Solidification of Containerless Undercooled Melts E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 996

Veröffentlichungsjahr: 2012

Beschreibung

All metallic materials are prepared from the liquid state as their parent phase. Solidification is therefore one of the most important phase transformation in daily human life. Solidification is the transition from liquid to solid state of matter. The conditions under which material is transformed determines the physical and chemical properties of the as-solidified body. The processes involved, like nucleation and crystal growth, are governed by heat and mass transport.
Convection and undercooling provide additional processing parameters to tune the solidification process and to control solid material performance from the very beginning of the production chain.
To develop a predictive capability for efficient materials production the processes involved in solidification have to be understood in detail.
This book provides a comprehensive overview of the solidification of metallic melts processed and undercooled in a containerless manner
by drop tube, electromagnetic and electrostatic levitation, and experiments in reduced gravity.
The experiments are accompanied by model calculations on the influence of thermodynamic and hydrodynamic conditions that control
selection of nucleation mechanisms and modify crystal growth development throughout the solidification process.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Preface

List of Contributors

Chapter 1: Containerless Undercooling of Drops and Droplets

1.1 Introduction

1.2 Drop Tubes

1.3 Containerless Processing Through Levitation

1.4 Summary and Conclusions

Acknowledgments

References

Chapter 2: Computer-Aided Experiments in Containerless Processing of Materials

2.1 Introduction

2.2 Planning Experiments

2.3 Operating Experiments

2.4 Data Reduction, Analysis, Visualization, and Interpretation

2.5 Conclusion

Acknowledgments

References

Chapter 3: Demixing of Cu–Co Alloys Showing a Metastable Miscibility Gap

3.1 Introduction

3.2 Mechanism of Demixing

3.3 Demixing Experiments in Terrestrial EML and in Low Gravity

3.4 Demixing Experiments in a Drop Tube

3.5 Spinodal Decomposition in Cu–Co Melts

3.6 Conclusions

Acknowledgments

References

Chapter 4: Short-Range Order in Undercooled Melts

4.1 Introduction

4.2 Experiments on the Short-Range Order of Undercooled Melts

4.3 Conclusions

Acknowledgments

References

Chapter 5: Ordering and Crystal Nucleation in Undercooled Melts

5.1 Introduction

5.2 Nucleation Theory—Some Background

5.3 Liquid Metal Undercooling Studies

5.4 Coupling of Ordering in the Liquid to the Nucleation Barrier

5.5 Conclusions

Acknowledgments

References

Chapter 6: Phase-Field Crystal Modeling of Homogeneous and Heterogeneous Crystal Nucleation

6.1 Introduction

6.2 Phase-Field Crystal Models

6.3 Homogeneous Nucleation

6.4 PFC Modeling of Heterogeneous NuCleation

6.5 Summary

Acknowledgments

References

Chapter 7: Effects of Transient Heat and Mass Transfer on Competitive Nucleation and Phase Selection in Drop Tube Processing of Multicomponent Alloys

7.1 Introduction

7.2 Model

7.3 Effect of Transient Heat and Mass Transfer on Nucleation and Crystal Growth

7.4 Competitive Nucleation and Phase Selection in Nd–Fe–B Droplets

7.5 Summary

Appendix 7.A: Extended Model of Nonstationary Heterogeneous Nucleation

References

Chapter 8: Containerless Solidification of Magnetic Materials Using the ISAS/JAXA 26-Meter Drop Tube

8.1 Introduction

8.2 Drop Tube Process

8.3 Undercooling Solidification of Fe–Rare Earth (RE) Magnetostriction Alloys

8.4 Undercooling Solidification of Nd–Fe–B Magnet Alloys

8.5 Concluding Remarks

Acknowledgment

References

Chapter 9: Nucleation and Solidification Kinetics of Metastable Phases in Undercooled Melts

9.1 Introduction

9.2 Thermodynamic Aspects and Nucleation of Metastable Phases

9.3 Metastable Phase Formation from Undercooled Melts in Various Alloy Systems

9.4 Summary and Conclusions

References

Chapter 10: Nucleation Within the Mushy Zone

10.1 Introduction

10.2 Incubation Time

10.3 Cluster Formation

10.4 Transient Development of Heterogeneous Sites

10.5 Comparing Critical Nucleus Development Mechanisms

10.6 Concluding Remarks

Acknowledgments

References

Chapter 11: Measurements of Crystal Growth Velocities in Undercooled Melts of Metals

11.1 Introduction

11.2 Experimental Methods

11.3 Summary and Conclusions

Acknowledgments

References

Chapter 12: Containerless Crystallization of Semiconductors

12.1 Introduction

12.2 Status of Research on Facetted Dendrite Growth

12.3 Twin-Related Lateral Growth and Twin-free Continuous Growth

12.4 Containerless Crystallization of Si [33]

12.5 Summery and Conclusion

12.6 Appendix 12.A: LKT Model

12.A.1 Wilson–Frenkel Model

References

Chapter 13: Measurements of Crystal Growth Dynamics in Glass-Fluxed Melts

13.1 Introduction

13.2 Methods and Experimental Set-Up

13.3 Growth Velocities in Pure Ni

13.4 Growth Velocities in Ni3Sn2 Compound

13.5 Crystal Growth Dynamics in Ni–Sn Eutectic Alloys

13.6 Opportunities with High Magnetic Fields

13.7 Summary

Acknowledgments

References

Chapter 14: Influence of Convection on Dendrite Growth by the AC+DC Levitation Technique

14.1 Convection in a Levitated Melt

14.2 Static Levitation Using the Alternating and Static Magnetic Field (AC + DC Levitation)

14.3 Effect of Convection on Nucleation and Solidification

References

Chapter 15: Modeling the Fluid Dynamics and Dendritic Solidification in EM-Levitated Alloy Melts

15.1 Introduction

15.2 Mathematical Models for Levitation Thermofluid Dynamics

15.3 Thermoelectric Magnetohydrodynamics in Levitated Droplets

15.4 Concluding Remarks

Acknowledgments

References

Chapter 16: Forced Flow Effect on Dendritic Growth Kinetics in a Binary Nonisothermal System

16.1 Introduction

16.2 Convective Flow in Droplets Processed in Electromagnetic Levitation

16.3 The Model Equations

16.4 Predictions of the Model

16.5 Quantitative Evaluations

16.6 Summary and Conclusions

Acknowledgments

References

Chapter 17: Atomistic Simulations of Solute Trapping and Solute Drag

17.1 Introduction

17.2 Models of Solute Trapping

17.3 Solute Drag

17.4 MD Simulations

17.5 Implications for Dendrite Growth

Acknowledgments

References

Chapter 18: Particle-Based Computer Simulation of Crystal Nucleation and Growth Kinetics in Undercooled Melts

