High-Temperature Measurements of Materials

advances in materials research

11

advances in materials research Series Editor-in-Chief: Y. Kawazoe Series Editors: M. Hasegawa

A. Inoue

N. Kobayashi

T. Sakurai

L. Wille

The series Advances in Materials Research reports in a systematic and comprehensive way on the latest progress in basic materials sciences. It contains both theoretically and experimentally oriented texts written by leading experts in the f ield. Advances in Materials Research is a continuation of the series Research Institute of Tohoku University (RITU). 1

Mesoscopic Dynamics of Fracture Computational Materials Design Editors: H. Kitagawa, T. Aihara, Jr., and Y. Kawazoe

7

Advanced Materials Characterization for Corrosion Products Formed on the Steel Surface Editors: Y. Waseda and S. Suzuki

2

Advances in Scanning Probe Microscopy Editors: T. Sakurai and Y. Watanabe

8

Shaped Crystals Growth by Micro-Pulling-Down Technique Editors: T. Fukuda and V.I. Chani

3

Amorphous and Nanocrystalline Materials Preparation, Properties, and Applications Editors: A. Inoue and K. Hashimoto

9

Nano- and Micromaterials Editors: K. Ohno, M. Tanaka, J. Takeda, and Y. Kawazoe

4

Materials Science in Static High Magnetic Fields Editors: K. Watanabe and M. Motokawa

5

Structure and Properties of Aperiodic Materials Editors: Y. Kawazoe and Y. Waseda

6

Fiber Crystal Growth from the Melt Editors: T. Fukuda, P. Rudolph, and S. Uda

10

Frontiers in Materials Research Editors: Y. Fujikawa, K. Nakajima, and T. Sakurai

11

High-Temperature Measurements of Materials Editors: H. Fukuyama, Y. Waseda

12

Oxide and Nitride Semiconductors Processing, Properties and Applications Editors: T. Yao, S.-K. Hong

Hiroyuki Fukuyama Yoshio Waseda (Eds.)

High-Temperature Measurements of Materials With 125 Figures

ABC

Professor Hiroyuki Fukuyama Professor Dr. Yoshio Waseda Tohoku University, Institute of Multidisciplinary Research for Advanced Materials 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan E-mail: [email protected], [email protected]

Series Editor-in-Chief: Professor Yoshiyuki Kawazoe Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Series Editors: Professor Masayuki Hasegawa Professor Akihisa Inoue Professor Norio Kobayashi Professor Toshio Sakurai Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Professor Luc Wille Department of Physics, Florida Atlantic University 777 Glades Road, Boca Raton, FL 33431, USA

Advances in Materials Research ISSN 1435-1889 ISBN 978-3-540-85917-8

e-ISBN 978-3-540-85918-5

Library of Congress Control Number: 200893 61 4 5 © Springer Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by SPi using a Springer LATEX macro package Cover: eStudio Calmar Steinen SPIN: 12254330 57/3180/SPI Printed on acid-free paper 987654321 springer.com

Preface of Series by the Editor-in-Chief

This book titled High-Temperature Measurements of Materials is the 11th volume in the series “Advances in Materials Research” edited by Professors Hiroyuki Fukuyama and Yoshio Waseda. The book is composed of nine chapters, the contents of which try to solve the currently important basic problems in high-temperature melts from scientific basics to real application fields related to industrial problems. The contents of this present volume are expected to contribute to solving the most important problem of sustainability of the global society by applying recent high-level scientific and engineering researches in materials science. Recent important contributions from large scale experimental facilities from over the world are collected and explained in detail in this book, where we can now measure clearly the important behaviors of high-temperature melts which have never been observed by traditional experimental techniques. This book also introduces recent advancements in computer simulations, which have become very effective not only in explaining experimentally observed phenomena but also in predicting materials properties and processes which are difficult to be measured for the materials under high-temperature environments. As the series editor, I thank Dr. Claus Ascheron of Springer-Verlag, who always has interest in and kindly takes care of our research activity to encourage publication in this series of books. Sendai, July 2008

Yoshiyuki Kawazoe

Preface

A variety of industries – information technology, aerospace, automobile, and basic and new materials manufacturing – need technological innovations, which bring high-value-added and high-quality products at low cost not only because of global competition, but also because of the perspective of environmental consciousness and regulation. Thermophysical properties of hightemperature melts are indispensable for numerical simulations of material processes such as semiconductor and optical crystal growth of the melt, and casting of super-high-temperature alloys for jet-engine turbine blades, in addition to welding in automobile manufacturing. Recent developments in process modeling provide 3D unsteady analysis of melt convection, temperature, and heat flux distribution, which enables us to predict product quality. In fact, 3D process visualization using computer modeling helps us to understand complicated phenomena occurring in the melt and to control the process. Accurate data are necessary to improve the modeling, which costeffectively engenders high-quality products. However, crucial obstacles render measurements of thermophysical properties difficult at elevated temperatures because of high chemical reactivity and fluidity of melts. Substantial and persistent challenges have been made to ascertain the precise thermophysical properties of high-temperature melts. This book describes the new techniques and latest developments in the measurements of atomic structure, density, surface tension, viscosity, heat capacity, thermal and mass diffusivity, thermal conductivity, emissivity, and electrical conductivity of high-temperature melts. In addition to up-to-date improvements in conventional techniques, some new attempts are introduced to open a new scientific field, that is, physics of high-temperature melts. Space-related organizations such as Japan Aerospace Exploration Agency (JAXA) and German Aerospace Center (DLR) demonstrate the importance of noncontact measurements and microgravity environments. Recent progress in levitation techniques enables high-precision measurements of various properties of stable and deeply undercooled metallic melts. An electrostatic levitation apparatus is specially designed not only

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Preface

to measure thermophysical properties but also to study the short-range order of metallic melts in combination with a synchrotron radiation facility: SPring-8 (Japan Synchrotron Radiation Research Institute, JASRI). A noncontact measurement of thermal conductivity was developed using an electromagnetic levitator incorporating a superconducting magnet (High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University). An electromagnetically levitated droplet behaves as a hard sphere in a static magnetic field. The oscillation and convection of the droplet are suppressed because of the Lorentz force, which enables measurement of its true thermal conductivity using modulation laser calorimetry. Utilization of microgravity conditions provides an ideal environment without a fluid flow driven by buoyancy force. Diffusion coefficients of metallic melts have been determined using the shear cell technique under microgravity using a sounding rocket. As described earlier, this book is a unique compilation of information related to recent advances in high-temperature measurements and thermophysical properties data. The editors earnestly hope that this book is a useful guide for the scientists and engineers who are working in the field of materials science and processing, and that it is attractive to students interested in the physics of high-temperature melts. The editors thank Yoshimasa Ito and Miwa Sasaki at the Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University for preparing TeX manuscripts and figures. The editors gratefully acknowledge the encouragement and patience of Dr. Claus Ascheron of Springer-Verlag. Sendai July 2008

Hiroyuki Fukuyama Yoshio Waseda

Contents

1 Measurement of Structure of High Temperature and Undercooled Melts by using X-Ray Diffraction Methods Combined with Levitation Techniques Tadahiko Masaki, Akitoshi Mizuno, and Masahito Watanabe . . . . . . . . . .

1

1.1 1.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Electrostatic Levitator for the Structural Analysis by X-Ray Diffraction Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Viscosity and Density Measurements of High Temperature Melts Yuzuru Sato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Viscosity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Capillary Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Oscillating Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Rotating Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Density Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Archimedean Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Pycnometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Manometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Maximum Bubble Pressure Method . . . . . . . . . . . . . . . . . . . . . 2.3.5 Sessile Drop Method and Levitation Method . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 21 26 28 29 31 32 33 34 36 36

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3 Marangoni Flow and Surface Tension of High Temperature Melts Taketoshi Hibiya and Shumpei Ozawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Marangoni Effect on High-Temperature Melts . . . . . . . . . . . . . . . . . . . 3.2.1 Definition of Marangoni Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Electron Beam Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Methods for Measuring Surface Tension: Oscillating Drop Method Using Electromagnetic Levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Surface Tension of Molten Silicon: Influence of Oxygen on Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Surface Tension of Molten Iron and Iron-based Alloy . . . . . . . . . . . . . 3.8 Thermodynamic Approach for Adsorption of Oxygen at Melt Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 39 41 44 46 47 49 54 56 56 57

4 Diffusion Coefficients of Metallic Melts Measured by Shear Cell Technique Under Microgravity and on the Ground Shinsuke Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 4.2

4.3

4.4

4.5 4.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design of Shear Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Principle of Shear Cell Technique . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Minimization of Shear Convection . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Minimization of Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Structure of the Shear Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Diffusion Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Evaluation of Mean Square Diffusion Depth . . . . . . . . . . . . . . Quantitative Measurement of Shear Convection and Correction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Short-Time Diffusion Experiments . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Time Dependence of Mean Square Diffusion Depth . . . . . . . . 4.4.3 Influence of Shear Convection . . . . . . . . . . . . . . . . . . . . . . . . . . Correction Method for the Determination of Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1g-Diffusion Measurements with Stable Density Layering . . . . . . . . . 4.6.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Effect of Density Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 62 64 65 66 66 66 67 68 68 70 71 71 72 72 73 76

Contents

4.7

Microgravity Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Utilization of Microgravity Environment . . . . . . . . . . . . . . . . . 4.7.2 Microgravity Diffusion Experiments in Foton-M2 . . . . . . . . . 4.8 Temperature Dependence of the Diffusion Coefficients . . . . . . . . . . . . 4.9 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 77 77 79 80 82 83

5 Thermal Diffusivity Measurements of Oxide and Metallic Melts at High Temperature by the Laser Flash Method Hiroyuki Shibata, Hiromichi Ohta, and Yoshio Waseda . . . . . . . . . . . . . . . . 85 5.1 5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 A Brief Background of the Present Requirement for the Thermal Property Measurements of High Temperature Materials . . . . . . . . . . 86 5.3 Experimental Procedures and Theoretical Basis for the Laser Flash Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Selected Examples of Thermal Diffusivities of Oxide Melts . . . . . . . . 94 5.5 Selected Examples of Thermal Diffusivities of Metallic Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Emissivities of High Temperature Metallic Melts Masahiro Susa and Rie K Endo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 6.2 6.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Definition of Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Measurement Techniques for Emissivities . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.1 Method Based on Wien’s Formula . . . . . . . . . . . . . . . . . . . . . . 112 6.3.2 Method Based on Optical Constants . . . . . . . . . . . . . . . . . . . . 113 6.3.3 Method Based on Direct Measurements of Radiation Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3.4 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Emissivity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.4.1 Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.4.2 Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4.3 Semiconducting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.4.4 Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7 Noncontact Thermophysical Property Measurements of Metallic Melts under Microgravity Ivan Egry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Microgravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

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7.3 7.4

Containerless Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.4.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.4.2 Density and Thermal Expansion . . . . . . . . . . . . . . . . . . . . . . . . 139 7.4.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.4.4 Viscosity and Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8 Noncontact Laser Calorimetry of High Temperature Melts in a Static Magnetic Field Hiroyuki Fukuyama, Hidekazu Kobatake, Takao Tsukada, and Satoshi Awaji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1 8.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Theory of Modulation Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.1 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.2 Thermal Conductivity and Emissivity . . . . . . . . . . . . . . . . . . . 153 8.2.3 Verification of the Assumptions of Conduction-Dominated Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.4 Verification of the Model and Sensitivity Analysis . . . . . . . . . 159 8.2.5 Emissivity Determination from Cooling Curve . . . . . . . . . . . . 163 8.3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4.1 Motion of the Silicon Droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4.2 Temperature Response and Phase Difference . . . . . . . . . . . . . 164 8.4.3 Isobaric Molar Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.4.4 Hemispherical Total Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.4.5 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9 Noncontact Thermophysical Property Measurements of Refractory Metals Using an Electrostatic Levitator Takehiko Ishikawa, Paul-Fran¸cois Paradis . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.2 Electrostatic Levitation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.3 Thermophysical Property Measurements . . . . . . . . . . . . . . . . . . . . . . . 177 9.3.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3.2 Surface Tension and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.3.3 Experimental Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.4 Results of Thermophysical Property Measurements of Refractory Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.4.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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9.4.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.4.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

List of Contributors

Satoshi Awaji Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan, [email protected] Ivan Egry Institute for Materials Physics in Space, German Aerospace Center (DLR) 51170 Cologne German, [email protected]

Hiroyuki Fukuyama Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan, [email protected] Taketoshi Hibiya Graduate School of System Design and Management, Keio University 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8528, Japan, [email protected]

Rie K Endo Department of Metallurgy and Ceramics Science, Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan, [email protected]

Hidekazu Kobatake Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan, [email protected]

Paul-Francois Paradis Central Reseach Institute Mitsubishi Materials Corporation 1002-14 Mukohyama, Naka, Ibaraki 311-0102, Japan, [email protected]

Takehiko Ishikawa Japan, Aerospace Exploration Agency (JAXA) 2-1-1 Sengen, Tsukuba, Ibaraki 305-8505, Japan, [email protected]

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List of Contributors

Tadahiko Masaki Shibaura Institute of Technology 3-7-5 Toyosu, Koto-ku, Tokyo 135-8548, Japan, t [email protected] Akitoshi Mizuno Department of Physics, Gakushuin University 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan, [email protected] Hiromichi Ohta Faculty of Engineering, Ibaraki University 4-12-1 Nakanarusawa-machi, Hitachi, Ibaraki 316-8511, Japan, [email protected] Shumpei Ozawa Department of Aerospace Engineering, Tokyo Metropolitan University 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan, [email protected] Yuzuru Sato Department of Metallurgy, Tohoku University 6-6-02 Aramaki, Aoba-ku, Sendai 980-8579, Japan, [email protected] Hiroyuki Shibata Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan, [email protected]

Masahiro Susa Department of Metallurgy and Ceramics Science, Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan, [email protected] Shinsuke Suzuki The Institute of Scientific and Industrial Research, Osaka University 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan, [email protected] Takao Tsukada Department of Chemical Engineering, Tohoku University 6-6-07 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan, [email protected] Masahito Watanabe Department of Physics, Gakushuin University 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan, [email protected]

Yoshio Waseda Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan, [email protected]

1 Measurement of Structure of High Temperature and Undercooled Melts by using X-Ray Diffraction Methods Combined with Levitation Techniques Tadahiko Masaki, Akitoshi Mizuno, and Masahito Watanabe

1.1 Introduction The knowledge of the microscopic feature of matter is of paramount importance in materials science. In particular, the information about the atomic configuration is essential for the understanding of the characteristic properties of disordered matter. Therefore, a huge amount of efforts has been devoted to the development of experimental techniques coupled with X-ray or neutron diffraction techniques [1–3] to study the structure of liquids. In this decade, intense and high-energy X-ray beam sources, especially synchrotron radiation facilities, have emerged, and can be used for diffraction experiments of disordered matter. Compared to former experimental facilities, they enabled us to perform the highly precise investigation of the structure of liquids in a much shorter time. Although the methods and facilities for diffraction experiments have been improved rapidly, the sample handling techniques of high-temperature liquids have not been advanced so much because of the difficulty in the selection of crucible materials. In the case of liquid metals, several ceramics (e.g., fused silica, sintered alumina, sapphire, graphite, boron nitride) have been used for crucibles. Despite the adoption of these various materials, the maximum temperature of the experiments has been limited due to the corrosion of the crucible. Levitation techniques use a variety of external forces (e.g., aerodynamic [4], acoustic [5], electromagnetic [6], and electrostatic [7]) to hold a small amount of material in space without a crucible. When a levitated sample is in its liquid phase, it takes a spherical shape because the lack of crucible minimizes the surface free energy. In particular, a great deal of attention has been given to the measurement of the thermophysical properties of extremely high-temperature melts and the study of solidification phenomena from deeply undercooled liquids.

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The levitation technique can also be a very elegant way to handle liquid samples on diffraction experiments. For these experiments, they offer many advantages. As the liquid sample is held containerless, there is no need to subtract the diffraction contribution of a crucible from the total scattering intensity. Hence, this reduces by half the measurement time because it is not necessary to measure the diffraction of the empty cell. In addition, the symmetrical shape (nearly spherical) of the sample enables us to evaluate easily the correction of absorption and multiple scattering. Moreover, the most important advantage is the fact that the liquid samples under containerless conditions can easily reach deep undercooled states because the heterogeneous nucleation sites, which are usually the inner wall of the crucible, are absent. The observation of the structure of undercooled liquids can be realized by these techniques. Up-to-date, several levitation techniques have been applied to the diffraction experiments for the structural analysis of liquid matters, as shown in Table 1.1, in which the typical works of respective levitation techniques are summarized. The aerodynamic levitation is simple, yet useful for such experiments [8–25]. In this method, a sample is levitated in a conical nozzle at the location of the minimum potential well of the gas flow. Since the sample is small (1–3 mm diameter), this levitator is well suited for with the application to a high-energy synchrotron radiation X-ray beam. Moreover, this process can be applied to several types of materials under various atmospheres, though this cannot be used under vacuum conditions. So far, X-ray diffraction experiments of several high-temperature melts (e.g., liquid boron, alumina) have been performed by this technique and provided the static structure factors of these liquids [8–25]. Recently, using this method, liquid structures of glassforming alloys and ceramics have been studied together with the discussion of total and partial structure factors with the aid of the computer simulations [11, 12]. The electromagnetic levitation is another technique applicable to diffraction experiments. In this method, a sample of conductive material is levitated in a RF coil. The high frequency current of the coil induces an eddy current in the metallic sample and the electromagnetic force is induced for the levitation. The levitated sample is positioned at a stabilized point, which depends on the shape of the coil and on the electromagnetic properties of the sample. Since the sample size is large (6–8 mm diameter), this method is especially well suited to neutron scattering experiments. Schenk et al. [29] applied this levitation method to neutron and X-ray scattering experiments of equilibrium and non-equilibrium liquid metals. Recently, Watanabe et al. [26] studied the structure of liquid silicon by using this technique. It is also possible to levitate matter by applying electrostatic forces, under an active feedback system, to charged samples by electronic emission [39–43]. Electrostatic levitation is extremely attractive for X-ray diffraction experiments for several reasons. The size of the levitated sample (1–2 mm diameter) is suitable for the diffraction of high-energy X-rays from synchrotron radiation source, taking into account the X-ray absorption coefficient and the atomic

1 Electrostatic Levitation for Liquid Structural Analysis

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Table 1.1. Liquid structural analysis by using diffraction methods coupled with levitation techniques Levitation

Material

Structural analysis

Ref

CNL

Si Si Si CuZr, NiZr CuZr Co80 Pd20 Y2 O3 , Al2 O3 , B, Si, CoNi, Ni Al2 O3 Al2 O3 Al2 O3 , ZrO Al2 O3 MgAl2 O4 , CaAl2 O4 , Al2 O3 BaB2 O3 Mg2 SiO4 , CuZr Y2 O3 Y2 O3 YAG SiO2 Si Si Si Ni, Fe, Zr Ti Fe, Zr, Ni, Al65 Cu25 Co10 Al–Cu, Al–Ni AuCu, PdCuSi, Si, CoPd Co80 Pd20 Co–Pd, Au–Cu–Co Ni–V Nd–Fe–B Ti–Fe–Si–O Si Ni, Ti Ti39.5 Zr39.5 Ti21 TiZrNi Ba–Ge

XRD XRD IXS XRD XRD XRD XRD, NS IXS XRD EXAFS XRD XRD, NS, IXS XRD XRD XRD, NS AXRD EXAFS XRD XRD XRD ED-XRD NS NS NS XRD EXAFS EXAFS EXAFS XRD XRD ED-XRD XRD XRD XRD XRD XRD

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

EML

ESL

CNL Conical nozzle levitation, EML Electromagnetic levitation, ESL Electrostatic levitation, XRD X-ray diffraction, NS Neutron scattering, ED-XRD Energy dispersive X-ray diffraction, IXS Inelastic X-ray diffraction, AXRD Anomalous X-ray diffraction, EXAFS Extended X-ray absorption fine structure

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scattering factor of typical high-temperature metallic melts. In addition, as the charged liquid sample is levitated between pairs of electrodes, the sample is free from any obstacle, such as the nozzle or the coil in other levitators. Moreover, to avoid electrical breakdown on the application of a high voltage between electrodes, electrostatic levitators have to be operated either under pressurized atmospheres (∼0.4 MPa) or under high vacuum. The high vacuum conditions are particularly advantages for X-ray diffraction because there is no need to consider the scattering from the ambient gas. Recently, Gangopadhyay et al. [44] used such levitators for X-ray diffraction in a synchrotron radiation facility and observed the static structure factors and solidification behavior of several metallic melts. Electrostatic levitation has also been applied by Aoki et al. [45] to neutron diffraction experiments. They successfully measured the diffraction pattern of sintered alumina at room temperature. Although the validity of electrostatic levitation for diffraction experiments has been recognized, the previous electrostatic levitators exhibited limitations for precise measurements. In particular, the limitation was present for the observable range of the diffraction angle, which affects the resolution of the data obtained through a Fourier transform. The atomic configuration of liquids in real space can be investigated from the radial distribution function, g(r), which is obtained by a Fourier transformation of S(Q) as follows:  ∞  1 S(Q) − 1 Q sin Qr dr, (1.1) g(r) = 1 + 2 2π ρr 0 where ρ is the number density, S(Q) is the static structure factor, and Q is the momentum transfer. The g(r) obtained from diffraction experiments involves experimental errors, essentially because of the limited Q range of S(Q) due to the wavelength of X-rays and the available range of 2θ, and the diverted tail of the direct beam. In addition, the previously developed electrostatic levitators could be used only in large beam source facilities (e.g., synchrotron radiation facilites [44] or nuclear reactors [45]). Therefore, this situation restricts the opportunities for experiments because of the limited machine time. Recently, we developed an electrostatic levitator for X-ray diffraction measurements with a high applicability to various beam sources [46]. Since the system is very compact, it can be utilized coupled not only with the diffractometer at the high-energy X-ray diffraction beamline, BL04B2 of the synchrotron radiation facility, SPring-8 [47], but also with a laboratory X-ray system (RIGAKU RINT). The present electrostatic levitator was designed for the X-ray diffraction measurements by a two-axis diffractometer with slits collimation coupled with a germanium detector or a proportional counter. The scattering intensity for each scattering angle, 2θ, was acquired by the counter with the step-scan method. The high-energy X-ray beam from the synchrotron source of over 100 keV is a very attractive probe for the liquid structure analysis compared with the laboratory X-ray sources. The structure of liquid 3d or 4d metals can be measured by using the high-energy X-rays due to the high penetration of incident X-ray to samples. In addition, since

1 Electrostatic Levitation for Liquid Structural Analysis

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the momentum transfer, Q = 4π sin θ/λ (2θ, scattering angle; λ, wavelength of incident X-rays), is proportional to the X-ray energy, the static structure factor, S(Q), in sufficiently wide Q range can be obtained from the measurement of diffraction pattern with small scattering angles. On the other hand, the laboratory X-ray source can be used for diffraction experiments of lighter materials, such as silicon. Since the laboratory X-ray source is free from the restriction of user time of the facility, preliminary or challenging experiments can be performed with trial and error. Therefore, the laboratory X-ray experiments complement synchrotron X-ray ones. In this report, we describe the electrostatic levitation system for the observation of high temperature liquid structures and present the results of a preliminary application to the atomic structure analysis by X-ray diffraction measurements.

1.2 Electrostatic Levitator for the Structural Analysis by X-Ray Diffraction Technique The design of the present apparatus was based on an electrostatic levitator developed by Rhim et al. [7]. However, the present levitator was optimized for the liquid structure analysis of high-temperature melts by X-ray diffraction technique. The apparatus consists of a vacuum chamber, a sample position control system, and a sample heating source. The sample, charged by electronic emission, is levitated by applying an electrostatic field (typically 20–30 kV cm−1 for metallic materials) between two electrodes. To prevent the electrical breakdown, the electrodes are contained in a chamber that is evacuated to a level of vacuum lower than 1 × 10−4 Pa by a turbo molecular pump attached directly to the side of the chamber. Figures 1.1 and 1.2 illustrate the side and top views of the chamber, respectively. The chamber has a cylindrical shape (height, 200 mm; diameter, 200 mm) and comprises several view ports. A thin sapphire window (thickness, 0.5 mm; diameter, 17 mm) enables the incident X-ray beam to reach the sample. With the use of a rectangular and curved beryllium window, the intensity of the X-rays diffracted by the levitated sample can be detected over a wide angle. The available range of 2θ is −5◦ to 80◦ , which is wider than that previously reported [44]. Sufficiently wide −1 Q range (Q ∼ 11.5 ˚ A ) can be obtained even for laboratory X-ray source (Mo Kα). Five silica glass windows, located on the top of the chamber, are used, along with mirrors inside the chamber, for the position control system and the sample observation by a camera. A ZnSe window (or lens) in the middle of the top plate was used for the sample heating by a CO2 laser (wavelength, 10.6 μm; max. power, 240 W). A glass window on the side of the chamber was employed for the temperature measurement by a single-color pyrometer. Two valves located on the top plate acted as an air lock that is equipped to insert samples without breaking the vacuum.

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j A i

k l e B

b

f c

d

a

g

h

Fig. 1.1. Side view of the chamber for electrostatic levitation. a, levitated sample; b, upper electrode; c, lower electrode; d, side electrodes; e, ceramic support; f, mirrors; g, positioning rod; h, solenoid; i, beam expander; j, He–Ne laser; k, position detector; l, ZnSe window; A, beam path of heating CO2 laser; B, beam path of positioning He–Ne laser a

e

f

A

d d B b D e

c

Fig. 1.2. Top view of the chamber for electrostatic levitation. a, sapphire window for the incident X-ray beam; b, glass window for the pyrometer; c, beryllium window for diffracted X-rays; d, mirrors for the He–Ne laser for positioning in X-Z directions; e, mirrors for the He–Ne laser for positioning in Y direction; f, mirror for CCD camera; A, path of incident X-rays; B and C path of He–Ne lasers; D, path of pyrometer

1 Electrostatic Levitation for Liquid Structural Analysis

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The design of the electrodes is of utmost importance for electrostatic levitators. In our levitator, there are two main electrodes for vertical and horizontal control and four side electrodes for additional horizontal control. The main electrodes consist of two parallel disks. The upper electrode (40 mm diameter), which is connected to a high voltage amplifier, is suspended from the top plate by using insulating ceramic rods. The upper electrode has a spherical end, which generates a concave electrostatic field that helps the position of sample to stabilize laterally. In addition, a hole (3 mm diameter) in its center is available for the sample heating by the CO2 laser. The lower electrode (20 mm diameter) is electrically grounded and possesses a hole, which is available for sample handling by a positioning rod. This rod can be moved up and down from the outside of the chamber to set the initial position of sample. Four small spherical electrodes are distributed around the lower electrode for additional control of the sample position along the horizontal direction. To maintain stable sample levitation, the voltage of the electrodes is controlled actively coupled with a position sensing system, a computer, and highvoltage DC amplifiers. The position of the levitated sample is detected by two sets of position sensors and associated He–Ne lasers. In each set, the expanded He–Ne laser beam (10 mm diameter) illuminates the sample and its shadow is projected on the position detector, which is located on the opposite side of the He–Ne laser. The computer receives from the position detectors an electric signal that corresponds to the sample position. It then calculates the control signal by using a PID control scheme and sends the proper information to the voltage amplifier that changes the voltage of the electrodes. By continuing this sequence at a feedback rate of 1,000 Hz, the sample can maintain a fixed position. The position control system used in this study is similar to that reported in this literature [7], though the optical paths for position sensing are modified because of the constraint of X-ray scattering facility. The He–Ne lasers and the position detectors are located on the top plate, and therefore, the optical paths of the lasers are bent twice by mirrors. This optical configuration offers a wide observation view of the sample as well as the miniaturization of the chamber. These enable us to set up the experimental configuration easier at the synchrotron radiation facility.

1.3 Experimental Electrostatic levitation is, in principle, applicable to a wide variety of materials because all charged materials can be levitated by the action of electrostatic forces. For the first experiment, zirconium was selected and the structural analysis of its liquid phase was carried out by high-energy X-ray diffraction measurement at SPring-8, which is the third generation synchrotron radiation facility in Japan. Similar experiments were performed for molten silicon and alumina samples by using a laboratory X-ray source.

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For the present experiments, the typical sample size was about 2 mm in diameter. Spherical zirconium samples were prepared in the following manner: 99.5 mass% pure zirconium wire was cut into 30–32 mg pieces; these pieces were melted with a diode laser (wavelength, 808 nm; max. power, 200 W) in a glove box filled with purified argon; the melted pieces took a spherical shape spontaneously because of the surface tension. Spherical samples of silicon and alumina were made similarly. Heating is the most delicate task on the electrostatic levitation of sample since the sample charge has a tendency to decrease due to the evaporation of absorbed gas or metallic oxides from its surface. In particular, heating of the sample from room temperature must be done carefully because the discharge of sample starts at a temperature of about 800 K. However, the charge can be increased through electronic emission if the sample reaches a temperature at which thermionic emission dominates (∼1,500 K in the case of metals). To overcome these difficulties, the “hot launch” method [48] was used. To initiate levitation, the sample was heated for the removal of surface oxides. When it reached ∼1,500 K, at which thermionic emission dominates, high voltage was applied to the electrodes and the feedback control system was activated. Once levitated, the sample could then be brought to temperatures beyond the melting point or be maintained under undercooled conditions for hours. The aforementioned method was adopted for the levitation of liquid zirconium and alumina. However, low melting point materials (e.g., silicon) have a tendency to stick to the positioning rod and, therefore, the heated sample must be tossed while heating. In the present apparatus, a small solenoid, which created vibrations, was fixed at the lower part of the positioning rod. The solenoid was activated remotely during the monitoring of the sample temperature. The temperature of the sample was measured with the use of a single-color pyrometer. The emissivity of the sample is necessary to obtain the exact value of the temperature. However, the emissivity strongly depends on the sample condition, the sample size, the focus of collimation lens of the pyrometer, and the transparency of the window of the chamber. In the present research, the emissivity calibrated referred to the melting points of zirconium and silicon. The undercooled liquid state was established simply by decreasing the laser power. After a certain depth of undercooling was reached, the sudden recalescence of the sample, due to the release of the latent heat of fusion, was observed on solidification. Therefore, the undercooled liquid state was confirmed by monitoring the signal of the pyrometer. The laser power was controlled to keep the temperature of the sample constant. The high-accuracy measurement of the liquid structure is one of the major purposes of this research. The two-axis diffractometer is the most typical instrument used nowadays for the diffraction experiments. The X-ray source was selected by considering the absorption coefficient of the material. For zirconium, which has a high absorption coefficient, high-energy X-rays (113 keV) from the BL04B2 of SPring-8 were used. For silicon and alumina, the laboratory X-rays from a Mo rotation target were sufficient to carry out preliminary

1 Electrostatic Levitation for Liquid Structural Analysis

9

diffraction experiments. The size of the incident beam was 0.7 mm in width and 2.0 mm in height for the synchrotron radiation X-rays, and 3 × 3 mm2 for the laboratory X-rays. The incident beam was collimated by the slit and delivered to the vacuum chamber through a sapphire window. The angular dependence of the intensity of the X-rays diffracted from the sample was measured in transmission geometry by a germanium detector or a proportional counter with a graphite monochrometor. Slit collimation eliminated the scattering from the windows on the chamber. This configuration is very helpful in performing the precise measurements of the diffraction only from the sample. The intensity of diffracted X-rays was acquired in each diffraction angle by the step scan method. The diffraction data was collected over a 2θ range of 0.3◦ –25◦ for measurements by synchrotron radiation and 0.5◦ –80◦ for measurements by −1 laboratory X-ray one. The obtained Q range of S(Q) was 0.3–24.7 ˚ A in the −1 former case and 0.08–11.4 ˚ A for the latter case. The duration of acquisition of data for each diffraction angle was greater than 5 s, which is sufficiently long with high statistics. To obtain the static structure factor of liquids, data correction of the absorption, background, polarization, and multiple scattering must be performed [1–3]. For the laboratory experiments, the cross-sectional area of the sample is smaller than that of the X-ray beam, and the correction of absorption and polarization can be performed by normal methods [2]. Though the width of incident X-rays (0.7 mm) is narrower than the sample diameter (2 mm) in the synchrotron radiation experiments, the influence of total angular dependence of the absorption and geometric factor was negligibly small, because diffraction experiments could be performed with rather small scattering angles for high-energy X-ray experiments. In addition, it is worth mentioning that the absorption coefficient itself is very small for high-energy X-rays (mass absorption coefficient is 0.673 cm2 g−1 for zirconium [49]). This implies that the contribution of absorption correction is extremely small. The contribution of multiple scattering has been mentioned for the structural analysis of disordered matter not only for the neutron scattering experiments but also for X-ray diffraction measurements. In the present study, the sample was spherical and quite small. We evaluated the contribution of double scattering compared with that of the single scattering by the method reported by Warren [1]. The ratio of the double scattering to the single scattering in the present case was less than 1%, and therefore, the contribution was neglected. After the correction of the absorption [49], background, and multiple scattering, the contribution of Compton scattering [50] was subtracted and then the X-ray static structure factor [51], S(Q), was derived from the corrected coherent intensity, I(Q), based on the equation  2  2     (1.2) I(Q) =  f (Q)  [S(Q) − 1] + f (Q) , where the angular brackets represent averages over all atoms and f (Q) is the atomic form factor [52].

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1.4 Results and Discussion The zirconium, silicon, and alumina samples were representative materials for metals, semiconductors, and ceramics, respectively. All these samples could be levitated successfully in their molten states and could be maintained at a fixed temperature for more than 1 h, which was sufficient for the measurement of X-ray diffraction. The fluctuation of the sample position was less than 0.1 mm for all materials during measurements. The diffracted X-rays were counted for 5 s for each diffraction angle. In addition, in the case of the laboratory X-ray facility, the area of the incident X-ray beam (3 × 3 mm2 ) was much larger than the cross-sectional area of the sample. Therefore, the sample fluctuation did not affect the diffraction data. The diffraction patterns for all three materials are shown in Figs. 1.3–1.5. As can be seen in Figs. 1.3 and 1.4, diffraction −1 from the empty chamber (background) was almost negligible at Q > 1 ˚ A . This was attributed to the use of vacuum and proper shields to the detector.

Intensity(arb.)

5 4 3 2125K(mp)

2 2035K(mp-90)

1 back ground

0 0

5

10 2θ(deg)

15

20

Fig. 1.3. High-energy synchrotron X-ray diffraction pattern of liquid and undercooled zirconium and background

Intensity(count)

3000 2000 1000

Si back ground

0 0

20

40 2θ(deg)

60

80

Fig. 1.4. X-ray diffraction pattern of liquid silicon at the melting point and background

1 Electrostatic Levitation for Liquid Structural Analysis

11

4

1.2x10

1.0 I(count)

0.8 0.6 0.4 0.2 0.0

0

20

40 2θ(deg)

60

80

Fig. 1.5. X-ray diffraction pattern of liquid alumina at the melting point 5

4 MC

S(Q)

3

2125K(m.p.)

2

2035K(m.p.-90K)

1

0

0

5

10

15

20

Q(A-1)

Fig. 1.6. Static structure factor of normal and undercooled liquid zirconium obtained from high-energy X-ray diffraction experiment and Monte Carlo simulation (2,125 K); normal, melting point; undercooled, the temperature of 90K below than the melting temperature

This small error from the background correction in the present experiments is remarkably different from the case of conical nozzle levitation. Thus the combination of containerless conditions and high vacuum enabled us to obtain the very reliable liquid structure. The static structure factor S(Q) of liquid samples can be obtained from the diffraction intensity. The static structure factors shown in Figs. 1.6–1.8 demonstrate that we have succeeded in performing precise observations of the liquid structure with the present electrostatic levitation apparatus, not only for synchrotron radiation X-rays but also for laboratory X-ray source. The liquid structures of the materials investigated in this study have been measured

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1.0 0.8 0.6 0.4 0.2 0.0

0

2

4

-1

6

8

10

Q(A )

Fig. 1.7. Static structure factor of liquid silicon at the melting point 2.0

S(Q)

1.5 1.0 0.5

0.0

0

2

4

6

8

10

12

-1

Q(A )

Fig. 1.8. Static structure factor of liquid alumina at the melting point

by means of other types of levitators coupled with neutron or X-ray scattering. Our experimental results are in good agreement with those previously published(Zr, [29, 31]; Si, [26–28, 39]; Al2 O3 , [16, 18]). However, the quality of data, such as the Q range of S(Q) and low background, was significantly improved. For example, the S(Q) of liquid zirconium was observed in the range of Q = 0.5–20.0 ˚ A, which was wider than that of a previous research [44]. To demonstrate the quality of the obtained S(Q), the effective pair potential, ueff (r), was deduced for liquid zirconium based on the Modified Hypernetted Chain approximation [53] as follows: ueff (r)/kB T = g(r) − 1 − c(r) − ln g(r) + BHS (r, η),

(1.3)

where c(r) is the direct correlation function, kB is the Boltzmann constant, and T is the temperature. The c(r) was calculated from S(Q) as follows:  

1 1 1− Q exp(−iQ · r) dQ (1.4) c(r) = 2πρr S(Q) The reliability of S(Q) in the small Q region is quite important for this calculation because the Fourier transform of the term, 1/S(Q), must be calculated.

1 Electrostatic Levitation for Liquid Structural Analysis

13

10 8

u(r)/kBT

6 4 2 0 0

2

4

6

8

10

12

r(A) Fig. 1.9. Effective pair potential of liquid zirconium (2,125 K) based on modified hypernetted chain approximation

In addition, high-Q data are necessary to reduce the truncation error in the Fourier transform of S(Q) on the evaluation of g(r) and c(r), which is necessary for the precise determination of the repulsive part of ueff (r), as shown in (1.3). The BHS (r, η) is the bridge function of the hard sphere fluid and η is the packing fraction. For the conventional estimation of BHS , the η of liquid zirconium was taken as 0.46, which is generally adopted for the packing fraction at the melting point of liquid metals [54]. The ueff (r) obtained is shown in Fig. 1.9. The validity of obtained ueff (r) was investigated by a Monte Carlo (MC) simulation [55, 56], in which the obtained ueff (r) was employed. The temperature of MC was 2,125 K. The S(Q) derived from MC is in good agreement with the experimental results, as can be seen in Fig. 1.6. The obtained ueff (r) is widely applicable to the evaluation of not only the static properties but also the dynamic properties. For example, transport properties, such as self-diffusion and viscosity coefficients, can be estimated from the combination of ueff (r) and molecular dynamics (MD) simulation. The viscosity coefficient of liquid zirconium has been already measured by the oscillation drop method coupled with the electrostatic levitator [57]. The detail analysis for the viscosity coefficients by the MD simulation by using this ueff (r) is in progress. The evaluation S(Q) of liquid silicon was performed for the laboratory X-ray source. The Q range of S(Q) was rather small for the calculation of g(r) by the Fourier transform shown in (1.1). Nevertheless, the present S(Q) for liquid silicon was sufficiently good as the structure data. For example, the g(r) of liquid silicon was calculated by introducing the present S(Q) into the Reverse Monte Carlo (RMC) simulation [58]. The obtained coordination number of nearest neighbors was 5.9, which agrees well with previous research [35].

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The present S(Q) of liquid alumina shows a better agreement with the results of molecular dynamics simulation [59] compared with the experimental S(Q) reported by Krishnan et al. [59]. This can be explained by the fact that the reliability of our S(Q) in the low Q region is much better than the experiment reported in [59]. It was previously reported that the structure of liquid alumina depends on the ambient atmospheric oxygen concentration [59]. No such behavior was found in our data. Furthermore, our preliminary high-energy X-ray diffraction experiments using a conical nozzle did not show such a behavior. This may suggest that the structural difference of liquid alumina in oxidizing and reducing condition is still open to question. Diffraction experiments on liquid silicon and alumina by synchrotron radiation X-ray source are being planned in a near future. It is desired that different types of materials can be successfully studied by using a single apparatus. For the diffraction experiments, this feature is very advantageous because a common optical set-up and background calibrations can be used. We developed such an electrostatic levitator which is applicable to a wide variety of materials and X-ray sources. Furthermore, we confirmed that the quality of structure data for high-temperature liquids by the presnt apparatus is much better than that in previously published literatures. We believe that we present a reliable apparatus for the structure study of hightemperature liquids and undercooled ones. The present apparatus will be able to perform the experimental analysis of high-temperature melts with high precision and will contribute to the fundamental understanding of the essential feature of liquids in normal and undercooled states.

References 1. B.E. Warren, X-Ray Diffraction (Addison-Wesley, Reading, MA, USA, 1969) 2. Y. Waseda, The Structure Non-Crystalline Materials (McGraw-Hill, NY, USA, 1980) 3. H.E. Fisher, A.C. Barnes, P.S. Salmon, Pep. Prog. Phys. 69, 233 (2006) 4. J.K.R. Weber, S. Krishnan, S. Ansell, S.D. Hixson, T.C. Nordine, Phys. Rev. Lett. 84, 3622 (2000) 5. T.G. Wang, J. Fluid Mech. 308, 1 (1996) 6. I. Egry, A. Diefenbach, W. Dreier, J. Piller, Int. J. Thermophys. 22, 569 (2001) 7. W.-K. Rhim, S.K. Chang, D. Barber, K.F. Man, G. Gutt, A. Rulison, R.E. Spjut, Rev. Sci. Instrum. 64, 2961 (1993) 8. S. Ansell, S. Krishnan, J.J. Felten, D.L. Price, J. Phys. Condens. Matter 10, L73 (1998) 9. N. Jaske, L. Hennet, D.L. Price, S. Krishnan, T. Key, E. Artacho, B. Glorieux, A. Pasturel, M.-L. Saboungi, Appl. Phys. Lett. 83, 4734 (2003) 10. A. Alatas, A.H. Said, H. Sinn, E.E. Alp, C.N. Kodituwakku, B. Reinhart, M.-L. Saboungi, D.L. Price, J. Phys. Chem. Solids 66, 2230 (2005) 11. A. Mizuno, T. Kaneko, S. Matsumura, M. Watanabe, S. Kohara, M. Takata, Mater. Sci. Forum 561–565, 1349 (2007)

1 Electrostatic Levitation for Liquid Structural Analysis

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12. A Mizuno, S. Matsumura, M. Watanabe, S. Kohara, M. Takata, Mater. Trans. 46, 2799 (2005) 13. S. Krishnan, S. Ansell, D.L. Price, J. Non-Crystalline Solids 250–252, 286 (1999) 14. S. Krishnan, D.L. Price, J. Phys. Condens. Matter 12, R145 (2000) 15. H. Sinn, B. Glorieux, L. Hennet, A. Altas, M. Hu, E.E. Alp, F.J. Bermejo, D.L. Price, M.-L. Saboungi, Science 299, 2047 (2003) 16. S. Ansell, S. Krishnan, J.K.R. Weber, J.J. Felten, P.C. Nordine, M.A. Beno, D.L. Price, M.-L. Saboungi, Phys. Rev. Lett. 78, 464 (1997) 17. C. Landron, L. Hennet, D. Thaudiere, D.L. Price, G.N. Greaves, Nucl. Instrum. Methods Phys. Res. B 199, 481 (2003) 18. C. Landron, L. Hennet, T.E. Jenkins, G.N. Greaves, J.P. Coutures, A.K. Soper, Phys. Rev. Lett. 86, 4839 (2001) 19. L. Hennet, I. Pozdnyakova, V. Cristiglio, S. Krishnan, A. Bytchkov, F. Albergamo, G.J. Cuello, J.-F. Brun, H.E. Ficher, D. Zanghi, S. Brassamin, M.-L. Saboungi, D.L. Price, J. Non-Crystalline Solid, 353, 1705 (2007) 20. S. Matsushita, M. Watanabe, A. Mizuno, S. Kohara, J. Am. Ceram. Soc. 90, 742 (2007) 21. A. Mizuno, S. Kohara, S. Matsumura, M. Watanabe, J.K.R. Weber, M. Takata, Mater. Sci. Forum 539–543, 2012 (2007) 22. L. Cristiglio, L. Hennet, G.J. Cuello, I. Pozdnyakova, A. Bytchkov, P. Palleau, H.E. Fischer, D. Zanghi, M.-L. Saboungi, D.L. Price, J. Non-Crystalline Solid 353, 993 (2007) 23. L. Hennet, D. Thiaudiere, C. Landron, J.-F. Berar, M.-L. Saboungi, D. Matzen, D.L. Price, Nucl. Instrum. Methods Phys. Res. B 207, 447 (2003) 24. C. Landron, X. Launay, J.C. Rifflet, P. Echegut, Y. Auger, D. Ruffier, J.P. Coutures, M. Lemonier, M. Gailhanou, M. Bessiere, D. Bazin, H. Dexpert, Nucl. Instrum. Methods Phys. Res. B 124, 627 (1997) 25. Q. Mei, C.J. Benmore, J.K.R. Weber , Phys. Rev. Lett. 98, 057802 (2007) 26. M. Watanabe, M. Adachi, T. Morishita, K. Higuchi, H. Kobatake, H. Fukuyama, Faraday Discuss. 136, 279 (2007) 27. K. Higuchi, K. Kimura, A. Mizuno, M. Watanabe, K. Katayama, K. Kuribayashi, Meas. Sci. Technol. 16, 381 (2005) 28. H. Kimura, M. Watanabe, K. Izumi, T. Hibiya, D. Hollad-Moritz, T. Schenk, K.R. Bauchspeiss, S. Schneider, I. Egry, K. Funakoshi, M. Hanfland, Appl. Phys. Lett. 78, 604 (2001) 29. T. Schenk, D. Holland-Moritz, V. Simonet, R. Bellissent, D.M. Herlach, Phys. Rev. Lett. 89, 075507 (2002) 30. D. Hollad-Moritz, O. Heinen, R. Bellissent, T. Schenk, Mater. Sci. Eng. A 449– 451, 42 (2007) 31. D. Holland-Moritz, T. Schenk, V. Simonet, R. Bellissent, P. Convert, T. Hansen, D.M. Herlach, Mater. Sci. Eng. A 375–377, 98 (2004) 32. J. Brillo, A. Bytchkov, I. Egry, L. Hennet, G. Mathiak, I. Pozdnyakova, D.L. Price, D. Thiaudiere, D. Zanghi, J. Non-Crystalline Solids 352, 4008 (2006) 33. I. Egry, J. Non-Crystalline Solids 250–252, 63 (1999) 34. G. Jacobs, I. Egry, Phys. Rev. B 59, 3961 (1999) 35. G. Jacobs, I. Egry, D. Holland-Moritz. D. Platzek, J. Non-Crystalline Solid 232–234, 396 (1998) 36. C. Notthoff, B. Feuerbacher, H. Franz, D.M. Herlach, D. Hollad-Moritz, Phys. Rev. Lett. 86, 1038 (2001)

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37. T. Volkmann, J. Strohmenger, J. Gao, D.M. Herlach, Appl. Phys. Lett. 85, 2232 (2004) 38. O. Heinen, D. Holland-Moritz, D.M. Herlach, K.F. Kelton, J. Cryst. Growth 286, 146 (2006) 39. T.H. Kim, G.W. Lee, B. Sieve, A.K. Gangopadhyay, R.W. Hyers, T.J. Rathz, J.R. Rogers, D.S. Robinson, K.F. Kelton, A.I. Goldman, Phys. Rev. Lett. 95, 085501 (2005) 40. G.W. Lee, A.K. Gangopadhyay, K.F. Kelton, R.W. Hyers, T.J. Rathz, J.R. Rogers, D.S. Robinson, Phys. Rev. Lett. 93, 037802 (2004) 41. K.F. Kelton, G.E. Lee, A.K. Gangopadhyay, R.W. Hyers, T.J. Rathz, J.R. Rogers, M.B. Robinson, D.S. Robinson, Phys. Rev. Lett. 90, 195504 (2003) 42. G.W. Lee, A.K. Gangopadhyay, T.K. Croat, T.J. Rathz, R.W. Hyers, J.R. Rogers, K.F. Kelton, Phys. Rev. B 72, 174107 (2005) 43. A. Ishikura, A. Mizuno, M. Watanabe, T. Masaki, T. Ishikawa, S. Kohara, J. Am. Ceram. Soc. 90, 738 (2007) 44. A.K. Gangopadhyay, G.W. Lee, K.F. Kelton, J.R. Rogers, A.I. Goldman, D.S. Robinson, T.J. Rathz, R.W. Hyers, Rev. Sci. Instrum. 76, 073901 (2005) 45. H. Aoki, P.-F. Paradis, T. Ishikawa, T. Aoyama, T. Masaki, S. Yoda, Y. Ishii, T. Itami, Rev. Sci. Instrum. 74, 1147 (2003) 46. T. Masaki, T. Ishikawa, P.-F. Paradis, S. Yoda, J.T. Okada, Y. Watanabe, S. Nanao, A. Ishikukra, K. Higuchi, A. Mizuno, M. Watanabe, S. Kohara, Rev. Sci. Instrum. 78, 026102 (2007) 47. S. Kohara, K. Suzuya, Y. Kashihara, N. Matsumoto, N. Umesaki, I. Sakai, I. Nucl. Instrum. Methods Phys. Res. Sect. A 467, 1030 (2001) 48. T. Ishikawa, P.-F. Paradis, S. Yoda, Rev. Sci. Instrum. 72, 2490 (2001) 49. S. Sasaki, X-ray Absorption Coefficients of the Elements (Li to Bi, U). KEK Report 90-16 (National Laboratory for High Energy Physics, Japan, 1991) 50. J.H. Hubbell et al., J. Phys. Chem. Ref. Data, 4, 471 (1975) 51. D. Waasmaier, A. Kirfel, Acta Crystallogr. A 51, 416 (1995) 52. T.E. Faber, J.M. Ziman, Philos. Mag. 11, 153 (1965) 53. M.W.C. Dharma-wardana, G.C. Ares, Phys. Rev. B 28, 1701 (1983) 54. M. Shimoji, Liquid Metals (Academic Press, London, 1977) 55. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953) 56. W.W. Wood, F.R. Parker, J. Chem. Phys. 40, 517 (1981) 57. P.-F. Paradis, T. Ishikawa, S. Yoda, Int. J. Thermophys. 23, 825 (2002) 58. R.L. McGreevy, L. Pusztai, Mol. Simul. 1, 359 (1988) 59. S. Krishnan, L. Hennet, S. John, T.A. Key, P.A Madden, M.-L. Saboungi, D.L. Price, Chem. Mater. 17, 2262 (2005)

2 Viscosity and Density Measurements of High Temperature Melts Yuzuru Sato

2.1 Introduction Since the viscosity and density are most fundamental properties for any fluids, many efforts to obtain reliable values have been made. However, the measurements are not so easy, especially at high temperature in molten state. The high temperature melts are typically classified into molten metals, molten salts, and molten oxides. They appear in many industrial processes, for example, steelmaking, nonferrous metallurgy, aluminum smelting, foundry, glass making, etc. The adaptable methods for the measurements should be chosen carefully by considering some physical and chemical properties of the melt. Iida published the review on the properties including viscosity and density of molten metals [1], and the comparison among the viscosities of molten iron reported by many researchers showed considerable difference of several dozen percent. The viscosity value is in considerably wide range depending on the groups of the melts, for example, in general low for molten metals and high for molten silicates, including slag and glass, and the difference reaches more than ten orders by reflecting the difference in the melt structure. On the other hand, density is mainly depending on atomic mass and not so different to each other because of not so big difference in molar volumes of the components. Various methods for viscosity and density measurement were also introduced [2] and also the viscometries were summarized [3].

