Metastable Solids from Undercooled Melts

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Author: Dieter Herlach | Dirk Holland-Moritz | Peter Galenko

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Metastable Solids from Undercooled Melts

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PERGAMON MATERIALS SERIES

Serious Editor: Robert W. Cahn FRS Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, UK Vol. 1 CALPHAD by N. Saunders and A. P. Miodownik Vol. 2 Non-Equilibrium Processing of Materials edited by C. Suryanarayana Vol. 3 Wettability at High Temperatures by N. Eustathopoulos, M. G. Nicholas and B. Drevet Vol. 4 Structural Biological Materials edited by M. Elices Vol 5 The Coming of Materials Science by R. W. Cahn Vol. 6 Multinuclear Solid-State NMR of Inorganic Materials by K. J. D. MacKenzie and M. E. Smith Vol. 7 Underneath the Bragg Peaks: Structural Analysis of Complex Materials by T. Egami and S. J. L. Billinge Vol. 8 Thermally Activated Mechanisms in Crystal Plasticity by D. Caillard and J. L. Martin Vol. 9 The Local Chemical Analysis of Materials by J. W. Martin

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Dieter M. Herlach, Peter Galenko and Dirk Holland-Moritz Institut für Materialphysik im Weltram, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Köln, Germany Edited by

Robert Cahn

Amsterdam ● Boston ● Heidelberg ● London ● New York ● Oxford Paris ● San Diego ● San Francisco ● Singapore ● Sydney ● Tokyo Pergamon is an imprint of Elsevier

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Pergamon is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (⫹44) (0) 1865 843830; fax (⫹44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is availabe from the Library of Congress ISBN-13: 978-0-08-043638-8 ISBN-10: 0-08-043638-2 For information on all Pergamon publications visit our website at books.elsevier.com Printed and bound in Germany 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

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Contents Preface Acknowledgements

ix xi

CHAPTER 1 Introduction

3

CHAPTER 2 Experimental Approach to High Undercoolings 2.1. Transient and Stationary Undercooling 2.2. Droplet Dispersion and Emulsification Method 2.3. Short- and Long Drop-Tube Processing 2.3.1 Short Drop Tubes 2.3.2 Long Drop Tubes 2.4. Processing of Bulk Melts by Melt Fluxing 2.5. Containerless Processing Through Levitation 2.5.1 Acoustic Levitation 2.5.2 Levitation by Stationary Magnetic Fields 2.5.3 Electromagnetic Levitation 2.5.4 Electrostatic Levitation 2.6. Containerless Processing in Space References CHAPTER 3 Physics of Undercooled Liquids 3.1. Thermodynamics 3.2. Structural Ordering in Undercooled Melts 3.2.1 Models for the Short-Range Order in Undercooled Melts 3.2.2 Scattering Theory 3.2.3 Experiments on the Short-Range Order in Metallic Melts 3.2.4 The Short-Range Order of Liquid Si 3.3. Magnetic Ordering in Liquid State 3.4. Kinetic and Transport Properties References v

9 9 11 13 13 18 19 20 20 26 33 42 47 52

59 59 67 68 76 79 95 97 103 105

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CHAPTER 4 Solid–Liquid Interface 4.1. Structural Order at the Interface 4.2. Interfacial Energy Under Local Equilibrium Conditions 4.2.1 The Negentropic Model by Spaepen and Thompson 4.2.2 Investigations Using Molecular Dynamics and Density Functional Theory 4.2.3 Experimental Results on the Solid–Liquid Interfacial Energy Under Local Equilibrium Conditions 4.2.4 The Energy of the Interface Between Structurally Complex Solids and their Melts 4.3. Diffuse Interface Theory 4.3.1 Physical Interpretation of the Diffuse-Interface Region 4.3.2 Phase-Field Models for Nucleation 4.3.3 Sharp Interface Versus Diffuse Interface References CHAPTER 5 Nucleation 5.1. Nucleation Theories 5.1.1 Homogeneous Nucleation 5.1.2 Heterogeneous Nucleation 5.1.3 Diffuse Interface Theory of Nucleation 5.2. Transient Nucleation 5.3. Statistics of Nucleation 5.4. Nucleation in Alloys 5.5. Magnetic Contributions to Crystal Nucleation 5.5.1 The Magnetic Contribution to the Driving Force for Crystal Nucleation 5.5.2 The Magnetic Contribution to the Solid–Liquid Interfacial Energy 5.6. Experimental Results on Undercooling and Nucleation 5.6.1 Homogeneous Versus Heterogeneous Nucleation 5.6.2 Nucleation in Undercooled Melts 5.6.3 Structural Dependence of Nucleation Behaviour 5.6.4 Undercooling of Magnetic Melts References

115 115 116 117 121 124 126 133 133 136 136 138

145 145 145 152 154 160 161 163 165 165 167 169 169 170 172 180 189

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CHAPTER 6 Crystal Growth in Undercooled Melts 6.1. Kinetics of the Advancement of a Solid–Liquid Interface 6.2. Departures of Local Equilibrium 6.3. Stability Analysis 6.4. Sharp-Interface Model 6.4.1 Growth in a Pure System 6.4.2 Solidification in a Binary System 6.4.3 Superlattice Structures in Intermetallics 6.5. Phase-Field Model 6.6. Transition from Faceted to Non-Faceted Growth 6.7. Experimental Data and Model Predictions 6.7.1 First Experiments 6.7.2 Measurements on Pure Nickel 6.7.3 Measurements on Dilute Ni᎐B and Ni᎐Zr Alloys 6.7.4 Measurements on Intermetallic Compounds 6.7.5 Measurements on Semiconductors 6.7.6 Effect of Convective Flow and Solute Diffusion 6.7.7 Influence of Local Non-equilibrium on Rapid Dendritic Growth References

vii

197 197 204 216 227 227 234 241 244 247 256 256 258 259 261 263 265 270 273

CHAPTER 7 Cooperative Growth in Undercooled Polyphase Alloys 7.1. Eutectic Growth Theory 7.2. Eutectic Morphology Transition 7.3. Stable and Metastable Monotectic Alloys 7.4. Peritectic Alloys References

283 283 294 303 307 310

CHAPTER 8 Metastable Solid States and Phases 8.1. General Conditions for the Formation of Metastable Solids 8.2. Supersaturated Solid Solutions 8.3. Formation of Metastable Crystalline Phases 8.4. Phase Selection Through the Solidification Kinetics

317 317 320 323 333

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8.5. Metallic Glasses 8.6. Grain-Refined Materials References

335 339 354

CHAPTER 9 Microstructure Selection Maps 9.1. Selection by Rapid Cooling 9.2. Selection by Undercooling 9.3. Selection by Droplet Size References

361 361 367 370 373

CHAPTER 10 Experiments in Reduced Gravity 10.1. Containerless Processing in Reduced Gravity 10.2. Experiments in Drop Tubes 10.2.1 Nucleation Studies on Glass-Forming Systems 10.2.2 Kinetics of Phase Selection 10.2.3 Microstructure Development 10.2.4 Liquid–Liquid Phase Separation 10.3. Electromagnetic Processing in Reduced Gravity 10.3.1 Thermophysical Properties 10.3.1.1 Thermal Expansion 10.3.1.2 Electrical Resistivity 10.3.1.3 Specific Heat and Thermal Conductivity 10.3.1.4 Surface Tension and Viscosity 10.3.2 Nucleation Investigations and Phase Selection 10.3.3 Measurements of Dendrite Growth Velocities References

377 377 379 379 380 382 386 389 389 389 390 392 394 397 401 403

CHAPTER 11 Conclusions and Summary

409

Appendix: List of Symbols

413

Index

425

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Preface This book aims to give an overview on the science of undercooled melts. It focuses on the characterization of the metastable state of an undercooled liquid and the description of crystal nucleation processes and subsequent growth mechanism as dendrite growth of single phase and eutectic/peritectic reactions of polyphase solidification. The consequences of short-range order and crystallization processes in undercooled melts are emphasized with respect to the formation of a great spectrum of metastable materials. It is demonstrated how nucleation preselects the crystallographic phase, stable or metastable, and crystal growth controls the evolution of various microstructures from the liquid state. The subject of the book will concentrate on metallic systems and, partially to some semiconductors, that show metallic like properties in the liquid state. The present seems an appropriate time for such a review. Both from experimental side as well as from theoretical developments essential progress has been achieved during the recent past. In particular, levitation methods like electromagnetic levitation or electrostatic levitation are meanwhile widely used to undercool bulk drops and droplets in diameter of some millimetres far below their melting temperatures. This is owing to the complete avoidance of heterogeneous nucleation on container walls that leads to undercoolings of melts, sometimes greater than the maximum undercoolings of about 20% of their respective melting temperatures. Such a large undercooling range even exceeds the early experimental findings of investigations by the droplet-dispersion method of the pioneering work by David Turnbull in the early 1950s. The experimental progress by levitation techniques demonstrates that large undercoolings are not only limited to very small particles but also be achieved on bulk samples. This fact enhances essentially the potential of investigations on the metastable state of undercooled melts and the non-equilibrium solidification of metastable solids. A freely suspended drop levitated by electromagnetic or electrostatic forces offers the essential benefit to combine levitation techniques with diagnostic facilities at research centres. In the present book, research is overviewed on levitation undercooled drops, which are directly studied, for example by muon spin rotation spectroscopy (at Paul Scherrer Institut Villigen, Switzerland), by synchrotron radiation (at European Synchrotron Radiation Facility, Grenoble) or by neutrons (at the Institut Laue Langevin, Grenoble, France). Moreover, by progress in high-speed camera technique, rapid solidification is directly recorded during non-equilibrium solidification of deeply undercooled melts. This technique allows for determining both morphology and dynamics of a rapidly propagating solid–liquid interface in an undercooled melt. ix

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In parallel to the rapid advancement of experimental investigation techniques, novel concepts have been developed to describe the metastable state of an undercooled melt and the non-equilibrium solidification processes leading to metastable phases from the state of an undercooled melt. Beyond the sharp interface models, which have been developed in the 1980s, phase field modelling has received the potential to quantitatively describe the kinetics of solidification of even multicomponent alloys of industrial interest. Nowadays, it is possible to include even changes of the heat and mass transport in front of the solid–liquid interface by forced convection in theoretical description of solidification phenomena. In this respect, the comparison of scientific results obtained on Earth and in Space, for example, on board of the International Space Station (ISS) that is currently in use leads to new horizons of solidification research. First results of microgravity research in Space are presented in this book. Taking together the progress in experimental investigations and theoretical description of solidification processes, the basis may be developed for quantitative modelling of solidification and the engineering of new materials with improved properties in the computer. The concept of virtual design of materials on the basis of experimentally verified physical models of solidification not only leads to concepts to explore new materials but also contains a great potential to make production routes in materials processing from the liquid much more efficient. Since 90% of all materials are produced from the liquid state as their parent phase already small steps of progress in materials production will essentially improve the economics of production routes both with respect to saving energy and environmental resources and the competitiveness of foundry and casting industry worldwide. At present this perspective is certainly still a vision for the future, however, with rapid progress. This book addresses students and scientists of the interdisciplinary field of materials science and research to give the physical concepts and tools to characterize and describe the formation of metastable solids from undercooled melts. Our hope is that it may help the readership to understand the development of science and technology of solidification of melts, a research area of long tradition, and may lead the readers towards new concepts within this exciting research field to fulfil the challenges of the future in science and technology of undercooled melts. Dieter Herlach Peter Galenko Dirk Holland-Moritz Köln, 2006

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Acknowledgements We express our gratitude to many students, colleagues, visiting scientists in our group of “Undercooling of Materials” of the Institute of Space Simulation of the German Aerospace Center, Köln, Germany, with whom we have studied and investigated the metastable state and non-equilibrium solidification of undercooled melts of metals, alloys and semiconductors. Many results of investigations presented in this book and much of its sense and content come from them. This outcome was only possible thanks to a friendly cooperation over many years, which has born novel and innovative ideas to progress in our understanding of undercooled melts, its short-range order, nucleation and crystal growth, all of them being essential to get insight into the background of the formation of metastable solids from the liquid state. It would be impossible to name everyone to whom we would like to convey our thanks. But, we would be ungracious not to mention Professor Feuerbacher, German Aerospace Center, Köln, Germany, for his continuous support; Professor Greer, University of Cambridge, UK; Professor Jou, Universita di Barcelona, Spain; Professor Kelton, Washington University, St. Louis, USA; Professor Kurz, Ecole Polytechnique Federale de Lausanne, Switzerland; Professor Spaepen, Harvard University, USA; and Professor Urban, Forschungszentrum Jülich, Germany, for many stimulating discussions and exciting ideas. We also acknowledge financial support by the German Aerospace Center, the Alexander von Humboldt-Foundation, Deutsche Forschungsgemeinschaft, the Europeans Space Agency and the European Communities. Last but not least, our special thanks are dedicated to Professor Robert Cahn, University of Cambridge, UK, for his continuous interest in our work, his great patience with the authors and the huge support to edit the present book.

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Introduction

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Chapter 1

Introduction During the past three decades rapid quenching methods have been successfully applied to produce metastable solids from the liquid state. A great variety of metastable materials have been discovered with physical properties, which make them suitable as new high-performance materials in mechanical and electrical engineering. To date, such metastable solids are even produced in an industrial scale for their application e.g. in steel production, design of electronic devices and aerospace applications. The method of rapid quenching of the melt is based upon the rapid heat transfer from the melt to a cooling substrate of good heat conductivity. To obtain the necessary cooling rates of the order of 106 K/s, the melt must be limited at least in one dimension to a thickness of less than 50 m. Such conditions are fulfilled e.g. in melt-spinning, splat-cooling and planar flow casting processes. From the technological point of view hot isostatic pressing and atomization production (powder metallurgy) routes are also interesting to produce materials with extraordinary properties. A systematic investigation of the physical processes underlying in such production processes is, however, difficult to perform. These methods imply experimental conditions not favourable for in situ diagnostics of rapid solidification. On the one hand, the steep temperature gradient externally imposed causes a strongly inhomogeneous melt that complicates a quantitative analysis. On the other hand, the solidification front is covered by the melt pool, which makes a direct observation of the solidification front almost impossible. Rapid solidification, however, is throughout not limited to rapid quenching. If the melt is undercooled below its melting temperature, a great driving force for crystallization is accumulated due to the Gibbs free energy difference, which increases with undercooling. In this way, rapid solidification is realized even if a bulk melt is slowly cooled. The solidification velocities obtained through largely undercooling the melt are comparable or even higher than in the case of rapid quenching. The present work will deal with such experiments. It will be demonstrated, in detail, that in this way rapid solidification processes will become accessible for direct experimental investigations. The state of an undercooled melt corresponds to a non-equilibrium state of metastable nature. Within thermodynamic considerations, the Gibbs free energy of an undercooled melt is always higher than that of the stable solid. With increasing undercooling driving forces also for solidification of metastable solids can also 3

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occur so that the number of possible solidification paths increases with undercooling. But the phenomenon of undercooling implies an activation threshold for crystallization whose origin is from the activation energy necessary to form a critically sized nucleus that can lower its energy by growing further. Calorimetric investigations on undercooled melts allow the determination of the Gibbs free energy difference between undercooled liquid and solid and, hence, create the preconditions for a quantitative description of the thermodynamics of solidification. In fact, the thermodynamics is able to predict the driving forces for solidification from the undercooled melt into different solid phases – stable or metastable. However, it cannot explain the phenomenon of undercooling. A complete description requires the additional regard of nucleation phenomena. The interface between a solid nucleus in an undercooled liquid implies an activation barrier that leads to the undercoolability of a melt. With increasing undercooling the Gibbs free energy difference increases and exceeds the energy barrier so that a cluster with instantaneous solid-like structure can transform to a solid nucleus. The maximum undercooling observed in the experiment, therefore, contains important information about the nucleation mechanism. The basic concept of classical nucleation theory will be introduced. In particular, the question of homogeneous versus heterogeneous nucleation will be discussed based upon experimentally determined maximum undercoolings. Nucleation is the process initiating solidification. It is continued by subsequent growth. The conditions as underlying in undercooled melts of pure metals, miscible and dilute alloys lead to dendritic growth. On the basis of the solutions of the heat and mass-transport equations and the stability analysis of the solid– liquid interface, the current theories of dendritic growth correlate the undercooling to the dendrite-growth velocity. New experiments of containerless processing of liquid metals and alloys will be presented which open up an undercooling range of up to 25% of the melting temperatures in the investigated metals. The levitation techniques are combined with suitable diagnostic means to observe primary crystallization events and to measure the dendrite growth velocities over an extended range of undercooling. The analysis of these experiments gives evidence that during the rapid propagation of dendrites into deeply undercooled melts deviations from local equilibrium at the solid–liquid interface occur. At extreme undercooling values, the crystals begin to grow with the initial nominal composition of the liquid in a segregation-free growth mode. It has been further suggested that the solute diffusion field near the rapidly propagating solidification front has no time to completely relax to local thermodynamic equilibrium. Thus, to describe rapid crystal growth in deeply undercooled liquids, current models of crystal growth include deviations from local equilibrium both at the interface and in the bulk liquid.

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Introduction

5

Dendrite growth is a single-phase growth process. Eutectic growth involves the interacting nucleation and co-operative growth of two or more solid phases within one liquid phase. The first successful attempts to describe this more complicated growth mechanism are due to the pioneering work by Jackson and Hunt. Recent experiments on eutectic alloys, however, have shown that this model is only applicable for small undercoolings of solidification of regular or lamellar eutectic structure. The non-equilibrium state of an undercooled melt gives the potential of the formation of metastable solids. Various metastable solid states can be formed from the undercooled melt. During rapid crystal growth, supersaturated solid solutions are solidified showing an extension of the equilibrium solubility. Metastable phases of a crystalline structure different from the structure of the stable solid are produced by seeding externally the crystallization of the undercooled melt with a nucleation trigger of proper crystallographic structure. A special case of the solid is the quasicrystalline state discovered in 1984. Quasicrystals do not exhibit longrange translational but exhibit long-range orientational order of fivefold symmetry. These systems are of special interest for investigations of homogeneous nucleation. Early considerations by Frank already pointed out that an icosahedral short-range order in the undercooled melt should be energetically favoured, and has been confirmed by computer simulation experiments. Such an icosahedral short-range order in the undercooled melt is similar to the short-range order in quasicrystalline solids. This means that the interfacial energy between an icosahedrally ordered nucleus and the undercooled melt should be relatively small, and thus the activation energy to form critical quasicrystalline nuclei should be reduced. In fact, this has been recently evidenced by undercooling experiments on quasicrystalline systems. Quasicrystals only show long-range orientational order, whereas metallic glasses do not show long-range order. To produce the structurally disordered state of an amorphous metal, nucleation in the undercooled melt must be completely avoided. In general, metallic systems show large nucleation frequences so that very high cooling rates, of the order of 106 K/s, are required to suppress nucleation. The nucleation frequency scales with the reduced glass transition temperature. Recently, metallic alloys have been discovered with relatively high-reduced glasstransition temperatures. These alloys can be transformed to the glassy state even in sample diameters of up to 14 mm at extremely low cooling rates of a few K/s. A further interesting feature of these glassy alloys is a wide supercooled region between the glass transition and the crystallization temperature. This opens up the possibility to investigate characteristic properties in the very highundercooled state. Finally, metastability is also present in grain-refined materials. The excess free energy of the undercooled melt can be used by the system to build up very fine

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grains with diameter of 1 m or even less. In principle, two different mechanisms may lead to grain refinement: (i) copious nucleation, in particular, as it is the case of homogeneous nucleation or (ii) by secondary reactions such as dendrite desintregation either by partial remelting or breakage. A quantitative modelling of all non-equilibrium solidification phenomena requires an accurate knowledge of thermophysical properties such as the specific heat or the viscosity in the undercooled melt regime. So far, only scarce information is available of such properties in the equilibrium liquid state above the melting temperature but almost nothing is known about thermophysical parameters and their temperature dependence in the metastable regime of an undercooled melt. In this context, future experiments of containerless processing including microgravity experiments are described which promise measurements of thermodynamic properties as a function of temperature over an extended range of undercooling.

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Chapter 2

Experimental Approach to High Undercoolings 2.1. 2.2. 2.3.

Transient and Stationary Undercooling Droplet Dispersion and Emulsification Method Short- and Long Drop-Tube Processing 2.3.1 Short Drop Tubes 2.3.2 Long Drop Tubes 2.4. Processing of Bulk Melts by Melt Fluxing 2.5. Containerless Processing Through Levitation 2.5.1 Acoustic Levitation 2.5.2 Levitation by Stationary Magnetic Fields 2.5.3 Electromagnetic Levitation 2.5.4 Electrostatic Levitation 2.6. Containerless Processing in Space References

9 11 13 13 18 19 20 20 26 33 42 47 52


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Chapter 2

Experimental Approach to High Undercoolings 2.1. TRANSIENT AND STATIONARY UNDERCOOLING

The tendency to undercool a melt depends strongly on the catalytic potency of heterogeneous nucleation sites and its nucleation kinetics. For this reason, there are, in principle, two different methods to obtain an undercooled melt: (i) reduction or even elimination of heterogeneous nucleation sites, or, alternatively, (ii) rapid cooling to bypass heterogeneous nucleation kinetically. The method of rapid cooling e.g. by melt spinning or splat cooling is nowadays state of the art and is applied even at an industrial scale to produce metastable materials from undercooled melts. In rapid quenching a melt high cooling rates are employed to cool and undercool the melt in a transient way. Typical quenching rates of the order of 106 K/s are achieved by melt spinning or splat cooling in the production of thin ribbons or splats with thickness of 10–100 m. Higher cooling rates are possible by laser or electron beam surface re-solidification at which a thin surface layer (a few m) of a material is melted and due to the rapid heat subtraction through the bulk material beneath the melted layer at the surface resolidifies quickly at rates as high as 108 K/s. Even though in all of these techniques of rapid quenching the melt is in direct contact with a crystalline solid material that can act as heterogeneous nucleation site, very large undercoolings are achieved because the heat removal from the melt occurs so rapidly that the characteristic time of cooling becomes comparable with the time needed for the formation of critical crystal nuclei. In such a case, the melt is undercooled in a transient way. The quenching rate can even be so high that nucleation is kinetically circumvented. The melt is transiently undercooled to the glass transition temperature at which the viscosity becomes so large that the undercooled melt falls out of local thermal equilibrium and freezes into a glassy state. Nowadays, there is a broad variety of techniques of rapid quenching to produce ribbons, thin wires, powders and sprays and re-solidified films at the surface of bulk material. An excellent overview of such techniques is given by Cahn [2.1]. The techniques of rapid quenching have the advantage that large amounts of metastable material are produced in a continuous way. However, during rapid quenching, there is a large temperature gradient across the solidifying melt and that the solidification front itself is concealed by the melt pool or substrate and therefore is not accessible for direct observation. These disadvantages are overcome by techniques of slowly cooling but large undercooling the melt prior to 9

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solidification, e.g. by containerless processing in alternating electromagnetic fields. Owing to the large undercoolings achieved in this way a high driving force for crystallization occurs, which leads to rapid solidification of the undercooled melt despite the very small cooling rates. In undercooling experiments, cooling rates are of the order of 1–100 K/s, which is very small compared with rapid quenching techniques. Independent of the method for undercooling, either transient undercooling by rapid quenching or stationary undercooling by deep undercoolings, the dominating growth mode of the crystal in the melt is the formation of dendrite pattern provided a crystal is formed and there is no glassy solidification. Even though in conventional casting undercooling is not very large, dendrite growth modes are of essential importance in such production routes in industrial processes since here constitutional effects lead to dendrite growth as the dominant morphology of the solid–liquid interface. Figure 2.1 gives an overview of undercooling methods, both including techniques of transient undercooling by rapid quenching of a melt, e.g. using melt

Figure 2.1. Techniques of solidification of melts: left top: laser surface re-solidification; right top, melt spinning of thin ribbons; left bottom: levitation melting of drops; right bottom: casting of melts; centre: formation of dendrites in solidification of melts.

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Experimental Approach to High Undercoolings

11

spinning, surface resolidification and stationary undercooling e.g. by applying containerless processing of drops of metals and alloys in alternating electromagnetic fields. More or less independent of the technique the undercooling achieved prior to solidification is the factor controlling the development of various crystallographic structures by nucleation phenomena and the evolution of a large spectrum of microstructures by non-equilibrium conditions during rapid growth of the primarily formed crystallites. Under the conditions of undercooling, a negative temperature gradient and, in alloys, a concentration gradient are established in front of the solid–liquid interface. These lead to destabilizing of the interface and cause dendritic growth morphologies. Even in industrial production routes of melt casting in foundry industry, dendrites are formed mostly by constitutional undercooling effects. Since the crystallographic structure and dendritic microstructures of metallic alloys predetermine the physical properties of the as-solidified material, such as mechanical properties like strength and plasticity, electrical properties like electrical and heat conductivity, magnetic properties like hysteresis and energy product, a detailed understanding of the physical mechanism during non-equilibrium solidification of undercooled melts is of enormous importance. Systematic investigations of phase selection by various nucleation phenomena and different growth mechanisms are therefore not only of fundamental interest but also of practical relevance in improving properties of cast materials and optimizing production routes in foundry industry.

2.2. DROPLET DISPERSION AND EMULSIFICATION METHOD

The method of transient undercooling by rapid quenching is not suitable for detailed fundamental investigations of the processes occurring during the early stage of crystallization. The alternative way, the reduction of heterogeneous nucleation of high catalytic potency, offers the benefit not only to obtain large undercoolings at even moderate cooling rates, but also the advantage to measure the undercooling of the melt directly prior to solidification. Heterogeneous nucleation can occur at the free surface of the melt by the formation of metal oxides. Note that metal oxides are often more stable than the respective metallic material. As an example, Al2O3 shows a melting temperature of 2300 K, which is much higher than the melting temperature of pure Al of 933 K. Heterogeneous nucleation can also arise within the volume of the melt owing to the presence of foreign phases or impurities. A simple, however very effective method to reduce heterogeneous nucleation is to divide a macroscopic melt into many small particles. If the heterogeneous nucleation “motes” of density ⫽ n/vp (n, the number of motes; vp,

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the volume of the particle) are randomly distributed in the melt, the probability WP(n) of finding a volume segment (particle) containing n motes is given by the Poisson distribution WP ( n) =

nn exp( − v p ). n!

(2.1)

Accordingly, the probability WP(0) of finding a “mote-free” volume segment rises very rapidly with increasing subdivision of the melt: WP (0) = exp( −v p ).

(2.2)

Similar considerations hold for nucleation motes at the surface of the sample if surface segments replace the volume segments. The principle of isolation of heterogeneous nucleation motes by subdivision of the melt is schematically shown in Figure 2.2 for a two-dimensional system. This method has been originally developed by Vonnegut [2.2] and has been improved by Turnbull, who applied this technique to investigate the undercooling behaviour of a whole series of pure metals [2.3–2.5]. Surprisingly, he found a relatively well-defined unique relative undercooling ⌬T/TE 0.2 for all materials investigated, despite the fact that the studied metals were different in physical nature. This had led to the speculation that at such a relative undercooling the limit of homogeneous nucleation may be reached [2.6]. Later on, the method of isolation of heterogeneous nucleation motes had been extended by Turnbull [2.4], Rasmussen [2.7] and Perepezko [2.8–2.10], by embedding the dispersed particles into an inert emulsion liquid. In this way, not only volume motes but also surface motes can be deactivated. This has led to an extension of the maximum undercooling to ⌬T/TE > 0.2, contradicting the speculation that at ⌬T/TE 0.2 the limit of homogeneous nucleation is reached. In particular, very large undercoolings have been observed by applying this method for Bi, In, Hg and

Figure 2.2. Principle of isolation of heterogeneous nucleation motes in the volume of a macroscopic melt by its subdivision into many small particles.

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13

Pb [2.10]. The largest relative undercooling of ⌬T/TE 0.58 has been found for Ga [2.11]. The extension of such experiments to high-melting metals has been reported recently. The silicate oils, used as carrier liquid for the previous experiments, have been thereby replaced by salt melts [2.12] and fine ceramic powders [2.13]. The analysis of these undercooling experiments within nucleation theory has yielded the result that, despite the large undercoolings obtained it is heterogeneous nucleation at interfaces that limits the undercoolability and not homogeneous nucleation. An exception is perhaps a study on Hg resulting in a relative undercooling of ⌬T/TE 0.33, which scales with the volume of the droplets and has been attributed to the onset of homogeneous nucleation [2.5]. 2.3. SHORT- AND LONG DROP-TUBE PROCESSING

The drop-tube technique is employed to cool and solidify small molten droplets, which fall containerlessly down a tube that can be evacuated and backfilled with processing gases such as He, Ar or others. It is convenient to distinguish between two categories of tubes – short and long – which reflect the type of the experiment that can be performed. In short drop tubes, a liquid jet of material is produced that disperses into many small droplets. In long drop tubes, individual drops with size ranging in few millimetres are undercooled and solidified during free fall. 2.3.1 Short drop tubes The principle of volume/surface separation of heterogeneous nucleation sites forms the basis in drop-tube and atomization experiments. Here, a thin liquid jet of a metal disperses into small droplets (Rayleigh instability of a thin liquid jet). They undercool and solidify during the free fall containerlessly. Most of the drop tubes process a spray of droplets of various size groups, with the diameter in the range of 50–1000 m. Therefore, diagnostics of individual droplets during free fall is not possible. This technique is employed to study undercooling and nucleation phenomena [2.14–2.16] to investigate the evolution of grain-refined microstructures [2.17, 2.18] and to produce metastable crystalline materials and metallic glasses [2.14–2.16, 2.19, 2.20]. Figure 2.3 illustrates the experimental set-up of a drop tube with length of 8 m at the German Aerospace Center (DLR) in Cologne [2.21]. The drop tube is made of stainless-steel components all of which are compatible with the requirements of ultra-high vacuum (UHV) technique. The drop tube is evacuated before each experiment by a turbomolecular pump to a pressure of 10⫺7 mbar and, subsequently, is backfilled with purified He or He᎐H2 gas of high thermal conductivity.

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Figure 2.3. Schematic view of the DLR drop tube; the drop-tube technique combines rapid cooling of small particles and reduction of heterogeneous nucleation by containerless processing and by dispersion of the melt into a spray of small droplets.

The processing gas is purified as it passes an oxysorb system and a liquid-nitrogen cold trap before it enters the tube. The sample (mass of several grams) is contained in a crucible e.g. of fused silica. The crucible has a small bore at its lower end. The sample material is melted inductively. After all material turns into liquid its

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15

temperature is measured by a two-colour pyrometer and subsequently forced by Ar pressure of 2 bar through the small bore. A thin liquid jet is formed that disperses into small droplets of different size owing to the Rayleigh instability. The droplets cool, undercool and solidify during free fall in time of 1 s under the conditions of reduced gravity in containerless state. Droplets of size ranging from 50 to 1000 m are formed. They are collected at the bottom and grouped into various size groups by screening in sieves of different mesh sizes for further analysis. Since the droplet diameter scales with the cooling rate at which the droplets cool down during falling down the tube, drop tubes are quite suitable to study statistical processes of phase selection and their temperature–time-transformation behaviour. The cooling rate of the falling droplets can be calculated on the basis of the heat transfer to the environment by heat radiation and thermal conductivity in the environmental gas. The heat balance of a falling drop is given by H + QH = 0,

(2.3)

where ⌬H is the heat content of the sample and ⌬Q the heat transferred from the sample to the environment. Provided no phase transformation of first order takes place the heat content of the liquid drop can be written as H = ss l C pl T ,

(2.4)

where s ⫽ d3/6 is the volume of the droplet with diameter d, l the mass density of the melt, Cplthe specific heat of the liquid and T a temperature differi ence. The heat flow Q through the surface of the droplet O ⫽ d2 is determined by radiation of heat and heat transfer to the environmental He-gas of pressure of 1000 mbar:

(

i

)

Q = O ⎡⎣ hm (T − TR ) + SB T 4 − TR4 ⎤⎦

(2.5)

with O the surface area of the droplet, hm the heat-transfer coefficient for heat conductivity in the gas, TR the ambient temperature (room temperature), the total hemispherical emissivity and SB the Stefan–Boltzmann constant. The heat transferred to the environment during the time from the formation of the drop (t ⫽ 0 s) until nucleation sets in (t ⫽ tn) is given by tn

i

QH = ∫ Q dt . 0

(2.6)

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Metastable Solids from Undercooled Melts i

For an average cooling rate, T = −dT/dt , which is supposed to be constant, the integral of Eq. (2.6) can be replaced by QH =

1 i

T

Tn

i

∫ Q dT ,

(2.7)

To

where ⌬QH is the heat, which is transferred from the sample to the environment, during the sample is cooling fromi a temperature To to a temperature Tn at constant i cooling rate T . The cooling rate T is estimated by substituting Eqs. (2.4) and (2.7) into Eq. (2.3) so that i

T=

6 d l C pl T

Tn

∫ ⎡⎣h (T − T ) + (T m

R

SB

To

4

)

− TR4 ⎤⎦ dT

(2.8)

with T ⫽ T0⫺Tn. The heat-transfer coefficient hm for spheres falling through a gas is estimated from the Nusselt number N, the Reynold number, R and the Prandtl number, P [2.22]: N = 2.0 + 0.6 R1/ 2 P1/ 3 .

(2.9)

Equation (2.9) combines the characteristic properties of the environmental gas. C pgas gas hm d V gas gas d N = gas , R = , P = gas gas

gas

(2.10)

with the properties of the gas. gas is the heat conductivity, gas the viscosity, gas the density, Cpgas the specific heat, and Vgas the stream velocity. Using Eq. (2.10) the heat-transfer coefficient is computed as 2.0 gas 0.6 gas ⎛ V gas gas ⎞ hm = + gas ⎟⎠ d d ⎜⎝

1/ 2

⎛ C pgas gas ⎞ ⎜ gas ⎟ ⎠ ⎝

1/ 3

.

(2.11)

If it is assumed that a pressure of 150 mbar is used to force the liquid sample through the nozzle at the lower end of the crucible, an average velocity Vav of the falling drop is calculatedd by Vav ⫽ (Vo + Vmax)/2 4 m/s, where Vo ⫽ (2 p/l )1/2 ⱕ 2 m/s. The average velocity Vav is set equal to the stream velocity of the gas, Vgas. Equation (2.10) is used in Eq. (2.8) to calculate the cooling rate as a function of droplet diameter by computing the integral over the temperature range To ⬎ T ⬎ Tn,, resulting in T=

a b + 1.5 + c 2 d d

(2.12)

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with T

a=

12.0 gas n (T − TR ) dT l C pl T T∫o T

3.6 gas Bo n b= l l (T − TR ) dT C p T T∫o c=

6 SB l C pl T

Tn

∫ (T

4

)

− TR4 dT

To

⎛ V gas gas ⎞ Bo = ⎜ ⎝ gas ⎟⎠

1/ 2

⎛ C pgas gas ⎞ ⎜ gas ⎟ ⎝ ⎠

1/ 3

.

In the literature [2.23], the cooling rate as a function of droplet diameter is often evaluated by a power law, such that Eq. (2.12) can be approximated by i

i

T ≡ T o ( d[m])

(2.13)

i

with T o a normalizing constant and a fractionalized exponent, both of which can be inferred by fitting Eq. (2.13) into Eq. (2.12). For instance, using the characteristic parameters for the glass-forming alloys Ni᎐Zr and Pd᎐Cu᎐Si one gets [2.21] i ⎛ 1 ⎞ T ≅ 2.37 × 108 ⎜ ⎝ d [m] ⎟⎠

1.95

⎛ 1 ⎞ T ≅ 7.16 × 10 ⎜ ⎝ d[m] ⎟⎠

1.95

i

8

[K/s]

for Cu − Zr

[K/s]

for Pd − Cu − Si.

For a drop falling in a gas atmosphere the equation of motion reads V 2 dv 3 =g − Dr rel dt g 4 d

(2.14)

where g is the gravitational acceleration, g the gas density, ⌬ ⫽ – g with the density of the sample, Vrel the relative velocity between the droplet and the environmental gas and Dr the drag coefficient usually taken as the empirically determined value for a hard sphere. The drag force FD exerted on the droplet by the gas is given by

FD =

Dr Ac gVrel 2 4

(2.15)

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where Ac is the area normal to the freestream, which increases the residual gravitational force experienced. For small drop tubes in length less than 10 m the drag force is small compared with the gravitational force. At a free falling distance of L ⫽ 10 m the final velocity of the droplet before touching the bottom is estimated to be Vrel ⫽ (2sg)1/2 ⫽ 31 m/s. A more detailed analysis of heat transport in falling drops, including heat and mass transport in the interior of the droplets was reported recently [2.24]. 2.3.2 Long drop tubes Long drop tubes are generally ⬎50 m in height, and individual drops are processed. To study the effects of microgravity on solidification in earthbound laboratories they exploit the fact that a body falling freely in vacuo experiences zero gravity. There are two such facilities: a 105-m drop tube at NASA Marshall Space Flight Center, described by Rathz et al. [2.26] and a 47-m drop tube at the Nuclear Research Centre at Grenoble [2.27]. In experiments using such facilities, the tube is evacuated and single droplets 1–5 mm in diameter are melted by an electron beam (pendant drop technique) or electromagnetic levitation. After release, the droplet is monitored by Si or InSb photodiodes along the length of the tube, which enable the recalescence event to be detected. The time of flight before this event is measured and used with a heat flow model and the initial droplet temperature to infer the undercooling achieved at nucleation ⌬Tn. With this method, Hofmeister et al. [2.28] estimate that the uncertainties in the measurement of the time of release and the initial droplet temperature result in an error of ⫾50 K. Processing of drops under high vacuum [2.26] or even UHV [2.27] reduces surface oxidation of the molten samples as a possible source of heterogeneous nucleation. On the other hand, only high-melting metals, e.g. refractory metals, can be processed since cooling is only by radiation (cf. Eq. (2.5)), which is efficient at high temperatures. The undercoolings achieved for pure metals seem to be highly reproducible. Lacy et al. [2.29] found the mean undercooling in niobium to be 525 ⫾ 8 K with a maximum of 535 K. They associated this nucleation event to the formation of NbO on the droplet surface because the nucleation temperature corresponded with the melting temperature of this oxide. These results show that high-vacuum conditions are not sufficient to avoid heterogeneous nucleation due to surface oxidation, but UHV may lead to an improvement. In fact, the highest absolute undercooling was measured on droplets processed in the Grenoble drop tube. Vinet et al. [2.30] report a maximum undercooling of 900 K for Re. They attribute the very high undercooling achieved to the UHV environment. The high value of undercooling together with the observation of polycrystalline microstructure in the as-solidified sample was assumed to be due to homogeneous nucleation in this experiment. They used the undercooling result to

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Experimental Approach to High Undercoolings Rapid Cooling

Slow Cooling

Melt Spinning

Dispersion

Atomizing

Flux Immersion

19

Levitation

Laser Annealing Drop Tube

Figure 2.4. Experimental methods to undercool metallic melts; undercooling is either obtained by rapid cooling or by the reduction of heterogeneous nucleation. The drop-tube technique combines both principles of rapid cooling of small particles and reduction of heterogeneous nucleation [2.25].

determine the solid–liquid interfacial energy by an analysis within homogeneous nucleation theory. Drop-tube experiments are complementary to levitation experiments. In both techniques the samples are processed containerlessly. While levitation experiments allow measuring the whole history of undercooling and solidification, drop tubes offer the possibility of statistical analysis of nucleation and crystal growth as a function of droplet size and cooling rate. Figure 2.4 gives a schematic overview of the different techniques used for undercooling of metallic systems [2.25]. 2.4. PROCESSING OF BULK MELTS BY MELT FLUXING

Large undercoolings are, however, not limited by the dispersion of the melt into small particles but can even be achieved for bulk melts by using the melt fluxing technique. Bardenheuer and Bleckmann [2.31] were the first to employ this technique to undercool bulk melts of Ni and Fe. Undercoolings of ⌬T/TE 0.18 have been observed for melts of several grams. The successful use of the melt fluxing technique has been confirmed by experiments on other metals and alloys.

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A particular interesting result has been reported by Turnbull et al. [2.32]. A fully amorphous Pd40Ni40P20 sphere (Ø 1 cm) was formed after fluxing the melt in B2O3 and cooling and undercooling it at a rate of a just a few K/s. The glassy material prepared in such a way shows a very pronounced thermal stability against crystallization in comparison with rapidly quenched materials of the same alloy [2.33, 2.34]. Nowadays, a whole series of bulk metallic glass-forming alloys are produced by casting [2.35]. Flemings and his co-workers [2.36] combined the melt fluxing method with electromagnetic levitation for undercooling bulk melts of transition metals. Here, the melt (in mass of several grams) is embedded into an inorganic glass. The whole arrangement is placed into an induction coil. These experiments do not only allow any measurements of undercooling but are appropriate to measure the solidification velocity and to investigate the microstructure evolution as a function of undercooling. 2.5. CONTAINERLESS PROCESSING THROUGH LEVITATION

A freely suspended drop without any contact to a solid or liquid medium is generated by employing levitation techniques. Levitation of bulk samples offers the unique possibility of undercooling massive samples, which remain accessible not only for direct observation but also for external stimulation of nucleation. An overview of levitation techniques applied for containerless processing of droplets in diameter ranging from a few m to 1 cm is given in Ref. [2.37]. The current state of acoustic levitation is described. We introduce the concepts of acoustic levitation, static magnetic levitation, electromagnetic and electrostatic levitation, which are used for containerless undercooling and solidification of metals and semiconductors. Air pressure, magnetic, electromagnetic and electrostatic forces are used to compensate for the gravitational force acting on the individual drop with size of several millimetres. 2.5.1 Acoustic levitation In 1934, King predicted that the acoustic radiation pressure of a standing wave is so large that the resultant acoustic force could compensate the gravitational force acting on a small sphere leading to levitation [2.38]. Ultrasonic levitators have been in use for various experimental studies [2.39]. The basic principle involves the generation of a steady-state force arising from a high-intensity sound field and of sufficient magnitude to counteract the gravitational force. Non-linear acoustic theory [2.40] predicts that in high-intensity standing wave sound fields, samples with mass densities large compared with the surrounding gas will be levitated at

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21

Reflector Light sphere z

z

z

Heavy sphere

+ g − n λ 2

+ − Pressure

Stable Unstable

Velocity

Force

FL = FO sin(2kz)

Piston sound emitter

Figure 2.5. Levitation of spheres in an acoustic standing wave: pressure p, gas displacement velocity V, and force FL are shown along symmetry axis of single axis levitator.

acoustic pressure nodal positions, which correspond to the minimum of the force potential well. In order to achieve stable acoustic levitation, sample dimensions must be significantly less than the sound wavelength. Thus, sample sizes from sub-millimetres to millimetres may be levitated in a proper gas medium using resonant or interfering sound fields with frequencies ranging from 1 to 100 kHz. An example of a single-axis acoustic levitator is shown in Figure 2.5. The pressure, gas displacement velocity and force along the symmetry axis of the levitator are also shown. Using non-linear acoustic theory, the force acting on a small particle due to the acoustic radiation pressure in a sound field can be calculated [2.41]. Assuming a plane standing wave with a gas displacement velocity VDP = VO sin( kz ) cos( t )

(2.16)

p = pO cos(kz )sin( t )

(2.17)

a pressure

and assuming that the density of the sample, , is much larger than the density of the gas, g, ( ⬎⬎ g), the levitation force, FL, is expressed as [2.42] 5 FL = (r 2 )g VO2 ( kr ) sin( 2kz ) 6

for kr 0

( vertical stability )

∂ 2x B 2 ( r ) > 0; ∂ 2y B 2 ( r ) > 0

( horizontal stability ).

(2.34)

According to Eq. (2.34) the energy increases in all directions from an equilibrium point satisfying Eq. (2.29). This is equivalent to the condition of stability. These equations of stability can be expressed in terms of the field on the axis of the solenoid: B⬘( z ) 2 + B( z ) B ⬙( z ) > 0

( vertical stability )

B⬘( z ) − 2 B( z ) B ⬙( z ) > 0

( horizontal stability ).

2

(2.35)

Mathematically, the reason why diamagnets can be levitated in spite of Earnshaw’s theorem is that the energy depends on the field strength B(r), which unlike any of its components does not satisfy Laplace’s equation and so can possess a minimum. Physically, the diamagnet violates the conditions of the theorem because its magnetization m is not fixed but depends on the magnetic field (cf. Eq. (2.27)) and is directed opposite to the magnetic field. The latter condition means that there is always a repulsive and not an attractive interaction between a diamagnetic substance and a magnetic field. Macroscopically, this is expressed by Lenz’s rule; that is if an electrically conductive material as a metal is moved inside a magnetic field or such a sample is placed within an alternating magnetic field, eddy currents are induced inside the sample whose magnetic moment is opposite to the direction

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of the magnetic field. This will be treated in more detail in Section 2.5.3. Microscopically, this is because diamagnetism originates in the orbital motion of electrons and so is dynamical. An analogy to that on a macroscopic scale is the stability of a permanent magnet levitated above a concave upwards bowl-shaped base of type I superconductor. A superconductor represents collective motion of quasiparticles (electrons coupled as Cooper pairs below the superconducting transition temperature) and therefore is an ideal diamagnet. A permanent magnet above the superconductor is repelled by the field of the image it induces. If the magnet moves sideways, the image gets closer, so that the energy increases and the magnet flows back to the equilibrium position [2.52, 2.53]. In fact, static magnetic fields are used to levitate diamagnetic substances. According to Eq. (2.30) the product of the magnetic field and its field gradient in the vertical direction must be equal to the product of and g divided by to compensate the gravitational force acting on a sample with density and diamagnetic susceptibility . Fields strong enough to lift diamagnetic materials became available during the mid-20th century by constructing Bitter magnets and superconducting solenoids. In 1937, Braunbeck levitated small beads of graphite in a vertical electromagnet [2.54]. Graphite has the highest / ratio of 8 ⫻ 10⫺5 cm3/g. Beaugnon and Taylor lifted water and a number of organic substances [2.55]. Later on, liquid hydrogen and helium were levitated [2.56]. The most spectacular experiment was performed by Berry and Geim. They levitated a living frog in a solenoid of a central field of 20 T [2.51]. The dipole induced in the levitated frog can be regarded as equivalent to a current circulating in a loop embracing it. In this case the current is estimated to be 1.5 A, which of course is the sum of microscopic currents in the individual atoms and molecules of this living system. The creature was not electrocuted and did not suffer from noticeable biological effects. As an example, for water the magnetic susceptibility ⫺8.8⫻10⫺6, mass density ⫽ 1000 kg/m3, and the equilibrium condition of Eq. (2.30) yields the required product of field and field gradient as B(z) B⬘(z) ⫽ ⫺1400.9 T2/m. Such experimental conditions can be achieved by a Bitter magnet of geometry shown in Figure 2.10. The operation of a Bitter electromagnet for levitation experiments consumes a large amount of power, i.e. in the order of 4 MW [2.51]. But it is emphasized that with the field of a persistent current in a superconducting magnet, levitation is maintained without supplying any energy. The measured field profile is shown in Figure 2.10. The region of stable levitation condition is located between z1 and z2. At the inflection point of the vertical component of the magnetic field B(z)/Bo at which d2B(z)/dz2 ⫽ 0, indicated by zi ⫽ 78 mm, the field is B(zi) ⫽ 0.63Bo and the gradient of the field at zi is ⫺8.15Bo T/m, from which the required central field is calculated to be Bo ⫽ 16.5 T. The region of stable levitation is rather narrow. The spatial extent of this region is typically a small fraction

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31

Coil 2

B(z)/ Bo

Coil 1

1.0 280 mm

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0.8 0.6 0.4 z1

0.2 40 mm 150 mm 410 mm

0

zi z2

50 100 150 200 250 300 350 z (mm) Stable zone

Figure 2.10. Left: Geometry of coils in a Bitter electromagnet used for levitating diamagnetic objects. The currents in the two coils are equal. The region of stable levitation is marked with a black dot, at which a living frog was suspended (see inset on the right). Right: Profile of magnetic field on the axis of a Bitter magnet showing the stable zone of levitation between z1 ⫽ 69 mm and z2 ⫽ 86 mm of the arrangement as indicated on the left [2.51].

of the magnet’s size – just 2 cm around zi in the present case. The field strength has to be carefully adjusted to compensate for gravity. A very attractive advantage of levitation of diamagnetic bodies and material is that – unlike any other known or feasible technique – the suspension is distributed uniformly over the bulk because levitation force is acting on each individual atom or molecule. In fact, for a homogeneous material in a field with profile B2z, gravity is cancelled at the level of individual atoms and molecules. Thus, gravitationally driven macroscopic phenomena in bulk systems, such as fluid flow in melts due to convection can be essentially reduced on Earth without going into Space. Apparently, stable levitation of diamagnetic materials is feasible in strong static magnetic fields, but according to Earnshaw’s theorem this should be not possible for paramagnetic and ferromagnetic materials. However, it is known that even paramagnetic materials can be levitated in a stable position if they are placed in a stronger paramagnetic medium such as ferrofluid or liquid oxygen, which makes them effectively diamagnetic [2.57]. There is an alternative way to levitate paramagnetic substances in a static magnetic field. Here, the intrinsically unstable equilibrium of paramagnets in static magnetic field is stabilized by repulsive forces from a diamagnet placed in direct vicinity to the paramagnet. In fact, it was demonstrated that the forces created by diamagnetic materials of magnetic susceptibility 10⫺5 are sufficient to stabilize levitation of even strong ferromagnetic materials as Nd᎐Fe᎐B magnets over distances up to several millimetres under Earth gravity conditions, even though they decay rapidly with distance as 1/r5 [2.58].

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Equation (2.28) gives the energy E(r) of a magnetic material, which experiences the gravitational force and an interaction with static magnetic field. Consider a paramagnetic or a ferromagnetic sample placed in a cylinder of diamagnetic material within a strong magnetic field. The total energy of the sample in the position of stable equilibrium is given by E ⫽⫺ MB(r) + mgz + Edia, where Edia is the energy of diamagnetic interaction of the sample with the cylinder. Close to the equilibrium position at the filed axis [2.51, 2.59] EEo + [ mg − MB⬘( z ) ] z + K v z 2 + K h r 2 + Cr 2 + with (2.36)

K v ( z ) = − MB ⬙( z ) / 2 K h ( z ) = − M ⎡⎣ B⬘( z ) 2 − 2 B( z ) B ⬙( z ) ⎤⎦ /8 B( z ).

For the geometry as shown in Figure 2.11, C ⫽ 45o | |M2/16D5 with D being the inner diameter of the cylinder. If there is no diamagnetic cylinder (C ⫽ 0), the position of the sample is always unstable, i.e. Kv and Kh are negative at any place. The diamagnetic interaction allows the energy E to have a minimum (Kv ⬎ 0 and Kh + C ⬎ 0), which leads to C ⬎ MB⬘(z)2/8B(z) just above the point of a maximum field gradient (B⬘⬘(z) ⫽ 0). Levitation is easiest in the region where the field gradient B⬘(r) is strongest; this is in the upper or lower region of the inner bore of the magnet. A coordinate L is introduced, on which the field changes, B⬘⫽B/L, the optimum levitation point is defined by B⬘⬘(z) ⫽ 0, L varies between R and 1.2R for long and short solenoids. If the magnetic sample to be levitated possess a sphere-like geometry with diameter d and a remnant field Br, then M ⫽ (4/4o)Brd3 and the condition for levitation is given by

(

A LBr2 d 3 / o g

)

1/ 5

>D>d

(2.37)

with A 1.92. Both, calculations and experiments give evidence that a ferromagnetic sample in size of several millimetres and with a remnant magnetization of about 1 T (Nd᎐Fe᎐B hard magnet) can be levitated with a clearance gap, D⫺d, of several millimetres using a solenoid of inner bore 10 cm and strongly diamagnetic Bi or graphite as material for the cylinder [2.58]. Even though levitation by stationary fields has not been applied for undercooling experiments so far, there is remarkable progress in this field. In particular, the large magnetic fields available nowadays by superconducting solenoids make the technique of diamagnetic levitation in stationary fields attractive. It offers very interesting capabilities since it not only provides the condition of containerless

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33

z 0

r

Stability functions (MB) 1 −1 0

R

Kv

Kn

−R C>0 Bi cylinder

Levitating magnet −2R

Figure 2.11. A Nd᎐Fe᎐B hard magnet, 4 mm high and 4 mm in diameter is levitated at the axis of a solenoid of inner radius R 10 cm and length 2R in vertical orientation. The levitation of the magnetic sample in the static magnetic field of the solenoid with field strength of about 100 G is stabilized by a bismuth cylinder ( ⫽ ⫺1.5 ⫻ 10⫺4) with inner diameter D ⫽ 8 mm. The photograph shows the top view of the levitating magnet. The right-hand plot shows the stability functions Kv and Kh calculated for a solenoid with a height of twice its radius (solid curves). Diamagnetic interaction C between ferromagnetic sample and diamagnetic cylinder shifts the horizontal stability function Kh to the left (dashed curve) and a small region of positive ⌬E emerges above the point where Kv ⫽ 0 (taken from Ref. [2.58]).

processing of small drops but also allows independent control of levitation force and heating if combined it with an additional heating source as, e.g. a high-power laser. Moreover, this technique provides levitation of each individual atom. Presuming homogeneity in the sample as usually present in melts and liquids the diamagnetic levitation can lead to a force-free sample quite comparable to the condition in reduced gravity. 2.5.3 Electromagnetic levitation For metallic systems the most suitable technique for freely suspending spheres of diameter up to 1 cm is the electromagnetic levitation technique. The principle of

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electromagnetic levitation is based on the induction of eddy currents in an electrically conducting material if the material experiences a time-dependent magnetic field B (Lenz rule): • E = −B/t

(2.38)

where E is the electrostatic field. The eddy currents produce a magnetic moment m that is opposite to the primary field. This leads to a diamagnetic repulsion force Fr = −( mB )

(2.39)

between the primary field and the sample. If the repulsion force Fr is equal in amount and opposite in direction to the gravitational force, Fr ⫽ mgg, the sample is levitated, where mg is the mass of the sample and g the gravitational acceleration. Electromagnetic levitation can be used to levitate metallic and even semiconducting samples. A characteristic feature of electromagnetic levitation is that both levitation and heating of the sample are always occurring simultaneously. This offers the advantage that no extra source of heating is required to melt the material, but it is associated with the disadvantage that levitation and heating can be controlled independently only in a very limited range. Consider a conically shaped levitation coil consisting typically of 5–8 windings and 1–2 counterwindings at its top to stabilize the position of the sample, an example is shown in Figure 2.12. If an alternating current is flowing through the coil, the alternating electromagnetic field will induce eddy currents within an electrically conducting system. The eddy currents lead to a repulsion force against the primary field, and simultaneously they produce heat. Assuming symmetrical coil loops and a sample diameter much smaller than the diameter of the coil windings, and replacing the eddy currents distributed over the entire surface near region of the sample by a single-loop circuit, the levitation force FL and the power absorption P are given by [2.60] FL Fg

=

P=

3G ( x ) ( B ) B, 2 g o

3rH ( x ) ( BB ) , el 02

(2.40)

(2.41)

where , el, and r are the mass density of the sample, the electrical conductivity, the magnetic permeability, and the radius of the sample, respectively. o is the

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35

z

bn Winding n Sample r

zn z

Winding 2

b2

Winding 1

b1 z2 z1

0

Figure 2.12. Typical conical levitation coil: the coordinates bn denote the radii of loops and the coordinates zn denote the heights of the loops above a reference level.

permeability of free space. B is the magnetic field strength and the radian frequency of the magnetic field. The functions G(x) and H(x) are defined by 3 sinh( 2 x ) − sin( 2 x ) , 2 x cosh( 2 x ) − cos( 2 x )

(2.42)

x [sinh( 2 x ) + sin( 2 x )] − 1. cosh( 2 x ) − cos( 2 x )

(2.43)

G( x) = 1 −

H ( x) =

The theory of electromagnetic levitation has been generalized allowing for analytical calculations of the field distribution independent of the ratio of sample and coil diameter [2.61, 2.62]. Taking into account the specific sample material parameters and applying Eqs. (2.40)–(2.43), the minimum power absorption needed to compensate for the gravitational force Fg can be calculated. As an example, Figure 2.12 shows a typical conically shaped levitation coil consisting of five windings and two counterwindings

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at the top. The coordinates zn give the height above a reference level while the coordinates bn give the radii of the loops. The magnetic field strength and the magnetic field gradient can be evaluated using the elementary rules of electromagnetism. The force acting on a spherical sample, anti-parallel to the gravitational force along the symmetry axis of the coil, is expressed as a function of the z-coordinate by [2.63, 2.64].

FLz = 1.5I 2 r 3G ( x )∑ n

bn2 ⎡bn2 + ( z − zn ) 2 ⎤ ⎣ ⎦

3/ 2

∑n ⎡

bn2 ( z − zn )

2 2 ⎤ ⎣bn + ( z − zn ) ⎦

5/ 2 .

(2.44)

In Figure 2.13, the calculated field strength (solid line) and the field gradient (dotted line) are plotted as a function of the position of a spherical sample with diameter of 5 mm. The assumed coil design is indicated by the cross sections of the loop and a coil current of Io ⫽ 400 A is used for the calculations [2.65]. In an equivalent way the power absorption of the sample as a function of the z-coordinate is computed by bn2 bn2 3r P= H ( x )∑ ∑n 2 5/ 2 2 3/ 2 el n ⎡b 2 + z − z ⎤ ⎡ b + ( z − z )2 ⎤ ( ) n n ⎣ n ⎦ ⎣ n ⎦

(2.45)

Field Gradient [Tesla/m] -12 Position of the Sample z [mm]

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-8

20

-4

0

4

8

12

0.08

0.12

Strength

15 x x

10 5

Gradient

0

x

-5

x x

x

-10

x x

-15 -20

Io= 400A

-0.12

-0.08

0 0.04 -0.04 Magnetic Field [Tesla]

Figure 2.13. Strength (solid line) and gradient (dashed line) of the magnetic field as a function of the sample position of a typical levitation coil developed for undercooling experiments on gold [2.65]. The dashed area gives the range of positions in which the sample can stably be levitated. A coil current of Io ⫽ 400 A has been assumed.

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37

where I is the coil current. Under equilibrium conditions the sample approaches a temperature, given by the heat balance under vacuum conditions (heat transfer to the environment by radiation only) i

P = Qr

(2.46)

i

where Q denotes the heat transfer from the sample to the environment via heat radiation. According to Planck’s law the heat transfer by radiation is given by i

(

Qr = 4r 2 SB T 4 − TR4

)

(2.47)

where SB is the Stefan–Boltzmann constant, the integral surface emissivity, T the sample temperature and TR the ambient temperature. The temperature control of electromagnetically levitated samples requires a separate action of P and FL as far as possible. As a consequence of Eqs. (2.41)–(2.42) the essential difference between P and FL is that the functions G(x) and H(x) have different characteristics with respect to the frequency of the alternating electromagnetic field: FL depends on the product (B●ⵜ)B while P is proportional to B2. Hence, temperature control is possible within a limited range by choosing a proper frequency of the alternating field and by the movement of the sample along the symmetry axis of a conically shaped coil. In the lower regions of the coil the windings are tighter, thus the magnetic field and power absorption are greater than in the upper region of the coil with lower field strength. By increasing the power the sample is lifted up into regions of larger field gradients and smaller magnetic field strength and cools down. Using coils of suitable geometry, controlled temperature variation is possible by some hundred degrees Kelvin. By changing the sample position in the levitation coil due to a variation of the current through the coil the temperature of a Ni sample (Ø 8 mm) may be altered within a range of approximately 600⬚ Kelvin. It is assumed that the sample is placed into a levitation coil with five windings and one counterwinding. An alternating electrical current at 300 kHz powers the coil. More details on this analysis of temperature control may be taken from Ref. [2.37]. As can seen from Figure 2.14 the force function G saturates at high frequencies (q ⬎ 10), while the power function monotonically increases with q. From the condition that the levitation force has to compensate for the gravitational force to levitate a sample, minimum power absorption is required, which in turn determines a minimum temperature of the sample that cannot be underpassed without fulfilling the levitation condition. The minimum temperatures, which can be reached on electromagnetically levitated transition metals and noble metals are in most cases such that the molten sample cannot be cooled below the equilibrium melting temperature or even in the undercooling range. It is expected that melts containerlessly

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Metastable Solids from Undercooled Melts 15

1.0 G(q)

0.8

10 0.6

H(q)

0.4 5 0.2

0

5

10

Power Function H(x)

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Force Function G(x)

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15

Parameter q

Figure 2.14. Force function G and power function H in dependence of the parameter q ⫽ r/ with the skin depth and r the radius of the sample.

processed in high-purity environment undercool by amounts of the order of 20% of their melting temperatures, i.e. ⌬T/TE 0.20 [2.66]. This means that with the exception of a few refractory metals such as W or Nb there is no chance of undercooling metals and their alloys below their melting temperatures in containerless state if electromagnetic levitation is applied under the conditions of environmental vacuum. The situation may change if electromagnetic positioning is used under reduced gravity in space (cf. Section 2.6). Under terrestrial conditions, additional cooling by forced convection of noble gases has to be used to supply sufficient heat transfer from the sample to the environment and to ensure in this way an undercooling below the melting point. Using forced convection by He or He᎐H2 gas of good thermal conductivity, undercooling experiments on transition metals and their alloys become possible. A schematic view of an electromagnetic levitation chamber for containerless undercooling and solidification experiments is shown in Figure 2.15 [2.67]. The levitation coil together with the sample (Ø 6 mm) is placed within an UHV chamber, which can be backfilled with gases such as He or He᎐H2 mixture. The gases are purified by an oxysorb system and, additionally, by passing them through a liquid-nitrogen cold trap. The sample is processed within the levitation coil, which is powered by a high-frequency generator. The maximum power-output of the rf-generator is 24 kW. The frequency can be changed in the range between 300 kHz and 1.2 MHz. Temperature control in a limited range is possible by varying the sample position along the symmetry axis of the levitation coil and by using forced convection due to cooling gases. The temperature of the sample is measured by means of a two-colour pyrometer with an absolute accuracy of ⫾3 K and

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39

Contactless temperature measurements by a two-colour pyrometer (100 Hz)

24 kW Sample RF

Levitation coil

Fast relative T-measurements by a photo-diode (1 MHz)

0.3 -1 MHz Crystallization trigger needle

Figure 2.15. Electromagnetic levitation chamber for containerless undercooling and solidification of metals.

a sampling rate of 100 Hz. However, such a frequency is not sufficient to monitor the rapid temperature rise during recalescence. Therefore, a high-speed photosensing device has been developed to record relative temperature changes during rapid solidification with a frequency of 1 MHz [2.68]. Solidification of the undercooled melt can be externally initiated by touching the lower bottom of the sample with a crystallization trigger needle. The levitation chamber can be combined with a drop calorimeter to measure the enthalpy of the melt as a function of undercooling [2.69]. This allows to determine the specific heat from the temperature derivative of the measured enthalpy ⌬H(T ). The electromagnetic levitation technique has been developed to measure the mass density and the thermal expansion of metallic liquid drops [2.70]. Here, an interesting result has been observed for pure Ni. If the measured mass density is extrapolated to lower temperatures a crossover point is detected at an undercooling of ⌬T ⫽ 480 K at which the mass density of the undercooled liquid is equal to the mass density of the solid. This should give rise to a constraint for the maximum achievable undercooling similar to the entropy catastrophe (Kauzmann paradox). Nowadays, electromagnetic levitation technique is not only widely used for investigations of the undercooling and nucleation behaviour of metallic samples [2.71, 2.72], but also for the determination of thermophysical properties of undercooled melts [2.72–2.74]. The undercoolings obtained by this method on bulk melts of diameter up to 1 cm are comparable with those achieved by the droplet dispersion technique [2.70, 2.72, 2.75–2.77]. These observations indicate that

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apparently surface-induced heterogeneous nucleation is the dominant process initiating solidification. Large undercoolings are throughout not limited to small particles but can also be achieved on bulk melts, provided that heterogeneous nucleation induced by container walls is completely eliminated and surfaceinduced heterogeneous nucleation is reduced by processing the drops in a highpurity environment. The freely suspended drop within a levitation coil is specially suited to directly observe the crystallization kinetics of an undercooled drop. Photodiodes are used to measure the rapid temperature change during recalescence, which is due to the rapid release of the heat of crystallization during fast crystal growth in undercooled melts. The photodiodes are arranged perpendicular to the symmetry axis of the coil. The solidification of the sample is externally nucleated by a trigger needle, which is touching the surface of the sample at its bottom. Solidification starts at this point and the dendrites propagate isotropically through the volume of the sample. A small part of the equatorial sample surface is imaged by an optical system on the sensitive area of the photodiodes. The growth velocity, V, is obtained by dividing the solidification pathway, ⌬s, by the measured time, ⌬t, needed by the solidification front to propagate through the observation window, V ⫽ ⌬s/⌬t [2.68]. Thereby, it is assumed that the solidification front is approximated by the envelope of the dendrite tips. This assumption is justified at medium and particularly at high undercoolings, where many small dendrites are formed, but it is violated at small undercoolings where only a few thick dendrites are propagating through the melt. Accordingly, the scatter and uncertainty of the measurements by using the photodiodes are increasing with decreasing undercooling. To measure the dendrite growth velocity at small and medium undercoolings with high reliability and reproducibility, we developed a capacitance proximity sensor (CPS) [2.78, 2.79]. The set-up is schematically illustrated in Figure 2.16. It consists of a nucleation trigger needle made of the same material as the sample. The needle is part of a resistance–capacitance (RC) electrical circuit whose resonance frequency is measured. If the needle is touching the sample and initiates solidification, the capacitance of the RC-circuit changes abruptly. The time, t1, of initiating crystallization is measured by a sudden change of the output signal of the RC circuit with a time resolution of 1 s. The counterpart of the triggering point at the opposite side of the sample is focused by an optical system on the sensitive area of a photodiode. As soon as the central dendrite arrives at the top of the sample at time, t2, the signal of the photodiode rapidly increases. The growth velocity, V, is then obtained by dividing the height, Do, of the as-solidified sample by the time difference ⌬t ⫽ t2⫺t1:V ⫽ Do/⌬t. In addition to photodiode and CPS, also high-speed digital camera technique is applied to observe the propagation of the solidification front [2.80, 2.81]. The

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Experimental Approach to High Undercoolings "Photodiode"

t2

Sample

P(t) t1

V=

d2 d1

Pyrometer (± 3 K)

ΔT = TL − TN

Photodiode

Semi-transparent mirror

d2 − d1

V=

D0 t2−t1

S(t)

Photodiode

t2 − t1

41

"CPS"

HFCoil

S

D0

P

P(t) t t1

t2

Transient recorder

t

ΔC

1 cm

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Sensing trigger

R

t1

t2

t CPS

R(t)

Transient recorder

Figure 2.16. Experimental techniques to measure the dendrite growth dynamics in levitation undercooled samples: the photodiode method (left-hand side) and the capacitance proximity sensor technique, CPS (right-hand side).

Figure 2.17. Propagation of the solidification front (light grey) through an undercooled Ni melt (dark grey) for ⌬T ⫽ 90 K (a) and ⌬T ⫽ 140 K (b). Both sequences were recorded at a frame rate of 30 000 fps, for 90 K only each fifth image is shown. Triggering occurred at the bottom of the sample, which is to the left in this display. A small part (15%) of the sample’s bottom part is hidden by the coil.

camera enables observations of the solidification process at frame rates of up to 120000 frames per second (fps) at a resolution of 128⫻16 pixel [2.81]. For the dendrite growth velocities in undercooled Ni melt, frame rates of 30000 fps have proven to be a good choice for low- and medium undercoolings. This frame rate offers a higher resolution of 256⫻128 pixel. Figure 2.17 gives a sequence of images taken during solidification of a Ni sample undercooled by ⌬T ⫽ 90 K (upper part) and by ⌬T ⫽ 140 K (lower part), respectively [2.81]. It can be seen

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that the solidification front at the sample surface is of an irregular shape for ⌬T ⫽ 90 K, while at ⌬T ⫽ 140 K the front appears very smooth. 2.5.4 Electrostatic levitation Electrostatic levitation makes it possible to levitate conducting as well as nonconducting samples in diameter of a few millimetres under UHV conditions. The method of electrostatic levitation is based on Coulomb forces acting on charged particles placed within an electrostatic field. Different configurations of the electrodes have been used, ranging from single axis, two electrodes to quadrupole arrangements. Tetrahedrally arranged electrodes have been applied as well. In all of these arrangements the sample is located in an unstable position since the Coulomb potential is proportional to the reciprocal distance r: =−

dFc 1 ⬀ . dr r

(2.48)

This means the potential has no minimum (cf. Section 2.5.2). Stable positioning of the specimen, therefore, must be accomplished by a feedback control system that monitors the position of the object. The feedback control can be realized either by optical observation of the sample in combination with a computer-controlled regulation of the positioning system, or by sensing the sample position by capacitance changes and rearrangement of the voltage applied to the electrodes. As an example, Figure 2.18 shows the block diagram of the electrostatic levitation system developed by Rhim et al. [2.82]. Here the position of the sample is controlled by a CCD (charge coupled device) camera picking up an image of the object inside the chamber with a frame rate of 120/s. The position and velocity of the sample are analysed by a microcomputer, which controls the position and damping of the sample through electrostatic forces.

High Voltage Amplifiers Top Electrode Position Sensor

Sample

Micro-Computer

Bottom Electrode

Figure 2.18. Block diagram of an electrostatic levitation system.

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Levitators designed for terrestrial operation have to generate upward electrostatic force that compensates for the downward pull of the gravitational force. The principles of electrostatic levitation of conducting and non-conducting samples are described in detail by Felici [2.83]. A pair of horizontal, parallel electrodes is assumed. A conducting spherical sample of radius r is considered to be placed on the lower plate of the electrodes. The electrode charges the conducting sample. To calculate the force to lift off the sphere in an electrostatic field E one presumes an electrical charge Qel of the sample: Qel = 4 o Er 2 .

(2.49)

Here, is a constant, which is determined for a conductive sphere to be 1.645 [2.84]. If a homogeneous electrostatic field between the electrodes of distance d is supposed, then E⫽U/d holds with U the electrical voltage between the electrodes. The force FE acting on the sample during lift off is given by 2 FE = Qel E = 4 oU Lo r/ d 2

(2.50)

where ⫽ 0.832. If the force according to Eq. (2.47) is set equal to the gravitational force Fg ⫽ mg ⫽ (4/3)r3g acting on the sample of mass m and density and g the gravitational acceleration the voltage ULo needed to lift off the sample is calculated as: U Lo = d

gr . 3 o

(2.51)

Accordingly, the charge of the sample is determined by g or 5 4 oU a r 2 . Qa = = 4 d 3

(2.52)

As long as the sample keeps the same temperature its electrical charge remains unaltered. If the sample is placed in between the electrodes, the electrical voltage of the electrodes for stationary levitation is calculated if the levitation force FL⫽QaEL⫽QaUL/d is taken into account: UL = d

gr . 3 o

(2.53)

Comparing Eq. (2.50) with Eq. (2.48), we see that the levitation voltage UL ⫽ Ua differs by the factor from the voltage ULo to lift off the sample.

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As described above the sample is not in a stable position if it is moving into the middle of the electrodes. Therefore, the controlling system to keep the sample just in between the electrodes has to adjust the levitation voltage according to its instantaneous position. This leads to fluctuations of the levitation voltage around the average value given by Eq. (2.50). If the sample is heated its charge will be altered. The evaporation of atoms (positive ions) of the heated sample under vacuum conditions leads to an increase in the negative charge and is given by the evaporation rate W + [2.85]: W+ =

Pd 2ma k BT

(2.54)

where Pd is the vapour pressure and ma the molecular mass of the atomic species of the sample material. The thermal emission of electrons leads to an increase in the positive charge of the sample and counteracts the effect by evaporation of atoms. It is given by the Richardson–Dushman equation [2.86] W− =

⎛ ⬘ ⎞ AT 2 exp ⎜ − . e ⎝ k BT ⎟⎠

(2.55)

Here, A ⫽ 120/A cm2/K2 is the Richardson constant, e the elementary electronic charge and ⬘ the effective work potential. Taking into account the strength of the electrostatic field E, ⬘ is determined as ⬘ = −

e3 E 4 o

(2.56)

with the work potential for the respective material. The electrical charge of the levitated sample can be systematically altered by irradiation with photons. This process is described by the photo-electronic effect. The energy of the photon has to be equal or greater than the work potential of the electrons. In the case of metals is in the order of 10 eV. Taking into account the relation E = hν =

hc ,

(2.57)

light of wavelength ⬍ 100 nm is required to emit electrons by the photo-electronic effect from metals (h, Planck’s constant; c, velocity of light). This means light from a ultra-violet (UV) light source is suitable to change the electrical charge of the

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levitated sample. If the UV photons hit the material of the environmental electrodes they can lead also to an emittance of electrons from the electrode material. This effect, in turn, may disturb the regulation of the sample position by the controlling system. To reduce such disturbances, it is advantageous to use material for the electrodes of a high work potential. Figure 2.19 shows the assembly of an electrostatic levitator constructed for measurements of thermophysical properties of liquid samples and for undercooling experiments [2.84]. The vertical electrodes can be charged by a maximum voltage of ⫾20 kV with the upper plate being on negative potential. Such large voltages are necessary to produce a force to compensate the gravitational force. If a typical value for the electrical charge of Q ⫽ 10⫺9 Cb of a 3-mm metallic sphere of mass of 0.1 g is assumed, a voltage of 8 kV is needed to generate an electrostatic force sufficiently large to levitate the sample between the electrodes of distance of 8 mm. Simultaneously, the maximum slew rate of the voltage supply has to be of the order of 300 V/s or even higher to react quickly enough for the positioning control of the sample. Four additional electrodes stabilize the sample horizontally. For this purpose the horizontal electrodes are loaded with a maximum voltage of ⫾ 6 kV at a slew rate of 12V/s. The distance between the vertical electrodes is 8 mm whereas the distance of two opposite horizontal electrodes amounts to 23 mm. The sample is in an unstable position. Electrostatic levitation requires therefore a sophisticated sample position control system. The position of the sample is

Figure 2.19. Top view of an electrostatic levitator for undercooling experiments on drops of size of a few millimetres. The inset gives a side view of the arrangement of vertical and horizontal electrodes and the levitated sample.

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detected with two position-sensors (PS) arranged in one plane with axes perpendicular to each other. The PS consists of a photodiode of a large active area but having several electrical contacts (more than two) to collect the photocurrent. Two lasers placed just opposite to the respective PSs illuminate the PS. The lasers are operating at a light wavelength of 632 nm and at a power output of 7 mW. If the sample comes in the laser light beam a shadow is produced on the sensitive area of the PS as schematically shown in Figure 2.20. The light of the laser falling onto the active area of the PS produces a photocurrent Ip that flows to the electrical contacts. The photocurrent, which is collected by the several contacts, depends on the distance between the contact and the position of the area illuminated by the laser light. The combination of two PS and lasers arranged perpendicular to each other allow to determine the position of the sample in three-dimensional space. The photocurrent is amplified and recorded by a computer. The computer evaluates the recorded signals and governs itself the control of the position voltage of the electrodes. In this a way the sample can be kept in stable position between the electrodes. As soon as the sample is levitated in a stable position the high power-heating laser heats it up. This system consists of two diode lasers and an optical system for

Figure 2.20. Arrangement of the photo-sensing detector (PS) with its active area, which is illuminated by the laser beam (dark circle) and the shadow of the sample (light circle). There are four contacts I1, I2, I3, I4 at which the photo current is collected. Its respective value at the different contacts depends on the position (coordinate XL) of the illuminated area and the position of the shadow of the sample (coordinate XP).

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47

focussing each laser beam. An optical lens at the exit of the laser is necessary since the original laser beam of the diode laser is very divergent. The wavelength is 810 nm at which the laser operates and both diode lasers can produce a maximum power output of 60 W. The advantage of diode lasers compared to Nd᎐Yag laser is the simple handling of the heating power. It is easily controlled by the diode current while in the case of Nd᎐Yag-laser power, control is possible only by using complex optical systems to weaken the primary beam. For every levitation technique it is mandatory to measure the temperature of the sample contactlessly, i.e. by pyrometric means. The basis for pyrometric temperature measurement is Planck’s law. Accordingly, the spectral density of radiation of a blackbody, LS, is given by 2hc 2 LS = 5

⎡ ⎛ hc ⎞ ⎤ − 1⎥ ⎢exp ⎜ ⎝ k BTS ⎟⎠ ⎥⎦ ⎢⎣

−1

(2.58)

where h is the Planck’s constant, c the velocity of light, the wavelength of the radiation, kB the Boltzmann’s constant and TS the temperature of the blackbody. For the special case TS ⬍⬍ 1, Eq. (2.58) can be approximated by Wien’s radiation law: LS

⎛ hc ⎞ 2hc 2 exp ⎜ − . 5 ⎝ k BTS ⎟⎠

(2.59)

For grey bodies, such as metals, the radiation density, LG, is smaller than for a blackbody LS: LG = LS

(2.60)

with 0 ⬍ ⬍ 1 the total hemispherical emissivity of the grey body. Accurate temperature measurements by a pyrometer requires the knowledge of the emissivity, which in turn depends on the temperature and the wavelength; it may change also upon phase transformations of the sample, such as solidification. In particular, semiconductors as Si and Ge show an essential change in the emissivity upon crystallization and melting. To reduce the influence of a change of the emissivity on temperature measurements, often two colour pyrometers are applied. Here, the temperature of the sample is determined by the ratio of the radiation densities measured at two different wavelengths 1 and 2. 2.6. CONTAINERLESS PROCESSING IN SPACE

The application of electromagnetic levitation under gravity requires relatively high power absorption to levitate the sample against the gravitational force. This is

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necessarily accompanied by an equivalent large heating effect. Under vacuum conditions heat transfer takes place by radiation only. This will limit the applicability of the electromagnetic levitation technique under vacuum conditions to undercooling experiments on refractory metals and high-melting alloys. For instance, if nickel samples are levitated under high-vacuum conditions the heat transfer by radiation is capable of cooling the sample only to temperatures of about 1700 K, which is close to the melting temperature of this metal. Therefore, this metal cannot be undercooled in electromagnetic levitation experiments by processing the sample in high vacuum. Additional convective cooling by He gas or H2᎐He mixture gas makes the technique suitable for undercooling of transition metals such as Fe or Ni. In this a way the levitated Ni sample can be cooled to temperatures of T > 1000 K. Even thoroughly cleaned cooling gases are dirty compared with UHV conditions and have been shown to induce heterogeneous nucleation by metaloxide formation at the surface of the samples. Levitation forces inevitably introduce dynamic motion in the liquid. Its influence on the solidification process is largely unknown to date. At the same time, levitation forces lead to a deformation of the liquid drop, so any measurements relying on a particular sample shape become difficult in a gravity field. These limitations of the electromagnetic levitation technique for undercooling experiments are circumvented by using the special environment in space, where the positioning forces to compensate disturbing accelerations are about three orders of magnitude smaller than the levitation force to counteract the gravitational force under terrestrial conditions. A special instrument called TEMPUS has been designed to provide means of containerless processing in space [2.87]. TEMPUS comes from the German acronym Tiegelfreies Elektro-Magnetisches Prozessieren Unter Schwerlosigkeit. A schematic view is shown in Figure 2.21. Positioning and main heating are separated in TEMPUS by placing the sample into the superposition of a quadrupole and a dipole field. The fields are generated by specially designed copper coils surrounding the sample. Both coil systems are powered independently by two rf generators at different frequencies. This two-coil concept has led to a drastic increase in the heating efficiency of levitated drops compared to usual levitation on Earth [2.88]. The coil system is integrated in a UHV chamber equipped with turbomolecular pump and ion-getter pump the latter one serves for pressure measurements. The recipient can be backfilled with highpurity Ar and/or He᎐3.5% H2 processing gas. Contamination analysis can be performed with a quadrupole mass spectrometer. Sample rotations and oscillations are damped by an inhomogeneous DC magnetic field in the sample region. Solidification of the undercooled melt can be externally triggered by touching the sample with a nucleation trigger needle, which is an integral part of the sample

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Figure 2.21. Schematic view of the TEMPUS facility. All subsystems are shown with the exception of the radial temperature detector.

holder. The samples are transferred into the coil system from sample storage within ceramic cups or refractory metal cages. TEMPUS is equipped with pyrometers and video cameras. The sample is observed from two orthogonal views. From the top three infrared detectors measure the temperature at different wavelength bands with a frequency of 1 MHz. A video camera is included in the optical path for sample observation with a maximum frame rate of 480 Hz. From the side two different instruments can be installed, either a pyrometer specialized for measurements of the crystal growth velocity at rates up to 1 MHz (RAD), which is combined with a video camera with frame rates up to 400 Hz, or a high-resolution video camera (RMK) with special optics. The resolution is 10⫺4 for a 8-mm sample as required for measurements of the thermal expansion. TEMPUS was successfully flown by NASA Spacelab missions IML2 (International Microgravity Laboratory 1994) and MSL1/MSL1R (Materials Science Laboratory 1997). TEMPUS is especially suited to perform undercooling experiments on metals and alloys under microgravity. Different types of experiments can be conducted. Solidification experiments by undercooling and measuring multi-step recalescence profiles provide information on primary crystallization of metastable crystallographic phases. Measurements of the growth velocity as a function of undercooling

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are interesting with respect to the formation criteria for non-equilibrium microstructures. They also give insight into growth phenomena, where influences of convection and fluid flow play a role in, e.g. dendritic/eutectic growth behaviour and its influence on pattern formation in microstructure development. TEMPUS is also well suited to measure thermophysical properties both in the stable liquid and metastable undercooled regime [2.89]. The electromagnetic positioning of near-spherical samples in microgravity is suitable to measure the mass density of liquid metals with high accuracy. The change in diameter of a levitated sphere as a function of temperature is observed by an optical arrangement imaging the profile of the droplet [2.90]. Precursor experiments on Ni in terrestrial levitation experiments have demonstrated their feasibility. However, the deviation from spherical symmetry of the liquid sample due to the strong levitation fields limits the accuracy of these measurements on Earth, a problem which was solved by experiments in space [2.91]. The frequency of the positioning coil current depends on the inductivity of the whole system consisting of coil and sample. The inductivity and, consequently, the frequency will change if the electrical resistivity of the sample increases with temperature. A new method has been developed to apply this principle to measure the temperature change in the electrical resistivity of molten and undercooled droplets [2.92]. A method based upon an AC modulation of the heating coil current has been proposed [2.93] and tested [2.94] to measure the specific heat of undercooled melts processed in a microgravity environment, using the TEMPUS instrument. Power modulation is achieved by either a pure sinusoidal modulation of the heater rf-circuit voltage resulting in P (t ) = Po + Pav + P ( )sin( t + o ) + P ( 2 )sin( 2 t + o ).

(2.61)

Alternatively, the rf-circuit voltage is modulated with the square root of a sinus function thus omitting the ⌬Pav and ⌬P(2 ) terms. The critical parameter describing the temperature response of the sample is the Biot number, Bi, i.e. the ratio of external heat loss and internal heat transport. Under the conditions that Bi ⬍⬍ 1 and that the temperature variation is much smaller then the bias temperature To the temperature response to power modulation according to Eq. (2.61) is given by T (t ) = To + Tav [1 − exp( −t / 1 ) ] + f ( , 1 , 2 )Tm ( )sin( o + Δ) exp( −t/1 ) + f ( , 1 , 2 )Tm ( )sin( t + o + Δ).

(2.62)

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The equilibrium temperature To of the first term is given by the Stefan– Boltzmann law Po = A SB To4 . The second term describes an increase in the average temperature with the external relaxation time 1 and amplitude ⌬Tav as 1 =

Cp 4 A SB To3

,

Tav = Pav 1 . Cp

(2.63)

where Cp is the heat capacity, A, SB and are sample surface, Stefan–Boltzmann constant and total hemispherical emissivity, respectively. The external relaxation time is determined from the temperature response to a step function change in heating power. The third term of Eq. (2.62) represents an initial transient with the internal relaxation time, 2, as given by 2 =

Cp k 2 + kc

(2.64)

where kc is the convective heat-transfer coefficient and k2 the conductive heattransfer coefficient obtained from 4 k2 = 3R. 3

(2.65)

R is the radius of the sample, the thermal conductivity, 1 a geometrical factor and the conductively heated volume fraction. By an analysis of the measured temperature–time profile the ratio of the specific heat, Cp, and the emissivity can be determined according to Eq. (2.63) as a function of To. According to Eq. (2.65) this method possesses potential to determine heat transport by convection and heat conductivity, respectively. A modulation of the heating coil current also excites surface oscillations of a freely suspended liquid. The oscillating drop method is used to measure the surface tension and the viscosity of levitated drops [2.95]. According to Rayleigh’s formula, the frequency of the surface oscillations is related to the surface tension. If the radius R of a spherical droplet undergoes oscillations of the form R = Ro (1 + cos( t ) exp( −t ))

(2.66)

where is the amplitude of the oscillation, the frequency and the damping, then the frequency is given by

2 = (32) / (3m) with being the surface tension and m the mass of the drop.

(2.67)

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Equation (2.67) cannot be applied directly to oscillations of levitated liquid drops. The influence of the external electromagnetic and gravitational fields must be taken into account. These fields lead to a splitting of the single peak for the oscillation spectrum into five peaks, and a shift of the peak positions. A correction term was developed for electromagnetic levitation experiments [2.96]. It is advantageous, however, to perform such experiments under microgravity, where both fields are reduced essentially. On the basis of Kelvin’s work on the oscillation of viscous drops [2.97], the oscillating drop technique yields also the viscosity of the droplet. He developed an expression for the damping factor , =

20 Ro 3 m

(2.68)

where is the viscosity. Equation (2.68) is only correct for spherical drops in the absence of external fields. Accurate measurements of the viscosity applying the droplet oscillation technique are therefore limited to experiments under microgravity. The TEMPUS facility had its maiden flight on board of NASA’s spacelabmission International Microgravity Laboratory IML-2 in 1994. The technical operation of the device with all subsystems worked nominally during the entire mission of 14 days. Important scientific results have been obtained. The element Zr was melted and undercooled several times. Melting of Zr requires a temperature of higher than 2125 K, which means it was the highest temperature ever achieved in the spacelab [2.98]. TEMPUS was reflown on board of NASA’s spacelab-mission Materials Science Laboratory MSL-1 in 1997. Altogether 17 different experiments of 10 research groups were performed. Experimental results of relevance to the present topic of metastable phases have been obtained. Studies of nucleation statistics in the microgravity environment were conducted on Zr and analysed within nucleation theory [2.99]. For the first time dendrite growth velocities on metallic systems have been measured in space [2.100]. An advanced facility is under development to be accommodated on board of the International Space Station (ISS). REFERENCES

[2.1] Cahn, R.W. (1993) in: Rapidly Solidified Alloys, ed. Liebermann H.H. (Marcel Dekker, New York), p. 1. [2.2] Vonnegut, B. (1948) Journal of Colloid Science 3, 563. [2.3] Turnbull, D. (1950) Journal of Applied Physics 21, 1022.

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[2.4] Turnbull, D., and Cech, R.E. (1950) Journal of Applied Physics 21, 804. [2.5] Turnbull, D. (1952) Journal of Chemical Physics 20, 411. [2.6] Chalmers, B. (1967) Principles of Solidification (Wiley, New York). [2.7] Rasmussen, D.H., and Loper, C.R. (1975) Acta Metallurgica 23, 1215. [2.8] Rasmussen, D.H., Perepezko, J.H., and Loper, C.R. (1976) in Proceedings of the 2nd International Conference on Rapidly Quenched Metals, eds. Grant, N.J., and Giessen, B.C. (MIT Press, Cambridge, MA), p. 5. [2.9] Perepezko, J.H., Rasmussen, D.H., Anderson, I.E., and Loper, C.R. (1979) in Proceedings of the International Conference on Solidification and Casting of Metals (Metals Society, London), p. 169. [2.10] Perepezko, J.H., Mueller, B.A., and Ohsaka, K. (1987) in: Undercooled Alloy Phases, eds. Collings, E.W. and Koch, C.C. (Metallurgical Society, Warrendale, PA), p. 289. [2.11] Perepezko, J.H. (1984) Materials Science & Engineering 65, 125. [2.12] Mueller, B.A., and Perepezko, J.H. (1987, 1988) Metallurgical Transactions A 18, 1143; Materials Science & Engineering 98, 153. [2.13] Rasmussen, D.H., Javed, K., Appleby, M., and Witowski, R. (1985) Materials Letters 3, 344. [2.14] Bayuzick, R.J., Hofmeister, W.H., and Robinson, M.B. (1987) in: Undercooled Alloy Phases, eds. Collings, E.W., and Koch, C.C. (Metallurgical Society, Warrendale, PA), p. 207. [2.15] Shong, D.S., Graves, J.A., Ujiie, Y., and Perepezko, J.H. (1987) in: Materials Research Society Symposia Proceedings, eds. Doremus, R.H., Nordine, P.C. Vol. 87, p. 17. [2.16] Drehman, A.J., and Turnbull, D. (1981) Scripta Metallurgica 15, 543. [2.17] Elder, S.P., and Abbaschian, G.J. (1990) in: Principles of Solidification and Materials Processing, eds. Trivedi, R., Sekhar, A., and Mazumdar, J. (Oxford and IBH Publishing Co. Pvt Ltd, Delhi), p. 299. [2.18] Cochrane, R.F., Herlach, D.M., and Willnecker, R. (1993) in: Metastable Microstructures, eds. Banerjee, D., and Jacobson, L.A. (Oxford and IBH Publishing Co. Pvt Ltd, New Delhi), p. 67. [2.19] Gillessen F., and Herlach, D.M. (1988) Materials Science & Engineering 97, 147. [2.20] Cochrane, R.F., Evans, P.V., and Greer, A.L. (1988) Materials Science & Engineering 98, 99. [2.21] Gillessen, F. (1989) Ph.D. Thesis, Ruhr-Universität Bochum, Germany. [2.22] Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena (Wiley, New York). [2.23] Hug, W., Kallien, L., and Sahm, P.R. (1986) Gießereiforschung 38, 73. [2.24] Fransaer, J., Wagner, A.V., and Spaepen, F. (2000) Journal Applied Phyics 87, 1801. [2.25] Feuerbacher, B. (1989) Materials Science & Engineering Reports 4, 1. [2.26] Rathz, T.J., Robinson, M.B., Hofmeister, W.H., and Bayuzick, R.J. (1990) Review of Scientific Instruments 61, 3846.