18.1 Introduction

18.2 Solid–Liquid Interfaces in Nickel

18.3 Homogeneous Nucleation in Nickel

18.4 Crystal Growth

18.5 Conclusions

Acknowledgments

References

Chapter 19: Solidification Modeling: From Electromagnetic Levitation to Atomization Processing

19.1 Introduction

19.2 Electromagnetic Levitation

19.3 Impulse Atomization

19.4 Modeling

19.5 EML Sample

19.6 IA Particles

19.7 Conclusion

Acknowledgments

References

Chapter 20: Properties of p-Si-Ge Thermoelectrical Material Solidified from Undercooled Melt Levitated by Simultaneous Imposition of Static and Alternating Magnetic Fields

20.1 Introduction

20.2 Simultaneous Imposition of Static and Alternating Magnetic Fields

20.3 Experimental

20.4 Results and Discussion

20.5 Summary and Conclusions

References

Chapter 21: Quantitative Analysis of Alloy Structures Solidified Under Limited Diffusion Conditions

21.1 The Need for an Instrumented Drop Tube

21.2 Description of IA

21.3 Powder Characteristics

21.4 Quantification of Microstructure

21.5 Modeling

References

Chapter 22: Coupled Growth Structures in Univariant and Invariant Eutectic Solidification

22.1 Introduction

22.2 Historical Perspective and Background

22.3 Basic Theory of Eutectic Solidification

22.4 Eutectic Solidification Theory for Ternary Systems

22.5 Solidification Paths and Competitive Growth Considerations

22.6 Recent Developments, Emerging Issues, and Critical Research Needs

Acknowledgments

References

Chapter 23: Solidification of Peritectic Alloys

23.1 Introduction

23.2 Peritectic Equilibrium and Transformation

23.3 Peritectic Reactions in the Ternary System

23.4 Nucleation Studies

23.5 Growth

23.6 Conclusions

References

Index

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Preface

Metallic materials are prepared from the liquid state as their parent phase. The conditions under which the liquid solidifies determine the physical and chemical properties of the as-solidified material. In most cases time and energy consuming post-solidification treatment of the material is mandatory to obtain the final product with its desired properties and design performance. Therefore, efforts are directed towards virtual material design with computer assisted modelling. This can shorten the entire production chain - ranging from casting the shaped solid from the melt to the final tuning of the product in order to save costs during the production process. The goal is to fabricate novel materials with improved properties for specific applications. To date, metal production is the largest industry worldwide. In the European Union there are 417 700 enterprises with 5.1 Million employees. They correspond to 3.9% of the entire workforce and produce 244.4 billion EUR added value each year (European business – Facts and figures Eurostat 2007). Therefore, even small improvements in production efficiency for the metal industry may lead to large economic gains.

Computational materials science from the liquid state requires thermo-physical parameters measured with high accuracy and detailed knowledge of the physical mechanisms involved in the solidification process. In particular, these are crystal nucleation and crystal growth. Both of these processes are driven by an undercooling of the liquid below its equilibrium melting temperature to develop conditions where a driving force for the advancement of a solidification front is created. This gives access to non-equilibrium solidification pathways which can form metastable solids which may differ in their physical and chemical properties from their stable counterparts. Detailed modelling of solidification, both near equilibrium and far away from thermodynamic equilibrium, requires that the solidification process must be investigated in every detail.

In order to achieve the state of an undercooled melt, it is advantageous to remove heterogeneous nucleation sites which otherwise limit the undercoolability. The most efficient way to realize such conditions is containerless processing of melts. In such, the most dominant heterogeneous nucleation process, involving interaction with container walls, is completely avoided. Nowadays, electromagnetic and electrostatic levitation techniques have been developed for containerless undercooling and solidification of molten metals and alloys. A freely suspended drop gives the extra benefit to directly observe the solidification process by combining the levitation technique with proper diagnostic means. For instance, short range ordering as precursor of crystal nucleation has been investigated by using synchrotron radiation and neutron diffraction on containerless undercooled melts. Additionally, primary phase selection processes for rapid solidification of metastable phases has been observed in situ by energy dispersive X-ray radiation using synchrotron radiation of high intensity. Rapid growth of dendrites is observed on levitation undercooled melts by using video camera techniques characterized by high spatial and temporal resolution.

The application of containerless processing on Earth is limited since large levitation forces are needed to compensate for the gravitational force acting on the samples. The large levitation forces cause undesirable effects like externally induced stirring of the liquid or deformation of the liquid sample from sphere-like geometry. These are overcome when utilizing the special environment of reduced gravity. Here, the forces to compensate for g-jitter, the small random accelerations associated with spacecraft operation, are about three orders of magnitude less than levitation forces on Earth. Based upon such consideration a facility for containerless electro-magnetic processing in space called TEMPUS (Tiegelfreies Elektro-Magnetisches Prozessieren Unter Schwerelosigkeit) has been developed by DLR, the German Space Agency. It was constructed by the German aerospace industry and tested during several parabolic flight campaigns to demonstrate technical functionality. In a cooperation between DLR Space Agency and the US National Aeronautics and Space Administration (NASA), TEMPUS had its maiden flight under real space conditions on board the shuttle Columbia during the NASA Spacelab mission International Microgravity Laboratory IML 2 in 1994. The German – USA cooperation was handled on the principle “no exchange of funds” meaning that the facility was provided by DLR and flight opportunity was offered by NASA. The total experiment time during the 14 days mission was shared between US and German investigator teams. The mission was successful not only by demonstrating technical functionality of TEMPUS in Space but also obtaining interesting scientific results including high accuracy measurements of thermophysical properties and investigations of gravity related phenomena in solidification of undercooled metals and alloys. Later on, TEMPUS was flown again on Columbia during NASA spacelab missions Microgravity Space Laboratory 1 (MSL-1) and Microgravity Space Laboratory 1 reflight (MSL-1R) in 1997. A broad spectrum of science return from the TEMPUS spacelab missions are published in Materials and Fluids Under Microgravity, Lecture Notes in Physics, eds.: L. Ratke, H. Walter, B. Feuerbacher (1995) and Solidification 1999, Proceedings of symposia at the TMS Fall Meeting 1998, eds.: W. H. Hofmeister, J. R. Rogers, N.B. Singh, S. P. Marsh, P. W. Vorhees (1999).