2.2 Viscosity Measurement Viscosity of the liquid decreases with an increase of temperature. The temperature dependence obeys Andrade law, which is similar to Arrhenius behavior as shown in (2.1). E . (2.1) η = A exp RT

18

Y. Sato

The typical methods used for viscosity measurement are a capillary method, an oscillating method, a rotating method, etc. Although it is difficult to adapt them to some liquids with non-Newtonian behavior, there are almost no problems because most high temperature melts show Newtonian behavior. The capillary method, which is used as the prescriptive method at room temperature, is simple and very precise in relatively wide viscosity range, although the application at high temperature is limited due to the problem in the use of refractory materials for apparatus and the difficulty for detecting meniscus. However, the successful application was reported on high temperature molten salts [4]. The oscillating method, especially an oscillating cylinder method [5], is also precise and sensitive although the measurable viscosity is limited in lower range. The big advantage of oscillating method is in wide selection of the container material. Therefore, it is used for high temperature melts with low viscosity such as most of molten metals and many molten salts. The rotating method [6] is suitable generally for high viscosity range because it contains power train as the driven device with friction loss. The method is suitable for measuring the viscosity of molten glass and slag. 2.2.1 Capillary Method Principle of the capillary method is simple. The viscosity, η is expressed by well known Hagen–Poiseuille’s equation (2.2) in the case of vertical type of the capillary viscometer. η=

mρV 1 πr4 ρgh t− , 8(L + nr)V 8π(L + nr) t

(2.2)

where ρ is the density of liquid, r and L are the radius and length of the capillary, h is the effective height of the liquid column, V is the volume of the liquid that has flowed, g is the gravitational acceleration, t is the time for the flow out of liquid, m and n are constants. For an identical viscometer, the coefficients of right hand terms in (2.2) except t and ρ are constant; therefore (2.2) is rewritten to (2.3) by introducing kinematic viscosity, ν: ν = η/ρ = C1 t − C2 /t.

(2.3)

As the determination of the constants, C1 and C2 based on the apparatus dimensions in (2.2) are not realistic. They are usually calibrated by using a standard liquid of known kinematic viscosity through the measurement of the time for it to flow out of the liquid. Distilled water is most popular as a standard liquid for low viscosity due to the well known kinematic viscosity. Figure 2.1 shows a capillary viscometer made of quartz for high temperature molten salts [4]. The melt under measurement is sealed completely inside. Then the melt with relatively high vapor pressure is allowed even if it is molten AlCl3 with considerably high vapor pressure and strong hygroscopicity [7]. The inner diameter and the length of the capillary were about 0.4 and 80 mm, respectively. Therefore, Reynolds number in the capillary was less

2 Viscosity and Density Measurements of High Temperature Melts

19

Filtration chamber Quartz frit Connection tube to be sealed Hole for flow of melt Funnel Upper fiducial mark Timing bulb Lower fiducial mark

Capillary

Slot for thermocouple Outlet for melt of suspended level type Melt

Fig. 2.1. Capillary viscometer made of quartz

than 100 to maintain the stable layer flow. The inverse funnel at the end of capillary is a device to reduce the surface tension effect, which gives the error on the effective height. Typical capacity of a timing bulb is about 3 cm3 . The time for flowing out is measured as a time interval between the meniscus passes through two fiducial marks. The viscometer was made available by the combination with a transparent “Gold Furnace” in which the temperature uniformity within 0.5 K was kept around the viscometer and the meniscus detection was possible visually as shown in Fig. 2.2. The measurement can be repeated by taking the liquid back to the timing bulb from the bottom by rotating the whole apparatus together with the furnace. The reproducibility of the measurement was within 0.2 s, which was corresponding to 0.05%–0.2% of the flow time. The total error accompanied in the measurement was reported not exceeding 1% and the viscosity determination was highly reliable. The viscosities measured of alkaline halides are shown in Fig. 2.3. The results of fluorides were measured with oscillating viscometer [8]. The feature of the viscosities of molten alkaline halides are similar to each other, especially for common anion. As mentioned earlier, the capillary viscometer can be improved to be adapted to high temperature molten salts. However, it has limitation as being made of quartz and cannot be adapted to erosive melt such as molten fluorides, carbonates, or slag. Furthermore, it is hardly adaptable to molten metals because of the difficulties to remove trace of oxide film on the metals or the reactivity with active metals such as aluminum or magnesium.

20

Y. Sato

Adjusting screw

Quartz supporting tube with silica radiation baffles

Halogen lamp

Capillary viscometer

Thermocouple

Gold furnace

Axis of rotation

Steel frame

Li

l

Fig. 2.2. Whole apparatus of high temperature capillary viscometer

Cl

Li l C

l

Cs

Kl

KF

lN aB KBr

r

0.1

N

al

Li Na K Rb Cs

Na

Cl KC

log h

Rb

Li

F Rb

RbCsC l l

F Na

0.2

C Rb sF Br

Cs

Br

Br

Li

F

0.3

0

Fluoride Chloride Bromide Iodide

-0.1 0.8

1.0

1.1

T -1,

10-3K-1

0.9

Fig. 2.3. Viscosities of alkaline halides

1.2

1.3

2 Viscosity and Density Measurements of High Temperature Melts

21

2.2.2 Oscillating Method Principle of the oscillating viscometer is based on the internal friction between the hypothetical layers assumed in the liquid. The oscillation is created by giving the alternated rotational motion to axially symmetrical vessel containing liquid or vibrating the plate immersed in the liquid. The former of suspension type by a wire is popular and found to be precise. The amplitude of the rotational oscillation caused in the liquid is attenuated by the internal friction if no external force is applied and the attenuation goes along the rule of logarithmic decrement. However, to obtain the numerical solution of the viscosity based on the logarithmic decrement measured is very difficult. Therefore, the calibration method based on “Knappwost’s relationship” [9] has been used previously. In this method, some reference calibration liquids with precisely known viscosity and density are required. However, it is considerably difficult to find the adequate reference liquids at high temperatures, and also it is easily predicted that the calibration curve is not so reliable if considerably different liquids such as molten metals and organic liquids are used for the calibration of identical apparatus. Therefore, it is not expected that precise and reliable determination of the viscosity can be achieved by using the calibration method. On the other hand, noncalibration methods usually called as absolute methods are recently used in most cases. Their principle is to determine the viscosity based on the mathematical calculation using logarithmic decrement and the period of oscillation, together with other physical parameters such as the mass and density of the liquid, the moment of inertia, the inner diameter of the cylindrical vessel, etc. The absolute methods are, in general, very complicated mathematically. However, it is great advantage that no reference liquids are required. Roscoe’s equation [10] shown in (2.5) may be most popular absolute algorithm. It was confirmed that the equation gave good reproducibility to obtain the viscosity even if the parameters are changed, for example, considerable changing the logarithmic decrement and the period of oscillation by alternating the dimension of the crucible or the suspension wire. The equation for cylindrical vessel is described as follows: 2

1 Iδ , (2.4) η= πR3 HZ πρτ  





4R 1 3 9R a2 3 1R a0 − + + + +0 Z = 1+ 4H 2 πH p 8 4H 2p2 

45R a4 63 − + ···, − 128 64H 4p4  12  12 √ √ 1 + Δ2 + 1 1 + Δ2 − 1 − (1 + Δ) , a0 = (1 − Δ) 2 2  12  √  12 √ a2 1 + Δ2 + 1 1 + Δ2 − 1 + , a4 = √ , a2 = 2 2 1 + Δ2

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Y. Sato

p=

πρ ητ

 12 R,

Δ=

δ 2π

δ = ln

Ai Ai+1

 ,

H=

W , πR2 ρ

where η, δ, τ , A, I, R, W , and ρ are viscosity, logarithmic decrement, period of oscillation, amplitude of the oscillation, moment of inertia of whole suspension system, inner radius of crucible, mass, and density of the melt. To solve the equation analytically is difficult due to the recursion of the viscosity given in the aforementioned equation. Therefore, the successive approximation is used to obtain the viscosity numerically. In the actual calculation, the logarithmic decrement is given as a difference between the values obtained for the crucible containing melt and the empty crucible, respectively. As the parameters experimentally determined from the oscillation are the logarithmic decrement and the period of oscillation, it is important how to determine them. Previously, the logarithmic decrement had been determined through an observation in the amplitude of oscillation by using a lamp scale or a recorded profile on the film, etc. because PC was not available until 1970’s. Now the time interval measurement [5] is simpler than direct measurement of the amplitude of the oscillation and is sufficiently precise compared with others. An oscillatory curve with decay expressed by (2.5) is shown in Fig. 2.4. L and −L are the positions of the detectors. C is the deviation of the center of oscillation from the center of detectors. The period, τ , and the logarithmic decrement of oscillation, δ, are defined by (2.6) and (2.7). Y = A exp(−Bt) sin ωt + C,

(2.5)

τ = 2π/ω,

(2.6)

δ = 2πB/ω = Bτ,

(2.7)

Y

L C

t

-L

t1 t2 t3 t4 Fig. 2.4. Oscillating curve with decay and the time intervals measured in a period

2 Viscosity and Density Measurements of High Temperature Melts

23

where ω is an angular frequency and B is a constant related to the decay. Four time intervals, t1 through t4 , measured in a period were used for determining τ and δ. It is difficult to obtain analytical solutions for τ and δ if C is not zero. Following approximation procedure is available. At first, oscillatory curve without decay is assumed as expressed by (2.8). Therefore, the primary period of the oscillation, τ  is given in (2.9). Y = A sin ωt + C,

(2.8)

τ  = t1 + t2 + t3 + t4 .

(2.9)

The constants A and C in (2.8) are defined by (2.10) and (2.11), respectively, for each period. A = 2L/[sin{π(0.5 − t1 /τ  )} − sin{π(1.5 − t3 /τ  )}],

(2.10)

C = −A[sin{π(0.5 − t1 /τ  )} + sin{π(1.5 − t3 /τ  )}]/2.

(2.11)

A primary constant B  is calculated with (2.12) using a series of Ai , which is the amplitude obtained for sequential individual periods. And a primary logarithmic decrement, δ  , is given by (2.13). B  = (1/τ , ) ln(A/Ai+1 ),

(2.12)

δ = B  τ  .

(2.13)

This set of first approximation values, δ  and τ  includes a considerable error depending on the logarithmic decrement as shown in Fig. 2.5. The error can be simulated by using a sample data of the time intervals, which was 1.0 Logarithmic decrement 0.0

Period

-1.0

Log (Error/%)

1st Approx. -2.0 -3.0 -4.0 2nd Approx.

-5.0 -6.0 -7.0

0.001

0.01

0.1

Logarithmic decrement

Fig. 2.5. Error contained in primary and secondary approximation values through the approximation method

24

Y. Sato

Y

t⬘

L C

t

-L t Fig. 2.6. Relation between true and apparent periods in the wave with decay

generated with given parameters of fixed C of 10 mm, 2L of 100 mm, and various B, although they are relatively worse in condition than that of the real measurement. The deviations of the first approximate values, δ  and τ  , from true values were about from 10−1.5 % to 10−0.5 % for the practical range of logarithmic decrement in the measurement. They are only available when the decay is impractically small. The effect of decay on the oscillation was taken into account in the next step. The oscillatory curve with decay and the relationship between the primary periods given by (2.9) and the true period are shown in Fig. 2.6. A primary period, τ , is always longer than a true period, τ . The true period can be determined if a difference (τ  − τ ) is estimated precisely. Figure 2.7 shows the overlapped consecutive half period with decay. The curve is apparently distorted sine wave and is vertically similar to the consecutive wave. The difference, (τ  − τ ), is apparently equal to ta from Fig. 2.6, although ta is not measurable. However, the following relationship in (2.14) is helpful. t1,i − t1,i+1 = ta + tb ,

(2.14)

ta can be determined if the measurable left hand term, (t1,i − t1,i+1 ), is reasonably divided into ta and tb . The times, at which a differential coefficient of a wave equal to zero, are the same for any period as shown in Fig. 2.7. Therefore, the left hand term in (2.14) was divided at the time mentioned above to distribute the left hand term into ta and tb by assuming that the ratio of ta to tb is equal to the ratio of divided parts. As a result, a second approximate value of τ  is obtained as (2.15). τ  = τ  − ta = τ  − (t1,i − t1,i+1 ) tan−1 (ω/B)/π,

(2.15)

2 Viscosity and Density Measurements of High Temperature Melts

25

Y

L C ta 1 -1 w w tan ( B )

tb p w

-

t

1 -1 w w tan ( B )

Fig. 2.7. Overlapped consecutive half waves with decay

where the parameter, τ  is taken from (2.9) and ω/B = 2π/δ is taken from δ  in (2.13), which are the first approximate values. The latest period, τ  , is used to calculate the second approximate value of logarithmic decrement through (2.10)−(2.13) again. The second approximate values of the period of oscillation and the logarithmic decrement are sufficiently precise. The error accompanied by the approximation is about from 10−4.5 % to 10−3.5 % as also shown in Fig. 2.5 in the same conditions earlier. Therefore, this set of second approximate values was employed for calculating the viscosity. An example of the suspension and oscillating system [5] is shown in Fig. 2.8. A cylindrical vessel is suspended by a wire through a mirror and an inertia disk made of aluminum. The rotational force is given to the inertia disk electromagnetically. The oscillation is started by removing the rotational force and is attenuated by the viscosity resistance without the change of period of oscillation. The reflected light from a laser passes through two photo detectors to measure the time intervals. Figure 2.9 shows an example of the viscometer. The uniform temperature profile along the crucible is very important, especially at high temperature to avoid the convection flow and inhomogeneity in the melt. The furnace in the figure consists of three independent heaters, and many thermal shielding plates made of refractory metal are installed above and under the crucible suspended in an inner tube to obtain good temperature profile. Additionally, it is also important to keep the temperature of suspension wire constant, because the modulus of rigidity varies with temperature and affects the measurement. As the vessel with simple geometry is used in this method, there is a great advantage. It is the wide flexibility on the choice of materials for the vessel. The vessel made of metal or various refractory ceramics are available for molten salts and molten metals for high temperatures. It should be noted that the viscosity range measurable in the

26

Y. Sato Data processing computer (PC-9801)

wire(Pt-13%Rh) mirror(Al)

Time Counter

photo transistor inertia disc laser light

oscillation initiator

colimator lens

W rod convex lens

laser source

Crucible

Fig. 2.8. Suspension and detection system in the oscillating viscometer

oscillating vessel method is limited to low viscosity, typically less than about 30 mPa s due to the principle of the method. Therefore, the oscillating method cannot be applied to high viscosity melts such as molten slag or glass. However, most of molten salts and molten metals have low viscosity typically less than 10 mPa s and the oscillating method is successfully adaptable to them. The viscosities of molten metals and semiconductors measured with the oscillating method are shown in Fig. 2.10. The viscosities of molten metals generally show relatively low values and good Arrhenian behavior, including semiconductors was obtained. In general, viscosity and activation energy are high as the melting temperature increases. 2.2.3 Rotating Method As the viscosity is originated from the friction between the hypothetical layer in the liquid, torque measurement is principle and the method is prevailing. Therefore, the measurable torque created by the liquid between coaxial inner cylindrical column and outer cylinder with different angular velocities is

2 Viscosity and Density Measurements of High Temperature Melts

9

Fig. 2.9. Whole view of the oscillating viscometer for high temperature

Log (Viscosity, h / mPa.s)

1.00 0.80

Fe

0.60

Ni

Fe Co Ni Au Cu Ag Zn Al Cd Pb Sb Bi Sn Tl In Ga Si Ge InSb GaSb

Co Au Ag Zn Cd Tl

Cu

0.40

Bi

Pb

Sn

0.20 InSb

In

Al Sb

0.00 GaSb

Ga

−0.20 Si

−0.40 0.40

Ge

0.60

0.80

1.00

1.20

1.40

1.60

1.80

(Temperature, T)−1 / 10−3 K−1

Fig. 2.10. Viscosities of molten metals and semiconductors

2.00

27

28

Y. Sato

determined to obtain the viscosity of the liquid. In the rotating viscometer, the viscosity is represented by following (2.16). η=

T (r12 − r22 ) , 4πhr12 r22 (ω1 − ω2 )

(2.16)

where T , r, h, and ω are torque, radius of cylinders, length of cylinders, and angular velocity of cylinders. Suffixs 1 and 2 mean outer and inner cylinders. Two different types of the apparatus are used for the rotating method. One consists of a rotating inner cylindrical column and a fixed outer cylinder which allows infinite radius. In this type, the rotating inner column accepts the roles of both creating and measuring torque. Therefore, measurable range of the torque is, in general, in considerably large viscosity due to the mechanical friction loss along the route from the column to the device of measurement and the value obtained is generally not so precise. This type of the viscometer is relatively simple for the construction of apparatus, which makes the measurement easy and most of the apparatus commercially available are of these types. It is adaptable to measure the viscosity not only of the liquid in small crucible but also for large scale of the liquid such as a big pool of molten glass. Another one is more essential and precise. It consists of rotating outer cylinder and fixed inner column that detects the torque. The feature of the type of viscometer is of higher precision than former type due to nonfriction loss for torque measurement, although the dimensions of inner and outer cylinder should be fixed. For the cases of rotating inner or outer cylinder, ω1 = 0 or ω2 = 0 in (2.16), respectively. However, the equation can be adapted only for semiinfinite length of cylinders. Therefore, calibration method is usually used for determining the apparatus constant regarding to the dimensions at various angular velocities by means of standard viscosity liquids such as silicon oils with various viscosities. An example for the outer crucible rotating viscometer is shown in Fig. 2.11 [6].

2.3 Density Measurements As density is the most fundamental property for all the matter in any state and temperature, many methods were propounded in the long history. Typical methods for high temperature melts [2] are an Archimedean method, a pycnometric method, a manometric method, a maximum bubble pressure method, a sessile drop method, a levitation method, a dilatometric method, a float method, a γ-ray transmission method, etc. Among them, the Archimedean method is the most popular and precise one for any liquids even at high temperatures although it is considerably traditional. The levitation method is recently developed for mainly molten metals. It has an advantage of no contamination from the container and limitless in the experimental temperature

2 Viscosity and Density Measurements of High Temperature Melts

29

(a)Differential transformer

Differential transformer

To the torsion wire

Torsion wire

Secondary coil

Oil damper

Coil holder

Thermocouple Water-cooled brass cylinder

Pt-20Rh bob MoSi2 heating element

Primary coil Core

(b)Crucible & Bob

To the rod

10 Molten slag 2

Pt-20Rh crucible Crucible supporter

10

Alumina tube Gas inlet Rotating axis (alumina tube)

4

28 32

28 32 /mm

Water-cooled brass cylinder To the motor

Fig. 2.11. Rotating viscometer for silicate melts To balance

To balance

Sample Crucible Liquid metal sample Buoyant liquid

Sinker Liquid metal sample

(a)

(b)

Fig. 2.12. Principle of direct and indirect Archimedean method

because of the containerless method. Additionally, the maximum bubble pressure method and the levitation method are basically same to those used for also surface tension measurement. 2.3.1 Archimedean Method In the Archimedean method, the buoyancy worked on the immersed sinker in the liquid is measured by using a balance. The buoyancy measurement is relatively easy and considerably precise by using a recent electric balance. It consists of direct and indirect methods as shown in Fig. 2.12 [1]. On the

30

Y. Sato

one hand, in the former, the sinker with known volume is immersed into the immiscible liquid. On the other hand, in the latter, liquid or solid sample contained in a basket is immersed in the liquid with known density. In general, the former is more popular and is widely used due to the simplicity in the experimental setup. The sinker should be made of refractory materials at high temperature such as platinum or tungsten for molten salts, molten slag, or molten glass, and the ceramics for molten metals. For using the combination of molten metal and ceramics sinker, heavy weight should be attached to immerse the sinker by taking account of high density metal. In the case of using electric balance, as a force, F acting on the balance is a product of the gravitational acceleration g and the reading of the balance w, which is corresponding mass, w = F/g. Therefore, the buoyancy force on the sinker with volume V in a medium with density ρ is given as −ρgV . The readings of the balance for the sinker in air and liquid, wair and wliq , are W − ρair V and W − ρliq V , respectively. W is real mass of the sinker. In the case of using suspension wire, additional force caused by surface tension, Fsurf = 2πrγ cos θ appears for immersing in liquid. r, γ, and θ are radius of wire, surface tension of the liquid and contact angle, respectively. Then, the following (2.17) is derived: Δw = wair − wliq = (ρliq − ρair )V − Fsurf /g.

(2.17)

Wetting condition of smaller contact angle decreases apparent buoyancy and vice versa. The radius of suspension wire should be as fine as possible to decrease the surface tension effect. However, to use very fine wire is not easy at high temperature, and the surface tension effect by Fsurf is an essential error source. Therefore, a popular solution is using two sinkers with different volumes and same radius of suspension wire, which contacts the surface of liquid to cancel the surface tension effect. Equation (2.18) is obtained for using large and small sinkers: ρliq − ρair =

Δwlarge − Δwsmall . Vlarge − Vsmall

(2.18)

For this purpose, replacing two sinkers alternately was frequently used at low temperature, or for molten salts and slag, which are not so sensitive against atmosphere. However, to replace the sinkers is very difficult at high temperatures and the control of atmosphere is almost impossible. It is a big disadvantage for oxidizable molten metals. Another way is to use the moving one sinker with two humps vertically to make pseudo condition like two sinkers [11]. But the moving one sinker may introduce position error and need complex devices. Therefore, another apparatus was developed [12]. It contains two sinkers that are suspended from independent balances and are immersed into the melt simultaneously as shown in Fig. 2.13. In this method, replacing the sinkers is not necessary, then closed system can be achieved to control the atmosphere. It is a big advantage for high temperature melts such as atmosphere sensitive molten metals.

2 Viscosity and Density Measurements of High Temperature Melts 1 2 3

4

5 6 7

31

1. Electric balabce (for small sinker) 2. Electric balance (for large sinker) 3. Moving unit (Balance) 4. Pt-Rh ( or Ta) wire 5. Furnace 6. Tungsten weight 7. small sinker 8. Large sinker 9. Crucible 10. Zirconium case (BN) 11. Alumina tube 12. Moving unit (Tube) 13. Thermocouple 8 9 10 11

13

12

Fig. 2.13. Archimedean densitiometer with dual sinkers

2.3.2 Pycnometric Method This method is based on the measurement of the mass with fixed volume by using a pycnometer with known capacity. The pycnometer is made of refractory materials such as ceramics, boron nitride, or graphite and machined to desired shape and dimension. The solid sample with enough volume contained in the pycnometer is melted and the excess volume more than the capacity of pycnometer is let brim over. The remaining melt in the pycnometer is brought out after solidification to measure the mass for determining the density. This method is adequate for molten metals with large surface tension to avoid the leakage through the gap between the body and the cap at high temperature, although the pycnometric method is used as simple device for the measurement of density of liquid or solid at room temperature. This method has an advantage for oxidizable molten metal due to easy atmosphere control and has relatively high precision. However, the experimental procedure is cumbersome because only one density at certain temperature can be obtained in one experiment. Several experiments are required to obtain the temperature dependence for one composition of the sample. And also the experimental handling at high temperature is not necessarily easy. In this method, the choice of adequate material and the machining precision are important. The capacity should be calibrated by means of the liquid with well known density such as mercury at room temperature, and also determined at high temperature using

32

Y. Sato φ 0.8mm

through hole

12mm

6mm screw

Fig. 2.14. Pycnometer made of refractory material

precise data of expansion coefficient, because the direct determination of the capacity at high temperature is difficult and error-prone. Figure 2.14 shows an example of pycnometer made of boron nitride or high purity alumina for measuring the density of molten silicon [13]. 2.3.3 Manometric Method As the height of liquid column under gravitational acceleration is proportional to the liquid density and the pressure applied, density is obtained by measuring the height of liquid column and the pressure. As the direct measurement of the height is difficult, the difference between the heights of two columns connecting in the shape of U-type tube is usually measured. The advantage of using U-type tube is the cancellation of surface tension effect by using uniform inner radius of the tube. For this method, in many cases, two U-type tubes connected in atmospheric path are used. One U-type tube contains high temperature sample melt and other contains reference liquid with well known density at certain temperature. The density of the sample melt is expressed as in (2.19). ρmelt = ρref

Δhref , Δhmelt

(2.19)

where Δh is the height difference in a U-type tube, and the suffixs “melt” and “ref” mean the sample melt and reference liquid. Figure 2.15 shows an example of the apparatus [14]. In this method, significant difficulty is in the detection of the meniscus. A transparent furnace up to about 1,000◦ C was

2 Viscosity and Density Measurements of High Temperature Melts

33

Vac.

Ar

A

D

B

E

C

F

A: Hg reservoir B: Gold furnace C: Molten sample D: Thermocouple E: Thermostat F: Standard sample

Fig. 2.15. Apparatus of manometric method for molten salts

used in the apparatus shown in the figure to make the detection of the meniscus easy. In this apparatus, the material of U-type tube is made of quartz because of the transparency and the easiness of glass blowing to observe the meniscus. Furthermore, the advantage of this method is the closed system to make atmosphere control easy. However, the method is hardly adaptable to molten metals and silicate melts because the small amount of oxide film remaining adheres on the quartz tube and makes the observation difficult, and the molten oxides or other erosive melts attack the quartz. The molten salts without erosive property for quartz such as molten chlorides are preferable for the method. 2.3.4 Maximum Bubble Pressure Method This method is usually adapted to density measurement although it is essentially used for surface tension measurement. The principle is to measure the hydrostatic pressure similar to the manometric method. A nozzle of thin tube is immersed in the melt and inert gas is sent to the nozzle very slowly. Then, a small bubble is formed at the end of the nozzle made of refractory materials as shown in Fig. 2.16. The pressure in the bubble is summation of hydrostatic pressure of the melt and the maximum pressure caused by the surface tension of the bubble when the radius of the bubble is minimum as indicated in (2.20). Pmax = Psurf + ρgh,

(2.20)

34

Y. Sato

2r

Bubble capillary

h

A bubble Fig. 2.16. Bubble formed at the end of capillary immersed in the liquid

where Pmax and Psurf are maximum pressure and the pressure caused by the surface tension. As the hydrostatic pressure is proportional to the immersion depth, the gradient of the relation between the depth and the pressure is a product of gravitational acceleration and the density of the melt as shown. This method is relatively simple and the feature is in easy atmospheric control. Furthermore, it is adaptable to any high temperature melts such as molten metals and molten salts by choosing adequate material for the nozzle. However, it is noted that the measurement may be often interfered by the oxide film in the case of molten metals due to the use of thin nozzle. 2.3.5 Sessile Drop Method and Levitation Method Both these methods are based on the volume measurement of the liquid with known mass and are mainly adapted to metallic liquids that have high surface tension and make spherical droplet on the substrate or under levitation, although these methods are also used for surface tension measurement. The former’s advantage is the relatively simple apparatus and the easy availability to stable droplet under controlled atmosphere. The latter’s advantage is essentially contactless with container and is also actually limitless in increasing temperature as a result of containerless measurement. In the sessile drop method, the substrate is made of refractory material such as the ceramic, which is chosen for not to react with the melt droplet. The profile of the droplet as shown in Fig. 2.17a is taken by using a digital camera and the volume is calculated based on the sectional integration [1]. The problem of this method is uncertainty in the degree of coaxial symmetry of the droplet and in the magnification of the image recorded from real dimension. Therefore, two axial shooting must be necessary to correct the volume calculation and also precise calibration by using reference sample with known shape and dimension is required. A modified one is called as constrained drop method, in which a cylindrical crucible with sharp edge is often used as shown in Fig. 2.17b [15]. This has some advantages in determining the volume. The

2 Viscosity and Density Measurements of High Temperature Melts

X

35

Liquid metal specimen Z Z

X

(a)

Liquid metal specimen

Edge

Vessel

(b) Fig. 2.17. Sessile drop and constrained drop for density measurement

sample melt with excess volume is contained in the crucible. The convex shape on the edge is expected to be coaxial and the calculation of the volume is relatively easy by summating the volume in the crucible and the volume of the convex part, although the problem in the magnification still remains. The levitation methods are classified by the levitation force, for example, electromagenetic, electrostatic, acoustic, and aerodynamic levitations. Former two methods are available under vacuum. Figure 2.18 shows an example of electrostatic levitation [16]. In these methods, small sample, typically 2–3 mm in diameter, is required to maintain near sphere form to make levitation and calculation easy. The major problem of the methods is uncertainty in the temperature measurement because only the optical thermometer is available. Additionally, the rapid oscillation of the droplet and the positional instability accompanied by the levitation make the volume determination difficult. To solve the problems, the temperature is usually calibrated by the recalessence at melting point for pure metals and average of the volumes obtained from many photo images are used. The superior advantage of the levitation method of limitless temperature was demonstrated in the density measurement of molten tungsten by the electrostatic levitation method [16].

36

Y. Sato Pyrometer He-Ne Laser

Pyrometer

Top electrode

YAG Laser Beam

He-Ne Laser

Rotation detector CO2 Laser beams(3) Sample Side electrodes(4)

TelePhoto Camera1

Oscillation detector

Position sensor Beam Splitter

Bottom electrode

Position sensor TelePhoto Camera2

Fig. 2.18. Electrostatic levitation method for thermophysical properties measurement

2.4 Summary Viscosity and density are widely recognized to be the fundamental and important properties for any fluids. However, measurement is more difficult as the temperature increases. Therefore, the data available are not necessarily reliable and much effort to obtain precise values was made by developing new methods and apparatus. As the principles of the measurements are generally well known, new techniques are desired for obtaining reliable values. The methods mentioned in this chapter are not necessarily enough, but it is great pleasure for the author if this chapter is of some help for researchers to study the viscosity and density of high temperature melts.

References 1. T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals (Clarendon Press, Oxford, 1988) 2. R.A. Rapp (ed.), Techniques of Metals Research Vol. IV, Physicochemical Measurements in Metals Research Part 2 (Interscience, New York, 1970) 3. R.F. Brooks, A.T. Dinsdale, P. Quested, Meas. Sci. Tech. 16, 354 (2005) 4. Y. Sato, M. Fukasawa, T. Yamamura, Int. J. Thermophys. 18, 1123 (1997) 5. Y. Sato, K. Sugisawa, D.A Oki, T. Yamamura, Meas. Sci. Tech. 16, 363 (2005) 6. S. Sukenaga, N. Saito, K. Kawakami, K. Nakajima, ISIJ Int. 46, 352 (2006) 7. Y. Sato, Y. Matsuzaki, M. Uda, A. Nagatani, T. Yamamura, Electrochemistry 67, 568 (1999) 8. T. Ejima, Y. Sato, S. Yaegashi, T. Kijima, E. Takeuchi, K. Tamai, J. Jpn. Inst. Metals, 51, 328 (1987)

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9. A. Knappwost, Z. Metallk. 39, 314 (1948) 10. R. Roscoe, Proc. Phys. Soc. 72, 576 (1958) 11. H. Sasaki, E. Tokizaki, K. Terashima, S. Kimura, J. Cryst. Growth 139, 225 (1994) 12. Y. Sato, Y. Anbo, K. Yanagase, T. Yamamura, in CD-ROM Reprint Volume for the 17th European Conference on Thermophysical Properties, 2005 13. Y. Sato, T. Nishizuka, K. Hara, T. Yamamura, Y. Waseda, Int. J. Thermophys. 21, 1463 (2000) 14. Y. Sato, K. Kobayashi, T. Ejima, J. Jpn. Inst. Metals 43, 97 (1979) 15. K. Mukai, Z. Yuan, Mater. Trans. JIM 41, 323 (2000) 16. T. Ishikawa, P.-F. Paradis, S. Yoda, Space Utiliz. Res. 21, 42 (2005)

3 Marangoni Flow and Surface Tension of High Temperature Melts Taketoshi Hibiya and Shumpei Ozawa

3.1 Introduction Marangoni flow plays an important role in the heat and mass transport for highly value-added high-temperature processes, such as crystal growth, welding, casting, and electron beam melting. For silicon single crystal growth, the effect of the oscillatory Marangoni flow on the introduction of growth striation was discussed by Chen and Wilcox for the first time in 1972 [1]. The existence of the Marangoni flow within molten silicon was proved through microgravity experiments in space on board a sounding rocket in 1983 by Eyer et al. [2], who found formation of growth striation in single crystals even under microgravity, where buoyancy-driven flow was suppressed. To explain the Marangoni effect at the melt surface, surface tension is essential. Keene [3] discussed the oxygen contamination in the surface tension measurement and recommended the use of a levitation technique, which is a containerless process and assures the contamination-free condition from measurement devices. It is well known that flow direction in the weld pool is dependent on surface contamination and that this is related to weldability [4, 5]. Flow direction is controlled by the temperature coefficient of surface tension for molten steels; contaminants are oxygen and sulfur. In the electron beam button melting system, the Marangoni flow is dominant because of intense heating at the melt surface [5]. In this chapter, surface tension of high temperature metallic melts is discussed from the viewpoint of the Marangoni effect in the value-added high temperature processes, particularly from the viewpoint of the effect of oxygen and sulfur. Theoretical treatment for oxygen adsorption is also discussed.

3.2 Marangoni Effect on High-Temperature Melts 3.2.1 Definition of Marangoni Flow The Marangoni effect appears on the melt surface where (a) the surface tension gradient exists parallel to the melt surface (Fig. 3.1a), or (Fig. 3.1b) where the

40

T. Hibiya and S. Ozawa

Gas σ small

Δ TH

Gas σ large

σ large

σ small

σ large

Temp Low Liquid

Liquid

Δ TV Temp High

a

b

Fig. 3.1. Definition of the Marangoni effect; (a) thermocapillary effect and (b) the classical Marangoni effect [6]

temperature gradient is normal to the surface. In the latter case, temperature is distributed homogeneously at the melt surface, but the surface temperature is lower than that inside, as shown in Fig. 3.1b. The former is called the thermocapillary or solutocapillary effect, depending on the origin of surface tension difference at the melt surface. The latter, case (b), is a classic definition of the Marangoni effect. For case (a), because of shear stress at the melt surface, a melt surface with large surface tension pulls that with small surface tension. Beneath the surface, flow takes place due to viscosity of the melt. The surface tension gradient is caused not only by difference in temperature, but also by that in concentration or adsorption of impurities. They are called thermocapillary and solutocapillary flow, respectively. For case (b), thermocapillary flow takes place locally, due to fluctuation of temperature at the melt surface with nominally homogeneous temperature. To compensate volume lost by this flow, it flows upward from the bottom at a high temperature. This flow supplies high temperature melt from the bottom. Therefore, once this flow is generated, it maintains the temperature difference at the melt surface to enable continuous flow. The balance of forces at the liquid surface is written as follows to describe a Marangoni flow: −μl

∂σ ∂T ∂σ ∂Ci ∂σ ∂Γ ∂u ∂u − μg = + + . ∂z ∂z ∂T ∂x ∂Ci ∂x ∂Γ ∂x

(3.1)

Here, μl and μg are the dynamic viscosity for liquid and gas phases. u is the velocity; z and x are the axes of the ordinate; T , σ, Ci , and Γ are the temperature, surface tension, concentration of solute at the liquid surface, and number of adsorbed atoms per unit area, respectively. The second term of the right hand equation is related with solute, whose mass transport is rapid between the adsorbed layer and the bulk liquid. The third term corresponds to a condition where atoms or molecules exist only at the interface, or mass transport between the adsorption layer and the bulk liquid is very slow. For example, for molten silicon case, the second term can be neglected, since number of adsorbed oxygen is supposed to be equilibrated with that of oxygen

3 Marangoni Flow and Surface Tension of High Temperature Melts

41

in melt; Ci can be written using Γ through the equilibrium constant. The Magnitude of Marangoni flow is written as follows, depending on the driving force of the flow: ∂σ ΔC · l ∂σ ΔΓ · l ∂σ ΔT · l + + . (3.2) Ma = ∂T μl · κ ∂C μl · κ ∂Γ μl · κ Here, l and κ are the characteristic length of the system and thermal diffusivity, respectively. For thermocapillary flow, the magnitude of the flow is expressed with the first term of the right side equation. 3.2.2 Crystal Growth Almost all melt crystal growth processes are accompanied by a free surface of melts; the existence of a surface tension gradient causes the Marangoni flow. A typical example for the Marangoni flow is that of float zone crystal growth. The existence of the Marangoni flow for the float zone growth of silicon and its effect on crystal quality, that is, the introduction of growth striation, were predicted for the first time by Chen and Wilcox in 1972 [1]. For the float zone case, the free surface is comparatively larger than volume and temperature difference between the hottest part of the molten zone and solid/liquid interfaces caused by the Marangoni flow. This was proven during silicon crystal growth experiments conducted by Eyer et al. under microgravity [2]. Since then, a study of the Marangoni flow of molten silicon has been carried out extensively by the Freiburg group [7,8]. A Japanese group also investigated the Marangoni flow on an experimental basis, particularly from the viewpoint of fluid dynamics under microgravity conditions [9]. Lan and Kou carried out precise work on the Marangoni flow of molten silicon numerically [10]. Since oscillatory Marangoni flow degrades the quality of grown crystals, as shown in Fig. 3.2 [11], the critical Marangoni number is one of the main concerns of Marangoni flow study. For the float zone configuration, a flow mode changes from axisymmetric to three-dimensional steady flow at the fist critical Marangoni number, M ac1 . Above the second critical Marangoni number, M ac2 , flow becomes oscillatory. Cr¨oll et al. experimentally determined M ac2 using partially confined float zone geometry [7]. For the Czochralski growth case, the formation of a network pattern has been observed at the melt surface, and this has been attributed to the existence of a thermal inverse layer [12–14]. A network pattern was reported for the first time for fluoride and oxide melts, as shown in Fig. 3.3. This inverse layer causes not only formation of a cell structure, known as the Benard–Rayleigh cell, but also formation of the cell pattern caused by the Benard–Marangoni–Rayleigh mechanism [15, 16]. For the Czochralski silicon growth, as shown in Fig. 3.4, oxygen O in the melt is provided through the dissolution of a crucible wall made of fused quartz (SiO2 ) and evaporates easily as SiO. The concentration of O in the silicon melt is highest at the crucible wall and then most diluted near the growing crystal. The amount of adsorbed oxygen is believed to exhibit a similar gradient to

42

T. Hibiya and S. Ozawa

feed

solidified last melt zone

grown crystal

seed

10mm Fig. 3.2. Growth striation of silicon single crystal grown under microgravity, where buoyancy flow is suppressed but the Marangoni flow is dominant [11]

Fig. 3.3. Cell structures on the inverse layer of the Czochralski melt [14]

3 Marangoni Flow and Surface Tension of High Temperature Melts

43

Fig. 3.4. Marangoni and buoyancy effects within the Czochralski melt of silicon [6]

Fig. 3.5. Network pattern observed at the Czochralski silicon melt. Arrows show surface flow direction [17]

that of concentration. Since at the crucible wall there exist three phases, that is, SiO2 (solid), Si (melt), and O2 (gas), the system is univariant according to calculation of Gibbs’ number-of-freedom f as shown in 3.3: f = c − p + 2 = 1,

(3.3)

where c is the number of independent components and p is the number of phases. Concentration of O in the silicon melt must be saturated at the crucible wall at a given temperature under equilibrium condition. Melt surface temperature is highest at the crucible wall and lowest at the crystal, where it exhibits the melting temperature of silicon, such as 1,693 K. Thus, there is a surface tension gradient due to temperature, oxygen concentration, and oxygen adsorption [6]. As shown in Fig. 3.5, a cellular pattern appears at the surface of a silicon melt 700 mm in diameter and 200 mm deep during crystal growth; Fig. 3.5

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T. Hibiya and S. Ozawa

was observed using an IR(infrared)-camera [17]. The cell shape shows hexagonal symmetry and is about 100 mm across. Formation of this cell is due to the existence of a thermal inverse layer several centimeters thick [14]. Within the thermal inverse layer, the melt is heated from the bottom side and is cooled at the surface by radiation. Thus, both Rayleigh and Marangoni mechanisms should be taken into account in the formation of the cell, that is, a Rayleigh–Marangoni–Benard cell. The pattern moves from the crucible wall to the crystal at about 10 mm s−1 . Cell motion was damped when a vertical magnetic flux density of 50 mT was applied. This suggests that cell flow is driven by a strong buoyancy flow beneath the inverse layer, whose direction is inbound. The application of a magnetic field, however, damped this flow and thus the motion of the cell pattern [6]. At the same time, a magnetic field also elongated the hexagonal pattern along the radial direction. This suggests that the surface tension-driven flow, which is caused by difference of temperature and that of amount of oxygen adsorption (oxygen concentration) between the crucible wall and the crystal, contributes to the formation of the hexagonal pattern and that this radial flow cannot be easily damped, because the radial flow component due to the aforementioned gradient would be exceedingly strong compared with that in the azimuthal direction. For the EMCZ (electromagnetic Czochralski) configuration (magnetic flux density of 0.1 T and electric current of 8 A), a flow pattern was highly twisted toward the azimuthal direction and moves clockwise slowly [6]. This is due to a strong forced flow induced by the Lorentz force, which resulted from coupling an applied magnetic field and an electric current passing between an inserted electrode and a growing crystal. Recently, Kalalev simulated crystal growth of a 400 mm diameter silicon ingot considering the stress at the free surface due to the Marangoni effect, so as to understand and improve the silicon crystal growth process [18].

3.3 Welding In a TIG (tungsten inert gas) welding process for steel, the shape of a weld pool is a key factor in determining weldability [4]. Because of a strong temperature difference between the center and rim of the pool surface, Marangoni flow takes place at the melt surface. The center of the melt surface is heated by a torch, whereas at the rim a melt coexists with the solid; the temperature of the melt is lowest at the rim. The temperature coefficient of surface tension depends on contamination, particularly that due to oxygen and sulfur. Since a temperature coefficient is negative in a reducing atmosphere, the direction of the Marangoni flow is outbound; the flow takes place from the center of the pool to the rim, as shown in Fig. 3.6a [19]. The pool is shallow. However, if the surface tension displays a positive temperature coefficient due to contamination, the Marangoni flow takes place from the rim to the center of the pool; the shape of the pool is deep and good weldability is expected, as shown

3 Marangoni Flow and Surface Tension of High Temperature Melts

45

Fig. 3.6. Relationship between temperature coefficient of molten steel and weld pool shape [19]

in Fig. 3.6b. This positive temperature coefficient of surface tension has been reported for systems adsorbed with the group-16 elements, such as oxygen, sulfur, etc. [20–22]. Sulfur also affects the temperature coefficient of surface tension for stainless steel [4, 5]. Surface tension indicates the maximal value at a given temperature; this is due to desorption with increase in temperature (Fig. 3.6c). In such cases flow is complicated within a weld pool. Lu et al. experimentally and quantitatively studied the effect of O2 and CO2 in the Ar-based shielding gas on formation of oxide layer, weld pool shape, and flow structure [23]. In this case, O in the liquid pool is supplied from O2 or CO2 and the melt surface is coated with an oxide layer. Even though the surface is coated with an oxide layer, Marangoni flow was observed to exist and to affect the pool shape. When the O2 or CO2 concentration is below 0.6 vol.% in Ar gas, the oxide layer is thin. This thin oxide layer on the pool peripheral area is easily destroyed and the weld pool surface is exposed to the shielding gas; thus, Marangoni flow takes place. When the oxygen in the weld pool is below 100 ppm (O2 or CO2 content in the shielding gas is below 0.2 vol.%), the direction of the Marangoni convection at the quasi-free surface is outbound; the pool is wide and shallow, as shown Fig. 3.7a. When the oxygen content is over 100 ppm (O2 or CO2 is over 0.2 vol.%), the direction of Marangoni convection changes from outward to inward; the weld pool shape is deep and narrow, as shown in Fig. 3.7b. With a further increase in oxygen concentration in the melt (O2 or CO2 is over 0.6 vol.%), a thick, rigid oxide layer is formed at the peripheral area of the liquid pool surface. However, in the pool center area, inward Marangoni convection still exists. This inward convection transports the hot melt from the surface to the bottom. As a result, the weld shape becomes wide again but convex toward the bottom, as shown in Fig. 3.7c.