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[2.27] Vinet, B., Cortella, L., Favier, J.J., and Desre, P. (1991) Applied Physics Letters 58, 97. [2.28] Hofmeisater, W.N., Robinson, M.B., and Bayuzick, R.J. (1986) Applied Physics Letters 49, 1342. [2.29] Lacy, L.L., Robinson, M.B., and Rathz, T.J. (1981) Journal of Crystal Growth 51, 47. [2.30] Tournier, S., Vinet, B., Pasturel, A., Ansara, I., and Desre, P.J. (1998) Physical Review B 57, 3340. [2.31] Bardenheuer, P., and Bleckmann, R. (1939) Mitteilungen Kaiser Wilhelm Institut für Eisenforschung 21, 201. [2.32] Kui, H.W., Greer, A.L., and Turnbull, D. (1984) Applied Physics Letters 45, 615. [2.33] Kui, H.W., and Turnbull, D. (1985) Applied Physics Letters 47, 796. [2.34] Gillessen, F., Herlach, D.M., and Feuerbacher, B. (1988) Zeitschrift für Physikalische Chemie 156, 129. [2.35] Johnson, W.L., Inoue, A., and Liu, C.T. (eds.) (1999) Materials Research Society Symposium Proceedings, vol. 554. [2.36] Wu, Y., Piccone, T.J., Shiohara, Y., and Flemings, M.C. (1987) Metallurgical Transactions A 18, 915. [2.37] Herlach, D.M., Cochrane, R.F., Egry, I., Fecht, H.J., and Greer, A.L. (1993) International Materials Review 38, 273. [2.38] King, L.V. (1934) Proceedings of the Royal Society London Ser. A. 147, 212. [2.39] Trinh, E.H., Robey, J., Arce, A., and Gaspar, M. (1987) in: Materials Processing in the Reduced Gravity Environment of Space, Vol. 87, eds. Doremus, R.H., and Nordine, P.C. (Materials Research Society Symposium Proceedings), p. 57. [2.40] Landau, L.D., and Lifschitz, E.M. (1978) Course of Theoretical Physics, vol. 6, Fluid Mechanics (Pergamon Press, Oxford). [2.41] Magill, J., Capone, F., Beukers, R., Werner, P., and Ohse, R.W. (1987) High Temperature – High Pressure 19, 461. [2.42] Gorkov, L.P. (1962) Soviet Physica Doklady 6, 773. [2.43] Gao, J.R., Cao, C.D., and Wei, B. (1999) Advances in Space Research 24, 1293. [2.44] Trinh, E.H. (1985) Review of Scientific Instruments 56, 2059. [2.45] Ohsaka, K., Trinh, E.H., and Glicksman, M.E. (1990) Journal of Crystal Growth 106, 191. [2.46] Ansell, S., Krishnan, S., Weber, J.K.R., Felten, J.J., Nordine, P.C., Beno, M.A., Price, D.L., and Saboungi, M.L. (1997) Phyical Review Letters 78, 464. [2.47] Xie, W.J., and Wei, B. (2001) Chinese Physical Letters 18, 68. [2.48] Xie, W.J., and Wei, B. (2001) Applied Physics Letters 79, 881. [2.49] Xie, W.J., Cao, C.D., Lü, Y.J., and Wei, B. (2002) Physical Review Letters 89, 104304–104311. [2.50] Earnshaw, S. (1842) Transactions Cambridge Philosophical Society 7, 97. [2.51] Berry, M.V., and Geim, A.K. (1997) European Journal of Physics 18, 307. [2.52] Arkadiev, V. (1947) Nature 160, 330. [2.53] Sasslow, W.M. (1991) American Journal of Physics 59, 16.

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[2.54] Braunbeck, W. (1937) Zeitschrift für Physik 112, 735. [2.55] Beaugnon, E., and Tournier, R. (1991) Nature 349, 470. [2.56] Weilert, M.A., Whitaker, D.L., Maris, H.J., and Seidel, G.M. (1996) Physical Review Letters 77, 4840. [2.57] Ikezoe, Y. et al. (1998) Nature 393, 749. [2.58] Geim, A.K., Simon, M.D., Boamfa, M.I., and Heflinger, L.O. (1999) Nature 400, 323. [2.59] Simon, M.D. et al. (1997) American Journal of Physics 65, 286. [2.60] Rony, P. R. (1964) in: Transactions of International Vacuum Metallurgy Conference, ed. Cocca, M.A. (Science Press, Boston), p. 54. [2.61] Lohöfer, G. (1989) SIAM Journal of Applied Mathematics, 49, 567. [2.62] Lohöfer, G. (1993) SIAM Journal of Applied Mathematics, 51, 495. [2.63] Fromm, E., and Jehn, H. (1965) Zeitschrift für Metallkunde, 56, 599. [2.64] Fromm, E., and Jehn, H. (1967) Zeitschrift für Metallkunde, 58, 566. [2.65] Sauerland, S. (1993), Ph.D Thesis, RWTH Aachen, Germany. [2.66] Herlach, D.M. (1991) Annual Review of Materials Science 21, 21. [2.67] Herlach, D.M., Willnecker, R., and Gillessen, F. (1984) Proceedings of the 5th European Symposium on Materials Scinces under Microgravity, ESA SP-222 (Schloß Elmau), p. 399. [2.68] Schleip, E., Willnecker, R., Herlach, D.M., and Görler, G.P. (1988) Materials Science & Engineering 98, 39. [2.69] Barth, M., Joo, F., Wei, B., and Herlach, D.M. (1993) Journal of NonCrystalline Solids 156–158, 398. [2.70] Shiraishi S.Y., and Ward, R.G. (1964) Canadian Metallurgy Quarterly 3, 117. [2.71] Amaya, G.E., Patchett, J.A., and Abbaschian, G.J. (1983) in: Grain Refinement in Castings and Welds, eds. Abbaschian, G.J., and David, A. (Metallurgical Society AIME, Warrendale, PA), p. 51. [2.72] Schade, J., McLean, A., and Miller, W.A. (1987) in: Undercooled Alloy Phases,ed. Collings, E.W., and Koch, C.C. (Metallurgical Society, Warrendale, PA), p. 233. [2.73] Arpaci, E. (1984) Ph.D. Thesis, Freie Universität Berlin, Germany. [2.74] Keene, B.J., Mills, K.C., Kasama, A., McLean, A., and Miller, W.A. (1986) Metallurgical Transactions 17B, 159. [2.75] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1986) Applied Physics Letters 49, 1339. [2.76] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1988) Materials Science & Engineering 98, 85. [2.77] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1986) Proceedings of the 6th European Symposium on Materials Sciences under Microgravity, ESA SP-256 (Bordeaux), p. 339. [2.78] Eckler, K., Kratz, M., and Egry, I. (1993) Review of Scientific Instruments 64, 2639. [2.79] Eckler, K., and Herlach, D.M. (1994) Materials Science & Engineering A 178, 159. [2.80] Matson, D.M. (1998), in: Solidification 1998, ed., Chu, M.G. (Proceedings of the 1998 TMS Annual Meeting, February 15–19, San Antonio, TX), p. 233.

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[2.81] Herlach, D.M., Funke, O., Gandham, P., and Galenko, P. (2004) in: Solidification Processes and Microstructures: A Symposium in Honor of Wilfried Kurz, eds. Rappaz, M., Beckermann, C., and Trivedi, R. (Proceedings of the TMS annual meeting, March, Charlotte, NC), p. 277. [2.82] Rulison, A.P.J., and Rhim, W.K. (1994) Review of Scientific Instruments, 65, 695. [2.83] Felici, N.-J. (1966) Revue General de l’Electricité, 75, 1145. [2.84] Meister, T. (2001) Ph.D. Thesis, Ruhr-Universität Bochum, Germany. [2.85] Lewin, G. (1965) Fundamentals of Vacuum Science and Technology (McGraw-Hill, New York), p. 53. [2.86] Dekker, A.J. (1962), Solid State Physics (Prentice-Hall, Englewood Cliffs, N.Y.), p. 221. [2.87] Piller, J., Knauf, R., Preu, P., Lohöfer, G., and Herlach, D.M. (1986) Proceedings of the 6th European Symposium on Materials Sciences under Microgravity, ESA SP-256 (Bordeaux), p. 437. [2.88] Herlach, D.M., Willnecker, R., and Lohöfer, G. (1989) German Patent DE 3639973 C2. [2.89] Herlach, D.M., Cochrane, R.F., Egry, I., Fecht, H.-J., and Greer, A.L. (1993) International Materials Reviews 38, 273. [2.90] Damascke, B., Oelgeschläger, D., Ehrich, E., Diertzsch, E., and Samwer, K. (1998) Review of Scientific Instruments 69, 2110. [2.91] Damaschke, B., Samwer, K., and Egry, I. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale), p. 43. [2.92] Lohöfer, G., and Egry, I. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale), p. 83. [2.93] Fecht, H.-J., and Johnson, W.L. (1991) Review of Scientific Instruments 62, 1299. [2.94] Wunderlich, R.K., Lee, D.S., Johnson, W.L., and Fecht, H.-J. (1997) Physical Review B 55, 26. [2.95] Egry, I., Lohöfer, G., Schneider, S., Seyhan, I., and Feuerbacher, B. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale), p. 15. [2.96] Sauerland, S., Lohöfer, G., and Egry, I. (1990) Thermochimica Acta 218, 445. [2.97] Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability (Dover, New York). [2.98] Team TEMPUS (1996) in: Materials and Fluids Under Low Gravity, eds. Ratke, L., Walter, H., and Feuerbacher, B. (Springer, Berlin), p. 233. [2.99] Hofmeister, W., Morton, C.M., Robinson, M.B., and Bayuzick, R.J. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale), p. 75. [2.100] Barth, M., Holland-Moritz, D., Herlach, D.M., Matson D., and Flemings, M.C. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale), p. 83.

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Chapter 3

Physics of Undercooled Liquids 3.1. 3.2.

Thermodynamics Structural Ordering in Undercooled Melts 3.2.1 Models for the Short-Range Order in Undercooled Melts 3.2.2 Scattering Theory 3.2.3 Experiments on the Short-Range Order in Metallic Melts 3.2.4 The Short-Range Order of Liquid Si 3.3. Magnetic Ordering in Liquid State 3.4. Kinetic and Transport Properties References

59 67 68 76 79 95 97 103 105


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Chapter 3

Physics of Undercooled Liquids 3.1. THERMODYNAMICS

Equilibrium thermodynamics describes the macroscopic state of a system in terms of a few thermodynamic functions and variables. It can also define the direction of a change in state. If, for example, the pressure p and the temperature T are chosen as experiment parameters, the Gibbs free energy G is the thermodynamic potential that describes the state of the system. If we are dealing with a non-equilibrium state such as an undercooled melt, the question arises whether the basic concepts of thermodynamics can still be applied to describe the state. Within the statistical interpretation, the concept of macroscopic thermodynamics can only be applied to ergodic systems. Turnbull demonstrated that a liquid can be kept in an undercooled state over extended periods of time [3.1] such that it can be considered an ergodic system [3.2]. In pure metals, the Gibbs free energy G depends exclusively on the pressure p and the temperature T. At constant pressure it is expressed by the relation G (T ) = H (T ) − TS (T ),

(3.1)

where H is the enthalpy and S the entropy of the system. Figure 3.1 shows in a schematic way the temperature dependence of G at constant pressure for the liquid phase L and various solid phases and . According to Eq. (3.1), the slope of the respective G(T) curves is determined by the entropy of the various states, –S. Since the liquid state has higher entropy than the solid phases, the G(T) curves intersect at characteristic temperatures, which define the melting temperatures of the different solid states. At high temperatures, the Gibbs free energy of the liquid is smaller than that of all solid phases. Consequently, the liquid state is stable in this temperature regime. When the melt is cooled below the melting temperature, the solid phase becomes stable. However, there may be other solid phases including the phase and the glassy state, which are metastable. To generate a thermodynamic driving force for solidification of the metastable phase, an undercooling is necessary exceeding the temperature differences between the equilibrium melting point and the melting point of the metastable solid: T TE–TE. A special case occurs if the melt is undercooled to the glass transition temperature Tg at which the undercooled melt freezes into the metastable amorphous phase. The crystallization of a solid phase from the liquid state requires a thermodynamic 59

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Figure 3.1. Gibbs free energy as a function of temperature for the liquid phase (L) and two different solid phases, (stable) and (metastable). At the glass temperature Tg, the undercooled melt freezes into the amorphous state. The crossing points of the various G(T ) curves define the melting temperatures of the respective phases.

driving force, which comes from the difference between GS and GL, the free energies of the solid and the liquid, respectively. The Gibbs free energy difference can be expressed by GLS (T ) = H LS (T ) − T S LS (T )

(3.2)

TE

H LS (T ) = H f − ∫ C p (T )dT

(3.2a)

T

S LS (T ) = S f −

TE

∫ T

C p (T ) T

(3.2b)

dT

H f = TE S f ,

(3.2c)

where the differences of enthalpy and entropy, HLS and SLS , are determined by enthalpy of fusion Hf, entropy of fusion Sf and the difference in the specific heat Cp between liquid and solid state. TE is the equilibrium melting temperature. Equation (3.2) can be rewritten as GLS (T ) =

H f T TE

with T= TE–T the undercooling.

TE

TE

T

T

− ∫ C p (T )dT + T ∫

C p (T ) T

dT ,

(3.3)

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Physics of Undercooled Liquids

61

Different models have been applied to describe the Gibbs free energy difference GLS as a function of temperature. The simplest approach is the linear approximation proposed by Turnbull [3.3], which assumes a vanishing difference in the specific heat at the equilibrium melting temperature, i.e. Cp = 0. G is then proportional to the undercooling T: GLS (T ) =

H f TE

T .

(3.4)

In the case that Cp 0 but independent of temperature, Eq. (3.3) gives GLS (T ) =

H f ΔT TE

− C p ⎡⎣T − T ln (TE /T )⎤⎦ .

(3.5)

Jones and Chadwick [3.4] suggest to use the specific heat difference at the melting point so that Cp = Cp(TE). For small undercoolings, the power series expansion (up to first order) of the logarithmic term in Eq. (3.5) yields GLS (T ) =

H f T TE

⎡ C p (TE ) ⎤ − T 2 ⎢ ⎥ ⎣ TE + T ⎦

(3.6)

Hoffmann [3.5] derives an expression for GLS(T) supposing a linear temperature dependence for the enthalpy difference, HLS = Hf (T – T)/(TE – T). Taking into account the condition HLS = 0 for T = T , it follows for small undercoolings that Cp(T) = 2HfT/TE2 = 2SfT/TE and for the Gibbs free energy difference: GLS (T ) =

H f T T . TE TE

(3.7)

According to Hoffmann’s model, the difference in the specific heat becomes Cp = 2Sf at T = TE. This condition may hold for covalent systems but is a rather poor approximation for metallic systems. Thompson and Spaepen [3.6] assume a proportionality between Cp and the entropy of fusion. The proportionality constant is defined by the condition that at a characteristic temperature Tog the difference of the entropy SLS vanishes: S LS (T og ) = S f −

TL

∫

T og

C p dT = 0. T

(3.8)

This temperature corresponds to the so-called ideal glass transition temperature Tog. Kauzmann [3.7] postulated that this temperature gives an ultimate limit for the

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undercoolability of a melt. This idea is based upon the consideration that a crystal should always show lower entropy than a liquid. If Cp is assumed to be constant, Eq. (3.8) can be used in order to determine Cp: C p =

S f ⎛T ⎞ ln ⎜ E ⎟ ⎝T ⎠ og

=

H f ⎛T ⎞ TE ln ⎜ E ⎟ ⎝T ⎠

.

(3.9)

og

Together with Eq. (3.5) this gives −1

⎡ ⎛ T ⎞⎤ GLS (T ) = S f (TE − T ) − ⎢ln ⎜ E ⎟ ⎥ S f ⎢⎣ ⎝ Tog ⎠ ⎥⎦

⎡ ⎛ ⎛ TE ⎞ ⎞ ⎤ ⎢TE − T ⎜1 + ln ⎜ ⎟ ⎟ ⎥ . ⎝ T ⎠⎠ ⎦ ⎝ ⎣

(3.10)

Thompson and Spaepen further assumed that the ideal glass temperature can be approximated by TogTE/3. This yields GLS (T ) =

H f T ⎡ 2T ⎤ ⎥ ⎢ TE ⎣ TE + T ⎦

(3.11)

An empirical expression similar to Eq. (3.11) has been obtained by Singh and Holtz [3.8]: GLS (T ) =

H f T ⎡ 7T ⎤ ⎥. ⎢ TE ⎣ TE + 6T ⎦

(3.12)

The different expressions for the Gibbs free energy difference GLS(T) discussed above are based on assumptions, which can be – if at all – only partly interpreted within physical models. Recently, a model for the temperature dependence of the Gibbs free energy difference has been developed based upon the free volume model of liquids [3.9, 3.10]. The state of the liquid is described by a quasilattice whose lattice sites are occupied by either atoms or defects [3.11, 3.12]. It is expected that the liquid differs from the crystal by an enhanced density of defects reflecting the lower mass density of a liquid in comparison with the crystalline solid. The formalism to describe the thermodynamic state of an alloy is used to get an expression for the Gibbs free energy difference whereby the defects are identified as the “solute” atoms. This yields GLS (T ) =

H f T TE

−

C p (TE )T 2 ⎡ T ⎤ ⎢1 − 6T ⎥ . 2T ⎣ ⎦

(3.13)

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The essential advantage of Eq. (3.13) as well as of the former expressions is that the Gibbs free energy difference can be calculated without the need to measure the temperature dependence of thermodynamic properties in the undercooled melt regime. Such measurements are difficult to perform, since a metallic melt (in contrast to, e.g., oxide glass systems) freezes very rapidly once cooled below the melting temperature. While the heat of fusion can be determined by differential thermal analysis (DTA) or differential scanning calorimetry (DSC), only a few measurements of the specific heat in the undercooled melt regime have been reported. However, if such data are available, the difference in the Gibbs free energy GLS(T) can be determined by using Eq. (3.2) on an experimental basis. Measurements of the specific heat as a function of undercooling have been reported for the pure elements Hg, In, Bi and Sn and Ga [3.13, 3.14]. In all these measurements, a monotonic increase in the specific heat with undercooling is observed. Various empirical expressions have been proposed to fit these data [3.15, 3.16], which differ only in the number of terms regarded in the power series expansion of Cp(T). As the analysis of these measurements indicates, the linear approximation according to Eq. (3.4) describes the Gibbs free energy difference GLS if the undercooling is not too large (see Figure 3.2). At large undercoolings, the expression based upon the free volume theory (Eq. (3.13)) leads to a better description of GLS(T) in agreement with a previous analysis [3.16]. In contrast, the predictions of the model by Thompson and Spaepen apparently to deviate from the GLS(T) behaviour calculated using measured data of the specific heat. The increase of the specific heat with undercooling is not yet well understood. Chemical short-range ordering effects can be ruled out to be responsible for such an increase in pure metals. First attempts to describe such behaviour are based on the assumption of thermal activation of defects in undercooled melts [3.17]. In alloys too, the Gibbs free energy difference GLS can be determined if data of the enthalpy of fusion Hf and the specific heat Cp(T) are available. Up to now, measurements of the specific heat in the undercooled melt regime have been reported in particular for glass-forming systems, since these alloys tend to be undercooled easily. Data are available for Au77Ge13.6Si9.4 [3.18], Au81.4Si18.6 [3.19], Mg85.5Cu14.5 [3.20], Pd80Si20 [3.14] and Au53.2Pb27.5Sb19.3 [3.21]. In contrast to pure metals, the specific heat of these alloys changes at the equilibrium melting temperature. During cooling from the liquid to the undercooled melt regime, the increase of the specific heat with undercooling is more pronounced than in the case of pure metals. This indicates that eventually chemical short-range ordering effects could have an essential influence on the temperature dependence of the specific heat.

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Figure 3.2. (a) The specific heat difference, Cp, and (b) the Gibbs free energy difference GLS as a function of temperature T for the pure metal In. The dots give the experimental results of Cp (a) [3.13] and the values of GLS (b) calculated on the basis of the measured specific heat using Eq. (3.3). The lines represent the predictions of the various models for the temperature dependence of GLS.

Investigations of the Gibbs free energy difference as a function of undercooling are of special interest for metallic glasses that show a pronounced difference between the glass transition temperature Tg and the crystallization temperature Tx. At Tx the amorphous phase transforms into the crystalline state. In the temperature range TgTTx, the system is in a highly undercooled viscous state. Therefore, such systems are quite suitable for measuring simultaneously the specific heat both above the glass temperature and below the melting temperature. In this way, a wide temperature range in the undercooled melt regime becomes accessible for these investigations. Such measurements have been reported for the easy glassforming systems Pd40Ni40P20 [3.22] and Pd17.5Cu6Si16.5 [3.23]. Figure 3.3 shows in the upper part the specific heat difference in the temperature range TgTTE. The dots represent the experimentally determined values

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Figure 3.3. (a) The temperature dependence of the specific heat difference Cp between crystalline and undercooled liquid state for Pd77.5Cu6Si16.5 glass-forming alloy [3.23]. The dots give the experimentally determined data of an amorphous sample above the glass temperature and the values of the liquid below the eutectic melting temperature. Tog denotes the ideal glass temperature at which SLS = 0. (b) The Gibbs free energy difference GLS (T ) as calculated on the basis of the measured specific heat data. The dashed line represents the prediction of the Dubey and Ramanchandrarao model.

measured above the glass temperature and below the melting temperature. The lower part reveals the temperature dependence of the Gibbs free energy difference GLS(T ) as calculated on the basis of the measured specific heat data. The thin line represents the prediction of the Dubey and Ramachandrarao model. Apparently, there is good agreement between this model and the experiments. It is concluded that the model based upon the free volume theory yields a good description of the driving force for crystallization as a function of undercooling. The application of the free volume model on alloy systems has to incorporate the influence of concentration on the enthalpy and entropy of mixing. These contributions can be easily determined if measured data of thermodynamic properties are available. If not, an excess of free energy has to be taken into account due to chemical short-range ordering effects in the undercooled alloy melt, which might be estimated within the model of associates [3.24]. The description of the specific

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heat difference also allows determining the ideal glass temperature Tog according to Eq. (3.2b) at which the entropy of the undercooled liquid becomes equal to the entropy of the solid. For PdCuSi, one obtains Tog = 537 K. This is comparable with the ideal glass temperature Tog = 529 K determined by investigations of the relaxation behaviour of the binary glassy alloy Pd82Si18 [3.25]. It is noteworthy that the considerations of Kauzmann with respect to the ultimate limit of undercoolability of the liquid have been recently extended to the maximum superheating of the solid phase [3.26]. In contrast to the phenomenon of undercooling, superheating effects of solids are very seldom observed. This is because the free surface of a sample forms an ideal nucleation site for the melting process. Consequently, melting normally sets in at the equilibrium melting temperature. However, under certain circumstances the phenomenon of superheating of the solid state can be detected. Daeges et al. [3.27] have demonstrated on Ag particles that if the surface of the droplets is coated with a material of higher melting temperature than that of the particle material, superheatings of several tens of degrees are observed. On the basis of measured data of the specific heat of crystalline and liquid Al including the temperature range of the undercooled melt, Fecht and Johnson [3.26] calculated the temperature dependence of the entropy for both states. Figure 3.4a gives the results of these calculations. Analogous to the Kauzmann temperature T = Tog = 225 K TE = 933 K at which S disappears a second instability temperature Tis = 1280 K above TE (the so-called inverse instability temperature) is found at which S also vanishes. The regard of the concentration dependence of Tis, Tog and To (To is defined by the condition G(c,T) = 0) leads to the surprising result that a triple point at T* and c* does exist at which liquid, glass and crystal coexist. At concentrations c c* even a “re-entrant” behaviour is presumed, at which a liquid if cooled transforms first to an ordered crystalline state before undergoing a second transition from the crystalline to a glassy state (cf. Figure 3.4b). Such behaviour is throughout comparable with the “re-entrant” behaviour of dilute magnetic systems. In these systems, if cooled from high temperatures, first a transition from the magnetically disordered paramagnetic to an ordered ferromagnetic (or ferrimagnetic) state occurs, which is followed by a second transition from the magnetically ordered to a disordered spinglass state at low temperatures [3.28]. The physical origin of both the structural and the magnetic the re-entrant behaviour may be explained by frustration effects. In the first case of the structural re-entrant behaviour, different structural crystalline states competing with each other may be responsible for the frustration effect [3.29], while in the second case of the magnetic re-entrant behaviour competing ferromagnetic and antiferromagnetic exchange interaction may cause the frustration effect [3.30]. The considerations of the structural re-entrant behaviour are of fundamental interest in treatment of the

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Entropy (J/mol K)

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(a)

(b)

Al T

40

Liquid

Δ Sf

L

s Ti

20

To S

0 − 20

Crystal Tog

TE

s

Tog

T*

Ti

Glass 0 200

600 1000 1400 Temperature (K)

c

c*

Figure 3.4. (a) Entropy of liquid and crystalline aluminium as a function of temperature in stable and metastable regions. Sf is the entropy of fusion. (b) Metastable phase diagram indicating a triple point at which crystal, glass and liquid coexist.

glass formation of metallic systems [3.31]. A more detailed discussion of this aspect is, however, beyond the scope of the present book.

3.2. STRUCTURAL ORDERING IN UNDERCOOLED MELTS

Till the early 1950s, it was believed that the short-range order in undercooled metallic melts resembles that of the corresponding crystalline phases, which nucleate from the melt [3.32–3.37]. This hypothesis was motivated by the experimental observation that some physical properties such as the mass density and specific heat do not essentially change when the system is transformed from liquid to solid or vice versa. Both the mass density and specific heat change only by a few percent during the liquid–solid transformation. Thus, Scheil [3.38] speculated that the atomic short-range order of dense packed structures of hard spheres should be the same in the liquid and solid phases. But in contrast to the mass density and specific heat there are other properties that undergo a drastic change during melting such as the viscosity or the atomic diffusion. To understand the dramatic change of these properties during the phase transformation, Sioul [3.39] postulated that the structure of the liquid should be deduced from that of the solid within the theory of plasticity. The solid at temperatures in close vicinity to the melting temperature is described as a crystal with a high density of lattice distortions. During melting, a

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fluctuating network is formed consisting of distortions that destroy the translational symmetry of the crystal. Such similarity of the short-range order in the solid and the undercooled melt implies a low energy of the interface between the nucleus of the solid phase and the liquid and therefore a low activation threshold for nucleation. This picture had to be completely revised after it was experimentally observed by Turnbull [3.3] that various metallic melts can be undercooled to a great extent, which is in contradiction to the assumption of a similar short-range order in the liquid and the crystalline phase. Moreover, for metallic systems the experimentally determined entropy of fusion is significantly larger than that predicted by models assuming a similar short-range order in liquid and solid state [3.40]. Obviously, metallic melts possess their own characteristic short-range order that is usually independent of that of the corresponding solid phases. 3.2.1 Models for the short-range order in undercooled melts The simplest models for the short-range order of undercooled melts analyse the stability of small clusters consisting of a few atoms. It is assumed that such types of clusters are predominantly formed in melt that are energetically favourable. For metallic systems, the interaction of the atoms is usually described by potentials with spherical symmetry consisting of an attractive and a repulsive term, like for instance the Mie potentials: ⎡⎛ ⎞ 1 ⎛ ⎞ 2 ⎤ V ( r ) = K ⎢⎜ 1 ⎟ − ⎜ 2 ⎟ ⎥ . ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠

(3.14)

In this formula, r denotes the distance between the atoms. 1, 2, K, v1 and v2 are constants. The special case of v1 = 6 and v2 = 12 is known as Lennard-Jones potential. These interaction potentials exhibit a minimum at the atomic distance rmin. The development of short-range order in undercooled metallic melts is schematically outlined in Figure 3.5. At its beginning, dimers (b) of atoms with a distance rmin will be formed in the melt. The number of bonds per atom is increased and, consequently, the energy reduced if one atom is attached to the dimers such that triangular clusters (c) are build up. A significant reduction of the energy of the system can be obtained by attachment of one more atom on top of the triangle under formation of a regular tetrahedron (d). The tetrahedron is the simplest cluster with a comparatively high density of packing and, consequently, low energy. The process of aggregation will continue such that larger polytetrahedral clusters consisting of several tetrahedra are formed (e–f), leading to a further

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Figure 3.5. Development of short-range order in undercooled metallic melts.

Figure 3.6. An icosahedron consisting of 13 atoms arranged in fivefold symmetry.

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reduction of the energy of the system. Out of 20 of such tetrahedra, an icosahedron (g) can be formed under a slight deformation of the originally regular tetrahedra [3.41]. The icosahedron (Figure 3.6) is one of the platonic bodies [3.42]. It consists of 13 atoms and has six fivefold symmetry axes. Icosahedral symmetry is incompatible with the translational invariance of crystals, e.g. it is impossible to fill space by packing of exclusively icosahedral clusters [3.43, 3.44]. In 1952, Frank [3.40] postulated that the short-range order in undercooled metallic melts is based upon such icosahedral aggregates, because an icosahedron possesses the lowest energy of all cluster consisting of 13 atoms, assuming an atomic interaction according to a Lennard-Jones potential. Its energy is about 8.4% lower than that of other clusters with also 13 atoms and a face-centred cubic (fcc)- or hexagonal close-packed (hcp) structure, as shown by Benson and Shuttleworth [3.45]. By further attachment of atoms, larger polytetrahedral aggregates such as the dodecahedron (h) are formed. Hoare and Pal suggested different sequences for the growth of these polytetrahedral clusters. If interactions according to a LennardJones potential are assumed, polytetrahedral short-range order is energetically favourable even for larger clusters (up to the largest investigated cluster size of 70 atoms) as compared with those short-range structures that correspond to the most common crystal structures [3.46]. Hence, for systems with a nearly spherical symmetry of the atoms (e.g. noble gases and metals), an icosahedral shortrange order should prevail in undercooled melts. In several experiments on free clusters of noble gases and metals, cluster spectra that exhibit a preferred formation of aggregates with “magic” atom numbers (13, 55, 147, 309, 561, ...) [3.47–3.50] were observed. These numbers correspond to the number of atoms in a simple icosahedron (13) and the Mackay icosahedra [3.51] of higher order with a completely filled shell (55, 147, ...). However, cluster spectra of Sb aggregates are indicative of a preference for clusters with a cubic structure [3.52]. In the case of the semi-metal Sb, however, a pronounced degree of covalent bonds must be supposed. Therefore, the assumption of non-directional bonds, which is the basis for the prediction of an icosahedral short-range order, appears unjustified for Sb. The energetic preference for the formation of icosahedral clusters with a completely filled shell is confirmed by calculations of the energy of atomic clusters as a function of the number of cluster atoms, utilizing Morse potentials [3.53]. They deliver local minima of the cluster energy at numbers of atoms that correspond to the magic numbers. Farges et al. [3.54] investigated free argon clusters consisting of 20–50 atoms by electron diffraction. Molecular dynamics simulations of clusters using

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Lennard-Jones potentials accompanied these experimental studies. The calculations predict a polyicosahedral structure of the aggregates, which means that the clusters are built from several partly overlapping icosahedra. The experimentally determined diffraction patterns are in good agreement with calculated diffraction patterns of the theoretically predicted polyicosahedral clusters. Although free atomic clusters are not directly comparable with clusters formed in undercooled melts, these observations outline the energetical favouritism of an icosahedral short-range order in small aggregates. While the arguments presented above for the preference of an icosahedral shortrange order in liquids are based upon an analysis of the energy of small clusters, other studies investigate larger systems. As suggested in 1932 by Menke [3.55], simple geometrical models aim to describe a melt by a dense random packing of hard spheres. In the works by Bernal [3.56, 3.57] and Scott [3.58, 3.59] mechanical models for liquids are constructed by filling identical hard spheres (e.g. bearing balls) in a container. The walls of the containers are roughened to suppress the formation of densely packed periodic structures (crystallization). By a subsequent mechanical compaction of the filling, a dense random packing of hard spheres is produced. The structure created by this procedure is fixed by sealing with wax or paint. To analyse the structure of the dense random packing of hard spheres the coordinates of every single ball are measured by disassembling the model ball by ball. Bernal analysed such a dense random packing of hard spheres by determining the types of polyhedra outlined by bonds between neighbouring balls [3.56, 3.57]. 86% of these polyhedra are tetrahedra, only 6% are octahedra and the rest are larger polyhedra (mainly deltahedra). Hence, the dense random packing of hard spheres is characterized by a polytetrahedral short-range order. A large number of the tetrahedra found in Bernal’s model form polytetrahedral aggregates with the structure of a pentagonal bipyramid (Figure 3.7). This cluster consists of seven atoms and possesses one fivefold symmetry axis [3.41]. Slightly distorted pentagonal bipyramids are part of icosahedral clusters [3.41]. Similar results were obtained in later works by Finney [3.60] and Ichikawa [3.61]. If a dense random packing of hard spheres is relaxed in a soft potential, the fraction of polytetrahedra with fivefold symmetry increases significantly [3.62, 3.63]. This highlights that the formation of an icosahedral short-range order, which is a special type of polytetrahedral short-range order, is favoured by soft interaction potentials. The prediction of an icosahedral short-range order in undercooled melts is confirmed by several molecular dynamics computer calculations on Lennard-Jones liquids [3.64–3.69]. For the first time, Steinhardt et al. [3.64] reported the establishment of an icosahedral short-range order in a monatomic Lennard-Jones liquid

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Figure 3.7. Pentagonal bipyramid consisting of seven atoms.

under periodic boundary conditions. While the simulation exhibited no icosahedral short-range order in the melt at temperatures above the melting point, first indications of an icosahedral ordering were observed at a relative undercooling of about 10%. The degree of icosahedral short-range order defined by an order parameter that depends upon the bond orientations in the Lennard-Jones liquid increases with increasing undercooling of the melt and is the more distinctive the slower the melt is cooled. In similar simulations, Nosé and Yonezawa [3.65, 3.67] observed a pronounced icosahedral short-range order in the undercooled LennardJones liquid as well. With the onset of crystallization of the melt into a crystalline fcc structure, the icosahedral order parameter suddenly drops. In a recent work, the temperature dependence of the short-range order of liquid Al was studied by molecular dynamics simulations [3.70]. For these calculations, a generalized, energy-independent, non-local pseudo-pair potential [3.71, 3.72] was utilized. Once again an icosahedral short-range order was found to prevail in the liquid, which becomes more pronounced with decreasing temperature. After the discovery of quasicrystalline solids by Shechtman et al. [3.73, 3.74], the topic of icosahedral short-range order attracted special attention because it characterizes these peculiar solids as well. Quasicrystals exhibit very sharp spots in diffraction experiments with a half width that is often comparable with that of good classical crystals. This is indicative of a highly ordered structure. Quasicrystalline phases are characterized by a so-called quasi-lattice showing no translational invariance [3.75]. The quasi-lattice possesses an icosahedral shortrange order and a long-range orientational order. While classical crystals are built up by a space-filling packing of identical unit cells, the non-periodic quasi-lattices can be constructed by a space-filling packing of at least two types of suitably chosen building units according to special matching rules [3.76–3.79]. By choice of rigid construction principles, strongly deterministic quasi-lattices with vanishing

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entropy are produced [3.76, 3.77], while less rigid matching rules lead to less strongly ordered structures with a non-vanishing entropy (random tiling models [3.78, 3.80]). Further information on quasicrystals can be found, for instance, in Refs. [3.44, 3.79, 3.81, 3.82]. The above considerations indicate that an icosahedral order is energetically advantageous. Therefore, the question arises why there exist only solids with an icosahedral short-range order, but no macroscopic icosahedral aggregates. An analysis of the icosahedral symmetry gives the answer to this question. The interatomic distance between two neighbouring atoms on the surface of an icosahedron consisting of equally large atoms is about 5% bigger than the distance between the surface atoms and the central atom [3.83]. Hence, the icosahedron does not consist of regular tetrahedra but of slightly distorted ones. This incompatibility of the energetically favoured perfect tetrahedral short-range order with the demand for global filling of space is called “space frustration”. When higher shells of the icosahedron are filled up, the gap of packing is increased more and more. While icosahedral structures exhibit the lowest energy at small cluster sizes, due to increasing space frustration it becomes disadvantageous compared with clusters with an fcc or hcp structure and the same number of atoms when the clusters reach a critical size. The critical cluster size can be estimated under the assumption of atomic interaction potentials according to Eq. (3.14). For such potentials with v1 = 14 and v2 = 7, v1 = 12 and v2 = 6, or v1 = 10 and v2 = 6, calculations delivered that up to a cluster size of approximately 1000 atoms icosahedral aggregates are energetically favourable compared to clusters of an fcc-type short-range order and a spherical surface [3.53]. This result is in good agreement with high-resolution electron-microscopical investigations performed on Au clusters [3.84]. These experiments showed phase fluctuations between icosahedral and fcc structure at a cluster size of 20 Å, which corresponds to a cluster size of about 600 atoms. Obviously, at this cluster size the energy of a cluster with icosahedral symmetry is nearly the same as that of an aggregate with an fcc structure. The considerations outlined above are with regard to monatomic systems. For alloys the situation is more complex. While for monatomic systems icosahedral clusters are characterized by high density of packing, for alloys with a large difference in the atomic radii of the components the formation of clusters with a different symmetry may be favourable for similar reasons that account, for instance, for the stability of solid Laves phases [3.85]. However, as long as the atomic radii of the alloy components are similar, it can be assumed that again an icosahedral short-range order will be advantageous. A small concentration of slightly smaller or larger atoms should even stabilize the icosahedral clusters,

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because placing a smaller atom at the centre of an icosahedron or a larger atom on its surface will decrease space frustration. In the molecular dynamics calculations by Jónsson and Andersen [3.66] a binary Lennard-Jones liquid that consisted of 1500 atoms and contained 20% of A atoms that are 25% larger than the B atoms was investigated. In a deeply undercooled liquid, 138 partly interpenetrating and interconnected icosahedra were found. Of all the atoms, 61% were located in an icosahedral environment. Maret et al. investigated melts of the quasicrystal-forming alloy Al80Mn20 [3.86] and Al80Ni20 [3.87] by neutron diffraction at temperatures above the melting temperature. From these experiments, the atomic pair potentials could be determined and were subsequently used for molecular dynamics calculations [3.88]. In the case of AlMn, these calculations exhibited some indications of icosahedral ordering in the melt even above the liquidus temperature TL, while for the Al80Ni20 melt no icosahedral ordering could be observed above TL. In the undercooled regime (T = 0.8TL), the calculations showed a pronounced degree of icosahedral short-range order for AlMn, and for AlNi they revealed the onset of icosahedral ordering. In contrast, molecular dynamics calculations from Dasgupta et al. [3.89] on a binary Lennard-Jones liquid consisting of an equal number of atoms, but with one species having a radius 1.6 times larger than the other, exhibited no indication for the development of an icosahedral short-range order in the undercooled melt. Ronchetti and Cozzini [3.90] pointed out that these contradicting findings might be a result of the difference in the atomic radii and the composition of the simulated binary systems. They studied the structure of 13-atom clusters in a binary Lennard-Jones liquid consisting of large atoms of type A and smaller atoms of type B as a function of the ratio rA/rB of the atomic radii and of the composition cB of the B atoms. For rA/rB = 1.25 it turned out that icosahedral clusters are dominant in the composition ranges 8/13 62 at.% cB 4/13 30 at.% while in the composition range around 50 at % no favoured structure was observed. With an increasing ratio rA/rB, the composition ranges, in which icosahedral clusters are dominant shrink. For rA/rB = 1.6, the icosahedral symmetry is preferred only for 69 at.% cB 7.7 at.%. In a composition interval 1/13 7.7 at.% cB 5/13 38 at.%, clusters consisting of 10 atoms are preferred; outside this range no dominant structure was found. The molecular dynamics investigations of Jónsson and Andersen [3.66] and Maret et al. [3.88] were carried out for systems that have a composition (cB 30 at.%) and a rA/rB-ratio (rA/rB-ratio 1.25) at which icosahedral clusters are favourable according to the calculations by Ronchetti and Cozzini [3.90]. In both cases, the development of such a short-range order was reported in the undercooled Lennard-Jones liquid. The results of Dasgupta et al. [3.89] also are in good

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agreement with those of Ronchetti and Cozzini [3.90], because for the parameters rA/rB = 1.6 and cB = 50 at.% icosahedral clusters are predicted to be energetically disadvantageous. Although for alloys the situation is not uniform, it must be stressed that a great variety of metallic alloys are characterized by a ratio of the atomic radii smaller than 1.25. This becomes obvious as a large number of metallic elements exhibit an atomic radius between 1.2 and 1.5 Å, for instance, the elements Al, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Ta, W, Re, Os, Ir, Pt, Au, U, Np and Pu. Among these are the most commonly used components of technologically relevant metallic alloys. For all alloys composed of these elements, the ratio of the atomic radii of the constituents is smaller than 1.25. Of course, beside the alloys consisting of the elements above listed, many other alloys that are characterized by a low difference of the atomic radii exist as well. This highlights the fact that icosahedral short-range order plays an important role in melts of a very large number of metallic alloys. The following considerations are only concerned with this type of alloys. However, it has to be emphasized that the idea of icosahedral short-range order is based upon the assumption of nearly spherical soft interaction potentials. This is justified for most purely metallic systems and noble gases. However, for systems characterized by directional covalent bonds, such as semiconductors, metal–metalloid alloys or ceramics, this assumption may be questionable. For example, neutron scattering experiments on pure sulphur lead to satisfactory description of the results obtained from melts just above the melting temperature if an arrangement of single atoms in rings of 8 coordinated atoms is assumed. This short-range order is broken at temperatures of 200 K above the melting temperature. It is then replaced by chain-like arrangements of atoms [3.91]. Interestingly enough, Bichara et al. [3.92] found two different coordinations of atoms in Te melts coexisting with each other, one with a small coordination number 2 and another of higher coordination. By neutron-scattering experiments on binary STe melts, a miscibility gap in the liquid state above the liquidus temperature was detected [3.93]. Both for STe and Te aggregates, coordination numbers of 2 and 3 were found. In the recent past, aerodynamic levitation was extensively used to perform diffraction experiments on undercooled melts of non-metallic substances such as B [3.94], Si [3.95] and Al2O3 [3.96, 3.97]. A review of these measurements and their discussion is given in Ref. [3.98]. None of these studies gave indications for icosahedral short-range order in the liquid, since in these non-metallic compounds and elements, the strongly directional bonds control the short-range order in the liquid. The present book concentrates on metals and their alloys. They are characterized by metallic bonding that is more or less isotropic in nature. As one example

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for a system with directional bonds, the short-range order of melts of pure Si is discussed in Section 3.2.4. 3.2.2 Scattering theory Before we discuss results of scattering experiments on liquids, the theoretical background of such diffraction experiments is briefly outlined for the example of neutron scattering. This short introduction is limited to nuclear-scattering processes. A detailed treatment of scattering theory is provided, for instance, in Refs. [3.99, 3.100]. We consider a sample consisting of N atoms. rj(t) denotes the position vector of the atom j as a function of the time t and bj the neutron-scattering length of this atom (j = 1, . . ., N). Neutron radiation that is irradiating the specimen is scattered at the sample, such that an incident neutron of wave vector k has the wave vector k after a scattering event. The difference of the wave vectors k and k defines the scattering vector Q : Q = k k . The partial differential scattering cross section of the specimen describes the number of neutrons with a final energy between E and E + dE that are scattered per unit time into the small solid angle d devided by the flux of the incident neutrons. For the above case, it is given by: d 2 1 k = ∑ b j b j ∫ exp(iQrj (0)) exp(iQrj (t )) exp(−it )dt d dE h k j j

(3.15)

In this formula “. . .” denotes the thermal average. For large systems, the products bjbj in Eq. (3.15) can be expressed by averaged values: d 2 1 k = ∑ b j b j ∫ exp(iQrj (0)) exp(iQrj (t )) exp(−it )dt d dE h k j j

(3.16)

where

()

2

b j b j = b , 2 b j b j = b ,

j j j = j.

(3.17)

The partial differential scattering cross section can then be split into two contributions, the coherent contribution ⎛ d 2 ⎞ ⎜⎝ d dE ⎟⎠

= c

1 k c ∑ exp(iQr j (0)) exp(iQr j (t )) exp(−it )dt h k 4 j j ∫

(3.18)

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and the incoherent contribution ⎛ d 2 ⎞ 1 k i ⎜⎝ d ΩdE ⎟⎠ = h k 4 ∑ ∫ exp(iQr j (0)) exp(iQr j (t )) exp( − it )dt . j i

(3.19)

Here, c and i denote the coherent and the incoherent scattering cross sections. They are defined as

()

(

2

c = 4 b ,

( ) ).

i = 4 b 2 − b

2

(3.20)

Within the formalism of van Hove [3.101] the scattering function S(Q,) is given by S ( Q, ) =

1 ∑ Nh j ′ j ∫

exp(iQr j (0)) exp(iQr j (t )) exp( − it )dt

(3.21)

and the time-dependent pair correlation function by g ( r, t ) =

1 ( 2 )3 N

∑∫ j j

exp(iQr j (0)) exp(iQr j (t )) exp( − iQr)d Q.

(3.22)

The scattering function is linked with the coherent partial differential scattering cross section: S (Q,) =

4 k ⎛ d 2 ⎞ . N c k ⎜⎝ d dE ⎟⎠ c

(3.23)

By Fourier backtransformation of the scattering function the time-dependent pair correlation function is obtained: g (r,t ) =

h S (Q,) exp ( − i(Qr −t ))d Q d . (2 ) 4 ∫

(3.24)

The structure factor S(Q) is defined by:

S (Q) = ∫ S (Q,)d (). −

(3.25)

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It is linked by Fourier transformation with the (static) pair-correlation function g(r): S (Q) = 1 + ∫ (g (r) − N ) exp (iQr)d r.

(3.26)

For isotropic materials, e.g. for liquids or amorphous solids, S and g are only dependent on the absolute values Q = |Q| and r = |r|: S (Q ) = 1 +

4

( g ( r ) − )sin(Qr )rdr. Q ∫0

(3.27)

Here, N denotes the particle density. The pair correlation function g(r) describes the probability of finding an atom at a distance r from another arbitrarily chosen atom. For alloys, apart from the topological short-range order also a possible chemical short-range order must be considered. Therefore, partial structure factors and the corresponding partial-pair correlation functions are determined. In the following treatment, suppose a binary alloy melt A–B with concentrations of the A and B atoms cA and cB, respectively. Hence, the average value of the scattering lengths and squares of the scattering lengths are given by b = c A bA + cB bB,

2 2 2 A B b = c bA + c bB .

(3.28)

Within the Bhatia–Thornton formalism [3.102], three partial structure factors SNN, SCC and SNC are defined such that A B 2(b A − b B )b ⎛ d ⎞ c c (b A − b B ) b S CC (Q ) + S NC ( Q ) . ⎜⎝ d ⎟⎠ = 2 S NN (Q ) + 2 2 c b b b 2

2

(3.29)

The partial structure factor SNN(Q) describes solely the topological short-range order of the system, SCC(Q) the chemical short-range order and SNC(Q) the correlation of particle density and chemical composition. An equivalent description provides the Faber–Ziman formalism [3.103], according to which the three partial structure factors SAA, SBB and SAB describe the contributions of the different atomic pairs A–A, B–B and A–B to the total structure factor: 2 B2 ⎡ A 2b 2 ⎤ 2c A c B b A b B ⎛ d ⎞ 2 c 2 c b 2 B A = + + Q Q Q ( ) ( ) ( ) + ( − ) ⎢ ⎥. S S S b BB AA AB b b ⎜⎝ d ⎟⎠ 2 2 2 ⎥⎦ c b b ⎣⎢ b

(3.30)

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3.2.3 Experiments on the short-range order in metallic melts Despite the fact that the idea of icosahedral short-range order in metallic melts is more than 50 years old, up to the recent past only a few experimental results on the short-range order in metallic melts were available, even in the stable regime at temperatures above the melting point. In the 1960s, Ruppersberg et al. [3.104–3.106] performed X-ray diffraction studies on stable melts of some pure metals (Cu, Fe, Al). The experimentally determined radial distribution functions could not be explained by assuming a short-range order in the liquid similar to that of the crystalline phases [3.104–3.106]. To explain the experimental results, Ruppersberg et al. suggest that such a crystal-like short-range order might be heavily disturbed, for instance, by dislocations or voids [3.104–3.106]. Supposing that the structure of a metallic glass resembles that of an undercooled metallic melt, investigations by Cargill [3.107] on amorphous NiP alloys support the idea of the formation of a polytetrahedral short-range order in undercooled metallic liquids. Cargill showed that the radial distribution function of amorphous Ni76P24 is in good agreement with that of the dense random packing of hard spheres as constructed in the mechanical model by Finney [3.60]. Waseda [3.108] performed diffraction studies on a large number of metallic melts at temperatures above the melting point. A rather qualitative description of the measured structure factors of most1 of the investigated melts of pure metals is possible in the framework of a structural model obtained by solving of the Percus–Yevick equation [3.109] under the assumption of hard sphere potentials [3.108]. The description of the experimental results within this model, however, implies a low density of packing of only 0.45 [3.108]. This value is considerably smaller than the densities of packing of 0.63–0.64 that were determined from models based on dense random packing of hard spheres [3.60, 3.110], and are inconsistent with the typical experimentally determined densities of metallic melts. At temperatures above the melting temperature the short-range order of quasicrystal-forming AlPdMn and AlMnCr melts was investigated by neutron scattering [3.111–3.113]. The measured structure factors S(Q) can be explained by assuming icosahedral clusters prevailing in the liquid, for which the transition metal atoms are located in the centre of the icosahedra. All other models presuming clusters of cubic-face-centred and hexagonal-dense-packed structure lead to systematic deviations from the measured scattering factors. Also a model based on 1

A different behaviour is observed for the structure factors of molten Zn, Cd, Hg, Ga, Sn, Sb and Bi. Some of these elements are semi-metals or are located below semiconductors or isolators in the periodic table of the elements. Therefore, it is reasonable to assume that the short-range order in melts of these elements is influenced by directional bonds.

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the Percus–Yevick equation under assumption of hard sphere potentials delivered no reasonable description of the measured structure factors. The first direct evidence of icosahedral short-range order in a pure metal comes from X-ray diffraction investigations by Reichert et al. [3.114] on pure liquid Pb adjacent to an Si (001) surface. Lead clusters were captured onto the Si interface and orientationally ordered. The breaking of the symmetry due to the attachment of the clusters at the Si surface partly cancels the thermal averaging over all cluster orientations, which is usually associated with diffraction studies. This enables direct detection of a fivefold orientational order, while conventional diffraction studies deliver only the isotropic pair correlation functions, which contain no direct information on the bond angles. Using high-intensity synchrotron radiation, diffraction spectra were recorded with the azimuth angle varying in the plane of the SiPb interface. Five diffraction maxima in the range 0–180 of the azimuth angle were clearly observed, giving evidence of fivefold symmetry of the Pb clusters captured onto the Si (001) surface. These studies prove the local icosahedral symmetry of Pb clusters but only in the direct neighbourhood of the Si (001) surface. They do not allow investigations of the short-range order in the bulk melt, or in the undercooled melt, which is essential for a study of nucleation of the solid phase in the parent liquid state. In a rather indirect way, studies of the dependence of the maximum undercoolability of metallic melts on the structure of the primarily nucleating solid phase provide information on the short-range order in undercooled melts [3.115, 3.116]. These studies reveal that the undercoolability of melts forming quasicrystalline and polytetrahedral phases, which are based on polytetrahedral structural units, is lower than that of melts forming phases with non-complex crystalline structures. This may be explained by a similarity of the short-range order in the melt and in polytetrahedral solid phases (compare the detailed discussion in Sections 4.2.4 and 5.6.3). Direct investigations on the short-range order of undercooled melts, e.g. by diffraction techniques, are difficult to perform because the preparation of the metastable state of a deeply undercooled melt and its conservation for times long enough to perform diffraction studies of good quality requires considerable experimental effort. To deeply undercool a melt below its melting point for extended periods of time, heterogeneous nucleation at foreign phases (e.g. crucible walls, impurities of the sample material or surface oxides, etc.) must be suppressed as far as possible. As shown in Chapter 2, this requires the (quasi)containerless processing of the melts under high-purity conditions. Owing to advances both in the field of the containerless processing techniques (compact, transportable levitation facilities) and structure determination techniques (synchrotron radiation sources of high intensity and modern detector systems), direct experiments on the short-range order of deeply undercooled melts by diffraction of synchrotron – or

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neutron–radiation or X-ray absorption spectroscopy (EXAFS) recently became feasible. X-ray absorption spectroscopy on pure Pd undercooled by a dispersion technique shows a different short-range order in solid and undercooled melt, which is consistent with the hypothesis of icosahedral short-range order in the liquid phase [3.117]. Recently, the containerless processing technique of electromagnetic levitation was combined with neutron scattering and energy-dispersive diffraction of synchrotron radiation to study the short-range order of undercooled and stable melts of a large variety of pure metals and alloys [3.118–3.122]. The neutron-scattering experiments were performed at the Institut Laue Langevin (ILL) and the X-ray diffraction experiments at the European Synchrotron Radiation Facility (ESRF), both in Grenoble. A different group combined the electrostatic levitation technique with angle dispersive diffraction of synchrotron radiation to investigate the shortrange order of stable and undercooled metallic melts [3.123, 3.124]. The top part of Figure 3.8 shows the structure factor S as a function of the scattering vector Q for Ni melts at T = 1435, 1465, 1605 and 1905 K, T = 1765 and 1905 K (melting point TLNi = 1728 K) measured by neutron diffraction and Figure 3.9 the corresponding pair correlation functions g(r) determined from S(Q) by Fourier transformation [3.118]. Similar investigations were performed for Fe melts at T = 1670, 1730, 1750, 1830 and 1870 K (melting temperature TLFe = 1811 K) (middle parts of Figures 3.8 and 3.9) and Zr melts at T = 1830, 1890, 2135 and 2290 K (melting point TLZr = 2125 K) (bottom parts of Figures. 3.8 and 3.9) [3.118]. Moreover, the short-range order of Co melts at T = 1800 and 1670 K (melting temperature TLCo = 1668 K) was investigated by energy dispersive diffraction of synchrotron radiation [3.119]. Recent results by Lee et al. [3.124] on the structure factor of liquid Ni determined by a combination of the containerless processing technique of electrostatic levitation with the diffraction of monochromatic synchrotron radiation are in excellent agreement with the neutron diffraction data shown in Figure 3.8. Also, the structure factor of liquid Ti was investigated by X-ray diffraction [3.124] and neutron scattering [3.122]. For all measured structure factors of Ni, Fe, Zr and Co melts a shoulder on the right-hand side of the second oscillation is observed. It is visible at temperatures above the melting point and becomes more pronounced as T is decreased. In earlier diffraction experiments [3.104–3.106, 3.108] on stable melts of pure metals, this important feature was not resolved. Interatomic distances and coordination numbers are simple quantitative parameters of short-range order directly obtained from the pair-distribution functions g(r). The nearest neighbour distances rn1 (first maximum of g(r)) and the second neighbour distances rn2 (second maximum of g(r)) determined from the diffraction experiments on liquid Zr, Ni, Fe and Co at different temperatures are

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Figure 3.8. Structure factor S as a function of the scattering vector Q determined by elastic neutron scattering for liquid Ni at T = 1435, 1465, 1605, 1765 and 1905 K, for liquid Fe at T = 1670, 1730, 1750, 1830 and 1870 K and for liquid Zr at T = 1830, 1890, 2135 and 2290 K [3.118].

compiled in Table 3.1. Moreover, the nearest-neighbour coordination number Z of the liquids is inferred from the area under the first maximum of the radial distribution function 4 Nr2g(r). Here, N denotes the atomic density. For the liquids of Fe, Ni, Zr and Co, the nearest-neighbour distances rn1, remain essentially unchanged while the second neighbour distances tend to slightly decrease on lowering the temperature. The nearest-neighbour coordination number Z is of the order of 12, independent of the metal investigated though they form different crystallographic phases in the solid: bcc for Fe, Zr and fcc for Ni and Co, with coordination numbers Z = 8 and 12, respectively. The measured coordination numbers of the melts exhibit a tendency to increase if T is decreased. This temperature evolution of Z is attributed to the thermal expansion of the liquids, which is indicated by the very small change of the ratio of Z over the bulk density N as a function of T (Table 3.1). The Z values (Z ~ 12) found for liquid Fe, Zr, Ni and Co are characteristic of several types of densely packed short-range order, e.g. icosahedral, fcc and hcp.

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Figure 3.9. Pair correlation functions g(r) determined from the structure factors shown in Figure 3.8 for melts of Ni, Fe and Zr at different temperatures [3.118].

To obtain further information on the short-range order in the liquids, S(Q) was simulated in the regime of large Q vectors using a method described in Refs. [3.111, 3.112]. It is assumed that the melt contains one dominant type of isolated tightly bound clusters. The simulations were performed for the following cluster structures: icosahedral, dodecahedral, fcc, hcp and bcc. A dodecahedron (Figure 3.10) is a larger cluster of icosahedral symmetry consisting of 33 atoms, which can be constructed from the icosahedron (13 atoms) by placing atoms on all the 20 triangular faces of the icosahedron. Thus, it represents larger polytetrahedral aggregates. For the sake of simplicity, it is assumed that the melt consists of only one type of these clusters. Furthermore, only intracluster contributions to the scattered intensity are considered while long-range intercluster contributions are neglected. This simplification is justified in the regime of large Q vectors where the contributions from the less tightly bound longer intercluster distances are damped out

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Table 3.1. Nearest (rn1) and second (rn2) neighbour distances and coordination numbers, Z, inferred from the diffraction experiments on Ni, Fr, Zr and Co melts as a function of the temperature T [3.118, 3.119]. The number densities N of the liquids [3.125–3.128] and the ratios Z/ N are also listed here. Element

T(K)

N (at./Å3)

g(r) rn1 (Å)

Ni Ni Ni Ni Ni Fe Fe Fe Fe Fe Zr Zr Zr Zr Co Co

1905 1765 1605 1465 1435 1870 1830 1750 1730 1670 2290 2135 1890 1830 1800 1670

0.0790 [3.125] 0.0806 [3.125] 0.0826 [3.125] 0.0842 [3.125] 0.0846 [3.125] 0.0751 [3.126] 0.0754 [3.126] 0.0761 [3.126] 0.0763 [3.126] 0.0769 [3.126] 0.0409 [3.127] 0.0412 [3.127] 0.0417 [3.127] 0.0418 [3.127] 0.0790 [3.128] 0.0803 [3.128]

2.48 0.02 2.48 0.02 2.49 0.02 2.49 0.02 2.49 0.02 2.55 0.02 2.55 0.02 2.55 0.02 2.55 0.02 2.55 0.02 3.12 0.02 3.12 0.02 3.12 0.02 3.13 0.02 2.53 0.03 2.53 0.03

Z

Z/ N (Å3/at.)

11.2 0.5 11.5 0.5 11.9 0.5 12.1 0.5 12.3 0.5 12.3 0.5 12.3 0.5 12.4 0.5 12.5 0.5 12.6 0.5 11.9 0.5 12.0 0.5 12.2 0.5 12.2 0.5 12.1 0.5 12.5 0.5

142 6 143 6 144 6 144 6 145 6 164 7 163 7 163 7 164 7 164 7 291 12 291 12 293 12 292 12 153 6 156 6

rn2 (Å) 4.53 0.05 4.50 0.05 4.46 0.05 4.43 0.05 4.43 0.05 4.55 0.05 4.54 0.05 4.53 0.05 4.52 0.05 4.52 0.05 5.74 0.05 5.73 0.05 5.72 0.05 5.71 0.05 4.5 0.1 4.6 0.1

Figure 3.10. Dodecahedron. This cluster consists of 33 atoms and is characterized by icosahedral symmetry. It is constructed from a simple icosahedron of 13 atoms (black) by placing further atoms (grey) densely on each of the 20 triangular faces of the icosahedron.

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by thermal motions. Under these assumptions, the structure factor at large Q is given by [3.111, 3.112] S (Q ) = 1 +

cclust .

N

∑

N b 2 i, j (i j )

bib j

(

sin Q rij Q rij

) exp ⎛⎜ − 2Q ⎜⎝

2

rij2 ⎞ ⎟. ⎟⎠ 3

(3.31)

Here, N denotes the number of atoms in the cluster, rij the mean distance between the atoms i and j and rij2 its mean thermal variation that determines the Debye–Waller factor exp( 2Q2 rij2/3). bi is the coherent neutron-scattering length of atom i and b 2 is the average of the squares of the coherent scattering length of all cluster atoms. This simulation method has the advantage that it depends on three free parameters only. These are the shortest mean distance, r0, of cluster atoms (the smallest of all the rij), its mean thermal variation, r02 and the concentration, cclust., of cluster atoms in the melt. For a given cluster structure, all intracluster distances rij can be directly calculated from r0. To estimate the mean thermal variations rij2 from the value, r02, at the shortest intracluster distance, we assume rij2 = r02 rij2/r02 [3.110, 3.111]. The three free parameters r0, r02 and cclust are adjusted such that a good agreement with the experimentally determined S(Q) is obtained especially at large Q values. The results of such simulations for liquid Zr undercooled by 290 K are presented in Figure 3.11 together with the experimentally determined S(Q) in the regime of large Q. Liquid Zr crystallizes into a solid bcc structure of coordination number Z = 8 upon cooling. Nevertheless, for a bcc-like short-range order no reasonable description of the experimental results could be obtained. Also, if a short-range order of fcc structure is assumed, neither the position nor shape of the second oscillation of S(Q) is well described, while for an icosahedral short-range order a good fit of the experimental data is achieved (Figure 3.11). The fit for a short-range order consisting of hcp clusters resembles that for fcc clusters and is therefore not shown in Figure 3.11. The simulation parameters are listed in Table 3.2. The better agreement of the simulation for icosahedral aggregates compared to that for fcc- or hcp-type clusters is understood when considering the individual contributions of each interatomic distance within these clusters. The distance of the central atom to the atoms on the shell (r0) and the nearest neighbour distance on the shell (r1) are the same in fcc/hcp clusters but differ of ~5% in icosahedral clusters. The contributions from both these distances are the major ones to the total simulation and are shown in the inset of Figure 3.11. These two intra-cluster distances explain the better fit of the experimental data obtained assuming the preference of icosahedral clusters, in particular, the asymmetric

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Figure 3.11. Measured structure factor, S(Q), of a Zr melt at T = 1830 K in the range 3.7 Å 1 Q 11.5Å 1 (dots), and simulated S(Q) under assumption of clusters with different symmetries prevailing in the liquid: icosahedra (———), bcc (-----) and fcc clusters (……). The parameters used for the simulations are compiled in Table 3.2). The inset shows the simulation under assumption of icosahedral clusters (———) together with the contributions to the structure factor resulting from the two nearest neighbour distances in an icosahedron: the distance, r0, between the central atom and an atom on the shell (……) and the distance, r1, between two neighbouring atoms on the shell (-----).

Table 3.2. Parameters used for the simulations of the structure factors of Zr melts. Short-range order

T (K)

r0 (Å)

r02 (Å2)

cclust.

bcc fcc Icosahedral Dodecahedral Dodecahedral Dodecahedral Dodecahedral

1830 1830 1830 1830 1890 2135 2290

2.86 3.09 2.996 3.05 3.05 3.05 3.05

0.012 0.042 0.041 0.040 0.041 0.043 0.045

0.85 0.85 0.90 0.91 0.89 0.83 0.81

shape of the large oscillation at about 4.5 Å with the characteristic shoulder on its right-hand side. The first interatomic distances r0 and r1 obtained from the simulation parameters are consistent with the intermediate value of rn1, the position of the maximum of the first oscillation of g(r), which includes both r0 and r1.

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We have been dealing up to now with isolated clusters with not more than 13 atoms, but we may wonder about the nature of short-range order at larger distances. For this purpose, simulations based on a larger cluster of icosahedral symmetry, the dodecahedron, were also performed (Figure 3.12). When comparing the results obtained for the two clusters with icosahedral symmetry it is obvious that the assumption of dodecahedral clusters leads to an even better description of the measured S(Q). This may indicate that a short-range order consisting of larger polytetrahedral aggregates (such as dodecahedra) prevails in the liquid. Figure 3.12 shows the results of simulations of S(Q) of Zr melts assuming dodecahedral short-range order at other temperatures also. In all cases, good agreement with the measured S(Q) was obtained, even for T TL. The simulation parameters are listed in Table 3.2. Obviously, r02 must be decreased and cclust must be increased to obtain a good fit of the experimental data if T is lowered. The change of r02 corresponds to the temperature dependence of the Debye–Waller factor, whereas that of cclust. indicates that the icosahedral short-range order becomes more and more pronounced if T is decreased. Similar conclusions as for the Zr melts were drawn from simulations of S(Q) of Fe, Ni and Co liquids [3.118, 3.119, 3.129]. For stable and undercooled liquid Ti, indications of a distorted icosahedral short-range order are reported from diffraction studies with synchrotron radiation [3.124]. It has been suggested that the distortions of the icosahedral short-range order may result from some angular

Figure 3.12. Simulations of the structure factor, S(Q), for Zr melts at different temperatures under assumption of a short-range order consisting of dodecahedral clusters (solid lines) together with the experimental results (dots). The parameters used for the simulations are compiled in Table 3.2.

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dependence of the atomic interaction potentials in early transition metals. However, the shape of the structure factor of stable and undercooled liquid Ti measured in a different investigation [3.122] by neutron scattering differs from that shown in Ref. [3.124]. In the latter study, similar to the other metallic melts, an excellent fit of the experimental data is obtained under assumption of an undistorted icosahedral short-range order based on dodecahedral aggregates [3.122]. To summarize the results on the short-range order in melts of pure metals, all diffraction studies on liquids of Fe, Ni, Zr, Co and Ti unambiguously demonstrate the preference of icosahedral (or at least distorted icosahedral) short-range order of fivefold symmetry in pure liquid metals. This confirms a more than 50-year-old hypothesis by Frank [3.40]. The icosahedral short-range order differs from the structure of the corresponding crystalline phases. However, it shows some similarity with the short-range order in quasicrystalline solid phases or the structurally closely related crystalline approximant phases. Such phases appear in alloys but were not discovered so far in pure metals. Therefore, the question arises on the short-range order of molten alloys forming quasicrystals or their approximants. Short-range order in alloys concerns both topological and chemical short-range ordering effects. All the quasicrystal-forming alloy systems discussed here possess only a moderate difference of the atomic radii such that an icosahedral short-range order should also be favourable in the liquid state from the theoretical point of view. Intensive investigations on the topological and chemical short-range order in this special kind of melts were performed for Al13(Co,Fe)4 alloys as a function of the temperature [3.120, 3.130]. The binary phase Al13Fe4 is a polytetrahedral crystal of monoclinic structure [3.131], which is an approximant of quasicrystalline solid phases. There are two modifications of Al13Co4. Both are polytetrahedral phases and one of them is isostructural with Al13Fe4 [3.132–3.134]. The structure factors measured by neutron diffraction for five Al13(Co,Fe)4 alloy melts with a different Co/Fe ratio at (a) T = TL + 150 K and (b) T = TL–150 K are depicted in Figure 3.13 [3.120, 3.130]. Four oscillations of S(Q) are observed in the investigated range of wave vectors. For Al13Co4, the second oscillation exhibits a shoulder on its right-hand side. If Co is substituted by Fe the shoulder becomes more pronounced and, finally, for the Fe-rich alloys a maximum arises at the place of the shoulder such that the second oscillation is split. In addition to this, a prepeak rises with increasing Fe content at small momentum transfer. This is considered the signature of medium-range chemical order and attributed to correlations between minority atoms [3.135]. All the characteristic features of S(Q) become better defined at lower temperatures. For Al13(Co,Fe)4 melts it could be shown that Fe substitutes for the Co isomorphously [3.120]. This enables varying the neutron-scattering contrast of the transition metal (TM) component. Since more than the minimum number of three

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Figure 3.13. Structure factors S(Q) measured by elastic neutron scattering for Al13(Co,Fe)4 melts of composition Al13Fe4, Al13Co1Fe3, Al13Co2Fe2, Al13Co3Fe1 and Al13Co4 at (a) T = TL + 150 K and (b) T = TL 150 K [3.120].

independent measurements with different scattering contrast have been performed for the Al13(Co,Fe)4 melts, the equation system given by Eq. (3.29) can be solved to determine the three partial structure factors within the Bhatia-Thornton formalism [3.102]. In a similar way, with Eq. (3.30), partial structure factors according to the Faber-Ziman formalism [3.103] can be calculated. As outlined in Section (3.2.2), the Faber–Ziman structure factors SAlAl(Q), SAlTM(Q) and STMTM(Q) describe the contributions to the total structure factor S(Q), which result from the three different types of atomic pairs (AlAl, AlTM and TMTM). Within the Bhatia–Thornton formalism the partial structure factor SNN(Q) describes solely the topological short-range order of the system, SCC(Q) the chemical short-range order and SNC(Q) the correlation of number density and chemical composition. As an example, the partial structure factors determined for Al13(Co,Fe)4 melts at T = TL + 150 K are shown in Figure 3.14. It is noteworthy also that SNN(Q) exhibits the characteristic shoulder on the right-hand side of the second oscillation. This highlights the fact that this feature already visible in S(Q) results from the topological short-range order of the liquid.

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Figure 3.14. Partial structure factors determined within the (a) Bhatia–Thornton and (b) Faber–Ziman formalism for Al13(Co,Fe)4 liquids at T = TL + 150 K.

As for melts of pure metals, this shoulder is considered as a first indication of an icosahedral short-range order prevailing in the liquid. As in the case of the melts of pure metals, for a further analysis of the topological short-range order in Al13(Co,Fe)4 melts, the Bhatia–Thornton structure factors SNN were simulated at large scattering vectors (Q 4 Å 1) according to the method described in Refs. [3.11-3.113] under assumption of different dominating types of short-range order (icosahedral, dodecahedral, bcc, fcc and hcp) prevailing in the liquid. The results of these simulations of SNN(Q) for Al13(Co,Fe)4 melts at T = TL + 150 K under assumption of dodecahedral (———), icosahedral (-----) and fcc-like shortrange order (……) together with the experimentally determined SNN(Q) (circles) at large wave vectors (Q > 4 Å 1) are depicted in Figure 3.15. The parameters used for the fits are r0 = 2.54 Å, r02 = 0.026 Å2, cclust. = 0.65 for dodecahedral short-range order, r0 = 2.53 Å, r02 = 0.029 Å2, cclust. = 0.73 for icosahedral short-range order and r0 = 2.58 Å, r02 = 0.030 Å2, cclust. = 0.70 for fcc-type

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Figure 3.15. Bhatia–Thornton partial structure factor SNN(Q) determined for Al13(Co,Fe)4 melts at T = TL + 150 K in the momentum transfer range 4 Å 1 Q 13 Å 1 (dots), and calculated structure factors by assuming clusters with different symmetries prevailing in the melt: dodecahedra (r0=2.54 Å, r02>=0.026 Å2, cclust. = 0.65, ———), icosahedra (r0 = 2.53 Å, r02 = 0.029 Å2, cclust. = 0.73, -----) and cubooctahedra (r0 = 2.58 Å, r02 = 0.030 Å2, cclust. = 0.70, ……).

short-range order. For a short-range order with bcc structure it was not possible to obtain a reasonable fit of the measured SNN(Q) assuming physically sound atomic radii. The simulation for clusters with hcp structure is nearly identical to that obtained for fcc-type short-range order and is therefore not shown in Figure 3.15. Under the assumption of an fcc- or hcp-type short-range order neither the position of the second maximum of SNN(Q) nor its half-width is in agreement with the experimental results. Similar as for melts of pure metals the assumption of clusters with icosahedral symmetry (icosahedra and dodecahedra) gives the best description of the measured SNN(Q) for the alloy Al13(Co,Fe)4 also. Among these two clusters for dodecahedra, a slightly better agreement with the experimentally determined SNN(Q) is obtained than for simple icosahedra of 13 atoms. This indicates that a topological short-range order consisting of larger polytetrahedral aggregates such as dodecahedra prevails in the liquid. The partial-pair distribution functions are calculated by Fourier transformation from the partial structure factors. The three Bhatia–Thornton pair correlation functions gNN(r), gCC(r) and gNC(r) are shown in Figure 3.16 for the different temperatures and the three Faber–Ziman pair distribution functions gAlAl(r), gAlTM(r) and gTMTM(r)

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Figure 3.16. Bhatia–Thornton pair correlation functions determined for Al13(Co,Fe)4 melts at different temperatures.

in Figure 3.17. From the partial pair correlation functions the mean distances of nearest neighbours rNN and the mean distances of AlAl, AlTM and TMTM neighbours rAlAl, rAlTM and rTMTM were determined. These are compiled in Table 3.3. From the partial radial distribution functions cA4 N r2gAB(r) (with A, B = N, Al, TM) the corresponding partial coordination numbers ZNN, ZAlAl, ZAlTM and ZTMTM were calculated by integration over its first maximum (Table 3.3). Here N(r) denotes the atomic density and cA the concentration of the component A. As for melts of pure metals, no significant temperature dependence of the nearest-neighbour distances is observed. Together with the decrease of ZNN with increasing temperature, this once again indicates that thermal expansion is mainly governed by the temperature dependence of the coordination numbers and not by a change of the nearest-neighbour distances. The Bhatia–Thornton pair correlation function gCC is characterized by a remarkable minimum at r 2.55 Å. Its depth increases with falling temperature of the melt. This minimum is a signature of a chemical short-range order prevailing in the liquid, which is characterized by an affinity for the formation of AlTM

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Figure 3.17. Faber–Ziman pair correlation functions determined for Al13(Co,Fe)4 melts at different temperatures.

Table 3.3. Mean distances of nearest neighbours and coordination numbers inferred from the partialpair correlation functions for the different pairs of atoms for Al13(Co,Fe)4 melts at different temperatures. ZNN denotes the average coordination number of the melt without distinguishing between the atomic species, while the coordination numbers, ZAB, that are determined within the Faber–Ziman formalism denote the number of A atoms surrounding a B atom. Temperature (K)

rNN (Å)

rAlAl (Å)

rAlTM (Å)

rTMTM (Å)

ZNN

ZAlAl

ZAlTM

ZTMTM

TL + 150 K TL + 30 K TL 60 K TL 150 K

2.67 2.67 2.67 2.67

2.78 2.78 2.78 2.80

2.60 2.60 2.60 2.58

2.48 2.51 2.47 2.50

12.08 12.26 12.39 12.53

9.69 9.82 9.97 10.12

9.42 9.65 9.73 9.77

1.94 1.90 1.87 1.87

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nearest neighbours. This also follows from the Faber–Ziman pair distribution functions. The shape of gTMTM(r) differs significantly from gAlAl(r) and gAlTM(r). While the first maximum of gAlAl(r) and gAlTM(r), like that of gNN(r), is considerably larger than the second one, for gTMTM(r) the amplitude of the first maximum is similar in height or, at high temperatures even smaller than the amplitude of the second maximum. This shows that in Al13(Co,Fe)4 melts, it is unlikely to find TM atoms in the first coordination shell of another TM atom. The amplitude of the first peak of gTMTM(r) rises with increasing temperature, contrary to the evolution of the second maximum. Obviously, the number of TM atoms in the first coordination shell of a TM atom decreases with falling temperature whereas the concentration of TM atoms in the second coordination shell increases. In a quantitative way, this tendency is expressed by the temperature dependence of the partial coordination numbers (Table 3.3). ZNN, ZAlAl and ZAlTM increase with decreasing temperature, while ZTMTM is falling. The chemical short-range order observed for Al13(Co,Fe)4 melts is reminiscent of that of the corresponding solid phases. At an atomic distance of approximately 2.5 Å, the radial distribution functions of the crystalline Al13Co4 phases exhibit a marked maximum for AlCo pairs, a significantly lower number of AlAl pairs and a negligible number of CoCo neighbours [3.136]. CoCo pairs prefer larger atomic distances of approximately 4.5 Å [3.136]. An interpretation for this observation is provided by the pair potentials of the AlCo system that were calculated by Phillips et al. [3.137]. At an atomic distance of approximately 2.5 Å the AlAl potential is repulsive, while the AlCo potential and also the CoCo potential show a minimum. Thus, the energy of the whole system is minimized if the number of AlAl nearest neighbours is reduced and if preferentially energetically favourable AlCo pairs are formed instead. This also results in a reduction of the number of CoCo nearest neighbours. Moreover, the CoCo potential exhibits a minimum at 4.4 Å that lies significantly below the value of the other pair potentials at the same atomic distance. This minimum occurs nearly at the same position as the second maximum of gCoCo(r). The above arguments explain the stability as well as the short-range order of the solid phases in Al13Co4 [3.136]. The same arguments explain the experimental findings on the chemical short-range order of Al13(Co,Fe)4 melts. The results obtained for the Al13(Co,Fe)4 melts are in excellent agreement with those from experiments on the short-range order of other Al-based alloys, which form quasicrystalline and polytetrahedral solid phases. Structure factors determined by neutron scattering on stable AlPdMn [3.112] and AlMnCr melts [3.112] and on undercooled and stable Al74Co26 [3.121], Al65Cu25Co10 [3.138] and Al60Cu34Fe6 liquids [3.139] are best described if an icosahedral short-range order is assumed to prevail in the melt, such that the Mn, Cr, Co or Fe atoms are located in the centre of the icosahedra and are surrounded preferentially by Al atoms. Moreover, EXAFS studies on Al65Cu25Co10 melts [3.140, 3.141] confirm the results on the chemical short-range order in quasicrystal forming Al-based alloys.

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These experiments clearly show that a chemical short-range order exists in the melt for which the Co atoms are preferentially surrounded by Al atoms. This also implies that there are significantly fewer Cu atoms in the first coordination shell around a Co atom than the alloy composition suggests. Hence, all studies on the short-range order of liquid Al-based alloys forming quasicrystalline or polytetrahedral solid phases clearly show, just as for melts of pure metals, the existence of an icosahedral topological short-range order, which was postulated by Frank [3.40] for metallic melts. Moreover, this kind of alloy melts is characterized by a chemical short-range order with a preferred formation of AlTM pairs. A decrease in the temperature leads to an enhancement of both the topological and the chemical short-range order. Similar results on the topological short-range order were also obtained from X-ray diffraction studies on quasicrystal-forming Ti39.5Zr39.5Ni21 alloy melts undercooled by use of an electrostatic levitator [3.123]. All these studies unambiguously prove that an icosahedral short-range order prevails in the investigated metallic melts, independent of the corresponding solid structure (Ni, Co: fcc; Fe, Zr: bcc; Al60Cu34Fe6, Ti39.5Zr39.5Ni21: icosahedral quasicrystals; Al13(Co,Fe)4: polytetrahedral phases; Al65Cu25Co10, Al74Co26: decagonal quasicrystals). Such a common observation for different alloys and metals, using different levitation and scattering techniques, demonstrates that the development of icosahedral short-range order is extremely widespread, perhaps universal, in metallic melts. 3.2.4 The short-range order of liquid Si Different from metallic elements, solid Si is characterized by covalent bonding of the atoms. Liquid Si, however, exhibits metallic character. As an additional peculiarity, Si has a higher density in the liquid state than in the solid state. Nevertheless, a significant role of covalent bonding must be expected for molten Si, so the assumption of radial symmetric interaction potentials, which is key to the concept of icosahedral short-range order in melts, is not justified. Owing to the unusual behaviour of Si at the solid–liquid phase transition, the study of the shortrange order of Si-melts is of high scientific interest. Moreover, the understanding of the short-range order in Si melts is of fundamental importance for the optimization of techniques for growing large single Si crystals that are basis material for the production of a large variety of electronic components. Preceding investigations [3.142–3.144] on the density of Si melts delivered contradicting results. While Ref. [3.141] reports a discontinuity of the density in liquid Si close to the melting point, other measurements [3.142, 3.143] showed no discontinuous behaviour of the density of molten Si from temperatures well above the melting temperature down to the deeply undercooled regime.

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Figure 3.18. Structure factor of Si-melts at different temperatures [3.146].

In the past few years, diffraction investigations on liquid Si were performed by different research groups [3.95, 3.145-3.147, 3.178]. Figure 3.18 shows structure factors S(Q) measured for Si melts by energy dispersive diffraction of synchrotron radiation in the broad temperature regime 1405 K T 1895 K (melting temperature TL = 1693 K) [3.146]. One characteristic feature of the structure factor of liquid Si is a shoulder2 on the high-Q side of the first maximum of S(Q) that becomes more pronounced with falling temperature. While there is a controversy concerning the temperature dependence of the coordination numbers that were inferred from the experimental results obtained by the different groups, all studies [3.95, 3.145–3.147, 3.178] deliver values in the range 4.9 Z 6.4. These values are considerably smaller than those determined for melts of metals (Z 12). This clearly shows that the short-range order in liquid Si is different from that in metallic melts. It is also noteworthy that the coordination numbers measured for liquid Si are larger than those of crystalline Si (Z = 4), which is consistent with the higher density of molten Si. 2

To avoid any misunderstandings, it must be stressed that this shoulder on the first maximum of S(Q) is different from the characteristic shoulder observed for metallic melts on the second maximum of S(Q).

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Coordination numbers of Z 6 are in agreement with the assumption of a short-range order consisting of clusters with a space-centred tetragonal A5 structure [3.148, 3.149]. Such an A5 structure is also found in solid state for other elements of Group IV, e.g. for white tin and at high pressures for Ge. Moreover, the diffraction experiments on liquid Si gave no indication of any discontinuity of the nearest-neighbour distances of the coordination number at the melting point as suggested by the density measurements of Ref. [3.142]. 3.3. MAGNETIC ORDERING IN LIQUID STATE

It is commonly believed that the existence of magnetic long-range order as present in, for example, solid ferromagnets is restricted to the solid state of matter. There are, in principle, two reasons that can explain this experimental experience: (i) Within the Heisenberg model, the magnetic exchange energy between the ions carrying a local magnetic moment is responsible for magnetic ordering. The exchange energy strongly depends on the interatomic distance. This energy leads to magnetic long-range order if the temperature decreases below the Curie temperature. At this temperature range, the magnetic exchange energy is larger than the thermal energy of the individual magnetic moments of the atoms. Hence, because of the mass density of solid matter is larger than in the liquid or much larger than in the gaseous state, the solid state is the most favourable for the occurrence of strong magnetic interaction and, thus, the occurrence of long-range magnetic order. (ii) All magnetic metallic alloys show Curie temperatures, which are smaller than the melting temperatures of the respective substances. Even though large thermally driven oscillation amplitudes of atoms in a liquid may hinder the establishment of permanent long-range magnetic correlation, there is no rigorous argument against the occurrence of magnetic ordering in liquids. In many cases, the density change on the melting of a solid is only a few per cent, which means that the interatomic distance essentially does not change upon melting. In fact, there is a rather “exotic” liquid system that does show magnetic ordering. This is the superfluid A1 phase of 3He. 3He atoms are fermions that form a Fermi liquid at very low temperatures. Below T = 2.7 mK, the superfluid state occurs owing to the formation of pairs consisting of two 3He atoms, which are coupled via a BCS-like mechanism (BCS: Bardeen–Cooper–Schrieffer coupling of quasiparticles in the superconducting state) in a fashion similar to the Cooper pairs in superconducting systems. In contrast to the Cooper pairs (singlet state S = 0, antiparallel spins), a coupled 3He pair corresponds to a triplet state of spin S = 1

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T[K]

with parallel alignment of the spins. Such a pair carries a magnetic moment. Longrange ordering of these moments leads to the existence of an internal effective field, which has been theoretically predicted [3.150] and experimentally observed [3.151]. These findings confirm that magnetic ordering is not restricted to the solid state and can occur even in liquid systems. All magnetic elements possess Curie temperatures that are far below the melting or liquidus temperatures. But metallic elements and even semiconductors can be undercooled considerably through the application of techniques of, e.g. containerless processing. The question arises, therefore, whether a material can be undercooled below its paramagnetic–ferromagnetic transition temperature TC (Curie temperature) to study magnetic ordering in a liquid system. The binary alloy system exhibiting the smallest difference between the Curie temperature and liquidus temperature, TC–TL, is CoPd at a composition of about Co80Pd20. This system offers the additional advantage that their components Co and Pd are completely miscible forming a solid solution over the entire concentration range [3.152]. Confusion due to chemical clustering and/or segregation as observed in, e.g. undercooled eutectic alloys [3.153] and eutectic alloys in the mushy zone regime [3.154], and their possible consequences on the magnetism of melts [3.155] can therefore be excluded in studies of CoPd melts. Figure 3.19 shows a temperature–time profile as measured on a levitationundercooled sample of Co80Pd20 [3.156]. The dashed lines represent liquidus and solidus temperatures while the dashed-dotted lines give the Curie temperatures of

Co80Pd20

1700 1600 1500

TL TS ΔT = 320 K

1400

Temperature

Ch003.qxd

1300

TcS

1200

TcL dT= TcS - TcL = 20 K

1100 40

80

2000

2040

Time t [s]

Figure 3.19. Temperature–time profile as measured on levitation undercooled Co80Pd20 alloy. The dashed lines represent the ferromagnetic Curie temperature of the solid material and the respective Curie temperature of the liquid ferromagnet the latter being 20 K smaller.

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solid and liquid temperature of the liquid ferromagnet. The latter was obtained by measurements of the temperature dependence of the magnetic susceptibility on the levitation-undercooled alloy [3.157]. It is obvious that the sample could be undercooled to temperatures quite close to the Curie temperature. It is noteworthy that in the case of CoPd alloys, even the hypercooling limit can be exceeded both by levitation experiments [3.158] as well as by fluxing the sample in a glass slag [3.159]. The onset of magnetic ordering in the undercooled liquid sample was clearly shown by a simple experiment. The Co80Pd20 alloy was undercooled in an electromagnetic levitation coil. In a short distance from the levitation coil, a strong SmCo permanent magnet was placed. At temperatures far above the Curie temperature, the liquid sample was symmetrically positioned in the levitation coil. But if the temperature was decreased, on approaching the Curie temperature TC, an attractive force between undercooled liquid and external magnet arose that drew the sample out of the symmetry axis of the levitation coil. The liquid state of the sample was unambiguously proven by the steep temperature rise during recalescence if the sample was crystallized in a subsequent step [3.156]. A modified Faraday balance was utilized to measure the magnetic susceptibility as a function of the temperature in the liquid undercooled regime. Figure 3.20 shows the reciprocal magnetic susceptibility as a function of temperature, measured for a Co80Pd20 alloy in the solid (crosses) and undercooled liquid (dots) state. The inset shows the rapid increase in the magnetization of the liquid sample if the

Figure 3.20. Reciprocal magnetic susceptibility as a function of temperature measured for a Co80Pd20 alloy in the solid (crosses) and undercooled liquid (dots) state. The inset gives the rapid increase of the magnetization of the liquid sample if the temperature approaches the Curie temperature.

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temperature approaches the Curie temperature [3.157]. The liquid sample shows a behaviour similar to the solid sample. The slope of both lines is similar. Apparently, the magnetic moments of the liquid and solid ferromagnet are comparable, indicating similar magnetic moments in the liquid and solid material. The only difference is that the extrapolation of the data to 1/ = 0 leads to a Curie temperature of the liquid sample about 20 K smaller than that of the solid. The similar behaviour of the liquid and solid ferromagnet is understood if it is assumed that the Curie temperature is mainly determined by exchange interaction, which depends on the interatomic distance. This result is not surprising since the change of the mass density of CoPd is only 4% during melting. Also, the relaxation times of the spin system are much shorter than the structural relaxation times. For a rough value, structural relaxation times of the atoms can be estimated from the Debye frequency to be in the order of 10 13 s, whereas the relaxation times of the spin system are in the order of 10 16 s if the Heisenberg uncertainty relation for electrons are taken into account. This means that the dynamic behaviour of the magnetic spin system is nearly independent of whether the spins carrying the magnetic moments are placed in a liquid or solid environment. The results of the measurements of the macroscopic volume susceptibility using the method of the Faraday balance are in agreement with measurements of the microscopic susceptibility applying the muon spin rotation (SR) technique [3.160]. A polarized beam of muons ( +) is produced from the decay of free positrons + ( ): +→ + + e + + (e : electron; +: antineutrino). A momentum of the muons of about 70 MeV/c is sufficient to stop the muons in levitated Co80Pd20 drops of diameter ~6 mm. The weak interaction decay of the precessing muons (decay time: 2.2 s) is anisotropic with respect to the muon spin, therefore time-resolved detection of the high-energy decay positrons yields the spin-precession frequency fL. A plastic scintillation counter (1 mm NE 110) mounted into the muon beam as a start-trigger allows determining the muon lifetime and measuring the precession frequency of the muons. The decay positrons are detected with three scintillation counters (3 mm NE 110) mounted at three different angles (0, 90, 180) in the plane of the muon precession. More details on the muon spin spectroscopy are given in Ref. [3.160]. Figure 3.21 shows the temperature dependence of the muon spin precession frequency fL (squares) and the transverse spin relaxation rate t (triangles) in undercooled liquid Co80Pd20 measured at temperatures T TC in an external magnetic field Bext = 75 mT. At temperatures T 1500 K, fL approaches a value of fLext = 9.98 MHz = constant, which corresponds to the Larmor frequency fL measured in the external field Bext only. When the temperature drops below 1500 K, fL shows a continuously progressing decrease from fLext leading to systematic deviations and a rapid increase of fLext fL 0 (Knight shift) in the vicinity of the Curie temperature TCL. However, the relaxation rate t steeply rises when the temperature

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101

10

0.8 0.7

Frequency [MHz]

9.95 0.6 9.9

0.5

μSR with external field of 750G

0.4

9.85

0.3 9.8 0.2 9.75 1200

1300

1400

1500

1600

0.1 1700

Temperature [K] 50 40 1/frequency

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30 20

Tc(l)=1252K

10 0 1250

1300 1350 Temperature [K]

1400

Figure 3.21. Upper diagram: Muon-spin precession frequency fL (squares) and the transverse spin relaxation rate t(triangles) as a function of temperature, measured on Co80Pd20 sample. Lower diagram: reciprocal Muon-spin precession frequency 1/fL = SP as a function of temperature; the reciprocal precession frequency scales with the magnetic susceptibility.

approaches the Curie temperature. The relaxation rate t of the signal increases by about 500%, which indicates large spin fluctuations caused by the onset of spontaneous magnetization. Within a model, the measured Larmor precession frequency fL of the muon spin is correlated to the internal field B that is experienced by the muon spin [3.161].