At present, an advanced Electro-Magnetic Levitator (EML) facility is under development by a common effort between the DLR Space Agency and the European Space Agency ESA. The EML is constructed by ASTRIUM and is scheduled for accommodation on board the International Space Station ISS in 2013. Meanwhile, several international investigator teams of scientists from the member states of ESA, USA and Japan are preparing experiments dedicated to be performed in Space using the EML multi-user facility on the ISS. In parallel to the experimental work, modelling and theoretical evaluation of solidification processes are planned. In particular, understanding the importance of gravity-driven phenomena like changes in heat and mass transport by forced convection is a central part of these solidification investigations.

These developments both on the experimental and on the theoretical side stimulated the editors of the present book to collect the state of solidification research as far as it is directly correlates to solidification of containerless undercooled melts. These attempts were supported by our colleagues who contributed to the scientific content of the present book. We appreciate their efforts and cooperation in delivering high quality articles to this book. Most of the authors of the book are members of an international Topical Team on Containerless Undercooling and Solidification of Melts (SOL-EML) sponsored by the European Space Agency. We thank ESA for this support and for their vision to bring together experts in this field from all over the world - with membership coming from Europe, North America, Japan and also from China and India. In these latter countries, enormous efforts are undertaken at present to set up a materials science programme in space. In particular, German partners and colleagues benefitted from priority programmes focused on solidification research on undercooled melts in the Earth laboratory, which were financed by the German Research Foundation DFG. This support is greatly appreciated as well. Last but not least the editors are very grateful to Dr. Martin Graf from WILEY – VCH for pleasant and efficient cooperation during the entire course of preparing and editing the present book.

Dieter Herlach and Douglas Matson

List of Contributors

Mark Asta University of California Department of Materials Science and Engineering 210 Hearst Memorial Mining Building, Room 384 Berkeley, CA 94720 USA

Sven Binder Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

Krishanu Biswas Indian Institute of Technology Department of Materials Science and Engineering Faculty Building, Room 407 Kanpur 208016 India

Valdis Bojarevics University of Greenwich School of Computing and Mathematical Sciences Old Royal Naval College Park Row London SE10 9LS UK

Georg Ehlen Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

and

Institut für Festkörperphysik Ruhr-Universität Bochum Universitätsstraße 150 44780 Bochum Germany

Jan Fransaer University of Leuven Department of Metallurgy and Materials Engineering (MTM) Kasteelpark Arenberg 44 - box 2450 3001 Heverlee Belgium

Peter K. Galenko Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

and

Institut für Festkörperphysik Ruhr-Universität Bochum Universitätsstraße 150 44780 Bochum Germany

Charles-André Gandin MINES ParisTech CEMEF UMR 7635, CNRS 06904 Sophia Antipolis France

Jianrong Gao Northeastern University Key Laboratory of Electromagnetic Processing of Materials P.O. Box 314 3-11 Wenhua Road Shenyang 110004 China

László Gránásy BCAST Brunel University Uxbridge Middlesex UB8 3PH UK

and

Research Institute for Solid State Physics and Optics P.O. Box 49 1525 Budapest Hungary

Alain Lindsay Gree University of Cambridge Department of Materials Science and Metallurgy Pembroke Street Cambridge CB2 3QZ UK

Tsuyoshi Hamada Yokohama National University Graduate School of Environment and Information Sciences 79-7 Tokiwadai Hodogaya-ku Yokohama 240-8501 Japan

Hani Henein University of Alberta Department of Chemical and Materials Engineering 7th Floor, Electrical & Computer Engineering Research Facility (ECERF) 91017-116 Street Edmonton, Alberta T6G 2V4 Canada

Dieter M. Herlach Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

Dirk Holland-Moritz Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt (DLR) Linder Höhe 51170 Köln Germany

Jürgen Horbach Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt (DLR) Linder Höhe 51170 Köln Germany

and

Heinrich Heine-Universität Düsseldorf Institut für Theoretische Physik der Weichen Materie Universitätsstraß e 1 40225 Düsseldorf Germany

Jeff J. Hoyt McMaster University Department of Materials Science and Engineering 1280 Main Street Hamilton, Ontario L8S4LS Canada

Robert W. Hyers University of Massachusetts Mechanical and Industrial Engineering 160 Governors Drive Amherst, MA 01003-2210 USA

Arash L. Ilbagi University of Alberta Department of Chemical and Materials Engineering Department 7th Floor, Electrical & Computer Engineering Research Facility (ECERF) 91017-116 Street Edmonton, Alberta T6G 2V4 Canada

Yuko Inatomi Institute of Space and Astronautical Science, JAXA 3-1-1 Yoshinodai, Sagamihara Chuo-ku Kanagawa 229-8510 Japan

Andrew Kao University of Greenwich School of Computing and Mathematical Sciences Old Royal Naval College Park Row London SE10 9LS UK

Alain Karma Northeastern University Department of Physics 360 Huntington Ave Boston, MA 02115 USA

Kenneth F. Kelton Washington University in St. Louis Department of Physics One Brookings Drive St. Louis, MO 63130-4899 USA

Matthias Kolbe Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt (DLR) Linder Höhe 51147 Köln Germany

Mikhael Krivilyov Udmurt State University Department of Physics Laboratory of Condensed Matter Physics Universitetskaya 1 426034 Izhevsk Russia

Philipp Kuhn Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt (DLR) Linder Höhe 51170 Köln Germany

Kazuhiko Kuribayashi Japan Aerospace Exploration Agency The Institute of Space & Astronautical Science 3-1-1 Yoshinodai, Sagamihara Chuo-ku Kanagawa 229-8510 Japan

Wolfgang Löser Leibniz-Institut für Festkörper und Werkstoffforschung, IFW Helmholtzstraße 20 01069 Dresden Germany

Douglas M. Matson Tufts University Department of Mechanical Engineering 200 College Avenue Medford, MA 02155 USA

Hideaki Nagai National Institute of Advanced Industrial Science and Technology AIST Tsukuba Central 5 Tsukuba, Ibaraki 305-8565 Japan

Ralph E. Napolitano Iowa State University Division of Materials Science and Engineering 3273 Gilman Hall Ames, IA 50011-2300 USA

Takeshi Okutani Yokohama National University Graduate School of Environment and Information Sciences 79-7 Tokiwadai Hodogaya-ku Yokohama 240-8501 Japan

Shumpei Ozawa Tokyo Metropolitan University Department of Aerospace Engineering Hino-shi Tokyo 1901-065 Japan

Koulis Pericleous University of Greenwich School of Computing and Mathematical Sciences Old Royal Naval College Park Row London SE10 9LS UK

Tamás Pusztai Research Institute for Solid State Physics and Optics P.O. Box 49 1525 Budapest Hungary

Roberto E. Rozas Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt (DLR) Linder Höhe 51170 Köln Germany

and

Heinrich Heine-Universität Düsseldorf Institut für Theoretische Physik der Weichen Materie Universitätsstraß e 1 40225 Düsseldorf Germany

