓 𝑉⁄(Δ𝑆𝑓 𝑉0 )

K

Keyword Index Columnar Zone

A Absolute Stability Activation Energy Adaptive Mesh Additive Manufacturing Alloy Dendrite Alloy Types Amorphous State Amplification Rate of Instability Anisotropy of Growth Anisotropy Parameter Antitrapping Current Atomic Density Atomic Structure Attachment Kinetics Averaging Method

149 17 67 1, 149, 173 67, 149, 277, 285 1 149 51, 149, 265 17 17 67, 325 17, 325 17, 255 17, 149, 325 213

B Back Diffusion Banding Bond Number Boundary Conditions Coupling Boundary Conditions Flux Boundary Layer Branching Eutectic Bridgman Method Bulk Metallic Glass (BMG)

133, 313 149, 173 17 223 223 51, 67, 133, 309, 313 109 1 17

Casting Cell-Dendrite Transition Cells Cellular Automaton Chemical Potential Classical Nucleation Theory Cluster Cluster Size Colloids Columnar-to-Equiaxed Transition

Constitutional Undercooling Constitutional Undercooling (Criterion) Constrained Growth Contact Angle Continued Fraction Continuous Casting Convection Cooling Curve Cooling Rate Cooperative Growth Coupled Zone (CZ) Criterion Critical Nucleus Critical Nucleus Size Crystallisation Entropy Crystallography of Dendrites Curvature Curvature Undercooling Cylindrical Coordinates

67, 173, 265, 285, 325 17 67, 277 1, 173 67, 173 17, 67, 173 1, 17, 67, 173 109 109, 173 67, 149, 285 17, 251 17 17 67 1, 17, 67, 109, 223, 241, 297 17, 67, 109, 149, 265, 277, 285, 297 223

D

C Capillarity Capillary Length Cast Iron (Fe-C)

Competitive Growth Composite Concentration Gradient

1, 67, 133, 173 1, 67, 173 109 51, 109, 133, 149, 223 51, 67, 109, 173, 223 51, 149

1, 109, 241 51, 67, 285 1, 17, 109, 133, 173, 341 1 149, 173 51, 67, 149 67, 173 67, 149, 241 17, 251 17, 251 17 17 173

Deformation Dendrite Arm Thickening Dendrite Detachment of Arms Dendrite Trunk Dendrite Trunk Spacing Density Functional Theory (DFT) Differential Equation Diffuse Interface Diffusion around Dendrite Tip Diffusion around Equiaxed Grains Diffusion Coefficient Diffusion Coupled Growth Diffusion in Liquids

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1, 17, 67 17 67 67 67, 285 17 223 17, 67, 325 67, 149 173 17, 67, 133, 149, 173, 259 109 17, 133, 255, 277, 325

350

Fundamentals of Solidification 5th Edition

Diffusion Limited Diffusionless Transformation Direct Metal Laser Melting (DMLM) Directional Growth Directional Growth Equation Directional Solidification Dislocation Mechanism Distribution Coefficient Driving Force Constitutional Undercooling Driving Force Crystallisation Dynamic Molecular Model

67 149 173 51, 67, 109, 149, 173, 223, 285 223 1, 67, 109, 173 17 1, 51, 133, 149, 173 17, 51, 67 259 17

E EBSD Embedded Atom Method (EAM) Entropy of Fusion Epitaxial Single Crystal Deposition Equiaxed Dendrite Equilibrium Equilibrium Melting Range Equivalent Boundary Layer Error Function Eutectic Al-Si Eutectic Coupled Zone Eutectic Crystallography Eutectic Curvature Undercooling Eutectic Dendrites Eutectic Diffusion-Coupled Growth Eutectic Divorced Eutectic Equiaxed Eutectic Faceted/Non-Faceted Eutectic Fibrous Eutectic Growth Eutectic Interdendritic Eutectic Spacing Eutectoids Exponential Integral Extended Fraction of Grains Extremum Growth Criterion

51, 173 17 17, 109, 325 173 67, 285 133, 313 133 51, 133, 223, 309, 313 213 1, 17, 109, 173, 341 109, 173 109 109 109 109 109 173 17, 109, 173 109, 297 1, 109, 149, 297 1 109, 149, 297 109, 149 67, 213, 277 173 67, 173, 285