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A positive muon carries a permanent spin S = 1/2 which precesses in a field B with the frequency fL: f L = g gL B

(3.32)

gis the gyromagnetic ratio, 130 MHz/T in the present case. Since fL can be measured with high accuracy, the SR technique is an excellent method to determine the internal field B at the place of an implanted muon. In ferromagnetic materials, the internal field consists of several contributions: B = Bext + Bdem + Bdip + BL + BF

(3.33)

with Bext an applied external field, Bdem the demagnetization field, Bdip the dipolar field, BL the Lorentz field, and BF the conduction-electron-contact or Fermi field, respectively. The dipolar field Bdip is averaged to zero due to the fast diffusive motion of the muons in the liquid sample. The demagnetization field Bdem is approximated to zero because of the sphere-like geometry of the levitated liquid drop. The Lorentz field is given by BL(T) = o/3 Ms(T) with Ms the macroscopic saturation magnetization of the sample. At temperatures T TC the saturation magnetization is zero so that BL = 0. Thus, Eq. (3.33) is reduced for our measurements of the muon precession frequency in the undercooled drops to

(

2 f L = L g Bext + BF (T )

)

(3.34)

According to the model used for the analysis, the time-averaged Fermi field is proportional to the magnetization M(T ) = (T )Bext. Thus, = L(T ) gBext is proportional to the paramagnetic susceptibility (T ) [3.161]. In the lower part of Figure 3.21, the inverse deviation of the precession frequency ( f ) 1, from the precession frequency f(T ) in the applied magnetic field Bext, is plotted as a function of T. The dependence of ( f ) 1 on temperature clearly shows a linear relation, indicative of a classical Curie–Weiss behaviour of the liquid metallic magnet. The results of the present measurements using the muon as a microscopic sensor are in agreement with the experimental determination of the macroscopic magnetic susceptibility as a function of temperature by means of a Faraday balance [3.157], leading to the same Curie temperature of the liquid ferromagnet of TC = 1250 K 5 K and the same magnetic moment inferred from the slope of the ( f ) 1 versus temperature correlation. So far, with the exception of one work [3.162], it was not possible to undercool CoPd alloys below their Curie temperature [3.163]. In contrast, studies of the concentration dependence and the nucleation statistics of CoPd melts suggest that the onset of magnetic ordering in the undercooled melts stimulates crystal

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nucleation [3.164] (compare Sections 5.5 and 5.6.4). Magnetic ordering may initiate crystallization of undercooled melts in some analogy to magnetically driven phase transformations in solids as, e.g. the –-transformation in pure Fe or FeNi alloys. 3.4. KINETIC AND TRANSPORT PROPERTIES

The temperature dependence of the viscosity in undercooled metallic melts is by far an unresolved problem. Often, an Årrhenius ansatz is used with a fixed activation energy for viscous flow, G: ⎛ ΔG ⎞ (T ) = 0 exp ⎜ − . ⎝ k BT ⎟⎠

(3.35)

0 is a constant parameter. Such an Årrhenius-like behaviour should be able to describe the viscosity of systems with strong covalent bonding, where the viscous flow requires the break of directional bonds. In contrast, in metallic systems one expects more or less pronounced isotropic bonding. For these systems the Vogel– Fulcher–Tammann expression has been proposed for the description of the viscosity: ⎛ A ⎞ (T ) = o exp ⎜ ⎟. ⎝ T − To g ⎠

(3.36)

Equation (3.36) implies that the viscosity diverges if the temperature approaches the ideal glass temperature Tog. A physical interpretation of the Vogel– Fulcher–Tammann expression has been given within the free-volume model. Here, the viscosity is correlated with the Doolittle equation [3.165]:

(

)

= 0 exp K v*f /v f ,

(3.37)

where vf = Tvm (T Tog), T is the thermal expansion coefficient and vm the molar volume. v*f represents the excess free volume in the undercooled melt, while K is a geometrical factor of order unity. The ideal glass temperature Tog is the temperature at which the free volume vanishes. The description of the temperaturedependent viscosity within the free-volume model is based upon the assumption that the free volume is rearranged between the atoms without a change of the free energy [3.166]. The Vogel–Fulcher–Tammann expression should therefore be valid for systems with isotropic bonding such as noble gases and metals. If the atomic dynamics are described in the framework of the mode coupling theory [3.167–3.169], the existence of a critical point TMK is assumed. At this point, flow processes freeze such that below TMK thermally activated hopping processes

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dominate the atomic transport. TMK usually lies above the glass transition temperature Tg [3.170]. For temperatures T TMK, the mode coupling theory predicts a potency law for the temperature dependence of the viscosity [3.167, 3.171]: (T ) = 0 (T − TMK )

−ν

,

(3.38)

with an exponent v 2. This model is supported by measurements of the diffusion coefficients in metallic melts (above the melting temperature) performed under microgravity (g) conditions. In these conditions, the disturbing influence of convection-driven flow on atomic diffusion is essentially reduced so that the diffusion coefficient can be measured with much better accuracy than under terrestrial conditions. Frohberg et al. [3.172] were able to demonstrate that under such conditions, the diffusion coefficient of Sn-melts follows a power law in temperature with an exponent v 2 [3.172]. The experimental determination of the temperature dependence of the viscosity in metallic melts in the undercooled melt regime is difficult to perform. On the one hand, the nucleation frequency in metallic undercooled melts is relatively high, for example, in comparison with oxide glasses, so that sufficient time is not available to conduct such measurements. On the other hand, the conventional methods for measurements of the viscosity imply a direct manipulation of the state of an undercooled melt by, e.g. the rotating disc whose torsion is measured when immersed into the melt. This will initiate heterogeneous nucleation limiting the undercoolability of the melt. Recently, a new approach has been proposed to measure the viscosity in undercooled metallic melts by investigating the damping behaviour of surface oscillations of freely suspended levitated drops [3.173]. Only very few measurements of the viscosity in metallic melts have been reported to date [3.174]. To determine the viscosity around the glass temperature – in this particular temperature range a very pronounced change of the viscosity with temperature is expected – the creep behaviour of metallic glasses has been studied [3.175]. Below the glass temperature, an Årrhenius-like behaviour is found. An extrapolation of those data into the regime of the undercooled melt, i.e. above the glass temperature is, however, doubtful. If interpolation is tried between data of the viscosity measured in the stable liquid regime above the melting temperature and those measured around the glass temperature, neither Årrhenius nor Vogel–Fulcher–Tammann expressions deliver satisfactory description of the experimental results. In the case of Pd82Si18, for which experimental data are available both above TL and in the vicinity of Tg, none of the expressions discussed above is able to predict the temperature dependence of the viscosity by a unique relationship [3.176, 3.177].

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REFERENCES

[3.1] [3.2] [3.3] [3.4] [3.5] [3.6] [3.7] [3.8] [3.9] [3.10] [3.11] [3.12] [3.13] [3.14] [3.15] [3.16] [3.17] [3.18] [3.19] [3.20] [3.21] [3.22] [3.23] [3.24] [3.25] [3.26] [3.27] [3.28] [3.29] [3.30] [3.31] [3.32] [3.33]

Turnbull, D. (1956) Solid State Physics 3, 225. Johnson, W.L. (1986) Progress in Materials Science 30, 81. Turnbull, D. (1950) Journal of Applied Physics 21, 1022. Jones, D.R.H., and Chadwick, G.A. (1971) Philosophical Magazine 24, 995. Hoffmann, D.J. (1958) Journal of Chemical Physics 29, 1192. Thompson, C.V., and Spaepen, F. (1979) Acta Metallurgica 27, 195. Kauzmann, W. (1948) Chemical Review 42, 219. Singh, H.B., and Holtz, A. (1983) Solid State Communications 45, 985. Frenkel, J. (1955) Kinetic Theory of Liquids (Dover, New York). Hirai, N., and Eyring, H. (1958) Journal of Applied Physics 29, 810. Ramachandrarao, P., Cantor, B., and Cahn, R.W. (1977) Journal of Materials Science 12, 2488. Dubey, K.S., and Ramachandrarao, P. (1984) Acta Metallurgica 32, 91. Perepezko, J.H., and Paik, J.S. (1984) Journal of Non-Crystalline Solids 61–62, 113. Chen, H.S., and Turnbull, D. (1968) Acta Metallurgica 16, 369. Smithells, C.J. (1983) Smithells Metals Reference Book, ed. Brandes, E.A. (Butterworths, London). Wachter, H.J. (1989) Ph.D. Thesis, Universität Stuttgart, Germany. Ågren, J. (1988) Physical Chemistry of Liquids 18, 123. Chen, H.S., and Turnbull, D. (1968) Journal of Chemical Physics 48, 2560. Chen, H.S., and Turnbull, D. (1967) Journal of Applied Physics 38, 3646. Sommer, F., Bucher, G., and Predel, B. (1980) Journal de Physique Colloque 41, 563. Lee, M.C., Fecht, H.J., Allen, J.L., Perepezko, J.H., Ohsaka, K., and Johnson, W.L. (1988) Materials Science & Engineering 87, 301. Evans, P.V., Garcia-Escorial, A., Donovan, P.E., and Greer, A.L. (1987) Materials Research Society Symposia Proeedings 57, 239. Gillessen, F., and Herlach, D.M. (1990) Journal of Non-Crystalline Solids 117/118, 555. Sommer, F. (1982) Zeitschrift für Metallkunde 73, 78. van den Beukel, A., and Radelaar, J. (1983) Acta Metallurgica 31, 419. Fecht, H.-J., and Johnson, W.L. (1988) Nature 334, 50. Daeges, J., Gleiter, H., and Perepezko, J.H. (1986) Physics Letters A 119, 79. Herlach, D.M., Kästner, J., Heller, A., and Wassermann, E.F. (1983) Journal of Magetism and Magnetic Materials 40, 98. Dubois, J.M. (1988) Journal of Less Common Metals 145, 309. Huang, C.Y. (1985) Journal of Magnetism and Magnetic Materials 51, 1. Samwer, K. (1988) Physics Reports 161, 1. Eyring, H. (1936) Journal of Physical Chemistry 4, 283. Eyring, H., and Hirschfelder, J.O. (1937) Journal of Physical Chemistry 41, 249.

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Chapter 4

Solid–Liquid Interface 4.1. 4.2.

Structural Order at the Interface Interfacial Energy Under Local Equilibrium Conditions 4.2.1 The Negentropic Model by Spaepen and Thompson 4.2.2 Investigations Using Molecular Dynamics and Density Functional Theory 4.2.3 Experimental Results on the Solid–Liquid Interfacial Energy Under Local Equilibrium Conditions 4.2.4 The Energy of the Interface Between Structurally Complex Solids and their Melts 4.3. Diffuse Interface Theory 4.3.1 Physical Interpretation of the Diffuse-Interface Region 4.3.2 Phase-Field Models for Nucleation 4.3.3 Sharp Interface Versus Diffuse Interface References

115 116 117 121 124 126 133 133 136 136 138


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Chapter 4

Solid–Liquid Interface 4.1. STRUCTURAL ORDER AT THE INTERFACE

The energy of a solid–liquid interface LS is governed by the interplay of the shortrange order in the liquid phase with the structure of the solid phase. As outlined in Section 3.2, most metallic melts are characterized by an icosahedral short-range order. Owing to the fact that an icosahedral short-range order is incompatible with the translational invariance of crystalline phases, for metallic systems a comparatively high energy of the interface between an undercooled liquid and a nucleus of a non-complex crystalline phase that is formed in the melt is predicted [4.1]. The question of the influence of the short-range order in solid and liquid phases on the solid–liquid interfacial energy and finally on the undercoolability of the melt attracted new attention with the discovery of quasicrystalline solids [4.2–4.4] by Shechtman et al. [4.5]. The (icosahedral) quasicrystals are a peculiar kind of solids exhibiting an icosahedral short-range order in the solid state. Owing to the similarity of the icosahedral short-range order in metallic melts and in quasicrystalline solids, a low energy of the interface between a melt and a quasicrystalline solid nucleus is predicted as compared with that of an interface between a melt and a non-polytetrahedral crystal [4.1]. As discussed later in Chapter 5, this results in a small activation energy for the formation of a critical nucleus of the quasicrystalline phase and consequently a low undercoolability of quasicrystal-forming melts. Polytetrahedral crystalline phases, e.g. the Frank–Kasper phases [4.6], play a special role among the crystalline structures. These phases are characterized by a complex structure with a large unit cell (e.g. 162 atoms for (Al,Zn)49Mg32) and polytetrahedral symmetry elements inside this unit cell. They are frequently found in quasicrystal forming alloy systems and in many cases they are considered as approximants [4.3, 4.4] of quasicrystals. Because these phases exhibit a polytetrahedral short-range order, the same arguments as discussed before for the quasicrystalline phases may be applied also for the polytetrahedral crystalline phases. Hence, a low solid–liquid interfacial energy is also predicted for this special kind of crystals [4.1]. The qualitative arguments outlined above motivate detailed theoretical and experimental investigations on the structure dependence of the energy of the solid–liquid interface, which will be the topic of this chapter. Section 4.2 is 115

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concerned with the solid–liquid interfacial energy under local equilibrium conditions. Later, in Section 5.1.3 the interfacial energy under non-equilibrium conditions will be discussed, which is relevant for the formation of a solid nucleus from the non-equilibrium state of an undercooled melt. 4.2. INTERFACIAL ENERGY UNDER LOCAL EQUILIBRIUM CONDITIONS

The solid–liquid interfacial energy LS is the free enthalpy that is required for the formation of a solid–liquid interface per unit area [4.7]. It depends on an enthalpic and entropic contribution. The enthalpic term considers that the atoms are bound on all sides inside the bulk crystals, while there are partly non-saturated bonds for atoms at the interface. The entropic contribution is due to the lower entropy of interface atoms as compared with atoms in the melt. Skapski [4.8] suggested the simplest estimation of the solid–liquid interfacial energy. In this ansatz LS is expressed by the difference of the energy SV of the phase boundary between solid and vacuum and the energy LV of the interface between liquid and vacuum: LS ⫽ SV⫺ LV. Ewing [4.9] estimated the entropic contribution to the solid–liquid interfacial energy based on experimentally determined radial distribution functions. For the calculation of the enthalpic term the energy of all bonds between an atom at the phase boundary with its nearest neighbours in the solid and in the liquid phase was compared with the energy of all bonds between an atom inside the bulk solid forms with its neighbours. This delivers the following expression for the energy of the solid–liquid interface: LS =

1 2/ 3

vm

( LH f + S * TE ) .

(4.1)

where vm is the molar volume. The estimations delivered L ⫽ 2.5 ⫻ 10⫺9 and the entropy of the interface S* has values between 2 ⫻10⫺8 and 3.1⫻10⫺8 J/K for melts of the metals Li, Na, Au, Ag and Al. Miedema and den Broeder [4.10] approximated S* by fitting of Eq. (4.1) to values for solid–liquid interfacial energies that were determined from measurements of the maximum undercoolability of melts of pure metals. Owing to the fact that the influence of heterogeneous nucleation (cf. Section 5.1.2) was neglected in this calculation, the determined value of S* ⫽ 5⫻10⫺8 J/K should be considered as a lower limit of S*. The above simple models provide expressions for the energy LS of the interface between liquid and solid phases that are independent of the solid structure. However, the negentropic model by Spaepen and Thompson, which is described

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Solid–Liquid Interface

117

below as well as new molecular dynamics simulations and studies in the framework of density functional theory (DFT), (cf. Section 4.2.2) incorporate structural aspects into the theoretical models for the energy of the solid–liquid phase boundary. 4.2.1 The negentropic model of Spaepen and Thompson By definition, the solid–liquid interfacial energy is the difference of the Gibbs free energy of a system with a solid–liquid interface and of a hypothetical system, for which the Gibbs free energy jumps discontinuously at the phase boundary from the value of the bulk solid, G S , to the value of the bulk liquid, G L . The tilde (~) over the symbols indicates that the corresponding physical quantity is normalized to one single atom for simplification of the following derivations. G S and G L are the sum of an enthalpic and of an entropic contribution: G S = H S − T SS

and

G L = H L − T SL .

(4.2)

Here, H S and H L denote the enthalpies of the bulk solid and of the bulk liquid phase. SS and SL are the entropies of these phases. Under equilibrium conditions, which means at the melting point TE, follows: G S (TE ) = H S (TE ) − TSS (TE ) = G L (TE ) = H L (TE ) − TSL (TE ).

(4.3)

As schematically shown in Figure 4.1, within the phase boundary, the enthalpy and the entropy change continuously from H S and SS in the bulk solid to the values H L and SL in the bulk liquid. The solid–liquid interfacial energy LS at the melting point is the shaded area between the curves H ( x ) and S ( x ) TL, which are functions of the distance x perpendicular to the interface (cf. Figure 4.1).1 The enthalpy of fusion is given by H f = H L (TE ) − H S (TE ) and the entropy of fusion by S f = SL (TE ) − SS (TE ) = H f /TE . S f is the sum of two contributions. The vibrational contribution Svib is a result of the fact that there is a larger local volume available for thermal oscillations of atoms around their average positions in metallic melts than for the solid phase. The configurational contribution Scf (ls) results from the number of possible atomic configurations in the liquid and the assumption of negligible configurational entropy of the solid. In order to simplify the problem, Spaepen and Thompson [4.11–4.13] assume a negligible density deficit in the interface, which means that the phase boundary has the same density as the liquid phase. This assumption is consistent with results 1

In the reference system with a discontinuous change of the Gibbs free energy the corresponding area is vanishing.

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Figure 4.1. Schematic representation of the enthalpy H ( x ) and of the product S ( x ) TE of entropy and melting temperature at the solid–liquid interface as a function of the distance x, perpendicular to the interface at T ⫽ TE.

of positron-annihilation experiments, performed during crystallization of metallic glasses. These are indicative of a small number of vacancy-like defects in the specimen during the early stages of crystallization [4.14]. The assumption of an insignificant density deficit in the interface region implies that the density changes directly at the solid surface – similar as in the hypothetic reference system – from its value in the bulk melt, L, towards the value in the bulk solid, S (Figure 4.2). Because in first approximation the enthalpy of a metallic system is mainly dependent on the density, the enthalpy of a system with interface changes similar to that in the reference system. Hence, the enthalpic contribution to the solid–liquid interfacial energy is negligible. The solid–liquid interfacial energy is then dominated by the loss of entropy within the solid– liquid interface, when going from the bulk liquid to the bulk solid. As pointed out by Turnbull [4.15], the energy of the interface between solid and its melt is consequently of “negentropic” origin. Therefore, the model for the solid–liquid interfacial energy developed by Spaepen and Thompson is called “negentropic model”. If a negligible density deficit in the interface is assumed, the vibrational contribution to the interfacial entropy is insignificant, because the atomic volume that is correlated with the density mainly governs this quantity. Therefore, of all relevant thermodynamic quantities, only the configurational entropy exhibits a decisive change inside the solid–liquid interface. The configurational entropy of the interface is due to the adaptation of the liquid structure to that of the solid. The boundary conditions given as a result of the contact with the solid phase lead to a reduction in the number of possible configurations in the interface as compared

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Figure 4.2. Schematic view of the development of density , enthalpy and of the product of melting temperature TE and entropy as a function of the distance, perpendicular to the interface that is assumed in the negentropic model by Spaepen and Thompson [4.11, 4.12, 4.13]. i denotes the solid–liquid interfacial energy under the assumption that its enthalpic contribution is negligible.

with the number of configurations in the bulk liquid. Scf (i ) denotes the difference in the configurational entropy of interface and solid per atom. The development of density, enthalpy and the product of melting temperature and entropy at the phase boundary is schematically shown in Figure 4.2. As discussed before, the solid–liquid interfacial energy is defined as the difference of the Gibbs free energies of a system with interface and of a reference system without interface. Hence, the solid–liquid interfacial energy i per interface atom at the melting point is given by [4.11] i (TE ) = TE [Scf (ls) − Scf (i )]

(4.4)

The number Ni of atoms in the interface per area unit is usually different from that in the uppermost solid layer NS. At the melting point the solid–liquid interfacial energy s per atom of the uppermost solid layer is given by [4.11] s(T E ) =

Ni TE ⎡⎣Scf (ls) − Scf (i ) ⎤⎦ . NS

(4.5)

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The dimensionless solid–liquid interfacial energy, also called -factor, is defined as :=

s (TE ) . H f

(4.6)

Different from LS, the -factors are independent of the temperature. They, however, significantly depend on the structure of both the solid and the liquid phase. Therefore, they are suitable parameters to describe the structural dependence of the solid–liquid interfacial energy. The change Scf (i ) of the configurational entropies at the interface equates the configurational entropy Scf (i ) of the interface per atom, because the configurational entropy of the crystal is negligible. Hence, we obtain [4.11–4.13] =

N i ⎡ Scf (ls) − Scf (i ) ⎤ ⎢ ⎥ S f N S ⎢⎣ ⎥⎦

(4.7)

where Scf (i ) is expressed [4.12]

N Scf (i ) = Scf (1) 1 Ni

(4.8)

Here Scf (1) denotes the configurational entropy of the first atomic layer of the phase boundary per atom and N1 the number of atoms within this layer. For metals, typical values for S f and Scf (ls) are S f 1.2kB and Scf (ls) 1kB [4.11]. For the determination of Scf (1) as well as of the ratios N1/Ni and Ni/NS, a model of the solid–liquid interface is required. Spaepen and Thompson [4.11– 4.13] suggested to model the interface by randomly dense packing of hard spheres such that (1) tetrahedral short-range order is preferred, (2) octahedral short-range order is forbidden, and (3) the density is maximized. The first two construction rules based upon Frank’s [4.16] prediction of a polytetrahedral short-range order in metallic melts. As outlined in Section 3.2, this hypothesis has meanwhile been experimentally confirmed. Therefore, a liquid-like short-range order based on tetrahedra is assumed to exist in the solid–liquid interface, while an octahedral short-range order typical of crystals is ruled out. The third rule is necessary to minimize the energy of the interface.

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The atomic arrangement constructed under these assumptions on top of an fcc(111) layer do not exhibit translational order in the first liquid layer parallel to the interface, whereas the structural order decreases only slowly in the direction perpendicular to the interface [4.11]. Ni/NS and N1/Ni can be directly inferred from the construction of the solid– liquid interface by counting of the atoms. The configurational entropy Scf (1) of the first atomic layer of the interface depends on the number Zcf of possibilities to build up the interface in accordance with the construction rules ( Scf (1) ⫽ kB/N1 · ln(Zcf)). It can be approximated by analysing the structure of the first interfacial layer as drawn out by Spaepen and Thompson for fcc, hcp and bcc phases ( Scf (1) ⫽ 0.096kB) [4.11–4.13]. The exact solution of the geometrical problem by Elser [4.17] delivers the slightly different value Scf (1) ⫽ 0.113kB. Together with the values for Ni/NS and Ni /Nl calculated in Ref. [4.11] for fcc and hcp structures (Ni/NS ⫽ 1.10; Ni /N1 ⫽ 1.46) and in Ref. [4.13] for bcc structures (Ni/NS ⫽ 0.923; Ni /N1 ⫽ 1.23), this results in fcc/hcp⫽0.85 for fcc or hcp crystals [4.1] and bcc ⫽ 0.70 for bcc structures. If is known, by using Eq. (4.6), the energy of the solid–liquid interface per unit area is expressed as LS =

(

S f TE N L vm2

)

13

.

(4.9)

Here, vm denotes the molar volume and NL the Avogadro’s number. 4.2.2 Investigations using molecular dynamics and density functional theory When modelling the energy of the interface between a solid and a liquid phase in the framework of the negentropic model, the solid–liquid interface is constructed by packing of hard spheres. The hard sphere approximation leads to a significant simplification of the problem. For metallic systems, however, hard sphere potentials do not provide a very realistic description of the atomic interactions. To compensate this flaw, in the negentropic model further modelling assumptions are made such that a polytetrahedral short-range order in the interface is compelled, which follows from the theoretical and experimental findings outlined in Section 3.2. Owing to the strong simplifications the negentropic model by Spaepen and Thompson will provide only a rough approximation of the solid–liquid interfacial energy in metallic systems. In other approaches, LS is calculated by molecular dynamics simulations or in the framework of density functional theory (DFT). Some results of such

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Table 4.1. Results of different calculations of the dimensionless solid–liquid interfacial energy for the (111), (110) and (100) layers of crystals with fcc structure and for the (110) layers of solids with bcc structure. Authors

Method

Potential

Fcc (111)

Spaepen and Thompson [4.1] McMullen and Oxtoby [4.22] Broughton and Gilmer [4.29] Curtin [4.21]

DPHS DFT MD DFT

Marr and Gast [4.23]

DFT

Ohnesorge et al. [4.24]

DFT

Davidchack and Laird [4.30] Hoyt et al. [4.31]

MD MD

HS HS LJ LJ HS HS AS HS LJ HS Ni-EAM

0.85 0.87 0.36 0.45 0.43 0.48 0.44 0.18 0.24 0.48 0.55

(110)

Bcc (100)

(110) 0.70

0.37

0.35 0.45 0.48 0.46

0.20 0.28 0.54 0.57

0.24 0.30 0.52 0.59

Abbreviations: HS, hard sphere potential; LJ, Lennard–Jones potential; AS, adhesive sphere potential; EAM, embedded atom method; DPHS, dense packing of hard spheres; DFT, density functional theory; MD, molecular dynamics simulation.

investigations on the dimensionless solid–liquid interfacial energies for different crystal surfaces of phases with fcc or bcc structure are compiled in Table 4.1.2 For the investigations based on DFT [4.18], it is assumed that the density S(r) of the solid phase is a function of the position r and shows the symmetry of solid. On the other hand, the density L of the liquid is set spatially constant [4.19, 4.20]. It is assumed that the thermodynamic potentials, e.g. the free energy F, are solely functionals of the density (r) [4.18]. Within the interface region the density (r) changes continuously from the density value S(r) of the bulk solid phase to the density value, L, of the liquid phase. Then, (r) is given by minimization of the function = (( r )) − L = F (( r )) − L ∫ ( r ) dr + pv ,

(4.10)

where is the grand potential, L the grand potential of the liquid, L the chemical potential of the liquid, p the pressure and v the volume [4.21]. If O denotes the surface of the phase boundary, the energy of the interface between solid and liquid is expressed as LS = 2

. O

(4.11)

If in the original work the values for the solid–liquid interfacial energy are published in different units, from these the dimensions less solid–liquid interfacial energy defined by Eq. (4.9) was calculated to allow a comparison of the results.

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The function (r) that minimizes is determined by variation techniques. This requires additional assumptions and approximations, which are the reasons for the differences of the various approaches to model the solid–liquid interfacial energy using DFT found in the literature [4.21–4.24]. Information on the short-range order in the liquid phase enters into the density functional theoretical calculations via correlation functions. In all the approaches discussed here [4.21–4.24] the correlation functions are estimated in the framework of the Percus–Yevick approximation [4.25]. For metallic melts, the description of the measured structure factors using this approximation is not satisfactory [4.26–4.28]. Especially, the fact that the short-range order in undercooled metallic liquids is based on polytetrahedral aggregates is not explicitly considered in any of the presently known density functional theoretical approaches. Depending on the used atomic interaction potentials and approximations the various density functional theoretical calculations deliver substantially different results on the solid–liquid interfacial energy (cf. Table 4.1). Molecular dynamics simulations of the interface between solid and liquid enable to treat aspects related to the short-range order both in the liquid and solid phase when estimating the interfacial energy. Similar to DFT, the results of the molecular dynamics simulations are strongly dependent on the choice of the interaction potentials, which explains the broad spread of values for the solid–liquid interfacial energy determined in the different studies [4.29–4.31] (cf. Table 4.1). In the work of Broughton and Gilmer [4.29] as well as in Davidchack and Laird [4.30], Lennard–Jones and hard sphere potentials were utilized. Hoyt et al. [4.31] employed interaction potentials determined by the embedded atom method (EAM) for Ni [4.32] to estimate the solid–liquid interfacial energy for Ni and its anisotropy. These calculations deliver solid–liquid interfacial energies of (LS111) ⫽ 308.90 mJ/m2, (LS110 ) ⫽ 317.53 mJ/m2 and (LS110 ) ⫽ 329.47 mJ/m2, respectively, for the (111), (110) and (100) layers of Ni. According to Eq. (4.9), with an entropy of fusion SfNi ⫽ 10.12 J/(mol K) [4.33] and with a molar volume vmNi ⫽ 7.09⫻106 m3/mol that is calculated from the density of solid Ni at the melting point (TL) ⫽ 8.28 g/cm3 [4.34], this corresponds to dimensionless solid–liquid interfacial energies (111) ⫽ 0.55, (110) ⫽ 0.57 and (100) ⫽ 0.59, respectively. The comparison of the -factors compiled in Table 4.1 shows drastic differences of the solid–liquid interfacial energies calculated using different models. All studies, in which the energy of the solid–liquid interface is estimated for different crystal layers are indicative of a quite low anisotropy of the solid–liquid interfacial energy. The results of the various approaches are however, non-uniform concerning the relative values of the -factors obtained for different crystal layers. The energy of the interface between crystals with bcc structure and their melts was only determined within the negentropic model by Spaepen and Thompson [4.1, 4.11–4.13]

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and by the DFT calculations by Marr and Gast [4.23]. Both estimations were performed for the (110) layer that is the most densely packed layer in the bcc structure. The former study delivers a significantly smaller -value for bcc crystals than for the most densely packed layer in fcc structures, the (111) layer. On the other hand, Marr and Gast obtained no substantial differences of the dimensionless solid–liquid interfacial energies of solids with fcc or bcc structure. 4.2.3 Experimental results on the solid–liquid interfacial energy under local equilibrium conditions We will now compare the solid–liquid interfacial energies predicted by the above models with experimental findings. In this section, exclusive experiments will be discussed that were performed under local equilibrium conditions, which means that the measurements of the solid–liquid interfacial energy are drawn out at the melting point or in the temperature interval between solidus and liquidus temperature. Solid–liquid interfacial energies under non-equilibrium conditions, as in the case of a nucleus of a solid phase within an undercooled melts, are treated in Section 5.6.3. One method to measure the energy of the solid–liquid interface at the melting point is based on the determination of the dihedral angle, ϑ, of the liquid groove that is formed at the grain boundary between two neighbouring grains, which are in contact with the corresponding melt [4.35, 4.36]. If the energy gb of the grain boundary is known, the solid–liquid interfacial energy is given as LS =

gb 2 cos(ϑ/2)

.

(4.12)

Because metallic melts are not transparent, the dihedral angle cannot be measured in situ. Therefore, the samples are rapidly quenched after the two-phase equilibrium was established. Subsequently, they are metallographically investigated in the solid state. Some change in the shape of the formerly liquid grooves during cooling and complete solidification cannot be excluded with absolute certainty. To distinguish during the postmortem analysis the areas that were liquid in the two-phase equilibrium from the solid ones, usually eutectic or peritectic alloys are investigated for which the composition of the liquid differs from that of the primary solid phase. Hence, the system does not consist of one component and consequently the interfacial energy may be influenced not only by topological effects but also by chemical ones. Other techniques use the reduction of the melting point of small particles due to the Gibbs–Thomson effect in order to determine the energy of the interface

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between solid and melt [4.37]. The reduction of the melting point Tr is coupled with the radius of curvature Rc and the Gibbs–Thomson coefficient GT:

Tr =

2 GT Rc

(4.13)

With GT and with the entropy of fusion per volume unit, S f , the solid–liquid interfacial energy is expressed by LS = GT S f . (4.14) Both techniques can be combined, if the Gibbs–Thomson coefficient is determined by an analysis of the shape of the liquid grooves that are formed at a grain boundary located within a temperature field of known gradient [4.36, 4.38]. This allows the calculation of LS without any knowledge of the grain boundary energy. The measurements of the solid–liquid interfacial energy using of the techniques described above are prone to considerable errors ranging typically between 15 and 30% of the measured value. Most of the investigations also neglect effects of crystal anisotropy. Some experimentally determined energies of the interface between metallic solids of fcc, hcp or bcc structure and their corresponding melts are listed in Table 4.2 together with the temperatures T at which the measurements were performed [4.36, 4.37, 4.39–4.41]. In this table, only such measurements are considered, for which the compositions of the liquid and the solid phase (second column of Table 4.2) show only a moderate difference. Table 4.2. Solid–liquid interfacial energies LS determined by different methods at temperature T and dimensionless solid–liquid interfacial energy calculated from LS according to Eq. (4.9). Metal Solid phase; liquid phase Al Al Al Au Cd Pb Tl Zn

Al97.5Cu2.5; Al82.7Cu17.3 Al81.1Mg18.9; Al62.6Mg37.4 Al; Al97Ni3 Au; Au * Pb; Pb99.96Sb0.04 * *

Crystal Structure

Method

LS(T ) (J/m2)

T (K)

Ref.

fcc fcc fcc fcc hcp fcc bcc hcp

GBS GBS GBS RM GBA GBA GBA GBA

0.160 0.149 0.172 0.270 0.087 0.076 0.067 0.123

831 723 913 1337 594 600 577 693

0.68 0.72 0.66 0.89 0.67 0.95 0.93 0.64

[4.39] [4.40] [4.39] [4.37] [4.41] [4.36] [4.41] [4.41]

Abbreviations: GBA, measurement of the dihedral angle at grain boundary grooves; GBS, determination of the shape of the liquid groove at the grain boundary; RM, reduction of melting point. * The solid–liquid interfacial energies were determined by measuring the dihedral angles at grain boundaries for different eutectic alloys [4.41].

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The dimensionless solid–liquid interfacial energies were calculated from the measured interfacial energies using Eq. (4.9). They are listed in the last column of Table 4.2. For all -factors 0.64 ⬍ ⬍ 0.95. When comparing the results predicted by the different models (Table 4.1) with the experimental ones (Table 4.2), it becomes obvious that only the model of Spaepen and Thompson [4.11–4.13] and the calculations by McMullen and Oxtoby [4.22] yield -factors that lie within the range of measured values for . Of all discussed approaches, these two models appear to be best suited to describe the solid–liquid interfacial energy of metallic systems forming solid phases with fcc structures. The -factors determined in the molecular dynamics simulations by Hoyt et al. [4.31] are slightly lower than the measured dimensionless solid–liquid interfacial energies. All other approaches predict significantly lower energies of the interface between solid and melt than experimentally observed for metals with fcc, hcp or bcc structure. The new methods employing DFT and the molecular dynamics simulations are highly interesting approaches to calculate the solid–liquid interfacial energy, which delivered first promising results for colloidal systems that can be described by hard sphere potentials [4.30, 4.42]. In their present form, most of these models, however, are not suitable for a quantitative description of the solid–liquid interfacial energy of metallic melts. Surprisingly the oldest calculation employing DFT [4.22], which uses less realistic hard sphere potentials and some strong simplifications, gives the best agreement with the experimental results for metallic melts. Among all discussed models, only for the molecular dynamics simulation by Hoyt et al. [4.31] an interaction potential was employed that was specially developed for metallic systems. Nevertheless, this approach still slightly underestimates the energies of the solid–liquid interface. Hence, for a quantitative calculation of the solid–liquid interfacial energies of metallic systems a further improvement of the models using DFT or molecular dynamics is necessary. Especially the use of realistic interaction potentials is required. Despite its simplicity, the negentropic model by Spaepen and Thompson [4.11–4.13] delivers a surprisingly good description of the measured solid–liquid interfacial energies. This approach was recently modified in order to calculate the energy of the solid–liquid interface also for structurally complex solids.

4.2.4 The energy of the interface between structurally complex solids and their melts The analytical estimation of the interfacial energy by Spaepen and Thompson [4.1, 4.11–4.13] was possible for simple crystalline solids with fcc, hcp or bcc structure because for these crystal structures there are only three different types of atomic

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neighbourhoods in the first interfacial layer. Complex crystal structures, however, are characterized by a larger variety of different atomic surroundings. Moreover, quasicrystalline phases do not even show a translational invariance of the solid structure. Because of these reasons an analytic treatment of the problem appears impossible for structurally complex solids. To determine the energy of the interface between a structurally complex solid phase and its melt, in a recent work a numerical approach was chosen for the construction of the solid–liquid interface and for the determination of the configurational entropy [4.43–4.46]. The method can be applied to all metallic systems for which the model assumption of a polytetrahedral short-range order in the solid–liquid interface is justified. To the present date the following solid phases were analysed: the icosahedral quasicrystalline I-phase in Al᎐Pd–Mn, the polytetrahedral phases -Al13Fe4, -Al5Fe2 and ⬘-Ni-V and the structurally complex, hard magnetic phase -Nd2Fe14B1. While the I-, - and -phase are based upon icosahedral structural units, the polytetrahedral ⬘-phase is a Frank–Kasper phase [4.6] consisting of Z14 coordination polyhedra for which the central atom is surrounded by 14 nearest neighbours. The -phase is a non-polytetrahedral structurally complex phase. The numerical calculations were drawn out employing personal computers. The interface between a section of the most dense packed (quasi)crystal plane and the liquid is built up using the same construction principles as in the model by Spaepen and Thompson [4.1, 4.11–4.13] (cf. Section 4.2.1). A large number of possibilities exist to construct the first interfacial layer. The numerical algorithm that was developed computes all these possibilities, which in the following are called “configurations”. For simplicity we do not distinguish between different atomic species, which means that only the topological structure of the solid is considered. This simplification appears to be justified especially for the investigated Al-based alloys forming quasicrystalline and polytetrahedral phases. Experiments on the undercoolability of such alloy melts show that their nucleation behaviour is dominated by the topological structure of the primarily formed solid phase, while the chemical composition of the alloy is of minor importance as long as phases of the same topological structure nucleate primarily [4.43] (cf. Section 5.6.3). Moreover, neutron scattering investigations [4.47, 4.48] and EXAFS experiments [4.49, 4.50] on melts of Al-based alloys forming polytetrahedral phases are indicative of a similar chemical short-range order in the liquid and the solid phase (cf. Section 3.2.3). Therefore, for these systems a possible influence of the chemical short-range order on the solid–liquid interfacial energy can be neglected. Nd2Fe14B1 consists of more than 82% of Fe atoms, which suggests also for this alloy a small impact of the chemical short-range order on the energy of the solid–liquid interface.

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Table 4.3. Numerically calculated -factors (calc) and data determined by modelling of the solid–liquid interface for different solid phases [4.43, 4.45, 4.46]. Phase I ⬘-Ni-V -Nd2Fe14B1 -Al13Fe4 -Al5Fe2 bcc fcc/hcp

System size

Ni/N1

Ni/NS

Scf (1)

Scf (i )

calc

83 ⱕ NS ⱕ 145 60 ⱕ NS ⱕ 135 60 ⱕ NS ⱕ 135 104 ⱕ NS ⱕ 234 72 ⱕ NS ⱕ 128 analyt. calc. [4.13,4.17] analytical calculation [4.12, 4.17]

1 1 1 1 1 1.23 [4.13]

0.50 0.59 0.60 0.55 0.67 0.923 [4.13]

0.19kB 0.27kB 0.26kB 0.15kB 0.23kB 0.113kB [4.17]

0.19kB 0.27kB 0.26kB 0.15kB 0.23kB 0.092kB

0.34 0.36 0.37 0.39 0.43 0.70

1.46 [4.11, 4.12]

1.10[4.11, 4.12]

0.113kB [4.17]

0.077kB

0.85

The numerical calculations of the solid–liquid interfacial energy were performed for different sections of the solid basis layers consisting of various numbers NS of atoms (see Table 4.3). From the number of non-identical configurations Zcf , which were determined by the algorithm and the number N1 of atoms in the constructed first interfacial layer, the configurational entropy Scf (1) per atom in the first interfacial layer is calculated using Scf (1) ⫽ kB /N1 ln(Zcf), which is valid for systems that are large enough. The operation of the numerical algorithm is illustrated for the example of a section of the fivefold plane of the I-phase in Al᎐Pd–Mn as shown in Figure 4.3. In this figure open circles mark the atoms of the upper solid layer. At the beginning of the calculation all possible sites to place an atom onto the (quasi)crystal layer under the formation of tetrahedra are determined, which are plotted as squares in Figure 4.3. Only these sites are possible locations of interface atoms that fulfil the first construction rule (cf. Section 4.2.1). The aim of the further calculations is to compute all possibilities to occupy the potential locations of the interface atoms that are in accordance with the construction rules. This is drawn out by an iterative procedure. For every atomic configuration during the run of this procedure each site is in one of the three following states: (a) occupied (b) not occupied (c) not defined. Sites are in the state “occupied” if interface atoms were placed on these sites in accordance with the construction principles. In the case of a “not occupied” site in

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Figure 4.3. Section of a fivefold layer of the I-phase in Al᎐Pd–Mn used for the construction of the solid–liquid interface. The squares mark the possible sites to place an interface atom on top of the solid layer [4.43].

the given configuration of atoms, the occupation of this site is not possible without violating the construction rules. The state “not defined” characterizes all sites, the final state of which was not determined untill the present step of the algorithm. The first four steps of the iterative construction of one exemplary possibility of building the first interfacial layer are depicted in Figure 4.4. During the first step (ni ⫽ 1) of the iteration one starting configuration is defined such that the starting atom occupies one site and all other sites are in the state “not defined”. Consequently, the number Zcf of configurations in the initial step is one (Zcf (ni⫽1) ⫽ 1). Subsequently, the final states of the “not defined” sites are determined step by step. The procedure is based upon the idea that for each configuration – in order to achieve a high density of packing in the interface – every site that is newly occupied in step (ni⫹1) of the iteration must be inside a ball of the radius around a site occupied in the preceding step (ni). depends on the structure of the solid and has to be chosen such that the density is maximized.

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Figure 4.4. The first four steps of the iterative construction of the first interfacial layer on top of a section of a fivefold layer of the I-phase in Al᎐Pd–Mn [4.43, 4.45].

In detail, during step (n⫹1) of the iteration the following procedure is drawn out for all Zcf (ni) configurations (in Figure 4.4 this is shown only for one of these configurations for reasons of lucidity) stored during step ni. In case that a configuration has no “not defined” sites its calculation is completed and the configuration is stored in a result file and statistically analysed with respect to the total number Zcf of configurations and the average number (N1) of occupied sites per configuration. If, however, for instance the configuration k still has “not defined” sites, then first all “not defined” sites are determined that are inside the balls of the radius around the sites occupied during the (ni)th step. Depending on the number of undefined sites inside the balls there exist several possibilities to occupy these sites, but only some of them are in accordance with the construction rules. A newly configuration calculated in step (ni⫹1) of the iteration from the configuration k is withdrawn, if octahedral symmetry or an insufficient density is detected. Moreover, it is verified that all atoms are at least separated by a minimum distance dmin to account for the size of the atoms. Finally, all Zcfk(ni⫹1) configurations that match with the construction rules are stored in a file to undergo an identical treatment in step (ni⫹2). Owing to the fact that this procedure is executed for all (ni) configurations stored during step n, the number of configurations existing after the (ni⫹1)th step of the iteration is given by Zcf ( ni + 1) =

Zcf ( ni )

∑Z k =1

k cf

( ni + 1).

(4.15)

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Figure 4.5. One possibility to construct the first interfacial layer on top of the section of the fivefold layer of the I᎐phase in Al᎐Pd᎐Mn that is shown in Figure 4.3. The open circles mark atoms of the solid phase, the filled circles interface atoms and the crossed squares “not occupied” sites [4.43, 4.45].

For clarity it must be stressed that in the case depicted in Figure 4.4 starting from the one initial configuration (step ni ⫽ 1), in step (ni ⫽ 2) two different configurations were computed, of which only one is shown in the figure. From these two configurations, five more were determined in the third step (ni ⫽ 3), of which again only one is plotted in Figure 4.4. The iteration stops when all configurations have no more “not defined sites”. One example of such a possibility to build up the first solid–liquid interfacial layer on the section of the fivefold plane shown in Figure 4.3 is depicted in Figure 4.5. Owing to the fact that in every step all allowed possibilities of occupation and non-occupation of the sites in the neighbourhood of the sites occupied in the

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preceding step are considered, it is guaranteed that at the end of the iteration all possible configurations were found. In the example presented above, one starting configuration was defined in step (ni ⫽ 1), in which one arbitrarily chosen site was occupied by an atom. However, other possibilities of constructing the first interfacial layer exist, where this site is not occupied. Therefore, additional calculations were performed using an initial configuration with the starting site unoccupied. Both calculations together deliver all possible configurations Zcf. In addition, the average ratio N1/NS is directly inferred by counting of the number of occupied sites in the determined configurations. Unlike for fcc, hcp and bcc phases [4.11–4.13], the density of the first interfacial layer is already so low for the -, -, ⬘-, - and I-phase that a second interfacial layer cannot be constructed. The voids between the interface atoms are filled with atoms that belong already to the liquid phase since they are not directly influenced by the structure of the solid phase. Thus, the interface consists only of one layer (Ni ⫽ N1; Scf (i ) ⫽ Scf (1) ). From these results, Scf (i ) , and by use of Eq. (4.7), the dimensionless solid–liquid interfacial energy, calc, is determined for the different phases. Results from this analysis together with those from the analysis by Spaepen and Thompson [4.11–4.13] for fcc, hcp and bcc structures are summarized in Table 4.3. Obviously, the -values calculated for the polytetrahedral phases are significantly lower than those estimated for non-complex crystalline phases with fcc, hcp or bcc structures. The icosahedral quasicrystalline I-phase exhibits the lowest -value of all these phases. A small -factor, which is comparable with those of the polytetrahedral phases, is determined also for -Nd2Fe14B1. A common characteristic of all these phases is the complex crystal structure. The geometrical constraints resulting from this complex crystal structure when occupying the interface sites with atoms lead to low values of Ni/NS and consequently small -factors. Analysing the data compiled in Table 4.3, it turns out that the differences in calc are mainly a result of a different number Ni of atoms in the interface, while the variation of the estimated configurational entropy is less decisive. For the Al-based alloys the same structure dependence of as that shown in Table 4.3 was inferred from undercooling experiments (cf. Section 5.6.3). This indicates that the simple interfacial model is able to describe the fundamental physical effects that determine the energy of the solid–liquid interface. One model assumption is the idea of a polytetrahedral short-range order in the melt. Therefore, the agreement between theory and experiment further supports the hypothesis of a polytetrahedral short-range order in undercooled melts.

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4.3. DIFFUSE INTERFACE THEORY

Arising from phase transformations, a classic free-boundary problem introduces a model of phase interface with zero thickness. Within this problem, a sharp discontinuity in properties or a jump of fluxes and thermodynamic functions occurs across the interface. The sharp-interface model has been a successful description of numerous physical phenomena in various systems [4.51–4.53]. However, the sharp-interface model has difficulties in describing situations when interfacial thickness becomes comparable with the characteristic length of the considered phenomenon, and when a topology of the interface becomes complicated or multiply connected. To avoid these difficulties in the sharp-interface model, an alternative model with a finite interfacial thickness was suggested for explaining phase transformations [4.54]. Historically, the first formulation of basic principles of diffuse interfaces was given by Poisson, Maxwell, and Gibbs [4.55–4.57], who suggested to consider an interface as an area with finite thickness in which steep but smooth transition of physical properties of phases occurs. Lord Rayleigh, van der Waals, and Korteweg [4.58–4.60] applied thermodynamical principles to develop gradient theories for the interfaces with non-zero thickness. Through the last century, ideas of diffuse interface given by these authors [4.55–4.60] were refined and applied in many physical phenomena (see overviews in Refs. [4.61, 4.62]). The formalism of diffuse interface has been widely applied to the phase transformations in condensed media. Borrowing the formalism of the Landau theory of phase transitions [4.63], the first introduction of the diffuse interface to the theory of phase transformations was made by Landau and Khalatnikov [4.64], who labelled the different phases by an additional order parameter to describe anomalous sound absorption of liquid helium. In its well-known form, the formal variational approach has been established by Ginzburg and Landau for phase transitions from the normal to the superconducting phase [4.65]. On the basis of this approach, the diffuse-interface models with order parameters have been developed by Halperin et al. [4.66] for the theory of critical phenomena, and by Allen and Cahn [4.67] for antiphase domain coarsening. 4.3.1 Physical interpretation of the diffuse-interface region The diffuse-interface model has been also developed for the description of phase transformations of the first order, especially, for the solidification phenomenon. In the theory of solidification, the diffuse-interface model developed uses the introduction of the order parameter in the form of a phase-field variable [4.68–4.70]. The phase field has a constant value in phases, e.g. ⫽ 0 for unstable liquid

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phase which is transforming into the solid phase with ⫽ 1. Between these phases in the interfacial region, the phase field, , changes steeply but smoothly from 0 to 1. The molecular dynamic simulations demonstrate that the transition between a fully ordered crystal to a completely disordered liquid occurs within several molecular layers within which the thermal motion of the particles destroys the order [4.71, 4.72]. This allows to introduce the continous scalar field . The mathematical requirement of such a smooth but rapid behaviour for is a balance between two effects: an increase in energy associated with states intermediate between the initial metastable liquid phase and the final solid phase and energy cost associated with large values of the gradient of . In numerical solutions, introduction of allows one to avoid the explicit interface tracking and locate the interface at ⫽ 1/2 [4.73, 4.74]. As a particular case, the phase-field model is reduced to the sharp interface limits [4.75, 4.76] and adopts the major models of sharp interface (such as Hele–Shaw type models and classical or modified Stefan problem) itself. Consequently, the phase field is considered as an order parameter, which is introduced to describe the moving interfacial boundary between initially unstable phase and final phase. The scalar order parameter is introduced on the basis of DFT by Mikheev and Chernov [4.77].3 The interfacial region itself and its motion are depicted by a damped wave that represents the probability of finding an atom at a particular location. Figure 4.6 demonstrates schematic representation of the interfacial region within several molecular layers in which the phase-field ⫽ ∫ G ( r ) dV is given by the density wave amplitude G (cm⫺3). As the solid–liquid interface is reached from the bulk solid, the density wave smears out, disappearing in the melt when G(z) ⫽ 0 and the density wave is a constant (the latter is assumed to be not much different from the solid crystalline density, c). The density distribution ( r ) (cm⫺3) near the interface is presented as (r) = ∑ ( z − Vt )exp(iGr),

(4.16)

G

3

Density functional theory (Hohenberg and Kohn, 1964 [4.93]; Kohn and Sham, 1965 [4.94]; Callaway and March, 1984 [4.95]) provides a framework for the calculation of the electronic structure and total energy of any solid-state or atomic/molecular system. DFT focuses on the behaviour of distribution functions or probability densities in media (liquids or gases near surfaces, two-phase regions, periodic solids, and so forth) the properties of which vary in space. Of particular interest is the singlet probability distribution function, usually called the density, which gives the probability of observing a particle at a point in space. This approach to elucidating the thermodynamics of inhomogeneous fluids and solids is often called density functional theory.

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135

Figure 4.6. Diffuse interface via wave density between crystalline solid (left) and liquid (right) due to the density wave approach of Mikheev and Chernov [4.77]. (a) Atomic order in solid and disorder in liquid. Ordering process proceeds simultaneously in several future lattice planes in between. (b) Number density (z) assuming average solid crystalline density, S, is equals to that of the liquid. The density amplitude, G(z), increases with the interface motion to the right (shown by bold arrows).

where the wave amplitude G(z) decreases from 1 to 0 when coordinate z normal to the interface increases from ⫺⬁ in the solid to ⫹⬁ in the liquid. In Eq. (4.16), V is the interface growth velocity, t the time, and G are reciprocal lattice vectors over which the sum is taken. These vectors for the long-range order in solid and short-range order in the liquid are assumed to be the same. At equilibrium, V ⫽ 0, the extension of crystalline order from the solid to liquid occurs via short-range atomic interactions. At small driving force (small V ), the density profile can be reduced to the equilibrium one (for V ⫽ 0). A solid crystal surface induces melt ordering in its vicinity, “exciting” the density waves most relevant to the surface and the liquid. Since the liquid short-range order potentially includes all density waves of the solid, waves characterized preferentially by different G vectors are induced by each crystal surface. The thermal motion of atoms dynamically builds up and eventually destroys the planes at the interfaces (arrows in Figure 4.6a). The ordering atomic flux building up the crystalline planes increases the density wave amplitude, G (symbolized by bold arrows in Figure 4.6b). The density wave approach of Mikheev and Chernov to propagation of a spatially diffuse interface allows the calculation of kinetic coefficients of crystal

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growth and yields predictions for the crystalline anisotropy in the kinetic coefficients. We will consider the kinetics of interface advancing in Section 6.1. 4.3.2 Phase-field models for nucleation The first phase-field model for nucleation within the diffuse interface has been developed by Cahn [4.78] to the case of the two-dimensional nucleation on the solid–liquid interfaces. Considering the diffuse interface of a growing crystal, he formulated a condition for interface advancing by nucleation mechanism. By considering a free energy for step advancing with two-dimensional nucleation, Cahn described the transition from lateral mechanism to continuous mechanism of crystal growth (Section 6.6). Owing to the fact that the vapour–liquid and crystal–liquid interfaces have a thickness of several molecular layers [4.79–4.81], the classical nucleation theory (see Section 5.1) might be revised. Indeed, precision condensation experiments demonstrated that the measured and calculated nucleation rates differ by several orders of magnitude [4.82], and problems arise in interpreting the experiments on crystal nucleation [4.83]. Therefore, the diffuse-interface approach using the phase-field methodology has been applied to describing the nucleation phenomena in condensed media. Using the free energy functional of a Ginzburg–Landau type, Gránásy et al. [4.84, 4.85] developed the phase-field methodology for the nucleation and bulk crystallization in pure and binary systems. First, it has been shown that the phase-field model for nucleation can be a useful tool to describe quantitatively crystal nucleation in two and three dimensions. Second, quantitative agreement has been achieved with atomistic simulations (with Lennard–Jones system [4.86]) and experiments (ice–water system [4.87]) for the phase-field modelling of nucleation in a pure system. 4.3.3 Sharp interface versus diffuse interface A transfer from the diffuse interface to the sharp interface can be made by the reduction of the interfacial region with a finite thickness to the interface with zero thickness. In this case, the interface between liquid and solid is treated as a jump discontinuity and as an object of infinitesimal thickness. To derive the equation of motion for the sharp interface, let us consider a one-dimensional interface (as a smooth continuous line) in a two-dimensional solidifying system. Dividing the interface into a finite number of points, one may further assume that the point with a number “i” may move for the period t by the small normal displacement zi with the velocity Vi = zi /t .

(4.17)

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137

For the fixed positions zi, one may assume a free energy G({zi}) for such restricted interface. The probability for this configuration is

(

)

P { zi } =

exp( −G/k BT )

∑( ( i

)

−G { zi } k BT

)

,

(4.18)

where the sum iexp(⫺G({zi})/kBT) is taken over the points i and plays a role of position function. The probability flux JP for the conserved total probability P may be considered as a difference between a drift ViP and the constitutive relation Di P/ zi: JP ⫽ ViP⫺Di P/ zi (where Di is a diffusion constant in i-space). Assuming local equilibrium within the spatial scale i along the interface (i.e., with i→⫾1), JP ⫽ 0, one obtains ViP ⫽ Di P/ zi. Then, with Eqs. (4.17) and (4.18), we arrive at zi/ t ⫽⫺(Di/kBT) G/ zi. Therefore, in the continuum limit i → 0, one gets z (u )

G = − M z (u) , t

z (u )

(4.19)

where u denotes a position on the interface, z(u) is the normal displacement, / z the variational derivative, and Mz(u) the mobility of the sharp interface dependent on the position u. Note that G is considered in this equation represents the Gibbs free energy per unit surface element. Equation (4.19) is the time-dependent Ginzburg–Landau equation [4.88–4.90] that has been successfully applied to the solidification problems. In particular, a motion of the interface with the kinetic transition “order–disorder” has been described using Eq. (4.19) by Brener and Temkin [4.91]. It is simple to show that from Eq. (4.19) one may obtain a local equation of the interface advancing. Writing the Gibbs free energy as the sum of the contributions from the liquid and solid phases plus contribution from the free energy of the interface itself, one can get an explicit local equation. This yields [4.89] d 2 LS ⎞ ⎪⎫ 1⎛ z ⎪⎧ ≡ Vn = M z () ⎨GLS (u ) − ⎜ LS + ⎬. t R⎝ d 2 ⎟⎠ ⎭⎪ ⎩⎪

(4.20)

Equation (4.20) has been derived from initially postulating an existence probability (4.18) of the sharp interface having some restricted configuration and free energy G in two dimensions. Consequently, this equation is applicable to the motion of the interface in two-dimensional space with normal velocity Vn, anisotropic kinetic coefficient dependent on local orientation (given by angle of orientation between the normal vector to the interface and the considered crystallographic

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axis), surface free energy LS dependent on local orientation of the interface, jump GLS = GL − GS of free energy density between liquid “L” and solid “S ” phases at position u along the interface,4 and the curvature given by ⫺R⫺1 ⫽ d/du. At equilibrium, i.e. with Vn ⫽ 0, for the finite radius Rc of curvature one gets GLS = Rc −1 LS + d 2 LS d 2 . (4.21)

(

)

Equation (4.21) determines the crystal shape accordingly to the Wulff rule [4.92]. With the finite velocity Vn, the interface can move, e.g., at (a) large driving force GLS and finitebut small mobility Mz, or (b) large mobility Mz and very small driving force GLS . REFERENCES

[4.1] Nelson, D.R., and Spaepen, F. (1989) in: Solid State Phys., Vol. 42, eds. Ehrenreich, H., Seitz, F., and Turnbull, D. (Academic Press, New York), p. 1. [4.2] Janot, C. (1992) Quasicrystals a Primer (Oxford University Press, Oxford). [4.3] Kelton, K.F. (1993) International Materials Review 38, 105. [4.4] Kelton, K.F. (1994) in: Intermetallic Compounds, 1, eds. Westbrook, J.H., and Fleischer, R.L. (Wiley, Chichester). [4.5] Shechtman, D., Blech, I., Gratias, D., and Cahn, J.W. (1984) Physical Review Letters 53, 1951. [4.6] Frank, F.C., and Kasper, J.S. (1958) Acta Crystallographica 11, 184. [4.7] Gibbs, J.W. (1932) The Collected Work of J. Willard Gibbs, Vol. I, Thermodynamics (Longmans and Green, London). [4.8] Skapski, A.S. (1956) Acta Metallurgica 4, 576. [4.9] Ewing, R.H. (1972) Philosophical Magazine 25, 779; (1971) Journal of Crystal Growth 11, 221. [4.10] Miedema, A.R., and den Broeder, F.J.A. (1979) Zeitschrift für Metallkunde 70, 14. [4.11] Spaepen, F. (1975) Acta Metallurgica 23, 729. [4.12] Spaepen, F., and Meyer, R.B. (1976) Scipta Metallurgica 10, 257. [4.13] Thompson, C.V. (1979) Ph.D. Thesis, Harvard University, USA. [4.14] Chen, H.S., and Chuang, S.Y. (1974) Physica Status Solidi (A) 25, 581. [4.15] Turnbull, D. (1964) in: Physics of Non-Crystalline Solids, ed. Prins, J.A. (North-Holland, Amsterdam), p. 41. [4.16] Frank, F.C. (1952) Proeedings of the Royal Society London A 215, 43. [4.17] Elser, V. (1984) Journal of Physics A 17, 1509. [4.18] Mermin, N.D. (1965) Physical Review 137, A1441. [4.19] Ramakrishnan, T.V., and Youssouff, M. (1979) Physical Review B 19, 2775.

Note that the Gibbs free energy density GLS is considered for the infinite liquid or solid without influence from the interface curvature.

4

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[4.20] Haymet, A.D.J., and Oxtoby, D.W. (1981) Journal of Chemical Physics 74, 2559; (1982) 76, 6262. [4.21] Curtin, W.A. (1989) Physical Review B 39, 6775. [4.22] McMullen, W.E., and Oxtoby, D.W. (1988) Journal of Chemical Physics 88, 1967. [4.23] Marr, D.W., and Gast, A.P. (1993) Journal of Chemical Physics 99, 2024. [4.24] Ohnesorge, R., Löwen, H., and Wagner, H. (1986) Physical Review E 80, 4801. [4.25] Percus, J.K., and Yevick, G.J. (1958) Physical Review 110, 1. [4.26] Schenk, T. (2002) Ph.D. Thesis, Ruhr-University Bochum, Germany (in German). [4.27] Simonet, V., Hippert, F., Klein, H., Audier, M., Bellissent, R., Fischer, H., Murani, A.P., and Boursier, D. (1998) Physical Review B 58, 6273. [4.28] Simonet, V. (1998) Ph.D. Thesis, Université de Paris-Sud U.F.R Scientifique d’Orsay, France (in French). [4.29] Broughton, J.Q., and Gilmer, G.H. (1986) Journal of Physical Chemistry 84, 5749 and 5759. [4.30] Davidchack, R.L., and Laird, B.B. (2000) Physical Review Letters 85, 4751. [4.31] Hoyt, J.J., Asta, M., and Karma, A. (2001) Physical Review Letters 86, 5530. [4.32] Foiles, S.M., Baskes, M.I., and Daw, M.S. (1986) Physical Review B 33, 7983. [4.33] Lide, D.R. (ed.) (1996) CRC Handbook of Chemistry and Physics, 77th Edition (CRC Press, Boca Raton, FC). [4.34] Shiraishi, S.Y., and Ward, R.G. (1964) Canadian Metallurgy Q 3, 117. [4.35] Smith, C.S. (1948) Transactions of AIME 175, 15. [4.36] Nash, G.E., and Glicksman, M.E. (1971) Philosophical Magazine 24, 577. [4.37] Samples, J.R. (1971) Proceedings of Royal Society London A 324, 339. [4.38] Gündüz M., and Hunt, J.D. (1985) Acta Metallurgica 33, 1651. [4.39] Maraşli, N., and Hunt, J.D. (1996) Acta Metallurgica 44, 1085. [4.40] Gündüz, M., and Hunt, J.D. (1989) Acta Metallurgica 37, 1839. [4.41] Mondolfo, L.F., Parisi, N.L., and Kardys, G.J. (1980) Materials Science Engineering 68, 249. [4.42] Marr, D.W., and Gast, A.P. (1994) Langmuir 10, 1348. [4.43] Holland-Moritz, D. (1998) International Journal of Non-Equilibrium Processing 11, 169. [4.44] Holland-Moritz, D. (1999) in: Proceedings of the 6th International Conference on Quasicrystals, Tokyo, 1997, eds. Takeuchi, S., and Fujiwara, T. (World Scientific, Singapore), pp. 293–296. [4.45] Holland-Moritz, D. (1999) Journal of Non-Crystalline Solids 250–252, 839. [4.46] Holland-Moritz, D. (2000) in: Materials Research Society Symposium Proceedings 580, eds. Gonis, A., Turchi, P.E.A., and Ardell, A.J. (Materials Research Society, Warrendale, PA), pp. 245–250. [4.47] Holland-Moritz, D., Schenk, T., Simonet, V., Bellissent, R., Convert, P., and Hansen, T. (2002) Journal of Alloys Compounds 342, 77.

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[4.48] Schenk, T., Simonet, V., Holland-Moritz, D., Bellissent, R., Hansen, T., Convert, P., and Herlach, D.M. (2004) Europhysics Letters 65, 34. [4.49] Egry, I., Jacobs, G., and Holland-Moritz, D. (1999) Journal of NonCrystalline Solids 250252, 820. [4.50] Holland-Moritz, D., Jacobs, G., and Egry, I. (2000) Materials Science & Engineering A 294–296, 369. [4.51] Ockendon, J.R., and Hodgkins, W.R. (eds.) (1975) Moving Boundary Problems in Heat Flow and Diffusion (Oxford University Press, Oxford). [4.52] Wilson, D.G., Solomon, A.D., and Boggs P.T. (eds.) (1978) Moving Boundary Problems (Academic Press, New York). [4.53] Friedman, A. (1982) Variational Principles and Free-Boundary Problems, (Wiley, New York). [4.54] Caginalp, G. (1986) Archives for Rational Mechanics and Analysis 92, 205. [4.55] Poisson, S.D. (1831) Nouvelle Thèorie de l’Action Capillaire (Bachelier, Paris). [4.56] Maxwell, J.C. (1876) Capillary action, in: The Scientific Papers of James Clerk Maxwell, Vol. 2 (Dover, New York, 1952), p. 541. [4.57] Gibbs, J.W. (1876) Transactions of the Connecticut Academy 3, 108. [4.58] Rayleigh L. (1892) Philosophical Magazine 33, 209. [4.59] Waals van der, J.D. (1979) Journal Statistical Physics 20, 179; translation from the original work of 1893. [4.60] Korteweg, D.J. (1901) Archive Nèel. Science Exactes Nature Series II 6, 1. [4.61] Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford). [4.62] Rowlinson, J.S., and Widom, B. (1989) Molecular Theory of Capillarity (Clarendon, Oxford). [4.63] Landau, L.D. (1937) JETP 7, 19; see also Haarter, D. (ed.) (1965) Collected Papers of L.D. Landau (Pergamon Press, Oxford), p. 193. [4.64] Landau, L.D., and Khalatnikov, I.M. (1954) Doklady Akademii Nauk SSSR 96, 469; see also ter Haar, D. (ed.) (1965) Collected Papers of L.D. Landau (Pergamon Press, Oxford), p. 626. [4.65] Ginzburg, V.L., and Landau, L.D. (1950) JETP 20, 1064; see also Haarter, D. (ed.) (1965) Collected Papers of L.D. Landau (Pergamon Press, Oxford), p. 546. [4.66] Halperin, B.I., Hohenberg, P.C., and Ma, S.-K. (1974) Physical Review B 10, 139. [4.67] Allen, S.E., and Cahn, J.W. (1979) Acta Metallurgica 27, 1085. [4.68] Fix, G.J. (1983) in: Free Boundary Problems: Theory and Applications, eds. Fasano, A., and Primicerio, M. (Pitman, Boston), p. 580. [4.69] Collins, J.B., and Levine, H. (1985) Physical Review B 31, 6119. [4.70] Langer, J.S. (1986) in: Directions in Condensed Matter Physics, eds. Grinstein, G., and Mazenko, G. (World Scientific, Philadelphia, PA), p. 165. [4.71] Bonissent, A. (1983) in: Crystals. Growth, Properties and Applications, eds. Chernov, A.A. and Müller-Krumbhaar, H. (Springer, Berlin), p. 1. [4.72] Eerden van der, J.P. (1995) in: Science and Technology of Crystal Growth, eds. van der Eerden, J.P., and Bruinsma, O.S.L. (Kluwer, Dordrecht), p. 15.

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[4.73] Chen, L.Q. (2002) Annual Review Materials Research 32, 113. [4.74] Boettinger, W.J., Warren, J.A., Beckermann, C., and Karma, A. (2002) Annual Review of Materials Research 32, 163. [4.75] Caginalp, G. (1989) Physical Review A 39, 5887. [4.76] Caginalp, G., and Socolovsky, E.A. (1991) Journal of Computational Physics 95, 85. [4.77] Mikheev, L.V., and Chernov, A.A. (1991) Journal of Crystal Growth 112, 591. [4.78] Cahn, J.W. (1960) Acta Metallurgica 8, 554. [4.79] Braslau, A., Pershan, P.S., Swislow, G., Ocko, B.M., and Als-Nielsen, J. (1988) Physical Review A 38, 2457. [4.80] Lairf, B.B., and Haymet, A.D.J. (1992) Chemical Review 92, 1819. [4.81] Nyquist, R.M., Talanquer, V., and Oxtoby, D.W. (1995) Journal of Chemical Physics 99, 2865. [4.82] Schmitt, J.L., Zalabski, R.A., and Adams G.W. (1983) Journal of Chemical Physics 79, 4469. [4.83] Laaksonen, A., Talanquer, V., and Oxtoby, D.W. (1995) Annual Review Physical Chemistry 46, 489. [4.84] Gránásy, L., Pusztai, T., and Hartmann, E. (1996) Journal of Crystal Growth 167, 756. [4.85] Gránásy, L., Börzsö, T., and Pusztai, T. (2002) Journal of Crystal Growth 237–239, 1813. [4.86] Báez, L.A., and Clancy, P.J. (1995) Journal of Chemicals Physics 102, 8138. [4.87] Butorin, G.T., and Skripov, V.P. (1972) Kristallographiya 17, 379 [(1972) Soviet Physics, Crystallography 17, 322]. [4.88] Mettiu, H., Kitahara, K., and Ross, J. (1976) Journal of Chemical Physics 65, 393. [4.89] Müller-Krumbhaar, H., Burkhardt, T., and Kroll, D. (1977) Journal of Crystal Growth 38, 13. [4.90] Lifshitz, E.M., and Pitaevskii, L.P. (1981) Physical Kinetics. Course Theoretical Physics, Vol 10, (Pergamon Press, Oxford, UK). [4.91] Brener, E., and Temkin, D. (1983) Kristallografiya 28(1), 18; ibidem 28(2), 242. [4.92] Wulff, G. (1901) Zeitschrift für Kristallographie und Mineralogie 34, 449. [4.93] Hohenberg, P., and Kohn, W. (1964) Physical Review B 136, 864. [4.94] Kohn, W., and Sham, L.J. (1965) Physical Review A 140, 1133. [4.95] Callaway, J., and March, N.H. (1984) Solid State Physics 38, 135.


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Chapter 5

Nucleation 5.1.

Nucleation Theories 5.1.1 Homogeneous Nucleation 5.1.2 Heterogeneous Nucleation 5.1.3 Diffuse Interface Theory of Nucleation 5.2. Transient Nucleation 5.3. Statistics of Nucleation 5.4. Nucleation in Alloys 5.5. Magnetic Contributions to Crystal Nucleation 5.5.1 The Magnetic Contribution to the Driving Force for Crystal Nucleation 5.5.2 The Magnetic Contribution to the Solid–Liquid Interfacial Energy 5.6. Experimental Results on Undercooling and Nucleation 5.6.1 Homogeneous Versus Heterogeneous Nucleation 5.6.2 Nucleation in Undercooled Melts 5.6.3 Structural Dependence of Nucleation Behaviour 5.6.4 Undercooling of Magnetic Melts References

145 145 152 154 160 161 163 165 165 167 169 169 170 172 180 189


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Chapter 5

Nucleation 5.1. NUCLEATION THEORIES

5.1.1 Homogeneous nucleation The first attempts to describe nucleation processes are due to Volmer and Weber [5.1], who treat the condensation of a supersaturated vapour. This model has been later on improved and extended by Becker and Döring [5.2]. Fisher and Turnbull applied the basic concepts of this theory to the phase transition liquid – solid [5.3]. Consider a liquid with its essentially random motion of atoms, driven by temperature. At any time atoms will approach each other statistically to distances comparable to the interatomic spacing in the solid, forming solid-like clusters. At temperatures above the melting temperature the clusters will decay spontaneously. At the melting temperature, the free energy of the liquid is equal to the free energy of the solid. As the liquid is undercooled below the melting temperature the free energy of the solid becomes smaller than for the liquid. Hence, the formation of a solid-like cluster implies an energy gain by the transformation of a volume fraction of the system from the liquid undercooled state of higher energy to the solid state of lower energy. On the other hand, the formation of a solid-like cluster also means that energy is needed to build up an interface between solid and liquid. Assuming for simplicity a sphere like geometry of the clusters, the energy balance during the formation of a cluster of radius r can be written as the sum of two contributions 4 (5.1) G ( r ) = − r 3GLS + 4 r 2 LS , 3 with LS ⬎ 0 the interface energy and GLS (T ⬍ TE) ⬎ 0 the Gibbs free energy difference per unit volume (cf. Section 3.1). Since the dependence of the volume contribution (~r3) on the radius differs from that of the interface contribution (~r2), G(r) will pass through a maximum. The function according to Eq. (5.1) is plotted in Figure 5.1 showing separately the volume and the interface contributions. G exhibits a maximum with the height G* at a critical radius r*, which forms an activation barrier against crystallization and therefore is crucial for the undercooling behaviour of a melt. For clusters of size smaller than r* the interface-to-volume ratio is large, so the interface contribution dominates. Such a cluster is called an embryo; it is unstable and decays spontaneously. On the other hand, a cluster in size larger than r*, termed a nucleus, can lower its free energy by 145

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Figure 5.1. The energy balance G(r) during the formation of a solid-like cluster in the undercooled melt as a function of the cluster radius r. G* is the activation energy for the formation of a nucleus of critical size r*.

growing and becomes stable at r0 = 1.5r*. Once it has grown beyond r0, it represents the stable form of the system, and will initialize crystallization. The critical nucleus size r* is obtained by differentiation of Eq. (5.1) with respect to r: r* =

LS . GLS

(5.2)

The activation energy to form a nucleus of critical size r* is calculated by substituting Eq. (5.2) into Eq. (5.1): G * =

163LS 2 . 3GLS

(5.3)

Equations (5.2) and (5.3) can be used to estimate the critical nucleus size r* and the activation energy G* as a function of undercooling. As an example, applying the linear approximation for the Gibbs free energy difference GLS (Eq. (3.4)), and using numerical values of LS = 0.45 J/m2 and Hf = 2.9⫻105 J/kg a critical nucleus size of r* 2.2 nm and an activation energy G* 58 eV are obtained for a Ni melt undercooled by 300 K. The high activation energy for the formation of critical nuclei explains why metallic melts can be undercooled by substantial amounts. On the other hand, the estimation of the critical nucleus size indicates that the critical nucleus is very small containing only a few hundred atoms. This also means that the critical nucleus shows a very pronounced curvature, which may influence the interfacial energy (cf. Section 5.1.3).

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Below the melting temperature the liquid is in a metastable state. The phase transformation implies indeed an energy gain, but to initiate the phase transformation energy is needed to form critical nuclei. Volmer and Weber assume that the number N n of clusters containing n atoms per unit volume, is given by the Boltzmann statistics ⎛ G ( n) ⎞ N n = N exp ⎜ − . ⎝ k BT ⎟⎠

(5.4)

N = N L /vm is the total number of atoms in the undercooled melt per unit volume, with NL denoting Avogadro’s number and m the molar volume. G(n) is the energy required forming a cluster containing n atoms. kB denotes the Boltzmann constant. N n continuously decreases with increasing cluster size n. This means that the probability to find small clusters is higher than the probability to find large clusters. However, because G(n) decreases at large clusters sizes n ⬎ n* (or r ⬎ r*) and even becomes negative at n ⬎ n0 (or r ⬎ r0) (cf. Figure 5.1), Eq. (5.4) suggests a steep increase in N n with growing cluster size in the regime n ⬎ n* (see thin line in Figure 5.2). This is not reasonable. The resulting contradiction is circumvented by the postulation that all critical clusters n ⬎ n* become critical nuclei which are taken out of the ensemble of the clusters. This is equivalent to a cluster distribution function, which is interrupted at the critical cluster size n* (cf. Figure 5.2). However, to keep the stationary distribution of the cluster all atoms removed by the critical clusters have to be replaced by an equivalent number of atoms, which are continuously added to the system (quasi-stationary cluster distribution).

Figure 5.2. Cluster distribution function N n as a function of the cluster size n, according to the Volmer–Weber [5.1] and the Becker–Döring theory [5.2].

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The Volmer–Weber theory further assumes that a cluster as soon as it becomes critical in size grows to a nucleus by the addition of atoms from the melt. In reality, however, there is a probability that even a postcritical cluster can shrink back. This is understandable by the fact that a critical cluster is in an unstable state (cf. Figure 5.1). Becker and Döring proposed a cluster distribution function as shown in Figure 5.2. The particle conservation law requires that the integral over the cluster distribution function must converge so that it indeed exceeds n* but vanishes at large cluster sizes. This means that the shaded areas in Figure 5.2 must be equal. Becker and Döring postulated that the nucleation frequency Iss corresponds to the rate at which clusters larger than the critical size n* are growing. This must be equal to the difference of the two rates at which clusters containing n* atoms grow into supercritical clusters containing n* + 1 atoms, and the inverse rate corresponding to the reversion of aggregates containing n* + 1 atoms into those containing n* atoms. Thus, the steady-state nucleation frequency can be expressed by the following equation: I (t ) = K n+* N n* (t ) − K n−* +1 N n* +1 (t ). (5.5) N n* and N n* +1 denote the number of clusters containing n* and n* + 1 atoms, respectively. K n+* is the conversion rate for clusters of size n* to size n*+1 and K n−* +1 the corresponding opposite rate for the conversion from size n*+1 to n*. Under the assumption that the number of atoms organized in clusters is small in comparison with the total number of atoms, Becker and Döring obtain ⎛ G * ⎞ I ss = K n+* N z exp ⎜ − . ⎝ k BT ⎟⎠

(5.6)

The only difference from the Volmer–Weber theory comes through the Zeldovich factor ⎛ G * ⎞ z = ⎜ *2 ⎝ 3k Tn ⎟⎠ B

1/ 2

1/ 2 ⎛ GLS ⎞ = ⎜ , * ⎝ 6k Tn ⎟⎠

(5.7)

B

which takes into account the postcritical clusters n ⬎ n*. Analytical [5.4] and numerical [5.5] studies indicate that the Zeldovich factor varies between 0.01 and 0.1. The transition of an atom from the liquid state to the solid nucleus needs atomic diffusion [5.6]. Turnbull and Fisher [5.3] treated this problem on the basis of a concept shown in Figure 5.3. The liquid and the solid nucleus states are represented by a double-wall potential. The barrier corresponds to the activation energy for interatomic diffusion, Ga, in the liquid state. The asymmetry of both potentials is

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Figure 5.3. Double-wall potential for the interpretation of diffusion-controlled addition of single atoms to the nucleus.

given by the Gibbs free energy difference GLS. In this model, the atomic attachment is a thermally activated process such that the attachment rate can be described by a Boltzmann ansatz ⎛ Ga ⎞ K n+* = 4 n*2 / 3 f v exp ⎜ − . ⎝ k BT ⎟⎠

(5.8)

Here f denotes the vibration frequency of the atoms in the liquid state, for which f = kBT/h (h: Planck’s constant) is assumed. The factor 4n*2/3 considers that the attachment ozf further atoms to the critical nucleus is only possible at its surface. Then the nucleation rate is given by ⎛ Ga ⎞ ⎛ G * ⎞ I ss (T ) = 4 n*2 / 3 Nf v z exp ⎜ − exp ⎜ − . ⎟ ⎝ k BT ⎠ ⎝ k BT ⎟⎠

(5.9)

According to Eq. (5.9) the nucleation rate Iss is governed by two exponential terms. At high temperatures in the vicinity of the melting temperature (i.e. at small undercoolings) the second exponential term containing the activation energy for the formation of critical nuclei dominates. At low temperatures, in particular in the vicinity of the glass temperature (i.e. at high undercoolings) the first exponential term regarding the atomic diffusion process controls the nucleation frequency. If in first approximation a temperature-independent interfacial energy and activation energy G are assumed, and the linear approximation (3.4) of the Gibbs free

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Metastable Solids from Undercooled Melts ΔHa =0 ΔG* log Iss

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ΔHa ΔG*

=1

ΔHa ΔG*

Tg =

1 T 4 E

1 T 3 E

= 2.2

1 2 T T 2 E 3 E

TE T

Figure 5.4. The temperature dependence of the nucleation rate Iss with and without taking into account thermally activated diffusion. G* is the activation energy for nucleation, and Ha is the enthalpic contribution to the activation energy for atomic diffusion.

energy is utilized, one obtains for the temperature dependence of the nucleation rate the curves shown schematically in Figure 5.4. It is interesting to note that even if the temperature dependence of the diffusion term is neglected the nucleation frequency passes through a maximum which is located at a temperature of T = TE /3. With increasing influence of the temperature dependence of the diffusion term, the maximum in the nucleation rate versus temperature relation is shifted to higher temperatures, while the maximum nucleation frequency decreases. The growing influence of the diffusion is demonstrated in Figure 5.4 by the increasing enthalpic contribution, Ha, to the activation energy for atomic diffusion. For a quantitative estimate of nucleation rate, it is often assumed that Ga corresponds to the activation energy for atomic self-diffusion. The diffusion coefficient D is correlated with the viscosity of the melt, , via the Einstein–Stokes relation

D=

k BT 1 = da2 f j , 3 da (T ) 6

(5.10)

with da a typical atomic diameter and fj a characteristic atomic jump frequency: ⎛ Ga ⎞ f j = f o exp ⎜ − . ⎝ k BT ⎟⎠

(5.11)

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The prefactor fo denotes the vibration frequency of the atoms and is of the order of magnitude of 1012–1013 Hz comparable to the Debye frequency. The Einstein– Stokes relationship correlates with the macroscopic magnitude of the viscosity, , with the microscopic magnitude of the atomic diffusion coefficient, D. Equation (5.11) is physically interpreted in the framework of the free volume model [5.7]. In particular, it is supposed that both the shear viscosity and the atomic diffusion are governed by the same relaxation processes of free volume, as presented by microscopic hollow regions, the so-called voids [5.8]. This consideration leads to the conclusion that the validity of the Einstein–Stokes relation can be assumed in the first approximation for liquid systems [5.9]. However, this is not longer true for frozen-in liquids, the metallic glasses. Comparative studies of both the viscosity and the atomic diffusion demonstrate that for each local density fluctuation in the microscopic regions more atomic jumps arise, which cause atomic diffusion in comparison with those, which cause a viscous flow of the metallic glassy state [5.10]. Diffusion and viscous flow are thermally activated processes, which become important at high temperatures. It is, however, interesting to note that local atomic rearrangements in “open” structures of amorphous solids are not limited by atomic diffusion throughout, but can even occur via atomic tunnelling [5.11]. Such tunnelling processes are observed at low temperatures – if atomic diffusion is frozen-in – as low-energy excitations. They become apparent in the specific heat [5.12] as well as in the thermal conductivity as scattering centres for thermal phonons [5.13]. The density of the lowenergy excitations scales with the amount of frozen-in free volume in metallic glasses [5.14]. Assuming the validity of the Einstein–Stokes relation for undercooled melts, the expression for the nucleation frequency can be rewritten by replacing the diffusion term by the viscosity, as already pointed out by Becker [5.6]. The combination of Eqs. (5.9) – (5.11) yields ⎛ G * ⎞ ⎛ G * ⎞ 8n*2 / 3 k B T N Γ z hom I SS (T ) = exp = exp − K V ⎜⎝ − k T ⎟⎠ . ⎜⎝ k T ⎟⎠ da3 (T ) B B

(5.12)

The prefactor hom V

K

8n*2 / 3 k B T N Γ z = = (T ) da3 (T ) K

(5.13)

in Eq. (5.12) mainly depends on the viscosity (T). Turnbull [5.15] estimated that the factor K is in the order of 1036 N m⫺5. Its temperature dependence is negligible as

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compared with the exponential term. This gives an approximation for the nucleation frequency I ss (T )

⎛ G * ⎞ −3 −1 1036 m s . exp ⎜ − (T ) ⎝ k BT ⎟⎠

(5.14)

Since the viscosity changes by about 15 orders of magnitude in the temperature range between the glass – and the melting temperature a quantitative estimation of the viscosity will be crucial for a quantitative determination of the nucleation frequency and its temperature dependence. For the calculation of the nucleation rate the activation energy G* to form critical nuclei has to be considered. G* depends on the interfacial energy LS and the free energy difference GLS. The solid–liquid interfacial energy is treated in Chapter 4. As discussed in Chapter 2, GLS can readily be determined if data of the specific heat in the undercooled melt regime are available. 5.1.2 Heterogeneous nucleation In the case of homogeneous nucleation, the activation energy to form a critical nucleus depends on the interface energy LS and the Gibbs free energy difference GLS. Thus, the nucleation threshold depends exclusively on the thermodynamic properties of the respective material. Homogeneous nucleation is, therefore, an intrinsic process. In practise, heterogeneous nucleation plays the more dominant role. Here, also foreign phases such as container walls, metal oxides at the surface of the melt and even “motes” in the volume of the melt participate in the nucleation process. Heterogeneous nucleation is consequently an extrinsic process and can be influenced by the experimental conditions. Volmer was the first who described the formation of a heterogeneous nucleus on a substrate [5.16]. According to Figure 5.5, the equilibrium of the interfacial tensions is given by LS = CS + LC cos(ϑ ) .

(5.15)

The indices C, S, L denote crystal nucleus, substrate and undercooled melt. It is also obvious from Figure 5.5 that the volume of a heterogeneous nucleus of critical size r* is smaller than that of a homogeneous nucleus of the same critical radius. The reduction of the volume depends on the wetting angle ϑ, and is given by 1 f (ϑ ) = ( 2 − 3 cos ϑ + cos 3ϑ ) . 4

(5.16)

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Figure 5.5. Growth of a heterogeneous nucleus onto a substrate. The nucleus cap is stabilized through the equilibrium of the interfacial tensions between substrate, crystal nucleus, and undercooled melt. ϑ is the contact angle.

The function f(ϑ) is smaller than unity for every contact angle ϑ ⬍ 180⬚. Correspondingly, the activation energy to form a critical heterogeneous nucleus is reduced by the factor f(ϑ): * * Ghet = Ghom f (ϑ ),

0 ≤ f (ϑ ) ≤ 1.

(5.17)

According to Eq. (5.17), the wetting behaviour of the melt on the substrate influ* ences the activation energy Ghet , and thus the undercooling. The lower the wetting angle the lower will be the activation threshold. The special case ϑ = 180⬚ means complete non-wetting. Here, the substrate will not influence the nucleation * * barrier at all so that f(ϑ)=1 and Ghet . The other extreme case f(ϑ)=0 = Ghom means complete wetting so that the activation energy is zero. Here, the melt will not undercool, and will solidify near the equilibrium melting point T = TE by epitaxial growth on the substrate. Complete wetting will be observable in the particular case if small hollow regions or cracks at the surface of a container are present. The solid phase of the sample can be stabilized at such a hollow region at temperatures even above the melting temperature [5.17]. During the cooling of a melt such an inclusion of solid phase can act as an ideal heterogeneous nucleation site. Similar conditions are given, e.g. in laser surface resolidification experiments. Also in this example, crystallization occurs by epitaxial growth on the substrate without stationary undercooling and nucleation. The formalism for the derivation of the homogeneous nucleation rate can also be applied to the case of heterogeneous nucleation. The only differences to homogeneous nucleation are the changed activation energy G* and a reduced number of atoms that may act as starting points for nucleation. In the case of homogeneous nucleation, nucleation can start at each atom. Nucleation is limited in the case of heterogeneousnucleation on atoms that are located at the interface to the substrate. The number N of these atoms per unit volume is reduced by a factor ⬍1 as

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compared with the number N of all atoms per unit volume: N = N Thus, the expression for the heterogeneous nucleation rate reads I

het ss

(T ) =

8n*2 / 3 k B T N z da3 (T )

* ⎛ Ghet ⎞ exp ⎜ − ⎝ k BT ⎟⎠

⎛ G * f (ϑ ) ⎞ = K Vhom exp ⎜ − k BT ⎟⎠ ⎝

(5.18)

⎛ G ∗ f (ϑ ) ⎞ = KV exp ⎜ − . k BT ⎟⎠ ⎝ For a model system of a reduced glass temperature Trg = Tg/TE = 0.6 and N = 10⫺6 NL, Eq. (5.18) predicts a change in the maximum nucleation rate by more than 20 orders of magnitude, if the wetting factor f (ϑ) changes from 1 to 0.2. This example shows the drastic influence of heterogeneous nucleation on the nucleation rate. 5.1.3 Diffuse interface theory of nucleation In the most simple approaches to describe the nucleation of solid phases in undercooled melts, it is assumed that the interfacial energy is independent of the temperature and that its constant value corresponds to that at the melting point. It was already pointed out by Turnbull [5.18] that this assumption leads to unrealistically high pre-exponential factors KV in the nucleation rate equation (5.18), if the results of undercooling experiments on Hg are evaluated. Hence, it must be assumed that the solid–liquid interfacial energy LS is temperature-dependent [5.18, 5.19]. Owing to the fact that in the model by Spaepen and Thompson LS is purely of entropic origin, a linear temperature dependence of the energy of the solid–liquid interface per surface unit, which is based on Eq. (4.9), is suggested [5.20]: LS =

S f T 1/3

( N L vm2 )

.

(5.19)

The modelling of the solid–liquid interface in the framework of the negentropic model, which has been described in Sections 4.2.1 and 4.2.4, but also the molecular dynamics computer simulations [5.21] and the models based on density functional theory [5.22–5.25] have exhibited that the solid–liquid interface is not absolutely sharp, but has a finite thickness which lies in the order of atomic distances (Figure 5.6).

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Figure 5.6. Schematic plot of a spherical solid cluster in an undercooled melt, which has a diffuse interface of thickness .

If the interface is curved and if is not negligible as compared with the radius of curvature, the curvature of the interface together with the finite interface thickness must be considered by applying diffuse interface approaches (cf. Section 4.3) when calculating LS [5.19, 5.26, 5.27]. Equation (5.19), however, is strictly only valid for a planar interface. This is the case at the melting point TE,1 at which the solid phase is in equilibrium with the liquid, but not for the interface between a nucleus of a solid phase and an undercooled melt. Nevertheless, as discussed below, for moderate undercoolings and consequently small deviations from the equilibrium, Eq. (5.19) gives an acceptable description of the temperature dependence of the interfacial energy. Therefore, and due to the fact that, except all parameters of Eq. (5.19) are experimentally accessible, this simple equation is frequently employed for the analysis of undercooling experiments. In many cases, the error resulting from the using Eq. (5.19) is of the same order of magnitude as the error resulting from using other approximations and from measurement errors. For a more precise estimation of the solid–liquid interfacial energy under nonequilibrium conditions, the curvature of the nucleus together with the finite thickness of the interface must considered [5.19, 5.27–5.29]; this will be the topic of the following discussion. The radius of the core of a spherical cluster of the solid phase within the undercooled melt, which has the structure of the bulk solid, is denoted by rs. This solid core of the cluster is surrounded by the interfacial region of thickness , such that 1

The radius of a critical nucleus converges against infinity, if the temperature converges from below against TE.