Sumanta Samal Indian Institute of Technology Department of Materials Science and Engineering Faculty Building Kanpur 208016 India

Olga Shuleshova Leibniz-Institut für Festkörper- und werkstoffforschung, IFW Helmholtzstr. 20 01069 Dresden Germany

György Tegze Institute for Solid State Physics and Optics Wigner Research Centre for Physics P.O. Box 49 1525 Budapest Hungary

Gyula I. Tóth Institute for Solid State Physics and Optics Wigner Research Centre for Physics P.O. Box 49 1525 Budapest Hungary

Damien Tourret MINES ParisTech CEMEF UMR 7635, CNRS 06904 Sophia Antipolis France

and

Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

Thomas Volkmann Institut für Materialphysik im Weltraum Deutsches Zentrum für Luft- und Raumfahrt Linder Höhe 51147 Köln Germany

Chao Yang Northeastern University Key Laboratory of Electromagnetic Processing of Materials P.O. Box 314 3-11 Wenhua Road Shenyang 110004 China

Hideyuki Yasuda Osaka University Graduate School of Engineering Department of Adaptive Machine Systems Suita Osaka 565-0871 Japan

Yikun Zhang Northeastern University Key Laboratory of Electromagnetic Processing of Materials P.O. Box 314 3-11 Wenhua Road Shenyang 110004 China

Zongning Zhang Northeastern University Key Laboratory of Electromagnetic Processing of Materials P.O. Box 314 3-11 Wenhua Road Shenyang 110004 China

Chapter 1

Containerless Undercooling of Drops and Droplets

Dieter M. Herlach

1.1 Introduction

Containerless processing of droplets has a long traditional experience. In his work Discorsi e Dimostrazioni Matematiche intorno a due nuove scienze published in 1639, Galileo Galilei describes experiments in which materials of different specific mass density were dropped down to ground from the leaning tower of Pisa to demonstrate that bodies of different mass fall with same velocity if friction in the air is neglected. In 1799, it was reported that a drop tower was used to produce lead shots by containerless solidification of liquid droplets during free fall. Liquid lead was pressed through a sieve at the top of the drop shaft to produce droplets of unique size, which solidified during free fall. The conditions of reduced gravity during free fall favored an ideal sphere-like geometry of the droplets upon solidification.

If a droplet is containerless solidified, often the liquid cools down below the equilibrium melting temperature prior to solidification. By using containerless processing methods, large undercoolings can be achieved since heterogeneous nucleation on container walls is completely avoided that is otherwise initiating crystallization of the melt. Nowadays, a great variety of techniques are applied for containerless undercooling. One distinguishes between drop tubes for containerless solidification of a spray of droplets, drop towers to process individual drops during free fall, and levitation techniques. Small drop tubes are quite suitable to study the statistics of phase and microstructure formation of particles on size less than 1 mm. The droplets are solidifying during free fall inside the drop tube. Thus, drop tubes are in house facilities to study solidification under reduced gravity conditions. For instance phase selection diagrams can be constructed such that they are describing the formation of competing phases in dependence of the droplet size, or the cooling rate since the droplet size directly correlates to the cooling rate [1]. Large drop tubes in height up to 150 m enable solidification of individual drops in size up to several millimeters. They are used to study the glass-forming ability of metallic alloys [2]. The temperature profile of drops falling under ultrahigh-vacuum conditions is recorded by a set of photodiodes arranged along the dropt tube. In such a way phase selection of refractory alloy systems is studied as a function of undercooling [3].

Drop towers and drop shafts are differing from drop tubes in such that experiment facilities are falling and samples can be studied under reduced gravity conditions for a period of 4.5 s at a falling distance of 150 m (drop tower in Bremen) and 9 s at falling distance of 500 m (drop shaft in Hokkaido), respectively. In all drop tubes and drop towers, it is difficult if not at all impossible to perform in situ diagnostics of solidification of metallic drops.

Levitation techniques offer the great potential not only to containerless undercool and solidify drops in size up to 10 mm but they can also be combined with proper diagnostic means and allow for even stimulate solidification of freely suspended drops externally at various undercooling levels. A simple quasilevitation technique was frequently used to undercool a liquid metal or alloy by embedding it into a denucleation agent. In such a way, contact to the solid container is avoided, and in most cases the melt fluxing agent removes heterogeneous motes on the surface of the molten drop [4]. However, this technique is limited by the need to avoid chemical reactions between fluxing agent and liquid metal. Electromagnetic levitation was developed for containerless undercooling and solidification of metallic systems. The eddy currents induced by an alternating, inhomogeneous electromagnetic field create a secondary field that is opposite to the primary one. Thus, the eddy currents will create a repulsive force. If a properly designed coil is used and the coil current is adjusted, the repulsive force compensates the gravitational force and the sample is electromagnetically levitated. The eddy currents induced by alternating electromagnetic field cause at the same time heating the sample. Coupling of levitation and heating gives the advantage that no extra heating source is required, however, leads to the disadvantage that temperature control is only possible in a range at elevated temperature since levitations needs a minimum power absorption to guarantee a freely suspended drop [5]. This boundary condition is circumvented by applying electrostatic levitation. Here, a sample in diameter of 2–3 mm is electrically charged up and levitated in a strong electrostatic field. In most cases a laser is used to heat the sample [6]. Whereas the electromagnetic levitation is a self-stabilizing method, the electrostatic levitation needs a sophisticated sample positioning and a real-time electrostatic field control, since the sample is always in an unstable position (Earnshow theorem). Other methods like aerodynamic and acoustic levitation are frequently used for organic substances and oxides. They are not favorable techniques to undercool high melting metals. On the one side, a liquid metal changes at high temperatures the local levitation conditions, and more seriously, some residual amounts of oxygen in the environmental processing gas leads to the formation of metal oxides at the surface of the metallic drop. Sine metal oxides are in most cases thermodynamically more stable than the parent metal, they act as heterogeneous nucleation sites and limit the accessible undercooling range. Therefore, these techniques are not further dealt with in the present book.