F Faceted Interface Far-Field Boundary Condition Fibrous Eutectic Fick's Law Finite Difference Method Finite Element Method Flake Graphite Fluid Flow Flux Balance

Form Factor Fourier Numbers Fourier Series Fourier’s First Law Fragmentation Freckles Free Growth Freezing Range Front Tracking Method

17 223 109, 297 223, 325 67, 325 67, 325 17 67, 173 17, 51, 67, 109, 149, 213, 265, 277, 285, 309 251 313 223, 265 51 17, 67 67 17 67 17, 213, 325

G Gas Turbine Blade Gibbs Free Energy Gibbs-Thomson Effect Gibbs–Thomson Effect Glass Grain Boundary Grain Growth Restriction Factor Grain Refiner Grain Size Graphite Flake Graphite Nodular Growth Constrained Growth Steps Gulliver-Scheil Equation

1, 67, 173 17, 51, 149, 241, 251, 325 1, 17, 173, 265, 297 51, 149 1, 17, 173, 265, 297 67, 173 173 17 173 17 17, 109 67, 173, 265, 285, 325 17 133, 149

H Habit Heat Affected Zone (HAZ) Heat Diffusion Heat Flux

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17 173 213, 265 1, 17, 173, 223, 265

Foundations of Materials Science and Engineering Vol. 103 Heat Treatment Hemispherical Needle Heterogeneous Nucleation Homogenisation Hypereutectic Composition Hypoeutectic Composition

133 67, 109, 277, 285 17, 173, 241 133, 223 173 109, 173

Interface Curvature Interface Energy Interface Energy Anisotropy Interface Instability Interface Kinetics Interface Perturbation Interface Structure Interface Temperature Interface Tension Interface Width Invariant Equilibrium Irregular Ivantsov Function

1 17 17, 173 109 109 1, 173 17, 173 1 241 109, 173 133 133, 321 67, 109, 133, 149, 173 1, 17, 149, 241 1, 17, 67 17, 67, 241, 325 1, 51, 109 251, 325 51, 149, 223, 265 17, 255 109 17, 241 325 109 17, 109, 173 67, 173, 277

K Kinetics

17, 173, 213, 251

L Lamellae-Fibre Transition Lamellar Lamellar Eutectics Laser Beam Laser Metal Forming (LMF)

Lateral Diffusion Lever Rule Liquid Liquidus Slope Local Equilibrium

I Ice Icosahedron Impingement of Grains In Situ Composites Inclined Interface Instability Inner-Equiaxed Zone Inoculation Instabilities Interaction Parameter Interdendritic Interdendritic Precipitation Interdendritic Segregation Interface Cellular

Laser Treatment Latent Heat of Fusion

109 109, 297 109, 297 149, 173 173

Local Solidification Time Low Angle Grain Boundary

351 149, 173 1, 17, 173, 255 109, 149, 223 133, 313 67, 149 1, 149, 241, 259, 313 51, 67, 109, 149, 241 67 51

M Macro-Segregation Marginal Stability Mass Balance Mechanical Equilibrium Mechanical Properties Melt Spinning Metastable - Stable Dendrite Transition Metastable Eutectic Micro-Macroscopic Modelling Micro-Segregation Microscopic Solvability Miller Indices Monte-Carlo (MC) Simulation Morphologies Morphology Morphology of Crystal (Habit) Moving Boundary Problem Mushy Zone

1, 67, 133 51, 67, 173, 277 67, 133, 313 109, 241 1, 17, 67, 133 1, 149 173 173 213 133, 149 67, 109, 149, 173, 285 17, 255, 325 17 1, 67, 173, 265 109, 173 17 213 1, 67, 133

N Navier-Stokes Needle Crystal No-Slip Condition Nodular Cast Iron Non-Equilibrium Non-Equilibrium Distribution Coefficient Non-Equilibrium Solidification Non-Faceted Interface Nucleation Radius Nucleation Rate Nucleation Steady State

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67 67, 277 67 17, 109 133, 173, 259 149, 259 259 17, 109, 173, 259 17 17, 149 17