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the whole cluster together with the interface has radius rl = rs + (Figure 5.6). Outside of the sphere of radius rl all atoms belong to the liquid phase. Between rs and rl, the enthalpy H(r) and the entropy S(r) change gradually from the values in the bulk solid phase (HS and SS) at r ⱕ rs to those of the liquid phase (HL and SL) and r ⱖ rl. Different from the assumptions used when deriving Eq. (5.3), in real systems the transition from the liquid to the solid phase is not sharp but occurs in the spatial interval between rs and rl. The formation of a cluster with the solid radius rs and the corresponding interface in the undercooled melt results in a change of the Gibbs free energy: ⬁

4 r 2 G ( rs ) = ∫ v m (G ( r ) − GL ) dr.

(5.20)

0

G(r) = H(r)⫺TS(r) denotes the Gibbs free energy of the system as a function of the radius r, with G(r) = GS for r ⬍ rs and G(r) = GL for r ⬎ rl. The radius rs* of the solid core of a critical nucleus corresponds to that radius rs, for which G (rs) is maximum. Hence, the activation energy for nucleation is given by G * = G (r*s ). If G * is equated with expression (5.3) derived for a sharp ( → 0) interface, a definition of the solid–liquid interfacial energy LS of a curved, diffuse interface is obtained: LS

1/ 3 2 ⎛ 3 *⎞ =⎜ GLS G ⎟ . ⎝ 16 ⎠

(5.21)

A frequently used estimation of the effect of the curvature of the nucleus on the solid–liquid interfacial energy was derived by Tolman [5.26]: R

LS . 1 + 2 T /Ri

(5.22)

In this formula R denotes the energy of a curved solid–liquid interface with a radius of curvature of Ri and LS the solid–liquid interfacial energy of a planar interface, which was discussed in Section 4.2. T is called Tolman length and may be interpreted as thickness of the interface. Gránásy [5.30] determined the Tolman length for the nucleation processes occurring during condensation of nonan in the framework of a semiempirical model based on the van der Waals/Cahn–Hilliard theory. This modelling predicts a pronounced temperature dependence of the Tolman length, which can even be negative. In a simple model, Spaepen [5.19] assumes a temperature-independent thickness of the interface, an incompressible system and a roughly similar volume of all atoms in the system. The entropy and the enthalpy in the interface are

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Figure 5.7. Schematic plot of the enthalpy H(r) and of the product S(r)T of entropy and temperature at the interface between a nucleus of the solid phase and an undercooled melt (bold lines). Moreover, the assumed approximation for H(r) and S(r)T within the ansatz by Spaepen [5.19] is shown (thin lines). Within the interface of thickness , H and ST are approximated by constant average values Hi and SiT. Outside the interface region they are described by constant values corresponding to those of the bulk solid and the bulk liquid phase.

approximated by constant average values Si and Hi (Figure 5.7). Moreover, it is supposed that Hi = Hi⫺HL and Si = Si⫺SL are independent of the temperature. GLS is expressed by the linear approximation as suggested by Turnbull (3.4). This gives the following expression for the solid–liquid interfacial energy [5.19]: 2/3

⎞ ⎞ ⎛ H i + SiT H f ⎛ H i + SiT T H i + Si T T − + 1⎟ . LS = 1/ 3 ⎜ − + 1⎟ ⎜ + S f TE S f TE 4 vm ⎝ S f TE TE ⎠ TE ⎠ ⎝ 1/ 3

(5.23) The ratio Si/Sf is linked with the parameter , which is inferred from modelling of the solid–liquid interface within the model by Spaepen (cf. Sections 4.2.1 and 4.2.4), with the interatomic distance da and with the thickness of the interface:

S i d a = . S f

(5.24)

An approximation of follows also from modelling of the interface [5.19]: da

Ni . N1

(5.25)

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It is furthermore supposed that – similar to that in negentropic model of Spaepen and Thompson [5.20, 5.31, 5.32] (cf. Section 4.2.1) – the density deficit at the interface is negligible, and hence Hi 0. Therefore, Eq. (5.23) can be simplified LS

H f ⎛ d a T T ⎞ = 1/3 − + 1⎟ ⎜⎝ ⎠ TE TE 4 vm

1/3

⎛ ⎜ ⎝

⎞ daT T d aT + − +1 ⎟ TE TE TE ⎠

2/3

.

(5.26)

This expression for the solid–liquid interfacial energy depends only on two parameters and , which cannot be measured directly. The estimation of these parameters requires a model for the structure of the interface, like the negentropic model, which was described in Sections 4.2.1 and 4.2.4. For the cases of T = TE or for the limit → 0 Eq. (5.26) converges against the expression (4.9), which was derived in the framework of the negentropic model by Spaepen and Thompson [5.20, 5.31, 5.32] for an interface under equilibrium conditions. Spaepen re-evaluated experiments by Turnbull [5.18] on the volume dependence of the undercoolability of Hg droplets within this diffuse interface approach. A thickness of the interface = 1.46da is assumed, which is motivated by Eq. (5.25) and the results of the modelling of the solid–liquid interface on top of a (111) layer of an fcc crystal according to the model of Spaepen and Thompson [5.20, 5.31] (cf. Table 4.3). While in the original analysis of Turnbull under assumption of a temperature-independent SL, a good agreement with the experimental data is only obtained with unrealistically high pre-exponential factors KVhom in the nucleation rate equation (5.12) [5.18], this is achieved with physically reasonable values for KVhom in the new evaluation [5.19]. In a similar diffuse interface approach by Gránásy [5.27–5.29], the entropy S(r) and the enthalpy H(r) are approximated by the step functions SSt(r) and HSt(r), respectively. At radius RS, SSt(r) jumps from SS to SL, while at radius RH, HSt(r) jumps from HS to HL (Figure 5.8). RS and RH are chosen such that the integrals of S(r) and SSt(r) and of H(r) and HSt(r), respectively, are identical when integrating from r = 0 to r = ⬁. Hence, in this model the thickness of the interface in given by = RS⫺RH. Then, the activation energy for nucleation is given by [5.27] G * =

4 3 ⎛ GLS ⎞ GLSY ⎜ 3 ⎝ H LS ⎟⎠

(5.27)

with

(

)

3 + 2 1 − GLS H LS ⎛ GLS ⎞ 2 1 + 1 − (GLS H LS ) 1 Y⎜ − + . = 3 2 ⎟ GLS H LS ⎝ H LS ⎠ (GLS H LS ) (GLS H LS ) (5.28)

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Figure 5.8. Schematical plot of the enthalpy, H(r), and of the product of entropy and temperature, S(r)T, at the interface between a nucleus of the solid phase and an undercooled melt (bold lines). In the model of Gránásy [5.27–5.29], H(r) and S(r)T are approximated by the plotted step functions (thin lines).

and

=

GLS H LS

(5.29)

If the thickness, , of the interface is assumed to be temperature-independent, this thickness can be calculated from the solid–liquid interfacial energy at the melting point LS(TE). Using expression (4.9) for LS(TE), which was derived in the framework of the negentropic model by Spaepen and Thompson is given by =

1/3

vm . N L1/3

(5.30)

Owing to the approximation of the thermodynamic functions by step functions, the interface thickness, , defined by the above equation is essentially a mathematical quantity that should not be interpreted as the thickness of the physical interface, which is usually larger. Also in this model, the solid–liquid interfacial energy defined by Eq. (5.21) converges for T → TE against the values predicted by formula (4.9) that was derived under equilibrium conditions. Gránásy [5.28, 5.29] calculated the ratio LS(T)/LS(TE) as a function the temperature in the framework of his model using the linear approximation for GLS (Eq. (3.4)). For relative undercoolings T/TE ⬍ 0.4 a nearly linear decrease of LS with decreasing temperature was observed. However,

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in this temperature regime the slope of LS(T)/LS(TE) is smaller than that predicted by Eq. (5.19) (LS(T)/LS(TE) = T/TE). For relative undercoolings of 20%, which are typical of undercooling experiments at pure metals, the interfacial energy calculated according to Eq. (5.19), which neglects the curvature of the solid–liquid interface, is about 10% smaller than that estimated in the framework of the diffuse interface model by Gránásy [5.28, 5.29]. If GLS is not approximated by Eq. (3.4), which is based on the assumption that Cp(T) = 0, but by supposing a non-vanishing difference of the specific heat of solid and liquid phase, the function LS(T)/LS(TE) lies below the curve calculated for Cp(T) = 0 at temperatures T ⬍ TE [5.29]. If Cp(T) Sf /2 is assumed, which lies within the range of values typical of metallic melts [5.33], in the wide temperature range 0.3TE ⬍ T ⱕ TE the function LS(T)/LS(TE) calculated using the diffuse interface model is nearly identical as that estimated employing Eq. (5.19) [5.29]. Recently, a phase field model for the solid–liquid interfacial energy under nonequilibrium conditions was developed by Gránásy et al. [5.34]. Using this model, results of undercooling experiments [5.35, 5.36] on Ni᎐Cu melts of different compositions in the full compositional range (0 at.% Cu ⱕ c ⱕ 100 at.% Cu) were reevaluated [5.34]. For this analysis homogeneous nucleation and 0.6 was assumed. This -value is close to the molecular dynamics result of Ref. [5.37]. A good agreement with the experimental results was obtained in the full compositional range. This has been interpreted as an indication for homogeneous nucleation [5.34]. More recent experimental results on the maximum undercoolability of pure Cu, however, are in contradiction of this interpretation. Two different working groups [5.38, 5.39] observed significantly deeper undercoolings (T = 310 K [5.38] or T = 352 K [5.39]) by use of two different undercooling techniques (electromagnetic levitation and melt fluxing) than observed in [5.35, 5.36] for samples of similar size and at similar cooling rates. Obviously, the increase of the maximum undercoolability is due to a more efficient reduction of heterogeneous nucleation in the newer experiments. Consequently, heterogeneous nucleation must be assumed for the evaluation of the measurements of Refs. [5.35, 5.36], as performed in the original analysis [5.35]. This also implies that the -factor must be larger than the assumed value of 0.6. For instance, -values as predicted by the negentropic model [5.20, 5.31, 5.32] appear more consistent with the experimental findings. 5.2. TRANSIENT NUCLEATION

The derivation of the nucleation rate described in the previous section is based on the assumption of a quasi-stationary cluster distribution function according to Eq. (5.4). This implies that the distribution function can follow rapidly enough

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changes in the system so that at all times the cluster distribution function has its instantaneous equilibrium value. During rapid cooling, however, this precondition can be violated. In such experiments, the experimental time as given by the high cooling rate can be so short that sufficient time does not remain to establish the dynamic equilibrium in the disturbed cluster system. This leads to deviations from the quasi-stationary cluster distribution function and to transient effects. The calculations of the transient nucleation presume in most cases a system that does not contain clusters after the disturbance at the time t = 0, thus Nn,t=0 = 0 [5.17]. The evolution of the cluster distribution as a function of time can be then derived by the application of the formalism of reaction kinetics (cf. Eq. (5.5)) yielding the net rate of the cluster formation with time t: ⭸N = I n −1,t − I n,t . ⭸t

(5.31)

An incubation time is needed before the first critical nucleus is formed. The evolution of the cluster distribution takes place with a characteristic time, the transient . There were several attempts to solve Eq. (5.31) analytically to determine the transient nucleation rate or the transient . Analytical solutions have been found for the limiting cases of t → 0 [5.40] and t → ⬁ [5.41], which lead to different expressions for the transient nucleation rate. All models show a dependence of the transient on the atomic diffusion coefficient. For the estimation of the transient nucleation rate, often an expression developed by Kashiev [5.42] is used: ⬁ ⎤ ⎡ I t (t ) = I ss ⎢1 + ∑ ( −1) m exp( − m 2t / ) ⎥ , ⎦ ⎣ m =1

(5.32)

with ~ D⫺1. The assumption of the validity of Eq. (5.32) is supported by numerical calculations of the solution of Eq. (5.31) for isothermal conditions [5.5] as well as for continuous cooling conditions [5.43]. The calculations of the transient nucleation rate are based on the assumption of homogeneous nucleation, but can readily be transferred to the case of heterogeneous nucleation [5.44]. Figure 5.9 shows the evolution of the number of nuclei as a function of time under isothermal conditions. Homogeneous and heterogeneous nucleation with and without transient effects are considered. 5.3. STATISTICS OF NUCLEATION

The nucleation rate Iss according to Eq. (5.18) can be rewritten, if the validity of the linear approximation (Eq. (3.4)) for the Gibbs free energy difference,

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Steady state homogeneous nucleation Number of nuclei

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Transient homogeneous nucleation

Number of heterogeneous nucleation sites Steady state heterogeneous nucleation

Transient heterogeneous nucleation

τ : transient time τ

Time

Figure 5.9. The evolution of the number of nuclei as a function of time. Homogeneous and heterogeneous nucleation with and without taking into account transient effects are considered.

GLS = Sf T, and the negentropic model for the interface energy, LS, according to Eq. (5.19) is used: ⎛ CT 2 ⎞ I ss (T ) = KV exp ⎜ − , ⎝ T 2 ⎟⎠

(5.33)

with C=

16 S f 3 f (ϑ ) 3k B N L

.

For homogeneous nucleation, the pre-exponential factor becomes KV 10 39 m⫺3 s⫺1, because all atoms in the specimen are potential nucleation sites ( N = N = NL/m). In the case of heterogeneous nucleation only those atoms that are in contact with the foreign phase are potential nucleation sites. This leads to a reduction in N and therefore in KV. Skripov applied Poisson statistics to determine KV and C from the distribution of the experimentally measured undercoolings [5.45]. Under non-isothermal conditions (cooling rate dT/dt ⫽ 0) the probability P for nucleation in a sample of volume within the temperature interval between T and T+ T is given by P (1, T + T ) = T

⎛ T v I (T ) ⎞ v I ss (T ) exp ⎜ − ∫ ss dT ⎟ . dT / dt ⎝ TE dT / dt ⎠

(5.34)

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Equations (5.33) and (5.34) deliver the cumulative distribution function [5.46, 5.47] T ⎡ ⎛ CT 2 ⎞ ⎤ v FDF (T ) = 1 − exp ⎢ − K exp V ∫ ⎜⎝ − T 2 ⎟⎠ dT ⎥ . ⎢⎣ dT / dt TE ⎥⎦

(5.35)

In the temperature range near the liquidus temperature, and thus far above the glass-transition temperature, the temperature dependence of the nucleation rate is controlled by the exponential factor containing the activation energy for nucleation, but not by the temperature dependence of the viscosity. Under such circumstances, KV is essentially independent of T, and Eq. (5.35) can be simplified to ⎡ ⎛ CT 2 ⎞⎤ v Kv FDF (T ) = 1 − exp ⎢ − exp ⎜⎝ − T 2 ⎟⎠ ⎥ . * ⎥⎦ ⎢⎣ ( dT / dt ) d − G ( k BT ) dt

(

)

(5.36)

According to Eq. (5.36) a plot of ln(-ln(1-FDF(T)) versus T2/T2 gives a straight line with the slope ⫺C, and the intercept b: ⎤ ⎡ v Kv ⎥, b = ln ⎢ * ⎢⎣ ( dT dt ) d − G ( k BT ) dT ⎥⎦

(

)

(5.37)

with

(

) = 2C T T + T

d − G * ( k B T ) dT

T 3

2

.

By means of this, the prefactor KV and the activation energy G* to form critical nuclei are deduced from C and b, which in turn are obtained from the straight-line fits. Since, KV and G* depend essentially on the nucleation mode, homogeneous or heterogeneous, a statistic analysis of undercooling behaviour opens up the possibility to study the physical nature of nucleation. 5.4. NUCLEATION IN ALLOYS

All considerations made so far are applicable to pure metals only. In alloy systems, the concentration must be taken into account as an additional degree of freedom in the calculation of the activation energy to form critical nuclei. It is expected that the concentration influences both the interfacial energy LS as well as the Gibbs free energy difference GLS.

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The Gibbs free energies of the liquid and the solid phase, GL and GS, are not only a function of the temperature, but also of the chemical composition. Also the entropy of fusion, Sf, depends on the chemical composition. Moreover, the composition of a solid nucleus in an undercooled melt usually differs from that of the melt. The most probable composition of the nucleus is the one for which the activation energy G* is minimum. According to Eq. (5.3), G* depends on the solid–liquid interfacial energy LS and on the driving force for solidification, GLS. In the negentropic model for the energy LS of the solid–liquid interface (cf. Chapter 4.2.1) one decisive composition-dependent parameter is the entropy of fusion Sf [5.48]. According to Richard’s rule [5.49, 5.50] the entropy of fusion of most metallic systems (with the exception of e.g. some systems forming chemically ordered solid phases) can be approximated by the gas constant RG: Sf RG. This suggests a small composition dependence of Sf and consequently of LS [5.32, 5.48]. Therefore, often the simplified assumption is made that the nucleus has that composition for which GLS is maximum [5.32, 5.48]. For the evaluation of GLS the Gibbs free energies GL and GS of liquid and solid phase must be known as a function of temperature and composition. The same methods may be utilized to determine GL and GS, which have been developed for the calculation of phase diagrams (CALPHAD methods) [5.51–5.54] and which based on works by Kaufman and Bernstein [5.51]. In most CALPHAD approaches an analytical expression based on a subregular solution model is used to approximate the thermodynamic functions of the different phases. The coefficients of the expressions are determined such that a good fit to experimental data, such as calorimetric data, phase equilibria or partial free energies of the components, is obtained. If a binary A–B alloys can be described by the regular solution model, the concentration of the A atoms cNA in a solid nucleus formed in an undercooled liquid of nominal composition c0A can be estimated according to a model of Thompson and Spaepen [5.32, 5.48] at a given nucleation temperature TN. If the liquidus line decreases with increasing concentration cA, as in the case shown in Figure 5.10 (otherwise the atomic species A and B must be exchanged in the following description), cNA is given by c N (T N ) = A

A

A A c 0 + (1 − c 0 ) exp ⎡⎣(T L − T N

c0 . A A A A B A R G T N ) ( S f − S f ) + (T L T N ) ln ( c 0 (1 − c S ) c S (1 − c 0 )) ⎤⎦

(5.38) Here, cSA denotes the equilibrium concentration of the A atoms in the solid phase at the liquidus temperature TL, which is inferred from the equilibrium phase diagram.

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Figure 5.10. Composition cNA of a solid nucleus in an undercooled liquid of nominal composition c0A as a function of the temperature.

SfA and SfB are the entropies of fusion of the pure components A and B. The dependence of the concentration of the solid nucleus on the nucleation temperature is schematically shown in Figure 5.10. With increasing undercooling the difference of cNA from the equilibrium value cSA increases. At the nucleation temperature TN, the driving force GLS for the formation of a critical nucleus of composition cNA is given by [5.32, 5.48] G LS (T N ) = (T L − T N ) S Af + R G T N ln (c 0A c NA ) − R G T L ln (c 0A c SA) .

(5.39)

5.5. MAGNETIC CONTRIBUTIONS TO CRYSTAL NUCLEATION

The onset of magnetic ordering in undercooled magnetically ordering liquids (cf. Section 3.3) will be accompanied by a change of the thermodynamic quantities of the liquid and the solid phase. Hence, magnetic contributions may influence the driving free energy for nucleation and the solid–liquid interfacial energy. This in turn will affect the nucleation behaviour of magnetically ordering systems. Such contributions were discussed in a recent work [5.55]. 5.5.1 The magnetic contribution to the driving force for crystal nucleation The magnetic contribution, Gmag , to the driving force for nucleation, GLS , can be estimated in the framework of molecular field theory of magnetism [5.56, 5.57]. + The magnetic free energy Fmag of a single spin J (the superscript + always indicates that the quantity is normalized to one single spin) in an external magnetic

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field B is defined by e

− F +mag

=

J

∑

e

− g L B B J z

=

e

J z =− J

g L B B(J +1/2 )

e

−e

g L B B/ 2

− g L B B ( J +1/22 )

−e

− g L B B/ 2

,

(5.40)

with = (kBT)⫺1, gL the Landé’s g-factor and B the Bohr magneton [5.57]. For condensed matter, the magnetic contribution to the Gibbs free energy of one spin + + can be approximated by the magnetic free energy: Gmag Fmag . The magnetic contribution Smag to the entropy of the system of NJ spins is given by ⭸ F +mag ⭸

(5.41)

+ N J ⭸ F mag N J = g L B JBJ ( g L B JB ) . v ⭸B v

(5.42)

S mag = N J k B

2

and the magnetization is given by M 0 (T , B ) = −

Here BJ ( gL BJB) denotes the Brillouin function and v the sample volume. In molecular field theory an effective field Beff is defined as Beff = B + KMF M

(5.43)

with KMF a coupling constant and B the external field. With Eq. (5.42) this gives an expression for the magnetization: M (T ) = M 0 (T , Beff ) = M 0 (T , B + K MF M (T )),

(5.44)

which has to be solved self-consistently. In the molecular-field theory the Curie temperature is expressed by [5.56] N J g L2 B J ( J + 1) K MF . 3k Bv 2

TC =

(5.45)

If the Curie temperature of a given material is known, the above formula allows calculation of the coupling constant KMF. Then, numerical solution of Eq. (5.44) gives the magnetization M(T) as a function of the temperature and Eq. (5.43) gives the effective field Beff as a function of temperature. Finally, with Eq. (5.40) the free energy Fmag(T) Gmag(T) is calculated as a function of temperature. As an example, the magnetic contribution to the difference Gmag between the Gibbs free energies of solid and liquid Co is estimated. The Curie temperature of

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solid Co is TCS = 1394 K [5.58]. Based on measurements of the magnetic susceptibility of liquid Co as a function of the temperature using a Faraday balance, the Curie temperature of liquid Co is estimated at TCL = TCS⫺6 K [5.59]. When the temperature dependence of the saturation magnetization, Ms(T), of fcc Co [5.59, 5.60] is modelled with the molecular field theory, the best fit is obtained with J = 1/2 [5.59]. The same value of J results from molecular field fits to data for Fe and Ni [5.56, 5.61, 5.62]. The value J = 1/2 has been interpreted in terms of the itinerant electron model of magnetism [5.61]. The extrapolation of measurements [5.60] of the saturation magnetization Ms of fcc-Co to 0K gives a value of Ms(0K) = 1478 G [5.58], which corresponds to a magnetic moment per atom of p = 1.75 B [5.58]. With the experimentally determined value of Landé’s g-factor (gL = 2.17) for Co and J = 1/2, this gives a number j = p /(JgL) = 1.61 spins per atom [5.58]. If the sample consists of N atoms, the number of spins in the sample is given by NJ = Nj. If there is no external magnetic field (B = 0), by numerically solving Eqs. (5.44) and (5.43), (5.42) and (5.40) with the above parameters and using G mag = jG+mag gives magnetic Gibbs free energies per atom G mag , L and G mag , S , magnetic entropies per atom Smag , L and Smag , S (from Eq. (5.41)), magnetic enthalpies per atom H mag , L and H mag , S (from H mag = G mag + T Smag ) and magnetizations ML and MS of solid and liquid Co that are depicted in Figure 5.11. For both the liquid and solid phase, the magnetic entropies Smag , L and Smag , S have a constant value of Smax = j ln(2J + 1)kB 1.12kB at temperatures above the Curie temperature of the respective phase, while at temperatures below TC the entropies decrease with decreasing temperature due to magnetic ordering. The (negative) magnetic contributions to the enthalpy H mag , L and H mag , S of the solid and liquid phase vanish at temperatures above TCL and TCS, respectively, and their absolute values increase if the temperature is decreased below the Curie temperature. The differences in Gibbs free energy, G mag = G mag , L ⫺ G mag , S , entropy, Smag = Smag , L ⫺ Smag , S , and enthalpy, H mag = H mag , L ⫺ H mag , S , between liquid and solid are plotted in Figure 5.12. At temperatures T ⬎ TCS there is no magnetic contribution to the difference of the Gibbs free energies. However, if the temperature is decreased below TCS, G mag steeply rises to its maximum value at TCL. A further decrease in the temperature leads to a slight decrease in G mag . 5.5.2 The magnetic contribution to the solid–liquid interfacial energy To estimate the magnetic contribution, mag, to the solid–liquid interfacial energy, we consider a solid nucleus inside a liquid at a temperature TCL ⬍ T ⬍ TCS. Because the nucleus is ferromagnetically ordered while the surrounding liquid is still in the paramagnetic regime gives rise to a magnetic contribution, mag, to the solid–liquid interfacial energy. As a rough estimate of mag a simple broken bond

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Figure 5.11. Magnetic contributions G mag , L and G mag , S to the Gibbs free energies per atom; magnetic contributions Smag , L and Smag , S to the entropies per atom; magnetic contributions H mag , L and H mag , S to the enthalpies per atom; and magnetizations ML and MS in liquid (L) and solid (S) Co [5.55].

Figure 5.12. Differences in the magnetic contributions to the Gibbs free energy G mag = G mag , L ⫺ G mag , S , entropy Smag = Smag , L ⫺ Smag , S , and enthalpy

H mag = H mag , L ⫺ H mag , S between liquid and solid Co [5.55].

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model is chosen. For a more precise estimate of mag, effects of the topological structure of the solid–liquid interface, which differs from that of the solid as shown in Sections 4.2.1 and 4.2.4, should to be considered. If it is assumed that the solid nucleus has an fcc structure (e.g. pure Co or Co᎐Pd alloys), the coordination number of the atoms inside the nucleus is Z = 12. For an atom at a (111) surface three of the 12 bulks nearest neighbours are missing. Therefore, the magnetic enthalpy of one surface atom is increased by ⫺ 3/12H mag , S . Here, H mag , S denotes the magnetic contribution to the enthalpy of the solid phase that was discussed in the preceding section. This gives an increase of the solid– liquid interfacial energy per surface area due to magnetism of

H mag , s (5.46) . 2/3 4 N 1/3 L vm Because of the temperature dependence of H mag , S (Figure 5.11), this contribution increases when the temperature is decreased to TCL. Since at TCL the liquid phase begins to order ferromagnetically, the magnetic contribution to the solid–liquid interfacial energy decreases if the temperature is further lowered. For pure Co this leads to a maximum value of mag, which is less than 1% of its classical value, according to Eq. (5.19). This demonstrates that the magnetic contribution to the solid–liquid interfacial energy is negligible. mag = −

5.6. EXPERIMENTAL RESULTS ON UNDERCOOLING AND NUCLEATION

5.6.1 Homogeneous versus heterogeneous nucleation The classical nucleation theory gives a concept to describe homogeneous nucleation. From the experimental point of view it is interesting to detect homogeneous nucleation by avoiding heterogeneous nucleation. Under constant experiment conditions the homogeneous nucleation sets the ultimate limit for the undercoolability of a melt, since the activation energy to form critical homogeneous nuclei is always larger than that for heterogeneous nucleation. To evidence homogeneous nucleation the following criteria must be fulfilled: (i)

Similar to heterogeneous nucleation in the volume, the homogeneous nucleation rate scales with the volume of the sample. (ii) In contrast to heterogeneous volume nucleation, the homogeneous nucleation shows a different time characteristic in the evolution of clusters or nuclei. This is due to the fact that the volume of a homogeneous nucleus is larger than that of a heterogeneous nucleus and therefore requires more time to be formed. A direct consequence is a different dependence of the homogeneous and heterogeneous nucleation rate on the cooling time.

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(iii) As a result of (ii), transient effects should be more pronounced in the case of homogeneous nucleation in comparison with heterogeneous nucleation. (iv) In the case of homogeneous nucleation a high density of potential nucleation sites in the order of Avogadro’s number exist, since every atom can form the starting point for homogeneous nucleation. In alloys of small crystal growth velocity, homogeneous nucleation should lead to the formation of a correspondingly high number of crystallites (grain-refined materials). (v) In the case of homogeneous nucleation the microstructure should be free of texture. (vi) Under isothermal conditions the timely evolution of nuclei should saturate in the case of heterogeneous nucleation earlier than in the case of homogeneous nucleation (cf. Figure 5.9). (vii) As a result of (vi) the number of nuclei formed depends on time even if the evolution of heterogeneous nuclei became time independent by the saturation of all heterogeneous nucleation sites. (viii) The Avrami analysis of the crystallization process should yield an exponent n = 4 for homogeneous nucleation, while for heterogenous nucleation in the volume and on the surface, n = 3 and n = 2 are expected.

5.6.2 Nucleation in undercooled melts In undercooling experiments by Turnbull [5.18] on Hg dispersions volume nucleation was detected such that the necessary condition for homogeneous nucleation is fulfilled. The maximum relative undercoolability was T/TE = 0.34. In another work significantly larger undercoolings T/TE = 0.38 were measured for liquid Hg also by use of a dispersion technique, which may be indicative of heterogeneous volume nucleation in Turnbull’s experiments on liquid Hg [5.63]. Perepezko reports results of maximum undercoolings of Sn᎐Bi droplets for various concentrations obtained by the application of the droplet emulsion technique [5.64]. An analysis of these experimental results within the nucleation theory leads to the conclusion of homogeneous nucleation. Figure 5.13 shows results of experimentally determined undercoolings by the application of the droplet dispersion technique [5.65] and electromagnetic levitation technique [5.35, 5.66] for Cu᎐Ni (a) and Fe᎐Ni (b) alloys. In the case of Cu᎐Ni the results can be interpreted if heterogeneous nucleation of a unique catalytic potency factor f (ϑ) = 0.16 [5.67] for the droplets (diameter 10 m), and f (ϑ) = 0.19 [5.66] for the electromagnetically processed drops (diameter 6 mm) is assumed. The comparison of the results obtained by the droplet dispersion technique with those obtained by the electromagnetic levitation technique shows similar undercoolings despite the fact that the samples differ in size by more than two orders of magnitude. This finding

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Figure 5.13. Phase diagrams of Cu᎐Ni (upper panel) and Fe᎐Ni (lower panel) depicting results of undercooling experiments. The open dots give the maximum undercoolings as obtained by the droplet dispersion technique, while the closed circles represent undercooling results obtained from levitation experiments on bulk samples. The lines correspond to the analysis within nucleation theory.

hints on heterogeneous nucleation on the surface of the samples by e.g. metal oxides formed at the surface of the samples. The comparable undercoolings lead to the conclusion that large undercoolings can also be achieved on bulk melts and are not limited on small droplets. This result is confirmed by undercooling investigations on Fe᎐Ni alloys (cf. Figure 5.13).

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The undercoolings measured on Ni-rich Fe᎐Ni alloys processed by the electromagnetic levitation technique can be described by heterogeneous nucleation of the same catalytic potency factor f(ϑ) = 0.19 as in the case of the Cu᎐Ni alloys. This observation is attributed to the formation of NiO on the surface of the sample. NiO is stable up to temperatures of 2240 K. Therefore, an oxide layer may be formed due to the reaction of the sample with oxygen impurities in the environmental gas atmosphere. NiO is solid even in the temperature range where the metallic sample is molten (T ⬎ TL 1726 K). The solid NiO layer at the surface of the sample, in turn, might form the heterogeneous nucleation site. As outlined in Section 5.3, the pre-exponential factor KV in the nucleation rate equation can be estimated by a statistical analysis of the probability distribution function of the nucleation temperatures measured in undercooling experiments [5.45]. This allows identifying a heterogeneous nucleation mechanism, if the order of magnitude of KV is significantly below the order of magnitude typical of homogenous nucleation ( KVhom 1039 s⫺1 m⫺3). Such a statistical analysis of the undercoolability of Zr melts of different purity, which were containerlessly processed by means of the electrostatic and the electromagnetic levitation technique as well as in a drop tube delivered pre-exponential factors ranging between 108 and 1013 s⫺1m⫺3 depending on the purity of the sample material and the undercooling technique [5.47]. These values are by more than 26 orders of magnitude smaller than the pre-exponential factor typical of homogeneous nucleation ( KVhom 1039/s⫺1 m⫺3). This clearly indicates a heterogeneous nucleation mechanism even for the samples of highest purity. The decisive influence of impurities in the sample material on the nucleation behaviour is demonstrated by the increase in the maximum undercoolability and of the prefactor KV, if sample material of 99.995% purity is used instead of 99.95% purity. A statistical analysis of the results of drop tube experiments on Nb droplets, for which maximum relative undercoolings T/TE 0.16 were obtained, gave preexponential factors of KV 1031⫺1034 s⫺1 m⫺3 depending on the method of data analysis [5.47]. Although these values are larger than those measured for Zr, they still lie significantly below the value for homogeneous nucleation. Moreover, a statistical analysis of the nucleation behaviour of Co50Pd50 and Co70Pd30 melts undercooled by application of the electromagnetic levitation technique delivered pre-exponential factors KV 1023/s/m/ [5.68, 5.69]. Obviously in this case also heterogeneous nucleation prevails, despite the fact that large relative undercoolings T/TL 0.21 were measured. These investigations will be described in more detail in Section 5.6.4. 5.6.3 Structural dependence of nucleation behaviour In Section 4.2 we have presented several models to describe the energy of the interface between a solid phase and its melt. According to Eq. (5.3) the solid–liquid

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interfacial energy has a decisive impact on the nucleation behaviour of solid phases from undercooled liquids. In this section, results of experimental studies on the maximum undercoolability of metallic melts are presented and compared with the theoretical results on the solid–liquid interfacial energy. The discussion in the preceding chapters shows that for most undercooling experiments performed at moderate cooling rates heterogeneous nucleation is the most likely nucleation mechanism, even if very high levels of undercooling are obtained. Nevertheless, owing to the lack of information on f (ϑ) and , undercooling experiments are frequently evaluated under the simplified assumption of homogeneous nucleation to estimate solid–liquid interfacial energies. The values determined in this way are lower limits of the real solid–liquid interfacial energies. Results of undercooling studies on a variety of pure metals are compiled in Table 5.1 [5.35, 5.36, 5.39, 5.63, 5.70, 5.71] in which deep undercoolings were achieved. By using Eqs. (5.48), (5.12), (5.3), (5.19) and (3.13) the lower limits of the solid–liquid interfacial energy LS at the nucleation temperature TN were estimated assuming homogeneous nucleation. The lower limits of the dimensionless solid–liquid interfacial energies are determined with Eq. (5.19) (Table 5.1). Similar to the -factors determined under local equilibrium conditions (Table 4.2), the lower limits of the -factors inferred from undercooling studies for meltsforming crystals with fcc and bcc structures are consistent with the theoretical values predicted by the negentropic model of Spaepen and Thompson [5.20, 5.31, 5.32] and by the density functional theoretical work of McMullen and Oxtoby

Table 5.1. Results of undercooling experiments [5.35, 5.36, 5.39, 5.63, 5.70, 5.71] on pure metals and lower limits of the solid–liquid interfacial energy SL and of the -factor calculated using Eqs. (5.48), (5.12), (5.3), (5.19) and (3.13). Material

Stable phase

Ag Bi Co Cu Fe Ga Hg Ni Pb Sn

fcc A7* fcc fcc bcc A11* A10 fcc fcc A7

Method TE (K) TN (K) MF Disp. MF MF EML Disp. Disp. EML disp. disp.

1235 544 1768 1358 1811 303 234 1728 600 505

985 317 1385 1004 1487 128 155 1387 447 318

T/TE 0.20 0.42 0.22 0.26 0.18 0.58 0.38 0.20 0.26 0.37

tN (m3 s) LS(T) (J/m2) ~10⫺3 ~10⫺14 ~10⫺5 ~10⫺4 3⫻10⫺6 ~10⫺14 ~10⫺14 3⫻10⫺6 ~10⫺14 ~10⫺14

ⱖ0.143 ⱖ0.083 ⱖ0.277 ⱖ0.244 ⱖ0.211 ⱖ0.0302 ⱖ0.0301 ⱖ0.261 ⱖ0.043 ⱖ0.069

Ref.

ⱖ0.65 ⱖ0.82 ⱖ0.67 ⱖ0.80 ⱖ0.60 ⱖ1.77 ⱖ1.05 ⱖ0.58 ⱖ0.72 ⱖ0.84

[5.70] [5.63] [5.71] [5.39] [5.36] [5.63] [5.63] [5.35] [5.63] [5.63]

Note: Different techniques were utilized to undercool the liquids: dispersion technique (Disp.), electromagnetic levitation (EML), and melt fluxing (MF). * Instead of the stable phase, some metastable modifications solidify from deeply undercooled melts [5.63, 5.74, 5.75].

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[5.23]. They are significantly larger than the -factors estimated by all other models [5.22, 5.24, 5.25, 5.21, 5.72, 5.37] as discussed in Section 4.2. Moreover, the large relative undercoolings and the large -factors observed for the elements Bi, Ga, Hg and Sn that are not characterized by an fcc, hcp or bcc structure are noteworthy. The numerical modelling of the solid–liquid interfacial energy described in Section 4.2.4 predicts considerably lower -factors for quasicrystalline phases and also for some structurally complex crystals as for non-complex crystal structures. Therefore, experimental studies of the nucleation behaviour of such structurally complex solids are of special interest. The first investigations on the nucleation of quasicrystalline phases from the melt were performed by Bendersky and Ridder [5.73]. Liquid droplets of an Al86Mn14 melt were solidified during free fall under vacuum in an electrohydrodynamic atomization facility. The diameters of the droplets ranged from 20 nm to 2 m and are correlated inversely with the cooling rate that covered a range between 104 and 106 K s⫺1. By transmission electron microscopy investigations, an extremely high density of quasicrystalline grains of the order of 1024/m3 was detected in the smallest droplets. These huge densities of grains are indicative of a high nucleation rate and a low activation threshold for the formation of a critical cluster of the I-phase. Similar studies on Al88Mn12 droplets of larger size were done by using of a drop tube with a length of 2.5 m [5.74–5.76] (cf. also Section 10.2.2). Droplets of 20–1000 m in diameter were containerlessly solidified during free fall under a He protective atmosphere. They were separated into different particle sizes by sieving. The cooling rates are estimated to range from 800 K/s for the 1 mm droplets to 3⫻105 K/s for the 20 m particles. The droplet sizes in each group were investigated by X-ray diffraction and differential scanning calorimetry (DSC) to determine the phases formed during solidification. At a small cooling rate only stable crystalline phases were solidified while at elevated cooling rates also the metastable quasicrystalline phases were formed: the decagonal quasicrystalline T-phase2 at medium cooling rates and the icosahedral quasicrystalline I-phase at high cooling rates. While icosahedral quasicrystals are quasiperiodic in all three directions [5.77–5.80], decagonal quasicrystals are quasiperiodic in two dimensions and translational invariant in the third dimension [5.79–5.81]. To analyse the nucleation behaviour, a temperature-time transformation diagram was plotted for all these phases that describe the experimental findings [5.76]. This 2

Because of historic reasons the decagonal phase in Al–Mn is called T-phase, while decagonal quasicrystalline phases in other alloy systems are usually called D-phases.

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evaluation was based upon the classical nucleation theory assuming homogeneous nucleation of the quasicrystalline phases. The interfacial energy between the nucleus and the melt was expressed according to Eq. (5.19), with the structure-dependent factor as a free parameter. The best fit to the experimental results was achieved with I=0.36 for the icosahedral I-phase and of D = 0.45 for the decagonal T-phase. Significantly larger -factors were determined experimentally for crystals with fcc, hcp or bcc structure (cf. Tables 4.2 and 5.1) and also calculated in the framework of the negentropic model by Spaepen and Thompson (fcc/hcp = 0.85 and bcc = 0.70). Hence, the predicted small energy of the interface between nuclei of quasicrystalline phases and the melt is experimentally confirmed. Moreover, the -factor determined for icosahedral I-phase, which is quasiperiodic in three dimensions, is lower than that of the decagonal T-phase that shows quasiperiodicity in two directions only. Holzer and Kelton [5.82] investigated the kinetics of the amorphous to icosahedral phase transition in Al75Cu15V10 alloys by means of isothermal and nonisothermal DSC measurements and isothermal measurements of the electrical resistivity. By means of an analysis of the experimental results in the framework of classical nucleation theory and by taking transient effects into account, the interfacial energy LS between the quasicrystalline nucleus and the amorphous phase was estimated resulting in extremely low values of 0.002 ⱕ LS ⱕ 0.015 J/m2. These are significantly lower than the typical energies of the interface between solids with non-complex structure and their melts (cf. Tables 4.2 and 5.1). The drop tube and atomisation investigations described above provided indirect information on the nucleation behaviour of quasicrystalline phases in undercooled melts. However, the nucleation temperatures were not determined directly. Systematic experiments on the structure dependence of the maximum undercoolability, in which the maximum undercoolings were directly measured, were performed by applying the containerless processing technique of electromagnetic levitation for a large variety of Al-based alloys [5.83–5.88]. The investigated alloy systems Al᎐Cu᎐Fe [5.89–5.92], Al᎐Cu᎐Co [5.93–5.95], Al᎐Ni᎐Co [5.95–5.98] and Al᎐Pd᎐Mn [5.99–5.102] form several quasicrystalline and polytetrahedral phases as well as non-complex crystalline phases. For these alloy systems already small changes of the alloy composition lead to a change of the phase selection during solidification from the undercooled melt which allows studying the nucleation behaviour of a large number of phases under similar conditions. Among the investigated phases are the icosahedral quasicrystalline I-phases in Al᎐Cu᎐Fe [5.83–5.85, 5.103] and Al᎐Pd᎐Mn [5.85, 5.86], the decagonal quasicrystalline D-phases in Al᎐Cu᎐Co [5.83–5.85, 5.103], Al᎐Ni᎐Co [5.85] and Al᎐Co [5.85, 5.86], the polytetrahedral phases -(Al,Cu)13Fe4 and -Al5Fe2 in the alloy system Al-(Cu᎐)Fe [5.84, 5.85] and the cubic -phase with CsCl-structure in Al᎐Cu᎐Co [5.84, 5.85].

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Figure 5.14. Temperature–time profile of a levitation undercooling experiment on an Al72Pd21Mn7 sample, for which the maximum undercooling was obtained [5.85, 5.86].

As an example, Figure 5.14 shows the temperature–time profile of a levitation undercooling experiment on an Al72Pd21Mn7 alloy, which forms an icosahedral quasicrystalline I-phase. First, the sample is molten and subsequently the melt is cooled and undercooled below its liquidus temperature TL. At the temperature TN the nucleation sets in. During solidification the I-phase is formed, as concluded from the subsequent analysis of the solidification products by scanning electron microscopy, transmission electron microscopy and X-ray diffraction [5.86]. The release of heat of fusion leads to a temperature rise up to the melting temperature of the I-phase. This temperature rise is called recalescence. After the recalescence the solidified specimen is cooled to lower temperatures. The undercooling T obtained during this experiment is determined from the measured difference of the nucleation and the liquidus temperature T = TL⫺TN. By use of the electromagnetic levitation technique maximum relative undercoolings of TI/TLI 0.1 were achieved for the I-phases, both in Al᎐Cu᎐Fe and in Al᎐Pd᎐Mn [5.83–5.85, 5.103], while for the D-phases in Al᎐Cu᎐Co, Al᎐Ni᎐Co and Al᎐Co values of TD/TLD 0.15 [5.83–5.85, 5.103] were obtained. In case of the crystalline polytetrahedral phases relative undercoolings of T/TL 0.13 were measured for the -phase in Al(᎐Cu)᎐Fe and of T /TL 0.14 for the

-phase in Al᎐Fe [5.84, 5.85]. The maximum undercoolings reached for the crystalline -phase in Al᎐Cu᎐Co are highest (T /TL 0.25) [5.84, 5.85]. All results of the undercooling studies on the Al-based alloys are summarized in Table 5.2. The undercooling behaviour of the investigated phases is described by the relation T I /TL I < T /TL < T /TL < T D /TL D < T /TL

(5.47)

Obviously, the maximum relative undercoolings are strongly dependent on the structure of the primarily nucleating solid phases. The experimentally determined maximum undercoolings were analysed in the framework of classical nucleation theory under the assumption of homogeneous

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Table 5.2. Melting temperatures TL of the primarily formed phases, nucleation temperatures TN, relative undercoolings T/TL, solid–liquid interfacial energies SL(TN) and structure-dependent -factors determined from results of experiments on the maximum undercoolability of various Al-based alloys [5.83–5.86]. Alloy Al60Cu34Fe6 Al58Cu34Fe8 Al72Pd21Mn7 Al13Fe4 Al62Cu25.5Fe12.5 Al5Fe2 Al74Co26 Al75Ni14.5Co10.5 Al64Cu22Co14 Al65Cu25Co10 Al67Cu21Co12 Al65Cu20Co15

Primary phase

TL (K)

TN (K)

T/TL

LS(TN) (J/m2)

I-phase I-phase I-phase -phase -phase

-phase D-phase D-phase D-phase D-phase D-phase ß-phase

1098 1130 1137 1420 1258 1430 1440 1360 1274 1259 1293 1485

1000 1030 1012 1250 1085 1225 1230 1169 1084 1055 1083 1120

0.09 0.09 0.11 0.12 0.14 0.14 0.15 0.14 0.15 0.16 0.16 0.25

0.09⫾0.01 0.09⫾0.01 0.10⫾0.01 0.16⫾0.01 0.15⫾0.01 0.18⫾0.02 0.16⫾0.01 0.11⫾0.01 0.11⫾0.01 0.11⫾0.01 0.116⫾0.01 0.17⫾0.01

0.28⫾0.03 0.28⫾0.03 0.32⫾0.03 0.33⫾0.03 0.36⫾0.03 0.37⫾0.03 0.40⫾0.03 0.44⫾0.04 0.46⫾0.04 0.48⫾0.04 0.49⫾0.04 0.63⫾0.06

nucleation ( f (ϑ) = 1, = 1) [5.83–5.85]. According to this model, the maximum undercooling achievable under certain experiment conditions depends on the volume of the sample and the experimental time. Assuming that only one nucleation event is needed for the onset of crystallization the product of the nucleation rate Iss(TN) times the volume of the sample times the experimental time tN should be equal to 1 at the nucleation temperature TN. In the case that during undercooling more than one nucleation event occurs this product is greater than unity. Thus, it holds in general that I ss (TN )v t N ≥ 1,

(5.48)

TN and tN are directly inferred from the measured temperature-time profiles. If the energy of the interface between undercooled melt and solid nucleus is described according to the negentropic model of Spaepen and Thompson [5.20, 5.31, 5.32, 5.104] (Eq. (5.19)), the dimensionless solid–liquid interfacial energy can be estimated from the measured maximum undercoolings by using Eqs. (5.48), (5.12), (5.3), (3.10) and (3.36), because all other parameters are either experimentally determined or they can at least be approximated by suitable models [5.85]. The results of this analysis are compiled in Table 5.2 for the different alloys. Dimensionless interfacial energies I 0.3 for the I-phases in Al᎐Cu᎐Fe [5.83–5.85, 5.103] and Al᎐Pd᎐Mn [5.85, 5.86], 0.33 for the -phase in Al᎐Fe and Al᎐Cu᎐Fe [5.84, 5.85], 0.37 for the -phase in Al᎐Fe [5.84, 5.85], D 0.45 for the D-phases in Al᎐Cu᎐Co [5.83–5.85, 5.103], Al᎐Ni᎐Co [5.85] and

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Al᎐Co [5.85, 5.86] and 0.63 for the cubic ß-phase in Al᎐Cu᎐Co [5.84, 5.85] were determined. Hence, for the structure-dependent factors of the different phases the following relation is obtained [5.85]: I 0.3 ⬍ 0.33 ⬍ 0.37 ⬍ D 0.45 ⬍ 0.63. (5.49) This relation is in good agreement with the results of the modelling of the solid–liquid interface (cf. Section 4.2.4), which has given the same sequence of the -factors (cf. Table 4.3). Again, especially the icosahedral quasicrystalline phases and the polytetrahedral phases are characterized by low -factors, while the noncomplex crystalline ß-phase exhibits a large -value. The above analysis is based on the idealized assumption of homogeneous nucleation. Indeed, the extremely high densities of the quasicrystallites found in rapidly quenched quasicrystal forming alloys are indicative of homogeneous nucleation of the quasicrystalline phases during rapid quenching of the melts [5.73, 5.84]. Although the results of rapid quenching experiments cannot be directly compared with those of undercooling studies at moderate cooling rates, they indicate that the case of homogeneous nucleation can be achieved more easily for quasicrystalline phases than for structurally non-complex phases, presumable due to the lower solid–liquid interfacial energies. If, however, heterogeneous nucleation is prevailing, which is likely especially for the non-complex cubic -phase, the -values listed in Table 5.2, as discussed above, are lower limits of the dimensionless solid–liquid interfacial energies. Nevertheless, the main conclusions and the sequence (5.49) remain valid even if heterogeneous nucleation is present in the levitation-undercooling experiment, provided the catalytic potency f(ϑ) is nearly the same for all the different phases. The activation energy for the formation of critical nuclei, which controls the maximum undercooling, depends in the third power on the interfacial energy LS but only linearly on the catalytic potency factor f(ϑ) in the case of heterogeneous nucleation. If the observed changes of G* with the variation of primary nucleating phase were exclusively attributed to a corresponding alteration of f(ϑ), unreasonably large differences in the data for f(ϑ) have to be presumed [5.84], with particularly low values for the quasicrystalline phases. The latter assumption, however, appears unlikely because the complex, non-periodic structure of the quasicrystalline phases implies a bad fit of heterogeneous nucleants of crystalline structure to a nucleus of a quasicrystalline phase and consequently a low catalytic potency for nucleation of quasicrystalline phases. For these reasons, it appears reasonable to assume that for the quasicrystalforming alloys the observed structure dependence of the maximum undercoolability is dominated by the structure dependence of LS, as predicted by modelling of the solid–liquid interfacial energy (cf. Section 4.2.4)

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It is noteworthy that the dimensionless solid–liquid interfacial energy is mainly governed by the topological structure of the solid, whereas the influence of the chemical structure is of minor importance [5.85]. For example, the -factors determined for D-phases lie closely around the value of 0.45 for all investigated alloy compositions in three different alloy systems (Al᎐Cu᎐Co, Al᎐Ni᎐Co and Al᎐Co) [5.85]. A similar result is obtained for the I-phases in Al᎐Pd᎐Mn and Al᎐Cu᎐Fe [5.85]. While the influence of chemical effects on the nucleation behaviour appears to be small, measurements of the growth velocities of polytetrahedral phases during solidification of undercooled melts [5.105] show a pronounced composition dependence on the growth behaviour. This difference may be explained by the fact that crystal growth is dominated by the kinetics of the attachment of liquid atoms to the solid phase, while for nucleation processes the attachment kinetics enter only into the pre-exponential term of the nucleation rate function (Eq. (5.12)). Because the investigated polytetrahedral phases are chemically highly ordered, chemical diffusion processes control the attachment kinetics, which explains the strong concentration dependence of the growth velocity. A similar dependence of the maximum undercoolability of the structure on the solid phase primarily formed from the undercooled melt as that found for the Albased alloys was observed by Kelton et al. [5.106] for quasicrystal-forming Ti᎐Zr᎐Ni melts by in-situ diffraction experiments with synchrotron radiation. In this work the nucleation behaviour of the icosahedral quasicrystalline I-phase, of a polytetrahedral phase with C14-structure and of the -(Ti,Zr) solid solution with hcp structure was studied by employing the electrostatic levitation technique. For the I-phase a maximum relative undercooling of TI/TLI = 0.11 ⫾ 0.02 was obtained. This value is in good agreement with the maximum relative undercoolings measured for the I-phases in Al᎐Pd᎐Mn and Al᎐Cu᎐Fe. Similar to the polytetrahedral phases in the Al-based alloys, the C14-phase shows a slightly higher relative undercooling of TC14/TLC14 = 0.15 ⫾ 0.01, while for the -phase considerably larger relative undercoolings of T/TL = 0.19 ⫾ 0.01 were observed. The latter value is typical of phases with a non-complex structure. For TicFe94-cSi4(SiO2)2 alloys with 67 at.% ⬍ c ⬍ 70 at.% the primary solidification of the polytetrahedral (1/1)-approximant phase is observed at comparatively small relative undercoolings of T1/1/TL1/1 0.1 in undercooling experiments utilizing the electromagnetic levitation technique. For Ti-rich alloy melts, which primarily form structurally non-complex phases, significantly higher relative undercoolings of T/TL 0.18 were obtained [5.107]. Summarizing, a large variety of undercooling experiments performed on metallic melts indicate a pronounced dependence of the maximum undercoolability on the structure of the solid phases that are primarily formed from the undercooled melt. These observations are explained by the structure dependence of the

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solid–liquid interfacial energy, which is predicted by modelling of the solid-liquid interface (cf. Section 4.2.4). Melts exhibit a smaller undercoolability if quasicrystalline phases or polytetrahedral solid phases are nucleating than observed for the nucleation of structurally non-complex crystals. This can be explained by a polytetrahedral short-range order prevailing in the melt (cf. Section 3.2.3) and its influence on the solid–liquid interfacial energy. 5.6.4 Undercooling of magnetic melts The occurrence of magnetic ordering in fluids has been discussed in Section 3 of Chapter 3. It attracts broad attention both from fundamental aspects as well as from the application point of view. To combine the properties of magnetic matter with those of a fluid phase the so-called ferrofluids have been developed [5.108]. These are multiphase systems consisting of magnetic solid particles of nanometersize dispersed into a non-magnetic carrier fluid. From the theoretical side magnetic ordering in fluid matter is discussed recently with respect to the order–disorder transition [5.109] and the spatial structures of ferromagnetic liquids [5.110]. From the experimental side ferromagnetism in a single-phase fluid was observed in liquid 3He [5.111]. Concerning metallic systems the Curie temperatures TC of all known ferromagnetic metallic materials are lower than their liquidus temperatures TL. This is why ferromagnetism is not observed in stable metallic melts. However, metallic melts can be undercooled significantly below TL. As discussed in Section 3 of Chapter 3 undercooling experiments on Co᎐Pd alloys have shown unambiguously the onset of magnetic ordering if the temperature of the undercooled melt is approaching the Curie temperature of the respective alloys. No undercoolings below TCL were reported with the exception of one study [5.112]. In this study, small drops with diameter of approximately 2 mm were undercooled in a high frequency electromagnetic levitation apparatus. Undercoolings of about 10 K below the Curie temperature of the liquid Co80Pd20 alloy were achieved. Simultaneously, magnetic properties were measured. For the analysis of the nucleation behaviour of Co᎐Pd melts, it is assumed that the onset of nucleation is described by Eq. (5.48). Co᎐Pd is a solid solution. It solidifies very rapidly from the undercooled melt with crystal growth rates of the order of 30 m s⫺1 at undercoolings of about 300 K [5.113]. In the case of Co᎐Pd alloys it is therefore justified to assume that one nucleation event initiates solidification. Once the first nucleation process occurs solidification takes place by the growth of the crystalline phase into the undercooled melt, i.e. the product of Eq. (5.48) should be set to unity. In a typical levitation undercooling experiment the cooling rates are small so that steady-state nucleation can be assumed and transient effects can be neglected. Using

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Eq. (5.18) the temperature dependence of the nucleation rate is calculated, taking into account the characteristic data for the thermodynamic properties of the Co80Pd20 alloy. The nucleation rate was computed for homogeneous nucleation and for heterogeneous nucleation. The catalytic potency factor f(ϑ) = 0.28 for heterogeneous nucleation was chosen such that the experimental result of undercooling as observed in levitation undercooled drops of Co82Pd18 alloy of diameter 6 mm is described (cf. black dot) [5.71]. The results of the calculations are shown in Figure 5.15. The horizontal lines represent different experimental conditions: a 6 mm drop of 1 g in mass, cooled from the liquidus temperature to the nucleation temperature within a time of 100 s (straight line), a drop of 1.5 mm diameter in mass of 10 mg, cooled with the same cooling rate and a small drop of 60 m in diameter of 1 g mass which was rapidly cooled from the liquidus temperature to the nucleation temperature within a time of 1 s. The intersection points of these lines with the curves of the temperature-dependent nucleation rates define the undercoolings achievable under the specified experiment conditions. For the sample 6 mm in diameter, the intersection point of the horizontal line with the curve of homogeneous nucleation rate defines the ultimate limit of undercooling under such conditions to be Thom = 474 K, corresponding to a temperature of the undercooled melt far below the Curie temperature of Co82Pd18 alloy. This means that if

Figure 5.15. Homogeneous (solid curve) and heterogeneous (dashed curve) nucleation rate Iss as a function of temperature T for Co82Pd18. The solid circle gives the result of a levitation experiment on a sample of 6 mm diameter cooled at a rate of 30 K/s resulting in a maximum undercooling of T = 335 K. TCS and TL represent the Curie temperature of solid Co82Pd18 and the liquidus temperature, respectively [5.71].

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heterogeneous nucleation can be completely avoided the sample should be undercooled far below the magnetic transition temperature TC. Such a high undercooling has, however, not yet been achieved. From the computed curves of the nucleation rates one can estimate that under same experiment conditions the undercooling may increase by about 10 K if the droplet diameter is decreased from 6 to 1.5 mm keeping the experiment time constant and by about 100 K if the size of the sample is further lowered to 60 m at simultaneously enhanced cooling rate or decreased experiment time. But such a small sample cannot be processed in the levitation device. Moreover, direct measurements by contactless pyrometry become very difficult at such small sample size. In another approach, to achieve undercoolings below the Curie temperature a third element was alloyed to the binary Co᎐Pd samples to reduce the difference between the liquidus temperature and the Curie temperature. In doing so some boundary conditions have to be taken into account. First, the ternary alloy should be completely miscible to avoid any problems with poly-phase solidification in the undercooled melt. It is well known that in particular eutectic systems the liquidus temperature is essentially reduced by the addition of an alloy component. But the concentration of the alloying element has to be kept below the solubility limit to ensure single-phase solidification. Second, the third element added to binary Co᎐Pd samples should be small in atomic size and/or should possess a suitable electronic structure to ensure that the magnetic interaction of the spins is not weakened. The higher the magnetic exchange energy the higher the Curie temperature expected. Table 5.3 gives a survey of the results on various ternary Co᎐Pd-based alloys [5.69]. Results are only considered for those alloys that solidify unambiguously by single-phase crystallization. The results of investigations on various ternary Co᎐Pd-based alloys are plotted in Figure 5.16. The values of alloys which do not solidify in a single-phase but in a two-phase crystallization process are also included. The full symbols represent the results of single-phase solidification type while the open symbols give the data for multi-phase ones. From Figure 5.16 it can be inferred that the addition of a third element as C, Al, Nd and Sm leads to an increase in the relative Curie temperature. However, as described previously, the alloys of multi-phase solidification type are excluded for further undercooling experiments. Undercooling investigations by electromagnetic levitation on the alloys containing Nd and Sm gave only small values of undercooling because of the strong tendency of Nd and Sm to form solid oxides, which act as heterogeneous nucleation sites. The only ternary alloy system, which gave useful undercooling results, is the Co᎐Pd᎐C system. But even in this case no undercooling was achieved below the Curie temperature despite the fact that the

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Table 5.3. Results of DTA experiments on ternary Co82Pd18-cMc alloys with M = C, Nd, Sm. Co82Pd18-cCc c=0 c = 0.5 c=1 c=2 c=3 c=4 Co82Pd18-cNdc c = 0.5 c=1 Co82Pd18-cSmc c = 0.5 c=1

TCS (K)

TS (K)

TL (K)

TrC =TC/TL

1284 1274 1264 1266 1269 1275

1580 1575 1566 1550 1553 1556

1626 1623 1619 1612 1605 1600

0.789 0.784 0.780 0.785 0.790 0.796

1293 1305

1573 1583

1631 1645

0.792 0.793

1291 1305

1574 1582

1631 1638

0.791 0.796

Note: TC , Curie temperature; TS , solidus temperature; TL, liquidus temperature; Trc = TC /TL relative Curie temperature.

Figure 5.16. Relative Curie temperatures of the solid phase of four different ternary alloys on the basis of Co᎐Pd-containing aluminium (diamonds), carbon (circles), neodynium (squares) and samarium (triangles). The data are obtained from DTA measurements and are plotted as a function of the relative Co/Pd concentration. Such a graph was chosen to show all results on a uniform scale in a uniform manner. The closed symbols denote single phase and the open symbols the multiphase solidification.

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difference between liquidus and Curie temperature was slightly reduced from 342 K (Co82Pd18) to 325 K (Co82Pd14C4). This difference amounts only to 17 K, but even though small it is larger than the difference between Curie temperature TCL and minimum nucleation temperature TN as measured for binary Co82Pd18 alloy, TN⫺TC 6 K. Assuming the same nucleation behaviour in binary and ternary alloys one would expect a possible undercooling of Co᎐Pd᎐C alloy below the Curie temperature apparently in contradiction to the experimental findings. Studies of the composition dependence of the undercoolability of Co᎐Pd melts by electromagnetic levitation experiments [5.71] and by melt fluxing technique [5.114] show that the relative undercooling T/TL depends on the composition on the Co-rich side of the phase diagram at which the nucleation temperature is near to TCL. Figure 5.17 (upper part) exhibits the Co-rich side of the phase diagram of Co᎐Pd as based on differential thermal analysis (DTA) data [5.65, 5.114, 5.115]. Also shown are the minimum nucleation temperatures measured in undercooling experiments on bulk samples employing the electromagnetic levitation technique and the melt fluxing technique [5.68, 5.71, 5.113, 5.114]. A good description of the experimental results at 50 at.% Co is obtained if a catalytic potency factor of f(ϑ) = 0.35 is assumed. Supposing that classical nucleation theory can be applied and that f(ϑ) is independent of the Co concentration, as observed for other solid solutions such as Ni᎐Cu [5.35, 5.65], it is possible to predict nucleation temperatures for all other alloy compositions [5.55] using Eqs. (5.48), (5.18), (5.3), (5.39), (5.38), (5.19) and (3.36). As shown in Figure 5.17 (dashed curve), the nucleation temperatures TNclass calculated by this classical approach are only in agreement with the measured nucleation temperatures for c ⬍ 70 at.% Co, where TC is smaller than TN. At high Co concentrations,when the temperature approaches TC , nucleation occurs at higher temperatures than predicted. To investigate this behaviour in more detail undercooled melts of Co100-cPdc alloys at four different compositions (c = 18, 25, 30, 50) were statistically studied with respect to their undercooling behaviour. These concentrations cover both ranges: one in which the nucleation temperature TN is close to TCL (c ⱕ 25) and the other in which TN is significantly higher than TCL (c ⱖ 30) [5.68]. Each sample was undercooled approximately 100 times under the same experimental conditions. The distributions of the measured relative undercoolings are plotted in Figure 5.18. The average levels of undercooling Ta, the average relative undercoolings Ta /TL and the half-width, W, of the distribution functions are compiled in Table 5.4 for the different alloys. The results are analysed within the model by Skripov [5.45]. For homogeneous nucleation the pre-exponential factor KV 1039 /m3/s, because all atoms in the specimen are potential nucleation sites ( N = NL/m). In

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Figure 5.17. Upper part: Co-rich side of the Co᎐Pd phase diagram. The liquidus (TL, asterisks) and solidus (TS, diamonds) temperatures as well as the Curie temperatures of solid samples (TCS, triangles) as determined by DTA investigations. Moreover, the nucleation temperatures as obtained from undercooling experiments on drops in mass of 0.5⫺1 g, using melt fluxing technique (squares) and electromagnetic levitation (circles) are shown. The dashed curve labelled TNclass shows nucleation temperatures calculated according to classical nucleation theory with a catalytic potency factor f(ϑ) = 0.35, while the full curve labelled TNmag shows the nucleation temperatures calculated if the model is extended by a magnetic contribution to the driving force for nucleation [5.55]. Lower part: Dependence of the pre-exponential factor K on the concentration as determined from the statistical nucleation analysis of results of undercooling experiments on four different Co᎐Pd alloys (solid circles). The uncertainty of the results is indicated by the bars. The magnitude of KV drastically changes at a concentration at which the curve of TNclass crosses the line of TCS [5.68].

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Figure 5.18. Measured distributions of the relative undercoolings T/TL and the probability density functions, P, of T/TL (solid lines) calculated within a model by Skripov for (a) Co50Pd50, (b) Co70Pd30, (c) Co75Co25 and (d) Co82Pd18 alloy [5.68].

the case of heterogeneous nucleation only those atoms that are in contact with the foreign phase are potential nucleation sites. This leads to a reduction in N and therefore in KV. According to Eq. (5.36) a plot of ln(⫺ln(1⫺FDF(T)) versus T 2/T 2 gives a straight line with slope ⫺C and intercept b. KV and the product of · f (ϑ)1/3 are deduced from C and b obtained from the straight-line fits. The values are given in Table 5.4. The corresponding P(1,T + T) functions calculated from Eq. (5.34) are plotted in Figure 5.18 (cf. solid lines). For the Co82Pd18 and Co75Pd25 alloys KV is 7–11 orders of magnitude higher than that for the alloys with higher Pd content (Co70Pd30 and Co50Pd50). The error in calculation of KV value was analyzed in detail in Ref. [5.46] by Monte Carlo simulations. For 100 experimental cycles, KV has a statistical error of about ⫾3 orders of magnitude in case of KV-values of 1039 m⫺3s⫺1 and of less than ⫾2 orders of magnitude for KV-values of 1021 m⫺3 s⫺1. Thus, the differences in KV of 7–11 orders of magnitude inferred by analyzing the undercooling results of the alloys with a low Pd composition (TN TCL) compared

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Table 5.4. Characteristic data of the alloys, data obtained from undercooling experiments and from their statistical analysis within a nucleation model of Skripov [5.45]. Alloy

TL (K) T (K) T/TL W (K) Sf (J/mol K)

Co82Pd18 Co75Pd25 Co70Pd30 Co50Pd50

1625 1572 1554 1531

328.0 329.5 326.5 326.0

0.202 0.209 0.210 0.213

5.5 5.0 8.5 8.5

7.47 7.47 7.47 7.23

b

⫺C

53.34 60.24 35.72 36.60

3.435 4.273 2.549 2.704

KV (s⫺1 m⫺3) f(ϑ)1/3 3⫻1030 4⫻1033 5⫻1022 1⫻1023

0.61 0.67 0.55 0.57

Note: TL, Liquidus ⫺temperature; Tav = TL ⫺ TN, average undercooling with TN the nucleation temperature; Tav/TL, the relative undercooling; W, Half width of the distribution function; Sf, entropy of fusion; b and C, parameters as inferred from the nucleation analysis; KV, pre-exponential factor of the nucleation rate according to Eq. (5.18) and f(ϑ)1/3, the product of the structural factor in the interfacial energy times the catalytic potency factor f(ϑ) of 1/3 power.

to those with a higher Pd composition (TN ⬎ TCL) are well outside the error limits of the experimental results. The statistical analysis of the undercooling results indicates that the nucleation mechanism changes within the very narrow compositional range of 25 ⬍ c ⬍ 30 at which the nucleation temperature approaches the Curie temperature of the liquid. Since all alloys are prepared by an identical procedure from the same batches of raw materials and processed in the same levitation facility, the concentration and nature of the heterogeneous nucleation sites in both alloys should be similar ( N depends only linearly on the alloy composition). Therefore impurities cannot explain the observed drastic change in KV. Moreover, Co᎐Pd is miscible in the entire composition range. Thus, an anomalous change of the nucleation behaviour of Co᎐Pd melts occurs, if the concentration of the alloys is altered. The lower part of Figure 5.17 shows the change in the pre-exponential factor KV as inferred from the statistical analysis of the undercooling experiments. It is obvious that the change in the nucleation behaviour as indicated by the sharp decrease in KV occurs in the compositional range at which the Curie temperature TC cannot be accessed by undercooling. Apparently, the undercooling is limited by the onset of magnetic ordering just above TC. If magnetic effects are tentatively assumed, the larger KV of the Co-rich compositions is not necessarily an indication for a larger N . N should be a constant in case of a unique nucleation process. A change in N as indicated if the nucleation temperature approaches the Curie temperature strongly suggests that effects that are not taken into account in the statistical analysis of nucleation rate as e.g. magnetic contributions to the activation energy G*, may play an essential role. Magnetic contributions may influence LS and/or GLS. Recently a thermodynamical model based on mean field theory of magnetism was developed to calcu-

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late such magnetic contributions [5.55] (cf. Section 5.5). The estimations of the magnetic contribution to LS using a broken bond approach have shown that this contribution is negligible compared with the classical contribution according to Eq. (5.19). As shown for pure Co, the magnetic contribution to GLS, however, is of significant magnitude. To describe the nucleation behaviour in the system Co᎐Pd, the magnetic contri bution Gmag (T, cN) = G mag (T, cN)NL/m to the difference of the Gibbs free energies of solid and liquid phase per unit volume is calculated at the composition cN of the solid nucleus. This calculation is made in the same way as described in Section 5.5.1 for pure Co. (T, c ) G N is then added to the classical contribution mag GLS = GLS/m, calculated from Eq. (5.39). The estimations are based on the following assumptions of the composition dependence of the parameters: The magnetic moment of Co᎐Pd alloys per atom was measured as a function of the composition for c ⬎ 78 at.% Co [5.60]. These results imply p (c) = 1.75 − 0.926 (1 − c) B .

(5.50)

The composition dependence of TCS is inferred by interpolation between the values measured by DTA for various alloys [5.114, 5.115, 5.116] (see the corresponding black line in Figure 5.17). For pure Co TCL lies 6 K below TCS [5.60], while for Co80Pd20 the difference amounts to 18 K [5.117] (see Figure 3.20). Based on these results a linear composition dependence of TCSL = TCS⫺TCL is assumed: TC SL (c) = TC S (c) − TC L (c) = (1 − c) 60 + 6 K.

(5.51)

The nucleation temperatures TNmag calculated by including the magnetic contribution to GLS are shown in Figure 5.17 (solid line). The fit to the experimental results is satisfactory in the full investigated compositional range 50 at.% Co ⱕ c ⱕ 100 at.% Co. Adding a magnetic contribution is essential to obtain a fit to the data over this entire composition range. For the calculation of the magnetic contribution it is necessary to consider that the composition cN of the solid nucleus differs from the nominal composition c0 of the melt. cN is estimated according to Eq. (5.38). The importance of the composition of the solid nucleus is particularly clear in the composition range 70 at.% Co ⬍ c ⬍ 75 at.% Co, where the experimentally observed nucleation temperatures are higher than those calculated with the classical approach (TNclass) and also above TCS. As shown in Figure 5.12 for pure Co at a temperature above TCS, there is no magnetic contribution to the driving force for nucleation. That for Co᎐Pd alloys there is a magnetic contribution in this composition range is a consequence of the solid nucleus having a higher Co content than the melt, as computed from Eq. (5.38).

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There are two discontinuities in the slope of TNmag versus composition. The one at 68 at.% Co corresponds to TCS(cN). The other at 72 at.% Co corresponds to TCL(cN), which is also where the maximum magnetic contribution is reached (see Figure 5.12). According to this model an undercooling into the ferromagnetic regime of the melt is not totally excluded. It only means that if the temperature is approaching the ferromagnetic transition temperature there will be some magnetic contributions to the activation energy to form nuclei of critical size. These are, however, of finite magnitude. Summarizing the results of investigations on magnetic Co᎐Pd alloys magnetic contributions can be neglected in the determination of the solid–liquid interfacial energy, however, they can alter the Gibbs free energy by about 5%, which in turn leads to a measurable change in the maximum undercoolability by about 20–30 K.

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[5.20] Spaepen, F., and Meyer, R.B. (1976) Scipta Metallurgica 10, 257. [5.21] Broughton, J.Q., and Gilmer, G.H. (1986) Journal of Physical Chemistry 84, 5749, 5759. [5.22] Curtin, W.A. (1989) Physical Review B 39, 6775. [5.23] McMullen, W.E., and Oxtoby, D.W. (1988) Journal of Chemical Physics 88, 1967. [5.24] Marr, D.W., and Gast, A.P. (1993) Journal of Chemical Physics 99, 2024. [5.25] Ohnesorge, R., Löwen, H., and Wagner, H. (1986) Physical Review E 80, 4801. [5.26] Tolman, R.C. (1948) Journal Chemical Physics 16, 758; (1949) 17, 118; (1949) 17, 333. [5.27] Gránásy, L. (1994) Materials Science & Engineering A 178, 121. [5.28] Gránásy, L. (1993) Journal of Non-Crystalline Solids 162, 301. [5.29] Gránásy, L., Pusztai, T., and Hartmann, E. (1996) Journal of Crystal Growth 167, 756. [5.30] Gránásy, L. (1998) Journal of Chemical Physics 109, 9660. [5.31] Spaepen, F. (1975) Acta Metallurgica 23, 729. [5.32] Thompson, C.V. (1979) Ph.D. Thesis, Harvard University. [5.33] Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., and Kelley, K.K. (1973) Selected Values of the Thermodynamic Properties of Binary Alloys (ASM, Metals Park, OH). [5.34] Gránásy, L., Börzsönyi, T., and Pusztai, T. (2002) Physical Review Letters 88, 206105. [5.35] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1988) Materials Science & Engineering 9, 85. [5.36] Willnecker, R. (1988) Ph.D. Thesis, Ruhr-University Bochum, Germany (in German). [5.37] Hoyt, J.J. Asta, M., and Karma, A. (2001) Physical Review Letters 86, 5530. [5.38] Li, D., Eckler, K., and Herlach, D.M. (1996) Journal of Crystal Growth 160, 59. [5.39] Dragnevski, K.I., Cochrane, R.F., and Mullis, A.M. (2004) Materials Science & Engineering A 375–377, 479. [5.40] Probstein, R.F. (1950) Journal of Chemical Physics 19, 619. [5.41] Wakeshima, H. (1954) Journal of Chemical Physics 22, 1614. [5.42] Kashiev, D. (1969) Surface Science 18, 389. [5.43] Kelton, K.F., and Greer, A.L. (1985) in: Rapidly Quenched Metals, ed. Steeb, S., and Warlimont, H. (Elsevier Science Publishers, Amsterdam), p. 223. [5.44] Greer, A.L., Evans, P.V., Hamerton, R.G., Shangguan, D.K., and Kelton, K.F. (1990) Journal of Crystal Growth, 99, 39. [5.45] Skripov, V.P. (1977) in: Current Topics in Mater. Science, Crystal Growth and Materials, Vol. 2, eds. Kaldis, E., and Scheel, H. (North Holland, Amsterdam), p. 23. [5.46] Hofmeister, W.H., Morton, C.W., and Bayuzick, R.J. (1998) Acta Materialia 46, 1903.

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[5.47] Morton, C.W., Hofmeister, W.H., Bayuzick, R.J., and Robinson, M.B. (1994) Materials Science & Engineering A 178, 209. [5.48] Thompson, C.V., and Spaepen, F. (1983) Acta Metallurgica 31, 2021. [5.49] Gaskell, D.R. (1981) Introduction to Metallurgical Thermodynamics (Hemisphere, New York). [5.50] Porter, D.A., and Easterling, K.E. (1992) Phase Transformations in Metals and Alloys, 2nd Edition (Chapman & Hall, London). [5.51] Kaufman, L., and Bernstein, H. (1970) Computer Calculations of Phase Diagrams (Academic Press, New York). [5.52] Kaufman, L. (1986) Proceedings of the International Conference on Alloy Phase Diagrams (American Society of Metals, Metals Park, OH), p. 59. [5.53] Chuang, Y.Y., and Chang, Y.A. (1987) Metallurgical Transactions A 18A, 733. [5.54] Bormann, R. (1994) Materials Science & Engineering A 178, 55. [5.55] Holland-Moritz, D., and Spaepen F. (2004) Philosophical Magazine 84, 957. [5.56] Kittel, Ch. (1976) Introduction to Solid State Physics (Wiley, New York). [5.57] Ashcroft, N.W., and Mermin, N.D. (1981) Solid State Physics (Saunders College, Philadelphia, PA/Holt-Saunders Japan, Tokyo). [5.58] Tebble, R.S., and Craik, D.J. (1969) Magnetic Materials (Wiley, London). [5.59] Reutzel, S. and Herlach, D.M. (2004) Materials Science & Engineering A375–377, 552. [5.60] Crangle, J., and Parsons, D. (1960) Proceedings Royal Society London A255, 509. [5.61] Hellwege, K.H. (1976) Einführung in die Festkörperphysik (Springer, Berlin). [5.62] Chikazimi, S. (1964) Physics of Magnetism (Wiley, New York). [5.63] Perepezko, J., and Paik, J.S. (1982) in: Novel Materials and Techniques in Condensed Matter, eds. Crabtree, G.W., and Vashishta, P. (Elsevier, Amsterdam), p. 57–88. [5.64] Perepezko, J.H. (1980) in: Rapid Solidification Processing: Principles and Technology II, eds. Mehrabian, R., Kear, B.H., Cohen, M. (Claitor’s Baton Rouge, Los Angeles), p. 56. [5.65] Cech, R.E. and Turnbull, D. (1951) Journal of Metals 191, 242. [5.66] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1986) Applied Physics Letters 49, 1339. [5.67] Thompson, C.V., and Spaepen, F. (1982) in: Materials Processing in the Reduced Gravity Environment of Space, ed. Rindone, G.E. (North Holland, Amsterdam), p. 603. [5.68] Schenk, T., Holland-Moritz, D., and Herlach, D.M. (2000) Europhysics Letters 50, 402. [5.69] Herlach, D.M., Holland-Moritz, D., Willnecker, R., Herlach, D., and Maier, K. (2003), Philosophical Transactions Royal Society London A 361, 497. [5.70] Powell, G.L.F. (1965) Journal of Australian Institute of Metallurgy 10, 223.

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[5.71] Herlach, D.M., Holland-Moritz, D., Schenk, T., Schneider, K., Wilde, G., Boni, O., Fransaer, J., and Spaepen, F. (1999) Journal of Non-Crystalline Solids 250–252, 271. [5.72] Davidchack, R.L., and Laird, B.B. (2000) Physical Review Letters 85, 4751. [5.73] Bendersky, L.A., and Ridder, S.D. (1986) Journal Materials Research 1, 405. [5.74] Yoon, W., Paik, J.S., LaCourt, D., and Perepezko J.H., (1986) Journal Applied Physics 60, 3489. [5.75] Bosio, P.L., Defrain, A., and Epelboin, I. (1966) Journal de Physique (Paris) 27, 61. [5.76] Herlach, D.M., Gillessen, F., Volkmann, T., Wollgarten, M., and Urban, K. (1992) Physical Review B 46, 5203. [5.77] Shechtman, D., Blech, I., Gratias, D., and Cahn, J.W. (1984) Physical Review Letters 53, 1951. [5.78] Shechtman, D., and Blech, I. (1985) Metallurgical Transactions A 16, 1005. [5.79] Kelton, K.F. (1993) International Matererials Review 38, 105. [5.80] Kelton, K.F. (1994) in: Intermetallic Compounds, Vol. 1, ed. Westbrook, J.H., and Fleischer, R.L. (Wiley, New York). [5.81] Bendersky, L. (1985) Physical Review Letters 55, 1461. [5.82] Holzer, J.C., and Kelton, K.F. (1991) Acta Metallurgica 39, 1833. [5.83] Holland-Moritz, D., Herlach, D.M., and Urban, K. (1993) Physical Review Letters 71, 1196. [5.84] Holland-Moritz, D., Schroers, J., Herlach, D.M., Grushko, B., and Urban, K. (1998) Acta Materialia 46, 1601. [5.85] Holland-Moritz, D. (1998) International Journal of Non-Equilibrium Processing 11, 169. [5.86] Schroers, J. (1997) Ph.D. Thesis, Rheinisch-Westfälische Technische Hochschule Aachen, Germany (in German). [5.87] Holland-Moritz, D. (2001) Materials Science & Engineering A 304–306, 108. [5.88] Holland-Moritz, D. (2002) in: Quasicrystals, eds. Suck, J.-B., Schreiber, M., and Häussler, P. (Springer, Berlin), p. 233–248. [5.89] Gayle, F.W., Shapiro, A.J., Biancaniello, F.S., and Boettinger, W. (1992) Metallurgical Transactions 23A, 2409. [5.90] Faudot, F., Quivy, A., Calvayrac, Y., Gratias, D., and Harmelin, M. (1991) Materials Sciente & Engineering A 133, 383. [5.91] Gratias, D., Calvayrac, Y., Devaud-Rzepski, J., Faudot, F., Harmelin, M., Quivy, A., and Bancel, P.A. (1993) Journal of Non-Crystalline Solids 153 & 154, 482. [5.92] Grushko, B., Wittenberg, R., and Holland-Moritz, D. (1996) Journal of Materials Research 11, 2177. [5.93] Grushko, B. (1993) Journal of Non-Crystalline Solids 153 & 154, 489. [5.94] Grushko, B. (1993) Phase Transitions 44, 99.

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[5.95] Grushko, B., and Holland-Moritz, D. (1997) Materials Science & Engineering A 226–228, 999. [5.96] Gödecke, T., and Ellner, M. (1997) Zeitschrift für Metallkunde 88, 5. [5.97] Grushko, B., and Holland-Moritz, D. (1997) Journal of Alloys & Compounds 262–263, 350. [5.98] Gödecke, T., and Ellner, M. (1997) Zeitschrift für Metallkunde 88, 5. [5.99] Beeli, C., Nissen, H.-U., and Robadey, J. (1991) Philosophical Magazine Letters 63, 87. [5.100] Yokoyama, Y., Miura, T., Tsai, A.P., Inoue, A., and Masumoto, T. (1992) Materials Transactions JIM 33, 97. [5.101] Audier, M., Durand-Charre, M., and de Boissieu, M. (1993) Philosophical Magazine B 68, 607. [5.102] Gödecke, T., and Lück, R. (1995) Zeitschrift für Metallkunde 86, 109. [5.103] Urban, K., Holland-Moritz, D., Herlach, D.M., and Grushko, B. (1994) Materials Science & Engineering A178, 294. [5.104] Nelson, D.R., and Spaepen, F. (1989) in: Solid State Physics 42, eds. Ehrenreich, H., Seitz, F., and Turnbull, D. (Academic Press, New York), p. 1. [5.105] Schroers, J., Holland-Moritz, D., Herlach, D.M., and Urban, K. (2000) Physical Review B 61, 14500. [5.106] Kelton, K.F., Gangopadhyay, A.K., Lee, G.W., Hannet, L., Hyers, R.W., Krishnan, S., Robinson, M.B., Rogers, J., and Rathz, T.J. (2002) Journal of Non-Crystalline Solids 312–314, 305. [5.107] Croat, T.K., Kelton, K.F., Holland-Moritz, D., Rathz, T.J., and Robinson, M.B. (1999) in: Materials Research Society Symposium Proceedings, Vol. 553, eds. Dubois, J.-M., Thiel, P.A., Tsai, A.-P., and Urban, K. (Materials Research Society, Warrendale, PA), p. 43. [5.108] Rosensweig, R.E. (1985) Ferrohydrodynamics (Cambridge University Press, New York). [5.109] Nijmeijer, M.J.P., and Weis, J.J. (1995) Physical Review Letters 75, 2887. [5.110] Groh, B., and Dietrich, S. (1997) Physical Review Letters 7, 749. [5.111] Lang, T., Moyland, P.L., Sergatskov, D.A., Adams, E.D., and Takano, Y. (1996) Physical Review Letters 77, 322. [5.112] Albrecht, T., Bührer, C., Fähnle, M., Maier, K., Platzek, D., and Reske, J. (1997) Applied Physics A 65, 215. [5.113] Volkmann, T., Wilde, G., Willnecker, R., and Herlach, D.M. (1998) Journal of Applied Physics 83 3028. [5.114] Wilde, G. (1997) Ph.D. Thesis, Technische Universität Berlin, Germany (in German). [5.115] Görler, G.P., Private communication. [5.116] Schneider, K. (1997) Diploma Thesis, University of Köln, Germany (in German). [5.117] Reske, J., Herlach, D.M., Keuser, F., Maier, K., and Platzek, D. (1995) Physical Review Letters 75, 737.


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Chapter 6

Crystal Growth in Undercooled Melts 6.1. 6.2. 6.3. 6.4.

Kinetics of the Advancement of a Solid–Liquid Interface Departures of Local Equilibrium Stability Analysis Sharp-Interface Model 6.4.1 Growth in a Pure System 6.4.2 Solidification in a Binary System 6.4.3 Superlattice Structures in Intermetallics 6.5. Phase-Field Model 6.6. Transition from Faceted to Non-Faceted Growth 6.7. Experimental Data and Model Predictions 6.7.1 First Experiments 6.7.2 Measurements on Pure Nickel 6.7.3 Measurements on Dilute NiB and NiZr Alloys 6.7.4 Measurements on Intermetallic Compounds 6.7.5 Measurements on Semiconductors 6.7.6 Effect of Convective Flow and Solute Diffusion 6.7.7 Influence of Local Non-Equilibrium on Rapid Dendritic Growth References

197 204 216 227 227 234 241 244 247 256 256 258 259 261 263 265 270 273


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Chapter 6

Crystal Growth in Undercooled Melts 6.1. KINETICS OF THE ADVANCEMENT OF A SOLID–LIQUID INTERFACE

An important problem, which is arising in considering the solidification problems, concerns the kinetics of the advancement of the solid–liquid interface. The motion of the interface might be considered for various degrees of the driving force and various kinetic coefficients for the advancing interface. In this section, we consider recent developments in evaluation of the kinetic coefficients for solidification of metallic systems. The description of kinetic effects during rapid crystal growth in deeply undercooled melts can be explained using the rate theory [6.1, 6.2]. It is assumed that the attachment of single atoms from the liquid to the crystallization front takes place via atomic diffusion processes over an activation threshold, which is built up by the energy potential of the surrounding atoms. Usually, it is assumed that the energy barrier corresponds to the activation energy for atomic diffusion. Calculating the rates with which the atoms undergo a transition from the liquid into the solid state, RS, and vice versa the backward reaction, RL, the growth velocity, V, results one has V = da ( RL − RS ) ,

(6.1)

where da is assumed to be the typical interatomic spacing. To illustrate this situation an asymmetric double well potential is considered schematically as shown in Figure 6.1. The asymmetry is given by the difference of ~ =G ~ G ~ , and the barthe chemical potentials between solid and liquid state, G LS L S rier height corresponds to the activation energy for atomic jumps, Ga. The transition rates RL and RS can be evaluated assuming a thermal activation of single atoms across the barrier according to an Årrhenius law. Thus, Eq. (6.1) can be rewritten as ⎡ ⎛ G LS ⎞ ⎤ ⎛ G LS ⎞ ⎤ ⎛ Ga ⎞ ⎡ 1 exp = 1 − exp V = da f v exp ⎜ − − − V ⎥ ⎢ ⎢ 0 ⎜⎝ k T ⎟⎠ ⎜⎝ − k T ⎟⎠ ⎥ , ⎝ RG T ⎟⎠ ⎢⎣ ⎥⎦ ⎥⎦ ⎢⎣ B B

(6.2)

where fv characterizes a typical atomic vibration or thermal frequency of single atoms in the liquid state of the order of 10111013 Hz. 197

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~ and a barrier height of Figure 6.1. An asymmetric double-well potential with asymmetry of G LS Ga. The temperature T is assumed to be below the equilibrium melting point TE. The asymmetry energy represents the difference of the free energies between the solid and the liquid states, while the barrier height is assumed to be the activation energy for atomic diffusion. The rates with which single atoms migrate from the solid into the liquid and vice versa are denoted by RS and RL, respectively.

Generally, Eq. (6.2) adopts the factor V0 that represents the speed limiting the attachment of atoms at the interface. In the recent theories, this speed has been merely assumed to be defined by thermal fluctuations or diffusion jumps of atoms. However, recent molecular dynamics (MD) simulation results exhibit more complicated behaviour for the atomic ensemble around the interface leading to the reexamination of the classic models of interface kinetics. Starting from the classic models of crystal growth we analyse these novel results, which follow from the recent kinetic models and results of atomistic simulations. We consider V0 as a temperature-dependent factor in Eq. (6.2) representing ~ . the maximum growth velocity at infinite thermodynamic driving force G LS ~ ~ Eq. (6.2) is linear in GLS at GLS/kBT 1, therefore, by linearization of this equation one gets V = (V0 /k BT ) GLS = k Tk ,

(6.3)

where Tk = TET is the interfacial kinetic undercooling relative to the melting point TE. This equation is true for flat surfaces.1 Consequently, the kinetic coefficient k in Eq. (6.2) is the constant of proportionality between the velocity V of the Equation (6.3) can also be obtained from Eq. (4.19) for flat interface, R → , and with the ~ = G, where is an atomic volume. thermodynamic driving force G LS 1

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Crystal Growth in Undercooled Melts

199

Table 6.1. Models for kinetic mechanisms of crystal growth. Models

Characteristics

Contradictions

Model of Wilson [6.4] and Frenkel [6.5]

The timescales for diffusive atomic motion govern growth kinetics for pure metals. The term V0 in Eq. (6.4) is related to atomic diffusivity D of the liquid phase with the temperature dependence of V0 by D ~ exp(H f /kBT), so that V0 Da/2, where a is a thickness of the adjacent liquid layer to the crystal surface and the mean free path for the process. Interface advancement requires significant structural rearrangement as atoms transform from liquid to crystal and becomes negligibly small at low temperatures

Contradictions with experiment: (a) pure metal crystals grow readily even at low temperatures, i.e. even at effectively zero values of diffusivity D; (b) the maximum interface velocity predicted by Eq. (6.2) assuming V0 as the atomic diffusion speed is much less than the maximum growth velocity measured on pure nickel samples at large undercoolings [6.6]

Model of Turnbull [6.7, 6.8]

The crystal growth kinetics for pure metals is governed by the frequency of collisions of liquid atoms with the crystal-growth surface. The speed of sound VS is an upper limit for the term V0 in Eq. (6.4), so that the kinetic co efficient is limited by k VS H f /(kBTE2)

Contradictions with molecular dynamics (MD) simulation data and density-functional theory: the values of k are substantially smaller than the upper bound given by the speed VS

solid–liquid interface and the interfacial undercooling Tk. After linearization of Eq. (6.2), the kinetic coefficient k is given by k =

V0 H f k BTE2

,

(6.4)

where H f is the latent heat per atom. The classic kinetic mechanisms of crystal growth can be divided into two main groups: growth limited by the diffusion jumps of atoms and growth limited by atomic collisions. These are both allowed to compute V0 in Eq. (6.4) and systematized in Table 6.1. The complexity of experimentation with metallic systems has put forward the theoretical predictions and atomistic simulation into one of the forefronts of recent investigations [6.3]. Particularly, the density-functional theory and Lennard–Jones

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MD simulations have been developed intensively for theoretical modelling of the crystal growth kinetics. The MD simulations provided ability to verify the mechanisms of crystal growth and compare their predictions for kinetic coefficient of growth with the previous models [6.4, 6.5] and the experimental data. The linear melt growth kinetic coefficient k has also been calculated on the basis of a simple extension of the modern density-functional theory of freezing, taking into account of the strong subsurface ordering of the melt. The result of the densityfunctional theory has been expressed via easily measured parameters of bulk crystal and melt in agreement with existing data for crystal growth kinetics. The outcomes from atomistic and density-functional theory are summarized below. The first work on MD simulations has been published by Broughton et al. [6.9] in which the MD equivalent of a directional solidification experiment has been represented. These authors considered, in particular, the crystallization of a Lennard–Jones liquid as the motion of the (100) fcc interface, which has been measured for a range of temperatures below the melting point. Variation of the velocity V allowed one to determine the velocity V–interfacial undercooling Tk relationship from which the kinetic coefficient k has been extracted. One of the main MD results found by Broughton et al. is the demonstration of remaining high crystallization velocities even in the temperature regimes where the liquid diffusivity vanishes, D 0. This result contradicts with the prediction of the Wilson–Frenkel model [6.4, 6.5], therefore the modification of V0 has been proposed in the form [6.7] V0 = (fada/d)(3kBT/m)1/2, where m is the atomic mass, (3kBT/m)1/2 the average thermal velocity, da the interplanar spacing, d the distance an atom must move to reach the crystal, and fa the fraction of atomic collisions with the growth surface that are effective for crystallization. Substituting this expression into Eq. (6.4) gives the following expression for the kinetic coefficient: k =

f a da ⎛ 3k BT ⎞ d ⎜⎝ m ⎟⎠

1/ 2

H f k BTE2

.