The special environment of reduced gravity during parabolic flight and in Space offers the great advantage that the forces to compensate disturbing accelerations are by orders of magnitude smaller than the force needed to compensate the gravitational force on Earth. Moreover, in case of electromagnetic processing the stirring of the melt due to the eddy currents are much reduced. The German Space Agency Deutsche Agentur für Raumfahrtangelegenheiten DARA, now Deutsches Zentrum für Luft- und Raumfahrt – Raumfahrtagentur (DLR Space Agency) – has developed an electromagnetic levitator for the use in reduced gravity. It applies a new technical concept such that two different frequency generators operating at different frequencies power a coil for positioning by a quadrupole field and, separately from that, a coil that produces a dipole field for efficient heating [7]. This concept was mandatory to develop a levitator for the usage in Space since it increased the efficiency in energy consumption of high-frequency generators for levitation from 1 to 2% (conventional high-frequency generators) to more than 30%. This device, called TEMPUS (German acronym for containerless processing in reduced gravity, Tiegelfreies Electro-Magnetisches Prozessieren Unter Schwerelosigkeit) was successfully tested in the realistic environment in Space by three NASA Spacelab missions, IML2 (1994), MSL1, and MSL1R (1997). At the same time very interesting results were obtained in measuring thermophysical properties of liquid metals and alloys even in the metastable regime of the undercooled melt, and in investigating phase selection and dendrite growth in reduced gravity [8]. Basing upon the success of TEMPUS, DLR, and ESA are currently developing in a common effort, an electromagnetic levitator (EML) as a multiuser facility on board the International Space Station (ISS). Thanks to the national agencies and the European Space Agency (ESA), several international researcher teams are preparing experiments using the EML on board the ISS. These experiments are divided into four different classes: (i) solidification, (ii) measurements of surface tension and viscosity, (iii) measurements of thermodynamic properties, and (iv) measurements of the mass density and thermal expansion. In the present book we concentrate on solidification comprising both experimental research in drop tubes and levitation devices on Earth and some specific experiments in Space. These experimental works are escorted by theoretical works as mesoscopic modeling of dendrite growth and atomistic modeling of attachment kinetics of atoms from liquid to solid.

In the present chapter, facilities for containerless solidification of undercooled melts are introduced. Their technical concepts are described and some exemplary results are demonstrated as obtained from experiments using the various devices.

1.2 Drop Tubes

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 in size of a few millimetres are undercooled and solidified during free fall.

1.2.1 Short Drop Tubes

Sample material in mass of several grams is melted in a crucible, which contains a small bore at its lower side. By using Ar gas at overpressure, the liquid metal is pressed through the bore of the crucible. A thin liquid jet of a metal is formed and it disperses into small droplets (Rayleigh instability of a thin liquid jet). The small droplets undercool and solidify during the free fall containerlessly in reduced gravity. This technique is employed to study undercooling and nucleation phenomena [9–11], to investigate the evolution of grain-refined microstructures [12, 13], and to produce metastable crystalline materials and metallic glasses [9–11, 14, 15].

Figure 1.1 illustrates the experimental setup of a drop tube in length of 14 m (free fall time 1.4 s) at the German Aerospace Center (DLR) in Cologne [16]. The drop tube is made of stainless steel components all of which are compatible with the requirements of ultrahigh vacuum (UHV) technique. The drop tube is evacuated before each experiment to a pressure of approximately 10−7 mbar and, subsequently, backfilled with high purity He or He–H2 gas of high thermal conductivity. The processing gas is purified as it passes a chemical oxygen absorption system and a liquid nitrogen cold trap. The sample material in a crucible of, for example, fused silica, is melted inductively. After all the material is liquid, its temperature is measured by a two-color pyrometer and subsequently forced by Ar pressure of 2 bars through the small bore. The droplets solidified during free fall through the drop tube they are collected at the bottom of the drop tube and are sorted by meshes in different size groups ranging from 50 to 1000 µm diameter. Since the droplet diameter scales with the cooling rate at which the droplets cool down, drop tubes are quite suitable to study statistical processes of phase selection and their temperature–time–transformation behavior.

Figure 1.1 (a) 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; (b) droplets of Pd40Cu10Ni30P20 alloy as solidified in the drop tube and sorted in different size groups [17]. The drops do not show all sphere-like geometry since the alloy is an easy glass forming system that is characterized by a high viscosity. This leads to cylindrical shape of the fragmented portion of the liquid metal that undercools and solidifies at large undercoolings at which the viscosity of the melt has essentially increased.

Figure 1.2 shows the volume fractions of the various phases formed in drop tube processed Al88Mn12 alloy as a function of droplet diameter [1]. Quasicrystalline phases of fivefold symmetry were discovered as a new class of solid-state matter in between of crystalline and amorphous solids in melt spun ribbons of Al88Mn12 alloy [18]. Depending on the preparation conditions, an icosahedral 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 at highest cooling rate. At medium droplet size, T-phase and supersaturated Alss solid solution are found. The mass fraction of Alss phase increases with droplet size (decreasing cooling rate) on the expense of T-phase. At largest droplet size of drops in the order of about 1 mm in diameter, also the equilibrium intermetallic phase Al6Mn is crystallized. Calculations of nucleation–kinetics plots reproduce the experimentally observed phase-selection behavior of drop tube processed Al88Mn12 alloy [19].

Figure 1.2 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 [1].

Drop tube experiments are also used to determine the formation of different phases selected kinetically by the cooling rate. Temperature–time–transformation (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 Avrami analysis [20] is utilized that describes the time t necessary to produce a mass fraction X = 10−3, which is barely detectable by experimental diagnostics (X-ray diffraction, optical and electron microscopy), of the equivalent phases formed at a certain undercooling. It is given by

(1.1)

with Iss the steady-state nucleation rate and V the crystal growth rate. The crystal growth velocity in quasicrystal forming alloys is extremely sluggish. This is because it requires short-range diffusion of the various atomic species to arrange them in a correct way at the solid–liquid interface to form the complex structure of quasicrystalline phases [21]. The propagation of the solidification front into the undercooled melt is essentially driven by the kinetic undercooling of the interface. Under such circumstances, the speed of the solidification front is estimated by the rate theory leading to

(1.2)

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−1, the intermetallic Al6Mn phase disappears, while the quasicrystalline T-phase progressively forms. Further increasing the cooling rate to 1× 104 K s−1 leads to solidification of the quasicrystalline I-phase. In order to avoid the nucleation of quasicrystalline phases and in particular the crystalline Al-phase, very large cooling rates greater than 106 K s−1 are needed. This is in accordance with the observation that quasicrystalline phases nucleate quite easily in undercooled melts and the formation of amorphous phases in quasicrystal forming alloys during rapid cooling of a liquid is very difficult. Figure 1.3 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 [19].

Figure 1.3 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 [19]. 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 [22].