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T. Hibiya and S. Ozawa

Fig. 3.7. The effect of oxygen on a weld pool shape. (a) Oxygen concentration of Ar-shielding gas is 0.1 vol% (30 ppm in the melt), (b) 0.4 vol% (150 ppm), (c) 0.9 vol% (200 ppm) [23]

3.4 Electron Beam Melting Electron beam button melting (EBBM) is used for processing titanium, refractory metals, and refractory alloys for aerospace applications. The process is also used for preparation of thin films for microelectronic chips. Buttonshaped material c.a. 50 mm in diameter contained within a cold crucible is melted by using an electron beam. The configuration of the melt is similar to that of a weld pool. Lee et al. examined the Marangoni effect for the EBBM process not only from the viewpoint of heat and mass transport in this process but also from that of texture of the solidified material, that is, secondary dendrite arm length [24]. They paid particular attention to the effect of surfactant sulfur on Ni-based IN718 alloy; the temperature coefficient of surface tension is dependent on the amount of sulfur.

3 Marangoni Flow and Surface Tension of High Temperature Melts

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3.5 Methods for Measuring Surface Tension: Oscillating Drop Method Using Electromagnetic Levitation Surface tension is measured by several methods, such as (a) capillary rise method, (b) maximum drop method, (c) maximum bubble pressure method, (d) sessile drop method, (e) pendant drop method, (f) drop weight method, and (g) oscillating drop method [25]. For surface tension measurements of high temperature melts, such as molten metals, alloys, and semiconductors, data for the ambient atmosphere must be attached, because surface tension is sensitive to surfactants of the group-16 elements, particularly oxygen. Temperature range for measurement must also be extended, because uncertainty for the temperature coefficient is reduced by measurement in the wide temperature range. For these purposes, an oscillating drop method using levitation can supply a solution, because the containerless technique assures contamination-free conditions from the crucible wall and the wide temperature range, including undercooled condition due to the fact that there is neither a contamination source nor a nucleation site. Oxygen partial pressure can be controlled only by using electromagnetic levitation, although contamination-free conditions and a wide rage of temperature measurements can be realized using electrostatic levitation. For electrostatic levitation, it is very difficult to control the ambient atmosphere, because discharge takes place between electrodes, when introducing gas for oxygen partial pressure control. For the oscillating drop method, surface tension can be obtained through the Rayleigh’s equation as follows [26]: 3 2 πν M. (3.4) 8 Here, σ is the surface tension. ν and M are the oscillation frequency and the mass of the droplet, respectively. The above ideal relationship is valid only under microgravity [27], whereas frequencies of m = 0, ±1, and ±2 oscillations for the l = 2 mode are not degenerated but split into five peaks, when the oscillation frequency of a rotating droplet is observed from the top of the rotation axis on earth as shown in Figs. 3.8a and 3.8b. Here the droplet is no longer spherical but is deformed into an egg shape due to gravitational acceleration as well as magnetic forces. Thus, surface tension must be calculated using the Cummings and Blackburn correction, including the frequency of translational mode νt as follows [28]:  2 z 2  1  2 0 2 2 ν = , (3.5) ν − νt 1.9 − 1.2 5 m=−2 2,m a  g 3M z0 = . , a= 3 2 2 8π νt 4ρπ σ=

Here, ν2,m is the frequency of m = 0, ±1, and ±2 oscillations for the l = 2 mode. The νt is the oscillation frequency of the translational mode, that is,

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Fig. 3.8. Frequency of surface oscillation observed on earth (a) and in microgravity obtained by a parabolic flight of a jet aircraft (b). Multiple frequencies are observed on earth; m = 0, ±1, and ±2 frequencies split. All oscillations for the l = 2 mode were degenerated and a single peak was observed in microgravity Table 3.1. Identification rule for the oscillations

Area R+ R−

m=0

m = ±1

m = ±2

Rotation

Yes Yes No

Yes Yes Yes

No No Yes

No No Yes

movement of gravitational center. a is the equilibrium radius of the droplet. g and ρ are the gravitational acceleration and the density of the droplet, respectively. The frequency of the m = 0, ±1, and ±2 oscillations can be identified based on the rule shown in Table 3.1. However, both the frequencies of the m = ±2 oscillation and of the rotation of elliptically deformed droplets appear in the R− signals, that is, the difference in radius along the x and y axes, because a droplet is deformed into a twofold symmetric shape in both cases. If there is no droplet rotation, m = +2 and m = −2 exhibit the same frequency for m = |2|; a single peak appears. When the droplet rotates, frequencies for m = +2 and m = −2 oscillations appear separately apart by two times of rotation frequency [29]. In another word, the real frequency of the m = |2| oscillation is located just between the frequencies of m = +2 and m = −2 oscillations. The difference in the m = +2 and m = −2 is featured with the temporal phase difference between two oscillations, ΔΨ . This is derived from the hemispherical harmonics to describe the shape of the droplet. It causes apparent rotation, which exhibits a much higher frequency than that of real droplet rotation. Thus, analysis of oscillation frequency becomes complicated. The height of the m = +2 and m = −2 oscillations depends on the phase difference, ΔΨ ; usually the height of these peaks is uneven. When a droplet

3 Marangoni Flow and Surface Tension of High Temperature Melts

49

shows real rotation and the phase difference is ΔΨ = 0 or π, both m = +2 and m = −2 oscillation peaks exhibit the same height. However, when ΔΨ = +π/2 or −π/2, only a single peak appears, even though the droplet really rotates. The peak appears at the frequency shifted from that for m = |2|. Thus, it is possible to misidentify the frequency for m = |2| oscillations. To avoid the error, it is advisable to identify the two peaks for m = +2 and m = −2 oscillations [30, 31]. By superimposing a weak static magnetic field parallel to the axis of the levitation coil axis, the rotation axis of a droplet is constrained so that it is parallel to that of the coil, and mutation is also suppressed. Otherwise, the rotation axis and the rotation rate exhibit time dependence. The application of a magnetic field would make it possible to clearly identify the m = |2| frequency and improve uncertainty of surface tension. However, the application of a strong magnetic field damps surface oscillation and makes it almost impossible to identify frequencies. Furthermore, the application of the magnetic field appears to shift m = 0 and ±1 frequencies higher. On the other hand, the application of a strong magnetic field enables us to measure thermal conductivity and density of metallic melts easily, because a strong magnetic field can suppress flow and surface oscillation, that is, a droplet behaves as if it were solid and axisymmetry of the droplet shape is assured [32, 33]

3.6 Surface Tension of Molten Silicon: Influence of Oxygen on Surface Tension For molten silicon, Keene collected and evaluated a great deal of published data [3]. As shown in Fig. 3.9, the surface tension of molten silicon has been classified into two groups; the larger the surface tension is, the larger the temperature coefficient is [34–54]. This classification is attributed to oxygen contamination from measurement devices and atmosphere. Keene advised measuring surface tension using a contamination-free environment, such as the use of levitation. Huan et al. and Mukai et al. measured surface tension and its temperature coefficient as a function not only of temperature but also of oxygen partial pressure of an ambient atmosphere P o2 [55, 56]. Figures 3.10 and 3.11 show the measurements carried out by Mukai et al. They are written using equations as follows: σ/mNm−1 = 831 − 29.5 ln(1 + 3.88 × 1010 P o2 2 ) 1

1 2

814 − 30.1 ln(1 + 3.06 × 1010 P o2 ) 1 2

793 − 30.6 ln(1 + 2.47 × 1010 P o2 ) 1 2

774 − 31.0 ln(1 + 1.01 × 1010 P o2 )

at 1, 693 K, at 1, 723 K, at 1, 753 K, at 1, 773 K,

(3.6)

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Fig. 3.9. Surface tension of molten silicon reviewed by Keene [3] 1 ∂σ /Nm−1K−1 = −0.90 + 0.370 ln(1 + 6.62 × 1010 P o2 2 ) ∂T 1 − 0.387 ln(1 + 8.22 × 109 P o2 2 )

(3.7)

(1, 693 K < T < 1, 773 K, P o2 < P o2 sat ). Comparing Keene’s review with Mukai’s measurements, it is clear that low surface tension of ca. 700×10−3 N m−1 (See Fig. 3.9) corresponds to that measured at high oxygen partial pressure; the oxygen partial pressure of ambient atmosphere exceeds the equilibrium oxygen partial pressure for SiO2 formation, and the surface was oxidized. Mukai suggests that surface tension-like values can be obtained even above the equilibrium oxygen partial pressure for SiO2 formation. In Fig. 3.10a, kinks correspond to the formation of SiO2 at the melt surface. Oxygen partial pressure was not controlled quantitatively for any of the measurements in Fig. 3.9. Hardy and Kingery changed the ambient atmosphere qualitatively, such as the use of hydrogen and helium [34, 43]. According to Mukai’s measurements, the excess amount of adsorbed oxygen atoms Γ0 was calculated to be 2.1 × 10−6 mol m−2 at 1,693 K and at P o2 of 10−21 − 5 × 10−20 MPa using the following equation, that is, the Gibbs adsorption isotherm: ΓO = −2(1/RT )(∂σ/∂ ln P o2 ).

(3.8)

3 Marangoni Flow and Surface Tension of High Temperature Melts

51

860

Po2 >Po2sat

Po2
1693K Po2sat = 1.32 ×10-20 MPa

Surface Tension: s (mN/m)

840 820

1723K Po2sat = 4.27 ×10-20 MPa

800

1753K Po2sat =1.32 ×10-19 MPa

780 760 740

Observed 1693K 1753K

1723K 1773K

Po2sat = 2.75×10-19 MPa 1773K

720 -26

-24

-22

-20

-18

-16

-14

log(Po2/MPa) Fig. 3.10. Surface tension of molten silicon as a function of oxygen partial pressures at various temperatures. Kinks near the P o2 = 10−18 –10−19 MPa correspond to the formation of SiO2 at the melt surface [56]

Temperature Coefficient of Surface Tension:

0 Sessile drop method BN substrate

−0.2 −0.4 −0.6 1773K Po2sat = 2.75×10−19 MPa

−0.8 1693K

−1 −26

Po2sat = 1.32×10−20 MPa

−24

−22

−20

−18

−16

−14

log(Po2/MPa)

Fig. 3.11. Temperature coefficient of surface tension of molten silicon as a function of oxygen partial pressure of the ambient atmosphere. Even above the equilibrium P o2 for SiO2 formation, the temperature coefficient of surface tension is obtained [56]

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Fig. 3.12. Oxygen partial pressure dependence of the Marangoni flow velocity observed as tracer particle velocity using X-ray radiography under microgravity. Actual velocity would be almost one order of magnitude larger than that of tracer particle velocity [59]

Here, R and T are the gas constant and temperature, respectively. The excess amount of adsorbed oxygen of ΓO = 2.1 × 10−6 mol m−2 corresponds to the fact that one out of every ten atoms is oxygen on the melt surface; this suggests that the oxygen concentration is extremely high at the surface [57], whereas the bulk concentration of oxygen was calculated to be as low as ca. 0.001 mass% (8.5 × 1017 atoms cm−3 or 18 ppm) [58]. As shown in Fig. 3.11, it should be noted that the temperature coefficient was never converted from negative to positive, although the absolute value of temperature coefficient decreased with increase in oxygen partial pressure and the temperature-coefficient-like value was obtained even above the equilibrium P o2 for SiO2 formation. This is consistent with the result of Marangoni flow visualization under microgravity as shown in Fig. 3.12 [59]. Marangoni flow velocity decreased with increase in P o2 and flow stopped at P o2 inlet = 10−2 MPa due to a SiO2 film formation. Oxygen partial pressure at the inlet condition of P o2 inlet = 10−2 MPa far exceeds that for equilibrium oxygen partial pressure for SiO2 formation defined at the inlet condition of P o2 inlet ≈ 10−3 MPa. it was Through crystal growth experiments under the various Po2 condition,  made clear that a surface tension temperature coefficient ∂σ/∂T  of less than 0.25 × 10−3 N m−1 K−1 was not realistic; the corresponding oxygen partial pressure was 4.9 × 10−4 MPa defined at the inlet condition. Above this value the melt surface was coated with a thin SiO2 film [60].

3 Marangoni Flow and Surface Tension of High Temperature Melts

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As mentioned earlier, surfactants such as oxygen or sulfur are recognized as playing an important role in describing Marangoni flow through the surfactant effect on surface tension and its temperature coefficient. This means that surface tension must be reported to include data of oxygen partial pressure or concentration of oxygen or sulfur in the melt. However, it is very difficult to ascertain the real oxygen partial pressure at the melt surface besides that in the bulk atmosphere, whose oxygen partial pressure can be measured using an oxygen sensor. Ricci’s group has tried to determine the relationship between real oxygen partial pressure at the melt surface and effective oxygen partial pressure in the bulk atmosphere [61]. This has been clarified through considering not only chemical equilibria for oxidation of melts but also oxygen transport under stationary conditions. The effective oxygen pressure, that is, oxygen partial pressure in the bulk atmosphere, is specific for each metal/metal oxide system and is several orders of magnitude larger than equilibrium oxygen partial pressure determined by chemical thermodynamics. Ratto et al. derived the equation, which shows the relationship between the actual oxygen partial pressure at the melt surface P o2 surface and that at inlet of the systems P o2 inlet for molten silicon under gas flow conditions [62]. Actual oxygen partial pressure at the melt surface was also described as a function of oxygen partial pressure at the outlet of the system. Equation (3.9) is useful in practically controlling the oxygen partial pressure: saturation = POsurface + PSi POinlet 2 2

SiO,SiO  2 j=reaction

αj Kj

αj 1 + Pe POsurface . 2 1 + P e/|Ψj |

(3.9)

Here, two chemical equilibria are considered as follows: Si + αj O2 = SiO2αj α1 = 1/2, α2 = 1, for j = 1, Si(l) + 1/2O2 (g) → SiO(g), and for

j = 2,

Si(l) + O2 (g) → SiO2 (s).

Kj is the dimensionless equilibrium constant for the jth reaction, Ψj is the dimensionless diffusion constant of oxygen, and P e, Peclet number, is defined as the ratio of flow to diffusion of oxygen. Ratto’s model was experimentally validated by Azami and Hibiya [63]. There was a magnitude difference of over tenfold in oxygen partial pressure between Huan’s measurement [55] and Mukai’s measurement [56]; Huan et al. reported the higher oxygen partial pressure, such as 10−6 to 10−5 MPa, and Mukai the lower, such as 10−24 to 10−14 MPa for similar surface tension values. Azami also reported higher oxygen partial pressure for visualization of the Marangoni flow, that is, 10−8 to 10−4 MPa. Differences in the oxygen partial pressure reported by various researchers were universally understood by employing Ratto’s interpretation, because a lower P o2 was obtained at the outlet than that at the inlet.

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Fig. 3.13. Dependence of temperature coefficient of iron on oxygen concentration [20]

3.7 Surface Tension of Molten Iron and Iron-based Alloy For molten iron, the temperature coefficient for surface tension was observed to change from negative to positive along with increases in oxygen concentration as shown in Fig. 3.13 [20]. Ozawa et al. measured the surface tension of molten SUS304 stainless steel (8%Ni-18%Cr–Fe) by the oscillating droplet method using electromagnetic levitation in a wide temperature range including the undercooled condition of 170 K [64]. Measurement was carried out on two types of ambient atmosphere, that is, low P o2 of 10−28 MPa measured at the exit of the gas flow system and high P o2 of 10−21 MPa. The low P o2 atmosphere was prepared by a flow of 5% H2 –Ar mixture, whereas the 5% H2 –He mixture was used for lowering temperature. Pure Ar or He gas flow was employed to prepare high P o2 condition; residual O2 concentration was <0.1 ppm for both Ar and He. As shown in Fig. 3.14, a negative temperature coefficient was observed in the low P o2 atmosphere, whereas in the high P o2 case a positive temperature coefficient was observed at a lower temperature (1,620–2,070 K), and the temperature coefficient changed into a negative value above 2,070 K. This would be attributed to desorption of oxygen from the melt surface. It is noteworthy that temperature dependence showed almost the same behavior above 2,070 K. McNallan and Debroy theoretically calculated surface tension for molten iron and 18% Cr–Fe stainless steel as a function of both temperature and sulfur concentration [5], as shown in Fig. 3.15. As oxygen does, sulfur significantly affects surface tension and temperature coefficients. It should be noted that, for Fe and Fe-based alloys, surface tension decreased with increases in oxygen and/or sulfur concentration, and that the temperature coefficient changed

3 Marangoni Flow and Surface Tension of High Temperature Melts

55

Surface tension: σ / 10−3 N.m−1

2400 2200

Ar-5%H2 / He-5%H2(Po2:10−9−10−15Pa)

s = −0.3063(T−1694)+1676

2000 1800 1600 1400 1200 Melting point

1000 1500

1600

Ar / He (Po2:10−2Pa) s = 0.3896(T−1694)+1423 1700

1800 1900 2000 Temperature / K

2100

2200

2300

Fig. 3.14. Surface tension of molten SUS304 stainless steel measured by the oscillating droplet method using electromagnetic levitation [64]

Cr -S 8% -1 Fe

00

Fe-S

0. PE 02 RC EN T

180 0 2

TE 200 MP 260 ER 0 AT UR E. K

300 0

.03

Fe-S

0. 0 SU 1 LF UR

0.

N/m SURFACE TENSION. 1.75 1.50 1.25

2.00

Fe-18%Cr

Fig. 3.15. The effect of sulfur concentration on surface tension of molten Fe and 8Ni–18Cr–Fe stainless steel (results of calculations) [5]

from negative to positive. However, as mentioned in the previous section, silicon does not display a positive temperature coefficient. This difference is attributed to that of the solubility of oxygen between molten silicon and iron. For molten silicon, the solubility limit is 0.001 mass%, whereas for iron, solubility is in the order of 0.1 mass%. The solubility of oxygen in molten silicon is two orders of magnitude smaller than that in molten iron. This is why

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the amount of adsorbed oxygen is so high for molten silicon, that is, SiO0.1 for estimated chemical composition at the melt surface, even for low bulk oxygen concentration; the melt surface is fully coated with SiO2 prior to the occurrence of temperature coefficient inversion from negative to positive.

3.8 Thermodynamic Approach for Adsorption of Oxygen at Melt Surface As discussed in the previous sections, adsorption of oxygen at the melt surface plays a significant role in high temperature processes such as silicon crystal growth and welding, because adsorption not only decreases surface tension but also changes the temperature coefficient of surface tension. However, surface structure and mechanism of adsorption have not yet been clarified sufficiently. This is attributed to difficulties in controlling PO2 during surface tension measurement. Molten silver is drawing interest as a model liquid material for studying the Marangoni flow of low-Pr-number fluids, such as molten metals, alloys, and semiconductors [65], because silver cannot be oxidized at elevated temperatures, yet can sustain bare surfaces. Also molten silver shows inversion of a temperature coefficient of surface tension. Bernard and Lupis measured surface tension of silver as a function of oxygen partial pressure and tried to understand the surface structure adsorbed with oxygen [66]. By applying the monolayer approximation, it has been clarified that the surface of molten silver is saturated with oxygen when each adsorbed oxygen atom occupies an area of 3.3 ± 0.5 × 10−1 nm2 . This corresponds to the formation of the two-dimensional compound of Ag3 O; the ratio of the amount of metal to that of oxygen is in the same order as that of the silicon case, that is, SiO0.1 . Sangiorgi et al. also measured surface tension of molten silver as a function of oxygen partial pressure and found a surface area per oxygen atom to be 8.8 × 10−2 nm [67]. They suggested the presence of a “corrugated” surface structure of Ag–O with the oxygen ions protruded towards the gas phase. Yoshikawa et al. introduced the “associate compound” model to understand the adsorption of oxygen at the melt surface of Fe–Cr, Ag–Au, and Ag–Cu melts [68]. An observed decrease in surface tension was systematically analyzed using the Belton equation. Because the surface tension of oxide melt is smaller than that of metals, it is reasonable to understand that oxygen adsorption decreases surface tension by the formation of associate oxides at the melt surface.

3.9 Perspective Development of high technology society requires products fabricated by highly value-added high-temperature process, such as crystal growth of semiconductor, fine casting of turbine blades for jet engines, welding of automobiles,

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etc. Numerical modeling with high-performance computers is used to improve the quality of products and to shorten the turn-around-time for production. Marangoni flow plays a key role in heat and mass transport for crystal growth and welding; the driving force of the Marangoni flow is surface tension differences at the melt surface. Since thermophysical properties of molten materials are required to be measured with small uncertainty for use in large scale numerical calculation, there is significant demand for improvement or new developments in measurement technology. An oscillating drop method using a levitation technique is an elegant means of measurement; particularly electromagnetic levitation is suitable for molten metals, alloys and semiconductors. Oxygen partial pressure can be controlled for electromagnetic levitation. Nowadays, surface tension must be measured under controlled oxygen partial pressure, because surface tension of molten metals, alloys, and semiconductors are sensitive to the oxygen partial pressure of an ambient atmosphere. Electromagnetic levitation techniques have been improved upon recently to a great extent, because this technique has been used in microgravity conditions on board the space shuttle and will be used in the International Space Station. Using modern measurement technologies, highly reliable thermophysical properties including surface tension will be prepared. This technique supplies not only basic data for engineering science but also for the new field of physics and chemistry of high temperature melt surfaces, which is indeed a cutting edge science.

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45. V.P. Elyutin, V.I. Kostikov, V.Y. Levin, Fiz. Khim. Poverkh Yavleni Rasp., ed. by V.N. Eremenko, (Naukova Dumka, Kiev, 1971) (in Russian) pp 153–156 46. N.K. Dzhemilev, S.I. Popel, B.V. Tsarevskii, Poverkh Yavlen v Rasp. i Voz iz Nikh Tverd Faz. Nal’chik 1965 (in Russian) pp 306–311 47. Y.M. Shashkov, T.P. Kolesnikova, Rus. J. Phys. Chem. 37, 747 (1963) 48. F.N. Tavadze, M.G. Kekus, D.V. Khantadze, T.G. Tsverdvase, Poverkh Yavleniya Rasp., ed. by V.N. Eremenko, (Naukova Dumka, Kiev, 1968) pp 159–162 49. G.P. Khilya, Y.I. Vashchenko, Dopov Akad Nauk Ukr RSR Ser B, 35, 69 (1973) 50. P. Keck, W. van Horn, Phys. Rev. 91, 512 (1953) 51. P.V. Geld, M.S. Petrushevski, Izv AN SSSR Otd, Tkhn Nauk Met i Toplivo 3, 160 (1960) 52. V.I. Kositkov, A.S. Tarabanov, Metod Isssl i Svoistva Gran Raz Kontakt Faz, AN UKR SSR, ed. by V.N. Eremenko, (Naukova Dumka, 1977) pp 79–90 53. S.-I. Chung, K. Izunome, A. Yokotani, S. Kimura, Jpn. J. Appl. Phys. 34, L631 (1995) 54. M. Przyborowski, T. Hibiya, M. Eguchi, I. Egry, J. Cryst. Growth 151, 60 (1995) 55. X. Huang, S. Togawa , S.-I. Chung, K. Terashima, J. Cryst. Growth 156, 52 (1995) 56. K. Mukai, Z. Yuan, K. Nogi, T. Hibiya, ISIJ Int. 40, S148 (2000) 57. T. Hibiya, S. Nakamura, K. Mukai, Z.-G. Niu, N. Imaishi, S. Nishizawa, S. Yoda, M. Koyama, Phil. Trans. Royal Soc. London A 356, 899 (1998) 58. H. Hirata, K. Hoshikawa, J. Cryst. Growth 106, 657 (1990) 59. T. Azami, S. Nakamura, T. Hibiya, J. Electrochem. Soc., 148, G185 (2001) 60. T. Hibiya, Y. Asai, M. Sumiji, T. Kojima, Cryst. Res. Technol. 38, 619 (2003) 61. E. Ricci, E. Arato, A. Passerone, P. Costa, Adv. Colloid Interface Sci. 117, 15 (2005) 62. M. Ratto, E. Ricci, E. Arato, J. Cryst. Growth 217, 233 (2000) 63. T. Azami, T. Hibiya, J. Cryst. Growth 233, 417 (2001) 64. S. Ozawa, K. Morohoshi, T, Koda, T. Hibiya, ISIJ Spring Meeting, 2008 65. T. Hibiya, K. Nagafuchi, S. Shiratori, N. Yamane, S. Ozawa, Adv. Space Res. 41, 2107 (2008) 66. G. Bernard, C.H.P. Lupis, Metall. Trans. 2, 2991 (1971) 67. R. Sangiorgi, M.L. Muolo, A. Passerone, Acta Metall. 30, 1597 (1982) 68. T. Yoshikawa, K. Yamamoto, T. Tanaka, K. Morita, J. High Temp. Soc. (in Japanese) 32, 289 (2006)

4 Diffusion Coefficients of Metallic Melts Measured by Shear Cell Technique Under Microgravity and on the Ground Shinsuke Suzuki

4.1 Introduction Diffusion coefficients in liquid metals are very important parameters for both material processing and physics of liquid. However, we have not yet had enough amounts of accurate diffusion data to discuss the diffusion model in liquid metals, because several kinds of convection bother measurements. The sources of convections expected in diffusion experiments are classified into the followings: (1) natural convection caused by gravity, (2) Marangoni convection caused by difference of the surface tension, (3) convection induced by mechanical force, etc. As a result of existence of convection, the measured diffusion coefficients show values higher than the real values and the measured values are scattered. Until now several attempts were done to suppress natural convection during diffusion experiments. One of them is to perform diffusion experiments under microgravity conditions where no buoyancy force is generated. In 1980s– 1990s Frohberg et al. performed diffusion experiments using the long capillary technique as explained later in the space shuttle missions [1–3] and demonstrated: both measured values of diffusion coefficients obtained under microgravity in the space shuttle; their scattering are smaller than that obtained on the ground. Furthermore, the temperature dependence of the diffusion coefficients was found to be proportional to the square of the temperature. Afterward Yoda et al. [4] and Itami et al. [5] performed concerning experiments in another space mission and obtained results that have consistency with the results of Frohberg’s group. These results showed the effectiveness of microgravity conditions to suppress natural convection and are helpful to discuss diffusion mechanism. Microgravity experiments are very costly and the number of flight opportunity is limited. Even if we have a flight opportunity, there still remains a risk of failure either in rocket launches or in hardware. Therefore, an experimental technique for accurate diffusion measurements on the ground is strongly desired. To suppress buoyancy-driven convection, it is known that

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a strong magnetic field is effective. Recently, the results of diffusion experiments in a strong magnetic field have been reported and showed agreement with microgravity experiments [6–8]. Another possibility of diffusion experiments on the ground is a method using a stable density layering. The diffusion axis is set parallel to the gravity vector and the samples are set such that the density increases monotonously downwards during the whole diffusion time. In such a case, even if forces are generated by temperature gradients and try to start convection rolls, the force generated by the density gradient can cancel them mostly. The results with a stable density layering on the ground [9–14] showed that the obtained concentration curves in four parallel experiments are lying over one another and each of them can be fitted well with the theoretical equation, and the calculated diffusion coefficients agree well with that obtained in former μgexperiments. The temperature dependence of the diffusion coefficients in 1gexperiments follows the power law with n  2. Although the principle of the arrangement for a stable density layering seems very simple, it is not until the completion of the shear cell technique that the stable density layering was realized, as explained in Sect. 4.2.1. In June 2005, the research group including the author performed diffusion experiments in the Russian recovery satellite Foton-M2 successfully using the shear cell technique. At the same time, the group developed the diffusion experiment technique on the ground, expanded the potential of the diffusion research, and changed the concept of microgravity experiments. This chapter introduces the followings, with showing the results of Foton-M2 mission as examples: 1. Design concept of the shear cell for both reliable performance under μg conditions and minimization of several kinds of convections 2. Quantification and correction of the shear-induced convection 3. Diffusion experiments with a stable density layering under 1g conditions 4. Diffusion experiments under μg conditions 5. Perspective of future in the field of diffusion measurement in liquid metals

4.2 Design of Shear Cell 4.2.1 Principle of Shear Cell Technique Most of the diffusion experiments have been performed using the long capillary method, where the liquid diffusion sample is held in a one-dimensional fixed thin capillary through the whole deffusion experiment. Recently, the shear cell method is replacing the long capillary as a standard measurement tool. A diffusion experiment with the shear cell technique is shown in Fig. 4.1, which is the case of the shear cell specially developed for Foton missions (named Foton shear cell) [10]. A lot of discs with thin holes pile up and the holes

heat up + homogenization Tdiff

diffusion

63

cooling

final shearing

TL

initial shearing

temperature [a.u.]

4 Diffusion Coefficients of Metallic Melts

time [a.u.]

graphite felts

F

reservoir PU

Rc

φd

H

cell no. 1 2 3

sample A intermediate cell

10

sample B 20 reservoir PL graphite felts (a) set-up

PL

PL

(b) melting and (c) initial shear (d) diffusion (e) final shear (f) cooling and homogenization (start of diffusion) (end of diffusion) solidification

Fig. 4.1. Path of a diffusion experiment with the Foton shear cell

form capillaries. By rotating the discs we can form and segment the samples. Because of this mechanism two kinds of samples are connected only during the diffusion phase. Therefore, the shear cell technique has several advantages compared with the long capillary method: 1. No necessity to correct diffusion coefficients during the heating and the cooling phase 2. No necessity to correct diffusion coefficients for the effect of volume expansion and shrinkage 3. Possibility of homogenization before the diffusion phase starts (e.g. in case of monotectic alloys) 4. No influence of segregation during solidification Especially the advantage (4) is the most important to expand the variation of sample selection into multicomponent systems, which usually show the segregation problem and are necessary to realize the stable density layering in ground experiments.

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In spite of these advantages, after the invention of the shear cell technique by Nachtrieb et al. [15], the amount of the measurement results with using the shear cell technique is not so large because its complicated structure can cause experimental failure or additional convection, for example, misalignment, leakage of liquid samples, not smooth rotation of cells, shear-induced convection, averaging effect inside the cells, etc. From these reasons, several attempts were done to improve the reliability and to avoid the error sources. One of them is the shear cell developed for the diffusion experiments onboard ISS by JAXA [16]. Another is the Foton shear cell with which space experiments have been already successfully done [14]. The following sections explain the development of the shear cell technique with showing the case of the Foton shear cell as an example. 4.2.2 Minimization of Shear Convection Convection takes place during shearing because of the friction between the liquid sample in a cell and the moving cell wall. This convection is called shear convection. To minimize the shear convection, the cell size, the shearing mode, the rotation method, and the rotation speed should be optimized. There are several reports about the effect of these parameters on the intensity of the shear convection in a diffusion experiment using the shear cell technique. Arnold et al. [17] and Oda et al. [18] investigated the effect of the shear convection by numerical simulations. Arnold et al. concluded that the aspect ratio of the cell length H (Fig. 4.1f) and the diameter d (Fig. 4.1f) of a capillary is important; for H/d > 3, the measurement is unaffected by the shearing action; for 1 < H/d < 3, convective mixing exists but the disturbance is small enough to be corrected well; for H/d < 1, convective mixing is contributing strongly. If the cell length H is longer, the averaging effect of AAS(mentioned later) will be larger, but can be corrected. Considering these results and the restriction to minimize the whole shear cell length and weight for the flights, the Foton shear cell employed the cell size of H = 3 mm, d = φ1.5 mm. The shearing mode is also critical for the shear convection. Mathiak et al. [19] investigated this problem by numerical simulations. Griesche et al. [20] observed the shear convection of colored water in a transparent model of shear cell made of PlexiglasTM in parabolic flight experiments. They showed that the initial shearing mode with an intermediate cell has a smaller penetration depth than the mode with two half columns. There was no large difference between the comb shearing and the helical shearing for the final shearing process. Considering these results, the Foton shear cell employs the mode with an intermediate cell as the initial shearing and the comb shearing as the final shearing mode. If it is needed, the initial shear mode with two half columns is also possible by changing the set-up of the Foton shear cell. The smoothness of the rotation is also critical to avoid vibration and convection. From this point of view a stepping motor, which causes vibration, is not preferable, although it is advantageous for the positioning of the cells.

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The Foton shear cell employed a DC-motor with a specially designed amplifier circuit that controls the tachometer voltage (rotation speed). Positioning is done with software that detects the rapid increase of the motor current at the end of a shear process and then switches off the motor. The discs are pressed together by elastically compressed graphite felts. The compression force F in Fig. 4.1a was set to 2 N, which is large enough to avoid leakage of liquid sample as mentioned later. With this compression force, the friction force between the discs is small enough to guarantee a smooth rotation of the discs/cells. The intensity of the shear convection is proportional to the shear speed [18]. On the other hand, this kind of motor regulation system has a low torque at a low speed. Thus the shearing speed of the discs was set between 0.5 and 1 mm s−1 . 4.2.3 Minimization of Free Surfaces In shear cell experiments, misalignment of cells and scratches on the cell wall cause free surfaces on the liquid diffusion samples. If the free surface has a large gradient of surface tension, Marangoni convection takes place, which also disturbs the concentration profile in diffusion experiments [21]. To avoid this problem and a leakage of liquid sample materials, the graphite parts have been machined with a tolerance of about 10 μm. Another technique for the minimization of free surfaces is setting reservoirs with sufficient high pressure on both ends of the capillary (Fig. 4.1a). At the set-up the volume of solid samples in the capillaries must be about 20% smaller than the capillary capacity; otherwise the expanded sample material opens the gap between the discs and molten sample material leaks out through the gap. The material in the reservoir should have identical compositions to that in the capillary. Pressure on the material is given by elastically compressed graphite felts in the reservoir. At the liquidus temperature of the sample, the compressed elastic graphite felts press onto the molten metal and fill the capillary completely. To minimize free surfaces, it is required to keep a sufficient pressure PL (Fig. 4.1) on the liquid in the capillary during the diffusion phase. Hence, the diameter of the reservoir must be much larger than that of the capillary so that there is only a small reduction in the spring force of the felts. At the initial shearing one of the reservoirs is separated from the capillary, but the other remains in the initial position. The curvature radius RC (Fig. 4.1d) of the free surface at the corner in the capillary can be estimated by γ/PL , where γ is the surface tension of the molten sample. A too high pressure could cause leakage of the molten sample from the capillary. Considering also the requirement of the smooth rotation as mentioned earlier, the reservoir pressures were set at 10–30 kPa in case of the Foton shear cell. RC is estimated as 20–60 μm, for example, in case of Sn at 300◦ C, γ = 560 mN m−1 [22].

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4.2.4 Structure of the Shear Cell The materials of the main parts of the shear cell should not react with the melt, be finely machinable, heat proof, and be lubricant. The shear cell of JAXA is made of glassy carbon [16] and the Foton shear cell [10] is made of graphite. A shear cell unit usually has several diffusion capillaries so that several parallel experiments can be performed simultaneously, for example, three capillaries in a set of JAXA shear cell and four capillaries in a set of Foton shear cell. In case of the Foton shear cell, the whole shear cell consists of an outer fixed tube, an inner rotatable tube, 20 discs, two reservoir units, and an axle. Each capillary inside the discs has 60 mm length and 1.5 mm diameter. The shear cell is heated up by a Mo-wire isolated by ceramic tubes and mounted around the outer fixed tube. Ceramic fiber sheets are wound around the outer fixed tube for thermal isolation and for damping of mechanical shocks at launching. Six shear cells can be operated in six separate chambers in the AGAT-furnace (Foton-M2 mission). The holders on both ends of the shear cell are fixing the shear cell inside the chamber. The motor rotates the inner tube and thereby every second disc (comb shearing).

4.3 Experimental Procedure 4.3.1 Diffusion Experiments By varying the combination of the samples and the position of the intermediate cell, the Foton shear cell can perform several types of diffusion experiments: 1. Interdiffusion (diffusion between semi-infinite sources) (Fig. 4.2a) 2. Thick layer diffusion (diffusion into double semi-infinite liquids from a planar source) (Fig. 4.2b) 3. Thick layer diffusion (diffusion into a single semi-infinite liquid from a planar source) (Fig. 4.2c) Each sample part is weighed before insertion in order to check filling of the capillary after the experiment. To investigate the reproducibility of measurements, all the four capillaries (named A, B, C, and D) are filled with samples the same composition from the same charge and the experiments are simultaneously performed. An example of experimental procedures is shown in Fig. 4.1, which was used in the AGAT-furnace facility (Foton-M2 mission). The 1g and low-g (parabolic flight) diffusion experiments are performed by using the Foton shear cell installed in a vacuum chamber. After evacuating the chamber the furnace is heated up to the diffusion temperature. After a 1-h homogenization phase, the initial shear process is performed at a constant shear speed v0 to start the diffusion process. The

4 Diffusion Coefficients of Metallic Melts

(b)

(c)

0h

0

(a)

67

0h

x

x

x

x0

: sample A : sample B Fig. 4.2. Possible diffusion types with the Foton shear cell. (a) Interdiffusion (diffusion between semi-infinite sources), (b) thick layer diffusion (diffusion into double semi-infinite liquids from a planar source), and (c) thick layer diffusion (diffusion into a single semi-infinite liquid from a planar source)

axial temperature gradient ΔT should be kept small. In case of the Foton-M2 v0 = 0.5 mm min−1 , ΔT = 100 K m−1 . At the end of the diffusion time, the final shear is started at a constant speed, and then the heater is switched off. After cooling down each sample piece is pressed out of the cell mechanically, weighed, and dissolved in acid solutions for the analysis byAAS, ICP-MS. The concentration of the components in each cell is plotted with an error of ±2% at the center position of each cell and then the concentration profiles can be obtained. 4.3.2 Evaluation of Mean Square Diffusion Depth Here the cases of the diffusion of an impurity component in an alloy into a pure material are described. The concentration profile for the interdiffusion between two semi-infinite sources can be described: 

x − x0 c0 √ . (4.1) c(x, t) = efrc 2 2 Dt The concentration profile for the thick layer diffusion is fitted with the thick layer solution: 

 

c0 h−x h+x √ √ c(x, t) = + erf . (4.2) erf 2 2 Dt 2 Dt

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Here D is the diffusion coefficient, t is the diffusion time, c0 is the initial concentration of the impurity in the alloy side, x is the distance from the edge of the diffusion sample in (4.1) (Fig. 4.2a) and in (4.2) (Fig. 4.2c) and the distance from the center of the intermediate cell in (4.2) (Fig. 4.2b) and x0 is the position of the interface. The initial thickness of the planar source is either 2h in case of Fig. 4.2b or h in case of Fig. 4.2c. Because x2 meas includes contributions from the shear convection and the averaging effect, as mentioned later, (4.3) has to be dealt carefully. The details of the calculation of D from x2 meas is described in Sect. 4.5. The fitting parameters were h, x0 , and the product Dt, which can be described by a measured mean square diffusion depth: x2 meas = 2Dt.

(4.3)

By fitting the following functions to the concentration profiles obtained by, for example, AAS, the measured mean square diffusion depths x2 meas can be determined.

4.4 Quantitative Measurement of Shear Convection and Correction Method If a new type of shear cell is developed, it is recommended to investigate the shear convection quantitatively before measurements of diffusion coefficients. It has been reported that the mass transport caused by the shear convection consists of three parts: convection during the initial shearing, relaxing convection just after the initial shearing, and during the final shearing [23]. The relaxation time is reported to be a few seconds, and the total time of the shear convection is about 10 s, which is independent from the diffusion time. Therefore, several methods to investigate the shear convection were proposed. Here two kinds of experiments under conditions, where the buoyancy convection and the Marangoni convection are suppressed, are introduced. One method is short-time diffusion experiments, where the contribution of the shear convection to the mass transport is larger than that of the diffusion process. The other is measurements of the time dependence of x2 meas , where the contribution from the diffusion process is proportional to the diffusion time and the additional mass transport is independent from the diffusion time. Thus extrapolation to the initial time (t = 0) yields the contribution of the additional mass transport. 4.4.1 Short-Time Diffusion Experiments For short-time diffusion experiments in low gravity, parabolic flight in an airplane is available, when the shear convection lasts shorter than 20 s. Using the Foton shear cell, short-time diffusion experiments were performed on ESA’s parabolic flight campaign (PFC31) by Airbus A300 [23].

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Two parallel experiments were performed simultaneously with two capillaries in the shear cell. One capillary was prepared for interdiffusion between Sn47 In53 alloy (sample A) and Sn (sample B), the diffusion type shown in Fig. 4.2a. The other capillary was prepared for diffusion of Sn47 In53 (sample A) from a thick layer into Sn (sample B), the diffusion type shown in Fig. 4.2b. Each shear process took 2.5 s and the completely overlapped position of the capillaries and the intermediate cell remained for 10 s. The diffusion time tdiff was defined as the time from the half-overlapped position at the initial shear to the half-overlapped position at the final shear, which was 12.5 s. A low gravity parabola lasted about 20 s with a residual gravity of ±0.05g. Within one of the low gravity parabolas, all experiments were performed from the start of the initial shearing to the end of the final shearing. The experiments were performed at 250◦ C. The concentration profiles are shown in Fig. 4.3. Similar short-time diffusion experiments can be performed also under 1g conditions with a stable density layering. The diffusion type was thick layer diffusion from Sn90 In10 (sample A) into pure Sn (sample B), the diffusion type shown in Fig. 4.2c. The density increases from Sn90 In10 to Sn monotonically

(a) c0

t=0

after diffusion

x0

(b) c0

t=0

after diffusion

2h

Fig. 4.3. Obtained In concentration profiles after short-time diffusion experiments under low-g at 250◦ C for 12.5 s: (a) interdiffusion between In53 Sn47 -Sn, (b) thick layer diffusion In53 Sn47 into Sn

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Fig. 4.4. Measured mean square diffusion depth for different times and temperatures. The slope gives 2D and the offset represents the mean square diffusion depth caused by the convection and the averaging effect [25]. (a) shows the magnified short time range of (b)

[24]. The diffusion temperature was 275◦ C and the diffusion time tdiff was 14 s. The obtained value of x2 meas is shown in Fig. 4.4. 4.4.2 Time Dependence of Mean Square Diffusion Depth The time dependence of the measured mean square diffusion depth has been reported [26, 27]. The additional transport except the pure diffusion process can be determined by extrapolation of the measured mean square diffusion depth as a function of the diffusion time to t = 0. The Foton-shear cell was also investigated by this method, using a stable density layering under 1g-conditions [25]. The configuration was the thick

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71

layer diffusion (Fig. 4.2c), where the sample A was SnBi3 or SnIn10 and the sample B was Sn. The diffusion experiments were performed at several different temperatures for several different diffusion times. For each condition four parallel experiments were performed simultaneously. After the concentration measurements, x2 meas was determined (Fig. 4.4). The extrapolation from the measured values to the y-intercept gave x2 add = 1 ∼ 5 × 10−6 m2 . Half of the inclination of the line represents the diffusion coefficient at each temperature, which corresponds to the result determined using another correction method (4.6). 4.4.3 Influence of Shear Convection The obtained x2 meas is theoretically the sum of the values of the mean square diffusion depths of different origins under the assumption that the additional mass transports can be also interpreted as additional diffusion processes: x2 meas = x2 diff + x2 add = x2 diff + x2 aver + x2 shear .

(4.4)

Here x2 diff is the contribution from diffusion (=2Dtdiff ) and x2 aver is the contribution from the mixing effect inside a single cell. This mixing is induced by the convection after the final shear process, by the segregation effects during the solidification and finally by the integrating analysis of AAS or ICP. x2 shear is the contribution from the initial and the final shear convection and can be calculated by subtraction of x2 diff and x2 aver from x2 meas . If we consider x2 shear as a quasi-prolongation of a diffusion time, the extended time tshear is calculated by tshear =

x2 shear . 2D

(4.5)

For the calculations, diffusion coefficients of D = 2.1 × 10−9 m2 s−1 (250◦ C) and 2.3 × 10−9 m2 s−1 (275◦ C) were used according to the μg-reference data in the D1-mission [2]. As a result, the shear convection turned out to be small. The mean square diffusion depths caused by the shear convection were in the order of 10−7 m2 , which is much smaller than 1% of the typical value x2 diff ≈ 10−4 m2 in a standard diffusion experiment using the Foton shear cell. Nevertheless, this value can be used for the correction of diffusion experiments.

4.5 Correction Method for the Determination of Diffusion Coefficients By using these results, a correction method of these influences was suggested for the determination of D-values in long time diffusion experiments using (4.4):

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1 2 x meas − x2 shear − x2 aver , 2tdiff 

1 H2 2 2 = , x meas − x shear − 2tdiff 12

D=

(4.6)

where x2 meas is the result of any long time diffusion experiment and x2 shear is the result of the corresponding short time diffusion experiment, that is, at the same temperature and in the same diffusion system under the same gravity conditions. Here x2 aver was approximated as H 2 /12 and depends only on the cell geometry under the assumption that tdiff is sufficiently long [12]. For the Foton shear cell, H (the thickness of a disc) is 3 mm and x2 aver is 7.5 × 10−7 m2 . In the present investigation of a short time experiment, x2 shear could be determined because x2 diff could be calculated from 2Dtdiff and D was known from μg-experiments. Nevertheless, we can use (4.6) in the following way by denoting the short time data with a suffix 0. Then from (4.4) we have H2 . 12 Inserting the results in (4.6) in the generally applicable form x2 shear/0 = x2 meas/0 − 2Dtdiff/0 −

D=

x2 meas − x2 meas/0  .  2 tdiff − tdiff/0

(4.7)

(4.8)

This equation allows to eliminate the disturbing shear convection and the averaging effect of shear cell diffusion experiments by the additional performance of a short time diffusion experiment (zero-experiment). This is advisable if an accuracy of <2–3% is envisaged. If a higher deviation is accepted, it will be normally not necessary to use the correction with (4.8). Then (4.6) with x2 shear ≈ 0 could be sufficient.

4.6 1g-Diffusion Measurements with Stable Density Layering 4.6.1 Experimental To realize the stable density layering, the sample materials should be selected so that the density increases monotonously parallel to the gravity vector during the whole diffusion time. It must be carefully checked before the diffusion that the density gradient correspond to the concentration gradient during the diffusion time and also increases monotonously parallel to the gravity vector. Until now, experiments using the samples of PbAg–Pb, PbGa–Pb, SnBi– Sn, SnIn–Sn, InSn–In, SnBiIn–Sn, SnSb–Sn, and AlNi–Al were performed successfully. The experimental conditions are listed in Table 4.1.