(6.5)

In addition to the MD simulations based on the Broughton et al. technique of directional solidification simulations, the other atomistic methods have been developed. The free solidification technique [6.10–6.12] deals with the thermostats applied to the entire system (bulk undercooled system considered) and the slope of velocity versus interface undercooling again provides a measure of k. An interatomic potential method based upon pseudopotential perturbation theory [6.13] involves imposing high pressures to the atomistic systems in the direction normal to the boundary. The entire system is maintained at the zero-pressure melting point (as in the free solidification technique) but the actual melting point is varied due to the imposed pressure through the Clausius–Claperyon relation.

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201

1500.0

Interface velocity (cm/s)

1000.0

Finite-k fluctuation method Free solidification method Imposed pressure method

500.0

0.0 −500.0 −1000.0 −1500.0 −30.0

−20.0

−10.0

0.0

10.0

20.0

30.0

Undercooling (°K)

Figure 6.2. Comparison of predictions from different MD methods for the interface velocity V. Data are shown for the [100] growth direction of solidifying Ni crystals. For each data set, the computed value of the kinetic coefficient agrees to within numerical uncertainty. The value of k has been obtained from the slope of calculated V(TK) function.

The velocity V versus interfacial undercooling Tk relationship can be found using coexistence of the solid and liquid in the pressure–temperature diagram and the undercooling is expressed in terms of pressure. Fluctuation methods can be implemented in equilibrium MD simulations, and provide an independent check of values derived from the non-equilibrium atomistic simulations. For example, the kinetic coefficient k can be extracted in these methods by tracking changes in the total number of solid atoms arising from the pressure fluctuations during constantvolume MD simulations [6.14]. Asta et al. [6.15] presented a comparison between results from the above described fluctuation, free solidification and imposed pressure methods for crystallization of Ni. They demonstrated an equivalence between the different MD methods for computing the kinetic coefficient k (Figure 6.2). The value of k for Ni is found to be 0.39 ± 0.04 m/(s/K), which is about five times smaller than the upper bound2 given by the collision-limited model (see Table 6.1). However, the

2

The upper limit in growth kinetics given by the collision-limited model of Coriell and Turnbull is k = VS H f /(kBTE2) 1.22.0 m/(s/K) for nickel crystals growing in its own melt, which has the speed of sound VS = 20004000 m/s.

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relatively small value of the kinetic coefficient (in comparison with Coriell and Turnbull’s upper bound) is consistent with experimental measurements for Pb [6.16] and P [6.17]. The magnitude of the computed kinetic coefficient via various MD simulation methods (Figure 6.2) is also in agreement with the model of Broughton et al. [6.9] for the Lennard–Jones system. A comparison has been made with the modified model proposed by Jackson [6.18]: instead of Eq. (6.5), the following kinetic expression k =

f a da ⎛ 3k BT ⎞ d ⎜⎝ m ⎟⎠

1/ 2

H f

⎛ H f ⎞ exp ⎜ k T ⎟ k BTE2 ⎝ B ⎠

(6.6)

has been proposed. Figure 6.3 shows a comparison of the solid–liquid interface velocity V obtained from MD simulations for elemental metals versus the velocity predicted by the linear law (Eq. (6.3)) with the kinetics given by Eq. (6.6). The results shown in the figure indicate that the model of Broughton provides a good prediction of the [100] and [110] kinetic coefficient for the metals. However, the predictions for [111] direction required that the anisotropy of the crystalline planes is known (see discussion and explanation on 144 of Ref. [6.19]). 105

v (cm/s, simul.)

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104

Ni(FBD) 100 Ni(FBD) 100 Ni(FBD) 110 Ni(VC) 110 Au(VC) 100 Au(VC) 110

Ag(VC) 100 Ag(VC) 110 Cu(FBD) 100 Cu(FBD) 110

103

102 2 10

103

104

105

v (cm/s, BGJ)

Figure 6.3. Comparison of the Broughton–Gilmer–Jackson (BGJ) model (slightly modified by Jackson [6.18] in the kinetic coefficient, Eq. (6.6)) with free solidification MD simulations for various metals. The MD results represent results for [100] and [110] growth directions.

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The density-functional theory-based model of Mikheev and Chernov [6.3] gives a physical interpretation of the diffuse-interface. It also yields predictions for the crystalline anisotropy in k, which appears to be in rather close agreement with recent simulation results for k in each of the low-index interface orientations [6.17, 6.20]. Indeed, the ordering within a two-phase diffuse zone is associated with the release and dissipation of latent heat of crystallization, which is compensated by a gain in chemical potential. According to the model of Mikheev and Chernov [6.3], dissipation occurs within the diffuse interface, induced by ordering along all reciprocal lattice vectors G. In such a case, the energy dissipation balance per unit area is described by k BT 2

∑∫D

−1 G

2 J L dz = SV GLS ,

(6.7)

G −s

where L and S are the average densities in liquid and crystal, respectively. The diffusivity DG, which characterize the atomic arrangement rate in the G wave and can be expressed via a frequency of thermal motion of particles, H(kz), in the G density wave, is described by [6.21] DG = H S ( k x ) /k x2 .

(6.8)

In Eq. (6.8), S(kx) is the maximum of the structure factor at the wavenumber kx, while H is the half-width at half-height of the dynamic structure factor S(k,) at the same wavenumber k = kx.3 Therefore, from the equality (Eq. (6.7)) of the dissipated free energy and the free energy gain with diffusivity (Eq. (6.8)), one can find the linear law (Eq. (6.3)) with the kinetic coefficient given by k =

H f H S ( k x )b . k BTE2 N rl AS

(6.9)

Here Nrl is the number of reciprocal lattice vectors in the minimal set (i.e. Nrl = 8 for fcc) and b the correlation length in the liquid, i.e. the inverse half width of S(k) As is shown in Ref. [6.19], the typical frequency, H, of atomic motion reaches a minima, H(kx), at the same wave vectors, k = G, when the conventional static structure factor of the liquid, S(k), reaches a maxima. Experimentally, the frequency H of atomic motion in simple liquids is determined from experimental energy spectrum of thermal neutrons scattered by a simple liquid and resulting in the dynamic structure factor of this liquid. This dynamic factor shows sharp minimum at the neutron scattering directions where static structure factor is maximal. Also, the frequency H gives the relaxation time (kx) = H1 of density fluctuations given by the inverse half width of the dynamical structure factor S(k = kx,). 3

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evaluated at the main peak, AS the factor governing the anisotropy of k and is given as AS = N rl−1

b

∑

G = Gl

.

(6.10)

G

Here, the widths G are related to the correlation length by G = cos( G)b, when G is the angle between the vector G and the normal to the solid–liquid interface. As an example, for the low-index growth directions in fcc materials, the anisotropy factors are A100 1.7, A110 2.4 and A111 2.2. The Mikheev–Chernov model predicts a ratio 100/111 = 1.41 in agreement with the MD simulation results (cf. Figure 6.3). Also, the complete correct hierarchy for the kinetic coefficients for the (111), (100) and (110) faces of Au, Cu and Ni might be predicted from the model of Mikheev and Chernov. However, absolute values for these coefficients, following from the Mikheev–Chernov model, are approximately two times lower than those from MD simulations. This discrepancy may be understood and eliminated if one takes into account the vibration frequency, H, in Eqs. (6.8) and (6.10), making the dissipation rate in Eq. (6.7) and the kinetic coefficient higher within the “semi-solid” diffuse interface as compared to the bulk liquid [6.22]. Thus, the discrepancy is attributed to the assumption in the Mikheev–Chernov model that bulk liquid properties describe the dynamics within the solid–liquid interface region. 6.2. DEPARTURES OF LOCAL EQUILIBRIUM

In the absence of gradients of temperature, pressure and matter, a system exists in a homogeneous state that might be the equilibrium state, reaching the local minimum of free energy (the local maximum for entropy). When the system undergoes solidification, even temperature is below the equilibrium melting point for a substance or below the liquidus temperature for alloys, the system might be considered by local thermodynamic equilibrium for the case of slow interface advancing. In this case, the local volumes within the system reach local minima in free energy. From the statistical point of view, the local volumes can be represented within the system by their own atomic ensembles, and local equilibriumness is characterized by the statistical distribution function given by the first-order term of its expansion [6.23]. In their classic form, the theory of solidification [6.24] and the theory of crystal growth [6.2] assume a local equilibrium at the interface and bulk phases, which is an excellent approximation for many metallic systems solidifying at small interface velocities. With the increase in the solidification velocity due to increase in

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undercooling in the liquid, the condition of local equilibrium is absent at the solid–liquid interface. By an extension of classic theory, the local equilibrium condition was relaxed by taking into account a kinetic interface undercooling and deviations from chemical equilibrium at the interface. Under the assumption of collision-limited growth [6.8], a kinetic undercooling Tk can be determined using the linear approximation for the driving force of crystallization such as (see Chapter 4) Tk =

k BTE T ⎛ V − Vs ⎞ ln ⎜ H f ⎝ Vs ⎟⎠

(6.11)

where Vs is the speed of the sound. At small undercoolings, the series expansion of the logarithmic term can be cut off after the first term yielding Tk V/k, where k = H f Vs /(kBTE2) the kinetic growth coefficient. However, the speed of sound for the attachment of atoms to the interface is only the limiting speed for the advancement of the interface, and the atomic ensemble around the interface exhibits more complicated behaviour leading to the lower value of k than it is given by the collision-limited growth theory. Therefore, in the linear approximation, deviation from local equilibrium due to atomic kinetics at the interface is expressed by the following expression: Tk = V/ k

(6.12)

with the kinetics defined by the coefficient k =

H f k BTE2

V0

(6.13)

where the speed V0 is given by the following models (i) Modified Broughton–Gilmer–Jackson model [6.18] (developed from the MD simulations of crystallization of a Lennard-Jones liquid) V0 =

fda ⎛ 3k BT ⎞ ⎜⎝ m ⎟⎠

1/ 2

⎛ H f ⎞ exp ⎜ ⎟ ⎝ k BTE ⎠

(6.14)

(ii) Density-functional theory-based model of Mikheev and Chernov [6.3]: V0 =

H S ( k z )b . N1 AS

These are confirmed by the atomistic simulations.

(6.15)

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Figure 6.4. Near-equilibrium solidification of an alloy of composition co with strong solute partitioning (left) and rapid solidification with complete solute trapping (right). cS* and cL* denote the compositions of solid and liquid at the interface.

Deviations from local equilibrium at the solidification front are due to kinetic effects in pure systems and in alloys, additionally due to the dependence of the partition coefficient on the solidification velocity. Therefore, concerning alloys, besides a kinetic undercooling of the interface due to the atomic attachment, kinetics also due to deviations from the local chemical equilibrium can occur. The first experimental evidence for deviations from chemical equilibrium has been reported by Baker and Cahn [6.25, 6.26]. These authors have given convincing evidence that in the eutectic system of ZnCd with a retrograde solidus line the maximum equilibrium solubility can be remarkably exceeded by rapid solidification. In general, the solid state shows a lower solubility of solute in solvent compared with the liquid state. As a consequence, solute is piled up in front of the solid–liquid interface during solidification under near-equilibrium conditions. A concentration gradient develops perpendicular to the solidification front, which changes locally the conditions of crystallization and may eventually lead to the constitutional undercooling of the liquid in front of the solid–liquid interface. This situation is schematically illustrated in Figure 6.4 (left). If, however, the velocity of the advancing solid–liquid interface increases and becomes comparable with the atomic diffusive speed VDI the atoms of the solute can no longer escape from the solidification front and are trapped in the solid. The concentration of solute in the solid state exceeds the equilibrium concentration and the

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alloy becomes supersaturated due to the effect of solute trapping4 caused by the increasing solidification velocity V. To explain the behaviour of solute redistribution at the interface [6.25, 6.26], several models of solute trapping were suggested [6.27–6.29]. The process of solute trapping formulated in the work of Baker and Cahn [6.25, 6.26] has been described in detail by Aziz and Kaplan [6.30], who derived the nonequilibrium solute partitioning function k(V,c) for both diluted and concentrated alloys in the form k (V , T , cL* ) =

ke + V/VDI , 1 − (1 − ke )cL* + V/VDI

(6.16)

where the driving force for the solute redistribution at the interface is given by ⎛ (B − A ) ⎞ ke (cL* , cS* , T ) = exp ⎜ − ⎟⎠ RG T ⎝

and (c, T ) = (c, T ) − RG T ln c.

is the chemical potential. The symbol refers to the differences of the thermodynamic potentials in the liquid and the solid state with respect to components A and B. The validity of the Aziz model has been evidenced by laser surface resolidification experiments on Si-containing Bi impurities [6.31]. Simultaneously, during melting, and solidification the transient electrical conductivity has been measured. These measurements allow one to determine the growth velocities [6.32], while Rutherford-backscattering experiments on the rapidly solidified samples allow to determine the concentration profile. The interrelation of the results of both investigations leads to a correlation of the velocity and the partition coefficient as depicted in Figure 6.5. The “best-fit” partition coefficients versus interface velocity shown in Figure 6.5 was found for VDI = 32 m/s, i.e. comparison [6.31] for the solute partitioning was given for the small and moderate growth velocity in the range V/VDI = 0.0625 to 0.4375. The same results for moderate velocities have been summarized in Ref. [6.33] by the concentration-depth profiles obtained from Rutherford backscattering spectrometry. Therefore, a good comparison can be made for the small and 4

The term “solute trapping” has been introduced to define the processes of solute redistribution at the interface, which are accompanied by (i) the increase of chemical potential [6.25] and (ii) the deviation of the partition coefficient for solute distribution towards unity at very high velocities from its equilibrium value (independently of the sign of the chemical potential) [6.30].

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Figure 6.5. The dependence of the partition coefficient k(V) on the velocity V. Squares, experimentally data determined by laser pulse re-solidification experiments on SiBi alloy [6.31], line: prediction by the Aziz model [6.30] (Eq. (6.16) in the dilute limit cL* → 0).

moderate solidification velocity once the deviation from chemical local equilibrium is taken into account at the interface. Miroshnichenko [6.34] points out the fact that in the splat cooling process for the binary Al, Mg, AlMn, CuSn and CuSb systems cooled at a rate of more than 106 K/s the concentration in the core of dendrites equals the initial chemical composition of the alloys. The concentration in the dendritic cores changed discontinuously at a cooling rate equal to 106 K/s. This phenomenon can be determined as a transition to chemically partitionless or diffusionless solidification [6.35, 6.36] with the complete solute trapping, k(V) = 1, at some finite solidification velocity.5 Giving a detailed discussion on “diffusive velocity at the interface” in Aziz’s model, Cook and Clancy [6.37] investigated solidification kinetics of Lennard–Jones heterostructures using non-equilibrium MD simulations. For unconstrained growth 5 Analytical conditions for diffusionless phase transition have been formulated first by Temkin [6.35]. In his model, Temkin assumed that the diffusionless transition occurs if the time for irreversible transfer of a particle from metastable (liquid) phase into symmetric (crystalline) phase becomes less than the time for diffusional exchange of places in the metastable phase. In essence, Temkin assumed that the diffusion process is absent during the phase transition and he avoided the problem of complete solute trapping with the finite interface velocity. The analysis of the Temkin’s model and Monte-Carlo simulations shows that in the system with the infinite diffusion speed in bulk the diffusionless growth occurs with the zero diffusion coefficient, i.e. with the absence of the diffusion process itself [6.200].

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on the (100) orientation, these authors presented evidence for solute segregation in a binary A–B system while the velocity of the solid–liquid interface was at its slowest. When the interface attained its steady-state re-growth value of 4 m/s, complete solute trapping was found: the liquid ahead of the interface and the solid behind it had the identical chemical composition equal to the initial (nominal) content. The situation was similar to that shown in Figure 6.4 (right): at higher solidification velocities partitionless crystallization took place leading to a solid being far from chemical equilibrium. Comparing their results with the existing solute redistribution models,6 Cook and Clancy concluded that none of the models of solute segregation predict the complete solute trapping observed in the MD simulations. In contrast with the results of experiments [6.34] and MD simulation [6.37] in which the transition to the complete solute trapping occurs at a finite solidification velocity, Eq. (6.16) gives the complete solute trapping that occurs only in the limit of infinite velocity, i.e. k(V → ) = 1. Aziz and co-workers [6.38–6.40] provided experimental tests of various kinetic models for rapid solidification of a SiAs alloy. Measurements in liquid and resolidifying layers of the SiAs alloys by pulsed-laser melting have provided data on the temperature–velocity relationship and solute trapping of a planar solid– liquid front [6.39]. The main result of investigations of these authors is the absence of solute drag effect in rapid solidification and a good description of the experimental data for solute partitioning by means of the continuous growth model (CGM) in the formulation of Aziz and Kaplan [6.30]. However, the predictions of the CGM for solute trapping deviate significantly from the experimental data at high growth velocities V, in the range of 2 m/s (see e.g., Figure 9 in Ref. [6.40]). Consequently, the Aziz model of solute trapping is confirmed for small and moderate interface velocities and the model exhibits disagreement with experimental data at higher velocity when complete solute trapping occurs in alloy systems. The disagreements of results of experiments and of MD simulations with the models of solute redistribution, which assume the deviation from equilibrium at the interface only, can be attributed to the increasing influence of the local nonequilibrium solute diffusion around the interface at high solidification velocity. In the case of rapid solidification of the Si-9 at.%As alloy with planar interface [6.40], one may predict the steep transition to chemically partitionless solidification with the complete solute trapping using the factor essential influence of the local non-equilibrium solute bulk diffusion near the interface at finite solidification velocity [6.41]. 6

For solidification of LennardJones alloys along the (100) orientations, Cook and Clancy [6.37] compared results of their MD simulations with the continuous growth model of Aziz [6.29, 6.30] and of Jackson et al. [6.27].

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It has been shown [6.42, 6.43] that taking the relaxation of the diffusion flux J to its local equilibrium steady-state value one may assume local non-equilibrium in the solute diffusion field. Such a description leads to the model including the relaxation constitutive equation D

J (r, t ) + J (r , t ) + D c = 0

t

(6.17)

for the diffusion flux with the relaxation in the scale of the characteristic time D . Within the local equilibrium limit, D → 0 , we obtain from Eq. (6.17) to the standard classic Fickian description J (r) = −D c with the instantaneous response of the flux J to the existence of the solute gradient c in the point characterized by the radius vector r at time t. Equation (6.17) is known as the Maxwell–Cattaneo equation in the context of heat transport7 and it implies the finiteness of the solute diffusion propagation with the finite speed VD = ( D / D )1/ 2 . Combining this equation with the balance law

c + ⋅ J = 0,

t

(6.18)

we obtain to the hyperbolic partial differential equation D

2c c + = D 2 c. 2

t

t

(6.19)

Equation (6.19) combines the dissipative (by diffusion) and propagative (by the wave mechanism) solute redistribution in bulk system. A solution of Eq. (6.19) predicts the propagation of concentration disturbances with the finite diffusive speed VD, i.e. the sharp front of the concentration profile moves into bulk with the speed VD. We will describe the propagation of the diffusion profile in the task with a dot source by a potency cF = constant, which is located in an initial moment t = 0 at a point x = 0 with the initial homogeneous distribution of solute c0. Then, we have c(t,0) = cF, c(0,x) = c(t,) = c0, and ( c/ t)t=0 = 0. Under these conditions,

7 Equation (6.17) has been introduced first by James Clerk Maxwell in 1867 [6.201] and proposed further by Carlo Cattaneo in 1948 [6.202] for a damped version of Fourier‘s law by introducing a heat-relaxation term. It has been also used by P. Vernotte (1958) [6.203] in a problem of the speed of propagation of thermal signals. Therefore, on some occasions, Eq. (6.17) is called as Maxwell–Cattaneo–Vernotte equation or equation of Cattaneo and Vernotte.

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the solution of Eq. (6.19) under the assumption of constant values of D and D is described in [6.44]: (i) For the region beyond the diffusion front, 0 x/t VD: ⎛ x⎞ ⎛ x⎞ t c(t , x ) = c0 + (cF − c0 ) exp ⎜ − ⎟ + (cF − c0 ) ⎜ ⎟ ∫ f (t , x ) dt ⎝ la ⎠ ⎝ la ⎠ t = x/VD

(6.20)

with f (t , x ) =

exp( −t/ 2 D )

(t

2

− x 2 /VD2

)

1/ 2

⎡ (t 2 − x 2 /VD2 )1/ 2 ⎤ I1 ⎢ ⎥, 2 D ⎣ ⎦

(ii) For the point at the diffusion front, x/t = VD: ⎛ x⎞ ⎛ t ⎞ c(t , x ) = c0 + (cF − c0 ) exp ⎜ − ⎟ ≡ c0 + (cF − c0 ) exp ⎜ − ⎝ 2 D ⎟⎠ ⎝ la ⎠

(6.21),

(iii) For the region ahead of the diffusion front, VD x/t , c(t , x ) = c o .

(6.22)

Here la = (D D )1/ 2 is the characteristic scale of the solute diffusion layer and I1 is a modified Bessel function of the first order. In contrast with the concentration profile described by the partial differential equation of parabolic type (which gives the propagation of a solute with infinite speed VD → ) the profile according to Eq. (6.20) has a sharp front with discontinuity described by Eq. (6.21) moving with the speed VD. The front as given by Eq. (6.21) divides the space into a region where the diffusion occurs and a region in which the diffusion is absent (Eq. (6.22)). This simple illustration of Eqs. (6.20)–(6.22) may lead to the following conclusion. As soon as the interface velocity V becomes comparable by magnitude with the speed VD, it is necessary to take into account the relaxation of the solute diffusion flux at its steady state (similarly to Eq. (6.17)) and to describe the problem of solute redistribution using equation of type (6.19). Thus, the local non-equilibrium can be taken into account at the interface by introducing of solute diffusion speed VDI = D/da and in bulk phases by introducing the bulk solute diffusion speed VD = (D/D)1/2. Taking the solute diffusion speed VD as a limiting speed of solute profile, one can show [6.42, 6.43] that when the interface velocity V approaches the speed VD, the concentration in the liquid reaches the initial concentration c0 of a binary system.

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In this case, constitutional undercooling ahead of the interface becomes equal to zero [6.45] and an alloy may solidify by partitionless at the finite solidification velocity V = VD. As we have already mentioned, experiments [6.25, 6.26, 6.34] have provided evidence that the initial composition during rapid solidification at some finite solidification velocity V has been reached. Analytical results [6.42–6.45] have qualitatively explained these experimental data on rapid solidification of binary systems. In the case of local non-equilibrium solute diffusion described by the Maxwell–Cattaneo model (6.7), one may suggest a generalized function for solute partitioning at the interface. Within the approximation of a dilute alloy and assuming the absence the diffusion process D = 0 at V VD, solute partitioning function takes the form [6.46] ⎧ ⎡1 − (V/V )2 ⎤ k + V/V D DI ⎦ e ⎪⎪ ⎣ , k = ⎨ 1 − (V/V )2 + V/V D DI ⎪ ⎪⎩ 1,

V/VD < 1,

(6.23)

V/VD ≥ 1.

In the local equilibrium limit VD → , the solute partitioning function (6.13) transfers into the function (6.16) for the dilute alloys, CL 1. In addition, Eq. (6.23) gives the complete solute trapping at the finite velocity V = VD, and quantitatively describes experimental data on solute trapping in the SiAs alloy [6.39, 6.40] for all regions of the solidification velocities investigated (see Figure 1 in Ref. [6.46]). However, the function (6.23) gives solute partitioning for averaged quantitative description of the SiAs alloys. As an extension of Eq. (6.23), one may consider the solute partitioning function for both diluted and concentrated alloys. This yields [6.47] ⎧ (1 − V 2 /VD2 )ke + V/VDI , ⎪ k = ⎨ (1 − (V/VD ) 2 )[1 − (1 − ke )cL* ] + V/VDI ⎪1, ⎩

V/VD < 1,

(6.24)

V/VD ≥ 1.

With the dilute alloys, cL* 1, Eq. (6.24) transforms into Eq. (6.23). Using the analytical solution [6.48] for the steady-state alloy solidification, the concentration at the planar interface is given by cL* = c0 / k with V VD , and cL* = c0 with V VD . Therefore, Eq. (6.24) can be rewritten as

(

)

⎧ 1 − (V / VD ) 2 [ ke + (1 − ke )co ] + V / VDI ⎪ , k=⎨ 1 − (V/VD ) 2 + V/VDI ⎪ ⎩1,

V / VD < 1, V / VD ≥ 1.

(6.25)

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Equation (6.25) describes the scheme presented in Figure 6.4 and recovers the following important cases for local non-equilibrium solidification. The dilute limit for local non-equilibrium at the interface [6.29] is recovered with c0 1 and VD → . The concentrated binary system for local non-equilibrium at the interface [6.30] is recovered with VD → . The dilute limit for local non-equilibrium at the interface and bulk liquid [6.46] is recovered with c0 1. Figure 6.6 exhibits deviations of theoretical predictions including deviation from equilibrium at the interface (Eq. (6.16)) at high solidification velocities. Introducing the deviation from equilibrium both at the interface and bulk liquid allows one to describe the whole set of experimental data. The complete solute trapping given by Eq. (6.25) occurs for Si4.5 at.%As for VD = 2.5 m/s and for Si9.0 at.%As for VD = 2.1 m/s (Table 6.2) in satisfactory agreement with experiment. An additional possibility to demonstrate the influence of deviation from local equilibrium in bulk liquid is by considering the interface response functions. As seen in Figure 6.6, the predictions of the model according to Eq. (6.16) begin to disagree with experimental data in the region 1.5 V (m/s) 2 of solidification velocities. One may note that at the same solidification velocity, i.e. below about V = 2 (m/s), the “interface temperature–velocity” relationship also exhibits a clear deviation from experimental data (see Figure 11 in Ref. [6.40]). One may attribute 1.2 Partition coefficient, k(V,C0)

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1

2

1

0.8 1'

2'

0.6 0.4 0.2

- Si - 4.5 at% As - Si - 9.0 at% As

0 0

0.5

1

1.5

2

2.5

3

Interface velocity, V (m/s)

Figure 6.6. Solute partitioning versus interface velocity for experimental data on solidification of SiAs alloys [6.40]. Curves 1 and 2 given by Eq. (6.16) describe experiment at small and moderate solidification velocities. Curves 1 and 2 given by Eq. (6.25) show the ability to describe experiment for a whole region of investigated solidification velocities for both alloys. Data for calculations are given in Table 6.1.

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Table 6.2. Interface diffusion speed VDI and bulk diffusion speed VD for binary systems used in calculations of the partitioning function k(V,c0) at the planar solid–liquid interface. Equilibrium partition coefficient is ke = 0.3 [6.40]. Equation

Binary system

Interface diffusion speed VDI (m/s)

Bulk diffusion speed VD (m/s)

Reference

(6.16)

Si4.5 at.%As

0.46 0.37 0.46 0.37 0.75

2.7

[6.39] [6.40] [6.39] [6.40] [6.46]

0.8 0.8

2.5 2.1

[6.47] [6.51]

Si9.0 at.%As (6.23) (6.25)

Si4.5 at.%As and Si9.0 at.%As Si4.5 at.%As Si9.0 at.%As

this deviation to the increasing influence of the local non-equilibrium solute diffusion around the interface. Analysis based on extended irreversible thermodynamics [6.49] gives an expression for the slope of the liquidus line in the kinetic phase diagram of local non-equilibrium steady-state solidification. As extension of previous analysis [6.50], this yields [6.51] ⎧ m ⎧⎪ ⎡ ⎡ ⎛k⎞ ⎛ k ⎞⎤ V ⎤ ⎫⎪ e ⎪ ⎨1 − k ⎢1 − ln ⎜ ⎟ ⎥ + (1 − k ) ⎢ln ⎜ ⎟ + (1 − k ) ⎥ ⎬ , VD ⎥⎦ ⎪⎭ ⎝ ke ⎠ ⎥⎦ ⎪ 1 − ke ⎪⎩ ⎢⎣ ⎢⎣ ⎝ ke ⎠ m(V ) = ⎨ ⎪ me ln ke ⎪ 1− k , e ⎩

V / VD < 1 V / VD ≥ 1. (6.26)

Equation (6.26) shows that the velocity-dependent liquidus slope m(V) has a constant value at V VD due to the absence of the solute diffusion ahead of the interface at finite velocity, when complete solute trapping occurs in the alloy which begins to proceed at V = VD. Within the local equilibrium limit VD → , Eq. (6.16) reduces to the function m(V ) =

me 1 − ke

⎧⎪ ⎛ k ⎞ ⎫⎪ ⎨1 − k + ln ⎜ ⎟ ⎬ , ⎝ ke ⎠ ⎭⎪ ⎩⎪

(6.27)

which adopts solute drag8 and deviation from local equilibrium at the interface [6.30, 6.50]. Taking the velocity-dependent liquidus slope m(V), the equation for 8

Solute-drag effect arises due to impurity resistance to motion of the grain boundaries [6.204] or interface boundaries [6.205]. The moving interface drags behind itself an impurity atmosphere with it, which in turn creates the resistance to motion of the interface.

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the response functions “temperature–velocity” and “concentration–velocity” at the planar interface holds TI = TE + m(V )cL* − V/ k ,

(6.28)

cL* = c0 /k (V ),

(6.29)

and

where k is the kinetic growth coefficient of atomic attachment to the interface. As it follows from the results of calculation (Figure 6.7) the model with local non-equilibrium at the interface and in bulk liquid (solid curve) predicts the resulting interface response function “temperature versus velocity” in quantitative agreement with experimental data over the whole range of velocities investigated. One may also note that the resulting solid curve gives evidence of the transition from the regime with solute drag (solid curve is coincident with the dashed-dotted curve at small and moderate growth velocities) to the regime in which the solute drag is absent (solid curve is coincident with the dashed curve at high growth velocities). The presence of the solute drag at small and moderate velocities can be interpreted as the existence of the solute profile developed ahead of the interface. The absence of the solute drag at high growth velocity can be interpreted as the disappearance of the solute profiles ahead of the interface. 1600 Interface temperature T1 (K)

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Si - 9 at.% As 1500

1400

1300

1200 0

0.5

1 1.5 2 2.5 Interface velocity V (m/s)

3

3.5

Figure 6.7. Predictions for the interface temperature – velocity relationship (Eqs. (6.28)–(6.29)) obtained for Si-9 at.%As system [6.41]. Dashed curve: model without solute-drag effect m(V) = [me/(1ke)]{1k[1ln(k/ke)]} and Eq. (6.16) with c0 1. Dashed-dotted curve: model with solute-drag (Eqs. (6.27) and (6.16)) with c0 1. Solid curve: Eqs. (6.26) and (6.23). Experimental data are taken from Ref. [6.40].

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Even though direct experimental evidence for the existence of local nonequilibrium in the solute diffusion field is absent till now in rapid solidification of alloys, the above examples confirm the possible existence of deviations from local equilibrium at the rapidly moving interface and within bulk phases during rapid solidification. Therefore, we will formulate the main conditions for rapid solidification under local non-equilibrium conditions. (i) The crystals may grow very fast from the undercooled melt due the largely increased driving force for the advancement of the interface accumulated in this way. Under the conditions of rapid solidification, deviations from local equilibrium may occur which have to be incorporated into the theory of crystal growth. (ii) For the case of solidification of pure system, the deviation from local equilibrium may be present at the interface. It can be considered as the kinetic undercooling of the interface, which drives atomic attachment from liquid to solid. (iii) In an alloy, the consequences of deviations from local chemical equilibrium are threefold. First, the partition coefficient becomes dependent on the growth velocity. Second, the liquidus and solidus lines approach close to each other. For these cases it will be sufficient to introduce into the theory the deviation from local equilibrium at the interface only. Third, in the extreme case if the solidification velocity is comparable to the atomic diffusive speed in bulk liquid the partition coefficient k(V) becomes unity and the liquidus and solidus lines coincide. Such conditions are of special importance in the preparation of metastable supersaturated solutions. At solidification velocities larger than the atomic diffusive speed in bulk partitionless crystallization takes place leading to the formation of a solid being far from chemical equilibrium. 6.3. STABILITY ANALYSIS

In the solidification of liquids, an initial solid–liquid interface is subject to unstable growth, which leads to various crystal patterns such as cellular, dendritic, banded, fractal. morphologies [6.2, 6.24, 6.52–6.54]. Typical pattern developed upon morphological instability is shown in Figure 6.8. A general scheme of changing the crystal morphology with the interface velocity can be considered in the example of one-phase solidification, i.e. when the liquid transforms into solid without precipitation of additional phases. Figure 6.9 shows schematically the steady-state growth morphologies that form in a liquid, as a result of morphological instability at a given interface velocity V in single-phase solidification. With a small velocity, an initially smooth interface remains planar up to a velocity equal to the critical velocity VC defined by the constitutional undercooling [6.55]. Above VC, the smooth interface becomes unstable and the interface exhibits a steady cellular morphology. By further increasing the velocity, a surface of cells may become unstable owing to the development of dendritic patterns.

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Figure 6.8. Typical morphology of a two-dimensional dendrite. Planar front

0

Cells

VC

Dendrites

Planar front

Cells

VA

V

Figure 6.9. Morphological diagram for solidification of binary systems illustrating the microstructural transitions “planar front” → “cellular structure” → “dendrites” → “cellular structure”→ “planar front”, with the increase in the solidification velocity V. Here VC is the velocity given by the criterion of constitutional undercooling, and VA is the velocity for absolute morphological stability of the interface.

At high interface velocity, dendritic patterns degenerate by the appearance of rapidly moving cells. A demarcation line at VA divides the regions between the interface instability, V VA, and the absolute stability, V VA, where the planar interface is morphologically stable against small perturbations of its form. This demarcation is usually known as the critical velocity VA for absolute stability of the planar interface. The sequence of growth morphologies (Figure 6.9) is well known from experiments on directional solidification and solidification in the undercooled state [6.56]. It has been demonstrated in computational modelling [6.57] of crystal growth as well.

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The theory of morphological stability was developed first by Mullins and Sekerka [6.58], who considered the stability of a spherical particle grown in a supersaturated solution, and stability of the planar front during directional solidification of a binary liquid [6.59]. In investigating the evolution of small harmonic perturbations of the planar interface [6.59], they provided a rigorous basis of linear morphological (in)stability at low solidification velocity. Particularly, Mullins and Sekerka introduced a concept of marginal stability for the wavelength of perturbation, which gives neutral stability of the plane when the amplitude of perturbation does not change with time. Behind the critical velocity of the marginal stability, the increase in the amplitude of perturbation with time may lead to cellular or dendritic interfaces. The theory [6.59] gave rise to a great number of investigations of morphological transformations due to linear instability of interfaces and non-linear behaviour of unstable interfaces. These are presented in an overview by Coriell and McFadden [6.60] and a monograph by Davis [6.61]. In its classic form [6.60], the theory assumes a local equilibrium at the interface, which is an excellent approximation for many metallic systems solidifying at small interface velocities. At large driving force for the advancement of the interface, and with the increase in its velocity, the analysis of Mullins and Sekerka can be modified. Trivedi and Kurz [6.62] extended the analysis of Mullins and Sekerka [6.59] for the case of rapid solidification, and introduced the stability functions to be dependent on the interface velocity. By taking into account the velocitydependent coefficient of solute redistribution (partitioning function), they developed an analytical model [6.63] of microstructure formation under directional solidification, ranging from low interface velocity up to high velocity VA of absolute morphological stability. This situation is in contrast to, e.g. directional solidification experiments in which the heat from the melt and the interface is transferred via the solidified part of the sample and the crucible walls to the environment. Under such conditions, the growth behaviour is constrained by the heat extraction during the experiment. Figure 6.10 illustrates schematically the different conditions for free dendritic growth in an undercooled melt (right) and for the constrained growth in directional solidification (left). The quantitative description of the heat and mass redistribution is based upon the thermal and mass transport equations. If transient effects in the development of the temperature and concentration fields are neglected, i.e. stationary conditions, the stationary thermal and diffusion equations hold for the one-dimensional case of a planar interface propagating with the velocity V in the z-direction perpendicular to the solidification front: 2 c +

V c =0 D z

(6.30a)

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Figure 6.10. A comparison of the conditions for the heat flow and the resulting morphologies of the solidification front in constrained growth (left-hand side) and free dendritic growth (right-hand side). GT is the temperature gradient ahead of the interface. Arrows indicate the direction of growth with the velocity V and the direction of removing of the latent heat released at the interface during growth.

2 TL +

V TL =0 aL z

(6.30b)

2 TS +

V TS =0 aS z

(6.30c)

where c is the concentration, D the diffusion coefficient in the liquid state, and aL and aS are the thermal diffusivities in the liquid and the solid state. Diffusion in the solid is neglected, since the diffusion coefficient in the solid is smaller by several orders of magnitude compared to the liquid state. The solutions of the differential Eqs. (6.30) yield an exponential decay of temperature and concentration ahead of the interface. Tiller et al. [6.55] showed that heat and mass transport in the concept of constitutional undercooling can lead to an instability of the solidification front even at directional solidification (positive temperature gradient in the liquid). This situation occurs if the increase in the liquidus temperature, due to the change of concentration at the solid–liquid interface, overcompensates the externally constrained temperature field. This model has been later on extended by Mullins and Sekerka [6.58, 6.59], taking into account additionally the stabilizing effect due to the Gibbs–Thomson effect. They showed that for both a spherical and a planar interface, considering the stabilizing effect of the Gibbs–Thomson effect leads to absolute stability of the solidification front if the solidification velocity approaches a critical value. At this velocity, the destabilizing effect due to the negative temperature gradient and due to the concentration gradient is compensated by the stabilizing effect due to the interface energy contribution. For the derivation of the condition of absolute stability, the interface is superposed by a perturbation

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of infinitesimal amplitude. Then, in first approximation the disturbance of the temperature and the concentration gradients by the perturbation can be neglected. For the interface of a sphere of radius R, the following expression in spherical coordinates is used to describe the perturbation of the interface: r = R + pYl , m (ϑ, )

(6.31)

where p is the amplitude of the perturbation and Yl,m the single spherical harmonics of mode l, m. The solution of the transport equations for the distorted system taking into account the curvature effect yields an expression for the dynamic behaviour of the amplitude, p(t)/ t. In the case p(t)/ t 0 the interface will be morphologically unstable, and the perturbation will propagate. If, however,

p(t)/ t 0, the perturbation will decay. Consequently, the condition p(t)/ t = 0 marks the transition to absolute stability. In contrast to absolute stability, relative stability exists if the propagation velocity of the perturbation is much lower than the propagation velocity of the solidification front, hence p1 p(t)/ t = R1 R(t)/ t. For pure metals, the growing sphere-like crystallite reaches absolute stability at a radius R 7r* [6.58] and relative stability at R 36r* [6.59] (r* is the critical nucleus size). In the case p R, the interface can be assumed to be planar. The investigations on the stability criteria of a planar interface are easier to be performed. Here, the perturbation of the interface can be described by z = p (t )sin( k x x )

(6.32)

where kx = 2 / is the wave number and the wavelength of the perturbation. The z-coordinate is perpendicular to the interface while the x-coordinate is parallel to the interface. The criterion of stability of a planar interface is also obtained from the condition p1 p(t)/ t = 0. In advanced stability analysis of Trivedi and Kurz [6.62] the result of computations lead to the following marginal stability criterion: S () = TE GT k P 2 − [ L L L + S S S + mc c ] = 0

(6.33)

with

L = L /( L + S ) and S = S /( L + S )

being the averaged thermal conductivities of the liquid and solid, respectively. L and S are the thermal conductivities of the liquid and solid phase. L, S and C are the thermal gradients in liquid and solid state, and the concentration gradient

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in the liquid phase, respectively. GT is the Gibbs–Thomson coefficient. Diffusion in the solid is neglected. The stability functions are defined by L = [ L − (V/aL )] /( S S + L L ),

(6.34a)

S = [ S − (V/aS )] /( S S + L L ),

(6.34b)

c = [ c − (V/D )] / ( c − (V/D )(1 − k E ))

(6.34c)

with L = (V / 2aL ) + [(V / 2aL ) 2 + k x 2 ]1/ 2 , S = −(V/ 2aS ) + [(V/ 2aS ) 2 + k x 2 ]1/ 2 , c = (V/ 2 D ) + [(V/ 2 D ) 2 + k x 2 ]1/ 2 . k is the partition coefficient. The main difference in the treatment by Trivedi et al. to the previous work by Mullins and Sekerka is the introduction of the stability functions that deviate from unity at large undercoolings. Differentiation of Eq. (6.33) with respect to kx yields, with regard to dS/dkx = 0, the velocity needed to reach absolute stability. In the case of the pure thermal stability and assuming the thermal diffusivities of solid and liquid are equal, aL = aS = a, the velocity to reach absolute stability of the growing dendrite is given by VAT =

a 2 H f K LGT TE

.

(6.35)

If the growth velocity V approaches the solidification speed VAT according to Eq. (6.35) the planar interface becomes stable. Taking into account the respective material parameters of pure Ni one calculates a critical velocity to reach absolute stability to be VA 2000 m/s. This corresponds approximately to the speed of sound and is much higher than the highest crystal growth velocities measured in undercooled Ni melts [6.64–6.67]. Even if kinetic effects in the interface undercooling on the stability analysis are taken into account [6.68, 6.69], absolute stability cannot be expected at the largest undercooling of T = 480 K, as reported so far for pure Ni [6.70]. If, however, heat transport is taken into account, not only in the undercooled melt but also into the solidified part of the sample the velocity to reach absolute stability can be substantially decreased. Such conditions are supporting,

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Figure 6.11. Optical micrograph of longitudinal through-thickness section of a melt-spun ribbon of Ni18 at.% B alloy [6.74]. The ribbon was produced with the spinning velocity of 120 m/s and with the thickness of 35 m. The crystal microstructure exhibits a transition from planar interface with solute segregation-free to cellular-dendritic patterns. The transition proceeds due to decrease in the interface velocity from V VA down to V VA. The bottom of micrograph corresponds to the boundary to the wheel surface.

for example, in melt-spinning or splat-cooling experiments. Figure 6.11 demonstrates the longitudinal cross-section of a ribbon produced by melt spinning. The transition from the macroscopically smooth interface to the cellular dendritic microstructures is clearly visible. This transition occurs with the decrease of interface velocity V below critical velocity VA for absolute stability of the planar front. Coriell and Sekerka [6.71] pointed out that under the conditions of rapid solidification a planar interface may be observed even if the velocity is less than VA. On the one hand, the time needed to establish the perturbation can be less than the crystallization time, which is of the order of 10 s for pure Ni undercooled by 300 K [6.66]. On the other, absolute stability through the observation of a planar interface may be simulated, if the sample dimension is less than the wavelength of the perturbation. In fact, a planar interface has been detected in atomized particles with dimension of a few m [6.72, 6.73]. For binary systems, the previous analysis of pattern formation and morphological stability [6.62, 6.63] can be extended for the case when the local non-equilibrium in bulk phases may play an essential role in rapid solidification (see Section 6.2). Particularly, a deviation from local equilibrium in solute diffusion may act on the rapid advancement of the solid–liquid interface because the interface velocity V can be of the order of or even greater than the solute diffusion speed VD in bulk phases. The diffusion speed [6.42] VD 0.110 m/s. In modern experiments on solidification of undercooled droplets, the interface velocity approaches [6.75] V 10100 m/s. Thus, the undercooling of liquids is sufficient for detecting solidification with the interface velocity comparable to or even larger than the diffusion speed.

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The analysis shows that the deviations from local equilibrium in bulk undercooled liquid and at the interface drastically affect both the solute diffusion and the interface kinetics [6.48]. Therefore we summarize recent results on the linear stability analysis for a rapidly moving interface under local non-equilibrium solute diffusion. In the presence of local non-equilibrium solute diffusion, it has been found [6.76] that the marginal stability criterion is described by 2 ⎪⎧ k + L L L + S S S − me c c = 0, V < VD , S ( k x ) = ⎨ GT x 2 V ≥ VD . ⎪⎩GT k x + L L L + S S S ,

(6.36)

where the functions L and S of thermal stability are defined by Eqs. (6.34a, 6.34b) and are similar to those derived by Trivedi and Kurz [6.62]. However, the function c of concentrational stability differs from the corresponding function derived previously in Ref. [6.62]. For the rapidly moving interfaces with velocity VVD, the function C is defined by ⎧ c − V/[D (1 − (V/VD ) 2 )1/ 2 ] , ⎪ c = ⎨ c − (1 − k )V/[D (1 − (V/VD ) 2 )1/ 2 ] ⎪0 ⎩

V < VD , V ≥ VD .

From this it follows that within the local equilibrium limit, VD→, one obtains the result given by Eq. (6.34c), which is similar to the special case of Trivedi and Kurz. Therefore, for VD → , Eq. (6.36) transforms to the criterion of marginal stability obtained in Ref. [6.62] on the basis of a local equilibrium approach to solute diffusion transport (Eq. (6.33)). The introduction of the finite diffusion speed VD into the model leads to the qualitative result, which is related to the transition to completely partitionless solidification. As Eq. (6.36) shows, with the finite interface velocity V VD, the solute diffusion ahead of the rapid interface is absent, and the morphological stability is defined by the stabilizing force GTkx2, due to surface energy, and the contribution LLL + SSS of temperature gradients, L and S. Using criterion (6.36), one can analyse qualitatively two different situations for solidification when (i) the latent heat is removed from the interface through the solid crystal phase (Figure 6.10, left), and (ii) the latent heat is removed from the interface inside the undercooled liquid phase (Figure 6.10, right). In case (i), with LLL +

SSS 0, the total heat flux is directed from the interface to the solid phase and the positive temperature gradient, in addition to the surface energy, stabilizes the form of the interface. In this case, with V VD, the morphological stability depends

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on the destabilizing action of the force mCC and the stabilizing force GTkx2+ LLL+ SSS. With the interface velocity V VD, solidification leads to the chemically partitionless pattern. A destabilizing action on the front is absent and the interface itself remains linearly stable against any small interfacial perturbation. In case (ii), one obtains LLL + SSS 0, and the temperature gradient destabilizes the interface. Therefore, if the absolute morphological stability is not reached by the steady balance GTkx2 = LLL + SSS, the interface is unstable, and the resulting interface may exhibit a cellular-dendritic pattern. Assuming the marginal stability hypothesis (suggested in Ref. [6.77]) one can compute a characteristic size for the solidified microstructure.9 A characteristic size R, selected by crystal microstructure in solidification (e.g. the dendrite tip radius), is related to the critical wavelength of interface perturbation as R = . Using this equality and assuming equality of thermophysical parameters of the liquid and solid, one can obtain a characteristic size R from Eq. (6.36). This yields ⎧⎛ GT /k x ⎪⎜ ⎪⎜⎝ m(V )c − 0.5 L L L + S S S ⎪ R=⎨ 1/ 2 ⎞ ⎪⎛ GT /k x ⎪⎜ ⎟ , ⎪⎩ ⎜⎝ −0.5 L L L + S S S ⎟⎠

(

(

)

⎞ ⎟ ⎟⎠

1/ 2

,

V < VD , (6.37) V ≥ VD .

)

In Eq. (6.37), the case V VD is true only for solidification in an undercooled liquid, i.e. when the temperature gradient is negative, T 0. For the case of solidification in the positive temperature gradient, T 0, the absolute morphological stability takes place at the interface velocity V smaller than the diffusion speed VD in the liquid. In Eq. (6.37) the following variables are introduced: ⎛ 1 ⎞ * = 1 / 4 , L = 1 − ⎜1 + ⎝ * P 2 ⎟⎠

( ) 2

T

c = 1 +

−1 / 2

⎛ 1 ⎞ , S = 1 + ⎜1 + 2 ⎝ * P ⎟⎠

−1 / 2

(6.38)

c

2k ⎛ 1 − V 2 / VD2 ⎞ 1 − 2k − ⎜ 1 + * P 2 ⎟⎠ ⎝

1/ 2

,

(6.39)

c

9

The marginal stability hypothesis [6.77] has been formulated to select a growth mode for stable steady-state dendritic tip. A main idea of the hypothesis was the assumption that the radius of a stable dendritic tip is exactly adjusted and defined by the scale of the characteristic wavelength extracted from the Mullins and Sekerka stability analysis [6.59].

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where PT = VR/(2a) and PC = VR/(2D) are the thermal and solutal Péclet numbers, respectively. With S = 0 in Eq. (6.36), one can find the criterion of absolute morphological stability for the planar interface given by the velocity VA. For non-isothermal solidification of a binary alloy, this yields aL D ⎧ T c ⎪VA + VA = TT + k Tc , ⎪ GT GT VA = ⎨ ⎪V T = aL T , T ⎪⎩ A GT

V < VD , (6.40) V ≥ VD ,

where VTA and VCA are the velocities of thermal and chemical absolute stability, respectively, TT = TQ = H f /Cp is the thermal undercooling and is equal to the unit of undercooling TQ, and TC = (k1)mc0 /k is the constitutional undercooling, which is necessary for solidification with the planar interface on the scale of solute diffusion. Additionally, TC is the non-equilibrium temperature interval of solidification between liquidus and solidus lines in the kinetic diagram of rapid solidification with planar interface. The criterion VTA = aLTT /GT in Eq. (6.40) is the same as that obtained by Trivedi and Kurz [6.62] using the advanced model for large growth velocities. The criterion VCA = DTC/(GT k) = D(k1)mc0/(GT k 2) in Eq. (6.40) is similar to that obtained by Mullins and Sekerka [6.59] for the case of small growth velocities, and re-derived by Trivedi and Kurz [6.62] for the case of rapid solidification. In addition, by introducing the finite speed VD in the model we reach a qualitative result. At the finite velocity V VD, due to the absence of the solute diffusion the interval between non-equilibrium liquidus and solidus lines is equal to zero, TC = 0. These lines converge in the kinetic phase diagram with V VD [6.48]; the absolute stability of the planar interface is defined only by the undercooling TT and relation between the thermal diffusivity aL and Gibbs–Thomson parameter GT related to the capillary effects. During isothermal solidification of a binary alloy, the stability of the planar interface is defined by the velocity VA =

m(V ) D ( k − 1)c0 < VD . GT k 2

(6.41)

This expression coincides with the expression given for the case of local equilibrium solute diffusion transport at VD → . However, a final form of the function VA(c0) is defined by the functions of solute partitioning, k(V), and the velocitydependent slope of liquidus line, m(V), in the kinetic phase diagram. The behaviour

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of these functions is rather different for the cases of local equilibrium and local nonequilibrium solute diffusion [6.48, 6.51]. For a quantitative comparison of the model predictions we have chosen experimental results on interface stability during rapid solidification of a SiSn alloy as presented by Hoglund and Aziz [6.78]. These authors measured a critical concentration of Sn for interface breakdown in a steady-state solidification after pulsed laser melting. The model predictions for the function c0(V) are compared quantitatively with experimental results [6.78]. As shown in Figure 6.12, the predictions of the model for interface stability with the local non-equilibrium diffusion (Eqs. (6.23), (6.26) and (6.41)) are consistent with the experiment. At the concentration Sn = 0.02 atomic fraction (see the extreme right experimental point in Figure 6.12, the discrepancy between the model with local equilibrium solute diffusion (dashed curve in Figure 6.12) and experiment gives the value of 38.90%. At the same concentration the alloy, the present model (solid curve in Figure 6.12) gives the discrepancy of 16.93% with experiment. Consequently, even better comparison with

0.1 Critical concentration (atomic fraction)

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Bulk Si data SOS data

0.01

VD 0.001 0.1

1

10

100

Interface velocity (m/s)

Figure 6.12. Critical concentration c0 above which a planar interface is unstable. Experimental points correspond to solidification of the SiSn alloy [6.78]. Circles: measurements performed on bulk single crystal Si(100); squares: measurements using Sn-implanted Si-on-sapphire (SOS) samples; curves: model predictions for interfacial absolute stability: dashed: local equilibrium diffusion and solute-drag effect (Eqs. (6.16), (6.27) and (6.41)); solid: with the local non-equilibrium diffusion (Eqs. (6.23), (6.26) and (6.41)), dashed-dotted line: V = VD is the limiting velocity for the absolute interface stability.

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the available experimental data can be obtained by using the present model of local non-equilibrium solidification. The stability analysis determines the criteria according to which a perturbation at the interface becomes stable or unstable. The question of the evolution of such a perturbation as a function of time led to discussions on dendritic growth. 6.4. SHARP-INTERFACE MODEL

Classical free boundary problems arise because the phase interface is considered to have zero thickness in the models of phase transformations. In this context, sharp discontinuities in properties, or jumps of fluxes and thermodynamic functions, occur across the interface. Sharp-interface models have provided successful descriptions of many physical phenomena in various systems [6.79–6.81]. In the theory of solidification the free boundary problem within the sharp-interface formulation has been established by Stefan. He has developed a model for the rate of melting of the polar ice-caps and icebergs. This problem remains as one of the biggest alloy solidification problems. Heat must be conducted from the oceans to the melting interface to provide the latent heat of melting and salt must be supplied as well since the equilibrium concentrations of salt in the liquid and solid differ. A method of solution for the general Stefan problem and its exact mathematical solution has been found for the local equilibrium solidification [6.82]. Various solutions for the Stefan problem with approximate, semi-analytical, and numerical solutions were obtained in applied sciences (for the overview, see Ref. [6.83]). In this section, two sharp-interface problems will be formulated. First, an appropriate solution of the dendritic solidification problem in a pure system will be given. Second, dendritic solidification of an alloy under non-equilibrium conditions will be considered as an extended formulation of the classic Stefan problem. 6.4.1 Growth in a pure system For pure systems, the heat transport determines the conditions for the propagation of the solidification front at small and moderate undercooling. For the growth at high undercoolings, kinetic effects play an important role in determining the growth shape and velocity of a crystal. Therefore, the sharp-interface model for pure system is formulated as follows. A transport in bulk is described by

T = a 2 T .

t

(6.42)

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The conditions at the interface, which moves with the velocity Vn and the temperature TI = TL = TS are defined by the Stefan balance − aL TL + aS TS = TQVn ,

(6.43)

and the velocity-dependent Gibbs–Thomson condition 2 ⎡

2d ( n ) ⎤ TI = TE + TQ ∑ ⎢ d ( n ) + ⎥ K i − Vn / k ( n ).

i2 ⎦ i =1 ⎣

(6.44)

In the above equations, the velocity Vn is directed along the nomal n to the interface, TQ is the adiabatic temperature of solidification defined by Thyp =TQ = H f /Cp, TE is the equilibrium temperature of solidification, Ki = 1/Ri are the mean curvatures of the interface with radii Ri, and indexes “L” and “S ” denote liquid and solid, respectively. The functions of anisotropic capillary d(n) and anisotropic kinetics (n) are defined by d ( n ) = d0 ac ⎡⎣1 + c ( nx4 + ny4 + nz4 ) ⎤⎦

(6.45a)

k ( n ) = k 0 ak ⎡⎣1 − c ( nx4 + ny4 + nz4 ) ⎤⎦

(6.45b)

with ac = 1 − 3 c , ak = 1 + 3 k , c =

4 c 4 k , k = . 1 − 3 c 1 + 3 k

(6.45c)

Here d0 = 0TE/(TQ H f ) is the capillary length, which is typically on the order of 1 nm, 0 is the mean value of the interfacial energy, ko the averaged kinetic coefficient of growth, c and k are the strengths of crystalline anisotropy, and the angle between the normal to the interface and the direction of growth. The sharp-interface model described by Eqs. (6.42)–(6.45) has been solved in various approximations. Particularly, these equations have been solved for the dendritic problem (for the overviews, see Refs. [6.84–6.87). First, Papapetreou [6.88] and, later on, Ivantsov [6.89] demonstrated that parabolic isotropic interfaces form stationary solutions for the transport equations (6.42)–(6.43). In the case of the isothermal solution, i.e. the temperature at the interface corresponds to the equilibrium melting temperature, TI = TE, the shape of a purely thermal

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dendrite is given by a paraboloid of revolution with the thermal undercooling TT = TIT [6.90, 6.91]10: TT =

H f C pl

PeT exp( PeT ) E1 ( PeT ) ≡ TQ Iv( PeT )

(6.46)

with E1 the first exponential integral function E1 =

∫

PeT

exp( − z ) dz , z

PeT = VR/(2aL) is the thermal Péclet number, R the radius at the tip of a dendrite, and Iv the Ivantsov function. The effect of curvature and kinetic effect, which reduce the melting temperature TE at the tip of a dendrite, lead to the non-isothermal case. From boundary condition (6.44), one can get the corresponding curvature and kinetic undercoolings. These are 2 ⎡

d (n ) ⎤ 2TQ , TR = ⎢ d (n ) + ⎥

2 ⎦ R ⎣ Tk = V/ k (n ).

(6.47)

The capillary undercooling stabilizes the interface and causes the decrease the tip radius at small tip radii. The total undercooling T, measured in the experiment consists of three contributions: T = TT + TR + Tk .

(6.48)

Equation (6.48) correlates the undercooling T to the product of velocity V and dendrite tip radius R. Thus an infinite number of possible growth modes exist. Either large dendrites grow slowly or small dendrites grow fast at a fixed undercooling. This is illustrated in Figure 6.13, where the solidification velocity V is plotted versus the dendrite tip radius R at fixed undercooling T for a pure system. As can be seen the curve for the non-isothermal Ivantsov solution (taking into 10

Ivantsov [6.89] and also Horvay and Cahn in 1961 [6.206] have found solutions for seven main shapes of crystals that satisfy the balance conditions at the interface under heat diffusion. These solutions were found for the quasi-equilibrium conditions of isotropic growth in a non-stationary regime (the growth velocity decreases as a square root of time) or in a steady-state regime of motion with a constant velocity along a selected coordinate direction (dendritic problem). Particularly, Horvay and Cahn found general steady-state solution in a form of a crystal with the shape of elliptical paraboloid.

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Figure 6.13. Dendrite growth velocity V as a function of dendrite tip radius R at constant undercooling T. Lines: isothermal and the non-isothermal Ivantsov solutions: open square: represents the extremum growth condition; closed circle: operation point for the growing dendrite as predicted by marginal stability analysis. For comparison, the open circle represents the operation point as measured on the transparent system succinonitrile [6.90].

account the curvature undercooling and neglecting the kinetic undercooling) passes through a maximum. At small dendrite tip radii the curve is cut off at a radius that corresponds to the critical nucleus size r*. For a long time, it has been assumed that the dendrites will grow with maximum velocity (extremum criterion) [6.91–6.93]. This fixes the operation point of the growing dendrite at the maximum in the V(R) relationship (cf. open square in Figure 6.13) and proposes a dependence of the velocity on undercooling according to a power law V ~ T 2. First measurements of the growth velocity as a function of undercooling seem to confirm such a dependence [6.64, 6.65]. But measurements of the growth velocity as a function of undercooling with an improved accuracy of the measurements indicate rather a dependence of V(T) according to a power law V ~ T 3 [6.66, 6.67] that contradicts the prediction of the extremum criterion. Direct measurements of both the growth velocity V and the dendrite tip radius R are not possible in metallic systems. For such investigations, transparent model systems are needed, in which V and R can be measured simultaneously. Glicksman et al. [6.94] report such measurements on the so-called pseudometallic system of succinonitrile (this system shows an entropy of fusion of comparable magnitude with metallic systems). The result of these experiments is represented by the open circle in Figure 6.13, which is located in a radius far above the extremum point.

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This observation has led Langer and Müller-Krumbhaar [6.77] to the suggestion that the operation point of the growing dendrite may be defined by the marginal stability analysis. The marginal stability hypothesis is based upon the consideration that dendrites (large radius R) approach the conditions for a planar interface. This is, however, unstable. Therefore, a critical dendrite radius Rms must exist, at which the tip of the dendrite loses its stability and begins to split. Accordingly, the region of stability extends between the extremum point and the marginally stable radius Rms. Furthermore, it is argued on the basis of experimental observations that during the growth of dendrites of parabolic shape they begin to develop sidebranches. These side-branches will broaden the temperature field leading to an enlargement of the effective tip radius in comparison with the original needle-like dendrite. Consequently, at large dendrite tip radii the curve approaches the marginal stability point Rms at which the dendrite tip becomes morphologically unstable (cf. solid circle in Figure 6.13). According to the condition of stability R Rms it is proposed that the operation point of the growing dendrite will be at R = Rms. This is in agreement with the experimental result obtained by Glicksman et al. [6.94] (cf. solid and open circles in Figure 6.13). The marginal stability analysis allows the calculation of the dendrite tip radius at its operation point under the assumption Rms = i, where i is defined as the critical wavelength for the Mullins and Sekerka instability of a planar interface. The marginal stability analysis leads to the following relation for the stable growth mode [6.77] * ≡

2 d0 a VR 2

(6.49)

where * is the stability parameter. Once the stability parameter is defined, one may compute the dendrite tip radius using Eq. (6.49) (see Table 6.3). The criterion of marginal stability leads to a constant value * = 1/(4 2). The experimental investigations on succinonitrile [6.94] as well as on Ni and CuNi alloys [6.95] indicate a good agreement between experiment and theory, if * = 1/(4 2) is used for the calculations of the dendrite growth velocity as a function of undercooling. Even though the marginal stability hypothesis is in good agreement with experiment, it leads to the instability of the dendritic tip due to the considered isotropic growth [6.85–6.87]. It is based on the erroneous assumption that steady-state growth is possible for a continuum of velocities and tip radii. The introduction of models with crystalline anisotropy [6.96–6.104] has made it possible to solve the steady-state growth equations (6.42)–(6.45) without uncontrolled approximation on the absence of anisotropy of growth. In these solutions, anisotropy has a critical role: there is a discrete spectrum of possible steady-state solutions, only the

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Table 6.3. Comparison of the stability conditions given by the marginal stability hypothesis and microscopic solvability theory (a1 = 1/0.42 and a2 = 0.3 as it follows from asymptotic analysis of Brener and Melnikov [6.87]). Marginal stability hypothesis, * = 1/(4 2) 0.025 Small Péclet numbers, RT 1 [6.77]

R=

d0 * PT

Microscopic solvability theory, * = *(c) (kinetics is not included)

Arbitrary Péclet numbers, RT 1 or RT 1 [6.62, 6.63]

R=

d0 , * PT (1 − n)

⎛ 1 ⎞ n = ⎜1 + ⎝ * P 2 ⎟⎠

Small Péclet numbers, RT 1 [6.103 6.105]

R=

d0 a1 7c / 4 PT

Arbitrary Péclet numbers, RT 1 or RT 1 [6.87]

R=

d0 (1 + a2 c PT2 ) a1 7c / 4 PT

−1 / 2

T

one with the largest velocity being linearly stable against dendrite tip splitting [6.96–6.102]. Under such circumstances, a discrete set of form-conservation solutions for the growth of dendrites is obtained; among those only solution to the maximum growth velocity is linearly stable against the splitting of the dendrite tip. It has been proved numerically [6.85] and analytically [6.87]. This solution (known as solvability condition) also gives a unique method to determination for the radius of the dendrite tip. This is valid for small and large Péclet numbers [6.100] in pure metals as well as in alloys [6.101]. Interestingly enough, an expression for the dendrite tip radius, which is equivalent to Eq. (6.49), results from solvability theory. The essential difference is that the stability parameter depends on an anisotropy parameter (see Table 6.3), which takes into account the direction of growth of a crystal along preferred orientations. Numerical simulations allow a quantitative correlation between the stability and the anisotropy parameter [6.102, 6.106]. The introduction of anisotropy allowed one to predict a correct shape of dendritic tip and the formation of dendritic secondary branches. In Figure 6.14, the shape of a dendrite around its tip and side-branches of the main stem are shown. It can be seen that fourfold anisotropy leads to the so-called “fins” origination and the side-branching appears consistent with this crystalline anisotropy. First, Brener [6.107] predicted analytically that the outer shape of the fins with fourfold symmetry sliced along the main axis of the growth direction show a behaviour that is asymptotically described by z = ax 5 / 3 ,

(6.50)

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Figure 6.14. Top: shape of the nearly parabolic tip of a fourfold anisotropic dendrite. The shape can be represented by a fourth-order correction to the parabolic shape (Eq. (6.50)). Bottom: typical dendrite growth of a xenon dendrite developed by the fourfold crystalline symmetry [6.108]. Close to the tip, fins grow by the power law z = bx5/3. Far from the tip side-branches grow due to thermal fluctuations on the anisotropic (axisymmetric) shape.

where the tip is placed at the origin and z is the height measured from the tip. The maximum thickness of these fins is predicted to behave asymptotically as z = bx5/2. Bisang and Bilgram [6.108] found that for xenon dendrites, power law (6.50) was a good fit even quite close to the tip. Hence, the power law offers one of the ways to characterize the tip size and location. Second, in the limit of small fourfold anisotropy of d(n) in Eq. (6.45), Ben Amar and Brener [6.109] found that the lowest-order term is proportional to cos(4φ), where φ is the rotational angle around the z-axis. Thus, at least close to the tip, the tip shape could be reasonably well described by z = ztip +

( x − xtip ) 2 2R

− A4 cos( 4φ )

( x − xtip ) 4 R3

,

(6.51)

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where (xtip, ztip) is the location of the tip and A4 = 1/96 is independent of anisotropy strength c [6.110], or dependent to second order on c, namely, A4 = c +12c2 [6.111]. Detailed shape measurements of the tips of three-dimensional NH4Cl dendrites were provided in a dendrite growth from supersaturated aqueous solution [6.112]. They have shown that the fourth-order fit (6.51) appears to provide the most robust description of the tip shape even though the power law according to Eq. (6.50) may give a good description of the tip shape. Third, the side-branching of the dendrites is due to the stochastic noise amplification along the anisotropic interface. The stochastic noise as thermal fluctuations are inherently present in real materials because the kinetic energy of atoms or molecules fluctuates by an amount being proportional to kBT. In most situations, these fluctuations are small and can be neglected, however, anisotropic interfaces are very sensitive to the noise. According to current theories, the formation of secondary branches behind the dendrite tip should be the result of the selective amplification of the thermal fluctuations along the sides of dendrites [6.113, 6.114]. However, experimentally observed side-branches have much larger amplitudes than explicable by thermal noise in the framework of the axisymmetric approach [6.114]. Therefore, Brener and Temkin [6.115] considered non-axisymmetric dendrite growth in the presence of anisotropy presence with the thermal fluctuations along the sides of a dendrite. They have shown that noise-induced wave packets generated in the tip region grow in amplitude, and spread and stretch as they move down the sides of the dendrite producing a train of side-branches. The amplitude grows exponentially as a function z = bx5/3 (as in Eq. (6.50) z is the distance from the dendrite tip). The main result of the analysis [6.115] was that the crystalline anisotropy amplifies thermal fluctuations which should, in this case, be large enough to account for the experimentally observed side-branching. Investigating the shape of the tip and the formation of side-branches of xenon dendrites, Bisang and Bilgram [6.108] concluded that thermal noise initiates the formation of side-branches in agreement with the analysis of Brener and Temkin [6.115]. The scaling behaviour of the three-dimensional dendrite found from the experiments [6.116] has been found in good agreement to analytical predictions of Brener and Temkin [6.117]. Numerical study of side-branching has also been in agreement with the Brener and Temkin predictions [6.118]. 6.4.2 Solidification in a binary system Modern techniques of experiments and advanced technologies of melting and solidification of metallic systems allow us to reach deep undercoolings, large temperature and concentration gradients, and high velocities of the phase transformations (Chapter 2). For example, in modern experiments [6.75, 6.119] interface

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velocities in the range of 10–100 m/s are measured during solidification of melts undercooled in a wide range of 10–400 K. For such large velocities of phase transformation, deviations from local thermodynamic equilibrium are likely to be present at the advancing interface. For the solidification processes, the first models for crystal growth [6.120, 6.121], which take into account deviations from local equilibrium at the solid–liquid interface due to solute trapping and kinetics of atomic rearrangement from the non-equilibrium liquid to solid crystal phase, were developed in the last two decades. A wide application of these models in describing the pattern formation in rapidly solidified materials lead to the conclusion that local non-equilibrium only at the interface may predict the velocity and microstructural features at small and moderate driving forces of solidification satisfactorily. At larger driving forces and higher solidification rates, several contradictions of the model predictions with experimental data were found. These contradictions are summarized as follows. (i)

Thickness of the solute boundary layer. As the crystal velocity V increases, the solute boundary layer shrinks, and at the higher solidification velocities its computed thickness approaches physically meaningless values lower than one interatomic distance [6.122]. (ii) Solute partitioning function k (V ) dependent on solidification velocity V and taking into account the deviation from local equilibrium only on the interface diverges from experimental data at higher growth velocities [6.46]. (iii) Solute-drag effect. In some cases, introducing this effect into the model of rapid solidification allows us to describe satisfactory experimental data [6.122]. However, a question about how many solute-drag effects should be introduced into the model still was open [6.123–6.126]. (iv) Sharp break point in the kinetic curves. After a sudden rise in solidification velocity, the rapid solidification mechanism of dendrite growth is found to change abruptly in alloys [6.126]. In contrast to numerous experimental data, the models of dendrite solidification [6.120, 6.121], which are taking into account deviations from local equilibrium at the interface, predict the gradual smooth increase of the solidification velocity with increasing undercooling. These contradictions can be avoided by taking into account deviations from local equilibrium both at the interface and in the solute diffusion near the interface (Section 6.2). Solidification of metallic liquids can be so fast that the interface velocity V is of the order of or even greater than the diffusion speed VD in bulk liquid. In this case, the approximation of local equilibrium may become unacceptable for description of solute diffusion in rapid solidification. Therefore, one should

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take into account finiteness of a speed for solute diffusion in bulk and at the interface for description of rapid solidification [6.42, 6.43]. A system of governing equations for rapid solidification under local nonequilibrium solute diffusion is described by [6.127, 6.128] the following (i)

transport in bulk

T

2c c = a 2 T , D 2 + = D 2 c,

t

t

t

(6.52)

(ii) conditions at the interface − aL TL + aS TS = TQVn , − DL cL = (cL − cS )Vn + D

[(cL − cS )Vn ]

t

TI = TE + m(Vn )cL + d ( ) K − Vn / k ( ), cS = k (Vn )cL ,

(6.53) (6.54)

The functions of anisotropic capillary d( ) and anisotropic kinetics k( ) are described by Eqs. (6.45). Particularly, from Eqs. (6.45), for two-dimensional space and cubic crystal symmetry these are defined by d ( ) = d0 [1 + c cos 4 ], k ( ) = k 0 [1 − k cos 4 ],

(6.55)

The velocity-dependent partition coefficient k(Vn) and the velocity-dependent slope of the liquidus line m(Vn) in the kinetic phase diagram are described by Eqs. (6.25) and (6.26), respectively. The system of equations (6.52)–(6.54) has been resolved for planar and nonplanar rapidly moving interfaces [6.48, 6.76, 6.129]. According to these solutions, the interface moving with the velocity V equal to or greater than the diffusion speed VD cannot change the concentration or create the concentration profile ahead of itself due to the fact that solute diffusion vanishes at V = VD. Therefore, solidification with V VD proceeds without solute redistribution ahead of the interface. It is in consistent with the first experiments on transition to diffusionless (completely chemically partitionless) solidification from the liquid melts in alloys and binary systems (see Refs. [6.2, 6.25, 6.34, 6.130, 6.131]). One of the remarkable features of rapid solidification is the existence of nonequilibrium morphological transitions in crystal patterns [6.132]. Equations (6.52)–(6.54) can also be used for analytic and numeric investigation of morphological transitions during rapid solidification of alloys. Here, the transitions in crystal morphology are considered in an example of isothermal solidification of

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undercooled binary liquid. A change of morphology is described using analytic solutions [6.76] and computational modelling [6.44, 6.57]. The change of the crystal morphology with the interface velocity can be seen in the example of single-phase solidification, i.e. when the liquid transforms into solid without precipitation of additional phases. Figure 6.9 shows schematically the steady-state growth morphologies which form in a liquid as a result of morphological instability at given interface velocity V, in single-phase solidification. The sequence of growth morphologies as shown in Figure 6.9 is well known from experiments [6.56, 6.133]. To test the possibility of transitions at high solidification rates, Eqs. (6.52)–(6.55) have been solved numerically for isothermal approximation of solidification by a finite-difference technique on a uniform computational grid. The details of numeric approximations are given in Ref. [6.44]. As soon as an initially planar interface becomes morphologically unstable, a cellular-dendritic pattern is selected. If the thickness of the solute diffusion layer ahead of the interface is enough to produce secondary branches due to morphological instability of the cellular interfaces, the dendritic pattern appears (Figure 6.15a).

Figure 6.15. Modelling of two-dimensional solidification of Cu30 at.%Ni alloy [6.44]: (a) dendrite patterns at V = 0.012VA; (b) dense cellular structure at V = 0.025VA; (c) almost chemically partitionless solidification at VA VVD (grey colour in solid shows the content cS different from the initial concentration c0); (d) diffusionless solidification at V VD.

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With increasing of undercooling, high growth velocity leads to shorter solute diffusion layer ahead of the interface and shorter wavelength of morphological instability of the interfaces. In this case, the transition from dendritic pattern to the dense cellular pattern occurs (Figure 6.15b). Cells and dendrites gradually degrade in their development as soon as deviation from equilibrium becomes large. Figure 6.15c demonstrates the degeneration of cellular array as the initial undercooling in a system increases. When the length of diffusional layer becomes shorter than the tip radius of dendrite or cell, the cellular-dendritic pattern does not grow. Beyond the absolute stability limit, V VA, the planar front is stable. Owing to stochastic noise in the growth kinetics, the crystal solid phase has no homogeneity in chemical composition. The solidification proceeds by the partition mechanism (see grey bands in solid phase (Figure 6.15c) that show chemical composition different from the initial nominal concentration c0). With the interface velocity greater than the solute diffusion speed in bulk liquid, V VD, the solidification with the initial composition occurs, cL = cS = c0. A system solidifies partitionless (Figure 6.15d) with the absence of the solute diffusion ahead of the interface. A transition from chemically diffusion controlled to diffusionless growth is considered as a process accompanied by the non-isothermal morphological transition of the solute diffusion-limited growth to thermally controlled growth of crystal patterns. A first explanation for such a transition has been presented by Kurz and Fisher [6.53] using the models of dendrite growth [6.120, 6.121]. They claimed that if dendrites reach the velocity VA of absolute chemical stability for solute diffusion a transition from a mostly solutal to a thermal growth is occurring with the sudden increase of the dendrite tip radius. It can be found that the models [6.120] predict a more gradual transition to thermal patterns than observed in experiments on rapid growth of dendrites [6.75, 6.131]. Introduction of the local non-equilibrium solute diffusion into the model leads to the conclusion that the complete transition from diffusion-limited to thermally controlled growth proceeds with the sharp change in the behaviour of the dendrite velocity and dendrite tip radius [6.127, 6.128]). Referring to the solution of Eqs. (6.52)–(6.55), one can extend the model of rapid dendritic growth to the case of local non-equilibrium solidification. The dendrite tip radius R and the dendrite tip velocity V as the main characteristics of primary dendrites are obtained from the steady-state model of an axisymmetric dendrite growth in undercooled binary system [6.129]. Hence, the model equations of the dendrite tip are summarized as follows. The final balance of the various undercoolings contributions to the dendrite growth is given by T = TT + TC + TN + TR + TK

(6.56)

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where TT = ( H f /Cp)Iv(PeT) is the thermal undercooling, TN = (mem)c0 is the undercooling arising due to the shift of equilibrium liquidus from its equilibrium position in the kinetic phase diagram of steady-state solidification, TR = 2d0 H f /(RCp) is the curvature undercooling due to the Gibbs–Thomson effect, and TK = V/k is the kinetic undercooling. Note that if the dendrite tip velocity V is equal to or greater than the solute diffusion speed VD the constitutional undercooling TC is equal to 0: Iv( Pec ) ⎧ , with V < VD , ⎪k v 1 − (1 − k ) Iv( Pec ) , TC = ⎨ ⎪ 0, V ≥ VD . ⎩

(6.57)

In Eqs. (6.56)–(6.57), the thermal Péclet number and solutal Péclet number are given by PeT = VR/(2a) and PeC = VR/(2D), respectively. The velocity-dependent non-equilibrium interval v of solidification in Eq. (6.57) is described by ⎧mc ( k − 1) / k , with V < VD , v = ⎨ 0 , V ≥ VD . ⎩0

(6.58)

In Eqs. (6.57) and (6.58), non-equilibrium solute partitioning coefficient k and the slope m of the liquidus line in the kinetic phase diagram are described by Eqs. (6.25) and (6.26), respectively. The liquid concentration cL* at the tip of paraboloid of revolution is given by c0 ⎧ , with V < VD , ⎪ c = ⎨1 − (1 − k ) Iv( Pec ) ⎪c , V ≥ VD . ⎩ 0 * L

(6.59)

The condition of the absence of constitutional undercooling at V VD follows from the analytical solution of the model of solute diffusion applicable for rapid solidification [6.48, 6.76, 6.129]. In this case, the liquidus and solidus lines are merging into one line in the kinetic phase diagram of alloy, and the non-equilibrium interval of solidification becomes exactly zero (Eq. (6.57)). Therefore, the critical point V = VD is considered as a finite velocity at which the complete solute trapping, k = 1 occurs (Eq. (6.25)), and solidification proceeds with the initial (nominal) composition c0 (Eq. (6.59)). At this point, V = VD, a transition from the solutal and thermal dendrites ends with the onset of the diffusionless solidification and beginning of the purely thermal growth of dendrites.

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The balance of undercoolings (6.56) represents the only equation for two variables, namely, the dendrite tip velocity V and dendrite tip radius R. The second equation for these two variables can be obtained from the stability analysis for selection of the stable mode of the dendrite tip. This analysis gives the following expressions: ⎧ 7/4 ⎡ 1 ⎤ aC p v k C ( Pec ) ⎥ , ⎪a3 c ⎢ T ( PeT ) + 2 d0 a ⎪ DH f [1 − (1 − k ) Iv( Pec ) ] ⎥⎦ ⎢⎣ 2 = ⎨ VR 2 ⎪ 7 / 4 0 c T ( PeT ), with V < VD , ⎪ ⎩ 2 (6.60) where are the stability functions dependent on Péclet numbers PeT and Pec, and c is the strength of anisotropy of the surface tension at the solid–liquid interface. Using the local equilibrium solute diffusion model, V VD, Brener and Temkin found the expressions for the stability functions as [6.134, 6.135]

T ( PeT ) =

1 2 , c ( Pec ) = , 2 1 + a1 PeT 1 + a2 Pec2

(6.61)

which hold for arbitrary Péclet numbers, anisotropy of surface tension, and when the kinetics of atomic attachment to the interface is neglecting. In Eqs. (6.60)–(6.61), the numeric coefficients, a3 = 1/0.42, a1 = 0.3, a2 = 0.6, can be taken in agreement with the asymptotic analysis [6.87]. Even though Eqs. (6.61) have been derived for the conditions of relatively small growth velocity, V VD, these represent so far an unique ability to find the rapid dendrite growth with the crystalline anisotropy for a binary system. Therefore, in addition to Eq. (6.56), one may take Eqs. (6.60)–(6.61) as the second equation to obtain velocity V and radius R for the dendritic tip. Figure 6.16 shows computed undercooling contributions using Eqs. (6.56)–(6.61). It can be seen that the constitutional undercooling TC changes abruptly at T(V = VD) with the onset of diffusionless solidification. At TC = T(V = VD), the transition from solute to thermal dendrites proceeds with the existence of a sharp break in the kinetic curve at V = VD [6.127, 6.128]. The transition from the chemically partition growth to the diffusionless solidification has a clear physical meaning: when the interface moves with the velocity V equal to or greater than the solute diffusion speed VD, the concentration cannot be changed ahead of the interface and solidification proceeds by the segregationfree crystallization.

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Figure 6.16. Semi-logarithmic plot of the individual undercooling contributions: thermal undercooling TT, undercooling TN arising due to the shift of equilibrium liquidus from its equilibrium position in the kinetic phase diagram of rapid solidification, curvature undercooling TR, kinetic undercooling TK, and constitutional undercooling TC. a b c d

Figure 6.17. Schematic illustration of L21 superlattice structure. In the case of Ni2TiAl intermetallic alloy, a and b denote the lattice sites occupied by Ni atoms, while c and d correspond to the places occupied by Ti and Al atoms, respectively. In the disordered B2-phase the occupation of c and d are randomly distributed among Ti and Al atoms whereas in the completely disordered bcc structure chemical order is totally lacking.

6.4.3 Superlattice Structures in Intermetallics Superlattice structures appear in intermetallic phases, which are often formed in alloys besides the solid solutions. The intermetallics differ from solid solutions in which they show chemical order in their crystalline structure. The chemical order in a crystallographic structure of intermetallics is referred to its superlattice structure. Figure 6.17 gives a schematic illustration of the L21 superlattice

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structure, which occurs for example, in the well-known ternary intermetallic alloy Ni2TiAl [6.136]. The degree of chemical order is described by introducing an order parameter w. Because of the complexity of some intermetallic phases w may depend on the system under consideration. In the case of binary A–B alloys with lattices sites a and b, the order parameter is defined as the concentration cAa of alloy component A on sites of sublattice a minus the corresponding concentration cAb: w = cAa cAb. An intermetallic compound melts either congruently as a pure metal or incongruently as an alloy with solute redistribution in front of the interface. As an example the phase of NiAl at a composition of 50 at% Al melts congruently while the phase of Ni3Al compound melts at a composition of 23.5 at% by a peritectic reaction. The mechanical properties of intermetallic phases differ essentially from those of solid solutions because of their chemical order. In particular, NiAl and TiAl intermetallics are interesting as high-temperature materials of high yield strength and excellent corrosion resistance. They are therefore widely used by aerospace industry. A disadvantage, however, is their brittleness at low temperatures, which makes mechanical working of such compounds very difficult. A solution this problem is the formation of intermetallics by rapid solidification processing such that a high density of antiphase boundaries is produced. By rapid solidification a disordered superlattice structure is primarily crystallized, which forms multidomains of chemically ordered regions in a second reaction step. Each domain is separated from its neighbouring domains by antiphase boundaries. Superlattice structures transform from a chemically ordered state to a chemically disordered state at a critical temperature Tc. Tc can be either below the liquidus temperature, TL, or above TL. In the latter case, Tc is a virtual transition temperature since the liquid state is always chemically disordered. The order parameter decreases from 1 at T = 0 K with increase in the temperature. At the transition temperature Tc, the order parameter w drops either discontinuously to w = 0 or falls continuously to w = 0. In the former case, there is a phase transformation of first order whereas in the latter case it is of second order. To describe a transition from ordered to disordered growth of undercooled intermetallic compounds, we refer to the model developed by Boettinger and Aziz [6.137] for binary alloys. This model gives an expression for the velocitydependent interface temperature TI which is dependent on the order parameter w, yielding ⎞ dTI RT 2 ⎛ 1 1 f (w) h( w ) . =− I ⎜ + H ⎝ VS f ( w ) + g ( w ) VDI f ( w ) + g ( w ) ⎟⎠ dV

(6.62)

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Equation (6.62) describes the transition from a sluggish interface motion at small growth velocities V VDI to a highly mobile interface at large growth velocities V VDI. The functions f, g and h are defined as follows: f ( w ) = −( M + 1 + (1 + w ) M lnM ), g(w) = −

w 2 RTI ⎛ 1 + w ⎞ ⎛ M (1 + w ) ⎞ M ln M ln ⎜ , ⎜ ⎟ ⎝ 1 − w ⎟⎠ H f ⎝ w ⎠

⎛ M (1 + w ) ⎞ h( w ) = w 2 ln ⎜ , ⎝ 1 − w ⎟⎠

(6.63a) (6.63b) (6.63c)

where M = exp(W1h/RGTI). RG is the gas constant and W1 the thermodynamic parameter for the Gibbs free energy of the solid. This parameter is correlated to the (negative) heat of mixing Hmix and the equilibrium long-range order parameter we by 1 = 4H mix /(1 + we2 ).

(6.63d)

Equation (6.63d) is valid for intermetallic binary compounds of equiatomic composition which show a (virtual) second-order transition from an ordered to a disordered state with a transition temperature Tc above the melting temperature TL. Under this assumption the critical transition temperature Tc and the critical growth velocity Vc for disorder trapping are given by T0 = −

i , 2R

⎡T ⎤ Vc = VDI ⎢ cr − 1⎥ . ⎣ TL ⎦

(6.64a) (6.64b)

As seen from Eq. (6.64), the higher the value of Tcr the larger the critical velocity at which disorder trapping occurs. Since Tcr scales with the heat of mixing one expects low critical velocities in the case of alloys showing a small Hmix. This model has been extended for order–disorder transitions of first order and has been applied even for ternary alloys [6.138]. The kinetic undercooling Tk according to Eq. (6.47) gives the contribution of the atomic attachment kinetics to the interface undercooling TI. Here, one may distinguish between two different cases. If the atomic attachment kinetics is limited by the atomic vibration frequency, which is of the order of 1013 Hz, the speed of sound Vs gives the ultimate limit of growth kinetics. The dynamics of the solidification front is collision-limited. But if the atomic attachement kinetics is controlled

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by atomic diffusion processes the atomic diffusion speed VD sets the upper limit of the growth velocity. The dynamics of the solidification front is diffusion-limited. The formation of a superlattice structure of intermetallics is thought to require atomic short-range diffusion. As a consequence, one may expect a slow growth kinetics of intermetallics since the atomic diffusion speed is about three orders of magnitude smaller than the speed of sound. In fact, measurements of growth velocity as a function of undercooling on intermetallics confirm slow growth kinetics of intermetallic melts over a considerable range of undercooling [6.139]. 6.5. PHASE-FIELD MODEL

In addition to the sharp-interface methods, the diffuse-interface methodology has been developed as well (Section 4.3). Over the last decades, the diffuse-interface methods have emerged as powerful computational approaches to simulate morphological development during solidification. To those methods, the phase-field method is related, which is widely developed presently [6.19, 6.140–6.142]. This method has the well-known advantage that it avoids to explicitly track a sharpphase boundary by making the interface region between solid and liquid spatially diffuse over some finite thickness. The phase-field model replaces the sharpinterface model by the solution in the entire computational domain of coupled partial differential equations of heat and mass transport and for an auxiliary variable that keeps track of the interface. The parameter , the phase field, is introduced to indicate the phase. is a continuous variable that takes constant values in the bulk phases, say 1 in the solid and 0 in the liquid, and increases from 0 to 1 over a thin layer, the diffuse interface (Figure 6.18). From the formal point of view, the phase-field equations incorporate automatically the Gibbs–Thomson equation (shift from the equilibrium due to curvature effect), anisotropy and interface kinetics asymptotically for a sufficiently thin diffuse interface. A starting point of the phase-field model is the energy functional or the entropy functional of the Ginzburg–Landau form. For the total entropy S of an entire system of volume v one uses [6.143] ⎡ 2 2 2 2 2 2⎤ ⎥ dv. S = ∫ ⎢ s( e , c , ) − e e − c c − 2 2 2 ⎥⎦ v ⎢ ⎣

(6.65)

Here e, c, and are constants for the energy, concentration and phase-field, respectively. In the functional (6.65), the gradient terms are used to describe a spatial inhomogeneity within the fields. It is logical to include gradient terms in Eq. (6.65) because of the existence of steep gradients within the diffuse interface.

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Figure 6.18. Schematic representation of the diffuse-interface (above) and the phase-field variable variation along some direction n to the interface (below).