1.2.2 Long Drop Tubes

Long drop tubes are generally in excess of 50 m high and individual drops are processed. They exploit the fact that a body falling freely in vacuo experiences zero gravity, to study the effects of microgravity on solidification in earthbound laboratories. There are two such facilities: a 105-m drop tube at NASA Marshall Space Flight Center, described by Rathz et al. [23], and a 47-m drop tube at the Nuclear Research Center at Grenoble [24]. 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 estimate the undercooling achieved at nucleation ΔTn. Processing of drops under high vacuum [23] or even UHV [24] reduces surface oxidation of the molten samples as a possible source of heterogeneous nucleation. On the other hand only high melting metals as, for example, refractory metals can be processed since cooling is only by radiation, which is efficient at high temperatures exclusively. Lacy et al. found the mean undercooling in niobium to be 525 ± 8 K with a maximum of 535 K [25]. They associated this nucleation event with the formation of NbO on the droplet surface because the nucleation temperature corresponded to 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. report a maximum undercooling of 900 K for Re [3]. The high value of undercooling together with the observation of polycrystalline microstructure in the as-solidified sample was taken to assume homogeneous nucleation to be present in this experiment. They used the undercooling result to estimate 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.

1.3 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 bulk samples, which remain accessible not only for direct observation but also for external stimulation of nucleation. The current state of electromagnetic and electrostatic levitation is described.

1.3.1 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 schematic of electromagnetic levitation is illustrated in Figure 1.4. The principle of 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)

Figure 1.4 Schematics of electromagnetic levitation; the red arrows indicate the magnetic field of the levitation coil. The blue arrows represent the eddy currents induced within the sample by the alternating electromagnetic field. The eddy currents, in turn, produce a magnetic field that is opposite to the primary magnetic field due to a phase shift by π between the electrical current in the coil and the induced eddy currents. This leads to a repulsive force indicated by the green arrow. At properly designed coil geometry and coil current, the repulsive force compensates the gravitational force [27].

(1.3)

with E the electrostatic field. For a nonuniform magnetic field, the eddy currents induced in a sample produce a magnetic dipole moment m that is opposite to the primary field B. This leads to a diamagnetic repulsion force Fr

(1.4)

between the primary field und 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. mg denotes the mass of the sample and g the gravitational acceleration. Electromagnetic levitation can be used to levitate metallic and even semiconducting samples. However, electromagnetic levitation of semiconductors requires either doping with a metallic element to increase the electrical conductivity or preheating the pure semiconductor to a temperature of about 1000 K by a laser or by a graphite susceptor within the levitation coil so that the intrinsic conduction is sufficiently increased to electronically couple the sample to the alternating external field. 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.

According to Rony [26], the mean force on an electrically conductive nonferromagnetic sample is determined by

(1.5)

Here, r denotes the radius of the sphere-like sample, µo the permeability of vacuum. The function G(q) is calculated as

(1.6)

q is the ratio of the sample radius and the skin depth

(1.7)

ω, σ, and µ are the angular frequency of the electrical current, the electrical conductivity, and the magnetic permeability of the sample, respectively. According to Eq. (1.5) the levitation force scales with the gradient of the magnetic field. To optimize levitation, it is therefore crucial to design properly the geometry of the levitation coil and optimize the function G(q). This function is plotted versus q in Figure 1.5 (dashed line). Consequently, the efficiency of electromagnetic levitation is adjusted by the parameters of the frequency of the alternating electromagnetic field, the sample size, and the electrical conductivity of the sample. For a vanishing conductivity (q → 0), G(q) becomes zero and levitation is not possible. For G(q → ∞) G(q) is approaching saturation.

Figure 1.5 Efficiency of the function G(q) (dashed line) which is proportional to the levitation force Fem, and the function H(q) (solid line) which is proportional to the power absorption P [27].

To levitate a sample of masse m, the gravitational force Fg has to be compensated by the electromagnetic levitation force Fem

(1.8)

where ρ denotes the mass density of the material. The z-component of the force follows as

(1.9)

For a given magnetic field and sample size, the levitation force is determined by the skin depth d and the mass density m. The mean power absorption P is calculated according to Roney as

(1.10)

with

(1.11)

H(q) is the efficiency of the power absorption as illustrated by the solid line in Figure 1.5. For vanishing electrical conductivity no power is absorbed by the sample. On the other hand for an ideal conductor no ohmic losses occur so that H(q) converges to zero. The function H(q) passes through a maximum at q ≈ 2.

The concept developed by Rony has been extended and applied by Fromm and Jehn to calculate both the levitation force and the power absorption for a levitation coil that is approximated by different single loops being parallel to each other [28, 29].

The temperature control of electromagnetically levitated samples requires a separate action of P and Fem as far as possible. The essential difference between P and FL is that the functions G(q) and H(q) have a different characteristics with respect to the frequency of the alternating electromagnetic field: Fem depends on the product (B·)B, while P is proportional to B2 (cf. Eqs. (1.5) and (1.10)). Hence, temperature control is possible within a limited range by choosing a proper frequency of the alternating field and by a movement of the sample along the symmetry axis of a conically shaped coil. In the lower regions of the coil, the windings are tighter, and thus the magnetic field and power absorption are greater than that 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 several 100 K. 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 K. It is assumed that the sample is placed into a levitation coil with six windings and two counter-winding at its top. An alternating electrical current at 300 kHz powers the coil. More details on this analysis of temperature control may be taken from reference [30].

Under equilibrium conditions, the sample approaches a temperature that is given by the balance of heat produced in the sample and loss of the heat of the sample to the environment. For a containerlessly processed droplet, the loss of heat is possible by heat radiation, and – if an environmental gas atmosphere is present – by heat conduction and convection, in the environmental gas. Thus, the balance is given by

(1.12)

According to Planck's law, the heat transfer by radiation is given by

(1.13)

where denotes the Stefan–Boltzmann constant, the total emissivity of the sample, A the surface area of the sample, T the temperature of the sample, and To the ambient temperature. Since the Stefan–Boltzmann constant is very small, heat transfer by radiation becomes important only at elevated temperatures of T > 1000 K, but increasing then rapidly because of the fourth power of the T dependence. This means that in most cases radiation cooling is not sufficient to cool and undercool a metallic sample below its melting temperature. Therefore, cooling by an environmental gas is employed. The heat transfer by conduction can be approximated by

Figure 1.6 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 [31]. 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.

(1.14)

where λeff is an effective heat transfer coefficient including both heat conduction and heat transport by convection in the surrounding gas atmosphere.

A schematic view of an electromagnetic levitation chamber for containerless undercooling and solidification experiments is shown in Figure 1.7 [32]. The levitation coil together with the sample (Ø ≈ 6 mm) is placed within an ultrahigh-vacuum chamber, which can be backfilled with gases such as He or He–H2 mixture. The gases are purified by an oxygen absorption 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 radio-frequency 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 using forced convection with cooling gases. The temperature of the sample is measured by means of a two-color pyrometer with an absolute accuracy of ±3 K and a sampling rate up to 1 kHz. Solidification of the undercooled melt can be externally initiated by touching the sample with a crystallization trigger needle.