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Table 4.1. 1g-diffusion experiments performed during the Foton projects and obtained mean square diffusion depths and diffusion coefficients Samplea (A)PbAg5 (B)Pb

Typeb

Gravity vector

References

2.3 2.3 2.0 2.3 3.6 5.2 2.0 2.5 5.8 5.8 2.1 2.3 2.7 3.5 2.8 9.9 10.3 3.0 2.4 7.2 −−

[13]

2.9

[29]

3.1 −−

[10, 11] [28]

−x +x

630 (12,600)

+x

730 (18,800) Reference

3.81 ± 0.12 3.9 ± 0.29

(a) (b)

+x +x

(A)SnBi1.5 (B)Sn

(a)

−x

(A)SnBi3 (B)Sn

(a)

−x

(b) (a)

−x +x

(A)SnIn3 Bi5 (b) (B)Sn (A)SnSb3 (a) (B)Sn (A)AlNi (a) (B)Al

ΔD/D [%]

400 600 800 500

+x

(A)SnIn10 (B)Sn

Dc ± ΔDd ×109 [m2 · s−1 ] 2.05 ± 0.05 3.63 ± 0.09 7.69 ± 0.16 2.65 ± 0.06 2.27 ± 0.08 3.57 ± 0.19 5.45 ± 0.11 2.35 ± 0.06 2.4 ± 0.14 2.38 ± 0.14 2.41 ± 0.05 4.35 ± 0.10 7.62 ± 0.20 2.31 ± 0.08 2.44 ± 0.07 2.74 ± 0.27 2.6 ± 0.27 3.73 ± 0.11 5.82 ± 0.14 8.62 ± 0.62 (Bi)4.26 (In)4.99 6.80 ± 0.20

(b)

(A)PbGa5 (B)Pb

Tdiff (◦ C) (tdiff [s]) 350 (28,800) 500 (21,600) 670 (12,600) 450 (5,104) 350 (12,600) 500 (21,600) 630 (12600) 300 (28,800) Reference 300 (28,800) 500 (16,200) 800 (9,000) 300 (12,600) 275 (28,800) Reference (21,600) (21,600) (14,400) (21,600)

[14]

[10, 11] [28] [6] [11, 12]

[12] [9, 11] [28] [6] [9, 11]

[29]

a

(A), and (B) are the sample parts shown in Fig. 4.2 (a), (b), and (c) are the sample types shown in Fig. 4.2 c Average value of D obtained from the four capillaries A–D d Full statistical errors from four capillaries including temperature error b

4.6.2 Data Analysis It is generally known from the previous results that if convection exists in the liquid sample, the measured concentration curve has wave-like deviations from the fitting function, the reproducibility of the diffusion measurements is low and the diffusion coefficient becomes larger than that in the case without

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Fig. 4.5. Bi concentration profiles in four parallel experiments with SnBi3 –Sn (at 300◦ C for 8 h) with the error of ΔC/C = 2% dependent on the performance of the AAS equipment [12]

convection. To check the influence of the buoyancy-convection on the diffusion measurements, the obtained data were investigated from these three points of view [11]. (1) Comparison with fitting function As a typical example, Fig. 4.5 shows Bi concentration profiles in four parallel experiments with SnBi3 -Sn at 300◦ C. In the other experiments in Table 4.1, the concentration profiles are similar to that shown in Fig. 4.5. All data were well described by the fitting function (coefficient of determination R2 > 0.995). This means that there was no indication of buoyancy-driven convection. This profile can be also approximately described by a thin layer profile using 

−x2 c0 h , (4.9) exp c(x, t) =  4D(t + t0 ) πD(t + t0 ) where t0 = h2 /6D is the time offset at t = 0, which gives the same mean square diffusion depth as the real start profile. Because of the linearity of the relationship between x2 and ln c, (4.9) can be used to show the absence of nondiffusion-like convection (Fig. 4.6). (2) Reproducibility of diffusion experiments As shown in Figs. 4.5 and 4.6, the four concentration curves gave about the same diffusion coefficient. The reproducibility can be investigated by the

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Fig. 4.6. “Linear plot” of Bi concentration profiles in four parallel experiments with SnBi3 –Sn (at 300◦ C for 8 h) [12]

standard deviation ΔD among the four parallel diffusion experiments. Since the ΔD -values only represent the reproducibility under nearly equal external conditions, all statistical errors should be included there except those from the temperature measurement. Hence, for the total statistical error ΔD in Table 4.1, a standard temperature deviation was additionally included on the basis of the relation D = AT 2 by the equation   2 N

ΔD  1  Dn − D 2 = + (2σtemp ) , (4.10) D N − 1 n=1 D where the number of the capillaries N is four in the Foton shear cell, Dn is the diffusion coefficient individually measured for the nth capillary, and D is the average value among the Dn -values and the standard temperature deviation σtemp is estimated as 1%. Thus, we arrive at relative total statistical errors of ΔD/D = 2–5%, which is indeed lower than that in the Foton-12 mission, as described in Table 4.1. Garandet et al. [28] analyzed all expected error sources in the diffusion measurements in Foton-12. They assumed a maximum error ΔD/Dμg of 6% for the SnBi1.5 –Sn interdiffusion experiment at 300◦ C, 10% for the SnIn1 –Sn interdiffusion experiment at 300◦ C, and 7.5% for the AlNi1.5 –Al interdiffusion experiment at 700◦ C. It seems remarkable that the total statistic errors in the 1g-experiments are less than the error values estimated in [28], also for the concentration measurements, as can be seen from the reproducibility. It was considered that the reduction of the total statistic errors on the ground is the

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result of the upgrade of the shear cell from the Foton-12 model. From these new investigations, no evidence of disturbance of convective flow was detected in the 1g-measurements. Only the experiments of SnIn10 –Sn at 800◦ C have a relative large total statistical error of 7.2%. This has not yet been explained clearly. There might have been a malfunction of the AAS procedure. (3) Comparison with reference data Even if the criteria mentioned earlier are satisfied, there still remains possibility that the diffusion measurement was bothered by convection. The experiments of PbAg–Pb demonstrated that the density gradient antiparallel to the gravity vector also delivered high reproducibility and concentration curves, which agree well with the fitting function [14]. Therefore, it is necessary to compare the obtained diffusion coefficient with reference data, such as that obtained in a magnetic field or under microgravity. Although it seems a paradox that the 1g-experiments requires reference data from μg-experiments, if we have a few μg-data, it is very helpful to check much more amount of 1g-data. For the diffusion of Bi in Sn, and In in Sn, the results of 1g-measurement in a magnetic field are available. The reference values are shown in Table 4.1. This comparison shows that the 1g values using stable density layering agree well with both the μg-reference data and the results of 1g-measurements in a magnetic field. 4.6.3 Effect of Density Layering The results obtained using stable density layering indicate that buoyancy convection was largely suppressed in these 1g-experiments by using the condition dρ − → − e→ (4.11) x · g > 0, dx where − e→ x is the unit vector in the x-direction. Strong buoyancy convection is normally caused by a density gradient antiparallel to the g-vector, that is, the left side in (4.11) is negative. Since dρ − → ∂ρ ∂c − − → ∂ρ ∂T − − → − e→ e→ e→ (4.12) x· g = x· g + x· g dx ∂c ∂x ∂T ∂x and ∂ρ/∂T is normally negative, the condition for dρ/dx can be met by an upwards temperature gradient ∂T /∂x and/or a positive contribution from the solutal term in (4.12). The 1g-experiments shown in this section are designed to meet both conditions. Since it is difficult to arrange the temperature gradient exactly antiparallel to the g-vector, we have to consider horizontal temperature gradients, say ∂T /∂y. Such gradients produce forces that try to start convection rolls, too. Then it can be shown that there are always convection rolls, but a sufficient

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negative solutal density gradient can strongly reduce the velocity of the rolls so that the influence on the determination of the real atomic diffusion coefficient is below a certain required level. It depends on the material parameters in (4.12) whether the condition can be fulfilled at least in the main part of the diffusion zone. It can normally not be met in the capillary parts away from the diffusion zone, since there c is constant and dT /dx should be small.

4.7 Microgravity Experiments 4.7.1 Utilization of Microgravity Environment Nowadays various kinds of facilities are available for utilization of low gravity or microgravity environments, such as drop towers/shafts, airplanes for parabolic flight, sounding rockets, recovery satellites, space shuttles, and the space station ISS. Duration of reduced gravity, cost, preparation time, success rate, and quality of gravity are different from each other. The duration of usual diffusion experiments is from several dozens of minutes to several hours. Therefore, until now diffusion experiments under microgravity were carried out in the space shuttles or in recovery satellites that fly about two weeks in the orbit. Some experiments in the space station MIR [30] and in sounding rockets have been also reported [31]. The diffusion experiments under microgravity are listed up in [16, 32]. Now the diffusion experiments are prepared for JEM (Japanese Experiment Module) on ISS [16]. It is also pointed out that the residual gravity and the g-jitter also bother diffusion measurements. We have to pay attention to the quality of gravity [30]. Recently μg-diffusion experiments were carried out in the mission FotonM2 [14, 25, 33]. The following sections introduce several parts of the diffusion experiments in Foton-M2 and concerning experiments. 4.7.2 Microgravity Diffusion Experiments in Foton-M2 In the Foton-M2 mission, μg-diffusion experiments were performed in six shear cell units with a total of 24 capillaries (four capillaries A–D in each shear cell) in the AGAT-facility. The mission procedure of the Foton-M2 is shown in Fig. 4.7. The Soyuz-U rocket was launched on 30 May 2005 from Baikonur Cosmodrome in Kazakhstan, and the Foton capsule was put into the lowearth orbit. During 2 weeks of the flight, the experiments were performed in one shear cell after another cell. During the experiments it was possible to control the AGAT via the payload control center ESRANGE located in Kiruna, Sweden, in the Arctic Circle. The downlink was aquired from the satellite, which contained the temperature in the shear cell, torque of the motor during both the initial and the final shear, and the vacuum pressure. After the data analysis it was confirmed that all of the experiments had performed normally. On 16 June 2005 the capsule landed in Kazakhstan. The sample and the

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μg-experiments in orbit (June 2005)

Soyuz-U was launched from Baikonur Cosmodrome (May 2005)

Recovery to earth Kasakhstan (June 2005)

Fig. 4.7. Mission procedure of Foton-M2. The pictures were provided by ESA

Fig. 4.8. Sb concentration profiles in four parallel 1g-experiments and in a μgexperiment with SnSb5 –Sn (at 630◦ C for 12,600 s) with the error of ΔC/C = 2% dependent on the performance of the AAS equipment

data were collected from the capsule and analyzed, as explained later. Experimental procedure is same as described in Sect. 4.3. Figure 4.8 shows Sb concentration profiles in four parallel 1g-experiments and in a μg-experiment with SnSb5 –Sn (at 630◦ C for 12,600 s), which demonstrate agreement between 1g and μg-experiments.

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4.8 Temperature Dependence of the Diffusion Coefficients

diffusion coefficient Dx109 [m2/s]

The temperature dependence of the diffusion coefficients in liquid metals has been discussed for a long time. One of the descriptions is the power law. The results of molecular dynamics simulations [34] showed a temperature dependence of D = AT n , with n = 1.7–2.3. Several results of μg-experiments (self-diffusion of Sn (FSLP [1], MSL-1 [4, 5]), In, Pb, and Sb (D2 [3]), interdiffusion of Sn-In (D1 [2])) showed D = AT 2 , which is considered as a special case of the power law. The function D = AT n was fitted to the measurement results of the experiment of In in Sn, Bi in Sn (Fig. 4.9), Ag in Pb (Fig. 4.10), Ga, and Pb207 (Fig. 4.11) using weighted least squares according to ΔD. In every case, the 10

Sn-SnIn In in Sn(1g) [9]

8

Sn-SnBi

In in Sn

Bi in Sn(1g) [11] Bi in Sn(µ g, Foton-M2) [25]

6

Previous reference data

Sn-SnInBi

In in Sn(1g, magnetic field) [6] In in Sn(µ g, Foton-12) [28] Bi in Sn(1g, magnetic field) [6] Bi in Sn(µ g, Foton-12) [28]

4 2 0

Bi in Sn

0

200

400 600 800 temperature T[K]

1000 1200

Fig. 4.9. Temperature dependence of diffusion coefficients of Bi in Sn and In in Sn

μ

μ

D=1.4x10 −15 T

2.2

Fig. 4.10. Temperature dependence of diffusion coefficients of Ag in Pb. 1g-data [13], and μg-data [29]

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measured diffusion coefficient Dx10 9 [m2/s]

10 Experiments with shear cell[14]: 9 : Ga in Pb (1g) : Fitting curve for Ga in Pb (1g) 8 : Ga in Pb (μg, Foton−M2) 7 : Pb in Pb (μg, Foton−M2) 6 Reference with long capillary[3]: : Fitting curve for Pb in Pb (μg, D2) 5 4 3 2 1 0 0 200 400 600 800 T[K]

1000

Fig. 4.11. Temperature dependence of diffusion coefficients of Ga and Pb207 in Pb

temperature dependence is described better with a fitting parameter n between 1.5 and 2.4 than with a fixed n of 2. This suggests the possibility of the power law with a variable exponent according to the measured system. But the number of the fitting points is so small that this discussion does not exclude the possibility of the T 2 -law. In any case it was confirmed that the temperature dependence can be well described by the power law. But the power law has not yet been explained conclusively with a clear physical model. This is still under work.

4.9 Perspectives Since Frohberg’s successful experiments in the space shuttle, it has been believed that the space environment is the best condition for diffusion measurements. However, the measurement technique for 1g-experiments with a stable density layering was developed and demonstrated high reliability. Now we have to change the concept about the reason why we need space environments. Nowadays the aims of diffusion experiments in the space can be classified into two categories [11]; one is the experiments that can be performed also on the ground and the data will be used as reference values to check the reliability of the ground experiments, and the other is that the experiments that cannot be performed on the ground, for example, a diffusion pair that does not have a stable density layering. If exact diffusion measurements are required, the combination of μgconditions and the shear cell is always helpful. But both the shear cell and μg-conditions are not always available and necessary. Here the diffusion types were classified as shown in Table 4.2 that shows in which cases the shear cell

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Table 4.2. Classification of diffusion types and possible conditions Diffusion type

Selfdiffusion Impurity diffusion

In elements In alloys In elements

In alloys Interdiffusion

Combination of inter and self-diffusion

a b

In binary alloys In multicomponent alloys (>2 components) In binary alloys In multicomponent alloys (>2 components)

Possibility of stabilization by density gradient

Segregation (in general)

Possible gravity conditions

Possible techniques

No (or low) No Yes (at sufficient high concentrations) No (at low concentration) If “yes” If “no” If “yes” If “no” No (because of drag effects)

No Yes If “no” If “yes”

μg μg μg, 1g μg, 1g

SCa, LCb SC SC, LC SC

No

μg

SC, LC

Yes Yes Yes Yes Yes

μg,1g μg μg, 1g μg μg

SC SC SC SC SC

If “yes” If “no” No (because of drag effects)

Yes Yes Yes

μg, 1g μg μg

SC SC SC

SC shear cell LC long capillary

can be substituted by the classical long capillary method or in which cases diffusion experiments can be performed also under 1g-conditions. If a sufficient stabilization is possible by density gradients, there is the possibility to perform impurity diffusion, interdiffusion in binary alloys, and the combination of inter and self-diffusion under 1g conditions. Only in case of no segregation we can use the long capillary, but such a case is rare and the shear cell is usually needed. However, if no sufficient stabilization is possible, we need μg-conditions, for example, in the case of self-diffusion in elements or impurity diffusion at low concentration. In the case of interdiffusion with more than three components, because of drag effects it is not possible to obtain a stable density layering, and hence the shear cell and μg-conditions are necessary. Usual alloys for commercial use have more than three components. Also if reliable reference data are needed, this is up to now only by μg-diffusion data. A more exact quantitative discussion about the necessary density layering arrangement to avoid the buoyancy convection is a future subject. More exact thermodynamic data of the samples are required for the discussion.

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For a more detailed discussion on the diffusion mechanism, more measurements are required. The influence of the atomic weight, the atomic radius, and the electronic structure of the tracer element on the diffusion coefficients is significant. From these points of view, the next experiments will be planned.

4.10 Summary For accurate diffusion experiments both under μg and 1g, the highly completed shear cell should be developed and optimized for considering the minimization of shear convection, minimization of free surfaces, and a reliable operation. The short time diffusion experiments under low-g and 1g conditions and the measurement of the time dependence of x2 meas are helpful to evaluate the mass transport caused by the shear convection quantitatively. To determine the diffusion coefficient, the correction method proposed here can be applied. The 1g-experiments of In in Sn, Bi in Sn, Sn in In, Ga in Pb, Ag in Pb, and Ni in Al with the Foton shear cell using a stable density layering showed a good agreement between the concentration profile and the fitting function, a high reproducibility with a small deviation among parallel experiments, and a good agreement with μg-reference data. Thus the high reliability of the 1g D-values is demonstrated. The μg-experiments in Foton-M2 were performed successfully. It is also a fact that the apparatuses for μg-experiments were graded up and highly completed through a lot of the 1g-experiments. On the basis of these results, the diffusion types were classified according to the reliability of 1g-measurement. It is expected that the experimental method in this study can deliver a large amount of exact diffusion data.

Acknowledgement The author thank the Germany Federal Ministry of Education and Research, BMBF/ German Aerospace Center (DLR) for the financial support for the Foton-missions under national registration numbers 50WM0048 and 50WM0348, and ESA for giving the opportunity to perform the space and the parabolic flight experiments. Collaboration with Prof. G. Frohberg and Dr. K.-H. Kraatz (TU Berlin) for the development of the Foton-shear cell, 1g-experiments with a stable density layering and the Foton experiments are gratefully acknowledged. Collaboration and discussion with Dr. A. Griesche (DLR), Dr. G. M¨ uller-Vogt, Dr. R. Rosu (Univ. Karlsruhe) and Dr. T. Itami (Hokkaido Univ.), and Dr. T. Masaki (JAXA, present Shibaura Inst. Tech.) are gratefully acknowledged.

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References 1. G. Frohberg, in Fluid Sciences and Materials Sciences in Space, ed. by H.U. Walter. Thermophysical Properties (Springer, Berlin, 1987), p. 159 2. G. Frohberg, K-H. Kraatz, H. Wever, A. Lodding, H. Odelius, Defect Diffus. Forum 66, 295 (1989) 3. G. Frohberg, in Scientific Results of the German Spacelab Mission D-2, (WPF/DLR, Cologne, 1995), pp. 275–287 4. S. Yoda, H. Oda, T. Oida, T. Masaki, M. Kaneko, K. Higashino, J. Jpn. Soc. Micrograv. Appl. 16, 111 (1999) 5. T. Itami, H. Aoki, M. Kaneko, M. Uchida, A. Shisa, S. Amano, O. Odawara, T. Masaki, H. Oda, T. Ooida, S. Yoda, J. Jpn. Soc. Micrograv. Appl. 15, 225 (1998) 6. V. Botton, P. Lehmann, R. Bolcato, R. Moreau, Energy Conversion Manage. 43, 409 (2002) 7. G. Mathiak, A. Griesche, K-H. Kraatz, G. Frohberg, J. Non-Crystalline Solids 205–207, 412 (1996) 8. Y. Inatomi, T. Miyake, K. Kuribayashi, J. Jpn. Soc. Micrograv. Appl. 20, 160 (2003) 9. S. Suzuki, K-H. Kraatz, G. Frohberg, Ann. N. Y. Acad. Sci. 1027, 169 (2004) 10. S. Suzuki, K-H. Kraatz, G. Frohberg, Micrograv. Sci. Technol. 16-1, 120 (2005) 11. S. Suzuki, K-H. Kraatz, G. Frohberg, J. Jpn. Soc. Micrograv. Appl. 22, 165 (2005) 12. S. Suzuki, K-H. Kraatz, G. Frohberg, R. Ro¸su, G. M¨ uller-Vogt, J. NonCrystalline Solids 357, 3300 (2007) 13. S. Suzuki, K-H. Kraatz, G. Frohberg, R. Ro¸su, W. Wendl, G. M¨ uller-Vogt, Ann. N. Y. Acad. Sci. 1077, 380 (2006) 14. S. Suzuki, K-H. Kraatz, G. Frohberg, Micrograv. Sci. Technol. 18(3/4), 82 (2006) 15. N.H. Nachtrieb, J. Petit, J. Chem. Phys. 24, 746 (1956) 16. T. Masaki, T. Fukazawa, S. Matsumoto, T. Itami, S. Yoda, Meas. Sci. Technol. 16, 327 (2005) 17. W. Arnold, D. Matthiesen, J. Electrochem. Soc. 142, 433 (1995) 18. H. Oda, S. Yoda, T. Nakamura, T. Masaki, N. Koshikawa, S. Matsumoto, A. Tanji, M. Kaneko, Y. Arai, K. Goto, N. Tateiwa, J. Jpn. Soc. Micrograv. Appl. 16, 33 (1999) 19. G. Mathiak, G. Frohberg, W.A. Arnold, Proceedings of 2nd European Symposium on Fluids in Space, Naples, Italy, April, p. 369 (1996) 20. A. Griesche, K-H. Kraatz, G. Frohberg, G. Mathiak, R. Willnecker, Proceedings of 1st International Symposium on Microgravity Research and Applications in Physical Science and Biotechnology, ESA SP-454, Sorrento, Italy, September, pp. 497–503 (2000) 21. H. M¨ uller, G. M¨ uller-Vogt, Cryst. Res. Technol. 38, 707 (2003) 22. C.J. Smithells, in Metals Reference Book, vol. 2, 3rd edn. (Elsevier, Amsterdam, 2004), p. 699 23. S. Suzuki, K-H. Kraatz, A. Griesche, G. Frohberg, Micrograv. Sci. Technol. 15(1), 127 (2005) 24. P-E.Berthou, R. Tougas, Metal. Trans. 1, 2978 (1970) 25. S. Suzuki, K-H. Kraatz, G. Frohberg, Micrograv. Sci. Technol. 18(1), 155 (2006)

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26. G. M¨ uller-Vogt, R. K¨ oßler, J. Cryst. Growth 186, 511 (1998) 27. M. Uchida, Y. Watanabe, S. Matsumoto, M. Kaneko, T. Fukazawa, T. Masaki, T. Itami, J. Non-Crystalline Solids 312–314, 203 (2003) 28. J.P. Garandet, G. Mathiak, V. Botton, P. Lehmann, A. Griesche, Int. J. Thermophys. 25, 249 (2004) 29. S. Suzuki, K-H. Kraatz, G. Frohberg, R. Rosu-Pflumm, W. Wendl, G. M¨ uller-Vogt, to be published 30. R.W. Smith, Micrograv. Sci. Technol. 11/2, 78 (1998) 31. T. Itami, A. Mizuno, H. Aoki, M. Kaneko, T. Fukuzawa, A. Tanji, Y. Arai, K. Goto, Y. Yamaura, N. Tateiwa, M. Koyama, T. Morita, H. Kawasaki, M. Fujishima, T. Masaki, S. Yoda, Y. Nakamura, N. Koshikawa, T. Nakamura, A. Ogiso, J. Jpn. Soc. Micrograv. Appl. 3, 198 (1999) 32. T. Itami, Jpn. Soc. Micrograv. Appl. 19, 185 (2002) 33. A. Griesche, M-P. Macht , S. Suzuki, K-H. Kraatz, G. Frohberg, Scripta Mater. 57, 477 (2007) 34. M. Shimoji, T. Itami, Atomic Transport in Liquid Metals, p. 250, Trans. Tech. Aedermannsdorf (1986)

5 Thermal Diffusivity Measurements of Oxide and Metallic Melts at High Temperature by the Laser Flash Method Hiroyuki Shibata, Hiromichi Ohta, and Yoshio Waseda

5.1 Introduction The importance of heat transfer properties such as thermal conductivity or thermal diffusivity of various materials at high temperature is strongly emphasized, in parallel with recent progress in surface technology for several electronic devices. Such importance has been well recognized in many pyrometallurgical processes related to plant design and accurate control of continuous casting in steelmaking. For example, heat transfer properties of molten salts are essential to design applications to heat transfer fluids for fusion reactors, breeder reactors, and thermal energy storage systems. Then, thermal property data of molten salts with sufficient reliability are strongly required to select an optimum composition of salt mixture for the desired condition [1]. We also need thermal property data of molten iron at elevated temperature and continuous casting powder melts consisting of various oxide components; SiO2 , CaO, MgO, Al2 O3 , etc. for further improving the present continuous casting process for steel [2]. In producing single crystals supplied for devices of semiconductor compounds such as GaAs and GaP, using Czochralski method, the components of high vapor pressure of P and As are likely to diffuse from the master melt, causing the original compositions to vary. To reduce such trouble, boron oxide melts have widely been employed as liquid capsules to encase the semiconductor master melt [3]. It is necessary to minimize the temperature gradient in the melt by accurate temperature control for producing high quality singlecrystals with a low dislocation density. Although the thermal diffusivity of a liquid capsule material is one of the important properties, no report is available on the value of thermal diffusivity of molten boron oxide within the best knowledge of the present authors. On the other hand, thermal diffusivity measurements of high temperature melts are still far from complete in various cases, arising mainly from onset convective heat flow, heat leakage to the container, and mixed contributions of radiative and conductive heat transfer components. For this reason, most

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values for melts compiled in the TPRC handbook [4] are estimated from measured values of respective substances in the solid state. Therefore, some differences are frequently observed between the simulated results and experimental data. The laser flash method (frequently referred to as laser pulse method), first proposed by Parker and his colleagues [5], has been widely employed as one of the most versatile techniques for measuring thermal diffusivity of various substances at high temperature. Particularly, the laser flash method with a three layered cell was successfully developed for determining the thermal diffusivity values of high temperature melts of both oxides and metals [6–8]. The main purpose of this chapter is to describe the recent developments of the laser flash method and its successful application to the thermal diffusivity measurements of oxide and metallic melts at temperatures above 1,000 K, together with some selected examples of the thermal diffusivity values by precisely separating the radiative component from measured temperature response signals.

5.2 A Brief Background of the Present Requirement for the Thermal Property Measurements of High Temperature Materials For unidirectional heat flow in an isotropic medium, the following Fourier–Biot equation holds with the thermal conductivity λ, J = −λ∇T,

(5.1)

where J is the quantity of heat flowing in unit time through unit area under influence of a temperature gradient ∇T by conduction. For the transient condition where the temperature changes with time t, (5.1) becomes ∇(λ∇T ) = Cp ρ(dT /dt),

(5.2)

where Cp is the specific heat at constant pressure and ρ is the density. In cases where the thermal properties of λ, Cp , and ρ are treated as constants, independent of both position and temperature, (5.2) can be written as follows: ∇2 T = (Cp ρ/λ)(dT /dt)

(5.3)

∇2 T = α−1 (dT /dt),

(5.4)

or

where α = λ/Cp ρ is the quantity called thermal diffusivity, since all diffusivity processes can be represented by an equation similar to (5.3), and since it has the same dimensions as diffusion coefficient. Usually, measurements of thermal diffusivity or conductivity are made under conditions in which the assumption that thermal properties of the sample are independent of temperature and position in the sample is well justified. Of course, particular attention should

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Table 5.1. A comparison for steady state measurement with transient state measurement for thermal conductivity, λ, and thermal diffusivity, α [9] Steady state measurement Transient state measurement Obtained value

λ

λ or α

Mathematical basis

Equation (5.1)

Equations (5.2) or Eqs.(5.1) and (5.2)

Boundary conditions Constant in time

Varies with time and position

Temperature distribution

Stationary in time at all points

Varies with time and position

Measured quantities

J and ∇T

T vs. t or J and T vs. t

be given to measurements at temperatures close to any phase transformation, where apparent heat capacity Cp shows strong temperature dependence. The thermal diffusivity values can be determined by any of the experimental methods based on (5.1)–(5.4). The experimental methods for measuring thermal diffusivity can be classified into two main categories: steady state measurement and transient state measurement, respectively. Comparisons of these two methods for measuring thermal diffusivity are summarized in Table 5.1, along the line suggested by Taylor [9]. The experimental uncertainties due to heat leak, such as radiation and convection, can be generally reduced when experimental time is decreased, and much shorter times are convenient for transient experiments. Much shorter times can be used in transient-state experiments than when a steady state distribution has to be established. Steady-state experiments also have a theoretical disadvantage in measurements on a fluid mixture, because the existence of the temperature gradient in the sample for a long period may cause the separation of the components due to thermal diffusion. Improved electronic devices that have allowed temperature change to be accurately determined over short time intervals have led to an increase of interest in transient state measurements for determining thermal diffusivity particularly at high temperatures. Owing to these factors, the transient measurement such as the laser flash method has gained a dominant position at the present time. Some characteristic features of the laser flash method currently being used for thermal diffusivity measurements of various substances including melts at high temperatures will be given in Sects. 5.4 and 5.5.

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5.3 Experimental Procedures and Theoretical Basis for the Laser Flash Method The laser flash apparatus for measuring thermal diffusivity of various substances at variety of conditions has already been described [5,10]. One typical example is given in Fig. 5.1 using the schematic diagram of the experimental arrangement in the vertical line-up. Figure 5.2 shows a photograph of the facility at the Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University. The front surface of a sample is irradiated by a Nd glass (or ruby) laser beam with a pulse duration of the order of 1 ms. The temperature response signals at the back surface of the sample are detected by using an infrared detector (e.g., InSb: effective wavelength 2–5.5 μm) without mechanical contact. The sample is usually heated up to the desired temperature, within ±2 K controlled by a thermocouple installed near the sample cell and holder, using an additional heating element such as a tungsten mesh heater under vacuum of less than 2 × 10−3 Pa or a platinum wire heater under the air atmosphere. The three layered cell assembly for oxide melts is illustrated in Fig. 5.3 [7]. In this case, the liquid sample is sandwiched with two platinum plates. A platinum crucible containing a small amount of a liquid sample is placed on the alumina pipe. The platinum crucible size (corresponding to the third layer) Pulse laser Al2O3 pipe Micrometer Al2O3 pipe Fe2O3 powder Pt heater Pt crucible Sample liquid Pt crucible Al2O3 pipe I.R.

Infrared detector

Mirror Si lens

Fig. 5.1. A laser flash apparatus at high temperature

5 Thermal Diffusivity of Melts

Neodymium glass laser

Concave lens

Vaccum chamber

InSb infrared detector Vacuum pump

Fig. 5.2. Photograph of the laser flash apparatus set at Tohoku University

Pulse laser

Pulse laser

Micrometer Mullite tube Upper plate Molten sample α

Platinum crucible (upper plate) Molten sample

Δl l2

Platinum crucible (lower plate)

Lower plate

Infrared detector T(l2+Δl)

T(l2) t

(a)

t

(b)

Fig. 5.3. The cell system for oxide melts in the three-layered laser flash method

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is 0.2 mm thick, 10 mm deep, and 19 mm in diameter. Another relatively small platinum crucible of 0.2 mm thickness, 10 mm depth, and 14 mm diameter is employed as the first layer and it is centered to the end of the alumina tube by ceramic cement. The top platinum crucible is located just right above the bottom one to consist of a three-layered cell. The alumina tube is connected to a holder on a three axial movable arm for adjusting the sample thickness. A typical flow of a measuring process is as follows. The inner and outer platinum crucibles were separated several times during the course of the experiments to check that there was no bubble in the sample melt. The alumina pipe was lowered vertically by a micrometer. Then, a laser pulse was irradiated on the top surface of the inner crucible (the first layer). The temperature response signals were recorded by means of an infrared detector focused on the back surface of the outer crucible (the third layer) through a gold mirror. Another temperature response signals were also measured after varying slightly the thickness of a sample melt. The variation of the sample thickness can be determined by reading a micrometer scale attached on the holder. In this type apparatus, the relative difference of sample thickness l in two sets of the measurements was fixed to be 0.2 mm, on the basis of the previous results on high temperature melts at various conditions using the three-layered laser flash method [7, 8]. After measurement, the upper platinum crucible was raised to again check visually that there were no bubbles in the sample melt. From the results of measurements by applying the three-layered cell filled with distilled water, and molten carbonate salts, it was found that the experimental uncertainties were within ±5%. As easy and reliable data processing method has also been developed to estimate the thermal diffusivity value of a sample melt at high temperature using the three layered laser flash method [6]. This data processing method is based on a point that the initial time region of the temperature response curves is unlikely to be affected by heat loss by conduction from the sample to the holder and by radiation. The essential points are summarized later for convenience of discussion. The temperature response of the three-layered cell at the initial time region of the temperature response curve can be given in the following form, analogous to the approach for a two-layered cell proposed by James [11]: √ (η1 + η2 + η3 )2 ∂ ln(T t) =− , (5.5) ∂(1/t) 4 √ where ηi is li / αi , T is temperature rise, and t is time after irradiating a laser pulse. The thickness and thermal diffusivity value √ of the ith layer are denoted by li and αi , respectively. Thus, a plot of ln(T t) against 1/t for (5.5) is considered to show a straight line with slope of β = −(η1 + η2 + η3 )2 /4. The thermal diffusivity value of α2 of the second layer (a sample melt in the present case) can readily be estimated, when the thermal diffusivity values of

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the first and third layers (platinum in the present case) and the thickness of all three layers are provided. Let us consider an infinite slab as shown in Fig. 5.3 consisting of three layers. At the initial time region, the temperature response of the back surface of the third layer can be described as in (5.5). Then, we can obtain the following useful relations:  √ (5.6) l2 / α2 = 2 −β(l2 ) − η1 − η3 ,  √ (l2 + l)/ α2 = 2 −β(l2 +l) − η1 − η3 ,

(5.7)

where, l2 is the thickness of a sample melt in the first measurement, l is the relative change of sample thickness corresponding to the difference between two measurements obtained by lifting √ up the inner crucible (the first layer in the present case), β(l2 ) = ∂ ln(T(l2 ) t)/∂(1/t) and β(l2 +l) = √ ∂ ln(T(l2 +l) t/∂(1/t) are the slopes of temperature response Tl2 for the first measurement and Tl2 +l for the second measurement. The relation between the sample thickness l2 and the thermal diffusivity value of α2 for sample is expressed as a solid line in Fig. 5.4 [7]. For a given l, the similar relation between l2 and α2 is also described as a dashed line in Fig. 5.4. The intersection of these two lines provides the resultant thermal diffusivity value of α2 and its thickness l2 . The temperature response curve is estimated at one thickness l2 as shown in Fig. 5.5a. The same procedure is applied for a thickness l2 + l. The two response curves are converted as provided in Fig. 5.5b and the respective gradients β(l2 ) and β(l2 +l) are estimated by the least squares method. The values of η1 , η3 and l are given, the values of α2 and l2 to satisfy (5.6) and (5.7) are calculated as the intersection (see Fig. 5.4).

a2

Sample thickness: l 2 Sample thickness: l2 + Δl

0

0

l2

Fig. 5.4. Relationship of α2 as a function of thickness estimated from (5.6) and (5.7)

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(a)

T

Tmax

0 0 ta tb

t

ln(T t1/2)

(b)

1/ta

1/t

1/tb

Fig. 5.5. Principle of data processing from experimental output of T vs. t. (a) Determination of ta and tb in the range 0.2–0.4Tmax . (b) Plot of l/t vs. ln(T t1/2 ) to determine slopes β1 and β2

In practice, to derive more accurate values from temperature response curves, numerical iterative procedure combined with a solver of heat transfer equations is employed using the derived values by this method as an initial trial values, because (5.5) is exact only for short duration after laser pulse flashed. Detailed information for the data processing with high accuracy has been given in [7]. It may also be worth mentioning from recent simulated results by applying the finite element method that the heat leakage through a side wall of a platinum crucible is able to reduce less than 5%, when only the values of sample thickness and its variation are carefully selected [12]. In addition, from the results of systematic measurements with samples having different diameters larger than that of the laser beam, it was proved that the shape of the three-layered cell presently used reduces heat leakage to the sample container and the radial heat flow was well-confirmed to be insignificant [6–8]. It is worthy of note that another three-layered cell has been developed for keeping the shape of metallic melt uniform for a given thickness. A schematic diagram of the devised cell is illustrated in Fig. 5.6 [13]. This cell system

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Pulse laser Hole

Carbon fixture

Mullite or alumina tube

Sapphire plate Molten metals Infrared detector

Fig. 5.6. The cell system for metallic melts in the three-layered laser flash method [13]

consists of two sapphire plates and ceramic (alumina or mullite) tube. In other words, alumina or mullite tube is sandwiched between two sapphire plates. Sapphire is known to be transparent for energies produced from Nd glass or ruby laser source as well as energies of infrared ray. The advantage of this sample cell is to vary the sample thickness easily by changing the thickness of ceramics tube, depending upon thermophysical properties of the desired sample. For reducing the volume expansion when the metal sample is melted and residual gas inside the sample, four small holes with size of 0.5 mm in diameter are made in the upper sapphire plate. As shown in Fig. 5.6, this threelayered cell is fixed by the carbon corn. Any leakage of molten metal from this cell system was not detected, probably because of relatively low wettability between cell materials (alumina or mullite) and molten metal sample. The following comments may be worthy of note for further application of this devised cell. Two kinds of sample cell systems were tested. One consists of a mullite tube (thickness, 0.8–1.5 mm; inner diameter, 9 mm; outer diameter, 13 mm) sandwiched by two sapphire plates (thickness, 1.0 mm; diameter, 13 mm). Another is a cell with an alumina tube (thickness, 0.8–1.5 mm; inner diameter, 9 mm; outer diameter, 13 mm) and two sapphire plates (thickness, 1.0 mm; diameter, 13 mm). The mullite tube cell was successively used in the thermal diffusivity measurements of iron, cobalt, and nickel. However, it should be kept in mind that mullite is not a suitable material to contain metallic melt at temperature above 1,700 K for expected time span (about 3 h). In the case of iron melt, we could not obtain sufficiently reliable temperature response signals when using an alumina tube. This is considered mainly arising from a problem for alumina in degassing the residual gas inside the iron sample. In the cases of cobalt and nickel, there were no experimental problems associated with neither the mullite nor alumina tubes [13].

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5.4 Selected Examples of Thermal Diffusivities of Oxide Melts Boron oxide is widely used as a liquid capsule material to encase molten semiconductors for reducing a trouble due to diffusion of high vapor pressure components. The laser flash method with a three-layered cell has been applied to measure the thermal diffusivity data of boron oxide melt at intervals 50–100 K over the temperature range of 1,000–1,500 K [8]. With different water contents, Figure 5.7 shows the results for boron oxide melts containing 0.08 mol% In2 O3 and molten boron oxide containing a small amount of P2 O5 , respectively. In all cases, the thermal diffusivity values of boron oxide melt increase as the temperature rises. Changes in thermal diffusivity, which are not prominent, systematically decrease as the contents of In2 O3 and P2 O5 increase. It can be qualitatively understood that the results are attributed to an increase of the anharmonicity of boron oxide melt resulting from the addition of other oxide, which causes the mean free path of phonons to decrease as compared with the values for pure boron oxide melts [14]. Of course, some further studies are required to obtain the definite answer about these experimental results. It may also be noted that unless a forced dehydrating treatment such as bubbling

(a) α / 10 −7 m 2 s −1

9 7

1.B 2 O 3 2.B 2 O 3 +P 2 O 5 (0.03mol%) 3.B 2 O 3 +P 2 O 5 (0.10mol%) 4.B 2 O 3 +P 2 O 5 (1.00mol%)

H 2 O(120ppm)

1 2 3 4

5 3 1

α / 10 −7 m 2 s −1

(b)

9 7

B 2 O 3 :H 2 O(120ppm) B 2 O 3 +In 2 O 3 (0.08mol%):H 2 O(120ppm) B 2 O 3 :H 2 O(500ppm)

5 3 1 1000

1100

1200

1300

1400

1500

T/K Fig. 5.7. Thermal diffusivity of molten boron oxide (a) containing H2 O and In2 O3 as a function of temperature. (b) Effect of the P2 O5 content on thermal diffusivity of molten boron oxide (containing 120 ppm water) [8]

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5

a/10-7m2s-1

4 3 2 1 0 1500 1520

pure LN LN+2mol% MgO LN+5mol% MgO 1540 1560 T/K

1580

1600 1620

Fig. 5.8. Thermal diffusivity values of molten LiNbO3 with doping MgO as a function of temperature [16]

is employed, the water content of the boron oxide melts does not change even if samples are heated up to the temperature range between 1,000 and 1,500 K. Lithium niobate (LiNbO3 ) is one of the most promising nonlinear optical materials, because of its excellent optical properties [15]. Thermal diffusivity values of LiNbO3 melt are important with respect to the solid–liquid interface for making a large size and subgrain free single crystal. Figure 5.8 shows the results of molten LiNbO3 at 1,523–1,604 K obtained by the laser flash method [16]. This includes the results of molten LiNbO3 with 0, 2, and 5 mol% MgO. All data are found in the relatively narrow range around 3×10−7 m2 s−1 , which is approximately half the value reported for LiNbO3 crystal at 773 K [17]. A positive temperature dependence of thermal diffusivity value of LiNbO3 melt is found and its increase is in the order of 5% as the temperature raises from 1,523 to 1,604 K. Their compositional variation of thermal diffusivity is not detected by doping with MgO, although some variation has been reported in density, surface tension, and shear viscosity [18, 19]. This implies that the fundamental mechanism of thermal diffusion in LiNbO3 melt is unchanged both by temperature change or by doping with MgO up to 5 mol%. The thermal diffusivity measurements for continuous casting powder (hereafter referred to as CC powder) for steel were systematically made by the three-layered laser flash method. The results of thermal diffusivity values are given in Fig. 5.9a, using the results of CC powder melts containing titanium oxide and iron oxide as an example. The reference composition of CC powder is 35.6% SiO2 , 19.9%, 17.1% CaF2 , 10.1% Na2 O, 9.3% MgO, and 7.7% Al2 O3 in mass%. The measured values similar to the Fig. 5.9 case were obtained for 19 cases, although there are differences in detail [12]. The results of Fig. 5.9a clearly indicate a slightly positive temperature coefficient of thermal diffusivity of molten CC powders in the temperature range presently investigated. However, the present results are somewhat spread in certain temperatures and

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8 (a)

(a)

6

a / 10-7m2s-1

4 2

TiO2 0.7% TiO2 2.6%

TiO2 4.9% TiO2 7.4% TiO2 9.6%

T.Fe 0.4% T.Fe 1.2% T.Fe 2.6%

0 (b)

(b)

6 4 2

0 1350 1400 1450 1500 1550 1600 1400 1450 1500 1550 1600 T/K Fig. 5.9. Thermal diffusivity values of continuous casting powders for steel containing titanium oxide and iron oxide. (a) Radiative component is included. (b) Radiative component is excluded [12]

such scattering is considered beyond the experimental uncertainty. This is attributed to the effect of radiative heat transfer component involved in a molten sample. Although the initial time region of the temperature response curve is known not to be severely affected by the radiative heat transfer [11], the radiative component should be separated from measured temperature response signals. This is particularly true for the higher temperature measurements. The absorption coefficient of samples of interest is needed for quantitative discussion of the radiative heat transfer in high temperature substances. It should be mentioned that the absorption coefficient data are essentially required for the semitransparent media, in comparison with the transparent and opaque cases. The absorption coefficients of three CC powder samples containing titanium oxide and iron oxide are given in Fig. 5.10 as a function of wave length together with that of the hemispherical emissive power of blackbody at 1,573 K [21]. It is also very interesting that the absorption coefficients of CC powder samples in the range between 1 × 10−6 and 4 × 10−6 m−1 is found not to be significantly affected by the change in CaO/SiO2 ratio or the addition of zirconium oxide and hafnium oxide to the reference composition [21]. These results suggest that the correction by the transparent body approximation is sufficiently accepted for these CC powder samples with respect to the radiative heat transfer behavior. In other words, the contribution from the radiative component should be explicitly considered only for cases containing titanium oxide and iron oxide. In samples for which the transparent body approximation is valid, a liquid layer appears to be transparent to radiative heat transfer where no radiation

3000 1573K Reference T.Fe 1.2% 2000 k / m -1

0

10

TiO2 9.6%

5

1000

0

2

6 4 l/10 -6 m

8

97

Hemispherical emissive power, 1010 Wm-3

5 Thermal Diffusivity of Melts

10

0

Fig. 5.10. Absorption coefficient of continuous casting powders for steel containing titanium oxide and iron oxide together with the hemispherical spectral emissive power of blackbody at 1,573 K [20]

Δl= 0.2 mm l2(mm) 0.3

a/10-7m2s-1

6

0.2

5

0.1

4

Given value of thermal disffusivity for calculation

1350

1400

1450 1500 T/K

1550

1600

Fig. 5.11. Apparent thermal diffusivity indicating radiative heat flow theoretically calculated under the transparent body approximation for a sample of thermal diffusivity of 4 × 10−7 m2 s−1

is absorbed or emitted from a liquid layer. In this condition the radiation from a platinum plate is considered to be dominant. With respect to this subject, Ohta et al. [22] have systematically estimated the effect of radiative component in the transparent body approximation by computing the apparent thermal diffusivity values for a case of α2 = 4 × 10−7 m2 s−1 using the finite difference method and the results are illustrated in Fig. 5.11. The essential point of Fig. 5.11 is to suggest that the apparent thermal diffusivity values increase with increasing temperature and sample thickness by the contribution due to radiative component in cases where the transparent body approximation is valid. These results also suggest that the radiative component induces an

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increase of about 20% in thermal diffusivity for the measurement where the sample thickness of 0.1 mm (for the first measurement) with its variation of 0.2 mm. As the coefficients for correction are summarized in the following form as a function of the absolute temperature T , the measured thermal diffusivity values of α0 should be corrected with the value of l2 . α = cα0 ,

(5.8)

c = 2.549 × 10−7 T 2 − 4.504 × 10−4 T + 1.244 for l2 = 0.1 mm, c = 6.986 × 10−7 T 2 − 1.466 × 10−4 T + 1.883 for l2 = 0.2 mm, c = 1.253 × 10−6 T 2 − 2.675 × 10−3 T + 2.606 for l2 = 0.3 mm.

(5.9)

On the other hand, both the absorption and emission of the radiative component in a liquid layer should be included when the transparent body approximation is not well accepted as for CC powder samples. With respect to this subject, Darby [23] has reported fundamental equations by considering all modes of radiative heat transfer. Ohta et al. [22] computed the results for the gray body approximation where the spectral absorption coefficient of the semitransparent media is reduced to the mean absorption coefficient κm on the basis of Darby equations using the control volume method [24]. In this computation, the temperature response signals are estimated as a function of κm from which the thermal diffusivity values can be obtained. Figure 5.12 shows the results for a case of thermal diffusivity of 4×10−7 m2 s−1 [22]. The validity and usefulness of these results may be confirmed by suggesting the following points. Good agreement is found with the value proposed by Rosseland [25], who suggests for the condition of κl > 1, in the higher absorption region

a/10−7m2s−1

l = 0.3mm 573K Rosseland approximation

4.5 Transparent approximation

4

κ l >1

Given value of thermal diffusivity

101

102

103

104

k /m−1

105

106

Fig. 5.12. Apparent thermal diffusivity including radiative heat flow theoretically calculated under the gray body approximation for a sample of thermal diffusivity of 4 ×10−7 m2 s−1 .