The entropy density S must contain a double well in the variable , which distinguishes the liquid and solid. A construction of functional (6.65) gives a quite general formulation of the phase-field equations based on the classic irreversible thermodynamics. These are of the following formfield variable variation along the direction n to the interface (below). The governing equation for energy density: ⎡

e ⎛ s ⎞⎤ = − ⋅ ⎢ M ee ⎜ + e2 2 e⎟ ⎥ , ⎝ e ⎠⎦

t ⎣

(6.66)

The governing equation for solute concentration: ⎡

c ⎛ s ⎞⎤ = − ⋅ ⎢ M cc ⎜ + c2 2 c⎟ ⎥ , ⎝ c ⎠⎦

t ⎣

(6.67)

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The governing equation for the phase field ⎛ s ⎞

= M ⎜ + 2 2 ⎟ .

t ⎝ ⎠

(6.68)

Here Mee, and Mcc are the diagonal terms in the matrix of mobilities for thermal and solute transport, and M the mobility of the diffuse interface. For a binary system, the interface mobility is assumed to be dependent on composition as M = (1 − c) M A + cM B 0, where MA and MB are the interface mobility for the transformation in pure systems consisting of A or B component, respectively. In various formulations of the phase-field model [6.144, 6.145], the mobilities of MA and MB are proportional to the atomic interface kinetic coefficient k and inversely proportional to the interface width , hence M k/. The phase-field method has been applied to the description of growth of complicated dendrite morphology in metallic systems. The first models [6.146, 6.147] have yielded the three-dimensional dendrite morphologies that resemble those seen in experiment, although they have not compared the results quantitatively with the microscopic solvability theory. More detailed phase-field calculations have been carried out by Karma and Rappel [6.148], and they have compared their results with those of the numerical solvability theory and experiments. They made their comparison using the “thin-interface” analysis [6.145] of the phase-field equations. The phase-field model via “thin-interface” analysis [6.145] has been developed for the interface thickness assumed to be small compared to the scale of the crystal but not smaller than the microscopic capillary length d0. The phase-field equation is derived variationally from a phenomenological free-energy functional F, and, for a pure system, is given by ( n )

F =− ,

t

(6.69)

in which ⎡ 2 (n) 2⎤ ⎥ dv, F = ∫ ⎢ f (T , ) + 2 ⎦ v ⎣ f =−

⎛ 2 ⎛ 2 ⎞ 23 5 ⎞ 1 − + T + ⎟ − T ⎜⎝ 2 ⎜⎝ 2 ⎟⎠ 3 5⎠

and T = (T − TE ) / TQ .

(6.70)

(6.71)

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Equation (6.71) has minima at = 1 and = +1 that corresponds to the liquid and solid phases, respectively. The anisotropic capillary (6.45a) is recovered for the interface energy LS by choosing (n) = LS (n ) / 0 ,

(6.72)

where n = − / is the vector field that corresponds to the normal to the boundary. Equation (6.69) is coupled to that of the diffusion field

T 1 = 2 T +

t 2 t

(6.73)

in such a way that the equations for the two fields reduce to those of the original sharp-interface model in the limit where the interface thickness is small compared to the principal radii of curvature of the boundary. Within the thin-interface analysis [6.145], the necessary relation for choosing the coupling parameter T and the function (n) is taken such that 1/k in Eq. (6.47) vanishes, thereby eliminating the kinetic undercooling at the interface. The entropy functional (6.65) and free-energy functional (6.70) assume local equilibrium in transport processes, being consistent with the basic hypothesis of the classic irreversible thermodynamics [6.23]. For rapid phase transformations, such as rapid solidification, the local equilibrium is missing both at the interface and within bulk phases (Section 6.2). Assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables (e, c, ) and the space of fast variables (q, J , / t ) , one can describe transformations within rapid diffuse-interface theory [6.149] (here q is the heat flux, J the solute diffusion flux and / t the rate of change of the phase field). The phasefield model with a relaxation of the diffusion flux can describe the phenomenon of an advancement of diffuse interfaces with higher velocities comparable with the solute diffusion speed. It has been shown that by choosing the concrete form of entropy (as the thermodynamic potential), one may recover the existing models based on the classic irreversible thermodynamics and analyse solidification under local non-equilibrium conditions [6.150]. 6.6. TRANSITION FROM FACETED TO NON-FACETED GROWTH

As is well known, faceted growth of crystals is characterized by the atomically smooth interfacial motion and the non-faceted growth might be characterized by the atomically rough interfaces [6.2]. Considering the mechanisms of crystal

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growth, Cahn [6.151] interpreted the transition from the atomically smooth growth to the atomically rough growth of crystal surfaces by the analysis of a driving force needed for interface advancement. Cahn [6.151] distinguished between two major mechanisms of crystal growth: (i)

the crystal-melt surface advances by the lateral motion of steps by one interplanar distance in height; and (ii) the crystal-melt surface advances normal to itself without additional steps, so that every element of surface is capable of continued change. The first mechanism is geometrical-lateral motion of steps versus motion of the whole surface normal to itself. The second mechanism is based on the time sequence of an element of surface and can be considered as a motion when a step passes versus continual change. Therefore Cahn termed the first mechanism as a non-uniform or lateral growth, and the second as a uniform or normal growth. Strongly faceting growth by the lateral mechanism can occur if atom transfer from liquid to solid may proceed at a few special sites on the interface (e.g. ledge or kink sites). In this case, the growth velocity at a given interfacial undercooling will depend on whether these ledges arise from surface nucleation, or from intersections of screw dislocations with the interface. In the normal growth, atom transfer can occur at any site on the interface and the growth velocity is expected to have a linear dependence on interfacial undercooling. Examination of the above two mechanisms of growth lead to the conclusion that when a sharp-interface is slightly inclined to a crystallographic direction, the interface may become the diffused one. Keeping the interface to follow the direction parallel to the low index of crystallographic planes, the interface may consist of large areas, which are bounded by the steps of height corresponding to an integral number of interplanar distances. This region of several interplanar layers might be considered as a transitive diffusive region in which the disordered atomic structure of a liquid transfers into the long-range order of a crystal (Figure 6.19). Therefore Cahn re-analysed the nature and energy of the steps, namely for spatially diffuse interfaces. By consideration of a free energy for step advancing with two-dimensional nucleation, Cahn described the transition from lateral mechanism to continuous mechanism of crystal growth formally using the Ginsburg–Landau energetic functional. According to him, the free energy for a step can be calculated 2⎫ ε ⎧ F = ∫ ⎨ f ( p ) + p ⎬ dx dy, 2 ⎭ ⎩

(6.74)

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Lateral growth

Transitional regime

Continuous growth

Figure 6.19. Scheme of atomic structure of crystal faces for different mechanisms of growth. During lateral growth (upper panel), the growth is possible by gradual motion of steps with velocity VS. In transient regime (middle panel), the growth of faces is possible by both step motion and atomic attachment to the interface (lower panel). Continuous growth proceeds due to atomic attachment giving the interface velocity VC.

where the kernel of the integral is a sum of the density f (p) of free energy and the non-local term proportional to the surface term given by the gradient p of the position p of the interface relative to a fixed plane in the crystal lattice. Obviously, F must tend to its minimum as soon as the reconstruction of the interface during lateral or normal growth proceeds. The density of free energy is described by f ( p ) = p FV + LS ( p ),

(6.75)

where Fv = ( H f /vmTE)T is the free energy change per unit volume, which acts as a driving force for formation of the new phase, H f , vm and TE the latent heat, molar volume of solid and melting temperature, respectively, T is the

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undercooling at a given point of the interface. The interface tension LS(p) in Eq. (6.75) is given by LS ( p ) = o ⎡⎣1 + g ( p ) ⎤⎦ ,

(6.76)

in which 0 is the minimum value of SL(p), 0g(p) the periodic part of LS(p) and g(p) the fractional change in surface tension (p) from the equilibrium value for the position given by p. Thus, Eq. (6.74) can be represented as the two-dimensional analogue of the Cahn and Hillard equation [6.152] for three-dimensional nucleation with (pFv + 0g) the two-dimensional equivalent to their quantity f, and (0/2) (p)2 equivalent to the gradient energy. Analyses of Eqs. (6.74)–(6.76) with the computations of the step energies and step widths for diffuse interfaces lead to the discrimination of the mechanisms of the advancing interface. It was convenient to distinguish according to the driving force Fv of the advancing crystal lattice with the interplanar distance a as follows. For small driving forces, 0 Fv 0gmax/a, classical-lateral growth mechanisms should be observed. This condition, expecting a range of driving force in which observable growth by classic two-dimensional nucleation is expected does not explicitly depend on the interface diffuseness. (ii) For the increased driving forces, 0gmax /a Fv 0gmax /a, the lateral growth mechanism has to be modified to take into account that the size of the critical nucleus (and the spacing of the possible spiral arms) is comparable with the step width. This region can be characterized by a gradual transition from classical lateral growth to uniform advancement of the interface normal to itself. (iii) For the large driving forces, Fv 0gmax /a, the interface can advance normal to itself without the benefit of the lateral motion of steps. (i)

Hence, Fv = 0gmax /a represents a driving force to permit uniform advancement of a diffuse interface. Such advancement does not need steps, and if in addition there are no diffusional barriers to motion, the interface will be able to move without thermal activation. Note that the lateral growth (i) occurs times smaller than it is necessary for the continuous growth (iii). Cahn et al. [6.153] verified the diffuse interface theory of Cahn [6.152]. They found the explicit approximate expression for the critical undercoolings of the transition. Lateral growth: T T * = BgDT

Continuous growth: T T * = T * = BgDT

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Here B is the empirical Turnbull’s constant that lies between 0.2 and 0.5 for nonpolymeric materials, and the diffuseness parameter gD is described for the sharp interfaces as gD → 1, and for the diffuse boundaries as gD = ( 4n3/8) exp( 2n/2), with n being the number of the lattice planes within the diffuse region. Therefore, on the basis of these outcomes, Cahn et al. [6.153] have presented a relationship “crystal growth velocity – undercooling”, which gives a smooth but rapid transition from lateral to continuous growth (Figure 6.20). They also verified the outcomes of various theories of crystal growth in comparison with experimental data on growth velocities for a set of materials (metals, inorganic compounds and nonmetallic elements). Particularly, it has been shown that the diffuse interface theory of Cahn is consistent with the experimental observations. To verify the idea that the transition from faceted growth (lateral mechanism) may proceed to non-faceted growth (normal mechanism) with the increase of undercooling, Peteves and Abaschian [6.154] investigated experimentally the kinetics of faceted growth for pure Ga as a function of the interface undercooling. They determined directly the solid–liquid interface undercooling and have done in situ observation of the interfacial conditions during the growth of crystalline gallium. Their study covered a range of 1032 104 m/s growth velocities at interface undercoolings ranging from 0.2 to 4.6K, corresponding to bulk undercoolings of about 0.253 K. Using the Seebeck technique, it has been found that the faceted (111) and (001) Ga interfaces grow at low undercoolings with either of the lateral growth mechanisms (two-dimensional nucleation growth or screw dislocation-assisted growth). At high undercoolings, the growth kinetics and growth modes deviate from the classical laws (that is observed at T T *, Figure 6.19) and follow a transitional region, eventually growing under linear kinetics (which is observed at T T *; Figure 6.19). The linear kinetics was observed for a threshold undercooling T *, beyond which the faceted interfaces (with atomically

Figure 6.20. Velocity–undercooling relationship predicted by the theory of Cahn [6.151, 6.153].

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rough structure) gradually become kinetically rough. In essence, Peteves and Abaschian confirmed experimentally the theoretical ideas of Cahn for the application of the formalism of diffuse interfaces to describe the transition in the crystal growth mechanisms, velocity–undercooling relationships and critical undercoolings for the transition from one mechanism to the other. Particularly, they defined the transition from the lateral to continuous growth as a smooth but rapid increase in crystal growth velocity in consistent with the prediction of Cahn’s theory. For most metals the transition from faceted to continuous growth occurs at low undercoolings, and continuous growth is ordinarily observed. For instance, germanium might have only observable continuous normal growth with the macroscopic smooth branches of dendritic patterns. However, the behaviour of germanium crystals and the crystals of dilute germanium alloys exhibit experimentally more complicated patterns. Devaud and Turnbull [6.155] and later Lau and Kui [6.156] found the twin-free dendritic structure and concluded that the crystallization mode in the undercooling regime was continuous normal growth. A group of authors [6.157–6.159] commonly observed that there is a transition from lateral to continuous growth at the critical undercooling T * and that the resultant grain size decreases with increasing undercooling. The measured values of T *, however, were quite different (T * = 300 K [6.156], T * = 30 K [6.157] and T * = 170 K [6.158]). Returning to the work of Devaud and Turnbull [6.155], one may suggest that as the undercooling is increased, the mechanism of crystal growth changes from stepwise lateral to continuous normal, enabling the formation of twin-free dendrites, and the grain refinement has its origin in the break-up of primarily formed dendrites. Therefore in the light of these suggestions for a transition in mechanisms and the existence of grain refinement, Evans et al. [6.160] re-examined the observations of Devaud and Turnbull on germanium. They employed the numerical model for spherical growth to estimate the critical interface temperature for the transition from stepwise lateral to continuous normal. The model has included a Crank–Nicholson finite-difference solution to the fully timedependent heat diffusion equation, using a moving grid point to track the interface and incorporating the effects of curvature and kinetics of the interface motion. The solution allows one to predict that an interface temperature Ti 1057 K is required for normal growth, and if localized recalescence occurs at a temperature above the Ti value, before relative instability has set in, there will be a transition in the kinetic growth mechanism to lateral stepwise (edgewise) growth. After such conclusion, Evans et al. [6.160] extended the numerical model to treat a Ge-0.39 at.% Sn alloy, for which the abscence of dendrite growth at undercooling T 255 K was noted by Devaud and Turnbull [6.155]. They successfully predicted the bulk undercoolings required in the alloy for the occurrence of dendrite growth and

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found evidence for a grain refinement effect, associated with a critical dendrite velocity. The critical velocity has been estimated as ~ 17 m/s and reduction in the undercooling for grain refinement brought about by the addition of 0.39 at.% Sn. A further understanding of the transition from the faceted to non-faceted growth has been performed in electromagnetic facility using high-speed camera observations of solidifying drops. Aoyama and Kuribayashi [6.162] investigated interface morphology from the high-speed camera observations of the recalescence front during rapid solidification of Si and Ge. Particularly, the growth behaviour of Si was found to be classified in three categories of lateral growth (low undercooling), isolated dendrite (moderate undercooling) and closer dendrite growth (high undercooling). In addition, Panofen et al. [6.163] measured the growth velocity of pure Si and interface morphology during solidification with a high-speed camera. As be seen in Figure 6.21, the growth mode of pure silicon can be divided into three regions. The first region is the faceted growth region, T 105 K. Figure 6.22a shows a picture of the surface of a liquid silicon sample solidifying at an undercooling of 50 K. Note that the solid material is brighter due to a higher emissivity. There has been seen a set of straight lines running on the surface of the silicon droplet. Aoyama and Kuribayashi [6.164] interpreted these lines as plate-like crystals that extend through the volume of the sample and cut the surface forming circles on it. The growth rate of these plates could not be measured since most of the crystallization takes place inside the liquid and the lines grow discontinuously at

Figure 6.21. Growth velocity versus undercooling measured with a high-speed camera during solidification of silicon [6.161]. Faceted growth has been observed clearly at the undercooling T 90 K. The transitional region extends from T = 90 to 110 K. Beyond the undercooling T = 110 K, only dendritic growth was found.

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(a) ΔT = 50 K

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(b) ΔT = 110 K (c) ΔT = 158 K (d) ΔT = 218 K (e) ΔT = 260 K (f) ΔT = 295 K

Figure 6.22. Solidification of silicon drops recorded with the high-speed camera at a frame rate of 15 000 s1 [6.165].

small undercooling T 70 K. The slow lateral growth after the first recalescence is due to the high entropy of fusion of silicon resulting in a heating of the liquid up to the melting point. Further growth can only occur through heat exchange with the surrounding gas atmosphere. The hypercooling temperature, Thyp, is the ratio of volumetric latent heat H f of fusion and the heat capacity Cv of liquid. In case of liquid silicon it amounts to Thyp = H f /Cv = 4.6122 109 (J/m3)/2.334 106(J/K/m3) = 1976 K, and at an undercooling of 50 K only about 4 vol.% of the sample solidifies during recalescence under non-equilibrium conditions. When the undercooling is further increased, a gradual transition towards dendritic growth is observed. The transitional region extends from T = 90 to 110 K. Beyond this limit only dendritic growth was found [6.161]. The region of dendritic growth is characterized by a front of recalescence running from the trigger point through the volume of the liquid to the opposing side. Figure 6.22b–f shows a solidifying sample at an undercooling of 110–295 K. The growth mode in this region of undercooling is clearly dendritic. At medium undercoolings (110–260 K) the growth front is continuous but shows some irregularities. With increasing undercooling the growth front becomes more and more smooth. At very high undercoolings 295 K (Figure 6.22f), the growth front shows no more irregularities. A crystal morphology of silicon samples after solidifying in electromagnetic facility has been also studied using metallographic methods [6.165]. For small undercoolings, the faceted crystal growth is observed (Figure 23a). Pronounced edges and faces are visible on a large scale. Twinned crystals are evident on the surface. Layer-by-layer growth can be concluded from these pictures with atoms attaching to laterally growing edges or kinks. Twin planes serve as preferential sites and re-entrant corners for the formation of a new layer [6.166, 6.167]. No dendrites are visible on the surface of the sample solidified at low undercoolings. At higher undercoolings, e.g. T = 255 K, the surface morphology changes drastically. Dendrites become visible on the surface originating from the trigger site as can be seen in Figure 6.23b. The solidification speed increased considerably

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Figure 6.23. (a) Surface morphology of a sample solidified at 86 K. undercooling. (b) Surface morphology of a sample solidified at 255 K undercooling. (c) Surface morphology of a sample solidified at 330 K undercooling.

resulting in a recalescence time of only ~1 ms compared to the timescale of ~30 ms at low undercoolings during faceted growth. An atomically rough surface is required for dendritic growth [6.61], and whether an interface is rough enough depends on the particular entropy of melting S. According to the works of Jackson [6.168, 6.169], an atomically rough interface might be observed for S 3R and an atomically flat surface might be seen for S 3R, where R is the gas constant. Because Jackson found his criteria for the transition using classic thermodynamics of Gibbs, these inequalities are true for the small deviations from equilibrium, i.e. at small undercoolings. Silicon has S f = H f TE = 5.06 104(J/mol)/1687(K) 3.3R (J/mol/K), and a change in the atomic surface is theoretically expected to be observed with increasing undercooling. This outcome is confirmed experimentally (cf. Figure 6.23a and Figure 6.23 b). At the highest undercoolings, e.g. T = 330 K, no more dendrites or faces are visible on the surface of the sample. The surface is homogeneous and further refined, as shown in Figure 6.23c. One may interpret this as a dendritic break-up and remelting of the truncated dendrite branches resulting in grain refinement of the sample as reported by Schwarz et al. [6.170]. As a final note, it can be mentioned that the phase-field methodology can be successfully extended to model the solidification of faceted materials. As we noted, this growth may occur at low undercooling for most metals. Therefore, Debierre et al. [6.171] solved the equations of the phase field in the thin-interface limit of Karma and Rappel for small undercoolings, i.e. with local equilibrium at the interface and with the capillarity. Because at small undercooling the crystal

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Figure 6.24. Needle dendrite growth in faceted manner with the cusp amplitude in contrast to the Ivantsov parabolic plate. Extracted from the results of two-dimensional modelling [6.171] using the phase-field model via a thin-interface analysis.

shape is determined mainly by the free energy LS of the solid–liquid interface, their approach consists of using an approximate LS plot with rounded cusps that can approach arbitrarily closely the true LS plot with sharp cusps that correspond to faceted orientations. The 2D growth of faceted needle crystals has been studied as a function of undercooling and the cusp amplitude c for a LS plot of the form LS = 0 [1+ c(|sin | + |cos |)], where 0 is the averaged value of the interfacial energy and the angle between the direction normal to the interface and some fixed crystalline axis. It has been shown that from the initial spherical particle placed into undercooled melt the faceted needle is growing different from the parabolic form near the needle parabolic dendrite (Figure 6.24). To compare the phase-field model predictions, Debierre et al. [6.171] have developed analytical theory of faceted needle growth, which predicts the tip growth velocity V and facet length . The theory predicts the scaling law ~ V1/2 observed both experimentally [6.172] and in the phase-field simulations given by Debierre et al. [6.171]. To predict the transition from faceted growth at low undercoolings to continuous growth at large undercoolings by the phase-field simulations, it is necessary to develop a model, which deals with the phase-field kinetics by including the kinetic relaxation time dependent on undercooling in addition to orientation. 6.7. EXPERIMENTAL DATA AND MODEL PREDICTIONS

6.7.1 First experiments The first quantitative measurements of large growth velocities as a function of undercooling were performed by Walker [6.64] on pure Ni. The technique used by Walker utilizes a sample melt embedded into a glass flux contained in vitreous silica boats. Dendrite growth velocities were measured by means of two photo diodes encapsulated by silica tubes and immersed into the liquid sample at a fixed

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Figure 6. 25. Crystal growth velocity V as a function of undercooling T measured on Cu [6.173].

distance to measure the time lag of recalescence between the two sensors. Colligan and Bayles [6.65] reported measurements of dendrite growth velocities on undercooled Ni samples, as determined by direct optical measurements as well as by high-speed cinematography. The propagating crystallization front was observed by detecting changes in sample surface brightness due to recalescence. The same technique as originally introduced by Walker was employed by Suzuki et al. [6.173] to measure growth velocities in undercooled Cu and Ag. All of these measurements indicate a relatively large scatter of the results. This has been attributed to the fact that the growing dendrites interfere with the mould wall. Hence reason it has been argued that only the largest growth velocities measured at fixed undercooling are of relevance (cf. solid line in Figure 6.25). A modified technique has been applied by Flemings et al. [6.174–6.176] to measure dendrite growth velocities as a function of undercooling on levitation processed drops of metals and alloys. In these experiments the samples were encased into a glassy flux and melted inductively within a levitation coil. When crystallization occurred spontaneously the temperature rise during recalescence was measured by means of a fast two-colour pyrometer with a time resolution of 10 s.

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Metastable Solids from Undercooled Melts 103

μ=∞

Fe-5wt%Ni

μ=1.53 m/sK

102

V (m/s)

103

V (m/s)

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102

100

10−1

10−1 bcc

10−2

fcc

10−3 10−4 0 10

µ=1.55 m/sK

fcc bcc

10−3 101

102 ΔT (K)

103

μ=∞

101

100

10−2

Fe-10wt%Ni

10−4 0 10

101

102

103

ΔT (K)

Figure 6. 26. Crystal growth velocities V as a function of undercooling T for two different FeNi alloys. Dots: measured data; lines: theoretical calculations for bcc and fcc growth [6.174].

The recalescence times were divided by the sample diameter to obtain the solidification speed. Direct fast cinematography was employed as well. The following were the disadvantage of this method (i) the nucleation point is unknown, from which the solidification front starts to propagate into the volume of the melt, and (ii) the solidification occurs spontaneously so that it cannot be externally controlled. This leads to measurements that cover only a part of the accessible undercooling range (cf. Figure 6.26). 6.7.2 Measurements on pure nickel Figure 6.27 shows measurements of dendrite growth as a function of undercooling on pure Ni using photosensing (squares) and capacity proximity sensor (closed circles) technique. For further details of measurement technique the reader is referred to Chapter 2. The experimental results obtained by applying photosensing technique reveal a large scatter compared with those taken by the capacity proximity sensor, in particular at small undercooling. Considerable growth velocities up to 70 m/s are measured at the highest undercooling values of T 325 K. In an intermediate undercooling range 100 K T 200 K the experimental results can be well described within current dendrite growth theories provided an interface undercooling is taken into account. At small undercoolings T 100 K systematic deviations occur, whose origin has not yet clarified. Probably, they are

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Figure 6. 27. Dendrite growth velocity V of Ni as a function of undercooling T, determined with different experimental set-ups. At undercoolings below 100 K the C-sensor method yields data with considerable reduced scatter compared to those obtained with the photodiode technique. The solid line gives the prediction of dendrite growth model [6.120] if an interface undercooling is taken into account.

caused by fluid flow effects in the undercooled melt induced by convection and electromagnetically stirring in levitation experiments, since they occur at growth velocities comparable with the fluid flow velocities. Space experiments are in progress to investigate this problem, since under-microgravity fluid flow is very much reduced. At undercoolings larger than a critical undercooling T * 200 K systematic deviations also are observed that might be associated with a change of the growth mode. The origin of this change, however, needs further investigations. 6.7.3 Measurements on dilute NiB and NiZr alloys The corresponding results obtained on dilute alloys of NiB are shown in Figure 6.28 [6.131]. The behaviour of the solidification velocity as a function of undercooling for both alloys of concentrations Ni99.3B0.7 and Ni99B1, respectively, are quite dissimilar to the pure metal. This indicates the serious influence of constitutional effects on the crystallization kinetics of alloys. For undercoolings less than a critical undercooling T *, which depends sensitively on the alloy concentration, the growth velocities of the alloys are much smaller than those of the pure metal. At the critical undercooling T * = 214 K (Ni99.3B0.7) and T * = 267 K (Ni99B1) the growth velocity rises sharply. The thin solid lines represent the predictions of dendrite growth theory taking into account deviations from local equilibrium at the interface. One can see that a quantitative description of the experimental results is possible if the

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conditions of local equilibrium at the interface are relaxed by taking into account an interface undercooling and a velocity-dependent partition coefficient. In the framework of dendrite growth theory, the sharp increase in the dendrite growth velocity at the critical undercooling can be interpreted as a transition from diffusion to thermally controlled solidification. At undercoolings larger than the critical undercooling the samples solidify almost partitionless with the consequence that supersaturated solid solutions are formed. This has been confirmed by autoradiography investigations of the boron distribution in alloys undercooled by different amounts prior to solidification [6.177]. It has been pointed out that besides solute trapping solute drag effects may also play a role in rapid solidification of alloys. The crystal growth model incorporating solute drag is based on the consideration that the free energy G provides the driving force for crystallization but a part of G is consumed by the solute drag effect in driving solvent–solute redistribution in the interface [6.31]. Taking into account such a solute drag effect in the analysis of the measurements of crystal growth velocities on dilute NiB alloys leads to an even better agreement up to critical undercooling T * (cf. thick lines in Figure 6.28). This undercooling is considered in Ref. [6.131] as the beginning of intensive solute trapping. At large undercoolings the transition from solutal to thermal dendrites is completed with the onset of the chemically partitionless growth of the thermal dendrites [6.127]. Important parameters in modelling dendrite growth in alloys in particular with respect to solute trapping are the atomic diffusive speed VDI and the chemical diffusion coefficient DL. Both quantities have been measured for dilute Ni99Zr1 alloys applying laser surface resolidification experiments in combination with Rutherford

Figure 6. 28. Dendrite growth velocities as a function of undercooling. Measured values: squares, Ni99.3B0.7; circles, Ni99B1; solid lines, calculated within dendrite growth theory without solute drag (thin lines) and with solute drag (thick lines).

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Figure 6.29. Dendrite growth velocity as a function of undercooling for Ni99Zr1 alloy. The critical undercooling T * marks the transition from chemically to thermally controlled growth.

backscattering analysis of the concentration profiles in the as-solidified surface layers [6.178]. In addition, the dendrite growth velocity V was measured as a function of undercooling T on the same alloy of Ni99Zr1. Figure 6.29 depicts the results of these measurements. To evaluate the importance of the atomic diffusive speed at the interface in modelling of dendrite growth of dilute Ni99Zr1 alloy the dendrite growth velocity was calculated within dendrite growth theory changing the numerical values of VDI. To obtain numerical values for the theoretical prediction for V(T), the values of VDI and DL are taken from independent laser resolidification experiments on thin films. The best-fit values of VDI = 26 m/s and DL = 2.7 109 m2/s were used. As illustrated by Figure 6.29 modelling of dendrite growth velocity as a function of undercooling leads to agreement with the experimental results up to critical undercooling T * [6.179]. At the undercoolings T = T *, the solidification mechanism is changed abruptly which can be described by the existence of deviations from local equilibrium both at the interface and solute diffusion field. Following the local non-equilibrium model (Section 6.3.2), the completely partitionless growth proceeds at the region T T *. Therefore, to predict the growth kinetics in a whole region of undercooling and solidification velocity measured in experiments, it is possible to apply the model of local non-equilibrium solidification for rapid dendritic growth of alloys. 6.7.4 Measurements on intermetallic compounds Another interesting feature that may be observed in rapid solidification of undercooled intermetallic alloys is the phenomenon of disorder trapping. Boettinger and Aziz [6.180] have developed a model which analyses disorder trapping in rapidly

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Metastable Solids from Undercooled Melts 2.8

FeSi

2.4 μ=4.2 m/ sK

V (m/s)

2.0

μ=0.014 m/ sK

1.6 1.2 0.8 Measured Theory

0.4 0

100

200

300

CoSi 5 4 V (m/s)

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3

μ=0.021 m/ sK

2 Measured Theory

1 0

100

200

300

ΔT (K)

Figure 6. 30. Measured dendrite growth velocities V as a function of undercooling T for FeSi (upper panel) and CoSi (lower panel) intermetallic alloys (squares). The thin line corresponds to the prediction of dendrite growth theory if an interface undercooling due to collision-limited attachment kinetics is assumed, while the thick lines represent the theoretical predictions if a large interface undercooling due to short-range diffusion-limited attachment kinetics is supposed. Note the large increase of V at T 300 K in the case of CoSi intermetallic compound (marked by the arrows), which may indicate the onset of disorder trapping.

solidified intermetallic alloys in close analogy to the treatment of solute trapping. Disorder trapping during rapid solidification of intermetallic compounds may improve the mechanical properties of such materials. It has been experimentally observed by laser surface resolidification [6.181, 6.182], by rapid quenching using melt spinning technique [6.183] and by drop tube processing [6.184]. Figure 6.30 shows the results of measurements of the growth velocity as a function of undercooling for intermetallic FeSi and CoSi alloys at equiatomic concentration. Growth velocities up to 2.5 m/s (FeSi) and 4 m/s (CoSi) are observed at maximum undercooling of T = 300K [6.185, 6.186], which are much less than that observed

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in pure Ni. The analysis of the results within dendrite growth theory and the assumption that the interface undercooling is due to the collision-limited attachment kinetics (k = 4.2 m/s K for both alloys) yields the thin solid lines in Figure 6.30. Apparently, the prediction of theory leads to a large overestimation of the experimental results under those conditions. If, however, the atomic diffusive speed VDI is used to calculate the interface undercooling an excellent agreement is found (cf. thick solid lines in Figure 6.30). This means that in these alloys growth is short range diffusion limited leading to a large interface undercooling. Such a behaviour is characteristic for growth of solid having chemical short-range order, as in an intermetallic compound [6.187]. In this case the atoms must themselves sort out onto the various sublattices. The analysis of the growth velocity undercooling behaviour of the FeSi intermetallic compound leads to the conclusion that in the accessible undercooling range the growth velocity was not sufficiently large to cause disorder trapping. In the case of CoSi intermetallic compound a similar behaviour as found in FeSi is observed. However, in contrast to FeSi, a rapid increase in the solidification velocity becomes apparent at the largest undercooling T 300 K, which may indicate the onset of disorder trapping. It should be noticed that there are two preconditions to obtain disorder in the as-solidified intermetallic products: (i) the growth velocity has to reach a critical value, V VDI [(Tc /TS)1] with Tc the critical temperature for the order–disorder transition, and TS the solidus temperature of the compound, and (ii) the cooling rate is sufficiently high to avoid reordering after solidification. The unambiguous evidence of disorder trapping requires further analysis of the as-solidified structures. X-ray diffraction may help to find out whether the superlattice peaks disappeared, and electron microscopy investigations may reveal the occurrence of antiphase domain grain boundaries indicating disorder trapping. Similar investigations of the growth dynamics were conducted on undercooled melts of intermetallic compounds of NiAl [6.188] and FeGe alloys [6.189]. 6.7.5 Measurements on semiconductors There are two idealized mechanisms for crystal growth from the melt. (i) Systems with rough S/L interfaces are thought to grow by the continuous growth mechanisms, in which atom transfer from liquid can occur at any site on the interface. (ii) Strongly faceting materials of high entropy of fusion may exhibit edgewise growth, in which the interface is atomically smooth except for the presence of ledges or steps, and in which atom transfer can only occur at a few special sites on the interface. Jackson’s factor [6.190], J, which depends on the entropy of fusion and the number of nearest neighbours in the plane of the interface, can be used for a rough prediction of the growth mode under normal solidification conditions: J 2 for continuous growth related to metals, and J 3 for edgewise growth

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such as in most non-metals. In contrast, considering solidification into an undercooled melt, Cahn et al. [6.151] have suggested a different model featuring a transition from edgewise growth (EG) to continuous growth (CG) when the driving force for crystallization is raised. Ge and dilute Ge-based systems are particularly suited for exploring the effects of undercooling on solidification modes, since the faceted structure is formed under normal conditions, and kinetic roughening of the S/L interface is expected to appear when a driving force threshold is exceeded. Figure 6.31 shows the measured crystal growth velocity V as a function of undercooling T measured on pure Ge and dilute GeSn alloys [6.166]. Two salient features are exhibited for pure Ge: first, in the range T Tc 300 K, the growth velocity is very small and increases slowly. In this region, the crystal grows in the edgewise mode. Second, at Tc = 300 K, a change takes place in the temperature dependence of the growth behaviour. In the range T Tc 300 K, the velocity rapidly increases with T. Such behaviour is typical for the continuous growth mode of dendritic solidification as described for gallium by Peteves et al. [6.154]. Adding a small amount of 0.39 at.% Sn to pure Ge leads to a drastic enhancement of the crystal growth velocity by a factor of 5 at the largest undercooling. Such a behaviour is quite unusual at least for metallic systems in which the VT curves for alloys are shifted towards higher undercoolings relative to the pure metal because of the existence of constitutional undercooling. Further increasing the Sn concentration up to 2 at.% slightly reduces this effect. The increase in the growth velocity by the addition of 0.39 at.%Sn to pure Ge implies a larger value of the atomic attachment factor f = 0.045, which gives the

Figure 6. 31. Crystal growth velocity, V, as a function of undercooling, T, for pure Ge (triangles), a very dilute Ge99.61Sn0.39 (closed circles) and a dilute Ge98Sn2 (squares) alloy. The points and curves correspond to measured values and predictions based on the sharp-interface theory of dendrite growth [6.166].

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probability that an atom from the liquid is captured by the solid upon a jump from the liquid to the interface. This means that the fraction of interface sites at which atomic attachment can occur increases by the addition of the metallic component, most probably because the presence of the solute can widen the S/L interface region and then exhibit a higher tendency to incorporate newly arriving atoms into crystals. This effect prevails the counteracting effect of the contribution of the constitutional undercooling TC, which causes a decrease in the growth velocity. This finding is supported by the results obtained for the alloy with the higher Sn content of 2 at.%. Even though f is further increased in comparison with the Ge99.61Sn0.39 alloy, the growth velocity is slightly reduced. This indicates that with increasing concentration of Sn the constitutional undercooling becomes more important. Similar investigations on pure silicon were recently conducted using electromagnetic levitation technique [6.162]. 6.7.6 Effect of convective flow and solute diffusion The Lipton–Kurz–Trivedi (LKT) [6.120] model of dendrite growth predicts the dendrite growth velocity V as a function of undercooling T in good agreement with experimental data for nickel solidification in the region of medium undercoolings 100 K T 200 K (see Figure 6.27). A modification to the LKT model, which takes into account the effect of forced convective flow caused by electromagnetic stirring, has been suggested [6.191]. The modified model predicts an increase in velocity when the flow is directed opposite to the dendrite growth. However, the effect of forced convective flow alone cannot still explain the measured data satisfactorily [6.192]. The additional reason for dendrite velocities higher than that predicted by the model might be due to the presence of small amounts of impurities, which may drastically influence the kinetics of solidification [6.193]. In the present section the results on modelling of dendritic solidification from undercooled melts processed by the electromagnetic levitation technique are discussed. To model the details of formation of dendritic we have used the phase-field model via “thin-interface” analysis [6.145], where the interface thickness is assumed to be small compared to the scale of the crystal but not smaller than the microscopic capillary length d0. The phase field and energy equations were taken from Ref. [6.194] with the momentum and continuity equations for the liquid taken from Ref. [6.195]. Furthermore, in the momentum equation, the Lorentz force caused by the alternating electromagnetic field has been introduced for an undercooled levitated droplet. A system of governing equations is described by –

energy conservation: TQ

T + (1 − )( U ⋅ )T = a 2 T + ,

t 2 t

(6.77)

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–

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continuity of the liquid phase: ⋅ ⎡⎣(1 − ) U ⎤⎦ = 0,

–

momentum transfer:

(1 − )(U ⋅ ) U = − –

(6.78)

⎤ ⎡ 1− 1− + FLZ + ⋅ ⎢ (1 − )V ⋅ U ⎥ + FD , ⎦ ⎣

(6.79)

phase-field evolution: ( n )

⎛

(n) ⎞ F 2 . = ⋅ 2 (n) + ∑ (n) − ⎜

t

( w ) ⎟⎠ w = x , y , z w ⎝

(

)

(6.80)

In Equations (6.77)–(6.80), is the phase-field variable ( = 1 is the liquid phase and = 1 is for the solid phase), = (1+)/2 is the fraction of the solid phase ( = 0 is for the liquid and = 1 is for the solid), U is the fluid flow velocity in the liquid, x, y, z are the Cartesian coordinates, t is the time, is the density, is the dynamic viscosity, and p the pressure. The dissipative force FD in the Navier–Stokes equation (6.79) is taken from Ref. [6.195]. Furthermore, in solution to Eq. (6.79), the Lorentz force has been averaged in time: FLZ |B|2/(4 SD), where |B| = B0 exp[(rR0)/SD] is the modulus of the magnetic induction vector, B0 is the time-averaged value of the magnetic induction, r is the radial distance of a droplet of radius R0, sd = [2/(el0)]1/2 is considered as the skin depth for the eddy currents induced by the alternating magnetic field in the droplet, which decreases for a short distance at which the modulus of magnetic induction |B| decays exponentially (where is the frequency of the applied current, el the electric conductivity, and 0 is the magnetic permeability). The free energy F is defined as F(T,) = f()+(TTE)g()/TQ, where TE is the equilibrium temperature of solidification. By including the double-well function f() = 2/2+4/4 and the odd function g() = 23/3+4/5 itself, the free energy F is constructed in such a way that a tilt of an energetic well controls the coupling for T and . The time (n) of the phase-field kinetics and the thickness (n) of the anisotropic interface are given by ⎡ d a (n) ⎤ (n) = 0 ac (n)ak (n) ⎢1+ a2 T 0 c ⎥, a 0 ak (n) ⎦ ⎣

(n) = 0 ac (n),

(6.81)

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where 0 is the time scale for the phase-field kinetics, 0 the parameter of the interface thickness with 0 = d0/a1 and a1 = (5/8)21/2. The second term in brackets of Eq. (6.81) for (n) defines a correction a2 = 0.6267 for the “thin-interface” asymptotic [6.194]. The anisotropy of interfacial energy is given by ac (n) =

⎡ ⎤ 4 c ( n ) = (1 − 3 c ) ⎢1 + ( nx4 + ny4 + nz4 ) ⎥ , 0 ⎣ 1 − 3 c ⎦

(6.82)

where (n) is the surface energy dependent on the normal vector n to the interface, 0 the mean value of the interfacial energy along the interface, and c the anisotropy parameter. The anisotropy of kinetics of atomic attachment to the interface is given by ak ( n) =

⎡ ⎤ 4 k ( n) = (1 + 3 k ) ⎢1 − ( nx4 + ny4 + nz4 ) ⎥ , 0 ⎣ 1 + 3 k ⎦

(6.83)

where (n) is the kinetic coefficient dependent on the normal vector n to the interface, 0 the averaged kinetic coefficient along the interface which is defined by 0 = (1/1001/110)/(2TQ), and k = (100110)/(1/100+1/110) is the kinetic anisotropy parameter in which 100 and 110 are the kinetic coefficients in the 100 and 110 direction, respectively. In Eqs. (6.80)–(6.83), the normal vector has the components (nx,ny,nz) defined by the gradients of the phase field as follows: nx4 + ny4 + nz4 = ⎡⎣( / x ) 4 + ( / y ) 4 + ( / z ) 4 ⎤⎦ . 4

(6.84)

Equations (6.77)–(6.84) have been solved numerically for the three-dimensional space by a finite-difference technique on a uniform computational grid. A multigrid algorithm has been used for resolving the equations of the phase field (6.80), heat transfer (6.77) and momentum (6.79), which have different spatial lengths and timescales of their dynamics. Parameters of modelling and material parameters for pure nickel are given in Ref. [6.191]. The morphological spectrum of interfacial crystal structures versus undercoolings has been obtained [6.193]. In the modelling, the spectrum of the crystal structures exhibits a change from grained crystals at very small undercoolings (T 0.15TQ) to dendritic patterns at intermediate undercoolings (0.1TQ T 1.0TQ) to grained crystals again at high undercoolings (T 1.0TQ). Within the range of intermediate undercoolings, the shape of dendrites is dictated by the preferable crystallographic direction (which is 100-direction for the case of Ni). Figure 6.32(a) shows the dendritic crystal growth, which has been dictated by the

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Figure 6.32. (a) Dendritic pattern growing into a stagnant pure nickel melt at an undercooling T = 0.30TQ K. The pattern has been simulated on a grid of size 4503 nodes. The details of modelling are given in Refs. [6.197, 6.198]. (b) Growth of nickel dendrite under convective flow at an undercooling of T = 0.30 TQ K and the fluid flow velocity U0 = 0.7 (m/s) imposed at the top surface of the computational domain. Growth velocity of the upstream branch is pronounced in comparison with the down-stream branch due to forced convection. The dashed lines around the dendrite indicate the flow velocity vectors in the vertical cross-section. The pattern has been simulated on a grid of size 230 230 330 nodes.

preferable crystallographic directions (these are 100-directions for nickel for all region of undercooling). Solidification under the influence of forced convective flow in a droplet produces dendritic growth pronounced in the direction opposite to that of the far-field flow velocity U0 (see Ref. [6.191]). The present results of modelling also confirm this outcome: by imposing the fluid flow, the growth becomes pronounced in the direction opposite to the flow as shown in Figure 6.32(b). For these structures growing in a stagnant melt and also with the melt flow (Figure 6.31), we compared the results for dendrite growth velocity V in pure Ni versus undercooling T quantitatively. The results of phase-field modelling exhibit an increase in the velocity of the upstream dendritic branch (Figure 6.32b). As soon as the thermal boundary layer shrinks ahead of the upstream branch due to the flow, the heat of solidification is removed faster and the growth velocity enhances. The enhanced dendrite velocity due to the melt flow decreases the discrepancy between theory and experimental data at small undercoolings. Hence, the predictions of the present phase-field modelling with the new experimental data [6.196] for growth kinetics of nickel dendrites were compared. The measurements were performed for dendritic growth velocity V as a function of undercooling T = TE T, measured experimentally for the melted drop (TE is the melting temperature and T the actual temperature of the drop). Solidification of the melt was triggered in the range of 30 K T 260 K.

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2000

Dimensionless dendrite tip radius, r

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1200

800

400

0 0

30

60

90

120

150

180

210

240

Undercooling, DT (K)

Figure 6.33. Dendrite tip radius as a function of undercooling. Solid line: pure Ni, dashed line for Ni 0.01 at.% of impurity; dashed-dotted line: Ni with 0.05 at.% of impurity.

The comparison of these new data with the solution of Eqs. (6.77)–(6.84) confirms that convection alone cannot describe the experimental results satisfactorily. An additional reason for the discrepancy that still exists might be the presence of small amounts of impurities [6.192]. Therefore, we have used the sharp-interface model [6.127, 6.128] to evaluate the influence of solute diffusion on dendrite growth kinetics. As can be seen in Figure 6.33, the dendrite tip radii for pure Ni and, respectively, for Ni with impurities differ significantly. For example, for Ni with 0.01 at.% of impurity, the transition from solute diffusion-limited growth to thermally controlled growth occurs in the range 30 K T 130 K (Figure 6.34). In this range, “thin” alloy dendrites grow rapidly in comparison with the “thick” thermal dendrites. Consequently, we have found that small amounts of impurities in nickel can lead to an enhancement of the growth velocity but with a temperature characteristic being different from that of the effect by fluid flow. This allows to discriminate between both contributions and model them separately by means of the phase-field modelling of dendrite solidification with convective flow and the sharp-interface model of dendritic growth of a binary system. Therefore, using the results of the present phase-field modelling for pure nickel (Eqs. (6.77)–(6.84)),

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100

Dendrite growth velocity (m/s)

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- Experimental data - Phase-field modeling without convection - Phase-field modeling plus effects from thermal convection and solute diffusion

10

1.0

0.1 10

100 Undercooling (K)

300

Figure 6.34. Comparison of data from experiment (open squares), phase-field modelling for pure Ni without flow (stars) and final data (black squares) with the effect from thermal convection and solute diffusion.

and the sharp-interface model [6.127, 6.128] for Ni + 0.01 at.% of impurity, we determined the final growth velocity as follows: V (T ) = V Ni (T ) + VConv (T ) + VDif (T ),

(6.85)

where VNi(T) is the velocity obtained from the phase-field modelling without convection of the liquid phase, VConv(T) the increase in velocity due to convection estimated from the phase-field modelling with convection, and VDif (T) the increase in velocity due to presence of a small amount of impurities as computed within the sharp-interface model. The increase in VConv(T) and VDif (T) was calculated relative to the velocity VNi(T) given by the phase-field modelling of dendrite solidification without convective flow. Figure 6.34 shows the final comparison of the modelling data and experimental results for growth velocity of nickel dendrites. We obtained a good agreement with experimental data provided that both convection and solute diffusion are taken into consideration. 6.7.7 Influence of local non-equilibrium on rapid dendritic growth It has been noted for NiB and NiZr alloys that by taking into account local non-equilibrium at the interface (by introducing the interfacial solute diffusion

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speed VDI) leads to an agreement up to the critical undercooling T* (Figures 6.28 and 6.29), consistent with the beginning of intensive solute trapping. At high undercoolings, the local non-equilibrium model (Eqs. (6.56)–(6.61)) predicts that the transition from solutal and thermal dendrites to thermal dendrites proceeds by the existence of a sharp break in the kinetic curve for the interface velocity equals to the solute diffusion speed in bulk liquid, i.e. at V = VD. At this point, the transition to thermal dendrites is completed with the onset of the chemically partitionless growth. This prediction is in good agreement with solidification studies on CuNi and NiB alloys [6.127, 6.128] and has a clear physical meaning: when the interface moves with the velocity V equal to or greater than the solute diffusion speed VD, the concentration cannot be changed ahead of the interface and solidification proceeds by the diffusionless solidification. Figure 6.35 demonstrates the ability to describe experimental data in the whole region of undercooling if local non-equilibrium is assumed at the interface (using the diffusion speed VDI at the interface) and also in the bulk liquid (using the bulk diffusion speed VD). In addition to investigations of the “velocity–undercooling” relationship, measurements of parameters of dendrite microstructure lead to the result that the dendrite tip radius R also shows a break point at the critical undercooling, consistent with the sharp change in the dendritic growth kinetics. Schwarz [6.199]

Figure 6.35. Dendrite tip velocity versus undercooling in Cu30 at.%Ni alloy. Experimental data (open squares) are taken from Ref. [6.126]. Curve is calculated within the three-dimensional model for dendrite growth [6.128].

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Figure 6.36. Dendrite tip radius versus undercooling in Ni-0.5 at.% Zr alloy. Experimental data (solid squares) are taken from Ref. [6.199]. The solid line gives the prediction of the three-dimensional model for dendrite growth [6.127, 6.128].

has provided an experimental test on the consistency of the sharp breaks in both the curves of “VT ” and “RT ”. He conducted experiments with rapid quenching of Ni0.5 at.% Zr droplets in Ga52In25Sn13 bath immediately upon rapid solidification during recalescence. Possible remelting of a microstructure of droplet due to recalescence has been excluded by rapid quenching of the solidifying droplet and the primary dendritic microstructure has been analysed by Schwarz [6.199]. While experimenting with the quenched samples, he has found that the dendrite tip radius R is changing abruptly at fixed undercooling (see solid squares in Figure 6.36). This critical undercooling for abrupt change of the dendrite tip radius is equal to the critical undercooling T * at which a sharp break in the kinetic curve “velocity–undercooling” appears at a velocity V comparable to the atomic diffusive speed VD, i.e. V = VD in the Ni0.5 at.% Zr alloy. The predictions of the model [6.127, 6.128] are shown in Figure 6.36. They are in a good agreement with experimental data of Schwarz [6.199] on dendritic tip radii R. Thus, the abrupt change of growth kinetics at V = VD proceeds with the completion of the transition from the solute diffusion-limited growth to the purely thermally controlled growth. It leads to the diffusionless dendritic growth with the sharp change in the relationship “dendrite tip radius–undercooling”. Therefore, including deviations from local equilibrium both at the interface and bulk liquid, the transition to diffusionless growth in association with the morphological transition from the solute diffusion-limited to thermally controlled growth of dendrites is considered as good agreement with experimental data. The local

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non-equilibrium model predicts the abrupt change of growth kinetics with the breakpoints in both the “velocity–undercooling” relationship and the “tip radius–undercooling” relationship. This breakpoint occurs at critical undercooling T * and solidification velocity V = VD (Figures 6.35 and 6.36) for the onset of the diffusionless growth of crystals. REFERENCES

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[6.189] Biswas, K., Gandham, P., Chattopadhyay, K., Volkmann, T., Funke, O., Holland-Moritz, D., and Herlach, D.M. (2004) Materials Science & Engineering A 375–377, 464. [6.190] Jackson, K.A. (1984) Materials Science & Engineering 65, 7. [6.191] Galenko, P., Funke, O., Wang. J., and Herlach, D.M.(2004) Materials Science Engineering A 375–377, 488. [6.192] Herlach, D.M., Funke, O., Phanikumar, G., and Galenko, P. (2004) in: Solidification Processes and Microstructures, eds. Rappaz, M., Beckermann, C., and Trivedi, R. (TMS, Warrendale, PA), p. 277. [6.193] Galenko, P.K., Herlach, D.M., Funke, O., and Phanikumar, G. (2004) in: Solidification and Crystallization, ed. Herlach, D.M. (Wiley-VCS, Weinheim), p. 52. [6.194] Bragard, J., Karma, A., Lee, Y.H., and Plapp, M. (2002) Interface Science 10(2–3), 121. [6.195] Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., and Tong, X. (1999) Journal of Computational Physics 154, 468. [6.196] Funke, O., Phanikumar, G., Galenko, P.K., Chernova, L., Reutzel, S., Kolbe, M., and Herlach D.M. (2006) Journal of Crystal Growth, in press. [6.197] Galenko, P.K., and Herlach, D.M. (2004) in: Non-equilibrium Solidification: Modelling for Microstructure Engineering of Industrial Alloys. ESA MAP Report No.AO 99-023, ed. Herlach, D.M. (European Space Agency, Noordwijk, Holland), p. 14. [6.198] Nestler, B., Danilov, D., and Galenko, P. (2005) Journal of Computational Physics 207, 221. [6.199] Schwarz, M. (1998) Kornfeinung durch Fragmentierung von Dendriten, Ph.D. Thesis, Ruhr-Universität Bochum. [6.200] Jackson, K.A. et al. (2004) Journal of Crystal Growth 271, 481. [6.201] Maxwell, J.C. (1867) Philosophical Transactions of the Royal Society 157, 49. [6.202] Cattaneo, C. (1948) Atti del Seminario della Università di Modena 3, 33. [6.203] Vernotte, P. (1958) Comptes Rendus de l’Academie des Sciences, Paris 246, 3154. [6.204] Cahn, J.W. (1962) Acta Metallurgica 10, 789. [6.205] Hillert, M., and Sundman, B. (1976) Acta Metallurgica 24, 731. [6.206] Horvay, G., and Cahn, J. W. (1961) Acta Metallurgica 9, 695.

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Chapter 7

Cooperative Growth in Undercooled Polyphase Alloys 7.1 Eutectic Growth Theory 7.2 Eutectic Morphology Transition 7.3 Stable and Metastable Monotectic Alloys 7.4 Peritectic Alloys References

283 294 303 307 310


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Chapter 7

Cooperative Growth in Undercooled Polyphase Alloys 7.1 EUTECTIC GROWTH THEORY

Dendritic growth is a single-phase growth mechanism and has been comparatively well investigated. Eutectic growth involves the interacting nucleation and cooperative growth of two or more solid phases within one liquid phase. Since eutectic alloys play a very important role in materials production, it has attracted extensive attention. From the experimental point of view, different eutectic microstructures have been found, which are classified as regular lamellar, regular rod and irregular or anomalous eutectic. The theoretical analysis of eutectic growth requires the simultaneous solution of the heat transport equations for both the solid phases and the liquid phase. For the sake of solvability, the current eutectic growth theories are mainly aimed at regular lamellar eutectic (Figure 7.1). Eutectic growth is a typical diffusion-controlled process. Therefore, the solution of this problem depends on how to analyse the diffusion field in front of the solid–liquid interface. The purpose of these analyses is to establish the relationship between undercooling ⌬T, growth velocity V and interlamellar spacing S. The concept of all of the theoretical models originated from Zener’s [7.1] and Brandt’s [7.2] work on the pearlite growth during eutectoid transformation in carbon steels. Tiller [7.3] was the first who attempted to solve the eutectic growth problem in a similar way. Hillert [7.4] made an important contribution by presenting a rigorous analysis for the mathematical model of Brandt. On the basis of these progresses, Jackson and Hunt [7.5] developed the classical lamellar eutectic growth theory. However, the Jackson–Hunt model is valid only at small undercoolings. Recently, Trivedi et al. [7.6] presented an updated modification applicable to rapid solidification processes. This model has been further extended by Kurz and Trivedi [7.7] to take into account the non-equilibrium interface kinetics effect. Figure 7.1 shows the solute diffusion pattern of the liquid phase in front of a growing eutectic grain. At a certain undercooling ⌬T, the equilibrium concentrations of and are c* and c*, respectively, which are far from the eutectic concentration ce. Therefore, the liquid near the phase is rich in solute component B, whereas the phase depletes the neighbouring liquid of this solute. Since the grain size is far larger than the interlamellar spacing S, it can be assumed that diffusion proceeds in two dimensions: along the growth direction (the z-axis), and transverse to the eutectic lamellae (the x-axis). 283

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Figure 7.1. Eutectic phase diagram and growth model [7.10].

CB

Z

α

1 Ce

x θα

b C1α

θβ β

C1β X

(a) Interface shape

(c) Transverse distribution T

CB * C1α

Tr

α C1α

Te

Ce

β

Tc ΔT

C1β

Tr

C∗1β

Tc Z

(b) Iongitudinal distribution

X (d) Interface undercooling

Figure 7.2. (a) Interface shape, (b and c) solute distribution profiles, and (d) interface undercooling.

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Cooperative Growth in Undercooled Polyphase Alloys

285

The longitudinal and transverse solute distribution profiles of the liquid phase are illustrated in Figure 7.2. Apparently, the liquid concentration in front of the phase decreases and that before the phase increases to the bulk composition ce as the distance from the solid–liquid interface extends. However, the alternating nature of lamellar eutectic structure leads to a cosinusoidal mode of solute distribution in the transverse direction. Similar to dendritic growth, the bulk undercooling ⌬T of a eutectic melt consists of four terms, i.e. constitutional undercooling ⌬TC, curvature undercooling ⌬TR, thermal undercooling ⌬TT and kinetic undercooling ⌬TK: ΔT = ΔTC + ΔTR + ΔTT + ΔTK .

(7.1)

In most cases of lamellar eutectic growth, the thermal and kinetic undercooling are negligible, hence ΔT = ΔTC + ΔTR .

(7.2)

Because regular eutectic usually does not involve faceted growth, the growth front of a eutectic grain can be considered an isothermal solid–liquid interface. Figure 7.2d illustrates the profile of Eq. (7.2). To derive an analytical expression for Eq. (7.2), it is necessary to solve the diffusion field and obtain the interface curvature of lamellae. For the sake of simplicity, Jackson and Hunt assumed that (i) the lamellar eutectic grows under steady state, and (ii) the solid–liquid interface is planar. Thus, the diffusion equation according to the coordinate system of Figure 7.1, which moves synchronically with the eutectic growth front, reads as follows: ∂ 2 c ∂ 2 c V ∂c = 0. + + ∂x 2 ∂z 2 D ∂z

(7.3)

If the initial alloy composition is cA, which may deviate from the eutectic concentration ce, the boundary conditions are c z →⬁ = cA , ∂c ∂x

= x=0

∂c ∂x

(7.4) = 0.

(7.5)

x = S + S

Furthermore, the conservation of matter at the interface requires ∂c ∂z ∂c ∂z

=− z=0

= z=0

V Δc 0 ≤ x ≤ S , D

V Δc D

S ≤ x ≤ S + S .

(7.6) (7.7)

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As shown in Figure 7.1, ⌬c = cL* ⫺ cS* = cL*(1 ⫺ k) and ⌬c = cL* ⫺ cS* = cL*(1 ⫺ k). When undercooling is small, Jackson and Hunt’s third assumption, ⌬c = c0 and ⌬c = c0, is reasonable. In such a case, the general solution is conveniently derived: ⎛ nx ⎞ c = c A + ∑ Bn cos ⎜ ⎟ exp( − n z ), ⎝ S + S ⎠ n

(7.8)

where 2 2 V ⎛ ⎛ V ⎞ ⎛ n ⎞ ⎞ ⎟ n = + ⎜⎜ ⎟ + 2 D ⎜⎝ ⎝ 2 D ⎠ ⎜⎝ S + S ⎟⎠ ⎟⎠

1/ 2

.

(7.9)

They further argued that the relation V/2D n/(S+S) is true for the usual case of lamellar eutectic growth. This fourth assumption is equivalent to supposing the eutectic solutal Péclet number Pe = SV/2D 2. Hence, the solution given by Eq. (7.8) reduces to ⎛ nx ⎞ ⎛ nz ⎞ ⎛ Vz ⎞ c = c A + B0 exp ⎜ − ⎟ + ∑ Bn cos ⎜ exp ⎜ − ⎟. ⎟ ⎝ D⎠ n ⎝ S + S ⎠ ⎝ S + S ⎠

(7.10)

The Fourier coefficients are specified by using Eq. (7.8): B0 = Bn =

c0 S − c0 S S + S

,

⎛ nS ⎞ 2 V ( S + S ) c0 sin ⎜ ⎟. 2 D ( n) ⎝ S + S ⎠

(7.11)

(7.12)

Up to here, the diffusion field is completely defined by Eq. (7.10). However, the liquid concentration varies apparently with both the transverse coordinate x and longitudinal coordinate z. To calculate the solute undercooling ⌬Tc, a reasonable treatment is to proceed with the determination of the average liquid concentrations at the interface in front of the and phases (z = 0): c =

1 S

1 c = S

S

∫ c z = 0 dx = cA + B0 + 0

2( S + S ) 2 V c0 Po , S D

S + S

∫

S

c z = 0 dx = c A + B0 −

2( S + S ) 2 V c0 Po , S D

(7.13)

(7.14)

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where 3 ⎛ nxS ⎞ ⎛ 1⎞ Po = ∑ ⎜ ⎟ sin 2 ⎜ ⎟. ⎝ S + S ⎠ n ⎝ n ⎠

(7.15)

If the absolute values of liquidus slopes for and phases are m and m, respectively, the average solutal undercoolings of the liquids adjacent to and phases can be written as ⎛ ⎞ 2( S + S ) 2 V ΔTc = m ⎜ c A − ce + B0 + c0 Po ⎟ , S D ⎝ ⎠

(7.16)

⎛ ⎞ 2( S + S ) 2 V ΔTc = m ⎜ ce − c A − B0 + c0 Po ⎟ . S D ⎝ ⎠

(7.17)

The next step is to calculate the average curvature undercooling ⌬TR. Referring to Figure 7.1, the solid–liquid interfaces can be specified by a certain curve equation z = z(x) in the z⫺x two-dimensional coordinate system. If represents the angle between the x-coordinate and the tangent line of the curve z = z(x) at an arbitrary location, the average curvature of the and interfaces are K =

K =

1 S

1 S

S

∫ 0

d 2 z/dx 2

(1 + (dz/dx) ) 2

S + S

∫

S

3/ 2

dx =

d 2 z/dx 2

(

1 + ( dz/dx ) 2

)

3/ 2

1 S

dx =

−

∫ cos d = − 0

1 S

0

sin , S

∫ cos d = −

sin S

(7.18)

.

(7.19)

The negative sign of curvature indicates that the interfaces are convex with respect to the positive direction of the z-axis. Consequently, the curvature undercoolings takes the form ΔTR = − Γ K = ΔTR = − Γ K =

Γ sin S Γ sin S

(7.20) .

(7.21)

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Here, and are the Gibbs–Thomson coefficients. Equation (7.2) can be analytically expressed by a combination of Eqs. (7.16), (7.17), (7.20) and (7.21): ⎛ ⎞ al 2( S + S ) 2 V ΔT = m ⎜ c A − ce + B0 + c0 P ⎟ + , S D ⎝ ⎠ S

(7.22)

⎛ ⎞ al 2( S + S ) 2 V ΔT = m ⎜ ce − c A − B0 + c0 P ⎟ + , S D ⎝ ⎠ S

(7.23)

where al = sin and al = sin. As mentioned earlier, the solid–liquid interfaces of regular lamellar eutectic are usually isothermal. Therefore, ⌬T = ⌬T = ⌬T. By combining Eq. (7.22) with Eq. (7.23) to eliminate the terms involving cA and B0, the following equation is derived: ΔT al = V S Ql + m S

(7.24)

with m=

m m m + m

; Q1 =

⎡ a1 sin a1 sin ⎤ P (1 + ) 2 c0 S ; = ; a1 = 2(1 + ) ⎢ + ⎥. m ⎥⎦ D S ⎢⎣ m

Equation (7.24) is frequently rewritten in another form: ΔT = K1 SV + K 2 / S

(7.25)

with Kl = mQl and K2 = mal. As an illustration, the profiles of Eq. (7.25) are plotted in Figure 7.3, where either the eutectic growth velocity V or undercooling ⌬T is kept constant. If the eutectic growth velocity V is fixed, undercooling exhibits a minimum as interlamellar spacing S varies. When undercooling remains unchanged, eutectic growth velocity V shows a maximum with the increase of interlamellar spacing S. It is apparent that Eq. (7.25) by itself cannot uniquely determine the ⌬T⫺V⫺S interrelationship. For this reason, Jackson and Hunt made their fifth assumption that lamellar eutectic prefers to grow under the extremum condition. That is to say, interface undercooling always takes its minimum for a given growth velocity, or equivalently eutectic growth velocity always takes its maximum at a certain undercooling. This condition requires d⌬T/dS = 0, or dV/dS = 0. A combination

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289

100

V = 10−3 m/s

ΔT (K)

80

60

40

Unstable

20 Stable 0

0

λm

2

Unstable 4

λM 6

8

10

λ (μm) 0.30

V (mm/s)

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0.26 ΔT = 7.5 K 0.22

0.18

0.14

0.10

0

2

4

6 λ

8

10

Figure 7.3. Profiles of the Eq. (7.26) ⌬T = K1SV + K2/S. The undercooling ⌬T as a function of interlamellar spacing S at fixed velocity V (upper part) and V as a function of S (lower part). For the calculations, the physical parameters of Co᎐Sb eutectic alloy have been used [7.8].

of the extremum condition and Eq. (7.25) leads to the classical results of Jackson and Hunt: 2SV = K 2 /K1 ,

(7.26a)

S ΔT = 2 K 2 ,

(7.26b)

ΔT /V = 4 K1 K 2 .

(7.26c)

2

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Figure 7.4. Interlamellar spacing S and growth velocity V as a function of undercooling ⌬T as calculated within the Jackson–Hunt model using the physical parameters of eutectic Co᎐Sb alloy [7.8].

The above equations indicate that S2V, S⌬T and ⌬T 2/V are constants during lamellar eutectic growth. Only two of these equations are independent. If an undercooling is given, the eutectic growth velocity and interlamellar spacing are uniquely determined by Eqs. (7.26b) and (7.26c), which is shown in Figure 7.4. Although the extremum condition has been confirmed to be equivalent to the marginal stability criterion [7.9], experimental investigations [7.10, 7.11] have shown that there is no sharp selection criterion for eutectic interlamellar spacing. Instead, a range of interlamellar spacings from the extremum m to a somewhat larger M (cf. Figure 7.3) are stable at a given growth velocity. The minimum observed spacing is usually equal to m, and the average spacing S is closer to m rather than located midway between m and M. Nevertheless, S2V has been proven to be constant under most experimental conditions, lending support to the Jackson–Hunt theory [7.10]. Under rapid solidification conditions, there are several process characteristics that are incompatible with Jackson–Hunt’s assumptions: 1. The interface undercooling becomes large during rapid solidification so that ⌬c and ⌬c deviate significantly from their values at the eutectic temperature. 2. The eutectic growth velocity can be large and consequently the small Péclet number assumption is no longer valid.

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3. Because of the non-equilibrium interface kinetics effect the solute partition coefficients and liquidus slopes do not remain constant. Trivedi et al. [7.6] extended the Jackson–Hunt model to the case of high Péclet numbers. The main conclusion is that S2V, S ⌬T and ⌬T 2/V are not constant but depend on the eutectic growth velocity and the Péclet number: 2SV = f1 (V , Pe)

(7.27a)

S ΔT = f 2 (V , Pe)

(7.27b)

ΔT /V = f 3 (V , Pe),

(7.27c)

2

where f1, f2 and f3 are functions of V and Pe. Since the case k = k = k provides more detailed physical insight into the problem of rapid eutectic growth, Kurz and Trivedi [7.7] carried out a further extension for the eutectic growth model according to Trivedi et al. [7.6] to take into account the non-equilibrium effect. The format of their final results is to replace the constants K1 and K2 in Eqs. (7.26) by K1v and K2v, which are determined by K1v =

mv Δc0v Po , Df f

⎛ Γ sin ⎞ K 2v = 2mv ∑ ⎜ i v i ⎟ ⎝ mi f i ⎠

(7.28) i = , .

(7.29)

Here, f and f stand for volume fractions of and phases. The quantities used in Eqs. (7.28) and (7.29) are characterized by the following expressions: Δc0v = 1 − k (V ), pn = 2n/Pe, f =

S , S + S

⎛ ΔGa ⎞ pn ⎛ 1⎞ Po = ∑ ⎜ ⎟ sin 2 ( nf ) , D = D0 exp ⎜ − , ⎝ n ⎠ ⎝ RG T ⎟⎠ 1 + pn2 − 1 + 2k (V ) 3

⎛ k − k (V )[1 − ln k (V ) /k ] ⎞ mv = m(V ) ⎜1 + ⎟⎠ , ⎝ 1− k ⎛ k − k (V )[1 − ln k (V ) / k ] ⎞ mv = m ⎜1 + ⎟⎠ , ⎝ 1− k ⎛ k − k (V )[1 − ln k (V ) / k ] ⎞ mv = m ⎜1 + ⎟⎠ . ⎝ 1− k

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Metastable Solids from Undercooled Melts k=0.9

λ2sV (μm3/s)

104

k=0.5 k=0.1

102

100

2

λ sV ≅ const if Pe>1

10−2 −1 10

100

101 102 Péclet number Pe

103

0.10 k=0.8

0.08

k=0.5 λs(μm)

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0.06 0.04

k=0.3

0.02 k=0.1 0.00 1

10 Growth velocity V (cm/s)

100

Figure 7.5. Results calculated within the eutectic growth model of Trivedi et al. [7.6] and Kurz and Trivedi [7.7]. S2V as a function of the Péclet number (a, upper part) and S as a function of growth velocity V (b, lower part). The equilibrium partition coefficient k is varied as a parameter.

Here, k is the equilibrium partition coefficient, k(V) the velocity-dependent partition coefficient according to the solute-trapping model by Aziz [7.12] (see Eq. (6.16)), D0 the pre-exponential factor of the diffusion coefficient, ⌬Ga the activation energy of atomic diffusion and RG the gas constant. According to the theory by Trivedi et al., the Jackson–Hunt analysis will cause significant error when the Péclet number is larger than unity, as shown in Figure 7.5a. Another important prediction by the extended eutectic growth model is that there is a maximum growth velocity beyond which regular lamellar eutectic growth is not possible (cf. Figure 7.5b). Although the theory by Trivedi et al. is the most successful model for rapid eutectic solidification to date, it is applicable only when regular lamellar eutectic growth is ensured. There are few experimental results on the variation of eutectic interlamellar spacing with growth rate in rapid solidification. Boettinger et al. [7.13] studied eutectic growth in the Ag᎐Cu system. Zimmermann et al. [7.14] investigated the

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Temperature

relation of interlamellar spacing S versus growth velocity V in the Al᎐Cu alloy by laser surface resolidification experiments. The results indicate that the modified theory by Trivedi et al. and Kurz and Trivedi fits the experimental data well, particularly in its prediction of a maximum velocity for eutectic growth. If this maximum velocity is exceeded banded structures are observed, which have been explained by oscillating growth modes [7.15]. It has been frequently observed that alloys deviating from eutectic concentration can be solidified in a way such that full eutectic microstructures are produced, whereas eutectic alloys sometimes exhibit primary phase formation under certain circumstances. To explain such behaviour of eutectic solidification, the concept of “coupled zone” was developed [7.16]. The coupled zone is a composition–temperature range in phase diagram within which cooperative growth is ensured. Figure 7.6 shows the two typical categories of coupled zone [7.16]: (a) symmetric

β-Dendrites

Eutectic

Te

α-Dendrites Composition

co

Growth velocity

(a) Symmetrical zone

Temperature

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β-Dendrites

Eutectic

Te

α-Dendrites Composition

co

Growth velocity (b) Skewed zone

Figure 7.6. Two typical kinds of coupled zone.

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zone, favouring regular lamellar eutectic and (b) skewed zone, corresponding to irregular eutectic. A competitive growth analysis is used to predict the coupled zone. In a binary eutectic system, three different kinds of microstructures are possible: (1) only eutectic, (2) primary dendrites plus eutectic and (3) primary dendrites plus eutectic. The essential idea of competitive growth is as follows: for an alloy with given composition c0 (⬍ce) or c0 (⬎ce) the preferred microstructure is the one that solidifies at the smallest undercooling for a same growth velocity. During directional solidification, the eutectic and primary phase growth velocities are related to the undercooling by the following equations: ΔTe = K eV 1/ 2 , ∇L D + K V 1/ 2 , V ∇ D ΔT = L + K V 1/ 2 . V ΔT =

(7.30a) (7.30b) (7.30c)

At the coupled zone boundaries, the interface temperatures of the eutectic and primary phases should be equal, hence ∇L D + ( K − K e )V 1/ 2 , V ∇ D ΔT − ΔTe = m (c0 − ce ) = L + ( K − K e )V 1/ 2 . V ΔT − ΔTe = m (c − c0 ) =

(7.31a) (7.31b)

Consequently, the locations of coupled zone boundaries are determined by the following relations between composition and boundary temperature: c0 = ce −

1 m

⎤ ⎡ ∇ L DK e2 K − K e (Te − T ) ⎥ + ⎢ 2 Ke ⎦ ⎣ (Te − T )

⎤ 1 ⎡ ∇ L DK e2 K − K e c = ce − (Te − T ) ⎥ + ⎢ 2 m ⎣ (Te − T ) Ke ⎦

(7.32a)

0

(7.32b)

where ⵜL is the liquid temperature gradient at the growth front; Ke, K and K are constants. 7.2 EUTECTIC MORPHOLOGY TRANSITION

In the previous section, we considered a model for lamellar eutectic growth. However, eutectic precipitation and growth exhibit various morphologies that go

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beyond the concept of “coupled zone” (Figure 7.6). These morphologies are shown and described in monographs [7.17–7.21]. Known morphologies of lamellar, rod and globular eutectics (Figure 7.7) can be described within the standard schemes of eutectic growth, consistent with the solute diffusion transport and equilibrium phase diagram of the alloy’s solidification [7.22]. Depending on the alloy’s composition, different types of lamellar or rod euetctics can be observed. For example, by choosing the system temperature below or above the eutectic temperature Te (Figure 7.8) one can observe cellular growth

Figure 7.7. Typical morphology of eutectic alloys: (a) lamellar pattern, (b) rod structure, (c) globular eutectics and (d) needle-like eutectics. 1.00 L β+L

α+L

0.95

(a)

(b)

(c)

(d)

T1 TE

0.90 Temperature

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0.80

α

α+β

β

0.75 cI

cE

0.70 0

0.2

0.4 0.6 Concentration of B

Figure 7.8. Phase diagram of eutectic type.

0.8

1

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of the primary solid phase (point a), melting of a lamellar eutectic structure at the temperature above eutectic temperature with the eutectic composition of an alloy (point b), growth of a hypoeutectic pattern (point c) and solidification with the eutectic lamellar pattern (point d). These are shown in Figure 7.9. A transition from the lamellar pattern to the rod pattern is favoured when the lead–lag distance between phases is large [7.21]. Convection may influence the transition from one lamellar spacing in the eutectic phase to another. This has been confirmed under microgravitational conditions (Figure 7.10; relevance of microgravity is overviewed in Chapter 10). As a result of space experiments, scientists are re-examining a classical theory on the formation of eutectics. The theory of Jackson and Hunt [7.5] assumes that there is no convection in the melt when the eutectic materials are processed in space. The theory works quite well on Earth, but an earlier rocket experiment produced a eutectic with rod spacing quite different from what was predicted by the classical

(a)

(b) 0.35

0.9

0.30

L

1.0

0.8

L

0.7

0.25

0.6

0.20

0.5 0.4

0.15

0.3

α

α

α

α

α

0.10 0.05

α

β

α

β

α

β

α

β

α

0.2 0.1 0.0

1.0

(c)

(d)

0.9 0.8

0.7

L

0.6

0.6

0.5

0.5

0.4

0.4

0.3

α β

α

β

α

β

α

β α

0.9 0.8

0.7

L

1.0

0.2 0.1 0.0

0.3

α

β

α

β

α

β

α

β

α

0.2 0.1 0.0

Figure 7.9. Modelled lamellar patterns [7.22] corresponding to the points indicated in phase diagram in Figure 7.8. Growing eutectic lamellar pattern are shown in (a), (c) and (d), and melting lamellar pattern is shown in (b). Arrows indicate the direction of motion of the eutectic fronts. The coloured scale shows the variation of the concentration in dimensionless units, normalized on the initial concentration.

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Figure 7.10. Al᎐AlCu eutectics processed in space (left) has a finer structure than the sample processed in the same facility on the ground (right).

theory. This was puzzling, but when the experiment was repeated in ground laboratories where a magnetic field was used to damp convection, experimenters got the same results. Scientists were faced with a paradox: a theory based on no convection worked fine when convection was present, but the theory did not work when convection was absent. As a result of experimentation with eutectic patterns in Bi᎐MnBi and InSb᎐NiSb systems, Favier [7.23] and Müller [7.24] suggested that the observed large differences in fibre spacing between 1 g and 1 g are caused by convection-driven changes in the elemental distribution within the solutal boundary layer ahead of the eutectic solidification front. Observation of the needle-like eutectic morphology (Figure 7.7d) is possible due to high solid–liquid surface tension for phase. It is observed at small or moderate solidification velocities. In this case, total morphology can be described by the socalled skeletal crystals (Figure 7.11, left) or sectorial eutectics (Figure 7.11, right). The lamellar/rod morphology degenerates with increasing velocity of solidification. A change in the composition of eutectics, occurring in the hypoeutectic (from point d to point c in Figure 7.8) or hypereutectic alloy, leads to the growth of primary dendrites of the leading phase. With the saturation of the interdendritic space by the second component, eutectic patterns are formed and grow coupled with the primary dendrites (Figure 7.12). Lamellar cells or dendrites were observed in experiments as well (Figure 7.13). The first observation of the eutectic cells has been explained by a motion of the curved interface of the two-phase solidification [7.30]. On studying a microstructure of interfaces in decanted Al᎐Cu and Sn᎐Zn alloys during their solidification, it has been observed that the boundaries of neighbour cells are merged with the

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Figure 7.11. Skeletal eutectic colony (left) and sectorial eutectic colony (right) [7.18].

Figure 7.12. Coupled growth of dendritic pattern (leading phase) and eutectic phase. Growth of primary dendrites and interdendritic eutectic pattern in Br4C-Heksachloretan system [7.25] (left) and in an iron-based alloy [7.18] (right).

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Figure 7.13. Eutectic cells in the transparent organic alloy CBr4⫺C2Cl6 grown during a directional solidification experiment [7.26] (left) and lamellar eutectic dendrites grown in an undercooled eutectic Ni᎐Si alloy [7.27] (right).

Figure 7.14. Eutectic cells in the phase-field simulations of a CBr4⫺C2Cl6 alloy grown by a directional solidification experiment [7.28].

profile of the curved interface. Direct observations of eutectic cell formation in transparent binary systems has been presented in Ref. [7.26] (Figure 7.13, left). Modelling of eutectic colony solidification (Figure 7.14) confirms that eutectic cells are formed due to instability of the steady-state interface. The cells themselves grow with the initially perturbed solid–liquid interface. Within the cells, lamellar eutectic phase grows. A transition to rapid eutectic dendrites has been observed in undercooled samples of eutectic Ni78.6Si21.4 alloys [7.27] (Figure 7.12, right). Large undercoolings up to ⌬T = 220 K were achieved by containerless processing of the melt. The microstructures were identified as eutectic structures formed with an envelope of the solidification front, which shows a dendrite-like morphology. This is the result of the existence of a negative temperature gradient (due to undercooling

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10 μm

(a) ΔT = 0 K

Lamellar

Anomalous

(b) ΔT = 127 K

(c) ΔT = 225 K

50 μm

10 μm

Figure 7.15. Eutectic growth morphologies at different undercoolings in Co74.5Sb25.5 alloy [7.29]: (a) lamellar eutectic (⌬T = 0 K), (b) mixed eutectic microstructures consisting of lamellar and anomalous eutectics (⌬T = 127 K) and (c) anomalous eutectic (⌬T = 225 K).

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Figure 7.16. A transition from supersaturated solid solution to eutectics in Al 35 wt.% Mg alloy [7.19]. The transition from one type of structure to another is visible as the sharp one. It is due to the sharp change of the mechanisms of growth.

ahead of the solid–liquid interface), which leads to a curved dendrite-like solidification front. A scheme of the eutectic transformations with increasing undercooling can be shown in the microstructures presented in Figure 7.15. The gradual transition from lamellar eutectic colony to the anomalous (or irregular) eutectic pattern is demonstrated. The next step in transformation might be the degeneration of eutectic precipitation. Indeed, Miroshnichenko observed that under high cooling rates (deep underccolings) the eutectic phase is suppressed completely due to the formation of a metastable supersaturated solid solution [7.19]. Figure 7.16 shows the transition from supersaturated solid solution to eutectic morphology with decrease in the solidification velocity. It is interesting to note that the rapidly solidified solid solution shows the initial (nominal) chemical composition of an alloy [7.19]. Formation of the supersaturated solid solution instead of the eutectic phase cannot be explained by continuing the liquidus and solidus lines into the metastable region in the phase diagram of an alloy. However, it can be attributed to the features of rapid solidification when the interface velocity is higher than the solute diffusion speed in bulk

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Anomalous

400 Eutectic growth velocity V (mm/s)

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Best fit 300

200 Jackson-Hunt (D=D(T)) ΔT∗=50 K

100

Vmax = 109 mm/s

Vmax = 63 mm/s TMK-model 319 K

0

0

100

200 300 Undercooling ΔT (K)

325 K 400

Figure 7.17. Measured growth velocity (closed circles: regular eutectic, open circles: anomalous eutectic) and calculated eutectic growth velocity (thick solid lines) as a function of undercooling in Co74.5Sb25.5 system [7.30]. Eutectic growth velocities are calculated within Jackson–Hunt model [7.5] assuming a temperature-independent and a temperature-dependent diffusion coefficient D, and Trivedi–Magnin–Kurz (TMK) model [7.6].

phases. In this case, the solidification proceeds by a diffusionless mechanism with the merging of the liquidus and solidus lines into one line in the kinetic phase diagram. As a consequence, the eutectic precipitation is suppressed at higher solidification velocity. With decreasing velocity (lower than the solute diffusion speed) the eutectic phase begins to form, as is shown by microstructure in Figure 7.16. To predict the scheme shown in Figure 7.15, quantitative measurements of the growth velocity and the interlamellar spacing as a function of undercooling have been performed on different eutectic alloys using the melt fluxing technique to undercool the samples. The recalescence time during solidification has been measured by a specially designed infrared photodiode device. This device was constructed and installed such that the whole image of the sample was focused onto the sensitive area of the photodiode. The velocity was obtained by dividing the sample diameter by the measured recalescence time. More experimental details are given in Ref. [7.30]. It has been found that interlamellar spacing S decreases with undercooling as predicted by theory. However, at a critical undercooling, ⌬T* 50 K, there is a transition from regular lamellar eutectic to irregular anomalous eutectic

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microstructure. Figure 7.17 shows as an example the measured growth velocity as a function of undercooling for the eutectic Co74.5Sb25.5 alloy together with calculations of the growth velocity within the Jackson–Hunt model and the extended theory by Trivedi et al. [7.31, 7.32]. It should be noted that the calculations within the eutectic growth theories make sense only for lamellar eutectic. When undercooling exceeds the critical undercooling ⌬T*, the experimental data correspond to anomalous eutectic growth. It is obvious that the eutectic growth theories are able to describe the experimental data in the undercooling range ⌬T ⬍ ⌬T*, while at larger undercoolings where anomalous eutectic occurs systematic deviations appear between experiment and theory. The anomalous eutectic growth velocity is essentially larger than growth velocities calculated by using the theories for lamellar eutectic growth. A characteristic feature of the model by Trivedi et al. is that it predicts a maximum growth velocity limit of 63 mm/s for Co74.5Sb25.5 lamellar eutectic at an undercooling of 319 K. In contrast, the experimentally determined maximum growth velocity is only 22 mm/s, which corresponds to an undercooling slightly below the critical undercooling. If a constant solute diffusion coefficient D is assumed, the Jackson–Hunt model gives a parabolic relation for the eutectic growth velocity as a function of undercooling, and no growth limit is expected. However, if the temperature dependence of the diffusion coefficient is taken into account, the Jackson–Hunt model also exhibits a maximum lamellar eutectic growth velocity of 109 mm/s at 325 K undercooling. Although the maximum growth velocities predicted by the various theories are quite different, the corresponding undercoolings are close to each other. This means that in the case of Co74.5Sb25.5 lamellar eutectic growth, the limiting growth velocity results from the temperature-dependent diffusion coefficient. Similar results have been obtained on other eutectic alloys such as Co᎐Sn, Ni᎐Si and Co᎐Mo [7.32–7.34 ].

7.3 STABLE AND METASTABLE MONOTECTIC ALLOYS

Monotectic alloys have a miscibility gap in the liquid state, which may affect the solidifying microsctructure. During the monotectic reaction L1 → L2 + S, a liquid L1 decomposes simultaneously into a solid S and another liquid L2 for a wide class of alloys [7.35–7.37] shown schematically in Figure 7.18. The equilibrium reaction occurs at a fixed temperature Tm and composition. It is similar to a eutectic reaction where, however, the liquid is decomposed in two solids. A typical solidifying microstructure of a monotectic system is shown in Figure 7.19. Depending on the properties of the alloy and solidification parameters (solidification velocity V and temperature gradient ⵜT), the following regimes of growth have been observed [7.38–7.42]: composite fibrous growth (regular

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TE

L1+L2 S+L1

S Tm

S+L2

CSm

m CL2

Cm

Figure 7.18. Typical phase diagram with monotectic reaction.

(b)

(a)

100 μm

Figure 7.19. Experimental solidifying microstructure with monotectic reaction [7.39]: (a) regular growth of fibres, and (b) irregular growth with droplets.

growth) and dendritic growth, where the L2 phase is deformed by droplets into the dendritic network (irregular growth). These are presented in Figure 7.19. Normally, a high thermal gradient to growth velocity ratio ⵜT/V leads to a stable planar solid–liquid interface and aligned fibrous microstructure [7.43] shown in Figure 7.19(a). In equilibrium, wetting conditions at the interfaces L1-L2, L1-S and L2-S play a main role in the type of monotectic reaction [7.44, 7.45]. Depending on surface energy , three types of monotectic reactions are possible: 1. The liquid L2 does not wet the solid (Figure 7.20a), S1-L2 ⬎ S1-L1 + L1-L2 and regular rod-like structures will not be formed in this type of reaction [7.44];

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Figure 7.20. Three types of monotectic solidification: (a) L2 does not wet the solid, wetting angle is = 180⬚; (b) L2 wets the solid, 0⬚ ⬍ ⬍ 180⬚ and (c) ideal wetting of L2 with the solid S, = 180⬚.

2. The liquid L2 wets the solid (Figure 7.20b), S1-L2 ⬍ S1-L1 + L1-L2, a surface energy balance is established at the triple-phase junction and steady-state growth similar to the eutectic can occur; 3. The liquid L2 wets the solid ideally (Figure 7.20c), steady-state growth will be then impossible and an irregular microstructure can be formed [7.46]. With the development of space science, monotectic solidification has become a more attractive subject. Solidification of monotectics (preferably, hypermonotectic alloys) under low-gravity or alternating gravity-level conditions has been considered in Ref. [7.47]. It was shown that monotectic solidification depends, in practice, on wetting conditions at the container walls or wetting conditions at the outer surface of the droplet in the case of containerless processing. It was also expected that microgravity conditions would result in a homogeneous dispersion of the liquid-product phase during monotectic reaction, but this has proven not to be the case [7.48]. Inhomogeneity may exist due to the presence of capillary forces on the solid–liquid interface, developing Marangoni convective flow, which leads to morphological instability of the interface and inhomogeneous distribution of phases during monotectic reaction. Phase-field modelling [7.49] of monotectic solidification with convection confirms interfacial instability due to Couette and Marangoni flow and, as a result, possible irregular distribution of phases. Although extensive work has been done on the directional solidification of monotectic alloys [7.39, 7.46], there are very few successful experiments on the rapid solidification of highly undercooled immiscible alloy melts. The crystal growth kinetics and microstructural characteristics of an undercooled Ni-5 wt.% Ag alloy has been investigated in melt fluxing processing [7.50]. During rapid solidification from undercooled liquid, the monotectic reaction may lead to a metastable microstructure. Figure 7.21 shows the solidification microstructures of undercooled Ni-5 wt.% Ag alloy. It has been observed that with no undercooling or very small undercoolings,

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(a) ΔT = 0 K

50 μm

(b) ΔT = 223 K

0.5 mm

Figure 7.21. Distribution of Ag phase in rapid solidification microstructures of undercooled Ni-5 wt.% Ag monotectic alloy: (a) ⌬T = 0 K, Ag particles are relatively homogeneously distributed; (b) ⌬T = 223 K, Ag particles agglomerate to produce macrosegregation at the sample bottom.

the Ag phase is fairly homogeneously dispersed in the ᎐Ni phase matrix. As the undercooling increases, the Ag phase shows an apparent tendency to coalesce and descend to the sample bottom. Unlike for small undercoolings, Ag agglomerations are also observed on the top of highly undercooled Ni-5 wt.% Ag monotectic samples, which must have been caused by the rapid growth of the solid ᎐Ni phase. These results suggest that the nucleation and separation of the liquid Ag phase occurs in preference to the solid ᎐Ni phase. Once the ᎐Ni phase appears, the advancing interface facilitates the coalescence of liquid Ag droplets. Figure 7.22 presents the measured growth velocities V of the ᎐Ni phase for different undercooled values ⌬T. At an undercooling of 291 K, the ᎐Ni phase achieves a growth velocity of 115 m/s. The solid line in Figure 4 shows the

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120

100

Calculated data Experimental data

80 V, m/s

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40

20

0

0

50

100

150 ΔT, K

200

250

300

Figure 7.22. Calculated and measured growth velocities of the ᎐Ni solid phase in undercooled Ni-5 wt.% Ag monotectic alloy [7.50].

calculated dendrite growth velocities according to dendric growth models [7.51, 7.52]. The predictions of the modes agree well with experiment for the growth of the ᎐Ni phase. Therefore, it is reasonable to infer that the ᎐Ni phase grows dendritically in undercooled Ni-5 wt.% Ag monotectic alloy melt. 7.4 PERITECTIC ALLOYS

Many binary phase diagrams exhibit peritectics, in which one solid phase reacts with a liquid phase on cooling to produce a second solid phase, i.e. + L . A typical peritectic diagram is shown in Figure 7.23. With a composition between cpL and cp, the phase nucleates and grows dendritically when the liquidus is crossed. At temperatures below Tp, the phase nucleates and grows rapidly around the phase. Once the phase is covered, the peritectic reaction effectively ceases because now the peritectic reaction can only take place by the much slower solidstate diffusion through the layer of phase. The solidification process then continues by the thickening of the solid as the temperature is further reduced. The rate of growth of both and depends on how easily the rejected B atoms can be removed from the interface. Below the peritectic temperature and near the triple L junction, B atoms can diffuse from the phase to the phase (C⬘L ⬎ C⬘L). At the front, the arrival of B atoms leads to melting of the phase, whereas

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Temperature

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Cpα C pβ

CpL

Tp C′Lα

C ′Lβ

Composition %B

Figure 7.23. A peritectic diagram showing definitions for composition Cp of the phase, composition Cp of the phase and composition CpL of the liquid phase at temperature Tp of the peritectic reaction. C⬘L and C⬘L are the liquid compositions at the and phases, respectively, below peritectic temperature Tp.

at the front the removal of B atoms leads to more rapid growth. Interface shape computed numerically for this case is shown in Figure 7.24. Features of peritectic equilibrium and near-equilibrium solidification are reviewed in Refs. [7.53, 7.54] Particularly, phase selection at low solidification velocity in technically important peritectic alloys has been considered in detail [7.54]. For directional solidification, it is assumed that the leading phase, i.e. phase/growth morphology, which grows at the highest interface temperature, is the kinetically most stable one. For a constant and positive temperature gradient through which a specimen is moved at constant velocity, a certain growth temperature corresponds to a given position in the sample (see Ref. [7.55] and Figure 7.25). For metastable to form at the velocity of the isotherm, Visotherm (Figure 7.25), nucleation of the phase in the melt just ahead of the growth front of must happen at the undercooling given by the difference between the equilibrium liquidus temperature and the growth temperature . Once the phase has nucleated, it passes through a transient regime during which its growth reaches steady state. Then the interface velocity equals the isotherm velocity and growth occurs at higher interface temperature, i.e. close to equilibrium. This approach, known as the maximum growth temperature criterion, uses steady-state growth theory for the calculation of the interface response functions (temperature, chemical composition and solidification velocity) for the competing phases (, and liquid phase) of cellular or dendritic morphology [7.55, 7.56]. The criterion of maximum growth temperature is true for low ⵜT/V (temperature gradient/solidification velocity) ratios. When the ⵜT/V is increased, more

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Growth direction L B atoms

L

Figure 7.24. Below the peritectic temperature Tp, the peritectic phase grows preferentially along the properitectic phase by solute diffusion in the liquid until the parent phase is engulfed. Top: a scheme of the ᎐᎐L junction during peritectic reaction [7.58]. Bottom: simulation of peritectic solidification [7.59].

complex microstructures can form in the two-phase region under steady-state or non-steady-state growth conditions. The formation of new phases occurs during the transient growth condition via nucleation on or ahead of the moving interface. The actual microstructure selection process is thus controlled by nucleation of the two phases, and by growth competition between the nucleated grains and the preexisting phase under non-steady-state conditions. For each phase, nucleation is supposed to occur instantaneously at a constant undercooling below the liquidus. If nucleation of the other phase at or ahead of the solid–liquid interface is not possible, the growing phase/morphology is considered to be stable for the chosen far-field composition and growth conditions (i.e. solidification velocity and temperature gradient). In this case, the maximum growth temperature criterion does not lead to the right answer and nucleation in the constitutionally undercooled zone ahead of the growth front has to be taken into account to determine the microstructure selection. This selection criterion is called nucleation and constitutional undercooling criterion [7.57]. Under rapid peritectic solidification conditions, phase selection occurs for cellular, dendritic and banded morphologies [7.54, 7.60]. In the undercooled state, the new metastable phase formation can occur during peritectic solidification. In situ

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Visotherm Vtransit Log V

Position

Teq Teq

x1 x2

Teq Teq

Tnucl.