Figure 1.7 Electromagnetic levitation chamber for containerless undercooling and solidification of metals.

Figure 1.8 depicts a typical temperature–time profile recorded contactless by a pyrometer during an undercooling experiment of an alloy. 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. After heating the sample to a temperature well above the liquidus temperature, 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 due to the rapid release of the heat of crystallization. During recalscence, solidification takes place far away from equilibrium and the undercooled melt acts as a heat sink. 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 near-equilibrium conditions. The plateau duration Δtpl is exclusively controlled by the heat transfer from the sample to the environment and is inferred from the measured temperature–time profile. Δtpl 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. By exceeding a critical undercooling, the solidification mode changes from coarse-grained dendriditic to grain-refined equiaxed microstructure. 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 break-up 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. The fragmentation process itself requires atomic diffusion in liquid phase. During the postrecalescence time, primary solidified dendrites coexist with interdendritic liquid. Therefore, the condition for dendrite break up is given if the dendrite break up time is smaller than the postrecalescence time. In the other case, the primary solidified dendrites survive leading to coarse-grained dendritic microstructures. The postrecalescence time is inferred from the measured temperature–time profiles while the dendrite break up time is calculated within a fragmentation model developed by Karma [33]. This model is verified by experiments on levitation-undercooled samples in which the microstructures are investigated as a function of undercooling prior to solidification. More details about the dendrite fragmentation process are given in [34].

Figure 1.8 Schematics of a typical temperature–time (T–t) profile obtained from an undercooling experiment using electromagnetic levitation. The undercooling ΔT and the plateau duration Δtpl are directly inferred from experimental profiles. If the time needed to break up a dendrite, Δtbu is smaller than the postrecalescence or plateau time, Δtpl dendrites will break up leading to a grain-refined equiaxed microstructure. In the other case, the undercooled melt will crystallize to a coarse-grained dendritic microstructure.

The crystallization needle is used to trigger externally solidification at preselected undercooling and well-defined position at the surface of the sample. In such a way the crystallization kinetics is investigated as a function of undercooling [35]. Figure 1.9 illustrates triggered nucleation of a metastable bcc phase of Fe-24at%Ni alloy. A trigger needle made of a Fe95Mo5 alloy is used since this alloy forms a stable bcc structure in the temperature range of the present experiment. The left peak represents a recalescence event as observed following spontaneous nucleation at 1472 K (ΔT = 278 K). An increase in temperature up to 1751 K during recalescence is found in good agreement with the equilibrium liquidus temperature of this alloy. The right peak was observed following solidification triggering with the Fe–Mo tip at a temperature of 1556 K (ΔT = 194 K). Obviously, the increase in temperature during recalescence ends at a temperature well below the equilibrium liquidus line, which points to a metastable bcc solidification product. Immediately following the recalescence peak, a weak hump is found in the cooling trace, which is due to a solid-state transformation of metastable bcc phase into stable fcc phase. This hump is missing in the temperature–time profile for the spontaneous nucleation. This confirms that during spontaneous crystallization fcc phase is nucleated, whereas triggered solidification leads to nucleation of metastable bcc phase, which however transforms into stable fcc phase during cooling of the sample to ambient temperature [36].

Figure 1.9 (a) Principle of nucleation triggering of a metastable bcc phase in Fe76Ni24 alloy by using a nucleation trigger made of Fe95Mo5 bcc phase (b). Two temperature–time profiles obtained during solidification of undercooled Fe76Ni24 alloy. Spontaneous crystallization of stable fcc phase (γ, blue line), and solidification of a metastable bcc phase upon triggering with the Fe95Mo5 nucleation trigger (green line).

The cooling rates in the order of 10–100 K s−1 in the above-described undercooling experiments on Fe–Ni alloys are not sufficient to conserve the primary solidified metastable bcc phase during cooling to ambient temperatures. If the cooling rate is increased up to 105–106 K s−1 the solid-state transformation of primary formed bcc phase into the stable fcc phase can be, however, avoided. This has been demonstrated in the early drop tube experiments by Cech [37] and Cech and Turnbull [38], and later on by atomization experiments [39, 40]. Meanwhile, an electromagnetic levitation chamber is used to combine it with external diagnostic means, for example, neutron scattering and X-ray scattering by synchrotron radiation [41]. In such a way, the primary crystallization of a metastable bcc phase in Ni–V alloys at large undercoolings was directly evidenced by in situ energy dispersive X-ray diffraction on levitation-processed undercooled melt using high-intensity synchrotron radiation at the European Synchrotron Radiation Facility [42].

Electromagnetic levitation is also frequently applied to measure the dendrite growth velocity as a function of undercooling. This will be the subject of a separate Chapter 11.

1.3.2 Electrostatic Levitation

Electromagnetic levitation requires sample material that is electrically conductive. Therefore, the application of electromagnetic levitation is restricted to metals and (doped) semiconductors. The advantage of electrostatic levitation is that levitation and heating is decoupled and the samples can be processed under conditions provided the vapor pressure of the processed material is small. However, there is a problem with the stability of the sample position. According to the theorem of Samuel Earnshow, it is not possible to levitate a charged sphere within a static electrostatic field [43].

Electrostatic levitation is based on the Coulomb forces acting on an electrically charged sample in a quasistatic electrical field [44]. A sample with a surface charge q and a mass m is levitated against gravity within a static electrostatic field as

(1.15)

is the unit vector in the z direction, that is, parallel to the electrostatic field. A stable position of the sample is based on a local potential minimum at for all directions in space.

(1.16)

The Maxwell equation for Gauss's law affords

(1.17)

Under vacuum conditions, . Hence, a potential minimum does not exist and a stable sample position under stationary conditions is not possible [27]. This means electrostatic levitation requires a sophisticated dynamic sample position and electrostatic field control. This became possible just since the 1990s of last century where high-voltage amplifiers were developed, which can be controlled with high slew rates of changing the voltage U, dU/dt > 400 V µs−1.

Figure 1.10 shows schematically the active sample positioning system. An electrically charged sample is levitated between two horizontal electrodes within a widened positioning laser beam filling the whole space between the electrodes. The sample shadow is detected by a two-dimensional photo-sensitive detector that gives information on the vertical and horizontal position of the sample. A real-time computer control algorithm developed by Meister [45] reads this information and adjusts instantaneously the voltage of the amplifier. In order to control the sample position in all three-dimensional directions, two positioning laser perpendicular to each other and an assembly of six electrodes are used. The arrangement of the electrodes is illustrated in Figure 1.11.