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Table 5.2. Mean absorption coefficient κm for samples containing TiO2 and iron oxide estimated from measured optical properties [21] Temperature(K) 1,373 Type of Powder Fe Fe TiO2 TiO2 TiO2

0.40% 1.20% 2.60% 4.90% 9.60%

1,473

1,573

κm 546 774 21 39 81

572 810 22 41 87

596 844 24 43 94

larger than 105 m−1 . When the absorption coefficient shows the value beyond 106 m−1 , it may safely be said that samples are considered opaque. In such case, the radiative contribution is insignificant. On the other hand, in the lower absorption region less than 102 m−1 , the results are found to be consistent with those of the transparent body approximation. The contribution from radiative component should be separated from measured thermal diffusivity values, at least, the gray body approximation or higher order (band) approximation in cases where the absorption coefficient is lying in the region between 102 and 104 m−1 . Samples containing titanium oxide and iron oxide are included in this category. The mean absorption coefficient of κm for five CC powder samples containing titanium oxide and iron oxide was determined from the measurements of spectral transmissivity and reflectivity in the wave length region between 4×10−7 and 1×10−5 m and the results are summarized in Table 5.2 [21]. Analysis for estimating the contribution of radiative component at high temperature measurements for CC powder samples has been made by considering the variation of the optical properties and provided information about the apparent thermal diffusivity values theoretically estimated for three cases of the transparent body approximation [20]. The wavelength dependence of the absorption coefficient for samples, as exemplified by the results of Fig. 5.10, is explicitly included in the band approximation, although this requires lengthy numerical computation. The theoretical details are given in [22]. The most significant aspects are to provide the variation of the apparent thermal diffusivity values when using three different approximations for estimating the radiative contribution as summarized in Table 5.3. The following points are worth of note. (1) The contribution from radiative component could be estimated by the transparent body approximation with the experimental uncertainty less than 4% for samples recognized as transparent. (2) The contribution from radiative component should be estimated, at least, by the gray body approximation, with the mean absorption coefficient for samples considered semi-transparent if we want to hold the experimental uncertainty down to less than 4%.

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Table 5.3. Apparent thermal diffusivity estimated from theoretical temperature response curves at 1,575 K by the three-layered laser flash method Type of Powder Reference Fe Fe TiO2 TiO2 TiO2

0.40% 1.20% 2.60% 4.90% 9.60%

Approximation Transparent Gray 4.90 (0.03) 4.90 4.90 4.90 4.90 4.90

(0.09) (0.11) (0.04) (0.04) (0.05)

Band 5.05

5.29 5.44 4.91 4.93 4.97

(0.02) (0.01) (0.04) (0.03) (0.03)

5.39 5.5 5.1 5.11 5.13

Given, thermal diffusivity, 4 × 10−7 m2 s−1 ; sample thickness and its variation, 0.2 and 0.2 mm. The numerical values in the parenthesis correspond to the ratio of deviation from the case estimated under the band approximation [20]

The thermal diffusivity values at high temperature were again estimated for CC powder samples for steel by applying the transparent body approximation or the gray body approximation, depending upon their optical properties characterized by the absorption coefficient behavior. The resultant thermal diffusivity values are given in Fig. 5.9b. It is clearly found from a comparison of the results of Fig. 5.9a that scattering detected in measured values is considerably reduced by correcting the contribution from radiative component, depending on both temperature and sample thickness. In addition, the thermal diffusivity values of continuous casting powders for steel are of the order to 4 ± 0.5 × 10−7 m2 s−1 and almost insignificant with the variation of both temperature and composition presently investigated.

5.5 Selected Examples of Thermal Diffusivities of Metallic Melts The laser flash method with a three-layered cell shown in Fig. 5.5 has been applied to obtain sufficiently reliable values of thermal diffusivity of molten iron, cobalt, and nickel at temperature above 1,700 K [13]. In this experimental condition with a devised cell, holding the sample of metallic melts, it is necessary to consider the effect of not only radiative heat loss but also conductive heat loss at interface between metallic melt and cell material. Regarding this subject, Nishi et al. [13] reported the computer simulation results using numerical analysis under the following eight assumptions to construct the theoretical heat transport equation with the appropriate boundary conditions and the initial conditions. (1) One dimensional heat flow and each layer is homogeneous (2) The whole cell is under adiabatic condition for the conductive heat flow

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(3) All thermophysical properties of three layers are given (4) The thermal contact resistance at the interface between layers is uniform and has same value at both the upper and the lower interfaces (5) A heat pulse is uniformly absorbed at the front surface (6) Radiative heat loss is proportional to the temperature difference, Tm − Te , where Tm is temperature rise in metallic melt and Te is steady-state temperature, respectively (7) No absorption of the energy in medium of the cell because the sapphire presently used is transparent to both the laser pulse and infrared (8) Radiative heat loss occurs only at the surface of the metallic melt The most important and significant points are given below for further work on the thermal diffusivity measurements of metallic melts at high temperature. Figure 5.13 shows a typical temperature response curve where the dimensionless parameter of thermal contact resistance at interface, R+ = λm R/lm = 102 , and Biot number, Y = 4εσTe3 lm /λm = 0.1, and the normalized temperature, T ∗ = T /Tmax . This result is demonstrated using the molten nickel case as an example. Here λm is thermal conductivity of metallic melts, R is the thermal contact resistance, lm is the sample thickness of metallic melts, ε is the thermal emissivity of the surface of metallic melts, and σ is the Stefan–Boltzmann constant, respectively. It is interesting to note that the decay part of the temperature response curve normalized by the maximum temperature rise, Tmax , is approximately given in the normalized time region between 8 and 12.

(5.10) T ∗ = TM exp −kt∗ ,

0.2 ln TM exp(-kt) *

0.1

ln T

*

0.0 −0.1 −0.2 −0.3 Calculated : molten nickel (R+ = 10 2, Y= 0.1)

−0.4 −0.5

0

5

t*

10

15

Fig. 5.13. Temperature response curve for molten nickel obtained by numerical simulation [13]

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where TM corresponding to the temperature when extrapolated the attenuation curve down to time at laser beam irradiation. As shown in the results of Fig. 5.13, the linear relationship between the logarithmic value of T ∗ and t∗ may be described in the following form: ln T ∗ = −kt∗ + ln TM .

(5.11)

The gradient k and the intercept ln TM provide information with respect to the effects of radiative and conductive heat loss at interface between melt sample and cell material. Similar calculations were made for various given conditions and the simulated results of the relationship between k and ln TM with conduction for the thermal resistance parameter R+ = 10n , n = 0.5−∞, and Biot number Y = 0 − 0.25 are illustrated in Fig. 5.14. This figure includes the experimental results of molten nickel obtained with different conditions and a point at the origin in Fig. 5.14 corresponds to the adiabatic condition. It has been intention of the present authors to suggest that most of the experimental results of k and ln TM for molten nickel are found to be situated near the curve of R+ = ∞. Small dispersion is likely to be attributed to the fluctuations in the signal-to-noise ratio detected in the high temperature measurements. Accordingly, the present authors maintain the view, from the results of Fig. 5.14 that the effect of the conductive heat loss at interface between metallic melt and cell material is negligibly small within the present experimental condition. In other words, thermal contact resistance at the interface between molten sample and cell material can be considered to

Molten nickel

: : : : :

0.20

Run1 Run2 Run3 Run4 Calculated

lnTM

0.15

R + = 10 n

0.10

n = 0.5

⬁ 0.05

1.0 2.0

0.00 0.00

0.01

0.02

0.03

0.04

0.05

k Fig. 5.14. Relationships between κ and ln TM of molten nickel. Symbols of open square, open circle, open triangle, and open inverted triangle denote the results obtained from the different runs [13]

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103

20

a /10-6 m2s-1

15

: Molten iron : Molten cobalt : Molten nickel

10

5

0 1700

1800

1900 T/K

Fig. 5.15. Thermal diffusivity of molten iron, cobalt, and nickel as a function of temperature [13]

be sufficiently large. Consequently, only the radiative heat leakage needs to be taken into account for the present experimental condition with the devised three-layered cell. Sufficiently reasonable reproducibility of the experimental thermal diffusivity data was confirmed by repeating the measurements under a given condition more than three times [13]. The thermal diffusivity values of molten iron, cobalt, and nickel were measured for iron in the temperature range from 1,808 to 1,868 K, for cobalt in the temperature range from 1,768 to 1,838 K, and for nickel in the temperature range from 1,728 to 1,928 K [13]. The results are summarized in Fig. 5.15. The thermal diffusivity value of molten iron at its melting point (1,808 K) is 12% smaller than the reported value of solid iron at 1,600 K [4]. The liquid value for cobalt with its melting point (1,768 K) is 21% smaller than that of solid cobalt at 1,700 K and for nickel case (with melting point 1,728 K) the liquid value is 36% smaller than that of solid nickel at 1,500 K [4]. Such distinct variation in thermal diffusivity of three metals should be attributed to a change in the structure from the long-range periodic atomic distribution to that with nonperiodicity on melting. It is also worth mentioning that the thermal diffusivity values of three metallic melts indicate slightly positive temperature dependence, although the origin of this behavior cannot be certainly identified at the present time. Thermal diffusivity values of molten three metals are summarized in the following equations (m2 s−1 ) [13].

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αFe = 4.51 × 10−9 (T − 1808) + 5.97 × 10−6 (1808 ≤ T /K ≤ 1868), αCo = 6.59 × 10−9 (T − 1768) + 6.14 × 10−6 (1768 ≤ T /K ≤ 1838), αNi = 6.61 × 10−9 (T − 1728) + 1.02 × 10−5 (1728 ≤ T /K ≤ 1928), (5.12) Thermal diffusivity values of molten germanium and silicon have also been obtained by the laser flash method with a three-layered cell, although silica glass, known to be transparent to Nd glass laser and infrared ray, was used in the measurement for silicon [26]. αGe = 1.40 × 10−8 (T − 1218) + 2.28 × 10−5 (1218 ≤ T /K ≤ 1398), αSi = 4.48 × 10−9 (T − 1685) + 2.23 × 10−5 (1685 ≤ T /K ≤ 1705). (5.13) A large number of attempts to produce metallic glasses have been reported by direct melt processing techniques such as chill block casting of ribbon and free jet spinning of wire. The quenching rate is of the order of 106 K s−1 . The glassy samples were usually produced in the form of ribbon with thickness of 30 μm or wire with diameter of 100 μm. Further, the addition of third elements that exhibit different properties, such as atomic size and crystalline symmetry to a binary alloy, is known to enhance drastically the ease of glass formation. Then some exceptional systems such as Pd–Cu–Si, Pd–Ni–P, and Pt–Ni–P have produced as cylindrical rods of 1–3 mm diameter in the glassy state at quenching rate of only 102 K s−1 [27, 28]. Inoue and his colleagues [29, 30] discovered new Zr-based alloy systems such as Zr–X–Y (X= Al, Y = Co, Ni, Cu), which show a wide super-cooled liquid region of more than 100 K and do not include any nonmetallic elements such as P and Si. Not only a cylindrical rods of several millimeters diameter, but also pipes with inner diameter of 10 mm and wall thickness of 2 mm and ingots with diameter or thickness larger than several centimeters [30, 31] have been recently fabricated with bulk metallic glass. These results prompt us to investigate thermal properties of these alloy melts because of their importance to estimate a temperature profile in a mold. A critical cooling rate to obtain glassy phase may also be a function of thermal diffusivity. Figure 5.16 shows the measured temperature response curve of molten Zr55 Al10 Ni5 Cu30 alloy at 1,218 K, as an example [32]. Measurements using the laser flash method with a three-layered cell were also made for molten Zr60 Al15 Ni25 and Zr65 Al7.5 Cu27.5 alloys for comparison. The thermal diffusivity values of molten Pd-based alloys which are known to show good glass formability were systematically measured by the laser flash method [33]. All these results for six alloy melts are summarized as a function of temperature in the following form (m2 s−1 ). αZr55 Al10 Ni5 Cu30 = 3.73 × 10−9 (T − 1154) + 4.53 × 10−6 (1154 ≤ T /K ≤ 1228), αZr60 Al15 Ni25 = 7.89 × 10−9 (T − 1170) + 5.01 × 10−6 (1170 ≤ T /K ≤ 1263), αZr65 Al7.5 Cu27.5 = 4.75 × 10−9 (T − 1180) + 5.39 × 10−6 (1180 ≤ T /K ≤ 1303), (5.14)

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Zr55Al10Cu30Ni5 in the liquid state at 1218K 0

200

400

600

t/ms Fig. 5.16. Measured temperature response curve of molten Zr55 Al10 Ni5 Cu30 alloy at 1,218 K [32] αPd40 Cu30 Ni10 P20 = 5.73 × 10−9 (T − 920) + 3.28 × 10−6 −9

αPd40 Ni40 P20 = 3.95 × 10

−6

(T − 1000) + 4.12 × 10

(920 ≤ T /K ≤ 1120), (1000 ≤ T /K ≤ 1120),

αPd40 Cu40 P20 = 6.67 × 10−9 (T − 1060) + 5.35 × 10−6 (1060 ≤ T /K ≤ 1120). (5.15)

Positive temperature dependence of the thermal diffusivity values is found for six alloys in the liquid state and the best glass forming alloy, such as Zr55 Al10 Ni5 Cu30 and Pd40 Cu30 Ni10 P20 , indicates the lowest value among the investigated alloys with respect to both Zr-based and Pd-based cases. The thermal diffusivity values of these six molten alloys are summarized in Fig. 5.17 as a function of normalized temperature, (T − Tl )/Tl , where Tl is the liquidus temperature. The thermal diffusivity value can be estimated at a liquidus temperature by extrapolating the obtained equations to that temperature, so that such values are plotted against the critical cooling rate of each alloy in Fig. 5.18 [32]. It is interesting to find that the lower the thermal diffusivity values of Zr-based alloys at liquidus temperature, the lower the critical cooling rate to obtain metallic glass phase becomes. Such behavior has also been confirmed in the Pd-based alloy case. Although any definite comment cannot be certainly obtained yet, the number of free electrons in molten alloys is likely to be close relationship between transport property and the critical cooling rate. The thermal diffusivity values of molten alloys may be one of the good indicators for predicting the glass forming ability of alloys of interest.

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6

-6

a /10 m s

2 -1

5 4 3

:Zr 65 Al 7.5 Cu 27.5 :Zr 60 Al 15 Ni 25 :Zr 55 Al 10 Ni 5 Cu 30 :Pd 40 Cu 40 P 20 :Pd 40 Ni 40 P 20 :Pd 40 Cu 30 Ni 10 P 20

2 1

0 0.0

0.1

0.2

0.3

0.4

0.5

(T - Tl) / Tl Fig. 5.17. The thermal diffusivity values of molten Zr-based and Pd-based alloys as a function of normalized temperature, (T − Tl )/Tl , where Tl is the liquidus temperature [32]

6

4

-6

a /10 m s

2 -1

5

3

:Zr 65 Al 7.5 Cu 27.5 :Zr 60 Al 15 Ni 25 :Zr 55 Al 10 Ni 5 Cu 30 :Pd 40 Cu 40 P 20 :Pd 40 Ni 40 P 20 :Pd 40 Cu 30 Ni 10 P 20

2 1 0

1

10

100 -l

R c / Ks

Fig. 5.18. Relationship between the estimated thermal diffusivity values at a liquidus temperature and the critical cooling rate for glass phase formation of molten Zr-based and Pd-based alloys [32]

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5.6 Summary The current state of the art concerning the laser flash method has been reviewed for determining the thermal diffusivity values of high temperature oxide and metallic melts in the temperature range between 1,000 and 2,000 K. To overcome some experimental difficulties at high temperature, a differential scheme without information of the absolute value of sample thickness was used in the oxide melt case with the help of a numerical technique to estimate the effect of radiative heat transfer in the cell system, where the absorption and emission behavior of the radiative component in the sample layer were fully considered. For metallic melt, a new simple cell system was developed to keep a precise value of sample thickness and the effect of conductive or radiative heat leak from sample was taken into account. The values of thermal diffusivity of molten boron oxide, lithium niobate, and various continuous casting powders were successfully determined by the differential three-layered laser flash method. The effect of radiative component in the cell system on the measured thermal diffusivity was systematically discussed by considering absorption and emission of radiation in the sample layer. The thermal diffusivity values of some metallic melts of Fe, Co, Ni, Ge, and Si were determined at temperature up to 1,700 K, with sufficient reliability by the laser flash method. This method was also systematically applied to molten Zr-based and Pd-based alloys, which show good glass formability. Although the number of results is limited, it would be interesting to find that the thermal diffusivity values of molten alloys provide a possible good indicator for predicting the glass forming ability of alloys of interest. These results clearly suggest that it would be very promising to extend the laser flash method with a three-layered cell to the thermal diffusivity measurements of other melts at high temperature and then its validity and usefulness can be confirmed on a wider base.

Acknowledgment The authors are deeply indebted to Professors Y. Tomota and S. Kitamura for their sustained encouragement of our projects on the thermal property measurements of materials. We are also grateful to Professors A. Inoue and J. Saida, and Drs. T. Nishi and N. Nishiyama for their collaboration on the metallic melt projects.

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References 1. R.J. Gale, D.G. Lovering (eds.), Molten Salt Technology (Plenum, New York, 1982; 1991) 2. K.C. Mills, in Proc. 4th Inter. Conf. on Molten Slag and Fluxes, Iron Steel Institute of Japan, 1992, p 405 3. Editorial Committee of Crystal Growth Society of Japan, Handbook for Crystal Growth, Crystal Growth Society of Japan, 1995 4. Y.S. Touloukian, C.Y. Ho, P.E. Liley (eds.), Thermophysical Properties of Matter, (Plenum, New York, 1971) 5. W.J. Parker, R.J. Jenkins, C.P. Butler, G.L. Abbott, J. Appl. Phys. 32, 1979 (1961) 6. H. Ohta, G. Ogura, Y. Waseda, M. Suzuki, Y. Yamashita, Rev. Sci. Insrum, 61, 2645 (1990) 7. Y. Waseda, M. Masuda, H. Ohta, in Proc. 4th Inter. Symp. on Advanced Nuclear Energy Research, Japan. Atomic Energy Research Institute, Tokai, Ibaraki, 1992, p 298 8. G. Ogura, I.K. Suh, H. Ohta, and Y. Waseda, J. Ceram. Soc. Jpn. 98, 320 (1990) 9. R.E. Taylor, High Temp. High Pressure 11, 43 (1979) 10. Y. Waseda, H. Ohta, Solid State Ionics 22, 263 (1987) 11. H.M. James, J. Appl. Phys, 51, 4666 (1980) 12. Y. Waseda, M. Masuda, K. Watanabe, H. Shibata, H. Ohta, and K. Nakajima, High Temp. Mater. Process 13, 267 (1994) 13. T. Nishi, H. Shibata, H. Ohta, Y. Waseda, Mater. Metallur. Trans. 34A, 2801 (2003) 14. T. Nakamura, Ceramics and Heat (Gihodo, Tokyo, 1985), p 81 15. A.A. Ballman, J. Am. Ceram. Soc. 48, 112 (1965) 16. H. Ogawa, H. Ohta, Y. Waseda, J. Cryst. Growth 133, 255 (1993) 17. R.A. Morgan, K.I. Kang, C.C. Hsu, C.L. Kiliopoulos, N. Peyghambarian, Appl. Optics 26, 5266 (1987) 18. Y. Zhou, J. Wang, P. Wangm, L. Tang, O. Zhu, Y. Wu, H. Tan, J. Cryst. Growth, 114, 87 (1991) 19. E. Tokizaki, K. Terashima, S. Kimura, J. Cryst. Growth 123, 121 (1992) 20. H. Ohta, M. Masuda, K. Watanabe, K. Nakajima, H. Shibata, Y. Waseda, Tetsuto-Hagane 80, 463 (1994) 21. H. Ohta, K. Watanabe, K. Nakajima, Y. Waseda, High Temp. Mater. Process, 12, 139 (1993) 22. H. Ohta, K. Nakajima, M. Masuda, Y. Waseda, in Proc. 4th Inter. Symp. On Slags and Fluxes, Iron and Steel Inst. Japan, Tokyo, 1992, p 421 23. M.I. Darby, High Temp. High Pressure, 15, 629 (1983) 24. E.R. Eckert, R.M. Drake Jr., Analysis of Heat Transfer (McGraw-Hill, Kougakusha, Tokyo, 1972), p 254 25. S. Rosseland, Theoretical Astrophysics, Claredon Press, Oxford, (1936); cited in a book of R. Siegel and J.R. Howel, Themal Radiation Heat Transfer (McGrawHill, Kougakusha, Tokyo, 1972), p 470 26. T. Nishi, H. Shibata, H. Ohta, Mater. Trans. 44, 2369 (2003) 27. H.S. Chen, Acta Metall. 22, 1504 (1974) 28. H.S. Chen, K.A. Jackson, Chapter 3 in Metallic Glasses, American Society for Metals, Metals park, Ohio, (1978), p 74

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29. A. Inoue, T. Zhang, T. Masumoto, Mater. Trans. JIM 31, 177 (1990) 30. A. Inoue, Bulk Amorphous Alloys, Preparation and Fundamental Characteristics, Trans. Tech. Pub., Zurich, (1998) 31. N. Nishiyama, A. Inoue, Mater. Trans. JIM 37, 1531 (1996) 32. H. Shibata, S. Nishihata, H. Ohta, S. Suzuki, Y. Waseda, M. Imafuku, J. Saida, A. Inoue, Mater. Trans. 48, 886 (2007) 33. T. Nishi, H. Shibata, H. Ohta, N. Nishiyama, A. Inoue, Y. Waseda, Phys. Rev. B 70, 174204 (2004)

6 Emissivities of High Temperature Metallic Melts Masahiro Susa and Rie K Endo

6.1 Introduction The mathematical modeling of heat flow in high temperature processes has been a useful means of obtaining more efficient process design and stricter process control. At high temperatures heat can be transferred by three mechanisms, that is, conduction, convection, and radiation. Analysis of heat transfer requires physical property data of a medium through which heat is transferred, relevant to the respective mechanisms: the thermal conductivity is indispensable to heat flux calculation by conduction from Fourier’s law, and the viscosity, density, and heat capacity are indispensable to heat flux calculation by convection. On the other hand, the emissivity plays a key role in heat transfer analysis by radiation because it quantifies how well a substance radiates energy in the form of light. The emissivity is more important in the analysis for materials processing involving metallic melts since the radiation contribution becomes more predominant as temperature rises. Thus, emissivity measurements have been conventionally attempted on various metallic melts at high temperatures, data of which have been published in the handbook [1], for example. However, the data have not been abundant enough for practical use in mathematical modeling, and measurements are still now being made continuously. This chapter focuses on emissivities of metallic melts and reviews recent measurement techniques and data for the emissivity, mainly on and after the publication of the handbook.

6.2 Definition of Emissivity The emissivity ε is a value to specify how well a real body radiates energy in comparison with a blackbody at the same temperature as the real body, and is defined using radiation intensities emitted from a real body (sample) and from a blackbody. However, the radiation intensity from a blackbody IB depends upon wavelength of radiation λ and temperature T and, furthermore, the

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radiation intensity from a real body IR depends upon direction of radiation as well. Thus, the emissivity should be strictly expressed as a function of wavelength, direction, and temperature. The most fundamental emissivity is directional spectral emissivity ε(λ, θ, ϕ, T ) that includes dependencies on wavelength, direction, and temperature, and is defined as ε(λ, θ, ϕ, T ) = IR (λ, θ, ϕ, T )/IB (λ, T ),

(6.1)

where θ is the polar angle measured from the normal to the surface of a sample and ϕ is the azimuth angle. Strictly, the radiation intensity as a function of wavelength is termed the spectral radiation intensity (or radiance) and the value of IB (λ, T ) is given by Planck’s law of radiation as follows: IB (λ, T ) = 2C1 /λ5 [exp(C2 /λT ) − 1] ,

(6.2)

where C1 and C2 are Planck’s first and second constants, respectively, 5.96 × 10−17 W m2 sr−1 and 1.44 × 10−2 m K. In particular, the spectral emissivity in a direction normal to the surface is termed normal spectral emissivity, which is often measured experimentally and is very important in temperature measurements using optical pyrometers (radiation thermometers). Integrations of (6.1) over all wavelengths and over all directions give directional total emissivity and hemispherical spectral emissivity, respectively, and integration of (6.1) over all wavelengths and all directions gives hemispherical total emissivity. The hemispherical total emissivity is indispensable to the calculation of energy loss from the surface and can also be measured. However, this direct measurement is rather difficult and the hemispherical total emissivity is usually estimated from the normal spectral emissivity. As a consequence, the following review focuses measurement techniques and data for normal spectral emissivity.

6.3 Measurement Techniques for Emissivities 6.3.1 Method Based on Wien’s Formula For small values of λT , that is, exp(C2 /λT ) 1, (6.2) for blackbody radiation can be approximated to the following: IB (λ, T ) = 2C1 /λ5 exp(C2 /λT ).

(6.3)

This equation is called Wien’s equation, which is accurate to within 1% for λT less than 3,000 μm K [2]. For real body (non-blackbody) at the same temperature as blackbody, the normal spectral radiation intensity IR could be expressed by Wien’s formula using an apparent or brightness temperature TA as IR (λ, T ) = 2C1 /λ5 exp(C2 /λTA ).

(6.4)

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23 mm Al2O3 crucible

19 mm

26 mm

18 mm

Cavity (1.6mm diameter, 16 mm depth)

Ta crucible

Fig. 6.1. Al2 O3 and Ta crucibles used by Ratanapupech and Bautista [5]

Thus, the normal spectral emissivity can be derived through (6.1) as ε(λ, T ) = IR (λ, T )/IB (λ, T ) = exp [(C2 /λ)(1/T − 1/TA )]

(6.5)

The normal spectral emissivity is sometimes determined based on (6.5) from measurements of true and apparent temperatures T and TA [3–6]. For example, Ratanapupech and Bautista [5] have determined the normal spectral emissivities of molten iron–nickel alloys over the complete range of composition for a wavelength of 645 nm. As shown in Fig. 6.1, they used two crucibles to hold molten samples: the outside crucible was made of tantalum and the inside crucible was made of alumina, which kept molten samples from chemical contamination. The tantalum crucible had a tantalum rod with a cavity (ca 1.6 mm diameter and 16 mm depth), which emitted blackbody radiation at the same temperature as the sample. The true and apparent temperatures T and TA were measured using an optical pyrometer on radiations from the cavity and the sample surface, respectively. The true temperature is sometimes measured using a thermocouple. Tomita et al. [6] have determined the normal spectral emissivity of titanium arc-melted on a water-cooled copper plate for a wavelength of 650 nm on the basis of (6.5), where the apparent temperature was measured using an optical pyrometer and the true temperature was measured using a W·5%Re–W·26%Re thermocouple sheathed in a silica tube. 6.3.2 Method Based on Optical Constants When a light beam is incident on a medium, several optical processes take place. Some of the light is reflected from the front surface, some is absorbed in the medium, and the rest is transmitted through the medium. The fractions of

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incident light that are reflected, absorbed, and transmitted by a material are termed the reflectivity r, absorptivity a, and transmissivity t, respectively. The sum of these values equal unity since all the incident light is either reflected, absorbed, or transmitted: r + a + t = 1.

(6.6)

For opaque materials such as metals, t = 0 and (6.6), reduces to r + a = 1.

(6.7)

When the system is in thermal equilibrium, Kirchhoff’s law applies, that is, a = ε. Thus, ε = 1 − r.

(6.8)

On the other hand, the normal spectral reflectivity r(λ, T ) can be expressed from Fresnel’s equation as     (6.9) r(λ, T ) = (n − 1)2 + k 2 / (n + 1)2 + k 2 , where n and k are the optical constants (refractive index and extinction coefficient, respectively) of material at wavelength λ and temperature T . Combination of (6.8) and (6.9) leads to   (6.10) ε(λ, T ) = 4n/ (n + 1)2 + k 2 . Accordingly, measurement of refractive index and extinction coefficient provides the normal spectral emissivity. The refractive index and extinction coefficient are usually determined simultaneously using ellipsometry. Ellipsometry is based on the measurement of the change in polarization state of a beam of light upon reflection from the sample, and can be conducted as a function of wavelength. The instrument basically consists of a light source, polarizer, sample, analyzer, and detector, although there are a number of ways that an ellipsometric measurement can be realized. As shown in Fig. 6.2, in the measurement the linearly polarized light is incident onto a sample at an angle of incidence φ1 ; as a result, the reflected light is elliptically polarized, and the polarization state is determined by the analyzer, providing the ellipsometry parameters Δ and Ψ , which characterizes the polarization state. The values of Δ and Ψ are then converted to the optical constants (n and k) of the sample using the angle of incidence φ1 and the refractive index of the ambient n1 (usually air, n1 ≈ 1) via (6.11-1) and (6.11-2) [7].  n2 − k 2 = n21 sin2 φ1 1 + tan2 φ1 (cos2 2Ψ − sin2 Δ sin2 2Ψ )/  (1 + sin 2Ψ cos Δ)2 (6.11-1) 2nk = n21 sin2 φ1 tan2 φ1 sin 4Ψ sin Δ/(1 + sin 2Ψ cos Δ)2

(6.11-2)

Usually, this calculation is conducted on a computer using sophisticated software that may correspond to more complicated situations of samples. Using

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ambient (n1)

φ1 linearly polarised

elliptically polarised

material (n,k)

Fig. 6.2. Reflection of linearly polarized light from material surface Air-tight chamber He-Ne laser

Gas in Polariser

Analyser Furnace Filter

Sample

Diaphragm Thermocouple

Gas out

Fig. 6.3. Normal spectral emissivity measurement using ellipsometer at high temperature [8]

this method, Tanaka et al. [8] have determined normal spectral emissivities of solid Cu–Ni alloys at high temperatures (Fig. 6.3), and Shvarev et al. [9] have determined normal spectral emissivities of Ni–Cr alloys in solid and liquid states. Measurements on liquid metals and alloys have been vigorously made by Krishnan and his co-workers [10–15]: They have applied ellipsometry to a metal droplet that is melted and suspended by electromagnetic levitation (Fig. 6.4), and produced data for Al [10–12], Fe [15], Ni [14, 15] etc for example. Electromagnetic levitation is very useful for preventing samples from chemical reactions with containers.

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D2

Sample

Electromagnetic Levitator Linear Polariser Laser Mirror

Fig. 6.4. Electromagnetic levitation with laser ellipsometry [14]

6.3.3 Method Based on Direct Measurements of Radiation Intensities According (6.1), the normal spectral emissivity can be obtained by measuring radiation intensities from the sample and the blackbody at the same temperature, and this method has very often been applied to emissivity measurement due to the simplicity in principle [16–19]. For example, Takasuka et al [18, 19] have determined the normal spectral emissivities of solid and liquid silicon at the melting point in the wavelength range 500–800 nm, using an experimental setup shown in Fig. 6.5 [18]: A silicon sample is held in a graphite crucible with cavities for blackbody radiation in a furnace. The temperature of the sample is measured with an optical pyrometer using blackbody radiation from a cavity. The normal radiation from the sample surface is led to a monochromator using the lens and optical fibre system, and its intensity is measured with a detector as a function of wavelength. The radiation intensity from the cavity is also measured similarly and both intensities are compared according to (6.1) to obtain the normal spectral emissivity. On occasion, these measurements are made in different experiments [16,17]. Here it should be noted that the term “radiation intensity” strictly means the spectral radiation intensity in W m−3 sr−1 derived from Planck’s law of radiation but not the output count (or voltage) from the detector simply. If the linear relationship between the intensity and the output count is guaranteed over a wide range of intensity, the emissivity can simply be derived as (output count from the sample)/(output count from the cavity) [16]; however, the relationship does not always obtain at any intensity, depending on the detector. Because of this, the output count is usually calibrated using blackbody radiation at various temperatures – this can be realized using fixed-point blackbody

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Optical Filter Monochromator

Quartz Window Pyrometer

Filter Quartz Lens Ar (1 atm) Carbon Crucible

Cavity Sample

Carbon Heater Heat Shield Chamber

Fig. 6.5. Direct measurement of radiation intensity [18]

with various fixed-point metals – or combination of blackbody radiation at a certain temperature and optical filters having various transmissivities. Recently, Watanabe et al. [20] have established an emissivity measurement system using a cold crucible furnace based upon the above principle, as shown in Fig. 6.6, followed by many measurements using this system [21–27]. In the cold crucible, eddy currents induced in a metallic sample by a high frequency current heat the sample, while its surroundings are kept at lower temperature because the cold crucible itself is water-cooled. Accordingly, the cold crucible enables only a metal sample to be melted, which can eliminate the effect of stray radiation on emissivity measurements. Furthermore, a repulsive force between the cold crucible and the molten sample prevents the melt from being in contact with the inner wall of the crucible, which helps to reduce chemical contamination of the sample. Using this method, Watanabe and his co-workers have made measurements on noble metals (Cu, Ag, and Au) [20,24, 26], transition metals (Fe, Co, and Ni) [25], and semi-conducting materials (Si and Ge) [21] in visible and near-infrared regions; however, these measurements have been limited only to the melting point of each substance due to the difficulty in temperature measurement. Afterwards, Hayashi et al. [23] and Tanaka et al. [27] have installed a two-colour pyrometer to this measurement system and determined normal spectral emissivities of Ag–Cu and Ni–Cu alloys, respectively, at temperatures higher than the liquidus temperatures. More recently, Fukuyama and his co-workers [28] have applied an electromagnetic levitation technique to emissivity measurements. This technique can

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Spectroscope

Prism Window Diaphragm

Camera Window

Window Sample

Induction coils

Ar- 5 %H2 (1 atm)

Cold crucible Graphite plate Air-tight chamber

Alumina tube Positioning rod

Fig. 6.6. Normal spectral emissivity measurement using cold crucible [20]

realize completely containerless melting and thereby eliminate chemical contamination of samples from containers. Another advantage in this technique enables the sample to be supercooled to a large extent. They have determined normal spectral emissivities of silicon in liquid state including supercooled state (1,553–1,797 K) over a wavelength range 550–1,600 nm, where temperature measurements were made using a two-colour pyrometer. They have also developed another technique for determining thermophysical properties including hemispherical total emissivity, which is covered in detail in Chap. 8 of this book. 6.3.4 Other Methods (1) Microsecond pulse heating The microsecond pulse heating system is basically a technique for heating samples; however, thermophysical property measurements using this technique have been highlighted as “Subsecond Thermophysics” since 1990 [29] and thus are outlined here. Pottlacher and his co-workers [30–38] have made a vigorous effort to establish the microsecond pulse heating method to apply to measurements of various thermophysical properties, including normal spectral emissivity. In

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5

2 1 4 3

Fig. 6.7. Spectral emissivity measurement for liquid metals [30]. 1 : tube-shaped sample, 2 : aperture, 3 : lens, 4 : optical fibre, 5 : optical fibre

this method, a metallic sample in the shape of tube (or wire) is resistively self-heated with a RCL discharge circuit at a heating rate 107 –109 K s−1 from room temperature to a temperature higher than the melting point, and during this heating cycle measurement of normal spectral emissivity is carried out. In their earlier work [30], this measurement was based on direct measurements of radiation intensities mentioned in Sect. 6.3.3: the radiation intensities of real body and blackbody were measured using radiations from the surface and the hole of a nickel tube (sample), as shown in Fig. 6.7, where note that the tube retains its shape into the liquid state owing to the very high heating rate. At the melting plateau, the ratio of the radiation intensity of sample surface ISR to that of blackbody IBR was normalized by the emissivity at the melting point ελ,M taken form the literature, and the emissivity in the liquid state ελ was represented by the following equation: ελ = ελ,M · (ISR /IBR ).

(6.12)

Consequently, they have determined the emissivity for liquid nickel for a wavelength of 850 nm in the temperature range 1,726–2,250 K. More recently, Pottlacher and his co-workers have combined the microsecond pulse heating system with ellipsometry and produced data of normal spectral emissivities for niobium [31, 32], tungsten [32], nickel [32], tantalum [32, 33], molybdenum [34], titanium alloy [35], nickel alloy [36], and palladium [37]. The readers can find a review about the pulse heating technique in [38].

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(2) Electrostatic levitation This system is basically a technique for levitating samples, and Rhim et al. [39,40] have applied this technique to measurements of various thermophysical properties including emissivity. In one of their studies [39], a silicon droplet (ca. 3 mm diameter) was electrostatically levitated in a vacuum of about 10−8 Torr and heated by a focused 1 kW xenon arc lamp until the sample was melted. After the lamp was blocked, the temperature of sample was continuously measured using a one-colour pyrometer to obtain the cooling curve of temperature vs. time. This cooling curve can also be described by the radiative heat transfer equation as follows. (m/M )CP (dT /dt) = −εT Aσ(T 4 − Ta4 ),

(6.13)

where m is the mass of sample, M is the molar mass of sample, CP is the molar heat capacity at constant pressure of sample, εT is the hemispherical total emissivity of sample, A is the surface area of sample, σ is the Stefan– Boltzmann constant 5.67 × 10−8 W m−2 K−4 , T is the temperature of sample, Ta is the ambient temperature, and t is the time. Values of m, M, A, and Ta are known from experimental conditions, and values of T and dT /dt can be derived from the cooling curve. Thus, the ratio of CP /εT can be determined, giving the value of εT with the help of the published value of CP . Other than liquid silicon, Rhim and his co-workers have determined hemispherical total emissivities of liquid nickel and zirconium at the melting points [40]. The electrostatic levitation technique is covered in detail in Chap. 9.

6.4 Emissivity Data This chapter summarises normal spectral emissivities for some common metals and alloys in the liquid state, which are important from practical and/or scientific viewpoints. 6.4.1 Noble Metals There have been considerable numbers of experimental investigations for the spectral emissivity of Cu, Ag, and Au in liquid states. Figure 6.8 shows the temperature dependence of the normal spectral emissivity of liquid copper around 650 nm. The temperature coefficients are positive except the value by Burgess; however, there are large discrepancies in the reported data. Figure 6.9 shows the emissivity of Cu as a function of wavelength at the melting point. Krishnan et al. have used an apparatus based on an electromagnetic levitation technique combined with rotating analyzer ellipsometry, and the other data included in Fig. 6.8 were measured based on the direct measurement methods. Even though Watanabe et al. and

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0.3

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ε

0.2

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5

4 8 10

2 7

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9

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650 nm

1800

2000

M.P.

0 1200

1400

1600

2200

T/K Fig. 6.8. Normal spectral emissivity of copper around 650 nm as a function of temperature; (650 nm): 1 Burgess [41], 2 Stubbs [42]; (660 nm): 3 Bidwell [43], 4 Burgess et al. [44], 5 Smith et al. [45], 6 Lange et al. [46]; (645 nm) 7 Stretz et al. [47], 8 Bonnell et al. [48], 9 Dokko et al.; (633 nm) 10 Krishinan et al. [49] 0.5 Cu

0.4

ε

0.3 0.2 0.1

1

3 2

0 400

500

600

700

800

900

l/nm

Fig. 6.9. Normal spectral emissivity of liquid copper at the melting point as a function of wavelength; 1 Watanabe et al. [24], 2 Tanaka et al. [50], 3 Krishnan et al. [49]

Tanaka et al. have used same cold crucible technique to eliminate the reaction between sample and a container, there are large differences between these data in Fig. 6.9, and the values reported by Tanaka show good agreement with those by Krishnan et al. These suggest the difficulty in emissivity measurements of liquid metals; the uncertainty remaining would be caused by the uneven surface of the sample etc. Figure 6.9 also suggests that the emissivity of Cu decrease drastically around 600 nm. This is because of the interband transition as well as free electron transition in the lower wavelength region. Figures 6.10 and 6.11 show normal spectral emissivities of liquid silver and gold, respectively, at the melting points. Both noble metals show negative wavelength dependencies.

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ε

2 1

0.05

0 400

500

600

700

800

900

l/nm

Fig. 6.10. Normal spectral emissivity of liquid silver at the melting point as a function of wavelength; 1 Watanabe et al. [24], 2 Krishnan et al. [49] 0.6 Au

0.5

ε

0.4 2

0.3 0.2

1

0.1 0 400

500

600

700

800

900

l/nm Fig. 6.11. Normal spectral emissivity of liquid gold at the melting point as a function of wavelength; 1 Watanabe et al. [24], 2 Krishnan et al. [49]

6.4.2 Transition Metals There have been many literature data of spectral emissivities and optical constants for transition metals at and near melting points, such as Fe [5, 43, 46, 51, 52], Ni [5, 44, 47, 49, 50], Co [53], Mo [54], and W [54]. In addition, Krishnan and his co-workers have actively measured emissivities for other metals, for example, Zr and Ti. In this section, spectral emissivities of Fe, Ni, and W are reviewed for representative examples. Figures 6.12 and 6.13 show normal spectral emissivities of Fe and Ni at 650 nm as functions of temperature. It is reported that Fe and Ni were easily oxidized at the surface which results in the higher emissivity values. Cagran et al. measured normal spectral emissivities of Mo and W by the microsecond pulse heating system with ellipsometer; heating technique which allowed the measurement on the materials having high melting points (Figs. 6.14 and 6.15). The emissivities of Mo increase with increase in

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0.6 2 1

ε

0.4

3

4 5

0.2 Fe 650 nm 0 1800

1900

2000

2100

T/K Fig. 6.12. Normal spectral emissivity of liquid iron at melting point at 650 nm as a function of temperature; 1 Dastur et al. [51], 2 Bidwell [43], 3 d’Entremont [52], 4 Ratanapupech et al. [5], 5 Lange et al. [46]

0.6

1 2

0.2

3

ε

0.4

Ni 650 nm 0 1700

1800

1900

2000

2100

T/K Fig. 6.13. Normal spectral emissivity of liquid nickel at melting point around 650 nm as a function of temperature ; 1 Lange et al. [47], 2 Ratanapupech et al. [5], 3 Bidwell [44] 0.45 684.5 nm 902 nm 1570 nm

Mo

ε

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0.35

0.3 3000

3500

4000

4500

T/K Fig. 6.14. Normal spectral emissivities of liquid molybdenum as functions of temperature reported by Cagran et al. [54]

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M. Susa and R.K. Endo 0.5 684.5 nm 902 nm 1570 nm

ε

0.4

0.3

W 0.2 3500

4000

4500

T/K

5000

Fig. 6.15. Normal spectral emissivities of liquid tungsten as functions of temperature reported by Cagran et al. [54]

0.3

1 2 3

ε

4

2

0.2 5 Si average over 1553 - 1797 K

0.1

1000

2000

l/nm

3000

Fig. 6.16. Normal spectral emissivities of liquid silicon as function of wavelength; 1 Shvarev et al. [55], 2 Takasuka et al. [56], 3 Tanaka et al. [22], 4 Kawamura et al. [28], 5 Watanabe et al. [21]

temperature for the entire measured wavelengths, whereas those for W show significant temperature dependence. 6.4.3 Semiconducting Materials There are many reports for emissivities of liquid Si (Figs. 6.16 and 6.17), because it is one of the most important material supporting the information technology in the modern society. The emissivity decreases with increasing wavelength (Fig. 6.16) and has almost no temperature dependence (Fig. 6.17). 6.4.4 Alloys There are few emissivity data reported for alloys comparing with pure metals. Schaefers et al. [59] reviewed emissivities for binary system such as Ni–Fe, Ce–Cu, Ni–Cr, and Ti–Al, and measured on Ti–V, Fe–V, and Fe–Nb systems

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1

0.3 6

2 5

ε

3

0.2

4

Si 0.1 1600

1700

1800

1900

2000

T/K

Fig. 6.17. Normal spectral emissivity of liquid silicon at 633 nm as a function of temperature; 1 Lampert [57], 2 Shvarev et al. (800 nm) [53], 3 Kawamura et al. [28], 4 Krishnan [58], 5 Lange et al. [47], 6 Takasuka [54] 0.5 0.4

ε

0.3 0.2 0.1 0

Ce-Cu 645 nm 0

20

40

60

80

100

at%Cu Fig. 6.18. Composition dependence of normal spectral emissivity of liquid Ce–Cu alloys at 645 nm [3]

at 547 and 650 nm. Recently, Hayashi et al. [23] and Tanaka et al. [50] measured emissivities for Ag–Cu and Cu–Ni alloys, respectively. Figure 6.18 shows emissivities of liquid Ce–Cu alloys at 645 nm measured using the integral blackbody comparison method. It is reported that the temperature dependence of emissivities can be neglected [3]. The value has peak at about 75%Cu, which corresponds to the congruent intermetallic phase Cu2 Ce. Figures 6.19 and 6.20 show spectral emissivities of Ag–Cu alloys. The reported values show negative temperature coefficient except for Ag-80 at%Cu and has local minimum in concentration dependence for 650 nm data. Figures 6.21 and 6.22 show normal spectral emissivities for Zr, Ni, and nickel based alloys. The alloys have larger emissivities at 633 nm than that for zirconium and nickel, and do not show significant temperature dependence. As reviewed earlier, there have been not enough emissivity data reported for liquid metals, epecially for alloys, because of the difficulty in measurements. In case of Cu and Si, relatively many data had been reported; however, there

126

M. Susa and R.K. Endo Ag-Cu 1300 nm

2

0.06

1

ε

3

0.05

4 6 5

0.04 1200

1300

1400

T/K

1500

Fig. 6.19. Temperature dependencies of spectral emissivities of Ag–Cu alloys measured at a wavelength of 1,300 nm; 1 Cu, 2 Ag-80 at%Cu, 3 Ag-60 at%Cu, 4 Ag40 at%Cu, 5 Ag-2 at%Cu, 6 Ag [23]

0.15 Ag-Cu 1 650 nm 2 1000 nm 3 1500 nm

ε

0.1

1 2 3

0.05 0

20

40

60

80

100

at%Cu Fig. 6.20. Spectral emissivities of Ag–Cu alloys for the wavelength of 650, 1,000, 1,500 nm at 1,373 K as a function of copper concentration [23]

0.5

1 Zr 2 Ni-75%Zr 3 Ni

2 3

ε

0.4 1

0.3

0.2

400

600

800

1000

l/nm Fig. 6.21. Wavelength dependencies of spectral emissivities for liquid Ni (1,800 K), Ni-75%Zr (1,300 K), and Zr (2,350 K) [58]

6 Emissivities of High Temperature Metallic Melts

127

0.45

0.4

1 2 3 4 5 6

4

ε

3 5

0.35

6

Zr Ni Ni-25%Sn Ni-32.5%Sn Ni-75% Zr Ni-40%Nb

2 1

0.3 1500

2000

2500

T/K Fig. 6.22. Plot of normal spectral emissivity at 633 nm as functions of temperature for zirconium, nickel, and nickel-based alloys [57]

are large discrepancies even in resent data. One thing can be said is that the temperature dependence of emissivities for liquid metals have no or extremely small positive temperature dependences. Further measurements and new approaches are expected in this field.