T d T d

Td

T T Position Solid

Log V

x1 x2 α

β

Visotherm

Liquid

Figure 7.25. Principle of phase selection in peritectic solidification using a maximum growth temperature criterion [7.55].

X-ray diffraction experiments during solidification of levitated Nd᎐Fe᎐B samples have been conducted [7.61]. Owing to the short measuring time the primary crystallization of a metastable phase that initiates the solidification of hard magnetic Nd2Fe14B1 phase was observed. The metastable phase dissolves subsequently and cannot be detected in the solidified sample at the ambient temperature. However, nucleation and existence of the metastable phase in liquid during solidification may identify metastable crystallization products in solidified samples [7.61]. REFERENCES

[7.1] Zener, C. (1946) Transactions of the AIME 167, 550. [7.2] Brandt, W.H. (1945) Journal of Applied Physics 16, 139. [7.3] Tiller, W.A. (1958) Liquid Metals and Solidification (ASM, Cleveland, Ohio), p. 276. [7.4] Hillert, M. (1957) Jernkontorets Annaler 141, 757. [7.5] Jackson, K.A., and Hunt, J.D. (1966) Transactions of the Metallurgical Society of AIME 236, 1129.

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[7.6] Trivedi, R., Magnin, P., and Kurz, W. (1987) Acta Metallurgica 35, 971. [7.7] Kurz, W., and Trivedi, R. (1991) Metallurgical Transactions 22A, 3051. [7.8] Wei, B. in: Unterkühlte Metallschmelzen: Thermodynamik, Keimbildung und Erstarrung, ed. Herlach, D.M., DLR IB-333-92/4 (Deutsche Forschungsanstalt für Luft- und Raumfahrt, Cologne). [7.9] Datye, V., and Langer, J.S. (1981) Physical Review B 24, 4155. [7.10] Trivedi, R. Mason, J.T., Verhoeven, J.D., and Kurz, W. (1991) Metallurgical Transactions 22A, 2523. [7.11] Seetharaman, R., and Trivedi, R. (1988) Metallurgical Transactions 19A, 2955. [7.12] Aziz, M.J. (1982) Journal of Applied Physics 53, 1158. [7.13] Boettinger, W.J., Shechtman, D., Schäfer, R.J., and Biancaniello, F.S. (1971) Metallurgical Transactions 15A, 55. [7.14] Zimmermann, M., Carrard, M., and Kurz, W. (1989) Acta Metallurgica et Materialia 37, 3305. [7.15] Karma, A., and Sarkissian, A. (1992) Physical Review Letters 68, 2616. [7.16] Wei, B., Herlach, D.M., Sommer, F., and Kurz, W. (1993) Materials Science & Engineering 173, 357. [7.17] Kurz, W., and Sahm, P.R. (1975) Gerichtet Erstarrte Eutektische Werkstoffe (Springer, Berlin). [7.18] Taran, Y. N., and Mazur, V.I. (1978) Structure of Eutectic Alloys (Metallurgia, Moscow). [7.19] Miroshnichenko, I.S. (1982) Quenching from the Liquid State (Metallurgia, Moscow). [7.20] Elliot, R. (1983) Eutectic Solidification Processing: Crystalline and Glassy Alloys (Butterworths, London). [7.21] Tiller, W.A. (1991) The Science of Crystallization: Macroscopic Phenomena and Defect Generation (Cambridge University Press, Cambridge, UK). [7.22] Danilov, D.A., Nestler, B., and Galenko, P.K. (2003) Adaptive Finite Element Simulations of Eutectic and Dendritic Growth (EUROMAT 2003, Lausanne, Switzerland). [7.23] Favier, J.J., and De Goer, J. (1984) Proceedings of the 5th European Symposium on Material Sciences under Microgravity Conditions, ESA SP222 (Schloss Elmau, Deutschland, November 1984), p. 127. [7.24] Müller, G., and Kyr, P. (1986) Scientific Results of the German Spacelab Mission D-1, Norderney, 27–29 August, 1986. [7.25] Hunt, J.D., and Jackson, K.A. (1967) Transactions of the Metallurgical Society of AIME 239, 864. [7.26] Hunt, J.D., and Jackson, K.A. (1966) Transactions of the Metallurgical Society of AIME 236, 843. [7.27] Goetzinger, R., Barth, M., and Herlach, D.M. (1998) Journal of Applied Physics 84, 1643. [7.28] Plapp, M., and Karma, A. (2002) Physical Review E 66, 061608.

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[7.29] B. Wei, D.M. Herlach, and F. Sommer, (1993) Journal of Materials Science Letters 12, 1774. [7.30] Weart, H.W., and Mack, J.D. (1958) Transactions of the Metallurgical Society of AIME 212, 664. [7.31] Wei, B., Herlach, D.M., Feuerbacher, B., and Sommer, F.(1992) Proceedings of the 1st Pacific Rim International Conference on Advanced Materials and Processing, eds. Shi, C., Li, H., and Scott, A. (Hangzhou, P.R. China), p. 263. [7.32] Wei, B., Herlach, D.M., Feuerbacher, B., and Sommer, F. (1993) Acta Metallurgica et Materialia 41, 1801. [7.33] D.M. Herlach, F. Sommer, and W. Kurz, (1994) Materials Science & Engineering 181/182, 1150. [7.34] Wei, B., and Herlach, D.M. (1995) Transactions of the Materials Research Society of Japan 14A, 639. [7.35] Sarson, S.C., and Charles, J.A. (1993) Materials Science Technology 9, 1049. [7.36] Ratke, L., and Diefenbach, D. (1995) Materials Science & Engineering Reports R 15, 263. [7.37] Predel, B. (1997) Journal of Phase Equilibria 18, 327. [7.38] Derby. B., and Favier, J.J. (1983) Acta Metallurgica 31, 1123. [7.39] Grugel, R.N., Longrasso, T.A., and Hellawell, A. (1984) Metallurgical Transactions A 15, 1003. [7.40] Grugel, R.N., and Hellawell, A. (1984) Metallurgical Transactions A 15, 1626. [7.41] Kamio, A., Kumai, S., and Tezuka, H. (1991) Materials Science Engineering A 146, 105. [7.42] Coriell, S.R., Mitchell, W.F., Murray, B.T., Andrews, J.B., and Arikawa, Y. (1997) Journal of Crystal Growth 179, 647. [7.43] Andrews, J.B., Sandlin, A.C., and Merrick, R.A. (1991) Advances in Space Research 11(7), 291. [7.44] Chadwick, G.A. (1965) British Journal of Applied Physics 16, 1095. [7.45] Cahn, J.W. (1977) Journal of Chemical Physics 66, 3667. [7.46] Grugel, R.N., and Hellawell, A. (1981) Metallurgical Transactions A 12, 669. [7.47] Andrews, J.B. (1993) in: Immiscible Liquid Metals and Organics. Proceedings of the International ESA Workshop (Bad Honnef, 1992), ed. Ratke, L. (DGM Informationsgesellschaft, Oberursel) p. 199. [7.48] Fredriksson, H. (1984) in: Proceedings of the RIT/EAS/SSC Workshop on the Effect of Gravity on the Solidification of Immiscible Alloys, ESA, SP219 (Jarva Krog, Sweden), p. 25. [7.49] Nestler, B., Wheeler, A.A., Ratke, L., and Stöker, C. (2000) Physica D 141, 133. [7.50] Wei, B., Herlach, D.M., Sommer, F., and Kurz, W. (1993) Materials Science & Engineering A 173, 357. [7.51] Lipton, J., Kurz, W., and Trivedi, R. (1987) Acta Metallurgica 35, 957.

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[7.52] Boettinger, W.J., Coriell, S.R., and Trivedi, R. (1988) in: Rapid Solidification Processing: Principles and Technologies IV, ed. Mehrabian, R. and Parrish, P. (Claitor’s, Baton Rouge, LA), p. 13. [7.53] Hunt, J.D., and Lu, S.-Z. (1994) in: Handbook of Crystal Growth, Vol. 2, Chapter 17, ed. Hurle, D.T.J. (North Holland, Amsterdam), p. 1112. [7.54] Kerr, H.W., and Kurz, W. (1996) International Materials Review 41, 129. [7.55] Gilgien, P., and Kurz, W. (1994) Materials Science & Engineering A 178, 171. [7.56] Umeda, T., Okane, T., and Kurz, W. (1996) Acta Materialia 44, 4209. [7.57] Hunziker, O., Vandyoussefi, M., and Kurz, W. (1998) Acta Materialia 46, 6325. [7.58] Hillert, M. (1979) Solidification and Casting of Metals (Metals Society, London), p. 81. [7.59] Nestler, B., and Wheeler, A.A. (2000) Physica D 138, 114. [7.60] Trivedi, R. (1995) Metallurgical Materials Transactions A 26, 1583. [7.61] Volkmann, T., Strohmenger, J., Gao, J., and Herlach, (2004) Applied Physics Letters 85, 2232.


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Chapter 8

Metastable Solid States and Phases 8.1 General Conditions for the Formation of Metastable Solids 8.2 Supersaturated Solid Solutions 8.3 Formation of Metastable Crystalline Phases 8.4 Phase Selection Through the Solidification Kinetics 8.5 Metallic Glasses 8.6 Grain-Refined Materials References

317 320 323 333 335 339 354


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Chapter 8

Metastable Solid States and Phases 8.1 GENERAL CONDITIONS FOR THE FORMATION OF METASTABLE SOLIDS

As described in Chapter 3 the driving force for crystallization of an undercooled melt is given by the Gibbs free energy difference between solid and undercooled liquid, GLS = GL GS. The fact that a metastable solid has a higher free energy than the stable solid leads to the consequence that the melting temperature of the metastable solid is lower than the melting temperature of the stable solid. A necessary precondition for the formation of a metastable solid is, therefore, that an undercooling larger than the difference between the melting temperature of stable and metastable solid, T Tst Tms, must be achieved to generate a driving force for the crystallization of the metastable phase. But even for undercoolings larger than Tst Tms the driving force for the crystallization of the stable solid is always larger than that of the metastable solid. Despite this fact, solidification of metastable solids can be understood if the nucleation process is taken into consideration. As discussed in Chapter 5 the activation threshold for the formation of critical nuclei depends both on the free energy difference and the interfacial energy, and in the case of heterogeneous nucleation, also on the catalytic potency factor. The effect of a lower free energy difference GLS for the formation of a metastable solid can be overcompensated by a corresponding decrease in LS and/or f(ϑ). Furthermore, the conditions of crystal growth will influence the final structure of the solidified product. These situations are schematically demonstrated in Figure 8.1. The crystallization of an undercooled melt takes place by a two-step process. Once nucleation has initiated solidification the subsequent growth of the solid phase leads to the release of the heat of fusion Hf. In the case of rapid crystal growth a steep rise of temperature will occur, called recalescence. Assuming that solidification during recalescence takes place under near-adiabatic conditions the volume fraction solidifying during recalescence under non-equilibrium conditions can be determined by fR =

H f C Pl

T .

(8.1)

The remaining part of the sample fP = 1 fR solidifies under equilibrium conditions at the melting temperature (in the case of pure metals and eutectics) or in the 317

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melting interval between liquidus (TL) and solidus temperature (TS), i.e. T0 = TLTS (in the case of alloys). In the special situation that the undercooling approaches the hypercooling limit the undercooled melt solidifies exclusively during recalescence so that fP = 0 and fR = 1. Under such conditions, H f =

TL

∫C

l P

(8.2)

(T ) dT ,

Thyp

L G MS S

a) MS

S

TE ΔG*

b)

MS

Tc c) V

TE S

MS

TE

S

TE

MS S

T

Figure 8.1. (a) Gibbs free energy G, (b) activation G* for the formation of critical nuclei and (c) crystal growth velocity V as a function of temperature for a liquid (L), a stable solid (S) and a metastable solid (MS) phase, respectively. Note that solidification of the metastable solid phase is favoured if the undercooling T exceeds TE – Tc.

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Metastable Solid States and Phases

319

where Thyp is the isenthalpic or hypercooling temperature. Presuming a temperatureindependent specific heat Cpl, Eq. (8.2) becomes Hf = Cpl(TE Thyp) = CplThyp with Thyp the hypercooling limit. So far, the hypercooling limit has not yet been unambiguously achieved for a pure metal. The hypercooling limit can be determined if data of Cpl(T) and Hf are available, or by measuring the length LP of the postrecalescence phase (plateau regime) as a function of undercooling and extrapolating the resulting correlation LP(T) to LP = 0 [8.1]. In this way the hypercooling limit for pure Ni has been determined as Thyp = 450 K. More recently, calorimetric investigations on PdCo alloys have shown small heat of fusion values [8.2]. From these measurements a small hypercooling limit of T = 250300 K is inferred depending on the concentration of the alloy, which is completely miscible over the entire concentration range. Undercoolings beyond the hypercooling of this alloy system has been observed by optical detection of the recalescence profile as well as by undercooling experiments of levitated drops [8.2, 8.3]. The different undercooling ranges are schematically illustrated in Figure 8.2 for an alloy of nominal composition c0. Nucleation at the temperature T1 and the subsequent recalescence leads to a temperature rise into the two-phase regime + L. Provided that the crystal growth velocity V VD (VD is the atomic diffusive speed) the temperature rise during recalescence ends at a maximum temperature T T0. As discussed in Chapter 6, rapid crystal growth leads to deviations from chemical equilibrium at the solid–liquid interface. In the extremum case that V VD segregation-free solidification is expected. In the postrecalescence period the remaining liquid transforms under equilibrium conditions so that segregation takes place. Here, the temperature–time characteristic is determined exclusively by heat

Figure 8.2. Eutectic phase diagram of an A–B alloy and the temperature–time profiles for hypocooling (left) and hypercooling (right) conditions of an alloy of nominal composition c0, and their consequences on the formation of a segregation-free solid.

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transfer to the environment in contrast to the recalescence phase where the undercooled melt acts as a heat sink. If, however, the undercooling approaches the hypercooling limit the temperature rise during recalescence does not exceed the temperature T0. The whole liquid then transforms under non-equilibrium conditions. The condition to reach the hypercooling limit has therefore a strong influence on the solidification of metastable solid phases such as supersaturated solid solutions or other metastable solids. 8.2 SUPERSATURATED SOLID SOLUTIONS

The conditions of rapid crystal growth have a strong impact on solute trapping such that at large solidification velocities supersaturated solid solutions can be formed. The constraints for the solidification of metastable supersaturated phases are twofold concerning both energetic or thermodynamic and kinetic preconditions. (i) The temperature of the sample during recalescence must remain below T0, which is defined by the condition that at this temperature the free energy of the crystal equals the free energy of the liquid GL(c, T) = GS(c, T). Only at temperatures T T0 does a finite driving force exist for chemical segregation. (ii) The solidification velocity V must exceed the atomic diffusive speed to trap solute into the solid with a concentration beyond the equilibrium solubility limit. To demonstrate the conditions for solute trapping and eventually complete partitionless solidification (k(V) = 1), Figure 8.3 illustrates possible solidification paths of an alloy of nominal composition c0 at different undercoolings prior to solidification. For simplicity, first a planar interface is assumed and kinetic interface undercooling is neglected, i.e. T I = TL and ciL = c0 (T I ) (T I and ciL are the temperature and concentration at the interface, respectively). If the melt is cooled from a temperature T TL(c0) first nuclei of phase of concentration cs may be formed in the temperature interval T0 = TL Te (Te is the eutectic temperature). Provided that the catalytic potency of the phase with respect to heterogeneous nucleation of the phase is small (e.g. if and phases have different crystallographic structures) the remaining melt will undercool until nucleation takes place at a temperature T1 Te. Under such conditions, the nucleation of phase can be avoided and the remaining undercooled melt will solidify into a supersaturated ss solid solution with a concentration corresponding to the metastable extension of solidus in the regime of the undercooled melt. Once crystallization has set in, the temperature will rise during recalescence in the temperature regime of the mushy zone where partial remelting and segregation can take place (cf. Figure 8.3b). At small cooling rate, the supersaturated solid ss will transform into a stable crystallite. The situation is completely different if the melt can be undercooled to temperatures T Thyp with Thyp being the isenthalpic or hypercooling temperature. In this

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321 ΔT>ΔThyp

ΔT 100 K [8.70]. The same effect has been proved to occur in Ag samples doped with 200 ppm oxygen [8.71]. A double-container experiment on Ag similar to that performed on Ni could, however, not confirm the pressureinduced crystallization. On the basis of this experimental result the model of copious nucleation has been rejected to be the origin of the grain refinement mechanism. Rather secondary effects such as recrystallization during the recalescence and postrecalescence period have been reported to be responsible for the onset of grain refinement. The driving force for recrystallization should be due to the deformation of primarily formed dendrites. Support for this assumption came from the observation on NiSi alloys that the residual stresses increased with undercooling up to the critical undercooling T* for grain refinement, but were suddenly relaxed at undercoolings larger than T* [8.72]. The investigations on the O2-doped Cu and Ag samples as well as on NiO systems [8.73] led to the speculations that the grain refinement process does occur exclusively in alloys. It was noted that the experiments on Ni and Co were conducted not on really pure metals but on samples, which contained small amounts of Ag impurities to ease the interpretation of the solidified microstructures (Ag segregates in NiAg alloys into the grain boundaries). Therefore, Ovsiyenko et al. [8.74] and Amaya et al. [8.75] repeated the early experiments by Walker, but on pure Ni samples. Ovsiyenko et al. observed a dendritic microstructure at undercoolings T 140 K, a grain-refined equiaxed microstructure at 140 K T 160 K, and a duplex structure at T 160 K. Opposite to these results the experiments by Amaya et al., who used electromagnetic levitation technique, revealed a continuous decrease in the grain size with undercooling. Also recent investigations on pure Fe samples do not show an abrupt grain refinement at undercoolings T 340 K [8.76]. The question whether the grain refinement is restricted to alloys therefore remains unanswered up to now. Kattamis and Flemings [8.77] report on grain refinement in Fe- and Ni-based alloys. The transition from coarse-grained structure to refined structure at a critical undercooling of T*175 K is accompanied by a change in the morphology. Below T* a dendritic morphology is observed, while above T* an equiaxed microstructure is found. Interestingly enough, the grain size of the equiaxed structure corresponds approximately to the distance of the dendrite side branches in the coarse structure. They concluded from these investigations that coarsening effects

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might be responsible for the formation of the refined equiaxed microstructure with the change in the interfacial energy as a driving force. In principle, two processes are assumed: the impingement of secondary dendrite side branches and subsequent coarsening. The first process has been directly observed on transparent model systems [8.78], whereby the dendrite side branches begin to remelt at their contact side to the dendrite stem during the recalescence phase. As a possible mechanism for this remelting process concentration changes in the interdendritic liquid have been cited, which are due to the concentration redistribution in front of the liquid–solid interface and cause a change in the concentration-dependent liquidus temperature. Then the dendrite fragments coarsen during the postrecalescence period leading to the grain-refined equiaxed microstructure. Such a fragmentation process may be favoured by non-equilibrium effects occurring during rapid solidification due to solute trapping and have been observed by Kattamis [8.79], Kattamis and Abbaschian [8.80] and Munitz and Abbaschian [8.81]. Kattamis [8.82] developed a coarsening model under the assumptions of diffusionless growth and adiabatic conditions during recalescence. This model predicts a quadratic dependence of the distance of the dendrite side branches (or the grain size in the grain-refined equiaxed structure) on the heating rate during recalescence, which has been experimentally found on Fe20 at.% Ni and Ni2 at.% Ag samples [8.83]. Further insight into constitutional effects in the grain refinement process may be inferred from investigations of the grain refinement process in dependence on the concentration. Such studies have been reported for NiSi [8.74] and NiCu [8.84]. The results show an increase in T* with concentration, which may reflect the larger undercooling necessary to reach the undercooling range in which solute trapping becomes important. In the case of NiCu alloys the increase in T* with the concentration reaches a maximum at 20 at.% Ni, which is followed by a decrease in T*. These studies confirm that non-equilibrium at the solid–liquid interface plays a role in the grain refinement process. Therefore, simultaneous measurements of both growth velocity and microstructure evolution as a function of undercooling might lead to an improvement in the understanding of this mechanism. Figure 8.17 shows such results obtained on Cu70Ni30 and Cu69Ni30B1 alloys [8.85]. Here, the dendrite growth velocities (upper panel) and the grain size (lower panel) are plotted versus the undercooling for the binary (left-hand side) and the ternary (right-hand side) alloy. In the undercooling range T T*, the experimentally determined growth velocities (dots) are well described by dendrite growth theory (cf. solid lines) assuming an atomic diffusive speed of 19 m/s. At the critical undercooling T* the temperature characteristics of the growth velocities change discontinuously. These critical undercoolings coincide with the critical undercooling for the onset of grain refinement. This coincidence is observed for

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Figure 8.17. Growth velocity V (upper panel) and grain size D (lower panel) as a function of undercooling T for Cu70Ni30 (left-hand side) and Cu69Ni30B1 (right-hand side). The solid lines in the V(T) curves give results of calculations of growth velocity [8.85].

both alloys despite the fact that T* is increased by 32 K due to the addition of B to CuNi. However, the growth velocity at T* is the same for both alloys. The critical velocity V* corresponds to the diffusive speed VD. This observation leads to the conclusion that a metastability of the interface itself becomes important and favours a fragmentation of primarily formed dendrites. This is supported by a reanalysis [8.86] of undercooling and solidification experiments on pure Ge and Ge doped with 0.39 at.% Sn [8.87]. Here the decrease in critical undercooling for grain refinement by adding 0.39 at.% Sn to Ge can be explained when assuming an unchanged critical velocity of 17.5 m/s for both systems. It is interesting to note that measurements of the growth velocity and grain diameter are related to rather different conditions of the undercooled melt. Dendrite growth velocities are measured during recalescence under non-equilibrium conditions, whereas measurements of the grain size and the determination of the microstructural transition from coarse-grained to grain-refined microstructure reflect processes of crystal formation during postrecalescence under near-equilibrium conditions in the

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range between liquidus and solidus temperature. At the same time it is found that the critical undercoolings T* for the microstructural transition coincides with the critical undercooling at which the temperature characteristics of the growth velocity versus undercooling relationship change abruptly (cf. Figure 8.17). From these observations one may suppose the existence of an influence of the primary dendritic growth during recalescence on the transformation of microstructural changes during postrecalescence period. Assuming that partitionless solidification begins to proceed at T T* one may speculate that solute trapping may enhance the instability of rapidly growing dendrites, which eventually leads to fragmentation of dendrites and a refined equiaxed microstructure. If fragmentation of primarily formed dendrites is supposed to be responsible for the grain refinement process the dendrite fragments should be in size of about 10–100 nm as expected from estimations of the dendrite tip radius and/or the distance of dendrite side branches as a typical length scale. However, the typical grain size in the refined microstructures is larger and reaches characteristic dimensions of a few micrometres in slowly cooled samples. Therefore, coarsening is very likely to occur. The coarsening process depends essentially on the coarsening time or the cooling rate with which the samples are cooled. Following coarsening models [8.88] the grain diameter dg of spherical grains depends on the coarsening time ts under isothermal and diffusion-limited conditions as dg3 (t s ) − do3 (t s = 0) = t s .

(8.11)

where and d0 are constants. d0 represents the grain size at time ts = 0, i.e. at the end of the recalescence period. Figure 8.18 exhibits results of drop tube experiments on Fe65Ni35 and Cu70Ni30 alloys [8.89, 8.90]. In drop tube experiments also containerless processing conditions are guaranteed, from which undercoolings larger than the critical undercooling for grain refinement can be expected. Furthermore, the solidification time can be systematically altered by a variation of the droplet size. The solidification time of the droplets of various diameters is estimated by heat balance calculations [8.91]. The experimental results of the grain diameter dg as a function of the solidification (or coarsening) time ts (dots) are well described by Eq. (8.11) if = 10 480 m3/s and d0 = 2.147 m for the FeNi alloy and = 4260 m3/s and d0 = 2.60 m for the CuNi alloy are assumed (cf. solid lines in Figure 8.18). Even though the drop tube investigations clearly reveal that the solidification time has a strong impact on the final grain size due to coarsening, the results also suggest that the spherical element diameter at solidification time ts = 0 is higher than 1 m. This means it is not comparable with the dendrite tip radius

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Metastable Solids from Undercooled Melts 8 Grain diameter dg (μm)

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8 Cu70Ni30

Fe65Ni35 7

7

6

6

5

5

4

4

3 3 2

0

0.01 0.02 0.03 0.04

0

0.04

Time ts (s)

0.08

Time ts (s)

Figure 8.18. Variation of spherical droplet diameter dg with local solidification time ts in Fe65Ni35 (left-hand side) and Cu70Ni30 (right-hand side) [8.90].

R (10–100 nm). However, according to the hypothesis that the grain refinement is caused by remelting and coarsening of primarily formed dendrites the dendrite tip radius is not the relevant length scale but rather the dendrite trunk dimension. Therefore, a new quantity should be introduced as a reference, such that rtt = Rtrunk/R. How this ratio depends on undercooling is not precisely known theoretically. In principle, rtt can be calculated by determining the amplification of thermal fluctuations on the dendrite sides, but this is very difficult and has not yet been done. The most reasonable way to estimate this ratio is to use the vertical distance behind the tip where the first side branch becomes visible, zs , and hence where the dendrite trunk starts. Assuming a parabolic tip, one obtains [8.92] rtt = ( 2 zs / R ) . 1/ 2

(8.12)

From experimental micrographs on a Cu70Ni30 alloy solidified at T = 195 K, which is just below the critical undercooling T* for grain refinement [8.85], the quantity zs can be inferred to be about 5 106 m. Calculations of the dendrite tip radius for the same alloy yield Rtip 5 107 m at an undercooling of 195 K, hence rtt 10. Such a value of rtt is of the same order as that measured on dendrites formed in undercooled xenon, where both the dendrite dimension and its growth dynamics are directly observed by proper optical diagnostics [8.93]. Taking into account this simple estimation the dendrite side branch spacing is throughout comparable with the initial grain size d0 measured by the drop-tube experiments on the same alloy.

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Meanwhile, many metallic alloys were investigated with respect to their microstructure evolution from the undercooled melt. Figure 8.19 gives an overview of the results of experiments on microstructure development upon undercooling [8.81]. It appears that the grain refinement through undercooling is a general phenomenon occurring in many metallic systems so far studied. Surprisingly, often not only one but even more microstructural transitions are observed if undercooling is increased. Careful reinvestigations on CuNi alloys demonstrate that first at small undercoolings a grain-refined equiaxed microstructure occurs followed by a coarse-grained dendritic microstructure at medium undercoolings, and at large undercoolings the grain-refined equiaxed microstructure reappears. This reminds of a re-entrant behaviour, which is also observed in other properties of solid matter as, e.g. in the magnetization of dilute magnetic alloys [8.94]. The development of various microstructures in Cu70Ni30 alloy with increasing undercooling is demonstrated in Figure 8.20 [8.95]. Different mechanisms have been proposed to explain the grain refinement. One of them is copious nucleation ahead of the solidification front induced by a pressure pulse [8.66, 8.96]. Copious nucleation may be favourable in systems showing a sluggish crystal growth behaviour; however, it can be excluded for pure metals and completely miscible alloys to be the origin of the grain refinement since a single nucleation event is sufficient to initiate and to complete crystallization, owing to the high solidification velocities in these systems (see Chapter 4). Another mechanism that has been considered is recrystallization after solidification [8.97]. This model could potentially play a role to explain different physical features of the two grain-refined equiaxed microstructures at small and large Fe-Ni Ni Ni-O Cu-S Cu-O Ni-Cu Ag Cu-Sn Cu Cu-Fe Ni-Ag

Equiaxed Dendritic 100 200 Undercooling ΔT [K]

300

Figure 8.19. Microstructures as a function of undercooling, results of experimental studies on various metallic systems investigated so far. Two different microstructural states are observed, a coarse-grained dendritic and a grain-refined equiaxed microstructure depending on the degree of undercooling achieved prior to solidification [8.81].

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undercoolings. Recently, a stability analysis of dendrite growth has been carried out resulting in the predictions of stable and unstable regions in a diagram showing undercooling versus alloy composition. If the grain refinement is associated with the occurrence of such instability, then two critical undercoolings are expected [8.95]. But it is not yet clear hether this model is in agreement with rigorous stability analyses of dendrite growth (see the review in Ref. [8.98]). A unique description of the physical origin of the grain refinement process has remained lacking up until now. On the basis of a previous work by Jackson et al. [8.99] a model was developed [8.95, 8.100] in which the transition between microstructures was assumed to result from dendrite fragmentation by remelting. This model describes microstructure evolution during solidification of undercooled melt as, e.g. in a levitation experiment (schematically shown in Figure 8.21). Figure 8.21 shows a typical temperature–time profile. During heating the sample melts in the interval between TL (liquidus temperature) and TS (solidus temperature) marked by a change in the slope of the temperature–time trace.

Figure 8.20. Grain size as a function of undercooling T. T1* and T2* denote the undercoolings of transition between microstructures. Typical examples of the microstructures are shown for samples undercooled by different amounts [8.95].

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Figure 8.21. Schematic temperature–time (T–t) profile obtained from an undercooling experiment using electromagnetic levitation. The undercooling T and the plateau duration tpl are directly inferred from experimental profiles.

After heating the melt moderately above TL, the sample is cooled and undercooled to a temperature TN at which nucleation is externally triggered. Crystallization then sets in, leading to a rapid temperature rise during recalescence. Dendrites form at the nucleation point and propagate rapidly through the volume of the melt. Once the temperature has reached a value between TL and TS, the remaining interdendritic melt solidifies during a “plateau phase” under quasistationary equilibrium conditions. The plateau duration tpl is exclusively controlled by the heat transfer from the sample to the environment and is inferred from the measured temperature– time profile as well as the undercooling. tpl is essentially an experimental control parameter, which can be varied by changing the cooling rate. After all the liquid is solidified the sample cools down to ambient temperature. The model assumes that the refinement of the microstructure is caused by remelting and coarsening of primarily formed dendrites. The transitional microstructures indicate the presence of sphere-like particles in the wake of a dendritic microstructure. This suggests that the sphere-like elements originate from the breakup of primary dendrites and their side branches by remelting. Physically, this process is driven by surface tension: the system attempts to minimize its solid/liquid interface area via heat and solute diffusion in the bulk phases. Dendrite breakup requires a characteristic time, tbu(T), which depends on the undercooling. According to this picture, one should observe a grain-refined

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equiaxed microstructure if tbu tpl, in which breakup occurs before the sample has had time to completely solidify, and a coarse-grained dendritic microstructure if tbu tpl. This implies that the undercoolings of transition between the two microstructures are determined by the relation tbu (T *) t pl .

(8.13)

The problem of determining T* is now reduced to calculating the breakup time. A simple model is used in which the dendrite trunk is stripped of its branches and treated as an infinitely long isolated solid cylinder embedded in a uniform liquid of temperature Tl and composition c0 equal to the nominal composition of the alloy [8.101]. This model obviously represents a very crude geometrical approximation of an actual dendrite trunk, but it contains the correct surface tension driving force for fragmentation. It is easy to show that such an idealized cylindrical trunk will break up into an array of quasispherical particles due to a shape instability similar to that of rod-like eutectic composites [7.102]. The classical Rayleigh instability of a liquid stream is a direct hydrodynamic analogue of this process. Consider a small cylindrically symmetric perturbation of the dendrite trunk of the form r ( z , t ) = R + k exp [ikz + ( kR) f ] + c.c.,

(8.14)

which is spatially periodic along the central z-axis of the trunk; r(z, t) denotes the perpendicular distance of the solid–liquid interface to the z-axis, R the unperturbed trunk radius, k the amplitude of the perturbation and (k, R) its amplification rate. A straightforward linear stability analysis yields the expression for dilute alloys [8.103], −1

⎡ 1 dD m c (1 − k E ) DT ⎤

( k , R) = 0 3 T f1 ( kR) ⎢ − l o ⎥ . H f /C P Dc ⎥⎦ R ⎢⎣ f 2 ( kR)

(8.15)

where DT denotes the thermal diffusivity taken to be equal in both phases, Dc the solute diffusivity in the liquid, ml the equilibrium liquidus slope, kE the equilibrium partition coefficient, Hf the heat of fusion, Cp the specific heat at constant pressure and d0 = Cp/Hf the capillary length. We have defined the functions 2 f1 ( kR) = kR ⎡1 − ( kR ) ⎤ K1 ( kR ) /K o ( kR ) , ⎣ ⎦

f 2 ( kR) = 1 + K 0 ( kR ) I1 ( kR ) / ⎡⎣ I 0 ( kR ) K1 ( kR )⎤⎦ ,

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where K0, K1, I0 and I1 denote the standard modified Bessel functions. The breakup time is assumed to be inversely proportional to the amplification rate of the fastest growing perturbation of wavevector kmax. At the level of a linear analysis, this time is also proportional to ln(R/kmax). We further assume that side branches perturb the trunk with an initial amplitude which scales with R, in which case this logarithmic factor becomes a constant of order unity independent of R. This yields the expression tbu (T ) 1 / ( k max ; R(T )).

(8.16)

It should also be noted that the supersaturation of the dendrite trunk resulting from solute trapping during growth provides an additional driving force for breakup (after growth) via diffusion in the solid. A crude estimate [8.100] indicates that this driving force is negligible at the lower undercooling transition (T1*), where growth is too slow for trapping to be significant, and is at the very most comparable in magnitude to capillarity at the higher undercooling transition (T2*). Hence, we suspect that the present results are not dramatically altered by the inclusion of this additional driving force. For the alloy composition, |mlc0(1 kE)DTCp/HfDc| 1 (i.e. breakup is dominated by solute diffusion in the liquid), in which case Eq. (8.16) reduces to 3 ( R(T ) ) ml c0 (1 − k E ) tbu (T ) 2 d0 Dc H f / C P 3

(8.17)

with kmaxR(T) 0.48. The trunk radius is correlated to the dendrite tip radius, Rtip(T), via a proportionality constant, z = R(T)/Rtip(T). This constant is determined by taking the ratio of the trunk radius, measured from micrographs of solidified samples, to the calculated tip radius at the same undercooling. This procedure yields an approximately constant value of z 20, which was used in all our calculations. Direct measurements of both tip and trunk radii during solidification of the transparent model system xenon indicate that z 18 independent of undercooling [8.93]. The tip radius is calculated following the same procedure as in Ref. [8.104], which is based on using a marginal stability criterion for the operating point of the dendrite tip [8.103]. This procedure offers a practical, albeit non-rigorous, substitute to using the now commonly accepted solvability theory, which remains difficult to carry out numerically in three dimensions. In previous studies [8.104, 8.105], this procedure was found to yield reasonably accurate predictions of dendrite growth velocities over a wide range of undercoolings (cf. Chapter 6). The curve tbu(T) in Figure 8.22 was plotted with calculations from Eq. (8.17) [8.95]. The breakup time sharply decreases with undercooling, passes through a

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Figure 8.22. Calculated dendrite breakup time versus undercooling tbu(T) curve (thick solid line), and measured plateau durations versus undercooling tpl(T) curves (thin solid lines), for three different cooling rates (see text). The arrows mark the transition temperatures (undercoolings) as determined from the microstructures of samples solidified from different undercoolings. They coincide well with the intersection points between the two curves at larger undercoolings.

minimum, rises very rapidly and finally goes through a maximum before falling down again. The occurrence of a minimum and a maximum in the tbu(T) curve is linked to the dependence of the dendrite tip radius on the undercooling. At a small undercooling, dendritic growth is controlled mainly by chemical diffusion and the tip radius decreases as the concentration gradient becomes steeper. As the undercooling is further increased, solute trapping sets in, leading to a decrease in the concentration gradient and an increase of the tip radius until the absolute stability of solutal dendrites is reached. Beyond this limit, dendritic growth is purely thermally controlled and the tip radius begins to fall again due to an increase in the thermal gradient. This non-monotonic variation of the tip radius with undercooling, which is crucial for the explanation of multiple transitions between microstructures, is a direct consequence of the aforementioned physical effects and is not an artefact of the approximate way in which this radius is calculated. The plateau durations, as inferred from temperature–time profiles, are plotted in Figure 8.22 as solid symbols and fitted graphically by thin solid lines which constitute the tpl(T) curves. The three thin solid lines correspond to three series of experiments, each performed at a different cooling rate. The squares represent the

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results of the experiments in which the sample was cooled in the levitation coil (low cooling rate 50 K/s). The circles represent the results obtained if the generator power was switched off and the sample cooled while placed on the sample holder (medium cooling rate 150 K/s). Finally, the triangles represent the results if the sample was quenched into a metallic bath of a GaInSn alloy, which is liquid at room temperature (large cooling rate 1000 K/s) [8.95]. According to the predictions of the model, the intersection points between the tbu(T) and tpl(T) curves define the transition undercoolings T1* and T2*. In particular, the latter should therefore depend on the plateau durations since at larger undercoolings the break-up time does not depend on undercooling as strongly as at lower undercoolings. Hence the upper critical undercooling changes with the rate at which the samples are cooled. As indicated by the arrows, which mark the transition temperatures (undercoolings) independently determined from the microstructures, there is an excellent quantitative agreement between the predictions of the model and the experimental findings as far as the higher critical undercoolings T2* are concerned. The lower critical undercoolings T1* are in less good agreement. In this undercooling regime systematic deviations are observed between experiment and theory. This is probably due to a change of heat and mass transport in front of the solid/liquid interface caused by fluid flow in samples electromagnetically processed under terrestrial conditions (electromagnetically induced stirring and natural convection). Such fluid flow directly influences the growth kinetics of dendrites if it becomes comparable with the solidification velocity at values 1 m/s in the range of small undercoolings [8.106]. Figure 8.22 suggests the existence of three microstructural transitions, two of which (T1*, T2*) are observed, but the transition at the smallest undercooling has not yet been experimentally verified. The alloy composition can also be used as an additional experimental parameter to test the model since it alters the dendrite breakup time. In this way experimentally determined microstructure selection maps can be constructed as demonstrated for the NiCu system, which are in agreement with the dendrite breakup model [8.107]. These maps give the ranges of concentration and undercooling to solidify grain-refined equiaxed microstructures. The dendrite breakup model for grain refinement occurring upon solidification from undercooled melts has additionally been corroborated by studies of the microstructure evolution in droplets of NiC [8.108] and FeCrNi [8.109] and by investigations of the texture of grain-refined and non-grain-refined CuNi [8.110] and FeCrNi [8.111] alloys. Moreover, the model predicts a fourth transition if the hypercooling limit Thyp is reached. At undercoolings T Thyp there is no postrecalescence plateau, hence Tpl = 0. Consequently, the equiaxed grainrefined microstructure should disappear again. In fact, this prediction has been confirmed by experimental studies on CoPd alloys that have been undercooled

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beyond the hypercooling limit [8.112]. If the undercooling passes through the hypercooling limit a microstructural transition from grain-refined to dendritic morphology is evidenced. The temperature characteristics of the growth kinetics do not show any significant alteration at this transition [8.113]. REFERENCES

[8.1] Barth, M., Joo, F., Wei, B., and Herlach, D.M. (1993) Journal of NonCrystalline Solids 158, 398. [8.2] Wilde, G., Görler, G.P., and Willnecker, R. (1996) Applied Physics Letters 69, 2995. [8.3] Wilde, G., Görler, G.P., and Willnecker, R. (1996) Applied Physics Letters 68, 2953. [8.4] Baker, J.C., and Cahn, J.W. (1971) in: Solidification (ASM, Metals Park, OH), p. 23. [8.5] Boettinger, W.J. (1982) in: Rapidly Solidified Amorphous and Crystalline Alloys, Materials Research Society Symposium Proceedings, eds. Kear, B.H., Giessen, B.C., Cohen, M., (North-Holland, New York Amsterdam Oxford), Vol. 8, p. 15. [8.6] Boettinger, W.J., and Coriell, S.R. (1986) in: Science and Technology of the Undercooled Melt, eds. Sahm, P.R., Jones, H., and Adam, C.M. (Martinus Nijhoff Publishers, Dordrecht/Boston/Lancaster), p. 81. [8.7] Fecht, H.J., and Perepezko, J.H. (1989) Metallurgical Transactions 20A, 785. [8.8] Fecht, H.J., Zhang, M.X., Chang, Y.A., and Perepezko, J.H. (1989) Metallurgical Transactions 20A, 795. [8.9] Allen, W.P., Fecht, H.J., and Perepezko, J.H. (1989) Scripta Metallurgica 23, 643. [8.10] Bosio, L., Defrain, A., and Epelborn, I. (1966) Journal de Physique 27, 61. [8.11] Bosio, L., Defrain, A., and Dupont, M. (1971) Journal of Chemical Physics 68, 542. [8.12] Bizid, A., Bosio, L., Curien, H., Defrain, A., and Dupont, M. (1974) Physica Status Solidi 23, 135. [8.13] Perepezko, J.H., and Paik, J.S. (1982) in: Novel Materials and Techniques in Condensed Matter, eds. Crabtree, G.W. and Yashishta, P. (NorthHolland, New York), p. 57. [8.14] Graves, A., and Perepezko, J.H. (1986) Journal Materials Science 21, 4215. [8.15] Yoon, W., Paik, J.S., LaCourt, D., and Perepezko, J.H. (1986) Journal of Applied Physics 60, 3489. [8.16] Cortella, L., Vinet, B., Desré, P.J., Pasturel, A., Paxton, A.T., and van Schilfgaarde, M. (1993) Physical Review Letters 70, 1469. [8.17] Cech, R.E. (1956) Transactions of the AIME 206, 585. [8.18] Cech, R.E., and Turnbull, D. (1956) Transactions of the AIME 206, 124. [8.19] Lacy, L.L., Rahtz, T.J., and Robinson, M.B. (1982) Journal Applied Physics 53, 682.

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[8.20] Evans, N.D., Hofmeister, W.H., Bayuzick, R.J., and Robinson, M.B. (1986) Metallurgical Transactions 17A, 973. [8.21] Perepezko, J.H., and Anderson, I.E. (1980) in: Synthesis and Properties of Metastable Phases (The Metallurgical Society of AIME, Warrendale, PA), p. 31. [8.22] Hoffmeyer, M.K., and Perepezko, J.H. (1988) Scripta Metallurgica et Materialia 22, 1143. [8.23] Herlach, D.M., Feuerbacher, B., and Schleip, E. (1991) Materials Science Engineering A 133, 795. [8.24] Schleip, E., Herlach, D.M., and Feuerbacher, B. (1990) Europhysics Letters 11, 751. [8.25] Chuang, Y.-Y., Chang, Y.A., Schmid, R., and Lin, J.-L. (1986) Metallurgical Transactions 17A, 1361. [8.26] Chuang, Y.-Y., Hsieh, C.-C., and Chang, Y.A. (1986) Metallurgical Transactions 17A, 1373. [8.27] Knapp, J.H., and Follstaedt, D.M. (1987) Physical Review Letters 58, 2454. [8.28] Xing, L.Q., Zhao, D.Q., and Chen, X.C. (1993) Acta Physica Sinica, 42, 465. [8.29] Kim, Y.-W., Lin, H.-W., and Kelly T.F. (1988) Acta Metallurgica 36, 2525. [8.30] Lin, H.-M., Kim, Y.-W., and Kelly, T.F. (1988) Acta Metallurgica 36, 2537. [8.31] Volkmann, T., Löser, W., and Herlach, D.M. (1996) International Journal of Thermophysics 17, 1217. [8.32] Löser, W., and Herlach, D.M. (1992) Metallurgical Transactions 23A, 1585. [8.33] Moir, S.A., and Herlach, D.M. (1997) Acta Materialia 45, 2827. [8.34] Schroers, J., Volkmann, T., Herlach, D.M., Allen D.R., and Perepezko, J.H. (1996) International Journal of Rapid Solidification, 9, 267. [8.35] Volkmann, T., Löser, W., and Herlach, D.M. (1997) Metallurgical and Materials Transactions, 28A, 461. [8.36] Schroers, J. (1995) Diploma Thesis, University of Köln, Germany. [8.37] Löser, W., Volkmann, T., and Herlach, D.M. (1994) Materials Science Engineering A 178, 163. [8.38] Volkmann, T., Löser, W., and Herlach, D.M. (1997) Metallurgical Materials Transactions A 28, 461. [8.39] Notthoff, C., Franz, H., Hanfland, M., Herlach, D.M., Holland-Moritz, D., and Petry, W. (2000) Review of Scientific Instruments 71, 3791. [8.40] Kaufman, L. (1986) Proceedings of the International Conference on Alloy Phase Diagrams (American Society of Metals, Metals Park, OH), p. 59. [8.41] Notthoff, C., Feuerbacher, B., Franz, H., Herlach, D.M., and HollandMoritz, D. (2001) Physical Review Letters 86, 1038. [8.42] Barin, I. (1995) Thermochemical Data of Pure Substances, Vol. 2, 3rd ed. (VCH, Weinheim, Germany). [8.43] Schmalzried, H., and Navrotzky, A. (1975) Festkörperthermodynamik: Chemie des festen Zustands (Verlag Chemie, Weinheim, Germany). [8.44] Frank, F.C., and Kasper, J.S. (1958) Acta Crystallographica 11, 184. [8.45] Lee, G.W., Gangopadhyay, A.K., Kelton, K.F., Hyers, R.W., Rathz, T.J., and Robinson, D.S. (2004) Physical Review Letters 93, 037802.

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[8.46] Luo, W.K., Sheng, H.W., Alamgir, F.M., Bai, J.M., He, J.H., and Ma, E. (2004) Physical Review Letters 92, 145502. [8.47] Blanke, H., and Köster, U. (1985) in: Rapidly Quenched Metals, eds. Steeb, S. and Warlimont, H. (Elsevier Science, Amsterdam), p. 227. [8.48] Köster, U., and Schünemann, U. (1993) in: Rapidly Solidified Alloys, ed. Liebermann, H.H. (Marcel Dekker, New York), p. 303. [8.49] Gillessen, F., Herlach, D.M., and Feuerbacher, B. (1988) Zeitschrift für Physikalische Chemie 156, 129. [8.50] Wachtel, E., Haggag, H., Gödecke, T., and Predel, B. (1985) Zeitschrift für Metallkunde 76, 120. [8.51] Herlach, D.M., and Gillessen, F. (1987) Journal of Physics F 17, 1635. [8.52] Gladys, A., Brohl, M., Schlawne, F., and Alexander, A. (1985) Zeitschrift für Metallkunde. 76, 254. [8.53] Drehman, A.J., Greer, A.L., and Turnbull, D. (1982) Applied Physics Letters 41, 716. [8.54] Kui, H.W., Greer, A.L., and Turnbull, D. (1984) Applied Physics Letters 45, 615. [8.55] Turnbull, D. (1969) Contemporary Physics 10, 473. [8.56] Inoue, A., Zhang, T., and Masumoto, T. (1993) Journal of Non-Crystalline Solids 156, 473. [8.57] Peker, A., and Johnson, W.L. (1993) Applied Physics Letters 63, 2342. [8.58] Inoue, A., Yavari, A.R., Johnson, W.L., and Dauskardt, R.H., eds. (2001) Supercooled Liquid, Bulk Glassy and Nanocrystalline States of Alloys, MRS Symposium Proceedings, (Materials Research Society, Warrendale, Pennsylvania), Vol. 644. [8.59] Gillessen, F. (1989) Ph.D. Thesis, Ruhr-Universität Bochum, Germany. [8.60] Birringer, R., Gleiter, H., Klein, H.-P., and Marquardt, P. (1984) Physics Letters A 102, 365. [8.61] Zhu, X., Birringer, R., Herr, U., and Gleiter, H. (1987) Physical Review B 35, 9085. [8.62] Boucharat, N., Rösner, H., Perepezko, J.H., and Wilde, G. (2004) Materials Science Engineering A 375–377, 713. [8.63] Chen, L.C., and Spaepen, F. (1988) Nature 336, 366. [8.64] Walker, J.L. (1959) in: The Physical Chemistry of Process Metallurgy, Part 2, ed. St. Pierre, G.R. (Interscience, New York), p. 845. [8.65] Walker, J.L. (1964) in: Principles of Solidification, ed. Chalmers, B. (Wiley, New York). [8.66] G. Horvay, G. (1965) International Journal of Heat and Mass Transfer 8, 195. [8.67] Glicksman, M.E. (1965) Acta Metallurgica et Materialia 13, 1231. [8.68] Powell, G.L.F. (1965) Journal of Australian Institute of Metallurgy 10, 223. [8.69] Powel, G.L.F., and Hogan, L.M. (1968) Transactions of the AIME 242, 2133. [8.70] Jones, B.L., and Weston, G.M. (1970) Journal Australian Institute of Metallurgy 15, 167. [8.71] Powell, G.L.F., and Hogan, L.M. (1969) Transactions of the AIME 245, 407.

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[8.72] Ovsiyenko, D.E., Maslov, V.V., and Alfinster, G.A. (1974) Russian Metallurgy 66, 66. [8.73] Jones, B.L., and G.M. Weston, G.M. (1970) Journal of Australian Institute of Metallurgy 15, 189. [8.74] Ovsiyenko, D.E., Maslov, V.V., and Alfinster, G.A. (1976) Russian Metallurgy 92, 92. [8.75] Amaya, G.E., Patchett, J.A., and Abbaschian, G.J. (1983) in: Grain Refinement in Castings and Welds, eds. Abbaschian, G.J. and David, S.A. (The Metallurgical Society of AIME, Warrendale, PA), p. 51. [8.76] Schleip, E., and Herlach, D.M. (1989) (unpublished results). [8.77] Kattamis, T.Z., and Flemings, M.C. (1966) Transactions of the AIME 236, 1523. [8.78] Jackson, K.A., Hunt, J.D., Uhlmann, D.R., and Seward, T.P. (1966) Transactions of the AIME 236, 149. [8.79] Kattamis, T.Z. (1970) Zeitschrift für Metallkunde 61, 856. [8.80] Kattamis, T.Z., and Mehrabian, R. (1974) Journal of Vacuum Science Technology 11, 1118. [8.81] Munitz, A., and Abbaschian, G.J. (1986) in: Undercooled Alloy Phases, eds. Collings, E.W. and Koch, C.C. (The Metals Society AIME, Warrendale, PA), p. 23. [8.82] Kattamis, T.Z. (1976) Journal of Crystal Growth 34, 215. [8.83] Kattamis, T.Z., and Skolianos, S. (1985) in: Proceedings of the 5th International Conference on Rapidly Quenched Metals, eds. Steeb, S. and Warlimont, H. (North-Holland, Amsterdam, Oxford New York Tokyo), p. 51. [8.84] Tarshis, L.A., Walker, J.L., and Rutter, J.W. (1971) Metallurgical Transactions 2, 2589. [8.85] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1990) Applied Physics Letters 56, 324. [8.86] Evans, P.V., Vitta, S., Hamerton, R.G., Greer, A.L., and Turnbull, D. (1990) Acta Metallurgica et Materialia 38, 233. [8.87] Devaud, G., and Turnbull, D. (1987) Acta Metalluirgica et Materialia 35, 957. [8.88] Thompson, C.V., Frost, H.J., and Spaepen, F. (1987) Acta Metallurgica et Materialia 35, 887. [8.89] Cochrane, R.F., Herlach, D.M., and Willnecker, R. (1993) in: Metastable Microstructures, eds. Banerjee, D and Jacobson, L.A. (Oxford & IBH Publishing Co., New Delhi), p. 67. [8.90] Cochrane, R.F., Herlach, D.M., and Feuerbacher, B. (1991) Materials Science & Engineering A 133, 706. [8.91] Cochrane, R.F. (1992) in: EUROMAT ‘91’, Advanced Processing, Vol. 1, eds. Clyne, T.W. and Withers, P.J. (The Institute of Materials, London), p. 18. [8.92] Karma, A. (1994) private communication. [8.93] Kaufmann, E. (2000) Ph.D. Thesis, Technical University Zürich, Switzerland. [8.94] Fischer. K.H. (1985) Physics and Statistical Solutions B 130, 13.

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[8.95] Schwarz, M., Karma, A., Eckler, K., and Herlach, D.M. (1994) Physical Review Letters 73, 1380. [8.96] Horvay, G. (1962) Proceedings of the 4th National Congress on Applied Mechanics (ASME, New York), p. 1315. [8.97] Mullis, A.M., and Cochrane, R.F. (1997) Journal of Applied Physics 82, 3783. [8.98] Langer, J.S. (1986) in: Chance and Matter, Lecture on the Theory of Pattern Formation, Proceedings of the Les Houches Summer School, eds. Souletie, J., Vannimenus, J., and Stora, R. (North-Holland, New York, 1987), p. 629. [8.99] Jackson, K.A., Hunt, J.D., Uhlmann, D.R., and Seward, T.P. (1966) Transactions of the Metallurgical Society of AIME 236, 149. [8.100] Karma, A. (1998) International Journal Non-Equilibrium Processing 11, 201. [8.101] Herlach, D.M., Eckler, K., Karma, A., and Schwarz, M. (2001) Materials Science & Engineering A 304–306, 20. [8.102] Cline, H.E. (1971) Acta Metallurgica 19, 481. [8.103] Lipton, J., Kurz, W., and Trivedi, R. (1987) Acta Metallurgica 35, 957; 35, 965. [8.104] Willnecker, R., Herlach, D.M., and Feuerbacher, B. (1989) Physical Review Letters 62, 2707. [8.105] Eckler, K., Cochrane, R.F., Herlach, D.M., Feuerbacher, B., and Jurisch, M. (1992) Physical Review B 45, 5019. [8.106] Eckler, K., and Herlach, D.M. (1994) Materials Science & Engineering A 178, 159. [8.107] Eckler, K., Schwarz, M., Karma, A., and Herlach, D.M. (1996) Materials Science Forum 215–216, 45. [8.108] Eckler, K., Norman, A.F., Gärtner, F., Greer, A.L., and Herlach, D.M. (1997) Journal of Crystal Growth 173, 528. [8.109] Moir, S.A., Eckler, K., and Herlach, D.M. (1998) Acta Materialia, 46, 4029. [8.110] Gärtner, F., Norman, A.F., Greer, A.L., Zambon, A., Ramous, E., Eckler, K., and Herlach, D.M. (1997) Acta Materialia 45, 51. [8.111] Gärtner, F., Moir, S.A., Norman, A.F., Greer, A.L., and Herlach, D.M. (1997) Materials Science & Engineering A 226–228, 307. [8.112] Willnecker, R., Görler, G.-P., and Wilde, G. (1997) Materials Science & EngineeringA 226–228, 439. [8.113] Volkmann, T., Herlach, D.M., Wilde, G., and Willnecker, R. (1998) Journal Applied Physics 83, 3028.

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Chapter 9

Microstructure Selection Maps 9.1. Selection by Rapid Cooling 9.2. Selection by Undercooling 9.3. Selection by Droplet Size References

361 367 370 373


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Chapter 9

Microstructure Selection Maps To follow the solidification pathway from the undercooled liquid up to the final structural state of an as-solidified material a sequence of morphological or phase instabilities are considered during transformation in metals and alloys. In this way, one may distinguish between phase selection due to nucleation or microstructure selection by the competition of growth and nucleation processes. An important part of experimental verification and industrially valuable applications for pattern selection is seen in construction of the microstructure selection map (MSM). The MSM has been introduced first by Gill and Kurz using experimental data [9.1] and computations [9.2] of microstructure formation in laser rapid resolidification. Now the investigations can be exactly viewed in the detailed MSM for single phase or even multiphase transformations in both liquid and solid states as well [9.3]. In the present chapter, we consider several interesting examples of microstructure selection controlled by both nucleation and growth of crystalline phases during processing by laser annealing of surfaces up to drops solidification. 9.1. SELECTION BY RAPID COOLING

Gill and Kurz investigated solidification of microstructures during laser treatment of specimens of Al᎐Cu alloys by experimental and computational methods. A series of microstructural observations have been made on Al᎐36 wt.% Cu (hypoeutectic alloy), Al᎐36 wt.% Cu and Al᎐44 wt.% Cu (hypereutectic alloys), and solidification velocities in the range of 10⫺2⫺2 m/s [9.1]. Calculations using solidification models have been carried out to predict formation of microstructures for the same Al᎐Cu alloys [9.2] processed by laser rapid solidification. Experimental investigations were performed by taking thin foils from the surface to study the resulting growth morphologies by using electronic microscope. Eutectic, dendritic, cellular, banded and planar front growth morphologies have been detected. The orientation of microstructural grains allows to correlate the morphologies with their local growth velocities V as shown in Figure 9.1 and as described by the expression V = Vb cos

(9.1)

where is the angle between the direction of grain growth and the vector of the scanning velocity Vb. 361

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Figure 9.1. Shape of the molten zone bath upon laser heating. In the central plane XZ of the remelted path, the local solidification velocity V is determined by the scanning velocity Vb of the laser beam and by the angle . w and h are the bath width and depth, respectively. Boundary surfaces ’s are indicated for choosing boundary conditions in numeric simulations of the heat transport during laser beam treatment of the specimens.

By combining the results of microstructure investigations [9.1] with earlier observations made on alloys with lower Cu concentration, it has been possible to establish an MSM diagram for Al᎐Cu alloys. This is demonstrated in Figure 9.2 in the coordinates “local growth velocity VS–alloy’s concentration”. Using the phase diagram for the Al᎐Al2Cu alloy and the experimentally determined map of Figure 9.2, the theoretically calculated MSM of Figure 9.3 has been obtained by computation of the possible regions of existence for cells, dendrites, eutectics, bands and plane front growth versus the alloy’s composition [9.2]. The rapid solidification models to compute microstructure evolution are used according to those described by Pierantoni et al. [9.4] and Gill et al. [9.5]. In particular, eutectic growth was modelled using the approach of Trivedi et al. [9.6], dendritic and high-velocity cellular growth by the approach of Kurz et al. [9.7] and the interface temperature of plane front was simply given by the solidus temperature.

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Microstructure Selection Maps

363 Al2 Cu

Al 10

Exptl. Vs max B 1

C

A

0.1

0.01

Ce 0.001 0

10

20

30

40

50

Wt % Cu α plane front

α cells

Regular eutectic

2λ - mode eutectic (hyper)

Bands

θ dendrites

2λ - mode eutectic (hypo)

Optical mode eutectic

Experimental transition

Figure 9.2. The experimentally determined MSM diagram for the Al᎐Al2Cu alloy under condition of laser rapid solidification with the thermal gradient of T 5 ⫻ 10 K/m. Here A denotes the eutectic to cellular transition, B denotes the transition from the eutectic growth mode to the plane front and the region C is unfilled because no experimental data is available.

Considering the banding phenomenon, Gill and Kurz assumed that the oscillatory growth mode for banding exists above velocity VA of absolute chemical stability given by the expression [9.8] VA =

v (V, T ) D (T ) , k (V, T )GT

(9.2)

where GT is the Gibbs–Thomson coefficient, and v, k and D are velocity/temperature-dependent solidification interval, solute partitioning function and solute diffusion coefficient, respectively. As a result of modelling the rapidly growing microstructures, the theoretical MSM is shown in Figure 9.3. Even though the computed MSM reproduces the microstructure well in comparison with the experimental MSM (Figure 9.2), it can be seen that

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Metastable Solids from Undercooled Melts Al2Cu

Al 100 PF 10 B 1 V (ms−1)

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C 0.1 E D 0.01

0.001 0

10

20

30

40

50

Wt% Cu

Figure 9.3. Computed MSM for the Al᎐Al2Cu alloy [9.2]. C, Cellular and dendritic ; D, dendritic ; E, regular eutectic; B, bands; PF, planar front.

the region of eutectic growth with oscillatory instability, observed experimentally between 0.3 and 2.0 m/s, has not been predicted by the modelling in a satisfactory manner. This inconsistency was due to the fact that the available model for unstable eutectic structure was not used in computations. However, the MSM diagrams shown in Figures 9.2 and 9.3 have been served by Gill and Kurz both as a test for rapid solidification models and for the predictions that are made by phase diagram calculations. The results of experimental study in multicomponent alloy systems also show well-defined correlation between microstructure and laser scanning velocity Vb, which is defined by the local solidification velocity according to Eq. (9.1). Particularly, the specific features of microstructure formation in a structural steel with 0.5 wt.% C [9.9] confirm the correctness of the analysis based on the scheme of Figure 9.1. In addition to investigations of Gill and Kurz, the structural steel has been studied for different thermal gradients T arising at the solid–liquid interface in a melted region during laser annealing [9.9]. Therefore, for given chemical content of the steel, the MSM has been constructed in the coordinates “temperature gradient T versus local solidification velocity VS”.

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365

Using experimental values of the laser scanning velocity in the range of Vb = 0.01–0.167 m/s, the thermal regimes during remelting and resolidification have been modelled. The three-dimensional finite-difference approximation of the heat transport equation shown schematically in Figure 9.4a has been applied. The special boundary conditions of the surface energies ’s (at the boundaries ’s, see Figure 9.1) were chosen to reach the steady state during solidification in the moving reference frame. The evolution of the thermal field (Figure 9.4b) allows the prediction of the gradient T at the solid–liquid interface and the velocity V. The experimental regimes were considered in the range of T = 105 to 4 ⫻ 107 K/s and V = 0.006–0.08 m/s (see thin dashed line in MSM, Figure 9.5). The microstructure is presented by dendritic and cellular pattern and has been observed in this range of parameters (stars in Figure 9.5). Using the idea that the structural steel might be considered as a quasibinary Fe᎐C alloy (carbon plays the main role in solute redistribution and formation of primary crystal microstructure), the characteristic size of the microstructure and chemical composition were computed in molten and two-phase zones during laser resolidification [9.9]. On the basis of the marginal stability criterion, the methodology of the local non-equilibrium model of solidification (see Chapter 6) has been used for predicting the characteristic size of the grains. The temperature gradient T has been used as a parameter in the calculation of the grain size (see Eq. (6.37)) for both the analytic solutions and numeric simulations of the microstructure [9.10]. The results of modelling are plotted in the computed MSM (Figure 9.5). Analysis of the MSM leads to the conclusion that the temperature gradient T is one of the major factors determining the morphology of the microstructure. Particularly, the transition from dendrites to cells happens due to significant increase in T in the molten zone with the increase in V and power density of the laser beam (Figure 9.6). It can also be seen in Figure 9.5 that in computing the microstructure of the structural steel, the numeric simulation and analytic calculation based on the local non-equilibrium model of solidification are well fitted within the model by Kurz–Giovanola–Trivedi (KGT) [9.7]. It is due to the fact that the deviation from local equilibrium in the solute diffusion field and solute trapping by the interface can be neglected at the velocities in the range of V ⱕ 3 m/s. For completion the microstructure development has also been modelled for laser resolidification, i.e. at high solidification velocities taking into account solute trapping effects, the results of which are enclosed in Figure 9.7. To construct the complete MSM, the structure was modelled up to velocities of V = 20 m/s compatible with the carbon diffusion speed VD in the steel. According to numeric modelling, the boundary presented by the red line in Figure 9.7 between

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Metastable Solids from Undercooled Melts z x

y Vb

(a)

Vb t2

t1

t3

0 0.5 z, mm

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1000 K 773 K

2

523 K

(b) 2.5 0

1

2

3 x, mm

4

5

6

Figure 9.4. (a) The calculated solid–liquid phase regions during laser surface resolidification processing. The area filled by red boxes on the top of the surface shows the action field of the laser beam moving across the surface with the velocity Vb. The sapphire boxes are cells with the liquid phase. Green boxes represent solid–liquid cells and the other fields are the cells with the solid phase. (b) The evolution of the temperature field below the surface is represented by different isotherms. After the initial period (t1) the temperature field arrives at a stationary form going from t2 to t3.

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367

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105 0.001

0.01

0.1 Vs, m/s

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Figure 9.5. Computed MSM in the coordinates thermal gradient T versus interface velocity VS for the structural steel as predicted by numeric and analytic modelling [9.10]. Squares, dendrites; triangles, cells; and circles, solidification front. The dash-dotted line is the boundary between dendrites and cells and the solid line is the boundary between cells and planar front. The dashed line represents the boundary between the perturbed and unperturbed front calculated by the KGT model [9.7]. The stars mark experimental data related to dendrites and cells. Thin dashed line demonstrates a feature of laser processing, i.e. increase in GT with the increase in Vb and VS, respectively (Eq. (9.1)).

the dendritic–cellular structure and the planar front has been determined. The MSM also includes the results of the analytical results computed within the KGT model (green line in Figure 9.7) and within the local non-equilibrium solidification (LNS) model as well (blue line in Figure 9.7). The microstructure shown along the perimeter of MSM demonstrates the modification of the morphology with the change of T and V. It can be seen that the results of both analytic models confluence at small solidification velocities V. At V ⬎ 3 m/s, the disagreement between KGT and LNS models has been evaluated by the value of the absolute stability for a planar interface as VA = 5.75 m/s for KGT model and VA = 7.04 m/s for LNS model. 9.2. SELECTION BY UNDERCOOLING

Electromagnetic levitation is successfully used for containerless solidification of a great variety of metals and alloys. The key parameter in this technique is the undercooling T, which controls the selection of the microstructure in droplets [9.13].

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Metastable Solids from Undercooled Melts C/Cθ, [−]: 0.500 0.611 0.722 0.833 0.944 1.056 1.167 1.278 1.389 1.500

Figure 9.6. Dendrites (left, T = 1 ⫻ 105 K/s and VS = 0.004 m/s) and cells (right, T = 2 ⫻ 107 K/s and VS = 0.084 m/s) with carbon redistribution (shown in a scale relative to initial concentration C0) in the molten zone computed within the numeric model of Ref. [9.11]. Calculations were made at domains of 279 ⫻ 549 nodes.

To produce MSM in coordinates “undercooling versus chemical composition” an electromagnetic rf-levitation facility was used for the processing of bulk Ni᎐Cu samples (droplets of 1 g mass), which were prepared from the elements (purity of 99.99%) by in situ melting under clean environmental conditions [9.14]. Large undercoolings of up to some 300 K were attained. The temperature was recorded by a two-colour pyrometer with an accuracy of ±3 K. Nucleation was externally simulated at a desired degree of undercooling by touching the sample with a trigger needle. The plateau duration following recalescence was inferred from the temperature–time profiles. The microstructures of the as-solidified samples were analysed by conventional optical microscopy. The experimentally determined MSM inferred from the microstructures of the levitation-processed Ni᎐Cu bulk samples is shown in Figure 9.8. For all alloy compositions investigated the following sequence was observed: grain refinement at low undercoolings, a coarse-grain dendritic structure at intermediate undercoolings and the re-entry of grain refinement at high undercoolings. The transitions between the microstructural regimes are quite sharp and take place within narrow undercooling intervals at critical values T*. The difference in grain size was as large as two orders of magnitude. The critical undercooling T1* depends only slightly on the composition, whereas the critical undercooling T2* monotonically increases with increasing Cu content. For pure Ni, a small but nevertheless abrupt

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Microstructure Selection Maps 0,27x0,27 μm2

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9

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7

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1.4 1.5

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105 0.001

16x16 μm2

0.01

0.1 Vs, m/s

16x16 μm2

1

16x16 μm2

10

0,27x0,27 μm2

Figure 9.7. MSM from small to high solidification velocities [9.12] constructed by the results of analytical solutions and numeric modelling. The symbols denote: squares, dendrites; triangles, cells; and circles, planar solidification front. Between the perturbed and unperturbed front (giving the transition from cells to planar front) the lines are: red line, results of numeric modelling [9.11]; blue line, calculation using the LNS (local non-equilibrium model of solidification) (Chapter 6); and green line, calculation using KGT model [9.7]. The pictures along the perimeter of MSM show the morphology and carbon distribution in the domain for the steady-state values of temperature gradient T and solidification velocity VS.

change of the average grain diameter from 0.6 to 0.4 mm was found at T = 53 K, as approximately predicted by the model of Karma [9.15, 9.16] (detailed description of the model for grain refinement is given in Chapter 8). It is suggested that grain-boundary migration, which is considerably faster in the pure metal compared to the alloys, renders the grain-refinement effect in pure Ni. On the basis of the dendrite breakup model [9.15, 9.16], the critical undercoolings T1* and T2* have been calculated as functions of the Cu content. These are shown in the theoretically predicted MSM (Figure 9.9). This, and the simplicity of the model, might explain the apparent discrepancies between experiment and theory. However, the essential features of the experimentally constructed MSM

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Figure 9.8. Experimental microstructure selection map for the Ni᎐Cu system from droplet levitation experiments showing the transition undercoolings T1* and T2*. The coarse-grained microstructures are dendritic, while the grain-refined microstructures are equiaxed.

(Figure 9.8) are undeniably reproduced. The same good comparison between the experimentally constructed and predicted MSM in coordinates “undercooling versus chemical composition” has been found in Ni᎐C droplets processed by electromagnetic levitation facility [9.17]. Thus, the MSM for levitation-processed Ni᎐Cu bulk samples is correctly described by the model of grain refinement [9.15, 9.16]. 9.3. SELECTION BY DROPLET SIZE

As observed in droplets [9.13], undercooling controls the nucleation of crystals and possible selection of crystallographically different phases. There may be a spread in nucleation temperatures even under nominally identical conditions, and consequently the results are best displayed on MSM (or, according to the terminology of some papers, in “microstructure predominance maps”). Such MSM have been constructed for binary alloys, in the coordinates “alloy composition versus droplet diameter” [9.14, 9.18, 9.19]. Therefore, in addition to the nucleation temperature as a key parameter determining the microstructure, the droplet diameter is chosen normally as the most practical variable. The results obtained in drop tube and atomization experiments corroborate each other using the droplet size as a determining parameter in phase selection. Similar to undercooling, the droplet size may determine selection of coarse-grained dendritic or grain-refined equiaxed microstructure. For instance, typical microstructures obtained in drop tube processed droplets of Ni 0.5 at.% C alloy are shown in Figure 9.10. In spray methods droplets of different size classes as obtained by sieving the produced powder through sieves of defined mesh size are analysed by optical and

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Figure 9.9. Microstructure selection map modelled for Ni᎐Cu droplets, for selected compositions. Below T1* and above T2*, the grain-refined equiaxed microstructure appears, while in the range T1* ⬍ T ⬍ T2* a coarse-grained dendritic microstructure is predicted in qualitative agreement with experiments.

Figure 9.10. Microstructures in the Ni 0.5 at.% C droplets produced in the drop tube [9.17]. (a) Dendritic microstructure. (b) Grain-refined microstructure.

electron metallographic methods. Normally, for the drop tube experiments more than 100 droplets are produced in most of the size fractions. In atomization experiments, the typical weight in each size fraction amounts to ⱖ100 g. For the Ni᎐Cu droplets processed by drop tube, four different types of microstructures have been found [9.14]: dendritic, grain-refined containing dendritic fragments, grain-refined without dendritic fragments and transitional microstructure in the form of loosely connected dendrites. For the Ni᎐Cu droplets processed by atomization, three microstructural types were observed for all size fractions [9.14]: dendritic, fine grains and dendrite fragments, and grain-refined microstructures. Therefore, for each Ni᎐Cu alloy composition, the fraction of each type of

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Figure 9.11. Microstructure predominance map of Ni᎐Cu droplets from spray methods – atomization and drop tube processing. A microstructure is taken to be predominant if it is occurs in >60% of the droplets for a given size and chemical composition.

microstructure as a function of the droplet size has been represented and the final MSM (the microstructure predominance map) has been constructed (Figure 9.11). It shows which predominant structure is selected as a function of droplet size for a given chemical composition. For the droplets processed by spray methods, inferences about droplet undercoolings by comparing chemical compositions can be misleading since the critical undercoolings T1* and T2*, which have been determined using electromagnetic leviation (see Figure 9.8), vary with composition. This can be taken into account, for instance, by plotting the data showing undercooling distributions with composition-dependent T1* and T2*. There are some features in the description of the grain refinement in droplets produced by spray methods. The plateau duration in droplets from spray methods is much shorter than in the droplet processed by levitation technique, because of both smaller droplet size and higher cooling rates. For the drop tube and atomization experiments, the calculated plateau durations are always considerably shorter than the dendrite breakup time calculated and predicted by the model of fragmentation of dendrites [9.15, 9.16]. Therefore, grain refinement should not to

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be expected in any drop tube processed or atomized droplets. This discrepancy can be resolved in two ways: (i) For the small droplets, the relationship “radius of dendrite trunk Rtrunk/dendrite tip radius R” may be significantly different from that in levitated samples. (ii) The size of the droplets may play a role. Whereas, the levitated droplets can be considered to be infinite to the spatial extent of the diffusion fields (solutal and thermal). This condition may not be met in the smallest droplets produced in the spray methods. Thus, the refinement effect due to dendrite breakup may also be observed in the droplets from spray methods, but its mechanism is probably not fully described by the model developed by Karma [9.15, 9.16]. REFERENCES

[9.1] Gill, S.C., and Kurz, W. (1993) Acta Metallurgica & Materialia 41, 3563. [9.2] Gill, S.C., and Kurz, W. (1995) Acta Metallurgica & Materialia 41, 139. [9.3] Boettinger, W.J., Coriell, S.R., Greer, A.L., Karma, A., Kurz, W., Rappaz, M., and Trivedi, R. (2000) Acta Metallurgica & Materialia 48, 43. [9.4] Pierantoni, M., Gremaud, M., Magnin, P., Stoll, D., and Kurz, W. (1992) Acta Metallurgica & Materialia 40, 1637. [9.5] Gill, S.C., Zimmermann, M., and Kurz, W. (1992) Acta Metallurgica & Materialia 40, 2895. [9.6] Trivedi, R., Magnin, P., and Kurz, W. (1987) Acta Metallurgica 35, 971. [9.7] Kurz, W., Giovanola, B., and Trivedi, R. (1986) Acta Metallurgica & Materialia 34, 823. [9.8] Merhant, G.J., and Davis, S.H. (1990) Acta Metallurgica & Materialia 38, 2683. [9.9] Galenko, P.K., Haranzhevskiy, E.V., and Danilov, D.A. (2002) Technical Physics 47(5), 48. [9.10] Haranzhevskiy, E.V., Danilov, D.A., Krivilyov, M.D., and Galenko, P.K. (2004) Materials Science & Engineering A 375–377, 502. [9.11] Galenko, P.K., and Krivilyov, M.D. (2000) Modelling Materials Science & Engineering 8, 67. [9.12] Krivilyov, M.D., Galenko, P.K., Haranzhevskiy, E.V., and Danilov, D.A., (2003) Selection of Crystal Patterns in Laser Resolidification Processing of a Structural Steel. Presentation in Congress “EUROMAT-2003,” (Lausanne, Switzerland, 1–5 September 2003). [9.13] Herlach, D. (1994) Materials Science & Engineering Reports R 12, 177. [9.14] Norman, A.F., Eckler, K., Zambon, A., Gärtner, F., Moir, S.A., Ramous, E., Herlach, D.M., and Greer, A.L. (1998) Acta Materialia 46, 3355. [9.15] Schwarz, M., Karma, A., Eckler, K., and Herlach D.M. (1994) Physical Review Letters 73, 1380. [9.16] Karma, A. (1998) International Journal of Non-Equilibrium Processing 11, 201.

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[9.17] Eckler, K., Norman, A.F., Gärtner, F., Greer, A.L., and Herlach D.M. (1997) Journal of Crystal Growth 173, 528. [9.18] Shao, G., and Tsakiropoulos, P. (1994) Acta Metallurgica & Materialia 42, 2937. [9.19] Zambon, A., Badan, B., Eckler, K., Gärtner, F., Norman, A.F., Greer, A.L., Herlach, D.M., and Ramous, E. (1998) Acta Materialia 46, 4657.

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Chapter 10

Experiments in Reduced Gravity 10.1. 10.2.

Containerless Processing In Reduced Gravity Experiments in Drop Tubes 10.2.1 Nucleation Studies on Glass-Forming Systems 10.2.2 Kinetics of Phase Selection 10.2.3 Microstructure Development 10.2.4 Liquid–Liquid Phase Separation 10.3. Electromagnetic Processing in Reduced Gravity 10.3.1 Thermophysical Properties 10.3.1.1 Thermal Expansion 10.3.1.2 Electrical Resistivity 10.3.1.3 Specific Heat and Thermal Conductivity 10.3.1.4 Surface Tension and Viscosity 10.3.2 Nucleation Investigations and Phase Selection 10.3.3 Measurements of Dendrite Growth Velocities References

377 379 379 380 382 386 389 389 389 390 392 394 397 401 403


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Chapter 10

Experiments in Reduced Gravity 10.1. CONTAINERLESS PROCESSING IN REDUCED GRAVITY

So far, containerless processing of undercooled melts in reduced gravity has been conducted either by processing small droplets during free fall in drop tubes on Earth or by the application of electromagnetic positioning and processing in Space (TEMPUS) during parabolic flight campaigns and NASA spacelab missions (IML2-1994, MSL1-1997). Drop tube experiments on Earth are easily performed and were first developed by David Turnbull [10.1]. Such drop tube experiments are devoted to studies of nucleation in glass-forming metallic alloys and, more recently, in the formation of metastable crystalline solids from the state of undercooled melts. They have also demonstrated their potential in investigating crystallization processes of various microstructure and phases as a function of cooling rate. But these experiments are related to the conditions of containerless processing and undercooling rather than those of reduced gravity. Real experiments of containerless processing under the conditions of reduced gravity in Space became accessible by the development of the TEMPUS facility for its use in the Spacelab during Shuttle missions of NASA. The application of electromagnetic levitation on Earth has serious drawbacks. It requires relatively high power absorption to levitate the sample, which is accompanied by an equivalent large heating effect. Under vacuum conditions heat transfer takes place by radiation only. This will limit the applicability of electromagnetic levitation technique under vacuum conditions to undercooling experiments on refractory metals and high melting alloys. Additional convective cooling of gas makes the technique suitable for undercooling of transition metals such as Fe or Ni. Even thoroughly cleaned cooling gases are dirty compared with ultra-high-vacuum conditions and have shown to induce heterogeneous nucleation by metal oxide formation on the surface of the samples [10.2]. Levitation forces on a molten material inevitably introduce dynamic motion in the liquid. Its influence on the solidification process is largely unknown to date. At the same time, levitating forces lead to a deformation of the liquid drop, so any measurements relying on a particular sample shape becomes difficult in a gravity field. These limitations of the electromagnetic levitation technique for undercooling experiments are circumvented by using the special environment in Space, where the positioning forces to compensate disturbing accelerations are about three orders of magnitude smaller than that under terrestrial conditions. TEMPUS 377

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provides means of containerless processing in Space [10.3]. TEMPUS and its technical specifications are described in Section 10.2. So far, TEMPUS was used to perform experiments during Spacelab missions of NASA and during parabolic flight campaigns of ESA and DLR. Different classes of experiments were conducted. TEMPUS is a powerful instrument to measure thermophysical properties of melts both in the stable regime above the melting temperature and in the metastable regime of the undercooled liquid below the melting temperature. Thermophysical parameters of high reliability are a necessary precondition for quantitative modelling of solidification and casting processes. The experimental determination of any transport properties is essentially influenced by gravity-driven heat and mass transport on Earth. Therefore, the reduced gravity environment offers unique opportunities for high accuracy measurements of thermophysical properties. A second class of experiments concerns investigations of solidification of undercooled melts. Experiments by undercooling and measuring multistep recalescence profiles provide information on primary crystallization of metastable crystallographic phases. Measurements of the growth velocity as a function of undercooling are interesting with respect to the formation criteria for non-equilibrium microstructures. They also give insight into growth phenomena, where influences of convection and fluid flow play a role in e.g. dendritic/eutectic growth and its influence on pattern formation in microstructure development. The TEMPUS facility had its maiden flight on board of NASA’s Spacelab mission International Microgravity Laboratory IML-2 in 1994. The technical operation of the device with all subsystems worked nominally during the entire mission of 14 days. Important scientific results have been obtained. The element Zr was melted and undercooled several times. Melting of Zr requires a temperature of more than 2125 K; this means it was the highest temperature ever achieved in the Spacelab [10.4]. TEMPUS was reflown on board of NASA’s Spacelab-mission Materials Science Laboratory MSL-1 in 1997. Altogether 17 different experiments of 10 research groups were performed. Experimental results of relevance to the present topic of metastable phases have been obtained. Studies of nucleation statistics in the microgravity environment were conducted on Zr and analysed using nucleation theory [10.5]. For the first time dendrite growth velocities on metallic systems have been measured in Space [10.6]. An advanced facility is under consideration to be accommodated on board of the International Space Station (ISS). An international community of scientists is presently preparing experiments to be performed onboard ISS using the multiuser facility materials science laboratory – electro-magnetic levitator (MSL-EML).

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10.2.1 Nucleation studies on glass-forming systems Drop tubes are frequently used to study glass formation in metallic alloys and phase transformation during solidification of undercooled melts. Fully glass spheres, in diameter ranging up to 1.5 mm of Pd77.5Cu6Si16.5 alloys were produced during free fall of liquid droplets in the 105-m drop tube at NASA Marshall Space Flight Center [10.7]. Drehmann and Turnbull [10.1] studied nucleation phenomena by drop tube processing of the glass formers Pd82Si18 and Pd83Si17. They found that individual droplets were either fully amorphous or fully crystalline. As illustrated in Figure 10.1, the variation with diameter of the fraction of amorphous droplets indicated a crystal nucleation probability scaling with droplet surface area rather than with droplet volume. This finding suggests that surface heterogeneous nucleation is the dominant process in the early stage of crystallization of Pd᎐Si alloys as it is further supported by the observation that careful drying of the environmental He gas in the drop tube decreased the nucleation probability. Similar experiments were performed on metallic glass-forming alloys of Au55Pb22.5Sb22.5 [10.8], Pd40Ni40P20 [10.9], Pd77.5Cu6Si16.5 [10.10, 10.11], Ni58.5Nb41.5 [10.12], Cu62Zr38 and Cu56Zr44 [10.13] and Fe40Ni40P14B6 [10.14]. The type of curve shown in Figure 10.1 are useful in revealing influences on nucleation. As shown by such investigations, melt superheat prior to undercooling and solidification is an important variable. For Cu᎐Zr alloys it was found that the glass-forming ability in drop tube processing significantly increased with the 1.0 Glassy Fraction X

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Figure 10.1. Glass fraction X of Pd82Si18 as function of droplet size d, for drop tube processing. The fitting to the experimental data (closed dots) implies heterogeneous surface nucleation [10.1].

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effect saturating for superheats of ⬎200 K. The improvement of glass-forming ability by drop tube processing of Cu᎐Zr alloys was evidenced by a critical cooling rate to avoid crystallization, being about two orders of magnitude smaller compared with critical cooling rates for glass solidification estimated from splat cooling experiments [10.15]. It is not yet clear whether the observed improvement of glass forming ability can be exclusively attributed to reduction of heterogeneous nucleation by containerless processing or could originated partly by reduction of heat and mass transport because of the lack of convection. The latter effect would lead to a more sluggish crystallization kinetics, which in turn will favour the avoiding crystal nucleation and reduction of growth of the crystals. 10.2.2 Kinetics of phase selection Drop tube experiments are also useful in measuring the phase selection during solidification of undercooled droplets as a function of their size. Because the droplet size scales directly with the cooling rate with which the droplets are cooled during free fall in an environmental gas, the temperature–time–transformation (TTT) behaviour is studied by investigating the solidified structures of droplets of various size classes. As an example, Figure 10.2 shows the volume fractions of the various phases formed in drop-tube-processed Al88Mn12 alloy as a function of droplet diameter [10.16]. Quasicrystalline phases of fivefold symmetry were discovered as a new class of solid-state matter in between crystalline and amorphous solids in melt-spun ribbons of Al88Mn12 alloy [10.17]. Depending on the preparation conditions, an icosahedral

10 5

1

T (K/s) 10 3

10 4

Equilibrium

Al88Mn12

I-Phase Volume fraction

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Al6Mn Al6Mn Alss

4 at% Mn 0

101

102

2.5 at% Mn

Al

103 d (µm)

Figure 10.2. Phase mixture in droplets of Al88Mn12 alloy as a function of droplet diameter. The large droplets crystallize a mixture of equilibrium Al6Mn phase and supersaturated solid solution Alss, while with decreasing droplet size (increasing cooling rate) quasicrystalline T- and I-phase are formed progressively [10.16].

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I-phase with quasiperiodicity in three dimensions, a decagonal T-phase with quasiperiodicity in two dimensions and periodicity in the third dimension and different crystalline phases are solidified in this alloy. The drop tube experiments reveal that the I-phase is formed far from equilibrium in the smallest droplets and at highest cooling rate. At medium droplet size, T-phase and supersaturated Alss-phase is found. The mass fraction of Alss-phase increases with droplet size (decreasing cooling rate) at the expense of T-phase. At largest droplet size of drops, of the order of 1 mm in diameter, also the equilibrium intermetallic phase Al6Mn is crystallized. Nucleation–kinetics plots reproduce the experimentally observed phase selection behaviour of drop-tube-processed Al88Mn12 alloy [10.18]. Drop tube experiments are also used to determine the formation of different phases selected kinetically by the cooling rate. TTT curves are constructed such that they show the kinetics of phase formation of the various phases individually involved in solidification of undercooled melts in multicomponent multiphase alloys. To do so the well-known Avrami analysis [10.19] is utilized that describes the time t necessary to produce a mass fraction X =10⫺3, which is hardly detectable by experimental diagnostics (X-ray diffraction, optical and electronic microscopy) of the equivalent phases formed at a certain undercooling. It is given by X = I ss v 3t 4 ,

(10.1)

where Iss is the steady-state nucleation rate (cf. Chapter 5) and V the crystal growth rate. The crystal growth in quasicrystal-forming alloys is extremely sluggish because it requires short-range diffusion of the various atomic species to arrange them correctly at the solid–liquid interface to form the complex structure of quasicrystalline phases [10.20]; the advancement of the solidification front into the undercooled melt is essentially driven by a kinetic undercooling of the interface. Under such circumstances the speed of the solidification front is estimated by the rate theory (cf. Chapter 6) leading to V=

⎛ GLS ⎞ ⎤ D⎡ ⎢1 − exp ⎜ ⎥. ao ⎣ ⎝ k BT ⎟⎠ ⎦

(10.2)

The TTT curves suggest an undercooling range of 150–200 K in drop tube processing. They predict a sequence of phase formation with the cooling rate as experiment parameter. At small cooling rates Al6Mn intermetallic and crystalline Al preferably solidify. At cooling rates exceeding 1000 K/s, the intermetallic Al6Mn phase disappears, while the quasicrystalline T-phase progressively forms. Further increasing the cooling rate to 1⫻104 K/s leads to solidification of the

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1100 1050 Temperature (K)

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800 750 AI I-phase

700 650 600

-6

10

-4

10

-2

10 10 Time (S)

0

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10

2

10

4

Figure 10.3. Temperature–time transformation diagrams of the various phases involved in the solidification of undercooled droplets of Al88Mn12 assuming a fixed volume fraction of X = 10⫺3 [10.16]. Critical cooling rates are also shown for the avoidance of crystallization of various phases. The solid triangle corresponds to the maximum undercoolability of the Al-phase in Al᎐Mn alloys as investigated by the droplet-dispersion technique [10.66].

quasicrystalline I-phase. To avoid the nucleation of quasicrystalline phases, and in particular the crystalline Al-phase, very large cooling rates greater than 106 K/s are needed. This is in accordance with the observation that quasicrystalline phases nucleate quite easily in undercooled melts (cf. Chapter 5) and the formation of amorphous phases in quasicrystal-forming alloys during rapid cooling of a liquid is very difficult. Figure 10.3 summarizes the TTT diagrams for the various phases formed from the undercooled melt of Al88Mn12 alloy taking into account the experimental results of the drop tube experiments [10.16]. 10.2.3 Microstructure development Another example to note the importance of drop tube experiments in phase selection of undercooled droplets can be seen in Nd᎐Fe᎐B alloys. The intermetallic compound Nd2Fe14B1 or -phase reveals extraordinary hard magnetic properties, which make Nd᎐Fe᎐B alloys suitable for materials as permanent magnets [10.21, 10.22]. Under equilibrium conditions -phase is formed via a

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peritectic reaction from pro-peritectic -Fe. Incomplete peritectic transformation leads to a high amount of soft-magnetic -Fe that reduces the quality of the magnetic Nd᎐Fe᎐B material. Undercooling may open up the possibility of primary solidification of -phase at the expense of -Fe. In fact, it is accomplished by melt spinning of thin ribbons [10.22] and inert gas atomization of very small droplets [10.23]. Undercooling of bulk melts of Nd᎐Fe᎐B alloys suffers from the extreme reactivity of Nd with oxygen forming very stable Nd2O3 oxides on the surface of the samples, which act as heterogeneous nucleation sites. Even in electromagnetic levitation experiments, which are conducted under a purified inert gas atmosphere, undercooling of Nd᎐Fe᎐B is very limited and solidification starts with the crystallization of the pro-peritectic -phase [10.24]. More recently, a method has been developed to dissolve Nd oxides by evaporation during electromagnetic levitation of bulk melts [10.25]. Using this method undercoolings up to 150 K are reached in levitation experiments, sufficiently high to observe three different solidification pathways [10.26]. The three metastable solidification paths are schematically depicted in Figure 10.4. Each solidification mode produces a characteristic microstructure. At small undercooling of T = 16 K (left), dendrites of -Fe are primarily formed, followed by peritectic formation of metastable -phase Nd2Fe17Bx (x 1) and -phase Nd2Fe14B. Nd2Fe17Bx decomposes at small undercoolings into -Fe and -phase owing to small cooling rates (100 K/s). At medium undercoolings of T = 60 K (middle) the hard magnetic -phase primarily solidifies from the melt and is the only solidification mode. At large undercoolings of T = 110 K (right), metastable -phase primarily solidifies and subsequently decomposes into -Fe and -phase. Due to the fact that the undercooling level achieved by drop tube processing should correlate to the droplet size in a statistical manner (it increases with reducing the droplet size), one can attribute the observed microstructure to the droplet size and can construct the phase selection map of an alloy of specific composition. Figure 10.5 shows the volume fractions of the various phases formed as a function of droplet size of Nd13Fe80.5B6.5 alloy. At large droplet size always a phase mixture of -Fe and metastable Nd2Fe17By is observed. With decreasing droplet size (i.e. increasing undercooling) the volume fraction of -Fe decreases and diminishes while hard magnetic -phase is progressively formed primarily from the undercooled melt at the expense of Nd2Fe17Bx (-phase). The physical mechanism underlying this process is that primary iron formation is suppressed due to its higher nucleation barrier and its more sluggish growth kinetics than the primary growth of Nd2F14B hard magnetic phase. In this way the peritectic reaction is circumvented. However, new metastable phases as the -phase also crystallize directly from the undercooled melt.