352

Fundamentals of Solidification 5th Edition

Numerical Modeling

17, 67

O Off-Eutectic Growth One-Domain Method Operating Point Operating Points Operating Range Organic Substance Ostwald Step Rule Outer-Equiaxed Zone Overlapping of Diffusion Fields

109, 173 213 17, 67, 109, 173 17, 67 109 17 17 173 149, 173

P Parabolic Coordinates Paraboloid Peclet Number Peritectic Perturbation Perturbation Method Phase Diagram Phase-Field Physical Properties Planar Solid/Liquid Interface Plate Polymers Powder Bed Fusion (PBF) Prandtl Number Precipitation Preferred Direction Preferred Growth Direction of Dendrite Prenucleation Primary Trunk Spacing Properties

223 67, 277 67, 109, 265, 285 109 51, 67 223, 265 1, 109, 133, 149, 173, 241 17, 67, 325 341 51, 109 223, 277 17 173 67 1, 67 1, 67, 173 67 17 67, 285 173, 341

Q Quasicrystal

17

R Rapid Growth Rapid Solidification Rayleigh-Plateau Instability Recalescence Rectangular Coordinates

149 1, 149, 173, 259, 297 67 1, 17, 173 223

Regular Repeatable Growth Defects Response Response (IR) Reynolds Number Ripening Root Rotation Boundary Roughness

109 17, 109 149 173 67 67, 133, 285 67, 133 17 17

S Satisfaction of Boundary Conditions Scheil-Gulliver Equation Schmidt Number Secondary Arm Spacing Secondary Dendrite Arm Spacing Secondary Dendritic Arm Spacing Segregation Segregation Homogenisation Segregation Intercellular Segregation Rapid Solidification Segregation Ternary Systems Selective Laser Melt (SLM) Short-Range Order Shrinkage Single Crystal Single Crystal Semiconductor Single-Phase Snow Flake Soft Impingement Solid Fraction Solid/Liquid Interface Solid/Liquid Interface Peritectic Solidification Path Solutal Solutal Undercooling Solute Boundary Layer Solute Redistribution Solute Supressed Nucleation (SSN) Solute Trapping Solutions of Differential Equation Solvability Condition Spacing Spherical Coordinates Spherulitic Growth Stagnant Film

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223 133, 149 67 67, 285 67 1 133, 149, 313 133, 313, 321 149 149 313 1, 173 17, 67, 173 1, 67 1, 67, 173 173 109, 173 67 173 133, 173 17, 241, 255 109 313 67, 285 109 17, 109, 223, 297 1, 133, 149 173 149, 259, 325 223 67, 109 109, 149 223 17 67

Foundations of Materials Science and Engineering Vol. 103 Steady-State Stiffness Stiffness of Interface Energy Substrate Superalloy Supersaturation Surface Tension Symmetry Boundary Conditions Synchrotron X-Ray Imaging

17, 109, 133, 173, 223, 297 17 17 17, 241 1, 67, 173 67, 223, 277 17, 173, 241 223 17

Weighted Residual Welding Wetting Angle Wulff Construction

Ternary Systems Thermal Dendrites Thermal Fluctuation Thermal History Thermal Length Three-Phase Junction Tip Temperature Titanium Boride Transients Final Transients Infinal Trijunction TTT Diagram Twin Plane re-Entrant Mechanism Twinned Dendrites Twist Boundary Two-Domain Method Two-Phase

X-Ray Imaging

149, 259 1, 51, 173, 277, 285 109, 313 1, 67, 149, 173, 259, 277 17, 51, 67, 265 17 67 109, 241 173 109 133, 309 109, 133, 309 241 17, 173 17 67 17 213 67

U Unconstrained Unconstrained Growth

67, 285 67, 285

V Vapour Velocity Viscosity Volume Fraction Volume Fraction Eutectic Volume Fraction Solid

17 1, 223 1, 17, 67 109, 297 109 133, 149, 173

W Wavelength of Instability

223, 297 1, 149, 173 17, 241 17

X

T T0 -Line Temperature Gradient

353

51, 67, 265

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17, 67, 241