Figure 1.10 Sketch of the fully automated active sample positioning system [45].

Figure 1.11 Arrangement of the electrodes of the electrostatic levitator. The distances and connections of the y-direction are analogues and not printed for a better visualization.

Two central electrodes arranged as a plate capacitor are surrounded by four electrodes in plane, which are cross-linked with the positioning lasers to push the sample in the central position. The forces acting in the z-direction, F(z), are the gravitational force, the force due to the electrical field, and the force between the sample and the grounded center electrodes. With the method of image charges, the force of a charged sphere between the electrodes can be determined by

(1.18)

with the position of the sample z, the distance of the electrodes dz, the charge q of the sample, the vacuum permittivity εo, and the number of reflections n. Neglecting multiple reflections, F(z) is approximated as

(1.19)

In the middle of the electrodes, the forces of the image charge acting on the sample are compensating each other. The equation of motion for the z direction is given by

(1.20)

The fields in the x- and y-direction are assumed to be between two parallel electrodes [45]

(1.21)

κ is a geometrical factor regarding the distance of the sample and the lateral electrodes.

For conducting an experiment using the electrostatic levitator, the sample in diameter of about 2–4 mm is placed at the lower electrode, which is grounded. The high-voltage power supply is switched on and electrostatic field between upper and lower electrode in the z-direction is built up. At the same time the sample is charged. Since the upper electrode is on negative potential, the surface of the sample is loaded with positive charge qi that is calculated as [46]

(1.22)

with r the radius of the sample and L = 1.645 a geometrical factor. The image charge of the bottom electrode dominates the initial levitation voltage. The force acting on a sample while lifting is given by

(1.23)

Combining Eqs. (1.22) and (1.23) yields the initial voltage for levitation

(1.24)

The charge of the sample in the beginning of the experiment is then

(1.25)

The voltage needed to keep the sample in the middle of the electrodes is calculated

(1.26)

The initial voltage is larger than the voltage that is needed to levitate the sample in the middle of the horizontal electrodes. For a constant initial voltage, the time is approximated which elapses until the sample hits the electrode. This time is used to estimate the minimum sampling rate required for positioning. For a silicon sample in diameter of 2 mm the sampling rate is 2 × 10−3 s [45].

Electrostatic levitation offers the advantage that positioning and heating are decoupled in contrast to electromagnetic levitation. Heating is realized in electrostatic levitation by an infrared laser. Increasing the temperature of the sample leads to an evaporation of surface atoms, which is useful for undercooling experiments since the evaporation cleans the surface and thereby reduces or even eliminates heterogeneous nucleation motes at the surface of the sample. On the other hand, the sample surface looses surface charge by evaporation. Therefore, the voltage has to be increased to keep the sample levitated. To facilitate recharging of the sample during levitation a focused ultraviolet light source with a high energy of several electronvolts (λ = 115–350 nm) is used. In addition to this procedure, the sample is also recharged at elevated temperatures by thermionic emission of electrons. More details about the electrostatic levitator build up and operated at DLR can be found in [27, 45, 47].

Figure 1.12 Force on a charged sample for a distance of the plates of 8 mm generated by the image charges for an applied voltage. In the middle of the electrodes, the forces of the image charges are compensating each other.

The electrostatic levitator is very suitable to study nucleation undercooling with special emphasis to homogeneous nucleation. To observe homogeneous nucleation, very large undercoolings have to be realized, since the onset of homogeneous nucleation gives the physical limit for maximum undercoolability of a melt. To realize such conditions, heterogeneous nucleation has to be eliminated. Electrostatic levitation under ultra-high-vacuum conditions is ideally suited for such experimental studies since heterogeneous nucleation on container walls is completely avoided and heterogeneous nucleation on surface motes is reduced or even eliminated due to self-cleaning of the surface by evaporation at elevated temperature.

In the following, nucleation undercooling studies on pure Zr are presented to demonstrate that how physically different nucleation processes are experimentally investigated. Figure 1.13 shows a temperature–time profile measured on pure Zr sample in the electrostatic levitator. First, the solid sample is heated up to its melting temperature, TL. In case of a pure metal as Zr, the sample melts congruently at TL. The small step in the melting plateau is due to the change in spectral emissivity when the solid transforms to the liquid. After complete melting, the liquid sample is heated to a temperature well above TL before cooling. During subsequent cooling, the liquid sample undercools well below TL. When spontaneous nucleation sets in at an undercooling ΔT = TL − Tn (Tn: nucleation temperature) the nucleated crystal rapidly grows due to a large thermodynamic driving force generated at such deep undercoolings. The rapid release of the heat of crystallization leads to a steep rise in temperature during recalescence. From such temperature–time profiles, ΔT is easily inferred since Tn is well defined by the onset of recalescence. After the entire sample has solidified, the next heating and cooling cycle is started.

Figure 1.13 Temperature–time profile measured on a zirconium drop levitated in an electrostatic levitator. The sample melts at TL = 2128 K. During undercooling nucleation sets in at TN = 1757 K. Subsequently, rapid crystal growth of β-Zr solid phase (bcc) leads to a steep rise of temperature during recalescence. The second recalescence event at 980 K is attributed to a transformation of solid β-Zr to solid α-Zr phase (hcp). The inset shows an enlargement of the recalescence profile with the nucleation temperature TN = 1757 K and the undercooling ΔT = 371 K.

Usually, the solidification of an undercooled metallic melt is a two-staged process. During recalescence, a fraction of the sample, fR, solidifies during recalescence under nonequilibrium condition. The remaining melt, fpr = 1 − fR, solidifies under near-equilibrium conditions during postrecalescence period. fR increases with the degree of undercooling and becomes unity, fR = 1 if ΔT = ΔThyp. The hypercooling limit, ΔThyp, is reached if the heat of fusion ΔHf is just sufficient to heat the sample with its specific heat Cp up to TL. In case of quasiadiabatic conditions, that is, if the amount of heat transferred to the environment is negligible compared to the heat produced during recalescence, the hypercooling limit is given by ΔThyp = ΔHf/Cp. In case of pure Zr, the hypercooling limit is estimated as ΔThyp = 359 K with ΔHf = 14 652 J mol−1 and Cp = 40.8 J mol K−1 [6]. With increasing undercooling, ΔT′ > ΔThyp, the postrecalescence plateau vanishes and TL will not be reached during recalescence. As can be seen from Figure 1.13, in this experiment an undercooling of ΔT = 371 K is measured, which is larger than ΔThyp.

Figure 1.14 shows the distribution functions of undercoolings measured in the electromagnetic levitator (Figure 1.14 left) and the electrostatic levitator (Figure 1.14