References 1. Y. Kawai, Y. Shiraishi, Handbook of Physico-chemical Properties at High Temperatures (The Iron and Steel Institute of Japan, Tokyo, 1988) 2. R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 4th edn (Taylor & Francis, New York, 2002) p 17 3. W. Dokko, R.G. Bautista, Metall. Trans. B 10B, 450 (1979) 4. W. Dokko, R.G. Bautista, Metall. Trans. B 11B, 309(1980) 5. P. Ratanapupech, R.G. Bautista, High Temp. Sci. 14, 269(1981) 6. A. Tomita, M. Susa, K. Nagata, Testu-to-Hagan´e 79, 1329 (1993) (in Japanese) 7. H.G. Tompkins, W.A. McGahan, Spectroscopic Ellipsometry and Reflectometry (Wiley, New York, 1999) p 63 8. R. Tanaka, T. Sato, M. Susa, Metall. Mater. Trans. A 36A, 1507 (2005) 9. K.M. Shvarev, N.I. Vnukovskii, B.A. Baum, P.V. Gelfd, High Temp. 20 540(1982) 10. S. Krishnan, C.D. Anderson, J.K.R. Weber, P.C. Nordine, W.H. Hofmeister, R.J. Bayuzick, Metall. Trans. A 24A, 67 (1993) 11. S. Krishnan, P.C. Nordine, Phys. Rev. B 47, 11780 (1993) 12. S. Krishnan, P.C. Nordine, Phys. Rev. B 48, 4130 (1993) 13. S. Krishnan, P.C. Nordine, Phys. Rev. B 49, 3161 (1994) 14. S. Krishnan, P.C. Nordine, J. Appl. Phys. 80, 1735 (1996) 15. S. Krishnan, K. Yugawa, P.C. Nordine, Phys. Rev. B 55, 8201 (1997) 16. T. Makino, H. Hasegawa, Y. Narumiya, S. Matsuda, T. Kunitomo, Trans. Jpn. Soc. Mech. Eng. 50, 2655 (1984) 17. K. Nagata, T. Nagane, M. Susa, ISIJ Int. 37, 399 (1997) 18. E. Takasuka, E. Tokizaki, K. Terashima, S. Kimura, Appl. Phys. Lett. 67, 152 (1995)

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19. E. Takasuka, E. Tokizaki, K. Terashima, S. Kimura, J. Appl. Phys. 81, 6384 (1997) 20. H. Watanabe, M. Susa, K. Nagata, Metall. Mater. Trans. A 28A, 2507 (1997) 21. H. Watanabe, M. Susa, H. Fukuyama, K. Nagata, High Temp. High Press. 31, 587 (1999) 22. R. Tanaka, M. Susa, High Temp. High Press. 34, 681 (2002) 23. M. Hayashi, M. Murata, H. Fukuyama, K. Nagata, Metall. Mater. Trans. B 33B, 47 (2002) 24. H. Watanabe, M. Susa, H. Fukuyama, K. Nagata, Int. J. Thermophys. 24, 223 (2003) 25. H. Watanabe, M. Susa, H. Fukuyama, K. Nagata, Int. J. Thermophys. 24, 473 (2003) 26. H. Watanabe, M. Susa, H. Fukuyama, K. Nagata, Int. J. Thermophys. 24, 1105 (2003) 27. R. Tanaka, M. Susa, in Proc of 7 th ECTP Conf., CD-ROM (2005) 28. H. Kawamura, H. Fukuyama, M. Watanabe, T. Hibiya, Meas. Sci, Technol. 16, 386 (2005) 29. A. Cezairliyan, G.R. Gathers, A.M. Malvezzi, A.P. Miiller, F. Righini, J.W. Shaner, Int. J. Thermophys. 11, 819 (1990) 30. E. Kaschnitz, G. Pottlacher, H. J¨ ager, Int. J. Thermophys. 13, 699 (1992) 31. A. Seifter, F. Sachsenhofer, S. Krishnan, G. Pottlacher, Int. J. Thermophys. 22, 1537 (2001) 32. A. Seifter, F. Sachsenhofer, G. Pottlacher, Int. J. Thermophys. 23, 1267 (2002) 33. G. Pottlacher, A. Seifter, Int. J. Thermophys. 23, 1281 (2002) 34. C. Cagran, B. Wilthan, G. Pottlacher, Int. J. Thermophys. 25, 1551 (2004) 35. M. Boivineau, C. Cagran, D. Doytier, V. Eyraud, M.-H. Nadal, B. Wilthan, G. Pottlacher, Int. J. Thermophys. 27, 507 (2006) 36. B. Wilthan, G. Pottlacher, Rare Metals 25, 592 (2006) 37. C. Cagran, G. Pottlacher, Platinum Metals Rev. 50, 144 (2006) 38. M. Boivineau, G. Pottlacher, Int. J. Mater. Prod. Technol. 26, 217 (2006) 39. W.K. Rhim, K. Ohsaka, J. Cryst. Growth 208, 313 (2000) 40. A.J. Rulison, W.K. Rhim, Metall. Mater. Trans. B 26B, 503 (1995) 41. G.K. Burgess, Bull. Nat. Bur. Stand. 6, 111 (1909) 42. C.M. Stubbs, Proc. Roy. Soc. (London) Ser. A 88, 195 (1913) 43. C.C. Bidwell, Phys. Rev. Ser. 2 3, 439 (1914) 44. G.K. Burgess, R.G. Waltenberg, Bull. Nat. Bur. Stand. 11, 591 (1915) 45. D.B. Smith, J. Chipman, Trans. AIME 194, 643 (1952) 46. von K.W. Lange, H.Schenck, Arch. Eisenhuettenw. 39, 611 (1968) 47. L.A. Stretz, R.G. Bautista, Temperature, Its Measuremant and Control in Science and Industry vol. 4, Part 1, ed. by H. Preston-Thomas, T.P. Murray, R.L. Shepard (Instrument Society of America, Pittsburgh, PA, 1972) p 489 48. D.W. Bonnell, J.A. Treverton, A.J. Valerga, J.L. Margrave, Temperature, Its Measurement and Control in Science and Industry, vol. 4, Part 1, ed. by H. Preston-Thomas, T.P. Murray, R.L. Shepard (Instrument Society of America, Pittsburgh, PA, 1972), p 483 49. S. Krishnan, G.P. Hansen, R.H. Hauge, J.L. Margrave, High Temp. Sci 26, 143 (1990) 50. R. Tanaka, M. Susa, in 17th European Conference on Thermophysical Properties, Collection of Contributions CD-ROM, Bratislava, Slovak Republic, 2005

6 Emissivities of High Temperature Metallic Melts 51. 52. 53. 54. 55. 56. 57. 58. 59.

129

M.N. Dastur, N.A. Gokcen, J. Metals 1, 665 (1949) J.C. d’Entremont, Trans. AIME 227, 482(1963) S. Krishnan, D. Basak, Int. J. Thermophys. 20, 1811 (1999) C. Cagran, G. Pottlacher, M. Rink, W. Bauer, Int. J. Thermophys. 26, 1001 (2005) K.M. Shvarev, B.A. Baum, P.V. Gel’d, Sov. Phys. Solid State 16, 2111 (1975) E. Takasuka, E. Tokizaki, K. Terashima, S. Kimura, J. Appl. Phys. 81, 6384 (1997) M.O. Lampert, J.M. Koebel, P. Siffert, J. Appl. Phys. 52, 4975 (1981) S. Krishnan, J.K.R. Weber, P.C. Nordine, R.A. Schiffman, R.H. Hauge, J.L. Margrave, High. Temp. Sci. 30, 137 (1991) K. Schaefers, M. R¨ oner-Kuhn, M.G. Frohberg, Int. J. Thermophys. 16, 997 (1995)

7 Noncontact Thermophysical Property Measurements of Metallic Melts under Microgravity Ivan Egry

7.1 Introduction Materials sciences provide the basis for most modern technologies, for example, IT and photovoltaics, aerospace and automotive applications, nanoand biotechnology, just to name a few. In recent years, the trend in developing new products moved from the conventional try-and-error approach to computer-based modeling. This has become possible due to the increase in computer power, but it is still hampered by an incomplete understanding of all mechanisms involved and by a lack of available thermophysical property data. To further advance the field, a multiscale and multidisciplinary approach is necessary, combining experimental and theoretical work with computer simulations on all length scales, starting from microscopic ab-initio calculations, covering the mesoscopic scale by, for example, phase field methods, and finally simulating the finishing steps, such as near net shape casting. Material scientists originally devoted most of their efforts to studying the solid state of materials, its microstructure, and its mechanical and thermal properties. However, in the last 10–20 years, a change in paradigm has taken place, and the importance of the liquid phase has been recognized. In fact, almost all industrially used materials have been molten in some processing step, specifically in casting. Solidification from the melt leaves its fingerprints in the final material, and hence it is of utmost importance to understand the properties of the molten state and its solidification behavior. The prominent feature of fluids, namely their ability to flow and to form free surfaces, poses the main difficulty in theoretically describing them. The physics of fluids is governed by the Navier–Stokes equation and by the ubiquitous presence of convection. In addition, when dealing with metallic materials, the high temperatures involved lead to experimental difficulties, the most trivial, but also most fundamental, being the availability of suitable containers. Consequently, the measurement of thermophysical properties of the liquid phase, in particular at high temperatures, is a very difficult task under terrestrial conditions.

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With the advent of microgravity platforms, a new tool has become available to study the properties of fluids. In the absence of gravity, density differences play no role; consequently, sedimentation effects and buoyancy driven convection can be minimized, which allows to study fluids and their solidification in a quiescent environment. A number of fundamental and seminal microgravity experiments have been performed since then, sheding new light onto old problems and, in some cases, rendering accepted textbook knowledge erroneous. An important breakthrough was the application of containerless processing techniques in the microgravity environment, which gave access to the metastable region of an undercooled melt and allowed measurements of growth velocities as function of undercooling, leading to a revision of dendrite growth and grain refining theories. In addition, containerless processing using electromagnetic levitation solved the problem of finding a suitable container for high-temperature, highly reactive metallic melts and allowed to measure thermophysical properties of these melts with a high precision.

7.2 Microgravity Microgravity is the term commonly used to describe weightlessness conditions onboard a spacecraft. Unfortunately, this term is quite misleading: first of all, gravity remains essentially unaltered, that is, the strength of the gravity field 300 km above the surface of the earth is still about 90% of its sea-level value. Second, the apparent weight is typically reduced by a factor of 103 , rather than 106 . The physical origin for weightlessness lies in the fact that the experiments are carried out in a frame of reference which is accelerated with respect to an inertial frame. Consequently, Newton’s equations of motion have to be supplemented by a fictitious force: s, Fs = −m¨ where s¨ is the acceleration of the frame of reference along its trajectory, s, with respect to an inertial system (e.g., the earth.) For a circular motion with angular frequency ω and radius R, like an orbiting satellite, this force is the centrifugal force: Fc = mω 2 R. If this force cancels the gravitational force, there is no net force on the body, it is weightless. In reality, a complete cancellation of the forces cannot be achieved, due to the spatial dependence of the earth’s gravitational field (tidal forces), the still existing air drag, and internal disturbances within the spacecraft (g-jitter). Weightlessness conditions can be achieved in drop tubes or drop towers, in parabolic flights, onboard sounding rockets, and on orbiting spacecrafts like the International Space Station, ISS.

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Table 7.1. Available microgravity platforms

Drop tower Aircraft Sounding rocket Space Station

g level(g0 )

μg duration(s)

payload volume(m3 )

10−3 − 10−6 10−2 − 10−3 10−4 − 10−6 10−3 − 10−6

1 − 10 ≈ 20 100 − 1,000 ∞

≈ 0.1 ≈ 1.0 ≈ 0.1 ≈ 1.0

Fig. 7.1. Temperature (left), heating control voltage (right) as function of time for containerless processing of a Cu–Co sample during a TEXUS sounding rocket flight

Table 7.1 shows the characteristics of the main facilities for microgravity studies available today, including the time interval available for a given measurement. The available free-fall time in a drop tube/drop tower scales with the square root of its height. It can be doubled if the payload is shot vertically upwards from the bottom of the tower, instead of dropping it from its top. This technically challenging catapult mode of operation has recently been introduced at the drop tower of ZARM in Bremen. An aircraft on a parabolic trajectory can provide a reduced gravity environment for about 20 s per parabola. Usually, up to 40 consecutive parabolas are performed per flight. Sounding rockets can provide free-fall periods between 5 and 20 min. As an example the experiment protocol for a CuCo sample to be processed in an electromagnetic levitation facility onboard a TEXUS sounding rocket is shown in Fig. 7.1. For many experiments, these times are not sufficient and it is necessary to perform the experiments on board a spacecraft.

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Liquids are the most sensitive of the phases to a gravitational force. This force causes an acceleration of all masses towards the center of gravitation, and it makes bodies fall and fluids flow downwards. In an inhomogeneous fluid system at rest, sedimentation will result. In addition, gravity can also induce flows in combination with a temperature or concentration gradient. This is called buoyancy driven convection and can lead to fluid flow instabilities, like the Rayleigh–B´enard instability [1]. Consequently, in the absence of gravity, none of these effects will occur. Specifically, this means the following • • •

No container is needed for positioning a fluid There is no sedimentation There is no buoyancy driven convection

In the absence of gravity, surface tension becomes the dominant force for a liquid system. It holds a liquid drop together and controls its wetting behavior. Therefore, capillarity phenomena become important in microgravity conditions. Variations in temperature or concentration along a free surface induce variations in the local surface tension, leading to surface tension-driven convection. This effect was discovered by Marangoni [2] and is named after him. In other words, Marangoni convection replaces the buoyancy driven Rayleigh– B´enard convection in microgravity. The absence of gravity therefore does not imply the absence of convection.

7.3 Containerless Methods Thermophysical properties of high temperature, and highly reactive, melts can be conveniently measured by containerless methods. These methods provide the purest environment possible. Since the surface of the liquid sample is not in contact with a wall, Marangoni convection will occur, if there is a temperature or concentration gradient along the surface. There are a number of containerless processing techniques, based on different levitation fields [3]. Acoustic and aerodynamic levitation techniques use standing waves or pressure gradients of a carrier gas, respectively. In electrostatic levitation, an electrostatic field is used to position a charged or polarized sample. In diamagnetic levitation, the molecular magnetic moments couple to an external dc magnetic field of typically 1–5 T. Electromagnetic levitation uses high frequency electromagnetic fields and exploits the Lorentz force to levitate electrically conducting specimen. Except for the latter, all other methods require an independent heating system in order to melt the sample, usually an infrared laser. Although there have been some early attempts to operate an acoustic levitator under microgravity conditions, up to date the only successful implementation of a microgravity levitator is TEMPUS [4, 5], based on the electromagnetic levitation principle. Thermophysical properties of levitated samples have been measured in TEMPUS during two Spacelab missions, IML-2 in 1994 [6] and MSL-1 in 1997 [7]. Later on, TEMPUS was

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flown regularly on parabolic flight campaigns. Its successor, MSL-EML, the Materials Science Lab – Electro-Magnetic Levitator, was flown in 2006 on a sounding rocket, and is presently being built as a second generation payload for ISS, to be flown in 2011. In the following, the principle of electromagnetic levitation will be briefly described. Levitation of electrically conducting samples is achieved by placing the sample into a high frequency alternating inhomogeneous electromagnetic field, produced by a levitation coil with a conical or cylindrical shape. This field B induces a current in the sample which, in turn, interacts with the field. Levitation is caused by the Lorentz force, F . Its magnitude can be expressed, to lowest order in a multipole expansion [8], as FL = −

∇B 2 4π 3 a Q(q), 2μ0 3 0

(7.1)

where μ0 is the magnetic permeability, a0 is the radius of the sample, q = a0 /δ is a dimensionless quantity, and δ is the skin depth, defined as follows:  ωρμ0 . (7.2) 1/δ = 2 Here, ρ is the electric conductivity of the sample and ω is the frequency of the alternating field. The function Q(q) is given by

 3 3 sinh(2q) − sin(2q) Q(q) = 1− . (7.3) 4 2q cosh(2q) − cos(2q) For stable levitation, this field has to cancel the gravitational field: 4π − → − → → FL = −Fg = − a30 ρ− g. (7.4) 3 Here, ρ is the density of the sample and g is the gravity vector. For a linear z magnetic field, ∂H ∂z = Hzz = const, the levitation force is given by [9] 2 , FL = 2πa30 z0 μ0 Hzz

(7.5)

where z0 is the equilibrium position of the droplet, measured downwards from the origin of the magnetic field, that is, from the equatorial plane of the coil. In contrast to other levitation techniques, electromagnetic levitation is intrinsically stable, that is, there is a restoring force for deviations from the equilibrium position in any direction. Consequently, a solid sample performs oscillations about its equilibrium position with a frequency which is determined by the “spring constant” of the field, and its mass. For the magnetic field introduced above, this frequency can be calculated and is given by [9] Ω2 =

3μ0 2 H , 2ρ zz

Ω2⊥ =

1 2 Ω , 4 

(7.6)

where Ω2 is the frequency for oscillations along the symmetry axis, and Ω2⊥ is the frequency for oscillations in a plane perpendicular to it.

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In addition, the sample may also rotate along any axis which, for a liquid sample, leads to a flattening and eventual fission of the drop, due to the centrifugal forces. This kind of instability is often encountered in practice, and has been studied theoretically by Gerbeth and coworkers [10, 11]. Under terrestrial conditions, the joint action of the gravitational and levitation force leads to a deformation of a liquid sample: the drop is elongated along the z-axis, that is, the direction of the gravity vector. As will be seen later, such a deformation is detrimental to thermophysical property measurements. On the other hand, the liquid sample remains essentially spherical in microgravity. In addition, electromagnetic fields induce fluid flows inside a liquid, conducting body. The calculation of the corresoponding flow pattern is a formidable task for magnetohydrodynamic calculations [12]. In particular, if these fields become turbulent, they render the measurement of viscosity impossible. This can only be avoided in microgravity, where only small positioning fields are required. Electromagnetic levitation not only provides positioning, it also enables inductive heating of the levitated sample. The power P absorbed by the sample due to ohmic losses of the induced currents is given by [9] P =

B 2 ω 4π 3 a H(q), 2μ0 3 0

with H(q) defined as 

9 sinh(2q) + sin(2q) H(q) = 2 q −1 . 4q cosh(2q) − cos(2q) The functions Q(q) and H(q) are shown in Fig. 7.2.

Fig. 7.2. Heating and positioning efficiencies of electromagnetic levitation

(7.7)

(7.8)

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7.4 Thermophysical Properties The measurement of thermophysical properties of liquid metals using electromagnetic levitation has become a routine task in the past few years, at least concerning surface tension and density measurements, and comprehensive reviews are available [14–16]. Concerning experiments under microgravity, the major event was the Spacelab mission MSL-1 in 1997 [7]. Since then, no long-duration microgravity campaign has taken place. However, a number of parabolic flights using electromagnetic levitation has been performed, and also one sounding rocket experiment. Consequently, most of the results discussed below have been obtained during MSL-1 and are now 10 years old. Where appropriate, they are supplemented by recent results obtained during parabolic flights. 7.4.1 Electrical Conductivity It is possible to measure the electrical conductivity of levitated melts using a noncontact, inductive method. The impedance of a coil surrounding the sample is influenced by the sample’s electrical conductivity. For spherical samples and homogeneous magnetic fields, as realized in microgravity, this relation is rather simple [17], whereas under terrestrial conditions, extensive mathematical and engineering effort is required to extract the required information from the measured impedance [18]. The complex impedance Z of the coil is a function of the oscillation frequency ω of the electromagnetic field, properties of the empty coil, and properties of the sample: → → → → r ,− α ) = R0 + iωL0 + ΔRs (δ, a0 , − r ,− α) Zcoil = Zcoil (ω, δ, a0 , − → → +iωΔLs (δ, a0 , − r ,− α ).

(7.9)

Here, R0 and L0 are resistance and inductivity of the empty circuit and ΔRs , ΔLs are the changes induced by the sample. These latter functions depend → r within on the skin depth, δ, the radius a0 of the sample, and its position − → the coil. The vector − α is meant to describe the shape of the sample as, for example, given by the coefficients of a series expansion in spherical harmonics. In a homogeneous magnetic field, and for a spherical sample, the dependence → → on − r and − α vanish. The complex impedance Ztot of the oscillatory circuit can be determined by measuring amplitudes of current I and voltage U , as well as their phase shift φ. The real and imaginary part of Ztot yield two equations for the two unknowns δ and a0 . For small ohmic losses in the circuit these are given by I0 C (R + Rs ) = cos(φ), L U0    I0 C L 2 ω − (LC)−1 + ωLs = sin(φ). L ω U0

(7.10) (7.11)

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Fig. 7.3. Apparent electrical resistivity of solid (lower curve) and undercooled liquid (upper curve) Co80 Pd20 . The increase below 1,300 K is due to magnetic effects, both in the solid and in the liquid state

For small skin depth δ  a0 , Rs has a relatively simple form, and (7.10) can be inverted to yield   ! A I0 a0 1− 1− 3 cos φ − B . (7.12) δ= 2 a0 U0 Here, A and B are two instrument constants, to be determined by calibration. Using (7.2) yields finally the electrical conductivity σ. As is evident from (7.2), the skin depth also depends on the magnetic permeability and is therefore sensitive to magnetic ordering effects. This was demonstrated experimentally on deeply undercooled Co80 Pd20 during the MSL-1 Spacelab mission and is shown in Fig. 7.3. The Curie temperature of the solid phase is around 1,250 K, and the steep rise in the apparent electrical resistivity reflects the onset of magnetic ordering. The same behavior is found for the liquid phase, at approximately the same temperature. This supports the assertion that undercooled liquid Co80 Pd20 becomes a magnetic liquid [19]. The electrical conductivity σ is of interest in its own, but it also allows to obtain the thermal conductivity λ through the Wiedemann–Franz relation, which is known to hold well for liquid metals [20]: λ = LσT.

(7.13)

Here, L is a universal constant, the so called Lorenz number, L = 2.44 W Ω K−2 . Thus, electrical conductivity measurements provide an alternative way, independent of convective effects, to determine the thermal conductivity.

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7.4.2 Density and Thermal Expansion Density and thermal expansion of levitated drops are determined by recording the visible cross section of the sample. Assuming rotational symmetry, the volume is calculated. Since the mass of the sample is known and does not change, this yields the density. Typically, a resolution of δV /V = 10−4 is required. This can be achieved using sub-pixel algorithms for edge detection, curve fitting of the shape, and statistical averaging [21]. Although such measurements can be performed on ground [22],the precision of the data is improved in microgravity due to the spherical shape of the sample. The volume of an axisymmetric sample can be calculated from  π 2 3 dϕ, (7.14) VP = π 3 0 where is the time-averaged, angle-dependent radius of the sample, 3 which, in the case of a sphere, reduces to a0 , yielding VP = 4π 3 a0 . The volume, VP , is given in pixel units and must be converted to physical units using a calibration body. This is the main source of error in this kind of measurements. Another difficulty arises from the edge detection algorithm. The brightness of the sample changes with temperature, and the edge detection must be independent of contrast changes and blooming effects. In terrestrial experiments, this problem is bypassed by using a shadowgraph technique [22], but the TEMPUS facility used in the microgravity experiments does not have this feature installed. During MSL-1, Samwer and coworkers have performed measurements on glass-forming metallic alloys [23]. For such systems, two interesting questions need to be addressed: A change in the slope of the density as a function of temperature, that is, a change in the thermal expansion, in the undercooled regime would indicate a liquid–liquid phase transition, for example, phase separation. The other interesting question is related to the glass transition itself: it has been speculated that, at this temperature, the density of the undercooled liquid would become equal to that of the crystalline solid. Unfortunately, the data of Samwer and coworkers do not extend close enough to the glass transition temperature to answer this question. More recently, the same group measured the density of liquid Si–Ge alloys with TEMPUS during parabolic flights [24]. They found good agreement with ground based data obtained in an electrostatic levitator, and an interesting, yet unexplained, anomaly in the thermal expansion around Si75 Ge25 . Their raw data for the volume of an Si25 Ge75 alloy are shown in Fig. 7.4. 7.4.3 Specific Heat A noncontact method developed by Fecht and Johnson [25] can be used to determine the specific heat in levitation experiments. It is a variant of noncontact modulation calorimetry, normally used in low temperature physics. The

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3.0x107

Sample: Si25Ge75

Volume(pixel3)

2.5x107 2.0x107 1.5x107 1.0x107 5.0x106

1400

1500

1600 1700 Temperature(K)

1800

1900

Fig. 7.4. Volume of liquid Si25 Ge75 as a function of temperature, measured during a parabolic flight with TEMPUS

heater power is modulated according to Pω (t) = ΔPω cos(ωt), resulting in a modulated temperature response ΔTω of the sample. A thermal model considering heat loss to the environment and heat conduction within the sample has been developed by Wunderlich [26]. It considers heat loss to the exterior (typically by radiation only if the experiment is carried out under vacuum), spatially inhomogeneous heating of the sample, and heat conduction within the sample. For small Biot numbers, Bi = kr /kc  1, where kr is the heat loss due to radiation and kc the heat loss due to heat conduction, adiabatic conditions are realized and quantitative modulation calorimetry is possible. Under such conditions, the following relation for ΔTω is obtained: "

2 !

2 !#−1/2 ω λ2 ΔPω ΔTω = 1+ 1+ , (7.15) ωcp λ1 ω where λ1 and λ2 are functions of kr and kc such that for Bi  1, λ2  λ1 . This allows to choose the modulation frequency between λ2 and λ1 such that λ2  ω  λ1 . Under these circumstances, a simple relation for the specific heat, cp , can be derived: cp =

1 ΔPω . ω ΔTω

(7.16)

The power input into the sample, ΔPω , cannot be measured directly. It is related to the current I flowing through the heating coil by a coupling coefficient GH , which has to be determined separately: ΔPω = GH Iω2

(7.17)

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Fig. 7.5. Modulation calorimetry on ZrAlCuNi alloy during MSL-1 Spacelab mission. The melting temperature of this alloy is 1,146 K. Modulation was carried out in the equilibrium liquid (1), as well as in the undercooled liquid (2,3). At point 4, solidification occurs from an undercooling level of 194 K

It should be noted that a harmonic modulation of the current with ω  leads to components in the power P with ω = 0, ω = ω  and ω = 2ω  . The static and frequency-doubled components can also be analyzed along the lines indicated above. Modulation calorimetry needs to be carried out under isothermal conditions, that is, the sample must be allowed to thermalize at a predefined temperature before the modulation signal can be applied. Therefore, such experiments cannot be performed during parabolic flights. Fecht and coworkers [27] applied this method during the MSL-1 Spacelab mission. The temperature modulation is shown in Fig. 7.5, where the quaternary alloy Zr65 Al7.5 Cu17.5 Ni10 was undercooled by 194 K, and modulation calorimetery could be performed in the equilibrium as well as in the undercooled liquid. More recently, Wunderlich et al. performed modulation calorimetry during a sounding rocket experiment on a Ti46 Al8 Nb alloy in the framework of the IMPRESS project [28]. It was possible to determine both the specific heat, and the total hemispherical emissivity in the liquid phase [29]. The results are shown in Table 7.2. 7.4.4 Viscosity and Surface Tension Viscosity and surface tension of levitated samples are conveniently measured by the oscillating drop technique [30]. Liquid samples perform oscillations around their equilibrium shape. In microgravity, this is a sphere and in that

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Table 7.2. Ti46 Al8 Nb specific heat capacity and total hemispherical emissivity in the liquid phase T (◦ C) −1

cp (J K εtot

−1

g

)

1570

1491

1468

1.14 0.44

1.06 0.36

1.03 0.34

case, simple formulae can be used to relate frequency ω and damping Γ of the oscillations to surface tension γ and viscosity η, respectively. They read ω2 =

32π γ 3 M

(7.18)

and 20π a0 η , (7.19) 3 M where M is the mass of the droplet and a0 its radius. Depending on the sample mass, typical frequencies ν = ω/2π lie in the range of 30–50 Hz. Under terrestrial conditions, the liquid drop is distorted by gravity and the compensating levitation field. As a consequence, the single frequency is split into up to five peaks. In addition, all peaks are shifted with respect to (7.18). This is due to the fact that the electromagnetic levitation field acts as an additional pressure term in the Navier–Stokes equation. It, therefore, leads to an apparent increase in surface tension. A correction formula to account for these effects on the frequency spectrum was developed by Cummings and Blackburn [9] who derived a sum rule that contains only measurable frequencies. It reads 1 2 32π γ = ω − 1.9Ω2tr − 0.3(Ω2tr )−1 (g/a0 )2 , (7.20) 3 M 5 m 2,m

1 2 Ω + 2Ω2⊥ . Ω2tr = 3 Γ =

Here Ω2tr is the mean translational frequency of the drop, and g is the gravitational acceleration. The sample oscillations are recorded with video cameras from the top, that is, along the symmetry axis. The frame rate must satisfy the Nyquist theorem to avoid aliasing and is typically 100–400 fps. The area of the cross section of the sample, its center of mass, and two perpendicular radii are calculated for each frame. A number of frames (256–4,096) is taken and the corresponding time series is Fourier transformed to yield the frequencies. From the time dependence of the center of mass the translational frequencies are derived, whereas the surface oscillation frequencies are contained in the temporal behavior of the two perpendicular radii. Because of their different symmetries, the oscillations corresponding to different m-values can be identified using selection rules, as derived by Egry et al. [30]. In Fig. 7.6, oscillation spectra of a gold–copper alloy are shown, recorded on ground and in microgravity during the IML-2 Spacelab mission [31]. As can

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Fig. 7.6. Frequency spectrum of an oscillating AuCu drop under 1g (top) and microgravity (bottom)

be seen in the ground-based spectra, both, a splitting of the single frequency into 5 peaks, and a shift to higher frequencies occur. By comparing measurements in microgravity with ground-based results, the validity of the Cummings correction has been confirmed experimentally. It is fair to say that this benchmark experiment under microgravity has led to a calibration of the ground-based measurements applying the oscillating drop technique, which has consequently become the accepted method for surface tension measurements of high-temperature liquid metals. Surface tension measurements using the oscillating drop technique are short duration experiments. Nowadays, they are routinely carried out during parabolic flight campaigns. For example, Wunderlich et al. [32, 33] have measured the surface tension of a number of industrially relevant multicomponent steels and nickel-based superalloys in the framework of ESA’s ThermoLab project [32]. A list of the investigated alloys is given in Table 7.3. Recently, there have been also surface tension measurements using the oscillating drop method under microgravity conditions in Japan, notably by the group of Nogi [34–36]. This group has either used the JAMIC drop tower [36] or a short 1.5 m drop tube [34, 35] and measured the oscillations during free fall of the drop. The oscillating drop technique also allows to measure the viscosity of liquids by measuring the decay of oscillations and using (7.19). Since no correction formula, equivalent to the Cummings correction, (7.20), exists, the application of the oscillating drop technique to viscosity measurements is restricted to microgravity conditions. Egry and coworkers [37] have measured the viscosity of a PdCuSi alloy during the MSL-1 mission. Their result is

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Table 7.3. Surface tension measurements of industrial alloys measured during parabolic flights in the framework of the ThermoLab project Alloy CMSX-4 IN738LC MM247LC C263 Low alloyed Steel Cr-alloyed Steel Cu-alloy Al-75 at% Ni Nickel Al-31.5 at% Ni Al-65 at% Ni

Tl (◦ C)

γ(Tl ) (N m−1 )

Year

1,384 1,335 1,368 1,368 1,480 1,477 1,067 1,410 1,455 1,250 1,550

1.85 1.85 1.86 1.74 1.61 1.76 1.15 1.63 1.71 1.16 1.65

2002 2003 2003 2003 2004 2004 2004 2004 2004 2005 2006

Fig. 7.7. Viscosity of Pd76 Cu6 Si18 measured during two different Spacelab missions. The data are fitted with an Arrhenius relation

shown in Fig. 7.7 and compared to terrestrial data by Lee et al. [38]. Two different datasets are displayed: data during the STS-83 mission were taken with no crew on board Spacelab, while data during STS-94 were recorded during nominal crew operations. Both data yield the same fit, the higher scatter in the STS-94 data is due to g-jitter. Although the data cover a temperature range of 400 K, they are not close enough to the glass transition temperature to distinguish between Arrhenius and Vogel–Fulcher behavior.

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7.5 Summary and Outlook In the preceding sections, we have demonstrated the potential of the combined use of the microgravity environment and containerless processing, in particular, electromagnetic levitation, for thermophysical property measurements. Because of the lack of flight opportunities, this potential has not been fully exploited yet. The facilities already operational on board the International Space Station, ISS, are of limited use for thermophysical property measurements. MSL-EML, the Materials Science Lab – Electro-Magnetic Levitator, has still to be built and will not become available before 2011. Until then, most thermophysical property measurements have to be carried out on board parabolic flights within 20 s of microgravity. This restricts the use of TEMPUS to surface tension measurements, as discussed earlier. For most other properties, the available microgravity period is not sufficient. Unfortunately, the Drop Tower at JAMIC has also been closed down. Using sounding rockets as a microgravity platform is attractive in terms of microgravity duration (about 5 min) and also in terms of microgravity quality (about 10−5 g0 ). However, there is only one flight every 2 years at best within the national German program, which is co-financed by and shared with ESA. Because of restrictions in payload mass and the available μg-time, only two samples can be processed, yielding about 160 s experiment time per sample, as shown, for example, in Fig. 7.1. Therefore, TEXUS is by far not as cost-effective as either parabolic flights or the ISS. When the MSL-EML on ISS becomes fully operational, experiment time will be virtually unlimited. This is the chance for performing systematic investigations varying the process parameters like, for example, temperature, cooling and heating rates, and composition of the ambient atmosphere, that is, the oxygen partial pressure. Generally speaking, the emphasis will shift from technology development to exploiting the technology for measurements of industrially and scientifically interesting (multicomponent) alloys. Even in this optimistic scenario, microgravity experiments will not replace a sound, ground-based measurement program. Microgravity experiments provide benchmarks for terrestrial measurements owing to their inherent higher precision. However, the huge amount of data required for numerical modeling of industrial processes by far exceeds the capabilities of spaceborne experiments. While most thermophysical properties can be measured in a containerless environment using, for example, MSL-EML, this is not the case for measurements of diffusion and thermal conductivity. Here, the application of an additional strong static magnetic field, reducing or even eliminating fluid flows [39, 40] may open up new possibilities.

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Acknowledgement This review is the result of continuous discussions with many colleagues involved in microgravity experiments, during the Spacelab missions, parabolic flights, or sounding rocket campaigns. Their fruitful comments are gratefully acknowledged. In particular, the author thanks B. Damaschke and R. Wunderlich for providing their data prior to publication.

References 1. Lord Rayleigh, Phil. Mag. 32, 529 (1916) 2. C. Marangoni, Nuovo Cim. 3, 97 (1878) 3. D. Herlach, R. Cochrane, I. Egry, H. Fecht, L. Greer, Int. Mater. Rev. 38, 273 (1993) 4. G. Loh¨ ofer, P. Neuhaus, I. Egry, High Temp. High Pressure 23, 333 (1991) 5. J. Piller, A. Seidel, M. Stauber, W. Dreier, Solidification 1999, ed. by W. Hofmeister, J. Rogers, N. Singh, S. Marsh, P. Vorhees (TMS, Warrendale, 1999) p 3 6. Team TEMPUS, Materials and Fluids under low Gravity, ed. by L. Ratke, H. Walter, B. Feuerbacher, (Springer, Berlin, 1996) 7. W. Hofmeister, J. Rogers, N. Singh, S. Marsh, P. Vorhees (ed.), Solidification 1999 (TMS, Warrendale, 1999) 8. G. Loh¨ ofer, Quarterly Appl. Math. 11, 495 (1993) 9. D. Cummings, D. Blackburn, J. Fluid Mech. 224, 395 (1991) 10. J. Priede, G. Gerbeth, IEEE Trans. Magnetics, 36, 349 (2000) 11. J. Priede, G. Gerbeth, IEEE Trans. Magnetics, 36, 354 (2000) 12. R.W. Hyers, Meas. Sci. Tech. 16, 394 (2005) 13. G. Loh¨ ofer, SIAM, J. Appl. Math., 49, 567 (1989) 14. Ivan Egry, C. Nieto de Castro, Chemical Thermodynamics, ed. by T. Letcher (Blackwell, Oxford, 1999) p 171 15. H. Suga, G. Pottlacher, I. Egry, Measurement of the Thermophysical Properties of Single Phases, Experimental Thermodynamics, vol VI, ed. by A. Goodwin, K. Marsh, W. Wakeham (Elsevier, Amsterdam, 2003) p 475 16. J. Brillo, G. Loh¨ ofer, F. Schmidt-Hohagen, S. Schneider, I. Egry, Int. J. Mater. Product Technol. 26, 247 (2006) 17. G. Loh¨ ofer, I. Egry, Solidification 1999, ed. by W. Hofmeister, J. Rogers, N. Singh, S. Marsh, P. Vorhees, (TMS, Warrendale, 1999) 18. T. Richardsen, G. Loh¨ ofer, Int. J. Thermophys. 20, 1029 (1999) 19. D. Platzek, C. Notthoff, D.M. Herlach, G. Jacobs, D. Herlach, K. Maier, Appl. Phys. Letts. 65, 1723 (1994) 20. K. Mills, B. Monaghan, B. Keene, Int. Mater. Rev. 41, 209 (1996) 21. E. Gorges, L. Racz, A. Schillings, I. Egry, Int. J. Thermophys. 17, 1163 (1996) 22. J. Brillo, I. Egry, Z. Metallkd. 95, 691 (2004) 23. B. Damaschke, K. Samwer, I. Egry, Solidification 1999, ed. by W. Hofmeister, J. Rogers, N. Singh, S. Marsh, P. Vorhees, (TMS, Warrendale, 1999) 24. B. Damaschke, private communication 25. H. Fecht, W. Johnson, Rev. Sci. Instrum. 62, 1299 (1991)

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26. R. Wunderlich, H. Fecht, Meas. Sci. Tech. 16, 402 (2005) 27. R. Wunderlich, R. Sagel, Ch. Ettel, H.-J. Fecht, D. Lee, S. Glade, W. Johnson, Solidification 1999, ed. by W. Hofmeister, J. Rogers, N. Singh, S. Marsh, P. Vorhees, vol 53 (TMS, Warrendale, 1999) 28. D.J. Jarvis, D. Voss, Mat. Sci. and Engng. A 413–414, 583–591 (2005) 29. R. Wunderlich, private communication 30. S. Sauerland, K. Eckler, I. Egry, J. Mat. Sci. Letters, 11, 330 (1992) 31. I. Egry, G. Loh¨ ofer, G. Jacobs, Phys. Rev. Letts. 75 4043 (1995) 32. R. Aune, L. Battezzati, R. Brooks, I. Egry, H. Fecht, J. Garandet, K. Mills, A. Passerone, P. Quested, E. Ricci, S. Schneider, S. Seetharaman, R. Wunderlich, B. Vinet, Microgravity Sci. Technol. XVI, 11 (2005) 33. K. Higuchi, H.-J. Fecht, R.K. Wunderlich, Adv. Eng. Mat. 9, 349 (2007) 34. H. Fujii, T. Matsumoto, T. Ueda, K. Nogi, J. Mater. Sci. 40, 2161 (2005) 35. T. Matsumoto, H. Fujii, T. Ueda, M. Kamai, K. Nogi, Meas. Sci. Tech. 16, 432 (2005) 36. H. Fujii, T. Matsumoto, S. Izutani, S. Kiguchi, K. Nogi, Acta Mat. 54, 1221 (2006) 37. I. Egry, G. Loh¨ ofer, I. Seyhan, S. Schneider, B. Feuerbacher, Appl. Phys. Lett. 73, 462 (1998) 38. S. Lee, K. Tsang, H. Kai, J. Appl. Phys. 70, 4842 (1991) 39. H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, S. Awaji, Appl. Phys. Lett. 90, 094102 (2007) 40. H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, S. Awaji, J. Appl. Phys. 104, 054901 (2008)

8 Noncontact Laser Calorimetry of High Temperature Melts in a Static Magnetic Field Hiroyuki Fukuyama, Hidekazu Kobatake, Takao Tsukada, and Satoshi Awaji

8.1 Introduction Numerical simulations are widely used for high value-added materials processing such as semiconductor crystal growth, casting of super high-temperature alloys for a jet-engine turbine blade, and for welding in automobile manufacturing [1, 2]. Process modeling involving a liquid-to-solid transition requires precise thermophysical properties of materials in the solid and liquid state at temperatures near their melting points. However, high-temperature materials such as liquid silicon are chemically reactive and are easily contaminated by their containers and contact materials. Therefore, it remains extremely difficult to measure the thermophysical properties of high-temperature liquids. Especially, the thermal conductivity of a high-temperature liquid is a difficult property to measure because of the existence of the buoyancy and Marangoni convections in the liquid. Not only from process modeling but also from a scientific perspective, thermal conductivity data of high-temperature metallic or semiconductor liquids are important to investigate whether the Wiedemann–Franz law [3] is applicable to them. Fecht et al. [4–7] developed modulation calorimetry for electromagnetically levitated metallic melts. The radio frequency (rf) coil’s power was modulated to provide sinusoidal heating to the sample melt. The heat capacities and hemispherical total emissivities of the melts were determined at higher temperatures. However, convections existing in the droplets make it difficult to measure the true thermal conductivity of the melts. Yasuda et al. [8] reported that motion of the center of gravity, surface oscillation, and convection of an electromagnetically levitated liquid metal were suppressed in a static magnetic field because of the Lorentz force resulting from interaction between the fluid flows and the static magnetic field. Based on the techniques described above [4–8], we devised noncontact modulation calorimetry, which incorporates a static magnetic field to realize measurement of the true thermal conductivity of a metallic melt in addition to measurements of the heat capacity and hemispherical total emissivity [9–11].

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An electromagnetic levitator is equipped in a superconducting magnet to hold a droplet statically. The high-temperature droplet is heated sinusoidally by laser irradiation on the top instead of using a modulated rf-coil heating; then, the temperature response is measured at the bottom of the droplet using a pyrometer. This technique enables us to measure the thermophysical properties, especially thermal conductivity, over a wide temperature range without contamination. This chapter presents an explanation of the theoretical background of this novel method to determine thermophysical properties such as heat capacity, thermal conductivity, and emissivity of a levitated droplet using an electromagnetic levitator with a static magnetic field. To confirm the fact that a static magnetic field actually suppresses the convection in a molten silicon droplet in the electromagnetic levitator, numerical simulations for convection in the droplet are demonstrated taking into account the buoyancy force, thermocapillary force, and electromagnetic force in the droplet. Finally, experimental details are presented for molten silicon as an example.

8.2 Theory of Modulation Calorimetry 8.2.1 Heat Capacity The principle of the conventional modulation calorimetry is described in detail elsewhere [12, 13]. Figure 8.1a presents a schematic illustration of the electromagnetic levitation apparatus incorporating a static magnetic field for laser modulation calorimetry. In the experiment, the upper part of an electromagnetically levitated droplet is heated periodically using a modulated light source, that is, a semiconductor laser (Fig. 8.1b); then the temperature variation at the lower part of the droplet attributable to heat flow from the upper part through the droplet is detected using a pyrometer. The simple heat flow model of this noncontact modulation calorimetry is presented in Fig. 8.2 [9]; it resembles that developed by Wunderlich and Fecht [7]. The sample droplet, which is levitated in the vacuum chamber, is heated using a modulated laser at a power of po (1 + cos ωt) per unit area [W m−2 ] from the top of the droplet. The temperature response is measured at the bottom of the droplet using a two-color pyrometer. The fraction of the irradiated surface area that is heated by the laser is defined as Sh ; the corresponding volume fraction is Vh . The heat balance equations for the system are as follows. For the laser irradiated part, dTh = Qh +αSh Apo (1+cos ωt)−Sh Aσ(Th4 −Ta4 )−Kc (Th −Tl ). dt For the laser nonirradiated part, Vh Cp

(1 − Vh )Cp

dTl = Ql − (1 − Sh )Aσ(Tl4 − Ta4 ) + Kc (Th − Tl ). dt

(8.1)

(8.2)

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151

dc magnetic field semiconductor laser rf coil

0.0 s

silicon droplet

laser spot

1.0 s

2.0 s

aperture

3.0 s

pyrometer (a)

(b)

mirror

Fig. 8.1. (a) The electromagnetic levitation apparatus incorporating a static magnetic field for noncontact laser modulation calorimetry; (b) a top view of the levitated liquid silicon during modulation heating in a static magnetic field

Laser irradiated part: Sh, Vh

ShKr Radiative heat transfer

Kc

Conductive heat transfer

Heat bath

Radiative heat transfer

Laser non-irradiated part: (1-Sh), (1-Vh)

(1-Sh)Kr

Fig. 8.2. Heat flow model in this noncontact modulated laser calorimetry

In those equations, Cp [J K−1 ]denotes the isobaric heat capacity, T [K] is the absolute temperature, Ta [K] is the ambient temperature, Q [W] is the power input from the rf-coil, α is the absorptivity, A [m2 ] is the surface

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area of the droplet,  is the hemispherical total emissivity, σ [W m−2 K−4 ] is the Stefan–Boltzmann constant, and Kc [W K−1 ] is the thermal conductance for conductive heat transfer from the laser-irradiated part to the nonirradiated part. Subscripts h and l, respectively, denote the laser irradiated and nonirradiated parts. In Fig. 8.2, Kr [W K−1 ] is the thermal conductance for radiative heat transfer from the sample surface to the heat reservoir in vacuum (Kr = 4AσTo3 , To is the initial temperature of the sample). The sample temperature is expressed as the sum of the initial temperature To , the average increase in temperature ΔTdc (dc component), and the modulation amplitude md (ac component) as ΔTac md T = To + ΔTdc + ΔTac .

(8.3)

The modulation amplitude is expressed in terms of the phase difference φ between the laser irradiation and the temperature response as md = ΔTac cos(ωt − φ). ΔTac

(8.4)

md Under the condition of To ΔTdc , ΔTac , the following linearization is reasonably applied for the radiative heat loss term in (8.1) and (8.2).   md ΔTdc ΔTac 4 md 4 4 . (8.5) +4 T = (To + ΔTdc + ΔTac )  To 1 + 4 To To

The heat balance between the power input from the rf-coil and radiation heat loss is given by the following equation when the sample droplet is maintained at the initial temperature. Q = Qh + Ql = Aσ(To4 − Ta4 ).

(8.6)

Substituting (8.3), (8.5), and (8.6) to (8.1) and (8.2), the ac components are expressed using the following equations. For the laser irradiated part: Vh Cp

md dΔTac,h md = αSh Apo cos ωt − 4Sh AσTo3 ΔTac,h dt md md −Kc (ΔTac,h − ΔTac,l ).

(8.7)

For the laser nonirradiated part: (1 − Vh )Cp

md dΔTac,l md = −4(1 − Sh )AσTo3 ΔTac,l dt md md +Kc (ΔTac,h − ΔTac,l ).