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Liquid

-Fe

Nd2Fe17Bx

Nd2Fe14B

Figure 10.4. Schematic summary of three different solidification pathways of hyperperitectic Nd13Fe80.5B6.5 alloy observed at three different undercoolings. Small undercooling T 16 K (left): primary -Fe formation, subsequent peritectic formation of Nd2Fe17Bx (-phase) and Nd2Fe14B (-phase); medium undercooling T 60 K (middle): primary Nd2Fe14B formation; large undercooling T = 110 K (right): primary Nd2Fe17Bx formation, decomposition of Nd2Fe17Bx into -Fe and (-phase) [10.27].

As far as the three crystalline phases are concerned, their growth kinetics are quite different. The iron phase has a very limited solubility of either Nd or B, i.e. a very small equilibrium partition coefficient. Therefore, its growth at undercoolings less than the critical undercooling for partitionless solidification requires much solute rejection. This will restrict its growth velocity to very small values (cf. Chapter 6). In contrast, Nd2Fe14B growth needs less solute rejection and, hence, can achieve a higher velocity. According to the criterion for phase selection at high growth velocities [10.27], it will be kinetically favoured if the nucleation is abundant. However, chemical order as present in the superlattice structures of intermetallic phases is also able to influence growth kinetics. An ordered intermetallic compound experiences a lower growth velocity than a disordered solid

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385

100

Droplet Fraction (%)

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80 60 40 20 0 2.0

1.6 1.2 1.0 0.80 0.63 0.34 0.25 0.10 Droplet Size (mm)

Figure 10.5. Primary phase selection map of Nd13Fe80.5B6.5 alloy, processed in a drop tube, droplet fraction as a function of droplet size, shadowed column = -Fe, gridded column = Nd2Fe17By, (-phase) and blank column = Nd2Fe14B (-phase) [10.28].

solution. Considering the chemically highly ordered structure of the Nd2Fe14B compound, its primary growth is also constrained at high undercoolings. As a result, the growth of the metastable Nd2Fe17Bx (x1) phase with its disordered structure will replace the growth of the ordered compound Nd2Fe14B at large undercoolings, i.e. at small droplet size. The growth conditions control the evolution of different microstructures and their volume fractions. As shown in Figure 10.5 the droplet size in drop tube experiments is an efficient experiment parameter for the production of droplet charges of varying volume fractions of the respective phases. For an alloy of stoichiometric composition of the hard magnetic phase Nd2Fe14B (Nd13Fe80.5B6.5), a mixture of -Fe and -phase is observed at large droplet size, while at medium droplet size -Fe completely disappeared with an essentially enhanced volume fraction of -phase and, finally, metastable -phase grows at the expense of -phase at the smallest droplet size. There is another characteristic feature in drop tube experiments concerning microstructure evolution. As discussed in Section 8.6, undercooling plays an essential role in the formation of microstructures of different morphology and grain size. In particular, the formation of grain-refined equiaxed microstructures from the undercooled melt was deduced to the fragmentation of dendrites, which grow during recalescence of a rapidly solidifying undercooled melt. The fragmentation process in turn needs solute diffusion in the interdendritic liquid region during the post-recalescence period tpl of solidification. Whether grain refinement occurs or not depends on the time duration of the post-recalescence period tpl. While the rapid increase in temperature during recalescence depends exclusively

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on the internal heat and mass transport of a solidifying droplet, tpl depends on the heat transfer from the drop to the environment. This is completely different in drop tube experiments compared with levitation experiments. Owing to the rapid heat transfer in particular at small droplet size in drop tube experiments, it can happen that the post-recalescence solidification period vanishes. This means that the primarily formed dendrites will not find an opportunity to fragment to form equiaxed grain-refined microstructures. In fact, it was observed that in semiconducting Ge even single crystals can be formed in drop-tube-processed samples at droplet size smaller than 300 m, while in levitation experiments with much smaller heat transfer conditions, i.e. smaller cooling rates, equiaxed grain-refined microstructures were observed in Ge drops at the largest undercoolings [10.28]. 10.2.4 Liquid–liquid phase separation Monotectics showing a miscibility gap in the liquid state have been subject to microgravity experiments for a long time. They attracted attention as model systems to study nucleation, growth and coagulation of a liquid phase L1 within the environment of a liquid L2 as the parent phase. Moreover, monotectics offer fascinating opportunities to produce fine dispersed materials for various applications [10.29]. A stable monotectic melt leads to liquid–liquid phase separation under thermodynamic equilibrium conditions at temperatures above the liquidus temperature. However, there are also metallic alloy systems showing miscibility gap in the metastable region of the undercooled melts. Such alloys offer the advantage that the driving force for crystallization of an undercooled melt is used to rapidly freeze in the instantaneous state of demixing at preselected temperatures and defined exposure times. Our particular interest is Co᎐Cu. This alloy combines a good electrical conductor (copper) with a strong ferromagnet (cobalt) of high Curie temperature. Recently, the phase diagram was reinvestigated with particular emphasis on the position of the binodal [10.30], it is shown in Figure 10.6. If the single phase melt is cooled and undercooled into the metastable miscibility gap to a temperature below the binodal liquid–liquid phase separation sets in. During liquid-phase separation the following processes have to be considered: ●

● ●

● ●

liquid–liquid decomposition L → L1 + L2 starting with nucleation of the minority phase in the environmental majority phase; growth of the nuclei either by diffusion or by convective diffusion; movement of the nucleated particles within the matrix liquid by Brownian, Stokes or Marangoni motion; particle motion due to fluid flow in the majority liquid; and rapid solidification of the undercooled melt.

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1800

L 1700

α - Co 1600 T [K]

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1500

L1 + L2 1400

TP

1300 0

20

40

60

80

100

at.% Cu

Figure 10.6. Phase diagram of Co᎐Cu with the metastable miscibility gap in the undercooled region: ♦, measured by DTA; ♦, concentration of the two separated phases from samples processed in an EML facility.

The individual contributions are of different importance for phase separation in containerlessly processed drops and droplets of Co᎐Cu alloy. In contrast to solidification of undercooled melts the nucleation of the minority liquid in the majority liquid is of homogeneous nature. Stokes sedimentation can be neglected for Co᎐Cu alloys since the mass density of both liquids L1(Co-rich) and L2 (Cu-rich) are comparable. Marangoni convection occurs if there are strong temperature gradients in the phase boundary. In small drops containerlessly processed the temperature gradients inside the drop are small. Consequently, Marangoni convection does not play a significant role. Therefore, only concurrent growth by collisions and diffusion has to be considered. For this situation, a simplified model has been developed [10.31]. As a result, the time dependence of the particle size is calculated as [10.32] ⎛ 3⎞ R(t ) = ⎜ ⎟ ⎝ 4 ⎠

1/ 3

(2 DSsst)

1/ 2

⎡2 ⎤ exp ⎢ c n0 ( 2 DSss t )3 / 2 t 3 / 2⎥ , ⎣15 ⎦

(10.3)

where D is the diffusion coefficient, Sss the supersaturation, c a collision constant and n0 the initial number density of particles. Eq. 10.3 allows for the calculation of the particle size under isothermal conditions after a time t. This equation can be applied both to samples processed in a drop tube and in electromagnetic levitation. To apply Eq. (10.3) the collision constant and the processing time to solidification is estimated leading to the relation R ( d ) = ad exp(bd p +4 ),

(10.4)

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Figure 10.7. The average droplet radius of L2 drops of the minority phase in undercooled Co᎐Cu droplets as a function of diameter d of droplets processed containerlessly during free fall in a drop tube [10.35].

where d is the diameter of the sample and a and b are numerical constants. For drop-tube-processed samples the average diameter of particles of minority phase L2 is computed as a function of the droplet size, which determines the cooling conditions. The results are shown in Figure 10.7. In fact, size, morphology and particle distribution depend very much on the processing conditions. This is illustrated by the micrographs of Figure 10.8, which show microstructures of Co᎐Cu alloys of intermediate composition undercooled into the metastable immiscibility gap by different techniques and subsequently rapidly solidified from the state of a deeply undercooled melt. A bulk sample of Co–41 at.%Cu was undercooled by 261 K by applying the melt flux technique in a DSC apparatus [10.33]. According to Figure 10.8(left), a serious macrosegregation pattern is formed. The Co-rich phase (dark coloured) is completely coagulated into a large sphere, which occupies most of the volume of the sample. Except for one large and many small Cu-rich spherulites (light coloured) embedded in the Co-rich L1 phase, most of the Cu-rich L2 phase is distributed at the outer surface of the sample. The density of liquid Co is slightly higher than that of liquid Cu leading to the tendency that the Cu-rich liquid moves upwards driven by the buoyancy force. The surface tension of Cu is smaller than that of Co [10.34]. Therefore, Cu-rich liquid has the

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Figure 10.8. Microstructures of Co᎐Cu alloy of intermediate composition rapidly solidified upon undercooling into the metastable miscibility gap, applying melt fluxing technique in DSC experiments (left), electromagnetic levitation technique (middle) and drop tube technique (right).

tendency to encapsulate the Co-rich liquid. A sample of Co41.8Cu58.2 undercooled by 207 K prior to solidification using electromagnetic levitation leads to a completely different microstructure [10.35]. There is also macrosegregation but the Co-rich phase (dark coloured) is distorted due to the electromagnetic stirring of the melt inside the levitation coil. The coalescence of the separated phase is strongly developed with the reduction of the interface energy as its driving force. Some Co-rich spheres fuse together and lose their original shapes. Applying solidification under reduced gravity during the free fall of droplets inside a drop tube changes the morphology of the microstructure of Co41.8Cu58.2 drastically. As obvious from Figure 10.8(right) many small spheres of Co-rich phase are homogeneously distributed inside the volume of the droplet. In the middle of the droplet the diameter of the Co-rich spheres is approximately 1.5 m while it increases to 3 m when going to the outer regions of the droplet. Drop tube experiments offer the easiest way to solidification experiments under reduced gravity. They possess a large potential to perform investigations concerning in particular solidification statistics since a large number of droplets of different size groups are produced during each single shot of such an experiment. However, they suffer from the fact that solidification of individual droplets cannot be directly observed during the course of the experiment. This disadvantage will be overcome by making use of electromagnetic levitation in flight missions of reduced gravity. 10.3. ELECTROMAGNETIC PROCESSING IN REDUCED GRAVITY

10.3.1 Thermophysical properties 10.3.1.1 Thermal expansion. The electromagnetic levitation technique has been used on Earth to measure the mass density of liquid Ni [10.36] and liquid Cu [10.37]. The change in diameter of a levitated sphere as a function of temperature has been observed by an optical arrangement photographing the profile of the

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drop. This allowed the measurement of the mass density and the thermal expansion over an extended undercooling regime. An interesting result has been obtained by extrapolating the experimental data of the undercooled liquid to lower temperatures. This extrapolation suggests that at an undercooling of about 480 K the mass density of the undercooled melt reaches the mass density of the solid. Since metals mostly show a lower mass density in the liquid state, a constraint for maximum undercooling could exist at this undercooling. The accuracy of such measurements improves if the shape of the liquid drop approaches the ideal geometry of a sphere as it is favoured by the conditions of an almost force-free liquid under reduced gravity. In fact, measurements of the mass density and thermal expansion were performed on board of the Space shuttle during the NASA Spacelab mission MSL1 using the TEMPUS facility. The containerlessly processed sample was imaged by a high resolution CCD camera with on-board recording of the data. After the mission the on board recorded pictures were analysed with digital image processing giving the volume of the sample as a function of time and temperature [10.38]. Six glass-forming alloys were investigated [10.39]. From the measurements of the volume as a function of temperature the volumetric thermal expansion th =

1 ∂v v(T = 273 K) ∂T

(10.5)

was determined. Collects the results of the measurements of the volumetric thermal expansion of glass-forming metallic alloys performed during MSL1 Spacelab mission are shown in Table 10.1. 10.3.1.2 Electrical resistivity. Consider a sample of spherical shape placed within a rf levitation coil, which is integrated into a free oscillating circuit. The frequency of Table 10.1. Thermal expansion of liquid alloys of glass-forming metals. Alloy Zr57Cu15.4Ni12.6Nb5Al10 Zr65Cu17.5Al7.5Ni10 Zr11Cu47Ti34Ni8 Zr60Cu18Al10Ni9Co3 Pd78Cu6Si16 Pd82Si18

th [10⫺5 K⫺1]

T range [K]

TL [K]

5.9⫾1.8 6.8⫾1.3 7.7⫾0.5 5.5⫾0.7 7.9⫾0.3 7.7⫾0.1

1073 1473 1003 1253 1108 1398 1108 1473 1193 1473 1253 1623

1092 1110 1115 1133 1033 1094

Note: The volume jump during solidification was measured to be less than 2%. The values measured under reduced gravity complement previous measurements on the same alloys at lower temperatures on Earth.

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the coil current will depend on the inductivity of the system consisting of coil and sample. The inductivity and, consequently, the frequency will change if the electrical resistivity of the sample increases with temperature. A method has been developed to use the alternating electromagnetic field of the heating coil of TEMPUS to measure the temperature change of the electrical resistivity el. The sample positioned inside the heating coil represents an additional, inductively coupled electric circuit (transformer) of resistance Rel. It increases the resistance Rec of the empty coil. At high angular frequencies , the ratio between radius r of the spherical sample inside a levitation coil and the skin depth sd(el, ) is high, the electrical resistance Rel is given by [10.40] ⎛ ⎞ 1 1 Rel (el , ) = const × r 3 ⎜ − 2 , ⎝ q(el , ) q (el , ) ⎟⎠

(10.6)

where const is a constant, and describes the geometry of the coil and q = r/ sd(el, ). Making use of the free oscillating heating circuit of TEMPUS and assuming in good approximation that all ohmic resistances are small compared to the inductive resistance, I C Rec + R el (el , ) ) = o cos( ) ( L Uo

(10.7)

holds. The current and voltage amplitude Io and Uo as well as the phase difference between Uo and Io are measured during the experiment [10.41]. Hence, by measuring of Uo, Io, and the frequency and by knowing of the electric circuit constants C, L and Rec as well as the coupling constant and the sample radius r, it is possible to determine the sample resistivity Rel. As a particularly interesting example, Figure 10.9 shows the specific electrical resistivity of solid and liquid Co80Pd20 alloy. At temperatures T ⬎ 1350 K for the solid sample and T ⬎ 1370 K for the liquid sample, the resistivity data of the alloy follow a linear temperature dependence as is well known for metals at high temperatures. However, if the temperature falls below 1350 K the resistivity of liquid and solid samples increase with decreasing temperature. This behaviour reflects the onset of long-range magnetic ordering when the temperature approaches the ferromagnetic Curie temperature of solid Co80Pd20. In this temperature range, the data points should no longer be interpreted as resistivities, because the applied inductive measurement technique reacts also very sensitively on the magnetic state of the sample as represented by the magnetic permeability, which changes drastically when the magnetically disordered

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Figure 10.9. Electrical resistivity el of Co80Pd20 in the solid and liquid state, together with corresponding mean values and linear fits (solid lines) as a function of temperature T [10.44].

paramagnetic state changes to a long-range magnetically ordered state on approaching the Curie temperature. 10.3.1.3 Specific heat and thermal conductivity. Modulation calorimetry as non-contact ac-calorimetry for measurements of highly reactive or metastable metallic liquids was proposed [10.42] and developed as an experimental tool for the TEMPUS facility [10.43]. The heater power is sinusoidally modulated. The specific heat is obtained from the measurement of the temperature response of the sample. Variation of the modulation frequency, mf, and analysis of the transient response following a step function change in the heater power allows the evaluation of thermal transport inside the sample, such as an effective thermal conductivity and the total hemispherical emissivity [10.44]. The heating coil works under steady-state conditions, i.e. there is a constant power input P0. Steady state conditions with respect to the temperature of the sample are reached at a temperature T0, where the absorbed power is equal to the heat transferred to the environment, i.e. P0 = Q. Under UHV conditions, Q = A SBT0 (A is the surface area of the sample, the emissivity, SB Stefan–Boltzmann constant, and T0 the ambient temperature). Consider now a modulation of the power input by adding a time-varying power modulation to the drop inside the heating

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coil of the form P (t) = P ocos2( t/2). This will give rise to temperature oscillations of the sample of the form

(T ) =

P 0 1 1 + 2 2 + 2 22 , 2 C P 1

(10.8)

where 1 2 are the external and the internal relaxation times, respectively. Under the condition 1 ⬎ 1 ⬎ 10 2 the relaxation times are given as

1 =

CP 4 Ate SBT03

(10.9)

and

2 =

CP with k + kc

4 k = tc 3 gR, 3

(10.10)

where CP is the heat capacity, k and kc are the conductive and convective heat transfer coefficients, respecitvelyy, g1 the geometrical factor, the conductively heated volume fraction, R the radius and tc the thermal conductivity of the sample. 1 is determined experimentally from the temperature response to a step function change in heating power input and 2 is inferred from the measurement of the phase shift by a variation of the modulation frequency, mf, according to −1

2 ⎡ ⎛ 1 ⎞ ⎤⎥ ⎢ ( mf , 1 , 2 ) = 1 + ⎜ − mf 2 ⎟ . ⎢ ⎝ mf 1 ⎠ ⎥ ⎢⎣ ⎥⎦

(10.11)

A representative temperature–time profile of an ac-calorimetry experiment using the TEMPUS facility is illustrated in Figure 10.10a. On melting ac-calorimetry is applied both in the stable and in the metastable undercooled melt. The sample is cooled stepwise to temperature plateaus during which the power of the heating coil is modulated leading to the oscillating ripple on the plateaus. The amplitude of the temperature oscillations is evaluated by averaging over all modulation cycles and a Fourier transform of the modulated signal. The external relaxation time 1 is inferred from the transient temperature signal following a change in heating power input. As indicated in Figure 10.10b, the ac-calorimetry leads to high-quality measurements of the specific heat (triangles), and the external relaxation time (triangles) both above and below the liquidus temperature of the glass-forming alloy Zr60Al10Cu18Ni9Co3. The error is less than 4%. The temperature dependence of CP is described by a linear increase in the specific heat with increasing undercooling

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(a)

(b)

Figure 10.10. (a): Experimental temperature–time profile measured during ac-calorimetry on a liquid levitated Zr᎐Ni sample, power modulation and temperature response [10.46]; (b): Specific heat (circles) and external relaxation time (triangles) as determined by ac-calorimetry on Zr60Al10Cu18Ni9Co3 alloy [10.47].

with a lower T⫺1 component. The external relaxation time 1 scales with temperature according to 1 ⬀ T⫺1/2T⫺3, where T1/2 dependence was taken from the te ⬀ (elT)1/2 dependence of the total hemispherical emissivity in the free electron model [10.45]. 10.3.1.4 Surface tension and viscosity. In the previous section temperature oscillations of a levitated drop have been discussed. Consider now surface oscillations of a freely suspended liquid drop excited by a modulation of the positioning field. The frequency of the surface oscillations of a liquid drop is related to the surface tension by Rayleigh’s formula [10.46]. If the radius R of a spherical droplet undergoes oscillations of the form R = R0 (1 + A0 cos( t ) exp( −t ) ) ,

(10.12)

where A0 is the amplitude of the oscillation, the angular frequency, and the damping factor. The angular frequency l of mode l is correlated with the surface tension via [10.47] l =

l (l − 1)(l − 2) , 3 m

(10.13)

where m is the mass of the sample. Measuring the frequency of the surface oscillation modes allows the determination of the surface tension [10.48]. Fourier analysis of the frequency spectrum delivers the oscillation frequencies of mode l. The electromagnetic and gravitational fields lead to a splitting of the single peak into five peaks according to five oscillation modes represented by the “quantum

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number” l. Although an approximate correction has been worked out to take into account the external fields acting on the liquid sample [10.49], it is advantageous to perform such experiments under the condition of reduced gravity, where both fields are negligible. Usually, if deviations from the ideal spherical shape are present, the fundamental l = 2 mode is split into three sidebands. On the other hand, if the shape of the sample corresponds to an ideal sphere, the l modes are degenerated so that all five peaks of the spectrum coincide. This allows the determination of from the frequency of the surface oscillation . In addition, the damping behaviour of the surface oscillations allows to determining the viscosity of the droplet according to an idea originally developed by Lord Kelvin [10.50]. He derived an expression for the damping factor : =

20 R0

, 3 m

(10.14)

where is the viscosity. This equation is only correct for spherical drops in the absence of external fields. Experiments under reduced gravity guarantee an almost force free liquid drop with a shape approaching that of an ideal sphere. They may be the only possibility to measure the viscosity by the oscillating drop technique. The oscillating drop technique was used in TEMPUS experiments during the NASA Spacelab missions IML2 (1994) and MSL1 (1997) to measure the surface tension and the viscosity for various pure metals and alloys [10.49, 10.51–10.53]. The raw data deliver a spectrum comprising surface and translational oscillations. A carefully designed filter to the Fourier spectra was designed to obtain a corrected time signal by removing the unwanted translational oscillations. An example of such a spectrum measured on pure liquid gold is shown in Figure 10.11. By fitting an exponential law to the envelope of the oscillations, a damping factor = 0.74 s⫺1 is determined. Inserting this value into Eq. (10.14) and taking the numerical data of R0 = 4 mm and m = 5 g, the viscosity of pure gold at temperature 1133 K was calculated as = 46 mPa s. Results of measurements of the surface tension and the viscosity of Co80Pd20 alloy are depicted in Figure 10.12. This alloy has a liquidus temperature of TL = 1610 K and a solidus temperature TS = 1565 K, and therefore only a small temperature interval of 45 K in which liquid and solid phases coexist. Hence this alloy is easy to handle in electromagnetic levitation. During MSL1 mission, the sample could be undercooled by ⬎340 K. At such a large undercooling the hypercooling of the alloy was exceeded and the Curie temperature Tc = 1257 K of the liquid alloy was approached [10.55]. Figure 10.12 shows the surface tension (left) and the viscosity (right) as a function of temperature both above and below the liquidus temperatures, TL, of Co80Pd20 alloy.

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Figure 10.11. Damping of surface oscillations for molten gold at 1133 K [10.49].

Figure 10.12. Surface tension (left) and viscosity (right) as a function of temperature for a Co80Pd20 alloy measured by the oscillating drop technique during MSL1 Spacelab mission [10.55]. Measurements of a liquid sample were possible to large undercooling values of 340 K. Such undercoolings exceed the hypercooling limit and the temperature of the undercooled melt approaches the Curie temperature of the liquid alloy.

Using Eqs. (10.13) and (10.14) to convert frequencies and damping factors into surface tension and viscosity, the temperature dependence of both quantities are expressed as Co᎐Pd = 1675–0.17 (T⫺16⫻10) (mN/m)

Co᎐Pd = 0.15 exp (9.37⫻10⫺20/kBT) (mPa s)

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The Boltzmann constant kB is expressed in terms of J/K and temperature T in K. The viscosities measured range from 5 to 30 mPa s, covering nearly one order of magnitude. 10.3.2 Nucleation investigations and phase selection Nucleation is a stochastic process. Investigations of nucleation statistics open up the possibility to gain insight into the physical nature of nucleation. For each heterogeneous or homogeneous nucleation site there is a range of undercoolings associated with the operative nucleation mechanism. If a single mechanism is responsible for nucleation under a given set of experiments the distribution of undercoolings measured for many discrete events of nucleation should obey a Poisson statistics. Studies of nucleation statistics can be realized either by measuring the undercooling of an ensemble of small droplets as e.g. in experiments on droplet dispersions or by a sequence of undercooling experiments on a single sample (cf. Chapter 4). The advantage of the latter method is that the undercooling of the sample is directly measured after each undercooling cycle. Terrestrial experiments on zirconium as a function of initial sample purity have been conducted using an electrostatic levitator (ESL) [10.54]. Both electrostatic levitation and electromagnetic positioning under reduced gravity allow for containerless undercoolings experiments under the condition of ultra-high-vacuum environment. Studies of nucleation statistics on pure Zr were performed using the TEMPUS facility during MSL1 Spacelab mission [10.55]. Liquid zirconium is a metal with very high melting temperature (2128 K) and is suitable for such investigations because it is a good solvent and shows a high solubility for contaminations. Oxides, nitrides and carbides which can act as heterogeneous nucleation sites do not exist in the melt and do not form on the surface of molten zirconium. Figure 10.13 shows results of undercooling experiments under different conditions of electromagnetic processing in TEMPUS. During cooling the heater power is set to the lowest value, and the primary power input to the sample comes from the positioning field. One sample was repeatedly melted and cooled to solidification, keeping all experimental variables constant except for the positioner power settings on cooling. Different positioner power supply settings were used. At the low positioner power (31 V) the cooling rate was 50 K/s and at the high positioner power (71 V) it was 48 K/s. Fluid flow velocities were calculated for these different experiment conditions as 5 and 27 cm/s [10.56]. These experiments of undercooling statistics with about 50 cooling cycles each reveal that there is no significant change in the nucleation behaviour in the range of flow conditions. The results obtained from the Space experiments can be directly compared with those of the terrestrial investigations in the ESL, if the different values of the volume and the cooling rates of the samples are taken into consideration. The

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Figure 10.13. Two distributions from experiments on zirconium using TEMPUS on MSL1 Spacelab mission. To change electromagnetic stirring inside the molten Zr, the power of the positioning field is altered. The low positioner distribution is in the laminar flow regime with maximum flows of 5 cm/s, while the high positioner cause maximum flow rates of 27 cm/s [10.59].

nucleation rate Iss of the samples can be scaled denoting v1 and v2 and T1 and T2 as the volumes and cooling rates of the sample processed in TEMPUS and in ESL, respectively. It then holds that I ss v1 I ss v2 = . T1 T2

(10.15)

The cooling rate in the ESL experiments is 240 K/s compared to 48 K/s in the MSL1 experiments. The mass of the sample processed in the TEMPUS facility was 1.18 g, while that of the sample processed in ESL was 0.14 g. For equal Iss values and a known undercooling in the ESL case where the undercooling is measured as T = 348 K, one can calculate the undercooling obtained by an identical mechanism for the larger sample at the smaller cooling rate by solving I ss (T = 348) =

v2 v1

T1 I ss (T2 ) = 414 I ss (T2 ). T2

(10.16)

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Assuming the Turnbull nucleation rate (see Chapter 4), the pre-exponential terms are equal and insensitive to temperature so that the Eq. (10.16) can be reduced to G1* G2* = + ln( 414), k BT1 k BT2

(10.17)

where G* is the activation energy to form clusters of critical size. If the solid–liquid interfacial energy is assumed to be constant over the temperature range, and the change in volume free energy is proportional to T2, then the computation yields T2 = 336 K. This value agrees within an uncertainty of a few Kelvins with the results of the measurements during the MSL1 mission. These experiments show that fluid flow with maximum velocities of 43 cm/s has apparently no effects on crystal nucleation in deeply undercooled melts of pure Zr. Nucleation is the first step of crystallization of an undercooled melt preselecting the crystallographic phase as stable or metastable. If a metastable phase is primarily nucleated solidification takes place in a two-step process. After the primary nucleation of the metastable phase this phase subsequently grows into the undercooled melt by a dendrite growth mode. After a delay time secondary nucleation of the stable can occur within the interdendritic liquid, which can lead either to a two phase mixture of solid and liquid or after remelting of the metastable phase to a solid consisting exclusively of the stable phase. The latter transformation is usually a slow process and can be at least partly avoided by rapid quenching the sample after the first recalescence event [10.56]. The mechanism for nucleation of the second phase and the delay time observed in this transformation appear to be dependent on solid movement and coalescence within the solid–liquid phase mixture probably the result of convective flow [10.57]. Phase selection in undercooled melts of Fe᎐Ni᎐Cr austenite steel alloys was investigated with a reduction of melt convection under reduced gravity making use of TEMPUS facility during MSL1 Spacelab mission [10.58]. Similar to the terrestrial experiments, double-recalescence events are observed in the temperature–time profiles during solidification of Fe᎐Nr᎐Cr austenite steels in the Space experiments. Examples are shown for two alloys Fe᎐12% Cr᎐16% Ni and Fe᎐16% Cr᎐12%Ni in Figure 10.14. In both cases a double recalescence is observed, indicating the primary crystallization of a metastable bcc phase (ferrit) of the austenite alloys before the stable fcc (austenite) phase is formed. Qualitatively, such a behaviour is similar throughout as in terrestrial experiments [10.61, 10.62]. However, both the critical undercoolings for the observation of doublerecalescence events and the delay time of nucleation of the second stable fcc

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Figure 10.14. Temperature–time profiles of recalescence measured on Fe᎐Ni᎐Cr alloys undercooled containerlessly in TEMPUS during MSL1 Spacelab mission: left: Fe᎐12% Cr᎐16%Ni, right: Fe᎐16% Cr᎐12%Ni [10.62].

Figure 10.15. Delay time measured during rapid solidification of Fe᎐Cr᎐Ni steel alloy under reduced gravity in Space (closed circles) and under terrestrial conditions (open boxes) with the T0bcc temperature of bcc phase [10.62].

phase after primarily formed metastable bcc phase depend apparently on the experiment conditions as shown in Figure 10.15. Significant deviations become obvious when the delay time of this transformation measured on ground is compared with that of the experimental results obtained by equivalent experiments performed during MSL1 mission in the TEMPUS facility. The delay time is enhanced if the gravitational driven mass and heat transport are essentially reduced in Space. This observation may support the

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hypothesis that convection could influence the formation of secondary nucleation sites of stable fcc phase in the interdendritic region of metastable bcc phase. But a detailed understanding of this mechanism is still lacking. Efforts to describe this phenomenon are in progress [10.59]. 10.3.3 Measurements of dendrite growth velocities The TEMPUS facility was also used to measure the dendrite growth velocity as a function of undercooling on pure nickel and dilute Ni᎐0.6 at.%C alloys during MSL1 Spacelab mission [10.60]. A metallic W-Re trigger needle coated at the Tipp with ZrO2 (to decrease the sticking tendency while touching the melt) was used for external stimulation of nucleation at selected undercoolings. The growth velocity was measured from recording the temperature–time profiles during recalescence in quite analogy to the measurements described in Chapter 2 for electromagnetic levitation on Earth. The results of investigations on pure Ni and dilute Ni99.4C0.6 are shown in Figure 10.16. For the dilute Ni᎐C alloy, g data on an extended undercooling range up to T = 305 K have been obtained, whereas the g-data points on Ni are restricted to

Figure 10.16. Dendrite growth velocity V versus undercooling T data on pure Ni (open symbols) and a dilute Ni᎐0.6 at.%C alloy (black symbols), obtained during the MSL1 Spacelab mission. Squares refer to triggered nucleation events with high accuracy of measurements, circles/dots to spontaneous nucleation events with reduced accuracy. An arrow marks the resolution limit of measurements of the pyrometer operating at a frequency of 100 Hz. Calculated curves according to the predictions of sharp interface model are plotted for Ni (dashed line) as well as for Ni᎐C (full line).

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the most interesting regime of small undercooling up to T = 85 K. The growth velocity as a function of undercooling was calculated within sharp interface model of dendrite growth (cf. Chapter 4) for pure Ni (see dashed line in Figure 10.16) and for the dilute Ni᎐C alloy (see solid line in Figure 10.16). Whereas at medium and large undercoolings the data are well described by dendrite growth theory, the experimental data measured at small undercoolings are located above the theoretical curves. Thus, either the reduction of fluid flow motion in liquid samples processed in TEMPUS during MSL-1R mission was not sufficient to eliminate the enhancement of growth velocity due to the change in heat and mass transport by forced convection induced externally by the alternating electromagnetic field of the TEMPUS coil system or the samples were contaminated by very small amounts of impurities, which can also lead to a slight increase of the growth velocity. Future experiments on board of the ISS are in preparation to determine quantitatively in separate way the change of growth dynamics by small amounts of solute and increase of mass and heat transport by forced convection. Since the MSL1 Spacelab mission of NASA in 1997 no opportunity was given to refly TEMPUS in Space up to now. An advanced Facility of an ElectroMagnetic Levitator (EML) is considered by the European Space Agency and the German Aerospace Center to be accommodated on board of the ISS in 2009. However, during the recent years it was successfully demonstrated that containerless undercooling and solidification experiments on drops of metals and alloys can even be performed during the short duration of about 20 s time of reduced gravity in parabolic flights. By using 40 subsequently flown parabolas during one-day flight it is possible to perform in direct sequence measurements of the growth dynamics as a function of undercooling. Such experiments were performed on Fe60Co40 alloy, in which the dendrite growth velocity was measured as a function of undercooling [10.61]. The results of the measurements in reduced gravity (open symbols) are compared with analogous measurements on Earth (closed symbols) as exhibited in Figure 10.17. The circles represent the growth velocity of primarily formed fcc phase, while the triangles give the corresponding data of growth velocity of primarily formed metastable bcc phase. The growth velocity of the primarily solidified fcc phase increases with increasing undercooling up to a critical value of undercooling at which the growth velocity abruptly drops to a lower value. Such a sudden drop of the growth velocity in Fe-based alloys can be associated with a change of primary crystallization of stable fcc to metastable bcc phase [10.62]. The results of the measurements of growth velocity as a function of undercooling in reduced gravity on the same alloy show the same order of growth velocities for both phases in the undercooling range from 30 to 240 K. But a comparison of both data sets yield the important result that the critical undercooling Tcrit for the onset of primarily

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Figure 10.17. Growth velocity as a function of undercooling measured on Fe60Co40 alloy on Earth (full symbols) and in reduced gravity (open symbols).

formed metastable bcc phase is decreased from about 90 K on Earth to 53 K in reduced gravity. Similar to the results of investigations on stainless-steel alloys during MSL1 Spacelab mission [10.62] it is concluded that the reduced critical undercooling for the primary crystallization of metastable bcc phase could be attributed to an increased lifetime of the metastable bcc phase and a retarded nucleation of the stable fcc phase in the mushy zone. A detailed understanding and theoretical modelling by taking into account convection in phase selection processes are, however, still lacking. REFERENCES

[10.1] Drehman, A.J., and Turnbull, D. (1981) Scripta Metallurgica 15, 543. [10.2] Notthoff, C., Feuerbacher, B., Frans, H., Herlach, D.M., and HollandMoritz, D. (2001) Physical Review Letters 86, 1038. [10.3] Piller, J., Knauf, R., Preu, P., Lohöfer, G., and Herlach, D.M. (1986) Proceedings of the 6th European Symposium on Materials Sciences under Microgravity, Vol. ESA SP-256 (Bordeaux), p. 437. [10.4] Team TEMPUS (1996) in: Materials and Fluids under Low Gravity, eds. Ratke, L., Walter, H., and Feuerbacher B. (Springer, Berlin), p. 233.

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[10.5] Hofmeister, W., Morton, C.M., Robinson, M.B., and Bayuzick, R.J. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 83. [10.6] Barth, M., Holland-Moritz, D., Herlach, D.M., Matson D.M., and Flemings, M.C. (1999) in: Solidification 1999, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 83. [10.7] Steinberg, J., Lord, A.E., Lacy, L.L., and Johnson, J. (1981) Applied Physics Letters 38, 135. [10.8] Lee, M.C., Kendall, J.M., and Johnson, W.L. (1982) Applied Physics Letters 40, 382. [10.9] Drehman, A.J., and Greer, A.L. (1984) Acta Metallurgica et Materialia 32, 323. [10.10] Kiminami, C.S., and Sahm, P.R. (1986) Acta Metallurgica et Materialia 34, 2129. [10.11] Gillessen, F., Herlach, D.M., and Feuerbacher, B. (1988) Journal of LessCommon Materials 145, 145. [10.12] Shong, D.S., Graves, J.A., Ujiie, Y., and Perepezko, J.H. (1987) Materials Research Society Symposium Proceedings 87, 17. [10.13] Gillessen, F., and Herlach, D.M. (1988) Materials Science & Engineering A 97, 147. [10.14] Dunst, A., Herlach, D.M., and Gillessen, F. (1991) Materials Science & Engineering A 133, 785. [10.15] Gillessen, F. (1989) Ph.D. Thesis, Ruhr-University Bochum, Germany. [10.16] Herlach, D.M., Gillessen, F., Volkmann, T., Wollgarten, M., and Urban, K. (1992) Physical Review B 46, 5203. [10.17] Shechtman, D., Blech, I., Gratias, D., and Cahn, J.W. (1984) Physical Review Letters 54, 1951. [10.18] Gillessen, F., and Herlach, D.M. (1991) Materials Science & Engineering A 134, 1220. [10.19] Mueller, B.A., Schaefer, R.J., and Perepezko, J.H. (1987) Journal of Materials Research 2, 809. [10.20] Avrami, M. (1939) Journal of Chemical Physics 7, 1103. [10.21] Schroers, J., Holland-Moritz, D., Herlach, D.M., and Urban, K. (2000) Physical Review B 61, 14500. [10.22] Croat, J.J., Herbst, J.F., Lee, R.W., and Pinkerton, F.E. (1984) Journal of Applied Physics 55, 2078. [10.23] Sagawa, M., Fujimura, S., Togawa, N., Yamamoto, H., and Matsuura, Y. (1984) Journal of Applied Physics 55, 2083. [10.24] Sellers, C.H., Hyde, T.A., Branagan, D.J., Lewis, L.H., and Panchanathan, V. (1997) Journal of Applied Physics 81, 1351. [10.25] Hermann, R., and Löser, W. (1988) Journal of Applied Physics 83, 6399. [10.26] Herlach, D.M., Volkmann, T., and Gao, J. (2003) German Patent No. 10106217.6. [10.27] Gao, J., Volkmann, T., and Herlach, D.M. (2002) Acta Materialia 50, 3003.

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[10.28] Herlach, D.M., Gao, J., Holland-Moritz, D., and Volkmann, T. (2004) Materials Science & Engineering A 375–377, 9. [10.29] Kerr, H.W., and Kurz, W. (1996) International Materials Review 41, 129. [10.30] Gao, J., Volkmann, T., Roth, S., Löser, W., and Herlach, D.M. (2001) Journal of Magnetism & Magnetic Materials 234, 313. [10.31] Li, D., and Herlach, D.M. (1997) Journal of Materials Science 32, 1437. [10.32] Ratke, L., and Diefenbach, S. (1995) Materials Science & Engineering Reports R 15, 263. [10.33] Cao, C., Letzig, T., Görler, G., and Herlach, D.M. (2001) Journal of Alloys & Compounds 325, 113. [10.34] Ratke, L. (1995) Materials Science & Engineering A 203, 399. [10.35] Egry, I., Herlach, D.M., Kolbe, M., Ratke, L., Reutzel, S., Perrin, C., and Chatain, D. (2003) Advances in Engineering Materials 11, 819. [10.36] Cao, C.D., Wei, B., and Herlach, D.M. (2002) Journal of Materials Science Letters 21, 341. [10.37] Eichel, R., and Egry, I. (1999) Zeitschrift für Metallkunde 90, 371. [10.38] Lu, X.Y., Cao, C.C., Kolbe, M., Wie, B., and Herlach, D.M. (2004) Materials Science & Engineering A 375–377, 1101. [10.39] Shiraishi, S.Y., and Ward, R.G. (1964) Canadian Metallurgy Quarterly 3, 117. [10.40] El-Mehairy, A.E., and Ward, R.G. (1963) Transactions of Metallurgical Society AIME 227, 1226. [10.41] Damaschke, B., Oelgeschläger, D., Ehrich, E., Dietzsch, E., and Samwer, K. (1998) Review of Scientific Instruments 69, 2110. [10.42] Damaschke, B., and Samwer, K. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., Vorhees, P. (TMS, Warrendale, PA), p. 43. [10.43] Lohöfer, G. (1994) International Journal of Engineering Science 32, 107. [10.44] Lohöfer, G., and Egry, I. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 65. [10.45] Fecht, H.-J., and Johnson, W.L. (1991) Review of Scientific Instruments 62, 1299. [10.46] Wunderlich, R., anf Fecht, H.-J. (1996) International Journal of Thermophysics 17, 1203. [10.47] Wunderlich, R.K., Sagel, R.A., Ettel, C., Fecht, H.-J., Lee, D.S., Glade, S., and Johnson, W.L. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 53. [10.48] Sievers, A.J. (1978) Journal of Optical Society America 68, 1505. [10.49] Team TEMPUS (1995), in: Materials and Fluids Under Low Gravity, eds. Ratke, L., Walter, H.U., and Feuerbacher, B. (Lecture Notes in Physics Springer), p. 233. [10.50] Lord Rayleigh (1879) Proceedings of the Royal Society London 29, 71. [10.51] Chandrasekhar, S. (1959) Proceedings of London Mathematical Society 3, 142.

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[10.52] Egry, I., Lohöfer, G., Neuhaus, P., and Sauerland, S. (1992) Journal of Thermophysics 13, 65. [10.53] Cummings, D.L., and Blackburn, D.A. (1991) Journal of Fluid Mechanics 224, 395. [10.54] Chandrasekhar, S. (1961) Hydrodynamics and Hydromagnetic Stability (Dover, New York). [10.55] Egry, I., Lohöfer, G., Schneider, S., Seyhan, I., and Feuerbacher, B.(1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 15. [10.56] Hyers, R.W., Trapaga, G., and Flemings, M.C. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 23. [10.57] Rösner-Kuhn, M., Hofmeister, W.H., Kuppermann, G., Morton, C.W., Bayuzick, R.J., and Frohberg, M.G. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 33. [10.58] Morton, C.W., Hofmeister, W.H., Rulison, A., and Watkins, J. (1998) Acta Materialia 46, 6033. [10.59] Hofmeister, W.H., Morton, C.M., Bayuzick, R.J., and Robinson, M.B. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 75. [10.60] Volkmann, T., Löser, W., and Herlach, D.M. (1997) Metallurgical Transactions 28A, 461. [10.61] Koseki, T., and Flemings, M.C. (1995) Metallurgical Transactions 26A, 2991. [10.62] Matson, D.M., Löser, W., and Flemings, M.C. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 75. [10.63] Hanlon, A.B., Hyers, R.W., and Matson, D.M. (in press) Proceedings of TMS. [10.64] Barth, M., Holland-Moritz, D., Herlach, D.M., Matson, D.M., and Flemings, M. (1999) in: Solidification, eds. Hofmeister, W.H., Rogers, J.R., Singh, N.B., Marsh, S.P., and Vorhees, P. (TMS, Warrendale, PA), p. 83. [10.65] Hermann, R., Löser, W., Lindenkreuz, G., Diefenbach, A., Zahnow, W., Dreier, W., Volkmann, T., and Herlach, D.M. (2004) Materials Science & Engineering A 375–377, 507. [10.66] Moir, S., and Herlach, D.M. (1997) Acta metallurgica et materialia 45, 2827.

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Chapter 11

Conclusions and Summary


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Chapter 11

Conclusions and Summary Investigations on slowly but deeply undercooled melts open up possibilities to study basic phenomena of the thermodynamics, nucleation and crystal growth. Such research work is of particular interest to improve the understanding of the formation of metastable solids, which are formed from the non-equilibrium state of an undercooled melt. Techniques have been described to achieve the state of an undercooled melt. Among these, methods of containerless processing are now widely used to perform undercooling experiments with the extra benefit that direct diagnostics becomes possible for in situ investigations of rapid solidification phenomena. Thermodynamic considerations lead to the conclusion that the undercooling achieved prior to solidification is the most important parameter for the solidification of metastable phases. The solidification process of an undercooled melt is initiated by nucleation. Thereby, nucleation preselects the crystallographic phase stable or metastable. Nucleation processes are described by the classical nucleation theory, which has been originally developed to predict the nucleation behaviour during the condensation of supersaturated vapours. It was extended to the phase transformation of liquid to solid and is often used to describe undercooling phenomena. The classical nucleation theory is able to describe in a phenomenological way the nucleation process in an undercooled melt. The advantage of this model is that it can be easily used to tractate experimental results. However, there are some problems related to the classical nucleation theory. A sharp interface between solid nucleus and melt is assumed, which is not necessarily true. A digression has to be made towards a more detailed study of interfacial thermodynamics as a function of interfacial width as recently performed using the approach of diffuse interfaces. Another difficulty arises from the uncertainty of the structure of the interface, which may be of relevance to explain different activation barriers for different solid phases. In the case of heterogeneous nucleation there is no theoretical approach to predict the catalytic potency factor f(ϑ) as a function of the structure and interatomic distances of the heterogeneous nucleation side. Therefore, the analysis of experimental results within nucleation theory, in particular, concerning the question of homogeneous versus heterogeneous nucleation may encompass some uncertainty. From the experimental side it is interesting to correlate the short-range order in the undercooled melt to the nucleation behaviour. Especially, such investigations merit attention with respect to studies of the nucleation behaviour of quasicrystal-forming alloys. 409

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Nucleation is the first process in solidification, which is followed by crystal growth. In the case of pure metals, miscible alloys and intermetallics, the condition of an undercooled melt imply dendritic crystal growth. Both theoretical concepts and experimental investigations of dendritic growth are relatively well developed. Nevertheless, some questions remain to be answered. Among these, a justification of the marginal stability hypothesis merits future work, in relation to the solvability criterion. Here, a possible dependence of the stability parameter on temperature and anisotropy parameter needs further attention. From the experimental side it would be interesting to clarify the role of fluid flow on dendritic growth in particular at low undercoolings. The conditions of containerless processing under microgravity conditions may be of some help to perform experiments in space to improve our understanding of such effects. Such investigations may also lead to an answering of the question whether the addition of small amounts of strongly partitioning elements may lead to an enhancement of solidification rates in comparison with those of a pure metal. While dendritic solidification concerns single-phase formation, eutectic growth comprises multiple, cooperative nucleation and growth processes. There is some progress in the theoretical description of eutectic growth made in the recent past, including non-equilibrium effects at the solid–liquid interface. However, these theoretical concepts are limited to the application of regular eutectic growth. Experimental investigations indicate that such growth modes are only present at small undercoolings. At larger undercoolings anomalous eutectic growth occurs whose origin is not well understood. It may be either a different growth mode or may be correlated to secondary effects due to e.g. aging effects. The state of an undercooled melt gives the potential to solidify metastable solids. There is a wide variety of such metastable states ranging from metastable crystalline phases via supersaturated and grain refined alloys to amorphous metals. A detailed understanding of the thermodynamics, the nucleation and crystal growth conditions can lead to a comprehensive understanding of the criteria for the formation of such metastable states. This in turn may lead to a predictive capability to describe metastable phase formation as a function of undercooling. Such a comprehensive description requires an accurate knowledge of the temperature dependence of thermophysical parameters in the metastable regime of an undercooled melt. Until now only less information on the thermophysical parameters as a function of undercooling is available. Here, future experiments in particular using the special conditions of microgravity may lead to some progress. These investigations would be not only of great importance to deliver numerical values for the thermophysical parameters with respect to the quantitative description of nucleation and crystal growth but are of interest to verify theoretical approaches to describe the temperature dependence of e.g. the viscosity and/or the specific

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heat. In particular, the viscosity changes by about 13 orders of magnitude in the temperature range between the melting and the glass temperature and different models have been proposed to describe the temperature dependence in this regime. But no experimental results are available to verify the different physical models. Future experiments on containerless processed drops, which are excited to surface oscillations may offer the unique possibility to investigate the viscosity over an extended range of undercooling. The present review tries to give a summary of the present state of art in the field of formation of metastable solids by non-equilibrium solidification of undercooled melts. From the foregoing discussion, it is apparent that even though much progress has been made to improve our understanding in this field many problems remain to be solved. A combined effort both from the theoretical and the experimental side can lead to an enhancement of fundamental understanding of metastable solidification. Such a progress is of high relevance for the defined and reproducible preparation of metastable solids from the undercooled melt.


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Appendix: List of the Symbols

LATIN SYMBOLS

aL aS A AS b bj B Bdem Bdip Beff Bext BF BL Bo c c* c* c* cA cB cclust ce cL cL* cN c0 cS cS* c’L c’L Cp d

Thermal diffusivity in liquid (m2/s) Thermal diffusivity in solid (m2/s) Constant in Vogel–Fulcher–Tammann ansatz (K) Factor of anisotropy of atomic kinetics Neutron scattering length (m) Neutron scattering length of atom j (m) Magnetic field (G) Demagnetisation field (G) Dipolar field (G) Effective magnetic field (G) External magnetic field (G) Fermi field (G) Lorentz field (G) Time-averaged alternating magnetic field (G) Concentration (at.%) Concentration at the triple point (at.%) Concentration of -phase in eutectic transformation (at.%) Concentration of -phase in eutectic transformation (at.%) Concentration of component A in an alloy (at.%) Concentration of component B in an alloy (at.%) Concentration of atom clusters in an undercooled melt (at.%) Eutectic concentration (at.%) Concentration in liquid (at.%) Concentration of liquid at the S/L interface (at.%) Concentration in a crystal nucleus (at.%) Initial alloy concentration (at.%) Concentration in solid (at.%) Concentration of solid at the S/L interface (at.%) Concentration of liquid at -phase in peritectic transformation (at.%) Concentration of liquid at -phase in peritectic transformation (at.%) Specific heat at constant pressure (J/mol/K) Average atomic distance in the construction of the liquid–solid interface(Å) 413

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da d(n) do D DL E E’ E1 f f, f() f(ϑ) fa fi fL fv F + Fmag FD FDF Fd FLZ g g() gAB(r) gD gL g(r) G G G+mag GL G L GS G S GSms G mag , L G mag ,S

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Interatomic spacing (m) Anisotropy of the capillarity (m) Capillary length (m) Diffusion coefficient (m2 s⫺1) Diffusion coefficient in liquid (m2 s⫺1) Electrostatic field (V/m) Neutron energy after scattering (eV) First exponential integral function Atomic attachment factor Volume fraction of and phases in eutectic microstructures Density of free energy (J/mol/kg3) Catalytic factor of heterogeneous nucleation Fraction of atoms effectively colliding with the S/L interface Atomic jump frequency (Hz) Muon spin precession frequency (Hz) Atomic vibration frequency (Hz) Free energy (J/mol) Magnetic free energy of a single spin (J/spin) Dissipative force in the Navier–Stokes equation (J/kg/m2) Distribution function Drag force (J/kg/m2) Lorentz force (J/kg/m2) Gravitational acceleration (m/s2) Fractional change of surface tension Partial pair correlation function Diffusive parameter Landé’s g factor Pair correlation function Gibbs free enthalpy (J/mol) Reciprocal lattice vector (m⫺1) Gibbs magnetic free enthalpy of a single spin (J/spin) Gibbs free enthalpy of the liquid phase (J/mol) Gibbs free enthalpy of the liquid phase per atom (J/atom) Gibbs free enthalpy of the solid phase (J/mol) Gibbs free enthalpy of the solid phase per atom (J/atom) Gibbs free enthalpy of a metastable solid phase (J/mol) Magnetic part of the Gibbs free enthalpy of the liquid phase per atom (J/atom) Magnetic part of the Gibbs free enthalpy of the solid phase per atom (J/atom)

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h hm H Hi HL H L H mag , L H mag ,S HS H S HSt(r) I(Q)meas I(Q)sample I I1 I(t) It(t) Iss I sshet Iv j J Jp J k k k’ ke kp k(V,c) kx,y,z kB K n+ K n+ KMF K KV

415

Planck’s constant (6.63⫻10⫺34 Js) Heat transfer coefficient Enthalpy (J/mol) Average of the enthalpy of the liquid–solid interface (J/mol) Enthalpy of the liquid phase (J/mol) Enthalpy of the liquid phase per atom (J/atom) Magnetic part of the enthalpy of the liquid phase per atom (J/atom) Magnetic part of the enthalpy of the solid phase per atom (J/atom) Enthalpy of the solid phase (J/mol) Enthalpy of the solid phase per atom (J/atom) Enthalpy approximated by a step function in dependence of the radius of the nucleus (J/mol) Intensity of scattered waves as a function of the scattering vector Intensity of the scattered radiation as a function of the scattering vector Electrical current (A) Modified Bessel function of first order Nucleation rate as a function of time t [m⫺3 s⫺1] Transient nucleation rate as a function of time t [m⫺3 s⫺1] Steady state nucleation rate [m⫺3 s⫺1] Steady state heterogeneous nucleation rate (m⫺3 s⫺1) Ivantsov function Number of spins per atom Spin Probability flux Diffusion flux (m s⫺1) Wave number Wave vector of the incident radiation (Å⫺1) Wave vector of the scattered radiation (Å⫺1) Equilibrium partition coefficient Wave number of perturbation (m⫺1) Non-equilibrium partition coefficient, depending on velocity V and concentration c Wave number in x, y, z direction (m⫺1) Boltzmann’s constant (1.380658⫻10⫺23 J/K) Rate of atomic attachment to clusters consisting of n atoms (s⫺1) Rate of atomic detachment from clusters consisting of n atoms (s⫺1) Coupling constant in the molecular field theory of magnetism (= KV(T )) Prefactor of the exponential in the nucleation rate (m⫺3 s⫺1)

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KVhom l la m m, me m(V) M M MA, MB Mcc, Mee ML Ms MS Mx,y,z n n n n* n0 nA ni nx,ny,nz N N N N1 Ni NJ NL Nn Nrl NS O p P

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Prefactor of the exponential term in the nucleation rate of homogeneous nucleation (m⫺3 s⫺1) Mean free path (m) Characteristic scale of the solute diffusion layer (m) Mass of a sample (kg) Liquidus slopes of ,-phase Slope of the liquidus line in the equilibrium phase diagram Non-equilibrium slope of the liquidus line depending on the growth velocity (K/at.%) Magnetization (Gauß) Mobility of the diffuse interface Interface mobility for the transformation in pure systems consisting of A or B components, respectively Diagonal terms in the matrix of mobilities for solutal and thermal transport Magnetization of the liquid phase (Gauß) Saturation magnetization (Gauß) Magnetization of the solid phase (Gauß) Interface mobility in x, y, z direction Number of atoms in a cluster, Avrami exponent Unit vector normal to the interface Vector field corresponding to the boundary = ⫺/| | Number of atoms in a nucleus of critical size Initial number of particles Number of atoms in a cluster Number of iteration steps in numerical calculations Components of the normal vector Total number of atoms Number of atoms per volume unit (at./m3) Number of heterogeneous nucleation sites per volume unit (at./m3) Number of atoms in the first layer of the liquid–solid interface [at./m2] Number of atoms in the liquid–solid interface (at./m2) Number of spins Avogadro’s number (6.0221367⫻1023 mol⫺1) Number of the clusters containing n atoms per unit volume Number of reciprocal lattice vectors Number of atoms within the solid surface layer at the liquid–solid phase boundary (at./m2) Surface (m2) Pressure (N/m2) Probability function

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p Pc Pd PT q Q Q1 Q2 Qel QH Qs r r r* r0 r1 rl rmin rn1 rn2 rs rij rX rs* R R0 Rel Rc RG RH RL Rms RS S

417

Magnetic moment per atom (B) Chemical Péclét number Vapour pressure (N/m2) Thermal Péclét number Heat flux Scattering vector (Å⫺1) Scattering vector of the first maximum of S(Q) (Å⫺1) Scattering vector of the second maximum of S(Q) (Å⫺1) Electrical charge (Cb) Heat content Scattering vector at the shoulder of S(Q) (Å⫺1) Radius (m) Radial vector Radius of a critical nucleus (m) Distance between the central atom and an atom within the surface layer of an icosahedron (Å) Distance between nearest neighbours within the surface layer of an icosahedron (Å) Radius of a spherical cluster of the solid phase in the melt including the boundary layer (m) Atomic distance at which the interatomic interchange potential V(r) becomes minimum (Å) Distance of nearest neighbours (Å) Distance of next nearest neighbours (Å) Radius of a spherical solid cluster in the melt (m) Distance between atoms i and j Radius of atomic species X (Å) Radius of the solid nucleus within the model of diffuse interface (m) Dendrite tip radius (m) Radius of a droplet Electrical resistance (Ohm) Curvature radius (m) Gas constant (= 8.31451 J/mol/K) Radius, at which HSt(r) changes from the value of the solid to that of the liquid phase within the model of diffuse interface (m) Atomic transfer rate from liquid to solid Critical dendrite tip radius according to the stability analysis by Mullins and Sekerka Atomic transfer rate from solid to liquid Entropy (J/mol/K)

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S* S, S(Q) SAB(Q)

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Entropy of the interface within the model of Ewing (J/mol/K) Half of interlamellar spacing of and phases in eutectic microstructures Structure factor as a function of the scattering vector Partial structure factor as a function of the scattering vector (cf. Section 3.2.3) Scf (1) Configurational entropy of the first layer of the liquid–solid interface per atom (J/at./K) Scf (i ) Configurational entropy of the liquid–solid interface per atom (J/at./K) Si Average of the entropy of the liquid–solid interface (J/at./K) SL Entropy of the liquid phase (J/mol/K) Entropy of the liquid phase per atom (J/at./K) SL Smag Magnetic part of the entropy of a sample consisting of NJ spins (J/K) Smag , L Magnetic part of the entropy of the liquid phase per atom (J/at./K) Smag , S Magnetic part of the entropy of the solid phase per atom (J/at./K) SS Entropy of the solid phase (J/mol/K) Entropy of the solid phase per atom (J/at./K) SS SSt(r) Entropy in dependence of the nucleus radius within diffuse interface theory (J/mol/K) t Time (s) tN Time of nucleation (s) T Temperature (K) T Cooling rate (K/s) T⬁ The temperature of the undercooled melt far from the interface (K) Tg Glass temperature (K) Tog Ideal glass temperature (K) Trg Reduced glass temperature=Tg/TL Tc Critical temperature for order–disorder transition TC Curie temperature (K) TCL Curie temperature of the liquid phase (K) TCS Curie temperature of the solid phase (K) Te Eutectic temperature (K) TE Equilibrium melting temperature (K) TI Temperature at the S/L interface [K] Tp Peritectic temperature (K) ms Tk Critical temperature defined by the condition G*ms (Tkms ) = G*st (Tkms ) (K) TL Liquidus temperature (K) TLms Virtual melting temperature of a metastable solid (K) TMK Critical temperature of the mode coupling theory (K) TN Temperature of nucleation (K) TNclass Nucleation temperature as calculated within nucleation theory (K)

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TNmag

419

Nucleation temperature taking into account a magnetic contribution in GLS (K) TNflux Nucleation temperature measured in melt fluxing experiments (K) Nucleation temperature measured in levitation experiments (K) TNlev TQ Unit of undercooling, hypercooling = Thyp, (K) TR Room temperature (K) TS Solidus temperature (K) U Electrical voltage (V) Uo Fluid flow velocity (m/s) v Volume of the sample (m3) v*f Excess free volume (m3) vf Free volume (m3) vm Mole volume (m3/mol) V Velocity of the solidification front (m/s) Vb Laser scanning velocity (m/s) Vn Velocity of the S/L interface along the normal n to the interface (m/s) Vo Speed limit (m/s) V(r) Atomic interaction potential as a function of distance r (J) Vrel Relative velocity VA Velocity for absolute morphological stability at the interface (m/s) VC Critical velocity defined by the constitutional undercooling or velocity for the continuous growth (m/s) VD Atomic diffusive speed (m/s) VDI Atomic diffusive speed on the interface (m/s) VDP Displacement velocity (m/s) VS Velocity for the steps motion or speed of sound (m/s) w Order parameter W Half-width of the distribution function of measured undercoolings (K) WP(n) Probability to find a volume segment containing n particles x, y, z Cartesian coordinates (m) Yl,m Spherical harmonics of mode l, m Z Coordination number Zcf Number of possibilities to construct the first solid–liquid interfacial layer Zcf(n) Number of configurations in iteration step n of interface modelling GREEK SYMBOLS

J

Dimensionless liquid–solid interfacial energy = s(TL)/ H f Sf /RG = Jackson–Hunt factor of dimensionless entropy of fusion

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p T (n) o i g gb LS mag s R c z GT c c r0 SD T T v Cp

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The position of the interface relative to a fixed plane in the crystal lattice Volumetric thermal expansion Kinetic growth coefficient depending on the normal vector to the interface Averaged kinetic growth coefficient along the interface Surface tension (J/m2) Liquid–solid interfacial energy per atom (J/atom) Gyromagnetic ratio Grain boundary interfacial energy (J/m2) Liquid–solid interfacial energy (J/m2) Magnetic part of the interfacial liquid–solid energy (J/m2) Liquid–solid interfacial energy per atom in the solid surface (J/atom) Energy of a curved liquid–solid interface with curvature radius R (J/m2) Damping parameter Collision constant Zeldovich factor Gibbs–Thomson coefficient = / S f Thickness of the liquid–solid interface (m) Cusp amplitude (m) Amplitude of perturbation (m) Fluctuation of r0 (Å) Skin depth (m) Tolman length (m) Brightness of a temperature interval (K) Non-equilibrium interval of solidification Difference of the specific heat of liquid and solid phases at constant pressure (J/mol/K) G(r) Change of the Gibbs free enthalpy during the formation of a solid cluster with radius r in an undercooled melt (J) G* Activation energy for the formation of a critical homogeneous nucleus (J) G * Activation energy for the formation of a critical homogeneous nucleus at diffuse liquid–solid interface (J) * Ghet Activation energy for the formation of a critical heterogeneous nucleus = G*f() (J) * Gms Activation energy for the formation of a critical nucleus of a metastable solid phase (J) Gst* Activation energy for the formation of a critical nucleus for the formation of a stable solid phase (J) Ga Activation threshold for atomic diffusion (J) G mag Magnetic contribution to the difference of Gibbs free enthalpy of liquid and solid phases per atom (J/atom)

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Gmag

421

Magnetic contribution to the difference of Gibbs free enthalpy of liquid and solid phases per volume unit = Gmag /vm (J/m3) GLS Difference of Gibbs free enthalpy of liquid and solid phases per mol (J/mol) GLS Difference of Gibbs free enthalpy of liquid and solid phases per volume unit = GLS/vm (J/m3) ms GLS Difference of Gibbs free enthalpy of liquid and metastable solid phases per volume unit = (GL⫺ GSms )/vm (J/m3) G (rs) Change of Gibbs free enthalpy during formation of a solid nucleus with diffuse interface thickness of d and with radius rs of the solid nucleus in an undercooled melt (J) G Activation energy per atom for atomic jumps in the Arrhenius ansatz of viscosity (J) Hf Enthalpy of fusion (J/mol) Enthalpy of fusion per atom (J/atom) H f Hi (= Hi⫺HL) (J/mol) Hmix Mixing enthalpy (J/mol) H mag Magnetic contribution to the difference of the enthalpies of liquid and solid phases per atom (J/atom) HLS Difference of free enthalpies of liquid and solid phases per mol (J/mol) Scf (i ) Difference of the configurational entropy of interface and solid per atom (J/at./K) Scf (ls) Difference of the configurational entropies of liquid and solid per atom (J/at./K) Sf Melting entropy (J/mol/K) SfA Melting entropy of the pure alloy component A (J/mol/K) S f Entropy of fusion per atom (J/at./K) S f Entropy of fusion per volume unit (J/m3/K) Si (= Si⫺SL) (J/mol/K) SLS Difference of the entropies of liquid and solid phases per mol (J/mol/K) Smag Magnetic contribution of the difference of the entropies of liquid and solid phases per atom (J/at./K) Svib Vibrational contribution to the entropy of fusion per atom (J/at./K) T = TL⫺TN: undercooling (K) Tav Average of undercooling (K) Tc Critical undercooling for the transition from facetted to non-facetted growth in semiconductors (K) Thom Undercooling predicted by homogeneous nucleation theory (K)

Emissivity

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1

2

c and k

te e ϑ

d

i

S

T B k L (n) o S v1 v2 v b c i T el L N S c

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Constant in Mie potentials Constant in Mie potentials Strengths of crystalline anisotropy Total hemispherical emissivity Viscosity (N s/m2), order parameter Equilibrium long-range order parameter Wetting angle (degree) Scattering angle (degree) Thermal conductivity (W/cm K) Wavelength (m) Distance over which an atom must move to join the crystal (m) Critical wavelength for the Mullins and Sekerka instability (m) Interlamellar spacing (m) Coupling parameter of phase and temperature fields Facet length (m) Chemical potential (J/mol) Bohr’s magneton (9.27410⫻10⫺24 J/T) Kinetic growth coefficient (m/s/K) Chemical potential of the liquid phase (J/mol) Anisotropic kinetic function Magnetic permeability Chemical potential of the solid phase (J/mol) Frequency (Hz) Exponent of the repulsive term in Mie potentials Exponent of the attractive term in Mie potentials Exponent in the power law of the viscosity (Eq. (2.35)) Factor denoting the reduction of the number of nucleation sites in the case of heterogeneous nucleation Correlation length in the liquid Function of chemical stability Factor denoting the reduction of nucleation sites of species i in the case of heterogeneous nucleation Function of thermal stability Mass density (kg/m3) Electrical resistivity ( cm) Mass density of liquid phase (kg/m3) Density of particles (at./Å3) Mass density of solid (kg/m3) Scattering cross-section (barn) Coherent scattering cross-section (barn)

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i SB el * L D df mf φ c T

423

Incoherent scattering cross-section (barn) Stefan–Boltzmann constant (= 5.67⫻10⫺12) (W/cm2 K4) Electrical conductivity (⫺1 m⫺1) Stability parameter Phase field Phase field variable and grand potential (J/mol) Grand potential of the liquid (J/mol) Magnetic susceptibility Transient (s) Relaxation time for the diffusion flux to its steady state value (s) Relaxation time of density fluctuations (s) Angular frequency (Hz) Modulation frequency of an alternating current Volume term of the free enthalpy of a cluster in the undercooled melt (J) Coulomb potential (J) Surface term of the free enthalpy of a cluster in the undercooled melt (J) Atomic volume Concentration gradient Thermal gradient


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Index Suffix ‘f’ after the page number indicates a reference to a figure or table caption. in space 47–52, see also TEMPUS instrument through levitation, in high undercoolings 20–47 Continuous growth model (CGM) 209 Convective flow and solute diffusion effect 265–270, see also Lipton–Kurz–Trivedi (LKT) model interfacial crystal structures versus undercoolings, morphological spectrum 267 dendrite tip radius as a function of undercooling 269f Cooperative growth in undercooled polyphase alloys 283–310 eutectic growth theory 283–303, see also Eutectic growth theory stable and metastable monotectic alloys 303–310, see also Stable and metastable monotectic alloys Coupled zone concept, in eutectic solidification 293–295 symmetrical zone 293f skewed zone 293f Crank–Nicholson finite-difference solution 252 Crystal growth in undercooled melts 197–273 Aziz model 207–208 Broughton–Gilmer–Jackson (BGJ) model, comparison 202f collision-limited model 201 convective flow and solute diffusion effect 265–270 crystal growth velocity V as a function of undercooling 257f departures of local equilibrium 204–216 energy density, governing equation 245 experimental data and model predictions 256–273 faceted to non-faceted growth transition 247–256, see also separate entry

Acoustic levitation 20–26 acoustic field, physical parameters of 23f Bitter electromagnet for 31f dendrite growth at small and medium undercoolings 40 dendrite growth dynamics in 41f electromagnetic levitation 33–42, see also Electromagnetic levitation technique electrostatic levitation 42–47 graphite in 30 high-speed digital camera technique in 40 levitation by stationary magnetic fields 26–33 levitation of diamagnetic bodies, advantage 31 magnetic fields to levitate 30 Nd᎐Fe᎐B hard magnet 33f single-axis acoustic levitator 25f Aggregation process 68 Årrhenius law 197 Avrami analysis 170, 381 Becker–Döring theory 147f, 148 Bessel function 211 Bhatia–Thornton formalism 78, 89–92 Binodal liquid–liquid phase separation 386 Bitter electromagnet 31f Bohr magneton 166 Boltzmann statistics 147 Brillouin function 166 Broughton–Gilmer–Jackson (BGJ) model 202f, 205 Cahn and Hillard equation 250 CALPHAD methods 164 Capacitance proximity sensor (CPS) 40–41 Clausius–Claperyon relation 200, 341 Containerless processing acoustic levitation 20–26, see also separate entry in reduced gravity 377–378

425

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Crystal growth in undercooled melts – continued Fickian description 210 first experiments 256–258 fluctuation methods 201 free solidification technique 200 interatomic potential method 200 interface temperature–velocity relationship 213, 215f intermetallic compounds, measurements on 261–263 kinetic mechanisms of crystal growth, models 199f Lennard–Jones MD simulations 200 local non-equilibrium on rapid dendritic growth, influence of 270–273 Mikheev–Chernov model 204 Ni᎐B and Ni᎐Zr alloys, measurements on 259–261 phase-field model via thin-interface analysis 246 phase-field model 244–247 pure nickel, measurements on 258–259 Rutherford-backscattering experiments 207 semiconductors, measurements on 263–265 sharp-interface model 227–244, see also separate entry solid–liquid interface, advancement, kinetics 197–204 solute partitioning versus interface velocity 213f stability analysis 216–227, see also under stability analysis theory of crystal growth 204 theory of solidification 204 Crystal nucleation, magnetic contributions to 165–169 solid–liquid interfacial energy, magnetic contribution to 167–169 to the driving force for crystal nucleation 165–167 Crystallization of an undercooled melt 317 Debye–Waller factor 85 Dendrite growth theory/Dendritic growth 263, 283 dendrite breakup model 369

dendrite solidification 235 local non-equilibrium on 270–273 velocity–undercooling relationship 271 Density functional theory (DFT) 117, 122 Diffusion 198 atomic diffusion speed 206, 216, 222, 235, 365 atomic diffusivity 199 bulk diffusion speed 210–214, 236, 238, 271, 301 concentrational profile due to diffusion 210, 211 diffusion jumps of atoms 199 diffusion-limited growth 238, 272 diffusion relaxation effect 210 diffusionless solidification 208, 236, 239, 240, 273 diffusionless phase transition 208, 236, 303 interfacial diffusion speed 206, 212–214, 260, 270 local non-equilibrium solute diffusion 210–212, 223, 236 solute diffusion 235, 240, 265 relaxation of the diffusion flux 210 Diffuse interface theory 133–138 density wave approach 135 diffuse-interface region, physical interpretation 133–136 formalism of diffuse interface 133 Hele–Shaw type models 134 Landau theory of phase transitions 133 Mobility of the 246 of nucleation 154–160 phase-field model for nucleation 136 sharp interface versus diffuse interface 136–138 sharp-interface model 133 Disorder trapping 261–263 Dodecahedron 84f Drop tubes experiments/techniques 379–389 in high undercoolings 13–19, see also Short- and long drop-tube processing kinetics of phase selection 380–382 liquid–liquid phase separation 386–389 microstructure development 382–386 nucleation studies on glass-forming systems 379–380

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Index partitionless solidification 384 solidification pathways 384f Dubey and Ramachandrarao model

65

Earnshaw’s theorem 27, 31 Einstein–Stokes relation 150–151 Electromagnetic levitation technique 33–47, see also Photo-electronic effect application 34 conical levitation coil 35f Electro-Magnetic Levitator (EML) 402 for containerless undercooling and solidification experiments 38f for mass density and thermal expansion measurements 39 for terrestrial operation 43 for undercooling experiments 45 limitations 48, 377 photodiodes in 40 photo-sensing detector (PS) for 46f principle 33 Electromagnetic processing in reduced gravity 389–403 ac-calorimetry 392–394 dendrite growth velocities, measurement 401–403 electrical resistivity 390–392 glass-forming metals, thermal expansion of liquid alloys of 390f nucleation investigations and phase selection 397–401 specific heat and thermal conductivity 392–394 surface tension and viscosity 394–397 thermal expansion 389–390 thermophysical properties 389–397 Electrostatic levitator (ESL) 397 Eutectic alloys 295, see also Skeletal crystals Al᎐AlCu eutectics 297f Bi᎐MnBi 297 CBr4᎐C2Cl6, eutectic cells in 299f Co74.5Sb25.5 alloy undercoolings, eutectic growth morphologies at 300f globular eutectics 295f InSb᎐NiSb systems 297 lamellar pattern 295f, 296f

427

needle-like eutectics 295f, 297 phase diagram 295f rod structure 295f Eutectic growth theory 283–294, see also Dendritic growth; Eutectic alloys; Pearlite growth as a diffusion-controlled process 283 binary eutectic system, possible microstructures 294 Co᎐Sb eutectic alloy 289–290f eutectic colony solidification 299f eutectic morphology transition 294–303 eutectic phase diagram and growth model 284f eutectic solidification, coupled zone concept in 293, see also Coupled zone eutectic transformations 301 extended eutectic growth model 292 in the Ag᎐Cu system 292–293 interface shape 284f interface undercooling 284f interlamellar spacing versus growth velocity in 293 irregular or anomalous eutectic structure 283 Jackson–Hunt model 283–288, see also Jackson–Hunt eutectic growth theory model microstructures classification 283 regular lamellar structure 283, 295 regular rod structure 283, 295 solute distribution profiles 284f supersaturated solid solution transition to eutectics 301f Experiments in reduced gravity 377–403, see also TEMPUS instrument containerless processing in reduced gravity 377–378 drop tubes, experiments in 379–389, see also separate entry electromagnetic processing in reduced gravity 389–403, see also separate entry Faber–Ziman formalism 78, 89–94 Faceted to non-faceted growth transition, and crystal growth in undercooled melts 247–256

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Faceted to non-faceted growth transition, and crystal growth in undercooled melts – continued atomic structure of crystal faces for different mechanisms of growth 249f crystal growth mechanisms 248 growth velocity versus undercooling 253f needle dendrite growth 256f truncated dendrite branches 255 velocity–undercooling relationship 251f, 252 Fickian description 210 Fluxing method 340 Fourier‘s law 210 Frank–Kasper phase 127, 333 Gauss’ theorem 26 Gibbs–Thomson effect/equation 124, 219, 221, 239, 244 Ginzburg–Landau equation 137, 244, 248 Glass-forming systems 335–339 nucleation studies on 379–380 Grain-refined materials 339–354 calculated dendrite breakup time versus undercooling 352f cavitation model in 342 coarsening models 342–345 copious nucleation in 347 Cu70Ni30 and Cu69Ni30B1 alloys 343–334f, 347 dendrite breakup model 349f, 352–353 dendrite growth velocities of 344 drop-tube experiments in 346 electromagnetic levitation technique for 342 equiaxed microstructure in 343 grain diameter as a function of undercooling 341 grain size as a function of undercooling 348f in Ag samples 341–342 in Fe- and Ni-based alloys 342 lower limit of grain diameter as a function of undercooling 340f microstructures as a function of undercooling 347f, 348 of O2-doped Cu and Ag samples 342 physical limit of 339

refining by pressure pulse 341 solidification of undercooled melts

340

Heisenberg model 97 Hele–Shaw type models 134 Heterogeneous nucleation 11 isolation methods 12 isolation methods, principle 12 High undercoolings, experimental approach to 9–52 bulk melts by melt fluxing processing 19–20 containerless processing in space 47–52 containerless processing through levitation 20–47, see also separate entry droplet dispersion and emulsification method 11–13 short- and long drop-tube processing 13–19, see also individual entry transient and stationary undercooling 9–11 High-speed digital camera technique 40 Hoffmann’s model 61 Hypercooling limit 319 Intermetallic compounds, measurements on 261–263 International Space Station (ISS) 378 Isenthalpic or hypercooling 320 Isothermal approximation in solidification 237 Ivantsov solution 229 Jackson–Hunt eutectic growth theory model 283–296, 302f, 303 assumptions 290–291 Kelvin’s work 52 Kinetics of solidification 209 Kurz–Giovanola–Trivedi (KGT) model

365

Landé’s g-factor 166–167 Laser annealing 364 Laser rapid solidification 361 Laser surface resolidification processing 366f Lennard–Jones liquid 72–74, 200, 202 Lenz’s rule 29 Levitation technique and MSM 372 Lipton–Kurz–Trivedi (LKT) model 265 Local non-equilibrium solidification 238, 369

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Index Magnetic ordering in liquid state 97–103, see also muon spin rotation (SR) technique BCS-like mechanism for 97 Curie temperature determination 98–100 Curie–Weiss behaviour of the liquid metallic magnet 102 Faraday balance usage in 99–100 Heisenberg model 97 magnetic long-range order 97 temperature–time profile as measured on levitation 98f Marangoni convective flow 305 Maxwell–Cattaneo equation/model 210, 212 Maxwell–Cattaneo–Vernotte equation 210 Melt fluxing technique 19 Melt spinning 9 Metallic glasses 335–339 eutectic phase diagram of 336f nucleation phenomena in 336 temperature–time nucleation plot 337f, 338f, 339f undercooling measurements of 338 Metallic melts, short-range order experiments on 79–95 chemical short-range for Al13(Co,Fe)4 melts 94 Dodecahedron 84f in alloys 88 liquid Si, short-range order of 95–97 Percus–Yevick equation 79, 80 x-ray absorption spectroscopy 81–82 Zr melts, simulations of the structure factors of, parameters used 86f, 87f Metastable crystalline phases, formation 323–333 analysis using classical nucleation theory 333 containerless processing technique fro 333 Fe69Cr31–xNix ternary alloy, pseudobinary phase diagram 327f Fe᎐Ni᎐Cr, metastable bcc phase in 327–328 Frank–Kasper phase 333 in Fe᎐Ni alloy 328 interpretation 329 metastable ferromagnetic bcc phase 323–325

429

Ni᎐V, phase selection diagram 332, 332f of ternary alloy Fe᎐Ni᎐Cr 326 spontaneous nucleation of a metastable phase 324 temperature–time profile 325–326 XRD and EDXD pattern 329–331 and metastable phases 323 Metastable solid states and phases 317–354, see also Grain-refined materials; Metastable crystalline phases; Metallic glasses; Phase selection through the solidification kinetics; Supersaturated solid solutions formation, conditions 317–320 Microstructure selection maps (MSM) 361–373 for the Al᎐Al2Cu alloy 364f from small to high solidification velocities 369f Ni᎐Cu droplets, microstructure predominance map of 372f Ni᎐Cu droplets, microstructure selection map modelled for 371f Ni᎐Cu system, microstructure selection map for 370f selection by droplet size 370–373 selection by rapid cooling 361–367 selection by undercooling 367–370 Mikheev–Chernov model 204 Molecular dynamics and density functional theory, of solid–liquid interface 121–124 interfacial energies for different crystal surfaces 122 -factors, comparison 123–126 Monotectic alloys, see under Stable and metastable monotectic alloys Monotectics 386 Monotectic solidification 305 Muon spin rotation (SR) technique 100–102 Navier–Stokes equation 266 Negentropic model 116 Spaepen and Thompson negentropic model 117–121 -factor 120–126 Non-linear acoustic theory 20–21

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Nucleation theories 145–160, see also Crystal nucleation; Undercooling and nucleation diffuse interface theory of nucleation 154–160 diffusion-controlled addition of single atoms to the nucleus, double-wall potential for 149f heterogeneous nucleation 152–154 heterogeneous nucleus growth onto substrate 153 homogeneous nucleation 145–152 homogeneous nucleation, formalism for the derivation of 153 interfacial energy 156–159 nucleation rate equation 154 temperature dependence of the nucleation rate 150f Tolman length for 156 undercooled melt, solid-like cluster in 146f undercooled melt, spherical solid cluster in 155 Nucleation 145–189, see also Nucleation theories and phase selection, investigations 397–401 homogeneous versus heterogeneous nucleation 169–170 in alloys 163–165, see also Richard’s rule; CALPHAD methods magnetic contributions to crystal nucleation 165–169, see also under Crystal nucleation nucleation theories 145–160, see also separate entry on glass-forming systems 379–380 statistics of 161–163 transient nucleation 160–161 undercooling and nucleation, experimental results 169–189, see also separate entry Nusselt number 16 Partitionless solidification 320 Pearlite growth 283 Péclet number 239–240, 286, 291

Percus–Yevick equation 79, 80 Peritectic alloys 307–310 constitutional undercooling criterion in 309 nucleation in 309 phase selection in 309–310, 310f solidification 309 and phase 307–309 Phase selection through the solidification kinetics 333–335 temperature–time nucleation (TTN) diagrams 334 Photo-electronic effect 44 Photo-sensing detector (PS) 46f Planck’s law 37, 47 Postrecalescence phase 319 Prandtl number 16 Quasi-stationary cluster distribution function 160–161 Quenching 9 Rapid cooling 9 Rapid quenching techniques 9 advantage 9 disadvantages 9 transient undercooling by 10–11 Rapid solidification 206, 216, 235, 242 conditions, process characteristics 290 undercooled intermetallic alloys 261 Rayleigh’s formula 51, 394 Recalescence 317, 342 Reynold number 16 Richard’s rule 164 Richardson–Dushman equation 44 Segregation-free crystallization 321f Semiconductors, measurements on 263–265 Sharp-interface model 227–244 anisotropic dendrite 233f cellular-dendritic pattern 237 crystal morphology change 237 dendrite tip, model equations 238 growth in a pure system 227–234 marginal stability hypothesis and microscopic solvability 232f solidification in a binary system 234–241

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Index superlattice structures in intermetallics 241–244 Short- and long drop-tube processing, in high undercoolings 13–19 DLR drop tube 14f flux immersion 19f lase annealing 19f levitation 19f long drop tubes 18–19 melt spinning 19f metallic melts undercool methods 19f rapid cooling 19f short drop tubes 13–18 slow cooling 19f Skeletal crystals 297 sectorial eutectic colony 298f skeletal eutectic colony 298f Solidification processes pattern 10f Solid–liquid interface 115–138 definition 119 diffuse interface theory 133–138, see also separate entry energy of, theoretical and experimental investigations, 115 Frank–Kasper phase 127 Gibbs–Thomson effect 124–126 icosahedral quasicrystalline I-phase 132 icosahedral short-range order 115 interface atoms, not defined location 128 interface atoms, not occupied location 128 interface atoms, occupied location 128 interface atoms, potential locations of 128 interfacial energy between structurally complex solids and their melts 126–132 interfacial energy under local equilibrium conditions 116–132, see also negentropic model interfacial energy under local equilibrium conditions, experimental results on 124–126 interfacial energy, numerical calculations 128 interfacial layer density, for fcc, hcp and bcc phases 132 interfacial layer, construction 130f, 131f investigations, using molecular dynamics and density functional theory 121–124

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I-phase 128f polytetrahedral crystalline phases at 115 structural order at the interface 115–116 Solute trapping 207, 209, 320, 321f Solute-drag effect 214 Space frustration 73 Spaepen and Thompson negentropic model 117–121 assumption 117 Splat cooling 9 Spray methods, in MSM 372 Stability analysis, crystal growth in undercooled melts 216–227 constitutional undercooling, heat and mass transport concept 219 heat and mass redistribution, quantitative description 218 interfacial absolute stability, model predictions 226f marginal stability for the wavelength of perturbation 218 marginal stability hypothesis 224, 231 morphological stability, theory of 218 solidification of binary systems, morphological diagram 217f two-dimensional dendrite 217f Stable and metastable monotectic alloys 303–310, see also Marangoni convective flow; Partitionless solidification 208, 209 Peritectic alloys monotectic solidification, types of 305, 305f phase diagram with monotectic reaction 304f phase-field modelling of 305 solidification microstructures 305, 306f solidification of 305 solidifying microstructure with monotectic reaction 304f ᎐Ni phase 306, 307f Stefan–Boltzmann law 15, 37, 51 Superlattice structures in intermetallics 241–244 intermetallic phases, mechanical properties 242 Supersaturated solid solutions 320–323 solidification, theoretical analysis 323

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Temkin’s model 208 Temperature–time nucleation (TTN) diagrams 334–339, 334f Temperature–time–transformation (TTT) behaviour 380–381 TEMPUS instrument 378, 391, 392, 395 in containerless processing in space 48–52 Theory of crystal growth 204 Theory of solidification 204 Thermal fluctuations 234 Thermodynamics of undercooled liquids 59–67 classical irreversible 247 differential scanning calorimetry (DSC) 63 differential thermal analysis (DTA) 63 extended irreversible 214, 247 Gibbs free energy expressions 59–65 specific heat as a function of undercooling measurements 63–65 Trivedi–Magnin–Kurz (TMK) model 302f Ultra-high vacuum (UHV) technique 13 Ultrasonic levitators 20 Undercooled liquids, physics 59–104 kinetic and transport properties 103–104, see also Vogel–Fulcher–Tammann expression 103 magnetic ordering in liquid state 97–103, see also separate entry methods to obtain 9 thermodynamics 59–67, see also under Thermodynamics, of undercooled liquids undercooled melts, structural ordering in 67–97, see also separate entry Undercooled melts, structural ordering in 67–97, see also Space frustration aerodynamic levitation in 75 atoms arranged in fivefold symmetry 69f based on icosahedral aggregates 69–73 Bernal’s model form polytetrahedral aggregates 71 Bhatia–Thornton formalism 78, 89 Faber–Ziman formalism 78 icosahedral short-range order in 71, 75 icosahedron consisting of 13 metallic melts, short-range order experiments on 79–95, see also separate entry

molecular dynamics calculation 74 Pentagonal bipyramid model 72f quasicrystalline phases 72–74 scattering theory 76–78 short-range order in 67 short-range order in, development of 69f short-range order in, models 68–76 Si-melts, structure factor of 96f special matching rules 72 Undercooling and nucleation, experimental results 169–189 droplet dispersion technique 170 electromagnetic levitation technique 170 homogeneous nucleation, idealized assumption 178 homogeneous versus heterogeneous nucleation 169–170 nucleation behaviour, structural dependence 172–180 nucleation in undercooled melts 170–172 of quasicrystalline phases 178 of Zr 172 on Ni-rich FeNi alloys 172 on pure metals 173f polytetrahedral phases during solidification 179 quasicrystalline phases, investigations 174 solid–liquid interfacial energy, numerical modelling of 174 temperature-time transformation diagram for 174, 176f undercooling of magnetic melts 180–189, see also separate entry Undercooling of magnetic melts 180–189 Co᎐Pd melts 180 Co-rich side of the Co᎐Pd phase diagram 185f single-phase crystallization 182 statistical analysis 187 two-phase crystallization process 182 Van der Waals/Cahn–Hilliard theory 156 Vogel–Fulcher–Tammann expression 103–104 Volmer–Weber theory 147f, 148 Wilson–Frenkel model

200

Metastable Solids from Undercooled Melts

Metastable Liquids

Metastable liquids

Solids far from Equilibrium

Snow Melts in Spring

Snow Melts in Spring

Industrial Crystallization of Melts

Solids far from equilibrium: growth, morphology, defects

Thermal Decomposition of Solids and Melts: New Thermochemical Approach to the Mechanism, Kinetics and Methodology

Snow Melts in Spring

Snow Melts in Spring

Snow Melts in Spring

Snow Melts in Spring

Studing thermodynamics of metastable states

Snow Melts in Spring

Snow Melts in Spring

Quantum Coherence: From Quarks to Solids

Murder Melts in Your Mouth

Murder Melts in Your Mouth

Murder Melts in Your Mouth

Properties of Glass-Forming Melts

Murder Melts in Your Mouth

Electrodynamics of Solids

Defects in solids

Diffusion in Solids

Engineering Mechanics of Solids

Quantum Theory of Solids

Mechanics of viscoelastic solids

Chemical Bonding in Solids

Thermoluminescence of solids

Molecular Organic Materials: From Molecules to Crystalline Solids

Metastable Solids from Undercooled Melts

Metastable Liquids

Metastable liquids

Solids far from Equilibrium

Snow Melts in Spring

Snow Melts in Spring

Industrial Crystallization of Melts

Solids far from equilibrium: growth, morphology, defects

Thermal Decomposition of Solids and Melts: New Thermochemical Approach to the Mechanism, Kinetics and Methodology

Snow Melts in Spring

Snow Melts in Spring

Snow Melts in Spring

Snow Melts in Spring

Studing thermodynamics of metastable states

Snow Melts in Spring

Snow Melts in Spring

Quantum Coherence: From Quarks to Solids

Murder Melts in Your Mouth

Murder Melts in Your Mouth

Murder Melts in Your Mouth

Properties of Glass-Forming Melts

Murder Melts in Your Mouth

Electrodynamics of Solids

Defects in solids

Diffusion in Solids

Engineering Mechanics of Solids

Quantum Theory of Solids

Mechanics of viscoelastic solids

Chemical Bonding in Solids

Thermoluminescence of solids

Molecular Organic Materials: From Molecules to Crystalline Solids

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