(8.8)

The modulation amplitude ΔTac,l and the phase difference are derived by solving (8.7) and (8.8). They are simplified under the condition of Kr /Kc ≤ 0.01 as  −1/2 1 αSh Apo 2 2 1 + 2 2 + ω τc (8.9) ΔTac,l = ωCp ω τr

8 Noncontact Calorimetry of High Temperature Melts

and cos φl =

τc ω



1 − ω2 τc τr

 1+

1 + ω 2 τc2 2 ω τr2

153

−1/2 ,

(8.10)

where τr [s] is the relaxation time for radiation from the droplet to heat reservoirs and τc [s] is the internal relaxation time for the thermal conductance of the droplet. These are defined as τr =

Cp Cp = Kr 4AσTo3

(8.11)

and Cp Vh (1 − Vh ). Kc The correction function, f , is defined as  −1/2 1 2 2 f = 1 + 2 2 + ω τc . ω τr τc =

(8.12)

(8.13)

The condition, ω 2 τr2 1 ω 2 τc2 , which satisfies f  1, is achieved using a proper choice of the modulation frequency. The heat capacity is therefore determined by the temperature amplitude from (8.9). Under this condition, the function ωΔTac,l has the maximum value as a function of the modulation frequency. At that modulation frequency, the phase difference is equal to −π/2, which is derived from the requirement of ∂f /∂ω = 0. Therefore, the heat capacity is determined experimentally from the temperature amplitude, with satisfaction of the requirement described earlier. The term αSh Apo in (8.9) represents the laser input power absorbed by an object. Here, the term is evaluated quantitatively by the products of the laser power and the normal spectral emissivity at a laser wavelength of the object. Assuming Kirchhoff’s law, the normal spectral emissivity is used as the absorptivity. In this study, the value of the normal spectral emissivity of liquid silicon is 0.223 at the laser wavelength (807 nm) measured at temperatures of 1,553 and 1,797 K with uncertainty of 5% [14]. The distribution of the laser intensity is Gaussian; the e−2 radius of the laser beam is 2 mm for a silicon droplet of 4 mm radius. Effects of the sample curvature on the absorptivity were ignored. The simple heat flow model described earlier is insufficient to determine the thermal conductivity; it is necessary to analyze the heat flow in the sphere more accurately. The theory of the measurements of the thermal conductivity and hemispherical total emissivity are explained in Sect. 8.2.2. 8.2.2 Thermal Conductivity and Emissivity Governing Equations and Boundary Conditions Figure 8.3 shows a schematic diagram of the modulated laser calorimetry in spherical coordinates used in our studies to determine thermal conductivity and emissivity

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rlaser

R

z x

q r y

Fig. 8.3. Modulated laser calorimetry in spherical coordinates

of molten materials simultaneously [9–11]. The upper part of an electromagnetically levitated droplet is heated periodically using a modulated laser; the temperature variation at the lower part of the droplet is detected using a pyrometer. By carrying out a similar experiment at various frequencies of the modulated laser ω, the relation of the phase difference between the modulated laser and the temperature variation, φ, to ω is obtained, where φ depends on the thermal conductivity, emissivity, and diameter of the droplet aside from ω. For analysis of the temperature variations in an electromagnetically levitated droplet whose upper part is irradiated using a modulated laser, the following are assumed: (1) the system is axially symmetric; (2) the thermophysical properties of the droplet are constant; (3) the droplet is opaque to the light source, that is, the light source is partially absorbed at the surface and the rest is reflected; (4) the distribution of laser intensity is Gaussian; (5) the heat loss from the droplet surface is radiation alone; and (6) the heat transfer in the droplet is governed solely by conduction. Under the aforementioned assumption, the unsteady-state heat conduction equation in the spherical coordinate system is expressed as 



 1 ∂ 1 ∂T ∂T ∂ 2 ∂T = κ r + sin θ ρcp,mass ∂t r2 ∂r ∂r r2 sin θ ∂θ ∂θ +Q(r, θ), (8.14) where cp,mass , κ, r, t, θ, and ρ respectively denote the isobaric mass heat capacity (specific heat), thermal conductivity, radial distance in the spherical coordinates, time, polar angle in the spherical coordinates, and density. In addition, Q(r, θ) in (8.14) is the heat generation rate attributable to electromagnetic induction heating. The boundary conditions are given by the following equations. At the droplet surface irradiated by the laser,

8 Noncontact Calorimetry of High Temperature Melts



−κ

∂T 2R2 sin2 θ = σ(T 4 − Ta4 ) − αIo (t) exp − s 2 ∂n rlaser

155

 (−n · elaser ). (8.15)

At the droplet surface without the light source, ∂T = σ(T 4 − Ta4 ). ∂n At the centerline, −κ

∂T =0 ∂θ and the initial condition is as follows.

(8.16)

−κ

(8.17)

T = To (r, θ).

(8.18)

Here, elaser and n, respectively, denote unit vector representing the incident direction of the light source and unit normal vector at the droplet surface. Rs and rlaser are radial distance from the centerline to the droplet surface and e−2 radius of laser beam, respectively. Furthermore, Io [W m−2 ] is the laser beam intensity at the centerline, which is related to the power of the modulated laser beam, Po (cos ωt + 1) [W], as Io (t) =

2 2 P (t) = 2 Po (cos ωt + 1). 2 πrlaser πrlaser

(8.19)

Therein, To (r, θ) is the initial temperature distribution in the droplet immediately before laser heating, as determined by considering the electromagnetically induced Joule heat. Simplified Model Basically, the thermal conductivity and emissivity of the droplet can be determined by fitting the experimental temperature response with the results obtained by solving (8.14)–(8.19) numerically if the thermophysical properties other than these parameters, that is, density and specific heat, are known. However, because (8.14) gives unsteady-state solutions, and because the boundary conditions for radiation at the droplet surface, (8.15) and (8.16), are nonlinear, they are too troublesome to use in a system of equations for determining thermophysical properties. Furthermore, Ta and To (r, θ) must be predetermined experimentally, but it is difficult to measure these values. Consequently, (8.14)–(8.18) are simplified as described later. When the upper part of the droplet is irradiated by the modulated laser beam, as presented in Fig. 8.3, the temperature at each point in the droplet, T (r, θ, t), exhibits increases in average temperature and modulation amplitude from the initial temperature; then reaches the stationary modulation state with certain constant average temperature and amplitude, depending on the laser beam’s power and frequency. Therefore, we consider only the temperature response at the stationary modulation state and express the temperature T (r, θ, t) as md (r, θ, t), T (r, θ, t) = To + ΔTdc (r, θ) + ΔTac

(8.20)

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md where ΔTdc (r, θ) is the increase in average temperature and ΔTac (r, θ, t) is the modulation amplitude. In addition, the initial temperature To is assumed to be uniform throughout the droplet. At the stationary modulation state md (r, θ, t) is expressed where the temperature varies sinusoidally with time, ΔTac in the following form: md in out ΔTac (r, θ, t) = ΔTac (r, θ) cos ωt + ΔTac (r, θ) sin ωt,

(8.21)

in (r, θ) ΔTac

out and ΔTac (r, θ), respectively, md (r, θ, t). Then, components of ΔTac

represent the in-phase and outwhere substituting (8.20) and (8.21) of-phase in (r, θ) into (8.14), the following steady-state linear equation systems for ΔTac out and ΔTac (r, θ) are obtained.



  in in 1 ∂ΔTac 1 ∂ ∂ 2 ∂ΔTac r + sin θ κ r2 ∂r ∂r r2 sin θ ∂θ ∂θ out = 0, −ρcp,mass ωΔTac

 κ

1 ∂ r2 ∂r

r2

out ∂ΔTac ∂r

 +

1 ∂ r2 sin θ ∂θ

sin θ

out ∂ΔTac ∂θ

(8.22)



in +ρcp,mass ωΔTac = 0.

(8.23)

Additionally, assuming that the increases in both the average temperature and md modulation amplitude, ΔTdc (r, θ) and ΔTac (r, θ, t), are much less than initial in out (r, θ) and ΔTac (r, θ) temperature To , and the boundary conditions for ΔTac are given as the following linear equations. At the droplet surface irradiated by the laser, the following pertain: 

in 2αPo 2R2 sin2 θ ∂ΔTac in = 4σTo3 ΔTac (−n·elaser ), (8.24) − 2 exp − s 2 −κ ∂n πrlaser rlaser −κ

out ∂ΔTac out = 4σTo3 ΔTac . ∂n

(8.25)

At the droplet surface without the light source, the following are true: −κ

in ∂ΔTac in = 4σTo3 ΔTac , ∂n

(8.26)

−κ

out ∂ΔTac out = 4σTo3 ΔTac . ∂n

(8.27)

At the centerline, −

in ∂ΔTac = 0, ∂θ

(8.28)

−

out ∂ΔTac = 0. ∂θ

(8.29)

and

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Equations (8.22) and (8.23) are solved with the boundary conditions in out (r, θ) and ΔTac (r, θ) (8.24)−(8.29) to determine the distributions of ΔTac in out in the droplet. Then, using ΔTac (r, θ) and ΔTac (r, θ), the distributions of the phase difference, φ(r, θ), in the droplet are obtainable using the following equation:

 out ΔTac . (8.30) φ(r, θ) = tan−1 in ΔTac Determination Procedure The experimental results for φ at various frequencies of the modulated laser ω are fitted with the numerical results obtained using the simplified model described earlier to determine the thermal conductivity and emissivity of the droplet simultaneously. Here, finite element method is used to solve (8.22)–(8.29) numerically for a droplet with arbitrary shape. To fit the experimental values of phase difference φ, the integral mean value of φ in (8.30) over the region corresponding to the spot area of the pyrometer used in the experiment is evaluated using the following equation: 

out average(ΔTac ) −1 . (8.31) φ = tan in ) average(ΔTac Therein, the following are true: in average(ΔTac )=

out average(ΔTac )=

1 Spyrometer 1 Spyrometer

 in ΔTac (r, θ)r sin θ

 (dr)2 + (r dθ)2 ,

(8.32)

S



out ΔTac (r, θ)r sin θ

 (dr)2 + (r dθ)2 (8.33)

S

In addition, Spyrometer is the spot area of the pyrometer. The numerical simulations shown above are incorporated into the Levenberg–Marquardt method, which enables us to compute the nonlinear least squares solutions, as a model function, to determine the thermal conductivity and emissivity of the molten droplet simultaneously from the experimental relation between φ and ω. The heat capacity and density of molten silicon are necessary to obtain the thermal conductivity and emissivity, where the heat capacity measured in this study (see Sect. 8.4.3) was used, and where the density cited from [15] was used. Here, both the heat capacity and the density are assumed to be constant during the modulation experiments. 8.2.3 Verification of the Assumptions of Conduction-Dominated Heat Transfer The mathematical model described earlier is based on the assumption that the heat transfer in the droplet is governed solely by conduction. In our studies, to realize such an assumption in the experiment, a static magnetic field was applied to an electromagnetically levitated molten droplet, and the suppression of convection in the droplet by the Lorentz force because of the interaction

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between the conductive fluid flow and the static magnetic field was investigated. Here, it is verified by numerical simulation whether convection in the molten silicon droplet is truly suppressed by applying a static magnetic field of 4 T used in the experiment. In the electromagnetic levitation (EML) system used to measure the thermophysical properties of molten materials, the alternating electric current in the rf coils induces an eddy current in the conductive material sample, which is melted by Joule heating from the current. In addition, the electromagnetic force caused by the interaction between the alternating magnetic field and the induced current lifts and deforms the molten droplet; moreover, it induces the magnetohydrodynamic (MHD) convection in the droplet. Aside from the MHD convection, the buoyancy convection, and the thermocapillary, Marangoni convection attributable to the temperature dependence of the surface tension on the melt surface exist in the droplet. Here, the MHD, buoyancy and Marangoni convections in the spherical molten silicon droplet with axial application of a static magnetic field are simulated numerically. In the numerical simulation for the EML system, the electromagnetic field in the system is first computed to obtain the electromagnetic force and the distribution of heat generation rate in the droplet; then the flow and temperature fields in the droplet are calculated. Figure 8.4 shows (a) the velocity profiles, (b) the stream lines, and (c) the isotherms in the molten silicon droplet, where a static magnetic field of

T = 1.0500

v max = 547.4 (a) velocity vectors

T = 1.0535 T = 1.052 (b) contours of (c) isotherms stream functions (Tmin = 1.0462, Tmax = 1.0539, (φmin = −13.80, φmax = 5.22, ΔT = 0.0005) Δφ = 2.0)

Fig. 8.4. Velocity and temperature fields in an electromagnetically levitated liquid silicon droplet while applying a magnetic flux density of 4 T

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4 T is applied axially. The physical properties of the molten silicon and processing parameters used in the calculations, except that of the laser power, are presented in Table 8.1. Here, the droplet surface is not irradiated using a laser beam as modulated laser calorimetry; therefore, the laser power is set as 0 W. The static magnetic field can suppress the flow perpendicular to the direction of the magnetic field, that is, the radial flow in the droplet, but cannot suppress the axial component of the flow. Therefore, convection cannot be suppressed completely and three longitudinal vortices appear in the droplet. Particularly, the intense flows, the Marangoni and MHD convections, remain near the surface of the equatorial part of the droplet. However, considering that the velocity of the MHD convection induced by the electromagnetic force is strong, for example, more than 10 cm s−1 , without a static magnetic field, and that fluid flow in the droplet is within the mild turbulent flow regime [16–19], it is readily apparent that the convection in the droplet is markedly suppressed by the applied static magnetic field. In Fig. 8.4, the velocity of the fluid flow near the centerline is less than 1 cm s−1 . In addition, the temperature distributions inside the droplet in the figure are the same as those without convection, that is, those of the conduction dominated model. From these results, it can be concluded that applying the static magnetic field of 4 T, which is used in the actual measurement, enables us to measure the thermal conductivity of molten silicon accurately. 8.2.4 Verification of the Model and Sensitivity Analysis Virtual measurement of the properties was carried out by numerical simulation to ascertain the reliability and accuracy of the model described above to determine thermal conductivity and emissivity of molten materials simultaneously. Here, the electromagnetic field in the electromagnetic levitator actually used in the experiment is analyzed numerically; then the temperature fields in the electromagnetically levitated droplet during modulated laser calorimetry are computed by solving (8.14) under the boundary and initial conditions, (8.15)–(8.18). In this case, it is assumed that the heat transfer in the droplet is governed by conduction alone because convection is suppressed by applying a static magnetic field. Figure 8.5a shows the calculated temperature response at the lower part of the droplet, that is, the integral mean value of temperature over the region corresponding to the spot area of the pyrometer, for ω/2π = 0.1 Hz, and additionally Fig. 8.5b shows the temperature distributions in the droplet for (i)–(iv) in Fig. 8.5a. The physical properties of the molten silicon and processing parameters used in the calculations are presented in Table 8.1. From those figures, it is apparent that the electromagnetically induced heterogeneous heat affects the isotherms in the droplet. By carrying out calculations similar to those in Fig. 8.5a at various frequencies of the modulated laser source ω, the relation between the phase difference φ and ω is obtained numerically as presented in Fig. 8.6. Then, the simplified mathematical model described in the previous section was adopted

(a) 50

1810

40

1800

T

30

(ii)

P

1790

(i)

20

1780

10

1770

0

1760

Temperature / K

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Laser power P / W

160

(iii) 0

50

100

150

200 (iv)

Time / s

(b)

(i) t = 190.0 s Tmin = 1790 K, Tmax = 1798 K ΔT = 0.84 K

(ii) t = 192.5 s Tmin = 1798 K, Tmax = 1810 K ΔT = 0.84 K 1811 K

1789 K (iii) t = 195.1 s Tmin = 1801 K, Tmax = 1809 K ΔT = 0.84 K

(iv) t = 197.6 s Tmin = 1790 K, Tmax = 1803 K ΔT = 0.84 K

Fig. 8.5. (Top) calculated temperature response at the lower part of the droplet, and (bottom) temperature distribution in the droplet for (i)–(iv)

for plots in Fig. 8.6, as shown by a fitting line in the figure. Consequently, the values 63.93 W m−1 K−1 and 0.321 were obtained respectively as thermal conductivity and emissivity. Comparing them to the input data in Table 8.1, the estimated value of thermal conductivity, 63.93 W m−1 K−1 , shows good agreement with the input data, 64.00 W m−1 K−1 ; it can be concluded that the present simplified model is appropriate for estimating the thermal conductivity of the electromagnetically levitated droplet. On the other hand, the

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Table 8.1. Physical properties and processing conditions used in calculations Physical properties of molten silicon Temperature coefficient of surface tension [N m−1 K−1 ] Viscosity [kg m−1 s−1 ] Density [kg m−3 ] Thermal expansion coefficient [K−1 ] Thermal conductivity [W m−1 K−1 ] Emissivity Specific heat [J kg−1 K−1 ] Electric conductivity [S m−1 ] Melting temperature [K]

−4.3 × 10−4 7.0 × 10−4 2,530 1.5 × 10−4 64 0.3 1,000 1.2 × 106 1,683

Operating conditions 8 × 10−3 375 200 4 323 9.56 2 × 10−3 2 × 10−3

Droplet diameter [m] Electric current in rf coil [A] Frequency of electric current in rf coil [kHz] Static magnetic flux density [T] Ambient temperature [K] Laser power [W] e−2 radius of semiconductor laser beam [m] Spot radius of pyrometer [m] 140 120

−φ/ο

100 80 60 numerical results fitting line

40 20 0 0

0.1

0.2

ω/

0.3

0.4

2 π /Hz

Fig. 8.6. Calculated relation between the phase difference and frequency

estimated value of emissivity, 0.321, is larger than that in Table 8.1: 0.3. The integral mean value of temperature over the region corresponding to the spot area of the pyrometer before being irradiated using a laser beam is 1,768 K. Consequently, the linearization expressed in (8.24) to (8.27) is approximately satisfied. Therefore, it is inferred that the discrepancy between the estimated value and the emissivity in Table 8.1 results from the fact that, actually, the initial temperature To is not constant throughout the droplet.

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-f/o

100 80 60 57.6 W.m−1.K−1 64.0 W.m−1.K−1 70.4 W.m−1.K−1

40 20

0 0

0.1

0.2

w / 2p /Hz

0.3

0.4

Fig. 8.7. Effect of variation in thermal conductivity on the relation between the phase difference and frequency 140 120

−φ/ο

100 80 60

0.24 0.30 0.36

40 20 0 0

0.1

0.2

0.3

0.4

ω / 2 π /Hz Fig. 8.8. Effect of variation in emissivity on the relation between the phase difference and frequency

Next, the sensitivity analyses of thermal conductivity and emissivity of molten silicon are carried out, where the effect of variations in these properties on the relation between φ and ω is investigated numerically. Figures 8.7 and 8.8 show the calculated relations between φ and ω for three different values of thermal conductivity and emissivity, respectively, where the values in Table 8.1 are varied by ±10% for thermal conductivity and by ±20% for emissivity. In the calculations for emissivity, the initial temperatures in the droplet before laser-heating are equalized in all three cases by adjusting the value of the rf current. The figure shows that it is found that the sensitivity of φ for thermal conductivity is significant over a wide range of frequencies, although the sensitivity for emissivity is significant only at a lower frequency region, that is, ω/2π < 0.05 Hz. Consequently, it is suggested from Figs. 8.7 and 8.8 that the simultaneous determination of thermal conductivity and emissivity of molten silicon by the present model must be performed from the relation

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between φ and ω measured over a wide range of ω; particularly, φ at low ω is necessary for precise determination of emissivity. 8.2.5 Emissivity Determination from Cooling Curve The emissivity of the sample is also evaluated from the cooling curve because of the radiation in vacuum. The change in sample temperature is expressed as the following equation after the laser irradiation ceases. Q − Aσ(T 4 − Ta4 ) dT = . dt Cp

(8.34)

The heat balance between the power input from the rf-coil and radiation heat loss is given by the following equation because T → To , dT /dt → 0 for t → ∞. Q = Aσ(To4 − Ta4 ).

(8.35)

Under the condition of To ΔT , the following linearization is applied for the radiative heat loss term in (8.34): T 4 = (To + ΔT )4  To4 + 4To3 ΔT.

(8.36)

Solving (8.34) with (8.35) and (8.36) yields ΔT = ΔTdc exp(−t/τr ).

(8.37)

By fitting the cooling curve with the exponential function shown above, the value of τr is determined. Consequently, the emissivity is determined using (8.11).

8.3 Experimental A single crystalline silicon cube (7×7×7 mm) was placed at the center of the rf-coil (15 kW, 200 kHz) in a vacuum chamber. Prior to the experiment, the chamber was filled with high-purity argon gas; it was subsequently evacuated to 10−3 Pa using a rotary pump in combination with a turbo-molecular pump. The silicon was preheated by irradiation of a semiconductor laser to a temperature at which the electric resistivity of the silicon was reduced sufficiently to levitate silicon using the applied electromagnetic force. A superconducting magnet with 220 mm bore was used to impose a static magnetic field. The strength of the static magnetic field was varied from 0 to 4 T to suppress convection in the liquid silicon. The initial temperature of the liquid silicon was controlled by the electric power supply to the rf coil. The top surface of the levitated silicon droplet was heated sinusoidally by a semiconductor laser through a function generator. The modulation frequency varied from 0.02 to 0.3 Hz, which is sufficiently wide to determine both thermal conductivity and emissivity experimentally, precisely, as suggested by the sensitivity analysis described in Sect. 8.2.4.

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The temperature response was measured at the bottom of the sample through a diaphragm with aperture diameters of 6 mm using a two-color pyrometer. The signals from the function generator and the pyrometer were recorded with a sampling interval of 20 ms. A laser apparatus (NBT-S140-mk II; JENOPTIK Laserdiode Japan Co. Ltd., Tokyo, Japan) equipped with a fiber-coupling type CW laser diode was used. The maximum output power was 140 W at a wavelength of 807 ± 3 nm. A calibrated laser power meter (FieldMate; Coherent Inc., Portland, OR, USA) evaluated the net laser power through the optical system, which consisted of an optical fiber, a corrective lens, and a glass window within uncertainty of ±2.4%. The two-color pyrometer (IR-CAQ2CS; Chino Corp., Tokyo, Japan), using the emissivity ratio of two wavelengths (1,350 and 900 nm) was used. The relative uncertainty of the pyrometer was ±0.5% of the temperature; the temperature resolution was 1 K. The pyrometer was calibrated using the melting point of the liquid silicon during solidification of the levitated liquid silicon. The laser power was turned off after the succession of the modulated laser calorimetry. Then the silicon liquid was cooled by radiation. The crystalline silicon grains appearing on the silicon droplet during solidification reflect the flow in the liquid silicon; a high-speed CCD camera was used to evaluate the effect of the static magnetic field on the convective flow.

8.4 Experimental Results 8.4.1 Motion of the Silicon Droplet Figure 8.9 shows a top view of the motion of the nucleated silicon grains on the levitated liquid silicon at 1 T. This picture was obtained by superimposing five consecutive images taken with a 0.024 s interval. The grains are arranged concentrically, suggesting that the silicon droplet rotated around the z-axis, similar to a solid sphere. Vertical convection was suppressed in the static magnetic field. The numerical simulation presented in Fig. 8.4 also reveals that MHD convection was reduced to less than 1 cm s−1 near the centerline of the silicon droplet at 4 T. 8.4.2 Temperature Response and Phase Difference Figure 8.10 depicts an example of the experimental temperature response of the noncontact modulated laser calorimetry for electromagnetically levitated liquid silicon. Initially, the silicon droplet temperature was kept at To , with balancing between the radiative heat loss and rf induction heating. Then, the laser sinusoidally heated the droplet. The average temperature of the droplet increased gradually by ΔTdc from To until the radiative heat loss was balanced with the sum of the rf induction heating and modulated laser heating; subsequently, the temperature reached an ac steady state. After the

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4 mm Fig. 8.9. Top view of the motion of silicon grains floating on the surface of the levitated liquid silicon under a static magnetic field of 1 T. This picture was obtained by superimposing five consecutive images taken at a 0.024 s interval 60

1930

T

ΔTac

1910

40 ΔTdc

P

30

1890

20 1870

Temperature / K

Laser power / W

0.2 Hz

0.1 Hz

50

10

To 0 0

100

200

300

1850

Time /s Fig. 8.10. Typical time dependence of the laser power (left-hand ordinate) and temperature response (right-hand ordinate) during modulated laser calorimetry for molten silicon

laser power was turned off, the sample temperature decreased according to an exponential cooling curve and reverted to To . From this cooling curve, the hemispherical total emissivity was also determined. Figure 8.11 shows that the relation between ωΔTac and ω was obtained from a series of modulation heating. The value of ωΔTac has a maximum around the ω, which gives φ = 90◦ . The isobaric heat capacity was obtained from this maximum value using (8.9). A curve fit of the numerically obtained φ − ω relation to the experimentally obtained data over the entire frequency

H. Fukuyama et al. 7.0

130

ωΔTac / K.rad.s−1

−φ

110

6.0 ωΔTac

90 70

−φ/ ο

166

5.0 50 4.0 0.00

0.10

0.20

30 0.30

ω/2 π /Hz Fig. 8.11. ωΔTac as a function of the modulation frequency is shown on the lefthand ordinate as open squares. The phase difference −φ is shown on the righthand ordinate, experimental data as solid circles; the solid line presents fitting to the experimental data using the numerical analysis explained in Sect. 8.2.2 for determination of the emissivity and thermal conductivity of liquid silicon

range is shown by a solid line in Fig. 8.11. The hemispherical total emissivity and thermal conductivity of liquid silicon were determined using this curvefitting process. 8.4.3 Isobaric Molar Heat Capacity Figure 8.12 shows the isobaric molar heat capacity of liquid silicon with literature data [20–23]. The clear temperature of the molar heat capacity was not observed. The average isobaric molar heat capacity of liquid silicon is 30 ± 5 J mol−1 K−1 at temperatures of 1,750–2,050 K. The experimental uncertainty presented in the above value is double the standard deviation for all the present data. The present result shows good agreement with the data reported by Kantor et al. [20], Yamaguchi and Itagaki [21], and Olette [22], which were measured using drop calorimetry method. The condition, Kr /Kc ≤ 0.01, is expected to be satisfied for justification of the heat capacity measurement, as explained in Sect. 8.2.1. However, it is difficult to estimate the volume heated directly by laser irradiation (Vh ) for the quantitative evaluation of Kc using (8.12). Here, the Biot number relevant to the value of Kr /Kc , which is defined as Bi =

4σTo3 , κ/R

(8.38)

is used for evaluation. The estimated Bi numbers are 0.019–0.031 under the present condition, depending on temperature. This value indicates that the experimental postulate that heat transfer to the external heat bath is much

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50

cp / J.mol−1.K−1

40 30

Ref.20 Ref.21 Ref.23 Average of present study

20 Ref.22 10 Melting point of Si

0 1600

1700

1800

1900

2000

2100

Temperature / K Fig. 8.12. The isobaric molar heat capacity of liquid silicon measured in a static magnetic field at 0.5 T (diamonds), 1.0 T (crosses), 2.0 T (squares), 3.0 T (triangles), and 4.0 T (circles) with data from the literature: Kantor et al., [20], Yamaguchi and Itagaki [21], Olette [22], NIST-JANAF [23]. The solid line represents the average of this study

smaller than heat transfer to the inner part, that is, the condition of Kr /Kc ≤ 0.01 is fundamentally satisfied. The correction function, f , which is expected to be nearly equal to unity, is also justified using (8.13) as follows. The external relaxation time, τr , is calculated from (8.11). Using τr and (8.10), the internal relaxation time, τc , is evaluated by fitting the frequency dependence of the phase difference. The values of τr and τc are 12 and 0.16 s, respectively, for the representative experimental condition as presented in Fig. 8.9. Consequently, the value of f has a maximum value of 0.99 at the frequency between 0.1 and 0.12 Hz, which indicates that the heat capacity can be measured to within 1% uncertainty. 8.4.4 Hemispherical Total Emissivity The hemispherical total emissivity of liquid silicon determined from the phase difference is presented in Fig. 8.13. Clear temperature dependence of the emissivity was not observed. The average emissivity is 0.27 ± 0.03 at temperatures of 1,750–1,910 K. The hemispherical total emissivity of liquid silicon was also obtained from the radiative cooling curve. Figure 8.14a shows the radiative cooling curve of the liquid silicon. By curve fitting of (8.37), the external thermal relaxation time attributable to the radiative cooling was determined to be 12.1 s. The emissivity determined from the radiative cooling curve is presented in Fig. 8.14b; the average value is 0.25 ± 0.04. This result agrees with data obtained from the phase difference (Fig. 8.13) within experimental uncertainty.

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ε

0.30

0.20

0.10 Melting point of Si

0.00 1650

1750

1850

1950

2050

Temperature / K Fig. 8.13. The hemispherical total emissivity of liquid silicon obtained from the phase difference measured in a static magnetic field at 0.5 T (diamonds), 1.0 T (crosses), 2.0 T (squares), 3.0 T (triangles), and 4.0 T (circles). The solid line represents the average of this study

(b)

0.40 0.30

1800

ε

Temperature / K

(a) 1805

ΔTdc

1795

0.20

0.10

To 1790

0

Melting point of Si

20

40

Time / s

60

0.00 1650

1750

1850

1950

2050

Temperature / K

Fig. 8.14. (a) An example of change in temperature of silicon droplet following turning off of the laser power. (b) The hemispherical total emissivity determined from the radiative cooling curve measured in a static magnetic field at 0.5 T (diamonds), 1.0 T (crosses), 2.0 T (squares), 3.0 T (triangles), and 4.0 T (circles). The solid line in (b) represents the average of this study

Both experimental uncertainties of the emissivity are double the standard deviation for all present data. 8.4.5 Thermal Conductivity Figure 8.15 shows the thermal conductivity of liquid silicon against the temperature for various static magnetic fields with data obtained from the relevant literature [24–32]. The apparent thermal conductivity decreases with increasing strength of the static magnetic field. The value converges on the value of 66 ± 11 W m−1 K−1 at 2 T or larger. The experimental uncertainty of

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100 90

0.5T

κ / W.m−1.K−1

80

1T

70 60

Present study 0.5T 1T Average (2-4T) ( 2T, 3T, 4T)

29

2-4T 31

26 24

32

50

30 27

40 Melting point of Si

30

(24) Yamamoto, (25 x) Takasuka, (26) Nishi, (27) Nagai, (28 ) Yamasue, (29) Cusack, (30) Glazof, (31) Sasaki, (32) Schnyders

20 1700

1800

1900

2000

2100

Temperature / K Fig. 8.15. Thermal conductivity of liquid silicon as a function of the temperature in a static magnetic field in 0.5 T (solid diamonds), 1.0 T (open diamonds), 2.0 T (squares), 3.0 T (triangles), and 4.0 T (circles) together with values measured using several methods: laser-flash method [24–26]; hot disk method [27], and transient hot wire method [28]. Calculated κ assuming the Wiedemann–Franz law from electrical conductivity is also presented [29–32]. The thermal conductivity measured using laser-flash technique is recalculated from the thermal diffusivity with cp and ρ used in this study

the thermal conductivity is double that of the standard deviation for all data greater than 2 T. No obvious temperature dependence of thermal conductivity was observed in the experimental temperature region. Our data measured greater than 2 T are in the high-temperature extended region of the literature thermal conductivities, which were measured [24–28] or calculated using the Wiedemann–Franz law [29–32]. The agreement of the present data with the Wiedemann–Franz law indicates that the electron contribution is dominant for thermal transport at the experimental temperature range. On the other hand, thermal conductivity measured using hot disk method [27] shows that the lowest value might be attributable to the influence of the interfacial resistance between the AlN insulator and the hot disk sensor.

8.5 Summary We developed a method of measuring the thermophysical properties, heat capacity, thermal conductivity, and emissivity of high-temperature molten droplets using the electromagnetic levitation technique. The method was based on modulated laser calorimetry where a static magnetic field was superimposed to suppress the melt convection in an electromagnetically levitated

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droplet. The modulated laser calorimetry was modeled to estimate the thermal conductivity and emissivity of the electromagnetically levitated droplet using a measured parameter, that is, the phase difference between the modulated laser and the temperature variations at various frequencies of the modulated laser. The experimental relation between the phase difference and frequency was fitted by the mathematical model to estimate the thermal conductivity and emissivity of molten silicon simultaneously. In addition, the numerical simulations for unsteady thermal field in the electromagnetically levitated droplet were carried out to demonstrate the validity of the proposed simplified model, then to investigate the sensitivity of the thermophysical properties to the relation between the phase difference and frequency. Numerical simulations for convection in the droplet were carried out to confirm the fact that a static magnetic field suppresses the convection in a liquid silicon droplet in the electromagnetic levitator. Here, the convection driven by the buoyancy force, thermocapillary force attributable to the temperature dependence of the surface tension on the melt surface, and electromagnetic force in the droplet were considered. Results show that applying a static magnetic field of 4 T can suppress the convection inside the droplet sufficiently to measure the thermal conductivity of the liquid silicon. The experimental results for liquid silicon are summarized as follows. The silicon droplet rotated around the z-axis similarly to a solid sphere: vertical convection was suppressed in the static magnetic field. The isobaric molar heat capacity of 30 ± 5 J mol−1 K−1 (1,750–2,050 K), the hemispherical total emissivity of 0.27 ± 0.03 (1,750–1,910 K), and the thermal conductivity of 66 ± 11 W m−1 K−1 (1,750–2,050 K). The convection in an electromagnetically levitated liquid silicon droplet is suppressed as sufficiently small for measurement of the thermal conductivity in a static magnetic field greater than 2 T.

Acknowledgements The authors thank Professors T. Hibiya (Keio University), M. Watanabe (Gakushuin University), H. Yasuda (Osaka University), H. Fecht (University of Ulm), I. Egry (DLR), R. K. Wunderlich (University of Ulm), and S. Ozawa (Tokyo Metropolitan University) for their helpful discussions and critical comments. The author (HF) appreciates financial support from the Japan Society for the Promotion of Science (Grants-in-Aid for Scientific Research), JFE 21st Century Foundation, and the Iron and Steel Institute of Japan. This study was subsidized by the Japan Keirin Association through its Promotion funds from KEIRIN RACE. It was also supported by the Mechanical Social Systems Foundation and the Ministry of Economy, Trade, and Industry of Japan. This work was performed at the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University.

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References 1. T. Hibiya, I. Egry, Meas. Sci. Technol. 16, 317 (2005) 2. M. Mito, T. Tsukada, M. Hozawa, C. Yokoyama, Y.R. Li, N. Imaishi, Means. Sci. Technol. 16, 457 (2005) 3. C. Kittel, Introduction to Solid State Physics, 7th edn. (Wiley, New York, 1996) p. 144 4. H.-J. Fecht, W.L. Johnson, Rev. Sci. Instrum. 62, 1299 (1991) 5. R.K. Wunderlich, H.-J. Fecht, Appl. Phys. Lett. 62, 3111 (1993) 6. R.K. Wunderlich, D.S. Lee, W.K. Johnson, H.-J. Fecht, Phys. Rev. B 55, 26 (1997) 7. R.K. Wunderlich, H.-J. Fecht, Meas. Sci. Technol. 16, 402 (2005) 8. H. Yasuda, I. Ohnaka, Y. Ninomiya, R. Ishii, S. Fujita, K. Kishio, J. Cryst. Growth 260, 475 (2004) 9. H. Fukuyama, H. Kobatake, K. Takahashi, I. Minato, T. Tsukada, S. Awaji, Meas. Sci. Technol. 18, 2059 (2007) 10. T. Tsukada, H. Fukuyama, H. Kobatake, Int. J. Heat Mass Trans. 50, 3054 (2007) 11. H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, S. Awaji, Appl. Phys. Lett. 90, 94102 (2007) 12. Y. Kraftmakher, Modulation calorimetry, Theory and Applications (Springer, Berlin, 2003) 13. P.F. Sullivan, G. Seidel, Phys. Rev. 173, 679 (1968) 14. H. Kawamura, H. Fukuyama, M. Watanabe, T. Hibiya, Meas. Sci. Technol. 16, 386 (2005) 15. K. Higuchi, K. Kimura, A. Mizuno., M. Watanabe, Y. Katayama, K. Kuribayashi, Meas. Sci. Technol. 16, 381 (2005) 16. J.H. Zong, B. Li, J. Szekely, Acta Astronautica 26, 435 (1992) 17. B.Q. Li, S.P. Song, Microgravity Sci. Technol. XI, 134 (1998) 18. V. Bojarevics, K. Pericleous, ISIJ Int. 43, 890 (2003) 19. R.W. Hyers, Meas. Sci. Technol. 16, 394 (2005) 20. P.B. Kantor, A.M. Kisel, E.N. Fomichev, Ukr. Fiz. Zh. 5, 358 (1960) 21. K. Yamaguchi, K. Itagaki, J. Therm. Anal. Cal. 69, 1059 (2002) 22. M. Olette, Compt. Rend. 244, 1033 (1957) 23. M.W. Chase Jr. (ed.), NIST-JANAF Thermochemical tables, 4th edn. (American Chemical Society and American Institute of Physics for the National Institute of Standards and Technology, Washington DC, 1998) 24. K. Yamamoto, T. Abe, S. Takasu, Jpn. J. Appl. Phys. 30, 2423 (1991) 25. E. Takasuka, E. Tokizaki, K. Terashima, S. Kimura, in Proc. the 4th Asian Thermo5hys. Properties Conf. B1d3, 89 (1995) 26. T. Nishi, H. Shibata, H. Ohta, Mater. Trans. 44, 2369 (2003) 27. H. Nagai, Y. Nakata, T. Tsurue, H. Minagawa, K. Kamada, E. Gustafsson, T. Okutani, Jpn. J. Appl. Phys. 39, 1405 (2000) 28. E. Yamasue, M. Susa, H. Fukuyama, K. Nagata, J. Cryst. Growth 234, 121 (2002) 29. N.E. Cusack, Rep. Prog. Phys. 26, 361 (1963) 30. V.M. Glazov, V.B. Kolftsov, V.A. Kurbatov, Sov. Phys. Semicond. 20, 1351 (1986) 31. H. Sasaki, A. Ikari, K. Terashima, S. Kimura, Jpn. J Appl. Phys. 34, 3426 (1995) 32. H.S. Schnyders, J.B. Van Zytveld, J. Phys. Condens. Matter 8, 10875 (1996)

9 Noncontact Thermophysical Property Measurements of Refractory Metals Using an Electrostatic Levitator Takehiko Ishikawa and Paul-Fran¸cois Paradis

9.1 Introduction The use of a containerless technique for materials processing has many technological and scientific advantages. The absence of a crucible allows the handling of chemically reactive materials such as molten refractory metals, alloys, or semiconductors, and eliminates the risk of sample contamination in overheated and in undercooled states (liquid phase below melting temperature). This offers excellent opportunities to characterize the structure of materials and to determine accurately their thermophysical properties in those states. The lack of a crucible also suppresses nucleation induced by the walls of a container (heterogeneous nucleation), thus increasing the possibility of producing new materials such as metallic glasses. Several levitation methods, including acoustic, electromagnetic, aerodynamic, and electrostatic have been applied for thermophysical property measurements. The electromagnetic levitation method has been most popularly used for metal samples because instrumentation is rather simple and compatible with high vacuum. Density, surface tension, electrical resistivity, spectral emissivity, and thermal conductivity of conductive materials are currently measured using this method [1–4]. An alternative method for metal samples processing and study is the electrostatic levitation, which uses Coulomb force between a charged sample and electrodes. This method is applicable to conductive as well as nonconductive samples but requires a high speed feedback control system to stabilize the sample position. Due to this technical difficulty, the development of electrostatic levitation method was slower than other methods. Rhim et al. from the Jet Propulsion Laboratory (JPL) developed a ground based electrostatic levitation system [5] and successfully levitated and melted refractory metals (zirconium [6] and titanium [7]) and semiconductors (silicon [8] and germanium [9]). This group also developed several techniques to measure thermophysical properties such as density [10], constant pressure heat capacity [11], surface tension, viscosity [12], and electrical resistivity [13]

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with the electrostatic levitation system. More recently, the Japan Aerospace Exploration Agency (JAXA) has improved the JPL technology to measure thermophysical properties of refractory metals whose melting temperatures are above 2,000 K [14]. This chapter briefly describes the electrostatic levitation system and the thermophysical property measurement techniques. Typical data of superheated liquid and undercooled materials are also reported.

9.2 Electrostatic Levitation System The electrostatic levitation method utilizes the Coulomb force between the sample and the surrounding electrodes to cancel the gravity force. A positively charged sample is levitated between a pair of parallel disk electrodes (top and bottom electrode), typically 10 mm apart, which are utilized to control the vertical position (z) of the specimen. The typical sample size is 2 mm in diameter with an electrical field of around 8–15 kV m−1 is necessary to levitate it against gravity. In addition, four spherical electrodes distributed around the bottom electrode are used for horizontal control (x and y). Since the electrostatic scheme can not produce a potential minimum, a feedback position control system is necessary. Figure 9.1 illustrates the hardware arrangement for the position control. Position sensing is achieved with a set of orthogonally disposed He–Ne laser (632.8 nm) that projects a sample image on a position sensor. The beam of a He–Ne laser is expanded and impinges Top electrode Band pass filter Lens He-Ne laser

Sample

Side electrodes

Position detector

Bottom electrode PC A/D High voltage amplifiers

PID calc.

Signal conditioner

D/A

Fig. 9.1. Schematic diagram of the sample position control system

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on a levitated sample. The size of the resulting sample shadow is optimized with a lens to cover the area of the sensor such that a good dynamic range is obtained. In addition, a polarization filter is used to optimize the laser intensity reaching the sensor. The sensor is equipped with a band-pass filter at 632.8 nm to eliminate the photon noise coming from sources other than the laser. The sample position information read by the sensor is then fed into a computer and analyzed by a program. The program uses PID servo algorithms for implementing the feedback system, thus allowing a sample to maintain a fixed position in time. The computer then inputs new values of voltages for each electrode. The feedback rate is 720 Hz for the z direction and 30 Hz for the horizontal directions. Figure 9.2 schematically depicts the electrostatic levitation furnace (ELF) currently being used at JAXA. It consists of a stainless steel chamber that is evacuated to a pressure around 10−5 Pa. The chamber houses the above mentioned electrode system (a pair of parallel disk electrodes and four spherical electrodes) in the center. Sample heating is achieved using two 100 W CO2 lasers emitting at 10.6 μm and a 500 W Nd:YAG laser emitting at 1.064 μm. The high power Nd:YAG laser is needed to melt materials with melting temperatures higher than 2,800 K (Mo, Ta, W, Re, and Os). (9) (2)

(5)

(9) (8)

(5) (7)

(7)

(4)

(11)

(1)

(13) (6)

(7) (14) (10) (3)

(12)

(6)

(11)

Fig. 9.2. Schematic view of the electrostatic levitation furnace and its diagnostic apparatus: (1) sample, (2) top electrode, (3) bottom electrode, (4) side electrodes, (5) He–Ne laser, (6) position detector, (7) CO2 laser beam, (8) Nd:YAG laser beam, (9) pyrometer, (10) CCD camera, (11) CCD camera with telephoto objective lens, (12) UV lamp, (13) beam Splitter, (14) power meter for oscillation detection

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T. Ishikawa and P.-F. Paradis Top electrode φ 25 mm

φ 10 mm

Bottom electrode

Fig. 9.3. Electric field between the top and bottom electrodes obtained by a numerical analysis

Proper arrangement for heating laser is important to prevent sample movement due to the photon pressures by heating lasers. Currently, the CO2 laser beams are divided into three beams and separated by 120◦ in a horizontal plane, while the Nd:YAG laser beam hits directly from the top, through a hole in the top electrode. Although this quasi-tetrahedral multiple beam configuration improves the sample position stability, it is not satisfactory for the thermophysical property measurements when the power of the Nd:YAG laser is very high. This problem is solved by making the top electrode smaller than the bottom electrode, to strengthen the horizontal restoring forces. Figure 9.3 shows the numerical analysis results of the electric field between the electrodes when the diameter of the top electrode is diminished. For the large electrode (left), the surfaces of the iso-electric potential are almost flat whereas those for the small electrode (right) exhibits a slope that generated a field gradient. The conical electrical field distribution resulting from this arrangement provides a horizontal field component and a natural restoring force towards the center [15]. Sample temperature data are measured using single-color pyrometers (0.90 and 0.96 μm, 120 Hz), equipped with a band stop filter (Rugate notch filter) at 1.064 μm to remove noise coming from the Nd:YAG laser. The levitated sample is observed by three charged-coupled-device cameras. One camera offers a view of both the electrodes and the sample. In addition, two black and white highresolution cameras, located at right angle from each other and equipped with telephoto objectives in conjunction with background lamps, provide magnified views of the sample. This also helps to monitor the sample position in the horizontal plane and to align the heating laser beams to minimize any photon induced effects on the sample. Four coils located below the bottom electrode generate a rotating magnetic field to control sample rotation.

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9.3 Thermophysical Property Measurements By combining noncontact diagnostics apparatus such as pyrometer or telephoto camera, several thermophysical properties can be measured with the electrostatic levitation furnace. Property measurements by containerless methods have several advantages compared with conventional methods. First, samples are free from the risk of contamination from the container and materials with melting points higher than that of crucibles (e.g., platinum or alumina) can be processed. Second, since nucleation from the container wall can be suppressed, molten samples can be maintained in deeply undercooled condition. 9.3.1 Density The density can be measured using an UV imaging technique. Once the sample is molten, it will be in spherical shape due to surface tension and the distribution of surface charge. Also, the molten sample is axi-symmetric since the sample is rotated in vertical axis by the rotating magnetic field. Because the sample is axi-symmetric and the mass is known, density(ρ) can be calculated as: 3m , (9.1) ρ= 4πr3 where m and r are the mass and the radius of the sample. The radiance temperature measured by the pyrometer is calibrated by using the recalescence of the sample (sudden temperature rise from the undercooled temperature to the melting temperature (Tm ) due to the release of latent heat of fusion). After cooling the sample, both the images and the cooling curve are recorded to determine the density. The recorded video images are digitized and matched to the cooling curve. Then, a JAXA developed program extracts the area from each image and calculates the density at each temperature. A good contrast between sample and background is important to get a precise result by the image analysis. Refractory metals become luminous at elevated temperatures and it is hard to get an excellent contrast with the background. Our solution to this problem is to use a high-pass filter at 450 nm in conjunction with an UV lamp (Hoya–Schott EX200W) and to observe the sample only in the UV range. Figure 9.4 clearly depicts the differences between a room temperature and an overheated liquid zirconium sample under different background conditions [16]. When a white light illumination source is used, a solid zirconium sample, originally appearing black on a white background (Fig. 9.4(a)), changes to a white and glowing sample while heated (Fig. 9.4(b)). If the sample is melted and overheated, it becomes so bright as if it makes the camera blind (Fig. 9.4(c)). If no background light is used, the solid sample is not so

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Fig. 9.4. Effect of the background on the appearance of a sample at several temperatures

visible on a black background (Fig. 9.4(d)). As it is heated, it becomes more visible (Figs. 9.4(e), 9.4(f)) but the blooming effect on the camera becomes more significant as the sample becomes brighter. With the proposed UV technique, the images keep to exhibit nearly constant contrast from the room temperature to well over the melting temperature (Figs. 9.4(g)–9.4(i)). 9.3.2 Surface Tension and Viscosity The surface tension and viscosity are determined by the drop oscillation method, for which the frequency of the surface oscillation of the levitated sample is measured around its equilibrium shape. In this method, a sample is molten and brought to a selected temperature. Then, a P2 (cos θ)-mode of drop oscillation is induced to the sample by superimposing a small sinusoidal electric field on the levitation field. Here, P2 (cos θ) is a Legendre polynomial of second-order. An oscillation detection system, illustrated in Fig. 9.5(a), measures the fluctuation of the vertical diameter of the molten sample with a sampling frequency of 4096 Hz. The transient signal that followed the termination of the excitation field is shown in Fig. 9.5(b). This signal is analyzed using an in-house written LabVIEWT M program. This is done for a few times at given temperature and repeated in several temperatures. Using the characteristic

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179

(a) Photo detector with vertical slit

Top electrode He-Ne laser

Sample shadow

Sample Bottom electrode

To position sensor

Beam Splitter

(b)

Amplitude (a.u.)

40

0

−40

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (s)

Fig. 9.5. Sample oscillation detection for surface tension and viscosity measurements

oscillation frequency, ωc , of this signal after correcting for nonuniform surface charge distribution, the surface tension γ can be found from the following equation 



Q2 8γ 2 1− {1 − f (γ, q, Y )} , (9.2) ωc = ρr03 64π 2 r03 γε0 where

 243.31γ 2 − 63.14q 2 γ + 1.54q 4 Y 2 f (γ, q, e) = 176γ 3 − 120q 2 γ 2 + 27γq 4 − 2q 6 

(9.3)

and r0 is the radius of the sample when a spherical shape is assumed, ρ is the liquid density, Q is the drop charge, ε0 is the permittivity of vacuum. The symbols q and Y are defined by q2 =

Q2 16π 2 r02 ε0

(9.4)

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and Y 2 = E 2 r0 ε0

(9.5)

respectively, and E is the applied electric field. The characteristic oscillation frequency ωc of molten refractory metal droplet (ca. 2 mm in diameter) ranges from around 180 to 240 Hz. Similarly, using the decay time τ given by the same signal, the viscosity η is found by [17] ρr02 . (9.6) 5τ The density data ρ are known by conducting measurements described in 9.3.1 and the radius of the sample r0 is determined by the image analysis of the recorded sample image during oscillation experiment. The drop charge Q can be calculated by η=

mg = QE,

(9.7)

where g is the gravitational acceleration. While measuring surface tension and viscosity, liquid samples are intentionally rotated to suppress the excitation of oscillation modes other than the P2 (cos θ)-mode or nonaxisymmetric oscillations. The characteristic oscillation frequency ωc and the decay time τ are dependent on the rotation rate of the liquid sample Ω, by the following equations:

2 19 Ω Δωc , (9.8) = ωc 21 ωc 

2  1 2 Ω 1 = 1− , (9.9) τ + Δτ τ 3 ωc where Δωc and Δτ are the respective deviations of the characteristic oscillation frequency and decay time due to rotation. Since the rotation rate of the liquid sample cannot be measured accurately, it is estimated by monitoring the aspect ratio of the sample (ratio of the horizontal radius and vertical radius) and by using the empirical equation calculated by Brown and Scriven [18] and experimentally confirmed by Rhim and Ishikawa [19]:

2

 Rhorizontal Ω Ω −2 + 1.2532 ≈ 1 + 1.476 × 10 Rvertical ωc ωc

3

4 Ω Ω −1.7877 + 3.7385 . (9.10) ωc ωc Using (9.8)–(9.10), relations between the aspect ratio and Δωc or Δτ can be estimated. If the aspect ratio of a rotating sample is less than 1.02, the reduced sample rotation rate (Ω/ωc ) is less than 0.13 and effects of the sample rotation to both the characteristic oscillation frequency and the decay time should not be greater than 1.5%. In our experiment, the aspect ratio of the

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levitated sample is always monitored and the sample rotation is controlled so that the aspect ratio does not exceed 1.02. Moreover, if the aspect ratios obtained by the image analysis are greater than 1.02, the oscillation data are discarded. 9.3.3 Experimental Uncertainties The experimental uncertainty for density measurements is derived from the respective uncertainty measurements for the mass and volume of samples. Because the uncertainty in mass is 0.1 mg, while a typical sample mass is 20 mg, the uncertainty can be estimated to be around 0.5%. The uncertainty of volume (ΔV /V ) can be calculated by 3Δr0 ΔV = , V r0

(9.11)

where Δr0 is the uncertainty in radius measurement by the image analysis. In our experiment, the average value of Δr0 is around 1 pixel, while r0 is around 160 pixels. Therefore, ΔV /V can be estimated to be around 1.9%, and the overall uncertainty of density measurement (Δρ/ρ) is estimated to be around 2%. According to (9.2), the uncertainty in surface tension measurement is mainly determined by those of ρ, r0 , and ωc . As described earlier, the uncertainties of ρ and r0 are 2 and 0.65%, respectively. The uncertainty of ωc induced by the FFT analysis is negligibly small (0.4%) and evaluated by considering the transformation error (less than 1 Hz) and the typical characteristic oscillation frequency (around 200 Hz). As a result, the uncertainty of surface tension measurements (Δγ/γ) can be estimated to be around 3% by the following equation: 

2

2 2

Δρ 3Δr0 Δωc Δγ ≈ + + . (9.12) γ ρ r0 ωc The uncertainty in viscosity measurements can be estimated by the uncertainties of ρ, r0 , and τ . The uncertainty of the decay time Δτ is estimated to be about 15%, which is mainly due to the sample motion with respect to the detector during drop oscillation. This determines the overall uncertainty of viscosity.

9.4 Results of Thermophysical Property Measurements of Refractory Metals 9.4.1 Density The density data for each metal are listed in Table 9.1 with literature values [2,6,10,14,17,20–60]. During these experiments, the density was measured

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Table 9.1. Density of refractory metals Metal Tm (K) Ti 1943

V 2183

ρ(Tm ) (103 kg m−3 )

dρ/dT (kg m−3 K−1 )

Temperature (K)

4.17 4.11 4.15 4.11 4.1 4.13 4.155 4.13 4.208 3.8

−0.22

1,680–2,060 1,943 1,943 1,943 1,943 1,943 1,943 1,993–2,373 1,650–2,000 1,943

Present work Eljutin [20] Allen [21] Maurakh [22] Eljutin [23] Ivaschenko [24] Seydel [25] Saito [26] Paradis [27] Peterson [28]

1,840–2,240 2,183 2,208 2,200–2,470 2,183 2,175–6,600

Present work [17] Allen [21] Maurakh [22] Saito [26] Eljutin [29] Seydel [25]

5,460 5,550 5734 6060 5300 5565

−0.226 −0.508 −0.49 −0.32

Reference

Ni 1728

7.89 7.89 7.81 7.81 7.78 7.77 7.92

−0.65 −0.67 −1.08 −0.87 −1.0 −1.42 −1.01

1,420–1,850 1,403–1,838 2,163–2,423

Present work [30] Chung [10] Saito [26] Saito and Sakuma [31] Saito and Sakuma [31] Allen [32] Loh¨ ofer [2]

Y 1796

4.15 4.15

−0.21

1,560–2,100

Present work Fogel [33]

Zr 2128

6.21 5.80 5.60 6.06 6.24 5.50

−0.27

1,850–2,750 2,128 2,128 2,108 1,700–2,300 2,125

Present work [14] Allen [21] Eljutin [29] Maurakh [22] Paradis [6] Peterson [28]

2,300–3,000 2,742 2,742 2,742 2,742

Present work [14] Allen [21] Ivaschenko [34] Eljutin [29] Shaner [35]

2,450–3,000 2,896 2,896 2,896 2,896-

Present work [36] Allen [21] Eljutin [29] Pekarev [37] Seydel [25]

Nb 2742

Mo 2896

−0.29

7.73 7.83 7.57 7.6 7.68

−0.39

9.11 9.35 9.10 9.33 9.10

−0.60

−0.54

−0.80

(continued)

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183

Table 9.1. (continued) Metal Tm (K)

ρ(Tm ) (103 kg m−3 )

dρ/dT (kg m−3 K−1 )

Temperature (K)

Reference

Ru 2607

10.75 10.9

−0.56

2,225–2,775 2,607

Present work [38] Allen [21]

Rh 2236

10.82 11.1 10.65 10.7 10.7 12.2

−0.76

−0.50

1,820–2,250 2,236 2,236 2,236–2,473 2,236 2,236–2,473

Present work [39] Allen [21] Eremenko [40] Mitko [37] Popel [41] Dubinin [37]

−0.77 −1.226

1,640–1,875 1,828–2,073

−1.169

1,828–1,973

Present work [42] Lucas [43] Eremenko [40] Allen [21] Vatolin [44] Martsenyuk [45]

Pd 1828

10.66 10.49 10.7 10.7 10.379 10.52

−0.90

Hf 2504

11.82 12.0 11.1 11.97 11.5

−0.55

2,300–2,700 2,236 2,504 2,504 2,504

Present work [14] Allen [21] Peterson [28] Ivaschenko [24] Arkhikin [46]

Ta 3,290

14.75 15.0 14.43 14.6

−0.85

2,650–3,420 3,290

Present work [47] Allen [21] Shaner [35] Berhault [48]

16.43 17.5 16.37 16.26 16.2 17.6

−1.08

Re 3459

18.65 18.7 18.0 18.9

Os 3306 Ir 2,719

W 3695

−1.3 3,290 3,125–3,707 3,693 3,693–8,000 3,693 3,693–5,340 3,693

Present work Allen [21] Seydel [25] Shaner [35] Berhault [48] Calverley [50]

−0.79

2,683–3,710 3,459 3,459 3,459

Present work Allen [21] Thevenin [51] Pekarev [37]

19.1 20.1 19.2

−1.16

2,670–3380 3,306 3,306

Present work [52] Allen [21] Vinet [53]

19.5 20.0 19.39 19.23 20.0

−0.85

2,300–3,000 2,719 2,719 2,723 2,719

Present work [54] Allen [21] Martsenyuk [55] Apollova [56] Gathers [57]

−0.97

(continued)

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T. Ishikawa and P.-F. Paradis

Table 9.1. (continued) Metal Tm (K)

ρ(Tm ) (103 kg m−3 )

dρ/dT (kg m−3 K−1 )

Temperature (K)

Reference

19.22 19.7 18.82 18.91 18.81 19.77 19.3 19.1

−0.96 −2.882

1,691–2,216 2,041 2,073 2,041–2,148

−2.4 −1.7 −1.3

2,041–2,473 2,041–5,100 2,095–4,500

Present work [58] Eremenko [37] Kozakevitch [37] Been [37] Martsenyuk [37] Dubinin [37] Hixson [59] Gathers [60]

Pt 2041

Density (kgm−3)

15500

15000

14500

Present work Allen Shaner Berhault

Tm

14000 2600

2800

3000

3200

3400

3600

Temperature (K) Fig. 9.6. Density of tantalum as a function of temperature

over large temperature ranges including regions above and below the melting temperature. The density, like that of alkaline metals, exhibited a linear behavior as a function of temperature. As an example, Fig. 9.6 shows the density of tantalum as a function of temperature, including literature values. Since our measurement covers a wide temperature, the temperature coefficient of density can be determined with higher accuracy. A simple relationship between the temperature dependence of the density of liquid metals and their boiling temperatures (Tb ) was proposed by Steinberg [61]. He collected liquid density data at the melting point and the temperature dependence of liquid density for 44 elements and found the following empirical relations

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185

−dρ/dT(10−4 kgm−3 K−1)

15 Pu Pb Ni Co

10

Cs Rb V

5

Nd

Re

Tl Cd

Se

Bi

Au

U Ag

Zn

Pt

Ir In

Pd

Te

Presentwork Steinberg

Ni Ru

Ge Ga Hf Zr Nb Ba Sr Pr Ce Y Mg La Li Be Ti

Al Si K

Na Ca 0

Mn Ta Fe Cu Rh Sn Cr Mo

Pd Os W

0

2

4

ρ00 /Tb

6

8

10

(kgm−3K−1)

Fig. 9.7. Correlation of dρ/dT with ρ00 /Tb for elements. Pt points by Ref. [61] falls outside of the range of figure (its ordinate is 28.8 and its abscissas is 6.05). The solid line is the best fit to data and the dashed lines represent the 20% error cone from Ref. [61]

−

ρ00 dρ ∝ , dT Tb

(9.13)

where ρ00 was the virtual density of the liquid at 0 K as determined by extrapolation from ρm and Tm with: dρ Tm . (9.14) dT Figure 9.7 illustrates the correlation of −dρ/dT with ρ00 /Tb . In his study, most of the metal elements followed the correlation, except for mercury and platinum. In Fig. 9.7, our measured data of refractory metals including that of platinum were also plotted. Our platinum data (−0.96 kg m−3 K−1 ), as well as those for other refractory metals, shows a good agreement with the Steinberg’s relation. ρ00 = ρm −

9.4.2 Surface Tension The measured values and literature data for surface tension and viscosity are listed in Table 9.2 [6, 20, 22, 28, 30, 32, 33, 37, 62–84]. Figure 9.8 shows

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T. Ishikawa and P.-F. Paradis

Table 9.2. Surface tension of refractory metals Metal Tm (K)

γ(Tm ) (10−3 N m−1 )

dγ/dT (10−3 N m−1 K1 )

Temperature (K)

Reference

1,557 1,510 1,650 1,427 1,390 1460 1,576 1,588

−0.16

1,750–2,050 1,943 1,943 1,973–2,323 1,943 1943 1,953 1,953

Present work [62] Eljutin [20] Alen [32] Maurakh [22] Peterson [28] Namba and Isobe [63] Tille and Kelly [64] Tille and Kelly [64]

1,739 1,809 1,758 1,778

−0.22 −0.39 −0.333 −0.38

1,553–1,963

Present work [30] Popel [37] Mitko [37] Allen [32]

Y 1796

804 872 610

−0.05 −0.09

1,830–2,070 1,796–2,023

Present work Sukhman [65] Fogel [33]

Zr 2128

1,500 1,459 1,512 1,480 1,400 1,411 1,430

−0.11 −0.24 −0.37

1,800–2,400 1,850–2,200 2,128 2,128 2,128 2,128

Present work [62] Paradis [6] Egry [66] Allen [32] Peterson [28] Shunk [67] Kostikov [68]

Nb 2742

1,937 1,900 1,827 1,839 2,040 1,853

−0.20

2,320–2,915 2,742 2,742 2,742 2,742 2,472

Present work [62] Allen [32] Flint [69] Ivaschenko [70] Arkhipkin [71] Eremenko [72]

Mo 2896

2,290 2,250 2,049 1,915 2,080 2,130 2,068

−0.26

2,650–3,000 2,896 2,896 2,896 2,896 2,896 2,896

Present work [73] Allen [32] Flint [69] Pekarev [74] Namba [37] Eljutin [37] Man [75]

Ru 2607

2,256 2,250 2,180

−0.24

2,450–2,725 2,607 2,607

Present work [38] Allen [32] Martensyuk [76]

Rh 2236

1,940 2,000

−0.30

1,860–2,380 2,236

Present work [39] Allen [32]

Ti 1943

Ni 1728

−1.075

1,728–2,473

(continued)

9 Thermophysical Property of Metals Using Electrostatic Levitator

187

Table 9.2. (continued) Metal Tm (K)

γ(Tm ) (10−3 N m−1 )

dγ/dT (10−3 N m−1 K1 )

Temperature (K)

Reference

1,940 1,915

−0.664

2,236 2,236–2,473

Eremonko [40] Gushchin [77]

Hf 2504

1,614 1,630 1,460 1,490

−0.10

2,220–2,670 2,504 2,504 2,504

Present work [14] Allen [32] Peterson [28] Kostikov [68]

Ta 3290

2,154 2,150 1,910 2,016 2,360 2,030

−0.21

3,143–3,393 3,290 3,290 3,290 3,290 3,290

Present work [78] Allen [32] Namba [63] Eremenko [72] Kelly [79] Kelly [79]

W 3695

2,477 2,300 2,500 2,200 2,316 2,300

−0.31

3,398–3,693 3,693 3,693 3,693 3,693 3,693

Present work [49] Calverley [50] Allen [32] Pekarev [74] Martsenyuk [80] Agaev [81]

Re 3459

2,710 2,700 2,610

−0.23

2,903–3,583 3,459 3,459

Present work [82] Allen [32] Pekarev [74]

Os 3306

2,480 2,500 2,400

−0.34

3,230–3605 3,306 3,306

Present work [52] Allen [32] Vinet [53]

Ir 2,719

2,230 2250 2,264 2,140

−0.17

2,373–2,833 2,720 2,720 2720

Present work [54] Allen [32] Apollova [83] Martensyuk [84]

1,800 1,869 1,673 1,740 1,699 1,800 1,746 1,865 1,707

−0.14

1,743–2,313 2,041 2,053 2,041 2,073 2,041 2,041–2,148 2,041 2,043

Present work [58] Quincke [37] Quincke [37] Eremenko [37] Kozakevitch [37] Allen [32] Dubinin [37] Kingery [65] Martsenyuk [84]

Pt 2041

−0.247

−0.307

188

T. Ishikawa and P.-F. Paradis

Surface tension (10−3 Nm−1)

1650

1600

Present work Allen Peterson Kostikov

1550

Tm

1500

1450 2200

2300

2400

2500

2600

2700

Temperature (K) Fig. 9.8. Surface tension of hafnium as a function of temperature

the surface tension of hafnium. The surface tension could be measured over large temperature ranges including the undercooled phase in these experiments, whereas measurements by other methods could be done only around the melting temperatures (Tm ). These tendency become clearer for higher Tm samples. Kasama et al. proposed an empirical equation on a physical model [85]. Based on this model, surface tension and its temperature dependence can be expressed as 2

%2 1 1 ρ 3$ 1 π 2 C 2 δ 2 Tm 3 (α + 1)ρ 3 − ρm , (9.15) γ= 2 2 NA M 3 ρm % 2 1 1 1 π 2 C 2 Tm Λδ 2 $ dγ −1 3 =− + ρ− 3 − 3(1 + α)ρm3 , 2(α + 1)2 ρ 3 ρm 2 dT 3 NA M 3 where NA is the Avogadro constant, M is the atomic number, and Λ is the temperature dependence of density (-dρ/dT ). In addition, C is a constant derived from Lindemann’s theory of melting (ranged from 2.8 × 1012 to 3.1 × 1012 ), δ is a ratio between the characteristic vibration frequency in the liquid phase and solid phase (estimated to be around 0.5), and α is a constant related to the distance, where an atomic attractive force is effective (ranged from 0.45 to 0.65). At the melting temperature, the temperature coefficient of the surface tension can be calculated by

9 Thermophysical Property of Metals Using Electrostatic Levitator

189

γmβm (10−7N/mK) 0

500

1000

1500

2000

2500

3000

0 Rb

Y

Cs

−0.1

Ba

Bi

dγ/dT (10−3 N/mK)

Hf

Sr

K

Na In

−0.2

Ga Nd Ca

Sb Sn Tl

La

Zr Pb Ti Ir Hg Cd

−0.3

−0.4

Li Pt Ag Zn Re Nb Ta Ru

Au

Ni

Pd

Pt

Cu Mn

Be Mg

Ce

Present work Allen Lucas Itami

Al

Rh

W Os Ni Cr Fe

Co

−0.5

−0.6

Fig. 9.9. Correlation between λm βm and dγ/dT . Open circles, open squares, and open diamonds represent data from [32], [86], [87], respectively. The solid line is the best fit to data

2 Λγm 2α + 1 Λγm dγ =− =K = Kγm βm , (9.16) dT 3 ρm α ρm −2(2α + 1) , K≡ 3α where γm is the surface tension at the melting temperature. This equation suggests that the temperature dependence of the surface tension is proportional to the product of the surface tension at the melting temperature by the thermal expansion coefficient. Validity of this formula for liquid metals is checked by using literature data [32, 86, 87] and our measurements. Results are shown in Fig. 9.9. Literature data of alkaline metals showed good agreements with (9.16), while those of transition metals exhibited scattering, some of them being far from the relation. However, our measured data show the same tendency as the alkaline metals. In particular, platinum and nickel data follow the relation, while those appearing in the literature do not. Temperature dependence of the surface tension is an important parameter to evaluate the magnitude of the thermo-capillary flow when the floating zone method or Czochralski method are used for single crystal growth of

190

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semiconductors. Based on our results and alkaline data, the temperature dependence of the surface tension of metals can be estimated if γm and the thermal expansion coefficients (βm ) are known. 9.4.3 Viscosity Table 9.3 shows the viscosity data for refractory metals [88–95], and Fig. 9.10 illustrates the viscosity of tungsten versus temperature. To our knowledge, the viscosity data of several elements (Y, Nb, Mo, Ru, Ta, W, Re, Os, and Ir) were the first to be reported. The viscosity measurement is a very unique capability of the electrostatic levitation among a variety of levitation methods. Table 9.3. Viscosity of refractory metals Metal η(Tm ) η(T ) = η0 exp(E/RT ) Temperature (K) Tm (10−3 Pa s) η0 E (K) (10−3 Pa s) (103 J mol−1 )

Reference

Ti 1943

4.4 2.2 5.2

0.033 (0.0034)

76.6 68

1,750–2,050

Present work Agaev [88] Eljutin [89]

Ni 1728

7.35 4.61 4.90 4.7

0.071 0.490 0.1663 0.265

66.6 32.2 50.2 41.3

1,553–1,963 1,728–2,023

Present work [30] Lucas [90] Brandes [88] Sato [91]

Y 1796

3.6

0.00287

106.5

1,830–2,070

Present work

Zr 2128

4.7 4.83 3.5 5.45

0.76

31.8

1,800–2,300 1,850–2,200 2,133 2,138

Present work [14] Paradis [6] Agaev [92] Elyutin [93]

Nb 2742

4.5

0.55

48.9

2,320–2,915

Present work [14]

Mo 2896

5.6

0.27

73

2,650–3,000

Present work [73]

Ru 2607

6.1

0.60

49.8

2,450–2,725

Present work [38]

Rh 2236

2.9 5

0.09

64.3

1,860–2,380 2,236

Present work [39] Demidovich [94]

Hf 2504

5.2 5.0

0.50

48.7

2,220–2,670 2,504

Present work [14] Agaev [88]

Ta 3290

8.6

0.004

213

3,143–3,393

Present work [78] (continued)

9 Thermophysical Property of Metals Using Electrostatic Levitator

191

Table 9.3. (continued) Metal η(Tm ) η(T ) = η0 exp(E/RT ) Temperature (K) Tm (10−3 Pa s) η0 E (K) (10−3 Pa s) (103 J mol−1 )

Reference

W 3695

6.9

0.11

128

3,398–3,693

Present work [95]

Re 3459

7.9

0.08

133

2,903–3,583

Present work [82]

Os 3306

4.2

0.00167

220

3,230–3,605

Present work [52]

Ir 2719

7.0

1.85

30.0

2,373–2,773

Present work [54]

Pt 2041

4.82 6.74

0.25 1.53

49.9 25.263

1,743–2,313 2,041–2,273

Present work [58] Zhuchenko [86]

Viscosity (10−3 Pa.s)

15

10

5

Tm

0 3300

3400

3500

3600

3700

3800

Temperature (K) Fig. 9.10. Viscosity of tungsten as a function of temperature

The relatively large uncertainty is mainly derived from the large uncertainty in τ . Moreover, the drop oscillation method assume that no external force is applied to the sample. In the case of our experiment, a strong electric field is applied and a feedback position control is used, which might affect the measurements.

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The effect of the feedback control is systematically being checked and improvement of the viscosity measurement technique is under way.

9.5 Summary Thermophysical properties of several refractory metals over wide temperature ranges in the undercooled as well as in the superheated states could be measured using the unique capabilities of the electrostatic levitation furnace. This method can measure the temperature dependence of surface tension and viscosity for refractory metals, which can not be measured by conventional methods. Another advantage offered by this technique is its ability to determine the thermophysical properties at the melting temperature without any risk of having a half-molten or solidified sample.

Acknowledgment Authors would like to express their gratitude to Dr. J. Yu, Dr. T. Aoyama, Ms. A. Ishikura, Mr. R. Ishikawa, Mr. R. Fujii, and Mr. N. Koike for help in some experiments. This work is partially supported by a Grant-in-Aid for Science Research (B) from the Japan Society for the Promotion of Science.

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53. B. Vinet, L. Magnusson, H. Fredriksson, J.P. Desre, J. Colloid Interface Sci. 255, 363 (2002) 54. T. Ishikawa, P.-F. Paradis, S. Yoda, Int. J. Thermophys. 26, 893 (2005) 55. P.S. Martensyuk, Yu.N.I. Ivaschtschenko, Ukr. Chim. Sh. 40, 431 (1974) 56. T.A. Apollova, E.L. Dubinin, M.M. Mitko, A.I. Chegodajev, L.L. Besuklandnikova, Izv. A.N. SSSR Met. 6, 55 (1982) 57. G.R. Gathers, J.W. Shaner, R.S. Hixson, D.A. Jung, High Temp. High Press. 11, 653 (1979) 58. T. Ishikawa, P.-F. Paradis, N. Koike, Jpn. J. Appl. Phys. 45, 1719 (2006) 59. R.S. Hixson, M.A. Winkler, Int. J. Thermophys. 14, 409 (1993) 60. G.R. Gathers, J.W. Shaner, W.M. Hodgson, High Temp. High Press. 11, 529 (1979) 61. D.J. Steinberg, Metall. Trans. 5, 1341 (1974) 62. P.-F. Paradis, T. Ishikawa, S. Yoda, Int. J. Thermophys. 23, 825 (2002) 63. S. Namba, T. Isobe, Sci. Papers I. P. C. R. 57, 51 (1963) 64. J. Tille, J.C. Kelly, J. Appl. Phys. 14, 717 (1963) 65. B.J. Keene, Int. Mater. Rev. 38, 157 (1993) 66. I. Egry, A. Diefenbach, W. Dreier, J. Piller, Int. J. Thermophys. 22, 569 (2001) 67. R. Shunk, M. Burr, Trans. AIME 55, 786 (1962) 68. V.I. Kostikov, B.D. Grigorjev, P.G. Arkhipkin, A.D. Agaev, Izv. Vyss. Uch. Sav. Chern. Met. 3, 25 (1972) 69. O. Flint, J. Nucl. Mater. 16, 260 (1965) 70. Yu.N. Ivaschenko, P.C. Marchenuk, Teplov. Vys. Temp. 11, 1285 (1973) 71. V.I. Arkhipkin, A.D. Agaev, G.A. Grigorev, V.I. Kostikov, Ind. Lab. (USSR) 39, 1340 (1973) 72. V.N. Eremenko, High Temp. 22, 705 (1984) 73. P.-F. Paradis, T. Ishikawa, N. Koike, Int. J. Refract. Met. Hard Mater. 25, 95 (2007) 74. V. Pekarev, Izv. Vyss. Uch. Sav. Tsvetn. Met. 6, 111 (1963) 75. K.F. Man, Int. J. Thermophys. 21, 793 (2000) 76. P.S. Martsenyuk, Yu.N. Ivashchenko, Adgez. Rasp. Paika Mater. 20, 15 (1988) 77. S.G. Gushchin, N.A. Vatolin, E.L. Dubinin, A.I. Timofeev, Ogneupory 12, 49 (1977) 78. P.-F. Paradis, T. Ishikawa, S. Yoda, J. Appl. Phys. 97, 053506 (2005) 79. J.C. Kelley, A. Calverley, SERL-Rep. 80, 53 (1959) 80. P.S. Martsenyuk, Yu.N. Ivashchenko, V.N. Eremenko, Tep. Vys. Temp. 12, 1310 (1974) 81. A.D. Agaev, Dissertation, Moscow Steel and Alloys Institute, 1973 82. T. Ishikawa, P.-F. Paradis, S. Yoda, Appl. Phys. Lett. 85, 5866 (2004) 83. T.A. Apollova, E.L. Dubinin, M.M. Mitko, A.I. Tshegodajev, L.L. Besukladnikova, Izv. A. N. Met. 6, 55 (1982) 84. P.S. Martsenyuk, Inst. Tekh. Probl. Mater. Akad. Nauk SSSR, 51 (1980) 85. A. Kasama, T. Iida, Z. Morita, J. Jpn. Inst. Met. 40, 1030 (1976) 86. T. Itami, in Condensed Matter-Liquid Transition Metals and Alloys Condensed Matter Disordered Solids, ed. by S.K. Srivastava, N.H. March (World Scientific, Singapore, 1995), pp. 123–250 87. L.D. Lucas, Tech l’Ing. 7, Form. M67 (1984) 88. L. Battezzatti, L. Greer, Acta. Metall. 37, 1791 (1989) 89. E.A. Brandes, G.B. Brook, in Smithells Metals Reference Book, 7th edn. (Butterworth-Heinemann, Burlington, 1992), pp. 14–17

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Index

γ-ray transmission method, 28 absorption, 2, 9, 98, 107 absorption coefficient, 8, 9, 96, 98–100 absorptivity, 114, 153 acoustic levitation, 134 activation energy, 26 adiabatic condition, 100 adsorption, 39, 40, 43, 56 aerodynamic levitation, 2, 134 Ag, 56, 79, 117, 120–122 Ag–Au, 56 Ag–Cu, 56, 117, 125, 126 aircraft, 133 Al, 73, 115 Al2 O3 , 3, 85, 95 Al65 Cu25 Co10 , 3 Al–Cu, 3 Al–Ni, 3 AlCl3 , 18 alkaline halides, 19 alkaline metals, 184, 189 alloys, 173 AlNi, 73 AlNi1.5 –Al, 75 AlNi–Al, 72 alumina, 1, 2, 8, 10, 14, 93, 177 aluminum smelting, 17 Andrade law, 17 anomalous X-ray diffraction, 3 Ar, 45 Archimedean method, 28, 29 Arrhenian behavior, 26

Arrhenius, 17, 144 As, 85 atomic attractive force, 188 atomic configuration, 1, 4 atomic form factor, 9 atomic scattering factor, 4 Au, 117, 120–122 Au–Cu, 142 Au-Cu-Co, 3 AuCu, 3 Avogadro constant, 188 B, 3 Ba–Ge, 3 BaB2 O3 , 3 Belton equation, 56 Benard–Marangoni–Rayleigh mechanism, 41 Benard–Rayleigh cell, 41 Bi, 75, 79 Biot number, 101, 140, 166 blackbody, 96, 111, 112, 116, 119 blackbody radiation, 116 boiling temperature, 184 Boltzmann constant, 12 boron nitride, 1, 31 boron oxide, 94, 107 brightness temperature, 112 buoyancy, 29, 39, 44, 132, 134, 149 buoyancy convection, 68, 74, 76, 81, 158 buoyancy force, 61, 150, 170 CaAl2 O4 , 3 CaF2 , 95

198

Index

CaO, 85 CaO/SiO2 ratio, 96 capillarity, 134 capillary method, 18 capillary rise method, 47 capillary viscometer, 19 casting, 39, 56, 149 Ce–Cu, 124, 125 centrifugal force, 132 ceramics, 1, 2, 10 characteristic oscillation frequency, 179, 180 chemical thermodynamics, 53 Co, 93, 100, 103, 107, 117, 122 CO2 laser, 5, 175 Co80 Pd20 , 3, 138 Co–Pd, 3 coherent intensity, 9 cold crucible, 46, 117, 121 Compton scattering, 9 computer simulations, 2 conduction, 111, 157 CoNi, 3 conical nozzle levitation, 3, 11 contact angle, 30 containerless, 2, 11, 34, 39, 47, 132, 134, 145, 173, 177 continuous casting, 85 continuous casting powder, 95, 100, 107 convection, 87, 111, 149, 150, 158, 163, 169, 170 convective flow, 164 cooling curve, 120, 163, 165, 167, 177 coordination number, 13 CoPd, 3 correction function, 153, 167 Coulomb force, 173, 174 critical cooling rate, 105 critical Marangoni number, 41 crystal growth, 39, 41, 56, 149 Cu, 117, 120, 121, 125 Cu2 Ce, 125 Cu–Ni, 115, 125 CuCo, 133 Cummings and Blackburn, 47, 142 Curie temperature, 138 CuZr, 3 Czochralski, 41 Czochralski method, 85, 189

decay time, 180 dendrite, 46 density, 17, 28, 36, 86, 111, 135, 137, 139, 154, 155, 157, 161, 173, 177, 180–182 density layer, 70 density layering, 62, 63, 72, 80, 81 desorption, 54 diffraction, 1 diffusion coefficient, 61, 63, 68, 71, 73, 75–77, 79, 82, 86 dilatometric method, 28 direct correlation function, 12 directional spectral emissivity, 112 dislocation, 85 disordered matter, 1 drop calorimetry method, 166 drop oscillation method, 178, 191 drop tower, 77, 133 drop tube, 133 drop weight method, 47 dynamic viscosity, 40 eddy current, 2, 117, 158 electric conductivity, 135, 137, 138, 161 electric resistivity, 138, 163 electrical resistivity, 173 electromagnetic field, 137, 158, 159 electromagnetic force, 2, 150, 158, 163, 170 electromagnetic induction heating, 154 electromagnetic levitation, 2, 3, 47, 54, 57, 115, 117, 120, 132, 134–136, 142, 145, 158, 169, 173 electromagnetic levitator, 150, 159, 170 electron beam button melting (EBBM), 46 electron beam melting, 39 electronic emission, 8 electrostatic force, 2 electrostatic levitation, 2, 3, 7, 8, 11, 35, 47, 120, 134, 173, 174, 177, 190, 192 electrostatic levitator, 4, 5, 14, 139 ellipsometer, 122 ellipsometry, 114, 115, 119, 120 EMCZ (electromagnetic Czochralski), 44 emission, 98, 107

Index emissivity, 8, 101, 111, 150, 154, 155, 157, 159, 161–163, 167, 170 EML, 3 energy dispersive X-ray diffraction, 3 equilibrium P o2 , 52 ESA, 68 ESL, 3 EXAFS, 3 extended X-ray absorption fine structure, 3 extinction coefficient, 114 Fe, 3, 54, 85, 93, 100, 103, 107, 115, 117, 122, 123 Fe–Cr, 56 Fe–Nb, 124 Fe–Ni, 113 Fe–V, 124 fixed-point blackbody, 116 float method, 28 floating zone method, 189 fluid dynamics, 41 fluorides, 19 Foton, 63, 66 Foton-M2, 62, 77, 82 foundry, 17 Fourier transform, 13 Fourier transformation, 4 Fourier’s law, 111 Fourier–Biot equation, 86 free electron, 105 free electron transition, 121 free surface, 41 Fresnel’s equation, 114 fused silica, 1 g-jitter, 77, 132, 144 Ga, 79 GaAs, 85 GaP, 85 Ge, 104, 107, 117, 173 germanium detector, 4, 9 Gibbs adsorption isotherm, 50 glass, 17 glass forming alloy, 2, 105 glass making, 17 glass transition, 139, 144 glass-forming metallic alloys, 139 graphite, 1, 31

199

graphite monochrometor, 9 gravitational acceleration, 47 gravity, 132, 134 gray body, 98–100 growth striation, 39, 41 Hagen–Poiseuille’s equation, 18 He–Ne laser, 7, 174 heat and mass transport, 39, 57 heat balance, 152, 163 heat capacity, 111, 120, 149, 150, 153, 157, 167, 173 heat transfer, 85, 92, 111, 157 hemispherical emissive power, 96 hemispherical spectral emissivity, 112 hemispherical total emissivity, 112, 118, 120, 141, 149, 152, 153, 165, 167, 170 heterogeneous nucleation, 2, 173 Hf, 183, 187, 188, 191 Hg, 185 high-energy synchrotron radiation X-ray, 2 high-energy X-ray, 1 high-energy X-ray diffraction, 14 high-temperature liquid, 14, 149 high-temperature melts, 2, 5, 17 hot launch, 8 hydrostatic pressure, 33 IML-2, 134, 142 IMPRESS project, 141 impurity diffusion, 81 In, 79 In2 O3 , 94 IN718, 46 inductive heating, 136 inelastic X-ray diffraction, 3 InSb, 88 InSn–In, 72 inter band transition, 121 interdiffusion, 66, 67, 69, 75, 79, 81 International Space Station, ISS, 57, 64, 77, 132, 135, 145 Ir, 183, 187, 190, 191 isobaric heat capacity, 151, 165 isobaric mass heat capacity, 154 isobaric molar heat capacity, 166, 170

200

Index

JAMIC drop tower, 143, 145 Japan Aerospace Exploration Agency (JAXA), 64, 174 Japanese Experiment Module, JEM, 77 Jet Propulsion Laboratory (JPL), 173 jet-engine turbine blade, 149 Joule heat, 155 kinematic viscosity, 18 Kirchhoff’s law, 114, 153 Knappwost’s relationship, 21 laser flash method, 86–88, 100, 104, 107 laser modulation calorimetry, 150 laser-pulse method, 86 latent heat of fusion, 177 Legendre polynomial, 178 Levenberg–Marquardt method, 157 levitation, 2, 39, 47, 57, 134 levitation force, 135 levitation method, 28, 34 Lindemann’s theory of melting, 188 liquid metals, 1, 61, 79, 115, 137, 184, 189 liquid structural analysis, 3 liquid structure, 2, 5, 8, 11 liquidus temperature, 105 lithium niobate, 107 long capillary, 61, 62, 81 long-range periodic atomic distribution, 103 Lorentz force, 44, 134, 135, 149, 157 Lorenz number, 138 low-Pr-number fluids, 56 magnetic field, 44, 62, 76, 135 magnetic flux density, 44 magnetic liquid, 138 magnetic ordering, 138 magnetic permeability, 135, 138 magnetohydrodynamic (MHD) convection, 158, 159, 164 manometric method, 28, 32 Marangoni convection, 61, 65, 68, 134, 149, 158 Marangoni effect, 40 Marangoni flow, 39, 41, 44, 52, 53, 57 mass absorption coefficient, 9 mass transport, 40, 68, 82

mathematical model, 157, 170 maximum bubble pressure method, 28, 33, 47 maximum drop method, 47 melting temperature, 161 metallic glass, 104, 105, 173 metallic melt, 4, 100, 111, 149 metals, 10, 17 metastable, 132 Mg2 SiO4 , 3 MgAl2 O4 , 3 MgO, 85, 95 microgravity, 39, 41, 52, 61, 62, 76, 77, 132–134, 136, 141, 143, 145 microsecond pulse heating, 118, 119, 122 Mo, 119, 122, 123, 175, 182, 186, 190, 191 modulated laser calorimetry, 159, 164, 169 modulation amplitude, 152 modulation calorimetry, 141, 149, 150 modulation frequency, 153 molar heat capacity, 166 molecular dynamics (MD) simulation, 13, 14, 79 momentum transfer, 4 monochromator, 116 Monte Carlo (MC) simulation, 13 MSL-1, 134, 137–139, 141 MSL-EML, Materials Science Lab – Electro-Magnetic Levitator, 135, 145 mullite, 93 multiple scattering, 2, 9 Na2 O, 95 natural convection, 61 Navier–Stokes equation, 131, 142 Nb, 119, 182, 186, 190, 191 Nd glass laser, 88, 93 Nd–Fe–B, 3 Nd:YAG laser, 175, 176 near net shape casting, 131 neutron diffraction, 1, 4 neutron scattering, 2, 3, 9 Newton’s equations of motion, 132 Newtonian, 18

Index Ni, 3, 93, 100, 103, 107, 115, 117, 119, 120, 122, 123, 125, 126, 182, 186, 189, 191 Ni–Cr, 115, 124 Ni–Cu, 117 Ni–Fe, 124 Ni–V, 3 Ni-75%Zr, 126 nickel alloy, 119, 127 nickel based alloy, 125 NiZr, 3 noble metals, 117, 120 non-Newtonian, 18 noncontact, 139 noncontact modulated laser calorimetry, 164 noncontact modulation calorimetry, 139, 149, 150 nonferrous metallurgy, 17 nonlinear optical materials, 95 normal spectral emissivity, 112, 114–116, 118–120, 153 nuclear reactors, 4 nucleation, 47, 173, 177 numerical analysis, 176 numerical modeling, 57, 145 numerical simulation, 64, 149, 150, 158, 170 Nyquist theorem, 142 optical constants, 114 optical processes, 113 optical pyrometers, 112 Os, 175, 183, 187, 190, 191 oscillating cylinder method, 18 oscillating drop method, 47, 57, 141, 143 oscillating method, 18, 26 oscillating vessel method, 26 oscillating viscometer, 19, 21 oscillation, 22 oscillation frequency, 47, 48, 137 oxidation, 53 oxide melts, 94 oxides, 17 oxygen, 39, 40, 44, 45, 47, 53 oxygen partial pressure, 49 oxygen sensor, 53 oxygen transport, 53

201

P, 85 P2 O5 , 94 packing fraction, 13 parabolic flight, 64, 66, 68, 77, 135, 137, 139, 143, 145 Pb, 73, 79 PbAg5 , 73 PbAg–Pb, 72, 76 PbGa5 , 73 PbGa–Pb, 72 Pd, 119, 183 Pd–Cu–Si, 104 Pd–Ni–P, 104 Pd-based alloy, 104, 105 PdCuSi, 3, 143 pendant drop method, 47 permittivity of vacuum, 179 phase difference, 48, 152–154, 157, 159, 166, 167, 170 physical properties, 161 Planck’s law of radiation, 112, 116 polarization, 9, 114 proportional counter, 4 Pt, 88, 177, 184, 185, 187, 189, 191 Pt–Ni–P, 104 pulse heating, 119 pycnometer, 31 pycnometric method, 28, 31 radial distribution function, 4 radiance, 112 radiation, 87, 97, 111, 116, 153, 155, 163, 164 radiation intensity, 112 radiative heat transfer, 96, 98 Ratto’s model, 53 Rayleigh, 47 Rayleigh–B´enard convection, 134 Rayleigh–B´enard instability, 134 Rayleigh–Marangoni–Benard cell, 44 Re, 175, 183, 187, 190, 191 real body, 111, 119 recalescence, 8, 177 reflection, 114 reflectivity, 99, 114 refractive index, 114 refractory metal, 25, 46, 173, 177, 180, 182, 185, 186, 190–192 relaxation time, 153

202

Index

Reverse Monte Carlo (RMC) simulation, 13 Reynolds number, 18 Rh, 183, 187, 191 Roscoe’s equation, 21 rotating magnetic field, 176, 177 rotating method, 18, 26 rotating viscometer, 28 rotation, 48 Ru, 182, 186, 190, 191 ruby laser, 93 salts, 17 sapphire, 1, 93, 101 sapphire window, 5 satellite, 62, 77 Sb, 79 scattering angle, 4 scattering intensity, 4 segregation, 63 self-diffusion, 13, 79, 81 semiconductor, 10, 56, 85, 149, 173 semiconductor laser, 150, 163 sensitivity, 170 sensitivity analysis, 159, 163 sessile drop method, 28, 34, 47 shear cell, 62, 64–68, 72, 80, 82 shear convection, 64, 65, 68, 71, 72, 82 shear stress, 40 Si, 2, 3, 8, 10, 13, 14, 39–41, 49, 56, 104, 107, 116–118, 120, 124, 125, 166–168, 170, 173 Si75 Ge25 , 139 Si–Ge, 139 silicon, 149, 150, 153, 157–159, 161, 163 single crystal, 39, 85, 95 single crystalline silicon, 163 single-color pyrometer, 5, 176 SiO, 41 SiO2 , 3, 41, 50, 85, 95 skin depth, 135, 137, 138 slag, 17 slits collimation, 4 Sn, 65, 69, 73, 79 Sn47 In53 , 69 Sn90 In10 , 69 Sn-In, 79 SnBi3 , 71, 73 SnBi3 –Sn, 74

SnBi1.5 , 73 SnBi1.5 –Sn, 75 SnBi–Sn, 72 SnBiIn–Sn, 72 SnIn1 –Sn, 75 SnIn3 Bi5 , 73 SnIn10 , 73 SnIn10 –Sn, 76 SnIn–Sn, 72 SnSb3 , 73 SnSb5 –Sn, 78 SnSb–Sn, 72 solidification, 1, 4, 8, 63, 131, 164 solubility, 55 solutocapillary, 40 sounding rocket, 39, 77, 133, 135, 137, 141, 145 Soyuz-U rocket, 77 space shuttle, 57, 61, 77, 80 Spacelab mission, 134, 137, 138, 141, 142 specific heat, 86, 139, 141, 154, 155, 161 spectral emissivity, 173 SPring-8, 4, 7, 8 stainless steel, 45, 54 standard deviation, 166, 168, 169 static magnetic field, 49, 149, 150, 158, 159, 163, 168–170 static structure factor, 4, 9, 11 steel, 39, 95 steelmaking, 17, 85 Stefan–Boltzmann constant, 101, 120, 152 Steinberg’s relation, 185 structural analysis, 2 STS-83, 144 STS-94, 144 sulfur, 39, 44–46, 53 superconducting magnet, 150, 163 supercooled, 118 surface free energy, 1 surface oscillation, 142, 149, 178 surface tension, 30, 33, 34, 39–41, 43, 46, 49, 54, 56, 65, 134, 137, 141–143, 145, 170, 173, 177–181, 185, 186, 188, 189, 192 surfactant, 47, 53 SUS304, 54 synchrotron radiation, 1, 4, 7, 9, 11, 14

Index Ta, 119, 175, 183, 187, 190, 191 tantalum, 184 telephoto camera, 177 temperature coefficient, 45 temperature coefficient of density, 184 temperature coefficient of surface tension, 39, 161 TEMPUS, 134, 139, 145 TEXUS sounding rocket, 133 thermal conductance, 153 thermal conductivity, 85, 86, 101, 111, 149, 150, 154, 155, 157, 159, 161, 162, 168, 170, 173 thermal contact resistance, 101 thermal diffusivity, 85–87, 91, 93, 94, 97–100, 103, 104, 107 thermal expansion, 139 thermal expansion coefficient, 161, 189, 190 thermionic emission, 8 thermocapillary, 40, 158 thermocapillary force, 150, 170 thermocouple, 113 ThermoLab project, 143 thermophysical property, 131, 136, 145, 173, 176 Ti, 3, 46, 122, 173, 182, 186, 191 Ti39.5 Zr39.5 Ti21 , 3 Ti46 Al8 Nb, 141 Ti–Al, 124 Ti–Fe–Si–O, 3 Ti–V, 124 TIG (tungsten inert gas), 44 titanium alloy, 119 TiZrNi, 3 transition metals, 117, 122, 189 transmissivity, 99, 114, 117 turbine blade, 56 two-axis diffractometer, 4 two-colour pyrometer, 117

203

uncertainty, 47, 57, 166–168, 181, 191 undercooled liquid, 1, 2, 8 undercooled melt, 132 UV imaging technique, 177 UV lamp, 177 V, 182 vapor pressure, 18 viscometer, 18, 19, 25 viscosity, 13, 17, 19, 22, 36, 40, 111, 136, 141–143, 161, 173, 178, 180, 181, 185, 190–192 Vogel–Fulcher behavior, 144 W, 119, 122, 124, 175, 183, 187, 190, 191 weld pool, 39, 44, 45 weldability, 39, 44 welding, 39, 44, 56, 149 wettability, 93 wetting, 30, 134 Wiedemann–Franz law, 138, 149, 169 Wien’s equation, 112 X-ray absorption coefficient, 2 X-ray diffraction, 1–5, 9, 10 X-ray scattering, 2, 7 xenon arc lamp, 120 Y, 182, 186, 190, 191 Y2 O3 , 3 YAG, 3 ZARM, 133 ZnSe window, 5 Zr, 3, 7–10, 12, 120, 122, 125–127, 173, 177, 182, 186, 191 Zr55 Al10 Ni5 Cu30 , 104 Zr60 Al15 Ni25 , 104 Zr65 Al7.5 Cu17.5 Ni10 , 141 Zr65 Al7.5 Cu27.5 , 104 Zr-based alloy, 104, 105 ZrO, 3